Heat transfer in beds for citrus fruits during forced convection cooling.

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Title:
Heat transfer in beds for citrus fruits during forced convection cooling.
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xiii, 165 leaves. : ill. ; 28 cm.
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English
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Baird, Carl Direlle, 1939-
Publication Date:

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Subjects / Keywords:
Citrus fruits -- Cooling   ( lcsh )
Citrus fruits -- Thermal properties   ( lcsh )
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bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 112-116.
Statement of Responsibility:
By Carl Direlle Baird.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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oclc - 14174264
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Full Text












HEAT TRANSFER IN BEDS OF CITRUS FRUITS
DURING FORCED CONVECTION COOLING









By

Carl Direlle Baird


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


UNIVERSITY OF FLORIDA
1973













ACKNOWLEDGMENTS


The author wishes to express his sincere appreci-

ation to Dr. E. A. Farber who served as chairman of his

supervisory committee. He has been a source of real

assistance, inspiration, and encouragement.

Appreciation is extended to the other members of

his supervisory committee: Professor J. M. Myers,

Agricultural Engineering, Dr. Z. R. Pop Stojanovic,

Mathematics, and Dr. F. L. Schwartz, Mechanical Engi-

neering, for their leadership, interest, and assistance

throughout his program. Appreciation is expressed to

Dr. R. B. Gaither, Chairman of the Mechanical Engineering

Department, and to each of the faculty members who con-

tributed to his graduate program.

He is indebted to the Department of Agricultural

Engineering for financial support of his research and to

Dr. E. T. Smerdon, Chairman, for his leadership, encour-

agement and untiring willingness to cope with the admini-

strative problems involved. Thanks are also due to each

of the many staff and faculty members who assisted in

numerous ways; especially to Dr. D. T. Kinard and Mr.

J. J. Gaffney who actively supported his research as project

leaders. Mr. Gaffney also provided indispensable help

with the computer programming.







For their cheerful sacrifices and loving patience,

the author dedicates this dissertation to his wife Judy

and daughters Sally Ann and Julia.


iii
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . .

LIST OF TABLES. . . .

LIST OF FIGURES . .

ABSTRACT . .. .

INTRODUCTION. . .. .

REVIEW OF LITERATURE. . ..

Studies on the Thermal Properties of Fruits and
Vegetables . .. .

Studies on Fruit and Vegetable Precooling .

CONDUCTION HEAT TRANSFER IN SINGLE CITRUS FRUITS .

Theory and Assumptions . .

Homogeneous Sphere. . .

Heat of Respiration ... ..


Page



S. vii

S. vii
* .v111

* xi

. 1

. 4


Analytical Solution. .


Numerical Solution for Heat Conduction in a Sphere
with Time-Dependent Boundary Conditions. .

TEMPERATURE DISTRIBUTION WITHIN A BED OF CITRUS DURING
COOLING . . .

Laplace Transform Solution of Simplified Model .

Numerical Solution of Realistic Model. .

Determination of Convective Heat Transfer Coef-
ficient and Thermal Diffusivity. . ..

Theory . .


r

r



r

r

r


. . .







TABLE OF CONTENTS (continued)



Development of Method . .

Experimental Evaluation of Thermal Diffusivity.

Design and Description of Experimental Facilities.


Basic Construction of Precooler


Refrigeration Components and Controls .

Example illustrating capacity of system. .

Components used for control. .

Evaporator coils . .

Air Temperature and Humidity. .

Air Flow . . .

Reheat Section . .

Experimental Test Procedures . .

Data and Results . .

Numerical Solution of Deep Bed Cooling .

Convective Heat Transfer Coefficient (h). .

Thermal Diffusivity () . .

Experimental Deep Bed Cooling Tests .

Discussion . . ..

Comparison of Laplace Transform Solution for
the Simplified Model and Numerical Solution
for the Realistic Model .. .

Convective Heat Transfer Coefficient .

Comparison of Experimental Data With Numer-
ical Solution . .

Experimental data. . .


. 64

. 67

. 67

. 68

. 69

. 70

. 72

. 72

.* 77

. 77

. 82

. 82

. 82

. 86


. 99


Analyses of the differences between experimen-
tal and theoretical data .

Evaporative cooling . .


