Citation
Modeling the evaporation and temperature distribution of a soil profile

Material Information

Title:
Modeling the evaporation and temperature distribution of a soil profile
Series Title:
Modeling the evaporation and temperature distribution of a soil profile
Creator:
Butts, Christopher Lloyd
Publisher:
Christopher Lloyd Butts
Publication Date:
Language:
English

Subjects

Subjects / Keywords:
Evaporation ( jstor )
Lysimeters ( jstor )
Moisture content ( jstor )
Soil temperature regimes ( jstor )
Soil water ( jstor )
Soils ( jstor )
Surface temperature ( jstor )
Surface water ( jstor )
Water temperature ( jstor )
Water vapor ( jstor )
City of Gainesville ( local )

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
024855629 ( alephbibnum )
20111916 ( oclc )

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Full Text












MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE








BY

CHRISTOPHER LLOYD BUTTS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1988


112- S














ACKNOWLEDGEMENTS

I would like to acknowledge, first and foremost, GOD's blessings

in giving me the talents and insights with which to complete this

dissertation. My wife, Sherry, has provided steadfast support and

encouragement without which this task would have been impossible. I

also thank my committee chairman, Dr. Wayne Mishoe, for his guidance

and encouragement and the free reign to conduct the research as I

deemed necessary. I thank my committee, Drs. James Jones, Khe Chau,

Hartwell Allen, Calvin Oliver and Mr. Jerome Gaffney, for their

guidance and support. To the Agricultural Engineering technical staff

in the fabrication and instrumentation of the weighing lysimeters

constructed for this project, I offer my thanks. I thank Bob Bush for

his diligent maintenance and operation of the weighing lysimeters.

Finally, I acknowledge the contribution to my well-rounded education of

my fellow graduate students, particularly, Bob Romero, Ken Stone, Matt

Smith, Ashim Datta and Kumar Nagarajan.









TABLE OF CONTENTS


Page


ACKNOWLEDGEMENTS . . . . . . . . . . . . ii

KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . v

ABSTRACT . . . ...... . ....... . . . . . .x

CHAPTERS

I. INTRODUCTION . . . . . . . . . . . 1

Problem Statement . . . . . . . . 1
Research goals and objectives . . . . . . 3

II. ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT 5


Introduction . . . . . .
Literature Review . . . .
Model Objectives . . . . .
Model development . . . .
Determination of Model Parameters
Numerical solution . . . .


III. EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND ENERGY


TRANSFER PROCESSES IN THE SOIL .

Introduction . . . . .
Objectives . . . . . .
Lysimeter Design, Installation a
and Construction . . .
Experimental Procedure . . .
Data Analysis . . . .
Experimental Results . . .
Summary . . . . . .


. . . . . . .o

. . . . . . .

d Calibration Design



. . . . . . .
. . . . ... .
. . . . . . .
. . . .
. . . . .


IV. MEASURING THERMAL DIFFUSIVITY OF SOILS . .. . . .

Introduction . . . . . . .. . . .
Objectives . . . . . . .. ....
Literature Review . . . . . . . . .
Procedure . . . . . . . . . . .
Results and Discussion . . . . . . ...
Conclusions . . . . . . . . . . .


5
6
19
20
30
43


66

66
67

67
85
89
90
96

134

134
135
135
141
143
148


. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
ooooooooo
oeooeooooo
oooooooooo
oooeoooooo
eoooooooo
oooooooeeo








V. MODEL ANALYSIS . . . . . . . . .. . .


158


Introduction . . . . . . . . .. . 158
Validation . . . . . ...158
Sensitivity Analysis . ..... . . . .. 170
Summary . . . . . . . . . . 176

VI. SUMMARY AND CONCLUSIONS . . . . . . . . . 208

BIBLIOGRAPHY . . . . . . . . .. . . . . 212

APPENDIX A ADI FORMULATION OF COUPLED HEAT AND MASS
TRANSFER MODEL . . . . . . . . . . . 219

APPENDIX B METEOROLOGICAL DATA FOR LYSIMETER EVAPORATION
STUDIES . . . . . . .... . . . . . 225

BIOGRAPHICAL SKETCH . . . . . . . . . . . 241







KEY TO SYMBOLS AND ABBREVIATIONS
Variable Definition
B empirical function for determination of Cer and Chr
dimensionlesss]
Cdr surface drag coefficient dimensionlesss]
Cer Dalton number, dimensionless mass transfer
coefficient
Chr Stanton number, dimensionless heat transfer
coefficient

Cpa specific heat of moist air [J'kg-1"K-1]
Cps specific heat of solid constituent of soil mixture
[J-kg-1.-oK-]

Cpw specific heat of water [J-kg-1'K-1]
Cpv specific heat of water vapor [J-kg-1-K-1]
Cs volumetric heat capacity of soil mixture (solid,
liquid and air phases) [J'kg-l1(-1]
Da diffusivity of water vapor in air [m2"s-1]
DL unsaturated hydraulic diffusivity of soil [m2.s-1]
Dt thermal diffusivity [m2.s-1]
Dv diffusivity of water vapor in soil [m2.s-1]
dwj height of volume of soil associated with node j. [m]
dxj width of soil cell j in the x direction (unity) [m]
dyj width of soil cell j in the y direction (unity) [m]
dzj vertical distance between nodes j and j+1. [m]
E(z,t) rate of phase change of liquid water to water vapor
@in the soil as a function of space (z) and time (t)
[kg H20)m(soil)s'








Variable Definition

Ej,n rate of phase change of liquid water to vapor in
soil cell j at time step n [kg(H20)'m(oi)*s" ]
ETa actual evapotranspiration [mm]

ETp potential evapotranspiration [mm]

ea water vapor pressure in ambient air [Pa]

eas saturated water vapor pressure in ambient air [Pa]
es(T) saturated vapor pressure at temperature, T [Pa]

evs(O) vapor pressure at water potential, 0 [Pa]
G sensible heat flux rate from the soil to the air
[W-m- ]

g gravitational acceleration [m's-2]

gj shape factor in the j-th principal axis for thermal
conductivity calculations dimensionlesss]

H vertical component of heat flux in the atmosphere
[J-m ]

hfg latent heat of vaporization of water [J3kg-1]
hh convection heat transfer coefficient for soil
surface [W -m'2K-i]

hm convection vapor transfer coefficient for soil
surface [ms- ]

K hydraulic conductivity [m0'm-2.s-1]

Kh, Km, Kw turbulent transfer coefficients for peat, momentum
and water vapor, respectively [m s ]

kij ratio of temperature gradient in i-th soil
constituent to the temperature gradient in the
continuous constituent (water or air) in the
direction of the j-th principal axis dimensionlesss]

LE latent energy transfer in the atmosphere [Jm-2]
n number of moles of gas present, used in ideal gas
law [mol]







Variable Definition
P precipitatior: flux of water at soil surface
[m H20)m- s-]

p atmospheric pressure [Pa]

Po reference atmospheric pressure [Pa]

Qv vertical flux density of water vapor in the
atmosphere [kg-m- 's- ]

q specific humidity of ambient air [kgapor/kgdry air]

qcj+l,n rate of heat flux conducted into node j from node
j+1 at time step n [W-m- ]

qej,n rate of latent heat loss due to evaporation at node
j and time step n [W'm" ]

qLj+1,n heat flux rate into node j from node j+1 via
movement of liquid at time step n [W'm-2]

qsj,n rate of sensible heat change of node j at time step
n [W'm-3]

qvj+1,n rate of heat carried into node j from node j+1 by
water vapor movement at time step n [W'm-2]
R universal gas constant [kg-m2-s-2.mol-1(-o-1]

Rw gas constant for water vapor, determined by dividing
the universal gas constant, R, by the molecular
weight of water [m2.s-2OK-1]
Re Reynolds number dimensionlesss]
Ri Richardson number dimensionlesss]
Rn net r diation flux incident upon soil surface
[W-m- ]
S soil porosity [m3m-3]

s slope of the saturated vapor pressure line
[kPa.oK-1]

Ta ambient air temperature [OK]

Tj,n soil temperature at node j and time step n [OK]
Tavgj (Tj+l,n + Tj,n)/2 [OK]







Variable Definition (continued)
t time [s]
U wind velocity [ms-l]
U* shear velocity [m-s-1]

V volume [m3]

WS Wind speed [mss-1]

xi volume fraction of the i-th soil component [m3m-3]
z depth below soil surface [m]
z0 surface roughness length [m]
a soil tortuosity dimensionlesss]

I- psychrometric constant [kPaCK-1]
7r diabetic influence function dimensionlesss]

soil water potential [m]
X thermal conductivity of soil composite at node j
[W-m-1.OK-1]
v kinematic viscosity [m2.s-1]

Pa density of moist air [kg'm-3]

Ps dry bulk density of soil [kg'm-3]
Pv water vapor concentration [kgH20'ms3il]

Pvj,n water va or concentration at node j and time step n
[kgH20'msoil]
Pva water va or concentration of ambient air
[kgH20'mairl

Pvs water vapor concentration in the air at saturation
[kgH20-m-lr]
Pw density of water [kg-m-3]
9 dimensionless water content


viii







Variable Definition (continued)
0 volumetric water content of soil [m3.m-3]

Ojn volumetric water content of soil at node j and at
time step n [mn3m-3]

Or residual volumetric water content [m3.m-3]
es volumetric water content at saturation [m3m-3]
7 shear stress [N-m-2]














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE

By

Christopher Lloyd Butts

December, 1988

Chairman: J. Wayne Mishoe
Cochairman: James W. Jones
Major Department: Agricultural Engineering

Evaporation of water from the soil is controlled by the transport

phenomena of energy and mass transfer. Most procedures for estimating

the loss of water from the soil assume that the soil is an isothermal

medium with evaporation of water occurring at the soil surface.

Estimating evaporation as a surface phenomenon independent of soil

hydraulic and thermal properties can lead to overestimation of the

amount of water lost from the soil.

Weighing lysimeters measuring 2 x 3 x 1.3 m3 were constructed

capable of detecting a change in weight equivalent to 0.02 mm of

water. Recorded data consisted of net radiation, soil temperature,

water content, load cell output, ambient air temperature, relative

humidity, windspeed and precipitation. Measured soil properties

included dry bulk density, porosity and thermal diffusivity.

A model describing the continuous distribution of heat, water and

water vapor within the soil was developed. The evaporation of water








was allowed to occur throughout the soil profile in response to the

assumed equilibrium between the liquid and vapor phases. Surface heat

and mass transfer coefficients were determined using equations based

upon equations of motion in the atmosphere. Thermal diffusivity of the

soil was measured and incorporated into the simulation.

Simulation results included temporal values of the cumulative and

hourly water loss from the soil, and spatial distribution of

temperature, water and water vapor in the soil. Experimental data

obtained from the weighing lysimeters were used to validate the model.

Prior to model calibration simulated soil temperatures were within 2 OC

of measured values and simulated cumulative evaporation was within ten

percent of measured water loss. Sensitivity analysis indicated that

calibration could be achieved by relatively small adjustments in the

values of the surface heat and mass transfer coefficients.

Experimental and simulated evaporation rates exhibited a diurnal

pattern in which maximum evaporation rates occurred approximately

midday then decreased to near zero at sunset. Under some conditions,

water continued to evaporate from the soil overnight at a rate of

approximately 0.03 mm/h while in some cases a net gain of water was

observed overnight.














CHAPTER I
INTRODUCTION

Problem Statement

The process by which water is lost from the soil has been the

object of considerable research over the years (Brutsaert, 1982).

Water is lost from the soil by runoff, drainage, evaporation and

transpiration. Evapotranspiration (ET) has been used to describe the

combined evaporative losses from the soil and plant. Potential evapo-

transpiration (ETp) has been defined as the amount of water

transferred from a wet surface to vapor in the atmosphere (Penman,

1948) and has been used as an estimate of the maximum amount of water

lost to the atmosphere from a given area. Estimates of actual

evapotranspiration (ETa) are frequently made by the use of coefficients

which vary according to ground cover and the region of the country.

The need for knowing ETa on a field basis arises for a variety of

reasons including design and selection of irrigation equipment,

optimization of irrigation management, and determination of water

requirements for agricultural and municipal entities (Heermann, 1986).

Regional estimates of ETa are necessary for hydrologic simulations

(Saxton, 1986).

Mathematical models have been developed to provide estimates of

both ETa and ETp. The use of evapotranspiration models has expanded in

recent years due to the advent of crop growth simulations requiring the

distribution of soil water as a component of the model. One such model

used to integrate the soil water and the plant water requirements was

1








2

developed by Zur and Jones (1981) and focused on estimating water

relations for crop growth. A soil water model which simulates ETa is

an integral part of the soybean production model, SOYGRO, developed by

Wilkerson et al. (1983). One of the primary uses of SOYGRO is to

determine the yield response of soybean to various environmental

conditions such as drought.

Even though ET models are currently used extensively, many

research needs remain. At best, crop coefficients are rough estimates

computed from previous experiments with the crop of concern. There is

a need to develop a better understanding of the variability of soil,

crop and environmental factors that determine crop coefficients.

Scientists need improved procedures for determining crop coefficients

for various cultural practices such as mulching, no-till and row

spacing.

One of the most pressing needs in the area of evapotranspiration

research is in the area of sensors which will detect soil water status

for a particular field (Heermann, 1986; Saxton, 1986). One of the

rising technologies for determining soil water status is the use of

remote sensing of soil temperature for determining soil water status

(Soer, 1980; Shih et al., 1986).

Evaporation from the soil is driven by the energy balance of the

surrounding environment. Because of the release of latent energy is

coupled with the sensible energy balance, it is necessary to consider

both components of the energy balance to completely describe the

evaporation process. Soil temperature has been found to influence many

processes in the soil. For example, soil temperature influenced

nodulation of roots of Phaseolus vulgaris L. (Small et al., 1968) and








3

root growth for various stages of development of soybean (Brouwer,

1964; Brouwer and Hoogland, 1964; Brouwer and Kleinendorst, 1967).

Blankenship et al. (1984) have shown that excessive temperatures in the

soil zone in which peanuts are produced (geocarposphere) can reduce

yields and increase susceptibility to disease, fungus and physical

damage. Unpublished data collected by Sanders (1988) for peanuts grown

in lysimeters with controlled soil temperatures indicated that the

maturation rate of peanuts were affected by soil temperature. The fact

that movement of water vapor in the soil in response to temperature

gradients can be significant was determined by Matthes and Bowen (1968)

and more recently by Prat (1986). Neglecting soil temperature in a

mass transfer model in the soil could cause significant errors in the

estimation of ETa from the soil if soil temperature were neglected in

the modeling analysis.

Research Goals and Objectives

An understanding of the processes of heat and mass transfer in the

soil is of interest to a variety of agricultural scientists,

particularly those involved in modeling agricultural systems and

developing management strategies. Evapotranspiration is a complex

process involving energy and mass transport in the plant and the soil

as well as the chemical processes such as photosynthesis and

respiration which require water from the soil and release water from

the reaction. It has been suggested by some researchers that the two

processes involving the plant and the soil be studied independently so

that greater insight might be obtained into each. Therefore, the goal

of this research is to study the process of evaporation from the soil

independent of transpiration for the purposes of furthering the









knowledge of the parameters affecting evaporation from the soil and

guiding future research. The specific objectives are:


1. to develop a model to simulate the evaporation of water from
the soil which couples the energy and mass transfer
processes;

2. to monitor evaporation of water and vertical distribution of
temperature and water in a sandy soil;

3. to determine the validity of the model by comparison of
simulated evaporation rates, soil temperatures and
volumetric water content to those measured in a sandy soil;

4. to determine the sensitivity of the evaporation process to
changes in various soil properties and environmental
conditions; and

5. to calibrate the model for local conditions using data
collected in Objective 2 utilizing information gained from
the sensitivity analysis.














CHAPTER II
ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT

Introduction

Models of varying complexity may be used to simulate or estimate

the amount of water which evaporates from the soil. The complexity of

the model should be determined by the projected use of the results and

the data available as input for the model. Empirical models generally

relate one or more parameters by regression analysis to evaporation

measured under various conditions. Variables or parameters typically

used in empirical models are evaporation pan data, air temperature, and

day length or solar radiation. Monthly estimates of evaporation for

relatively large areas are typically obtained from empirical models

(Jones et al., 1984).

Resistance analog models employ the concept of the electrical

analog to mass flow, where the flux of water vapor leaving the soil

surface is expressed as a potential (vapor pressure) difference divided

by a resistance. The resistance term for the mass flow is a function

of the boundary layer of the lower atmosphere and the mass transfer

characteristics of the soil (Camillo and Gurney, 1986; Jagtap and

Jones, 1986). These resistance models have the capability of

estimating actual evaporation from the soil on a daily or hourly basis.

Evaporation can also be estimated using mechanistic models

describing the conservation of mass, momentum, and energy in the lower

atmosphere and the soil. These models are the most detailed in their

derivation and have the advantage that they can provide substantial

5








6

insight into the actual processes describing the phenomenon of

evaporation. However, considerable data may be required as input to

provide reasonable estimates of the heat and mass transfer.

Literature Review

Empirical Models

Jones et al. (1984) reviewed some of the empirical procedures to

estimate crop water use noting that all require calibration for a

particular geographical region. Pruitt (1966) reported that serious

errors can occur when using pan evaporation data to estimate crop water

use, particularly under strong advective weather conditions. The pan

should be installed properly ensuring sufficient fetch surrounding the

pan.

The Thornwaite model, as presented by Jones et al. (1984), uses

monthly averages of air temperature and day length to estimate monthly

evapotranspiration (ET). Criddle (1966) summarized the Blaney-Criddle

model which predicts the actual water use by a crop from monthly

average air temperature and percent daylight and introduces crop

consumptive use coefficients. The Blaney-Criddle model, developed for

use in arid regions, greatly overestimates evaporation for the humid

climate of Florida during the summer months prompting Shih et al.

(1977) to replace the percent daylength with monthly net radiation to

account for the increased cloud cover.

Resistance Models

Resistance models presented in the literature are of the general

form

cpaPa (e(Ts) e(Td) )
E = Hf- R (2-1)









where:

E = evaporation rate [m-s-1]

Cpa = specific heat of air [J-kg-l'o-1]

Pa = density of dry air [kg'm-3]
hfg = latent heat of vaporization [J'kg-1]

= psychrometric constant [kPa*'K-1]

e(T) = saturated vapor pressure [kPa] at temperature T

Ts = soil surface temperature [OK]

Td = ambient dew point temperature [OK]
R = resistance to vapor movement from the soil to air

[kg-s-m-4]

The resistance term for the models presented by Conaway and Van Bavel

(1967), Tanner and Fuchs (1968), and Novak and Black (1985) represents

the resistance due to the laminar boundary layer. The boundary layer

resistance is a function of the wind speed, atmospheric instability,

and the surface roughness height. Their models are used to predict

evaporation from a well-watered bare soil surface. Jagtap and Jones

(1986) and Camillo and Gurney (1986) developed resistance terms to

include the boundary layer resistance in series with the resistance of

vapor flow in the soil. The soil resistance term is included since

after the soil surface dries, the water must change to vapor in the

soil below the surface then diffuse to the soil-atmosphere interface.

The soil resistance term developed by Jagtap and Jones was determined

by regression analysis as a function of cumulative evaporation, water

in the soil profile available for evaporation, and a daily running

average of the net radiation. The net radiation empirically accounted

for the heat flux into the soil, while the ratio of the cumulative








8

evaporation to the available water parameterized the thickness of dry

soil through which the water vapor diffused. The soil resistance term

used by Camillo and Gurney (1986) was a linear regression of the

difference between the saturated volumetric water content and the

actual soil water content. Both models were intended to provide daily

evaporation rates.

The resistance model estimates daily evapotranspiration relatively

well. However, the soil resistance term varies with time depending

upon the distance through which the water vapor must diffuse from below

the soil surface. One point that should be noted is that the vapor

pressure difference used in the analog models discussed is the

difference between the saturated vapor pressure at the temperature of

the soil surface and that of ambient air. In reality, evaporation

frequently occurs below the soil surface and the vapor pressure

gradient from the zone of evaporation may be substantially different

from that at the surface.


Mechanistic Models

Models derived from the basic physical relationships have the

advantage of providing estimates of evaporation on a relatively short

time scale. However, considerably more input data describing the

surface boundary conditions as well as the soil thermal and hydraulic

properties are required. Theoretical models have been developed from

four different perspectives (King, 1966; Goddard and Pruitt, 1966;

Fritschen, 1966; Penman, 1948). They are categorized as follows

1. a mass balance profile method,

2. a mass balance eddy flux method,








9

3. an energy balance method, or

4. a combination method.

The two mass balance methods (1 and 2 above) focus on the

equations of motion for the atmosphere and require accurate measurement

of velocity and temperature profiles or mass fluxes. The equations of

conservation of momentum, mass, and energy were used in the mass

balance methods to develop expressions for the rate of evaporation of

water and the shear stress between to vertical positions in the lower

atmosphere as





Qv = PaKw q (2-2)
az

S = p aKm aU (2-3)
az


H = cpaaKh aT (2-4)
az


where:

Qv = vertical flux density of water vapor [kg-m-2.s-1

Pa = density of air [kg'm-3]
7 = shear stress rate [N'm-2]

H = vertical flux of heat [J.m-2]

z = vertical distance [m]

q = specific humidity [kgapor/kgdry air]
U = velocity [m's-1]

Cpa = specific heat of dry air [J*kg-1K-l]








10
T = air temperature [OK]

Kh, Km, Kw = turbulent transfer coefficients for Peat, momentum
and water vapor, respectively [m' s- ]

Combining the equations 2-2 and 2-3 and rearranging the following

expression for evaporation is obtained.


Q V Kwm ~ (2-5)


The profile methods determine the rate of evaporation by analysis of

the vertical profile of various atmospheric variables such as specific

humidity or vapor pressure and wind velocity. King (1966) described

the procedures for determining the evaporation rate for adiabatic wind

profiles (neutral stability) and stratified conditions. One of the

basic assumptions used in the profile methods is that the turbulent

transfer coefficients of water vapor and momentum are equal. This

implies that the momentum and mass displacement thicknesses are the

same. The Richardson number (Ri) or the ratio of height (z) to the

Monin-Obukov length (L) are used as a measure of atmospheric

instability and can be determined by equations 2-6 and 2-7,

respectively.


aT
-T- (-- +r)
Ri = ---aU- (2-6)
au


L (2-7)
( kgH
cpaqaT
where:


= acceleration due to gravity [m-s-2]










r = adiabatic lapse rate [(K(m-1]
S 9.86 x 10-3

k = von Karman's constant dimensionlesss]
= 0.428

U* = shear velocity [mns-1]

For neutral conditions the familiar logarithmic velocity profile was

used. For unstable or diabetic conditions a variation of the

logarithmic profile was used or the KEYPS function (Panofsky, 1963) was

used to determine the velocity profile. King stressed that in using

profile methods to determine evaporation, one must take extreme care in

measuring wind velocity. He suggested that spatial averaging be used

for wind speed measurements near ground level and that 30- to 60-minute

averages be used considering the steady state assumptions made in

developing the equations for the profile equations. Corrections for

lower atmospheric instability caused by density gradients in the air

near the earth's surface must also be used.

Another method requiring only measurements of atmospheric
parameters is the eddy flux method (Goddard and Pruitt, 1966). The

method is based upon the turbulent transport equations of motion in

the atmosphere. The equations of turbulent motion in a fluid include

the transient random fluctuations in velocity, temperature, and mass

concentration of water vapor of the air. The following equations

describe the shear stress (T), sensible heat flux (H) and latent heat

flux (LE) due to the turbulent motion of the atmosphere

au
7 = PaKm,- awu (2-8)


H = cpaPaKh + cpaPaw'' (2-9)











LE = -hfg(paKw paw'q ) (2-10)
where:
T7 : time averaged turbulent fluctuation of the air
temperature [K(]
u7 : time averaged turbulent fluctuations pf the horizontal
component of the wind velocity [m-s"]
w : wind velocity in the vertical direction [m's-1]
-w : time averaged turbulent fluctuations of the vertical
component of the wind velocity [m-s-11

q : turbulent fluctuations in the specific humidity of the
air [kgvapor/kgdry air]

Assuming vertical gradients in temperature and absolute humidity
are insignificant as compared to the gradients in the direction as the
horizontal component of the wind, the latent and sensible heat can be
expressed as


H = CpaPaw'Ta (2-11)


LE = hfgPaw (2-12)



The equations are used as a basis for either the sensible or latent

heat flux in the atmosphere using measured values of the turbulent eddy
fluxes of heat and moisture. Specialized equipment was designed by the
Goddard and Pruitt to measure the parameters needed for the

calculations. However, the equipment was not reliable for low wind
speeds. This was attributed to the very small magnitudes of the
turbulent eddies for calm conditions.








13

Limitations of the mass balance (eddy flux and profile) methods

are the requirements for very sensitive equipment for measuring either

the vertical profiles or the vertical fluxes due to turbulence. King

(1966) questioned the practical use of the mass balance methods of

determination of the evaporation. Fritschen (1966) pointed out that

the mass balance methods do not integrate results over time, thus

requiring constant monitoring of meteorological parameters.

Another approach to estimate evaporation from the soil is the

energy balance method (Fritschen, 1966). The rate of change of

sensible heat in a control volume of air at the earth's surface is

equal to the rate at which sensible and latent heat are carried into

and out of the volume by the wind, the convection of sensible and

latent energy in the vertical direction, the energy transferred to the

soil and the net radiant exchange of energy. In the energy balance

method, it is assumed that the net transfer of sensible and latent heat

in the horizontal direction by the wind is negligible when compared to

the vertical movement of heat in the atmosphere. The equations of

motion are utilized as in the mass balance profile method with the

exception of using the similarity equation for the sensible heat flux

instead of the shear stress. A ratio of sensible to latent heat flux

(equations 2-4 and 2-2), referred to as the Bowen ratio (BR), is given

by

aT
H cpaPa Kh Jt
BR = (2-13)
8z


The energy balance at the soil surface is










Rn S G LE = 0 (2-14)



where:

Rn = net radiation incident upon soil surface [W*m-2]

S = soil heat flux [W-m-2]

G = sensible energy flux into atmosphere [W-m-2]

LE = flux of latent energy from soil [W-m-2]


An expression for the rate of evaporation of water from the soil can be

determined by substitution of equation (2-13) into (2-14) and

rearranging terms. This method is referred to as the Bowen ratio

equation and yields valid results for a wide range of conditions.

Fritschen states that it is imperative that efforts be undertaken to

assure that the assumption of no horizontal divergence of heat or

moisture be met in order for the Bowen ratio equation to yield

satisfactory estimates of the evaporation. Soil heat flux must be

measured as well, since omitting soil heat flux from the analysis could

lead to large errors.

In using the Bowen ratio, the soil surface temperature must be known as

in some of the other methods.

Penman (1948) used a combination of the energy and mass balance

methods with the objective of eliminating the need for surface

temperature. The underlying assumptions are the temperature of

evaporating surface is the same as the ambient air and the vapor

pressure is the saturated vapor pressure evaluated at the surface







15

temperature. It was also assumed that the net soil heat flux is zero

over a 24 h period. The resulting relationship
Pacpahfg (ea eas) + s(Rn G)
Ep = (- + s) (2-15)

where:

Ep potential evaporation [m3.m-2]

ea = vapor pressure of atmosphere [kPa]

eas = saturated vapor pressure at the air temperature [kPa]
h = surface vapor transfer coefficient [m's-1]

G = sensible heat flux in the atmosphere [W-m-2]

Rn = net radiation upon the soil surface [W-m-2]
s = slope of saturated vapor pressure line [kPa.'K-1]

7 = psychrometric constant [kPaOK-1]
provides an expression for the potential evaporation from the surface.

The boundary layer resistances account for advective conditions. The

Penman equation is the only method for estimating evaporation based

upon theoretical ideas and requires no highly specialized equipment.

Estimates of the evaporation have an accuracy of 5 to 10 percent on a

daily basis (Van Bavel, 1966). The disadvantage to the combination

method, as implemented by Penman, is that only daily estimates are able

to be determined and crop coefficients are necessary to estimate actual

evapotranspiration from various crops (Jones et al., 1984). The values

of the crop coefficients typically exhibit regional variation as well

as variation due to stage of crop growth.

Staple (1974) modified the Penman model to provide the upper

boundary condition for the isothermal diffusion of water in the soil.

The Penman model was modified by multiplying the saturated vapor







16

pressure by the ratio of vapor pressure at the soil surface to the

saturated vapor pressure at the same temperature (relative vapor

pressure). This incorporated the vapor pressure depression effect of

the soil water potential into the model. This seemed to match field

data for a clay loam soil. This model was not tested for more coarse

soils.

The Penman method eliminated the requirement of soil surface

temperature by neglecting the effects of soil heat flux at the expense

of being able only to predict the evaporation on a daily basis.

Various researchers have recently gone one more step in modeling the

evaporative loss of water from the soil by considering the movement of

water and heat below the soil surface (Van Bavel and Hillel, 1976;

Lascano and Van Bavel, 1983). Most of the research in which the

process of evaporation has been examined at this detailed level are

for a bare soil surface. This eliminated the complicating factors of

water removal by the plant, which surface temperature to use in

describing the driving potential for evaporation, and description of

the atmospheric boundary layer transfer coefficients.

Van Bavel and Hillel (1976) developed a model in which the partial

differential equations for water (liquid) and energy transfer in the

soil were considered to be independent processes. This approach

explicitly determined the soil heat flux for the boundary condition at

the soil surface as expressed in equation (2-14). The simultaneous

solution determined the actual evaporation from the soil as well as the

spatial distributions of water and temperature as a function of time.

The variation in hydraulic and thermal properties of the soil with time

was accounted for in this extended combination approach.







17
According to Fuchs and Tanner (1967) field and laboratory

observations indicated that the evaporation of water occurs in three

distinct stages. The first involves water evaporating from the soil-

atmosphere interface. As the soil surface dries, the water must

evaporate at a location below the soil surface then the vapor diffuse

to the surface. The combination methods of simulating evaporation from

the soil reproduce this process fairly well. However, the evaporation

is assumed to occur at the soil surface. This has the effect of

lowering the soil surface temperature, when in the case of water

evaporating below the dry soil surface, the soil surface temperature

would be higher due to the reduced thermal conductivity of the soil.

This error in soil surface temperature was noted by Lascano and Van

Bavel (1983). Lascano and Van Bavel (1983 and 1986) verified the model

of Van Bavel and Hillel (1976) using soil water content and temperature

data obtained from field plots. During the earlier study, the

simulated soil surface and profile temperatures were found to agree

quite closely over the range of 25 to 37 oC while the model
underpredicted surface temperature by 2 to 5 OC when the soil

temperature was above 37 OC. Distribution of soil water is simulated

to within the expected error of measurement. Similar results were

obtained during the 1986 experiment.

Movement of water primarily in the vapor phase in the soil has

been observed by several researchers. Taylor and Cavazza (1954) noted

that the measured diffusion coefficients were higher than that for

water vapor in air and suggested that transport was due the combined

effects of convection and diffusion within the soil pore spaces.

Schieldge et al. (1982) used the following equations of the







18
conservation of mass and energy to simulate the diurnal variations of
soil water and temperature.

ae a a8 a aT aK
-5 = --z (DO --) + -T (DT az-) + T- (2-16)

aT a aT
Cs at = ( )+ Q(O,T) (2-17)


where:
0 = volumetric water content [m3.m-3]
t = time [s]
z = depth below soil surface [m]
T = soil temperature [OK]
Cs = volumetriq heat capacity of soil mixture
[Jem-j3OK-i]
S= thermal conductivity of soil mixture [W-m-1OK-I]
K = unsaturated hydraulic conductivity [m's-1]

Do = hydraulic diffusivity [m2.s-1]

Sao0 D a VS9Pw
S- + PRT a80
Da = diffusivity of water vapor in air [m2.s-1]
P = atmospheric pressure [kPa]
v = ratio of atmospheric pressure (P) to partial
pressure of air
S = soil porosity [m3m-3]

g = acceleration due to gravity [m-s-1]
= soil matric potential [m]
Rw = ideal gas constant for water vapor

Pw = density of water vapor [kg'm-3]







19

DT = diffusivity of water (liquid and vapor) due to
temperature gradients [m*'s-10'1]

+ Da vSahon
Pw
Q(0,T) = heat flux within the spil due to movement of water,
liquid and vapor [W'm- ]
a = soil tortuosity factor (= 2/3)

f= coefficient of thermal expansion [kg'm-3"K-1]

S= vapor transfer coefficient dimensionlesss]
h = relative humidity

These equations were originally developed and presented by de Vries

(1958) with good agreement with field observations being achieved.

Diurnal fluctuations of water content were damped out below a depth of

4 cm. Particular attention was given to the magnitudes of DT DTL and

DTV. It was noted that the DT was small during the night but has a

significant contribution to the flow of water during the day. The

vapor component (DT-) was less than the liquid component (DTL) for

water contents greater than 0.30 m3/m3 and sensitive to changes in soil

temperature.

Jackson (1964) studied the non-isothermal movement of water using

equations (2-16) and (2-17) and concluded that classical diffusion

theory could be used satisfactorily if the relative vapor pressure is

greater than 0.97 with no modification to the diffusivity term.

However, if the relative vapor pressure is less than 0.97 then the

diffusivity must include the effect of the temperature gradient.

Model Objectives
The models presented thus far each have advantages and

disadvantages. The mechanistic models are based upon sufficient theory

that the diurnal fluctuations of the soil water and temperature








20

profiles can be observed. They also have the ability to provide

insight into the many processes involved in the evaporative loss of

water from the soil. Saxton (1986) stated that there is a need to

separate the evaporative loss of water from the soil from the losses

through the plant so that better understanding of the individual

processes can be achieved. The mechanistic models provide this

ability.

One might also note that in any of the models discussed

previously, conservation of water is discussed either in the vapor

phase, as in the mass balance methods, or the liquid phase, as in the

combination methods which consider the soil media. The vapor and the

liquid phases are not considered simultaneously.

The objectives in developing a new model were:

1. to account for the water changing from liquid to vapor
phase below the soil surface and diffusing to the
atmosphere;

2. to consider the overall mass continuity, specifically
include the liquid and vapor phases separately;

3. to account for the movement of water vapor in response
to temperature gradients in the soil; and

4. to simulate the diurnal variation of evaporation, soil
water and temperature profiles in response to conditions
at the soil surface.

Model Development

The soil for most purposes may be considered a continuous medium

in which the laws of conservation of mass and energy apply.

Application of the basic principles of thermodynamics to the soil

profile provide the basis for simulation of the temporal variation of

the distribution of water and temperature throughout the soil profile.

The following assumptions were made to simplify or clarify the







21

development of the model for coupled heat and mass transfer in the

soil.


1. The soil is unsaturated, therefore the soil water potential
is primarily due to osmotic and matric potential.

2. Water vapor behaves as an ideal gas.

3. The movement of water in the liquid and vapor phases occurs
due to concentration or potential gradients.

4. The liquid and vapor phases are in thermodynamic equilibrium
within the soil pore spaces.

Using the conservation of mass, the rate of change of the water

vapor within the soil air space was described in the model as

aPv a apv
t- = (Dv -- ) + Ev(z,t) (2-17)


where:

Pv = mass concentration of water vapor in the soil air
space [kg-m-3]

Dv = Diffusivity of water vapor in the soil air space
[m2.s-l]
= Da

Da = Diffusivity of water vapor in the air [m2.s-1]
a = soil tortuosity dimensionlesss]

Ev(z,t) = rate of phase change of water [kg'm-3-S-1]

t = time [s]

z = depth below the soil surface [m]

The time rate of change of the concentration of water vapor at any

point in the soil space is a function of the rate of water vapor

diffusing from other regions of the soil and the rate at which water

changes from the liquid to the vapor phase. The evaporation rate is







22

considered to occur throughout the soil profile in response to

temperature and water content conditions in the soil. The rate of

phase change of the water (Ev) is considered positive if the water

changes from the liquid to vapor (evaporation), while condensation is

indicated by a negative phase change rate.

The diffusivity of water vapor in the soil air spaces is assumed

to equal that in the atmosphere, however, Schieldge et al. (1982), as

well as others, have made use of a tortuosity factor (a) to account

for the increased path length of the interstitial spaces of the soil

through which the water vapor must traverse. The tortuosity factor

included the effect of the local evaporation and condensation of water

vapor as it passes through the regions of differing temperature and

water content in the soil.

The volumetric water content is typically used to express the
amount of liquid water present in the soil. The mass concentration of

water in the liquid phase is obtained by multiplying the volumetric

water content by the density of water. The time rate of change of the

mass concentration of water is increased by a the rate of diffusion of

water from surrounding soil and decreased by the rate of evaporation.

The mathematical expression of the conservation of mass for the water

is


at-W) = aQ EL(z,t) (2-18)

where:

0 = volumetric water content [m3m-3]

Pw = density of water [kg-m-3]

QL = diffusion of liquid water [kg-m-2 s-]










EL = rate of change of liquid water to water vapor
[kg-msils- ]
For the range of temperatures generally occurring in the soil, the
density of water may be considered constant. The volumetric flow rate
of water (qL) can be obtained by dividing the mass flow rate (QL) by
the density of water (Pw). The volumetric flow rate of water in an
unsaturated soil is governed by Darcy's Law as follows:

qL = k z- (2-19)
where:

qL = volumetric flow rate of liquid water [m3.m-2s-1]
k = unsaturated hydraulic conductivity [m-s-1]

S = soil water potential [m]
z = spatial dimension [m]
The relationship between soil water content and soil water potential is
unique for a given soil type. Therefore, the volumetric flux of water
can be expressed in terms of volumetric water content instead of soil
water potential by applying the chain rule of differentiation to
equation (2-19) as follows

L -k a (2-20)
The hydraulic diffusivity of the soil is defined as

DL M K -a (2-21)
a-
By substituting the definition for the hydraulic diffusivity (2-21) and
the volumetric flow rate in terms of the soil water content (2-20) the
equation for the conservation of mass for liquid water (2-18) becomes

aB a a( E(z,t) (2-22)
aT = az (DL -) Pw







24
The conservation equations for the liquid (2-22) and vapor (2-17)
both have terms relating to the rate of change from the liquid phase to
the vapor phase. The vapor phase change ( Ev(z,t) ) is based upon the
volume of air while the liquid phase change term ( EL(Z,t) ) is based
upon the volume of soil. The phase change terms are related by the
fraction of soil volume occupied by the air which can be determined by
subtracting the volumetric water content from the total soil porosity.
The relationship between the vaporization terms is expressed as follows


E(z,t) = EL(,t) = (S O)Ev(z,t) (2-23)
Rearranging equation (2-23) and solving for Ev(z,t) yields



Ev(z,t) = Ez,t) (2-24)

Substitution of equation (2-24) into the vapor continuity equation
(2-17) results in the following equations describing the conservation
of water vapor within the soil.

SPv a aPv Ev(z,t)
= (D t ) (S- ) (2-25)

The thermal energy equation describes the flow of heat within the soil
and includes the time rate of change of sensible heat in the soil
volume, the diffusion of heat due to temperature gradients, the
convection of heat due to diffusion of water and water vapor through
zones of variable temperature and the latent heat of vaporization. The
partial differential equation describing the rate of change of the
temperature of the soil used in the model was









aT 8 T 8 8apv aT
Cs -"- ( OF) + (PwCpwDL- + cpvDvz-- ) zT-

hfgE(z,t) j l (2-26)
where

X = thermal conductivity of soil mixture [Wnm-I.O-I]

T = soil temperature [OK]

hfg = latent heat of vaporization [J-kg-1]
Cs = volumetric heat capacity of soil mixture [J-m-1K-1]
= (1-S)Pscps + Opwcpw + (S-O)Pacpa
S = Soil porosity [m3m-3]

ps = dry bulk density of soil [kg'm-3]

Pa = density of moist air [kg-m-3]
cps = specific heat of solid portion of soil [J-kg-loK-1]

Cpw = specific heat of water [J-kg-l.'K-1]

Cpa = specific heat of moist air [J-kg-1oK-1]
Equations (2-22), (2-25), and (2-26) describe the conservation of
energy and mass within the soil profile and account for the vapor and
liquid water phases separately. However, those three equations contain
the three state variables, soil temperature, volumetric water content,
and vapor concentration as well as the unknown rate at which water is
changed from the liquid to the vapor phase. Another independent
equation was needed to adequately describe the soil mass and energy

system.
If the water vapor were in equilibrium with liquid water, the
vapor would have a partial pressure corresponding to the saturated
vapor pressure and would be a function of the liquid water temperature.
Assuming that water vapor behaves as an ideal gas, then the saturated







26

vapor density can be determined from the saturated vapor pressure using

the ideal gas law.


pV = nRT


(2-27)


where:

p: pressure of the gas

V: volume of gas

n: number of moles of gas present

R: ideal gas constant

T: absolute temperature of gas.

The ideal gas law may also be applied for the individual components of

a gas mixture with the pressure being the partial pressure of the

component gas and using the number of moles of the component gas.

Substitution for the molecular weight of water and solving for the

density of water vapor one obtains


es(T)
R, T


where:


(2-28)


Pvs = concentration of water vapor t saturated vapor pressure
per unit volume of air [kg-m" ]

es = saturated vapor pressure at given temperature [Pa]

T = temperature of free water surface [cK]

Rw = ideal gas constant for water vapor
= universal gas constant (R) divided by the molecular
weight of water
= 461.911 [m2.s-2K-1]

The above equation (2-28) is for the case in which the chemical

potential of the water is zero. However, due to matric and osmotic







27

forces, the water in the soil has a soil water potential less than

zero, thereby reducing the equilibrium vapor pressure from that of pure

water. The following relationship (Baver et al., 1972) can be used to

determine the saturated vapor pressure above a water surface with a

potential other than zero.


ev() = evs(T) exp(- -) (2-29)


where:

ev(0) = saturated vapor pressure of water with chemical
potential, 0 [Pa]

= soil water potential [m]

g = acceleration due to gravity [mns-2]
Using the ideal gas law, the vapor density over a water surface with a

chemical potential other than zero can be obtained by:


P = Pvs(T) exp(- -) (2-30)


A set of simultaneous equations (2-22, 2-25, 2-26, 2-30) describe

the conservation of thermal energy, water vapor and liquid water for

the soil continuum and formed the basis of a coupled mass and energy

model for the soil. The assumption of thermodynamic equilibrium

between the liquid and vapor states yields the constitutive

relationship expressed in equation (2-30) and provides a fourth state

equation to be used in the model.

A well-posed problem also includes boundary conditions and, in

the case of transient problems, the initial conditions must also be

prescribed. The system of governing equations was one-dimensional and

therefore required two boundary conditions. The first boundary is the







28
obvious soil-atmosphere interface. It was assumed that the thermal
capacitance of the soil at the surface was negligible when compared to
the magnitudes of the fluxes which occur. Therefore, the net flux of
energy must be zero at the soil surface. The ability of the soil to
maintain a significant rate of evaporation at the soil surface was
assumed to be small as well. This assumption required the water to
evaporate at a finite distance below the soil surface rather than at
the soil surface. The boundary conditions for the energy, vapor and
liquid continuity were


aT I0 Pv
(PwcpwDL -U + cpvDv --- )Tav
"x -- -z=O

= Rn hh(Tz=0- Ta) (2-31a)

-DL --= P (2-31b)


apv
-Dv = hm(pvO- Pva ) (2-31c)
z=0

where:

cpw = specific heat of water [J-kg-1.''1]
Cpv = specific heat of water vapor [J-kg-1.K-1]

DL = hydraulic diffusivity [m2s-1]
Dv = diffusivity of water vapor in soil [m2s-1]
hh = boundary layer heat transfer coefficient [W-m-2.K-1]
hm = boundary layer mass transfer coefficient [m-s-l]
P = precipitation or irrigation rate [m3.m-2s-1]

Rn = net radiation incident upon soil surface [WNm-2]
T = soil temperature [oK]










Tavg = average temperature [OK]
X = thermal conductivity of soil [W.m-l1.- 1]

Pva = mass oJ water vapor per unit volume of dry ambient air
[kg-m-J]

PvO = concentration of water vapor at the soil surface, z=0
[kg-m-3]
0 = volume of water per unit total soil volume [m3.m-3]

Ideally, for a semi-infinite medium, the flux of energy and mass
should be zero in the limit of depth (z) approaching infinity.

However, in anticipation of a numerical solution to the system of

partial differential equations, the boundary conditions were specified

at a depth of 1.5 meters. This depth was chosen by comparing the

error between the analytical and numerical solutions for conduction of

heat in a semi-infinite slab with constant uniform properties and a

uniform heat flux at the surface. This depth resulted in an error of

less than 0.1 OC at the lower boundary. The depth at which the

amplitude of the diurnal fluctuations in temperature is less than
0.1 OC is approximately 60 cm for a sandy soil (Baver et al., 1972).

The zero flux condition for the liquid and vapor continuity equations

represents an impermeable layer in the soil. This condition may or

may not physically exist in the field, but for most situations
encountered, the errors introduced into the solution at the depth of

chosen should be minimal. The boundary conditions used at the lower

boundary were

T- = 0 (2-32a)
az Zzo
800
o I = 0 (2-32b)
a zz0Z











z = 0 (2-32c)


The system defining the movement of water vapor, liquid water and

energy is defined by a system of partial differential equations (2-22,

2-25, 2-26) with boundary conditions (2-31a-c, 2-32a-c). The energy

equation is coupled to the continuity equations by the rate at which

the water is changed from liquid to vapor and through the movement of

sensible heat associated with the flux of water between zones of

differing temperatures. The conservation of water vapor is coupled to

the soil temperature indirectly in the calculation of saturated vapor

pressure. This coupling, along with the variation of the soil

properties with time and space, renders an analytical solution beyond

reach, therefore; a numerical solution was required.

Determination of Model Parameters

Solution of the governing equations for the energy and mass

balance for the soil requires knowledge of the properties relating to

the soils ability to diffuse heat, water, and water vapor. The other

parameters to be determined relate to the rate at which heat and water

vapor are dissipated from the soil surface to the air. In order to

determine a solution, relationships for determining thermal, hydraulic

and vapor diffusivities as well as the surface transfer coefficients

must be determined.

Diffusivity of Water Vapor

Diffusivity is the constant of proportionality relating flow to

the gradient in potential as stated in Fick's Law of molecular

diffusion. For the case of water vapor diffusing through air, the







31
diffusivity, or coefficient of diffusion, is a measure of the number of

collisions and molecular interaction between molecules of water vapor
and the other constituent components of the air (ASHRAE, 1979) and is a
function of temperature, total air pressure and partial pressure of
water vapor. Eckert and Drake (1972) presented the following equation

to calculate the diffusivity of water vapor in air.



Dva 2.302 ( (T X 10-5 (2-33)
va p


where

Dva = diffusivity of water vapor in air [m2.s-1]
p = atmospheric pressure [Pa]

Po = reference atmospheric pressure [Pa]
= 0.98 X 105 Pa
T = air temperature [OK]

T = reference air temperature [OK]
= 256 OK
However, for water vapor in the soil air space, the path is more
convoluted, thus increasing the probability of collisions with other
particles and lowering the kinetic energy of the water vapor molecules.
To account for the increased path length within the soil, a tortuosity
factor (a) has been introduced to reduce the effective coefficient of
diffusion in other porous materials. De Vries (1958) used a value for
the tortuosity of 0.667. Using this information the coefficient of
water vapor in the soil (Dv) becomes
Dv = Dva


= 1.535 (P T 1.81 X 10-5 (2-34)
~p ) To)









For the purposes of simulation, the barometric pressure (p) was assumed

to equal the standard pressure (po). The diffusivity then became a

function of time and space due to the temporal and spatial variation of

the soil temperature. Diffusivity was determined throughout the soil

profile by substitution of the soil temperature in equation (2-34).

Thermal properties

Very little literature exists regarding measurement of the thermal

properties of the soil. Soil composition as well as density and water

content affects the thermal properties of the soil (Baver, 1972). The

thermal diffusivity is the ratio of thermal conductivity to the product

of soil density and specific heat. Density is a property which can be

obtained as a function of depth at a given location by core samples.

Vries (1975) describes a method by which the volumetric heat

capacitance and the thermal conductivity of the soil can be calculated

based upon the volume fractions of the various soil constituents.

The volumetric heat capacitance is defined as the product of the soil

density and the specific heat and can be calculated from:



Cs = qCq + XmCm + XoC + ww + xaCa (2-35)



where

C = volumetric heat capacity [J-m-3.K-1]

x = volume fraction [m3.m-3]

q = quartz

m = mineral

o = organic







33

w = water (same as volumetric water content)

a = air

s = soil composite

The difficulty arises in estimating the various volume fractions of the

different components. For quartzitypic sands, approximately 70 to

80 percent of the solid constituent of the soil is quartz, with

generally less than 1 to 2 percent organic material and the remainder

consisting of other ninerals. According to information presented by

DeVries (1975), the volumetric heat capacity and density of quartz and

other minerals are very similar. For the purposes of this simulation,

the volume fraction of the organic material was assumed to be zero.

The thermal conductivity cannot be calculated in such a straight

forward manner. Thermal conductivity is defined as the constant of

proportionality in relating the heat flux by conduction to the

temperature gradient. Conduction of heat occurs due to physical

contact between adjacent substances. In a solid material such as steel

or concrete which is fairly homogeneous, the material conducting heat

can be considered continuous. However, the soil is a mixture of solid,

liquid and gaseous components and the area of physical contact between

soil particles is a function of the particle geometry. When the soil

is dry, the area of contact may be a single point. As the soil is

wetted, a thin film of water adheres to the soil particle and increases

the contact area between adjacent particles. This increase in area

accounts for a rapid change in the thermal conductivity as the soil is

initially wetted (Figure 2-1). Over a range of water contents between

dryness and saturation, the increase in thermal conductivity becomes

linear with increasing water content. As the soil approaches









saturation, the thermal conductivity begins leveling off to some

relatively constant value. Another complicating issue in calculating

the thermal conductivity of the soil is that the temperature gradient

in the solid and liquid portion of the soil can be significantly

different from that in the gaseous phase of the soil.

De Vries (1975) presented a method by which the thermal

conductivity could be calculated using a weighted average of the

various soil constituents where the weighting factors were a product of

the individual volume fractions and geometric factors as follows

kqXq + km xmA + k oX 0 + k X + kaxaa (2-36)
Sqxq + kmXm + koXo + kX + kaxa


where

S = thermal conductivity of the soil [W-m-1K-1]

\ = thermal conductivity of soil component [W'm-1.K-I1]
xi = volume fraction of soil component [m3m-3]

ki = geometric weighting factor dimensionlesss]
i = quartz (q), mineral (m), organic (o), water (w), air (a)

The geometric weighting factor depends upon geometric

configuration of the soil particles and the incorporated void space.
It represents the ratio of the spatially averaged temperature gradient

in the i-th soil component to the spatially averaged temperature

gradient of the continuous component of the soil (usually water). For

example, kq represents the space average of the temperature gradient
in the quartz particles in the soil to the space average of the

temperature gradient in the water. The geometric weighting factor can








be determined by

ki (kia + kib + kic) (2-37)

where
kij = the ratio of the temperature gradient in the i-th
component to the temperature gradient in the continuous
component in the direction of the j-th principal axis
i = quartz (q), mineral (m), organic (o), water (w), air (a)
j = a, b, or c for each of the three principal axis of the
particle
The ratio of the temperature gradients in the i-th component can be
determined by the following:



kj = 1 + 1 (2-38)
+ ( x -1

The shape factor for each principal axis (gj) can be approximated by
various empirical relationships depending upon the ratio of the unit
vectors (ua, ub, uc) of the principal axes of the soil components
(Table 2-1). The sum of the three shape factors must be unity.
In most cases, water is considered to be the continuous phase of
the soil in determining soil thermal conductivity. However, as the
soil dries and the film adhering to the surface of the soil particle
begins to break, making air the continuous phase. Equation (2-37) can
be used in these cases replacing the thermal conductivity of water (Aw)
with the thermal conductivity of air (Xa). De Vries (1975) noted that
the values for the thermal conductivity in the case of air being the
continuous phase were consistently low by a factor of approximately







36

1.25. This method yielded values of thermal diffusivity within ten

percent of those measured (de Vries, 1975). Extensive detail regarding

calculation of the thermal conductivity of the soil is given in

de Vries (1963).

The thermal properties were calculated using equation (2-35) for

the volumetric heat capacity and equations (2-36), (2-37) and (2-38) to

determine the thermal conductivity. Volume fractions of the various

soil constituents was determined from soil classification data and

knowledge of the soil bulk density and porosity. Shape factors used

for calculation of the thermal conductivity were for a typical sand

grain (Table 2-1).

Hydraulic Properties

The parameter governing the movement of liquid water in the soil

is a measurable property of the soil and is analogous to the thermal

diffusivity. The hydraulic diffusivity is a derived property of the

soil (i.e. not directly measured) and is defined as the hydraulic

conductivity divided by the specific water capacity of the soil (Baver

et al., 1972). The hydraulic conductivity is the constant of

proportionality for the diffusion of water in response to a gradient in

the soil water potential (Figure 2-2), while the specific water

capacity is the slope of the soil water retention curve. The

hydraulic conductivity (Figure 2-3) and the specific water capacity

(Figure 2-2) vary according to the soil composition as well as the soil

water potential. The measurement of hydraulic conductivity can be

accomplished by several methods, but most all require meticulous

control of the potential gradients and a great deal of time. This is

especially true if measurements are desired over a wide range of soil







37
water potential. Measurements of the soil water retention curve

require considerably less detail and are generally published for a

wide variety of soil types. Van Genuschten (1980) presented a method

by which the hydraulic conductivity could be calculated for unsaturated
soils using the soil water retention curve and the saturated hydraulic

conductivity. The Van Genuschten approach involves estimation of an

equation for the soil water retention curve of the form


9 = ( 10 r 1) m (2-39)
Ts r 1 + (a)n

where

9 = dimensionless water content

0 = volumetric water content [m3m-3]

r = residual volumetric water content [m3m-3]

es = volumetric water content at saturation [m3m-3]
= soil water potential [ m ]
m,a = regression coefficients
n = (1 m)

The coefficients m and a are determined by nonlinear least squares
regression of the soil water retention curve. That function is then

substituted into equations for the hydraulic conductivity presented by
Mualem (1976). The resulting expression for the diffusivity is



D(B) = (1 -m) k (0.5 I/m) [(1-e1/m)-m+ (1_01/m)m_2] (2-40)
am(Os-Or)


The Van Genuschten method yields a continuous function for the
hydraulic diffusivity which is highly desirable for numerical







38

simulations over the range of the water contents expected to occur in

the fields.

For modeling purposes, the Van Genuschten method was employed for

published potential-water content data for a Millhopper fine sand

(Carlisle, 1985). The water release curve usually varies with depth

due to the spatial variation of soil composition. Data for the a

single water release curve for a uniform soil profile was obtained by

averaging the volumetric water content for the specified water

potential over the A-i and A-2 horizons of a Millhopper fine sand. The

nonlinear regressions were then performed to yield the residual water

content, and the regression coefficients, a and n.

Surface Transfer Coefficients

The final parameters to be estimated are the surface heat and mass

transfer coefficients. Penman (1948) used an empirical wind function

to calculate the mass transfer coefficient based upon wind speed.


hm'- a WSb (2-41)

where

hm = surface mass transfer coefficient
WS = wind speed

a,b = empirical constants

Another approach is to use the equations of motion to describe mass and

energy transfer within the atmospheric boundary layer. Brutsaert

(1982) presented a detailed review of the equations of motion as

applied to the atmosphere. The basic assumption in these analyses is

that the boundary layers for momentum, energy and mass are similar.

Consequently, the surface sublayer becomes the area of most concern and







39

is defined as the fully turbulent region where the vertical turbulent

fluxes of mass and energy do not change appreciably from that at the

surface (Brutsaert, 1982). In other words, the vertical flux of mass
and energy is constant.

According to Brutsaert (1982), Prandtl introduced the use of the

logarithmic wind profile law into meteorology in 1932. This is an
approximation of the wind velocity profile in the surface sublayer



WS = In (-0) (2-42)



where

WS = wind speed [ m-s-1]

U* = shear velocity

1/2
= TI
P



z0 = surface roughness height [ m ]
k = von Karman constant dimensionlesss]

T = shear stress at the surface [N-m-21
p = density of air [kg-m-3]
According to Sutton (1953), for practical purposes, the shear velocity

can be estimated as


WS
U = (2-43)

The bulk mass transfer within the surface sublayer is defined by

Qm = CeyWSr (Pvs Pvr) (2-44)













where

Qm = vertical mass flux [kg'm-2.s-1]

WSr = wind speed at reference height, zr [m's-1]
Cer = Dalton Number; dimensionless mass transfer coefficient

Pvr = water yapor concentration at reference height, zr
[kg'm- ]

Pvs = water vapor concentration at soil surface [kg'm-3]
The surface mass transfer coefficient (hm) used in this model is

related to Cer by

hm = WS, Cer (2-45)


Brutsaert (1982) presented functions for the Dalton number in terms of

the drag coefficient and the dimensionless Schmidt number as


1/2
Cd,
Cer = (2-46)
-1/2
(B + Cdr)

where

B = function of dimensionless Schmidt number

Cer = Dalton number

Cdr = surface drag coefficient dimensionlesss]

2

WS








41

The empirical function, B, depends upon whether or not the surface is

hydrodynamically smooth or rough. In general, if the Reynolds number

based upon the shear velocity and the roughness height is less than

0.13 then the surface is considered smooth and B is determined using

Equation (2-47). The surface is rough if the Reynolds number is

greater than 2.0 and Equation (2-48) is used to determine the value

of B.



2
/3
B = 13.6 Sc 13.5 (2-47)

/4 2/
B = 7.3 Re 4 Sc 5.0 (2-48)

where

Sc = Schmidt number
Dv

Re = Reynolds number
z0 U*



The dimensionless heat transfer coefficient or Stanton number

(Chr) can be determined by substitution of the Prandtl number for the

Stanton number for the Schmidt number in equations (2-47) and (2-48)

above. The surface heat transfer coefficient can then be determined by


hh = Chr Pacpa WSr


(2-49)







42

Fuchs et al. (1969) used a method in which instability in the

lower atmosphere was accounted for in conjunction with the logarithmic

wind profile to calculate the surface mass transfer coefficient. This

involved using the KEYPS function (Panofsky, 1963) to determine the

curvature of the diabetic lapse rate (w) of the logarithmic wind

profile. The surface mass transfer coefficient was calculated from

-2
h = k2WS ( 7 + In (z + D) (2-50)
m Zo


where

hm = transfer coefficient of mass from surface to height, z,
in the air [m-s-1]

k = von Karman constant dimensionlesss]

WS = wind speed at height, z [m-s-1]

D = height displacement [m]
= d + z0

d = height above soil surface where wind velocity is zero
[m]
Z0 = roughness length [m]

z = height above the displacement height [m]

x = curvature of the diabetic lapse rate or diabetic
influence function dimensionlesss]

Fuchs et al. (1969) then calculated the surface heat transfer

coefficient using


hh = hmcpaPa


(2-51)









The diabetic influence function accounted for the transfer of air

movement in the vertical direction due to density gradients caused by

temperature gradients and is a function of the Richardson number

(Figure 2-4). Fuchs et al. (1969) stated that the roughness height

varied from 0.2 to 0.4 mm for a bare soil surface and accounted for a

small variation in the calculated heat transfer coefficient.

Therefore, for the purposes of this study, a roughness length of 0.3 mm

will be used Fuchs et al. (1969) also noted that the height of zero

wind velocity (d) was zero for a bare soil surface. This term was

employed in the approximate wind profiles to account for the fact that

wind does not penetrate full vegetative canopies and for practical

purposes the surface where the wind velocity is zero occurs at some

finite height above the soil surface (Brutsaert, 1982; Sutton, 1953).

Fuchs et al. (1969) compared transfer coefficients calculated using

equation (2-50) to that determined from field data for a bare sandy

soil and obtained fairly close agreement.

For the purposes of this model, the approach used by Fuchs et al.

(1969) to determine the surface mass (Equation 2-50) and heat

(Equation 2-51) transfer coefficients was employed. The diabetic

influence function was utilized to account for atmospheric instability

as proposed by Fuchs et al. (1969). A roughness height (Zo) of 0.3 mm

was utilized. The equations used in the formulation of this model and

the determination of parameters are summarized in Table 2-2

Numerical solution

The partial differential equations used to describe the mass and

energy balance in the soil must be solved numerically since analytical

solutions are not possible for the coupled nonlinear equations. Many







44

solution techniques utilize the finite difference form of the
differential equations. The differential operator in the continuous
domain becomes a difference operator in the discretized domain.
Difference operators are either forward, backward, or central
difference operators. The difference operators for the function of the
independent variable, x, [ f(x) ] are


af f(xj+l) f(xj)
forward: af = Af f(x ) f(xj) (2-52)
ax xj+1 xj

af f(xj) f(xj-1)
backward: = Vf = (2-53)
Xj xj-1
af
central: = 6f (2-54)

1 f(xj+1) f(xj) f(xj) f(xj-1)
T xj+1 xj xj xj-1



In expressing the differential equations in difference form, the
derivative with respect to time is accomplished using the forward
difference operator, while spatial derivatives are expressed using any
of the three difference operators. The continuous domain must first be
discretized or divided into several discrete regions such that the
finite difference approximation of the differential equation approaches
the differential equation in the limit of the grid spacing going toward
zero (Figure 2-5).
The partial differential equation defining the conservation of
energy within the soil profile (equation 2-26) stated that the change
in sensible heat over time is due to heat transferred by conduction
plus sensible heat carried by the diffusion of water in the liquid and







45

vapor phases from a region of differing temperature less the latent

energy required for evaporation of water. Examining this for the jth

node in the discretized domain, the above equation can be expressed in

terms of heat flux transferred (q) into node j from the surrounding

nodes as.


dx-dydwj)qsj = dx-dy (qcj+l,n + qcj-l,n + qLj+l,n

+ qLj-l,n + qvj+l,n + qvj-l,n)

(dx-dy-dwj)qej,n


(2-55)


The subscripts

s :
c :
e
L :
v
j :
j+l
j-1
n


in the above representation denote

change in sensible heat
heat conduction
latent heat
heat transfer due to diffusion of water (liquid)
heat transfer due to diffusion of water vapor
the current node
the node immediately following node j, and
the node immediately preceding node j.
the current time step


and


dx-dy :
dx-dyddwj :
dwj :

The change in

to n+1 is


cross-section area normal to z axis
volume of the node j
height of the cell for node j

sensible heat per unit volume of node j from time step n


Tj,rn+ Tj,
qsj = Cs dt


(2-56)


The heat flux conducted from nodes j+1 and j-1 to node j at a given

time step, n, is described by Fourier's law of heat conduction and is

expressed as follows









Xj
qcj+l,n= (Tj+i,n- Tj,n) (2-57)


qcj-l,n= (Tj-l,n- Tj,n) (2-58)
dzj-1

noting that dzj represents the difference between the depth of nodes
j+1 and j.
In developing the differential equations for this model, it is
assumed that mass movement is due to diffusion and obeys Fick's law of
diffusion. Therefore, the heat transferred by diffusion of water and
water vapor from node j+1 to j is expressed as

S(Oj+,n- Oj,n) (Tj+1,n+ Tj,n)2
qLj+l,n = PwCpwjDLj dz2 (2-59)


(Pvj+,n Pvj,n) (Tj+1,n+Tj,n)
qvj+1,n = cpvjDvj dz 2 (2-60)

and similarly for the heat flux transferred by mass diffusion from node
j-1 to node j as

(0j-l,n 1j,n) (Tj-l,n+Tj,n)
qLj-l,n = PwjCpwjDLj dzj-1 2 (2-61)


S(Pvj-l,n Pvj,n) (Tj-l,n+Tj,n)
qvj-l,n = cpvjDvj dzj- 2 (2-62)

The latent heat associated with the phase change of water for the
current time step (Ej,n) must be transferred per unit volume of soil
associated with node j and is determined by


hfgjEj,n


(2-63)


qej,n =










Substitution of the numerical expressions for the individual terms of
the energy balance (equation 2-55) and dividing by the cross-sectional
area normal to the z axis (dx-dy) yields



Tj,n+l-Tj,n Tj+l,n-Tj,n Tj-1,n Tj,n
dwjCs dt= J dz- + dzj_


(Oj+1,n-0j,n) (Tj+l,n+Tj,n)
+ PwjCpwjDLj dzj 2


(0j-i,n 0j,n) (Tj-l,n+Tj,n)
+ PwjcpwjDLj dzj- 2

(Pvj+1,n Pvj,n) (Tj+1,n+Tj,n)
+ cpvjDvj dzj 2

(Pvj-l,n Pvj,n) (Tj-l,n+Tj,n)
+ cpvjDvj dzj-. 2


dwj (hfg,jEj,n) (2-64)


The partial differential equation describing the diffusion of
water within the soil can be transformed to difference form in a
similar fashion as the energy equation. The change in water content
of a volume of soil for the jth node is caused by diffusion of water
from nodes j+1 and j-1 to node j less the amount of water changed to a
vapor phase. The numerical expression for the water continuity per
unit area becomes







48


(0j,n+l- Oj,n) 0 +1,n- 0j,n (Oj-l,n- Oj,n)
dwJ dt DLj dzj + dzj-l

Ej,n
dwj (2-65)
Pw,j


Similarly, the differential equation describing the conservation of
mass in the vapor state is transformed as

(Pvj,n+1 Pvj,n) (Pvj+l,n Pvj,n)
S- dt Dvj dzj
(2-66)
(Pvj-l,n Pvj,n) dwj Ej,n
+ -Dv dzj-1 + (Sj j,n)



Equations (2-64), (2-65), and (2-66) constitute the numerical
equivalents for the partial differential equations describing the
conservation of thermal energy and mass in the soil profile. However,
the conservation relationships have yet to be developed for the
boundaries at the soil surface and the bottom of soil profile. Since
there is no volume associated with the surface node (Fig. 2-1), there
is no storage capacity for energy or mass. Therefore, the flux of mass
or energy from the node directly below the surface (j=2) to the surface
added to the net flux due to solar radiation onto the surface and the
heat flux due to convection from the air to the surface must be zero.
This can be expressed directly in difference form as follows.









energy: (T2,n T,n) = Rnn + hhn(Tan T,n)
energy: --- (T2,n T, ) = Rn, + hhn(Tan T1,n)


+ PwlcpwlDL1


DL1
water: --


Dvi
vapor: -T-


(2-67)
(02,n 1l,n) (Pv2,n Pvl,n) (T2,n + Tl,n)
dzl + CpvlDv1 dz1 2


(02,n 91,n) + P = 0



(Pv2,n Pvl,n) + hm(Pva Pvl,n) = 0


(2-68)



(2-69)


: net solar radiation incident
at time step n [W-m'2]

: precipitation at time step n
: boundary layer mass transfer
step n [ms' ]


upon the soil surface

[m3.m-3.s-1]
coefficient at time


: boundary layer heat transfer coefficient at time
step n [W-m- 2.- ]
: air temperature at time step n [JK]
: ambient water vapor concentration at time step n
[kgm-3]


The boundary conditions at the lower boundary of the soil are

developed in a similar manner. The last node has volume and

capacitance for storage of energy and mass. A zero flux boundary

condition is used. There are two ways in which the zero-flux condition

can be represented. These will be developed for the water for the

purpose of illustration.


where:


P

hmn

hhn

Tan

Pvan







50
If the water content of the last node (j=nc) were equal to that of
the preceding node (j=nc-1), then no water would flow between the two
nodes. This would cause both of the last two nodes to act as one node

during the simulation. The other viewpoint would be that the sum of
the flows into the last node from the preceding node and an imaginary

node following must be zero.

