A MEMOIR ON CHEMICAL TRANSPORT
Applications to Soils and Crops
Allen R. Overman
Agricultural and Biological Engineering Department
University of Florida
Copyright 2006 Allen R. Overman
Table of Contents
Preface
List of Tables
List of Figures
1. Introduction
1.1 Mathematical Models
1.2 References
2. Convective Diffusion in a Capillary Tube
2.1 Background
2.2 Transport Equation
2.3 Steady State Solution
2.4 Transient Solution
2.5 Summary
2.6 References
3. Ion Transport in Porous Media
3.1 Background
3.2 Convective Diffusion across a Porous Diaphragm
3.3 Convective Diffusion through a Sand Column
3.4 Cation Transport through a Soil Column
3.5 Cation Transport across a Biological Membrane
3.6 Summary
3.7 References
4. Phosphorus Transport through Soil
4.1 Background
4.2 Phosphorus Kinetics in a Batch Reactor
4.2.1 Response in an Open Batch Reactor
4.2.2 Response in a Closed Batch Reactor
4.3 Phosphorus Transport in a Packed Column
4.3.1 Transport with Equilibrium Ion Exchange
4.3.2 Transport with Nonequilibrium Ion Exchange
4.3.3 Transport with Ion Exchange and Transformation
4.4 Summary
4.5 References
5. Nutrient Coupling Between Applied and Available
5.1 Background
5.2 Coupling of Applied and Available Soil Phosphorus
5.2.1 Response of Extractable Soil Phosphorus to Applied Phosphorus
5.2.2 Crop Response to Applied Phosphorus
5.3 Crop Response to Applied Nitrogen
5.4 Summary
5.5 References
Appendices
A.3 Simulation of Ion Transport in Soil
A.4 Simulation of Phosphorus Transport in Soil
Preface
During my childhood on a farm in eastern North Carolina I came up with many questions:
What was the origin of the seasons? How did farmers know when to plant crops? What was the
role of fertilizer and rainfall in crop production? What was the composition of plants? What
made plants (such as corn) at different locations and from year to year look essentially the same?
My parents advised me to go to school and learn about science to find some of these answers.
And so I did, some 22 years of formal education. Beginning with chemistry (my favorite subject
in high school), followed by college courses in math, chemistry, physics, engineering, and
agriculture. All of the courses seemed logical and useful until I encountered quantum mechanics.
What minds invented this stuff? Later I was to read that the eminent theoretical physicist John A.
Wheeler found himself really up a tree with this subject. Richard Feynman remarked in his
famous lecture notes on physics that nobody understands quantum mechanics! And these are two
major contributors to the subject. Over the years I have learned to accept the mysteries of nature
and to dine frequently on "humble pie." Albert Einstein has written that one who claims to hold
the truth is destined to be laughed at by the gods. And so I have continued a life long search for
answers to my childhood questions. The search can continue for as long as the mind remains
open to curiosity and is not closed by dogma or suppressed by authority.
The benchmark for the modern environmental era might be 1962 with the publication of
Rachael Carson's book Silent Spring, which called attention to negative impacts of some
chemicals in our environment and ecosystem. I was just beginning my PhD program in
Agricultural Engineering at N C State University at the time. So I decided to work with a soil
physical chemist on some fundamental processes related to chemical transport. This work
continued when he and I joined the Agronomy Department at the University of Illinois. I have
pursued this subject in greater depth during my career in Agricultural Engineering at the
University of Florida. The work has also led me into soil chemistry and soil fertility as related to
crop growth and yield.
This Memoir is an attempt to bring together my work in chemical transport in the soil and
into plants. A central theme of science for at least 400 years has been unification terrestrial and
celestial mechanics, electricity and magnetism, matter and motion, space and time. In my case
this centers on accumulation ofbiomass by photosynthesis (carbon fixation from the atmosphere)
and mineral elements from soil (rhizosphere). This necessarily leads to mathematical models to
describe and connect things together in a comprehensive framework. This work is full of
equations because the language of modern science is mathematics. The end goal is a
comprehensive description of the physical system. References have been included in each
chapter to provide some documentation for this work.
During my career I have been guided by the search for five characteristics: (1) patterns in
data, (2) mathematical relationships consistent with the patterns, (3) connections among various
components of the system, (4) consistency among various studies, and (5) mathematical beauty
in the models. This process has necessarily involved many failed attempts, but has been
sufficiently successful to sustain my continued search.
I am indebted to a number of people for inspiration and guidance. These include: W.P.
Hollowell (chemistry teacher), W.S. Lamm (4H advisor), George Blum (advisor), F.J. Hassler
(advisor), J. van Schilfgaarde (MS advisor), R.J. Miller (soil physical chemist, PhD advisor), T.P
Smith (engineer), W.G. Leseman (chemistry), L.C. Hammond (soil physics), C.C. Hortenstein
(soil chemistry), W.G. Blue (soil fertility), F.M. Rhoads (soil chemistry), R.L. Stanley (crop
science), O.C. Ruelke (crop science), F.G. Martin (statistics), E.J. Kamprath (soil chemistry),
G.W. Evers (crop science), and D.L. Robinson (crop science). I have drawn heavily from work
by W.E. Adams (soil fertility, USDAARS) and G.W. Burton (plant genetics, USDAARS).
I am particularly indebted to Stanley R. Wilkinson (soil fertility, USDAARS) with whom I
conducted cooperative research for more than 10 years, and to Richard V. Scholtz III (University
of Florida) who has been my cooperator beginning as a student in our program.
Finally, I thank my wife, Deanye, for many years of support and patience while I struggled
with ideas and data analysis. Our children have come to appreciate her loving support and
encouragement.
Allen R. Overman
July, 2006
List of Tables
Table 3.1 Summary of parameters for convective diffusion through a porous diaphragm.
Table 3.2 Summary of system parameters for cation transport through Lakeland fine sand.
Table 4.1 Dependence of rate coefficients on flow velocity for phosphorus transport through
Lakeland fine sand for equilibrium adsorption.
Table 4.2 Dependence of rate coefficients on flow velocity for phosphorus transport through
Lakeland fine sand for the LangmuirHinshelwood model.
Table 4.3 Dependence of rate coefficients k, and kr on flow velocity for phosphorus transport
through Lakeland fine sand for the extended LangmuirHinshelwood model.
Table 4.4 Numerical simulation of the response curves for the extended LangmuirHinshelwood
model of phosphorus transport in Lakeland fine sand at a flow velocity of v = 0.900 cm min1.
Table 5.1 Response of extractable soil phosphorus to applied phosphorus on Arredondo fine sand
at Gainesville, FL.
Table 5.2 Response of extractable soil phosphorus to applied phosphorus on Norfolk loamy fine
sand at Quincy, FL.
Table 5.3 Response of extractable soil phosphorus (Pex), plant phosphorus uptake (P,), vegetative
yield (Y), and plant P concentration (Pc) to applied phosphorus (P) for potato on Placid sand at
Hastings, FL.
Table 5.4 Response of extractable soil phosphorus (Pex), plant phosphorus uptake (Pu), biomass
yield (Y), and plant P concentration (Pc) to applied phosphorus (P) for a tall fescue/white clover
mixture on two soils in Northern Alabama.
Table 5.5 Summary of model parameters for response of biomass yield, plant P uptake, and
extractable soil P to applied P for a tall fescue/white clover mixture on two soils in Northern
Alabama.
Table 5.6 Response of biomass yield (Y), plant N uptake (N,), and plant N concentration (Ne) to
applied nitrogen (N) for tall fescue on Cahaba fine sandy loam at Tallassee, AL.
Table 5.7 Summary of logistic parameters for KY 31 and AU Triumph tall fescue on Cahaba fine
sandy loam at Tallassee, AL.
List of Figures
Figure 2.1 Steady state distributions of solute concentration (C CI)/(Co C,) vs. position (x/l) in
a circular straight bore capillary tube for various Peclet numbers (0). Curves drawn from Eq.
(2.3).
Figure 2.2 Steady state distribution of average solute concentration (C C,) /(Co C,) vs.
Peclet number (0). Curve drawn from Eq. (2.6).
Figure 3.1 Transient diffusion of KCl through a fitted glass filter. Line drawn from Eq. (3.11)
with a diffusion characteristic time of 123.4 h.
Figure 3.2 Transient convective diffusion of KC1 through a fritted glass filter. Line drawn from
Eq. (3.11) with appropriate Peclet numbers calculated from Eq. (3.15) and characteristic times
from Eq. (3.14).
Figure 3.3 Correlation of measured vs. calculated Peclet numbers (0). Data points from Table
3.1. Line is the 45 degree slope.
Figure 3.4 Correlation of measured vs. calculated characteristic time ratio (r /rD). Data points
from Table 3.1. Line is the 45 degree slope.
Figure 3.5 Correlation of the Peclet number (0) vs. flow rate (v) for convective diffusion
through Ottawa sand. Data points adapted from Overman (1975). Line drawn from Eq. (3.29).
Figure 3.6 Correlation of the dispersion coefficient (D) with pore velocity (v) for cation
transport through Lakeland fine sand. Data points are from Table 3.2. Curve drawn from Eq.
(3.40).
Figure 3.7 Correlation of the exchange coefficient (k) with pore velocity (v) for cation transport
through Lakeland fine sand. Data points are from Table 3.2. Lines drawn by visual inspection to
pass through the (0, 0) intercept.
Figure 3.8 Dependence of the exchange parameter (1k/v ) on feed concentration (Co) for cation
transport through Lakeland fine sand. Data points from Table 3.2.
Figure 3.9 Dependence of cation exchange capacity (Q) on feed concentration (Co) for cation
transport through Lakeland fine sand. Data points from Table 3.2.
Figure 3.10 Response curves for solution concentration (C*) and exchange concentration (q')
vs. time (t*) for cation transport through Lakeland fine sand for feed concentration Co = 0.01
mole L' and pore velocity v = 0.147 cm min'. Simulations are from Eqs. (3.41) and (3.42) at
position x* = 1.00 and with space step size of Ax* = 1.00.
Figure 3.11 Response curves for solution concentration (C*) and exchange concentration (q*)
vs. time (t*) for cation transport through Lakeland fine sand for feed concentration Co = 0.01
mole L' and pore velocity v = 0.147 cm min1. Simulations are from Eqs. (3.41) and (3.42) at
position x* = 1.00 and with space step size of Ax* = 0.20.
Figure 3.12 Response curves for solution concentration (C') vs. time (t*) for cation transport
through Lakeland fine sand for feed concentration Co = 0.01 mole L1 and pore velocity
v = 0.147 cm min1. Simulations are from Eqs. (3.41) and (3.42) at positions
x' = 0.2,0.4,0.6,0.8,1.0 and with space step size of Ax* = 0.2.
Figure 4.1 Dependence of the maximum rate parameter (rm) on soil mass (M) for phosphorus
kinetics in an open batch reactor with Lakeland fine sand by the extended Langmuir
Hinshelwood model. Curve drawn from Eq. (4.29).
Figure 4.2 Dependence of the distribution coefficient (K) on soil mass (M) for phosphorus
kinetics in an open batch reactor with Lakeland fine sand by the extended Langmuir
Hinshelwood model. Curve drawn from Eq. (4.30).
Figure 4.3 Dependence of reaction rate (r) on steady state solution phosphorus concentration (Cs)
for phosphorus kinetics in an open batch reactor with 250 g Lakeland fine sand in 500 ml
solution by the extended LangmuirHinshelwood model. Curve drawn from Eq. (4.31).
Figure 4.4 Correlation of the dispersion coefficient (D) with pore velocity (v) for phosphorus
transport through Lakeland fine sand. Data points are from Table 4.1. Curve drawn from Eq.
(4.50).
Figure 4.5 Dependence of the reaction rate coefficient (kr) and velocity/reaction rate ratio (v /k,)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.1. Line
drawn from Eq. (4.52); curve from Eq. (4.53).
Figure 4.6 Dependence of the distribution coefficient (K) and the overall reaction coefficient (k)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.1.
Curves drawn from Eqs. (4.57) and (4.58).
Figure 4.7 Dependence of the rate coefficients for adsorption (ka), desorption (ka), and reaction
(kr) on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.2.
Curves drawn from Eqs. (4.68) through (4.70).
Figure 4.8 Dependence of the distribution coefficient (K) and the overall reaction coefficient (k)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.2.
Curves drawn from Eqs. (4.71) and (4.72).
Figure 4.9 Dependence of the rate coefficients for decomposition (ks) and reaction (kr) on pore
velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.3. Curves
drawn from Eqs. (4.80) and (4.81).
Figure 4.10 Dependence of the overall reaction coefficient (k) on pore velocity (v) for a packed
bed reactor with Lakeland fine sand. Data from Table 4.3. Curve drawn from Eq. (4.82).
Figure 4.11 Response curves for solution concentration (C') vs. time (t') at space nodes
(z = 0.1, 0.2, 0.4, 0.6, 0.8,1.0) for phosphorus transport through Lakeland fine sand at a pore
velocity of v = 0.900 cm min1. Curves drawn Eq. (4.90).
Figure 4.12 Distribution curves for solution concentration (C') vs. depth (z') at various times
(t' = 1, 2, 5,10,15, 20, 25, 30) for phosphorus transport through Lakeland fine sand at a pore
velocity of v = 0.900 cm min'. Curves drawn Eq. (4.90).
Figure 4.13 Response curves for solution concentration (C'), adsorbed concentration (A"), and
transformed concentration (B*) vs. time (t*) at space node z* = 0.05 for phosphorus transport
through Lakeland fine sand at a pore velocity of v = 0.900 cm min1. Curves drawn Eq. (4.90).
Figure 4.14 Response curves for solution concentration (C'), adsorbed concentration (A*), and
transformed concentration (B*) vs. time (t*) at space node z' = 0.50 for phosphorus transport
through Lakeland fine sand at a pore velocity of v = 0.900 cm min'. Curves drawn Eq. (4.90).
Figure 5.1 Response of extractable soil phosphorus (Pex) to applied phosphorus (P) on
Arredondo fine sand at Gainesville, Florida. Data adapted from Sartain (1992). Curve drawn
from Eq. (5.1).
Figure 5.2 Response of extractable soil phosphorus (Pex) to applied phosphorus (P) on Norfolk
loamy fine sand at Quincy, Florida. Data adapted from Rhoads (1991). Curve drawn from Eq.
(5.4).
Figure 5.3 Response of extractable soil phosphorus (Pa) and plant P uptake (Pu) to applied
phosphorus (P) for potato on Placid sand at Hastings, Florida. Data adapted from Rhue et al.
(1982). Curves drawn from Eqs. (5.18) and (5.19).
Figure 5.4 Phase plot between plant P uptake (Pu) vs. extractable soil P (Pex) for potato on Placid
sand at Hastings, Florida. Data adapted from Rhue et al. (1982). Curve drawn from Eq. (5.22).
Figure 5.5 Response of vegetative biomass yield (Y), plant P uptake (Pu), and plant P
concentration (Pc) to applied phosphorus (P) for potato on Placid sand at Hastings, Florida. Data
adapted from Rhue et al. (1982). Curves drawn from Eqs. (5.23) through (5.25).
Figure 5.6 Phase plots between vegetative biomass yield (Y) and plant P concentration (Pc) vs.
plant P uptake (P,) for potato on Placid sand at Hastings, Florida. Data adapted from Rhue et al.
(1982). Curve drawn from Eq. (5.28); line from Eq. (5.29).
Figure 5.7 Phase plots between vegetative biomass yield (Y), plant P uptake (P,), and plant P
concentration (Pc) vs. extractable soil P (Pex) for potato on Placid sand at Hastings, Florida. Data
adapted from Rhue et al. (1982). Curves drawn from Eq. (5.32) through (5.34).
Figure 5.8 Response of extractable soil phosphorus (Pex) and plant P uptake (Pu) to applied
phosphorus (P) for tall fescue/white clover mixture on Wynnville sandy loam and Dickson silt
loam in Northern Alabama. Data adapted from Burmester and Adams (1984). Curves drawn
from Eqs. (5.36), (5.37), (5.48), and (5.49).
Figure 5.9 Phase plot between plant P uptake (P,) vs. extractable soil P (Pex) for tall fescue/white
clover mixture on Wynnville sandy loam and Dickson silt loam in Northern Alabama. Data
adapted from Burmester and Adams (1984). Curves drawn from Eqs. (5.40) and (5.52).
Figure 5.10 Response of biomass yield (Y), plant P uptake (P,), and plant P concentration (Pc) to
applied phosphorus (P) for tall fescue/white clover mixture on Wynnville sandy loam and
Dickson silt loam in Northern Alabama. Data adapted from Burmester and Adams (1984).