Page

. 59

. 62

. 63


. . 64















TABLE OF CONTENTS (continued)


Page

Radiation heat transfer. . ... 103

Conduction along the thermocouple wire 104

Insulating effect of rind. ... 104

Suggestions for Further Study . 108

Optimum design and operating conditions 108

Modification of numerical model .. 109

CONCLUSIONS. . .. .. ... .. 110

REFERENCES . .. ... .. 112

APPENDICES . . 117

Appendix A Computer Program and Sample Output
for Conduction in a Sphere with
Specified Surrounding Temperature. 117

Appendix B Computer Program and Sample Output
for Cooling a Deep Bed of Citrus
Fruit. . .. 121

Appendix C Graphical Results Showing Comparison
of Experimental Data with Numerical
Solution . . 134

BIOGRAPHICAL SKETCH. ... . 165













LIST OF TABLES


Table Page

1. Comparison of Rind and Juice Vesicle Components
of Marsh Grapefruit. . 7

2. Approximate Heat of Respiration of Oranges and
Grapefruit at Various Temperatures .. 9

3. Particle Shape Factors for Packed-Bed Correlation 52

4. Coefficients and Exponents for Emperical Heat
Transfer Correlations. .... 59

5. Coefficients, Exponents and Linear Correlation
Coefficients for Equation [84] Determined for
Forced-Air Cooling in Bulk Containers. ... .62

6. Thermal and Physical Properties of Valencia
Oranges and Marsh Grapefruit Used in Cooling
Tests . ... ... 91

7. The Effect of Increment Size on the Accuracy
of the Numerical Solution. ... .. 96


vii












LIST OF FIGURES


Figure Page

1. Internal temperature distribution in Marsh
grapefruit plotted from raw experimental
data recorded during transient cooling
test runs. ............... 16

2. Diagram indicating a heat balance on the
three types of nodes used in a sphere. 23

3. Flow diagram for digital computer program
of heat conduction in a sphere with
arbitrarily varying air temperature. 27

4. Comparison of analytical solution with
numerical solution for heat conduction in
a sphere . ..... 28

5. Control volume for heat balance on bed 31

6. Schumann's curves for temperature history of
solid for values of Y from 0 to 10 43

7. Schumann's curves for temperature history of
gas for values of Y from 1 to 10 .. 43

8. Flow diagram for the digital computer program
to solve deep bed heat transfer equations. 47

9. Magnitude and location of mass-average
temperature of Marsh grapefruit cooled in
agitated ice water .. 48

10. Unaccomplished temperature change and
location of mass-average temperature during
forced-air cooling . .48

11. Nomograph for evaluating elements of the
equation: MI = f3(GBi) 57

12. Elevation view of forced-air cooler. 65

13. Precooler refrigeration components and
controls . 66


viii






LIST OF FIGURES (Continued)


Figure Page

14. Bulk load of fruit showing the four levels
at which product temperature was meas-
ured. Dimensions A, B, C, and D are
indicated for each test. ... 74

15. Method used for placing thermocouples in
product. .. . . 75

16. Theoretical mass-average temperature re-
sponse of oranges to forced-air cooling
as predicted by numerical model. 78

17.-18. Theoretical mass-average and surface tem-
perature response of oranges to forced-
air cooling as predicted by numerical
model. . . .. 79

19. Theoretical mass-average and surface tem-
perature response of grapefruit to
forced-air cooling as predicted by
numerical model. .......... .. 81

20. Linear regression relating Nusselt number
and Reynolds number for forced-air
cooling of oranges in bulk loads 83

21. Linear regression relating Nusselt number
and Reynolds number for forced-air
cooling of grapefruit in bulk loads. 84

22. Linear regression relating Nusselt number
and Reynolds number for forced-air cooling
of oranges and grapefruit in bulk loads. 85

23. Experimental mass-average and surface
temperature response of oranges to forced-
air cooling. ... 87

24. Experimental mass-average temperature
response of oranges to forced-air cooling. 88

25. Experimental mass-average and surface
temperature response of oranges to forced-
air cooling. . ..... 89

26. Experimental mass-average;temperature
response of grapefruit to forced-air
cooling. ... . 90







LIST OF FIGURES (Continued)


Figure Page

27. Comparison of Laplace transform solution
of simplified model with numerical
solution of realistic model for a slow
cooling rate (25 fpm) . .. 92

28. Comparison of Laplace transform solution
of simplified model with numerical
solution of realistic model for a fast
cooling rate (400 fpm). .. 94

29.-31. Comparison of experimental and theoretical
temperature response of oranges to forced-
air cooling . . 100

32. Comparison of the temperature profile in a
homogeneous sphere with average proper-
ties of a grapefruit and the temperature
profile in a composite sphere with pro-
perties corresponding to the rind and
juice vesicles . 107