DLnc DLnc
Sz (On+c+l,n nc,n) = -- l(nc-l,n Onc,n) (2-70)


If the distance between nodes nc and nc-1 is the same as the distance
between the nodes nc and the imaginary node nc+1, then it follows that

Onc+l,n = Onc-1,n (2-71)


Substitution of 2-71 into 2-65 yields as the boundary condition for the

node, j=nc, the following

(Onc,n+1- Onc,n) DLnc
dwnc dt 2 d-- (Onc-1,n-nc,n)
(2-72)
nc,n
dwnc wnc
Pwnc

The same approach produces the following as the boundary condition for
the vapor continuity and energy equations, respectively.

(Pvnc,n+l- Pvnc,n) Dvnc
dwnc dt -= 2 (Pvnc-1-Pvnc)
(2-73)
dwnc Enc,n

(S Oj,n)

(Tnc,n+1-Tnc,n) (Tnc-l,n- Tnc,n)
dwnc Cs,nc dt = 2 fnc dznc-1










(nc-1,n Onc,n) (Tnc-l,n+ Tnc,n)
+ 2 PwncCpwncDLnc dznc-1

(Pvnc-l,n'Pvnc,n) (Tnc-l,n+ Tnc,n)
+ 2 CpvncOvnc dznc-- 2

+ dwnc Enc,n (2-74)




The conservation relationships provide sufficient information to

determine three of the desired quantities for the soil profile leaving

a fourth remaining unknown. The constitutive relationship requiring
that the water in the liquid phase be in thermodynamic equilibrium with

the vapor phase was used to provide the remaining equation. Since the

surface node provides only an interface between the soil and the

atmosphere, this equilibrium condition was not necessary for the

surface node. Under atmospheric conditions, the soil may not become
completely dry (0 % volumetric water content), but will reach some
moisture content which is in equilibrium with the atmosphere. This is

typically taken to be the same as the permanent wilting point of the
soil (0= -150 m) and is the same as the residual water content
described by van Genuschten (1980). As the soil surface dries, the
soil reaches this equilibrium moisture content and is assumed to act as
an interface between the air and the wet soil below.

The system of equations (2-64) through (2-69)and (2-72) through
(2-74) along with equation (2-30) represent an explicit solution to the
differential equations, that is, the values of the state variables at
the next time step are functions of those at the previous time step.
Using the explicit representation allows a simple algorithm to be used







52
during the solution phase of the system of equations and is equivalent

to an Euler integration in time (Conte and de Boor, 1980). The

disadvantage of the explicit solution is that the numerical error may

propagate through time and grow. The maximum time step is functionally

related to the boundary conditions and thermal properties of the soil.

The system of equations can also be changed such that all

derivatives are evaluated at the next time step in which none of the

state variables are known. This implicit representation requires a

relatively complicated iterative or matrix solution technique for the

system of equations. The advantage of implicit solution methods is

that the value of the state variables do not depend upon previous

values implying that the only source of error would be due to round-off

or truncation errors and would not propagate or grow with time and

would yield an unconditionally stable solution (Conte and de Boor,

1980).

Another technique which has some of the desirable characteristics

of both the explicit and implicit methods is the alternating direction

(ADI) method. This is accomplished by incrementally marching through

space in one direction (z=0 to zo) evaluating the derivatives

containing the previous node (j-1) at time, t + 1/2 dt, and those

containing the following node (j+1) at time, t. Then returning in the

opposite direction (z=zo to 0), evaluate the derivatives containing the

node j+1, at t=t+dt, and the derivatives containing j-1 at t=t+1/2dt.

This technique uses a relatively simple algorithm similar to that for

the explicit methods because all of the values of the state variables

used in estimating the state variable at the next time step are known

quantities. However, the number of equations to be evaluated is twice







53
that of the standard explicit methods. Increased stability is obtained

over the standard explicit methods but is less than that for the

implicit techniques. This implies that using the same grid spacing, a

larger time step can be used in the ADI methods than that for explicit

methods.

An ADI finite difference technique was used to solve the system of

equations for the soil profile due to the increased stability

characteristics over explicit methods. Appendix A contains a detailed

development of the numerical equations used in the ADI technique.

Hydraulic (equation 2-40) and thermal properties (equations 2-35 and

2-36) of the soil and the surface transfer coefficients (equations 2-50

and 2-51) were calculated at the beginning of each time step. A

general description of the solution algorithm is shown in Figure (2-6).

The energy and water equations were solved for the temperature and

volumetric water content, respectively at the next time step. The

equilibrium condition was used to determine the vapor concentration by

substitution of the values of soil temperature, water content, water

potential, and saturated vapor concentration in equation (2-30). The

vapor continuity equation was used to determine the evaporation rate.

The calculations were repeated using the new evaporation rate

until the absolute value of the maximum fractional change in any of the

variables was less than a prescribed convergence criterion.

The system of numerical equations was solved using computer code

written in FORTRAN77. A variable grid spacing was used throughout the

soil profile with the smaller mesh being located near the soil surface

due to expected large gradients in temperature and moisture content

once the soil surface begins to dry (Table 2-3). The program was then







54
run to simulate the response of the soil system under constant

radiation, ambient air temperature and relative humidity conditions for

a period of 72 hours using various time steps to evaluate the numerical

stability and numerical error characteristics of the procedure.

Thermal and hydraulic properties of the soil were assumed to be

constant for the duration of the simulation for an Millhopper fine

sand. The water content and temperature profiles were assumed to have

a uniform initial distribution as were density and soil porosity.

Overall mass and energy balances were calculated for the soil

profile during the simulation. Any residual in these balances would

constitute error arising from the numerical algorithm and was

accumulated on an hourly basis for analysis purposes. The sum of the

squares of the residuals for the energy and mass balances for the

simulation period are presented in Figures 2-7 and 2-8, respectively,

for the different time steps. The sum of the squares of the residuals

(SSR) for the energy balance decreased when the time step of was

increased from 30 to 60 s then remained fairly constant for larger time

steps. The square root of the SSR represents an estimate of the

standard error of the estimate of the total heat flux. For the time

steps of 60, 120, 300 and 600 s, this represents approximately one

percent of the cumulative soil heat flux. The standard error of the

mass balance was approximately two percent of the cumulative

evaporation over the 72 h simulation (600 s time step). It is expected

that as properties are varied with time, that the standard error would

increase due to increased gradients in soil temperature and water

content particularly near the surface. Therefore, a time step of 60 s

was utilized in the model validation and sensitivity analysis.









Table 2-1.



Object


Formulae for the geometric shape factors used in
calculating thermal conductivity based upon ratios
of the unit vectors of the principal axes.


Sphere

Ellipsoid of revolution
x=y=nz n = 0.1
n = 0.5
n = 1.0
n = 5.0
n = 10.

Elongated cylinder with
elliptical x-section


x = ny;


0.14 0.14


0.33


0.49
0.41
0.33
0.13
0.07


(n + 1)-1


0.33


0.49
0.41
0.33
0.13
0.07


n/((n +1)


0.33


0.02
0.18
0.33
0.74
0.86


0.00


0.72


Typical sand grain













Table 2-2. Summary of equations used in a model of heat and
mass transfer in the soil to determine state
variables and transport parameters.

Process or Parameter Equation No.

Conservation of mass (liquid) 2-22

Conservation of mass (vapor) 2-25

Conservation of energy 2-26

Vapor-Liquid equilibrium 2-30

Surface Boundary Conditions

Conservation of mass (liquid) 2-31b

Conservation of mass (vapor) 2-31c

Conservation of energy 2-31a

Lower Boundary Conditions

Conservation of mass (liquid) 2-32b

Conservation of mass (vapor) 2-31c

Conservation of energy 2-32a

Soil properties

Diffusivity of water vapor in soil (Dv) 2-34

Volumetric heat capacity (Cs) 2-35

Thermal Conductivity (X) 2-36

Hydraulic Diffusivity (DL) 2-40

Surface Mass Transfer Coefficient (hm) 2-50

Surface Heat Transfer Coefficient (hh) 2-51
















Table 2-3.


Mesh spacing used in grid generation for the coupled
heat and mass transfer model for sandy soils.


Cell Width
(cm)


0.0
1.0
1.0
1.0
1.0
1.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
10.0
10.0
10.0
10.0



10.0
10.0
10.0


Distance to
next node
(cm)


0.5
1.0
1.0
1.0
1.0
1.5
2.0
2.0
2.0
2.0
2.0
2.0
2.0
3.5
5.0
5.0
5.0
5.0
5.0
5.0
7.5
10.0
10.0
10.0
10.0



10.0
10.0


Node


Depth
(cm)


0.0
0.5
1.5
2.5
3.5
4.5
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
23.5
28.5
33.5
38.5
43.5
48.5
53.5
61.0
71.0
81.0
91.0


181.0
191.0
201.0








58


















15



12--

E
E





0


o 3

0)


0.0 0.1 0.2 0.3 0.4 0.5 0.6

Volumetric Water Content (m3.m-3)













Figure 2-1. Thermal conductivity of a typical sandy soil as a function
of volumetric water content (DeVries, 1975).









59

















104 0.010
Potential

1 \ _-- Capacity -/ 0.008 E
E 103 -
/ ooo8

S102
0

0.004 )
o /
10
0 /
L I 0.002 L
/ 0

1 0.000
-1 I I I 0.000
0.00 0.10 0.20 0.30 0.40

Volumetric Water Content ( cm3 / cm3 )














Figure 2-2. Soil water potential (tension) and specific water capacity
as a function of volumetric water content for a sandy soil.









60
















102

10 -- Diffusivity 10
~1 Conductivity o105
E 104
1 0-- 103
10-2 ---
0 102

o 10-4..
10-510-1
o 10-
10-6-- /*10-2 1
1 10 ..10-3

10-8 I 10o-4
0.00 0.10 0.20 0.30 0.40
Volumetric Water Content ( cm3/cm3)















Figure 2-3. Hydraulic conductivity and hydraulic diffusivity as a
function of soil water content for a typical sandy soil.


















































-1.50 -1.00 -0.50 0.00


0.50
0.50


Richardson Number ( Ri)














Figure 2-4. Relationship of the diabetic influence function and the
Richardson number.


0.5


0.0-1


-1.5
-2


2.00


I i ,


-0.5-


-1.0 -














H=f(To,hh,WS)
P Rn Qv=f(To,RH,hm,WS)


j=2

j=3





AZ j- j-l



i+l




nc-I

nc-


Figure 2-5. Schematic of discretized domain for evaporation and soil
temperature model.





















































Figure 2-6. Flowchart describing the solution algorithm to solve the
system of finite difference equations for a coupled heat
and mass transfer model.





















3.

0
2
o
S2-






E
4,
0
o
0
o" 1

E
C-


0


60 120
Time Step ( s )


300


600


Figure 2-7. Sum of squared residuals for the soil profile energy
balance after a 72 hour simulation using various time
steps.


1


_1_1111 _



















Sq


Sq


60 120
Time Step (


K


300


600


Figure 2-8. Sum of squared residuals for the soil profile mass balance
after a 72 hour simulation using various time steps.


0.06


0.04


0.02-


if fll


m\c~
i

T














CHAPTER III
EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND
ENERGY TRANSFER PROCESSES IN THE SOIL

Introduction

The process of evaporation of water from the soil is a complicated

process in which the dynamic processes of heat and mass transfer are

inter-related. Lysimetry has been defined as the observation of the

overall water balance of an enclosed volume of soil. Lysimeters have

been used for over 300 years to measure soil evaporation and crop water

use (Aboukhaled et al., 1982) and can be generally classified as either

weighing or non-weighing. As their names imply, weighing lysimeters

measure evaporation by monitoring the weight changes within the

enclosed soil volume while the non-weighing, or drainage lysimeters

determine evaporation by monitoring amount of water drained from the

bottom of the tank and the amount of rainfall upon the lysimeter. Most

drainage lysimeters are monitored on a seven to ten day cycle while the

weighing lysimeters can be monitored continuously to obtain hourly

evaporation rates.

The energy status of the soil can generally be determined by

measuring the temperature distribution over time. The energy flux at

the surface must be determined as well as latent heat transfer to

facilitate a complete energy analysis. Surface heat flux occurs as

radiant and convective heat transfer. The radiant energy impinging

upon the soil surface is either absorbed or reflected. The amount of

reflected and absorbed energy is primarily dependent upon the soil







67

surface. The reflectance and absorbtance change dramatically as a

function of water content. Convective heat losses from the surface are

generally estimated rather than measured using empirical functions to

determine surface heat transfer coefficients.

Objectives

The processes of energy and mass transfer in the soil are most

readily represented by the rate at which water is lost from the soil

(evaporation rate), the cumulative water loss (cumulative

evaporation), temperature, volumetric water content, and water vapor

density. All are temporal in nature, while the latter three are

functions of depth as well. The goal of the experimental design was to

provide data for the validation and calibration of the transport model

described in the previous chapter. Specific objectives were

1. to design a lysimeter and calibrate the instrumentation to
measure hourly evaporation rates,

2. to measure the vertical distribution of the soil temperature
and volumetric water content as function of time in the
lysimeter,

3. to measure ambient weather conditions to sufficiently
describe the boundary conditions for a coupled heat and
mass transfer model, and

4. to determine the physical characteristics of the soil
contained in the lysimeter.

Lysimeter Design. Installation and Calibration
Design and Construction

Weighing lysimeters were used in this research to meet the

objective of measuring hourly evaporation from a bare soil surface.

The weight of the lysimeter may be monitored by several different

methods. Mechanical measurement may be achieved by either supporting

the soil container with a lever and counterweight system or by







68

supporting the lysimeter directly by several load cells (Harrold, 1966;

Van Bavel and Meyers, 1961). The weight may also be detected by

supporting the soil container by a bladder filled with fluid and

monitoring the pressure of the fluid or the buoyancy of the lysimeter

container (Harrold, 1966; McMillan and Paul, 1961; King et al., 1956).

The hydraulic method has the disadvantages of being sensitive to

thermal expansion of the fluid and the high possibility of developing

leaks in the bladder and losing the fluid. The lever mechanism has the

advantage of requiring only one relatively low capacity load sensor

thus reducing instrumentation costs. The disadvantage of the counter-

balance system is requirement of extensive underground construction to

contain the lever apparatus. This would require disturbing the

surrounding soil and border area and may be considered undesirable in

some cases. The cost of additional supporting structures and

excavation could be considerable as well. Three to four high capacity

load cells are required to support the lysimeter for direct weighing.

The higher cost of the load cells for the direct weigh method may

offset the cost of the increased excavation required for the counter-

weight system. Direct-weigh lysimeters can be constructed such that a

high sensitivity can be achieved. Dugas et al. (1985) reported a

resolution of 0.02 mm of water for a direct weighing lysimeter with a

surface area of 3 m2. Most of the designs presented in the literature

have load cells which are difficult to access in the event of need for

maintenance or replacement.

A direct-weigh system was chosen because of its relatively simple

support system design and due to space limitations at the proposed

construction site at the Irrigation Research and Education Park (IREP)









located on the University of Florida campus in Gainesville, Florida

(Figure 3-1).

Aboukhaled et al. (1982) presented a relatively comprehensive

review of the existing literature regarding lysimeter construction and

design and noted the practical aspects of certain design

considerations. It was noted that the lysimeter annulus (gap area

between the soil container plus the area of the retaining wall) should

be minimized to reduce the effects of the thermal exchange between the

lysimeter soil mass and the surrounding air. Several researchers

(Aboukhaled et al., 1982; Black et al., 1968) made recommendations for

minimum lysimeter areas of 4 m2 for the primary purpose of reducing

the edge effect of the soil-to-soil gap between the lysimeter interior

and the surrounding fetch. Maintaining a 100:1 ratio of fetch to crop

height has been recommended to minimize border effects. The site

selected for the lysimeter construction satisfied the fetch

requirements for very short crops or bare soil. Dugas et al. (1985)

noted that the problems caused by soil-to-soil discontinuity should be

minimal if the lysimeter surface area were greater than 1 m2. Soil

depth should be sufficient so as not to impede root growth of crops to

be planted in the lysimeter.

Two soil containers were constructed using 4.8 mm (3/16 in.) steel

plate for the floor and side walls (Butts, 1985). All seams were

welded continuously to prevent water leaks and the entire container was

painted with an epoxy-based paint to prevent corrosion. The lysimeter

measured 305 cm long, 224 cm wide and 130 cm deep. This provided a

soil surface area of 6.8 m2 and a soil volume of 8.9 m3. Main floor

supports consisted of four 20-cm steel I-beams (W8x15) and extended the







70

entire length past the edge of the soil container (Figure 3-2). To

alleviate the need for underground access to the load cells supporting

the lysimeter, two 2-cm rods were inserted into both of the protruding

end of each floor joist and extended above the top of the lysimeter.

These rods were inserted through both webs of a 25-cm steel I-beam

(W10x26) extending parallel to the end of the lysimeter. The ends of

the rods were threaded such that nuts could be placed above and below

the upper I-beam. The lower nut prevented the hanger beam (W10x26)

from sliding to the bottom prior to installation in the ground. The

top nuts prevented the rods from sliding out of the hanger beam after

installation. This assembly formed a cradle by which the lysimeter was

supported and provided access to four 4.5 Mg capacity load cells

located under each end of the hanger beams.

A pit with reinforced concrete block retaining walls and a 10-cm

reinforced poured-in-place concrete floor was constructed for the

installation of the lysimeter at the IREP (Figure 3-2). The depth of

the pit was such that the bottom of the lysimeter tank was

approximately 15 cm above the pit floor and the tops of the retaining

wall and the lysimeter were flush. The pit floor was sloped toward the

center to drain water which might accumulate to a sump located

immediately outside of the North wall of the pit. Support columns were

incorporated into the retaining wall on which to place the load cells

supporting the lysimeter. The load cells could be installed or

removed by lifting the lysimeter with hydraulic automotive jacks from

ground level. Fill dirt was used to bring the surrounding soil up to

the elevation of the pit wall.

Prior to installing the lysimeters in the ground pits, two porous







71

ceramic stones (30 x 30 x 2.5 cm) were installed in the bottom of each

lysimeter and connected to vacuum tubing for the purpose of removing

excess water from the lysimeter. The lysimeters were then placed in

the pits and leveled using the top nuts of the supporting rods to

adjust the length of the rods and to maintain a fairly uniform load on

all of the rods.

The lysimeters were filled to a depth of 3 cm with a coarse sand

to provide free drainage of water to the ceramic stones. A Millhopper

fine sand excavated from a nearby site, was used to completely fill the

lysimeters. This soil was chosen because the profile was of fairly

uniform composition to a depth of 2 m and was typical of the local

sandy soils found in northern Florida and southern Georgia. After

loading the soil into the lysimeter, the soil was saturated with water

and allowed to stand overnight then removed by the vacuum system. This

provided settling to a density which might naturally occur in the field

after a long period of time.

Load cells manufactured by Hottinger Baldwin Measurements (HBM

Model USB10K) were then installed using a ball and socket connection

between the hanger beam and the load cell. A thin coat of white

lithium grease was applied to the surface of the ball and socket to

prevent corrosion of the contact surfaces. The output of the load

cells was a nominal 3 mV/V at full scale and could accept a maximum

supply voltage of 18 VDC. A regulated power supply provided 10 VDC

excitation voltage to each of the four load cells. Supply and output

voltages for each of the load cells was monitored using a single

channel digital voltmeter (Fluke Model 4520) and a multiple channel

multiplexer (Fluke Model 4506). Output was normalized using the ratio







72

of output voltage to input voltage (mV/V) to account for any variation

in supply voltage between load cells and time variation in the supply

voltage.

Lysimeter Calibration

Load cell calibration was performed by the manufacturer at the

factory for full load output of the cells. The manufacturer provided

load cell output (mV/V) at full scale load and no load. The load

carried by each load cell was expected to be near the full capacity of

each load cell with relatively small weight changes around that

reading. It was also noted that the calibration information provided

by the manufacturer was obtained under controlled atmospheric

conditions and using the factory-installed 3.3 m leads for measuring

the load cell output. The temperature sensitivity of the load cells

was very small compared to the changes in load cell output expected.

According the manufacturer's specifications, the effect of change in

load cell temperature upon the output was 0.08 percent of the load per

550C change in cell temperature. The installation site for the

lysimeters required the use of leads ranging in length from 37 to 53 m.

Therefore, an in-situ calibration was necessary.

The soil surface of the lysimeter was covered with a polyethylene

sheet to prevent weight loss due to evaporation during the calibration

procedure. A single weight having a mass of approximately 12 kg was

added to the lysimeter and the normalized output of each load cell was

recorded. A second weight was added, thus increasing the total weight

added to the system. The load cell output was again recorded. This

was repeated until the cumulative weight added to the lysimeter reached

175 kg (equivalent to 26 mm of water distributed over the lysimeter







73

surface). The normalized output was then recorded as the weights were

removed.

To account for unequal distribution of weight due to variations in

soil density and slope of the lysimeter, the above procedure was

performed such that weights were placed directly over a single load

cell and repeated for each load cell and by placing the weight in the

center of the lysimeter as well.

Soon after the load cells were installed and calibration

completed, one load cell from each lysimeter was damaged by lightning

and removed for repair. Since during the calibration procedure, the

normalized output of the individual load cells was recorded,

regressions for the remaining load cells for each lysimeter could be

determined. The calibration was repeated after the load cells were

repaired and installed. Regressions for each of the lysimeters were

obtained for the calibration data with three and four load cells

(Table 3-1). It was determined in both cases that the slopes were not

significantly different at the 99 % confidence level for the two

lysimeters and thereby allowing the same regression for both lysimeters

to be used. Since, interest was in changes in weight and not in

absolute weight, the intercept would not be significant. The

calibration curves used in data analysis are shown in Figure 3-3.

Resolution of the lysimeters based upon the specifications of the

voltmeter, and standard error of the regression parameters was

calculated to be approximately 0.02 mm of water.

Temperature Measurement

The vertical distribution of temperature within the soil was

measured by monitoring the output of ANSI Type T thermocouples (copper-







74

constantan) placed throughout the soil profile. Probes were

constructed using 2-cm PVC pipe with small holes drilled radially into

the pipe at specified intervals (Table 3-2). Thermocouple wire

(22 AWG) was fed into the top end of the PVC probe and the end of the

thermocouple protruding through the radial holes. Epoxy cement was

used to secure the thermocouples in place. After the thermocouples

were installed in the probe, the lower end of the probe was cut at an

angle then plugged with wood and epoxy. Two identical probes were

installed symmetrically about the center of the lysimeter to provide

some degree of redundancy and to obtain some idea of the uniformity of

the temperature distribution at a given depth in the soil. Shielded

multi-pair thermocouple extension cable was used to carry the

thermocouple signal to the Fluke multiplexer and digital voltmeter.

Soil Water Content Measurement

The measurement of the distribution of soil moisture was needed to

determine initial conditions for the model and to verify the subsequent

simulation of water movement within the soil. Several methods were

considered for the measurement of soil water content within the

weighing lysimeters. The method chosen had to provide a relatively

accurate measure of the volumetric soil water content as a function of

depth with minimal disturbance of the surrounding soil. The chosen

technique must also provide data for a relatively small vertical soil

volume and should be commercially available.

Soil water content is very difficult to measure accurately

especially when trying to do so with minimal disturbance of the

surrounding soil. The simplest method for determining soil water

content is the gravimetric method. Core samples are obtained from







75

various depths, and placed in a water tight container, weighed and

placed in a drying oven. The oven temperature should be maintained at a

constant temperature (1050C) and the sample remain in the oven until a

constant weight is obtained. The sample is then removed from the

oven, allowed to cool in a desiccator, then weighed. This information

provides the water content on a weight basis. If the dry bulk density

of the soil is known, the volumetric water content can then be

calculated. The gravimetric method is the method by which all other

methods are calibrated (Schwab et al., 1966). However, the sampling

procedure disturbs the surrounding soil and the flow of moisture until

the soil sample is returned to the sample area. Even then the oven

dried soil is not at the same conditions as the undisturbed soil.

Therefore, the gravimetric method is not an acceptable method for

continuous or regular monitoring of the soil water profile.

Other methods for measuring the soil water content are classified

as indirect methods because they measure some property or

characteristic of the soil which varies predominantly with water

content. One of the simplest is the tensiometer. This instrument is

constructed of a hollow porous ceramic cup connected to a Plexiglas

tube. The tube is filled with water, sealed and connected to a

pressure sensor, usually a manometer, vacuum gauge or vacuum

transducer. After all of the air has been removed from the column of

water the tensiometer is placed in the soil such that the porous cup is

located at the desired depth. The water potential of the soil is

generally less than the potential of the water within the tube and is

drawn out of the tube until the water in the tube comes to an

equilibrium tension (or vacuum) with that in the soil. An electronic







76

vacuum transducer or manometer is used to measure the equilibrium

tension on the water column. The tensiometer has a range of

application from 0 to approximately 85 kPa (Long, 1982; Rice, 1969).

This would cover approximately ninety percent of the range of moisture

contents encountered in the field (Schwab et al., 1966). Using an

electronic pressure transducer would allow for the tension to be

monitored continuously. Several disadvantages in using tensiometers

exist. The soil water release curve (tension vs. soil water content)

must be known for each soil in which the tensiometer is installed.

However, this information is needed for the determination of the

concentration of water vapor within the soil for modeling purposes as

well. A second difficulty is that the tensiometer is essentially

detecting water content at a single point within the soil profile, thus

requiring the installation of several tensiometers within a finite

volume of soil. Tensiometers also require frequent maintenance. In

situ calibration of tensiometers is not required since the soil water

release curve is a unique property of the soil allowing the researcher

to obtain a soil sample from the experimental site and determine the

water release curve in the laboratory.

Electrical resistance is another property which varies with soil

water content. Two electrodes are imbedded within a porus gypsum block

which, when placed in the soil reaches an equilibrium moisture content

with the soil. The resistance between the two electrodes is measured.

However, as the water evaporates from the gypsum block, dissolved salts

remain in the gypsum. This deposition of salts within the porous

material will become redissolved when rewetted and cause an incorrect

reading. Their range of applicability extends mainly over dry soil







77
conditions (100 to 1500 kPa tension).

Nuclear emission methods provide yet another means by which the

soil moisture can be measured indirectly. In this method, either the

transmission of gamma rays or back-scattering of neutrons is measured.

Fast neutrons are emitted from a neutron source and as they react with

the hydrogen nuclei present within the soil, they lose part of their

energy becoming slow neutrons. A detector is then used to count the

rate at which slow neutrons are back-scattered in the soil surrounding

the probe. The neutron probe requires a single probe containing both

the radiation source and the detector. Access is gained by

installation of a tube in the soil thus allowing for repeated readings

at the same location with minimal disturbance of the soil after the

initial installation. However; the volume of soil penetrated by the

neutrons increases as the soil dries out, thus increasing the volume

over which the subsequent water content represents. Individual

readings are influenced by density and by the amount of endogenous

hydrogen. Once a calibration curve has been developed for the local

conditions, volumetric water content can be determined to within 0.5 to

1.0 percent moisture by volume (Schwab et al., 1966). A separate

calibration curve for measurements near the surface may also be

required due to fast neutrons escaping from the soil surface and not

being reflected back to the detector. Tollner and Moss (1985) noted

low R-square values of 0.6 and 0.4 for the calibration curves for a

neutron probe at a depths of 46 cm and 20 cm, respectively.

The use of gamma attenuation methods has gained some popularity

in recent years due to the ability of the instrument to measure the

volumetric water content or bulk density of relatively small volumes of







78

soil. The gamma probe requires the installation of two parallel tubes

spaced approximately 30 cm apart. A radioactive source which emits

moderately high energy gamma rays (Cs 37) is placed in one tube while a

detector is placed in the second tube at the same depth. The intensity

of gamma radiation reaching the detector is monitored by the detector.

Ferraz and Mansell (1979) discuss the gamma radiation theory used in

measuring soil water content in great detail. Primarily, the use of

the gamma attenuation apparatus in which the gamma rays are collimated

so that the intensity of the beam is focused and a high intensity

source are used. The use of uncollimated radiation in the field

requires the use of lower intensity sources due to the increased

possibility of radiation exposure to the operator. Ferraz and Mansell

(1979) state that the gamma attenuation method may be used in-situ to

determine changes in volumetric water content quite accurately, but the

error associated with absolute moisture content can be quite high.

Ayers and Bowen (1985) used a gamma density probe designed for field

use to measure density of soil within a soil density box. Resolutions

of 0.016 g/cm3 for the wet bulk density were achieved in their

experiments.

The gamma attenuation method has several advantages over the

neutron method for measuring soil moisture. The volume of soil in

which the measurement is made remains constant regardless of water

content (Ferraz and Mansell, 1979) and consists of a band between the

source and detector probes approximately 5 cm deep (Ayers and Bowen,

1985). The disadvantage of the gamma method with respect to the

neutron method is the fact that a pair of tubes must be installed

instead of a single tube. However, the probes for the neutron and









gamma meters are typically the same diameter, thus allowing the use of

the neutron probe in the gamma access tube if necessary. The two

radiation methods will generally disturb less soil during installation

and will allow for recurring measurement at more frequent depth

intervals than with the tensiometers. Tensiometers also require

considerably more maintenance than do the radiation methods. The

tensiometers on the other hand can be monitored continuously by

automated data acquisition equipment while the radiation techniques

cannot. Extensive calibration is required for either of the radiation

measurement techniques. Based upon the desire to cause minimal

disruption of the soil profile and measure soil moisture at many points

within the soil, it was decided that a gamma attenuation technique

would be used to measure soil water content.

Gamma Probe Calibration

Extensive calibration curves were required to determine absolute

values of the soil water content using the gamma ray attenuation

method. While the soil was being loaded into the lysimeters, three

pairs of aluminum access tubes (5.1 cm O.D.) were installed in each

lysimeter (Figure 3-2). The tubes extended from the lysimeter floor to

23 cm above the soil surface. Parallel tube guides provided with the

Troxler 2376 Dual Probe Density Gauge were used to maintain parallelism

between the tubes as the lysimeter was filled.

The gamma probe primarily measures density of the test material;

therefore, it was necessary to measure the wet bulk density of the soil

corresponding to the depth at which the gamma readings were obtained.

A bulk density core sampler was used to obtain the soil samples for the

measurement of density and volumetric water content. The sampler







80

consisted of a series of removable brass rings of known volume within a

hollow cylinder. The cutting edge was tapered such that compaction of

the soil sample was minimized (Baver et al., 1972). The soil sample

was removed from the inner brass ring and placed in an aluminum soil

sample container and the lid closed to prevent moisture loss until the

samples from a single probe could be measured. Core samples were taken

at depth intervals of 10 cm from the soil surface to a depth of 71 cm.

Three probes were obtained from within the lysimeter to roughly

correspond to the locations of the paired access tubes. The soil

samples were weighed, dried, and reweighed. Wet density, dry density,

and volumetric water content were calculated for each sample. The soil

properties determined from each of the three probes were averaged for

each depth to account for the fact that the density between the access

tubes could not be measured. This procedure also provided information

regarding the areal uniformity of the soil with the lysimeter. The

dried soil was placed back in the holes in reverse order of their

removal so as to maintain the original soil profile. Care was taken

during subsequent sampling to avoid the soil core sites previously

sampled in the lysimeters.

The gamma readings were obtained in the following manner. The

meter was turned on for a minimum of twenty minutes prior to readings

being taken to allow for the circuits to stabilize. After warm-up, the

detector probe and Cs137 gamma source were placed in the tubes of a

calibration stand. The calibration stand consisted of two parallel

aluminum tubes with materials of known density. The standard materials

were polyethylene (1.06 g/cm3), magnesium (1.75 g/cm3), magnesium-

aluminum alloy (2.16 g/cm3), and aluminum (2.61 g/cm3). The source and








81

detector were placed in the stand level with the center of the

polyethylene standard and the amplifier gain of the meter was set so

that the peak count rate was achieved. The probes were then lowered to

correspond with the magnesium and a 4-minute count was taken. In the

calibrate mode of the timer, the meter indicates the count per minute

for the 4-minute interval. This was referred to as the standard count

and was used to normalize subsequent counts in the soil. The standard

count also provided a means by which to compare readings taken at

different times and accounted for variation in temperature of the gauge

and gain settings.

After the standard count was taken, the meter timer was set to

accumulate counts for one minute. The probes were placed in a pair of

tubes such that the source and detector were located at a depth

corresponding to the center of the core samples taken. A minimum of

two one-minute counts were recorded for each depth then the probe moved

to the next depth. After the last measurement at a depth of 66 cm was

obtained, the source and detector rods were removed from the access

tubes. Counts were taken at the same depths in all three set of access

tubes and the entire sampling process repeated in the second lysimeter.

Several attempts at this calibration procedure were made. Early

trials produced unacceptable ranges of scatter in the gamma probe

readings. At a later date, faulty electronic components were found and

expected to be the reason for the highly variable gamma probe readings.

After repair of the unit, the calibration procedure was repeated

yielding acceptable results.

According to Ferraz and Mansell (1979) the density of the test

material varies inversely with logarithm of the ratio of the intensity







82
of radiation observed in the test sample to the radiation intensity

measured in air. This relationship can be expressed as the count ratio

varying as a decreasing exponential with respect to the density of the

material. The count ratio is defined by the manufacturer as the ratio

of counts per minute observed for the test material to the counts per

minute observed in the magnesium standard (Troxler,1972). The observed

density of the soil obtained from the gamma measurement represents the

wet bulk density (Equation 3-1).



Pwet = A B ln(CR) (3-1)




where:

Pwet : wet bulk density of the soil [g-cm-3
A, B : constants of regression

CR : count ratio dimensionlesss]

count per min in test material
count per min in standard


If the dry bulk density is known, then the volumetric water content

can be calculated by


0 ( Pwet" dry) 3-2
Pw
where:

8 : volumetric water content [cm3cm-3]

-3
Pwet : wet bulk density of soil [gcm-3]

Pdry : dry bulk density of soil [gcm-3]







83

P : density ofyater [g-cm3]
= 1.00 [g.cm ]

Combining equations (3-1) and (3-2) yields a relationship for
volumetric water content in terms of the count ratio and the dry bulk

density of the soil.


6 = A + B-ln(CR) + C-Pdry (3-3)



It was conceivable that a calibration curve might be necessary for

each of the lysimeters and for various ranges of depth. Therefore,

linear regressions of the forms in Equations (3-1) and (3-3) were

determined for each lysimeter and for each of the depths. Statistical

testing revealed that a single regression could be used for all depths

and for both lysimeters for both the wet density and volumetric water

content. The final form of the calibration equation for determination

of volumetric water content was


0 = 1.52 0.33-1n(CR) 0.89-Pdry (3-4)



The coefficient of variation (R2) for the regression shown in

equation (3-4) was 0.888. Measured volumetric water content when

plotted against the water content estimated by the calibration

(equation 3-4) should lie about a line with a slope of unity and an

intercept of zero (Figure 3-4). Examination of Figure (3-4) indicated

that relatively large errors could be associated with the exact
estimate of volumetric water content. Therefore, the gamma probe







84

should be used to determine changes in volumetric water content during

the experiments and gravimetric sampling should be used to determine

initial conditions for the experiment.

Soil Bulk Density and Porosity

The calibration procedure for the gamma probe provided sufficient

data to determine the vertical and horizontal distribution of the soil

dry bulk density in each of the lysimeters. As mentioned previously,

repeated core samples were obtained at several depths dispersed over

the area of each of the lysimeters. Figure (3-5) shows the average dry

bulk density as a function of depth in each of the lysimeters. The

error bars represent the standard deviation of the measured bulk

density. The bulk density for both lysimeters was relatively constant

at 1.38 g/cm3 in the top 25 cm and increased to approximately 1.6 g/cm3

between 25 and 40 cm. The largest deviation in measured bulk density

occurred at the lower depths.

The literature (Baver, 1972; DeVries, 1975) indicates that the

porosity for sandy soils ranges from 0.50 to 0.55 cm3/cm3. The soil

porosity was a soil parameter required as a function of depth. Due to

the variation in soil density, especially at the depths from 25 to

40 cm, it was decided that the porosity for the soil in the lysimeters

should be determined as a function of depth.

The porosity is defined as the volume of pore space in the soil

per unit of bulk volume. An air pycnometer (Baver, 1972) was used to

measure the porosity of the soil. Four levels of soil bulk density

ranging from 1.2 to 1.6 g/cm3 were used for the tests. Three soil

samples at each level of bulk density were prepared from a soil sample

obtained from the weighing lysimeters. The volumetric water content







85
of each soil sample was 0.05 cm3/cm3. The mass of soil placed in the

50 cm3 sample cup of the air pycnometer was that required to achieve

the prescribed levels of dry bulk density. The test was performed for

the sample in the cup and the void volume recorded. The volume

occupied by the water in the sample was added to the measured void

volume to yield the total pore space in the soil. A total of three

measurements were made for each sample. The test was also repeated

for a single soil sample with the maximum density achievable

(1.64 g/cm3). The porosity was calculated for each measurement by

dividing the 50 cm3 sample volume into the void volume and were

averaged for each bulk density level (Table 3-3). The repeatability of

the experiment was indicated by the standard deviation of the porosity

for each of the samples, while the accuracy of the measurement was

indicated by the standard deviation for the average of all measurements

for a given density level. The porosity was found to vary linearly

with bulk density (Figure 3-6). The error bars shown in Figure 3-6

indicate the standard deviation of the soil porosity at each bulk

density level. The porosity measurements were then used in defining

the soil profile for the validation and calibration simulations.

Experimental Procedure

Experiments were designed to monitor the energy and mass transport

processes from a bare soil surface for two weighing lysimeters on a

continuous basis. Automatic data collection was performed by a Digital

Electronics Corporation (DEC) PDP-11/23 mini-computer equipped with an

IEEE-488 interface board and utilizing the RT-11 operating system. The

RT-11 operating system provided several system subroutines which were

primarily for scheduling the data collection as well as other time







86

manipulations. The main program and associated subroutines were

written in FORTRAN-66. Subroutines for initializing and retrieving

data through the IEEE interface were written in either assembly or

FORTRAN by Dr. J. W. Mishoe for previous research. All signals from

the sensors used were analog signals and were monitored on 50 channels

of a Fluke 4506 multiplexer and subsequently read via the IEEE

interface from a Fluke 4520 digital voltmeter. Data monitored by the

computer for each lysimeter were net radiation, soil temperature

distribution, and supply and output signals from each of the load

cells.

Net radiation was measured using net radiometers (WEATHERtronics,

Model 3035) mounted 85 cm above the soil surface of the lysimeter. The

net radiometers utilized blackened thermopiles as the sensing element

to detect the difference between incoming and outgoing radiation. A

positive millivolt signal was produced when the incident radiation was

greater than that being reradiated, while a negative signal indicated

that more energy was being radiated from the surface than impinging

upon the surface. Factory calibration curves were used to convert the

millivolt signals to net heat flux (W/m2). The millivolt signal was

read at two minute intervals and total millivolts and number of times

read were recorded on floppy diskette at ten minute intervals.

Thermocouples, supply voltage, and load cell output were recorded at

ten minute intervals as well.

Data files were closed on an hourly basis and new ones opened for

each hour. This minimized the risk of data loss due to power outages

or other computer malfunctions. The most data that would be lost that

had already been recorded would be that for one hour in the event that









the power was lost just prior to closing the data file. When power was

restored, the PDP-11/23 would automatically restart and begin the data

collection routine. Separate data files were maintained for each of

the two lysimeters.

Hourly values of the ambient dry bulb temperature and relative

humidity were measured within a standard weather shelter from the

adjacent weather station monitored and maintained by the Agricultural

Engineering Department. Sensors for measuring air temperature were

Type T thermocouples and instantaneous values were recorded hourly. A

Campbell Scientific CR-207 sensor was used to monitor relative

humidity and consists of a wafer whose electrical properties vary with

relative humidity. Manufacturer's literature states that the sensor

provides reliable information when used in non-condensing conditions

with a relative humidity between approximately 20 and 90 percent.

However, the wafer material tends to absorb moisture over time

decreasing the reliability of data obtained. To account for the lag

time of the sensor, data was averaged over an hour and recorded.

Relative humidity data was generated as well by assuming that the

minimum daily temperature was the dewpoint temperature for the day.

Linear interpolation between consecutive daily minimums to provide a

continuous estimate of the dewpoint temperature throughout the day.

Relative humidity could then be calculated based upon the estimated

dewpoint temperatures. Wind speed and direction were measured at a

height of 2 m within the same weather station. Meteorological data

were recorded hourly and uploaded daily to the VAX mainframe managed by

IFAS. Access to the hourly data was achieved through the AWARDS

(Agricultural Weather Acquisition Retrieval Delivery System).









Meteorological data, including relative humidity determined from the

estimated dewpoint temperature, are shown in APPENDIX B.

Tests were begun by an irrigation or rainfall event to ensure that

the upper layers of the soil were sufficiently wet so as to provide a

minimum of three to four days of evaporation data. Most frequently,

water was added to the soil the evening prior to beginning the test to

insure that the soil surface was wet and to allow some downward

distribution of water prior to the initiation of each test. Core

samples were obtained to determine the initial distribution of water

within the lysimeter. Samples were taken at vertical intervals of

5.1 cm for the first 30.5 cm then every 10.2 cm until a depth of

91.4 cm was reached. Gamma probe measurements were taken in each of

the three pairs of access tubes corresponding to the center of each of

the core samples. Actual depths of gamma probe measurements and core

samples are shown in Table 3-4. Subsequent gamma probe measurements

were made approximately every other day. More frequent readings were

not obtained in most cases due to the time required for each set of

readings (2 hrs.). This minimized disruption of the other data being

collected.

Obtaining the gamma probe readings required placing a scaffold

across the lysimeter to avoid disturbing the weight of the lysimeter

and compacting the soil. This, in turn, could have disrupted the

evaporation process due to shading of the soil surface as well as some

of the net radiation measurements. These periods when the integrity of

some of the data may have been questionable was marked in the hourly

data file by a flag entered via the keyboard. The flag was turned on

when gamma readings were being made on each lysimeter and turned off









again after the task was completed.

Data were collected continuously over a period extending from

November, 1986 to August, 1987. Data collection was interrupted on

several occasions due to equipment failures, power outages, equipment

maintenance, and instrument calibration. Most tests were three to four

days in length with a few extending to six or seven days.

Data Analysis

Data files containing the data recorded at ten-minute intervals

were concatenated into daily files. The daily files were then combined

to correspond to specific dates for the individual tests. Data

contained in the files consisted of day-of-year, time-of-day, the

normalized output for each load cell (mV/V), the 10-minute total net

radiometer output, the number of net radiometer readings during the

10-minute interval, the status flag, and twenty temperatures

corresponding to the depths indicated in Table 3-2.

The normalized output of the load cells was totaled and the

average total millivolt per volt for the hour was calculated. The

difference between successive hourly totals was used in the load cell

calibration equations to determine the hourly and cumulative

evaporation of water from the lysimeter. The output signal for the net

radiometers was totalled for the hour then an average rate of net heat

flux was determined using the manufacturer's calibration curves. The

two temperatures for each depth were averaged over the hour as well.

If the data flag had been turned on at any time during the hour, it was

assumed to have been on the entire hour. Reduced data files contained

the hourly values of julian date, time, cumulative evaporation (mm),

hourly evaporation (mm/hr), net radiation (W/m2), data flag, and




Full Text
Temperature (C) Temperature (C)
182
Figure 5-4. Simulated and experimental soil surface temperatures
during test 70 (January 5-8, 1987).


109
Figure 3-8. Hourly water loss from the University of Florida, IREP
weighing lysimeters for November 25 29, 1986.


123
Figure 3-22. Hourly water loss from the University of Florida, IREP
weighing lysimeters for February 3-5, 1986.


78
soil. The gamma probe requires the installation of two parallel tubes
spaced approximately 30 cm apart. A radioactive source which emits
137
moderately high energy gamma rays (Cs ) is placed in one tube while a
detector is placed in the second tube at the same depth. The intensity
of gamma radiation reaching the detector is monitored by the detector.
Ferraz and Mansell (1979) discuss the gamma radiation theory used in
measuring soil water content in great detail. Primarily, the use of
the gamma attenuation apparatus in which the gamma rays are collimated
so that the intensity of the beam is focused and a high intensity
source are used. The use of uncoilimated radiation in the field
requires the use of lower intensity sources due to the increased
possibility of radiation exposure to the operator. Ferraz and Mansell
(1979) state that the gamma attenuation method may be used in-situ to
determine changes in volumetric water content quite accurately, but the
error associated with absolute moisture content can be quite high.
Ayers and Bowen (1985) used a gamma density probe designed for field
use to measure density of soil within a soil density box. Resolutions
of 0.016 g/cm3 for the wet bulk density were achieved in their
experiments.
The gamma attenuation method has several advantages over the
neutron method for measuring soil moisture. The volume of soil in
which the measurement is made remains constant regardless of water
content (Ferraz and Mansell, 1979) and consists of a band between the
source and detector probes approximately 5 cm deep (Ayers and Bowen,
1985). The disadvantage of the gamma method with respect to the
neutron method is the fact that a pair of tubes must be installed
instead of a single tube. However, the probes for the neutron and


29
Tavg = average temperature [K]
A thermal conductivity of soil [W*m-1,(K-1]
/>va mass of water vapor per unit volume of dry ambient air
[kg-m-^]
pyo concentration of water vapor at the soil surface, 2=0
[kg-m-3]
6 = volume of water per unit total soil volume [m3*m'3]
Ideally, for a semi-infinite medium, the flux of energy and mass
should be zero in the limit of depth (z) approaching infinity.
However, in anticipation of a numerical solution to the system of
partial differential equations, the boundary conditions were specified
at a depth of 1.5 meters. This depth was chosen by comparing the
error between the analytical and numerical solutions for conduction of
heat in a semi-infinite slab with constant uniform properties and a
uniform heat flux at the surface. This depth resulted in an error of
less than 0.1 C at the lower boundary. The depth at which the
amplitude of the diurnal fluctuations in temperature is less than
0.1 C is approximately 60 cm for a sandy soil (Baver et al., 1972).
The zero flux condition for the liquid and vapor continuity equations
represents an impermeable layer in the soil. This condition may or
may not physically exist in the field, but for most situations
encountered, the errors introduced into the solution at the depth of
chosen should be minimal. The boundary conditions used at the lower
boundary were
= 0 (2-32a)
Z=Zq
36
3J_
3z
0
(2-32b)


28
obvious soil-atmosphere interface. It was assumed that the thermal
capacitance of the soil at the surface was negligible when compared to
the magnitudes of the fluxes which occur. Therefore, the net flux of
energy must be zero at the soil surface. The ability of the soil to
maintain a significant rate of evaporation at the soil surface was
assumed to be small as well. This assumption required the water to
evaporate at a finite distance below the soil surface rather than at
the soil surface. The boundary conditions for the energy, vapor and
liquid continuity were
(2-31a)
(2-31b)
(2-31c)
where:
Cpw = specific heat of water [J-kg'1*^'1]
CpV specific heat of water vapor [J*kg'l*K"l]
D[_ hydraulic diffusivity [m2*s_1]
Dv = diffusivity of water vapor in soil [m2*s"l]
hf, = boundary layer heat transfer coefficient [W*m'2*K_1]
hm = boundary layer mass transfer coefficient [m*sl]
P = precipitation or irrigation rate [m3*m'2*s-1]
Rn = net radiation incident upon soil surface [W*m"2]
soil temperature [K]
- X
3T
ST
z=0
99 ^/*v
(PwcpwL + cpvv tfz )^avg
Rn ^h(Tz=0 Ta)
n 30
-L ST
= P
Z=0
3pv
3v Sz~
z=0
= hm(pV0" hz )
T


Variable Definition
P
P
Po
Qv
q
qcj+l,n
qej,n
qLj+l,n
qsj,n
precipitation: flux of water at soil surface
S 1
atmospheric pressure [Pa]
reference atmospheric pressure [Pa]
vertical flux density of water vapor in the
atmosphere [kgnr^s12]
specific humidity of ambient air [kgvapor/kgdfy air!
rate of heat flux conducted into node j from node
j+1 at time step n [W*m'2]
rate of latent heat loss due to evaporation at node
j and time step n [W*m"3]
heat flux rate into node j from node j+1 via
movement of liquid at time step n [W*m'2]
rate of sensible heat change of node j at time step
n [W*m-3]
qvj+l,n
R
Re
rate of heat carried into node j from node j+1 by
water vapor movement at time step n [W*m"2]
universal gas constant [kgm2*s2*mol"1,0K~1]
gas constant for water vapor, determined by dividing
the universal gas constant, R, by the molecular
weight of water [m2,s'2,0K"l]
Reynolds number [dimensionless]
Ri Richardson number [dimensionless]
Rn net radiation flux incident upon soil surface
[W-nr2]
S soil porosity [m3*nr3]
s slope of the saturated vapor pressure line
[kPa-OK'1]
Ta ambient air temperature [K]
Tjjn soil temperature at node j and time step n [^C]
Tavgj (Tj+l,n + Tj,n)/2
vi i


Cumulative Evaporation (mm) Cumulative Precipitation (mm)
112
4--
3 +
2--
1 --
I M I M M I I I I I I I II 11 I t <1 I I I'M II I I'M IIMIIlllimilllllll'IIIIIIIIIH
I'll I III IMI ItU.lllililll I |:|i|,i,l¡Mii:l ll.lrl 1,1 III I l;l,l l l ll 1111 ni l,l
l'lili|ili|f|ifil:l Hl'M'|,| I l|l M l I f IM 11 l i l i I II I IiKI I 1,11 K ill
l|M¡l|Wll'1llKlilililil|ljKIll(ll!lilililtlil¡l,KllMl!l|liHlll!l1l|liil,l;l,l'lil¡li,l:,M,i|,|llilil,Ml!lil',l,l,l
ttimtim i t1 mm,i m mittj nw m miiii cm-i 0 ft i, umi i h i i i i k i
:H|I,IW;kK||i|iKM,|,IiIIiK!II:|'KIi>i;i-M'W,I|I I|M'M:I:>,I¡M.| l|l KilKI'KflKI III l
. l|,M|KillMil*l|l:>l'l|l|lil,l¡H:li/5MjM,i;MKI¡l'l|l¡l|llM¡l l'IM rijlll M:ljt||iM III l>l¡H
,l|l* Wjl!|MW;Kil'(|l,l|f;lil|liiii-M|l¡/ l|li|il(it|l|(iljlM|l I.Ml IKMKIiKMIf IHiU
If,11K K)fWlif+M l,lll,t;i'WM(lil (.1,1 IH;M,t l'(l|llii(:ll(.KM MiK,l;M*t¡H (¡I l.l 11 lllil
III Hint I l.KliMSlI I III,1)1 liMf/illl 1,1 KiI intiliin,ll i IIM 11 Cl i III l ll (I I t 1
itim fti mi\i(Him ni 11 (up n:i (hi it i ni i tin ti i Itiii i min tm
-1 1-
III limiillilllll Mill! nl 1111 m 11 mi 111111111111111111111111111
Figure 3-11. Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
December 15 19, 1986.


Hydraulic Conductivity ( cm/h)
60
Figure 2-3. Hydraulic conductivity and hydraulic diffusivity as a
function of soil water content for a typical sandy soil
Hydraulic Diffusivity (cm^/h )


Depth ( m ) Depth ( m )
110
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-9. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on November 25 and 26, 1986.


168
values. Both simulated and experimental daily evaporative losses
decreased each day as the soil surface dried. The model simulated the
initial high evaporation rate on the first morning of the test,
February 9, for both lysimeters quite well. However, the apparent
early morning burst of evaporation was not simulated on subsequent days
of the test. The excessive rates of evaporation simulated during this
experiment can be explained by examining the simulated soil surface
temperature and the water vapor density difference. Simulated surface
temperatures were well in excess of observed values with maximum
deviation of approximately 15 C (Figure 5-19). The high temperatures
in turn cause a large difference in the water vapor concentration
driving the flow of water vapor (Figure 5-20). Boundary conditions
which influenced the soil surface temperatures are the ambient air
temperature, net radiation and the wind speed. Net radiation was
relatively high throughout the duration of this test. The maximum net
radiant heat flux upon the soil surface was 450 W/m^ (Figure B-14)
with a minimum of approximately -75 W/m^. Air temperature and the wind
speed both affect the rate of heat transfer from the soil surface via
convection. Air temperature was 20C or less for most of the test
(Figure B-15), thus providing a large potential difference for heat
transfer between the soil and the atmosphere. Wind velocity directly
affects the surface heat transfer coefficient. The variation of wind
speed as function of time was not at all typical of that expected or as
compared to wind speeds recorded during tests 70 and 71. Since the
data for this test was very similar to that used in test 72, it was
highly questionable.


25
- hfgE(z,t)
(2-26)
where
\ = thermal conductivity of soil mixture [W*m'^,(^C'^]
T = soil temperature [K]
hfg latent heat of vaporization [J*kg-1]
Cs = volumetric heat capacity of soil mixture [J-m'^K'1]
= (l-S)psCpS + ^Pwcpw + (5~^)Pacpa
S = Soil porosity [m3*m"3]
ps = dry bulk density of soil [kg*m'3]
pa = density of moist air [kg*m-3]
cps specific heat of solid portion of soil [J-kg^*0K^]
CpW specific heat of water [J*kg"l*K"l]
Cpa = specific heat of moist air [J'kg"l*Kl]
Equations (2-22), (2-25), and (2-26) describe the conservation of
energy and mass within the soil profile and account for the vapor and
liquid water phases separately. However, those three equations contain
the three state variables, soil temperature, volumetric water content,
and vapor concentration as well as the unknown rate at which water is
changed from the liquid to the vapor phase. Another independent
equation was needed to adequately describe the soil mass and energy
system.
If the water vapor were in equilibrium with liquid water, the
vapor would have a partial pressure corresponding to the saturated
vapor pressure and would be a function of the liquid water temperature.
Assuming that water vapor behaves as an ideal gas, then the saturated


170
obtained from the literature for a Mi 11 hopper fine sand with similar
vegetative cover as that filling the lysimeters and not measured added
to the uncertainty of the values used in the model. The surface
transfer coefficients were implicated by the fact the model seemed to
simulate the evaporation of water when the soil was limiting the
movement of water from the soil but not when water was apparently
available at the surface with little or no resistance to diffusion into
the atmosphere.
Sensitivity Analysis
The experimental values of thermal diffusivity of the soil
obtained as described in chapter 4, were incorporated into the model to
insure that the thermal properties of the soil used in the model were
as close to those present in the lysimeter as possible. Thermal
conductivity was determined by multiplication of the experimental
values of the thermal diffusivity and the volumetric heat capacity.
Volumetric heat capacity was determined as previously described using
equation (2-35). Differences in the simulated evaporation rates and
soil temperature and water content using values of thermal conductivity
determined by the DeVries method and experimental data were insigni
ficant.
The data from the south lysimeter obtained during test 70
(January 5 to 10, 1987) was used as the calibration data set primarily
due to the quality of the evaporation data. The hourly evaporation
curve was smooth with few jumps and unexplainable spikes. The wind
speed, air temperature, net radiation and relative humidity data for
this particular test appeared to be high quality data as well.


223
The equations for the backward direction at the last node are
Backward direction: z = zo
j = nc
energy:
Tj,nfl
2*Fo*(l+KCl)Tj.1>n + (1 2*Fo*(l-KCl))Tj>n
hfg,jdt
'S,J
Ej,n
(A-10)
water:
(A-11)
d3,ml
(1 2*FL)0j>n + 2*FL*0j.1>n
dt
-J,n
vapor: (A-12)
Ej,n a dt ^vj,n+l (1 2*Fv)/5vj,n 2*Fv*/>vj-l,n
Equations A-13, A-14 and A-15 represent the conservation equations for
the interior nodes for the backward direction solution.
Backward direction: 0 < z < zo
1 < j < nc
energy:
(A-13)
Tj>n+1
Fo*(l+KC2) T
1 + Fo*(l-KC2) 1 j+l,n+l +
Fo*DZR*(l+KCl) T
+ 1 + Fo*(l-KC2) 'j-l.n"
1 Fo*DZR*(l-KCl) x
1 + Fo*(l-KZ) TJn
dt*hfgJ
Csj(l + Fo*(l-KC2)) hJn
water:
e3,r*l
FL (1 FL*DZR) n .
(1 + FL')' ^J+l,n+l + '(1 + FL)
FL*DZR dt
+ (1 + FL) Pw'Ji + 'Fiy ^.n
(A-14)
vapor:
(A-15)
^j,n = (jt U + Fv)Pvj,r>fr Fv*^vj+l,n+l
- (l-Fv*DZR)pvj>n- Fv*DZR*Pvj-l,n


CHAPTER VI
SUMMARY AND CONCLUSIONS
A model was developed using three partial differential equations
to describe the conservation of energy, water and water vapor in a one
dimensional soil system. It was assumed that the liquid and vapor
phases of the soil water were in equilibrium within the soil air space.
State variables for the system were the soil temperature, volumetric
water content, water vapor concentration, and the rate of change of
water from the liquid to the vapor phase. All state variables as well
as soil properties were variable with space and time. An alternating
direction finite difference technique was utilized to numerically solve
the system of equations subjected to surface boundary conditions which
varied with time. Transport properties for the soil were calculated
based upon relationships presented in the literature.
Two weighing lysimeters were constructed and filled with a uniform
soil profile of Mi11 hopper fine sand. The output of four 4545 kg
loadcells was monitored to determine changes in weight due to water
loss over time. Other data recorded for each lysimeter were soil
temperatures at various depths and net solar radiation impinging upon
the soil surface. Volumetric water content was measured over the depth
of the lysimeter at the beginning and end of evaporation trials using
core samples and a dual probe density gauge. The lysimeters were
estimated to have a sensitivity of 0.02 mm of water. Soil properties
measured for the soil contained in the lysimeter were
1. dry bulk density as a function of depth,
208


151
Table 4-
. Nomenclature and list of symbols (continued)
T : temperature
t : time
z : space coordinate
\ : thermal conductivity
Av : apparent enhancement to thermal conductivity due to
water vapor movement
6 : dimensionless temperature ratio
3ps
nj- : gradient of saturated vapor pressure with respect to
temperature


Temperature (C) Temperature (C)
121
Figure 3-20. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from January 11 to 14 1987.


48
*j,n)
fJ 31
n W,n- ^j,n (*j-l,rf *j,n)
u 3ij +
dWj -ji (2-65)
J pw,j
Similarly, the differential equation describing the conservation of
mass in the vapor state is transformed as
dw-i
(/vj.m-l Pvj,n)
31
(/Yj+l,n ^vj,n)
Jvj
QZ-
JVJ
(^vj-l,n £vj,n)
(2-66)
(sj ^j,n)
dwj EJ,n
Equations (2-64), (2-65), and (2-66) constitute the numerical
equivalents for the partial differential equations describing the
conservation of thermal energy and mass in the soil profile. However,
the conservation relationships have yet to be developed for the
boundaries at the soil surface and the bottom of soil profile. Since
there is no volume associated with the surface node (Fig. 2-1), there
is no storage capacity for energy or mass. Therefore, the flux of mass
or energy from the node directly below the surface (j=2) to the surface
added to the net flux due to solar radiation onto the surface and the
heat flux due to convection from the air to the surface must be zero.
This can be expressed directly in difference form as follows.


Surface Temperature (*C) Surface Temperature (C)
187
Julian Date
Figure 5-9. Comparison of soil temperatures measured at the soil
surface during test 71 (January 11 14, 1986) to
simulated values.


163
higher, more heat would have been partitioned into latent heat at the
surface rather than sensible heat gain. Note that the magnitude of the
discrepancy in surface temperatures corresponds fairly closely with the
discrepancy in cumulative evaporation during the first day.
Simulation of the change in water profile showed that water was
primarily lost from the surface with some redistribution of water
occurring overnight (Figure 5-5). The soil surface water content
decreased very rapidly from 15 and 17 percent for the north and south
lysimeters, respectively to approximately 11 percent after 26 h. The
soil surface water content was approximately the same each day at
noon; however, water had been moved from the soil below to maintain
that water content. The simulated profiles also showed some
redistribution of water downward in response to hydraulic gradients.
Simulated water profiles were in general agreement with the profiles
measured in the lysimeters at the conclusion of the test on January 9,
1987 (Figure 5-6). The experimental profiles showed very little
downward movement of water while the simulated profile approached a
more linear distribution. The final simulated water contents at the
soil surface were higher than those measured as anticipated due to the
lower cumulative water loss from the soil.
Comparison of data from test 71 to the simulation results showed
similar trends. Meteorological input data for the simulations are
shown in Figures B-9 and B-10. The simulated cumulative evaporation of
water from the lysimeters compared more favorably with measured water
loss (Figure 5-7). Both the model and measured evaporation indicated
that slightly over 1 mm of water had been lost from each of the two
lysimeters. However, the cumulative water loss from the south


14
Rn S G LE = O
(2-14)
where:
Rn -
net radiation incident upon soil surface [W*m"2]
S
soil heat flux [W*m'2]
G
sensible energy flux into atmosphere [W*m'2]
LE =
flux of latent energy from soil [W*nT2]
An expression for the rate of evaporation of water from the soil can be
determined by substitution of equation (2-13) into (2-14) and
rearranging terms. This method is referred to as the Bowen ratio
equation and yields valid results for a wide range of conditions.
Fritschen states that it is imperative that efforts be undertaken to
assure that the assumption of no horizontal divergence of heat or
moisture be met in order for the Bowen ratio equation to yield
satisfactory estimates of the evaporation. Soil heat flux must be
measured as well, since omitting soil heat flux from the analysis could
lead to large errors.
In using the Bowen ratio, the soil surface temperature must be known as
in some of the other methods.
Penman (1948) used a combination of the energy and mass balance
methods with the objective of eliminating the need for surface
temperature. The underlying assumptions are the temperature of
evaporating surface is the same as the ambient air and the vapor
pressure is the saturated vapor pressure evaluated at the surface


140
diffusivity is the thermally induced flow of water vapor. Cary (1966)
stated that thermally induced flow of water could occur to such an
extent that it was the predominant factor in mass transfer within the
soil, especially under dry soil conditions. Movement of water vapor in
the soil sample could induce errors in the measurement of the slope of
the transient temperature response if the evaporation or condensation
were occurring near to the temperature sensor due to the latent heat of
vaporization. Reidy and Rippen (1964) indicated that vapor movement
could be minimized by inducing relatively small temperature gradients.
A second transient method requires the use of a thermal
conductivity probe. The conductivity probe consists of a precision
resistance heating element and a small gauge thermocouple installed in
a stainless steel probe. The probe is inserted into the center of the
sample. A galvanometer is used to record the current passed through
the heating element and the thermocouple senses the temperature of the
sample. If the thermal conductivity probe were used in a moist porous
material, then care must be taken to prevent the vaporization of water
near the probe causing an increase in heat flux from the probe.
The steady state methods were not chosen due to the possible long
duration of a single test and the anticipated difficulties in
measuring heat flux and precise placement of the temperature sensors
within the sample. The transient methods were chosen because of the
short duration of the test and the lack of precise placement of the
temperature sensor. The short test duration may also reduce the effect
of vapor movement within the soil upon the determination of the thermal
diffusivity. The thermal conductivity probe was not used because of
the potential of inducing vapor flux in the vicinity of the probe. An


172
the cumulative water loss using the multiplier of 5 was slightly higher
than multiplier of 2 (Figure 5-22). Using a multiplier of 0.5 reduced
the total amount of water lost from the soil. After a 60-h simulation,
the cumulative water loss using the multiplier of 0.5, was
approximately 1 mm lower than the results for multipliers of 2.0 and
5.0. The point at which the cumulative evaporation curves first
separated for indicated the point at which the soil first became the
limiting component in the evaporation of water. This could better be
seen by comparing the evaporation rates for each of the multipliers
(Figure 5-23). The evaporation rates for the two highest multipliers
were identical. The curve for the multiplier of 0.5 first deviated
from the higher curve approximately 3 h after the beginning of the
simulation on January 5. All three curves were identical until
approximately noon, when the evaporation rate for the simulation for
the lowest multiplier decreased as the evaporation rate for the higher
multipliers continued to increase. This was due to the inability of
the soil to maintain sufficient upward flux of water to meet the
evaporative demand at the surface, therefore the rate of evaporation
decreased. Overnight the two curves rejoined, until evaporation began
again the third morning of the simulation. Hourly evaporation rates
coincided until approximately 1000 at which point soil with the lower
hydraulic diffusivity experienced its maximum evaporation rate then
began to decrease. Note the decreased time required for the peak
evaporation rate to occur on the second morning as compared to the
previous day. This was due to the decreased water available at the
soil surface each day (Figure 5-24). The simulated water content at
the soil surface indicated a diurnal pattern of drying during the day


Temperature (C) Temperature (C)
133
Julian Date
Figure 3-32. Soil temperatures measured at depths of 0, 15 and 80 cm
in weighing lysimeters located at the University of
Florida, 1REP from August 25 and 29, 1987.