Curves drawn from Eqs. (5.41) through (5.43) and (5.53) through (5.55).
Figure 5.11 Phase plots between biomass yield (Y) and plant P concentration (Pc) vs. plant P
uptake (P,) for tall fescue/white clover mixture on Wynnville sandy loam and Dickson silt loam
in Northern Alabama. Data adapted from Burmester and Adams (1984). Curves drawn from Eqs.
(5.46) and (5.58); lines from Eqs. (5.47) and (5.59).
Figure 5.12 Phase plots among biomass yield (Y), plant P uptake (P,), and plant P concentration
(Pc) vs. extractable soil P (Pex) for tall fescue/white clover mixture on Wynnville sandy loam and
Dickson silt loam in Northern Alabama. Data adapted from Burmester and Adams (1984).
Curves drawn from Eqs. (5.62) and (5.67) for yields, and (5.63) and (5.68) for plant P uptake.
Figure 5.13 Response of biomass yield (Y), plant N uptake (N,), and plant N concentration (Nc)
to applied nitrogen (N) for tall fescue on Cahaba fine sandy loam at Tallassee, Alabama. Data
adapted from Burmester and Adams (1984). Curves drawn from Eqs. (5.80) through (5.82).
Figure 5.14 Phase plots between biomass yield (Y) and plant N concentration (Nc) vs. plant N
uptake (Nu) for tall fescue on Cahaba fine sandy loam at Tallassee, Alabama. Data adapted from
Burmester and Adams (1984). Curve drawn from Eq. (5.83); line drawn from Eq. (5.84).
1. Introduction
1.1 Mathematical Models
Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to the articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve both analytical and numerical types.
This work is part of the triology of publications:
Mathematical Models of Crop Growth and Yield
A. R. Overman and R. V. Scholtz III (2002)
A Memoir on Chemical Transport: Applications to Soils and Crops
A. R. Overman (2006)
A Memoir on Crop Growth: Accumulation ofBiomass and Mineral Elements
A. R. Overman (2006)
The present memoir contains 268 equations, 13 tables, and 42 figures.
1.2 References
Aris, R. and M. Penn. 1980. The mere notion of a model. Mathematical Modeling 1:112.
Casti, J.L. 1997. Wouldbe Worlds: How Simulation is Changing the Frontiers of Science. John
Wiley & Sons. New York, NY.
Chandrasekhar, S. 1987. Truth and Beauty: Aesthetics and Motivations in Science. University of
Chicago Press. Chicago, IL.
Eyring, H. 1977. Men, mines, and molecules. Annual Reviews of Physical Chemistry 28:113.
1. Introduction
1.1 Mathematical Models
Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to the articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve both analytical and numerical types.
This work is part of the triology of publications:
Mathematical Models of Crop Growth and Yield
A. R. Overman and R. V. Scholtz III (2002)
A Memoir on Chemical Transport: Applications to Soils and Crops
A. R. Overman (2006)
A Memoir on Crop Growth: Accumulation ofBiomass and Mineral Elements
A. R. Overman (2006)
The present memoir contains 268 equations, 13 tables, and 42 figures.
1.2 References
Aris, R. and M. Penn. 1980. The mere notion of a model. Mathematical Modeling 1:112.
Casti, J.L. 1997. Wouldbe Worlds: How Simulation is Changing the Frontiers of Science. John
Wiley & Sons. New York, NY.
Chandrasekhar, S. 1987. Truth and Beauty: Aesthetics and Motivations in Science. University of
Chicago Press. Chicago, IL.
Eyring, H. 1977. Men, mines, and molecules. Annual Reviews of Physical Chemistry 28:113.
1. Introduction
1.1 Mathematical Models
Mathematical modeling covers a vast domain in the field of science and engineering. Models
include physical, chemical, and biological systems. It is common practice in engineering to
construct scale models of larger objects such as buildings, planes, ships, and other devices. In the
medical field animals may be used to evaluate chemicals and surgical procedures. Mathematical
models include both geometric and arithmetic. The focus here is on arithmetic models.
Arithmetic models include everything from regression models to ones derived from fundamental
principles. All of the models used in this work will be empirical in the sense that they arise out of
measurements and observations rather than from abstract postulates.
There is much controversy over the definition and application of mathematical models. The
interested reader is referred to the articles in the inaugural issue of the journal Mathematical
Modeling (Aris and Penn, 1980; Spanier, 1980) for some thoughts on the subject. In science new
ideas are sometimes greeted with intense hostility, as has occurred in the field of chemistry
(Jaffe, 1976). I consider myself a modeler in the sense of the eminent physical chemist Henry
Eyring (Eyring, 1977; Eyring and Caldwell, 1980). Models are developed to describe and explain
observations and measurements, to couple various phenomena, and to make predictions. While
no model has proven totally complete, high value is placed on consistency among various data
sets at different times, at different places, and by different observers (Pagels, 1982).
This memoir represents efforts over a period of nearly 50 years, from graduate student days
until the present. The flavor of my approach has been described in the book by Overman and
Scholtz (2002). The common theme has been unification, or what Chandrasekhar (1987) has
termed systemization. Some of the models described here are based on reductionism, while
others are of the more holistic type (Casti, 1997; Laughlin, 2006). Solutions to differential
equations involve both analytical and numerical types.
This work is part of the triology of publications:
Mathematical Models of Crop Growth and Yield
A. R. Overman and R. V. Scholtz III (2002)
A Memoir on Chemical Transport: Applications to Soils and Crops
A. R. Overman (2006)
A Memoir on Crop Growth: Accumulation ofBiomass and Mineral Elements
A. R. Overman (2006)
The present memoir contains 268 equations, 13 tables, and 42 figures.
1.2 References
Aris, R. and M. Penn. 1980. The mere notion of a model. Mathematical Modeling 1:112.
Casti, J.L. 1997. Wouldbe Worlds: How Simulation is Changing the Frontiers of Science. John
Wiley & Sons. New York, NY.
Chandrasekhar, S. 1987. Truth and Beauty: Aesthetics and Motivations in Science. University of
Chicago Press. Chicago, IL.
Eyring, H. 1977. Men, mines, and molecules. Annual Reviews of Physical Chemistry 28:113.
Eyring, H. and D. Caldwell. 1980. Inductive and deductive science. Mathematical Modeling
1:3340.
Jaffe, B. 1976. Crucibles: The Story of Chemistry. Dover Publications. New York, NY.
Laughlin, R.B. 2005. A Different Universe: Reinventing Physics from the Bottom Down. Basic
Books. New York, NY.
Overman, A.R. and R.V. Scholtz III. 2002. Mathematical Models of Crop Growth and Yield.
Marcel Dekker. New York, NY.
Pagels, H.R. 1982. The Cosmic Code: Quatum Physics as the Language ofNature. Simon &
Schuster. New York, NY.
Spanier, J. Thoughts about the essentials of mathematical modeling. Mathematical Modeling
1:99108.
2. Convective Diffusion in a Capillary Tube
2.1 Background
The first case to be considered is transport of a nonreactive solute through a straight bore
circular capillary tube. Transport includes two basic processes convective flow and molecular
diffusion. The question is whether transport can be described by the simple algebraic sum of
these two processes characterized by the transport parameters convective flow velocity, v, and
molecular diffusion coefficient, D. There is the possibility that the effective diffusion coefficient
for convective diffusion is coupled with the flow velocity.
2.2 Transport Equation
The equation for convective diffusion of a nonreactive solute in a uniform circular capillary
of radius a and length I under laminar flow is described by the transport equation (Taylor, 1953)
D a ++ vo 1 0=o (2.1)
9r 2 r ar aX2 a 2 at
where C is solute concentration, r is radial position from the center line of the tube, x is
longitudinal position from the entrance of the tube, t is the time variable, D is the diffusion
coefficient, and Vo is convective velocity along the centerline of the tube. In the case of extremely
low flow velocity and with a << 1, Overman (1965) proposed the simplified equation
02C OC OC
D v =0 (2.2)
x2 Ox Ot
where V is the average convective velocity of flow. Solutions to Eq. (2.2), C(x,t), will now be
obtained for initial and boundary conditions of C(x, 0) = Co, C(0, t) = Co, C(l, t) = C1.
2.3 Steady State Solution
The steady state distribution of solute in the capillary with D and v both constant is given by
(Overman and Miller, 1968)
CC, 1 exp(Ox/l)1 (2.3)
= 1 = (2.3)
Co C, exp(0) 
where 0 is the Peclet number defined by
0 = (2.4)
D
2. Convective Diffusion in a Capillary Tube
2.1 Background
The first case to be considered is transport of a nonreactive solute through a straight bore
circular capillary tube. Transport includes two basic processes convective flow and molecular
diffusion. The question is whether transport can be described by the simple algebraic sum of
these two processes characterized by the transport parameters convective flow velocity, v, and
molecular diffusion coefficient, D. There is the possibility that the effective diffusion coefficient
for convective diffusion is coupled with the flow velocity.
2.2 Transport Equation
The equation for convective diffusion of a nonreactive solute in a uniform circular capillary
of radius a and length I under laminar flow is described by the transport equation (Taylor, 1953)
D a ++ vo 1 0=o (2.1)
9r 2 r ar aX2 a 2 at
where C is solute concentration, r is radial position from the center line of the tube, x is
longitudinal position from the entrance of the tube, t is the time variable, D is the diffusion
coefficient, and Vo is convective velocity along the centerline of the tube. In the case of extremely
low flow velocity and with a << 1, Overman (1965) proposed the simplified equation
02C OC OC
D v =0 (2.2)
x2 Ox Ot
where V is the average convective velocity of flow. Solutions to Eq. (2.2), C(x,t), will now be
obtained for initial and boundary conditions of C(x, 0) = Co, C(0, t) = Co, C(l, t) = C1.
2.3 Steady State Solution
The steady state distribution of solute in the capillary with D and v both constant is given by
(Overman and Miller, 1968)
CC, 1 exp(Ox/l)1 (2.3)
= 1 = (2.3)
Co C, exp(0) 
where 0 is the Peclet number defined by
0 = (2.4)
D
2. Convective Diffusion in a Capillary Tube
2.1 Background
The first case to be considered is transport of a nonreactive solute through a straight bore
circular capillary tube. Transport includes two basic processes convective flow and molecular
diffusion. The question is whether transport can be described by the simple algebraic sum of
these two processes characterized by the transport parameters convective flow velocity, v, and
molecular diffusion coefficient, D. There is the possibility that the effective diffusion coefficient
for convective diffusion is coupled with the flow velocity.
2.2 Transport Equation
The equation for convective diffusion of a nonreactive solute in a uniform circular capillary
of radius a and length I under laminar flow is described by the transport equation (Taylor, 1953)
D a ++ vo 1 0=o (2.1)
9r 2 r ar aX2 a 2 at
where C is solute concentration, r is radial position from the center line of the tube, x is
longitudinal position from the entrance of the tube, t is the time variable, D is the diffusion
coefficient, and Vo is convective velocity along the centerline of the tube. In the case of extremely
low flow velocity and with a << 1, Overman (1965) proposed the simplified equation
02C OC OC
D v =0 (2.2)
x2 Ox Ot
where V is the average convective velocity of flow. Solutions to Eq. (2.2), C(x,t), will now be
obtained for initial and boundary conditions of C(x, 0) = Co, C(0, t) = Co, C(l, t) = C1.
2.3 Steady State Solution
The steady state distribution of solute in the capillary with D and v both constant is given by
(Overman and Miller, 1968)
CC, 1 exp(Ox/l)1 (2.3)
= 1 = (2.3)
Co C, exp(0) 
where 0 is the Peclet number defined by
0 = (2.4)
D
2. Convective Diffusion in a Capillary Tube
2.1 Background
The first case to be considered is transport of a nonreactive solute through a straight bore
circular capillary tube. Transport includes two basic processes convective flow and molecular
diffusion. The question is whether transport can be described by the simple algebraic sum of
these two processes characterized by the transport parameters convective flow velocity, v, and
molecular diffusion coefficient, D. There is the possibility that the effective diffusion coefficient
for convective diffusion is coupled with the flow velocity.
2.2 Transport Equation
The equation for convective diffusion of a nonreactive solute in a uniform circular capillary
of radius a and length I under laminar flow is described by the transport equation (Taylor, 1953)
D a ++ vo 1 0=o (2.1)
9r 2 r ar aX2 a 2 at
where C is solute concentration, r is radial position from the center line of the tube, x is
longitudinal position from the entrance of the tube, t is the time variable, D is the diffusion
coefficient, and Vo is convective velocity along the centerline of the tube. In the case of extremely
low flow velocity and with a << 1, Overman (1965) proposed the simplified equation
02C OC OC
D v =0 (2.2)
x2 Ox Ot
where V is the average convective velocity of flow. Solutions to Eq. (2.2), C(x,t), will now be
obtained for initial and boundary conditions of C(x, 0) = Co, C(0, t) = Co, C(l, t) = C1.
2.3 Steady State Solution
The steady state distribution of solute in the capillary with D and v both constant is given by
(Overman and Miller, 1968)
CC, 1 exp(Ox/l)1 (2.3)
= 1 = (2.3)
Co C, exp(0) 
where 0 is the Peclet number defined by
0 = (2.4)
D
For pure diffusion ( = 0) it can be shown by Taylor series expansion that Eq. (2.3) reduces to
the linear distribution
CC, _1 x (2.5)
Co Ct 1
Integration of Eq. (2.3) over the length of the capillary leads to a relation of average solute
concentration in the capillary tube, C, to the Peclet number of
C C, I
C C_ 1_ 1+L (2.6)
Co C 2 2
where L is the Langevin function (Debye, 1929; Eyring et al., 1964; Reitz and Milford, 1960;
Tanford, 1961) defined by
L(z) = coth z 1 (2.7)
z
where coth z is the hyperbolic cotangent defined by
cothz exp(z)+ exp(z) (2.8)
exp(z) exp(z)
It follows that L(oo, +oo) is bounded by 1 < L < +1 and passes through L(0) = 0.
A graph of Eq. (2.3) for 10 < 0 < +10 is shown in Figure 2.1. The corresponding graph of
Eq. (2.6) is shown in Figure 2.2. The validity ofEq. (2.6) has been verified by Overman and
Miller (1968) for a precision bore capillary of 1 mm diameter and 10 cm length for a flow rate of
v = 0.920 cm day"' using D20 (deuterium oxide) as tracer. The molecular diffusion coefficient
was assumed to be D = 2.37 cm2 day' (Wang et al., 1953), which provided a Peclet number of
0=3.88.
An element of symmetry may be noted in Figures 2.1 and 2.2. To establish these relations it
is convenient to introduce the linear transformations
x C C1
= and 
1 Co Ct
which transforms Eq. (2.3) to the form
,(0') =1 x (0(2.9)
exp(0) 1
For negative values of 0 Eq. (2.9) can be written as
exp(() 1
0 0(0, ~) = 1 (2.10)
exp(0) 1
Now Eq. (2.10) can be rewritten as
1 exp(O) 1 exp[O(1 )] exp(O) exp(04') 1
1 0(0, o = p = 1 = (0,1') (2.11)
exp(O) 1 1 exp(O) exp(O) 1
where I' = 1 It follows that symmetry involves transformations of both 4 and This
symmetry property follows from the linear nature of Eq. (2.2).
To establish the symmetry for average solute concentration Eq. (2.6) is rewritten as
c C) ' 1 = +L( (2.12)
Co C,C 2 2
Since it follows from Eqs. (2.7) and (2.8) that L(z) = L(+z), the symmetry relation for average
concentration becomes
(+0)+ (O) = + L+ +1+ L) =1 (2.13)
which is apparent from Figure 2.2.
2.4 Transient Solution
The boundaryvalue problem can be written as
Sa2C aC 8C
0
ax 2 x at
C(x,0) = Co (2.14)
x=O C(0, t) = C
x=l C(I, t)= C,
The solution to Eq. (2.14) is (Overman, 1968)
CC, 1exp(Ox/1)1
1
Co C] exp(O) 1
(2.15)
2exp[ 1( exp /2)2 2 +2 2 in "
In the limit as t > oo, Eq. (2.15) approaches the steady state solution given by Eq. (2.3). For the
case of pure diffusion (0 = 0) the transient solution reduces to (Carslaw and Jaeger, 1959)
CC x cosn 2 2Dt x16)
=1_2  exp[n2 E2 fsin nx (2.16)
Co C, I , nn l I 1
Average solute concentration C as a function of time is given by
CC, l'C C, d
Co C, rCo C,
= 1+L +2i exp  cost exp{ +n2 2] }
2 )2 2 7E2 12
S2J = (0/2)2+n 2 2
(2.17)
In the limit as t > oo, Eq. (2.17) approaches the steady state solution given by Eq. (2.6).