33.-52. Theoretical and experimental temperature
response of oranges to forced-air
cooling . 135

53.-62. Theoretical and experimental temperature
response of grapefruit to forced-air
cooling * 155













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



HEAT TRANSFER IN BEDS OF CITRUS FRUITS
DURING FORCED CONVECTION COOLING



By

Carl Direlle Baird

June, 1973



Chairman: Dr. E. A. Farber
Major Department: Mechanical Engineering



The removal of heat from fruits and vegetables as

soon after harvest as practical has long been recognized

as effective in retarding the ripening process and con-

trolling microbial processes. Presently, very little

citrus is being precooled. Hydrocooling of citrus has

been attempted, but was not satisfactory due to the

detrimental effect of the chemically treated water on the

surface appearance of the fruit. It is believed that

forced-air precooling of citrus would receive favorable

attention if additional basic information were available

on heat transfer characteristics of citrus in bulk.







Equations describing the heat transfer in a bed

of citrus fruit were derived. A numerical solution of

these equations was obtained and evaluated utilizing a

digital computer. Temperature was determined as a

function of 1) position within the individual fruit,

2) distance from inlet of bed, and 3) time. Variable

input data for the computer program include: diameter

of fruit, thermal diffusivity of fruit, specific heat of

fruit, density of fruit, initial fruit temperature, air

temperature, air flow rate, total weight of fruit, depth

of bed, and convective heat transfer coefficient. An

equation was developed for determining the average con-

vective heat transfer coefficient for a bed of citrus in

the form of a Nusselt number-Reynolds number relationship.

The constants in this equation were experimentally eval-

uated for oranges and grapefruit utilizing a method which

made use of the computer solution.

Fifteen experimental cooling tests were conducted

on oranges and grapefruit in bulk containers. Mass-

average and surface temperatures were measured at four

levels within the bed of citrus with air velocity ranging

from 15 to 400 fpm.

A comparison between the experimental and theoret-

ical results indicated good agreement except for surface

temperatures of grapefruit. The experimental surface

temperatures of grapefruit were consistently lower than

those predicted by the numerical solution. This was


xii







shown to be caused primarily by the low thermal conductiv-

ity of the thick rind. A solution was obtained for the

temperature gradient in a composite sphere which indicated

a much steeper temperature gradient through the rind than

through the juice vesicles.


xiii













INTRODUCTION


The removal of field heat from fruits and vegetables

prior to processing, transportation and storage has long

been recognized as effective in retarding the ripening

process and controlling microbial processes (21, 40, 41)1/.

The term "precooling" is commonly used to describe this

rapid cooling process (1, 2, 42). Maxie et al. (34) have

described the biological basis for prompt cooling, listing

a series of physiological, biochemical, pathological and

physical changes that are temperature-dependent in fruits.

Precooling is likely the most important of all the opera-

tions used in establishing and maintaining desirable, fresh,

and salable produce.

According to Guillou (22), the rate of various changes

such as moisture loss and the action of enzymes and micro-

organisms, are commonly doubled or tripled with each 10 F

rise in temperature. He estimated that some fresh fruits

and vegetables deteriorate as much in an hour at 90 F as

in a day at 50 F or in a week at 32 F.

According to the 1972 Citrus Summary (17), approximately

20% of Florida's citrus crop is consumed in the fresh form.



21/ Numbers in parentheses refer to numbered references.







The total value of this fruit is approximately $110 million

annually. This amounts to over 7% of the total value of

Florida's annual agricultural production.

At present most of the citrus fruits are not cooled

until they are placed in the refrigerated vehicle that will

carry them to market. The refrigeration systems on these

vehicles are not designed to remove great quantities of heat

rapidly; therefore, the temperature of the fruit is not

decreased to the desired level for a considerable period

after harvesting.

It has been conservatively estimated that 10% of all

fresh citrus fruits shipped are never consumed because of

waste and spoilage (48). The value of these fruits is over

$10 million.

A small quantity of citrus fruit is presently being pre-

cooled by hydrocooling and air cooling. Hydrocooling of

citrus has been tried by some producers, primarily due to

the successful hydrocooling of peaches and vegetables.

However, it soon became apparent that fungicide in the cool-

ing water was essential to hold decay to tolerable levels.