12
LE = -hgPa^w 5§- ¡w7^) (2-10)
where:
~V : time averaged turbulent fluctuation of the air
temperature [9(]
u7 : time averaged turbulent fluctuations of the horizontal
component of the wind velocity [m*s_1]
w : wind velocity in the vertical direction [m*s"l]
w7 : time averaged turbulent fluctuations of the vertical
component of the wind velocity [m*s-1]
qv : turbulent fluctuations in the specific humidity of the
air [kgvapor/kgdry air)
Assuming vertical gradients in temperature and absolute humidity
are insignificant as compared to the gradients in the direction as the
horizontal component of the wind, the latent and sensible heat can be
expressed as
cpaPaw'T
(2-11)
hfgPawX
(2-12)
The equations are used as a basis for either the sensible or latent
heat flux in the atmosphere using measured values of the turbulent eddy
fluxes of heat and moisture. Specialized equipment was designed by the
Goddard and Pruitt to measure the parameters needed for the
calculations. However, the equipment was not reliable for low wind
speeds. This was attributed to the very small magnitudes of the
turbulent eddies for calm conditions.


75
various depths, and placed in a water tight container, weighed and
placed in a drying oven. The oven temperature should be maintained at a
constant temperature (105C) and the sample remain in the oven until a
constant weight is obtained. The sample is then removed from the
oven, allowed to cool in a desiccator, then weighed. This information
provides the water content on a weight basis. If the dry bulk density
of the soil is known, the volumetric water content can then be
calculated. The gravimetric method is the method by which all other
methods are calibrated (Schwab et al., 1966). However, the sampling
procedure disturbs the surrounding soil and the flow of moisture until
the soil sample is returned to the sample area. Even then the oven
dried soil is not at the same conditions as the undisturbed soil.
Therefore, the gravimetric method is not an acceptable method for
continuous or regular monitoring of the soil water profile.
Other methods for measuring the soil water content are classified
as indirect methods because they measure some property or
characteristic of the soil which varies predominantly with water
content. One of the simplest is the tensiometer. This instrument is
constructed of a hollow porous ceramic cup connected to a Plexiglas
tube. The tube is filled with water, sealed and connected to a
pressure sensor, usually a manometer, vacuum gauge or vacuum
transducer. After all of the air has been removed from the column of
water the tensiometer is placed in the soil such that the porous cup is
located at the desired depth. The water potential of the soil is
generally less than the potential of the water within the tube and is
drawn out of the tube until the water in the tube comes to an
equilibrium tension (or vacuum) with that in the soil. An electronic


77
conditions (100 to 1500 kPa tension).
Nuclear emission methods provide yet another means by which the
soil moisture can be measured indirectly. In this method, either the
transmission of gamma rays or back-scattering of neutrons is measured.
Fast neutrons are emitted from a neutron source and as they react with
the hydrogen nuclei present within the soil, they lose part of their
energy becoming slow neutrons. A detector is then used to count the
rate at which slow neutrons are back-scattered in the soil surrounding
the probe. The neutron probe requires a single probe containing both
the radiation source and the detector. Access is gained by
installation of a tube in the soil thus allowing for repeated readings
at the same location with minimal disturbance of the soil after the
initial installation. However, the volume of soil penetrated by the
neutrons increases as the soil dries out, thus increasing the volume
over which the subsequent water content represents. Individual
readings are influenced by density and by the amount of endogenous
hydrogen. Once a calibration curve has been developed for the local
conditions, volumetric water content can be determined to within 0.5 to
1.0 percent moisture by volume (Schwab et al., 1966). A separate
calibration curve for measurements near the surface may also be
required due to fast neutrons escaping from the soil surface and not
being reflected back to the detector. Tollner and Moss (1985) noted
low R-square values of 0.6 and 0.4 for the calibration curves for a
neutron probe at a depths of 46 cm and 20 cm, respectively.
The use of gamma attenuation methods has gained some popularity
in recent years due to the ability of the instrument to measure the
volumetric water content or bulk density of relatively small volumes of


74
constantan) placed throughout the soil profile. Probes were
constructed using 2-cm PVC pipe with small holes drilled radially into
the pipe at specified intervals (Table 3-2). Thermocouple wire
(22 AWG) was fed into the top end of the PVC probe and the end of the
thermocouple protruding through the radial holes. Epoxy cement was
used to secure the thermocouples in place. After the thermocouples
were installed in the probe, the lower end of the probe was cut at an
angle then plugged with wood and epoxy. Two identical probes were
installed symmetrically about the center of the lysimeter to provide
some degree of redundancy and to obtain some idea of the uniformity of
the temperature distribution at a given depth in the soil. Shielded
multi-pair thermocouple extension cable was used to carry the
thermocouple signal to the Fluke multiplexer and digital voltmeter.
Soil Water Content Measurement
The measurement of the distribution of soil moisture was needed to
determine initial conditions for the model and to verify the subsequent
simulation of water movement within the soil. Several methods were
considered for the measurement of soil water content within the
weighing lysimeters. The method chosen had to provide a relatively
accurate measure of the volumetric soil water content as a function of
depth with minimal disturbance of the surrounding soil. The chosen
technique must also provide data for a relatively small vertical soil
volume and should be commercially available.
Soil water content is very difficult to measure accurately
especially when trying to do so with minimal disturbance of the
surrounding soil. The simplest method for determining soil water
content is the gravimetric method. Core samples are obtained from


236
m
\
E
T3
V
0)
CL
in
T5
c
34 35 36 37
Julian Date
Julian Date
Figure B-ll. Wind speed (top) and net radiation (top) data collected
during lysimeter evaporation studies conducted from
February 3 to 5, 1987.


235
Julian Date
Figure B-10. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from January 11 to 15,
1987.


97
indirect method, such as time domain reflectometry, in which the
sensors can remain in place and can be monitored via a data acquisition
system on a continuous basis similar to the measurement of other system
variables. Although the use of thermocouples for the measurement of
soil surface temperature did not appear to have caused significant
errors, the possibility exists. Errors could occur due to the
thermocouple becoming uncovered and exposed to direct sunlight or could
have become buried too deeply. It is suggested that in future
research, an infrared thermometer be permanently mounted on the
lysimeters to measure soil surface temperatures. This might also be
helpful when a crop is grown in the lysimeters for measurement of the
canopy temperature. Under a full crop canopy, the thermocouple could
be used with decreased possibility of errors due to direct radiation
impinging upon the sensor.
The overall performance of the weighing lysimeter system provided
data for the validation and calibration of a coupled soil water balance
model. The goals of monitoring hourly and cumulative evaporation as
well as soil temperature, water distribution and boundary conditions
were achieved.


Bulk
106
Figure 3-5.
Dry soil bulk density as a function of depth for two
lysimeters installed at the IREP, University of Florida,
Gainesville, FL.


90
average soil temperatures at various depths. The water content
profiles were generated and stored separately since they were not
monitored on a continuous basis as were other data.
Experimental Results
Experimental data showing cumulative and hourly water loss,
vertical distribution of water, and temperature of the air, soil
surface and the soil at depths of 15 and 80 cm are presented in Figures
3-7 through 3-32. If rainfall occurred during a test, then cumulative
precipitation is shown as well for that experiment. If rainfall was not
shown, none occurred during the test. Data are presented
chronologically according to calendar date.
The total water lost from the north and south lysimeters during
the period November 25 to 29, 1986 was approximately 6.0 and 3.5 mm,
respectively. A total of 20 mm of rain fell from just prior to
midnight on November 28 until 0400 on November 29 (Figure 3-7). Data
collection was discontinued for the north lysimeter shortly before the
rain began while data were monitored for the south lysimeter until the
morning of November 29. Soil conditions in the two lysimeters were
similar and water loss should have been similar as well. However,
discrepancies between the cumulative water loss from the two lysimeters
was evident. For instance, the south lysimeter indicated an increase
in weight every day during the experiment at approximately 1800, while
the north lysimeter indicated a slight increase in weight between the
hours of 0400 and 1000 on November 26 and 27. Examination of the
hourly evaporation rates (Figure 3-8) indicated this phase shift in
the evaporation of water from the soil. It also showed the highly
variable and erratic behavior of both lysimeters.


152
Table 4-2. Parameters for the empirical determination of the first
eigenvalue for solution to heat conduction equation for
a sphere, and infinite cylinder and an infinite slab.
Sphere
Infinite
Cylinder
Infinite
Slab
(l)co
7T
2.4048
tt/2
I
2.70
2.4500
2.24
s
1.07
1.0400
1.02


10
T air temperature [K]
Kfo, Kn,, Kw turbulent transfer coefficients for heat, momentum
and water vapor, respectively [m2*s_1]
Combining the equations 2-2 and 2-3 and rearranging the following
expression for evaporation is obtained.
k _g_
Si dz
K 3D-
"1 dz
(2-5)
The profile methods determine the rate of evaporation by analysis of
the vertical profile of various atmospheric variables such as specific
humidity or vapor pressure and wind velocity. King (1966) described
the procedures for determining the evaporation rate for adiabatic wind
profiles (neutral stability) and stratified conditions. One of the
basic assumptions used in the profile methods is that the turbulent
transfer coefficients of water vapor and momentum are equal. This
implies that the momentum and mass displacement thicknesses are the
same. The Richardson number (Ri) or the ratio of height (z) to the
Monin-Obukov length (L) are used as a measure of atmospheric
instability and can be determined by equations 2-6 and 2-7,
respectively.
where:
Ri
L
au
dz
+ r )
u*
(
k g H
Cpa^a^
(2-6)
(2-7)
9
acceleration due to gravity [m*s-2]


58
Volumetric Water Content (m^-m 3)
Figure 2-1. Thermal conductivity of a typical sandy soil as a function
of volumetric water content (DeVries, 1975).


Air Temperature ( C )
240
25
20
15
10
5
0
-5
40 41 42 43 44 45
Julian Date
Julian Date
Figure B-15. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from February 9 to 13, 1987.


BIOGRAPHICAL SKETCH
Christopher L. Butts was born September 19, 1957 in Knoxville, TN
while his father was attending the University of Tennessee. Chris
attended elementary and junior high school while living in the small
town, Estill Springs, TN. His family then moved to Virginia Beach, VA
where he attended First Colonial High school. After graduating from
high school in 1975, he attended Virginia Polytechnic Institute and
State University in Blacksburg, VA. Requirements for a Bachelor of
Science in Agricultural Engineering were completed in 1979, then a
Master of Science in 1981. After completion of the Master of Science,
he worked at the Veterans' Administration Medical Center in Salem, VA
as a hospital facilities engineer for a year, then worked as a research
engineer at the University of Georgia Coastal Plains Experiment
Station. The Butts family moved to Gainesville, FL to begin a program
to attain the Doctor of Philosophy Degree in Agricultural Engineering
at the University of Florida. Chris is currently employed by the U.S.
Department of Agriculture, Agricultural Research Service at the
National Peanut Research Laboratory in Dawson, GA.
241


9
3. an energy balance method, or
4. a combination method.
The two mass balance methods (1 and 2 above) focus on the
equations of motion for the atmosphere and require accurate measurement
of velocity and temperature profiles or mass fluxes. The equations of
conservation of momentum, mass, and energy were used in the mass
balance methods to develop expressions for the rate of evaporation of
water and the shear stress between to vertical positions in the lower
atmosphere as
Qv = PaKw dg
dz
(2-2)
* P#m JJL
dz
(2-3)
H = CpapaKh dl
K dz
(2-4)
where:
Qv
Pa
7
H
z
q
u
cpa
vertical flux density of water vapor [kgm'^s'1]
density of air [kg*nr3]
shear stress rate [N*m-2]
vertical flux of heat [J*m*2]
vertical distance [m]
specific humidity [kgvapor/kgdyy air]
velocity [nrs"1]
specific heat of dry air [J*kg-1*K"1]


162
the measured loss from the north lysimeter. However, since the model
did not simulate the initial high rate of evaporation observed for both
lysimeters, the discrepancy between simulated and observed water loss
increased on the second day, January 6. Beginning on January 7, the
model predicted the daily water loss very accurately for the north
lysimeter. This was possibly due to the water not redistributing
sufficiently toward the soil surface overnight to provide a wet surface
for evaporation. Thus water was evaporating below the surface and
diffusing through the soil until it reached the soil-atmosphere
interface. The increased resistance to vapor flow then reduced the
total water vapor leaving the soil. However, the water in the south
lysimeter had moved toward the surface and replenished the soil and
provided water for evaporation at the soil-atmosphere boundary.
Therefore, water evaporated very rapidly early in the morning until the
soil surface dried causing the evaporation rate to decrease. The model
did not predict this flush of water from the soil, thereby under
predicting the daily water loss from the south lysimeter. From January
7 to 10, the model predicted total water loss from the soil as well as
hourly water loss from the north lysimeter very well.
Agreement between the simulated and experimental soil surface
temperatures was very good throughout test 70 (Figure 5-4). The
simulated maximum surface temperature on January 5 was approximately
4 C and 2 C higher than the measured maximum for the north and south
lysimeters, respectively. Simulated temperatures were within 1 C of
measured surface temperatures for the duration of the test. The
excessive simulated surface temperatures were indicative of the reduced
evaporation which was simulated as well. Had the evaporation rate been


82
of radiation observed in the test sample to the radiation intensity
measured in air. This relationship can be expressed as the count ratio
varying as a decreasing exponential with respect to the density of the
material. The count ratio is defined by the manufacturer as the ratio
of counts per minute observed for the test material to the counts per
minute observed in the magnesium standard (Troxler,1972). The observed
density of the soil obtained from the gamma measurement represents the
wet bulk density (Equation 3-1).
Pmt A B ln(CR) (3-1)
where:
_3
/wet : wet bulk density of the soil [g*cm ]
A, B : constants of regression
CR : count ratio [dimensionless]
count per min in test material
count per min in standard
If the dry bulk density is known, then the volumetric water content
can be calculated by
where:
9 :
^wet :
pdry :
9
^ ^wet~ ^dry^
Pvi
volumetric water content [cm3*cm3]
wet bulk density of soil [g*cm ]
O
dry bulk density of soil [g*cm ]
( 3-2 )


216
Panofsky, H. A. 1963. Determination of stress from wind and
temperature measurements. Qrtrly. J. Roy. Met. Soc. 89:85.
Penman, H. L. 1948. Natural evaporation from open water, bare soil,
and grass. Proc. Royal Soc. London, Series A, 193:120-145.
Prat, M. 1986. Analysis of experiments of moisture migration caused
by temperature differences in unsaturated porous medium by means
of two-dimensional numerical simulation. Int. J. Heat and Mass
Transfer. 29(7):1033-1039.
Pruitt, W. 0. 1966. Empirical method of estimating evapotranspiration
using primarily evaporation pans. pp. 57-61. In Evapotranspiration
and its role in water resources management. Proc. Chicago, IL.
December 5-6. American Society of Agricultural Engineers, St.
Joseph, MI.
Rainey, L. J., J. H. Young and K. J. Boote. 1987. PEANUTPC: A user
friendly peanut growth simulation model. American Peanut Research
and Education Society Proceedings. 19:55.
Reidy, G. A. and A. L. Rippen. 1969. Methods for determining thermal
conductivity of foods. ASAE Paper No. 69-383. American Society
of Agricultural Engineers, St. Joseph, MI.
Sanders, T. H. 1988. Unpublished data relating geocarposphere
temperature to peanut maturity and yield. United States
Department of Agriculture, Agricultural Research Service, National
Peanut Research Laboratory, Dawson, GA.
Saxton, K. E. 1986. Evapotranspiration research priorities for
hydrology in the next decade. ASAE paper no. 86-2627. American
Society of Agricultural Engineers, St. Joseph, MI.
Schieldge, J. P., A. B. Kahle and R. E. Alley. 1982. A numerical
simulation of soil temperature and moisture variations for a bare
field. Soil Sci. 133(4):197-207.
Schwab, G. 0., R. K. Frevert, T. W. Edminster, and K. K. Barnes. 1966.
Soil and Water Conservation Engineering, pp. 128-138. John Wiley
and Sons, Inc., New York.
Shih, S. F., D. S. Harrison, A. G. Smajstrla, and F. S. Zazueta. 1986.
Using infrared thermometry data in soil water content estimation.
ASAE paper no. 86-2121. American Society of Agricultural
Engineers, St. Joseph, MI.
Shih, S. F., D. L. Myhre, J. W. Mishoe and G. W. Kidder. 1977. Water
management for sugarcane production in Florida Everglades. Vol
2:pp 995-1010. In Proc. Int. Soc. of Sugar Cane Tech., 16':"
Congress, Sao Paulo, Brazil.


143
of the copper were known, the convective heat transfer coefficient
could be calculated from the slope of the logarithmic plot of the
temperature ratio and the Fourier number.
Results and Discussion
The determination of the thermal diffusivity of each sample
required that the slope of the logarithm of the dimensionless
temperature ratio as a function of time be determined. The bath
temperature used in calculation of the temperature ratio was the time
averaged bath temperature over the length of the test. The logarithm
of the temperature ratio was then plotted against time (Figure 4-1) for
the sample. The slope of the line was obtained by linear regression
for the data excluding the initial and final transients. The values
for the regressions were 0.995 or greater This procedure was
conducted for each of the samples and the corresponding data for the
copper cylinder (Figure 4-2).
Using the solution of the transient heat conduction equation
(equation 4-4) and the transcendental equation for an infinite cylinder
(equation 4-6) the convective heat transfer coefficient was calculated
from the slope of the copper cylinder temperature response curve.
Values of the convective heat transfer coefficient ranged from 4400 to
5900 W/m^ K. Next, it was assumed that the Biot number for the soil
sample was infinite. Using the value of (ai), the radius of the
cylinder and the slope of the temperature response, an initial estimate
of the thermal diffusivity was determined from the following
(4-14)


13
Limitations of the mass balance (eddy flux and profile) methods
are the requirements for very sensitive equipment for measuring either
the vertical profiles or the vertical fluxes due to turbulence. King
(1966) questioned the practical use of the mass balance methods of
determination of the evaporation. Fritschen (1966) pointed out that
the mass balance methods do not integrate results over time, thus
requiring constant monitoring of meteorological parameters.
Another approach to estimate evaporation from the soil is the
energy balance method (Fritschen, 1966). The rate of change of
sensible heat in a control volume of air at the earth's surface is
equal to the rate at which sensible and latent heat are carried into
and out of the volume by the wind, the convection of sensible and
latent energy in the vertical direction, the energy transferred to the
soil and the net radiant exchange of energy. In the energy balance
method, it is assumed that the net transfer of sensible and latent heat
in the horizontal direction by the wind is negligible when compared to
the vertical movement of heat in the atmosphere. The equations of
motion are utilized as in the mass balance profile method with the
exception of using the similarity equation for the sensible heat flux
instead of the shear stress. A ratio of sensible to latent heat flux
(equations 2-4 and 2-2), referred to as the Bowen ratio (BR), is given
by
3T
H cPapa Kh gT
IF hfg Km dq
(2-13)
The energy balance at the soil surface is


41
The empirical function, B, depends upon whether or not the surface is
hydrodynamically smooth or rough. In general, if the Reynolds number
based upon the shear velocity and the roughness height is less than
0.13 then the surface is considered smooth and B is determined using
Equation (2-47). The surface is rough if the Reynolds number is
greater than 2.0 and Equation (2-48) is used to determine the value
of B.
%
B = 13.6 Sc J 13.5
% \
B = 7.3 Re 4 Sc 5.0
where
Sc Schmidt number
v
Dv
Re = Reynolds number
z0 U*
v
The dimensionless heat transfer coefficient or Stanton number
(Chr) can be determined by substitution of the Prandtl number for the
Stanton number for the Schmidt number in equations (2-47) and (2-48)
above. The surface heat transfer coefficient can then be determined by
(2-47)
(2-48)
hh = Chy. PgCpg WSp
(2-49)


161
overnight. Short periods in which condensation occurred were simulated
for both lysimeters during the early morning hours of January 9. A
maximum evaporation rate of 0.4 mm/h was simulated on the first day of
the test then decreased during subsequent days of the test. The
experimental maximum hourly evaporation rate of approximately 0.8 mm/h
for the north lysimeter and 0.9 mm/h for the south lysimeter occurred
on January 6. Since simulated evaporation rates lagged behind the
measured values, the measured daily maximum evaporation rate probably
occurred prior to the beginning of the test on January 5 and was most
likely to have been at least as high as those on the second day of the
test. It was noted that the lag time between the simulated and
experimental daily maximum evaporation rates was approximately the same
for both the north and south lysimeters. Initiation of water loss
appeared to occur at approximately the same time for both the
experimental water loss and the simulated results. However,
experimental data showed that very little time lapsed between the
initiation of evaporation and the occurrence of the peak water loss.
On the other hand, the simulation resulted in a very gradual increase
in the evaporation rate beginning at approximately the same time as the
experimental results.
Comparison of the simulated and experimental cumulative
evaporation for both lysimeters indicated that the most of the water
lost from the lysimeters during the day occurred during the initial
evaporation phase during the morning (Figure 5-3). The simulated water
loss during the first day corresponded fairly well with the measured
water loss for both the north and south lysimeters. At the end of
the first day, simulated water loss was approximately 0.5 mm less than


Cumulative Evaporation (mm) Cumulative Evaporation (mm)
190
Julian Date
Figure 5-12. Comparison of simulated and experimental cumulative
evaporation for both lysimeters during test 72 (February
3 5, 1987).


178
Table 5-1. Validation test names and inclusive dates.
Calendar Dates Julian Dates
Test Name
Beginning
Ending
Beginning
Ending
70
Jan. 5
Jan. 9, 1986
5
9
71
Jan. 11
Jan. 14, 1987
* 11
14
72
Feb. 3
Feb. 5, 1987
34
36
73
Feb. 9
Feb. 13, 1987
40
44


84
should be used to determine changes in volumetric water content during
the experiments and gravimetric sampling should be used to determine
initial conditions for the experiment.
Soil Bulk Density and Porosity
The calibration procedure for the gamma probe provided sufficient
data to determine the vertical and horizontal distribution of the soil
dry bulk density in each of the lysimeters. As mentioned previously,
repeated core samples were obtained at several depths dispersed over
the area of each of the lysimeters. Figure (3-5) shows the average dry
bulk density as a function of depth in each of the lysimeters. The
error bars represent the standard deviation of the measured bulk
density. The bulk density for both lysimeters was relatively constant
at 1.38 g/cm3 in the top 25 cm and increased to approximately 1.6 g/cm3
between 25 and 40 cm. The largest deviation in measured bulk density
occurred at the lower depths.
The literature (Baver, 1972; DeVries, 1975) indicates that the
porosity for sandy soils ranges from 0.50 to 0.55 cm3/cm3. The soil
porosity was a soil parameter required as a function of depth. Due to
the variation in soil density, especially at the depths from 25 to
40 cm, it was decided that the porosity for the soil in the lysimeters
should be determined as a function of depth.
The porosity is defined as the volume of pore space in the soil
per unit of bulk volume. An air pycnometer (Baver, 1972) was used to
measure the porosity of the soil. Four levels of soil bulk density
ranging from 1.2 to 1.6 g/cm3 were used for the tests. Three soil
samples at each level of bulk density were prepared from a soil sample
obtained from the weighing lysimeters. The volumetric water content


224
The backward form of the equations for the boundary conditions at the
soil surface are
Backward direction: z 0
j = 1
(A-16)
energy:
(A-17)
water:
dzj
^j,n+l = 0j+l,m-l+ J Dj7]pnfl
(A-18)
vapor:
The above equations were used to represent the partial differential
equations which describe the energy and mass transfer within the soil.
After a complete cycle through the nodes, from node 1 to node nc, then
from node nc back to node 1, all of the equations had been solved using
the most recent values of the state variables.


47
Substitution of the numerical expressions for the individual terms of
the energy balance (equation 2-55) and dividing by the cross-sectional
area normal to the z axis (dx*dy) yields
dWjCs
Tj,m-1-Tj,n
eft
Tj+l>n"Tj,n
dz-i
Tj-l,n Tj,n
n (5j+l,n"^j,n) (Tj+l,n+Tj,n)
+ ^wjcpwjDLj azj 2
n *j,n) (Tj-l,n+Tj,n)
+ ^wjcpwjDLj aij^[ 2
n (Pvj+l,n ^vj,n) (Tj+l,n+Tj,n)
+ cpvjDvj 3zj 2
^ (^vj-l,n /vj.n) (Tj-l,n+Tj,n)
+ CpvjDyj 2
- dwj (hfgjEjjn) (2-64)
The partial differential equation describing the diffusion of
water within the soil can be transformed to difference form in a
similar fashion as the energy equation. The change in water content
of a volume of soil for the jth node is caused by diffusion of water
from nodes j+1 and j-1 to node j less the amount of water changed to a
vapor phase. The numerical expression for the water continuity per
unit area becomes


27
forces, the water in the soil has a soil water potential less than
zero, thereby reducing the equilibrium vapor pressure from that of pure
water. The following relationship (Baver et al., 1972) can be used to
determine the saturated vapor pressure above a water surface with a
potential other than zero.
ev( evs(T) expf-^p) (2-29)
w
where:
ev(^) = saturated vapor pressure of water with chemical
potential, ip [Pa]
ip = soil water potential [m]
g acceleration due to gravity [m*s'2]
Using the ideal gas law, the vapor density over a water surface with a
chemical potential other than zero can be obtained by:
Pv = PvsCO exP(j^*T) (2-30)
w
A set of simultaneous equations (2-22, 2-25, 2-26, 2-30) describe
the conservation of thermal energy, water vapor and liquid water for
the soil continuum and formed the basis of a coupled mass and energy
model for the soil. The assumption of thermodynamic equilibrium
between the liquid and vapor states yields the constitutive
relationship expressed in equation (2-30) and provides a fourth state
equation to be used in the model.
A well-posed problem also includes boundary conditions and, in
the case of transient problems, the initial conditions must also be
prescribed. The system of governing equations was one-dimensional and
therefore required two boundary conditions. The first boundary is the


CHAPTER III
EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND
ENERGY TRANSFER PROCESSES IN THE SOIL
Introduction
The process of evaporation of water from the soil is a complicated
process in which the dynamic processes of heat and mass transfer are
inter-related. Lysimetry has been defined as the observation of the
overall water balance of an enclosed volume of soil. Lysimeters have
been used for over 300 years to measure soil evaporation and crop water
use (Aboukhaled et al., 1982) and can be generally classified as either
weighing or non-weighing. As their names imply, weighing lysimeters
measure evaporation by monitoring the weight changes within the
enclosed soil volume while the non-weighing, or drainage lysimeters
determine evaporation by monitoring amount of water drained from the
bottom of the tank and the amount of rainfall upon the lysimeter. Most
drainage lysimeters are monitored on a seven to ten day cycle while the
weighing lysimeters can be monitored continuously to obtain hourly
evaporation rates.
The energy status of the soil can generally be determined by
measuring the temperature distribution over time. The energy flux at
the surface must be determined as well as latent heat transfer to
facilitate a complete energy analysis. Surface heat flux occurs as
radiant and convective heat transfer. The radiant energy impinging
upon the soil surface is either absorbed or reflected. The amount of
reflected and absorbed energy is primarily dependent upon the soil
66


237
Julian Date
Figure B-12. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from February 3 to 5, 1987.


Temperature (C) Temperature (C)
127
40 41 42 43 44 45
. Julian Date
Figure 3-26. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from February 9 to 13, 1987.


147
locating the thermocouple near the center of the sample. These
precautions were both taken during the tests. Gaffney et al. (1980)
also demonstrated that conduction error did not significantly affect
the slope of the time-temperature curve.
A second source of error could arise from the movement of water
within the sample. The enhancement of the thermal conductivity due to
vapor movement in the soil occurs in response to concentration
gradients due to gradients in temperature within the sample and can be
calculated by (DeVries, 1975)
Dv 17 3ps
\ = hfg R^T dl
(4-17)
The apparent thermal conductivity is the sum of the conductivity for
conduction and the conductivity due to vapor movement (DeVries, 1975).
The apparent thermal diffusivity would be calculated by dividing the
apparent conductivity by the heat capacitance of the soil. Therefore,
the contribution of the vapor flow to the thermal diffusivity would be
Dtv
(4-18)
The temperature used in the calculation of the diffusivity due to vapor
movement was the time average temperature at the center of the sample.
Regression analysis was used to attempt to find the least squares fit
of the data using dry bulk density, volumetric water content and the


Net Radiation
229
Julian Date
Figure B-4. Wind speed (top) and net radiation (bottom) data for
lysimeter experiment conducted from December 15 to 19,
1986.


214
Ferraz, E. S. B. and R. S. Mansell. 1979. Determining water content
and bulk density of soil by gamma ray attenuation methods.
Bulletin 807 (Technical), Agricultural Experiment Stations,
Institute of Food and Agricultural Sciences, University of
Florida, Gainesville, FL.
Fritschen, L. J. 1966. Energy balance method, pp. 34-36. In
Evapotranspiration and Its Role in Water Resources Management.
Conference Proceedings, Chicago, IL. December 5-6, 1966. American
Society of Agricultural Engineers, St. Joseph, MI.
Fuchs, M. and C. B. Tanner. 1967. Evaporation from a drying soil.
J. Applied Meteoro!. 6:852-857.
Gaffney, J. J., C. D. Baird and W. D. Eshelman. 1980. Review and
analysis of the transient method for determining thermal
diffusivity of fruits and vegetables. ASHRAE Transactions,
86(2): 261-280.
Goddard, W. B. and W. 0. Pruitt. 1966. Mass transfer eddy flux
method, pp. 42-44. In Evapotranspiration and Its Role in Water
Resources Management. Conference Proceedings, Chicago, IL.
December 5-6, 1966. American Society of Agricultural Engineers,
St. Joseph, MI.
Harrold, L. L. 1966. Measuring evaporation by lysimetry. pp. 28-33.
In Evaporation and Its Role in Water Resources Management.
Conference Proceedings, Chicago, IL. December 5-6. American
Society of Agricultural Engineers, St. Joseph, MI.
Heermann, D. F. 1986. Evapotranspiration research priorities for the
next decade--irrigation. ASAE paper no. 86-2626. American
Society of Agricultural Engineers, St. Joseph, MI.
Idso, S. B., R. J. Reginato, R. D. Jackson, B. A. Kimball and F. S.
Nakayama. 1974. The three stages of drying of a field soil.
Soil Sci. Soc. Amer. Proc., 38:831-837.
Jackson, R. D. 1964. Water vapor diffusion in relatively dry soil: I.
Theoretical considerations and sorption experiments. Soil Sci.
Soc. Am. Proc. 28:172-176.
Jackson, R. D. 1973. Diurnal changes in soil water content during
drying, pp. 37-55. In Field Soil Water Regime, SSSA Special
Publication No. 5, SSSA, Madison, WI.
Jagtap, S. S. and J. W. Jones. 1986. Resistance type model of soil
evaporation. ASAE paper no. 86-2523. American Society of
Agricultural Engineers, St. Joseph, MI.


15
temperature. It was also assumed that the net soil heat flux is zero
over a 24 h period. The resulting relationship
Pacpa^fg (ea eas) + s(Rn )
' FTTT) (2'15)
potential evaporation [m3*m2]
vapor pressure of atmosphere [kPa]
saturated vapor pressure at the air temperature [kPa]
surface vapor transfer coefficient [m*s'l]
sensible heat flux in the atmosphere [W*nT2]
net radiation upon the soil surface [W*m-2]
slope of saturated vapor pressure line [kPa,0K_1]
psychrometric constant [kPa-^C1]
provides an expression for the potential evaporation from the surface.
The boundary layer resistances account for advective conditions. The
Penman equation is the only method for estimating evaporation based
upon theoretical ideas and requires no highly specialized equipment.
Estimates of the evaporation have an accuracy of 5 to 10 percent on a
daily basis (Van Bavel, 1966). The disadvantage to the combination
method, as implemented by Penman, is that only daily estimates are able
to be determined and crop coefficients are necessary to estimate actual
evapotranspiration from various crops (Jones et al., 1984). The values
of the crop coefficients typically exhibit regional variation as well
as variation due to stage of crop growth.
Staple (1974) modified the Penman model to provide the upper
boundary condition for the isothermal diffusion of water in the soil.
The Penman model was modified by multiplying the saturated vapor
where:
E(
e-
as
h
6
Rr
s
1


Relative Humidity ( % ) Air Temperature ( C )
230
Julian Date
Figure B-5. Measured relative humidity (bottom) and temperature
(top) of ambient air for evaporation tests conducted
from December 15 to 19, 1986.


KEY TO SYMBOLS AND ABBREVIATIONS
Variable Definition
B empirical function for determination of Cer and Chr
[dimensionless]
Cdr surface drag coefficient [dimensionless]
Cer Dalton number, dimensionless mass transfer
coefficient
Chr Stanton number, dimensionless heat transfer
coefficient
Cpa specific heat of moist air [J*kg-1-K_1]
Cne specific heat of solid constituent of soil mixture
K [ J*kg_1* K"^]
CpW specific heat of water [J*kg'l,{K"l]
CpV specific heat of water vapor [J'kg"1*^'1]
Cs volumetric heat capacity of soil mixture (solid,
liquid and air phases) [J'kg*1*^'1]
Da diffusivity of water vapor in air [m2*s_1]
Dl unsaturated hydraulic diffusivity of soil [m^s-1]
Dt thermal diffusivity [m2*s"l]
Dv diffusivity of water vapor in soil [m2,s_1]
dwj height of volume of soil associated with node j. [m]
dxj width of soil cell j in the x direction (unity) [m]
dyj width of soil cell j in the y direction (unity) [m]
dzj vertical distance between nodes j and j+1. [m]
E(z,t) rate of phase change of liquid water to water vapor
@in the soil as a function of space (z) and time (t)
tk9(H20)*m(soil)s' J
v


Predpitaon (mm/h)
238
Figure B-13. Precipitation during lysimeter evaporation studies
conducted from February 3 to 5, 1987.


Depth ( m ) Depth (
126
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-25. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on February 9 and 13, 1987.


3
root growth for various stages of development of soybean (Brouwer,
1964; Brouwer and Hoogland, 1964; Brouwer and Kleinendorst, 1967).
Blankenship et al. (1984) have shown that excessive temperatures in the
soil zone in which peanuts are produced (geocarposphere) can reduce
yields and increase susceptibility to disease, fungus and physical
damage. Unpublished data collected by Sanders (1988) for peanuts grown
in lysimeters with controlled soil temperatures indicated that the
maturation rate of peanuts were affected by soil temperature. The fact
that movement of water vapor in the soil in response to temperature
gradients can be significant was determined by Matthes and Bowen (1968)
and more recently by Prat (1986). Neglecting soil temperature in a
mass transfer model in the soil could cause significant errors in the
estimation of ETa from the soil if soil temperature were neglected in
the modeling analysis.
Research Goals and Objectives
An understanding of the processes of heat and mass transfer in the
soil is of interest to a variety of agricultural scientists,
particularly those involved in modeling agricultural systems and
developing management strategies. Evapotranspiration is a complex
process involving energy and mass transport in the plant and the soil
as well as the chemical processes such as photosynthesis and
respiration which require water from the soil and release water from
the reaction. It has been suggested by some researchers that the two
processes involving the plant and the soil be studied independently so
that greater insight might be obtained into each. Therefore, the goal
of this research is to study the process of evaporation from the soil
independent of transpiration for the purposes of furthering the


50
If the water content of the last node (j=nc) were equal to that of
the preceding node (j=nc-l), then no water would flow between the two
nodes. This would cause both of the last two nodes to act as one node
during the simulation. The other viewpoint would be that the sum of
the flows into the last node from the preceding node and an imaginary
node following must be zero.
Lnc ,/, ^ DLnc
clz^: l*nc+l,n *nc,n; = "3Fr£_i^nc-l*n
(2-70)
If the distance between nodes nc and nc-1 is the same as the distance
between the nodes nc and the imaginary node nc+1, then it follows that
^nc+l,n = ^nc-l,n (2-71)
Substitution of 2-71 into 2-65 yields as the boundary condition for the
node, j=nc, the following
dw,
nc
(^nc,m-r ^nc,n)
3E
Lnc rn n x
2 dznc_i (0nc-l,n-0nc,n)
(2-72)
dWr
-nc,n
nc /wnc
The same approach produces the following as the boundary condition for
the vapor continuity and energy equations, respectively.
(Pvnc,rH-r Pvnc,n) Dvnc
dwnc at = 2 znc_i (^vnc-r^vnc)
(2-73)
dwnc Enc,n
(sj ^j,n)
A n (^nc,rH-l^nc,n) o ^ (^nc-l,n_ ^nc,n)
dwnc Cs,nc dt = 2 Anc 3z^


52
during the solution phase of the system of equations and is equivalent
to an Euler integration in time (Conte and de Boor, 1980). The
disadvantage of the explicit solution is that the numerical error may
propagate through time and grow. The maximum time step is functionally
related to the boundary conditions and thermal properties of the soil.
The system of equations can also be changed such that all
derivatives are evaluated at the next time step in which none of the
state variables are known. This implicit representation requires a
relatively complicated iterative or matrix solution technique for the
system of equations. The advantage of implicit solution methods is
that the value of the state variables do not depend upon previous
values implying that the only source of error would be due to round-off
or truncation errors and would not propagate or grow with time and
would yield an unconditionally stable solution (Conte and de Boor,
1980).
Another technique which has some of the desirable characteristics
of both the explicit and implicit methods is the alternating direction
(ADI) method. This is accomplished by incrementally marching through
space in one direction (z=0 to zo) evaluating the derivatives
containing the previous node (j-1) at time, t + */2 dt, and those
containing the following node (j+1) at time, t. Then returning in the
opposite direction (z=zo to 0), evaluate the derivatives containing the
node j+1, at t=t+dt, and the derivatives containing j-1 at t=t+V;?dt.
This technique uses a relatively simple algorithm similar to that for
the explicit methods because all of the values of the state variables
used in estimating the state variable at the next time step are known
quantities. However, the number of equations to be evaluated is twice


Measured Volumetric Water Content (cmV cm^)
105
Predicted Volumetric Water Content (cm^/ cm^)
Using Gamma Probe Calibration
Figure 3-4. Comparison of measured volumetric water content to that
estimated using the gamma probe calibration.


91
Evaporation calculated from experimental measurements of
volumetric water content for the period from November 25 to 26 (Figure
3-9) was 5 and 6 mm of water from the north and south lysimeters,
respectively. The evaporation determined by changes in weight of the
lysimeters was 3.8 mm for the north lysimeter while 2.5 mm of water had
evaporated from the south lysimeter. Normally, the measurement of
volumetric water content would be more likely to have higher errors.
However, due to the close agreement between the evaporation determined
by changes in volumetric water content for the two lysimeters, the
determination of evaporation by weight loss from the lysimeters was
suspected to be in error. The data for the south lysimeter were most
highly suspect for this data set.
Temperatures followed a diurnal fluctuation as expected (Figure
3-10). The air temperature reached a maximum during mid-afternoon and
a minimum just prior to sunrise. It was interesting to note that the
peaks in air temperature lagged the peaks of soil temperature. This
was due to the fact that the primary source of sensible heat gain in
the air was by convection from surfaces instead of radiant heat
transfer. The lag time for the peaks within the soil increased as
expected with depth. The amplitude of the diurnal fluctuation also
diminished with depth due the thermal capacitance of the soil as
expected. The soil temperature at a depth of 80 cm varied less than
2C during the test.
The next set of data was recorded from December 15 to 19, 1986.
Similar problems appeared in this data set as had occurred in the
previous data (November 25 29). It was noted that the north
lysimeter initially lost approximately 1 mm more water than did the


174
to 8. The cumulative evaporation increased as the value of the
transfer coefficient multiplier increased (Figure 5-25). The
cumulative evaporation for the lowest values of transfer coefficients
matched the initial phase of the cumulative evaporation fairly well,
but then around sunset, the experimental data indicated a continued
slight water loss from the soil, while the simulation showed no water
loss overnight. Evaporation during the period following the first
night showed the model significantly under-predicted the cumulative
evaporation. The simulations for the multipliers of 2.0 and 5.0 over
predicted evaporation during the first day of the experiment, while at
the end of the second day, the cumulative evaporation simulated using
the multiplier of 2.0 matched experimental data fairly well. Since the
cumulative loss had been over-predicted on the first day but was
agreement by the end of the second day, indicated that the water loss
for the second day had been under-estimated. The error between
experimental and simulated cumulative evaporation remained
approximately the same from the first to second day using a multiplier
of 5.0. Examination of the hourly evaporation rates indicated that as
the multiplier for the surface transfer coefficients increase, the
hourly evaporation rates looked more like the experimental data (Figure
5-25) showing that early in the day, the boundary coefficients or the
method in which the boundary conditions for the transfer of vapor was
modeled was limiting the evaporation of water when water was available
at the surface. Increasing the heat and mass transfer coefficients
caused the evaporation rate to exhibit the type of behavior
demonstrated by the experimental rates. A high rate of evaporation


80
consisted of a series of removable brass rings of known volume within a
hollow cylinder. The cutting edge was tapered such that compaction of
the soil sample was minimized (Baver et al., 1972). The soil sample
was removed from the inner brass ring and placed in an aluminum soil
sample container and the lid closed to prevent moisture loss until the
samples from a single probe could be measured. Core samples were taken
at depth intervals of 10 cm from the soil surface to a depth of 71 cm.
Three probes were obtained from within the lysimeter to roughly
correspond to the locations of the paired access tubes. The soil
samples were weighed, dried, and reweighed. Wet density, dry density,
and volumetric water content were calculated for each sample. The soil
properties determined from each of the three probes were averaged for
each depth to account for the fact that the density between the access
tubes could not be measured. This procedure also provided information
regarding the areal uniformity of the soil with the lysimeter. The
dried soil was placed back in the holes in reverse order of their
removal so as to maintain the original soil profile. Care was taken
during subsequent sampling to avoid the soil core sites previously
sampled in the lysimeters.
The gamma readings were obtained in the following manner. The
meter was turned on for a minimum of twenty minutes prior to readings
being taken to allow for the circuits to stabilize. After warm-up, the
137
detector probe and Cs gamma source were placed in the tubes of a
calibration stand. The calibration stand consisted of two parallel
aluminum tubes with materials of known density. The standard materials
were polyethylene (1.06 g/cm3), magnesium (1.75 g/cm3), magnesium-
aluminum alloy (2.16 g/cm3), and aluminum (2.61 g/cm3). The source and


102
OFFICE a
DATA ACQUISITION EQUIPMENT

"SPAR" UNITS

A.
M
AWAROS
WEATHER
STATION
Figure 3-1. Site plan for weighing lysimeter installation at
University of Florida Irrigation Research and Education
Park.


62
H=f(TQlhh,WS)
\\\
Rn
\W
Qv=f(Ta,RH,hm,WS)
H
J*2
j = 3
H
AZ
J
i+l
nc-l
nc
Figure 2-5. Schematic of discretized domain for evaporation and soil
temperature model.


33
w = water (same as volumetric water content)
a air
s soil composite
The difficulty arises in estimating the various volume fractions of the
different components. For quartzitypic sands, approximately 70 to
80 percent of the solid constituent of the soil is quartz, with
generally less than 1 to 2 percent organic material and the remainder
consisting of other ninerals. According to information presented by
DeVries (1975), the volumetric heat capacity and density of quartz and
other minerals are very similar. For the purposes of this simulation,
the volume fraction of the organic material was assumed to be zero.
The thermal conductivity cannot be calculated in such a straight
forward manner. Thermal conductivity is defined as the constant of
proportionality in relating the heat flux by conduction to the
temperature gradient. Conduction of heat occurs due to physical
contact between adjacent substances. In a solid material such as steel
or concrete which is fairly homogeneous, the material conducting heat
can be considered continuous. However, the soil is a mixture of solid,
liquid and gaseous components and the area of physical contact between
soil particles is a function of the particle geometry. When the soil
is dry, the area of contact may be a single point. As the soil is
wetted, a thin film of water adheres to the soil particle and increases
the contact area between adjacent particles. This increase in area
accounts for a rapid change in the thermal conductivity as the soil is
initially wetted (Figure 2-1). Over a range of water contents between
dryness and saturation, the increase in thermal conductivity becomes
linear with increasing water content. As the soil approaches


Vapor Density Deficit (kg/m^)
( Pv1 ~ Pva )
198
Julian Date
Figure 5-20. Simulated difference between the water vapor concen
tration at the soil surface and ambient air during test
73 (February 9 13, 1987).


88
Meteorological data, including relative humidity determined from the
estimated dewpoint temperature, are shown in APPENDIX B.
Tests were begun by an irrigation or rainfall event to ensure that
the upper layers of the soil were sufficiently wet so as to provide a
minimum of three to four days of evaporation data. Most frequently,
water was added to the soil the evening prior to beginning the test to
insure that the soil surface was wet and to allow some downward
distribution of water prior to the initiation of each test. Core
samples were obtained to determine the initial distribution of water
within the lysimeter. Samples were taken at vertical intervals of
5.1 cm for the first 30.5 cm then every 10.2 cm until a depth of
91.4 cm was reached. Gamma probe measurements were taken in each of
the three pairs of access tubes corresponding to the center of each of
the core samples. Actual depths of gamma probe measurements and core
samples are shown in Table 3-4. Subsequent gamma probe measurements
were made approximately every other day. More frequent readings were
not obtained in most cases due to the time required for each set of
readings (2 hrs.). This minimized disruption of the other data being
collected.
Obtaining the gamma probe readings required placing a scaffold
across the lysimeter to avoid disturbing the weight of the lysimeter
and compacting the soil. This, in turn, could have disrupted the
evaporation process due to shading of the soil surface as well as some
of the net radiation measurements. These periods when the integrity of
some of the data may have been questionable was marked in the hourly
data file by a flag entered via the keyboard. The flag was turned on
when gamma readings were being made on each lysimeter and turned off


Net Radiation ( W/m2) yy¡nd
232
Julian Date
Figure B-7. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from January 5 to 10, 1987.


Depth ( m ) Depth (
184
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-6
Initial and final (simulated and experimental)
volumetric water content as a function of depth for test
70 (January 5-8, 1987).


202
Figure 5-24. Simulated diurnal variation of volumetric water content
of the soil surface for various multipliers for the
hydraulic diffusivity.


87
the power was lost just prior to closing the data file. When power was
restored, the PDP-11/23 would automatically restart and begin the data
collection routine. Separate data files were maintained for each of
the two lysimeters.
Hourly values of the ambient dry bulb temperature and relative
humidity were measured within a standard weather shelter from the
adjacent weather station monitored and maintained by the Agricultural
Engineering Department. Sensors for measuring air temperature were
Type T thermocouples and instantaneous values were recorded hourly. A
Campbell Scientific CR-207 sensor was used to monitor relative
humidity and consists of a wafer whose electrical properties vary with
relative humidity. Manufacturer's literature states that the sensor
provides reliable information when used in non-condensing conditions
with a relative humidity between approximately 20 and 90 percent.
However, the wafer material tends to absorb moisture over time
decreasing the reliability of data obtained. To account for the lag
time of the sensor, data was averaged over an hour and recorded.
Relative humidity data was generated as well by assuming that the
minimum daily temperature was the dewpoint temperature for the day.
Linear interpolation between consecutive daily minimums to provide a
continuous estimate of the dewpoint temperature throughout the day.
Relative humidity could then be calculated based upon the estimated
dewpoint temperatures. Wind speed and direction were measured at a
height of 2 m within the same weather station. Meteorological data
were recorded hourly and uploaded daily to the VAX mainframe managed by
IFAS. Access to the hourly data was achieved through the AWARDS
(Agricultural Weather Acquisition Retrieval Delivery System).


Evaporation Rate (mm/h)
201
Figure 5-23. Simulated hourly evaporation rates resulting from
calibration multipliers for 0.5, 2.0 and 5.0 for the
hydraulic diffusivity.


In 0
153
Time (s)
Figure 4-1. Semi-logarithmic plot of the dimensionless temperature
ratio as a function of time for a typical soil sample.


CHAPTER V
MODEL ANALYSIS
Introduction
Prior to using a model to simulate a given set of conditions, it
must be calibrated and validated. Validation of a model ensures that
the model responds to various stimuli in the same manner as the real
system would. This may be done by comparison of the simulated response
to experimental data or by analyzing the simulated response to changes
in various parameters used in the model. Model calibration requires
the use of experimental data so that parameters can be adjusted so that
the model simulates the response of the physical system as closely as
possible to a specified range of input conditions. The purposes of
this model analysis were to determine the validity of some of the
underlying assumptions in developing the model, to identify the
parameters most likely in need of calibration and to begin the process
of calibration for a limited set of experimental data.
Validation
Data used in the validation process were collected from
January 5, 1987 to February 13, 1987. Hourly ambient weather data were
not available from the AWARDS system during August, 1987 due to
equipment failures and were not included in the validation process.
Table 5-1 shows the test names and their inclusive dates used in
validation. The variation of soil water potential and the hydraulic
diffusivity with water content was generated using the Van Genuschten
method from data for a Mi 11 hopper fine sand published in Carlisle et
158


CHAPTER I
INTRODUCTION
Problem Statement
The process by which water is lost from the soil has been the
object of considerable research over the years (Brutsaert, 1982).
Water is lost from the soil by runoff, drainage, evaporation and
transpiration. Evapotranspiration (ET) has been used to describe the
combined evaporative losses from the soil and plant. Potential evapo
transpiration (ETp) has been defined as the amount of water
transferred from a wet surface to vapor in the atmosphere (Penman,
1948) and has been used as an estimate of the maximum amount of water
lost to the atmosphere from a given area. Estimates of actual
evapotranspiration (ETa) are frequently made by the use of coefficients
which vary according to ground cover and the region of the country.
The need for knowing ETa on a field basis arises for a variety of
reasons including design and selection of irrigation equipment,
optimization of irrigation management, and determination of water
requirements for agricultural and municipal entities (Heermann, 1986).
Regional estimates of ETa are necessary for hydrologic simulations
(Saxton, 1986).
Mathematical models have been developed to provide estimates of
both ETa and ETp. The use of evapotranspiration models has expanded in
recent years due to the advent of crop growth simulations rquiring the
distribution of soil water as a component of the model. One such model
used to integrate the soil water and the plant water requirements was
1


Evaporation Rota (mm/h) Evaporation Rota (mm/h)
196
Figure 5-18. Hourly simulated and experimental evaporation rates for
test 73 (February 9 13, 1987).


203
Figure 5-25. Cumulative evaporation resulting from the calibration
multipliers for the boundary heat (ty,) and mass (hm)
coefficients (Dl multiplier 2.0).


125
Julian Date
Figure 3-24. Hourly and cumulative water loss measured from February
9 13, 1987 using the University of Florida, IREP
weighing lysimeters.


Temperature (C) Temperature (C)
111
Julian Date
Figure 3-10.
Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from November 25 to 9, 1986.


Cumulative Evaporation (mm) Cumulative Evaporation (mm)
181
12
10
8
6
4
2
0
5 6 7 8 9 10
Julian Date
Figure 5-3. Comparison of simulated and experimental cumulative
evaporation rates for test 70 (January 5 10, 1987).


be determined by
35
= I (^ia + ^ib + ^ic) (2-37)
where
k^j = the ratio of the temperature gradient in the i-th
component to the temperature gradient in the continuous
component in the direction of the j-th principal axis
i = quartz (q), mineral (m), organic (o), water (w), air (a)
j = a, b, or c for each of the three principal axis of the
particle
The ratio of the temperature gradients in the i-th component can be
determined by the following:
(2-38)
The shape factor for each principal axis (gj) can be approximated by
various empirical relationships depending upon the ratio of the unit
vectors (ua, Uj,, uc) of the principal axes of the soil components
(Table 2-1). The sum of the three shape factors must be unity.
In most cases, water is considered to be the continuous phase of
the soil in determining soil thermal conductivity. However, as the
soil dries and the film adhering to the surface of the soil particle
begins to break, making air the continuous phase. Equation (2-37) can
be used in these cases replacing the thermal conductivity of water (Aw)
with the thermal conductivity of air (Xa). De Vries (1975) noted that
the values for the thermal conductivity in the case of air being the
continuous phase were consistently low by a factor of approximately
k
-1
w
%


175
occurred early in the day then decreased as the soil began to limit the
rate of movement of water vapor to the atmosphere.
Soil surface temperatures behaved as expected in response to the
changes in surface transfer coefficients. Using a decreased surface
transfer coefficients caused the diurnal variation of soil surface to
increase due to decreased heat transfer to the atmosphere and
conversion to latent heat (Figure 5-27). As the transfer coefficients
increased, the maximum soil surface temperature decreased. A
multiplier of 2.0 produced a lower maximum soil surface temperature
with maximum and minimum temperatures occurring at the same time as the
experimental data. Increasing the surface transfer coefficients
increased the effect of latent heat removal upon the soil surface
temperature. Note that the simulated soil surface temperature
initially decreased by approximately 2 C due to increased latent heat
transfer. This decrease caused the maximum soil temperature to occur
approximately two hours after the experimental maximum. On the second
and third days of the simulation, the rate of increase of the surface
temperature decreased at the same time that the maximum rate of
evaporation occurred.
The effect of surface transfer coefficients upon surface soil
water content was inverse to that of the hydraulic diffusivity. As the
transfer coefficients were increased, the amplitude of the diurnal
variation surface water content increased (Figure 5-28). This was due
to the increased rate of water removal from the soil caused by
increasing the mass transfer coefficient. The soil then redistributed
water toward the surface. The diurnal variation of water content
content penetrated the soil to a depth of approximately 5 cm using a


213
Brutsaert, W. 1982. Evaporation into the atmosphere: Theory,
history, and applications. D. Reidel Publishing Co., Dordecht,
Holland.
Butts, C. L. 1985. Weighing lysimeter at the U. F. Irrigation Park.
Florida Cooperative Extension Service Drawing No. SP-5125.
Institute of Food and Agricultural Sciences, Agricultural
Engineering Department, University of Florida, Gainesville, FL.
Camillo, P. J. and R. J. Gurney. 1986. A resistance parameter for
bare-soil evaporation models. Soil Sci. 141(2):95-105.
Carlisle, V. W., M. E. Collins, F. Sodek III, and L. C. Hammond. 1985.
Characterization data for selected Florida soils. Soil Science
Report No. 85-1, University of Florida, Institute of Food and
Agricultural Sciences, Soil Science Department Soil Science
Laboratory, Gainesville, FL. p. 169.
Conaway, J. and C. H. M. Van Bavel. 1967. Evaporation from a wet soil
surface calculated from radiometrically determined surface
temperatures. J. Applied Meteorol. 6:650-655.
Conte, S.D. and C. de Boor. 1980. Elementary numerical analysis: An
algorithmic approach. McGraw-Hill Book Co., New York, NY.
Criddle, W. D. 1966. Empirical methods of predicting
evapotranspiration using air temperature as the primary variable,
pp. 54-56. In Evapotranspiration and Its Role in Water Resources
Management. Conference Proceedings, Chicago, IL. December 5-6,
1966. American Society of Agricultural Engineers, St. Joseph, MI.
Decker, W. L. 1966. Potential evapotranspiration in humid and arid
climates, pp. 23-26. In Evapotranspiration and Its Role in Water
Resources Management. Conference Proceedings, Chicago, IL.
December 5-6, 1966. American Society of Agricultural Engineers,
St. Joseph, MI.
DeVries, D. A. 1958. Simultaneous transfer of heat and moisture in
porous media. Trans. Am. Geophys. Union. 39:909-916.
DeVries, D. A. 1975. Heat transfer in soils, pp. 5-28. In D. A. de
Vries and N. H. Afgan (ed.) Heat and mass transfer in the
biosphere I. Transfer processes in plant environment. John Wiley
and Sons, New York, NY.
Dugas, W. A., D. R. Upchurch, and J. T. Ritchie. 1985. A weighing
lysimeter for evapotranspiration and root measurements. Agron. J.
77:821-825.
Eckert, E. R. G. and R. M. Drake, Jr. 1972. Analysis of heat and Mass
transfer. McGraw-Hill Book Co., New York, NY.


8
evaporation to the available water parameterized the thickness of dry
soil through which the water vapor diffused. The soil resistance term
used by Camillo and Gurney (1986) was a linear regression of the
difference between the saturated volumetric water content and the
actual soil water content. Both models were intended to provide daily
evaporation rates.
The resistance model estimates daily evapotranspiration relatively
well. However, the soil resistance term varies with time depending
upon the distance through which the water vapor must diffuse from below
the soil surface. One point that should be noted is that the vapor
pressure difference used in the analog models discussed is the
difference between the saturated vapor pressure at the temperature of
the soil surface and that of ambient air. In reality, evaporation
frequently occurs below the soil surface and the vapor pressure
gradient from the zone of evaporation may be substantially different
from that at the surface.
Mechanistic Models
Models derived from the basic physical relationships have the
advantage of providing estimates of evaporation on a relatively short
time scale. However, considerably more input data describing the
surface boundary conditions as well as the soil thermal and hydraulic
properties are required. Theoretical models have been developed from
four different perspectives (King, 1966; Goddard and Pruitt, 1966;
Fritschen, 1966; Penman, 1948). They are categorized as follows
1. a mass balance profile method,
2. a mass balance eddy flux method,


Depth ( m ) Depth ( m )
132
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-31. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on August 25, 27, and 29,
1987.


137
homogeneous, isotropic solid with uniform initial temperature placed in
a moving fluid of constant and uniform temperature can be expressed in
terms of the dimensionless temperature ratio, the Biot (Bi) and Fourier
(Fo) numbers as follows (Equation 4-4)
TU.t) Tb
* T(z,0) T = n5> c"(z) exP('anFo) (44)
iL
where an is the ntn eigenvalue of the following transcendental
equations for the case
of a
sphere,
an infinite cylinder
and infii
slab.
sphere:
Bi
= 1 -
ancot(an)
(4-5)
infinite cylinder:
Bi
= an
Jl(an)/Jo(an)
(4-6)
infinite slab:
Bi
" an
tan(an)
(4-7)
The transcendental equation for the infinite cylinder (Equation 4-6)
contains zero and first order Bessel functions of the first kind (Jq
and Jj, respectively).
The solutions to the transient heat equation are often presented
graphically in the form of semi-logarithmic plots of the temperature
ratio as a function of the Fourier number. A family of curves is
presented for a particular position within the body for various values
of the reciprocal of the Biot number (Ozisik, 1980). The Biot number
is the ratio of the product of the surface heat transfer coefficient
(hfo) and some characteristic length of the sample (L) to the thermal
conductivity of the sample material (A).


11
r = adiabatic lapse rate [^nr1]
= 9.86 x 10-3
k von Karman's constant [dimensionless]
= 0.428
U* shear velocity [nrs"1]
For neutral conditions the familiar logarithmic velocity profile was
used. For unstable or diabatic conditions a variation of the
logarithmic profile was used or the KEYPS function (Panofsky, 1963) was
used to determine the velocity profile. King stressed that in using
profile methods to determine evaporation, one must take extreme care in
measuring wind velocity. He suggested that spatial averaging be used
for wind speed measurements near ground level and that 30- to 60-minute
averages be used considering the steady state assumptions made in
developing the equations for the profile equations. Corrections for
lower atmospheric instability caused by density gradients in the air
near the earth's surface must also be used.
Another method requiring only measurements of atmospheric
parameters is the eddy flux method (Goddard and Pruitt, 1966). The
method is based upon the turbulent transport equations of motion in
the atmosphere. The equations of turbulent motion in a fluid include
the transient random fluctuations in velocity, temperature, and mass
concentration of water vapor of the air. The following equations
describe the shear stress (r), sensible heat flux (H) and latent heat
flux (LE) due to the turbulent motion of the atmosphere
* K 3U
^a^m
- paw'u'
(2-8)
cpaPaKh fj"
+ CpaPaW''
(2-9)


Depth ( m ) Depth ( m )
129
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on August 12, 13, 18,
and 21, 1987.
Figure 3-28.


Depth ( m ) Depth ( m )
117
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-16. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on January 5, 7 and 9, 1987.


89
again after the task was completed.
Data were collected continuously over a period extending from
November, 1986 to August, 1987. Data collection was interrupted on
several occasions due to equipment failures, power outages, equipment
maintenance, and instrument calibration. Most tests were three to four
days in length with a few extending to six or seven days.
Data Analysis
Data files containing the data recorded at ten-minute intervals
were concatenated into daily files. The daily files were then combined
to correspond to specific dates for the individual tests. Data
contained in the files consisted of day-of-year, time-of-day, the
normalized output for each load cell (mV/V), the 10-minute total net
radiometer output, the number of net radiometer readings during the
10-minute interval, the status flag, and twenty temperatures
corresponding to the depths indicated in Table 3-2.
The normalized output of the load cells was totaled and the
average total millivolt per volt for the hour was calculated. The
difference between successive hourly totals was used in the load cell
calibration equations to determine the hourly and cumulative
evaporation of water from the lysimeter. The output signal for the net
radiometers was totalled for the hour then an average rate of net heat
flux was determined using the manufacturer's calibration curves. The
two temperatures for each depth were averaged over the hour as wel1.
If the data flag had been turned on at any time during the hour, it was
assumed to have been on the entire hour. Reduced data files contained
the hourly values of julian date, time, cumulative evaporation (mm),
hourly evaporation (mrn/hr), net radiation (W/m^), data flag, and


Cumulative Evaporation (mm)
131
Julian Date
Figure 3-30. Hourly and cumulative water loss measured from
August 25 29, 1987 using the University of Florida,
IREP weighing lysimeters.


Net Radiation ( W/m^)
226
Julian Date
Figure B-l. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from November 25 to 29, 1986.


228
Figure B-
25
20 -
N
-C
\
E
£.15
c
o
'o
;5.10
U
0)
u
CL
5--
329
J lij, i, i
330
331
332
Julian Date
333
334
335
3. Rainfall during lysimeter studies conducted from
November 25 to 29, 1986.


55
Table 2-1. Formulae for the geometric shape factors used in
calculating thermal conductivity based upon ratios
of the unit vectors of the principal axes.
Object
9x
9y
9z
Sphere
0.33
0.33
0.33
Ellipsoid of revolution
x=y=nz n = 0.1
0.49
0.49
0.02
n = 0.5
0.41
0.41
0.18
n = 1.0
0.33
0.33
0.33
n = 5.0
0.13
0.13
0.74
n = 10.
Elongated cylinder with
elliptical x-section
0.07
0.07
0.86
x = ny;
(n + l)"1
n/((n +1)
0.00
Typical sand grain
0.14
0.14
0.72


159
al. (1985) (Figure 5-1). The volumetric water content of the soil
that yielded specified values of soil water potential were tabulated
for various depths throughout the soil profile and corresponded to the
different horizons of the soil. The water content for a given water
potential was averaged for all depths to yield the relationship between
water potential and volumetric water content characteristic of a
uniform soil profile.
Hourly ambient air temperature, relative humidity, wind speed and
rainfall were used as input data for the boundary conditions of the
model. The relative humidity used as input data was generated using a
dewpoint temperature estimated from minimum daily temperatures as
described in Chapter III. Hourly net radiation incident upon each of
the lysimeter surfaces was included in the weather inputs as well. The
meteorological data describing the boundary conditions for each all
tests conducted are shown in Appendix B. Core samples were obtained to
determine the initial volumetric water content at the beginning of each
test. Subsequent moisture data were obtained either by core samples or
a dual probe density gauge. Due to the intrusive nature of the
moisture measurements, data were usually recorded only at the beginning
and end of each test. In test 72 only the initial profile was
obtained due to several days of rain ending the test prematurely. The
rainfall required that the evacuation system to be run during the rain
to prevent overloading the loadcells.
In general, the model predicted the diurnal pattern of the
cumulative and hourly evaporation fairly well. Experimental and
simulated maximum hourly evaporation rates occurred around midday for
all tests; however, measured maximum hourly evaporation rates usually


79
gamma meters are typically the same diameter, thus allowing the use of
the neutron probe in the gamma access tube if necessary. The two
radiation methods will generally disturb less soil during installation
and will allow for recurring measurement at more frequent depth
intervals than with the tensiometers. Tensiometers also require
considerably more maintenance than do the radiation methods. The
tensiometers on the other hand can be monitored continuously by
automated data acquisition equipment while the radiation techniques
cannot. Extensive calibration is required for either of the radiation
measurement techniques. Based upon the desire to cause minimal
disruption of the soil profile and measure soil moisture at many points
within the soil, it was decided that a gamma attenuation technique
would be used to measure soil water content.
Gamma Probe Calibration
Extensive calibration curves were required to determine absolute
values of the soil water content using the gamma ray attenuation
method. While the soil was being loaded into the lysimeters, three
pairs of aluminum access tubes (5.1 cm O.D.) were installed in each
lysimeter (Figure 3-2). The tubes extended from the lysimeter floor to
23 cm above the soil surface. Parallel tube guides provided with the
Troxler 2376 Dual Probe Density Gauge were used to maintain parallelism
between the tubes as the lysimeter was filled.
The gamma probe primarily measures density of the test material;
therefore, it was necessary to measure the wet bulk density of the soil
corresponding to the depth at which the gamma readings were obtained.
A bulk density core sampler was used to obtain the soil samples for the
measurement of density and volumetric water content. The sampler


114
Volumetric Water Content (cm^/cm^)
North Lysimeter
0.0
0.1
0.2
0.3
0.4
0.3
0.6
0.7
0.8
I
?
T
0.00
Hme Date
1000 12/15
AA 900 12/19
-i i|^
0.10
Volumetric Water Content (cm^/cm^)
South Lysimeter
ii
0.40
Figure 3-13. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on December 15 and 19, 1986


Evaporation Rate (mm/h) Evaporation Rate (mm/h)
191
Figure 5-13. Hourly simulated and experimental evaporation rates for
test 72 (February 3-5, 1987).