Overman (1968) has verified Eq. (2.17) for 0 = 3.88 for the experiment described above.
2.5 Summary
It appears from this analysis that transport of a nonreactive solute through a straight bore
capillary tube can be described by simple convective diffusion theory at low Peclet numbers,
with independent flow parameters v and D. The next step is to examine transport through porous
media with complex flow paths and at higher Peclet numbers. An excellent summary of
dimensionless numbers, including the Peclet number, can be found in Boucher and Alves (1959).
2.6 References
Boucher, D.F. and G.E. Alves. 1959. Dimensionless numbers. Chemical Engineering Progress
55:5564.
Debye, P. 1929. Polar Molecules. Chemical Catalog Co. New York, NY.
Eyring, H., D. Henderson, B.J. Stover, and E.M. Eyring. 1964. Statistical Mechanics and
Dynamics. John Wiley & Sons. New York, NY.
Overman, A.R. 1965. Net molecular transfer in a capillary with diffusion and laminar flow. Ph.
D. Thesis. North Carolina State University, Raleigh, NC. 64 p.
Overman, A.R. 1968. Transient convective diffusion in capillaries. J. Physical Chemistry
72:32863288.
Overman, A.R. and R.J. Miller. 1968. Convective diffusion in capillaries. J. Physical Chemistry
72:155158.
Reitz, J.R. and F.J. Milford. 1960. Foundations of Electromagnetic Theory. AddisonWesley
Publishing Co. Reading, MA.
Tanford, C. 1961. Physical Chemistry ofMacromolecules. John Wiley & Sons. New York, NY.
Taylor, G.I. 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc.
Royal Society ofLondon A219:186203.
Wang. J.H., C.V. Robinson, and I.S. Edelman. 1953. J. American Chemical Society 75:466470.
In the limit as t > oo, Eq. (2.15) approaches the steady state solution given by Eq. (2.3). For the
case of pure diffusion (0 = 0) the transient solution reduces to (Carslaw and Jaeger, 1959)
CC x cosn 2 2Dt x16)
=1_2  exp[n2 E2 fsin nx (2.16)
Co C, I , nn l I 1
Average solute concentration C as a function of time is given by
CC, l'C C, d
Co C, rCo C,
= 1+L +2i exp  cost exp{ +n2 2] }
2 )2 2 7E2 12
S2J = (0/2)2+n 2 2
(2.17)
In the limit as t > oo, Eq. (2.17) approaches the steady state solution given by Eq. (2.6).
Overman (1968) has verified Eq. (2.17) for 0 = 3.88 for the experiment described above.
2.5 Summary
It appears from this analysis that transport of a nonreactive solute through a straight bore
capillary tube can be described by simple convective diffusion theory at low Peclet numbers,
with independent flow parameters v and D. The next step is to examine transport through porous
media with complex flow paths and at higher Peclet numbers. An excellent summary of
dimensionless numbers, including the Peclet number, can be found in Boucher and Alves (1959).
2.6 References
Boucher, D.F. and G.E. Alves. 1959. Dimensionless numbers. Chemical Engineering Progress
55:5564.
Debye, P. 1929. Polar Molecules. Chemical Catalog Co. New York, NY.
Eyring, H., D. Henderson, B.J. Stover, and E.M. Eyring. 1964. Statistical Mechanics and
Dynamics. John Wiley & Sons. New York, NY.
Overman, A.R. 1965. Net molecular transfer in a capillary with diffusion and laminar flow. Ph.
D. Thesis. North Carolina State University, Raleigh, NC. 64 p.
Overman, A.R. 1968. Transient convective diffusion in capillaries. J. Physical Chemistry
72:32863288.
Overman, A.R. and R.J. Miller. 1968. Convective diffusion in capillaries. J. Physical Chemistry
72:155158.
Reitz, J.R. and F.J. Milford. 1960. Foundations of Electromagnetic Theory. AddisonWesley
Publishing Co. Reading, MA.
Tanford, C. 1961. Physical Chemistry ofMacromolecules. John Wiley & Sons. New York, NY.
Taylor, G.I. 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc.
Royal Society ofLondon A219:186203.
Wang. J.H., C.V. Robinson, and I.S. Edelman. 1953. J. American Chemical Society 75:466470.
List of Figures
Figure 2.1 Steady state distributions of solute concentration (C C/)/(Co Ci) vs. position (x/l) in
a circular straight bore capillary tube for various Peclet numbers (0). Curves drawn from Eq.
(2.3).
Figure 2.2 Steady state distribution of average solute concentration (C C) /(Co C,) vs.
Peclet number (0). Curve drawn from Eq. (2.6).
L \ \
0 
tC
a4'
0
a 4 
FQ 
6 /D
3. Ion Transport in Porous Media
3.1 Background
Transport through porous media represents complex flow through irregular flow paths due to
irregular boundaries between liquid and solid phases. Application of Eq. (2.1) to such cases is
not feasible. The question now is whether a simple onedimensional flow equation is adequate
for this case. The process of ion exchange between solution and soil particle surfaces is also
introduced.
3.2 Convective Diffusion across a Porous Diaphragm
It is assumed that transport by convective diffusion can be described by Eq. (2.2). The steady
state distribution is then given by
d2C _dC
D =0 (3.1)
dx2 dx
where C is solute concentration, x is travel distance, D is the effective molecular diffusion
coefficient in the pores, and V is average convective velocity of flow in the pores. According to
Eq. (2.3) the solution to Eq. (3.1) is (Mackie and Meares, 1956; Overman and Miller, 1968)
CC, exp(Ox/l)1
= 1 (3.2)
C, C, exp(O) 
where 0 is the Peclet number defined by
1v
0 = (3.3)
D
where 1 is the distance across the conductor.
Net transport, QN, of solute by convective diffusion at steady state can be described by
dC
QN = DA + QC (3.4)
dx
where D is the overall diffusion coefficient, A is total crosssectional area of the diaphragm, and
Q is volumetric flow rate of the solvent (water). Combination of Eqs (3.2) and (3.4) leads to
S C exp(0)C, (3.5)
exp(0) 1
3. Ion Transport in Porous Media
3.1 Background
Transport through porous media represents complex flow through irregular flow paths due to
irregular boundaries between liquid and solid phases. Application of Eq. (2.1) to such cases is
not feasible. The question now is whether a simple onedimensional flow equation is adequate
for this case. The process of ion exchange between solution and soil particle surfaces is also
introduced.
3.2 Convective Diffusion across a Porous Diaphragm
It is assumed that transport by convective diffusion can be described by Eq. (2.2). The steady
state distribution is then given by
d2C _dC
D =0 (3.1)
dx2 dx
where C is solute concentration, x is travel distance, D is the effective molecular diffusion
coefficient in the pores, and V is average convective velocity of flow in the pores. According to
Eq. (2.3) the solution to Eq. (3.1) is (Mackie and Meares, 1956; Overman and Miller, 1968)
CC, exp(Ox/l)1
= 1 (3.2)
C, C, exp(O) 
where 0 is the Peclet number defined by
1v
0 = (3.3)
D
where 1 is the distance across the conductor.
Net transport, QN, of solute by convective diffusion at steady state can be described by
dC
QN = DA + QC (3.4)
dx
where D is the overall diffusion coefficient, A is total crosssectional area of the diaphragm, and
Q is volumetric flow rate of the solvent (water). Combination of Eqs (3.2) and (3.4) leads to
S C exp(0)C, (3.5)
exp(0) 1
3. Ion Transport in Porous Media
3.1 Background
Transport through porous media represents complex flow through irregular flow paths due to
irregular boundaries between liquid and solid phases. Application of Eq. (2.1) to such cases is
not feasible. The question now is whether a simple onedimensional flow equation is adequate
for this case. The process of ion exchange between solution and soil particle surfaces is also
introduced.
3.2 Convective Diffusion across a Porous Diaphragm
It is assumed that transport by convective diffusion can be described by Eq. (2.2). The steady
state distribution is then given by
d2C _dC
D =0 (3.1)
dx2 dx
where C is solute concentration, x is travel distance, D is the effective molecular diffusion
coefficient in the pores, and V is average convective velocity of flow in the pores. According to
Eq. (2.3) the solution to Eq. (3.1) is (Mackie and Meares, 1956; Overman and Miller, 1968)
CC, exp(Ox/l)1
= 1 (3.2)
C, C, exp(O) 
where 0 is the Peclet number defined by
1v
0 = (3.3)
D
where 1 is the distance across the conductor.
Net transport, QN, of solute by convective diffusion at steady state can be described by
dC
QN = DA + QC (3.4)
dx
where D is the overall diffusion coefficient, A is total crosssectional area of the diaphragm, and
Q is volumetric flow rate of the solvent (water). Combination of Eqs (3.2) and (3.4) leads to
S C exp(0)C, (3.5)
exp(0) 1
for the boundary conditions C(x = 0) = Co and C(x = ) = Ci. Equation (3.5) contains two special
cases: pure diffusion and pure convective flow. For pure diffusion it can be shown by Taylor
series expansion of the denominator that net flow is given by
(Q > )= DA (C C) (3.6)
1
For pure convective flow with uniform concentration Co = C1, net flow is given by
Q, (Co = C,)= QCo (3.7)
Now consider an end chamber at x = 1 of constant volume V, which is many times the
volume of liquid in the pores of the diaphragm. Then to good approximation it is reasonable to
write
Q = V dC1 (3.8)
dt
provided that the liquid in V is well mixed. It is convenient to assume a quasisteady state
condition within the pores and write for the transport equation
VdC, = C, exp(O) C, (3.9)
dt exp() 1
The initial condition in the end chamber is assumed to be C, (t = 0) = C,. The final condition in
the end chamber will be
C" = lim C, = CO exp(9) (3.10)
The solution to Eq. (3.9) with these conditions becomes
S = exp (3.11)
Ct I t
where r is called the characteristic time, and is defined by
IV exp(O)l (3.12)
DA 0
By Taylor series expansion it can be shown that characteristic time for pure diffusion, ,D, at 0 =
0 is given by
IV
To =limr = (3.13)
oeo DA
It follows that time constants for convective diffusion and pure diffusion are related by
r exp(O) 1
exp (3.14)
TD 0
It may be noted that for 0 > 0, / r > 1 and that for 0 < 0, r / rT < 1. In other words,
characteristic time for convective diffusion may be enhanced or reduced over that for pure
diffusion depending on the direction and magnitude of the liquid flow rate.
The validity of the theory has been tested by Overman (1968). The experiment consisted of a
Biichner funnel with a fitted glass filter (5[t nominal pore size, 20 mm diameter, and 2 mm
thickness), with KCI used as the solute. An experiment was first performed for 0 = 0 to
determine the diffusion time constant of TD = 123.4h. Since the volume of the end chamber was
V= 14.7 cm3, this led to DA/I = 2.859 cm3 d"1, which can be used in calculations of Peclet
numbers from
1 1Q
0= (3.15)
D DA
for various volumetric flow rates. Equation (3.15) can be used to test the assumption that the
effective diffusion coefficient is independent of flow velocity for low Peclet numbers.
Response of Eq. (3.11) with time is shown in Figure 3.1 for pure diffusion with rD = 123.4h,
CO = 0, and C" = Co= 102 N. Results are shown in Figure 3.2 for convective diffusion, where
characteristic time appropriate for each flow rate is used. All results were found to lie on a single
line. A summary of parameters is listed in Table 3.1. Values in columns one and two are
estimated from Eqs. (3.15) and (3.10), respectively. There is close correlation between calculated
and measured values of 0, as shown in Figure 3.3. Values in columns 3 and 4 are estimated from
Eqs. (3.12) and (3.11), respectively, and shown in Figure 3.4. Correlation of calculated and
measured values is not as good as for Peclet numbers. Measured r was not available at 0 = 
5.313 due to irregularity in temperature control during the experiment.
Table 3.1 Summary of parameters for convective diffusion through a porous diaphragm.
0 Tr /TrD
Calc. Meas. Calc. Meas.
1.382 1.355 0.542 0.468
2.711 2.738 0.344 0.366
5.313 5.310 0.187 
7.440 7.298 0.134 0.135
3.3 Convective Diffusion through a Sand Column
The theory of convective diffusion is now applied to a column of inert sand (Overman,
1975). In this case solute concentration at x = 0 is maintained at Co, while solute is allowed to
accumulate in the upper chamber of volume V at x = 1. The boundary value problem can now be
written as
D 2C aC ac
0
ax2 ax at
C(x,0) = 0
x=0 C(0, t)= Co (3.16)
ac aC
x=l DA QC+V =0
ax at
C(,0) = 0
In this case D is effective diffusion coefficient, v is flux volumetricc flow rate/area, sometimes
referred to as Darcy velocity), e is void ratio of the sand column, A is total crosssectional area,
and Q is volumetric flow rate. The mathematical solution to the problem is made more complex
due to the flow boundary condition at x = 1. It is convenient to introduce the parameters
Sl=v lQ = Peclet number (3.17)
2D 2DA
EAl
h = A= volume ratio between soil pores and the end chamber (3.18)
V
Note the introduction of the factor 2 in Eq. (3.17) for convenience. It can be shown that the
steady state distribution is given by
C(x,oo) = C" = Co exp(20x/1) (3.19)
The solution to Eq. (3.16) can be shown to be (Overman, 1975)
1 = 2exp(0x/l) 'a, exp[(02 +a2 )Dt/r2]sina,x/ (3.20)
C n=l
where the expansion coefficients, a,, are defined by
a a, (a2 +02 + h)2 +(ha)2 (3.21)
a, 202 2 + )22 (3.21)+( )2 h (h+0)
a2 +0 (a2 + +hO)2+(ha) +h +
and a, are the roots of
(or + 02 + hO)tan a, = ha,
It should be noted that for convective diffusion Eq. (3.22) also contains an imaginary root
/ = i a (with i = V/i ) such that
(f2 + 02 + h)tanh 3 = hp (3.23)
where tanh represents the hyperbolic tangent. This root vanishes for pure diffusion. For pure
diffusion (0 = 0) Eq. (3.20) reduces to the well known solution (Carslaw and Jaeger, 1959;
Crank, 1956)
1 C =2 a, exp(aDt/cl2)sinanx/l (3.24)
Co n=l
where an are defined by
1 a +h2
a a (3.25)
a, a +h2 +h
and a, are the roots of
a, tan a, = h (3.26)
Values of a, in Eq. (3.26) are compiled by Carslaw and Jaeger (1959) for various values of h.
The validity of the convective diffusion theory for this case was established by Overman
(1975). The test cell consisted of crosssectional area of 11.04 cm2 and length of 1 cm. Volume
of the end chamber was 13.0 cm3. Ottawa sand was packed in the cell to a bulk density of 1.81 g
cm3 and void fraction of = 0.314. The solute was KC1. First, a diffusion experiment was
conducted to calibrate the system. Analysis of data for 24 < t < 72 h led to the regression
equation
1 C1 =1.122exp(0.0154t) (3.27)
Co
The volume ratio becomes h = (0.314)(11.04)(1.00)/13.0 = 0.266. This leads to the first root of
Eq. (3.26) of = 0.494. Since a2D / l2 = 0.0154 h1, it follows that the overall diffusion
coefficient is D = 0.477 cm2 day'. Since the molecular diffusion coefficient in solution is Do =
1.85 cm2 d"' (Robinson and Stokes, 1959), the ratio becomes D/Do = 0.258. Next, convective
diffusion experiments were conducted at various volumetric flow rates. Attention is focused on
the steady state results. For this case Eq. (3.19) reduces to
(3.22)
Cr = C0 exp(20)
for the upper chamber. According to Eq. (3.17) the Peclet numbers can be calculated from
20 =v = 2.096v (3.29)
D
In the experiments all convection velocities were negative at v = 0.50, 0.94, 1.92, 2.64, and
3.75 cm d'. From the steady state ratios given by Eq. (3.28), estimates were made of 20 
0.89, 2.11, 3.86, 5.50, and 7.67. Experimental results are shown in Figure 3.5, where the
line is drawn from Eq. (3.29). The data points are seen to lie very close to the line. In fact, linear
regression leads to the prediction equation
20 = 0.0036 + 2.056v r = 0.9989 (3.30)
with a correlation coefficient of r = 0.9989. This confirms the convective diffusion theory for
low Peclet numbers (0< 10).
