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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)
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.