Unfortunately, the chemically treated water had a detrimental

effect on the surface appearance of the fruit. Also, fruit

cooled with water seemed more susceptible to decay upon

warming than fruit which had received no cooling. Certain*

types of citrus, notably grapefruit, exhibited subsequent

chilling injury (28).







It is believed that forced-air precooling of citrus

would receive more favorable attention if additional basic

information were available on heat transfer characteristics

in beds of citrus fruit. Much of the research on precooling

has been done in production operations or in experimental

facilities which were not designed for obtaining basic heat

transfer data. Thus the objective of this investigation is

to obtain and test a mathematical model that can predict

the temperature distribution within a bed of citrus during

forced-air cooling. Other objectives incident to this

investigation are to devise methods for obtaining thermal

and physical properties for beds of citrus fruit.













REVIEW OF LITERATURE


Studies on the Thermal Properties of Fruits and Vegetables



The response in temperature of a body to a given input

or removal of energy in the form of heat is dependent upon

certain properties of the body. These properties govern

the storage and transfer of heat energy through the body.

Two of these properties are specific heat capacity (c),

BTU/lb-F, and thermal conductivity (k), BTU/hr-F-ft.

Another property, which is convenient to use since it con-

sists of a combination of three properties as they appear

in the heat conduction equation, is thermal diffusivity

(k/pc), ft2/hr.

Since most fruits and vegetables contain a high per-

centage of water it can be expected that their properties

will have values near that of water. Formulas for the

specific heat and thermal conductivities of fruits and

vegetables based on their percent water have been developed

(15).


c = p/100 + 0.2 (100-p)/100 BTU/lb-OF [l]

k = 0.32 /100 + 0.15 (100-p)/100 BTU/hr-ft-F [2]








where

c = specific heat

p = percent water

k = thermal conductivity

Gane (20) investigated the thermal diffusivity of fruits

and vegetables by using the graphical method of Gurney and

Lurie (23) for solutions to the conduction heat transfer equa-

tion for various shaped bodies. The temperature history of

of the center of fruits and vegetables was monitored with

thermocouple junctions during cooling and was used in con-

junction with these graphical solutions to estimate the ther-

mal diffusivity. He found the following average diffusivity

values (ft2/hr): apples -- 0.0049, oranges -- 0.0049,

grapefruit -- 0.0049,and squash -- 0.0044. He also found

the thermal conductivity of oranges and grapefruit to be 0.24

and 0.25 BTU/hr-ft-oF respectively.

Kethley et al. (30) conducted a series of cooling experi-

ments with eight different fruits and vegetables. They used

the same graphical methods of Gurnie and Lurie. In the

temperature range 80 to 32 F their values of diffusivity

ranged from 0.00535 to 0.00615 ft2/hr.

Turrell and Perry (50) found the specific heat by using

a calorimeter. They obtained values of 0.885 for Marsh

grapefruit, 0.892 for Eureka lemons, 0.840 for Valencia

oranges and 0.875 for Washington Navel oranges.

Several investigators have determined the thermal conduc-

tivity (k) of citrus by assuming it to be homogeneous. Turrell








and Perry (50) found the thermal conductivity averaged 0.23

BTU/hr-ft-F for Marsh grapefruit, Eureka lemons, Valencia

oranges and Washington Navel oranges when all varieties were

considered. Perry et al. (37) determined the following mean

values for thermal diffusivity: Valencia oranges -- 0.00368,

Washington Navel oranges -- 0.00415, Marsh grapefruit --

0.00355, and Eureka lemons -- 0.00417 ft /hr.

Smith et al. (47) developed a technique of evaluating

thermal diffusivity which corrects for deviation from the

conventional shapes. The technique incorporates a geometry

index (G) into the basic Fourier heat conduction equation for

a sphere. The geometry index was obtained from a measure of

two orthogonal areas of the shape. Using this technique

they determined an average thermal diffusivity of 0.00363

ft2/hr from 15 tests run on Marsh grapefruit. Bennett et al.

(7) used the same procedure and obtained an average thermal

diffusivity of 0.00355 ft /hr for five maturity groups of

Marsh grapefruit. This value was obtained from temperature

response at the approximate mass average point. Values cal-

culated from temperature response at the center and one-half

radius point were slightly lower, probably due to the non-

homogeneity of the product.

Bennett et al. (7) also made separate determinations for

the thermal properties of the rind and juice vesicle compo-

nents of Marsh grapefruit. Table 1 indicates average values

for each component. The thermal diffusivity determined here








Table 1. Comparison of Rind and Juice Vesicle Components
of Marsh Grapefruit.