3.4 Cation Transport through a Soil Column
The next step is to analyze solute flow through a soil column in which cation exchange
occurs between solution and soil particles. In the system for discussion exchange occurs between
the two monovalent cations K+ and NH4+, which have equal ionic mobility. The exchange model
of Hiester and Vermuelen (1952) is used, characterized by the kinetic equation
kc
C+BS< +B+CS (3.31)
kb
where C is solution concentration of the inflow cation, B is solution concentration of the outflow
cation, B S is exchange concentration of the outflow cation, C S is exchange concentration of
the inflow cation, kc is the rate coefficient for the inflow cation, and kb is the rate coefficient for
the outflow cation. The symbol S designates the exchange sites on the soil particles. Assuming
ion exchange to be controlled by the law of mass action, the rate equation for C can be written as
a(C S)
a = k C(B S) kbB(C S) (3.32)
8t
where t is the time variable. Uniform and constant ionic strength is assumed so that by electrical
neutrality
B + C = Co = constant (3.33)
for a given experiment. It is also assumed that cation exchange capacity Q of the soil remains
constant as well, so that
(3.28)
C S + B S = Q = constant
Using the definition q = C S, Eq. (3.32) becomes
q kcC(Q q) kb(CC)q (3.35)
at
For a bed initially free of C, the kinetic model can be written in dimensionless form
= a[C(1q*) (1C** (3.36)
with the dimensionless variables and parameters
t* = vt/l, q' = qQ, C' =C/Co, a = lkC C/, K =k kb
where vis average streaming velocity in the soil pores and / is the length of the column. In all
the experiments to be considered the distribution coefficient K is taken as unity.
A dispersedflow model is used to describe cation transport through the soil column
2C c_ C p a(C S) ac
x>o D v _V 0
ax2 ax C at at
8(C S)
a(c. kC(B S) kbB(C S)
at
C(x,O) = 0
C S(x,) = 0 (3.37)
x=0 C(0,t)= Co
B(0,t) =
x + oo C(x,t) > 0
where D is the dispersion coefficient (enhanced diffusion), p is bulk density of the soil, c is
porosity (void fraction) of the soil. By use of dimensionless variables and parameters, the system
can be described by
(3.34)
x2C* aC" aq*aC'
x >0 r P =0
ax2 ax at at*
B* =1C*
lq= 4c" 0 c"
at*
C'(x',0)= 0 (3.38)
q* (x',0)=
x =0 C*(0,t')=1
x* > Co(x*,t*) 0
where x' =x/l, t' =vt/l,C* =C/C, q* =q/Q, =D/lv, =pQ/sC,.
The cation transport theory was tested by Overman et al. (1980) with Lakeland fine sand
(thermic, coated typic Quartzipsamment). Experiments were conducted for several flow
velocities and feed concentrations. The two cations chosen were K+ and NH4+, which have equal
ionic mobility and hydrated radius. Simulations used the CrankNicolson finite difference
procedure. It was found that time step size was constrained by
At* < (3.39)
ap
to maintain numerical stability. Dimensions of the flow cell were 10 cm length and 4.8 cm
diameter. Flow rate was controlled with a peristaltic pump and samples were collected with a
fraction collector.
A summary of system parameters is given in Table 3.2. Dependence of the dispersion
coefficient (D) on average pore velocity (v) for Co 0 0.0100 mole L'1 is shown in Figure 3.6,
where the curve is drawn from
D=1.16v2 (3.40)
Clearly there is strong dependence of D on flow velocity, hence the term dispersion coefficient
in place of diffusion coefficient. This is caused by nonuniform velocity of liquid in the soil pores,
which enhances the rate of chemical mixing in soil solution due to concentration gradients. It
may be noted that the parameter Ik / v is independent of flow velocity for a fixed value of Co,
which is also illustrated in Figure 3.7. It follows that the exchange coefficient is a linear function
of flow velocity. However, the relationship between k and v is dependent on feed concentration
Co. This dependence is clearly illustrated in Figure 3.8. Similarly, dependence of cation
exchange capacity (Q) on Co is shown in Figure 3.9. These measurements lead to the estimates of
parameters a and p listed in the last two columns of Table 3.2.
Table 3.2 Summary of system parameters for cation transport through Lakeland fine sand.a
Co Q v k D Ik/l lkC0o/ pQ / sC
moles L' meq kg'1 cm min' L mole' min1 cm2 min1 L mole'
0.00131 1.69 0.132 4.0 0.001 303 0.40 5.69
0.00481 3.00 0.129 2.40 0.001 186 0.89 2.75
0.00499 3.25 0.132 2.65 0.001 201 1.00 2.87
0.00495 3.61 0.306 6.15 0.001 201 1.00 3.22
0.00550 3.21 0.503 10.1 0.001 201 1.10 2.57
0.00983 3.45 0.147 1.8 0.025 122 1.22 1.52
0.00969 3.48 0.297 3.6 0.100 121 1.21 1.54
0.00983 3.29 0.588 7.1 0.400 121 1.21 1.45
0.0453 4.13 0.132 1.0 0.070 75.8 3.43 0.402
0.0469 3.89 0.132 1.0 0.120 75.8 3.55 0.366
aData adapted from Overman et al. (1980).
Apparently k is not a measure of the true kinetic coefficients for cation exchange, since it is
dependent on flow velocity. It is more appropriate to refer to k as a global rate coefficient
(Smith, 1970). In fact, batch experiments with this system showed that the surface kinetics was
extremely fast (unpublished results). Dependence of parameters k and Q on Co remains
unexplained, but probably relates to the ionic double layer of the charged soil particles. This
question will not be explored further.
A simple numerical procedure is now demonstrated for this system. To simplify the analysis
the dispersion term is neglected (r = 0). Values for the parameters are assumed as follows:
1 = 10 cm, v = 0.147cm min', k = 1.8 Lmole' min', Co = 0.01 moleL', p = 1.66 gcm3
S= 0.376, Q = 0.345meq (100g)' = 0.345 105 molesg'
The dimensionless transport parameters assume the values
lkCo (10)cm(l.8)L mole' min' (0.01) mole 1.22
V (0.147) cmmin'
pQ (1.66) gcm3(0.345)10 moleg' 1.52
eC, (0.376)(102)moleL' (103) L cm3
Time step is constrained by At* <1/ a/3 = 1/(1.22)(1.52) = 0.53. The rate equations now become
q = =a[C (l q)( C )q*]=1.22[C 1q* )lC* )q*
at*
C' q' = G 1.52 q*
8t ox*
(3.41)
(3.42)
where G = aC* I x* is the gradient in C' between adjacent space nodes. Computations are
carried out for x' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0.
In the first appendix table (Table A3.1) calculations are carried out at x* = 1.00 for Ax* =
1.00, i.e. for one step to x = I = 10 cm. Results are shown in Figure 3.10. It can be seen that the
procedure is stable and appears to converge to C' = 1 and q* = 1. However, there is a question of
accuracy in the simulation due to the large step size of Ax = 10 cm. In subsequent tables (A3.2
through A3.7) computations are carried out forx' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0. Note that time
steps are At' < 0.1 for these cases. Results for x* = 1.00 are shown in Figure 3.11, which are
significantly different from those in Figure 3.10. It is concluded that the smaller step size is far
more accurate than the single step. A comparison of response for the various space nodes is
shown in Figure 3.12. In all cases q' < C' as expected. Accuracy of the simulation could
probably be enhanced even further by increasing the number of space nodes.
The transport model appears to describe ion transport rather well provided that the rate
coefficient k is taken as a linear function of flow velocity.
3.5 Cation Transport across a Biological Membrane
In passing it should be mentioned that the theory of cation transport across the membrane of
corn mitochondria by diffusion and osmosis has been reported (Overman et al., 1970). In this
case the volume within the membrane changes due to ion transport and changes in osmotic
pressure. This illustrates application of the convective diffusion theory to a biological system.
3.6 Summary
In the case of transport through porous media convective diffusion theory can be applied if
the diffusion coefficient is taken as dependent on flow velocity. This occurs because transport
has been treated as onedimensional flow when in fact it is multidimensional, and because the
velocity streamlines vary across the pores. It turns out that the rate coefficients for ion exchange
between solution and soil particle surfaces are also dependent on flow velocity. This occurs
because ion transfer between solution and particle surface is diffusion limited at the laminar flow
velocities used in the experiments. Due to the nonlinear nature of the chemical kinetics,
numerical solution of the transport equations is required. Careful attention must be given to time
steps in order to maintain numerical stability in the integration procedure. Accuracy in the
simulation is influenced by choice in the size of the space nodes.
3.7 References
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of Heat in Solids. Oxford University Press,
London.
Crank, J. 1956. The Mathematics ofDiffusion. Oxford University Press, London.
Hiester, N.K. and T. Vermeulen. 1952. Saturation performance of ionexchange and adsorption
columns. Chemical Engineering Progress 48:505516.
Mackie, J.S. and P. Meares. 1956. The sorption and diffusion of ethanol in a cation exchange
resin membrane. Discussions of the Faraday Society 21:111116.
where G = aC* I x* is the gradient in C' between adjacent space nodes. Computations are
carried out for x' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0.
In the first appendix table (Table A3.1) calculations are carried out at x* = 1.00 for Ax* =
1.00, i.e. for one step to x = I = 10 cm. Results are shown in Figure 3.10. It can be seen that the
procedure is stable and appears to converge to C' = 1 and q* = 1. However, there is a question of
accuracy in the simulation due to the large step size of Ax = 10 cm. In subsequent tables (A3.2
through A3.7) computations are carried out forx' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0. Note that time
steps are At' < 0.1 for these cases. Results for x* = 1.00 are shown in Figure 3.11, which are
significantly different from those in Figure 3.10. It is concluded that the smaller step size is far
more accurate than the single step. A comparison of response for the various space nodes is
shown in Figure 3.12. In all cases q' < C' as expected. Accuracy of the simulation could
probably be enhanced even further by increasing the number of space nodes.
The transport model appears to describe ion transport rather well provided that the rate
coefficient k is taken as a linear function of flow velocity.
3.5 Cation Transport across a Biological Membrane
In passing it should be mentioned that the theory of cation transport across the membrane of
corn mitochondria by diffusion and osmosis has been reported (Overman et al., 1970). In this
case the volume within the membrane changes due to ion transport and changes in osmotic
pressure. This illustrates application of the convective diffusion theory to a biological system.
3.6 Summary
In the case of transport through porous media convective diffusion theory can be applied if
the diffusion coefficient is taken as dependent on flow velocity. This occurs because transport
has been treated as onedimensional flow when in fact it is multidimensional, and because the
velocity streamlines vary across the pores. It turns out that the rate coefficients for ion exchange
between solution and soil particle surfaces are also dependent on flow velocity. This occurs
because ion transfer between solution and particle surface is diffusion limited at the laminar flow
velocities used in the experiments. Due to the nonlinear nature of the chemical kinetics,
numerical solution of the transport equations is required. Careful attention must be given to time
steps in order to maintain numerical stability in the integration procedure. Accuracy in the
simulation is influenced by choice in the size of the space nodes.
3.7 References
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of Heat in Solids. Oxford University Press,
London.
Crank, J. 1956. The Mathematics ofDiffusion. Oxford University Press, London.
Hiester, N.K. and T. Vermeulen. 1952. Saturation performance of ionexchange and adsorption
columns. Chemical Engineering Progress 48:505516.
Mackie, J.S. and P. Meares. 1956. The sorption and diffusion of ethanol in a cation exchange
resin membrane. Discussions of the Faraday Society 21:111116.
where G = aC* I x* is the gradient in C' between adjacent space nodes. Computations are
carried out for x' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0.
In the first appendix table (Table A3.1) calculations are carried out at x* = 1.00 for Ax* =
1.00, i.e. for one step to x = I = 10 cm. Results are shown in Figure 3.10. It can be seen that the
procedure is stable and appears to converge to C' = 1 and q* = 1. However, there is a question of
accuracy in the simulation due to the large step size of Ax = 10 cm. In subsequent tables (A3.2
through A3.7) computations are carried out forx' = 0.1, 0.2, 0.4, 0.6, 0.8, 1.0. Note that time
steps are At' < 0.1 for these cases. Results for x* = 1.00 are shown in Figure 3.11, which are
significantly different from those in Figure 3.10. It is concluded that the smaller step size is far
more accurate than the single step. A comparison of response for the various space nodes is
shown in Figure 3.12. In all cases q' < C' as expected. Accuracy of the simulation could
probably be enhanced even further by increasing the number of space nodes.
The transport model appears to describe ion transport rather well provided that the rate
coefficient k is taken as a linear function of flow velocity.
3.5 Cation Transport across a Biological Membrane
In passing it should be mentioned that the theory of cation transport across the membrane of
corn mitochondria by diffusion and osmosis has been reported (Overman et al., 1970). In this
case the volume within the membrane changes due to ion transport and changes in osmotic
pressure. This illustrates application of the convective diffusion theory to a biological system.
3.6 Summary
In the case of transport through porous media convective diffusion theory can be applied if
the diffusion coefficient is taken as dependent on flow velocity. This occurs because transport
has been treated as onedimensional flow when in fact it is multidimensional, and because the
velocity streamlines vary across the pores. It turns out that the rate coefficients for ion exchange
between solution and soil particle surfaces are also dependent on flow velocity. This occurs
because ion transfer between solution and particle surface is diffusion limited at the laminar flow
velocities used in the experiments. Due to the nonlinear nature of the chemical kinetics,
numerical solution of the transport equations is required. Careful attention must be given to time
steps in order to maintain numerical stability in the integration procedure. Accuracy in the
simulation is influenced by choice in the size of the space nodes.
3.7 References
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of Heat in Solids. Oxford University Press,
London.
Crank, J. 1956. The Mathematics ofDiffusion. Oxford University Press, London.
Hiester, N.K. and T. Vermeulen. 1952. Saturation performance of ionexchange and adsorption
columns. Chemical Engineering Progress 48:505516.
Mackie, J.S. and P. Meares. 1956. The sorption and diffusion of ethanol in a cation exchange
resin membrane. Discussions of the Faraday Society 21:111116.
Overman, A.R. 1968. Convective diffusion across a porous diaphragm. J. Physical Chemistry
72:41684171.
Overman, A.R. 1975. Ion transport through sand by convective diffusion. Plant and Soil 43:663
670.
Overman, A.R. and R.J. Miller. 1968. Convective diffusion in capillaries. J. Physical Chemistry
72:155158.
Overman, A.R., G.H. Lorimer, and R.J. Miller. 1970. Diffusion and osmotic transfer in corn
mitochondria. Plant Physiology 45:126132.
Overman, A.R., B.R. McMahon, R.L Chou, and F.C. Wang. 1980. Cation transport in packed
bed reactor of soil. Environmental Engineering Division, Proc. American Society of Civil
Engineers 106:267277.
Robinson, R.A. and R.H. Stokes. 1959. Electrolyte Solutions. Butterworths Scientific
Publications, London.
Smith, J.M. 1970. Chemical Engineering Kinetics. McGrawHill Book Co. New York, NY.
List of Figures
Figure 3.1 Transient diffusion of KCl through a fritted glass filter. Line drawn from Eq. (3.11)
with a diffusion characteristic time of 123.4 h.
Figure 3.2 Transient convective diffusion of KCl through a fritted glass filter. Line drawn from
Eq. (3.11) with appropriate Peclet numbers calculated from Eq. (3.15) and characteristic times
from Eq. (3.14).
Figure 3.3 Correlation of measured vs. calculated Peclet numbers (0). Data points from Table
3.1. Line is the 45 degree slope.
Figure 3.4 Correlation of measured vs. calculated characteristic time ratio (r / T ). Data points
from Table 3.1. Line is the 45 degree slope.
Figure 3.5 Correlation of the Peclet number (0) vs. flow rate (v) for convective diffusion
through Ottawa sand. Data points adapted from Overman (1975). Line drawn from Eq. (3.29).
Figure 3.6 Correlation of the dispersion coefficient (D) with pore velocity (v) for cation
transport through Lakeland fine sand. Data points are from Table 3.2. Curve drawn from Eq.
(3.40).
Figure 3.7 Correlation of the exchange coefficient (k) with pore velocity (v) for cation transport
through Lakeland fine sand. Data points are from Table 3.2. Lines drawn by visual inspection to
pass through the (0, 0) intercept.
Figure 3.8 Dependence of the exchange parameter (1k / v ) on feed concentration (Co) for cation
transport through Lakeland fine sand. Data points from Table 3.2.
Figure 3.9 Dependence of cation exchange capacity (Q) on feed concentration (Co) for cation
transport through Lakeland fine sand. Data points from Table 3.2.