49
X1
energy: (T2,n T¡>n) = Rnn + hj-,n(Tan Ti>n)
(2-67)
n (*2,n *l,n) (Pv2,n *>vl,n) (T2,n + Tl,n)
+ />wlcpwlDLl L1
water: (02,n *l,n) + P = 0 (2-68)
Dvl
vapor: (Pv2,n /Vl,n) + hm(/>va *vl,n) = 0 i2'69)
where:
Rnn
P
^mn
tyin
T
an
^van
net solar radiation incident upon the soil surface
at time step n [W*m'2]
precipitation at time step n [m3*m'3*s"l]
boundary layer mass transfer coefficient at time
step n [nrs"1]
boundary layer heat transfer coefficient at time
step n [W*m"2,0K_1]
air temperature at time step n [1C]
ambient water vapor concentration at time step n
[kg*m'3]
The boundary conditions at the lower boundary of the soil are
developed in a similar manner. The last node has volume and
capacitance for storage of energy and mass. A zero flux boundary
condition is used. There are two ways in which the zero-flux condition
can be represented. These will be developed for the water for the
purpose of illustration.


Depth ( m ) Depth ( m )
194
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm*5/cm^)
South Lysimeter
Figure 5-16
Simulated volumetric water content as a function of
depth at various times during test 72 (February 3-5,
1987).


22
considered to occur throughout the soil profile in response to
temperature and water content conditions in the soil. The rate of
phase change of the water (Ev) is considered positive if the water
changes from the liquid to vapor (evaporation), while condensation is
indicated by a negative phase change rate.
The diffusivity of water vapor in the soil air spaces is assumed
to equal that in the atmosphere, however, Schieldge et al. (1982), as
well as others, have made use of a tortuosity factor (a) to account
for the increased path length of the interstitial spaces of the soil
through which the water vapor must traverse. The tortuosity factor
included the effect of the local evaporation and condensation of water
vapor as it passes through the regions of differing temperature and
water content in the soil.
The volumetric water content is typically used to express the
amount of liquid water present in the soil. The mass concentration of
water in the liquid phase is obtained by multiplying the volumetric
water content by the density of water. The time rate of change of the
mass concentration of water is increased by a the rate of diffusion of
water from surrounding soil and decreased by the rate of evaporation.
The mathematical expression of the conservation of mass for the water
is
d 3Ql
(2-18)
where:
Pw
Ql
9
volumetric water content [m3*m~3]
density of water [kg*m-3]
-2 -1
diffusion of liquid water [kg*m *s ]


64
m
o
£>
O
3
y
35
v
cc
-o
£
o
3
cr
en
E
3
en
60 120 300
Time Step ( s )
600
Figure 2-7. Sum of squared residuals for the soil profile energy
balance after a 72 hour simulation using various time
steps.


CHAPTER IV
MEASURING THERMAL DIFFUSIVITY OF SOILS
Introduction
The temperature distribution of heat within a body is governed by
Fourier's law of conduction. For a one-dimensional body with no heat
generation, the partial differential equation describing the transient
temperature response subjected to various boundary conditions is
(Equation 4-1)
3T
It
Dt
dZJ
T?
(4-1)
The thermal diffusivity (D^) is the property of a material which
determines the rate at which heat is propagated. The thermal
diffusivity is defined as the ratio of the thermal conductivity to the
volumetric heat capacitance (Equation 4-2).
Dt = -4~ (4-Z)
A composite material such as soil may have thermal properties which
vary with space and time due variation of the composition with time and
space. The factors affecting the thermal diffusivity of the soil are
1. soil type,
2. soil water content,
3. soil bulk density and
4. possibly soil temperature.
134


81
detector were placed in the stand level with the center of the
polyethylene standard and the amplifier gain of the meter was set so
that the peak count rate was achieved. The probes were then lowered to
correspond with the magnesium and a 4-minute count was taken. In the
calibrate mode of the timer, the meter indicates the count per minute
for the 4-minute interval. This was referred to as the standard count
and was used to normalize subsequent counts in the soil. The standard
count also provided a means by which to compare readings taken at
different times and accounted for variation in temperature of the gauge
and gain settings.
After the standard count was taken, the meter timer was set to
accumulate counts for one minute. The probes were placed in a pair of
tubes such that the source and detector were located at a depth
corresponding to the center of the core samples taken. A minimum of
two one-minute counts were recorded for each depth then the probe moved
to the next depth. After the last measurement at a depth of 66 cm was
obtained, the source and detector rods were removed from the access
tubes. Counts were taken at the same depths in all three set of access
tubes and the entire sampling process repeated in the second lysimeter.
Several attempts at this calibration procedure were made. Early
trials produced unacceptable ranges of scatter in the gamma probe
readings. At a later date, faulty electronic components were found and
expected to be the reason for the highly variable gamma probe readings.
After repair of the unit, the calibration procedure was repeated
yielding acceptable results.
According to Ferraz and Mansell (1979) the density of the test
material varies inversely with logarithm of the ratio of the intensity


83
_3
p : density of water [g*cm ]
w = 1.00 [g-cnT3]
Combining equations (3-1) and (3-2) yields a relationship for
volumetric water content in terms of the count ratio and the dry bulk
density of the soil.
9 = A + B-ln(CR) + C-/>dry (3-3)
It was conceivable that a calibration curve might be necessary for
each of the lysimeters and for various ranges of depth. Therefore,
linear regressions of the forms in Equations (3-1) and (3-3) were
determined for each lysimeter and for each of the depths. Statistical
testing revealed that a single regression could be used for all depths
and for both lysimeters for both the wet density and volumetric water
content. The final form of the calibration equation for determination
of volumetric water content was
9 = 1.52 0.33"In(CR) 0.89-pdry (3-4)
The coefficient of variation (R^) for the regression shown in
equation (3-4) was 0.888. Measured volumetric water content when
plotted against the water content estimated by the calibration
(equation 3-4) should lie about a line with a slope of unity and an
intercept of zero (Figure 3-4). Examination of Figure (3-4) indicated
that relatively large errors could be associated with the exact
estimate of volumetric water content. Therefore, the gamma probe


6
insight into the actual processes describing the phenomenon of
evaporation. However, considerable data may be required as input to
provide reasonable estimates of the heat and mass transfer.
Literature Review
Empirical Models
Jones et al. (1984) reviewed some of the empirical procedures to
estimate crop water use noting that all require calibration for a
particular geographical region. Pruitt (1966) reported that serious
errors can occur when using pan evaporation data to estimate crop water
use, particularly under strong advective weather conditions. The pan
should be installed properly ensuring sufficient fetch surrounding the
pan.
The Thornwaite model, as presented by Jones et al. (1984), uses
monthly averages of air temperature and day length to estimate monthly
evapotranspiration (ET). Criddle (1966) summarized the Blaney-Criddle
model which predicts the actual water use by a crop from monthly
average air temperature and percent daylight and introduces crop
consumptive use coefficients. The Blaney-Criddle model, developed for
use in arid regions, greatly overestimates evaporation for the humid
climate of Florida during the summer months prompting Shih et al.
(1977) to replace the percent daylength with monthly net radiation to
account for the increased cloud cover.
Resistance Models
Resistance models presented in the literature are of the general
form
cpa£a (e(Ts) e(T(j) )
E
R
(2-1)


95
longer period of time on February 3 while the hourly evaporation rate
dropped off very sharply on January 11. Prior to the rain beginning on
February 5, the same maximum evaporation rate was attained indicating
vertical redistribution of water toward the surface occurred fast
enough to maintain the lower evaporation rate for a longer period of
time. Unfortunately, the rainfall prevented a final measurement of the
distribution of the water in the soil. The temperature of the soil
surface (Figure 3-23) followed the air temperature very closely during
the overcast weather until just prior to the rain beginning, then the
incident radiation of the soil caused the surface to continue
increasing even though the air increased at a much slower rate. After
the rain began, the soil surface cooled very rapidly and reached the
same temperature as the air.
The remaining data shown in Figures 3-24 to 3-32 show the same
general trends explained previously. Cumulative and hourly
evaporation rates for both lysimeters were similar. Some apparent
glitches in the data however, should be explained. The cumulative and
hourly rates of evaporation for the period from August 11 to 21
(Figure 3-27) show a rainfall of approximately 10 mm on August 12,
then an apparently very rapid evaporation of water on August 14. The
rapid loss of water was caused by activation of the vacuum system to
remove approximately 18 and 17 mm of water from the north and south
lysimeters respectively. The large dip and subsequent recovery on
August 13 was the instrumentation sensing the addition and removal of
the dual probe density gauge for moisture content. The distribution of
water in the soil (Figure 3-28) also indicates the rainfall as well.
It should also be noted that ambient weather data was not available


146
the temperature response of the copper cylinder increased. This was
due to the fact that the Biot number approached infinity, thus causing
the eigenvalue for an infinite Biot number to be used. Thermal
diffusivity was affected by almost a one to one correspondence by
changes in the slope of the soil sample temperature response.
Another source of analytical error would be using the solution for
an infinite cylinder instead of that for a finite cylinder. The
analytical solution for a finite cylinder was determined by the
principle of superposition of the solution for an infinite cylinder and
an infinite slab (Equation 4-15).
9 s *cyl*slab (4"15)
Substitution and rearranging terms yields a linear equation with the
slope of the line consisting of a linear combination of the eigenvalues
for a slab and cylinders.
0 = c
cyl
cylcslab exPt'Dt( ^
(4-16)
This procedure was incorporated into the analytical analysis and the
thermal diffusivity determined. No significant differences were
obtained.
Errors in temperature measurement would comprise the major source
of experimental errors. Conduction error occurs due the sensor passing
through regions of differing temperatures. Heat can then be conducted
from the soil sample to the water along the thermocouple. Gaffney
et al. (1980) stated that conduction error caused by the surrounding
fluid could be minimized by using 36 ga or smaller thermocouple and by


24
The conservation equations for the liquid (2-22) and vapor (2-17)
both have terms relating to the rate of change from the liquid phase to
the vapor phase. The vapor phase change ( Ev(z,t) ) is based upon the
volume of air while the liquid phase change term ( EL(z,t) ) is based
upon the volume of soil. The phase change terms are related by the
fraction of soil volume occupied by the air which can be determined by
subtracting the volumetric water content from the total soil porosity.
The relationship between the vaporization terms is expressed as follows
E(z,t) = EL(z,t) = (S 0)Ev(z,t) (2-23)
Rearranging equation (2-23) and solving for Ev(z,t) yields
EyU.t) (|Z j*)) (2-24)
Substitution of equation (2-24) into the vapor continuity equation
(2-17) results in the following equations describing the conservation
of water vapor within the soil.
dpy dPy Ev(z^)
85 Sz (Dv ST ) + "(S "-'W <225)
The thermal energy equation describes the flow of heat within the soil
and includes the time rate of change of sensible heat in the soil
volume, the diffusion of heat due to temperature gradients, the
convection of heat due to diffusion of water and water vapor through
zones of variable temperature and the latent heat of vaporization. The
partial differential equation describing the rate of change of the
temperature of the soil used in the model was


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
KEY TO SYMBOLS AND ABBREVIATIONS v
ABSTRACT x
CHAPTERS
I. INTRODUCTION . 1
Problem Statement 1
Research goals and objectives 3
II. ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT 5
Introduction 5
Literature Review 6
Model Objectives 19
Model development 20
Determination of Model Parameters 30
Numerical solution 43
III. EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND ENERGY
TRANSFER PROCESSES IN THE SOIL 66
Introduction 66
Objectives 67
Lysimeter Design, Installation and Calibration Design
and Construction 67
Experimental Procedure 85
Data Analysis 89
Experimental Results 90
Summary 96
IV. MEASURING THERMAL DIFFUSIVITY OF SOILS 134
Introduction 134
Objectives 135
Literature Review 135
Procedure 141
Results and Discussion 143
Conclusions 148
iii


231
Figure B-
Rainfall from December 15 to 19, 1986 during lysimeter
evaporation measurement.


167
occurring at a relatively high rate, but due to the low wind speed
corresponding to the maximum difference in water vapor density,
evaporation rates were relatively low. When the rapid increase in wind
speed occurred, the vapor density deficit at the soil surface was
declining but still relatively large. This caused the abrupt increase
in evaporation rate. This would tend to indicate that the model was
was relatively sensitive to wind speed due to the direct influence of
wind speed upon the surface transfer coefficients.
The initial water profile was somewhat different for test 72 when
compared to the previous test in that the soil surface had a higher
water volumetric water content than that below the soil surface (Figure
5-16). Drying the soil surface resulted in the same shape profile at
the end of the simulation as was typical of the initial conditions of
other tests thus far. No detectable differences in the simulated water
contents occurred below a depth of 20 cm indicating that all of the
water lost from the soil was evaporated from the top 20 cm of the soil
profile. Final soil water profile measurements were not obtained due
to heavy rains which prematurely ended the test.
The simulated evaporation of water from the lysimeters for
conditions recorded during test 73 exceeded the evaporation measured in
the weighing lysimeters (Figure 5-17). Initial high rates of water
loss observed for both lysimeters were simulated fairly well as
indicated by the comparison of the cumulative evaporation and the
hourly evaporation curves (Figure 5-17 and 5-18, respectively). The
measured evaporation rate decreased to about zero during the night
then increased again at sunrise; however, simulated rates of
evaporation decreased after sunset but remained small but positive


Thermal DiffusivUy (10 rn^/a)
157
2.0
Bulk Density * 1600 kg/m^
i.e --
1.2 -
0.8 --
0.4--
Experimental
DeVries
Quadratic
Quadratic + Vapor
o.o
+
+
0.00
0.03
0.10
0.1 s
0.20
0.2S
2.0
1.6 --
1.2 -
0.8 --
0.4 --
0.0
Bulk Density 1500 kg/m^
Experimental Quadratic
DeVries Quadratic + Vapor
0.00
0.09
0.10
0.15
0.20
0.29
2.0
1.6 --
1.2
0.8 -
0.4--
0.0
Experimental
DeVries
Quadratic
Quadratic + Vapor
Bulk Density 1300 kg/m^
+
+
0.00
0.09 0.10 0.19 0.20
Volumetric Water Content (cm^/cm^)
0.29
Figure 4-5. Comparison of experimental values of thermal diffusivity to
calculated thermal diffusivity by regressions and the
DeVries method.


ACKNOWLEDGEMENTS
I would like to acknowledge, first and foremost, GOD'S blessings
in giving me the talents and insights with which to complete this
dissertation. My wife, Sherry, has provided steadfast support and
encouragement without which this task would have been impossible. I
also thank my committee chairman, Dr. Wayne Mi shoe, for his guidance
and encouragement and the free reign to conduct the research as I
deemed necessary. I thank my committee, Drs. James Jones, Khe Chau,
Hartwell Allen, Calvin Oliver and Mr. Jerome Gaffney, for their
guidance and support. To the Agricultural Engineering technical staff
in the fabrication and instrumentation of the weighing lysimeters
constructed for this project, I offer my thanks. I thank Bob Bush for
his diligent maintenance and operation of the weighing lysimeters.
Finally, I acknowledge the contribution to my well-rounded education of
my fellow graduate students, particularly, Bob Romero, Ken Stone, Matt
Smith, Ashim Datta and Kumar Nagarajan.
ii


67
surface. The reflectance and absorbtance change dramatically as a
function of water content. Convective heat losses from the surface are
generally estimated rather than measured using empirical functions to
determine surface heat transfer coefficients.
Objectives
The processes of energy and mass transfer in the soil are most
readily represented by the rate at which water is lost from the soil
(evaporation rate), the cumulative water loss (cumulative
evaporation), temperature, volumetric water content, and water vapor
density. All are temporal in nature, while the latter three are
functions of depth as well. The goal of the experimental design was to
provide data for the validation and calibration of the transport model
described in the previous chapter. Specific objectives were
1. to design a lysimeter and calibrate the instrumentation to
measure hourly evaporation rates,
2. to measure the vertical distribution of the soil temperature
and volumetric water content as function of time in the
lysimeter,
3. to measure ambient weather conditions to sufficiently
describe the boundary conditions for a coupled heat and
mass transfer model, and
4. to determine the physical characteristics of the soil
contained in the lysimeter.
Lysimeter Design. Installation and Calibration
Design and Construction
Weighing lysimeters were used in this research to meet the
objective of measuring hourly evaporation from a bare soil surface.
The weight of the lysimeter may be monitored by several different
methods. Mechanical measurement may be achieved by either supporting
the soil container with a lever and counterweight system or by


Water Tension (
179
Figure 5-1.
Soil water potential and hydraulic diffusivity as a
function of volumetric water content for a Mi11 hopper
fine sand (Carlisle et al., 1985)
Hydraulic Diffusivity (m^/s)


72
of output voltage to input voltage (mV/V) to account for any variation
in supply voltage between load cells and time variation in the supply
voltage.
Lysimeter Calibration
Load cell calibration was performed by the manufacturer at the
factory for full load output of the cells. The manufacturer provided
load cell output (mV/V) at full scale load and no load. The load
carried by each load cell was expected to be near the full capacity of
each load cell with relatively small weight changes around that
reading. It was also noted that the calibration information provided
by the manufacturer was obtained under controlled atmospheric
conditions and using the factory-installed 3.3 m leads for measuring
the load cell output. The temperature sensitivity of the load cells
was very small compared to the changes in load cell output expected.
According the manufacturer's specifications, the effect of change in
load cell temperature upon the output was 0.08 percent of the load per
55C change in cell temperature. The installation site for the
lysimeters required the use of leads ranging in length from 37 to 53 m.
Therefore, an in-situ calibration was necessary.
The soil surface of the lysimeter was covered with a polyethylene
sheet to prevent weight loss due to evaporation during the calibration
procedure. A single weight having a mass of approximately 12 kg was
added to the lysimeter and the normalized output of each load cell was
recorded. A second weight was added, thus increasing the total weight
added to the system. The load cell output was again recorded. This
was repeated until the cumulative weight added to the lysimeter reached
175 kg (equivalent to 26 mm of water distributed over the lysimeter


104
Total Change in Load Cell Output ( mV/V )
Figure 3-3.
Linear regression of the load cell calibration data for
the weighing lysimeters at the Irrigation Research and
Education Park, Gainesville, FL.


200
Figure 5-22. Simulated cumulative evaporation resulting from
multipliers of 0.5, 2.0 and 5.0 for the hydraulic
diffusivity.


MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE
BY
CHRISTOPHER LLOYD BUTTS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
¡1 J5JE LIBRARIES
1988

ACKNOWLEDGEMENTS
I would like to acknowledge, first and foremost, GOD'S blessings
in giving me the talents and insights with which to complete this
dissertation. My wife, Sherry, has provided steadfast support and
encouragement without which this task would have been impossible. I
also thank my committee chairman, Dr. Wayne Mi shoe, for his guidance
and encouragement and the free reign to conduct the research as I
deemed necessary. I thank my committee, Drs. James Jones, Khe Chau,
Hartwell Allen, Calvin Oliver and Mr. Jerome Gaffney, for their
guidance and support. To the Agricultural Engineering technical staff
in the fabrication and instrumentation of the weighing lysimeters
constructed for this project, I offer my thanks. I thank Bob Bush for
his diligent maintenance and operation of the weighing lysimeters.
Finally, I acknowledge the contribution to my well-rounded education of
my fellow graduate students, particularly, Bob Romero, Ken Stone, Matt
Smith, Ashim Datta and Kumar Nagarajan.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
KEY TO SYMBOLS AND ABBREVIATIONS v
ABSTRACT x
CHAPTERS
I. INTRODUCTION . 1
Problem Statement 1
Research goals and objectives 3
II. ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT 5
Introduction 5
Literature Review 6
Model Objectives 19
Model development 20
Determination of Model Parameters 30
Numerical solution 43
III. EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND ENERGY
TRANSFER PROCESSES IN THE SOIL 66
Introduction 66
Objectives 67
Lysimeter Design, Installation and Calibration Design
and Construction 67
Experimental Procedure 85
Data Analysis 89
Experimental Results 90
Summary 96
IV. MEASURING THERMAL DIFFUSIVITY OF SOILS 134
Introduction 134
Objectives 135
Literature Review 135
Procedure 141
Results and Discussion 143
Conclusions 148
iii

V. MODEL ANALYSIS 158
Introduction 158
Validation 158
Sensitivity Analysis 170
Summary 176
VI. SUMMARY AND CONCLUSIONS 208
BIBLIOGRAPHY 212
APPENDIX A ADI FORMULATION OF COUPLED HEAT AND MASS
TRANSFER MODEL 219
APPENDIX B METEOROLOGICAL DATA FOR LYSIMETER EVAPORATION
STUDIES 225
BIOGRAPHICAL SKETCH 241
iv

KEY TO SYMBOLS AND ABBREVIATIONS
Variable Definition
B empirical function for determination of Cer and Chr
[dimensionless]
Cdr surface drag coefficient [dimensionless]
Cer Dalton number, dimensionless mass transfer
coefficient
Chr Stanton number, dimensionless heat transfer
coefficient
Cpa specific heat of moist air [J*kg-1-K_1]
Cne specific heat of solid constituent of soil mixture
K [ J*kg_1* K"^]
CpW specific heat of water [J*kg'l,{K"l]
CpV specific heat of water vapor [J'kg"1*^'1]
Cs volumetric heat capacity of soil mixture (solid,
liquid and air phases) [J'kg*1*^'1]
Da diffusivity of water vapor in air [m2*s_1]
Dl unsaturated hydraulic diffusivity of soil [m^s-1]
Dt thermal diffusivity [m2*s"l]
Dv diffusivity of water vapor in soil [m2,s_1]
dwj height of volume of soil associated with node j. [m]
dxj width of soil cell j in the x direction (unity) [m]
dyj width of soil cell j in the y direction (unity) [m]
dzj vertical distance between nodes j and j+1. [m]
E(z,t) rate of phase change of liquid water to water vapor
@in the soil as a function of space (z) and time (t)
tk9(H20)*m(soil)s' J
v

Variable Definition
cJ>n
ETa
ETP
ea
eas
es(T)
-vs
W
9
9j
H
hfg
hh
^m
rate of phase change of liquid water to vapor in
soil cell j at time step n [kg^jm^soil)*8"1]
actual evapotranspiration [mm]
potential evapotranspiration [mm]
water vapor pressure in ambient air [Pa]
saturated water vapor pressure in ambient air [Pa]
saturated vapor pressure at temperature, T [Pa]
vapor pressure at water potential, [Pa]
sensible heat flux rate from the soil to the air
[W-m']
gravitational acceleration [m*s-2]
shape factor in the j-th principal axis for thermal
conductivity calculations [dimensionless]
vertical component of heat flux in the atmosphere
[J*m]
latent heat of vaporization of water [J*kg_1]
convection heat transfer coefficient for soil
surface [W*m'^*K'l]
convection vapor transfer coefficient for soil
surface [m*s-l]
K hydraulic conductivity [mfj^m'^s"1]
Kh Km Kw turbulent transfer coefficients forJieat, momentum
and water vapor, respectively [m2*s_1]
k-y ratio of temperature gradient in i-th soil
constituent to the temperature gradient in the
continuous constituent (water or air) in the
direction of the j-th principal axis [dimensionless]
-2
LE latent energy transfer in the atmosphere [J*m ]
n number of moles of gas present, used in ideal gas
law [mol]
vi

Variable Definition
P
P
Po
Qv
q
qcj+l,n
qej,n
qLj+l,n
qsj,n
precipitation: flux of water at soil surface
S 1
atmospheric pressure [Pa]
reference atmospheric pressure [Pa]
vertical flux density of water vapor in the
atmosphere [kgnr^s12]
specific humidity of ambient air [kgvapor/kgdfy air!
rate of heat flux conducted into node j from node
j+1 at time step n [W*m'2]
rate of latent heat loss due to evaporation at node
j and time step n [W*m"3]
heat flux rate into node j from node j+1 via
movement of liquid at time step n [W*m'2]
rate of sensible heat change of node j at time step
n [W*m-3]
qvj+l,n
R
Re
rate of heat carried into node j from node j+1 by
water vapor movement at time step n [W*m"2]
universal gas constant [kgm2*s2*mol"1,0K~1]
gas constant for water vapor, determined by dividing
the universal gas constant, R, by the molecular
weight of water [m2,s'2,0K"l]
Reynolds number [dimensionless]
Ri Richardson number [dimensionless]
Rn net radiation flux incident upon soil surface
[W-nr2]
S soil porosity [m3*nr3]
s slope of the saturated vapor pressure line
[kPa-OK'1]
Ta ambient air temperature [K]
Tjjn soil temperature at node j and time step n [^C]
Tavgj (Tj+l,n + Tj,n)/2
vi i

Variable Definition (continued)
t
U
U*
V
US
*i
z
z0
a
7
7T
X
v
Pa
Ps
Pv
^vj,n
Pva
Pvs
Pw
0
time [s]
wind velocity [m*s_1]
shear velocity [nrs-1]
volume [m^]
Wind speed [m*s"l]
volume fraction of the i-th soil component [m3*m*3]
depth below soil surface [m]
surface roughness length [m]
soil tortuosity [dimensionless]
psychrometric constant [kPa,cK'l]
diabatic influence function [dimensionless]
soil water potential [m]
thermal conductivity of soil composite at node j
[W*nr**K"l]
kinematic viscosity [m2*s-1]
density of moist air [kg*m3]
dry bulk density of soil [kg*m'3]
water vapor concentration [kgH20*msoil]
water vapor concentration at node j and time step n
[k9H20*msoil]
water vapor concentration of ambient air
[k9H20*mair]
water vapor concentration in the air at saturation
[k9H20*mair]
density of water [kg*m~3]
dimensionless water content
viii

Variable Definition (continued)
1 volumetric water content of soil [m3*m"3]
0jjn volumetric water content of soil at node j and at
time step n [m3*m'3]
9r residual volumetric water content [m3*m'3]
6s volumetric water content at saturation [m3,m3]
r shear stress [N*m-2]
IX

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE
By
Christopher Lloyd Butts
December, 1988
Chairman: J. Wayne Mi shoe
Cochairman: James W. Jones
Major Department: Agricultural Engineering
Evaporation of water from the soil is controlled by the transport
phenomena of energy and mass transfer. Most procedures for estimating
the loss of water from the soil assume that the soil is an isothermal
medium with evaporation of water occurring at the soil surface.
Estimating evaporation as a surface phenomenon independent of soil
hydraulic and thermal properties can lead to overestimation of the
amount of water lost from the soil.
Weighing lysimeters measuring 2 x 3 x 1.3 m^ were constructed
capable of detecting a change in weight equivalent to 0.02 mm of
water. Recorded data consisted of net radiation, soil temperature,
water content, load cell output, ambient air temperature, relative
humidity, windspeed and precipitation. Measured soil properties
included dry bulk density, porosity and thermal diffusivity.
A model describing the continuous distribution of heat, water and
water vapor within the soil was developed. The evaporation of water
x

was allowed to occur throughout the soil profile in response to the
assumed equilibrium between the liquid and vapor phases. Surface heat
and mass transfer coefficients were determined using equations based
upon equations of motion in the atmosphere. Thermal diffusivity of the
soil was measured and incorporated into the simulation.
Simulation results included temporal values of the cumulative and
hourly water loss from the soil, and spatial distribution of
temperature, water and water vapor in the soil. Experimental data
obtained from the weighing lysimeters were used to validate the model.
Prior to model calibration simulated soil temperatures were within 2 C
of measured values and simulated cumulative evaporation was within ten
percent of measured water loss. Sensitivity analysis indicated that
calibration could be achieved by relatively small adjustments in the
values of the surface heat and mass transfer coefficients.
Experimental and simulated evaporation rates exhibited a diurnal
pattern in which maximum evaporation rates occurred approximately
midday then decreased to near zero at sunset. Under some conditions,
water continued to evaporate from the soil overnight at a rate of
approximately 0.03 mm/h while in some cases a net gain of water was
observed overnight.
xi

CHAPTER I
INTRODUCTION
Problem Statement
The process by which water is lost from the soil has been the
object of considerable research over the years (Brutsaert, 1982).
Water is lost from the soil by runoff, drainage, evaporation and
transpiration. Evapotranspiration (ET) has been used to describe the
combined evaporative losses from the soil and plant. Potential evapo
transpiration (ETp) has been defined as the amount of water
transferred from a wet surface to vapor in the atmosphere (Penman,
1948) and has been used as an estimate of the maximum amount of water
lost to the atmosphere from a given area. Estimates of actual
evapotranspiration (ETa) are frequently made by the use of coefficients
which vary according to ground cover and the region of the country.
The need for knowing ETa on a field basis arises for a variety of
reasons including design and selection of irrigation equipment,
optimization of irrigation management, and determination of water
requirements for agricultural and municipal entities (Heermann, 1986).
Regional estimates of ETa are necessary for hydrologic simulations
(Saxton, 1986).
Mathematical models have been developed to provide estimates of
both ETa and ETp. The use of evapotranspiration models has expanded in
recent years due to the advent of crop growth simulations rquiring the
distribution of soil water as a component of the model. One such model
used to integrate the soil water and the plant water requirements was
1

2
developed by Zur and Jones (1981) and focused on estimating water
relations for crop growth. A soil water model which simulates ETa is
an integral part of the soybean production model, SOYGRO, developed by
Wilkerson et al. (1983). One of the primary uses of SOYGRO is to
determine the yield response of soybean to various environmental
conditions such as drought.
Even though ET models are currently used extensively, many
research needs remain. At best, crop coefficients are rough estimates
computed from previous experiments with the crop of concern. There is
a need to develop a better understanding of the variability of soil,
crop and environmental factors that determine crop coefficients.
Scientists need improved procedures for determining crop coefficients
for various cultural practices such as mulching, no-till and row
spacing.
One of the most pressing needs in the area of evapotranspiration
research is in the area of sensors which will detect soil water status
for a particular field (Heermann, 1986; Saxton, 1986). One of the
rising technologies for determining soil water status is the use of
remote sensing of soil temperature for determining soil water status
(Soer, 1980; Shih et al., 1986).
Evaporation from the soil is driven by the energy balance of the
surrounding environment. Because of the release of latent energy is
coupled with the sensible energy balance, it is necessary to consider
both components of the energy balance to completely describe the
evaporation process. Soil temperature has been found to influence many
processes in the soil. For example, soil temperature influenced
nodulation of roots of Phaseolus vulgaris L. (Small et al., 1968) and

3
root growth for various stages of development of soybean (Brouwer,
1964; Brouwer and Hoogland, 1964; Brouwer and Kleinendorst, 1967).
Blankenship et al. (1984) have shown that excessive temperatures in the
soil zone in which peanuts are produced (geocarposphere) can reduce
yields and increase susceptibility to disease, fungus and physical
damage. Unpublished data collected by Sanders (1988) for peanuts grown
in lysimeters with controlled soil temperatures indicated that the
maturation rate of peanuts were affected by soil temperature. The fact
that movement of water vapor in the soil in response to temperature
gradients can be significant was determined by Matthes and Bowen (1968)
and more recently by Prat (1986). Neglecting soil temperature in a
mass transfer model in the soil could cause significant errors in the
estimation of ETa from the soil if soil temperature were neglected in
the modeling analysis.
Research Goals and Objectives
An understanding of the processes of heat and mass transfer in the
soil is of interest to a variety of agricultural scientists,
particularly those involved in modeling agricultural systems and
developing management strategies. Evapotranspiration is a complex
process involving energy and mass transport in the plant and the soil
as well as the chemical processes such as photosynthesis and
respiration which require water from the soil and release water from
the reaction. It has been suggested by some researchers that the two
processes involving the plant and the soil be studied independently so
that greater insight might be obtained into each. Therefore, the goal
of this research is to study the process of evaporation from the soil
independent of transpiration for the purposes of furthering the

knowledge of the parameters affecting evaporation from the soil and
guiding future research. The specific objectives are:
1. to develop a model to simulate the evaporation of water from
the soil which couples the energy and mass transfer
processes;
2. to monitor evaporation of water and vertical distribution of
temperature and water in a sandy soil;
3. to determine the validity of the model by comparison of
simulated evaporation rates, soil temperatures and
volumetric water content to those measured in a sandy soil;
4. to determine the sensitivity of the evaporation process to
changes in various soil properties and environmental
conditions; and
5. to calibrate the model for local conditions using data
collected in Objective 2 utilizing information gained from
the sensitivity analysis.

CHAPTER II
ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT
Introduction
Models of varying complexity may be used to simulate or estimate
the amount of water which evaporates from the soil. The complexity of
the model should be determined by the projected use of the results and
the data available as input for the model. Empirical models generally
relate one or more parameters by regression analysis to evaporation
measured under various conditions. Variables or parameters typically
used in empirical models are evaporation pan data, air temperature, and
day length or solar radiation. Monthly estimates of evaporation for
relatively large areas are typically obtained from empirical models
(Jones et al., 1984).
Resistance analog models employ the concept of the electrical
analog to mass flow, where the flux of water vapor leaving the soil
surface is expressed as a potential (vapor pressure) difference divided
by a resistance. The resistance term for the mass flow is a function
of the boundary layer of the lower atmosphere and the mass transfer
characteristics of the soil (Camillo and Gurney, 1986; Jagtap and
Jones, 1986). These resistance models have the capability of
estimating actual evaporation from the soil on a daily or hourly basis.
Evaporation can also be estimated using mechanistic models
describing the conservation of mass, momentum, and energy in the lower
atmosphere and the soil. These models are the most detailed in their
derivation and have the advantage that they can provide substantial
5

6
insight into the actual processes describing the phenomenon of
evaporation. However, considerable data may be required as input to
provide reasonable estimates of the heat and mass transfer.
Literature Review
Empirical Models
Jones et al. (1984) reviewed some of the empirical procedures to
estimate crop water use noting that all require calibration for a
particular geographical region. Pruitt (1966) reported that serious
errors can occur when using pan evaporation data to estimate crop water
use, particularly under strong advective weather conditions. The pan
should be installed properly ensuring sufficient fetch surrounding the
pan.
The Thornwaite model, as presented by Jones et al. (1984), uses
monthly averages of air temperature and day length to estimate monthly
evapotranspiration (ET). Criddle (1966) summarized the Blaney-Criddle
model which predicts the actual water use by a crop from monthly
average air temperature and percent daylight and introduces crop
consumptive use coefficients. The Blaney-Criddle model, developed for
use in arid regions, greatly overestimates evaporation for the humid
climate of Florida during the summer months prompting Shih et al.
(1977) to replace the percent daylength with monthly net radiation to
account for the increased cloud cover.
Resistance Models
Resistance models presented in the literature are of the general
form
cpa£a (e(Ts) e(T(j) )
E
R
(2-1)

7
evaporation rate [nrs"1]
specific heat of air [J*kg-1*K_1]
density of dry air [kg*nr3]
latent heat of vaporization [J kg-1]
psychrometric constant [kPa*K_1]
saturated vapor pressure [kPa] at temperature T
soil surface temperature [9<]
ambient dew point temperature [K]
resistance to vapor movement from the soil to air
[kg*Sm'4]
The resistance term for the models presented by Conaway and Van Bavel
(1967), Tanner and Fuchs (1968), and Novak and Black (1985) represents
the resistance due to the laminar boundary layer. The boundary layer
resistance is a function of the wind speed, atmospheric instability,
and the surface roughness height. Their models are used to predict
evaporation from a well-watered bare soil surface. Jagtap and Jones
(1986) and Camillo and Gurney (1986) developed resistance terms to
include the boundary layer resistance in series with the resistance of
vapor flow in the soil. The soil resistance term is included since
after the soil surface dries, the water must change to vapor in the
soil below the surface then diffuse to the soil-atmosphere interface.
The soil resistance term developed by Jagtap and Jones was determined
by regression analysis as a function of cumulative evaporation, water
in the soil profile available for evaporation, and a daily running
average of the net radiation. The net radiation empirically accounted
for the heat flux into the soil, while the ratio of the cumulative
where:
E
cpa =
Pa =
hfg =
7
e(T) =
TS -
Td -
R

8
evaporation to the available water parameterized the thickness of dry
soil through which the water vapor diffused. The soil resistance term
used by Camillo and Gurney (1986) was a linear regression of the
difference between the saturated volumetric water content and the
actual soil water content. Both models were intended to provide daily
evaporation rates.
The resistance model estimates daily evapotranspiration relatively
well. However, the soil resistance term varies with time depending
upon the distance through which the water vapor must diffuse from below
the soil surface. One point that should be noted is that the vapor
pressure difference used in the analog models discussed is the
difference between the saturated vapor pressure at the temperature of
the soil surface and that of ambient air. In reality, evaporation
frequently occurs below the soil surface and the vapor pressure
gradient from the zone of evaporation may be substantially different
from that at the surface.
Mechanistic Models
Models derived from the basic physical relationships have the
advantage of providing estimates of evaporation on a relatively short
time scale. However, considerably more input data describing the
surface boundary conditions as well as the soil thermal and hydraulic
properties are required. Theoretical models have been developed from
four different perspectives (King, 1966; Goddard and Pruitt, 1966;
Fritschen, 1966; Penman, 1948). They are categorized as follows
1. a mass balance profile method,
2. a mass balance eddy flux method,

9
3. an energy balance method, or
4. a combination method.
The two mass balance methods (1 and 2 above) focus on the
equations of motion for the atmosphere and require accurate measurement
of velocity and temperature profiles or mass fluxes. The equations of
conservation of momentum, mass, and energy were used in the mass
balance methods to develop expressions for the rate of evaporation of
water and the shear stress between to vertical positions in the lower
atmosphere as
Qv = PaKw dg
dz
(2-2)
* P#m JJL
dz
(2-3)
H = CpapaKh dl
K dz
(2-4)
where:
Qv
Pa
7
H
z
q
u
cpa
vertical flux density of water vapor [kgm'^s'1]
density of air [kg*nr3]
shear stress rate [N*m-2]
vertical flux of heat [J*m*2]
vertical distance [m]
specific humidity [kgvapor/kgdyy air]
velocity [nrs"1]
specific heat of dry air [J*kg-1*K"1]

10
T air temperature [K]
Kfo, Kn,, Kw turbulent transfer coefficients for heat, momentum
and water vapor, respectively [m2*s_1]
Combining the equations 2-2 and 2-3 and rearranging the following
expression for evaporation is obtained.
k _g_
Si dz
K 3D-
"1 dz
(2-5)
The profile methods determine the rate of evaporation by analysis of
the vertical profile of various atmospheric variables such as specific
humidity or vapor pressure and wind velocity. King (1966) described
the procedures for determining the evaporation rate for adiabatic wind
profiles (neutral stability) and stratified conditions. One of the
basic assumptions used in the profile methods is that the turbulent
transfer coefficients of water vapor and momentum are equal. This
implies that the momentum and mass displacement thicknesses are the
same. The Richardson number (Ri) or the ratio of height (z) to the
Monin-Obukov length (L) are used as a measure of atmospheric
instability and can be determined by equations 2-6 and 2-7,
respectively.
where:
Ri
L
au
dz
+ r )
u*
(
k g H
Cpa^a^
(2-6)
(2-7)
9
acceleration due to gravity [m*s-2]

11
r = adiabatic lapse rate [^nr1]
= 9.86 x 10-3
k von Karman's constant [dimensionless]
= 0.428
U* shear velocity [nrs"1]
For neutral conditions the familiar logarithmic velocity profile was
used. For unstable or diabatic conditions a variation of the
logarithmic profile was used or the KEYPS function (Panofsky, 1963) was
used to determine the velocity profile. King stressed that in using
profile methods to determine evaporation, one must take extreme care in
measuring wind velocity. He suggested that spatial averaging be used
for wind speed measurements near ground level and that 30- to 60-minute
averages be used considering the steady state assumptions made in
developing the equations for the profile equations. Corrections for
lower atmospheric instability caused by density gradients in the air
near the earth's surface must also be used.
Another method requiring only measurements of atmospheric
parameters is the eddy flux method (Goddard and Pruitt, 1966). The
method is based upon the turbulent transport equations of motion in
the atmosphere. The equations of turbulent motion in a fluid include
the transient random fluctuations in velocity, temperature, and mass
concentration of water vapor of the air. The following equations
describe the shear stress (r), sensible heat flux (H) and latent heat
flux (LE) due to the turbulent motion of the atmosphere
* K 3U
^a^m
- paw'u'
(2-8)
cpaPaKh fj"
+ CpaPaW''
(2-9)

12
LE = -hgPa^w 5§- ¡w7^) (2-10)
where:
~V : time averaged turbulent fluctuation of the air
temperature [9(]
u7 : time averaged turbulent fluctuations of the horizontal
component of the wind velocity [m*s_1]
w : wind velocity in the vertical direction [m*s"l]
w7 : time averaged turbulent fluctuations of the vertical
component of the wind velocity [m*s-1]
qv : turbulent fluctuations in the specific humidity of the
air [kgvapor/kgdry air)
Assuming vertical gradients in temperature and absolute humidity
are insignificant as compared to the gradients in the direction as the
horizontal component of the wind, the latent and sensible heat can be
expressed as
cpaPaw'T
(2-11)
hfgPawX
(2-12)
The equations are used as a basis for either the sensible or latent
heat flux in the atmosphere using measured values of the turbulent eddy
fluxes of heat and moisture. Specialized equipment was designed by the
Goddard and Pruitt to measure the parameters needed for the
calculations. However, the equipment was not reliable for low wind
speeds. This was attributed to the very small magnitudes of the
turbulent eddies for calm conditions.

13
Limitations of the mass balance (eddy flux and profile) methods
are the requirements for very sensitive equipment for measuring either
the vertical profiles or the vertical fluxes due to turbulence. King
(1966) questioned the practical use of the mass balance methods of
determination of the evaporation. Fritschen (1966) pointed out that
the mass balance methods do not integrate results over time, thus
requiring constant monitoring of meteorological parameters.
Another approach to estimate evaporation from the soil is the
energy balance method (Fritschen, 1966). The rate of change of
sensible heat in a control volume of air at the earth's surface is
equal to the rate at which sensible and latent heat are carried into
and out of the volume by the wind, the convection of sensible and
latent energy in the vertical direction, the energy transferred to the
soil and the net radiant exchange of energy. In the energy balance
method, it is assumed that the net transfer of sensible and latent heat
in the horizontal direction by the wind is negligible when compared to
the vertical movement of heat in the atmosphere. The equations of
motion are utilized as in the mass balance profile method with the
exception of using the similarity equation for the sensible heat flux
instead of the shear stress. A ratio of sensible to latent heat flux
(equations 2-4 and 2-2), referred to as the Bowen ratio (BR), is given
by
3T
H cPapa Kh gT
IF hfg Km dq
(2-13)
The energy balance at the soil surface is

14
Rn S G LE = O
(2-14)
where:
Rn -
net radiation incident upon soil surface [W*m"2]
S
soil heat flux [W*m'2]
G
sensible energy flux into atmosphere [W*m'2]
LE =
flux of latent energy from soil [W*nT2]
An expression for the rate of evaporation of water from the soil can be
determined by substitution of equation (2-13) into (2-14) and
rearranging terms. This method is referred to as the Bowen ratio
equation and yields valid results for a wide range of conditions.
Fritschen states that it is imperative that efforts be undertaken to
assure that the assumption of no horizontal divergence of heat or
moisture be met in order for the Bowen ratio equation to yield
satisfactory estimates of the evaporation. Soil heat flux must be
measured as well, since omitting soil heat flux from the analysis could
lead to large errors.
In using the Bowen ratio, the soil surface temperature must be known as
in some of the other methods.
Penman (1948) used a combination of the energy and mass balance
methods with the objective of eliminating the need for surface
temperature. The underlying assumptions are the temperature of
evaporating surface is the same as the ambient air and the vapor
pressure is the saturated vapor pressure evaluated at the surface

15
temperature. It was also assumed that the net soil heat flux is zero
over a 24 h period. The resulting relationship
Pacpa^fg (ea eas) + s(Rn )
' FTTT) (2'15)
potential evaporation [m3*m2]
vapor pressure of atmosphere [kPa]
saturated vapor pressure at the air temperature [kPa]
surface vapor transfer coefficient [m*s'l]
sensible heat flux in the atmosphere [W*nT2]
net radiation upon the soil surface [W*m-2]
slope of saturated vapor pressure line [kPa,0K_1]
psychrometric constant [kPa-^C1]
provides an expression for the potential evaporation from the surface.
The boundary layer resistances account for advective conditions. The
Penman equation is the only method for estimating evaporation based
upon theoretical ideas and requires no highly specialized equipment.
Estimates of the evaporation have an accuracy of 5 to 10 percent on a
daily basis (Van Bavel, 1966). The disadvantage to the combination
method, as implemented by Penman, is that only daily estimates are able
to be determined and crop coefficients are necessary to estimate actual
evapotranspiration from various crops (Jones et al., 1984). The values
of the crop coefficients typically exhibit regional variation as well
as variation due to stage of crop growth.
Staple (1974) modified the Penman model to provide the upper
boundary condition for the isothermal diffusion of water in the soil.
The Penman model was modified by multiplying the saturated vapor
where:
E(
e-
as
h
6
Rr
s
1

16
pressure by the ratio of vapor pressure at the soil surface to the
saturated vapor pressure at the same temperature (relative vapor
pressure). This incorporated the vapor pressure depression effect of
the soil water potential into the model. This seemed to match field
data for a clay loam soil. This model was not tested for more coarse
soils.
The Penman method eliminated the requirement of soil surface
temperature by neglecting the effects of soil heat flux at the expense
of being able only to predict the evaporation on a daily basis.
Various researchers have recently gone one more step in modeling the
evaporative loss of water from the soil by considering the movement of
water and heat below the soil surface (Van Bavel and Hi 11 el, 1976;
Lascano and Van Bavel, 1983). Most of the research in which the
process of evaporation has been examined at this detailed level are
for a bare soil surface. This eliminated the complicating factors of
water removal by the plant, which surface temperature to use in
describing the driving potential for evaporation, and description of
the atmospheric boundary layer transfer coefficients.
Van Bavel and Hillel (1976) developed a model in which the partial
differential equations for water (liquid) and energy transfer in the
soil were considered to be independent processes. This approach
explicitly determined the soil heat flux for the boundary condition at
the soil surface as expressed in equation (2-14). The simultaneous
solution determined the actual evaporation from the soil as well as the
spatial distributions of water and temperature as a function of time.
The variation in hydraulic and thermal properties of the soil with time
was accounted for in this extended combination approach.

17
According to Fuchs and Tanner (1967) field and laboratory
observations indicated that the evaporation of water occurs in three
distinct stages. The first involves water evaporating from the soil-
atmosphere interface. As the soil surface dries, the water must
evaporate at a location below the soil surface then the vapor diffuse
to the surface. The combination methods of simulating evaporation from
the soil reproduce this process fairly well. However, the evaporation
is assumed to occur at the soil surface. This has the effect of
lowering the soil surface temperature, when in the case of water
evaporating below the dry soil surface, the soil surface temperature
would be higher due to the reduced thermal conductivity of the soil.
This error in soil surface temperature was noted by Lascano and Van
Bavel (1983). Lascano and Van Bavel (1983 and 1986) verified the model
of Van Bavel and Hi11 el (1976) using soil water content and temperature
data obtained from field plots. During the earlier study, the
simulated soil surface and profile temperatures were found to agree
quite closely over the range of 25 to 37 C while the model
underpredicted surface temperature by 2 to 5 C when the soil
temperature was above 37 C. Distribution of soil water is simulated
to within the expected error of measurement. Similar results were
obtained during the 1986 experiment.
Movement of water primarily in the vapor phase in the soil has
been observed by several researchers. Taylor and Cavazza (1954) noted
that the measured diffusion coefficients were higher than that for
water vapor in air and suggested that transport was due the combined
effects of convection and diffusion within the soil pore spaces.
Schieldge et al. (1982) used the following equations of the

18
conservation of mass and energy to simulate the diurnal variations of
soil water and temperature.
89
ST =
a
IT
(D
89 ,
8z '
a
ST
(Dl
ai
ST
-) +
3K
ST~
(2-16)
c 3T -
cs at
a
8z
(X ) + Q(9,T) (2-17)
where:
9
=
volumetric water content [m3#m"3]
t
=
time [s]
z
s
depth below soil surface [m]
T
=
soil temperature [K]
cs
=
volumetric heat capacity of soil mixture
[ J *m-3* clC_13
\
=
thermal conductivity of soil mixture [W*m'l*K"l]
K
=
unsaturated hydraulic conductivity [nrs-1]
Da
=
hydraulic diffusivity [m2S_1]
=
30 Da uS^Vi 30
* ~ST~ + PRJT 89
Da
=
diffusivity of water vapor in air [m^s'1]
P
s
atmospheric pressure [kPa]
V
s
ratio of atmospheric pressure (P) to partial
pressure of air
S
=
soil porosity [m3*m"3]
9
=
acceleration due to gravity [nrs-1]
0
=
soil matric potential [m]
Rw
=
ideal gas constant for water vapor
P\i
=
density of water vapor [kg*m-3]

19
Dt
diffusivity of water (liquid, and,vapor) due to
temperature gradients [m
quid and Vi
i2.s-1.Qk-1]
Tb +
Da uSah$r)
Pvi
Q(0.T)
heat f!ux within the soil due to movement of water,
liquid and vapor [W*m'3]
a soil tortuosity factor 2/3)
fi a coefficient of thermal expansion [kgnr3,0^1]
r1 = vapor transfer coefficient [dimensionless]
h = relative humidity
These equations were originally developed and presented by de Vries
(1958) with good agreement with field observations being achieved.
Diurnal fluctuations of water content were damped out below a depth of
4 cm. Particular attention was given to the magnitudes of Dj Djl and
D-py. It was noted that the Dj was small during the night but has a
significant contribution to the flow of water during the day. The
vapor component (D-py) was less than the liquid component (Djl) for
water contents greater than 0.30 m3/m3 and sensitive to changes in soil
temperature.
Jackson (1964) studied the non-isothermal movement of water using
equations (2-16) and (2-17) and concluded that classical diffusion
theory could be used satisfactorily if the relative vapor pressure is
greater than 0.97 with no modification to the diffusivity term.
However, if the relative vapor pressure is less than 0.97 then the
diffusivity must include the effect of the temperature gradient.
Model Objectives
The models presented thus far each have advantages and
disadvantages. The mechanistic models are based upon sufficient theory
that the diurnal fluctuations of the soil water and temperature

20
profiles can be observed. They also have the ability to provide
insight into the many processes involved in the evaporative loss of
water from the soil. Saxton (1986) stated that there is a need to
separate the evaporative loss of water from the soil from the losses
through the plant so that better understanding of the individual
processes can be achieved. The mechanistic models provide this
ability.
One might also note that in any of the models discussed
previously, conservation of water is discussed either in the vapor
phase, as in the mass balance methods, or the liquid phase, as in the
combination methods which consider the soil media. The vapor and the
liquid phases are not considered simultaneously.
The objectives in developing a new model were:
1. to account for the water changing from liquid to vapor
phase below the soil surface and diffusing to the
atmosphere;
2. to consider the overall mass continuity, specifically
include the liquid and vapor phases separately;
3. to account for the movement of water vapor in response
to temperature gradients in the soil; and
4. to simulate the diurnal variation of evaporation, soil
water and temperature profiles in response to conditions
at the soil surface.
Model Development
The soil for most purposes may be considered a continuous medium
in which the laws of conservation of mass and energy apply.
Application of the basic principles of thermodynamics to the soil
profile provide the basis for simulation of the temporal variation of
the distribution of water and temperature throughout the soil profile.
The following assumptions were made to simplify or clarify the

21
development of the model for coupled heat and mass transfer in the
soil.
1. The soil is unsaturated, therefore the soil water potential
is primarily due to osmotic and matric potential.
2. Water vapor behaves as an ideal gas.
3. The movement of water in the liquid and vapor phases occurs
due to concentration or potential gradients.
4. The liquid and vapor phases are in thermodynamic equilibrium
within the soil pore spaces.
Using the conservation of mass, the rate of change of the water
vapor within the soil air space was described in the model as
3pv
IT
4z (Dv
Spy
~dz
+ Ev(z,t)
(2-17)
where:
fiy
mass concentration of water vapor in the soil air
space [kg*m'3]
Dv = Diffusivity of water vapor in the soil air space
[m2*s_1]
or Da
Da = Diffusivity of water vapor in the air [m2*s_1]
a = soil tortuosity [dimensionless]
Ev(z,t) rate of phase change of water [kg*m^r*s-1]
t = time [s]
z = depth below the soil surface [m]
The time rate of change of the concentration of water vapor at any
point in the soil space is a function of the rate of water vapor
diffusing from other regions of the soil and the rate at which water
changes from the liquid to the vapor phase. The evaporation rate is

22
considered to occur throughout the soil profile in response to
temperature and water content conditions in the soil. The rate of
phase change of the water (Ev) is considered positive if the water
changes from the liquid to vapor (evaporation), while condensation is
indicated by a negative phase change rate.
The diffusivity of water vapor in the soil air spaces is assumed
to equal that in the atmosphere, however, Schieldge et al. (1982), as
well as others, have made use of a tortuosity factor (a) to account
for the increased path length of the interstitial spaces of the soil
through which the water vapor must traverse. The tortuosity factor
included the effect of the local evaporation and condensation of water
vapor as it passes through the regions of differing temperature and
water content in the soil.
The volumetric water content is typically used to express the
amount of liquid water present in the soil. The mass concentration of
water in the liquid phase is obtained by multiplying the volumetric
water content by the density of water. The time rate of change of the
mass concentration of water is increased by a the rate of diffusion of
water from surrounding soil and decreased by the rate of evaporation.
The mathematical expression of the conservation of mass for the water
is
d 3Ql
(2-18)
where:
Pw
Ql
9
volumetric water content [m3*m~3]
density of water [kg*m-3]
-2 -1
diffusion of liquid water [kg*m *s ]

23
El rate of change of liquid water to water vapor
[kg-m^n-s'1]
For the range of temperatures generally occurring in the soil, the
density of water may be considered constant. The volumetric flow rate
of water (q|_) can be obtained by dividing the mass flow rate (Ql) by
the density of water (pw). The volumetric flow rate of water in an
unsaturated soil is governed by Darcy's Law as follows:
qL - k gf (2-19)
where:
qL volumetric flow rate of liquid water [m3*m"2*s-1]
k unsaturated hydraulic conductivity [m*s"l]
0 = soil water potential [m]
z = spatial dimension [m]
The relationship between soil water content and soil water potential is
unique for a given soil type. Therefore, the volumetric flux of water
can be expressed in terms of volumetric water content instead of soil
water potential by applying the chain rule of differentiation to
equation (2-19) as follows
q, = k
30 33
W W
(2-20)
The hydraulic diffusivity of the soil is defined as
DL
K
30
W
(2-21)
By substituting the definition for the hydraulic diffusivity (2-21) and
the volumetric flow rate in terms of the soil water content (2-20) the
equation for the conservation of mass for liquid water (2-18) becomes
33
3t
3_
3z
CDl
33
3z
1 E(z,t)
N
(2-22)

24
The conservation equations for the liquid (2-22) and vapor (2-17)
both have terms relating to the rate of change from the liquid phase to
the vapor phase. The vapor phase change ( Ev(z,t) ) is based upon the
volume of air while the liquid phase change term ( EL(z,t) ) is based
upon the volume of soil. The phase change terms are related by the
fraction of soil volume occupied by the air which can be determined by
subtracting the volumetric water content from the total soil porosity.
The relationship between the vaporization terms is expressed as follows
E(z,t) = EL(z,t) = (S 0)Ev(z,t) (2-23)
Rearranging equation (2-23) and solving for Ev(z,t) yields
EyU.t) (|Z j*)) (2-24)
Substitution of equation (2-24) into the vapor continuity equation
(2-17) results in the following equations describing the conservation
of water vapor within the soil.
dpy dPy Ev(z^)
85 Sz (Dv ST ) + "(S "-'W <225)
The thermal energy equation describes the flow of heat within the soil
and includes the time rate of change of sensible heat in the soil
volume, the diffusion of heat due to temperature gradients, the
convection of heat due to diffusion of water and water vapor through
zones of variable temperature and the latent heat of vaporization. The
partial differential equation describing the rate of change of the
temperature of the soil used in the model was

25
- hfgE(z,t)
(2-26)
where
\ = thermal conductivity of soil mixture [W*m'^,(^C'^]
T = soil temperature [K]
hfg latent heat of vaporization [J*kg-1]
Cs = volumetric heat capacity of soil mixture [J-m'^K'1]
= (l-S)psCpS + ^Pwcpw + (5~^)Pacpa
S = Soil porosity [m3*m"3]
ps = dry bulk density of soil [kg*m'3]
pa = density of moist air [kg*m-3]
cps specific heat of solid portion of soil [J-kg^*0K^]
CpW specific heat of water [J*kg"l*K"l]
Cpa = specific heat of moist air [J'kg"l*Kl]
Equations (2-22), (2-25), and (2-26) describe the conservation of
energy and mass within the soil profile and account for the vapor and
liquid water phases separately. However, those three equations contain
the three state variables, soil temperature, volumetric water content,
and vapor concentration as well as the unknown rate at which water is
changed from the liquid to the vapor phase. Another independent
equation was needed to adequately describe the soil mass and energy
system.
If the water vapor were in equilibrium with liquid water, the
vapor would have a partial pressure corresponding to the saturated
vapor pressure and would be a function of the liquid water temperature.
Assuming that water vapor behaves as an ideal gas, then the saturated

26
vapor density can be determined from the saturated vapor pressure using
the ideal gas law.
p V = n R T
(2-27)
where:
p: pressure of the gas
V: volume of gas
n: number of moles of gas present
R: ideal gas constant
T: absolute temperature of gas.
The ideal gas law may also be applied for the individual components of
a gas mixture with the pressure being the partial pressure of the
component gas and using the number of moles of the component gas.
Substitution for the molecular weight of water and solving for the
density of water vapor one obtains
es(T)
pvs = rw t (2'28)
where:
pys = concentration of water vapor at saturated vapor pressure
per unit volume of air [kg*m"3]
es = saturated vapor pressure at given temperature [Pa]
T = temperature of free water surface [^C]
Rw = ideal gas constant for water vapor
= universal gas constant (R) divided by the molecular
weight of water
= 461.911 [m2*s'2*K-1]
The above equation (2-28) is for the case in which the chemical
potential of the water is zero. However, due to matric and osmotic

27
forces, the water in the soil has a soil water potential less than
zero, thereby reducing the equilibrium vapor pressure from that of pure
water. The following relationship (Baver et al., 1972) can be used to
determine the saturated vapor pressure above a water surface with a
potential other than zero.
ev( evs(T) expf-^p) (2-29)
w
where:
ev(^) = saturated vapor pressure of water with chemical
potential, ip [Pa]
ip = soil water potential [m]
g acceleration due to gravity [m*s'2]
Using the ideal gas law, the vapor density over a water surface with a
chemical potential other than zero can be obtained by:
Pv = PvsCO exP(j^*T) (2-30)
w
A set of simultaneous equations (2-22, 2-25, 2-26, 2-30) describe
the conservation of thermal energy, water vapor and liquid water for
the soil continuum and formed the basis of a coupled mass and energy
model for the soil. The assumption of thermodynamic equilibrium
between the liquid and vapor states yields the constitutive
relationship expressed in equation (2-30) and provides a fourth state
equation to be used in the model.
A well-posed problem also includes boundary conditions and, in
the case of transient problems, the initial conditions must also be
prescribed. The system of governing equations was one-dimensional and
therefore required two boundary conditions. The first boundary is the

28
obvious soil-atmosphere interface. It was assumed that the thermal
capacitance of the soil at the surface was negligible when compared to
the magnitudes of the fluxes which occur. Therefore, the net flux of
energy must be zero at the soil surface. The ability of the soil to
maintain a significant rate of evaporation at the soil surface was
assumed to be small as well. This assumption required the water to
evaporate at a finite distance below the soil surface rather than at
the soil surface. The boundary conditions for the energy, vapor and
liquid continuity were
(2-31a)
(2-31b)
(2-31c)
where:
Cpw = specific heat of water [J-kg'1*^'1]
CpV specific heat of water vapor [J*kg'l*K"l]
D[_ hydraulic diffusivity [m2*s_1]
Dv = diffusivity of water vapor in soil [m2*s"l]
hf, = boundary layer heat transfer coefficient [W*m'2*K_1]
hm = boundary layer mass transfer coefficient [m*sl]
P = precipitation or irrigation rate [m3*m'2*s-1]
Rn = net radiation incident upon soil surface [W*m"2]
soil temperature [K]
- X
3T
ST
z=0
99 ^/*v
(PwcpwL + cpvv tfz )^avg
Rn ^h(Tz=0 Ta)
n 30
-L ST
= P
Z=0
3pv
3v Sz~
z=0
= hm(pV0" hz )
T

29
Tavg = average temperature [K]
A thermal conductivity of soil [W*m-1,(K-1]
/>va mass of water vapor per unit volume of dry ambient air
[kg-m-^]
pyo concentration of water vapor at the soil surface, 2=0
[kg-m-3]
6 = volume of water per unit total soil volume [m3*m'3]
Ideally, for a semi-infinite medium, the flux of energy and mass
should be zero in the limit of depth (z) approaching infinity.
However, in anticipation of a numerical solution to the system of
partial differential equations, the boundary conditions were specified
at a depth of 1.5 meters. This depth was chosen by comparing the
error between the analytical and numerical solutions for conduction of
heat in a semi-infinite slab with constant uniform properties and a
uniform heat flux at the surface. This depth resulted in an error of
less than 0.1 C at the lower boundary. The depth at which the
amplitude of the diurnal fluctuations in temperature is less than
0.1 C is approximately 60 cm for a sandy soil (Baver et al., 1972).
The zero flux condition for the liquid and vapor continuity equations
represents an impermeable layer in the soil. This condition may or
may not physically exist in the field, but for most situations
encountered, the errors introduced into the solution at the depth of
chosen should be minimal. The boundary conditions used at the lower
boundary were
= 0 (2-32a)
Z=Zq
36
3J_
3z
0
(2-32b)

dPy
ST
30
= 0 (2-32c)
z=z0
The system defining the movement of water vapor, liquid water and
energy is defined by a system of partial differential equations (2-22,
2-25, 2-26) with boundary conditions (2-31a-c, 2-32a-c). The energy
equation is coupled to the continuity equations by the rate at which
the water is changed from liquid to vapor and through the movement of
sensible heat associated with the flux of water between zones of
differing temperatures. The conservation of water vapor is coupled to
the soil temperature indirectly in the calculation of saturated vapor
pressure. This coupling, along with the variation of the soil
properties with time and space, renders an analytical solution beyond
reach, therefore; a numerical solution was required.
Determination of Model Parameters
Solution of the governing equations for the energy and mass
balance for the soil requires knowledge of the properties relating to
the soils ability to diffuse heat, water, and water vapor. The other
parameters to be determined relate to the rate at which heat and water
vapor are dissipated from the soil surface to the air. In order to
determine a solution, relationships for determining thermal, hydraulic
and vapor diffusivities as well as the surface transfer coefficients
must be determined.
Diffusivitv of Water Vapor
Diffusivity is the constant of proportionality relating flow to
the gradient in potential as stated in Fick's Law of molecular
diffusion. For the case of water vapor diffusing through air, the

31
diffusivity, or coefficient of diffusion, is a measure of the number of
collisions and molecular interaction between molecules of water vapor
and the other constituent components of the air (ASHRAE, 1979) and is a
function of temperature, total air pressure and partial pressure of
water vapor. Eckert and Drake (1972) presented the following equation
to calculate the diffusivity of water vapor in air.
(2-33)
where
Dva = diffusivity of water vapor in air [m^'s-^]
p = atmospheric pressure [Pa]
p0 = reference atmospheric pressure [Pa]
= 0.98 X 105 Pa
T = air temperature [K]
T0 reference air temperature [K]
= 256
However, for water vapor in the soil air space, the path is more
convoluted, thus increasing the probability of collisions with other
particles and lowering the kinetic energy of the water vapor molecules.
To account for the increased path length within the soil, a tortuosity
factor (a) has been introduced to reduce the effective coefficient of
diffusion in other porous materials. De Vries (1958) used a value for
the tortuosity of 0.667. Using this information the coefficient of
water vapor in the soil (Dv) becomes
D,
v
a D
va
X 10
-5
(2-34)