Specific Thermal Thermal
Heat Density Conductivity Diffusivity
BTU/lb-OF lb/ft3 BTU/hr-ft-oF x10-3 ft /hr


Juice
Vesicle 0.907 63.2 0.2716 4.56

Rind 0.829 36.1 0.1397 4.67




is considerably higher than that determined by assuming the

fruit to be homogeneous. It is interesting to note that the

thermal diffusivity of the rind and juice vesicle components

has values very near the same while the thermal conductivity

and density variation is much greater. This has been explained

by Perry et al. (37) that in a given material where density

changes because of changes in porosity, the thermal conduc-

tivity is approximately proportional to density, so that the

diffusivity remains about constant.

Bennett et al. (7) conducted similar tests on Valencia

oranges and found the combined thermal conductivity to be

0.248 BTU/hr-ft-OF.



Studies on Fruit and Vegetable Precooling


Presently, most citrus fruit is not cooled until it is

placed in the refrigerated vehicle that will carry it to








market. The refrigeration systems of these vehicles are not

designed to remove great quantities of heat rapidly; thus

the temperature of the fruit is not decreased to the desired

level for several days after harvesting. During this time

respiration reduces the quality, nutrient value and shelf

life of the fruit, thus indicating the importance of pre-

cooling.

During the respiration process oxygen from the air is

combined with carbon from the plant tissues to form various

decomposition products and eventually carbon dioxide and

water. In addition to this, the enzymes present act on

various substances in the plant tissues and gradually cause

changes in color, texture, and chemical composition which

mature fruit and may eventually cause serious deterioration.

During the respiration process heat is released vary-

ing in amount with the commodity and its temperature.

Table 2 illustrates the large difference between the heats

of respiration at high and low temperatures.

The respiration process is generally believed to cause

consumption of a simple hexose sugar glucose. With the aid

of an enzyme system and oxygen, glucose sugar is reduced to

carbon dioxide and water in accordance with the following

reaction:


C6H206 + 602 = 6CO + 6H20 + Heat
6 12 6 2 2 2








Table 2. Approximate Heat of Respiration of Oranges and
Grapefruit at Various Temperatures (26)




(BTU per 24 hours per ton of fruit)

Temp. (F) Sensible Latent Total
Heat Heat Heat


Oranges 32 900 63 963
40 1400 97 1497
60 5000 348 5348
80 8000 557 8557

Grapefruit 32 460 32 492
40 1070 74 1144
60 2770 193 2963
80 4180 291 4471


The enzyme acts as a catalyst but is not consumed during

the reaction. The effectiveness of the enzyme as a catalyst

is greatly reduced at low temperatures which causes a reduc-

tion in the heat of respiration when the temperature is

reduced.

Precooling usually refers to the rapid removal of heat

from a fruit or vegetable. In regards to what "pre" means,

Thevenot (49) stated: a) in a restrictive sense: previous

to transportation; b) in a wide sense: previous to transporta-

tion or to storing; c) in a very wide sense: previous to

transportation, storing, or processing. Redit and Hamer (42)

referred to precooling as the rapid cooling of a commodity

to a suitable transit or storage temperature soon after har-

vest, before it is stored or moved in transit.








A small quantity of citrus fruit is now precooled by

several methods. Of these hydrocooling and air cooling are

probably the only two that can be combined with the mechanized

methods of packing that are now used. However, some investi-

gations on hydrocooling of oranges have indicated that an

increase in respiration rate, over that of uncooled oranges,

occurred when they were allowed to return to room temperature.

Thus, hydrocooling can actually be detrimental to oranges(14).

On the other hand, Hopkins and Loucks (28) found that

air precooled oranges were outstanding in their keeping

quality and resistance to decay.

Sainsbury (43) investigated high velocity air as a means

of rapidly precooling fruits. In cooling experiments involv-

ing such fruits as cherries, apples, and apricots, predicta-

bility of the cooling behavior was established by identifying

the cooling rate (CR) for a given experiment.

Fruit temperature reduction
TD. TDi
CR = Time 14]
loge (TDi/TDf)


where

TD. = initial temperature difference between fruit and

air.

TDf = final temperature difference between fruit and air.

This equation for cooling rate (CR) reduces to
TD.
loge. TD
CR = T5]
Time








when the air temperature is maintained constant. The slope

of the line established by plotting loge (TDi/TDf) as a

function of time identifies the cooling rate (CR).