Figure 3.10 Response curves for solution concentration (C*) and exchange concentration (q*)
vs. time (t') for cation transport through Lakeland fine sand for feed concentration Co = 0.01
mole L' and pore velocity v = 0.147 cm min'. Simulations are from Eqs. (3.41) and (3.42) at
position x' = 1.00 and with space step size of Ax* = 1.00.
Figure 3.11 Response curves for solution concentration (C*) and exchange concentration (q')
vs. time (t*) for cation transport through Lakeland fine sand for feed concentration Co = 0.01
mole L' and pore velocity v = 0.147 cm min1. Simulations are from Eqs. (3.41) and (3.42) at
position x* = 1.00 and with space step size of Ax* = 0.20.
Figure 3.12 Response curves for solution concentration (C') vs. time (t') for cation transport
through Lakeland fine sand for feed concentration Co = 0.01 mole L' and pore velocity
S= 0.147 cm min'. Simulations are from Eqs. (3.41) and (3.42) at positions
x* = 0.2, 0.4, 0.6, 0.8,1.0 and with space step size of Ax' = 0.2.
1
D, 4
r~s~
B
O
Lii
cy/
~i ~a
10
S
/
F
/ '
4) /c
./
o j t
(Qw&)t
F(c ',6
l/
.el~)
0,/
L ctU)
p t:ts
i7V f I
:/ / ji
0 s y
V2 m L
s U) V i//C/U //0W4'
fi /
o
eI I
FYo 3,4
A4
~~'7
SI I
' 
e
0 I 1 I i
co, > &
6,al o!2a 6%)
7H '6 ,
i;
Li
8
,10
0
~i 31
X i~o 'D
/I
Ar~ A~I~~~ 
/
/i
/
r, N I'l.
0,4
g
I__ CIDIII~~3
/ / / /
vita / /i
0 / /
t
FB
4. Phosphorus Transport through Soil
4.1 Background
In this chapter focus is on transport of a solute (phosphorus) which undergoes convection,
diffusion, ion exchange, and chemical reaction. This subject is important in soil fertility and
resource management. Again the effective diffusion coefficient is related to flow velocity.
Dependence of kinetic coefficients on flow velocity is a key question to be examined. The
analysis includes both batch and packed bed studies. The first step is to clarify the kinetics of
phosphorus adsorption and reaction in the soil.
4.2 Phosphorus Kinetics in a Batch Reactor
The first step is to write a kinetic model of phosphorus adsorption from solution onto soil
particles. Since both liquid and solid phases of the system are involved, this is generally referred
to as heterogeneous kinetics. It is assumed that phosphate ions are adsorbed onto charged sites of
soil particles. Following the approach of enzyme kinetics (Mahler and Cordes, 1966) and of the
LotkaVolterra model of population dynamics (Kingsland, 1985), it seems logical to assume that
the rate of adsorption is proportional to the quantity of phosphate ions in solution and to the
quantity of sites for adsorption. It is also logical to assume by the concept of joint probability that
the rate of adsorption is related to the product of concentration in solution and concentration of
available sites for adsorption. This is known in chemistry as the law of mass action, and leads in
this case to overall second order kinetics. It is further assumed that the rate of desorption of
phosphate ions is proportional to the concentration of adsorbed ions. For a wellmixed
suspension in a batch reactor the kinetic model can be written as
dC= kSC+kdA (4.1)
dt
where t is time, S is concentration of available sites for adsorption, C is concentration of
phosphate ions in solution, A is concentration of adsorbed phosphate ions, k, is the rate
coefficient for adsorption, and kd is the rate coefficient for desorption. This is the famous
Langmuir equation for reversible adsorption (Paul, 1962). Since there is an irreversible loss of
available phosphorus in the soil suspension with time, it seems logical to add a surface reaction
to the kinetic model (Hinshelwood, 1940; Smith, 1970), so that the change of adsorbed ions with
time is given by
dA
=k SCkdAkA (4.2)
dt
where kr is the first order rate coefficient for reaction. Since there are three dependent variables
(C, A, S) a third equation is required to obtain a mathematical solution. It is now assumed that the
reaction scheme is described by
ka kr
C+S4 A F + S
kd
4. Phosphorus Transport through Soil
4.1 Background
In this chapter focus is on transport of a solute (phosphorus) which undergoes convection,
diffusion, ion exchange, and chemical reaction. This subject is important in soil fertility and
resource management. Again the effective diffusion coefficient is related to flow velocity.
Dependence of kinetic coefficients on flow velocity is a key question to be examined. The
analysis includes both batch and packed bed studies. The first step is to clarify the kinetics of
phosphorus adsorption and reaction in the soil.
4.2 Phosphorus Kinetics in a Batch Reactor
The first step is to write a kinetic model of phosphorus adsorption from solution onto soil
particles. Since both liquid and solid phases of the system are involved, this is generally referred
to as heterogeneous kinetics. It is assumed that phosphate ions are adsorbed onto charged sites of
soil particles. Following the approach of enzyme kinetics (Mahler and Cordes, 1966) and of the
LotkaVolterra model of population dynamics (Kingsland, 1985), it seems logical to assume that
the rate of adsorption is proportional to the quantity of phosphate ions in solution and to the
quantity of sites for adsorption. It is also logical to assume by the concept of joint probability that
the rate of adsorption is related to the product of concentration in solution and concentration of
available sites for adsorption. This is known in chemistry as the law of mass action, and leads in
this case to overall second order kinetics. It is further assumed that the rate of desorption of
phosphate ions is proportional to the concentration of adsorbed ions. For a wellmixed
suspension in a batch reactor the kinetic model can be written as
dC= kSC+kdA (4.1)
dt
where t is time, S is concentration of available sites for adsorption, C is concentration of
phosphate ions in solution, A is concentration of adsorbed phosphate ions, k, is the rate
coefficient for adsorption, and kd is the rate coefficient for desorption. This is the famous
Langmuir equation for reversible adsorption (Paul, 1962). Since there is an irreversible loss of
available phosphorus in the soil suspension with time, it seems logical to add a surface reaction
to the kinetic model (Hinshelwood, 1940; Smith, 1970), so that the change of adsorbed ions with
time is given by
dA
=k SCkdAkA (4.2)
dt
where kr is the first order rate coefficient for reaction. Since there are three dependent variables
(C, A, S) a third equation is required to obtain a mathematical solution. It is now assumed that the
reaction scheme is described by
ka kr
C+S4 A F + S
kd
4. Phosphorus Transport through Soil
4.1 Background
In this chapter focus is on transport of a solute (phosphorus) which undergoes convection,
diffusion, ion exchange, and chemical reaction. This subject is important in soil fertility and
resource management. Again the effective diffusion coefficient is related to flow velocity.
Dependence of kinetic coefficients on flow velocity is a key question to be examined. The
analysis includes both batch and packed bed studies. The first step is to clarify the kinetics of
phosphorus adsorption and reaction in the soil.
4.2 Phosphorus Kinetics in a Batch Reactor
The first step is to write a kinetic model of phosphorus adsorption from solution onto soil
particles. Since both liquid and solid phases of the system are involved, this is generally referred
to as heterogeneous kinetics. It is assumed that phosphate ions are adsorbed onto charged sites of
soil particles. Following the approach of enzyme kinetics (Mahler and Cordes, 1966) and of the
LotkaVolterra model of population dynamics (Kingsland, 1985), it seems logical to assume that
the rate of adsorption is proportional to the quantity of phosphate ions in solution and to the
quantity of sites for adsorption. It is also logical to assume by the concept of joint probability that
the rate of adsorption is related to the product of concentration in solution and concentration of
available sites for adsorption. This is known in chemistry as the law of mass action, and leads in
this case to overall second order kinetics. It is further assumed that the rate of desorption of
phosphate ions is proportional to the concentration of adsorbed ions. For a wellmixed
suspension in a batch reactor the kinetic model can be written as
dC= kSC+kdA (4.1)
dt
where t is time, S is concentration of available sites for adsorption, C is concentration of
phosphate ions in solution, A is concentration of adsorbed phosphate ions, k, is the rate
coefficient for adsorption, and kd is the rate coefficient for desorption. This is the famous
Langmuir equation for reversible adsorption (Paul, 1962). Since there is an irreversible loss of
available phosphorus in the soil suspension with time, it seems logical to add a surface reaction
to the kinetic model (Hinshelwood, 1940; Smith, 1970), so that the change of adsorbed ions with
time is given by
dA
=k SCkdAkA (4.2)
dt
where kr is the first order rate coefficient for reaction. Since there are three dependent variables
(C, A, S) a third equation is required to obtain a mathematical solution. It is now assumed that the
reaction scheme is described by
ka kr
C+S4 A F + S
kd
where F is concentration of"fixed" phosphorus in the soil. Since adsorption sites S are assumed
to be recycled in the sequence, this mechanism is referred to as heterogeneous catalysis. In other
words, the sites in the soil act as a catalyst for the irreversible reaction. Since the total number of
sites is fixed at So, the quantity of sites available for adsorption is given by
S = So A (4.3)
The kinetic scheme assumed here is an example of LangmuirHinshelwood kinetics (Overman
and Scholtz, 1999; Smith, 1970). Equations (4.1) through (4.3) describe the kinetics of the
chemical system.
Due to the SC product in Eqs. (4.1) and (4.2), this system of differential equations is referred
to as nonlinear, for which an analytical solution is not known. Numerical solution is generally
required. Since the system involves four parameters (k, kd, k,, So), this is not a trivial task. The
challenge is made even more daunting by the fact that only C is measured in the system.
4.2.1 Response in an Open Batch Reactor
It appears that a fresh approach is needed over the simple closed batch reactor. One possible
approach is to treat the system as an open batch reactor in which phosphate ions are added at a
steady rate r. Then the system of equations can be written as
dC
Sr ka SC +kd A
dt
dA = kSCkdAk,A (4.4)
dt
S = So A
Equation (4.4) constitutes a system of three equations in three unknowns. The trick now is to
examine the equations at steady state, where dC/dt = 0 = dA/dt. In this case the variables are
denoted with subscript s to designate steady state. Equation (4.4) now takes the form
0 = r k, S C, +kdA,
0 = kaS,C, kdA, k,A (4.5)
S, = So A,
which represents three algebraic equations in three unknowns. This greatly simplifies the
mathematical procedure. It can be shown that at steady state
r = rC (4.6)
K + C,
K+C,
A, SC (4.7)
K+C,
S, KSo (4.8)
K+C,
where K is the distribution coefficient and rm is maximum rate of phosphorus consumption at
high Cs, which are defined respectively by
K kd + kr (4.9)
k,
r, = krSo (4.10)
Note that Eq. (4.6) is similar to the MichaelicMenten relation in enzyme kinetics (Aiba et al.,
1965). This provides a rational basis for the Langmuir relation which has been assumed so
frequently in soil phosphorus experiments. Note that rm is controlled by kr and So, so that for kr
0 there is no net consumption of phosphorus, and that r, is linearly proportional to So (hence to
the amount of soil in the reactor). Both inferences can be subjected to experimental verification.
It follows from Eqs. (4.6) through (4.8) that the system can be reduced to the dimensionless
system of hyperbolic relations
C, /IK
r/r,, = (4.11)
"' 1+C, / K
C, I K
A /ISO (4.12)
S1+C,/K
1
S, /So = (4.13)
1+ C, IK
so that K and rm act as scaling factors for the variables C, and r. Note from Eqs. (4.11) through
(4.13) that for C/K = 1, r/r, = 1/2, A/So = 1/2, and S/So = 1/2.
Overman and Chu (1977ac) conducted experiments to verify Eq. (4.6). The soil was
Lakeland fine sand (thermic, coated typic Quartzipsamment), which is known to contain <5%
silt, <5% clay, and large amounts of aluminum and iron oxides (Hortenstine, 1966). For each run
of the open batch reactor, H3P04 was diluted to 500 ml with the pH adjusted to 5. Phosphate
solution was added to the reactor at a flow rate of 3.16 ml h' for a period of 6 h. A paddle wheel
stirrer was used to maintain the soil in suspension. A pH controller was used to maintain pH =
5.00 throughout the experiments. A check of electrodes before and after each experiment showed
no deterioration by the soil suspension. The soil/solution ratio was varied by adding 100, 150,
200, and 250 g of soil to 500 ml of solution. The rate of phosphorus injection r was varied to
achieve different levels of Ps. Plots of r vs. C, did follow the predicted hyperbolic relationship of
Eq. (4.6).
The role ofpH was next examined (Overman and Chu, 1977b). It was noted that upon
addition of soil to the reactor injection of acid from the pH controller was required to maintain
constant pH of the suspension. The rate of injection dropped rapidly with time as the system
approached steady state. This suggested release of OH as phosphorus was adsorbed. Perhaps the
mechanism involves anion exchange between H2P04 and OH'. Since fractionation of H3P04 into
H2PO4I, HP042, and P04"3 is pH dependent, this would influence the rate of adsorption onto the
soil particles. Since the measurements of solution phosphorus includes all three fractions, then
this effect would show up in k,, and therefore K. Experiments were conducted at pH = 2, 3, 5, 7,
and 8 by first adding either concentrated HCI or NaOH. Soil/solution ratio was constant at 250 g
soil in 500 ml of solution. Again phosphorus solution was injected at various rates to obtain plots
ofr vs. Ps, which followed hyperbolic patterns in all cases. It was found that 1/K vs. pH followed
essentially the same curve as the H2P041 fraction. This appeared to verify the mechanism of
anion exchange in accordance with earlier workers (Muljadi et al., 1966; Rajan et al., 1974).
Finally, a slow solution reaction was identified in the system (Overman and Chu, 1977c).
This introduced the modification
r' = r k'C, (4.14)
where k'is the first order rate coefficient for the solution reaction. Equation (4.6) is now
modified to the form
r' rm (4.15)
K+C,
This modification produced a linear plot between r, and the soil/solution ratio with an intercept
at the origin. A plot of 1/K vs. soil/solution ratio also produced a linear correlation, and not a
constant value as expected. Since this suggested an additional soil component in the system, the
model was later modified by Overman et al. (1983).
The model was next modified to incorporate the kinetic scheme illustrated below.
k. kf k,
C+Q<>A A+S<+B+Q B>C+S
kd k,
,rk,
F+S
where B represents concentration of transformed phosphate ions on the soil particles, Q is
concentration of sites for anion adsorption, S is concentration of sites for transformed ions, kfis
rate coefficient for forward transformation, kb is rate coefficient for backward transformation, ks
is rate coefficient for decomposition back to solution species, and kr is rate coefficient for
reaction to insoluble species. The last two steps are assumed to be irreversible. The kinetic
equations for an open batch reactor are now written as
dC
r k, QC + kdA+k,B
dt
dA
= k QC kdA kfSA + kbQB
dt
dB
d= kSA kbQB kB kB (4.16)
dt
Q = Qo A
S = So B
where Qo and So are total concentrations of surface sites for adsorption and transformation,
respectively. Equation (4.16) represents five equations for five variables (C, A, B, Q, S). Due to
the nonlinear nature of the three differential equations, an analytical solution has not been
obtained (Hommes, 1962). Numerical integration of Eq. (4.16) would be a very formidable task
since the system contains eight parameters. Again the trick is to examine the system for steady
state conditions (dC/dt = 0, dA/dt = 0, dB/dt = 0), for which Eq. (4.16) is replaced with the
system of algebraic equations
0 = r k, Q C, +kd A, +k,B,
0 = k,QC, kdA, kfSA, +kbQsBs
0 = kfSA, kbQsB kB, krB (4.17)
QS = Qo A,
S, = S B,
Equation (4.17) leads to quadratic equations in Qs and Ss, which can be approximated as
kd
Q d (4.18)
kd + ka C,
S kd (k + k + kb Qo (4.19)
Skd (kS +kr +kbQ) +kfk, QC,
It can be shown that net rate of consumption of phosphorus at steady state is given by
r = kBs = rC (4.20)
K+C,
where maximum rate rm and distribution coefficient K are defined by
r,  k krS (4.21)
k, kr + kfQ
K kd( k, + k, + kbo (4.22)
kk, +kk, kQo),
Dependence of r,, and K on soil/solution ratio in the reactor can be better understood by
rearranging Eqs. (4.21) and (4.22) to the forms
r, M (4.23)
kSo I H
1+1
M M
KK" 1 (4.24)
Ko K" M
1+
H
where the new parameters are defined by
H k, + k, (4.25)
k, Q 0
M
k,
KO =kd (4.26)
K = kd kb (4.27)
k, kk
and where M is the mass of soil in the reactor. For a uniform sample of soil, Qo and So should be
proportional to M. Equation (4.24) can be converted to the simple form
K = K(H+ fM (4.28)
where = k/kf.