32
For the purposes of simulation, the barometric pressure (p) was assumed
to equal the standard pressure (p0). The diffusivity then became a
function of time and space due to the temporal and spatial variation of
the soil temperature. Diffusivity was determined throughout the soil
profile by substitution of the soil temperature in equation (2-34).
Thermal properties
Very little literature exists regarding measurement of the thermal
properties of the soil. Soil composition as well as density and water
content affects the thermal properties of the soil (Baver, 1972). The
thermal diffusivity is the ratio of thermal conductivity to the product
of soil density and specific heat. Density is a property which can be
obtained as a function of depth at a given location by core samples.
Vries (1975) describes a method by which the volumetric heat
capacitance and the thermal conductivity of the soil can be calculated
based upon the volume fractions of the various soil constituents.
The volumetric heat capacitance is defined as the product of the soil
density and the specific heat and can be calculated from:
Cs xqcq + xCm + xoCo + xwCw + xaCa f2'36)
where
C
S
volumetric heat capacity [J-m'3*0^1]
X
=
volume fraction [m3*m~3]
q
=
quartz
m
s
mineral
0
=
organic

33
w = water (same as volumetric water content)
a air
s soil composite
The difficulty arises in estimating the various volume fractions of the
different components. For quartzitypic sands, approximately 70 to
80 percent of the solid constituent of the soil is quartz, with
generally less than 1 to 2 percent organic material and the remainder
consisting of other ninerals. According to information presented by
DeVries (1975), the volumetric heat capacity and density of quartz and
other minerals are very similar. For the purposes of this simulation,
the volume fraction of the organic material was assumed to be zero.
The thermal conductivity cannot be calculated in such a straight
forward manner. Thermal conductivity is defined as the constant of
proportionality in relating the heat flux by conduction to the
temperature gradient. Conduction of heat occurs due to physical
contact between adjacent substances. In a solid material such as steel
or concrete which is fairly homogeneous, the material conducting heat
can be considered continuous. However, the soil is a mixture of solid,
liquid and gaseous components and the area of physical contact between
soil particles is a function of the particle geometry. When the soil
is dry, the area of contact may be a single point. As the soil is
wetted, a thin film of water adheres to the soil particle and increases
the contact area between adjacent particles. This increase in area
accounts for a rapid change in the thermal conductivity as the soil is
initially wetted (Figure 2-1). Over a range of water contents between
dryness and saturation, the increase in thermal conductivity becomes
linear with increasing water content. As the soil approaches

34
saturation, the thermal conductivity begins leveling off to some
relatively constant value. Another complicating issue in calculating
the thermal conductivity of the soil is that the temperature gradient
in the solid and liquid portion of the soil can be significantly
different from that in the gaseous phase of the soil.
De Vries (1975) presented a method by which the thermal
conductivity could be calculated using a weighted average of the
various soil constituents where the weighting factors were a product of
the individual volume fractions and geometric factors as follows
X =
where
X
Ai =
*i -
^i =
i
The geometric weighting factor depends upon geometric
configuration of the soil particles and the incorporated void space.
It represents the ratio of the spatially averaged temperature gradient
in the i-th soil component to the spatially averaged temperature
gradient of the continuous component of the soil (usually water). For
example, kq represents the space average of the temperature gradient
in the quartz particles in the soil to the space average of the
temperature gradient in the water. The geometric weighting factor can
Wq + W + Wo Ww + kaVa ...
thermal conductivity of the soil [W*m^,0K"^]
thermal conductivity of soil component [W'nr1*0^1]
volume fraction of soil component [m3*m"3]
geometric weighting factor [dimensionless]
quartz (q), mineral (m), organic (o), water (w), air (a)
l

be determined by
35
= I (^ia + ^ib + ^ic) (2-37)
where
k^j = the ratio of the temperature gradient in the i-th
component to the temperature gradient in the continuous
component in the direction of the j-th principal axis
i = quartz (q), mineral (m), organic (o), water (w), air (a)
j = a, b, or c for each of the three principal axis of the
particle
The ratio of the temperature gradients in the i-th component can be
determined by the following:
(2-38)
The shape factor for each principal axis (gj) can be approximated by
various empirical relationships depending upon the ratio of the unit
vectors (ua, Uj,, uc) of the principal axes of the soil components
(Table 2-1). The sum of the three shape factors must be unity.
In most cases, water is considered to be the continuous phase of
the soil in determining soil thermal conductivity. However, as the
soil dries and the film adhering to the surface of the soil particle
begins to break, making air the continuous phase. Equation (2-37) can
be used in these cases replacing the thermal conductivity of water (Aw)
with the thermal conductivity of air (Xa). De Vries (1975) noted that
the values for the thermal conductivity in the case of air being the
continuous phase were consistently low by a factor of approximately
k
-1
w
%

36
1.25. This method yielded values of thermal diffusivity within ten
percent of those measured (de Vries, 1975). Extensive detail regarding
calculation of the thermal conductivity of the soil is given in
de Vries (1963).
The thermal properties were calculated using equation (2-35) for
the volumetric heat capacity and equations (2-36), (2-37) and (2-38) to
determine the thermal conductivity. Volume fractions of the various
soil constituents was determined from soil classification data and
knowledge of the soil bulk density and porosity. Shape factors used
for calculation of the thermal conductivity were for a typical sand
grain (Table 2-1).
Hydraulic Properties
The parameter governing the movement of liquid water in the soil
is a measurable property of the soil and is analogous to the thermal
diffusivity. The hydraulic diffusivity is a derived property of the
soil (i.e. not directly measured) and is defined as the hydraulic
conductivity divided by the specific water capacity of the soil (Baver
et al., 1972). The hydraulic conductivity is the constant of
proportionality for the diffusion of water in response to a gradient in
the soil water potential (Figure 2-2), while the specific water
capacity is the slope of the soil water retention curve. The
hydraulic conductivity (Figure 2-3) and the specific water capacity
(Figure 2-2) vary according to the soil composition as well as the soil
water potential. The measurement of hydraulic conductivity can be
accomplished by several methods, but most all require meticulous
control of the potential gradients and a great deal of time. This is
especially true if measurements are desired over a wide range of soil

37
water potential. Measurements of the soil water retention curve
require considerably less detail and are generally published for a
wide variety of soil types. Van Genuschten (1980) presented a method
by which the hydraulic conductivity could be calculated for unsaturated
soils using the soil water retention curve and the saturated hydraulic
conductivity. The Van Genuschten approach involves estimation of an
equation for the soil water retention curve of the form
0 =
(2-39)
where
6 = dimensionless water content
6 volumetric water content [m3*nr3]
6r = residual volumetric water content [m3*m'3]
0S = volumetric water content at saturation [m3*m'3]
ip soil water potential [ m ]
m,a = regression coefficients
n = (1 m)"1
The coefficients m and a are determined by nonlinear least squares
regression of the soil water retention curve. That function is then
substituted into equations for the hydraulic conductivity presented by
Mualem (1976). The resulting expression for the diffusivity is
D(0) = ks e(*5 Vto) [(i_0l/nym+ (l-0l/m)m-2] (2-40)
am(0s-0r)
The Van Genuschten method yields a continuous function for the
hydraulic diffusivity which is highly desirable for numerical

38
simulations over the range of the water contents expected to occur in
the fields.
For modeling purposes, the Van Genuschten method was employed for
published potential-water content data for a Mi11 hopper fine sand
(Carlisle, 1985). The water release curve usually varies with depth
due to the spatial variation of soil composition. Data for the a
single water release curve for a uniform soil profile was obtained by
averaging the volumetric water content for the specified water
potential over the A-l and A-2 horizons of a Mi11 hopper fine sand. The
nonlinear regressions were then performed to yield the residual water
content, and the regression coefficients, a and n.
Surface Transfer Coefficients
The final parameters to be estimated are the surface heat and mass
transfer coefficients. Penman (1948) used an empirical wind function
to calculate the mass transfer coefficient based upon wind speed.
hm = a WSb (2-41)
where
hm = surface mass transfer coefficient
WS wind speed
a,b = empirical constants
Another approach is to use the equations of motion to describe mass and
energy transfer within the atmospheric boundary layer. Brutsaert
(1982) presented a detailed review of the equations of motion as
applied to the atmosphere. The basic assumption in these analyses is
that the boundary layers for momentum, energy and mass are similar.
Consequently, the surface sublayer becomes the area of most concern and

39
is defined as the fully turbulent region where the vertical turbulent
fluxes of mass and energy do not change appreciably from that at the
surface (Brutsaert, 1982). In other words, the vertical flux of mass
and energy is constant.
According to Brutsaert (1982), Prandtl introduced the use of the
logarithmic wind profile law into meteorology in 1932. This is an
approximation of the wind velocity profile in the surface sublayer
WS = -Hi- In (-!-) (2-42)
K Zg
where
WS
=
wind speed [ m*s-1]
U*
=
shear velocity
r V2
v P '
ZO
=
surface roughness height [ m ]
k

von Karman constant [dimensionless]
To
s
shear stress at the surface [N*m~2]
P
s
density of air [kg*m'3]
According to Sutton (1953), for practical purposes, the shear velocity
can be estimated as
U*
WS
Iff
(2-43)
The bulk mass transfer within the surface sublayer is defined by
Qm = Ce,rWSr (Pys ^vr)
(2-44)

40
where
Qm vertical mass flux [kg-liras'1]
WSr wind speed at reference height, zr [m'S*1]
Cer = Dalton Number; dimensionless mass transfer coefficient
pvr water vapor concentration at reference height, zr
[kg*m'3]
pvs water vapor concentration at soil surface [kg*m-3]
The surface mass transfer coefficient (hm) used in this model is
related to Cer by
^m = wsr Cer
(2-45)
Brutsaert (1982) presented functions for the Dalton number in terms of
the drag coefficient and the dimensionless Schmidt number as
Cer
(B
Cd
V2
+ Cdr)
-1/2
(2-46)
where
B function of dimensionless Schmidt number
Cer = Dalton number
Cdr = surface drag coefficient [dimensionless]

41
The empirical function, B, depends upon whether or not the surface is
hydrodynamically smooth or rough. In general, if the Reynolds number
based upon the shear velocity and the roughness height is less than
0.13 then the surface is considered smooth and B is determined using
Equation (2-47). The surface is rough if the Reynolds number is
greater than 2.0 and Equation (2-48) is used to determine the value
of B.
%
B = 13.6 Sc J 13.5
% \
B = 7.3 Re 4 Sc 5.0
where
Sc Schmidt number
v
Dv
Re = Reynolds number
z0 U*
v
The dimensionless heat transfer coefficient or Stanton number
(Chr) can be determined by substitution of the Prandtl number for the
Stanton number for the Schmidt number in equations (2-47) and (2-48)
above. The surface heat transfer coefficient can then be determined by
(2-47)
(2-48)
hh = Chy. PgCpg WSp
(2-49)

42
Fuchs et al. (1969) used a method in which instability in the
lower atmosphere was accounted for in conjunction with the logarithmic
wind profile to calculate the surface mass transfer coefficient. This
involved using the KEYPS function (Panofsky, 1963) to determine the
curvature of the diabatic lapse rate (a) of the logarithmic wind
profile. The surface mass transfer coefficient was calculated from
_2
hm = k2WS ( % + In (2 + ) (2-50)
where
hm transfer coefficient of mass from surface to height, z,
in the air [m*s"l]
k = von Karman constant [dimensionless]
WS = wind speed at height, z [m*s"l]
D height displacement [m]
B d + zq
d = height above soil surface where wind velocity is zero
[]
zq roughness length [m]
z = height above the displacement height [m]
7T = curvature of the diabatic lapse rate or diabatic
influence function [dimensionless]
Fuchs et al. (1969) then calculated the surface heat transfer
coefficient using
h. = he a
h m pa a
(2-51)

43
The diabatic influence function accounted for the transfer of air
movement in the vertical direction due to density gradients caused by
temperature gradients and is a function of the Richardson number
(Figure 2-4). Fuchs et al. (1969) stated that the roughness height
varied from 0.2 to 0.4 mm for a bare soil surface and accounted for a
small variation in the calculated heat transfer coefficient.
Therefore, for the purposes of this study, a roughness length of 0.3 mm
will be used Fuchs et al. (1969) also noted that the height of zero
wind velocity (d) was zero for a bare soil surface. This term was
employed in the approximate wind profiles to account for the fact that
wind does not penetrate full vegetative canopies and for practical
purposes the surface where the wind velocity is zero occurs at some
finite height above the soil surface (Brutsaert, 1982; Sutton, 1953).
Fuchs et al. (1969) compared transfer coefficients calculated using
equation (2-50) to that determined from field data for a bare sandy
soil and obtained fairly close agreement.
For the purposes of this model, the approach used by Fuchs et al.
(1969) to determine the surface mass (Equation 2-50) and heat
(Equation 2-51) transfer coefficients was employed. The diabatic
influence function was utilized to account for atmospheric instability
as proposed by Fuchs et al. (1969). A roughness height (z0) of 0.3 mm
was utilized. The equations used in the formulation of this model and
the determination of parameters are summarized in Table 2-2
Numerical solution
The partial differential equations used to describe the mass and
energy balance in the soil must be solved numerically since analytical
solutions are not possible for the coupld nonlinear equations. Many

44
solution techniques utilize the finite difference form of the
differential equations. The differential operator in the continuous
domain becomes a difference operator in the discretized domain.
Difference operators are either forward, backward, or central
difference operators. The difference operators for the function of the
independent variable,
x, [ f(x)
] are
forward:
df
3x
o
III
f(*j+l) f(xj)
xj+l xj
(2-52)
backward:
3f
vxf -
f(Xj) f(Xj.!)
(2-53)
3x
xj XJ-1
central:
3f
35T
5xf
(2-54)
1 f(xj+l) f(xj) f(xj) f(Xj.!)
xj+l xj + xj xj-l
In expressing the differential equations in difference form, the
derivative with respect to time is accomplished using the forward
difference operator, while spatial derivatives are expressed using any
of the three difference operators. The continuous domain must first be
discretized or divided into several discrete regions such that the
finite difference approximation of the differential equation approaches
the differential equation in the limit of the grid spacing going toward
zero (Figure 2-5).
The partial differential equation defining the conservation of
energy within the soil profile (equation 2-26) stated that the change
in sensible heat over time is due to heat transferred by conduction
plus sensible heat carried by the diffusion of water in the liquid and

45
vapor phases from a region of differing temperature less the latent
energy required for evaporation of water. Examining this for the
node in the discretized domain, the above equation can be expressed in
terms of heat flux transferred (q) into node j from the surrounding
nodes as
dx*dy*dwj)qsj = dx-dy (qCj+l,n + + U.j-l.n + Qvj+^n + - (dx-dydwj)qej,n (2-55)
The subscripts in the above representation denote
s
c
e
L
v
j
j+1
j-1
n
and
change in sensible heat
heat conduction
latent heat
heat transfer due to diffusion of water (liquid)
heat transfer due to diffusion of water vapor
the current node
the node immediately following node j, and
the node immediately preceding node j.
the current time step
dx*dy : cross-section area normal to z axis
dx*dy*dwj : volume of the node j
dwj : height of the cell for node j
The change in sensible heat per unit volume of node j from time step n
to n+1 is
Tj,m-1 Tj,n
j = cs
(2-56)
The heat flux conducted from nodes j+1 and j-1 to node j at a given
time step, n, is described by Fourier's law of heat conduction and is
expressed as follows

46
9c j+1,n (Tj+l,n" Tj,n) (2-57)
Icd-l.nr ^-(Tj-l.n- Tj>n) (2-58)
dzj-l
noting that dzj represents the difference between the depth of nodes
j+1 and j.
In developing the differential equations for this model, it is
assumed that mass movement is due to diffusion and obeys Fick's law of
diffusion. Therefore, the heat transferred by diffusion of water and
water vapor from node j+1 to j is expressed as
qLj+l,n
n (^j+l,n *j,n)
PwjcpwjuLj 3zj
(Tj+l,n+ Tj,n)
2
(2-59)
9vj+l,n
n (Pvj+l,n Pvj,n)
cpvjvj azj
(Tj+l,n+Tj,n)
(2-60)
and similarly for the heat flux transferred by mass diffusion from node
j-1 to node j as
qLj-l,n
(*j-l,n *j,n)
^wjcpwjuLj dzj_i
(Tj-l,n+Tj,n)
2
(2-61)
qvj-l,n =
cpvjDvj
(^vj-l,n Pvj,n) (Tj-l,n+Tj,n)
3z"
ej-l
(2-62)
The latent heat associated with the phase change of water for the
current time step (Ej>n) must be transferred per unit volume of soil
associated with node j and is determined by
qej,n =
hfgjEj,n
(2-63)

47
Substitution of the numerical expressions for the individual terms of
the energy balance (equation 2-55) and dividing by the cross-sectional
area normal to the z axis (dx*dy) yields
dWjCs
Tj,m-1-Tj,n
eft
Tj+l>n"Tj,n
dz-i
Tj-l,n Tj,n
n (5j+l,n"^j,n) (Tj+l,n+Tj,n)
+ ^wjcpwjDLj azj 2
n *j,n) (Tj-l,n+Tj,n)
+ ^wjcpwjDLj aij^[ 2
n (Pvj+l,n ^vj,n) (Tj+l,n+Tj,n)
+ cpvjDvj 3zj 2
^ (^vj-l,n /vj.n) (Tj-l,n+Tj,n)
+ CpvjDyj 2
- dwj (hfgjEjjn) (2-64)
The partial differential equation describing the diffusion of
water within the soil can be transformed to difference form in a
similar fashion as the energy equation. The change in water content
of a volume of soil for the jth node is caused by diffusion of water
from nodes j+1 and j-1 to node j less the amount of water changed to a
vapor phase. The numerical expression for the water continuity per
unit area becomes

48
*j,n)
fJ 31
n W,n- ^j,n (*j-l,rf *j,n)
u 3ij +
dWj -ji (2-65)
J pw,j
Similarly, the differential equation describing the conservation of
mass in the vapor state is transformed as
dw-i
(/vj.m-l Pvj,n)
31
(/Yj+l,n ^vj,n)
Jvj
QZ-
JVJ
(^vj-l,n £vj,n)
(2-66)
(sj ^j,n)
dwj EJ,n
Equations (2-64), (2-65), and (2-66) constitute the numerical
equivalents for the partial differential equations describing the
conservation of thermal energy and mass in the soil profile. However,
the conservation relationships have yet to be developed for the
boundaries at the soil surface and the bottom of soil profile. Since
there is no volume associated with the surface node (Fig. 2-1), there
is no storage capacity for energy or mass. Therefore, the flux of mass
or energy from the node directly below the surface (j=2) to the surface
added to the net flux due to solar radiation onto the surface and the
heat flux due to convection from the air to the surface must be zero.
This can be expressed directly in difference form as follows.

49
X1
energy: (T2,n T¡>n) = Rnn + hj-,n(Tan Ti>n)
(2-67)
n (*2,n *l,n) (Pv2,n *>vl,n) (T2,n + Tl,n)
+ />wlcpwlDLl L1
water: (02,n *l,n) + P = 0 (2-68)
Dvl
vapor: (Pv2,n /Vl,n) + hm(/>va *vl,n) = 0 i2'69)
where:
Rnn
P
^mn
tyin
T
an
^van
net solar radiation incident upon the soil surface
at time step n [W*m'2]
precipitation at time step n [m3*m'3*s"l]
boundary layer mass transfer coefficient at time
step n [nrs"1]
boundary layer heat transfer coefficient at time
step n [W*m"2,0K_1]
air temperature at time step n [1C]
ambient water vapor concentration at time step n
[kg*m'3]
The boundary conditions at the lower boundary of the soil are
developed in a similar manner. The last node has volume and
capacitance for storage of energy and mass. A zero flux boundary
condition is used. There are two ways in which the zero-flux condition
can be represented. These will be developed for the water for the
purpose of illustration.

50
If the water content of the last node (j=nc) were equal to that of
the preceding node (j=nc-l), then no water would flow between the two
nodes. This would cause both of the last two nodes to act as one node
during the simulation. The other viewpoint would be that the sum of
the flows into the last node from the preceding node and an imaginary
node following must be zero.
Lnc ,/, ^ DLnc
clz^: l*nc+l,n *nc,n; = "3Fr£_i^nc-l*n
(2-70)
If the distance between nodes nc and nc-1 is the same as the distance
between the nodes nc and the imaginary node nc+1, then it follows that
^nc+l,n = ^nc-l,n (2-71)
Substitution of 2-71 into 2-65 yields as the boundary condition for the
node, j=nc, the following
dw,
nc
(^nc,m-r ^nc,n)
3E
Lnc rn n x
2 dznc_i (0nc-l,n-0nc,n)
(2-72)
dWr
-nc,n
nc /wnc
The same approach produces the following as the boundary condition for
the vapor continuity and energy equations, respectively.
(Pvnc,rH-r Pvnc,n) Dvnc
dwnc at = 2 znc_i (^vnc-r^vnc)
(2-73)
dwnc Enc,n
(sj ^j,n)
A n (^nc,rH-l^nc,n) o ^ (^nc-l,n_ ^nc,n)
dwnc Cs,nc dt = 2 Anc 3z^

51
0 n (^nc-l,n ^nc,n) (Tnc-l,n+ Tnc,n)
+ 2 /Jwrx^pwrKpLnc dz^-i ~2
o n (^vnc-ljiT^vnc,!!) (^nc-l,n+ ^nc,n)
+ 2 cpvncDvnc j
+ dwnc ^nc,n (2-74)
The conservation relationships provide sufficient information to
determine three of the desired quantities for the soil profile leaving
a fourth remaining unknown. The constitutive relationship requiring
that the water in the liquid phase be in thermodynamic equilibrium with
the vapor phase was used to provide the remaining equation. Since the
surface node provides only an interface between the soil and the
atmosphere, this equilibrium condition was not necessary for the
surface node. Under atmospheric conditions, the soil may not become
completely dry (0 % volumetric water content), but will reach some
moisture content which is in equilibrium with the atmosphere. This is
typically taken to be the same as the permanent wilting point of the
soil (^= -150 m) and is the same as the residual water content
described by van Genuschten (1980). As the soil surface dries, the
soil reaches this equilibrium moisture content and is assumed to act as
an interface between the air and the wet soil below.
The system of equations (2-64) through (2-69)and (2-72) through
(2-74) along with equation (2-30) represent an explicit solution to the
differential equations, that is, the values of the state variables at
the next time step are functions of those at the previous time step.
Using the explicit representation allows a simple algorithm to be used

52
during the solution phase of the system of equations and is equivalent
to an Euler integration in time (Conte and de Boor, 1980). The
disadvantage of the explicit solution is that the numerical error may
propagate through time and grow. The maximum time step is functionally
related to the boundary conditions and thermal properties of the soil.
The system of equations can also be changed such that all
derivatives are evaluated at the next time step in which none of the
state variables are known. This implicit representation requires a
relatively complicated iterative or matrix solution technique for the
system of equations. The advantage of implicit solution methods is
that the value of the state variables do not depend upon previous
values implying that the only source of error would be due to round-off
or truncation errors and would not propagate or grow with time and
would yield an unconditionally stable solution (Conte and de Boor,
1980).
Another technique which has some of the desirable characteristics
of both the explicit and implicit methods is the alternating direction
(ADI) method. This is accomplished by incrementally marching through
space in one direction (z=0 to zo) evaluating the derivatives
containing the previous node (j-1) at time, t + */2 dt, and those
containing the following node (j+1) at time, t. Then returning in the
opposite direction (z=zo to 0), evaluate the derivatives containing the
node j+1, at t=t+dt, and the derivatives containing j-1 at t=t+V;?dt.
This technique uses a relatively simple algorithm similar to that for
the explicit methods because all of the values of the state variables
used in estimating the state variable at the next time step are known
quantities. However, the number of equations to be evaluated is twice

53
that of the standard explicit methods. Increased stability is obtained
over the standard explicit methods but is less than that for the
implicit techniques. This implies that using the same grid spacing, a
larger time step can be used in the ADI methods than that for explicit
methods.
An ADI finite difference technique was used to solve the system of
equations for the soil profile due to the increased stability
characteristics over explicit methods. Appendix A contains a detailed
development of the numerical equations used in the ADI technique.
Hydraulic (equation 2-40) and thermal properties (equations 2-35 and
2-36) of the soil and the surface transfer coefficients (equations 2-50
and 2-51) were calculated at the beginning of each time step. A
general description of the solution algorithm is shown in Figure (2-6).
The energy and water equations were solved for the temperature and
volumetric water content, respectively at the next time step. The
equilibrium condition was used to determine the vapor concentration by
substitution of the values of soil temperature, water content, water
potential, and saturated vapor concentration in equation (2-30). The
vapor continuity equation was used to determine the evaporation rate.
The calculations were repeated using the new evaporation rate
until the absolute value of the maximum fractional change in any of the
variables was less than a prescribed convergence criterion.
The system of numerical equations was solved using computer code
written in F0RTRAN77. A variable grid spacing was used throughout the
soil profile with the smaller mesh being located near the soil surface
due to expected large gradients in temperature and moisture content
once the soil surface begins to dry (Table 2-3). The program was then

54
run to simulate the response of the soil system under constant
radiation, ambient air temperature and relative humidity conditions for
a period of 72 hours using various time steps to evaluate the numerical
stability and numerical error characteristics of the procedure.
Thermal and hydraulic properties of the soil were assumed to be
constant for the duration of the simulation for an Millhopper fine
sand. The water content and temperature profiles were assumed to have
a uniform initial distribution as were density and soil porosity.
Overall mass and energy balances were calculated for the soil
profile during the simulation. Any residual in these balances would
constitute error arising from the numerical algorithm and was
accumulated on an hourly basis for analysis purposes. The sum of the
squares of the residuals for the energy and mass balances for the
simulation period are presented in Figures 2-7 and 2-8, respectively,
for the different time steps. The sum of the squares of the residuals
(SSR) for the energy balance decreased when the time step of was
increased from 30 to 60 s then remained fairly constant for larger time
steps. The square root of the SSR represents an estimate of the
standard error of the estimate of the total heat flux. For the time
steps of 60, 120, 300 and 600 s, this represents approximately one
percent of the cumulative soil heat flux. The standard error of the
mass balance was approximately two percent of the cumulative
evaporation over the 72 h simulation (600 s time step). It is expected
that as properties are varied with time, that the standard error would
increase due to increased gradients in soil temperature and water
content particularly near the surface. Therefore, a time step of 60 s
was utilized in the model validation and sensitivity analysis.

55
Table 2-1. Formulae for the geometric shape factors used in
calculating thermal conductivity based upon ratios
of the unit vectors of the principal axes.
Object
9x
9y
9z
Sphere
0.33
0.33
0.33
Ellipsoid of revolution
x=y=nz n = 0.1
0.49
0.49
0.02
n = 0.5
0.41
0.41
0.18
n = 1.0
0.33
0.33
0.33
n = 5.0
0.13
0.13
0.74
n = 10.
Elongated cylinder with
elliptical x-section
0.07
0.07
0.86
x = ny;
(n + l)"1
n/((n +1)
0.00
Typical sand grain
0.14
0.14
0.72

56
Table 2-2. Summary of equations used in a model of heat and
mass transfer in the soil to determine state
variables and transport parameters.
Process or Parameter Equation No.
Conservation of mass (liquid) 2-22
Conservation of mass (vapor) 2-25
Conservation of energy 2-26
Vapor-Liquid equilibrium 2-30
Surface Boundary Conditions
Conservation of mass (liquid) 2-31b
Conservation of mass (vapor) 2-31c
Conservation of energy 2-31a
Lower Boundary Conditions
Conservation of mass (liquid) 2-32b
Conservation of mass (vapor) 2-31c
Conservation of energy 2-32a
Soil properties
Diffusivity of water vapor in soil (Dv) 2-34
Volumetric heat capacity (Cs) 2-35
Thermal Conductivity (X) 2-36
Hydraulic Diffusivity (D|_) 2-40
Surface Mass Transfer Coefficient (hm) 2-50
Surface Heat Transfer Coefficient (h^) 2-51

57
Table 2-3. Mesh spacing used in grid generation for the coupled
heat and mass transfer model for sandy soils.
Distance to
Node
Depth
Cell Width
next node
(cm)
(cm)
(cm)
1
0.0
0.0
0.5
2
0.5
1.0
1.0
3
1.5
1.0
1.0
4
2.5
1.0
1.0
5
3.5
1.0
1.0
6
4.5
1.0
1.5
7
6.0
2.0
2.0
8
8.0
2.0
2.0
9
10.0
2.0
2.0
10
12.0
2.0
2.0
11
14.0
2.0
2.0
12
16.0
2.0
2.0
13
18.0
2.0
2.0
14
20.0
2.0
3.5
15
23.5
5.0
5.0
16
28.5
5.0
5.0
17
33.5
5.0
5.0
18
38.5
5.0
5.0
19
43.5
5.0
5.0
20
48.5
5.0
5.0
21
53.5
5.0
7.5
22
61.0
10.0
10.0
23
71.0
10.0
10.0
24
81.0
10.0
10.0
25

91.0

10.0

10.0


3*4

18110

io!o

io!o
35
191.0
10.0
10.0
36
201.0
10.0


58
Volumetric Water Content (m^-m 3)
Figure 2-1. Thermal conductivity of a typical sandy soil as a function
of volumetric water content (DeVries, 1975).

Soil Water Potential ( cm )
59
Figure 2-2. Soil water potential (tension) and specific water capacity
as a function of volumetric water content for a sandy soil.
Specific Water Capacity ( cm *)

Hydraulic Conductivity ( cm/h)
60
Figure 2-3. Hydraulic conductivity and hydraulic diffusivity as a
function of soil water content for a typical sandy soil
Hydraulic Diffusivity (cm^/h )

Diabatic Influence Function ( 4> )
61
Figure 2-4. Relationship of the diabatic influence function and the
Richardson number.

62
H=f(TQlhh,WS)
\\\
Rn
\W
Qv=f(Ta,RH,hm,WS)
H
J*2
j = 3
H
AZ
J
i+l
nc-l
nc
Figure 2-5. Schematic of discretized domain for evaporation and soil
temperature model.

63
Figure 2-6. Flowchart describing the solution algorithm to solve the
system of finite difference equations for a coupled heat
and mass transfer model.

64
m
o
£>
O
3
y
35
v
cc
-o
£
o
3
cr
en
E
3
en
60 120 300
Time Step ( s )
600
Figure 2-7. Sum of squared residuals for the soil profile energy
balance after a 72 hour simulation using various time
steps.

Sum of Squared Residuals
65
Figure 2-8. Sum of squared residuals for the soil profile mass balance
after a 72 hour simulation using various time steps.

CHAPTER III
EQUIPMENT AND PROCEDURES FOR MEASURING THE MASS AND
ENERGY TRANSFER PROCESSES IN THE SOIL
Introduction
The process of evaporation of water from the soil is a complicated
process in which the dynamic processes of heat and mass transfer are
inter-related. Lysimetry has been defined as the observation of the
overall water balance of an enclosed volume of soil. Lysimeters have
been used for over 300 years to measure soil evaporation and crop water
use (Aboukhaled et al., 1982) and can be generally classified as either
weighing or non-weighing. As their names imply, weighing lysimeters
measure evaporation by monitoring the weight changes within the
enclosed soil volume while the non-weighing, or drainage lysimeters
determine evaporation by monitoring amount of water drained from the
bottom of the tank and the amount of rainfall upon the lysimeter. Most
drainage lysimeters are monitored on a seven to ten day cycle while the
weighing lysimeters can be monitored continuously to obtain hourly
evaporation rates.
The energy status of the soil can generally be determined by
measuring the temperature distribution over time. The energy flux at
the surface must be determined as well as latent heat transfer to
facilitate a complete energy analysis. Surface heat flux occurs as
radiant and convective heat transfer. The radiant energy impinging
upon the soil surface is either absorbed or reflected. The amount of
reflected and absorbed energy is primarily dependent upon the soil
66

67
surface. The reflectance and absorbtance change dramatically as a
function of water content. Convective heat losses from the surface are
generally estimated rather than measured using empirical functions to
determine surface heat transfer coefficients.
Objectives
The processes of energy and mass transfer in the soil are most
readily represented by the rate at which water is lost from the soil
(evaporation rate), the cumulative water loss (cumulative
evaporation), temperature, volumetric water content, and water vapor
density. All are temporal in nature, while the latter three are
functions of depth as well. The goal of the experimental design was to
provide data for the validation and calibration of the transport model
described in the previous chapter. Specific objectives were
1. to design a lysimeter and calibrate the instrumentation to
measure hourly evaporation rates,
2. to measure the vertical distribution of the soil temperature
and volumetric water content as function of time in the
lysimeter,
3. to measure ambient weather conditions to sufficiently
describe the boundary conditions for a coupled heat and
mass transfer model, and
4. to determine the physical characteristics of the soil
contained in the lysimeter.
Lysimeter Design. Installation and Calibration
Design and Construction
Weighing lysimeters were used in this research to meet the
objective of measuring hourly evaporation from a bare soil surface.
The weight of the lysimeter may be monitored by several different
methods. Mechanical measurement may be achieved by either supporting
the soil container with a lever and counterweight system or by

68
supporting the lysimeter directly by several load cells (Harrold, 1966;
Van Bavel and Meyers, 1961). The weight may also be detected by
supporting the soil container by a bladder filled with fluid and
monitoring the pressure of the fluid or the buoyancy of the lysimeter
container (Harrold, 1966; McMillan and Paul, 1961; King et al., 1956).
The hydraulic method has the disadvantages of being sensitive to
thermal expansion of the fluid and the high possibility of developing
leaks in the bladder and losing the fluid. The lever mechanism has the
advantage of requiring only one relatively low capacity load sensor
thus reducing instrumentation costs. The disadvantage of the counter
balance system is requirement of extensive underground construction to
contain the lever apparatus. This would require disturbing the
surrounding soil and border area and may be considered undesirable in
some cases. The cost of additional supporting structures and
excavation could be considerable as well. Three to four high capacity
load cells are required to support the lysimeter for direct weighing.
The higher cost of the load cells for the direct weigh method may
offset the cost of the increased excavation required for the counter
weight system. Direct-weigh lysimeters can be constructed such that a
high sensitivity can be achieved. Dugas et al. (1985) reported a
resolution of 0.02 mm of water for a direct weighing lysimeter with a
surface area of 3 m^. Most of the designs presented in the literature
have load cells which are difficult to access in the event of need for
maintenance or replacement.
A direct-weigh system was chosen because of its relatively simple
support system design and due to space limitations at the proposed
construction site at the Irrigation Research and Education Park (IREP)

69
located on the University of Florida campus in Gainesville, Florida
(Figure 3-1).
Aboukhaled et al. (1982) presented a relatively comprehensive
review of the existing literature regarding lysimeter construction and
design and noted the practical aspects of certain design
considerations. It was noted that the lysimeter annulus (gap area
between the soil container plus the area of the retaining wall) should
be minimized to reduce the effects of the thermal exchange between the
lysimeter soil mass and the surrounding air. Several researchers
(Aboukhaled et al., 1982; Black et al., 1968) made recommendations for
minimum lysimeter areas of 4 m2 for the primary purpose of reducing
the edge effect of the soil-to-soil gap between the lysimeter interior
and the surrounding fetch. Maintaining a 100:1 ratio of fetch to crop
height has been recommended to minimize border effects. The site
selected for the lysimeter construction satisfied the fetch
requirements for very short crops or bare soil. Dugas et al. (1985)
noted that the problems caused by soil-to-soil discontinuity should be
minimal if the lysimeter surface area were greater than 1 m2. Soil
depth should be sufficient so as not to impede root growth of crops to
be planted in the lysimeter.
Two soil containers were constructed using 4.8 mm (3/16 in.) steel
plate for the floor and side walls (Butts, 1985). All seams were
welded continuously to prevent water leaks and the entire container was
painted with an epoxy-based paint to prevent corrosion. The lysimeter
measured 305 cm long, 224 cm wide and 130 cm deep. This provided a
soil surface area of 6.8 m2 and a soil volume of 8.9 m3. Main floor
supports consisted of four 20-cm steel I-beams (W8xl5) and extended the

70
entire length past the edge of the soil container (Figure 3-2). To
alleviate the need for underground access to the load cells supporting
the lysimeter, two 2-cm rods were inserted into both of the protruding
end of each floor joist and extended above the top of the lysimeter.
These rods were inserted through both webs of a 25-cm steel I-beam
(W10x26) extending parallel to the end of the lysimeter. The ends of
the rods were threaded such that nuts could be placed above and below
the upper I-beam. The lower nut prevented the hanger beam (W10x26)
from sliding to the bottom prior to installation in the ground. The
top nuts prevented the rods from sliding out of the hanger beam after
installation. This assembly formed a cradle by which the lysimeter was
supported and provided access to four 4.5 Mg capacity load cells
located under each end of the hanger beams.
A pit with reinforced concrete block retaining walls and a 10-cm
reinforced poured-in-place concrete floor was constructed for the
installation of the lysimeter at the IREP (Figure 3-2). The depth of
the pit was such that the bottom of the lysimeter tank was
approximately 15 cm above the pit floor and the tops of the retaining
wall and the lysimeter were flush. The pit floor was sloped toward the
center to drain water which might accumulate to a sump located
immediately outside of the North wall of the pit. Support columns were
incorporated into the retaining wall on which to place the load cells
supporting the lysimeter. The load cells could be installed or
removed by lifting the lysimeter with hydraulic automotive jacks from
ground level. Fill dirt was used to bring the surrounding soil up to
the elevation of the pit wall.
Prior to installing the lysimeters in the ground pits, two porous

71
ceramic stones (30 x 30 x 2.5 cm) were installed in the bottom of each
lysimeter and connected to vacuum tubing for the purpose of removing
excess water from the lysimeter. The lysimeters were then placed in
the pits and leveled using the top nuts of the supporting rods to
adjust the length of the rods and to maintain a fairly uniform load on
all of the rods.
The lysimeters were filled to a depth of 3 cm with a coarse sand
to provide free drainage of water to the ceramic stones. A Hi11 hopper
fine sand excavated from a nearby site, was used to completely fill the
lysimeters. This soil was chosen because the profile was of fairly
uniform composition to a depth of 2 m and was typical of the local
sandy soils found in northern Florida and southern Georgia. After
loading the soil into the lysimeter, the soil was saturated with water
and allowed to stand overnight then removed by the vacuum system. This
provided settling to a density which might naturally occur in the field
after a long period of time.
Load cells manufactured by Hottinger Baldwin Measurements (HBM
Model USB10K) were then installed using a ball and socket connection
between the hanger beam and the load cell. A thin coat of white
lithium grease was applied to the surface of the ball and socket to
prevent corrosion of the contact surfaces. The output of the load
cells was a nominal 3 mV/V at full scale and could accept a maximum
supply voltage of 18 VDC. A regulated power supply provided 10 VDC
excitation voltage to each of the four load cells. Supply and output
voltages for each of the load cells was monitored using a single
channel digital voltmeter (Fluke Model 4520) and a multiple channel
multiplexer (Fluke Model 4506). Output was normalized using the ratio

72
of output voltage to input voltage (mV/V) to account for any variation
in supply voltage between load cells and time variation in the supply
voltage.
Lysimeter Calibration
Load cell calibration was performed by the manufacturer at the
factory for full load output of the cells. The manufacturer provided
load cell output (mV/V) at full scale load and no load. The load
carried by each load cell was expected to be near the full capacity of
each load cell with relatively small weight changes around that
reading. It was also noted that the calibration information provided
by the manufacturer was obtained under controlled atmospheric
conditions and using the factory-installed 3.3 m leads for measuring
the load cell output. The temperature sensitivity of the load cells
was very small compared to the changes in load cell output expected.
According the manufacturer's specifications, the effect of change in
load cell temperature upon the output was 0.08 percent of the load per
55C change in cell temperature. The installation site for the
lysimeters required the use of leads ranging in length from 37 to 53 m.
Therefore, an in-situ calibration was necessary.
The soil surface of the lysimeter was covered with a polyethylene
sheet to prevent weight loss due to evaporation during the calibration
procedure. A single weight having a mass of approximately 12 kg was
added to the lysimeter and the normalized output of each load cell was
recorded. A second weight was added, thus increasing the total weight
added to the system. The load cell output was again recorded. This
was repeated until the cumulative weight added to the lysimeter reached
175 kg (equivalent to 26 mm of water distributed over the lysimeter

73
surface). The normalized output was then recorded as the weights were
removed.
To account for unequal distribution of weight due to variations in
soil density and slope of the lysimeter, the above procedure was
performed such that weights were placed directly over a single load
cell and repeated for each load cell and by placing the weight in the
center of the lysimeter as well.
Soon after the load cells were installed and calibration
completed, one load cell from each lysimeter was damaged by lightning
and removed for repair. Since during the calibration procedure, the
normalized output of the individual load cells was recorded,
regressions for the remaining load cells for each lysimeter could be
determined. The calibration was repeated after the load cells were
repaired and installed. Regressions for each of the lysimeters were
obtained for the calibration data with three and four load cells
(Table 3-1). It was determined in both cases that the slopes were not
significantly different at the 99 % confidence level for the two
lysimeters and thereby allowing the same regression for both lysimeters
to be used. Since, interest was in changes in weight and not in
absolute weight, the intercept would not be significant. The
calibration curves used in data analysis are shown in Figure 3-3.
Resolution of the lysimeters based upon the specifications of the
voltmeter, and standard error of the regression parameters was
calculated to be approximately 0.02 mm of water.
Temperature Measurement
The vertical distribution of temperature within the soil was
measured by monitoring the output of ANSI Type T thermocouples (copper-

74
constantan) placed throughout the soil profile. Probes were
constructed using 2-cm PVC pipe with small holes drilled radially into
the pipe at specified intervals (Table 3-2). Thermocouple wire
(22 AWG) was fed into the top end of the PVC probe and the end of the
thermocouple protruding through the radial holes. Epoxy cement was
used to secure the thermocouples in place. After the thermocouples
were installed in the probe, the lower end of the probe was cut at an
angle then plugged with wood and epoxy. Two identical probes were
installed symmetrically about the center of the lysimeter to provide
some degree of redundancy and to obtain some idea of the uniformity of
the temperature distribution at a given depth in the soil. Shielded
multi-pair thermocouple extension cable was used to carry the
thermocouple signal to the Fluke multiplexer and digital voltmeter.
Soil Water Content Measurement
The measurement of the distribution of soil moisture was needed to
determine initial conditions for the model and to verify the subsequent
simulation of water movement within the soil. Several methods were
considered for the measurement of soil water content within the
weighing lysimeters. The method chosen had to provide a relatively
accurate measure of the volumetric soil water content as a function of
depth with minimal disturbance of the surrounding soil. The chosen
technique must also provide data for a relatively small vertical soil
volume and should be commercially available.
Soil water content is very difficult to measure accurately
especially when trying to do so with minimal disturbance of the
surrounding soil. The simplest method for determining soil water
content is the gravimetric method. Core samples are obtained from

75
various depths, and placed in a water tight container, weighed and
placed in a drying oven. The oven temperature should be maintained at a
constant temperature (105C) and the sample remain in the oven until a
constant weight is obtained. The sample is then removed from the
oven, allowed to cool in a desiccator, then weighed. This information
provides the water content on a weight basis. If the dry bulk density
of the soil is known, the volumetric water content can then be
calculated. The gravimetric method is the method by which all other
methods are calibrated (Schwab et al., 1966). However, the sampling
procedure disturbs the surrounding soil and the flow of moisture until
the soil sample is returned to the sample area. Even then the oven
dried soil is not at the same conditions as the undisturbed soil.
Therefore, the gravimetric method is not an acceptable method for
continuous or regular monitoring of the soil water profile.
Other methods for measuring the soil water content are classified
as indirect methods because they measure some property or
characteristic of the soil which varies predominantly with water
content. One of the simplest is the tensiometer. This instrument is
constructed of a hollow porous ceramic cup connected to a Plexiglas
tube. The tube is filled with water, sealed and connected to a
pressure sensor, usually a manometer, vacuum gauge or vacuum
transducer. After all of the air has been removed from the column of
water the tensiometer is placed in the soil such that the porous cup is
located at the desired depth. The water potential of the soil is
generally less than the potential of the water within the tube and is
drawn out of the tube until the water in the tube comes to an
equilibrium tension (or vacuum) with that in the soil. An electronic

76
vacuum transducer or manometer is used to measure the equilibrium
tension on the water column. The tensiometer has a range of
application from 0 to approximately 85 kPa (Long, 1982; Rice, 1969).
This would cover approximately ninety percent of the range of moisture
contents encountered in the field (Schwab et al., 1966). Using an
electronic pressure transducer would allow for the tension to be
monitored continuously. Several disadvantages in using tensiometers
exist. The soil water release curve (tension vs. soil water content)
must be known for each soil in which the tensiometer is installed.
However, this information is needed for the determination of the
concentration of water vapor within the soil for modeling purposes as
well. A second difficulty is that the tensiometer is essentially
detecting water content at a single point within the soil profile, thus
requiring the installation of several tensiometers within a finite
volume of soil. Tensiometers also require frequent maintenance. In
situ calibration of tensiometers is not required since the soil water
release curve is a unique property of the soil allowing the researcher
to obtain a soil sample from the experimental site and determine the
water release curve in the laboratory.
Electrical resistance is another property which varies with soil
water content. Two electrodes are imbedded within a porus gypsum block
which, when placed in the soil reaches an equilibrium moisture content
with the soil. The resistance between the two electrodes is measured.
However, as the water evaporates from the gypsum block, dissolved salts
remain in the gypsum. This deposition of salts within the porous
material will become redissolved when rewetted and cause an incorrect
reading. Their range of applicability extends mainly over dry soil

77
conditions (100 to 1500 kPa tension).
Nuclear emission methods provide yet another means by which the
soil moisture can be measured indirectly. In this method, either the
transmission of gamma rays or back-scattering of neutrons is measured.
Fast neutrons are emitted from a neutron source and as they react with
the hydrogen nuclei present within the soil, they lose part of their
energy becoming slow neutrons. A detector is then used to count the
rate at which slow neutrons are back-scattered in the soil surrounding
the probe. The neutron probe requires a single probe containing both
the radiation source and the detector. Access is gained by
installation of a tube in the soil thus allowing for repeated readings
at the same location with minimal disturbance of the soil after the
initial installation. However, the volume of soil penetrated by the
neutrons increases as the soil dries out, thus increasing the volume
over which the subsequent water content represents. Individual
readings are influenced by density and by the amount of endogenous
hydrogen. Once a calibration curve has been developed for the local
conditions, volumetric water content can be determined to within 0.5 to
1.0 percent moisture by volume (Schwab et al., 1966). A separate
calibration curve for measurements near the surface may also be
required due to fast neutrons escaping from the soil surface and not
being reflected back to the detector. Tollner and Moss (1985) noted
low R-square values of 0.6 and 0.4 for the calibration curves for a
neutron probe at a depths of 46 cm and 20 cm, respectively.
The use of gamma attenuation methods has gained some popularity
in recent years due to the ability of the instrument to measure the
volumetric water content or bulk density of relatively small volumes of

78
soil. The gamma probe requires the installation of two parallel tubes
spaced approximately 30 cm apart. A radioactive source which emits
137
moderately high energy gamma rays (Cs ) is placed in one tube while a
detector is placed in the second tube at the same depth. The intensity
of gamma radiation reaching the detector is monitored by the detector.
Ferraz and Mansell (1979) discuss the gamma radiation theory used in
measuring soil water content in great detail. Primarily, the use of
the gamma attenuation apparatus in which the gamma rays are collimated
so that the intensity of the beam is focused and a high intensity
source are used. The use of uncoilimated radiation in the field
requires the use of lower intensity sources due to the increased
possibility of radiation exposure to the operator. Ferraz and Mansell
(1979) state that the gamma attenuation method may be used in-situ to
determine changes in volumetric water content quite accurately, but the
error associated with absolute moisture content can be quite high.
Ayers and Bowen (1985) used a gamma density probe designed for field
use to measure density of soil within a soil density box. Resolutions
of 0.016 g/cm3 for the wet bulk density were achieved in their
experiments.
The gamma attenuation method has several advantages over the
neutron method for measuring soil moisture. The volume of soil in
which the measurement is made remains constant regardless of water
content (Ferraz and Mansell, 1979) and consists of a band between the
source and detector probes approximately 5 cm deep (Ayers and Bowen,
1985). The disadvantage of the gamma method with respect to the
neutron method is the fact that a pair of tubes must be installed
instead of a single tube. However, the probes for the neutron and

79
gamma meters are typically the same diameter, thus allowing the use of
the neutron probe in the gamma access tube if necessary. The two
radiation methods will generally disturb less soil during installation
and will allow for recurring measurement at more frequent depth
intervals than with the tensiometers. Tensiometers also require
considerably more maintenance than do the radiation methods. The
tensiometers on the other hand can be monitored continuously by
automated data acquisition equipment while the radiation techniques
cannot. Extensive calibration is required for either of the radiation
measurement techniques. Based upon the desire to cause minimal
disruption of the soil profile and measure soil moisture at many points
within the soil, it was decided that a gamma attenuation technique
would be used to measure soil water content.
Gamma Probe Calibration
Extensive calibration curves were required to determine absolute
values of the soil water content using the gamma ray attenuation
method. While the soil was being loaded into the lysimeters, three
pairs of aluminum access tubes (5.1 cm O.D.) were installed in each
lysimeter (Figure 3-2). The tubes extended from the lysimeter floor to
23 cm above the soil surface. Parallel tube guides provided with the
Troxler 2376 Dual Probe Density Gauge were used to maintain parallelism
between the tubes as the lysimeter was filled.
The gamma probe primarily measures density of the test material;
therefore, it was necessary to measure the wet bulk density of the soil
corresponding to the depth at which the gamma readings were obtained.
A bulk density core sampler was used to obtain the soil samples for the
measurement of density and volumetric water content. The sampler

80
consisted of a series of removable brass rings of known volume within a
hollow cylinder. The cutting edge was tapered such that compaction of
the soil sample was minimized (Baver et al., 1972). The soil sample
was removed from the inner brass ring and placed in an aluminum soil
sample container and the lid closed to prevent moisture loss until the
samples from a single probe could be measured. Core samples were taken
at depth intervals of 10 cm from the soil surface to a depth of 71 cm.
Three probes were obtained from within the lysimeter to roughly
correspond to the locations of the paired access tubes. The soil
samples were weighed, dried, and reweighed. Wet density, dry density,
and volumetric water content were calculated for each sample. The soil
properties determined from each of the three probes were averaged for
each depth to account for the fact that the density between the access
tubes could not be measured. This procedure also provided information
regarding the areal uniformity of the soil with the lysimeter. The
dried soil was placed back in the holes in reverse order of their
removal so as to maintain the original soil profile. Care was taken
during subsequent sampling to avoid the soil core sites previously
sampled in the lysimeters.
The gamma readings were obtained in the following manner. The
meter was turned on for a minimum of twenty minutes prior to readings
being taken to allow for the circuits to stabilize. After warm-up, the
137
detector probe and Cs gamma source were placed in the tubes of a
calibration stand. The calibration stand consisted of two parallel
aluminum tubes with materials of known density. The standard materials
were polyethylene (1.06 g/cm3), magnesium (1.75 g/cm3), magnesium-
aluminum alloy (2.16 g/cm3), and aluminum (2.61 g/cm3). The source and

81
detector were placed in the stand level with the center of the
polyethylene standard and the amplifier gain of the meter was set so
that the peak count rate was achieved. The probes were then lowered to
correspond with the magnesium and a 4-minute count was taken. In the
calibrate mode of the timer, the meter indicates the count per minute
for the 4-minute interval. This was referred to as the standard count
and was used to normalize subsequent counts in the soil. The standard
count also provided a means by which to compare readings taken at
different times and accounted for variation in temperature of the gauge
and gain settings.
After the standard count was taken, the meter timer was set to
accumulate counts for one minute. The probes were placed in a pair of
tubes such that the source and detector were located at a depth
corresponding to the center of the core samples taken. A minimum of
two one-minute counts were recorded for each depth then the probe moved
to the next depth. After the last measurement at a depth of 66 cm was
obtained, the source and detector rods were removed from the access
tubes. Counts were taken at the same depths in all three set of access
tubes and the entire sampling process repeated in the second lysimeter.
Several attempts at this calibration procedure were made. Early
trials produced unacceptable ranges of scatter in the gamma probe
readings. At a later date, faulty electronic components were found and
expected to be the reason for the highly variable gamma probe readings.
After repair of the unit, the calibration procedure was repeated
yielding acceptable results.
According to Ferraz and Mansell (1979) the density of the test
material varies inversely with logarithm of the ratio of the intensity

82
of radiation observed in the test sample to the radiation intensity
measured in air. This relationship can be expressed as the count ratio
varying as a decreasing exponential with respect to the density of the
material. The count ratio is defined by the manufacturer as the ratio
of counts per minute observed for the test material to the counts per
minute observed in the magnesium standard (Troxler,1972). The observed
density of the soil obtained from the gamma measurement represents the
wet bulk density (Equation 3-1).
Pmt A B ln(CR) (3-1)
where:
_3
/wet : wet bulk density of the soil [g*cm ]
A, B : constants of regression
CR : count ratio [dimensionless]
count per min in test material
count per min in standard
If the dry bulk density is known, then the volumetric water content
can be calculated by
where:
9 :
^wet :
pdry :
9
^ ^wet~ ^dry^
Pvi
volumetric water content [cm3*cm3]
wet bulk density of soil [g*cm ]
O
dry bulk density of soil [g*cm ]
( 3-2 )

83
_3
p : density of water [g*cm ]
w = 1.00 [g-cnT3]
Combining equations (3-1) and (3-2) yields a relationship for
volumetric water content in terms of the count ratio and the dry bulk
density of the soil.
9 = A + B-ln(CR) + C-/>dry (3-3)
It was conceivable that a calibration curve might be necessary for
each of the lysimeters and for various ranges of depth. Therefore,
linear regressions of the forms in Equations (3-1) and (3-3) were
determined for each lysimeter and for each of the depths. Statistical
testing revealed that a single regression could be used for all depths
and for both lysimeters for both the wet density and volumetric water
content. The final form of the calibration equation for determination
of volumetric water content was
9 = 1.52 0.33"In(CR) 0.89-pdry (3-4)
The coefficient of variation (R^) for the regression shown in
equation (3-4) was 0.888. Measured volumetric water content when
plotted against the water content estimated by the calibration
(equation 3-4) should lie about a line with a slope of unity and an
intercept of zero (Figure 3-4). Examination of Figure (3-4) indicated
that relatively large errors could be associated with the exact
estimate of volumetric water content. Therefore, the gamma probe

84
should be used to determine changes in volumetric water content during
the experiments and gravimetric sampling should be used to determine
initial conditions for the experiment.
Soil Bulk Density and Porosity
The calibration procedure for the gamma probe provided sufficient
data to determine the vertical and horizontal distribution of the soil
dry bulk density in each of the lysimeters. As mentioned previously,
repeated core samples were obtained at several depths dispersed over
the area of each of the lysimeters. Figure (3-5) shows the average dry
bulk density as a function of depth in each of the lysimeters. The
error bars represent the standard deviation of the measured bulk
density. The bulk density for both lysimeters was relatively constant
at 1.38 g/cm3 in the top 25 cm and increased to approximately 1.6 g/cm3
between 25 and 40 cm. The largest deviation in measured bulk density
occurred at the lower depths.
The literature (Baver, 1972; DeVries, 1975) indicates that the
porosity for sandy soils ranges from 0.50 to 0.55 cm3/cm3. The soil
porosity was a soil parameter required as a function of depth. Due to
the variation in soil density, especially at the depths from 25 to
40 cm, it was decided that the porosity for the soil in the lysimeters
should be determined as a function of depth.
The porosity is defined as the volume of pore space in the soil
per unit of bulk volume. An air pycnometer (Baver, 1972) was used to
measure the porosity of the soil. Four levels of soil bulk density
ranging from 1.2 to 1.6 g/cm3 were used for the tests. Three soil
samples at each level of bulk density were prepared from a soil sample
obtained from the weighing lysimeters. The volumetric water content

85
of each soil sample was 0.05 cm3/cm3. The mass of soil placed in the
50 cm3 sample cup of the air pycnometer was that required to achieve
the prescribed levels of dry bulk density. The test was performed for
the sample in the cup and the void volume recorded. The volume
occupied by the water in the sample was added to the measured void
volume to yield the total pore space in the soil. A total of three
measurements were made for each sample. The test was also repeated
for a single soil sample with the maximum density achievable
(1.64 g/cm3). The porosity was calculated for each measurement by
dividing the 50 cm3 sample volume into the void volume and were
averaged for each bulk density level (Table 3-3). The repeatability of
the experiment was indicated by the standard deviation of the porosity
for each of the samples, while the accuracy of the measurement was
indicated by the standard deviation for the average of all measurements
for a given density level. The porosity was found to vary linearly
with bulk density (Figure 3-6). The error bars shown in Figure 3-6
indicate the standard deviation of the soil porosity at each bulk
density level. The porosity measurements were then used in defining
the soil profile for the validation and calibration simulations.
Experimental Procedure
Experiments were designed to monitor the energy and mass transport
processes from a bare soil surface for two weighing lysimeters on a
continuous basis. Automatic data collection was performed by a Digital
Electronics Corporation (DEC) PDP-11/23 mini-computer equipped with an
IEEE-488 interface board and utilizing the RT-11 operating system. The
RT-11 operating system provided several system subroutines which were
primarily for scheduling the data collection as well as other time

86
manipulations. The main program and associated subroutines were
written in FORTRAN-66. Subroutines for initializing and retrieving
data through the IEEE interface were written in either assembly or
FORTRAN by Dr. J. W. Mishoe for previous research. All signals from
the sensors used were analog signals and were monitored on 50 channels
of a Fluke 4506 multiplexer and subsequently read via the IEEE
interface from a Fluke 4520 digital voltmeter. Data monitored by the
computer for each lysimeter were net radiation, soil temperature
distribution, and supply and output signals from each of the load
cells.
Net radiation was measured using net radiometers (WEATHERtronics,
Model 3035) mounted 85 cm above the soil surface of the lysimeter. The
net radiometers utilized blackened thermopiles as the sensing element
to detect the difference between incoming and outgoing radiation. A
positive millivolt signal was produced when the incident radiation was
greater than that being reradiated, while a negative signal indicated
that more energy was being radiated from the surface than impinging
upon the surface. Factory calibration curves were used to convert the
millivolt signals to net heat flux (W/m^). The millivolt signal was
read at two minute intervals and total millivolts and number of times
read were recorded on floppy diskette at ten minute intervals.
Thermocouples, supply voltage, and load cell output were recorded at
ten minute intervals as well.
Data files were closed on an hourly basis and new ones opened for
each hour. This minimized the risk of data loss due to power outages
or other computer malfunctions. The most data that would be lost that
had already been recorded would be that for one hour in the event that

87
the power was lost just prior to closing the data file. When power was
restored, the PDP-11/23 would automatically restart and begin the data
collection routine. Separate data files were maintained for each of
the two lysimeters.
Hourly values of the ambient dry bulb temperature and relative
humidity were measured within a standard weather shelter from the
adjacent weather station monitored and maintained by the Agricultural
Engineering Department. Sensors for measuring air temperature were
Type T thermocouples and instantaneous values were recorded hourly. A
Campbell Scientific CR-207 sensor was used to monitor relative
humidity and consists of a wafer whose electrical properties vary with
relative humidity. Manufacturer's literature states that the sensor
provides reliable information when used in non-condensing conditions
with a relative humidity between approximately 20 and 90 percent.
However, the wafer material tends to absorb moisture over time
decreasing the reliability of data obtained. To account for the lag
time of the sensor, data was averaged over an hour and recorded.
Relative humidity data was generated as well by assuming that the
minimum daily temperature was the dewpoint temperature for the day.
Linear interpolation between consecutive daily minimums to provide a
continuous estimate of the dewpoint temperature throughout the day.
Relative humidity could then be calculated based upon the estimated
dewpoint temperatures. Wind speed and direction were measured at a
height of 2 m within the same weather station. Meteorological data
were recorded hourly and uploaded daily to the VAX mainframe managed by
IFAS. Access to the hourly data was achieved through the AWARDS
(Agricultural Weather Acquisition Retrieval Delivery System).

88
Meteorological data, including relative humidity determined from the
estimated dewpoint temperature, are shown in APPENDIX B.
Tests were begun by an irrigation or rainfall event to ensure that
the upper layers of the soil were sufficiently wet so as to provide a
minimum of three to four days of evaporation data. Most frequently,
water was added to the soil the evening prior to beginning the test to
insure that the soil surface was wet and to allow some downward
distribution of water prior to the initiation of each test. Core
samples were obtained to determine the initial distribution of water
within the lysimeter. Samples were taken at vertical intervals of
5.1 cm for the first 30.5 cm then every 10.2 cm until a depth of
91.4 cm was reached. Gamma probe measurements were taken in each of
the three pairs of access tubes corresponding to the center of each of
the core samples. Actual depths of gamma probe measurements and core
samples are shown in Table 3-4. Subsequent gamma probe measurements
were made approximately every other day. More frequent readings were
not obtained in most cases due to the time required for each set of
readings (2 hrs.). This minimized disruption of the other data being
collected.
Obtaining the gamma probe readings required placing a scaffold
across the lysimeter to avoid disturbing the weight of the lysimeter
and compacting the soil. This, in turn, could have disrupted the
evaporation process due to shading of the soil surface as well as some
of the net radiation measurements. These periods when the integrity of
some of the data may have been questionable was marked in the hourly
data file by a flag entered via the keyboard. The flag was turned on
when gamma readings were being made on each lysimeter and turned off

89
again after the task was completed.
Data were collected continuously over a period extending from
November, 1986 to August, 1987. Data collection was interrupted on
several occasions due to equipment failures, power outages, equipment
maintenance, and instrument calibration. Most tests were three to four
days in length with a few extending to six or seven days.
Data Analysis
Data files containing the data recorded at ten-minute intervals
were concatenated into daily files. The daily files were then combined
to correspond to specific dates for the individual tests. Data
contained in the files consisted of day-of-year, time-of-day, the
normalized output for each load cell (mV/V), the 10-minute total net
radiometer output, the number of net radiometer readings during the
10-minute interval, the status flag, and twenty temperatures
corresponding to the depths indicated in Table 3-2.
The normalized output of the load cells was totaled and the
average total millivolt per volt for the hour was calculated. The
difference between successive hourly totals was used in the load cell
calibration equations to determine the hourly and cumulative
evaporation of water from the lysimeter. The output signal for the net
radiometers was totalled for the hour then an average rate of net heat
flux was determined using the manufacturer's calibration curves. The
two temperatures for each depth were averaged over the hour as wel1.
If the data flag had been turned on at any time during the hour, it was
assumed to have been on the entire hour. Reduced data files contained
the hourly values of julian date, time, cumulative evaporation (mm),
hourly evaporation (mrn/hr), net radiation (W/m^), data flag, and

90
average soil temperatures at various depths. The water content
profiles were generated and stored separately since they were not
monitored on a continuous basis as were other data.
Experimental Results
Experimental data showing cumulative and hourly water loss,
vertical distribution of water, and temperature of the air, soil
surface and the soil at depths of 15 and 80 cm are presented in Figures
3-7 through 3-32. If rainfall occurred during a test, then cumulative
precipitation is shown as well for that experiment. If rainfall was not
shown, none occurred during the test. Data are presented
chronologically according to calendar date.
The total water lost from the north and south lysimeters during
the period November 25 to 29, 1986 was approximately 6.0 and 3.5 mm,
respectively. A total of 20 mm of rain fell from just prior to
midnight on November 28 until 0400 on November 29 (Figure 3-7). Data
collection was discontinued for the north lysimeter shortly before the
rain began while data were monitored for the south lysimeter until the
morning of November 29. Soil conditions in the two lysimeters were
similar and water loss should have been similar as well. However,
discrepancies between the cumulative water loss from the two lysimeters
was evident. For instance, the south lysimeter indicated an increase
in weight every day during the experiment at approximately 1800, while
the north lysimeter indicated a slight increase in weight between the
hours of 0400 and 1000 on November 26 and 27. Examination of the
hourly evaporation rates (Figure 3-8) indicated this phase shift in
the evaporation of water from the soil. It also showed the highly
variable and erratic behavior of both lysimeters.