Guillou (22) reports that Newton's law of cooling will

predict quite well the average temperature progression of

fruits and vegetables during cooling. Newton's law is appli-

cable for an object placed in surroundings at a constant

lower temperature provided the resistance to heat transfer

is constant. The equation expressing this law using the

notation of Guillou is


dT
d = C (T-To) 161

where

t = time of exposure to surroundings

T = temperature of the object at time "t"

To = temperature of the surroundings

C ="cooling coefficient"

The integration of the equation by separation of variables

shows that


c 1 Tloge 1 71
t elog TlITo

where T1 equals the initial temperature of the object. This

equation for the "cooling coefficient" corresponds to the

equation of Sainsbury (43) for cooling rate (CR).

Bennett et al. (9 ) conducted tests on forced-air pre-

cooling of Florida citrus. These tests were conducted on








citrus in bulk containers through which they forced high

velocity air. They developed an equation to predict the

cooling rate. Since the inlet air temperature was not held

constant, the air temperature at the end of the run was used

as the reference temperature. They did not investigate the

temperature distribution within the bed.

Leggett and Sutton (33) also conducted precooling tests

on citrus fruits. They developed cooling curves and obtained

cooling coefficients for oranges and grapefruit in crates.

Both hydrocooling and air cooling tests were made for dif-

ferent crate spacings and arrangements.

Several investigators (5, 27, 36, 53, 29) have con-

ducted tests involving transport phenomena of heat and mois-

ture in beds of biological materials. Most of these investi-

gations involved analysis similar to that of Schumann (45)

and Furnas (19) who were evidently the first to solve the

differential equations governing heat transfer in a deep bed

of particles. For a system in which a fluid passes through

a prism of crushed material, Schumann developed a series of

temperature history curves subject to the following assump-

tions:



a) that the particles are small and no temperature

gradient exists within any particle at any time.

b) that no heat is produced by particles.

c) that no heat is transferred from particle to parti-

cle by conduction.




13


d) that the fluid is incompressible.

e) and that the thermal properties of the particles

are uniform throughout and do not vary with time.

Furnas found that Schumann's theoretical curves would pre-

dict the temperature behavior of beds of coke and iron

through which air was moving.













CONDUCTION HEAT TRANSFER IN SINGLE CITRUS FRUITS


Theory and Assumptions



Foundation for the theory of heat conduction is univer-

sally attributed to Fourier (18). It is apparent that

Fourier entertained a hope that practical use might be made

of his work. He stated, "It is easy to judge how much these

researches concern the physical sciences and civil economy,

and what may be their influence on the progress of the arts

which require the employment and distribution of heat."

Fourier's work was awarded a generous prize by the French

Royal Academy of Science in 1812. Fourier's work was treated

like a work of art for more than 100 years with no practical

use being made of it, mainly due to the difficulty encoun-

tered in obtaining practical solutions. However, in 1923

engineers Furney and Lurie (23) presented their solutions

based on theory but with an experimental character. Answers

to many practical problems of interest to engineers were

made readily accessible by means of the time-temperature

charts of Gurney and Lurie. Other such charts have been

developed by means of the experimental technique of the elec-

tric analog (25). A large collection of time-temperature








charts for conduction in various shapes with several dif-

ferent boundary conditions is given in a book by Schneider

(44).

A considerable amount of effort has been given by

researchers (24, 27, 38, 39, 46, 52) to correlate the

theoretical solutions for regular-shaped objects with exper-

imental data for biological materials that can be approxi-

mated by these regular-shapes.



Homogeneous Sphere


Most citrus fruits have the general shape of a sphere

and therefore could be expected to exhibit heat transfer

characteristics similar to that of a sphere. As noted in

the preceding section, citrus fruits are not composed of

homogeneous materials but consist of two main sections, the

rind and the juice vesicles. It was also noted that although

the thermal conductivity (k) and density (p) were quite

different for the two portions, their thermal diffusivities
k
( ) were surprisingly close to one another.

Bennett et al. (8) investigated the temperature distri-

bution in Marsh grapefruit, which has a rind that constitutes

30 to 40 percent of its total volume, and found that the

presence of the rind has very little effect on the tempera-

ture profile. This is indicated in Figure 1 which shows the

internal temperature distribution in Marsh grapefruit plotted

from raw experimental data recorded during transient cooling.
























LAL
0


,--60
LAJ
5 _
5 0


30


32



I 64


0.2 .0.4


0.6 0.8


RADIUS RATIO




Figure 1. Internal temperature distribution in Marsh
grapefruit plotted from raw experimental data
recorded during transient cooling test runs.
Dotted lines show discontinuity in temperature
gradient at interface.