Analysis of data by Overman et al. (1983) led to the parameter values krSo/M = 0.00107 mM
L' h1 g', Ko = 5.00 mM L',f= 0.05, and H = 50 g. These values lead to the prediction
equations
)M 0.00107M (4.29)
S H 50
1+ 1+
M M
K oH fM = 5.00 50 + 0.05M (4.30)
SH+M ) 50+M
Plots of Eqs. (4.29) and (4.30) are shown in Figures 4.1 and 4.2, respectively. For the case of M
= 250 g, Eq. (4.20) becomes
rrC, 0.223Cs (4.31)
K +C, 1.04 + C,
as shown in Figure 4.3. Equations (4.29) through (4.31) described data for the open batch reactor
rather well.
4.2.2 Response in a Closed Batch Reactor
The simple LangmuirHinshelwood model of phosphorus kinetics in a closed batch reactor
can be written as
dC
t>0 = kSC+kdA
dt
dA
S= kSCkdAkA
dt
S = S A (432)
t=0 C=Co
A= A =0
S = So
Due to the nonlinear nature of the differential equations, an analytical solution to this initial
value problem has not been found, and a numerical solution is required. A simple Euler method
of integration is chosen for the procedure in which discrete time steps At are used. The
corresponding increments for C and A for the nth time step, Atn, are estimated from
AC,, = At, (4.33)
/ dt "
AA, = At (4.34)
""\ dt "
in which the derivatives are evaluated at the current time step t,. Values of the variables at the
new time step tn+ are calculated from
t.+, = t, + At,
C+ = C, +AC (4.35)
An = A, + AA.
Sn+1 = So A,+1
The procedure starts with the initial condition t= 0, C = Co, A = 0, S = So. Values of At, are
chosen to achieve stability, convergence, and accuracy in the simulation, which can impose
severe limits on the time steps.
This problem presents a rather daunting challenge because of the four parameters (k,, kd, kr,
and So). In practice it is observed that the kinetics follows a two step process: rapid drop in
solution phosphorus concentration followed by a very slow decline. It appears that adsorption is
quite rapid compared to the surface reaction. In such cases the model is characterized by stiff
differential equations (Lapidus and Seinfeld, 1971). It is convenient to use a quasiequilibrium
assumption (kr 0 ) at the end of the rapid drop, so that
dC
t 0 = kaSCe + kdAe
dt
A, = Co Ce (4.36)
S, = So A
where subscript e designates equilibrium. From Eq. (4.36) it follows that
Ae = Co C = e (4.37)
K+C,
where the Langmuir coefficient K is defined by
K= kd (4.38)
k,
Note that Eq. (4.37) is similar to the MichaelisMenten hyperbolic relation for enzyme kinetics
(Laidler, 1965). The procedure is to estimate Ce from a time plot of C vs. t for each set of data. A
plot of (Co Ce) vs. Ce is then used to estimate So and K. Data for the slow decay can be used to
estimate k,r. Data from the initial rapid decay is used to estimate kaSo. A trialanderror approach
is then used to fine tune estimates ofka. Great care is called for in the numerical simulation to
maintain stability and accuracy, particularly during the initial stage when change in
concentration is rather rapid.
This procedure was used by Overman and Scholtz (1999) to analyze data from Burgoa et al.
(1990) with Myakka fine sand (sandy, siliceous, hyperthermic Aeric Haplaquod). Soil/solution
ratios consisted of 0.100, 0.125, 0.167, 0.250, 0.500 kg L'', with initial phosphorus
concentrations of 10.1, 20.3, 40.2, 62.4, 79.8, and 101.8 mg L1. The LangmuirHinshelwood
model was shown to describe the data rather well. Correlation of So with soil/solution ratio was
shown to be linear with a (0, 0) intercept, as expected. Analysis of data gave the parameters ka =
0.05 L mg' h1, kd = 0.25 h', kr = 0.005 hI, and So = 47 mg L1 for a soil/solution ratio ofM =
0.100 kg L'. This leads to kSo = 2.35 h1, so that the order of the kinetic coefficients is kSo >>
kd >> k. This explains the initial rapid decline followed by a very slow decline in solution
concentration. For M = 0.500 kg L', it was found that So = 220 mg L', so that kaSo = 11.0 h1.
This explains the strong sensitivity of response to soil/solution ratio in the reactor.
4.3 Phosphorus Transport in a Packed Column
In contrast to the batch reactor with rapid mixing of the soil suspension, flow in the packed
column occurs at Reynolds number below one laminarr flow). Average flow velocity through the
soil pores, v, is assumed to be uniform with depth and steady in time. Four basic processes are
included in the model: convective flow, dispersion (diffusion), ion exchange, and chemical
reaction. Chemical transport is treated at increasing levels of complexity of the model.
4.3.1 Transport with Equilibrium Ion Exchange
ka k,
In this case LangmuirHinshelwood kinetics is assumed as shown. C+> A F
kd
First order kinetics is assumed since the quantity of sites for exchange
is assumed large and to remain essentially constant. The rate of transfer of phosphorus from
solution to soil particles at a given depth can be described by
dC
kC+kdA (4.39)
ot
where k, is 1st order rate coefficient for adsorption, and kd is 1st order rate coefficient for
desorption. For very rapid kinetics, the quasiequilibrium assumption leads to
A = KC (4.40)
where K = ka/ka is the distribution coefficient. From this assumption, the rate of surface reaction
r can be written as
r = k,A = kKC = kC (4.41)
where the overall rate coefficient is defined as k = kr K.
The equation for phosphorus transport through the soil column can be written as
a2c ac ac
z>0 D vkC(I+K) 0
az2 z t
C(z,0) = 0 (4.42)
z=0 C(0,t)= Co
z o C(z,t)>0
where z is vertical distance from the inlet of the column, t is elapsed time since initiation of
chemical injection, D is the dispersion coefficient, and Co is the feed concentration of
phosphorus. The mathematical solution to Eq. (4.42) is given by (Overman et al., 1976)
z> 0 C = exp(2Z)erfc ZJ + ) + erfc V (4.43)
C, 2 [ 2 ) 2V
4.3 Phosphorus Transport in a Packed Column
In contrast to the batch reactor with rapid mixing of the soil suspension, flow in the packed
column occurs at Reynolds number below one laminarr flow). Average flow velocity through the
soil pores, v, is assumed to be uniform with depth and steady in time. Four basic processes are
included in the model: convective flow, dispersion (diffusion), ion exchange, and chemical
reaction. Chemical transport is treated at increasing levels of complexity of the model.
4.3.1 Transport with Equilibrium Ion Exchange
ka k,
In this case LangmuirHinshelwood kinetics is assumed as shown. C+> A F
kd
First order kinetics is assumed since the quantity of sites for exchange
is assumed large and to remain essentially constant. The rate of transfer of phosphorus from
solution to soil particles at a given depth can be described by
dC
kC+kdA (4.39)
ot
where k, is 1st order rate coefficient for adsorption, and kd is 1st order rate coefficient for
desorption. For very rapid kinetics, the quasiequilibrium assumption leads to
A = KC (4.40)
where K = ka/ka is the distribution coefficient. From this assumption, the rate of surface reaction
r can be written as
r = k,A = kKC = kC (4.41)
where the overall rate coefficient is defined as k = kr K.
The equation for phosphorus transport through the soil column can be written as
a2c ac ac
z>0 D vkC(I+K) 0
az2 z t
C(z,0) = 0 (4.42)
z=0 C(0,t)= Co
z o C(z,t)>0
where z is vertical distance from the inlet of the column, t is elapsed time since initiation of
chemical injection, D is the dispersion coefficient, and Co is the feed concentration of
phosphorus. The mathematical solution to Eq. (4.42) is given by (Overman et al., 1976)
z> 0 C = exp(2Z)erfc ZJ + ) + erfc V (4.43)
C, 2 [ 2 ) 2V
where erfc is the complementary error function (Abramowitz and Stegun, 1965). Steady state
distribution of solution phosphate is described by
C exp 1 (4.44)
Co 2 2D
which exhibits exponential decline of steady state solution concentration, Cs, with depth, z.
Transformed space and time variables are defined, respectively, by the linear transformations
Z 4kD (4.45)
V 2 2D
4kD" j72 1
T= + 4 2t (4.46)
T v2 4D 1+K
For the special case 4kD / 2 <<1, Eqs. (4.44) through (4.46) reduce to
C, _exp z (4.47)
Co
ZV z (4.48)
2D
2
T t (4.49)
4DI+K
The validity of the transport model was tested by Overman et al. (1976) for Lakeland fine
sand (thermic, coated typic Quartzipsamment). The soil column was packed to a bulk density of
1.73 g cm3 and porosity of 0.343. Feed concentration was 10 mg P L1 from KH2PO4. Flow rates
were run at pore velocities of 0.118, 0.256, 0.539, and 0.900 cm min'. Solution samples were
collected at depths ofz = 2, 4, 6, and 8 cm from a column of 10 cm length and total volume of
173 cm3. Measurements showed that the model described both steady state and transient
distributions rather well. However, it was found that the rate parameters were functions of flow
velocity, as shown in Table 4.1.
Table 4.1 Dependence of rate coefficients on flow velocity for phosphorus transport through
Lakeland fine sand for equilibrium adsorption.a
v, cm min1' 0.118 0.256 0.539 0.900
D, cm2 min1 0.014 0.031 0.097 0.242
k, min' 0.00459 0.00612 0.00749 0.00900
K 34.9 25.1 21.1 20.0
kr, min 0.000132 0.000244 0.000355 0.000450
v/k,,cm 890 1050 1520 2000
aData adapted from Overman et al. (1976).
The challenge is to explain results in Table 4.1. This means that the system is more
complicated than first visualized. It is concluded that k,, kd, and kr are not true kinetic
coefficients in the standard sense of chemical kinetics since they are functions of flow velocity.
Dependence of the dispersion coefficient is shown in Figure 4.4, where the curve is drawn from
D = 0.012 + 0.285v2 r = 0.99985 (4.50)
with a correlation coefficient ofr = 0.99985. Quadratic dependence on flow velocity agrees with
findings for cation transport through this same soil (see Table 3.2 and Figure 3.6). A plot ofkr
vs. v, as shown in Figure 4.5, exhibits a somewhat hyperbolic response described by the
empirical equation
kr a (4.51)
b, +v
where parameters ar and br are to be determined from the data. Equation (4.51) can be
rearranged to the linear form
V b 1
= ' +v= 709+1447V r= 0.9986 (4.52)
k, a, a,
with a linear correlation coefficient ofr = 0.9986. The linear result is also shown in Figure 4.5.
The curve in Figure 4.5 is drawn from
0.000690v
k, (4.53)
0.49 + v
It is next assumed that k, and kd vs. v also follow hyperbolic response described by
k = a (4.54)
b, +v
kd_ a (4.55)
bd +V
It follows from Eqs. (4.54) and (4.55) that the distribution coefficient should be described by
k a b +v
K= = a d (4.56)
kd ad ba +v
From a plot of K vs. v (Figure 4.6) it is estimated that the lower and upper limits on K are given
a a bd bd bd
K =17= , K =51 17 > =3
ab ab b, b, b.
By trialanderror, bc = 0.10 is chosen, which leads to the prediction equation
K=17 0.30+ (4.57)
LO.1 0+
The curve in Figure 4.6 is drawn from Eq. (4.57), which appears to describe the data reasonably
well. Based on this analysis dependence of k on v should be given by
k [0.0117vl 0.30 + (4.58)
S0.49 +V J0.10+v
as also shown in Figure 4.6, which exhibits reasonable agreement with the data.
A very significant inference can be drawn from this model analysis. According to Eq. (4.57)
the equilibrium distribution (at = 0) between adsorbed and solution phosphorus is
(A) =Ko =51 (4.59)
C equil
which indicates a moderately buffered system for phosphorus. Any reduction in solution
concentration would be accompanied by a shift in surface concentration to maintain equilibrium.
4.3.2 Transport with Nonequilibrium Ion Exchange
In this Section the restriction of equilibrium ion exchange is removed so that all the rate
coefficients can be estimated. The transport model can now be written as
z>0 D C Vk ,C+kdA =0
z2 dz at
aA
= kCkdAk,A
at
(4.60)
C(z,) = 0 (4.60)
A(z,0) = 0
z = 0 C(0, t) = C
z> oo C(z,t) 0
In this analysis the dispersion term will be neglected by assuming D = 0.
The steady state distributions ( C / at = 0 = aA / 8t) for this system are described by
z > 0 A, = KC,
(k> (4.61)
C, = Cexp z
where distribution coefficient K and overall rate coefficient k are defined, respectively, by
K = k (4.62)
kd + k,
k = Kk, (4.63)
This model predicts exponential decrease of steady state solution concentration with depth,
consistent with Eq. (4.47).
For the transient solution it is convenient to define the following dimensionless variables and
parameters:
t* = z = C C A A = a k= K2,
S, Cs A5, k K
where 1 is the length of the column. The transport equations now become
z >0 ++aC p(C'A*)}=0
za* at*
p(c" A)
t* (C
C*(z',0)=0 (4.64)
A'(z*,0)= 0
z* =0 C*(O,t') =
The mathematical solution becomes
t* < z* C*= = A
t' > z* A* = exp(apz* ) Jexp( py)10 4afz z*ydy (4.65)
0
C =exp(apz*)exp[ f(t'z' )I/o 4a2(t* z )+ A
where Jo is the modified Bessel function of first kind, zero order (Abramowitz and Stegun, 1965).
Note that the solution contains the integral of a Bessel function, which can be evaluated by
numerical procedures. There is a discontinuity (jump) in C* at t* = z* caused by neglect of the
dispersion term.
Analysis by Overman et al. (1978) showed that the kinetic model described both steady state
and transient response rather well for the experiment of Overman et al. (1976). It was found that
all three rate coefficients were dependent on flow velocity, as shown in Table 4.2. It again seems
reasonable to assume hyperbolic relations between the rate coefficients and flow velocity. This
leads to the equations
Sb 1
= b+V= 734+1352v r= 0.99966
kr ar ar
Sb, 1
=a + v = 0.0730 + 0.239v r = 0.9979
k, aa aa
(4.66)
(4.67)
Table 4.2 Dependence of rate coefficients on flow velocity for phosphorus transport through
Lakeland fine sand for the LangmuirHinshelwood model.a
v, cm min 0.118 0.256 0.539 0.900
k,, min1 1.1 2.0 2.7 3.1
kd, min1 0.03 0.08 0.13 0.15
kr, min' 0.00013 0.00024 0.00037 0.00046
K 36.5 24.9 20.7 20.6
k, min1 0.0047 0.0060 0.0077 0.0095
aData adapted from Overman et al. (1978).
The hyperbolic equations become
a,v 0.000740V
Sb, +v 0.543 +v
av 4.18v
k
b" +b 0.305 +
adv 0.300v
kd 0.915
bd + 0.915+v
(4.68)
(4.69)
(4.70)
where Eq. (4.70) has been obtained by visual inspection and with bd/b, = 3 as in Section 4.3.1.
Dependence of the rate coefficients on flow velocity is shown in Figure 4.7, where the curves are
drawn from Eqs. (4.68) through (4.70). Dependence of the distribution coefficient K and the
overall rate coefficient k on flow velocity can be written as
k k 0.915+ V1
K= a14
kd +k, kd 0.305 +
k = Kk [0.0104v [0.915]+v
0.543+;VJL 0.305 +V
which is shown in Figure 4.8 with the curves drawn from Eqs. (4.71) and (4.72). The kinetic
model also demonstrates the lag of A* behind C* during transient response.
(4.71)
(4.72)
In this case the equilibrium distribution (at v = 0) between adsorbed and solution phosphorus
is calculated to be
(A = K = 42 (4.73)
equil
which is in reasonable agreement with Eq. (4.59). The physical basis for Eqs. (4.68) through
(4.70) is left unresolved.
4.3.3 Transport with Ion Exchange and Transformation
In this case chemical kinetics is described by an extended LangmuirHinshelwood model
as shown below. For the packed column it is assumed
ka kf k,
that the quantity of sites for adsorption (Q) and transformation C < A <> B > C
kd kb
(S) are each constant. The transport model can be written as 1k k,
F
a2C ac ac
z>0 D v k,C+kdA+kB 0
az2 az t
aA
=kaCkdAkfA+kbB
at
aB
= kfA kB k,B k,B
at
C(z,0) = 0
A(z,O) = 0 (4.74)
B(z,0) = 0
z = 0 C(0,t)= C
z + oo C(z, t) 0
For this analysis, the dispersion term is assumed negligible (D = 0).