91
Evaporation calculated from experimental measurements of
volumetric water content for the period from November 25 to 26 (Figure
3-9) was 5 and 6 mm of water from the north and south lysimeters,
respectively. The evaporation determined by changes in weight of the
lysimeters was 3.8 mm for the north lysimeter while 2.5 mm of water had
evaporated from the south lysimeter. Normally, the measurement of
volumetric water content would be more likely to have higher errors.
However, due to the close agreement between the evaporation determined
by changes in volumetric water content for the two lysimeters, the
determination of evaporation by weight loss from the lysimeters was
suspected to be in error. The data for the south lysimeter were most
highly suspect for this data set.
Temperatures followed a diurnal fluctuation as expected (Figure
3-10). The air temperature reached a maximum during mid-afternoon and
a minimum just prior to sunrise. It was interesting to note that the
peaks in air temperature lagged the peaks of soil temperature. This
was due to the fact that the primary source of sensible heat gain in
the air was by convection from surfaces instead of radiant heat
transfer. The lag time for the peaks within the soil increased as
expected with depth. The amplitude of the diurnal fluctuation also
diminished with depth due the thermal capacitance of the soil as
expected. The soil temperature at a depth of 80 cm varied less than
2C during the test.
The next set of data was recorded from December 15 to 19, 1986.
Similar problems appeared in this data set as had occurred in the
previous data (November 25 29). It was noted that the north
lysimeter initially lost approximately 1 mm more water than did the

92
south lysimeter (Figure 3-11). However, the south lysimeter indicated
a gain of approximately 0.5 mm of water beginning at 1700 on December
15, while the north lysimeter continued to lose water until 1800. The
lysimeters then indicated a weight gain equivalent to approximately
2 mm of water between the hours of 0200 and 0500 on December 16. This
was confirmed by the AWARDS data indicating 2 mm of rain fell between
0200 and 0500. Water began evaporating at 0600 from the south
lysimeter, while the north lysimeter did not begin losing water until
approximately 1100. Erratic behavior was noted throughout the
remainder of the test for both lysimeters. The hourly evaporation
rates further indicate the high variability of the data (Figure 3-12).
The initial distribution of water ranged from a minimum of 18 percent
at the surface in both lysimeters to a maximum of 27 and 29 percent for
the south and north lysimeters respectively. Gravimetric samples could
only be obtained to a depth of 25 cm in both lysimeters due to the high
water content below that depth. Moisture content readings obtained by
the gamma probe were not reliable as measures of absolute moisture
content. Due to the erratic nature of the evaporation rates determined
by weight loss and insufficient data to calculate the evaporation from
the water content measurements, this data set was considered to be
unreliable as a measure of evaporation of water from the soil. Soil
temperatures followed the diurnal pattern as expected (Figure 3-14).
Maximum surface temperatures were typically 2 to 3C above the air
temperature and increased as the soil surface dried. The soil
temperature measured 80 cm below the surface was observed to remain
fairly constant during the four day test
The quality of the data obtained during tests conducted from

93
January 5 to 9 and January 11 to 14, 1987 was satisfactory. The
evaporation data was very smooth when compared to the data for the
previous two tests (Figures 3-15 and 3-18). The cumulative evaporation
exhibited maximum daily evaporation on the first or second day of the
test. The relatively small amount of water lost on the first day as
compared to that lost on the second day was due to the time at which
data collection began for the test. In both instances, data collection
did not begin until after initial measurement of the soil water profile
had been completed. Therefore, the water which had been lost prior to
beginning of data collection was not known. Maximum hourly evaporation
rates occurred just around noon on each day with each subsequent day
showing a lower peak evaporation rate than the previous day (Figures
3-15 and 3-18). The volumetric water distribution with depth in each
lysimeter (Figures 3-16 and 3-19) show that most of the water was lost
from the soil above 20 cm. For the two-day period between January 5
and 7, the water evaporated primarily in the soil shallower than 10 cm
with very little change in the water content of the soil below 20 cm
(Figure 3-16). The water then evaporated from between depths of 10 and
40 cm from January 7 to 9. The water content of the soil below 40 cm
remained constant. The south lysimeter lost less water from the upper
portion of the soil from January 5 to 7 than the north lysimeter.
Evaporation of water from the soil continued from the upper layers as
well as from the 20 to 40 cm area of the soil. Similar distributions
were noted for both lysimeters for the January 11 to 14 period. In the
north lysimeter, almost all of the water evaporated from the soil above
20 cm while the soil below 20 cm remained constant (Figure 3-19).
Water was lost from the soil to a depth of 40 cm. Note that the water

94
content at a depth of 80 cm decreased from the beginning of each test
to the end. Some could have been redistributed toward the surface, but
was more likely to have been distributed toward the bottom of the
lysimeter due to gravity. The soil at a depth below 60 cm had a water
content above the drained upper limit for the Mi11 hopper sand,
approximately 13 percent. The drained upper limit of a soil is the
highest water content at which the soil can exist without free drainage
of water due to gravity. Soil temperatures again exhibited the diurnal
fluctuation as expected with the amplitude of the variation diminishing
with depth until it had been effectively damped out at a depth of 80 cm
(Figures 3-17 and 3-20). Peak to peak variation of the temperature of
the soil surface was generally greater than that of the air
temperature. Close inspection of the surface temperature as compared
to the air temperature showed that the difference between the maximum
air temperature and surface temperature increased as the soil surface
became drier.
The data collected from February 3 to 5, 1987 were shown mainly to
indicate the evaporation which can occur with partially overcast
weather (Figures 3-21, 3-22, and 3-23). Note that only one full day of
measurement were taken before a rainfall of 35 mm occurred. Note that
approximately 2 mm of water (Figure 3-21) evaporated under partly
cloudy skies which was comparable to the amount of evaporation which
occurred under sunny conditions with comparable air temperatures on
January 11 and 12 (Figure 3-18). Examination of the hourly evaporation
rate (Figure 3-22) shows a maximum evaporation rate of 0.3 mm/h was
achieved on February 3, which was lower than the 0.6 mm/h peak noted on
January 11. However, the peak evaporation rate was maintained for a

95
longer period of time on February 3 while the hourly evaporation rate
dropped off very sharply on January 11. Prior to the rain beginning on
February 5, the same maximum evaporation rate was attained indicating
vertical redistribution of water toward the surface occurred fast
enough to maintain the lower evaporation rate for a longer period of
time. Unfortunately, the rainfall prevented a final measurement of the
distribution of the water in the soil. The temperature of the soil
surface (Figure 3-23) followed the air temperature very closely during
the overcast weather until just prior to the rain beginning, then the
incident radiation of the soil caused the surface to continue
increasing even though the air increased at a much slower rate. After
the rain began, the soil surface cooled very rapidly and reached the
same temperature as the air.
The remaining data shown in Figures 3-24 to 3-32 show the same
general trends explained previously. Cumulative and hourly
evaporation rates for both lysimeters were similar. Some apparent
glitches in the data however, should be explained. The cumulative and
hourly rates of evaporation for the period from August 11 to 21
(Figure 3-27) show a rainfall of approximately 10 mm on August 12,
then an apparently very rapid evaporation of water on August 14. The
rapid loss of water was caused by activation of the vacuum system to
remove approximately 18 and 17 mm of water from the north and south
lysimeters respectively. The large dip and subsequent recovery on
August 13 was the instrumentation sensing the addition and removal of
the dual probe density gauge for moisture content. The distribution of
water in the soil (Figure 3-28) also indicates the rainfall as well.
It should also be noted that ambient weather data was not available

96
during the August test due to equipment failure of the AWARDS data
system.
The data for the period from August 25 to 29 appeared to exhibit
some of the instability shown in the first two data sets (Figures 3-29
and 3-30). In contrast to the earlier data, the peak evaporation was
followed by a period of little or no evaporation then a gain of
approximately 1 mm of water possibly due to evaporation. The fact that
both lysimeters behaved similarly with very little time shift in events
indicated that the weight gains actually occurred. The measured water
profiles and soil temperature profiles are shown in Figures (3-31) and
(3-32), respectively for both lysimeters.
Summary
A system to measure evaporation of water from the soil as well as
the distribution of water and temperature in the soil as a function of
time was designed, installed and instrumented. The weighing lysimeters
were instrumented with load cells to support their full weight and a
single calibration curve was developed to analyze the weight data
collected on a continuous basis. The weighing lysimeter system was
capable of detecting hourly rates of water loss from the soil as low as
0.02 mm/h. Soil temperatures were recorded on a continuous basis
throughout the soil profile. A dual probe density gauge was utilized
to measure the distribution of water within the soil on a semi-daily
basis. The ambient weather conditions were obtained from a nearby
weather station.
Future experiments conducted in the weighing lysimeters should
utilize a non-intrusive method to measure the distribution of water
within the soil. A system utilizing tensiometers or some other

97
indirect method, such as time domain reflectometry, in which the
sensors can remain in place and can be monitored via a data acquisition
system on a continuous basis similar to the measurement of other system
variables. Although the use of thermocouples for the measurement of
soil surface temperature did not appear to have caused significant
errors, the possibility exists. Errors could occur due to the
thermocouple becoming uncovered and exposed to direct sunlight or could
have become buried too deeply. It is suggested that in future
research, an infrared thermometer be permanently mounted on the
lysimeters to measure soil surface temperatures. This might also be
helpful when a crop is grown in the lysimeters for measurement of the
canopy temperature. Under a full crop canopy, the thermocouple could
be used with decreased possibility of errors due to direct radiation
impinging upon the sensor.
The overall performance of the weighing lysimeter system provided
data for the validation and calibration of a coupled soil water balance
model. The goals of monitoring hourly and cumulative evaporation as
well as soil temperature, water distribution and boundary conditions
were achieved.

98
Table 3-1. Regression coefficients for weighing lysimeter
equations of the form... mm = a + b mV/V
No. of
Lysimeter Load Cells Regression Coefficients
North
South
Combined
3
4
3
4
3
4
a = 0.1688
a = 0.1688
a = 0.1346
a = 0.1345
a = 0.1572
a = 0.1572
b = 301.9807
b = 226.4855
b = 300.9117
b = 225.6838
b = 301.4969
b = 226.1227
calibration
R2
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985

99
Table 3-2. Vertical spacing of thermocouples for measurement of soil
temperature.
Thermocouple
Position No.
Depth
( cm )
1
0.0
2
1.0
3
2.0
4
4.0
5
8.0
6
20.0
7
40.0
8
60.0
9
80.0
10
100.0

100
Table 3-3. Experimental values of soil porosity as
bulk density of the soil.
Bulk Porosity (cm3/cm3)
Density
(g/cm3) Sample Rep #1 Rep # 2 Rep # 3
a function of dry
Sample
Average
1
0.555
0.556
0.556
0.556
1.20
2
0.555
0.556
0.554
0.555
3
0.554
0.555
0.555
0.555
1
0.516
0.515
0.515
0.515
1.30
2
0.514
0.515
0.515
0.515
3
0.514
0.514
0.515
0.514
1
0.477
0.477
0.478
0.477
1.40
2
0.473
0.474
0.474
0.474
3
0.473
0.473
0.473
0.473
1
0.402
0.403
0.404
0.403
1.60
2
0.406
0.407
0.407
0.407
3
0.403
0.403
0.400
0.402
1.64
1
0.390
0.391
0.391
0.391

101
Table 3-4. Depths below the soil surface for soil moisture core
samples and corresponding dual probe gamma readings.
Soil Sample Gamma Probe
Depth Depth
( cm ) ( cm )
0.0
-
5.1
2.5
5.1
-
10.2
7.6
10.2
-
15.2
12.7
15.2
-
20.3
17.8
20.3
-
25.4
22.9
25.4
-
30.5
27.9
30.5
-
40.6
35.6
40.6
-
50.8
45.7
50.8
-
61.0
55.9
61.0
-
71.1
66.0
71.1
-
81.3
76.2
81.3
_
91.4
86.4

102
OFFICE a
DATA ACQUISITION EQUIPMENT

"SPAR" UNITS

A.
M
AWAROS
WEATHER
STATION
Figure 3-1. Site plan for weighing lysimeter installation at
University of Florida Irrigation Research and Education
Park.

103
Masonry Block Retaining Wall
Hanger Rods
Concrete Slab
Ceramic Stones
Lysimeter Container
Thermocouple Probes
Floor Support Beams
Gamma Probe Access
Hanger Beam
Tubes (Pair)
Figure 3-2.
Schematic of lysimeter construction and support system.

104
Total Change in Load Cell Output ( mV/V )
Figure 3-3.
Linear regression of the load cell calibration data for
the weighing lysimeters at the Irrigation Research and
Education Park, Gainesville, FL.

Measured Volumetric Water Content (cmV cm^)
105
Predicted Volumetric Water Content (cm^/ cm^)
Using Gamma Probe Calibration
Figure 3-4. Comparison of measured volumetric water content to that
estimated using the gamma probe calibration.

Bulk
106
Figure 3-5.
Dry soil bulk density as a function of depth for two
lysimeters installed at the IREP, University of Florida,
Gainesville, FL.

107
Dry Bulk Density (g/cm^)
Figure 3-6
Soil porosity as a function of dry bulk density for a
Mi11 hopper fine sand.

Cumulative Evaporation (mm) Cumulative Precipitation (mm)
108
20
10-
5-r
ol
Iii.|i
+
T
i
1
I
!
Figure 3-7
Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
November 25 29, 1986.

109
Figure 3-8. Hourly water loss from the University of Florida, IREP
weighing lysimeters for November 25 29, 1986.

Depth ( m ) Depth ( m )
110
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-9. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on November 25 and 26, 1986.

Temperature (C) Temperature (C)
111
Julian Date
Figure 3-10.
Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from November 25 to 9, 1986.

Cumulative Evaporation (mm) Cumulative Precipitation (mm)
112
4--
3 +
2--
1 --
I M I M M I I I I I I I II 11 I t <1 I I I'M II I I'M IIMIIlllimilllllll'IIIIIIIIIH
I'll I III IMI ItU.lllililll I |:|i|,i,l¡Mii:l ll.lrl 1,1 III I l;l,l l l ll 1111 ni l,l
l'lili|ili|f|ifil:l Hl'M'|,| I l|l M l I f IM 11 l i l i I II I IiKI I 1,11 K ill
l|M¡l|Wll'1llKlilililil|ljKIll(ll!lilililtlil¡l,KllMl!l|liHlll!l1l|liil,l;l,l'lil¡li,l:,M,i|,|llilil,Ml!lil',l,l,l
ttimtim i t1 mm,i m mittj nw m miiii cm-i 0 ft i, umi i h i i i i k i
:H|I,IW;kK||i|iKM,|,IiIIiK!II:|'KIi>i;i-M'W,I|I I|M'M:I:>,I¡M.| l|l KilKI'KflKI III l
. l|,M|KillMil*l|l:>l'l|l|lil,l¡H:li/5MjM,i;MKI¡l'l|l¡l|llM¡l l'IM rijlll M:ljt||iM III l>l¡H
,l|l* Wjl!|MW;Kil'(|l,l|f;lil|liiii-M|l¡/ l|li|il(it|l|(iljlM|l I.Ml IKMKIiKMIf IHiU
If,11K K)fWlif+M l,lll,t;i'WM(lil (.1,1 IH;M,t l'(l|llii(:ll(.KM MiK,l;M*t¡H (¡I l.l 11 lllil
III Hint I l.KliMSlI I III,1)1 liMf/illl 1,1 KiI intiliin,ll i IIM 11 Cl i III l ll (I I t 1
itim fti mi\i(Him ni 11 (up n:i (hi it i ni i tin ti i Itiii i min tm
-1 1-
III limiillilllll Mill! nl 1111 m 11 mi 111111111111111111111111111
Figure 3-11. Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
December 15 19, 1986.

113
Figure 3-12. Hourly water loss from the University of Florida, IREP
weighing lysimeters for December 15 19, 1986.

114
Volumetric Water Content (cm^/cm^)
North Lysimeter
0.0
0.1
0.2
0.3
0.4
0.3
0.6
0.7
0.8
I
?
T
0.00
Hme Date
1000 12/15
AA 900 12/19
-i i|^
0.10
Volumetric Water Content (cm^/cm^)
South Lysimeter
ii
0.40
Figure 3-13. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on December 15 and 19, 1986

Temperature (C) Temperature (C)
115
Julian Date
Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from December 15 to
19, 1986.
Figure 3-14

Evaporation Rot. (mm/h) Cumulative Evaporation (mm)
116
Julian Date
Figure 3-15. Hourly and cumulative water loss measured from
January 5-9, 1987 using the University of Florida,
IREP weighing lyslmeters.

Depth ( m ) Depth ( m )
117
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-16. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on January 5, 7 and 9, 1987.

Temperature (C) Temperature (C)
118
Julian Date
Figure 3-17. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from January 5 to 9, 1986.

119
Julian Date
Figure 3-18. Hourly and cumulative water loss measured from
January 11 14, 1987 using the University of Florida,
IREP weighing lysimeters.

Depth ( m ) Depth ( m )
120
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-19. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on January 11 and 14, 1987.
1

Temperature (C) Temperature (C)
121
Figure 3-20. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from January 11 to 14 1987.

Cumulative Precipitation (mm)
122
Figure 3-21. Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
February 3-5, 1987.

123
Figure 3-22. Hourly water loss from the University of Florida, IREP
weighing lysimeters for February 3-5, 1986.

Temperature (C) Temperature (C)
124
30
25--
20--
15
North Lysimeter
Date: 2/3 2/5/87
Julian Date
Julian Date
Figure 3-23. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from February 3 to 5, 1987.

125
Julian Date
Figure 3-24. Hourly and cumulative water loss measured from February
9 13, 1987 using the University of Florida, IREP
weighing lysimeters.

Depth ( m ) Depth (
126
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-25. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on February 9 and 13, 1987.

Temperature (C) Temperature (C)
127
40 41 42 43 44 45
. Julian Date
Figure 3-26. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from February 9 to 13, 1987.

Evaporation Rate (mm/h) Cumulative Evaporation (mm)
128
Julian Date
Julian Date
Figure 3-27. Hourly and cumulative water loss measured from
August 12 21, 1987 using the University of Florida,
IREP weighing lysimeters.

Depth ( m ) Depth ( m )
129
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on August 12, 13, 18,
and 21, 1987.
Figure 3-28.

Temperature (C) Temperature (C)
130
Julian Date
20 4-* t -t~[ |t >-H . | t i + . | i | i i | . | t I
224 225 226 227 228 229 230 231 232 233 234
Julian Date
Figure 3-29. Soil temperatures measured at depths of 0, 15 and 80 cm
in weighing lysimeters located at the University of
Florida, IREP from August 12 to 21, 1987.

Cumulative Evaporation (mm)
131
Julian Date
Figure 3-30. Hourly and cumulative water loss measured from
August 25 29, 1987 using the University of Florida,
IREP weighing lysimeters.

Depth ( m ) Depth ( m )
132
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-31. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on August 25, 27, and 29,
1987.

Temperature (C) Temperature (C)
133
Julian Date
Figure 3-32. Soil temperatures measured at depths of 0, 15 and 80 cm
in weighing lysimeters located at the University of
Florida, 1REP from August 25 and 29, 1987.

CHAPTER IV
MEASURING THERMAL DIFFUSIVITY OF SOILS
Introduction
The temperature distribution of heat within a body is governed by
Fourier's law of conduction. For a one-dimensional body with no heat
generation, the partial differential equation describing the transient
temperature response subjected to various boundary conditions is
(Equation 4-1)
3T
It
Dt
dZJ
T?
(4-1)
The thermal diffusivity (D^) is the property of a material which
determines the rate at which heat is propagated. The thermal
diffusivity is defined as the ratio of the thermal conductivity to the
volumetric heat capacitance (Equation 4-2).
Dt = -4~ (4-Z)
A composite material such as soil may have thermal properties which
vary with space and time due variation of the composition with time and
space. The factors affecting the thermal diffusivity of the soil are
1. soil type,
2. soil water content,
3. soil bulk density and
4. possibly soil temperature.
134

135
DeVries (1975) presented methods for calculating the thermal
properties of the soil based upon the various volume fractions of the
solid, liquid and air constituents of the soil. These equations
estimated the specific heat and thermal conductivity of various soils.
However, extensive tests for a range of soil types as well as moisture
contents and density have not been conducted. The DeVries method has
been shown to estimate the thermal properties within ten percent for
limited soils. Tollner et al. (1984) compared the experimental values
of the thermal conductivity for potting soils consisting of various
mixtures of sand and pine bark to those calculated. They achieved an
average deviation between experimental values and values calculated by
the DeVries method of approximately fifteen to twenty percent.
Objectives
The objectives of this study were to determine a method by which
the thermal diffusivity of the soil could be obtained and to compare
the measured data with that calculated by the DeVries method. A third
objective was to determine a technique by which the measured data could
be used in a simulation of the energy transfer within the soil.
Literature Review
Methods of determining the thermal conductivity or diffusivity of
a material are based upon measuring either the steady-state or
transient temperature response to a particular set of initial and
boundary conditions. Steady-state or equilibrium methods are based
upon Fourier's law of conduction for a one-dimensional homogeneous,
isotropic material (Equation 4-3).
q
(4-3)

136
Applying the steady-state method requires the accurate measurement of
heat flux between two points. The spatial gradient of the temperature
must also be measured accurately as well. This indicates that the
position of the two temperature sensors must be known. Reidy and
Rippen (1969) described specific procedures possible for measuring the
thermal conductivity using steady-state methods in detail. Some of
the advantages of using equilibrium techniques are:
1. simple mathematical solutions are used,
2. small test samples are suitable, and
3. liquid, solid or dry granular materials can be used.
Measurement of the thermal conductivity by steady-state procedures has
the following disadvantages.
1. Thermally induced moisture migration in the sample may cause
errors in measurement due to the non-homogeneity produced.
2. Several hours may be required for the material to reach
equilibrium conditions.
3. Loss of heat from the ends of the sample may cause
significant errors in the measurement of heat flux.
Transient techniques are based upon the solution of the governing
partial differential equation for heat transfer in a homogenous,
isotropic sample. Transient methods require temperature measurement at
a given location within the sample over a period of time. Advantages
of the transient procedure over the steady-state method are that rapid
results may be obtained and no direct measurement of heat flux is
required.
The solution of the governing partial differential equation for a

137
homogeneous, isotropic solid with uniform initial temperature placed in
a moving fluid of constant and uniform temperature can be expressed in
terms of the dimensionless temperature ratio, the Biot (Bi) and Fourier
(Fo) numbers as follows (Equation 4-4)
TU.t) Tb
* T(z,0) T = n5> c"(z) exP('anFo) (44)
iL
where an is the ntn eigenvalue of the following transcendental
equations for the case
of a
sphere,
an infinite cylinder
and infii
slab.
sphere:
Bi
= 1 -
ancot(an)
(4-5)
infinite cylinder:
Bi
= an
Jl(an)/Jo(an)
(4-6)
infinite slab:
Bi
" an
tan(an)
(4-7)
The transcendental equation for the infinite cylinder (Equation 4-6)
contains zero and first order Bessel functions of the first kind (Jq
and Jj, respectively).
The solutions to the transient heat equation are often presented
graphically in the form of semi-logarithmic plots of the temperature
ratio as a function of the Fourier number. A family of curves is
presented for a particular position within the body for various values
of the reciprocal of the Biot number (Ozisik, 1980). The Biot number
is the ratio of the product of the surface heat transfer coefficient
(hfo) and some characteristic length of the sample (L) to the thermal
conductivity of the sample material (A).

138
(4-8)
The Fourier number is the ratio of the elapsed time to a characteristic
time for the sample and represents a dimensionless time scale (Equation
4-9). The characteristic time of the sample can be defined as the
square of the characteristic length of the sample (L^) divided by the
thermal diffusivity of the sample.
For values of the Fourier number greater than approximately 0.2, only
the first term of the infinite series solution is necessary to provide
and accurate expression for the temperature ratio as a function of
time. Thus, equation (4-4) can be simplified to
6{z) = Cj(z) exp(-aj jj- t)
(4-10)
Note that this expression is a simple exponential function. Taking the
natural logarithm of both sides of equation (4-10) yields
ln[(z)] = lntcjOs)] (af S )t (4-11)
which is a straight line that is a function of the thermal diffusivity
of the material. For various positions within the sample, the line

139
shifts only by its intercept with the slope remaining constant.
Therefore, accurate placement of the temperature sensor is not critical
to obtaining an accurate measure of the thermal diffusivity.
Gaffney et al. (1980) discussed possible sources of error in the
determination of thermal diffusivity using this technique. A major
source of error was in the determination of the first eigenvalue (aj).
Since the eigenvalues are functions of the Biot number it is imperative
that the Biot number be determined accurately. Many times the Biot
number has been assumed to approach an infinite value. Gaffney et al.
(1980) stated that the Biot number should be at least 200 for the error
introduced by the assumption of an infinite Biot number to be
negligible. If the Biot number is less than 200, then a finite value
for the Biot number should be used to determine aj. The error
introduced was given as
(4-12)
1
Luikov (1968) presented an empirical relationship for determining the
first eigenvalue within 0.1 % for a finite Biot number.
(4-13)
The constants, I, s and (aj),,, are functions of geometry and are
presented in Table 4-2 for the cases of a sphere, an infinite cylinder
and an infinite slab.
Another factor to be considered in the measurement of the thermal

140
diffusivity is the thermally induced flow of water vapor. Cary (1966)
stated that thermally induced flow of water could occur to such an
extent that it was the predominant factor in mass transfer within the
soil, especially under dry soil conditions. Movement of water vapor in
the soil sample could induce errors in the measurement of the slope of
the transient temperature response if the evaporation or condensation
were occurring near to the temperature sensor due to the latent heat of
vaporization. Reidy and Rippen (1964) indicated that vapor movement
could be minimized by inducing relatively small temperature gradients.
A second transient method requires the use of a thermal
conductivity probe. The conductivity probe consists of a precision
resistance heating element and a small gauge thermocouple installed in
a stainless steel probe. The probe is inserted into the center of the
sample. A galvanometer is used to record the current passed through
the heating element and the thermocouple senses the temperature of the
sample. If the thermal conductivity probe were used in a moist porous
material, then care must be taken to prevent the vaporization of water
near the probe causing an increase in heat flux from the probe.
The steady state methods were not chosen due to the possible long
duration of a single test and the anticipated difficulties in
measuring heat flux and precise placement of the temperature sensors
within the sample. The transient methods were chosen because of the
short duration of the test and the lack of precise placement of the
temperature sensor. The short test duration may also reduce the effect
of vapor movement within the soil upon the determination of the thermal
diffusivity. The thermal conductivity probe was not used because of
the potential of inducing vapor flux in the vicinity of the probe. An

141
adaptation of the method presented by Gaffney et al. (1980) for
measurement of thermal diffusivity in fruits and vegetables was
utilized.
Procedure
Sample containers were constructed of 102 mm lengths of 25 mm
(I.D.) copper pipe. A hole was drilled midlength of the sample
container to permit insertion of a 36 gauge ANSI Type T (copper -
constantan) thermocouple after soil was placed in the container. One
end of the sample container was closed by soldering a 30 mm square
copper sheet to the end of the tube. A second copper sheet was secured
over the remaining end with silicone adhesive after the sample
container had been filled with the soil.
Using a core sampler, approximately 4 kg of a Millhopper fine sand
was obtained over a depth of one meter. The soil was then dried in a
convection drying oven at a temperature of 125C for a period of 24 h
prior to any sample preparation. Soil samples were prepared by
measuring the amount of dry soil to obtain a dry bulk density of
1300 kg/m^. Water was added to the oven-dried soil to achieve a water
content of 2, 4, 6, 8, 10, 12 or 15 percent (weight basis). Sample
containers were then filled with the pre-measured soil using manual
vibration to prevent voids from occurring within the sample. After the
containers were filled with the soil, the 30 mm square sheet was placed
on the remaining open end and sealed with silicone adhesive. A
36 gauge thermocouple was then inserted into the center of the sample
in a radial direction. Silicone adhesive was used to seal the opening
through which the thermocouple was inserted to prevent leakage of water
during the test. Three samples were prepared at each moisture

142
content. All samples were allowed to stand overnight to equilibrate to
room conditions.
A water bath was agitated by forcing compressed air through a
perforated baffle located in the bottom of the tank. Air pressure was
maintained at 83 kPa for all tests. The water temperature was
maintained approximately 3 C below the initial temperature of the soil
samples by a set of cooling coils and a proportionally-controlled
heating element. The bath temperature was controlled to within 0.1
of the set point. Each sample was weighed. The initial temperature
of the sample was recorded. The sample was placed in the water bath
until the sample temperature was within 0.2 C of the water
temperature. Tests typically required two to four minutes to complete.
Bath and sample temperatures were recorded approximately every 1 to 2
seconds using a Fluke 4520 digital voltmeter controlled by a DEC
PDP-11/23 minicomputer. The sample was removed from the water, and the
exterior surface was dried and weighed to determine if arty of the
seals had leaked during the test. If leakage occurred, the container
was opened, emptied and a new sample prepared.
After testing, the samples were allowed to equilibrate overnight
to room temperature. Three repetitions of the test were conducted for
each soil sample. Tests were also conducted for soil bulk densities pf
1500 and 1600 kg/m3. A total of nine tests were conducted for each
water content and bulk density treatments.
It was necessary to determine the convective heat transfer
coefficient (h^) for each testing session. This was achieved by
recording the transient temperature response of a copper cylinder with
the same dimensions as the soil samples. Since the thermal properties

143
of the copper were known, the convective heat transfer coefficient
could be calculated from the slope of the logarithmic plot of the
temperature ratio and the Fourier number.
Results and Discussion
The determination of the thermal diffusivity of each sample
required that the slope of the logarithm of the dimensionless
temperature ratio as a function of time be determined. The bath
temperature used in calculation of the temperature ratio was the time
averaged bath temperature over the length of the test. The logarithm
of the temperature ratio was then plotted against time (Figure 4-1) for
the sample. The slope of the line was obtained by linear regression
for the data excluding the initial and final transients. The values
for the regressions were 0.995 or greater This procedure was
conducted for each of the samples and the corresponding data for the
copper cylinder (Figure 4-2).
Using the solution of the transient heat conduction equation
(equation 4-4) and the transcendental equation for an infinite cylinder
(equation 4-6) the convective heat transfer coefficient was calculated
from the slope of the copper cylinder temperature response curve.
Values of the convective heat transfer coefficient ranged from 4400 to
5900 W/m^ K. Next, it was assumed that the Biot number for the soil
sample was infinite. Using the value of (ai), the radius of the
cylinder and the slope of the temperature response, an initial estimate
of the thermal diffusivity was determined from the following
(4-14)

144
An estimate of the thermal conductivity was needed to determine the
Biot number. This was obtained by calculating the volumetric heat
capacitance of the soil based upon the mass average of the various
constituents of the soil as described by DeVries (1975). The thermal
conductivity was determined by rearranging equation (4-2) and
substitution of the values of thermal diffusivity and volumetric heat
capacitance. The Biot number for the sample could then be calculated
and used to determine aj. The error (e) in the thermal diffusivity
associated with calculation of the eigenvalues was calculated using
equation (4-12). If the absolute value of the error was greater than
107 then the thermal diffusivity was calculated using the new
eigenvalue. This iterative procedure was repeated until the error
calculated by substitution of the previous value of aj for (aj)*, was
less than 107.
The values of thermal diffusivity for each of the nine tests for
each water content and density treatment were averaged to determine the
mean value of the thermal diffusivity (Figure 4-3). The error bars
shown in Figure 4-3 indicated the 95 % confidence limits associated
with the experimental measurement and were approximately 5 percent of
the mean for each. Thermal diffusivity was found to increase with
increasing density (Figure 4-3). However, the relative change in
diffusivity from a density of 1300 to 1500 kg/m3 was larger than the
change from 1500 to 1600 kg/m3. The thermal diffusivity increased with
water content to a maximum then decreased as water content continued to
increase. Initially the soil pore spaces were filled predominantly
with air. Air has a relatively low thermal conductivity and volumetric
heat capacitance. Contact area between soil particles was also limited

145
to many small points. As the water content increased, a film was
formed around the soil particle causing a continuous contact surface.
Increasing the water content increased the heat capacitance and the
thermal conductivity as well by displacing some of the air from the
pore spaces. The increase in thermal conductivity and contact area
occurred at a faster rate than did the increase in thermal capacitance.
At some point, the rate of increase in contact area and thermal
conductivity becomes slower than that of heat capacitance; therefore,
the thermal diffusivity begins to decrease from the maximum.
Sources of experimental error arise from measurement errors and
analytical errors. Analytical errors can arise from errors in
determination of the slopes of the temperature response for the copper
cylinder and the soil sample and estimates of the volumetric heat
capacitance. Inclusion of points from the nonlinear portions of the
temperature response curve for either the copper or the soil samples
could cause a change in the slope. A miscalculation of heat
capacitance of the soil could influence the final determination of the
thermal diffusivity, also. To determine the relative influence of
these type errors the solution routine was run and varying each of the
parameters over a range experienced during the tests. The slope of the
temperature response of the copper cylinder and heat capacitance had a
minor effect upon the final value of the thermal diffusivity (Figure
4-4). A variation of plus or minus 50 % in the heat capacitance
exhibited a plus or minus 2 % change in the thermal diffusivity. A
thirty percent reduction in the slope of the temperature response of
the copper cylinder caused a 2 % increase in the thermal diffusivity.
The deviation in thermal diffusivity decreased to zero as the slope of

146
the temperature response of the copper cylinder increased. This was
due to the fact that the Biot number approached infinity, thus causing
the eigenvalue for an infinite Biot number to be used. Thermal
diffusivity was affected by almost a one to one correspondence by
changes in the slope of the soil sample temperature response.
Another source of analytical error would be using the solution for
an infinite cylinder instead of that for a finite cylinder. The
analytical solution for a finite cylinder was determined by the
principle of superposition of the solution for an infinite cylinder and
an infinite slab (Equation 4-15).
9 s *cyl*slab (4"15)
Substitution and rearranging terms yields a linear equation with the
slope of the line consisting of a linear combination of the eigenvalues
for a slab and cylinders.
0 = c
cyl
cylcslab exPt'Dt( ^
(4-16)
This procedure was incorporated into the analytical analysis and the
thermal diffusivity determined. No significant differences were
obtained.
Errors in temperature measurement would comprise the major source
of experimental errors. Conduction error occurs due the sensor passing
through regions of differing temperatures. Heat can then be conducted
from the soil sample to the water along the thermocouple. Gaffney
et al. (1980) stated that conduction error caused by the surrounding
fluid could be minimized by using 36 ga or smaller thermocouple and by

147
locating the thermocouple near the center of the sample. These
precautions were both taken during the tests. Gaffney et al. (1980)
also demonstrated that conduction error did not significantly affect
the slope of the time-temperature curve.
A second source of error could arise from the movement of water
within the sample. The enhancement of the thermal conductivity due to
vapor movement in the soil occurs in response to concentration
gradients due to gradients in temperature within the sample and can be
calculated by (DeVries, 1975)
Dv 17 3ps
\ = hfg R^T dl
(4-17)
The apparent thermal conductivity is the sum of the conductivity for
conduction and the conductivity due to vapor movement (DeVries, 1975).
The apparent thermal diffusivity would be calculated by dividing the
apparent conductivity by the heat capacitance of the soil. Therefore,
the contribution of the vapor flow to the thermal diffusivity would be
Dtv
(4-18)
The temperature used in the calculation of the diffusivity due to vapor
movement was the time average temperature at the center of the sample.
Regression analysis was used to attempt to find the least squares fit
of the data using dry bulk density, volumetric water content and the

148
diffusivity due to vapor movement as independent variables. The best
fit was achieved with the regression equation linear in soil density
and.thermal vapor diffusivity with first and second order terms for the
volumetric water content (R2 = 0.805). However, at the 95 % confidence
level, the coefficient for the thermal vapor diffusivity was not
significantly different than zero. Therefore, the regression
coefficients were obtained for the water content (first and second
order) and the bulk density (first order) with an equally high
correlation (R2 = 0.799). These regressions were compared to the
experimental data and the values calculated using the DeVries method
(Figure 4-5). Neither of the regressions nor the DeVries method
provided a suitable estimate for the thermal diffusivity for all ranges
of density and water contents tested.
Conclusions
A transient method was used to measure the thermal diffusivity of
a Millhopper fine sand at three dry bulk densities and seven volumetric
water contents. The 95 % confidence limits were within 5 percent of
the mean for each sample. It was critical to obtain a good estimate of
the slope of the time-temperature response curve of the test sample to
achieve accurate values of the thermal diffusivity. Estimating the
slope of the time-temperature response for the copper cylinder and the
specific heat of the soil were not critical to achieving consistent
results. A length to radius ratio of eight to one (8:1) was sufficient
to analyze these soil samples using the solution for the infinite
cylinder. Superposition of the infinite cylinder and slab solutions to
obtain the solution for a finite cylinder was not required. Apparent
changes in the thermal diffusivity due to vapor movement within the

149
sample were not significant at the 95 % confidence level. A least
squares regression with first and second order terms for the volumetric
water content and a first order term for the dry bulk density had an
of 0.799 but did not provide a satisfactory fit to the data for the
range of density and water content tested. Thermal diffusivity was
determined by calculations presented by DeVries (1975) but did not
compare favorably with experimental data. A two-way interpolation
would probably provide the best implementation of this data into a
model. This data might also be used to determine parameters for use in
the DeVries method.

150
Table 4
1. Nomenclature and list of symbols
|L
an : nin eigenvalue for transcendental equations for
solutions to transient conduction equations
Bi : Biot number
C : volumetric heat capacitance
1L
cn : coefficient for the ntn term of series solution of
transient conduction equation
Dt : thermal diffusivity
Dtv : apparent enhancement to thermal diffusivity due to
water vapor movement
Dv : diffusivity of water vapor in soil air space
Fo : Fourier number
hfg : latent heat of vaporization
hfo : surface heat transfer coefficient
Jq : zero order Bessel function of the first kind
: first order Bessel function of the first kind
L : characteristic length
1 : half thickness of infinite slab
m : slope of ln(0) vs time response of sample
q : heat flux
R : radius of cylinder
Rw : ideal gas constant for water vapor

151
Table 4-
. Nomenclature and list of symbols (continued)
T : temperature
t : time
z : space coordinate
\ : thermal conductivity
Av : apparent enhancement to thermal conductivity due to
water vapor movement
6 : dimensionless temperature ratio
3ps
nj- : gradient of saturated vapor pressure with respect to
temperature

152
Table 4-2. Parameters for the empirical determination of the first
eigenvalue for solution to heat conduction equation for
a sphere, and infinite cylinder and an infinite slab.
Sphere
Infinite
Cylinder
Infinite
Slab
(l)co
7T
2.4048
tt/2
I
2.70
2.4500
2.24
s
1.07
1.0400
1.02

In 0
153
Time (s)
Figure 4-1. Semi-logarithmic plot of the dimensionless temperature
ratio as a function of time for a typical soil sample.

In e
154
Time (s)
Figure 4-2. Semi-logarithmic plot of the dimensionless temperature
ratio versus time for the copper test cylinder.

155
Volumetric Water Content (cm^/cm^)
Figure 4-3. Experimental values of the thermal diffusivity of a
Mi11 hopper fine sand as a function of volumetric water
content at various dry bulk densities.

Fractional Change in
Thermal Diffusivity
156
Figure 4-4. Effects of changes in experimental parameters upon the
values of thermal diffusivity determined from experimental
data.

Thermal DiffusivUy (10 rn^/a)
157
2.0
Bulk Density * 1600 kg/m^
i.e --
1.2 -
0.8 --
0.4--
Experimental
DeVries
Quadratic
Quadratic + Vapor
o.o
+
+
0.00
0.03
0.10
0.1 s
0.20
0.2S
2.0
1.6 --
1.2 -
0.8 --
0.4 --
0.0
Bulk Density 1500 kg/m^
Experimental Quadratic
DeVries Quadratic + Vapor
0.00
0.09
0.10
0.15
0.20
0.29
2.0
1.6 --
1.2
0.8 -
0.4--
0.0
Experimental
DeVries
Quadratic
Quadratic + Vapor
Bulk Density 1300 kg/m^
+
+
0.00
0.09 0.10 0.19 0.20
Volumetric Water Content (cm^/cm^)
0.29
Figure 4-5. Comparison of experimental values of thermal diffusivity to
calculated thermal diffusivity by regressions and the
DeVries method.

CHAPTER V
MODEL ANALYSIS
Introduction
Prior to using a model to simulate a given set of conditions, it
must be calibrated and validated. Validation of a model ensures that
the model responds to various stimuli in the same manner as the real
system would. This may be done by comparison of the simulated response
to experimental data or by analyzing the simulated response to changes
in various parameters used in the model. Model calibration requires
the use of experimental data so that parameters can be adjusted so that
the model simulates the response of the physical system as closely as
possible to a specified range of input conditions. The purposes of
this model analysis were to determine the validity of some of the
underlying assumptions in developing the model, to identify the
parameters most likely in need of calibration and to begin the process
of calibration for a limited set of experimental data.
Validation
Data used in the validation process were collected from
January 5, 1987 to February 13, 1987. Hourly ambient weather data were
not available from the AWARDS system during August, 1987 due to
equipment failures and were not included in the validation process.
Table 5-1 shows the test names and their inclusive dates used in
validation. The variation of soil water potential and the hydraulic
diffusivity with water content was generated using the Van Genuschten
method from data for a Mi 11 hopper fine sand published in Carlisle et
158

159
al. (1985) (Figure 5-1). The volumetric water content of the soil
that yielded specified values of soil water potential were tabulated
for various depths throughout the soil profile and corresponded to the
different horizons of the soil. The water content for a given water
potential was averaged for all depths to yield the relationship between
water potential and volumetric water content characteristic of a
uniform soil profile.
Hourly ambient air temperature, relative humidity, wind speed and
rainfall were used as input data for the boundary conditions of the
model. The relative humidity used as input data was generated using a
dewpoint temperature estimated from minimum daily temperatures as
described in Chapter III. Hourly net radiation incident upon each of
the lysimeter surfaces was included in the weather inputs as well. The
meteorological data describing the boundary conditions for each all
tests conducted are shown in Appendix B. Core samples were obtained to
determine the initial volumetric water content at the beginning of each
test. Subsequent moisture data were obtained either by core samples or
a dual probe density gauge. Due to the intrusive nature of the
moisture measurements, data were usually recorded only at the beginning
and end of each test. In test 72 only the initial profile was
obtained due to several days of rain ending the test prematurely. The
rainfall required that the evacuation system to be run during the rain
to prevent overloading the loadcells.
In general, the model predicted the diurnal pattern of the
cumulative and hourly evaporation fairly well. Experimental and
simulated maximum hourly evaporation rates occurred around midday for
all tests; however, measured maximum hourly evaporation rates usually

160
occurred 1 to 2 h prior to the simulated maximum. Measured peak
evaporation rates were generally higher than the simulated values but
were shorter in duration than the simulated maximum evaporation rates.
Cumulative evaporation was under-estimated for three of the four
validation data sets. Simulated cumulative evaporation exceeded the
measured water loss during test 73. Since the simulated total water
loss from the soil was under-estimated, the simulated water content
profiles did not agree exactly with measured water content profiles.
However, simulated water profiles indicated movement of water toward
the surface in response to the evaporative demand at the surface as
well as downward redistribution due to gradients in soil water
potential. The simulated soil temperature profiles followed the trends
expected with the soil surface temperature exhibiting the maximum
amplitude in the diurnal cycle. The amplitude of the temperature wave
decreased with depth until a constant temperature was obtained at a
depth of approximately 60 cm. Maximum surface temperatures were
approximately 3 C higher than experimental maximum temperatures, while
simulated minimum surface temperatures were an average of 2 C lower
than measured. Simulated maximum and minimum soil surface temperatures
led ambient air temperatures as was observed in the experimental data.
The model was run using the values of net radiation, wind speed
(Figure B-7), relative humidity and air temperature (Figure B-8) as the
boundary conditions. Simulations were conducted for the north and
south lysimeters using measured values of the soil temperature and
water content as initial conditions. Simulated evaporation rates had a
diurnal variation with a maximum evaporation rate water occurring
around midday (Figure 5-2) then decreased to rates near 0 mm/h

161
overnight. Short periods in which condensation occurred were simulated
for both lysimeters during the early morning hours of January 9. A
maximum evaporation rate of 0.4 mm/h was simulated on the first day of
the test then decreased during subsequent days of the test. The
experimental maximum hourly evaporation rate of approximately 0.8 mm/h
for the north lysimeter and 0.9 mm/h for the south lysimeter occurred
on January 6. Since simulated evaporation rates lagged behind the
measured values, the measured daily maximum evaporation rate probably
occurred prior to the beginning of the test on January 5 and was most
likely to have been at least as high as those on the second day of the
test. It was noted that the lag time between the simulated and
experimental daily maximum evaporation rates was approximately the same
for both the north and south lysimeters. Initiation of water loss
appeared to occur at approximately the same time for both the
experimental water loss and the simulated results. However,
experimental data showed that very little time lapsed between the
initiation of evaporation and the occurrence of the peak water loss.
On the other hand, the simulation resulted in a very gradual increase
in the evaporation rate beginning at approximately the same time as the
experimental results.
Comparison of the simulated and experimental cumulative
evaporation for both lysimeters indicated that the most of the water
lost from the lysimeters during the day occurred during the initial
evaporation phase during the morning (Figure 5-3). The simulated water
loss during the first day corresponded fairly well with the measured
water loss for both the north and south lysimeters. At the end of
the first day, simulated water loss was approximately 0.5 mm less than

162
the measured loss from the north lysimeter. However, since the model
did not simulate the initial high rate of evaporation observed for both
lysimeters, the discrepancy between simulated and observed water loss
increased on the second day, January 6. Beginning on January 7, the
model predicted the daily water loss very accurately for the north
lysimeter. This was possibly due to the water not redistributing
sufficiently toward the soil surface overnight to provide a wet surface
for evaporation. Thus water was evaporating below the surface and
diffusing through the soil until it reached the soil-atmosphere
interface. The increased resistance to vapor flow then reduced the
total water vapor leaving the soil. However, the water in the south
lysimeter had moved toward the surface and replenished the soil and
provided water for evaporation at the soil-atmosphere boundary.
Therefore, water evaporated very rapidly early in the morning until the
soil surface dried causing the evaporation rate to decrease. The model
did not predict this flush of water from the soil, thereby under
predicting the daily water loss from the south lysimeter. From January
7 to 10, the model predicted total water loss from the soil as well as
hourly water loss from the north lysimeter very well.
Agreement between the simulated and experimental soil surface
temperatures was very good throughout test 70 (Figure 5-4). The
simulated maximum surface temperature on January 5 was approximately
4 C and 2 C higher than the measured maximum for the north and south
lysimeters, respectively. Simulated temperatures were within 1 C of
measured surface temperatures for the duration of the test. The
excessive simulated surface temperatures were indicative of the reduced
evaporation which was simulated as well. Had the evaporation rate been

163
higher, more heat would have been partitioned into latent heat at the
surface rather than sensible heat gain. Note that the magnitude of the
discrepancy in surface temperatures corresponds fairly closely with the
discrepancy in cumulative evaporation during the first day.
Simulation of the change in water profile showed that water was
primarily lost from the surface with some redistribution of water
occurring overnight (Figure 5-5). The soil surface water content
decreased very rapidly from 15 and 17 percent for the north and south
lysimeters, respectively to approximately 11 percent after 26 h. The
soil surface water content was approximately the same each day at
noon; however, water had been moved from the soil below to maintain
that water content. The simulated profiles also showed some
redistribution of water downward in response to hydraulic gradients.
Simulated water profiles were in general agreement with the profiles
measured in the lysimeters at the conclusion of the test on January 9,
1987 (Figure 5-6). The experimental profiles showed very little
downward movement of water while the simulated profile approached a
more linear distribution. The final simulated water contents at the
soil surface were higher than those measured as anticipated due to the
lower cumulative water loss from the soil.
Comparison of data from test 71 to the simulation results showed
similar trends. Meteorological input data for the simulations are
shown in Figures B-9 and B-10. The simulated cumulative evaporation of
water from the lysimeters compared more favorably with measured water
loss (Figure 5-7). Both the model and measured evaporation indicated
that slightly over 1 mm of water had been lost from each of the two
lysimeters. However, the cumulative water loss from the south

164
lysimeter was observed to be approximately 3 mm of water while the
north lysimeter had lost a total of approximately 4 mm of water at the
end of the second day. Meanwhile, the simulated cumulative evaporation
of water from the soil after the same period time was approximately
2 mm for the south lysimeter and 2.5 mm for the north lysimeter south
lysimeter. During the third day of the experiment, water loss from
both of the lysimeters was observed to be 0.5 mm or less while the
simulation showed approximately 1 to 1.5 mm of water evaporated from
the lysimeters. Measured maximum hourly evaporation rates occurred
approximately 2 to 4 h before simulated maximum rates (Figure 5-8).
The data for the south lysimeter was much smoother than that obtained
from the north lysimeter with most of the variation during the night
time hours. In most cases, the variability of the measured hourly
water loss from the north lysimeter occurred with subsequent equal and
opposite magnitudes with a zero net weight loss or gain. This could
have been wild animals, such as cats, crossing the lysimeter during the
night. Indications of the presence of animals was found on numerous
occasions during several of the experiments. Discrepancies between
simulated and measured hourly evaporation had similar characteristics
as those that occurred in test 70. The agreement between simulated
and experimental soil surface temperatures were not as close during
this test as was noted in test 70 (Figure 5-9). However, the simulated
surface temperatures corresponded with decreased evaporation. The
simulated daily maximum temperatures were 4 to 6 C above measured
values. If the evaporation rates had been higher, the latent heat
required would have reduced soil surface temperatures. Maximum
observed and simulated soil surface temperatures increased as the test

165
progressed and more water was lost from the soil surface. The
simulation indicated that the soil water content at the surface was
approximately 0.07 cm3/cm3 at 1200 on January 12, and decreased by
approximately 0.02 cm3/cm3 on January 13 and 14 (Figure 5-10). The
soil at a depth of 40 cm remained at a constant .095 cm3/cm3. The
simulation also indicated some vertical redistribution of water in the
profile. Measured water content profiles did not indicate the
occurrence of the vertical redistribution of water to the same extent
as the simulation (Figure 5-11). Comparison of the simulated water
profile to the measured profile showed that more water was lost from
the area above 5 cm for the simulation while the profile in the
lysimeters showed that water was lost from the top 20 cm.
The data for test 72 was recorded under partly cloudy, windy
conditions with the test being terminated prematurely due to rainfall.
Daily maximum net radiation during test 72 was approximately 400 W/m3
which was comparable to the net radiation levels measured during
previous tests (Figure B-ll). The variation of the net radiation
during test 72 was more irregular than in previous tests. Minimum net
radiation was higher than that of previous tests indicating the
presence of cloud cover during the day. According to field notes taken
during the experiment, gusty wind conditions prevailed during test 72.
However, the wind speed data (Figure B-ll) was not characteristic of
gusty wind conditions. Data indicated short periods of relatively high
gusts, but dropped to a constant speed of 0.5 m/s for long periods of
time. One would expect the wind speed data to exhibit characteristics
similar to that recorded during earlier tests. The long periods of
constant wind velocity were not characteristic of the previous tests

166
and raised some question as to its reliability. Relative humidity
ranged from approximately 50 to 100 percent and the air temperature
ranged from 7 to 22 C (Figure B-12). Approximately 2.5 and 2 mm of
water evaporated from the north and south lysimeters, respectively, on
February 3, 1987 (Figure 5-12). Simulated cumulative water loss was
within 0.3 mm of that measured from the south lysimeter after
approximately 10 hours had elapsed from the initiation of the test,
while simulated evaporation was 1 mm below measured water loss from the
north lysimeter. Experimental data indicated that the rate of water
loss from both lysimeters decreased at sunset to zero overnight (Figure
5-13). However, the simulation predicted evaporation to continue at a .
very slow rate between sunset on February 3 and sunrise on February 4
at which time the simulated evaporation rate increased. This caused
the simulated cumulative evaporation to surpass the observed cumulative
water loss for the south lysimeter. Simulated and observed evaporation
for the north lysimeter commenced at approximately the same time and at
the rate of evaporation. However, rate of water loss from the
lysimeters was observed to become zero at approximately 1500 on
February 4 while the simulated water loss continued at a relatively
high constant rate until near sunset at which time it approached zero
as well. Simulated soil surface temperature was always higher than
measured surface temperatures (Figure 5-14). The excessive simulated
surface temperature during the nighttime hours led to a small but
positive difference in the vapor concentration between the soil
surface and the atmosphere (Figure 5-15) and would explain the
continued water loss at night predicted by the model. The vapor
density deficit would also indicated that evaporation should be

167
occurring at a relatively high rate, but due to the low wind speed
corresponding to the maximum difference in water vapor density,
evaporation rates were relatively low. When the rapid increase in wind
speed occurred, the vapor density deficit at the soil surface was
declining but still relatively large. This caused the abrupt increase
in evaporation rate. This would tend to indicate that the model was
was relatively sensitive to wind speed due to the direct influence of
wind speed upon the surface transfer coefficients.
The initial water profile was somewhat different for test 72 when
compared to the previous test in that the soil surface had a higher
water volumetric water content than that below the soil surface (Figure
5-16). Drying the soil surface resulted in the same shape profile at
the end of the simulation as was typical of the initial conditions of
other tests thus far. No detectable differences in the simulated water
contents occurred below a depth of 20 cm indicating that all of the
water lost from the soil was evaporated from the top 20 cm of the soil
profile. Final soil water profile measurements were not obtained due
to heavy rains which prematurely ended the test.
The simulated evaporation of water from the lysimeters for
conditions recorded during test 73 exceeded the evaporation measured in
the weighing lysimeters (Figure 5-17). Initial high rates of water
loss observed for both lysimeters were simulated fairly well as
indicated by the comparison of the cumulative evaporation and the
hourly evaporation curves (Figure 5-17 and 5-18, respectively). The
measured evaporation rate decreased to about zero during the night
then increased again at sunrise; however, simulated rates of
evaporation decreased after sunset but remained small but positive

168
values. Both simulated and experimental daily evaporative losses
decreased each day as the soil surface dried. The model simulated the
initial high evaporation rate on the first morning of the test,
February 9, for both lysimeters quite well. However, the apparent
early morning burst of evaporation was not simulated on subsequent days
of the test. The excessive rates of evaporation simulated during this
experiment can be explained by examining the simulated soil surface
temperature and the water vapor density difference. Simulated surface
temperatures were well in excess of observed values with maximum
deviation of approximately 15 C (Figure 5-19). The high temperatures
in turn cause a large difference in the water vapor concentration
driving the flow of water vapor (Figure 5-20). Boundary conditions
which influenced the soil surface temperatures are the ambient air
temperature, net radiation and the wind speed. Net radiation was
relatively high throughout the duration of this test. The maximum net
radiant heat flux upon the soil surface was 450 W/m^ (Figure B-14)
with a minimum of approximately -75 W/m^. Air temperature and the wind
speed both affect the rate of heat transfer from the soil surface via
convection. Air temperature was 20C or less for most of the test
(Figure B-15), thus providing a large potential difference for heat
transfer between the soil and the atmosphere. Wind velocity directly
affects the surface heat transfer coefficient. The variation of wind
speed as function of time was not at all typical of that expected or as
compared to wind speeds recorded during tests 70 and 71. Since the
data for this test was very similar to that used in test 72, it was
highly questionable.

169
The validation process indicated that the model was simulating the
processes intended. Overall performance of the model to this point
were satisfactory. The general trends indicated by experimental data
of increasing daily maximum surface temperatures as daily evaporation
rates decreased were simulated by the model. Deficiencies which were
repeated during each of the tests, were the model's inability to
simulate the high rate of evaporation which occurred most mornings
after the soil had apparently been rehydrated by diffusion of water
from below or by condensation upon the surface from the atmosphere.
This discrepancy could have been caused possibly by errors in hydraulic
properties of the soil near the surface or by the surface transfer
coefficients. Incorrect values of hydraulic diffusivity could limit
the ability of the soil to replenish the water supply at the soil
surface. The fact that very little redistribution of water below a
depth of approximately 20 cm in the measured water profiles indicated
that the hydraulic diffusivity of the soil below the 20 cm depth was
less than that in the region between the soil surface and 20 cm.
Experiments conducted by Jackson (1973) indicated that a dry surface
layer could greatly reduce evaporation of water from the soil. This
was indicated by Jagtap (1986) during evaporation studies and
subsequent modeling. Jagtap's resistance model showed that the soil
resistance increased as cumulative evaporation increased. Jackson
(1973) also stated experimental measurements of hydraulic diffusivity
were generally accepted with deviations of up to a factor of two. A
variation of this magnitude could affect the simulated evaporation of
water significantly. This accepted level of error in values of
hydraulic diffusivity in addition to the fact that the data was

170
obtained from the literature for a Mi 11 hopper fine sand with similar
vegetative cover as that filling the lysimeters and not measured added
to the uncertainty of the values used in the model. The surface
transfer coefficients were implicated by the fact the model seemed to
simulate the evaporation of water when the soil was limiting the
movement of water from the soil but not when water was apparently
available at the surface with little or no resistance to diffusion into
the atmosphere.
Sensitivity Analysis
The experimental values of thermal diffusivity of the soil
obtained as described in chapter 4, were incorporated into the model to
insure that the thermal properties of the soil used in the model were
as close to those present in the lysimeter as possible. Thermal
conductivity was determined by multiplication of the experimental
values of the thermal diffusivity and the volumetric heat capacity.
Volumetric heat capacity was determined as previously described using
equation (2-35). Differences in the simulated evaporation rates and
soil temperature and water content using values of thermal conductivity
determined by the DeVries method and experimental data were insigni
ficant.
The data from the south lysimeter obtained during test 70
(January 5 to 10, 1987) was used as the calibration data set primarily
due to the quality of the evaporation data. The hourly evaporation
curve was smooth with few jumps and unexplainable spikes. The wind
speed, air temperature, net radiation and relative humidity data for
this particular test appeared to be high quality data as well.

171
The procedure for calibrating the model was to use multipliers to
vary the desired parameters of the model. This calibration focused
upon the effect of the hydraulic diffusivity of the soil (Dj_) and the
surface heat (h^) and mass (hm) coefficients primarily due to the
uncertainty of their values used during the validation phase of
modeling. A sensitivity analysis was performed by running the
simulation using several values of the multiplier for hydraulic
diffusivity, then selecting a value for the remaining calibration which
seemed to provide the least deviation between evaporation rates and
soil surface temperature. Since the calculation of the two surface
coefficients were derived from the same basic equations and are
proportional to each other, then a single multiplier was utilized for
both the heat and mass transfer coefficients. All other soil
properties such as heat capacity, thermal conductivity, soil water
potential and diffusivity of water vapor were allowed to vary according
to the temperature and water content of the soil as described in
Chapter II. This sensitivity analysis and calibration procedure was
used primarily to establish limits of values of the multipliers for
more precise calibration in future research efforts.
Sensitivity analysis for the effect of varying the hydraulic
diffusivity upon evaporation and soil temperature was conducted first.
Multipliers for the hydraulic diffusivity were 0.5, 2.0 and 5.0 and
shifted the relationship of diffusivity to water content vertically and
did not make any shifts with respect to water content (Figure 5-21).
Very little difference was seen in the simulated cumulative evaporation
using the hydraulic diffusivity multipliers of 2 and 5 until the
beginning of the third day of the simulation (January 7), at which time

172
the cumulative water loss using the multiplier of 5 was slightly higher
than multiplier of 2 (Figure 5-22). Using a multiplier of 0.5 reduced
the total amount of water lost from the soil. After a 60-h simulation,
the cumulative water loss using the multiplier of 0.5, was
approximately 1 mm lower than the results for multipliers of 2.0 and
5.0. The point at which the cumulative evaporation curves first
separated for indicated the point at which the soil first became the
limiting component in the evaporation of water. This could better be
seen by comparing the evaporation rates for each of the multipliers
(Figure 5-23). The evaporation rates for the two highest multipliers
were identical. The curve for the multiplier of 0.5 first deviated
from the higher curve approximately 3 h after the beginning of the
simulation on January 5. All three curves were identical until
approximately noon, when the evaporation rate for the simulation for
the lowest multiplier decreased as the evaporation rate for the higher
multipliers continued to increase. This was due to the inability of
the soil to maintain sufficient upward flux of water to meet the
evaporative demand at the surface, therefore the rate of evaporation
decreased. Overnight the two curves rejoined, until evaporation began
again the third morning of the simulation. Hourly evaporation rates
coincided until approximately 1000 at which point soil with the lower
hydraulic diffusivity experienced its maximum evaporation rate then
began to decrease. Note the decreased time required for the peak
evaporation rate to occur on the second morning as compared to the
previous day. This was due to the decreased water available at the
soil surface each day (Figure 5-24). The simulated water content at
the soil surface indicated a diurnal pattern of drying during the day

173
and rewetting overnight. The soil with the lowest hydraulic
diffusivity experienced the largest variation in surface water content.
The soil dried to the residual water content (0.005 cm^/an^) then
recovered during the night. The overnight rewetting was not sufficient
to raise the water content to the same level at which the simulation
had started. The maximum soil water content continued to decrease each
day as more water was removed from the lower depths. The amplitude of
the diurnal variation of surface soil water content decreased as the
hydraulic diffusivity increased indicating the increased capacity of
the soil to redistribute water toward the surface. The soil recovered
to the same water content during the night using multipliers of 2.0 and
5.0. This is the reason that the evaporation rates were identical for
the two calibration runs.
The fact that increasing the hydraulic diffusivity alone did not
achieve the required increase in the simulated evaporation indicated
that the hydraulic diffusivity was not the only parameter in the model
limiting evaporation. This was also supported by the fact that the
initial high rate of evaporation that was noted in the experimental
data did not occur in the simulation results even when the surface
water content was almost the same as on the previous days. The effect
of the boundary conditions upon the simulation of evaporation,
primarily the heat and mass transfer coefficients, was investigated.
During the sensitivity analysis for the heat and mass transfer
coefficients, the multiplier for hydraulic diffusivity was fixed at 2.0
and a single multiplier was applied to both surface transfer
coefficients. The multiplier was valued at 0.5, 2.0 and 5.0 and the
simulation run using the data for the south lysimeter from January 5

174
to 8. The cumulative evaporation increased as the value of the
transfer coefficient multiplier increased (Figure 5-25). The
cumulative evaporation for the lowest values of transfer coefficients
matched the initial phase of the cumulative evaporation fairly well,
but then around sunset, the experimental data indicated a continued
slight water loss from the soil, while the simulation showed no water
loss overnight. Evaporation during the period following the first
night showed the model significantly under-predicted the cumulative
evaporation. The simulations for the multipliers of 2.0 and 5.0 over
predicted evaporation during the first day of the experiment, while at
the end of the second day, the cumulative evaporation simulated using
the multiplier of 2.0 matched experimental data fairly well. Since the
cumulative loss had been over-predicted on the first day but was
agreement by the end of the second day, indicated that the water loss
for the second day had been under-estimated. The error between
experimental and simulated cumulative evaporation remained
approximately the same from the first to second day using a multiplier
of 5.0. Examination of the hourly evaporation rates indicated that as
the multiplier for the surface transfer coefficients increase, the
hourly evaporation rates looked more like the experimental data (Figure
5-25) showing that early in the day, the boundary coefficients or the
method in which the boundary conditions for the transfer of vapor was
modeled was limiting the evaporation of water when water was available
at the surface. Increasing the heat and mass transfer coefficients
caused the evaporation rate to exhibit the type of behavior
demonstrated by the experimental rates. A high rate of evaporation

175
occurred early in the day then decreased as the soil began to limit the
rate of movement of water vapor to the atmosphere.
Soil surface temperatures behaved as expected in response to the
changes in surface transfer coefficients. Using a decreased surface
transfer coefficients caused the diurnal variation of soil surface to
increase due to decreased heat transfer to the atmosphere and
conversion to latent heat (Figure 5-27). As the transfer coefficients
increased, the maximum soil surface temperature decreased. A
multiplier of 2.0 produced a lower maximum soil surface temperature
with maximum and minimum temperatures occurring at the same time as the
experimental data. Increasing the surface transfer coefficients
increased the effect of latent heat removal upon the soil surface
temperature. Note that the simulated soil surface temperature
initially decreased by approximately 2 C due to increased latent heat
transfer. This decrease caused the maximum soil temperature to occur
approximately two hours after the experimental maximum. On the second
and third days of the simulation, the rate of increase of the surface
temperature decreased at the same time that the maximum rate of
evaporation occurred.
The effect of surface transfer coefficients upon surface soil
water content was inverse to that of the hydraulic diffusivity. As the
transfer coefficients were increased, the amplitude of the diurnal
variation surface water content increased (Figure 5-28). This was due
to the increased rate of water removal from the soil caused by
increasing the mass transfer coefficient. The soil then redistributed
water toward the surface. The diurnal variation of water content
content penetrated the soil to a depth of approximately 5 cm using a

176
multiplier of 2.0 for the transfer coefficients and the hydraulic
diffusivity (Figure 5-29). The variation of water content exhibited
characteristics very similar to the variation of temperature with time
and space. The amplitude of the variation decreased with depth as was
delayed with depth as well. The water content at a depth of 5 cm
steadily decreased as time progressed with very little diurnal cycle of
drying and rewetting.
Summary
The model as originally presented required calibration to allow
the model to adequately describe the transport of water and energy in
the weighing lysimeters. The experimental data exhibited periods of
high rates of evaporation early in the morning immediately following
sunrise. The model as originally described adequately described soil
temperature. It also simulated the evaporation of water from the soil
during conditions in which the soil was the rate-limiting component of
the model, but failed to simulated adequately evaporation rates which
were characteristic of surface-limiting conditions. Possible reasons
investigated were distribution of water to replenish the water supply
for evaporation and the boundary heat and mass transfer coefficients.
The model also showed more downward movement of water than was
measured in the lysimeters. This was probably due to the existence of
two layers of soil with very different hydraulic diffusivity
properties. This was supported by the experimental measurements of
soil water content in which very little change in water content was
noted below a depth of approximately 20 cm and most of the reduction of
water content occurring between the soil surface and a 20 cm. The
simulation of the distribution of water would probably be improved by

177
incorporating at least two zones of soil with different hydraulic
properties. According to Carlisle et al. (1985), the A-l horizon of
the Mill hopper sand used in the model had a saturated hydraulic
diffusivity approximately twice that used in the uniform profile. If
this were incorporated into the model, water would be supplied to the
surface at a relatively high rate until the water content in the top
horizon decreased to the point of requiring water from below.
Calibration and sensitivity analysis of the surface transfer
coefficients indicated that the mechanism by which water is removed
from wet soil surface was not modeled adequately. By increasing the
mass transfer coefficient, the model approached simulating the initial
high rate of evaporation, but caused the soil surface temperature
showed a decrease because of the increased requirement of latent
energy. The inadequacy appeared to be accentuated following a light
dew where a slight film of water was on the soil surface. This water
was then intercepting radiant energy which would normally reach the
soil surface. Perhaps, the thin film of water directly on the surface
was evaporating due to direct absorption of the radiant energy onto the
soil surface resulting in no change in soil surface temperature.
Future improvements in the model should incorporate the ability
to utilize a soil profile with a minimum of two sets of hydraulic
characteristics and should investigate the mechanism of water loss from
the wet soil surface. One possible approach may be to assume that the
latent heat required to evaporate water from the first node of the
model is provided by the radiant energy at the surface rather than from
the sensible heat available in the soil.