1.0








The dotted lines show a discontinuity in the temperature

gradient at the interface between the juice vesicles and

rind.

On the basis of these findings, it will be assumed that

citrus fruits may be treated as homogeneous without signifi-

cant error in the temperature profile.



Heat of Respiration


The justification for not including the heat of respira-

tion as a heat source in the conduction equation is based on

the findings of Awberry (4) and Hood (27). Awberry calculated

the temperature excess at the center of an apple due to its

heat of respiration, after two hours of cooling and found

it to be 0.0232 C, while Hood calculated a value of 0.011 F

at the center of the cucumber after 30 minutes of cooling.

Both of these products have heats of respiration higher than

citrus, thus the effect of heat of respiration on citrus

during cooling is believed to be negligible. It should be

pointed out that the effect of heat of respiration should

not be neglected in some other cases such as storage at

elevated temperatures.



Analytical Solution


The governing equations for conduction heat transfer

from a homogeneous sphere without heat sources, initially








at a uniform te;. :.erature (T1) and surrounded by a fluid of

constant temperature (T ) are


T 32 2 T8
r- ( + [8]
at 3r2 r 3r


T(r,o) = T1 [9]



(k T) = h(T T ) [10]
3't r=a s


where

t = time, hr

r = radial distance from center, ft

a = radius of sphere, ft

T = temperature at r and t, F

a = thermal diffusivity, ft2/hr

k = thermal conductivity, BTU/hr-ft-F

The theory for the following solution of equations

[8], [9], and [10] was taken from Arpaci ( 3) and equation

[21] agrees with his solution.

Equation [ 8] can be written as:



T a (r2 'T
= (r 111]
at r2 rr r

and substituting 6 = T T into equations [9], [10], and

[11] gives:


8_ a (r 2 8
(t (r r) 112]








S(r,0) = e1



-k [a (a,t) ] = h6 (a,t)
dr


[13]



[14]


One more boundary condition is:


S(0,t) = finite or a (0,t) = 0
ar


115]


Using the transformation:


6 (r,t) = y (r,t)/r


[16]


reduces the spherical Laplacian to the cartesian Laplacian

whose solution is expressible in terms of circular functions.

We may now express equations [12] through [15] in

terms of Y by using equation [16]. The result is:


t a 2
ar


Y (r,0) = r


[173


Y(0,t) = 0,


-k (at) (h ) (a,t)
dr a


Hence the problem is reduced to a problem of cartesian geome-

try.

The product solution Y (r,t) = R(r) T (t) yields:


d2R 2
+ X2 R= 0
dr


dR(a) h 1
R(0) = 0, da+ ( ) R (a) = 0 [18]
dr k a








in r and equation [19] in time.


dT + ak2
dt


= 0


The solution of equation [18] is:


R (r) = A n (r),
n nfnl


n (r) = sinXnr


and the zeros of


(A a) cosA a = (1-Bi) sinA a


are the characteristic values, where Bi = ha/k.

The solution of equation [19] is:


2
T (t) = Ce n
n n


Thus the product solution becomes:


(r,t) =
n=l


2
b e n sinX r
n n


[20]


The initial value of equation [20] is:


r6 = 7
1
n=1l


b sinA r
n n


The coefficient b is obtained by the expansion of rO, into
n
a Fourier sine series evaluated from zero to a.


[19]







201 (sinA a A a cosX a

n n n n


Thus the unsteady temperature of the sphere is:



00 2
T(r,t) T sinA a X a cosX a -ax t sinX r
m n n n n n
T T Xa sinX a cosX a Ar
1 mn n n n n
n=l

[21]




Numerical Solution for Heat Conduction in a
Sphere with Time-Dependent Boundary Conditions



The analytical solutions for the heat conduction equa-

tion presented in the preceding section are based upon

assumed initial and boundary conditions which can seldom be

satisfied in practical applications. Especially in dealing

with many biological products, there is a need for informa-

tion when initial and boundary conditions may exist such as:

with time-varying environmental conditions or surface con-

ductances, with change-of-phase or mass transfer at the

boundary, or with initial object temperature distributions.

The only practical method for solving this type problem

is by numerical methods, such as those presented by Arpaci (3).

Since cooling in a deep bed involves time-dependent boundary

conditions, a numerical solution of the heat conduction

equation for a sphere surrounded by a fluid with arbitrarily

varying temperature will be derived.