Distributions for steady state ( C / at = 0, aA / at = 0, aB / at = 0) are described by
z > 0 B, = K A,
A, = K, C, (4.75)
C, = Co exp z
(V
where the distribution and overall rate coefficients are defined by
K = (4.76)
Sk, +k, +k,
K, k, (4.77)
kd + k kbK/
k = K,Kfk, (4.78)
Note that if either k, kf, or kr are zero, then the overall rate coefficient k is also zero. This seems
intuitively correct. This model also predicts exponential decrease in steady state solution
concentration with depth, consistent with the previous two models.
To facilitate analysis of the transient case, it is convenient to introduce the dimensionless
variables and parameters
z V t C C, A, B
Z t ,C =,A = ,B = B
I C, A, B,
a dK 1ka kb K2 lk lksKK
a k, 8b K, = K 
SvKa k vK .
a6 'a kaf VKf v
The transport problem can now be written in dimensionless form
z' >0 a +af(C' A) w(B' C')+ C =0
dz at
= 8(c' A*)y6(A' B')
at
B S(A B') (4.79)
at*
C* (z*,0) = 0, A' (z',0) = 0, B (z',0) = 0
z* =0 C*(O,t') =
This comprises a system of three partial differential equations in three unknowns, and is
theoretically solvable. Since an analytical solution has not been obtained, a numerical procedure
is used.
Analysis by Overman et al. (1980) showed that this model described both steady state and
transient data rather well for the experiment of Overman et al. (1976). It was shown that stability
in the numerical procedure imposed the constraints on step size At* of
1 1 1
At* < At* << ,At* <
s' + ay o +ap
Analysis of data obtained parameter values
k = 3.00 min', kd = 0.25 min', kf = 0.09 min', kb = 0.03 min1
with k, and kr functions of velocity as shown in Table 4.3. Dependence of ks and kr on flow
velocity is also shown in Figure 4.9, where the curves are drawn from
Table 4.3 Dependence of rate coefficients k, and k,r on flow velocity for phosphorus transport
through Lakeland fine sand for the extended LangmuirHinshelwood model.a
V, cm min1 0.118 0.256 0.539 0.900
k,, min1 0.011 0.031 0.052 0.063
kr, min1 0.00020 0.00041 0.00072 0.00093
Kf 2.18 1.47 1.09 0.96
Ka 10.9 10.1 9.76 9.64
k, min' 0.0048 0.0061 0.0077 0.0086
aData adapted from Overman et al. (1980).
k aV _0.175V (4.80)
Sb, +V 1.46 + v
k= arV 0.00204v (4
b, + v 1.04 + v
which provides reasonable fit of the data. A plot of the overall rate coefficient k vs. V is shown in
Figure 4.10, where the curve is drawn from
_av 0.00989V
k = (4.82)
b+v 0.143 +
Equation (4.82) appears to give a reasonable correlation between the two. Dependence of the
distribution coefficients Ka and Kf on velocity derives mainly from dependence ofks on flow
velocity.
Some practical inferences can now be drawn with this theory. For equilibrium conditions
(V = 0) where ks = 0 = kr, values of Kf and Ka become
K _3.0, K, 12.0
Skb 0.03 a kd 0.25
It follows from Eq. (4.75) that the ratio of surface to solution phosphorus at equilibrium is
A, +B, =[K.( +Kf)] 0 +=12.0(1+3.0)=48 (4.83)
Cs equil
or about 50:1. This value is in agreement with those of the other two models. From Eq. (4.75) the
steady state distribution of solution phosphorus can be described by
C, = Co exp kz = Co exp (4.84)
where zc = v/ k is defined as characteristic depth for the steady state distribution. For z/zc = 1,
C/Co = exp() = 0.368 : 1/3. It follows from Eq. (4.82) that dependence ofzc on flow velocity
is given by
v b
z = +v =101+14.5v cm (4.85)
k a a
which shows linear increase in characteristic depth with flow velocity. As flow velocity is
increased, the phosphorus distribution is pushed deeper into the soil, as expected. Note that as
v > 0, zc 101 cm, or about 1 m. This appears to agree with field observations (Overman et
al., 1976; Allhands et al., 1995).
A comment is in order about the decomposition step from B to C. It was found that
k, > 0 as v > 0. This result is essential to avoid perpetual motion with no net energy input at
v = 0. This result also suggests a shear mechanism of B by the flowing liquid.
It is informative to examine the adsorption/desorption process for the model. Does the
mathematical model hold for the case ofdesorption? Designate concentrations during desorption
as C, A, B'. Transport is assumed to be described by
BC> BC I p(c1A )/B C')=
z* >0 a + + a,(C A') ,o(B C')= 0
az' at*
aA = (cA')rs (A'B')
at
B* = (A B) (4.86)
at*
C(z',0)=1, A(z*,0)=1, B"(z',0)=1
z =0 C'(0, t*)=0
Assume the symmetry transformations between desorption and adsorption processes hold so that
C =1C*, A =A*, B =IB* (4.87)
It follows from Eq. (4.86) that
ac ac* aC aC* aA A* a aB aB*
az* az* at* at* at* at* at* at*'
C' A' =(C' A*), B C =(B' C*) A' B =(A B*)
With these substitutions, Eq. (4.86) becomes
z >0 + c+ap(C A')o(B C C)=0
az at*
A = p(c' A ) y A' B')
at*
B' = (A* B*) (4.88)
at*
C*(z*,0)= 0, A*(z',0)= 0, B*(z*,0)= 0
z'=0 C*(0,t')=1
This preserves the form of the model through the mathematical transformations, which is
essential if the model is to have general applicability. Note that symmetry holds by use of the
reduced variables C*, A*, and B*, all referenced to steady state conditions. In fact, this
consequence highlights the significance of the reduced variables in the mathematical theory.
We now illustrate a simple numerical procedure for simulation of phosphorus transport. The
dispersion term will be neglected for simplicity. For the case with = 0.900 cm min1 and = 10
cm, assume the parameter values k, = 3.00 min ', kd = 0.25 min1, kf 0.09 min1, kb = 0.03 min',
ks = 0.063 min', kr = 0.00093 min~, which leads to dimensionless parameters of
k k
K = 0.958, K, = 9.64,
/ kb+k,+k, kd +k kbKf
kd Lk k Lk2 LksK, KK
a = K = 7.74, f = = 3.46, = K = 0.306, 8 = = 1.04, co = = 6.46
ka v K kf vKf
The transport problem can now be written as
z >0 + +26.8(C A) 6.46(B' C)=
az' at*
A = 3.46(C' A) 0.318(A' B)
Bt= 1.04(A' B*) (4.89)
at'
C'(z*,0)= 0, A'* (z*,0) = 0, B*(z*,0) = 0
z*=0 C*(0,t*)=l
1 1 1
The time constraints become At* < = 0.96, At* < = 0.17, At* < = 0.030,
SP + ay co + a
which imposes a severe limit on time steps to maintain numerical stability. The convergence
condition can now be stated as C*(z*,t* >oo) 1, A*(z*,t* oo) >1,B*(z*,t* oo) 1.
Equation (4.89) can now be rearranged to write the rate equations as
C. = C= _C 26.8(C A')+ 6.46(B* C)
a8t' z*
A* 3.46(C* A')0.318(A* B) 4.90
t* (4.90)
B B =1.04(A' B')
at*
with C'(z*,0)= 0, A(z,0)= 0, B*(z*,0)= 0, C(0,t')=l
SC' 01
For the first space step use Az = 0.05, so for z* = 0.05 and t = 0, G = 20.0. The
8z 0.05
gradient is updated for each successive time step.
From Table A4.1 it is apparent that C', A, and B* are all converging to 1 as required, and
that the procedure remains numerically stable. Note that the time derivatives and the gradient are
all converging toward zero. At 25 pore volumes through the column (t'= 25.0) solution
concentration has reached C* = 0.999999+ at z* = 0.05. After 40 pore volumes C* exceeds
0.999999999 and is still rising ever so slowly. It appears safe to assume that C* converges to 1 as
time continues provided enough digits are carried in the computations to avoid round off error.
Other simulation values are given in Tables A4.2 through A4.9 for z* = 0.10, 0.20, 0.30, 0.40,
0.50, 0.60, 0.80, and 1.00, respectively. The importance of small step size of At* = 0.03 is
illustrated.
A summary of values for solution concentration (C*) vs. time (t*) at various depths (z*) is
listed in Table 4.4. These results can be illustrated graphically in two different ways. Response
curves (C* vs.t') are shown in Figure 4.11 for depths of z* = 0.10, 0.20, 0.40, 0.60, 0.80, and
1.00. Time lag increases with depth as expected. In all cases C',A*, and B* appear to converge
toward 1. Distribution curves (C* vs.z*) are shown in Figure 4.12 for times of t*= 2, 5, 10, 15,
20, 25, and 30. Note that t' = vt / indicates the number of pore volumes of liquid which has
passed through the column at z = 10 cm. Even after 30 pore volumes solution concentration has
still not reached steady state at all depths, which indicates the strong capacity of the soil to
adsorb phosphorus.
Insight into the kinetics of the system can be gained from a plot of C', A*,andB* vs.t*. This
response is shown in Figure 4.13 for for z* = 0.05 and Figure 4.14 for for z* = 0.50. Note that
C' 2 A* > B* for all times, as expected from the kinetic model. Lag of surface concentration
behind solution concentration is much greater at the more shallow depth due to the rapid rise of
C* with time. In fact, the jump in C* near t* = 0 is apparent. This is caused by neglect of the
dispersion term in the transport model. The effect is less noticeable at greater depths in the
column.
Similar simulations can be run at other flow rates with appropriate values for k, and kr.
Table 4.4 Numerical simulation of the response curves for the extended LangmuirHinshelwood
model of phosphorus transport in Lakeland fine sand at a flow velocity of 7 = 0.900 cm minl
t* C'
0.10
0.20
0.30
z
0.40
0.50 0.60 0.80 1.00
0.0000
0.7058
0.8327
0.9020
0.9431
0.9663
0.9804
0.9884
0.9931
0.9960
0.9976
0.9986
0.9991
0.9995
0.9997
0.9998
0.99990
0.99994
0.99996
0.99998
0.99999
0.999992
0.999995
0.999997
0.999998
0.9999993
0.9999996
0.9999998
0.0000
0.4554
0.6424
0.7675
0.8473
0.9012
0.9375
0.9599
0.9745
0.9840
0.9899
0.9936
0.9961
0.9975
0.9985
0.9991
0.9994
0.9996
0.9998
0.9999
0.99992
0.99995
0.99997
0.99998
0.99999
0.0000
0.1900
0.3478
0.4868
0.6026
0.6959
0.7754
0.8332
0.8769
0.9106
0.9347
0.9527
0.9662
0.9756
0.9825
0.9876
0.9911
0.9937
0.9955
0.9968
0.9978
0.9984
0.9988
0.9992
0.9994
0.999993 0.9996
0.999996 0.9997
0.999997 0.9998
0.99999990 0.999998 0.9999
0.99999993 0.999999 0.9999
0.99999995 0.999999 0.99993
0.0000
0.0740
0.1734
0.2857
0.3938
0.4942
0.5894
0.6689
0.7360
0.7932
0.8382
0.8743
0.9038
0.9262
0.9438
0.9577
0.9680
0.9760
0.9821
0.9867
0.9901
0.9927
0.9945
0.9959
0.9970
0.9978
0.9984
0.9988
0.9991
0.9994
0.9995
0.0000 0.0000 0.0000 0.0000
0.0270 0.0093 0.0032 0.0004
0.0807 0.0353 0.0149 0.0033
0.1550 0.0790 0.0383 0.0109
0.2381 0.1349 0.0726 0.0242
0.3257 0.2014 0.1178 0.0444
0.4169 0.2769 0.1740 0.0721
0.5012 0.3534 0.2358 0.1066
0.5789 0.4294 0.3020 0.1473
0.6506 0.5049 0.3720 0.1950
0.7117 0.5733 0.4396 0.2453
0.7643 0.6360 0.5049 0.2986
0.8101 0.6941 0.5690 0.3539
0.8472 0.7440 0.6267 0.4091
0.8780 0.7873 0.6794 0.4632
0.9041 0.8259 0.7282 0.5169
0.9243 0.8576 0.7706 0.5668
0.9406 0.8839 0.8072 0.6142
0.9540 0.9068 0.8401 0.6598
0.9642 0.9250 0.8674 0.7007
0.9723 0.9401 0.8907 0.7375
0.9788 0.9526 0.9109 0.7722
0.9837 0.9624 0.9274 0.8023
0.9875 0.9703 0.9410 0.8292
0.9905 0.9768 0.9527 0.8537
0.9927 0.9817 0.9617 0.8747
0.9944 0.9857 0.9693 0.8930
0.9958 0.9889 0.9756 0.9091
0.9968 0.9914 0.9806 0.9225
0.9976 0.9933 0.9846 0.9344
0.9982 0.9949 0.9879 0.9450
0.05
0.0000
0.0000
0.0004
0.0025
0.0068
0.0145
0.0270
0.0437
0.0654
0.0929
0.1246
0.1607
0.2011
0.2434
0.2882
0.3362
0.3829
0.4293
0.4767
0.5222
0.5651
0.6066
0.6446
0.6802
0.7150
0.7465
0.7752
0.8019
0.8249
0.8460
0.8656
4.4 Summary
By now the reader may be somewhat overwhelmed at the complexity of soil phosphorus
chemistry. In fact, the comments of Professor Toby Kurtz of the University of Illinois come to
mind in his talk at the 1981 Soil and Crop Science Society of Florida conference (Kurtz, 1981)
about the 'phosphorus merrygoround'. An indication of the mathematical complexity of the
field of chemical kinetics can be seen from the work of Henry Eyring and associates (Eyring and
Eyring, 1963; Eyring et al., 1964; Johnson et al., 1974).
In the present analysis the LangmuirHinshelwood model of heterogeneous kinetics has been
adopted, which is very similar to the model used in enzyme kinetics and appears to provide a
very rational description of the system. It incorporates the Langmuir approximation which has
been used extensively in soil chemistry. Much work remains to be done in this subject. For
example, the identity of the transformed species has not been established. This is a case of
working out the kinetics ahead of the mechanism, a procedure which is not uncommon in the
field of chemical kinetics.
Further comments on the surface kinetics of soil phosphorus is in order, particularly
dependence of the rate coefficients on flow velocity. It was concluded that the coefficients (k,,
kd, and kr) for the simple LangmuirHinshelwood model were all dependent on flow velocity.
This does not seem logical for kd and kr since these relate to surface processes. It could be
surmised that k, could relate to flow velocity if transfer of solution ions to the surface sites was
diffusion limited. After all, we found in Section 3.4 that the coefficients for cation exchange
were a linear function of flow velocity. Since it was pointed out that the surface kinetics of ion
exchange in a batch reactor is extremely fast, it could be concluded that ion transfer to the
surface is diffusion limited. In fact this was the rationale for development of the extended
LangmuirHinshelwood model of surface kinetics for soil phosphorus. In that case it was
expected that the rate coefficients kd, kf, and kb would definitely be independent of flow velocity.
Perhaps k, for adsorption would depend on velocity. However, preliminary efforts along those
lines failed. It was concluded that surface kinetics was slow enough that transfer from solution to
the particle surface was not diffusion limited. This meant that velocity dependence has to lie with
k, and/or kr. This assumption did account for velocity dependence of the overall rate coefficient k
and distribution coefficient K, which is confirmed in Figures 4.9 and 4.10. Dependence ofk, on
flow velocity is attributed to a shear mechanism; dependence ofk, on velocity remains
unexplained at this point. Further work on this point is needed.
The reader may feel that the models arrived at in this chapter are unduly complicated. For
further enlightenment of the subject of kinetics a book on enzyme kinetics is suggested (such as
Mahler and Cordes, 1966). Further insight into the theory of reaction rates can be found in
Eyring and Eyring (1963) and Eyring et al. (1964, 1980). In short, chemical kinetics is a rich and
complex subject in general, incorporating ideas from classical and quantum physics.
4.5 References
Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Aiba, S., A.E. Humphrey, and N.F. Millis. 1965. Biochemical Engineering. Adademic Press.
New York, NY.
4.4 Summary
By now the reader may be somewhat overwhelmed at the complexity of soil phosphorus
chemistry. In fact, the comments of Professor Toby Kurtz of the University of Illinois come to
mind in his talk at the 1981 Soil and Crop Science Society of Florida conference (Kurtz, 1981)
about the 'phosphorus merrygoround'. An indication of the mathematical complexity of the
field of chemical kinetics can be seen from the work of Henry Eyring and associates (Eyring and
Eyring, 1963; Eyring et al., 1964; Johnson et al., 1974).