178
Table 5-1. Validation test names and inclusive dates.
Calendar Dates Julian Dates
Test Name
Beginning
Ending
Beginning
Ending
70
Jan. 5
Jan. 9, 1986
5
9
71
Jan. 11
Jan. 14, 1987
* 11
14
72
Feb. 3
Feb. 5, 1987
34
36
73
Feb. 9
Feb. 13, 1987
40
44

Water Tension (
179
Figure 5-1.
Soil water potential and hydraulic diffusivity as a
function of volumetric water content for a Mi11 hopper
fine sand (Carlisle et al., 1985)
Hydraulic Diffusivity (m^/s)

Evaporation Rate (mm/h)
180
Figure 5-2.
Hourly simulated and experimental evaporation rates for
the north and south lysimeters during test 70 (January 5
- 10, 1987).

Cumulative Evaporation (mm) Cumulative Evaporation (mm)
181
12
10
8
6
4
2
0
5 6 7 8 9 10
Julian Date
Figure 5-3. Comparison of simulated and experimental cumulative
evaporation rates for test 70 (January 5 10, 1987).

Temperature (C) Temperature (C)
182
Figure 5-4. Simulated and experimental soil surface temperatures
during test 70 (January 5-8, 1987).

Depth ( m ) Depth (
183
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm**)
South Lysimeter
Figure 5-5. Simulated volumetric water content as a function of
depth for various times during the simulation.
(January 5-8, 1987).

Depth ( m ) Depth (
184
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-6
Initial and final (simulated and experimental)
volumetric water content as a function of depth for test
70 (January 5-8, 1987).

Cumulative Evaporation (mm) Cumulative Evaporation (mm)
185
Julian Date
Figure 5-7. Comparison of simulated and experimental cumulative
evaporation for both lysimeters during test 71 (January
11 14, 1987).

Evaporation Rate (mm/h) Evaporation Rate (mm/h)
186
Figure 5-8. Comparison of simulated and experimental hourly
evaporation rates for test 71 (January 11 14, 1987).

Surface Temperature (*C) Surface Temperature (C)
187
Julian Date
Figure 5-9. Comparison of soil temperatures measured at the soil
surface during test 71 (January 11 14, 1986) to
simulated values.

Depth ( m ) Depth (
188
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-10. Simulated volumetric water content as a function of
depth for various times during the simulation.
(January 11 14, 1987).

Depth ( m ) Depth ( m )
189
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-11. Initial and final (simulated and experimental)
volumetric water content as a function of depth for test
71 (January 11 14, 1987).

Cumulative Evaporation (mm) Cumulative Evaporation (mm)
190
Julian Date
Figure 5-12. Comparison of simulated and experimental cumulative
evaporation for both lysimeters during test 72 (February
3 5, 1987).

Evaporation Rate (mm/h) Evaporation Rate (mm/h)
191
Figure 5-13. Hourly simulated and experimental evaporation rates for
test 72 (February 3-5, 1987).

Temperature (C) Temperature (C)
192
Figure 5-14. Simulated and experimental soil surface temperatures
during test 72 (February 3-5, 1987).

Vapor Density Deficit (kg/m3)
( Pv1 Pva )
193
Julian Date
Figure 5-15.
Simulated difference between the water vapor concen
tration at the soil surface and ambient air during test
72 (February 3 5, 1987).

Depth ( m ) Depth ( m )
194
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm*5/cm^)
South Lysimeter
Figure 5-16
Simulated volumetric water content as a function of
depth at various times during test 72 (February 3-5,
1987).

Cumulative Evaporation (mm) Cumulative Evaporation (mm)
195
Figure 5-17. Cumulative simulated and experimental evaporation rates
for test 73 (February 9 13, 1987).

Evaporation Rota (mm/h) Evaporation Rota (mm/h)
196
Figure 5-18. Hourly simulated and experimental evaporation rates for
test 73 (February 9 13, 1987).

Temperature (C) Temperature (C)
197
Figure 5-19. Comparison of experimental and simulated soil surface
temperatures at the soil surface during test 73
(February 9-13, 1987).

Vapor Density Deficit (kg/m^)
( Pv1 ~ Pva )
198
Julian Date
Figure 5-20. Simulated difference between the water vapor concen
tration at the soil surface and ambient air during test
73 (February 9 13, 1987).

Hydraulic Diffusiviiy (m^/s)
199
Volumetric Water Content (cm3/em3)
Figure 5-21. Relationship of hydraulic diffusivity to water content
as used in the validation of the couple heat and mass
transfer model and calibration factors of 0.5, 2.0 and
5.0.

200
Figure 5-22. Simulated cumulative evaporation resulting from
multipliers of 0.5, 2.0 and 5.0 for the hydraulic
diffusivity.

Evaporation Rate (mm/h)
201
Figure 5-23. Simulated hourly evaporation rates resulting from
calibration multipliers for 0.5, 2.0 and 5.0 for the
hydraulic diffusivity.

202
Figure 5-24. Simulated diurnal variation of volumetric water content
of the soil surface for various multipliers for the
hydraulic diffusivity.

203
Figure 5-25. Cumulative evaporation resulting from the calibration
multipliers for the boundary heat (ty,) and mass (hm)
coefficients (Dl multiplier 2.0).

Evaporation Rate (mm/h)
204
Figure 5-26. Simulated hourly evaporation rates using a calibration
multiplier for Dl (2.0) and various values for the
surface transfer coefficient multipliers.

205
Figure 5-27. Simulated soil surface temperature resulting from
various values of surface transfer coefficient
multipliers.

206
Figure 5-28. Simulated surface water content using various
multipliers for the heat and mass transfer coefficient
and a multiplier for hydraulic diffusivity (2.0).

Simulated Water Content (cm^/cm^)
207
Figure 5-29. Simulated diurnal variation of soil water content at
depths of 0, 2 and 5 cm using a multiplier of 2.0 for
the hydraulic diffusivity (D^), heat (h^) and mass (hm)
transfer coefficient.

CHAPTER VI
SUMMARY AND CONCLUSIONS
A model was developed using three partial differential equations
to describe the conservation of energy, water and water vapor in a one
dimensional soil system. It was assumed that the liquid and vapor
phases of the soil water were in equilibrium within the soil air space.
State variables for the system were the soil temperature, volumetric
water content, water vapor concentration, and the rate of change of
water from the liquid to the vapor phase. All state variables as well
as soil properties were variable with space and time. An alternating
direction finite difference technique was utilized to numerically solve
the system of equations subjected to surface boundary conditions which
varied with time. Transport properties for the soil were calculated
based upon relationships presented in the literature.
Two weighing lysimeters were constructed and filled with a uniform
soil profile of Mi11 hopper fine sand. The output of four 4545 kg
loadcells was monitored to determine changes in weight due to water
loss over time. Other data recorded for each lysimeter were soil
temperatures at various depths and net solar radiation impinging upon
the soil surface. Volumetric water content was measured over the depth
of the lysimeter at the beginning and end of evaporation trials using
core samples and a dual probe density gauge. The lysimeters were
estimated to have a sensitivity of 0.02 mm of water. Soil properties
measured for the soil contained in the lysimeter were
1. dry bulk density as a function of depth,
208

209
2. soil porosity as function of dry bulk density and
3. thermal diffusivity as a function of volumetric water content
and bulk density.
Weather data for model validation was obtained from a weather station
adjacent to the lysimeters.
Simulations were conducted using the boundary and initial
conditions measured for the weighing lysimeters. Most often, the model
under-estimated the amount of water lost by evaporation from the soil.
Errors in simulated evaporation were due to the inability of the model
to simulate initial high rates of evaporation when the soil was wet and
evaporation was limited by the boundary conditions. Simulated soil
temperatures generally agreed with those measured in the lysimeters.
Timing of simulated maximum and minimum soil surface temperatures
coincided with measured temperatures. The simulated water profile
indicated some redistribution of water within the soil profile which
the experimental data did not appear to indicate. The shape of the
distribution of the simulated water profile was consistent with the
experimental profile. The model of coupled heat and mass transfer
within the soil appeared to simulate the overall transport processes
within the weighing lysimeters reasonably well. However, comparison of
simulated data to the experimental data suggested that calibration of
parameters used in the model was necessary.
Hydraulic properties of the soil affected the evaporation rate
indirectly by influencing the ability of the soil to transport water
from the wetter regions of the soil to the soil surface. The primary
effect of the hydraulic diffusivity was in the distribution of water.
Calibration and sensitivity analysis indicated that adjustments of the

210
hydraulic properties of the soil alone were not sufficient to
completely correct the model.
Additional sensitivity analyses focused upon the simultaneous
variation of the surface heat and mass transfer coefficients and their
effect upon evaporation and soil temperature. Increasing the heat and
mass transfer coefficients caused the simulated evaporation rates to
approximate the experimentally observed rates more closely. The
simulated evaporation rate reached a maximum earlier then rapidly
decreased with increased mass transfer coefficients. However, the
increased evaporation caused simulated soil surface temperatures to
deviate from those observed due to the increased latent energy require
ment. Modifying hydraulic diffusivity and the surface transfer
coefficients did not cause the model to completely simulate the
evaporation rates observed in the weighing lysimeters. Future
simulation studies should focus upon describing the mass transfer from
the soil surface after rewetting has occurred.
The results of this study indicated that the interaction between
the air and the soil surface govern the entire evaporation process.
The model was sensitive to variations in the input data. To achieve
accurate results from the simulation, it was imperative that the
meteorological data be accurate since these data provide the boundary
conditions driving the processes of heat and mass transfer below the
soil surface.
It is recommended that future experiments conducted in the
weighing lysimeters include instrumentation to continuously monitor the
soil water status at several depths. This would provide a better
measure of the distribution of water within the lysimeter as a function

211
of depth and time. A permanently mounted infrared thermometer on each
of the lysimeters would provide a means to continuously monitor soil
surface temperature as well as plant canopy temperatures in future
tests. Use of an infrared thermometer could eliminate the possibility
of erroneous surface temperature readings due to incorrect placement of
the thermocouple.
This model is not suitable for use on a personal computer in its
current state. The memory requirements, small time step, and small
grid spacing require the power and batch processing capability of a
mainframe computer. However, this model could be used to determine
coefficients for simpler models that assume that evaporation is a
surface phenomenon. For instance, this model might be used to
determine the soil resistance term for the models developed by Jagtap
and Jones (1986) and Camillo and Gurney (1986). This model might also
be modified for the purpose of simulating water loss from the soil
during plant canopy development. Modifications to the model would have
to account for shading of the soil surface during plant growth and for
extraction of water from the soil by the plant roots. This model could
be expanded yet further to a two dimensional model to determine water
distribution in the soil between rows during canopy development. Once
water loss from the soil becomes insignificant compared to that lost by
transpiration from the plant, a simpler two-dimensional model for
unsaturated flow could be used.

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APPENDIX A
ADI FORMULATION OF COUPLED HEAT AND MASS TRANSFER MODEL
The partial differential equations governing the heat and mass
transfer in the soil were presented in Chapter 2 along with their
explicit finite difference expressions. Boundary conditions at the
soil surface and the lower boundary were also presented. An
alternating direction finite difference technique requires that
numerical expressions be formulated for the case of indexing the node
number from the soil surface to the lower boundary, then a second set
for use when the index is being changed in the other direction. The
equations for the forward direction (increasing node number) are
formulated by evaluating the spatial derivatives from the previous node
(j-1) to the current node (j) at the next time step (t+dt) which has
already been determined. Then the derivatives involving the node ahead
of the current node (j+1) is evaluated at the current time step. The
resulting equation is then solved for the value of the current node (j)
at the next time step (t+dt). The opposite is done when marching in
the backward direction (decreasing node number). The formulation for
the forward and backward marching difference equations is presented
below. The following are definitions of constants used in the
equations for either direction.
hhdZl WZ1
Bl = t Bl =
Fo
Ajdt
cs,jdwjdzj
DLjdt
dwjdzj"
Fv
Dvj(it
dwjdzj
219

220
DZR =
dz-
HzT
j-1
The following are definitions for coefficients used in the equations
used when moving in the forward direction:
KC1 =
TX]
cpv,jDvj
+ "XT" Wj-l.rH-r Avj,n+l)
J
NicfM,jDL,j
KCZ 2 Aj ^J+l,n ffj,ni
cpv,jDvj
+ 2 Xj ^vj+l,n" Nj,n)
Forward direction: z = 0
j = 1
energy:
(A-1)
dz-
T 'J n Bi
]3,r*l- Aj(l KC2 + Bi)Rn,n+1 + (1 KC2 + Bi)1 a,rvt-l
1 + KC2
+ (1 KC2+ Bi) Tj+l*n+l
water:
dzj
5j,n+l = + Pr>+1
vapor:
B^m 1
^vj+l,rH-l = "(i +"ITJ ^rw-l + (T+Bt^) Avj+l.m-l
(A-2)
(A-3)
The conservation of energy, water and water vapor are formulated by
setting the time rate of change of the state variable equal to the sum
of the flows from the surrounding nodes minus any sink terms. Latent
heat associated with the phase change from liquid to vapor is a heat
sink term for the energy equation. The vapor equation was solved for
the rate of evaporation at each node and the vapor density was

221
calculated using the assumption of thermodynamic equilibrium between
the water adhered to the soil particle and the air in the soil air
space.
Forward direction: 0 < z < zo
energy:
Tj,nfl
+
water:
*j,n+l
vapor:
- (1-Fv)pvj n Fv*DZR*/>vj_1?rH.1
The equations for the lower boundary were formulated by applying a no
flux boundary condition. This was imposed by assuming that an
imaginary node existed one grid spacing beyond the last node (j=nc).
Applying the assumption of no flux across the last node, implied that
the state variable evaluated at the imaginary node (nc+1) had the same
value as that at the node preceding the last node (nc-1). This was
substituted into the conservation equation and yielded equations A-7 to
A-9.
1 N J N HU
(A-4)
Fo*(l+KC2) T 1 Fo*(l-KC2) T
= 1 + fo*DZF'(i-KCi) TJ+ln + 1 + Fo*DZR*(l-KCl) TJn
Fo*DZR*(l+KCl) T dt*hfg
1 + Fo*DZR*(l-KCl) Tj-l,m-l ~C'sj(T + Fo*bZft"*(T-^i)l Ejn
(A-5)
FL (1 FL) fl.
- n+TLW) *J+l,n + "(1 +FL*dZRy
FL*DZR
+ (1 + FL*DZR) "
dt
pw(l + FL*DZR) M
(A-6)
It- ^ + Fv*DZR)^vj,nfl_ Fv*/>vj+l,n

Forward direction: z = zo
j = nc
222
energy:
x 2*Fo*
(A-7)
Tj,nfl 1 + 2*Fo*
(1+KC1) T .1
(1-K'Cl) 'J-l.rH-1 +1'+ 2*Fo*(l-KCiy TJ>n
+ 0sJ(i + 2W(i-kCTJ')
water:
(A-8)
vapor:
(A-9)
jf C1 + 2*Fv)Pvj,m-l ^vj,n 2*Fv*pvj-i>m.i]
Similarly, moving in the backward direction, the derivatives
containing the properties at the node j+1 are evaluated at time step
n+1, while the others are evaluated at the current time step. The
following are the expressions for the constants KC1 and KC2 as
evaluated in the backward direction.
KC1 =
KC2 =

223
The equations for the backward direction at the last node are
Backward direction: z = zo
j = nc
energy:
Tj,nfl
2*Fo*(l+KCl)Tj.1>n + (1 2*Fo*(l-KCl))Tj>n
hfg,jdt
'S,J
Ej,n
(A-10)
water:
(A-11)
d3,ml
(1 2*FL)0j>n + 2*FL*0j.1>n
dt
-J,n
vapor: (A-12)
Ej,n a dt ^vj,n+l (1 2*Fv)/5vj,n 2*Fv*/>vj-l,n
Equations A-13, A-14 and A-15 represent the conservation equations for
the interior nodes for the backward direction solution.
Backward direction: 0 < z < zo
1 < j < nc
energy:
(A-13)
Tj>n+1
Fo*(l+KC2) T
1 + Fo*(l-KC2) 1 j+l,n+l +
Fo*DZR*(l+KCl) T
+ 1 + Fo*(l-KC2) 'j-l.n"
1 Fo*DZR*(l-KCl) x
1 + Fo*(l-KZ) TJn
dt*hfgJ
Csj(l + Fo*(l-KC2)) hJn
water:
e3,r*l
FL (1 FL*DZR) n .
(1 + FL')' ^J+l,n+l + '(1 + FL)
FL*DZR dt
+ (1 + FL) Pw'Ji + 'Fiy ^.n
(A-14)
vapor:
(A-15)
^j,n = (jt U + Fv)Pvj,r>fr Fv*^vj+l,n+l
- (l-Fv*DZR)pvj>n- Fv*DZR*Pvj-l,n

224
The backward form of the equations for the boundary conditions at the
soil surface are
Backward direction: z 0
j = 1
(A-16)
energy:
(A-17)
water:
dzj
^j,n+l = 0j+l,m-l+ J Dj7]pnfl
(A-18)
vapor:
The above equations were used to represent the partial differential
equations which describe the energy and mass transfer within the soil.
After a complete cycle through the nodes, from node 1 to node nc, then
from node nc back to node 1, all of the equations had been solved using
the most recent values of the state variables.

APPENDIX B
METEOROLOGICAL DATA FOR LYSIMETER EVAPORATION STUDIES
225

Net Radiation ( W/m^)
226
Julian Date
Figure B-l. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from November 25 to 29, 1986.

227
Julian Date
Figure B-2. Measured relative humidity (bottom) and temperature
(top) of ambient air for evaporation tests conducted
from November 25 to 29, 1986.

228
Figure B-
25
20 -
N
-C
\
E
£.15
c
o
'o
;5.10
U
0)
u
CL
5--
329
J lij, i, i
330
331
332
Julian Date
333
334
335
3. Rainfall during lysimeter studies conducted from
November 25 to 29, 1986.

Net Radiation
229
Julian Date
Figure B-4. Wind speed (top) and net radiation (bottom) data for
lysimeter experiment conducted from December 15 to 19,
1986.

Relative Humidity ( % ) Air Temperature ( C )
230
Julian Date
Figure B-5. Measured relative humidity (bottom) and temperature
(top) of ambient air for evaporation tests conducted
from December 15 to 19, 1986.

231
Figure B-
Rainfall from December 15 to 19, 1986 during lysimeter
evaporation measurement.

Net Radiation ( W/m2) yy¡nd
232
Julian Date
Figure B-7. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from January 5 to 10, 1987.

Relative Humidity ( % ) Air Temperature ( C )
233
Figure B-8. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from January 5 to 10, 1987.

Net Radiation ( W/m2) Wind
234
Julian Date
Figure B-9. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from January 11 to 15, 1987.

235
Julian Date
Figure B-10. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from January 11 to 15,
1987.

236
m
\
E
T3
V
0)
CL
in
T5
c
34 35 36 37
Julian Date
Julian Date
Figure B-ll. Wind speed (top) and net radiation (top) data collected
during lysimeter evaporation studies conducted from
February 3 to 5, 1987.

237
Julian Date
Figure B-12. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from February 3 to 5, 1987.

Predpitaon (mm/h)
238
Figure B-13. Precipitation during lysimeter evaporation studies
conducted from February 3 to 5, 1987.

239
Julian Date
Figure B-14. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from February 9 to 13, 1987.

Air Temperature ( C )
240
25
20
15
10
5
0
-5
40 41 42 43 44 45
Julian Date
Julian Date
Figure B-15. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from February 9 to 13, 1987.

BIOGRAPHICAL SKETCH
Christopher L. Butts was born September 19, 1957 in Knoxville, TN
while his father was attending the University of Tennessee. Chris
attended elementary and junior high school while living in the small
town, Estill Springs, TN. His family then moved to Virginia Beach, VA
where he attended First Colonial High school. After graduating from
high school in 1975, he attended Virginia Polytechnic Institute and
State University in Blacksburg, VA. Requirements for a Bachelor of
Science in Agricultural Engineering were completed in 1979, then a
Master of Science in 1981. After completion of the Master of Science,
he worked at the Veterans' Administration Medical Center in Salem, VA
as a hospital facilities engineer for a year, then worked as a research
engineer at the University of Georgia Coastal Plains Experiment
Station. The Butts family moved to Gainesville, FL to begin a program
to attain the Doctor of Philosophy Degree in Agricultural Engineering
at the University of Florida. Chris is currently employed by the U.S.
Department of Agriculture, Agricultural Research Service at the
National Peanut Research Laboratory in Dawson, GA.
241

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Chairman
Professor of Agricultural
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
is W. Jones, c
ifessor of Agri
ngineering
(airman
iultural
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Khe V. Chau
Associate Professor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Jerome J. Gaffneyu (
Associate Professor of
Agricultural Engineering

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Calvin C. Oliver
Professor of Mechanical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
L. H. Allen, Jr. / '
Professor of Agronomy
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the decree of Doctor of Philosophy.
December, 1988
Dean, College of Engineering
Dean, Graduate School

W1 UNIVERSITY of
I FLORIDA
The Foundation for The Gator Nation
Internet Distribution Consent Agreement
In reference to the following dissertation:
AUTHOR: Butts, Christopher
TITLE: Modeling the evaporation and temperature distribution of a soil profile /
(record number: 1125835)
PUBLICATION DATE: 1988
I, Christopher Ir.~Btrtts.~as copyright holder for theaforCmentiOrted dissertation, hereby
grant specific and limited.archive and. distribution rights to the Board df-Trustees of the
University of Florida and its agents. I authorizeAhe University of Florida to digitize and
distribute the dissertation described above for nonprofit, educational purposes via the Internet or
successive technologies.
This is a non-exclusive grant of permissions for specific offline and on-line uses for an
indefinite term. Off-line-uses-shall-be limited-to-those specittlty allowed by "Fair Use" as
prescribed by the terms of United States copyright legislation (cf, TitlaT7, U.S. Code) as well as
to the maintenance and preservation of a digital archive copy. Digitization allows the University
of Florida to generate image- and text-based versions as appropriate and to provide and enhance
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13 June 2008
Date of Signature



42
Fuchs et al. (1969) used a method in which instability in the
lower atmosphere was accounted for in conjunction with the logarithmic
wind profile to calculate the surface mass transfer coefficient. This
involved using the KEYPS function (Panofsky, 1963) to determine the
curvature of the diabatic lapse rate (a) of the logarithmic wind
profile. The surface mass transfer coefficient was calculated from
_2
hm = k2WS ( % + In (2 + ) (2-50)
where
hm transfer coefficient of mass from surface to height, z,
in the air [m*s"l]
k = von Karman constant [dimensionless]
WS = wind speed at height, z [m*s"l]
D height displacement [m]
B d + zq
d = height above soil surface where wind velocity is zero
[]
zq roughness length [m]
z = height above the displacement height [m]
7T = curvature of the diabatic lapse rate or diabatic
influence function [dimensionless]
Fuchs et al. (1969) then calculated the surface heat transfer
coefficient using
h. = he a
h m pa a
(2-51)


Temperature (C) Temperature (C)
130
Julian Date
20 4-* t -t~[ |t >-H . | t i + . | i | i i | . | t I
224 225 226 227 228 229 230 231 232 233 234
Julian Date
Figure 3-29. Soil temperatures measured at depths of 0, 15 and 80 cm
in weighing lysimeters located at the University of
Florida, IREP from August 12 to 21, 1987.


53
that of the standard explicit methods. Increased stability is obtained
over the standard explicit methods but is less than that for the
implicit techniques. This implies that using the same grid spacing, a
larger time step can be used in the ADI methods than that for explicit
methods.
An ADI finite difference technique was used to solve the system of
equations for the soil profile due to the increased stability
characteristics over explicit methods. Appendix A contains a detailed
development of the numerical equations used in the ADI technique.
Hydraulic (equation 2-40) and thermal properties (equations 2-35 and
2-36) of the soil and the surface transfer coefficients (equations 2-50
and 2-51) were calculated at the beginning of each time step. A
general description of the solution algorithm is shown in Figure (2-6).
The energy and water equations were solved for the temperature and
volumetric water content, respectively at the next time step. The
equilibrium condition was used to determine the vapor concentration by
substitution of the values of soil temperature, water content, water
potential, and saturated vapor concentration in equation (2-30). The
vapor continuity equation was used to determine the evaporation rate.
The calculations were repeated using the new evaporation rate
until the absolute value of the maximum fractional change in any of the
variables was less than a prescribed convergence criterion.
The system of numerical equations was solved using computer code
written in F0RTRAN77. A variable grid spacing was used throughout the
soil profile with the smaller mesh being located near the soil surface
due to expected large gradients in temperature and moisture content
once the soil surface begins to dry (Table 2-3). The program was then


Simulated Water Content (cm^/cm^)
207
Figure 5-29. Simulated diurnal variation of soil water content at
depths of 0, 2 and 5 cm using a multiplier of 2.0 for
the hydraulic diffusivity (D^), heat (h^) and mass (hm)
transfer coefficient.


23
El rate of change of liquid water to water vapor
[kg-m^n-s'1]
For the range of temperatures generally occurring in the soil, the
density of water may be considered constant. The volumetric flow rate
of water (q|_) can be obtained by dividing the mass flow rate (Ql) by
the density of water (pw). The volumetric flow rate of water in an
unsaturated soil is governed by Darcy's Law as follows:
qL - k gf (2-19)
where:
qL volumetric flow rate of liquid water [m3*m"2*s-1]
k unsaturated hydraulic conductivity [m*s"l]
0 = soil water potential [m]
z = spatial dimension [m]
The relationship between soil water content and soil water potential is
unique for a given soil type. Therefore, the volumetric flux of water
can be expressed in terms of volumetric water content instead of soil
water potential by applying the chain rule of differentiation to
equation (2-19) as follows
q, = k
30 33
W W
(2-20)
The hydraulic diffusivity of the soil is defined as
DL
K
30
W
(2-21)
By substituting the definition for the hydraulic diffusivity (2-21) and
the volumetric flow rate in terms of the soil water content (2-20) the
equation for the conservation of mass for liquid water (2-18) becomes
33
3t
3_
3z
CDl
33
3z
1 E(z,t)
N
(2-22)


was allowed to occur throughout the soil profile in response to the
assumed equilibrium between the liquid and vapor phases. Surface heat
and mass transfer coefficients were determined using equations based
upon equations of motion in the atmosphere. Thermal diffusivity of the
soil was measured and incorporated into the simulation.
Simulation results included temporal values of the cumulative and
hourly water loss from the soil, and spatial distribution of
temperature, water and water vapor in the soil. Experimental data
obtained from the weighing lysimeters were used to validate the model.
Prior to model calibration simulated soil temperatures were within 2 C
of measured values and simulated cumulative evaporation was within ten
percent of measured water loss. Sensitivity analysis indicated that
calibration could be achieved by relatively small adjustments in the
values of the surface heat and mass transfer coefficients.
Experimental and simulated evaporation rates exhibited a diurnal
pattern in which maximum evaporation rates occurred approximately
midday then decreased to near zero at sunset. Under some conditions,
water continued to evaporate from the soil overnight at a rate of
approximately 0.03 mm/h while in some cases a net gain of water was
observed overnight.
xi


51
0 n (^nc-l,n ^nc,n) (Tnc-l,n+ Tnc,n)
+ 2 /Jwrx^pwrKpLnc dz^-i ~2
o n (^vnc-ljiT^vnc,!!) (^nc-l,n+ ^nc,n)
+ 2 cpvncDvnc j
+ dwnc ^nc,n (2-74)
The conservation relationships provide sufficient information to
determine three of the desired quantities for the soil profile leaving
a fourth remaining unknown. The constitutive relationship requiring
that the water in the liquid phase be in thermodynamic equilibrium with
the vapor phase was used to provide the remaining equation. Since the
surface node provides only an interface between the soil and the
atmosphere, this equilibrium condition was not necessary for the
surface node. Under atmospheric conditions, the soil may not become
completely dry (0 % volumetric water content), but will reach some
moisture content which is in equilibrium with the atmosphere. This is
typically taken to be the same as the permanent wilting point of the
soil (^= -150 m) and is the same as the residual water content
described by van Genuschten (1980). As the soil surface dries, the
soil reaches this equilibrium moisture content and is assumed to act as
an interface between the air and the wet soil below.
The system of equations (2-64) through (2-69)and (2-72) through
(2-74) along with equation (2-30) represent an explicit solution to the
differential equations, that is, the values of the state variables at
the next time step are functions of those at the previous time step.
Using the explicit representation allows a simple algorithm to be used


150
Table 4
1. Nomenclature and list of symbols
|L
an : nin eigenvalue for transcendental equations for
solutions to transient conduction equations
Bi : Biot number
C : volumetric heat capacitance
1L
cn : coefficient for the ntn term of series solution of
transient conduction equation
Dt : thermal diffusivity
Dtv : apparent enhancement to thermal diffusivity due to
water vapor movement
Dv : diffusivity of water vapor in soil air space
Fo : Fourier number
hfg : latent heat of vaporization
hfo : surface heat transfer coefficient
Jq : zero order Bessel function of the first kind
: first order Bessel function of the first kind
L : characteristic length
1 : half thickness of infinite slab
m : slope of ln(0) vs time response of sample
q : heat flux
R : radius of cylinder
Rw : ideal gas constant for water vapor


2
developed by Zur and Jones (1981) and focused on estimating water
relations for crop growth. A soil water model which simulates ETa is
an integral part of the soybean production model, SOYGRO, developed by
Wilkerson et al. (1983). One of the primary uses of SOYGRO is to
determine the yield response of soybean to various environmental
conditions such as drought.
Even though ET models are currently used extensively, many
research needs remain. At best, crop coefficients are rough estimates
computed from previous experiments with the crop of concern. There is
a need to develop a better understanding of the variability of soil,
crop and environmental factors that determine crop coefficients.
Scientists need improved procedures for determining crop coefficients
for various cultural practices such as mulching, no-till and row
spacing.
One of the most pressing needs in the area of evapotranspiration
research is in the area of sensors which will detect soil water status
for a particular field (Heermann, 1986; Saxton, 1986). One of the
rising technologies for determining soil water status is the use of
remote sensing of soil temperature for determining soil water status
(Soer, 1980; Shih et al., 1986).
Evaporation from the soil is driven by the energy balance of the
surrounding environment. Because of the release of latent energy is
coupled with the sensible energy balance, it is necessary to consider
both components of the energy balance to completely describe the
evaporation process. Soil temperature has been found to influence many
processes in the soil. For example, soil temperature influenced
nodulation of roots of Phaseolus vulgaris L. (Small et al., 1968) and


Temperature (C) Temperature (C)
197
Figure 5-19. Comparison of experimental and simulated soil surface
temperatures at the soil surface during test 73
(February 9-13, 1987).


103
Masonry Block Retaining Wall
Hanger Rods
Concrete Slab
Ceramic Stones
Lysimeter Container
Thermocouple Probes
Floor Support Beams
Gamma Probe Access
Hanger Beam
Tubes (Pair)
Figure 3-2.
Schematic of lysimeter construction and support system.


Temperature (C) Temperature (C)
118
Julian Date
Figure 3-17. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from January 5 to 9, 1986.


56
Table 2-2. Summary of equations used in a model of heat and
mass transfer in the soil to determine state
variables and transport parameters.
Process or Parameter Equation No.
Conservation of mass (liquid) 2-22
Conservation of mass (vapor) 2-25
Conservation of energy 2-26
Vapor-Liquid equilibrium 2-30
Surface Boundary Conditions
Conservation of mass (liquid) 2-31b
Conservation of mass (vapor) 2-31c
Conservation of energy 2-31a
Lower Boundary Conditions
Conservation of mass (liquid) 2-32b
Conservation of mass (vapor) 2-31c
Conservation of energy 2-32a
Soil properties
Diffusivity of water vapor in soil (Dv) 2-34
Volumetric heat capacity (Cs) 2-35
Thermal Conductivity (X) 2-36
Hydraulic Diffusivity (D|_) 2-40
Surface Mass Transfer Coefficient (hm) 2-50
Surface Heat Transfer Coefficient (h^) 2-51


70
entire length past the edge of the soil container (Figure 3-2). To
alleviate the need for underground access to the load cells supporting
the lysimeter, two 2-cm rods were inserted into both of the protruding
end of each floor joist and extended above the top of the lysimeter.
These rods were inserted through both webs of a 25-cm steel I-beam
(W10x26) extending parallel to the end of the lysimeter. The ends of
the rods were threaded such that nuts could be placed above and below
the upper I-beam. The lower nut prevented the hanger beam (W10x26)
from sliding to the bottom prior to installation in the ground. The
top nuts prevented the rods from sliding out of the hanger beam after
installation. This assembly formed a cradle by which the lysimeter was
supported and provided access to four 4.5 Mg capacity load cells
located under each end of the hanger beams.
A pit with reinforced concrete block retaining walls and a 10-cm
reinforced poured-in-place concrete floor was constructed for the
installation of the lysimeter at the IREP (Figure 3-2). The depth of
the pit was such that the bottom of the lysimeter tank was
approximately 15 cm above the pit floor and the tops of the retaining
wall and the lysimeter were flush. The pit floor was sloped toward the
center to drain water which might accumulate to a sump located
immediately outside of the North wall of the pit. Support columns were
incorporated into the retaining wall on which to place the load cells
supporting the lysimeter. The load cells could be installed or
removed by lifting the lysimeter with hydraulic automotive jacks from
ground level. Fill dirt was used to bring the surrounding soil up to
the elevation of the pit wall.
Prior to installing the lysimeters in the ground pits, two porous


APPENDIX B
METEOROLOGICAL DATA FOR LYSIMETER EVAPORATION STUDIES
225


16
pressure by the ratio of vapor pressure at the soil surface to the
saturated vapor pressure at the same temperature (relative vapor
pressure). This incorporated the vapor pressure depression effect of
the soil water potential into the model. This seemed to match field
data for a clay loam soil. This model was not tested for more coarse
soils.
The Penman method eliminated the requirement of soil surface
temperature by neglecting the effects of soil heat flux at the expense
of being able only to predict the evaporation on a daily basis.
Various researchers have recently gone one more step in modeling the
evaporative loss of water from the soil by considering the movement of
water and heat below the soil surface (Van Bavel and Hi 11 el, 1976;
Lascano and Van Bavel, 1983). Most of the research in which the
process of evaporation has been examined at this detailed level are
for a bare soil surface. This eliminated the complicating factors of
water removal by the plant, which surface temperature to use in
describing the driving potential for evaporation, and description of
the atmospheric boundary layer transfer coefficients.
Van Bavel and Hillel (1976) developed a model in which the partial
differential equations for water (liquid) and energy transfer in the
soil were considered to be independent processes. This approach
explicitly determined the soil heat flux for the boundary condition at
the soil surface as expressed in equation (2-14). The simultaneous
solution determined the actual evaporation from the soil as well as the
spatial distributions of water and temperature as a function of time.
The variation in hydraulic and thermal properties of the soil with time
was accounted for in this extended combination approach.


148
diffusivity due to vapor movement as independent variables. The best
fit was achieved with the regression equation linear in soil density
and.thermal vapor diffusivity with first and second order terms for the
volumetric water content (R2 = 0.805). However, at the 95 % confidence
level, the coefficient for the thermal vapor diffusivity was not
significantly different than zero. Therefore, the regression
coefficients were obtained for the water content (first and second
order) and the bulk density (first order) with an equally high
correlation (R2 = 0.799). These regressions were compared to the
experimental data and the values calculated using the DeVries method
(Figure 4-5). Neither of the regressions nor the DeVries method
provided a suitable estimate for the thermal diffusivity for all ranges
of density and water contents tested.
Conclusions
A transient method was used to measure the thermal diffusivity of
a Millhopper fine sand at three dry bulk densities and seven volumetric
water contents. The 95 % confidence limits were within 5 percent of
the mean for each sample. It was critical to obtain a good estimate of
the slope of the time-temperature response curve of the test sample to
achieve accurate values of the thermal diffusivity. Estimating the
slope of the time-temperature response for the copper cylinder and the
specific heat of the soil were not critical to achieving consistent
results. A length to radius ratio of eight to one (8:1) was sufficient
to analyze these soil samples using the solution for the infinite
cylinder. Superposition of the infinite cylinder and slab solutions to
obtain the solution for a finite cylinder was not required. Apparent
changes in the thermal diffusivity due to vapor movement within the


69
located on the University of Florida campus in Gainesville, Florida
(Figure 3-1).
Aboukhaled et al. (1982) presented a relatively comprehensive
review of the existing literature regarding lysimeter construction and
design and noted the practical aspects of certain design
considerations. It was noted that the lysimeter annulus (gap area
between the soil container plus the area of the retaining wall) should
be minimized to reduce the effects of the thermal exchange between the
lysimeter soil mass and the surrounding air. Several researchers
(Aboukhaled et al., 1982; Black et al., 1968) made recommendations for
minimum lysimeter areas of 4 m2 for the primary purpose of reducing
the edge effect of the soil-to-soil gap between the lysimeter interior
and the surrounding fetch. Maintaining a 100:1 ratio of fetch to crop
height has been recommended to minimize border effects. The site
selected for the lysimeter construction satisfied the fetch
requirements for very short crops or bare soil. Dugas et al. (1985)
noted that the problems caused by soil-to-soil discontinuity should be
minimal if the lysimeter surface area were greater than 1 m2. Soil
depth should be sufficient so as not to impede root growth of crops to
be planted in the lysimeter.
Two soil containers were constructed using 4.8 mm (3/16 in.) steel
plate for the floor and side walls (Butts, 1985). All seams were
welded continuously to prevent water leaks and the entire container was
painted with an epoxy-based paint to prevent corrosion. The lysimeter
measured 305 cm long, 224 cm wide and 130 cm deep. This provided a
soil surface area of 6.8 m2 and a soil volume of 8.9 m3. Main floor
supports consisted of four 20-cm steel I-beams (W8xl5) and extended the


Soil Water Potential ( cm )
59
Figure 2-2. Soil water potential (tension) and specific water capacity
as a function of volumetric water content for a sandy soil.
Specific Water Capacity ( cm *)


Diabatic Influence Function ( 4> )
61
Figure 2-4. Relationship of the diabatic influence function and the
Richardson number.


100
Table 3-3. Experimental values of soil porosity as
bulk density of the soil.
Bulk Porosity (cm3/cm3)
Density
(g/cm3) Sample Rep #1 Rep # 2 Rep # 3
a function of dry
Sample
Average
1
0.555
0.556
0.556
0.556
1.20
2
0.555
0.556
0.554
0.555
3
0.554
0.555
0.555
0.555
1
0.516
0.515
0.515
0.515
1.30
2
0.514
0.515
0.515
0.515
3
0.514
0.514
0.515
0.514
1
0.477
0.477
0.478
0.477
1.40
2
0.473
0.474
0.474
0.474
3
0.473
0.473
0.473
0.473
1
0.402
0.403
0.404
0.403
1.60
2
0.406
0.407
0.407
0.407
3
0.403
0.403
0.400
0.402
1.64
1
0.390
0.391
0.391
0.391


Fractional Change in
Thermal Diffusivity
156
Figure 4-4. Effects of changes in experimental parameters upon the
values of thermal diffusivity determined from experimental
data.


99
Table 3-2. Vertical spacing of thermocouples for measurement of soil
temperature.
Thermocouple
Position No.
Depth
( cm )
1
0.0
2
1.0
3
2.0
4
4.0
5
8.0
6
20.0
7
40.0
8
60.0
9
80.0
10
100.0


V. MODEL ANALYSIS 158
Introduction 158
Validation 158
Sensitivity Analysis 170
Summary 176
VI. SUMMARY AND CONCLUSIONS 208
BIBLIOGRAPHY 212
APPENDIX A ADI FORMULATION OF COUPLED HEAT AND MASS
TRANSFER MODEL 219
APPENDIX B METEOROLOGICAL DATA FOR LYSIMETER EVAPORATION
STUDIES 225
BIOGRAPHICAL SKETCH 241
iv


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE
By
Christopher Lloyd Butts
December, 1988
Chairman: J. Wayne Mi shoe
Cochairman: James W. Jones
Major Department: Agricultural Engineering
Evaporation of water from the soil is controlled by the transport
phenomena of energy and mass transfer. Most procedures for estimating
the loss of water from the soil assume that the soil is an isothermal
medium with evaporation of water occurring at the soil surface.
Estimating evaporation as a surface phenomenon independent of soil
hydraulic and thermal properties can lead to overestimation of the
amount of water lost from the soil.
Weighing lysimeters measuring 2 x 3 x 1.3 m^ were constructed
capable of detecting a change in weight equivalent to 0.02 mm of
water. Recorded data consisted of net radiation, soil temperature,
water content, load cell output, ambient air temperature, relative
humidity, windspeed and precipitation. Measured soil properties
included dry bulk density, porosity and thermal diffusivity.
A model describing the continuous distribution of heat, water and
water vapor within the soil was developed. The evaporation of water
x


209
2. soil porosity as function of dry bulk density and
3. thermal diffusivity as a function of volumetric water content
and bulk density.
Weather data for model validation was obtained from a weather station
adjacent to the lysimeters.
Simulations were conducted using the boundary and initial
conditions measured for the weighing lysimeters. Most often, the model
under-estimated the amount of water lost by evaporation from the soil.
Errors in simulated evaporation were due to the inability of the model
to simulate initial high rates of evaporation when the soil was wet and
evaporation was limited by the boundary conditions. Simulated soil
temperatures generally agreed with those measured in the lysimeters.
Timing of simulated maximum and minimum soil surface temperatures
coincided with measured temperatures. The simulated water profile
indicated some redistribution of water within the soil profile which
the experimental data did not appear to indicate. The shape of the
distribution of the simulated water profile was consistent with the
experimental profile. The model of coupled heat and mass transfer
within the soil appeared to simulate the overall transport processes
within the weighing lysimeters reasonably well. However, comparison of
simulated data to the experimental data suggested that calibration of
parameters used in the model was necessary.
Hydraulic properties of the soil affected the evaporation rate
indirectly by influencing the ability of the soil to transport water
from the wetter regions of the soil to the soil surface. The primary
effect of the hydraulic diffusivity was in the distribution of water.
Calibration and sensitivity analysis indicated that adjustments of the


Variable Definition
cJ>n
ETa
ETP
ea
eas
es(T)
-vs
W
9
9j
H
hfg
hh
^m
rate of phase change of liquid water to vapor in
soil cell j at time step n [kg^jm^soil)*8"1]
actual evapotranspiration [mm]
potential evapotranspiration [mm]
water vapor pressure in ambient air [Pa]
saturated water vapor pressure in ambient air [Pa]
saturated vapor pressure at temperature, T [Pa]
vapor pressure at water potential, [Pa]
sensible heat flux rate from the soil to the air
[W-m']
gravitational acceleration [m*s-2]
shape factor in the j-th principal axis for thermal
conductivity calculations [dimensionless]
vertical component of heat flux in the atmosphere
[J*m]
latent heat of vaporization of water [J*kg_1]
convection heat transfer coefficient for soil
surface [W*m'^*K'l]
convection vapor transfer coefficient for soil
surface [m*s-l]
K hydraulic conductivity [mfj^m'^s"1]
Kh Km Kw turbulent transfer coefficients forJieat, momentum
and water vapor, respectively [m2*s_1]
k-y ratio of temperature gradient in i-th soil
constituent to the temperature gradient in the
continuous constituent (water or air) in the
direction of the j-th principal axis [dimensionless]
-2
LE latent energy transfer in the atmosphere [J*m ]
n number of moles of gas present, used in ideal gas
law [mol]
vi


40
where
Qm vertical mass flux [kg-liras'1]
WSr wind speed at reference height, zr [m'S*1]
Cer = Dalton Number; dimensionless mass transfer coefficient
pvr water vapor concentration at reference height, zr
[kg*m'3]
pvs water vapor concentration at soil surface [kg*m-3]
The surface mass transfer coefficient (hm) used in this model is
related to Cer by
^m = wsr Cer
(2-45)
Brutsaert (1982) presented functions for the Dalton number in terms of
the drag coefficient and the dimensionless Schmidt number as
Cer
(B
Cd
V2
+ Cdr)
-1/2
(2-46)
where
B function of dimensionless Schmidt number
Cer = Dalton number
Cdr = surface drag coefficient [dimensionless]


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Calvin C. Oliver
Professor of Mechanical
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
L. H. Allen, Jr. / '
Professor of Agronomy
This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the decree of Doctor of Philosophy.
December, 1988
Dean, College of Engineering
Dean, Graduate School


Variable Definition (continued)
1 volumetric water content of soil [m3*m"3]
0jjn volumetric water content of soil at node j and at
time step n [m3*m'3]
9r residual volumetric water content [m3*m'3]
6s volumetric water content at saturation [m3,m3]
r shear stress [N*m-2]
IX


I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Chairman
Professor of Agricultural
Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
is W. Jones, c
ifessor of Agri
ngineering
(airman
iultural
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Khe V. Chau
Associate Professor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Jerome J. Gaffneyu (
Associate Professor of
Agricultural Engineering


In e
154
Time (s)
Figure 4-2. Semi-logarithmic plot of the dimensionless temperature
ratio versus time for the copper test cylinder.


Relative Humidity ( % ) Air Temperature ( C )
233
Figure B-8. Measured and estimated relative humidity (bottom) and
measured temperature (top) of ambient air for
evaporation tests conducted from January 5 to 10, 1987.


164
lysimeter was observed to be approximately 3 mm of water while the
north lysimeter had lost a total of approximately 4 mm of water at the
end of the second day. Meanwhile, the simulated cumulative evaporation
of water from the soil after the same period time was approximately
2 mm for the south lysimeter and 2.5 mm for the north lysimeter south
lysimeter. During the third day of the experiment, water loss from
both of the lysimeters was observed to be 0.5 mm or less while the
simulation showed approximately 1 to 1.5 mm of water evaporated from
the lysimeters. Measured maximum hourly evaporation rates occurred
approximately 2 to 4 h before simulated maximum rates (Figure 5-8).
The data for the south lysimeter was much smoother than that obtained
from the north lysimeter with most of the variation during the night
time hours. In most cases, the variability of the measured hourly
water loss from the north lysimeter occurred with subsequent equal and
opposite magnitudes with a zero net weight loss or gain. This could
have been wild animals, such as cats, crossing the lysimeter during the
night. Indications of the presence of animals was found on numerous
occasions during several of the experiments. Discrepancies between
simulated and measured hourly evaporation had similar characteristics
as those that occurred in test 70. The agreement between simulated
and experimental soil surface temperatures were not as close during
this test as was noted in test 70 (Figure 5-9). However, the simulated
surface temperatures corresponded with decreased evaporation. The
simulated daily maximum temperatures were 4 to 6 C above measured
values. If the evaporation rates had been higher, the latent heat
required would have reduced soil surface temperatures. Maximum
observed and simulated soil surface temperatures increased as the test


136
Applying the steady-state method requires the accurate measurement of
heat flux between two points. The spatial gradient of the temperature
must also be measured accurately as well. This indicates that the
position of the two temperature sensors must be known. Reidy and
Rippen (1969) described specific procedures possible for measuring the
thermal conductivity using steady-state methods in detail. Some of
the advantages of using equilibrium techniques are:
1. simple mathematical solutions are used,
2. small test samples are suitable, and
3. liquid, solid or dry granular materials can be used.
Measurement of the thermal conductivity by steady-state procedures has
the following disadvantages.
1. Thermally induced moisture migration in the sample may cause
errors in measurement due to the non-homogeneity produced.
2. Several hours may be required for the material to reach
equilibrium conditions.
3. Loss of heat from the ends of the sample may cause
significant errors in the measurement of heat flux.
Transient techniques are based upon the solution of the governing
partial differential equation for heat transfer in a homogenous,
isotropic sample. Transient methods require temperature measurement at
a given location within the sample over a period of time. Advantages
of the transient procedure over the steady-state method are that rapid
results may be obtained and no direct measurement of heat flux is
required.
The solution of the governing partial differential equation for a


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TITLE: Modeling the evaporation and temperature distribution of a soil profile /
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44
solution techniques utilize the finite difference form of the
differential equations. The differential operator in the continuous
domain becomes a difference operator in the discretized domain.
Difference operators are either forward, backward, or central
difference operators. The difference operators for the function of the
independent variable,
x, [ f(x)
] are
forward:
df
3x
o
III
f(*j+l) f(xj)
xj+l xj
(2-52)
backward:
3f
vxf -
f(Xj) f(Xj.!)
(2-53)
3x
xj XJ-1
central:
3f
35T
5xf
(2-54)
1 f(xj+l) f(xj) f(xj) f(Xj.!)
xj+l xj + xj xj-l
In expressing the differential equations in difference form, the
derivative with respect to time is accomplished using the forward
difference operator, while spatial derivatives are expressed using any
of the three difference operators. The continuous domain must first be
discretized or divided into several discrete regions such that the
finite difference approximation of the differential equation approaches
the differential equation in the limit of the grid spacing going toward
zero (Figure 2-5).
The partial differential equation defining the conservation of
energy within the soil profile (equation 2-26) stated that the change
in sensible heat over time is due to heat transferred by conduction
plus sensible heat carried by the diffusion of water in the liquid and


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76
vacuum transducer or manometer is used to measure the equilibrium
tension on the water column. The tensiometer has a range of
application from 0 to approximately 85 kPa (Long, 1982; Rice, 1969).
This would cover approximately ninety percent of the range of moisture
contents encountered in the field (Schwab et al., 1966). Using an
electronic pressure transducer would allow for the tension to be
monitored continuously. Several disadvantages in using tensiometers
exist. The soil water release curve (tension vs. soil water content)
must be known for each soil in which the tensiometer is installed.
However, this information is needed for the determination of the
concentration of water vapor within the soil for modeling purposes as
well. A second difficulty is that the tensiometer is essentially
detecting water content at a single point within the soil profile, thus
requiring the installation of several tensiometers within a finite
volume of soil. Tensiometers also require frequent maintenance. In
situ calibration of tensiometers is not required since the soil water
release curve is a unique property of the soil allowing the researcher
to obtain a soil sample from the experimental site and determine the
water release curve in the laboratory.
Electrical resistance is another property which varies with soil
water content. Two electrodes are imbedded within a porus gypsum block
which, when placed in the soil reaches an equilibrium moisture content
with the soil. The resistance between the two electrodes is measured.
However, as the water evaporates from the gypsum block, dissolved salts
remain in the gypsum. This deposition of salts within the porous
material will become redissolved when rewetted and cause an incorrect
reading. Their range of applicability extends mainly over dry soil


18
conservation of mass and energy to simulate the diurnal variations of
soil water and temperature.
89
ST =
a
IT
(D
89 ,
8z '
a
ST
(Dl
ai
ST
-) +
3K
ST~
(2-16)
c 3T -
cs at
a
8z
(X ) + Q(9,T) (2-17)
where:
9
=
volumetric water content [m3#m"3]
t
=
time [s]
z
s
depth below soil surface [m]
T
=
soil temperature [K]
cs
=
volumetric heat capacity of soil mixture
[ J *m-3* clC_13
\
=
thermal conductivity of soil mixture [W*m'l*K"l]
K
=
unsaturated hydraulic conductivity [nrs-1]
Da
=
hydraulic diffusivity [m2S_1]
=
30 Da uS^Vi 30
* ~ST~ + PRJT 89
Da
=
diffusivity of water vapor in air [m^s'1]
P
s
atmospheric pressure [kPa]
V
s
ratio of atmospheric pressure (P) to partial
pressure of air
S
=
soil porosity [m3*m"3]
9
=
acceleration due to gravity [nrs-1]
0
=
soil matric potential [m]
Rw
=
ideal gas constant for water vapor
P\i
=
density of water vapor [kg*m-3]


Forward direction: z = zo
j = nc
222
energy:
x 2*Fo*
(A-7)
Tj,nfl 1 + 2*Fo*
(1+KC1) T .1
(1-K'Cl) 'J-l.rH-1 +1'+ 2*Fo*(l-KCiy TJ>n
+ 0sJ(i + 2W(i-kCTJ')
water:
(A-8)
vapor:
(A-9)
jf C1 + 2*Fv)Pvj,m-l ^vj,n 2*Fv*pvj-i>m.i]
Similarly, moving in the backward direction, the derivatives
containing the properties at the node j+1 are evaluated at time step
n+1, while the others are evaluated at the current time step. The
following are the expressions for the constants KC1 and KC2 as
evaluated in the backward direction.
KC1 =
KC2 =


119
Julian Date
Figure 3-18. Hourly and cumulative water loss measured from
January 11 14, 1987 using the University of Florida,
IREP weighing lysimeters.


92
south lysimeter (Figure 3-11). However, the south lysimeter indicated
a gain of approximately 0.5 mm of water beginning at 1700 on December
15, while the north lysimeter continued to lose water until 1800. The
lysimeters then indicated a weight gain equivalent to approximately
2 mm of water between the hours of 0200 and 0500 on December 16. This
was confirmed by the AWARDS data indicating 2 mm of rain fell between
0200 and 0500. Water began evaporating at 0600 from the south
lysimeter, while the north lysimeter did not begin losing water until
approximately 1100. Erratic behavior was noted throughout the
remainder of the test for both lysimeters. The hourly evaporation
rates further indicate the high variability of the data (Figure 3-12).
The initial distribution of water ranged from a minimum of 18 percent
at the surface in both lysimeters to a maximum of 27 and 29 percent for
the south and north lysimeters respectively. Gravimetric samples could
only be obtained to a depth of 25 cm in both lysimeters due to the high
water content below that depth. Moisture content readings obtained by
the gamma probe were not reliable as measures of absolute moisture
content. Due to the erratic nature of the evaporation rates determined
by weight loss and insufficient data to calculate the evaporation from
the water content measurements, this data set was considered to be
unreliable as a measure of evaporation of water from the soil. Soil
temperatures followed the diurnal pattern as expected (Figure 3-14).
Maximum surface temperatures were typically 2 to 3C above the air
temperature and increased as the soil surface dried. The soil
temperature measured 80 cm below the surface was observed to remain
fairly constant during the four day test
The quality of the data obtained during tests conducted from


39
is defined as the fully turbulent region where the vertical turbulent
fluxes of mass and energy do not change appreciably from that at the
surface (Brutsaert, 1982). In other words, the vertical flux of mass
and energy is constant.
According to Brutsaert (1982), Prandtl introduced the use of the
logarithmic wind profile law into meteorology in 1932. This is an
approximation of the wind velocity profile in the surface sublayer
WS = -Hi- In (-!-) (2-42)
K Zg
where
WS
=
wind speed [ m*s-1]
U*
=
shear velocity
r V2
v P '
ZO
=
surface roughness height [ m ]
k

von Karman constant [dimensionless]
To
s
shear stress at the surface [N*m~2]
P
s
density of air [kg*m'3]
According to Sutton (1953), for practical purposes, the shear velocity
can be estimated as
U*
WS
Iff
(2-43)
The bulk mass transfer within the surface sublayer is defined by
Qm = Ce,rWSr (Pys ^vr)
(2-44)


138
(4-8)
The Fourier number is the ratio of the elapsed time to a characteristic
time for the sample and represents a dimensionless time scale (Equation
4-9). The characteristic time of the sample can be defined as the
square of the characteristic length of the sample (L^) divided by the
thermal diffusivity of the sample.
For values of the Fourier number greater than approximately 0.2, only
the first term of the infinite series solution is necessary to provide
and accurate expression for the temperature ratio as a function of
time. Thus, equation (4-4) can be simplified to
6{z) = Cj(z) exp(-aj jj- t)
(4-10)
Note that this expression is a simple exponential function. Taking the
natural logarithm of both sides of equation (4-10) yields
ln[(z)] = lntcjOs)] (af S )t (4-11)
which is a straight line that is a function of the thermal diffusivity
of the material. For various positions within the sample, the line


Cumulative Precipitation (mm)
122
Figure 3-21. Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
February 3-5, 1987.


71
ceramic stones (30 x 30 x 2.5 cm) were installed in the bottom of each
lysimeter and connected to vacuum tubing for the purpose of removing
excess water from the lysimeter. The lysimeters were then placed in
the pits and leveled using the top nuts of the supporting rods to
adjust the length of the rods and to maintain a fairly uniform load on
all of the rods.
The lysimeters were filled to a depth of 3 cm with a coarse sand
to provide free drainage of water to the ceramic stones. A Hi11 hopper
fine sand excavated from a nearby site, was used to completely fill the
lysimeters. This soil was chosen because the profile was of fairly
uniform composition to a depth of 2 m and was typical of the local
sandy soils found in northern Florida and southern Georgia. After
loading the soil into the lysimeter, the soil was saturated with water
and allowed to stand overnight then removed by the vacuum system. This
provided settling to a density which might naturally occur in the field
after a long period of time.
Load cells manufactured by Hottinger Baldwin Measurements (HBM
Model USB10K) were then installed using a ball and socket connection
between the hanger beam and the load cell. A thin coat of white
lithium grease was applied to the surface of the ball and socket to
prevent corrosion of the contact surfaces. The output of the load
cells was a nominal 3 mV/V at full scale and could accept a maximum
supply voltage of 18 VDC. A regulated power supply provided 10 VDC
excitation voltage to each of the four load cells. Supply and output
voltages for each of the load cells was monitored using a single
channel digital voltmeter (Fluke Model 4520) and a multiple channel
multiplexer (Fluke Model 4506). Output was normalized using the ratio


173
and rewetting overnight. The soil with the lowest hydraulic
diffusivity experienced the largest variation in surface water content.
The soil dried to the residual water content (0.005 cm^/an^) then
recovered during the night. The overnight rewetting was not sufficient
to raise the water content to the same level at which the simulation
had started. The maximum soil water content continued to decrease each
day as more water was removed from the lower depths. The amplitude of
the diurnal variation of surface soil water content decreased as the
hydraulic diffusivity increased indicating the increased capacity of
the soil to redistribute water toward the surface. The soil recovered
to the same water content during the night using multipliers of 2.0 and
5.0. This is the reason that the evaporation rates were identical for
the two calibration runs.
The fact that increasing the hydraulic diffusivity alone did not
achieve the required increase in the simulated evaporation indicated
that the hydraulic diffusivity was not the only parameter in the model
limiting evaporation. This was also supported by the fact that the
initial high rate of evaporation that was noted in the experimental
data did not occur in the simulation results even when the surface
water content was almost the same as on the previous days. The effect
of the boundary conditions upon the simulation of evaporation,
primarily the heat and mass transfer coefficients, was investigated.
During the sensitivity analysis for the heat and mass transfer
coefficients, the multiplier for hydraulic diffusivity was fixed at 2.0
and a single multiplier was applied to both surface transfer
coefficients. The multiplier was valued at 0.5, 2.0 and 5.0 and the
simulation run using the data for the south lysimeter from January 5


Evaporation Rate (mm/h) Evaporation Rate (mm/h)
186
Figure 5-8. Comparison of simulated and experimental hourly
evaporation rates for test 71 (January 11 14, 1987).


113
Figure 3-12. Hourly water loss from the University of Florida, IREP
weighing lysimeters for December 15 19, 1986.


176
multiplier of 2.0 for the transfer coefficients and the hydraulic
diffusivity (Figure 5-29). The variation of water content exhibited
characteristics very similar to the variation of temperature with time
and space. The amplitude of the variation decreased with depth as was
delayed with depth as well. The water content at a depth of 5 cm
steadily decreased as time progressed with very little diurnal cycle of
drying and rewetting.
Summary
The model as originally presented required calibration to allow
the model to adequately describe the transport of water and energy in
the weighing lysimeters. The experimental data exhibited periods of
high rates of evaporation early in the morning immediately following
sunrise. The model as originally described adequately described soil
temperature. It also simulated the evaporation of water from the soil
during conditions in which the soil was the rate-limiting component of
the model, but failed to simulated adequately evaporation rates which
were characteristic of surface-limiting conditions. Possible reasons
investigated were distribution of water to replenish the water supply
for evaporation and the boundary heat and mass transfer coefficients.
The model also showed more downward movement of water than was
measured in the lysimeters. This was probably due to the existence of
two layers of soil with very different hydraulic diffusivity
properties. This was supported by the experimental measurements of
soil water content in which very little change in water content was
noted below a depth of approximately 20 cm and most of the reduction of
water content occurring between the soil surface and a 20 cm. The
simulation of the distribution of water would probably be improved by


227
Julian Date
Figure B-2. Measured relative humidity (bottom) and temperature
(top) of ambient air for evaporation tests conducted
from November 25 to 29, 1986.


205
Figure 5-27. Simulated soil surface temperature resulting from
various values of surface transfer coefficient
multipliers.


166
and raised some question as to its reliability. Relative humidity
ranged from approximately 50 to 100 percent and the air temperature
ranged from 7 to 22 C (Figure B-12). Approximately 2.5 and 2 mm of
water evaporated from the north and south lysimeters, respectively, on
February 3, 1987 (Figure 5-12). Simulated cumulative water loss was
within 0.3 mm of that measured from the south lysimeter after
approximately 10 hours had elapsed from the initiation of the test,
while simulated evaporation was 1 mm below measured water loss from the
north lysimeter. Experimental data indicated that the rate of water
loss from both lysimeters decreased at sunset to zero overnight (Figure
5-13). However, the simulation predicted evaporation to continue at a .
very slow rate between sunset on February 3 and sunrise on February 4
at which time the simulated evaporation rate increased. This caused
the simulated cumulative evaporation to surpass the observed cumulative
water loss for the south lysimeter. Simulated and observed evaporation
for the north lysimeter commenced at approximately the same time and at
the rate of evaporation. However, rate of water loss from the
lysimeters was observed to become zero at approximately 1500 on
February 4 while the simulated water loss continued at a relatively
high constant rate until near sunset at which time it approached zero
as well. Simulated soil surface temperature was always higher than
measured surface temperatures (Figure 5-14). The excessive simulated
surface temperature during the nighttime hours led to a small but
positive difference in the vapor concentration between the soil
surface and the atmosphere (Figure 5-15) and would explain the
continued water loss at night predicted by the model. The vapor
density deficit would also indicated that evaporation should be


171
The procedure for calibrating the model was to use multipliers to
vary the desired parameters of the model. This calibration focused
upon the effect of the hydraulic diffusivity of the soil (Dj_) and the
surface heat (h^) and mass (hm) coefficients primarily due to the
uncertainty of their values used during the validation phase of
modeling. A sensitivity analysis was performed by running the
simulation using several values of the multiplier for hydraulic
diffusivity, then selecting a value for the remaining calibration which
seemed to provide the least deviation between evaporation rates and
soil surface temperature. Since the calculation of the two surface
coefficients were derived from the same basic equations and are
proportional to each other, then a single multiplier was utilized for
both the heat and mass transfer coefficients. All other soil
properties such as heat capacity, thermal conductivity, soil water
potential and diffusivity of water vapor were allowed to vary according
to the temperature and water content of the soil as described in
Chapter II. This sensitivity analysis and calibration procedure was
used primarily to establish limits of values of the multipliers for
more precise calibration in future research efforts.
Sensitivity analysis for the effect of varying the hydraulic
diffusivity upon evaporation and soil temperature was conducted first.
Multipliers for the hydraulic diffusivity were 0.5, 2.0 and 5.0 and
shifted the relationship of diffusivity to water content vertically and
did not make any shifts with respect to water content (Figure 5-21).
Very little difference was seen in the simulated cumulative evaporation
using the hydraulic diffusivity multipliers of 2 and 5 until the
beginning of the third day of the simulation (January 7), at which time


Sum of Squared Residuals
65
Figure 2-8. Sum of squared residuals for the soil profile mass balance
after a 72 hour simulation using various time steps.