The difference equations which approximate the govern-

ing differential equations can be obtained by approximating

each term of the differential equation with its equivalent

finite difference expression or by performing a heat balance

method on each type of node in the solid. The heat balance

method will be used in this formulation. For a sphere there

are three types of nodes as indicated in Figure 2, surface

nodes, interior nodes, and center nodes.

The first law of thermodynamics applied to the interior

node gives:


qi-1 [4(r- ) 2+q i+[4T (r+)2

Tn+l-Tn
4 Ar 3 Ar 3 1 1 [2
pC -[ (r+-.) (r- ) At [221

where

Tn+1 denotes the temperature at the ith node at time n+l.
i
Now stating Fourier's law of conduction, we have:

Tn n Tn -T
K 1-1 I i+l 1
qi+l = K r i' i+l = k r


Substituting intoequation [221 results in:

Tn Tn Tn -n
T -T T n T
4Tk (Ar 2 i-1 i 4-k Ar 2 i+l i
(r--) + (r+--) r
Ar 2 Ar Ar 2 Ar

n+l n
4 Ar 3 Ar 3 iT i
pc [(r+) (r-) At




















Pi


Pi-1


j i+1 R








P--I
r rPi-I




















Figure 2. Diagram indicating a heat balance on the
three types of nodes used in a sphere








rearranging:


Tn+1 aA t Ar 2 +Arz2 n
i (Ar)2 2r 2r i


+ aA t (i Ar)2 Tn A+ ( r 2
(A2 2r i-1 2r i+l
Car)


where {


} 0 to insure convergence.


Now consider the surface node. The first law of thermo-

dynamics applied to the surface node gives:



qi_[4r (R-)r 2] + q o [4vR2 =

n+l -Tn
T -T2
4 (Rtr3 f
pc 3-[R -(R--2) At


Substitution for:


n n
T =-Tr
qi-1 = k Ar '


q = h(Tn-T')
conv 00i


gives:


n n
i-l- i (R-Ar2 n 4R2h]
k Ar [4r(R ) + h(T -Tk)[ 1


n+l_ n
4 3 Ar 3 i i
pc 3 R -(R--) At
3 2 at


rearranging:




25

(-Ar 2
n+l 3aAt 2( n
T T. +
i Ar 3 Ar 3r i-3


3R2haAt Tn +
3 Ar 3
k[R 3-(R--~) 3


r2
(R-) 2 2
3aAt 2 Rh Tn [24]
[R3_ 3 Ar



where {} >0 to insure convergence.

Applying the first law to the center node results in:

n+l n
Ar 2 4 Ar 3 1 i
qi- [4d(-) ] = pc 3 7 (--) 1t



substituting for:

Ti -Ti
qi+l = k r


gives:


i+lT i Ar2 4 Ar 3 Tnl. -Tn
k [4r ([ )2] = pc -7 (---) 1i
Ar 2 3 2
At


Simplifying and rearranging:


n+l 6aAt n 6aAt n
T. = T + {1 T" 1253
1 2 i+l 2 i3
(Ar)2 i (Ar)2

where } > 0 to insure convergence.

No similar solution was found in the literature for comparison.








A computer program was developed to evaluate equations

[23], [24], and [251 for temperature, T, as a function of

"r" and "t." The arbitrarily varying fluid temperature,

T is entered as input data. A flow diagram for the com-

puter program is given in Figure 3 and the computer program

is shown in Appendix A.

In order to test the accuracy of the numerical solution,

the air temperature, T was entered as a constant so that

the results could be compared with the exact solution

obtained in the preceding section. This comparison is

shown in Figure 4. The data used to generate these curves

are the following: radius = 0.113 ft, k = 0.256 BTU/hr-ft-

OF, h = 11.7 BTU/hr-ft2- F, a = 0.0051 ft /hr, c = 0.9

BTU/lb-OF.









Read: a, k, h, radius, number of node points in sphere,
initial temperature of fruit, time increment,
total cooling time, print frequency, test number,
arbitrarily varying air temperature.

Compute combinations of input data which remain constant.

Set initial conditions.

Compute air temperature from input data.


Compute surface temperature (Equation [23]).

Compute interior node temperatures, except center
S(Equation [24]).

Compute center temperature (Equation [25])


Yes


Print time,I
air temp,
fruit temp


Figure 3. Flow diagram for digital computer program of
heat conduction in a sphere with arbitrarily
varying air temperature.