In the present analysis the LangmuirHinshelwood model of heterogeneous kinetics has been
adopted, which is very similar to the model used in enzyme kinetics and appears to provide a
very rational description of the system. It incorporates the Langmuir approximation which has
been used extensively in soil chemistry. Much work remains to be done in this subject. For
example, the identity of the transformed species has not been established. This is a case of
working out the kinetics ahead of the mechanism, a procedure which is not uncommon in the
field of chemical kinetics.
Further comments on the surface kinetics of soil phosphorus is in order, particularly
dependence of the rate coefficients on flow velocity. It was concluded that the coefficients (k,,
kd, and kr) for the simple LangmuirHinshelwood model were all dependent on flow velocity.
This does not seem logical for kd and kr since these relate to surface processes. It could be
surmised that k, could relate to flow velocity if transfer of solution ions to the surface sites was
diffusion limited. After all, we found in Section 3.4 that the coefficients for cation exchange
were a linear function of flow velocity. Since it was pointed out that the surface kinetics of ion
exchange in a batch reactor is extremely fast, it could be concluded that ion transfer to the
surface is diffusion limited. In fact this was the rationale for development of the extended
LangmuirHinshelwood model of surface kinetics for soil phosphorus. In that case it was
expected that the rate coefficients kd, kf, and kb would definitely be independent of flow velocity.
Perhaps k, for adsorption would depend on velocity. However, preliminary efforts along those
lines failed. It was concluded that surface kinetics was slow enough that transfer from solution to
the particle surface was not diffusion limited. This meant that velocity dependence has to lie with
k, and/or kr. This assumption did account for velocity dependence of the overall rate coefficient k
and distribution coefficient K, which is confirmed in Figures 4.9 and 4.10. Dependence ofk, on
flow velocity is attributed to a shear mechanism; dependence ofk, on velocity remains
unexplained at this point. Further work on this point is needed.
The reader may feel that the models arrived at in this chapter are unduly complicated. For
further enlightenment of the subject of kinetics a book on enzyme kinetics is suggested (such as
Mahler and Cordes, 1966). Further insight into the theory of reaction rates can be found in
Eyring and Eyring (1963) and Eyring et al. (1964, 1980). In short, chemical kinetics is a rich and
complex subject in general, incorporating ideas from classical and quantum physics.
4.5 References
Abramowitz, M. and I.A. Stegun. 1965. Handbook of Mathematical Functions. Dover
Publications. New York, NY.
Aiba, S., A.E. Humphrey, and N.F. Millis. 1965. Biochemical Engineering. Adademic Press.
New York, NY.
Allhands, M.N., S.A. Allick, A.R. Overman, W.G. Leseman, and W. Vidak. 1995. Municipal
water reuse at Tallahassee, Florida. Trans. American Society ofAgricultural Engineers
38:411418.
Eyring, H. and E.M. Eyring. 1963. Modern Chemical Kinetics. Reinhold. New York, NY.
Eyring, H., D. Henderson, B.J. Stover, and E.M. Eyring. 1964. Statistical Mechanics and
Dynamics. Wiley. New York, NY.
Eyring, H., S.H. Lin, and S.M. Lin. 1980. Basic Chemical Kinetics. Wiley. New York, NY.
Hinshelwood, C.N. 1940. Kinetics of Chemical Change. Oxford University Press. London,
England.
Hommes, F.A. 1962. The integrated MichaelisMenten equation. Archives ofBiochemistry and
Biophysics 96:2831.
Hortenstine, C.C. 1966. Phosphorus fixation and phosphorus fractions in sandy soils. Proc. Soil
and Crop Science Society ofFlorida 26:136142.
Johnson, F.H., H. Eyring, and B.J. Stover. 1974. The Theory of Rate Processes in Biology and
Medicine. Wiley. New York, NY.
Kingsland, S.E. 1985. Modeling Nature. University of Chicago Press. Chicago, IL.
Kurtz, L.T. 1981. Dismounting from the phosphorus fixation merrygoround? Proc. Soil and
Crop Science Society of Florida 40:3540.
Laidler, K.J. 1965. Chemical Kinetics. McGrawHill. New York, NY.
Lapidus, L. and J.H. Seinfeld. 1971. Numerical Solution of Ordinary Differential Equations.
Academic Press. New York, NY.
Mahler, H.R. and E.H. Cordes. 1966. Biological Chemistry. Harper & Row Publishers. New
York, NY.
Muljadi, D., A.M. Posner, and J.P. Quirk. 1966. The mechanism of phosphate adsorption by
kaolinite, gibbsite, and pseudoboehmite. I. The isotherms and the effect of pH on adsorption.
J. Soil Science 17:212229.
Overman, A.R. and R.L. Chu. 1977a. A kinetic model of steady state phosphorus fixation in a
batch reactor. I. Effect of soil solution ratio. Water Research 11:771775.
Overman, A.R. and R.L. Chu. 1977b. A kinetic model of steady state phosphorus fixation in a
batch reactor. II. Effect ofpH. Water Research 11:777778.
Overman, A.R. and R.L. Chu. 1977c. A kinetic model of steady state phosphorus fixation in a
batch reactor. III. Effect of solution reaction. Water Research 11:779781.
Overman, A.R., R.L. Chu, and Y. Le. 1978. Kinetic coefficients for phosphorus transport in
packedbed reactor. J. Water Pollution Control Federation 50:19051940.
Overman, A.R., R.L. Chu, and W.G. Leseman. 1976. Phosphorus transport in a packed bed
reactor. J Water Pollution Control Federation 48:880888.
Overman, A.R., R.L. Chu, and B.R. McMahon 1983. A kinetic model of steady state phosphorus
fixation in a batch reactor. IV. Details of surface kinetics. Water Research 17:1519.
Overman, A.R., B.R. McMahon, and R.L. Chu. 1980. Velocity dependence of phosphorus
transport in a packedbed reactor. J. Water Pollution Control Federation 52:24712476.
Overman, A.R. and R.V. Scholtz. 1999. LangmuirHinshelwood model of soil phophorus
kinetics. Communications in Soil Science and Plant Analysis 30:109119.
Paul, M.A. 1962. Physical Chemistry. D. C. Heath & Co. Boston, MA.
Rajan, S.S.S. K.W. Perrott, and W.M.H. Saunders. 1974. Identification of phosphatereactive
sites of hydrous aluminum from proton consumption during phosphate adsorption at constant
pH values. J. Soil Science 25:438447.
Smith, J.M. 1970. Chemical Engineering Kinetics. McGrawHill Book Co. New York, NY.
List of Figures
Figure 4.1 Dependence of the maximum rate parameter (rm) on soil mass (M) for phosphorus
kinetics in an open batch reactor with Lakeland fine sand by the extended Langmuir
Hinshelwood model. Curve drawn from Eq. (4.29).
Figure 4.2 Dependence of the distribution coefficient (K) on soil mass (M) for phosphorus
kinetics in an open batch reactor with Lakeland fine sand by the extended Langmuir
Hinshelwood model. Curve drawn from Eq. (4.30).
Figure 4.3 Dependence of reaction rate (r) on steady state solution phosphorus concentration (Cs)
for phosphorus kinetics in an open batch reactor with 250 g Lakeland fine sand in 500 ml
solution by the extended LangmuirHinshelwood model. Curve drawn from Eq. (4.31).
Figure 4.4 Correlation of the dispersion coefficient (D) with pore velocity (v) for phosphorus
transport through Lakeland fine sand. Data points are from Table 4.1. Curve drawn from Eq.
(4.50).
Figure 4.5 Dependence of the reaction rate coefficient (kr) and velocity/reaction rate ratio (/k k,)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.1. Line
drawn from Eq. (4.52); curve from Eq. (4.53).
Figure 4.6 Dependence of the distribution coefficient (K) and the overall reaction coefficient (k)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.1.
Curves drawn from Eqs. (4.57) and (4.58).
Figure 4.7 Dependence of the rate coefficients for adsorption (ka), desorption (kd), and reaction
(kr) on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.2.
Curves drawn from Eqs. (4.68) through (4.70).
Figure 4.8 Dependence of the distribution coefficient (K) and the overall reaction coefficient (k)
on pore velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.2.
Curves drawn from Eqs. (4.71) and (4.72).
Figure 4.9 Dependence of the rate coefficients for decomposition (ks) and reaction (kr) on pore
velocity (v) for a packed bed reactor with Lakeland fine sand. Data from Table 4.3. Curves
drawn from Eqs. (4.80) and (4.81).
Figure 4.10 Dependence of the overall reaction coefficient (k) on pore velocity (v) for a packed
bed reactor with Lakeland fine sand. Data from Table 4.3. Curve drawn from Eqs. (4.82).
Figure 4.11 Response curves for solution concentration (C*) vs. time (t*) at space nodes
(z* = 0.1, 0.2,0.4,0.6,0.8,1.0) for phosphorus transport through Lakeland fine sand at a pore
velocity of v = 0.900 cm min'. Curves drawn Eq. (4.90).
Figure 4.12 Distribution curves for solution concentration (C*) vs. depth (z*) at various times
(t* = 1, 2,5,10,15,20,25,30) for phosphorus transport through Lakeland fine sand at a pore
velocity of v = 0.900 cm min'. Curves drawn Eq. (4.90).
Figure 4.13 Response curves for solution concentration (C*), adsorbed concentration (A*), and
transformed concentration (B') vs. time (t') at space node z* = 0.05 for phosphorus transport
through Lakeland fine sand at a pore velocity of v = 0.900 cm min. Curves drawn Eq. (4.90).
Figure 4.14 Response curves for solution concentration (C'), adsorbed concentration (A*), and
transformed concentration (B*) vs. time (t*) at space node z* = 0.50 for phosphorus transport
through Lakeland fine sand at a pore velocity of = 0.900 cm min. Curves drawn Eq. (4.90).
B^ ^11^ ..`. . .r
5'
0 do /Ir /
M01
O q
AA)
729 t,
o I
~~ 72
f
0
CS
t
cs~
'L
5
AN k
,3
I I I
(c3
e4J1
/B/P
0
o, op
ffPe2
4 OQP
0_
1
 4 ral, lirrm Ll
OA Ij
I\ 1  ) I
C
x
w^''
00
D _
Si
RI
O 
bJa i^ G 2 Y
T:i t ^i^
'Fi H^C
1/
TS
_ I
L LJJ~
0ol
A0
~ ~~
~0"
,rb 1l
_ )
/4(
l, cmw 'i'
Yim 4,
'11
pl 4
c 1i
1tbfrlO
) '/~r~
6 P8 q
4f
1... . ... ... ... . 
" /0
Fr/ H, al1
)
t L
Of 1
0
0
tI
/7,
* it
A.
/,T) 1 1 r '
`7t' e r
~~f 1~ ~
Ir
1 4 r
A A
e I/
V I' 3
t:
b 0
0 1Z
~i~:
i, 1
PZ, 4~ILle
5. Nutrient Coupling Between Applied and Available
5.1 Background
The first four chapters have dealt with chemical transport through capillary tubes, porous
media, and soil. Focus was first on nonreactive solutes, followed by transport of ions which
undergo either ion exchange or chemical reaction. In all cases, transport was described by means
of a differential equation (following a reductionist approach). It should be apparent that the
kinetics of such systems can be rather complex. Numerical methods are generally required to
obtain a solution.
A different approach is used in this chapter. A series of postulates are used to state various
mathematical relationships based on previous work (Overman and Scholtz, 2002). The two types
of equations employed are logistic and hyperbolic. These are used to derive other relationships
among system variables. Results are used to analyze and describe data, and to test predictions
made from the models. Finally, the chronological order in which the models were actually
developed is discussed to emphasize the emergent nature of the process of model development.
5.2 Coupling of Applied and Available Soil Phosphorus
Crop response to applied phosphorus is used to illustrate coupling among applied, soil
extractable, crop uptake, and crop biomass. A similar procedure is believed to apply for other
mineral elements such as nitrogen and potassium.
5.2.1 Response of Extractable Soil Phosphorus to Applied Phosphorus
The first step is to describe coupling between applied and extractable soil phosphorus. Data
from the literature are used for this purpose.
Study on Arredondo Fine Sand at Gainesville, Florida
Sartain (1992) conducted field measurements on response of extractable soil phosphorus to
applied phosphorus for Arredondo fine sand (loamy, siliceous, hyperthermic Grossarenic
Paleudult) at Gainesville, FL. Data are given in Table 5.1 and shown in Figure 5.1. Overman and
Scholtz (2002, p. 229) has shown that these data can be
Table 5.1 Response of extractable soil phosphorus to applied phosphorus on Arredondo fine sand
at Gainesville, FL.a
Applied P, g m2 0 10 20 40 80
Extractable P, mg kg1 151 192 218 313 429
aData adapted from Sartain (1992).
described with the logistic equation
5. Nutrient Coupling Between Applied and Available
5.1 Background
The first four chapters have dealt with chemical transport through capillary tubes, porous
media, and soil. Focus was first on nonreactive solutes, followed by transport of ions which
undergo either ion exchange or chemical reaction. In all cases, transport was described by means
of a differential equation (following a reductionist approach). It should be apparent that the
kinetics of such systems can be rather complex. Numerical methods are generally required to
obtain a solution.
A different approach is used in this chapter. A series of postulates are used to state various
mathematical relationships based on previous work (Overman and Scholtz, 2002). The two types
of equations employed are logistic and hyperbolic. These are used to derive other relationships
among system variables. Results are used to analyze and describe data, and to test predictions
made from the models. Finally, the chronological order in which the models were actually
developed is discussed to emphasize the emergent nature of the process of model development.
5.2 Coupling of Applied and Available Soil Phosphorus
Crop response to applied phosphorus is used to illustrate coupling among applied, soil
extractable, crop uptake, and crop biomass. A similar procedure is believed to apply for other
mineral elements such as nitrogen and potassium.
5.2.1 Response of Extractable Soil Phosphorus to Applied Phosphorus
The first step is to describe coupling between applied and extractable soil phosphorus. Data
from the literature are used for this purpose.
Study on Arredondo Fine Sand at Gainesville, Florida
Sartain (1992) conducted field measurements on response of extractable soil phosphorus to
applied phosphorus for Arredondo fine sand (loamy, siliceous, hyperthermic Grossarenic
Paleudult) at Gainesville, FL. Data are given in Table 5.1 and shown in Figure 5.1. Overman and
Scholtz (2002, p. 229) has shown that these data can be
Table 5.1 Response of extractable soil phosphorus to applied phosphorus on Arredondo fine sand
at Gainesville, FL.a
Applied P, g m2 0 10 20 40 80
Extractable P, mg kg1 151 192 218 313 429
aData adapted from Sartain (1992).
described with the logistic equation
5. Nutrient Coupling Between Applied and Available
5.1 Background
The first four chapters have dealt with chemical transport through capillary tubes, porous
media, and soil. Focus was first on nonreactive solutes, followed by transport of ions which
undergo either ion exchange or chemical reaction. In all cases, transport was described by means
of a differential equation (following a reductionist approach). It should be apparent that the
kinetics of such systems can be rather complex. Numerical methods are generally required to
obtain a solution.
A different approach is used in this chapter. A series of postulates are used to state various
mathematical relationships based on previous work (Overman and Scholtz, 2002). The two types
of equations employed are logistic and hyperbolic. These are used to derive other relationships
among system variables. Results are used to analyze and describe data, and to test predictions
made from the models. Finally, the chronological order in which the models were actually
developed is discussed to emphasize the emergent nature of the process of model development.
5.2 Coupling of Applied and Available Soil Phosphorus
Crop response to applied phosphorus is used to illustrate coupling among applied, soil
extractable, crop uptake, and crop biomass. A similar procedure is believed to apply for other
mineral elements such as nitrogen and potassium.
5.2.1 Response of Extractable Soil Phosphorus to Applied Phosphorus
The first step is to describe coupling between applied and extractable soil phosphorus. Data
from the literature are used for this purpose.
Study on Arredondo Fine Sand at Gainesville, Florida
Sartain (1992) conducted field measurements on response of extractable soil phosphorus to
applied phosphorus for Arredondo fine sand (loamy, siliceous, hyperthermic Grossarenic
Paleudult) at Gainesville, FL. Data are given in Table 5.1 and shown in Figure 5.1. Overman and
Scholtz (2002, p. 229) has shown that these data can be
Table 5.1 Response of extractable soil phosphorus to applied phosphorus on Arredondo fine sand
at Gainesville, FL.a
Applied P, g m2 0 10 20 40 80
Extractable P, mg kg1 151 192 218 313 429
aData adapted from Sartain (1992).
described with the logistic equation