19
Dt
diffusivity of water (liquid, and,vapor) due to
temperature gradients [m
quid and Vi
i2.s-1.Qk-1]
Tb +
Da uSah$r)
Pvi
Q(0.T)
heat f!ux within the soil due to movement of water,
liquid and vapor [W*m'3]
a soil tortuosity factor 2/3)
fi a coefficient of thermal expansion [kgnr3,0^1]
r1 = vapor transfer coefficient [dimensionless]
h = relative humidity
These equations were originally developed and presented by de Vries
(1958) with good agreement with field observations being achieved.
Diurnal fluctuations of water content were damped out below a depth of
4 cm. Particular attention was given to the magnitudes of Dj Djl and
D-py. It was noted that the Dj was small during the night but has a
significant contribution to the flow of water during the day. The
vapor component (D-py) was less than the liquid component (Djl) for
water contents greater than 0.30 m3/m3 and sensitive to changes in soil
temperature.
Jackson (1964) studied the non-isothermal movement of water using
equations (2-16) and (2-17) and concluded that classical diffusion
theory could be used satisfactorily if the relative vapor pressure is
greater than 0.97 with no modification to the diffusivity term.
However, if the relative vapor pressure is less than 0.97 then the
diffusivity must include the effect of the temperature gradient.
Model Objectives
The models presented thus far each have advantages and
disadvantages. The mechanistic models are based upon sufficient theory
that the diurnal fluctuations of the soil water and temperature


38
simulations over the range of the water contents expected to occur in
the fields.
For modeling purposes, the Van Genuschten method was employed for
published potential-water content data for a Mi11 hopper fine sand
(Carlisle, 1985). The water release curve usually varies with depth
due to the spatial variation of soil composition. Data for the a
single water release curve for a uniform soil profile was obtained by
averaging the volumetric water content for the specified water
potential over the A-l and A-2 horizons of a Mi11 hopper fine sand. The
nonlinear regressions were then performed to yield the residual water
content, and the regression coefficients, a and n.
Surface Transfer Coefficients
The final parameters to be estimated are the surface heat and mass
transfer coefficients. Penman (1948) used an empirical wind function
to calculate the mass transfer coefficient based upon wind speed.
hm = a WSb (2-41)
where
hm = surface mass transfer coefficient
WS wind speed
a,b = empirical constants
Another approach is to use the equations of motion to describe mass and
energy transfer within the atmospheric boundary layer. Brutsaert
(1982) presented a detailed review of the equations of motion as
applied to the atmosphere. The basic assumption in these analyses is
that the boundary layers for momentum, energy and mass are similar.
Consequently, the surface sublayer becomes the area of most concern and


211
of depth and time. A permanently mounted infrared thermometer on each
of the lysimeters would provide a means to continuously monitor soil
surface temperature as well as plant canopy temperatures in future
tests. Use of an infrared thermometer could eliminate the possibility
of erroneous surface temperature readings due to incorrect placement of
the thermocouple.
This model is not suitable for use on a personal computer in its
current state. The memory requirements, small time step, and small
grid spacing require the power and batch processing capability of a
mainframe computer. However, this model could be used to determine
coefficients for simpler models that assume that evaporation is a
surface phenomenon. For instance, this model might be used to
determine the soil resistance term for the models developed by Jagtap
and Jones (1986) and Camillo and Gurney (1986). This model might also
be modified for the purpose of simulating water loss from the soil
during plant canopy development. Modifications to the model would have
to account for shading of the soil surface during plant growth and for
extraction of water from the soil by the plant roots. This model could
be expanded yet further to a two dimensional model to determine water
distribution in the soil between rows during canopy development. Once
water loss from the soil becomes insignificant compared to that lost by
transpiration from the plant, a simpler two-dimensional model for
unsaturated flow could be used.


CHAPTER II
ENERGY AND WATER TRANSFER IN A SANDY SOIL: MODEL DEVELOPMENT
Introduction
Models of varying complexity may be used to simulate or estimate
the amount of water which evaporates from the soil. The complexity of
the model should be determined by the projected use of the results and
the data available as input for the model. Empirical models generally
relate one or more parameters by regression analysis to evaporation
measured under various conditions. Variables or parameters typically
used in empirical models are evaporation pan data, air temperature, and
day length or solar radiation. Monthly estimates of evaporation for
relatively large areas are typically obtained from empirical models
(Jones et al., 1984).
Resistance analog models employ the concept of the electrical
analog to mass flow, where the flux of water vapor leaving the soil
surface is expressed as a potential (vapor pressure) difference divided
by a resistance. The resistance term for the mass flow is a function
of the boundary layer of the lower atmosphere and the mass transfer
characteristics of the soil (Camillo and Gurney, 1986; Jagtap and
Jones, 1986). These resistance models have the capability of
estimating actual evaporation from the soil on a daily or hourly basis.
Evaporation can also be estimated using mechanistic models
describing the conservation of mass, momentum, and energy in the lower
atmosphere and the soil. These models are the most detailed in their
derivation and have the advantage that they can provide substantial
5


101
Table 3-4. Depths below the soil surface for soil moisture core
samples and corresponding dual probe gamma readings.
Soil Sample Gamma Probe
Depth Depth
( cm ) ( cm )
0.0
-
5.1
2.5
5.1
-
10.2
7.6
10.2
-
15.2
12.7
15.2
-
20.3
17.8
20.3
-
25.4
22.9
25.4
-
30.5
27.9
30.5
-
40.6
35.6
40.6
-
50.8
45.7
50.8
-
61.0
55.9
61.0
-
71.1
66.0
71.1
-
81.3
76.2
81.3
_
91.4
86.4


Hydraulic Diffusiviiy (m^/s)
199
Volumetric Water Content (cm3/em3)
Figure 5-21. Relationship of hydraulic diffusivity to water content
as used in the validation of the couple heat and mass
transfer model and calibration factors of 0.5, 2.0 and
5.0.


32
For the purposes of simulation, the barometric pressure (p) was assumed
to equal the standard pressure (p0). The diffusivity then became a
function of time and space due to the temporal and spatial variation of
the soil temperature. Diffusivity was determined throughout the soil
profile by substitution of the soil temperature in equation (2-34).
Thermal properties
Very little literature exists regarding measurement of the thermal
properties of the soil. Soil composition as well as density and water
content affects the thermal properties of the soil (Baver, 1972). The
thermal diffusivity is the ratio of thermal conductivity to the product
of soil density and specific heat. Density is a property which can be
obtained as a function of depth at a given location by core samples.
Vries (1975) describes a method by which the volumetric heat
capacitance and the thermal conductivity of the soil can be calculated
based upon the volume fractions of the various soil constituents.
The volumetric heat capacitance is defined as the product of the soil
density and the specific heat and can be calculated from:
Cs xqcq + xCm + xoCo + xwCw + xaCa f2'36)
where
C
S
volumetric heat capacity [J-m'3*0^1]
X
=
volume fraction [m3*m~3]
q
=
quartz
m
s
mineral
0
=
organic


Vapor Density Deficit (kg/m3)
( Pv1 Pva )
193
Julian Date
Figure 5-15.
Simulated difference between the water vapor concen
tration at the soil surface and ambient air during test
72 (February 3 5, 1987).


17
According to Fuchs and Tanner (1967) field and laboratory
observations indicated that the evaporation of water occurs in three
distinct stages. The first involves water evaporating from the soil-
atmosphere interface. As the soil surface dries, the water must
evaporate at a location below the soil surface then the vapor diffuse
to the surface. The combination methods of simulating evaporation from
the soil reproduce this process fairly well. However, the evaporation
is assumed to occur at the soil surface. This has the effect of
lowering the soil surface temperature, when in the case of water
evaporating below the dry soil surface, the soil surface temperature
would be higher due to the reduced thermal conductivity of the soil.
This error in soil surface temperature was noted by Lascano and Van
Bavel (1983). Lascano and Van Bavel (1983 and 1986) verified the model
of Van Bavel and Hi11 el (1976) using soil water content and temperature
data obtained from field plots. During the earlier study, the
simulated soil surface and profile temperatures were found to agree
quite closely over the range of 25 to 37 C while the model
underpredicted surface temperature by 2 to 5 C when the soil
temperature was above 37 C. Distribution of soil water is simulated
to within the expected error of measurement. Similar results were
obtained during the 1986 experiment.
Movement of water primarily in the vapor phase in the soil has
been observed by several researchers. Taylor and Cavazza (1954) noted
that the measured diffusion coefficients were higher than that for
water vapor in air and suggested that transport was due the combined
effects of convection and diffusion within the soil pore spaces.
Schieldge et al. (1982) used the following equations of the


210
hydraulic properties of the soil alone were not sufficient to
completely correct the model.
Additional sensitivity analyses focused upon the simultaneous
variation of the surface heat and mass transfer coefficients and their
effect upon evaporation and soil temperature. Increasing the heat and
mass transfer coefficients caused the simulated evaporation rates to
approximate the experimentally observed rates more closely. The
simulated evaporation rate reached a maximum earlier then rapidly
decreased with increased mass transfer coefficients. However, the
increased evaporation caused simulated soil surface temperatures to
deviate from those observed due to the increased latent energy require
ment. Modifying hydraulic diffusivity and the surface transfer
coefficients did not cause the model to completely simulate the
evaporation rates observed in the weighing lysimeters. Future
simulation studies should focus upon describing the mass transfer from
the soil surface after rewetting has occurred.
The results of this study indicated that the interaction between
the air and the soil surface govern the entire evaporation process.
The model was sensitive to variations in the input data. To achieve
accurate results from the simulation, it was imperative that the
meteorological data be accurate since these data provide the boundary
conditions driving the processes of heat and mass transfer below the
soil surface.
It is recommended that future experiments conducted in the
weighing lysimeters include instrumentation to continuously monitor the
soil water status at several depths. This would provide a better
measure of the distribution of water within the lysimeter as a function


177
incorporating at least two zones of soil with different hydraulic
properties. According to Carlisle et al. (1985), the A-l horizon of
the Mill hopper sand used in the model had a saturated hydraulic
diffusivity approximately twice that used in the uniform profile. If
this were incorporated into the model, water would be supplied to the
surface at a relatively high rate until the water content in the top
horizon decreased to the point of requiring water from below.
Calibration and sensitivity analysis of the surface transfer
coefficients indicated that the mechanism by which water is removed
from wet soil surface was not modeled adequately. By increasing the
mass transfer coefficient, the model approached simulating the initial
high rate of evaporation, but caused the soil surface temperature
showed a decrease because of the increased requirement of latent
energy. The inadequacy appeared to be accentuated following a light
dew where a slight film of water was on the soil surface. This water
was then intercepting radiant energy which would normally reach the
soil surface. Perhaps, the thin film of water directly on the surface
was evaporating due to direct absorption of the radiant energy onto the
soil surface resulting in no change in soil surface temperature.
Future improvements in the model should incorporate the ability
to utilize a soil profile with a minimum of two sets of hydraulic
characteristics and should investigate the mechanism of water loss from
the wet soil surface. One possible approach may be to assume that the
latent heat required to evaporate water from the first node of the
model is provided by the radiant energy at the surface rather than from
the sensible heat available in the soil.


Evaporation Rate (mm/h)
180
Figure 5-2.
Hourly simulated and experimental evaporation rates for
the north and south lysimeters during test 70 (January 5
- 10, 1987).


46
9c j+1,n (Tj+l,n" Tj,n) (2-57)
Icd-l.nr ^-(Tj-l.n- Tj>n) (2-58)
dzj-l
noting that dzj represents the difference between the depth of nodes
j+1 and j.
In developing the differential equations for this model, it is
assumed that mass movement is due to diffusion and obeys Fick's law of
diffusion. Therefore, the heat transferred by diffusion of water and
water vapor from node j+1 to j is expressed as
qLj+l,n
n (^j+l,n *j,n)
PwjcpwjuLj 3zj
(Tj+l,n+ Tj,n)
2
(2-59)
9vj+l,n
n (Pvj+l,n Pvj,n)
cpvjvj azj
(Tj+l,n+Tj,n)
(2-60)
and similarly for the heat flux transferred by mass diffusion from node
j-1 to node j as
qLj-l,n
(*j-l,n *j,n)
^wjcpwjuLj dzj_i
(Tj-l,n+Tj,n)
2
(2-61)
qvj-l,n =
cpvjDvj
(^vj-l,n Pvj,n) (Tj-l,n+Tj,n)
3z"
ej-l
(2-62)
The latent heat associated with the phase change of water for the
current time step (Ej>n) must be transferred per unit volume of soil
associated with node j and is determined by
qej,n =
hfgjEj,n
(2-63)


169
The validation process indicated that the model was simulating the
processes intended. Overall performance of the model to this point
were satisfactory. The general trends indicated by experimental data
of increasing daily maximum surface temperatures as daily evaporation
rates decreased were simulated by the model. Deficiencies which were
repeated during each of the tests, were the model's inability to
simulate the high rate of evaporation which occurred most mornings
after the soil had apparently been rehydrated by diffusion of water
from below or by condensation upon the surface from the atmosphere.
This discrepancy could have been caused possibly by errors in hydraulic
properties of the soil near the surface or by the surface transfer
coefficients. Incorrect values of hydraulic diffusivity could limit
the ability of the soil to replenish the water supply at the soil
surface. The fact that very little redistribution of water below a
depth of approximately 20 cm in the measured water profiles indicated
that the hydraulic diffusivity of the soil below the 20 cm depth was
less than that in the region between the soil surface and 20 cm.
Experiments conducted by Jackson (1973) indicated that a dry surface
layer could greatly reduce evaporation of water from the soil. This
was indicated by Jagtap (1986) during evaporation studies and
subsequent modeling. Jagtap's resistance model showed that the soil
resistance increased as cumulative evaporation increased. Jackson
(1973) also stated experimental measurements of hydraulic diffusivity
were generally accepted with deviations of up to a factor of two. A
variation of this magnitude could affect the simulated evaporation of
water significantly. This accepted level of error in values of
hydraulic diffusivity in addition to the fact that the data was


21
development of the model for coupled heat and mass transfer in the
soil.
1. The soil is unsaturated, therefore the soil water potential
is primarily due to osmotic and matric potential.
2. Water vapor behaves as an ideal gas.
3. The movement of water in the liquid and vapor phases occurs
due to concentration or potential gradients.
4. The liquid and vapor phases are in thermodynamic equilibrium
within the soil pore spaces.
Using the conservation of mass, the rate of change of the water
vapor within the soil air space was described in the model as
3pv
IT
4z (Dv
Spy
~dz
+ Ev(z,t)
(2-17)
where:
fiy
mass concentration of water vapor in the soil air
space [kg*m'3]
Dv = Diffusivity of water vapor in the soil air space
[m2*s_1]
or Da
Da = Diffusivity of water vapor in the air [m2*s_1]
a = soil tortuosity [dimensionless]
Ev(z,t) rate of phase change of water [kg*m^r*s-1]
t = time [s]
z = depth below the soil surface [m]
The time rate of change of the concentration of water vapor at any
point in the soil space is a function of the rate of water vapor
diffusing from other regions of the soil and the rate at which water
changes from the liquid to the vapor phase. The evaporation rate is


26
vapor density can be determined from the saturated vapor pressure using
the ideal gas law.
p V = n R T
(2-27)
where:
p: pressure of the gas
V: volume of gas
n: number of moles of gas present
R: ideal gas constant
T: absolute temperature of gas.
The ideal gas law may also be applied for the individual components of
a gas mixture with the pressure being the partial pressure of the
component gas and using the number of moles of the component gas.
Substitution for the molecular weight of water and solving for the
density of water vapor one obtains
es(T)
pvs = rw t (2'28)
where:
pys = concentration of water vapor at saturated vapor pressure
per unit volume of air [kg*m"3]
es = saturated vapor pressure at given temperature [Pa]
T = temperature of free water surface [^C]
Rw = ideal gas constant for water vapor
= universal gas constant (R) divided by the molecular
weight of water
= 461.911 [m2*s'2*K-1]
The above equation (2-28) is for the case in which the chemical
potential of the water is zero. However, due to matric and osmotic


45
vapor phases from a region of differing temperature less the latent
energy required for evaporation of water. Examining this for the
node in the discretized domain, the above equation can be expressed in
terms of heat flux transferred (q) into node j from the surrounding
nodes as
dx*dy*dwj)qsj = dx-dy (qCj+l,n + + U.j-l.n + Qvj+^n + - (dx-dydwj)qej,n (2-55)
The subscripts in the above representation denote
s
c
e
L
v
j
j+1
j-1
n
and
change in sensible heat
heat conduction
latent heat
heat transfer due to diffusion of water (liquid)
heat transfer due to diffusion of water vapor
the current node
the node immediately following node j, and
the node immediately preceding node j.
the current time step
dx*dy : cross-section area normal to z axis
dx*dy*dwj : volume of the node j
dwj : height of the cell for node j
The change in sensible heat per unit volume of node j from time step n
to n+1 is
Tj,m-1 Tj,n
j = cs
(2-56)
The heat flux conducted from nodes j+1 and j-1 to node j at a given
time step, n, is described by Fourier's law of heat conduction and is
expressed as follows


37
water potential. Measurements of the soil water retention curve
require considerably less detail and are generally published for a
wide variety of soil types. Van Genuschten (1980) presented a method
by which the hydraulic conductivity could be calculated for unsaturated
soils using the soil water retention curve and the saturated hydraulic
conductivity. The Van Genuschten approach involves estimation of an
equation for the soil water retention curve of the form
0 =
(2-39)
where
6 = dimensionless water content
6 volumetric water content [m3*nr3]
6r = residual volumetric water content [m3*m'3]
0S = volumetric water content at saturation [m3*m'3]
ip soil water potential [ m ]
m,a = regression coefficients
n = (1 m)"1
The coefficients m and a are determined by nonlinear least squares
regression of the soil water retention curve. That function is then
substituted into equations for the hydraulic conductivity presented by
Mualem (1976). The resulting expression for the diffusivity is
D(0) = ks e(*5 Vto) [(i_0l/nym+ (l-0l/m)m-2] (2-40)
am(0s-0r)
The Van Genuschten method yields a continuous function for the
hydraulic diffusivity which is highly desirable for numerical


Depth ( m ) Depth (
188
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-10. Simulated volumetric water content as a function of
depth for various times during the simulation.
(January 11 14, 1987).


54
run to simulate the response of the soil system under constant
radiation, ambient air temperature and relative humidity conditions for
a period of 72 hours using various time steps to evaluate the numerical
stability and numerical error characteristics of the procedure.
Thermal and hydraulic properties of the soil were assumed to be
constant for the duration of the simulation for an Millhopper fine
sand. The water content and temperature profiles were assumed to have
a uniform initial distribution as were density and soil porosity.
Overall mass and energy balances were calculated for the soil
profile during the simulation. Any residual in these balances would
constitute error arising from the numerical algorithm and was
accumulated on an hourly basis for analysis purposes. The sum of the
squares of the residuals for the energy and mass balances for the
simulation period are presented in Figures 2-7 and 2-8, respectively,
for the different time steps. The sum of the squares of the residuals
(SSR) for the energy balance decreased when the time step of was
increased from 30 to 60 s then remained fairly constant for larger time
steps. The square root of the SSR represents an estimate of the
standard error of the estimate of the total heat flux. For the time
steps of 60, 120, 300 and 600 s, this represents approximately one
percent of the cumulative soil heat flux. The standard error of the
mass balance was approximately two percent of the cumulative
evaporation over the 72 h simulation (600 s time step). It is expected
that as properties are varied with time, that the standard error would
increase due to increased gradients in soil temperature and water
content particularly near the surface. Therefore, a time step of 60 s
was utilized in the model validation and sensitivity analysis.


MODELING THE EVAPORATION AND TEMPERATURE
DISTRIBUTION OF A SOIL PROFILE
BY
CHRISTOPHER LLOYD BUTTS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
¡1 J5JE LIBRARIES
1988


Cumulative Evaporation (mm) Cumulative Evaporation (mm)
185
Julian Date
Figure 5-7. Comparison of simulated and experimental cumulative
evaporation for both lysimeters during test 71 (January
11 14, 1987).


73
surface). The normalized output was then recorded as the weights were
removed.
To account for unequal distribution of weight due to variations in
soil density and slope of the lysimeter, the above procedure was
performed such that weights were placed directly over a single load
cell and repeated for each load cell and by placing the weight in the
center of the lysimeter as well.
Soon after the load cells were installed and calibration
completed, one load cell from each lysimeter was damaged by lightning
and removed for repair. Since during the calibration procedure, the
normalized output of the individual load cells was recorded,
regressions for the remaining load cells for each lysimeter could be
determined. The calibration was repeated after the load cells were
repaired and installed. Regressions for each of the lysimeters were
obtained for the calibration data with three and four load cells
(Table 3-1). It was determined in both cases that the slopes were not
significantly different at the 99 % confidence level for the two
lysimeters and thereby allowing the same regression for both lysimeters
to be used. Since, interest was in changes in weight and not in
absolute weight, the intercept would not be significant. The
calibration curves used in data analysis are shown in Figure 3-3.
Resolution of the lysimeters based upon the specifications of the
voltmeter, and standard error of the regression parameters was
calculated to be approximately 0.02 mm of water.
Temperature Measurement
The vertical distribution of temperature within the soil was
measured by monitoring the output of ANSI Type T thermocouples (copper-


31
diffusivity, or coefficient of diffusion, is a measure of the number of
collisions and molecular interaction between molecules of water vapor
and the other constituent components of the air (ASHRAE, 1979) and is a
function of temperature, total air pressure and partial pressure of
water vapor. Eckert and Drake (1972) presented the following equation
to calculate the diffusivity of water vapor in air.
(2-33)
where
Dva = diffusivity of water vapor in air [m^'s-^]
p = atmospheric pressure [Pa]
p0 = reference atmospheric pressure [Pa]
= 0.98 X 105 Pa
T = air temperature [K]
T0 reference air temperature [K]
= 256
However, for water vapor in the soil air space, the path is more
convoluted, thus increasing the probability of collisions with other
particles and lowering the kinetic energy of the water vapor molecules.
To account for the increased path length within the soil, a tortuosity
factor (a) has been introduced to reduce the effective coefficient of
diffusion in other porous materials. De Vries (1958) used a value for
the tortuosity of 0.667. Using this information the coefficient of
water vapor in the soil (Dv) becomes
D,
v
a D
va
X 10
-5
(2-34)


142
content. All samples were allowed to stand overnight to equilibrate to
room conditions.
A water bath was agitated by forcing compressed air through a
perforated baffle located in the bottom of the tank. Air pressure was
maintained at 83 kPa for all tests. The water temperature was
maintained approximately 3 C below the initial temperature of the soil
samples by a set of cooling coils and a proportionally-controlled
heating element. The bath temperature was controlled to within 0.1
of the set point. Each sample was weighed. The initial temperature
of the sample was recorded. The sample was placed in the water bath
until the sample temperature was within 0.2 C of the water
temperature. Tests typically required two to four minutes to complete.
Bath and sample temperatures were recorded approximately every 1 to 2
seconds using a Fluke 4520 digital voltmeter controlled by a DEC
PDP-11/23 minicomputer. The sample was removed from the water, and the
exterior surface was dried and weighed to determine if arty of the
seals had leaked during the test. If leakage occurred, the container
was opened, emptied and a new sample prepared.
After testing, the samples were allowed to equilibrate overnight
to room temperature. Three repetitions of the test were conducted for
each soil sample. Tests were also conducted for soil bulk densities pf
1500 and 1600 kg/m3. A total of nine tests were conducted for each
water content and bulk density treatments.
It was necessary to determine the convective heat transfer
coefficient (h^) for each testing session. This was achieved by
recording the transient temperature response of a copper cylinder with
the same dimensions as the soil samples. Since the thermal properties


155
Volumetric Water Content (cm^/cm^)
Figure 4-3. Experimental values of the thermal diffusivity of a
Mi11 hopper fine sand as a function of volumetric water
content at various dry bulk densities.


dPy
ST
30
= 0 (2-32c)
z=z0
The system defining the movement of water vapor, liquid water and
energy is defined by a system of partial differential equations (2-22,
2-25, 2-26) with boundary conditions (2-31a-c, 2-32a-c). The energy
equation is coupled to the continuity equations by the rate at which
the water is changed from liquid to vapor and through the movement of
sensible heat associated with the flux of water between zones of
differing temperatures. The conservation of water vapor is coupled to
the soil temperature indirectly in the calculation of saturated vapor
pressure. This coupling, along with the variation of the soil
properties with time and space, renders an analytical solution beyond
reach, therefore; a numerical solution was required.
Determination of Model Parameters
Solution of the governing equations for the energy and mass
balance for the soil requires knowledge of the properties relating to
the soils ability to diffuse heat, water, and water vapor. The other
parameters to be determined relate to the rate at which heat and water
vapor are dissipated from the soil surface to the air. In order to
determine a solution, relationships for determining thermal, hydraulic
and vapor diffusivities as well as the surface transfer coefficients
must be determined.
Diffusivitv of Water Vapor
Diffusivity is the constant of proportionality relating flow to
the gradient in potential as stated in Fick's Law of molecular
diffusion. For the case of water vapor diffusing through air, the


57
Table 2-3. Mesh spacing used in grid generation for the coupled
heat and mass transfer model for sandy soils.
Distance to
Node
Depth
Cell Width
next node
(cm)
(cm)
(cm)
1
0.0
0.0
0.5
2
0.5
1.0
1.0
3
1.5
1.0
1.0
4
2.5
1.0
1.0
5
3.5
1.0
1.0
6
4.5
1.0
1.5
7
6.0
2.0
2.0
8
8.0
2.0
2.0
9
10.0
2.0
2.0
10
12.0
2.0
2.0
11
14.0
2.0
2.0
12
16.0
2.0
2.0
13
18.0
2.0
2.0
14
20.0
2.0
3.5
15
23.5
5.0
5.0
16
28.5
5.0
5.0
17
33.5
5.0
5.0
18
38.5
5.0
5.0
19
43.5
5.0
5.0
20
48.5
5.0
5.0
21
53.5
5.0
7.5
22
61.0
10.0
10.0
23
71.0
10.0
10.0
24
81.0
10.0
10.0
25

91.0

10.0

10.0


3*4

18110

io!o

io!o
35
191.0
10.0
10.0
36
201.0
10.0



96
during the August test due to equipment failure of the AWARDS data
system.
The data for the period from August 25 to 29 appeared to exhibit
some of the instability shown in the first two data sets (Figures 3-29
and 3-30). In contrast to the earlier data, the peak evaporation was
followed by a period of little or no evaporation then a gain of
approximately 1 mm of water possibly due to evaporation. The fact that
both lysimeters behaved similarly with very little time shift in events
indicated that the weight gains actually occurred. The measured water
profiles and soil temperature profiles are shown in Figures (3-31) and
(3-32), respectively for both lysimeters.
Summary
A system to measure evaporation of water from the soil as well as
the distribution of water and temperature in the soil as a function of
time was designed, installed and instrumented. The weighing lysimeters
were instrumented with load cells to support their full weight and a
single calibration curve was developed to analyze the weight data
collected on a continuous basis. The weighing lysimeter system was
capable of detecting hourly rates of water loss from the soil as low as
0.02 mm/h. Soil temperatures were recorded on a continuous basis
throughout the soil profile. A dual probe density gauge was utilized
to measure the distribution of water within the soil on a semi-daily
basis. The ambient weather conditions were obtained from a nearby
weather station.
Future experiments conducted in the weighing lysimeters should
utilize a non-intrusive method to measure the distribution of water
within the soil. A system utilizing tensiometers or some other


Depth ( m ) Depth ( m )
120
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 3-19. Measured vertical distribution of volumetric water
content of the soil in the lysimeters located at the
University of Florida, IREP on January 11 and 14, 1987.
1


107
Dry Bulk Density (g/cm^)
Figure 3-6
Soil porosity as a function of dry bulk density for a
Mi11 hopper fine sand.


215
Jones, J. W.t L. H. Allen, S. F. Shih, J. S. Rogers, L. C. Hammond, A.
G. Smajstrla and J. D. Martsolf. 1984. Estimated and measured
evapotranspiration for Florida climate, crops and soils. Bulletin
840, Agricultural Experiment Stations, Institute of Food and
Agricultural Sciences, University of Florida, Gainesville, FL.
Kimball, B. A. 1983. Canopy gas exchange with the soil, in
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ed. H. H. Taylor, W. R. Jordan, and T. R. Sinclair. ASA-CSSA-
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Kimball, B. A. and E. R. Lemon. 1972. Theory of soil air movement due
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measurement of evaporation from a bare soil. Soil Sci. Soc. Am.
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York.
Matthes, R. K., Jr. and H. D. Bowen. 1968. Steady-state heat moisture
transfer in an unsaturated soil. ASAE paper no. 68-354. American
Society of Agricultural Engineers, St. Joseph, MI.
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temperature. ASAE paper no. 87-1086. American Society of
Agricultural Engineers, St. Joseph, MI.
McMillan, W. D. and H. A. Paul. 1961. Floating lysimeter.
Agricultural Engineering 42(9):498-499.
Novak, M. D. and T. A. Black. 1985. Theoretical determination of the
surface energy balance and thermal regimes of bare soil. Bound.
Layer Meteorology. 33:313-333.
Ozisik, M. N. 1980. Heat Conduction. John Wiley and Sons, New York,
NY.


Cumulative Evaporation (mm) Cumulative Precipitation (mm)
108
20
10-
5-r
ol
Iii.|i
+
T
i
1
I
!
Figure 3-7
Cumulative water loss and precipitation obtained from
the University of Florida, IREP weighing lysimeters for
November 25 29, 1986.


206
Figure 5-28. Simulated surface water content using various
multipliers for the heat and mass transfer coefficient
and a multiplier for hydraulic diffusivity (2.0).


141
adaptation of the method presented by Gaffney et al. (1980) for
measurement of thermal diffusivity in fruits and vegetables was
utilized.
Procedure
Sample containers were constructed of 102 mm lengths of 25 mm
(I.D.) copper pipe. A hole was drilled midlength of the sample
container to permit insertion of a 36 gauge ANSI Type T (copper -
constantan) thermocouple after soil was placed in the container. One
end of the sample container was closed by soldering a 30 mm square
copper sheet to the end of the tube. A second copper sheet was secured
over the remaining end with silicone adhesive after the sample
container had been filled with the soil.
Using a core sampler, approximately 4 kg of a Millhopper fine sand
was obtained over a depth of one meter. The soil was then dried in a
convection drying oven at a temperature of 125C for a period of 24 h
prior to any sample preparation. Soil samples were prepared by
measuring the amount of dry soil to obtain a dry bulk density of
1300 kg/m^. Water was added to the oven-dried soil to achieve a water
content of 2, 4, 6, 8, 10, 12 or 15 percent (weight basis). Sample
containers were then filled with the pre-measured soil using manual
vibration to prevent voids from occurring within the sample. After the
containers were filled with the soil, the 30 mm square sheet was placed
on the remaining open end and sealed with silicone adhesive. A
36 gauge thermocouple was then inserted into the center of the sample
in a radial direction. Silicone adhesive was used to seal the opening
through which the thermocouple was inserted to prevent leakage of water
during the test. Three samples were prepared at each moisture


63
Figure 2-6. Flowchart describing the solution algorithm to solve the
system of finite difference equations for a coupled heat
and mass transfer model.


85
of each soil sample was 0.05 cm3/cm3. The mass of soil placed in the
50 cm3 sample cup of the air pycnometer was that required to achieve
the prescribed levels of dry bulk density. The test was performed for
the sample in the cup and the void volume recorded. The volume
occupied by the water in the sample was added to the measured void
volume to yield the total pore space in the soil. A total of three
measurements were made for each sample. The test was also repeated
for a single soil sample with the maximum density achievable
(1.64 g/cm3). The porosity was calculated for each measurement by
dividing the 50 cm3 sample volume into the void volume and were
averaged for each bulk density level (Table 3-3). The repeatability of
the experiment was indicated by the standard deviation of the porosity
for each of the samples, while the accuracy of the measurement was
indicated by the standard deviation for the average of all measurements
for a given density level. The porosity was found to vary linearly
with bulk density (Figure 3-6). The error bars shown in Figure 3-6
indicate the standard deviation of the soil porosity at each bulk
density level. The porosity measurements were then used in defining
the soil profile for the validation and calibration simulations.
Experimental Procedure
Experiments were designed to monitor the energy and mass transport
processes from a bare soil surface for two weighing lysimeters on a
continuous basis. Automatic data collection was performed by a Digital
Electronics Corporation (DEC) PDP-11/23 mini-computer equipped with an
IEEE-488 interface board and utilizing the RT-11 operating system. The
RT-11 operating system provided several system subroutines which were
primarily for scheduling the data collection as well as other time


34
saturation, the thermal conductivity begins leveling off to some
relatively constant value. Another complicating issue in calculating
the thermal conductivity of the soil is that the temperature gradient
in the solid and liquid portion of the soil can be significantly
different from that in the gaseous phase of the soil.
De Vries (1975) presented a method by which the thermal
conductivity could be calculated using a weighted average of the
various soil constituents where the weighting factors were a product of
the individual volume fractions and geometric factors as follows
X =
where
X
Ai =
*i -
^i =
i
The geometric weighting factor depends upon geometric
configuration of the soil particles and the incorporated void space.
It represents the ratio of the spatially averaged temperature gradient
in the i-th soil component to the spatially averaged temperature
gradient of the continuous component of the soil (usually water). For
example, kq represents the space average of the temperature gradient
in the quartz particles in the soil to the space average of the
temperature gradient in the water. The geometric weighting factor can
Wq + W + Wo Ww + kaVa ...
thermal conductivity of the soil [W*m^,0K"^]
thermal conductivity of soil component [W'nr1*0^1]
volume fraction of soil component [m3*m"3]
geometric weighting factor [dimensionless]
quartz (q), mineral (m), organic (o), water (w), air (a)
l


Temperature (C) Temperature (C)
115
Julian Date
Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from December 15 to
19, 1986.
Figure 3-14


Temperature (C) Temperature (C)
192
Figure 5-14. Simulated and experimental soil surface temperatures
during test 72 (February 3-5, 1987).


93
January 5 to 9 and January 11 to 14, 1987 was satisfactory. The
evaporation data was very smooth when compared to the data for the
previous two tests (Figures 3-15 and 3-18). The cumulative evaporation
exhibited maximum daily evaporation on the first or second day of the
test. The relatively small amount of water lost on the first day as
compared to that lost on the second day was due to the time at which
data collection began for the test. In both instances, data collection
did not begin until after initial measurement of the soil water profile
had been completed. Therefore, the water which had been lost prior to
beginning of data collection was not known. Maximum hourly evaporation
rates occurred just around noon on each day with each subsequent day
showing a lower peak evaporation rate than the previous day (Figures
3-15 and 3-18). The volumetric water distribution with depth in each
lysimeter (Figures 3-16 and 3-19) show that most of the water was lost
from the soil above 20 cm. For the two-day period between January 5
and 7, the water evaporated primarily in the soil shallower than 10 cm
with very little change in the water content of the soil below 20 cm
(Figure 3-16). The water then evaporated from between depths of 10 and
40 cm from January 7 to 9. The water content of the soil below 40 cm
remained constant. The south lysimeter lost less water from the upper
portion of the soil from January 5 to 7 than the north lysimeter.
Evaporation of water from the soil continued from the upper layers as
well as from the 20 to 40 cm area of the soil. Similar distributions
were noted for both lysimeters for the January 11 to 14 period. In the
north lysimeter, almost all of the water evaporated from the soil above
20 cm while the soil below 20 cm remained constant (Figure 3-19).
Water was lost from the soil to a depth of 40 cm. Note that the water


Evaporation Rate (mm/h) Cumulative Evaporation (mm)
128
Julian Date
Julian Date
Figure 3-27. Hourly and cumulative water loss measured from
August 12 21, 1987 using the University of Florida,
IREP weighing lysimeters.


Variable Definition (continued)
t
U
U*
V
US
*i
z
z0
a
7
7T
X
v
Pa
Ps
Pv
^vj,n
Pva
Pvs
Pw
0
time [s]
wind velocity [m*s_1]
shear velocity [nrs-1]
volume [m^]
Wind speed [m*s"l]
volume fraction of the i-th soil component [m3*m*3]
depth below soil surface [m]
surface roughness length [m]
soil tortuosity [dimensionless]
psychrometric constant [kPa,cK'l]
diabatic influence function [dimensionless]
soil water potential [m]
thermal conductivity of soil composite at node j
[W*nr**K"l]
kinematic viscosity [m2*s-1]
density of moist air [kg*m3]
dry bulk density of soil [kg*m'3]
water vapor concentration [kgH20*msoil]
water vapor concentration at node j and time step n
[k9H20*msoil]
water vapor concentration of ambient air
[k9H20*mair]
water vapor concentration in the air at saturation
[k9H20*mair]
density of water [kg*m~3]
dimensionless water content
viii


135
DeVries (1975) presented methods for calculating the thermal
properties of the soil based upon the various volume fractions of the
solid, liquid and air constituents of the soil. These equations
estimated the specific heat and thermal conductivity of various soils.
However, extensive tests for a range of soil types as well as moisture
contents and density have not been conducted. The DeVries method has
been shown to estimate the thermal properties within ten percent for
limited soils. Tollner et al. (1984) compared the experimental values
of the thermal conductivity for potting soils consisting of various
mixtures of sand and pine bark to those calculated. They achieved an
average deviation between experimental values and values calculated by
the DeVries method of approximately fifteen to twenty percent.
Objectives
The objectives of this study were to determine a method by which
the thermal diffusivity of the soil could be obtained and to compare
the measured data with that calculated by the DeVries method. A third
objective was to determine a technique by which the measured data could
be used in a simulation of the energy transfer within the soil.
Literature Review
Methods of determining the thermal conductivity or diffusivity of
a material are based upon measuring either the steady-state or
transient temperature response to a particular set of initial and
boundary conditions. Steady-state or equilibrium methods are based
upon Fourier's law of conduction for a one-dimensional homogeneous,
isotropic material (Equation 4-3).
q
(4-3)


knowledge of the parameters affecting evaporation from the soil and
guiding future research. The specific objectives are:
1. to develop a model to simulate the evaporation of water from
the soil which couples the energy and mass transfer
processes;
2. to monitor evaporation of water and vertical distribution of
temperature and water in a sandy soil;
3. to determine the validity of the model by comparison of
simulated evaporation rates, soil temperatures and
volumetric water content to those measured in a sandy soil;
4. to determine the sensitivity of the evaporation process to
changes in various soil properties and environmental
conditions; and
5. to calibrate the model for local conditions using data
collected in Objective 2 utilizing information gained from
the sensitivity analysis.


Cumulative Evaporation (mm) Cumulative Evaporation (mm)
195
Figure 5-17. Cumulative simulated and experimental evaporation rates
for test 73 (February 9 13, 1987).


218
Van Bavel, C. H. M. and L. E. Myers. 1962. An automatic weighing
lysimeter. Agricultural Engineering 43(10)s580-583, 586-588.
Van Genuschten, M. Th. 1980. A closed form equation for predicting
the hydraulic conductivity of unsaturated soils. Soil Sci. Soc.
Am. J. 44:892-898.
Van Wijk, W. R. (Ed). 1963. Physics of Plant Environment. North-
Holland Publishing Co., Amsterdam.
Wilkerson, G. G., J. W. Jones, K. J. Boote, K. T. Ingram and J. W.
Mishoe. 1983. Modeling soybean growth for crop management.
Transactions of the ASAE. 26(1):63-73.
Zur, B. and J. W. Jones. 1981. A model for the water relations,
photosynthesis and expansive growth of crops. Water Resources
Research. 17(2):311-320.


239
Julian Date
Figure B-14. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from February 9 to 13, 1987.


94
content at a depth of 80 cm decreased from the beginning of each test
to the end. Some could have been redistributed toward the surface, but
was more likely to have been distributed toward the bottom of the
lysimeter due to gravity. The soil at a depth below 60 cm had a water
content above the drained upper limit for the Mi11 hopper sand,
approximately 13 percent. The drained upper limit of a soil is the
highest water content at which the soil can exist without free drainage
of water due to gravity. Soil temperatures again exhibited the diurnal
fluctuation as expected with the amplitude of the variation diminishing
with depth until it had been effectively damped out at a depth of 80 cm
(Figures 3-17 and 3-20). Peak to peak variation of the temperature of
the soil surface was generally greater than that of the air
temperature. Close inspection of the surface temperature as compared
to the air temperature showed that the difference between the maximum
air temperature and surface temperature increased as the soil surface
became drier.
The data collected from February 3 to 5, 1987 were shown mainly to
indicate the evaporation which can occur with partially overcast
weather (Figures 3-21, 3-22, and 3-23). Note that only one full day of
measurement were taken before a rainfall of 35 mm occurred. Note that
approximately 2 mm of water (Figure 3-21) evaporated under partly
cloudy skies which was comparable to the amount of evaporation which
occurred under sunny conditions with comparable air temperatures on
January 11 and 12 (Figure 3-18). Examination of the hourly evaporation
rate (Figure 3-22) shows a maximum evaporation rate of 0.3 mm/h was
achieved on February 3, which was lower than the 0.6 mm/h peak noted on
January 11. However, the peak evaporation rate was maintained for a


86
manipulations. The main program and associated subroutines were
written in FORTRAN-66. Subroutines for initializing and retrieving
data through the IEEE interface were written in either assembly or
FORTRAN by Dr. J. W. Mishoe for previous research. All signals from
the sensors used were analog signals and were monitored on 50 channels
of a Fluke 4506 multiplexer and subsequently read via the IEEE
interface from a Fluke 4520 digital voltmeter. Data monitored by the
computer for each lysimeter were net radiation, soil temperature
distribution, and supply and output signals from each of the load
cells.
Net radiation was measured using net radiometers (WEATHERtronics,
Model 3035) mounted 85 cm above the soil surface of the lysimeter. The
net radiometers utilized blackened thermopiles as the sensing element
to detect the difference between incoming and outgoing radiation. A
positive millivolt signal was produced when the incident radiation was
greater than that being reradiated, while a negative signal indicated
that more energy was being radiated from the surface than impinging
upon the surface. Factory calibration curves were used to convert the
millivolt signals to net heat flux (W/m^). The millivolt signal was
read at two minute intervals and total millivolts and number of times
read were recorded on floppy diskette at ten minute intervals.
Thermocouples, supply voltage, and load cell output were recorded at
ten minute intervals as well.
Data files were closed on an hourly basis and new ones opened for
each hour. This minimized the risk of data loss due to power outages
or other computer malfunctions. The most data that would be lost that
had already been recorded would be that for one hour in the event that


BIBLIOGRAPHY
Aboukhaled, A. 1982. Lysimeters. FAO Irrigation and Drainage
Paper 39. Food and Agriculture Organization of the United
Nations, Rome.
ASHRAE. 1979. Mass transfer, pp. 3.1-3.16. In ASHRAE Handbook of
Fundamentals. American Society of Heating, Refrigerating and Air
Conditioning Engineers, New York, NY.
Ayers, P. D. and H. D. Bowen. 1985. Laboratory investigation of
nuclear density gauge operation. ASAE paper No. 85-1541.
American Society of Agricultural Engineers, St. Joseph, MI.
Baver, L. D., W. H. Gardner and W. R. Gardner. 1972. Soil physics.
John Wiley and Sons, Inc., New York, NY.
Black, T.A., G. W. Thurtell and C. B. Tanner. 1968. Hydraulic load
cell lysimeter, construction, calibration and tests. Soil. Sci.
Soc. Amer. Proc. 32:623-629.
Blaney, H. F. and W. D. Criddle. 1950. Determining water requirements
in irrigated areas from climatological and irrigation data.
USDA/SCS Rep. SCS-TP-96. U.S. Government Printing Office,
Washington, DC.
Blankenship, P. D., R. J. Cole, T. H. Sanders and R. A. Hill. 1984.
Effect of geocarposphere temperature on pre-harvest colonization
of drought-stressed peanuts by Aspergillus flavus and subsequent
aflatoxin contamination. Mycopathologea 85:69-74.
Boote, K. J., J. W. Jones and G. Bourgeois. 1987. Validation of
PNUTGRO, a crop growth simulation model for peanut. American
Peanut Research and Education Society Proceedings. 19:40.
Brouwer, R. 1964. Responses of bean plants to root temperatures:
I. Root temperatures and growth in vegetative stage. Annual
Report. Mededeling 235 van het I.B.S., pp. 11-22.
Brouwer, R. and A. Hoogland. 1964. Responses of bean plants to root
temperatures. II. Anatomical aspects. Annual Report. Mededeling
235 van het I.B.S., pp. 23-31.
Brouwer, R. and A. Kleinendorst. 1967. Responses of bean plants to
root temperatures. III. Interactions with hormone treatment.
Annual Report. Mededeling 341 van het I.B.S., pp. 11-28.
212


98
Table 3-1. Regression coefficients for weighing lysimeter
equations of the form... mm = a + b mV/V
No. of
Lysimeter Load Cells Regression Coefficients
North
South
Combined
3
4
3
4
3
4
a = 0.1688
a = 0.1688
a = 0.1346
a = 0.1345
a = 0.1572
a = 0.1572
b = 301.9807
b = 226.4855
b = 300.9117
b = 225.6838
b = 301.4969
b = 226.1227
calibration
R2
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985


139
shifts only by its intercept with the slope remaining constant.
Therefore, accurate placement of the temperature sensor is not critical
to obtaining an accurate measure of the thermal diffusivity.
Gaffney et al. (1980) discussed possible sources of error in the
determination of thermal diffusivity using this technique. A major
source of error was in the determination of the first eigenvalue (aj).
Since the eigenvalues are functions of the Biot number it is imperative
that the Biot number be determined accurately. Many times the Biot
number has been assumed to approach an infinite value. Gaffney et al.
(1980) stated that the Biot number should be at least 200 for the error
introduced by the assumption of an infinite Biot number to be
negligible. If the Biot number is less than 200, then a finite value
for the Biot number should be used to determine aj. The error
introduced was given as
(4-12)
1
Luikov (1968) presented an empirical relationship for determining the
first eigenvalue within 0.1 % for a finite Biot number.
(4-13)
The constants, I, s and (aj),,, are functions of geometry and are
presented in Table 4-2 for the cases of a sphere, an infinite cylinder
and an infinite slab.
Another factor to be considered in the measurement of the thermal


221
calculated using the assumption of thermodynamic equilibrium between
the water adhered to the soil particle and the air in the soil air
space.
Forward direction: 0 < z < zo
energy:
Tj,nfl
+
water:
*j,n+l
vapor:
- (1-Fv)pvj n Fv*DZR*/>vj_1?rH.1
The equations for the lower boundary were formulated by applying a no
flux boundary condition. This was imposed by assuming that an
imaginary node existed one grid spacing beyond the last node (j=nc).
Applying the assumption of no flux across the last node, implied that
the state variable evaluated at the imaginary node (nc+1) had the same
value as that at the node preceding the last node (nc-1). This was
substituted into the conservation equation and yielded equations A-7 to
A-9.
1 N J N HU
(A-4)
Fo*(l+KC2) T 1 Fo*(l-KC2) T
= 1 + fo*DZF'(i-KCi) TJ+ln + 1 + Fo*DZR*(l-KCl) TJn
Fo*DZR*(l+KCl) T dt*hfg
1 + Fo*DZR*(l-KCl) Tj-l,m-l ~C'sj(T + Fo*bZft"*(T-^i)l Ejn
(A-5)
FL (1 FL) fl.
- n+TLW) *J+l,n + "(1 +FL*dZRy
FL*DZR
+ (1 + FL*DZR) "
dt
pw(l + FL*DZR) M
(A-6)
It- ^ + Fv*DZR)^vj,nfl_ Fv*/>vj+l,n


Temperature (C) Temperature (C)
124
30
25--
20--
15
North Lysimeter
Date: 2/3 2/5/87
Julian Date
Julian Date
Figure 3-23. Ambient air and soil temperatures measured at depths of
0, 15 and 80 cm in weighing lysimeters located at the
University of Florida, IREP from February 3 to 5, 1987.


217
Small, J. 6. C., M. C. Hough, B. Clarke and N. Grobbelaar. 1968. The
effect of temperature on nodulation of whole plants and isolated
roots of Phaseolus vulgaris L. South African Journal of Science,
pp. 218-224.
Soer, G. J. R. 1980. Estimation of regional evapotranspiration and
soil moisture conditions using remote sensed crop surface
temperatures. Remote Sensing of Environment. 9:27-45.
Staple, W. J. 1974. Modified Penman equation to provide the upper
boundary condition in computing evaporation from soil. Soil Sci.
Soc. Am. Proc. 38:837-839.
Stephens, J. C. and E. H. Stewart. 1963. A comparison of procedures
for computing evaporation and evapotranspiration. Trans. Int.
Union of Geodesy and Geophysics, Berkeley, CA. 62:123-133.
Sutton, 0. G. 1953. Micrometeorology: A study of physical processes
in the lowest layers of the earth's atmosphere. McGraw-Hill Book
Co., New York.
Tanner, C. B. and M. Fuchs. 1968. Evaporation from unsaturated
surfaces: A generalized combination method. J. Geotech. Res.
73:1299-1304.
Taylor, S. A. and L. Cavazza. 1954. The movement of soil moisture in
response to temperature gradients. Soil Sci. Soc. Am. Proc.
18(4):351-358.
Thornwaite, C. W. 1944. Report of the committee on transpiration and
evaporation. Trans. Am. Geo. Union. 25:683-693.
Tollner, E. W. and R. B. Moss. 1985. Neutron probe vs tensiometer vs
gypsum blocks for monitoring soil moisture status. ASAE paper No.
85-2513. American Society of Agricultural Engineers, St. Joseph,
MI.
Tollner, E. W., B. P. Verma and S. Vandergrift. 1984. Thermal
conductivity of artificial potting soils. ASAE Paper No. 84-1086.
American Society of Agricultural Engineers, St. Joseph, MI.
Troxler Electronic Laboratories. 1972. Model 2376 two-probe density
gauge instruction manual. Troxler Electronic Laboratories, Inc.
P. 0. Box 5997, Raleigh, NC 27607.
Van Bavel, C. H. M. 1966. Combination (Penman type) methods, p. 48.
In Evapotranspiration and Its Role in Water Resources Management.
Proc. Chicago, IL. December 5-6. American Society of Agricultural
Engineers, St. Joseph, MI.
Van Bavel, C. H. M. and D. I. Hi 11 el. 1976. Calculating potential and
actual evaporation from a bare soil surface by simulation of
concurrent flow of water and heat. Ag. Meteor. 17:453-476.


Evaporation Rot. (mm/h) Cumulative Evaporation (mm)
116
Julian Date
Figure 3-15. Hourly and cumulative water loss measured from
January 5-9, 1987 using the University of Florida,
IREP weighing lyslmeters.


APPENDIX A
ADI FORMULATION OF COUPLED HEAT AND MASS TRANSFER MODEL
The partial differential equations governing the heat and mass
transfer in the soil were presented in Chapter 2 along with their
explicit finite difference expressions. Boundary conditions at the
soil surface and the lower boundary were also presented. An
alternating direction finite difference technique requires that
numerical expressions be formulated for the case of indexing the node
number from the soil surface to the lower boundary, then a second set
for use when the index is being changed in the other direction. The
equations for the forward direction (increasing node number) are
formulated by evaluating the spatial derivatives from the previous node
(j-1) to the current node (j) at the next time step (t+dt) which has
already been determined. Then the derivatives involving the node ahead
of the current node (j+1) is evaluated at the current time step. The
resulting equation is then solved for the value of the current node (j)
at the next time step (t+dt). The opposite is done when marching in
the backward direction (decreasing node number). The formulation for
the forward and backward marching difference equations is presented
below. The following are definitions of constants used in the
equations for either direction.
hhdZl WZ1
Bl = t Bl =
Fo
Ajdt
cs,jdwjdzj
DLjdt
dwjdzj"
Fv
Dvj(it
dwjdzj
219


7
evaporation rate [nrs"1]
specific heat of air [J*kg-1*K_1]
density of dry air [kg*nr3]
latent heat of vaporization [J kg-1]
psychrometric constant [kPa*K_1]
saturated vapor pressure [kPa] at temperature T
soil surface temperature [9<]
ambient dew point temperature [K]
resistance to vapor movement from the soil to air
[kg*Sm'4]
The resistance term for the models presented by Conaway and Van Bavel
(1967), Tanner and Fuchs (1968), and Novak and Black (1985) represents
the resistance due to the laminar boundary layer. The boundary layer
resistance is a function of the wind speed, atmospheric instability,
and the surface roughness height. Their models are used to predict
evaporation from a well-watered bare soil surface. Jagtap and Jones
(1986) and Camillo and Gurney (1986) developed resistance terms to
include the boundary layer resistance in series with the resistance of
vapor flow in the soil. The soil resistance term is included since
after the soil surface dries, the water must change to vapor in the
soil below the surface then diffuse to the soil-atmosphere interface.
The soil resistance term developed by Jagtap and Jones was determined
by regression analysis as a function of cumulative evaporation, water
in the soil profile available for evaporation, and a daily running
average of the net radiation. The net radiation empirically accounted
for the heat flux into the soil, while the ratio of the cumulative
where:
E
cpa =
Pa =
hfg =
7
e(T) =
TS -
Td -
R


Net Radiation ( W/m2) Wind
234
Julian Date
Figure B-9. Wind speed (top) and net radiation (bottom) data
collected during lysimeter evaporation studies
conducted from January 11 to 15, 1987.


Evaporation Rate (mm/h)
204
Figure 5-26. Simulated hourly evaporation rates using a calibration
multiplier for Dl (2.0) and various values for the
surface transfer coefficient multipliers.


68
supporting the lysimeter directly by several load cells (Harrold, 1966;
Van Bavel and Meyers, 1961). The weight may also be detected by
supporting the soil container by a bladder filled with fluid and
monitoring the pressure of the fluid or the buoyancy of the lysimeter
container (Harrold, 1966; McMillan and Paul, 1961; King et al., 1956).
The hydraulic method has the disadvantages of being sensitive to
thermal expansion of the fluid and the high possibility of developing
leaks in the bladder and losing the fluid. The lever mechanism has the
advantage of requiring only one relatively low capacity load sensor
thus reducing instrumentation costs. The disadvantage of the counter
balance system is requirement of extensive underground construction to
contain the lever apparatus. This would require disturbing the
surrounding soil and border area and may be considered undesirable in
some cases. The cost of additional supporting structures and
excavation could be considerable as well. Three to four high capacity
load cells are required to support the lysimeter for direct weighing.
The higher cost of the load cells for the direct weigh method may
offset the cost of the increased excavation required for the counter
weight system. Direct-weigh lysimeters can be constructed such that a
high sensitivity can be achieved. Dugas et al. (1985) reported a
resolution of 0.02 mm of water for a direct weighing lysimeter with a
surface area of 3 m^. Most of the designs presented in the literature
have load cells which are difficult to access in the event of need for
maintenance or replacement.
A direct-weigh system was chosen because of its relatively simple
support system design and due to space limitations at the proposed
construction site at the Irrigation Research and Education Park (IREP)


144
An estimate of the thermal conductivity was needed to determine the
Biot number. This was obtained by calculating the volumetric heat
capacitance of the soil based upon the mass average of the various
constituents of the soil as described by DeVries (1975). The thermal
conductivity was determined by rearranging equation (4-2) and
substitution of the values of thermal diffusivity and volumetric heat
capacitance. The Biot number for the sample could then be calculated
and used to determine aj. The error (e) in the thermal diffusivity
associated with calculation of the eigenvalues was calculated using
equation (4-12). If the absolute value of the error was greater than
107 then the thermal diffusivity was calculated using the new
eigenvalue. This iterative procedure was repeated until the error
calculated by substitution of the previous value of aj for (aj)*, was
less than 107.
The values of thermal diffusivity for each of the nine tests for
each water content and density treatment were averaged to determine the
mean value of the thermal diffusivity (Figure 4-3). The error bars
shown in Figure 4-3 indicated the 95 % confidence limits associated
with the experimental measurement and were approximately 5 percent of
the mean for each. Thermal diffusivity was found to increase with
increasing density (Figure 4-3). However, the relative change in
diffusivity from a density of 1300 to 1500 kg/m3 was larger than the
change from 1500 to 1600 kg/m3. The thermal diffusivity increased with
water content to a maximum then decreased as water content continued to
increase. Initially the soil pore spaces were filled predominantly
with air. Air has a relatively low thermal conductivity and volumetric
heat capacitance. Contact area between soil particles was also limited


220
DZR =
dz-
HzT
j-1
The following are definitions for coefficients used in the equations
used when moving in the forward direction:
KC1 =
TX]
cpv,jDvj
+ "XT" Wj-l.rH-r Avj,n+l)
J
NicfM,jDL,j
KCZ 2 Aj ^J+l,n ffj,ni
cpv,jDvj
+ 2 Xj ^vj+l,n" Nj,n)
Forward direction: z = 0
j = 1
energy:
(A-1)
dz-
T 'J n Bi
]3,r*l- Aj(l KC2 + Bi)Rn,n+1 + (1 KC2 + Bi)1 a,rvt-l
1 + KC2
+ (1 KC2+ Bi) Tj+l*n+l
water:
dzj
5j,n+l = + Pr>+1
vapor:
B^m 1
^vj+l,rH-l = "(i +"ITJ ^rw-l + (T+Bt^) Avj+l.m-l
(A-2)
(A-3)
The conservation of energy, water and water vapor are formulated by
setting the time rate of change of the state variable equal to the sum
of the flows from the surrounding nodes minus any sink terms. Latent
heat associated with the phase change from liquid to vapor is a heat
sink term for the energy equation. The vapor equation was solved for
the rate of evaporation at each node and the vapor density was


20
profiles can be observed. They also have the ability to provide
insight into the many processes involved in the evaporative loss of
water from the soil. Saxton (1986) stated that there is a need to
separate the evaporative loss of water from the soil from the losses
through the plant so that better understanding of the individual
processes can be achieved. The mechanistic models provide this
ability.
One might also note that in any of the models discussed
previously, conservation of water is discussed either in the vapor
phase, as in the mass balance methods, or the liquid phase, as in the
combination methods which consider the soil media. The vapor and the
liquid phases are not considered simultaneously.
The objectives in developing a new model were:
1. to account for the water changing from liquid to vapor
phase below the soil surface and diffusing to the
atmosphere;
2. to consider the overall mass continuity, specifically
include the liquid and vapor phases separately;
3. to account for the movement of water vapor in response
to temperature gradients in the soil; and
4. to simulate the diurnal variation of evaporation, soil
water and temperature profiles in response to conditions
at the soil surface.
Model Development
The soil for most purposes may be considered a continuous medium
in which the laws of conservation of mass and energy apply.
Application of the basic principles of thermodynamics to the soil
profile provide the basis for simulation of the temporal variation of
the distribution of water and temperature throughout the soil profile.
The following assumptions were made to simplify or clarify the


Depth ( m ) Depth (
183
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm**)
South Lysimeter
Figure 5-5. Simulated volumetric water content as a function of
depth for various times during the simulation.
(January 5-8, 1987).


Depth ( m ) Depth ( m )
189
Volumetric Water Content (cm^/cm^)
North Lysimeter
Volumetric Water Content (cm^/cm^)
South Lysimeter
Figure 5-11. Initial and final (simulated and experimental)
volumetric water content as a function of depth for test
71 (January 11 14, 1987).


149
sample were not significant at the 95 % confidence level. A least
squares regression with first and second order terms for the volumetric
water content and a first order term for the dry bulk density had an
of 0.799 but did not provide a satisfactory fit to the data for the
range of density and water content tested. Thermal diffusivity was
determined by calculations presented by DeVries (1975) but did not
compare favorably with experimental data. A two-way interpolation
would probably provide the best implementation of this data into a
model. This data might also be used to determine parameters for use in
the DeVries method.


160
occurred 1 to 2 h prior to the simulated maximum. Measured peak
evaporation rates were generally higher than the simulated values but
were shorter in duration than the simulated maximum evaporation rates.
Cumulative evaporation was under-estimated for three of the four
validation data sets. Simulated cumulative evaporation exceeded the
measured water loss during test 73. Since the simulated total water
loss from the soil was under-estimated, the simulated water content
profiles did not agree exactly with measured water content profiles.
However, simulated water profiles indicated movement of water toward
the surface in response to the evaporative demand at the surface as
well as downward redistribution due to gradients in soil water
potential. The simulated soil temperature profiles followed the trends
expected with the soil surface temperature exhibiting the maximum
amplitude in the diurnal cycle. The amplitude of the temperature wave
decreased with depth until a constant temperature was obtained at a
depth of approximately 60 cm. Maximum surface temperatures were
approximately 3 C higher than experimental maximum temperatures, while
simulated minimum surface temperatures were an average of 2 C lower
than measured. Simulated maximum and minimum soil surface temperatures
led ambient air temperatures as was observed in the experimental data.
The model was run using the values of net radiation, wind speed
(Figure B-7), relative humidity and air temperature (Figure B-8) as the
boundary conditions. Simulations were conducted for the north and
south lysimeters using measured values of the soil temperature and
water content as initial conditions. Simulated evaporation rates had a
diurnal variation with a maximum evaporation rate water occurring
around midday (Figure 5-2) then decreased to rates near 0 mm/h


36
1.25. This method yielded values of thermal diffusivity within ten
percent of those measured (de Vries, 1975). Extensive detail regarding
calculation of the thermal conductivity of the soil is given in
de Vries (1963).
The thermal properties were calculated using equation (2-35) for
the volumetric heat capacity and equations (2-36), (2-37) and (2-38) to
determine the thermal conductivity. Volume fractions of the various
soil constituents was determined from soil classification data and
knowledge of the soil bulk density and porosity. Shape factors used
for calculation of the thermal conductivity were for a typical sand
grain (Table 2-1).
Hydraulic Properties
The parameter governing the movement of liquid water in the soil
is a measurable property of the soil and is analogous to the thermal
diffusivity. The hydraulic diffusivity is a derived property of the
soil (i.e. not directly measured) and is defined as the hydraulic
conductivity divided by the specific water capacity of the soil (Baver
et al., 1972). The hydraulic conductivity is the constant of
proportionality for the diffusion of water in response to a gradient in
the soil water potential (Figure 2-2), while the specific water
capacity is the slope of the soil water retention curve. The
hydraulic conductivity (Figure 2-3) and the specific water capacity
(Figure 2-2) vary according to the soil composition as well as the soil
water potential. The measurement of hydraulic conductivity can be
accomplished by several methods, but most all require meticulous
control of the potential gradients and a great deal of time. This is
especially true if measurements are desired over a wide range of soil


145
to many small points. As the water content increased, a film was
formed around the soil particle causing a continuous contact surface.
Increasing the water content increased the heat capacitance and the
thermal conductivity as well by displacing some of the air from the
pore spaces. The increase in thermal conductivity and contact area
occurred at a faster rate than did the increase in thermal capacitance.
At some point, the rate of increase in contact area and thermal
conductivity becomes slower than that of heat capacitance; therefore,
the thermal diffusivity begins to decrease from the maximum.
Sources of experimental error arise from measurement errors and
analytical errors. Analytical errors can arise from errors in
determination of the slopes of the temperature response for the copper
cylinder and the soil sample and estimates of the volumetric heat
capacitance. Inclusion of points from the nonlinear portions of the
temperature response curve for either the copper or the soil samples
could cause a change in the slope. A miscalculation of heat
capacitance of the soil could influence the final determination of the
thermal diffusivity, also. To determine the relative influence of
these type errors the solution routine was run and varying each of the
parameters over a range experienced during the tests. The slope of the
temperature response of the copper cylinder and heat capacitance had a
minor effect upon the final value of the thermal diffusivity (Figure
4-4). A variation of plus or minus 50 % in the heat capacitance
exhibited a plus or minus 2 % change in the thermal diffusivity. A
thirty percent reduction in the slope of the temperature response of
the copper cylinder caused a 2 % increase in the thermal diffusivity.
The deviation in thermal diffusivity decreased to zero as the slope of


165
progressed and more water was lost from the soil surface. The
simulation indicated that the soil water content at the surface was
approximately 0.07 cm3/cm3 at 1200 on January 12, and decreased by
approximately 0.02 cm3/cm3 on January 13 and 14 (Figure 5-10). The
soil at a depth of 40 cm remained at a constant .095 cm3/cm3. The
simulation also indicated some vertical redistribution of water in the
profile. Measured water content profiles did not indicate the
occurrence of the vertical redistribution of water to the same extent
as the simulation (Figure 5-11). Comparison of the simulated water
profile to the measured profile showed that more water was lost from
the area above 5 cm for the simulation while the profile in the
lysimeters showed that water was lost from the top 20 cm.
The data for test 72 was recorded under partly cloudy, windy
conditions with the test being terminated prematurely due to rainfall.
Daily maximum net radiation during test 72 was approximately 400 W/m3
which was comparable to the net radiation levels measured during
previous tests (Figure B-ll). The variation of the net radiation
during test 72 was more irregular than in previous tests. Minimum net
radiation was higher than that of previous tests indicating the
presence of cloud cover during the day. According to field notes taken
during the experiment, gusty wind conditions prevailed during test 72.
However, the wind speed data (Figure B-ll) was not characteristic of
gusty wind conditions. Data indicated short periods of relatively high
gusts, but dropped to a constant speed of 0.5 m/s for long periods of
time. One would expect the wind speed data to exhibit characteristics
similar to that recorded during earlier tests. The long periods of
constant wind velocity were not characteristic of the previous tests


43
The diabatic influence function accounted for the transfer of air
movement in the vertical direction due to density gradients caused by
temperature gradients and is a function of the Richardson number
(Figure 2-4). Fuchs et al. (1969) stated that the roughness height
varied from 0.2 to 0.4 mm for a bare soil surface and accounted for a
small variation in the calculated heat transfer coefficient.
Therefore, for the purposes of this study, a roughness length of 0.3 mm
will be used Fuchs et al. (1969) also noted that the height of zero
wind velocity (d) was zero for a bare soil surface. This term was
employed in the approximate wind profiles to account for the fact that
wind does not penetrate full vegetative canopies and for practical
purposes the surface where the wind velocity is zero occurs at some
finite height above the soil surface (Brutsaert, 1982; Sutton, 1953).
Fuchs et al. (1969) compared transfer coefficients calculated using
equation (2-50) to that determined from field data for a bare sandy
soil and obtained fairly close agreement.
For the purposes of this model, the approach used by Fuchs et al.
(1969) to determine the surface mass (Equation 2-50) and heat
(Equation 2-51) transfer coefficients was employed. The diabatic
influence function was utilized to account for atmospheric instability
as proposed by Fuchs et al. (1969). A roughness height (z0) of 0.3 mm
was utilized. The equations used in the formulation of this model and
the determination of parameters are summarized in Table 2-2
Numerical solution
The partial differential equations used to describe the mass and
energy balance in the soil must be solved numerically since analytical
solutions are not possible for the coupld nonlinear equations. Many