r r 0. T constants. Blot number (= hr0/k ) simplification factors isobaric specific heat of dry air isobaric specific heat of ..... Composition : 100 water;30 sugar;3 agor. 9. Size :50.0 mm ..... Guide and Data Book (Applications). New York: ASHRAE.
W~irme-
und Stofffibertragung
Warme-undStoffiibertragung
12 (1979)261-268
Thermo-and Fluid Dvnamics 9 by Springer-Verlag ]979
Determination of Thermal Properties of Food Materials Theory and Experiments K. Badari
Narayana
and M.
V. Krishna
Murthy,
Madras
Abstract. Mathematical models are presented for predicting the time-temperature characteristics during air cooling of moist food products in the shape of infinite slab, infinite cylinder and sphere, taking into account the mass transfer effects at the surface. An experiment to determine the time-temperature characteristics of food products from which the thermal properties can be determined using the proposed mathematical model is described. The present analysis predicts faster cooling rates compared to an analysis which considers only pure convection heat transfer at the boundary. Time-temperature histories of some food models, fruits and vegetables are obtained experimentally. A nonlinear estimation procedure is used to match the experimentally obtained time-temperature histories with theoretical results to evaluate the unknown thermal properties of the products studied. Bestimmung
thermischer
StoffgrSBen
von Lebensmitteln
- Theorie und Experiment
Zusammenfassun~. Mit Hilfe eines mathematischen Modells lassen sich die Temperaturfelder w~ihrend der Luftkiihlung feuchter Lebensmittel in der Form einer Platte, eines Zylinders oder einer Kugel berechnen. In einem Versuch kSnnen dann die thermischen ZustandsgrSBen durch Vergleich mit dem Modell bestimmt werden. Die vorgelegte Untersuchung sagt hShere Kiihlungsraten voraus, verglichen mit dem Fall reiner Konvektionskiihlung an der Oberfl~iche. Die zeitlichenTemperaturverl~iufe in einigen Lebensmitteln, Frfichten und Gemiisen werden experimentell ermittelt. Indem man diese experimentellen Daten durch eine nichtlineare Vergleichsmethode an die theoretischen Werte anpaBt, lassen sich die nicht bekannten StoffgrSBen der Produkte ermitteln.
Nomenclature
T constants Blot number
a,b,c Bi
CI,C2,C c P c
(= hr0/k )
simplification
3
factors
difference between theoretical perimental value enthalpy of air enthalpy of saturated air
s
h
hf
w
humidity ratio of air number of thermocouple locations thermal diffusivity reference thermal diffusivity
isobaric specific heat of dry air
F
time relative humidity
Subscripts
g
k k' P Ps
effective thermal conductivity diffusion coefficient vapour pressure of the unsaturated air vapour pressure of the wetted surface
R
radius radius vector half slab thickness,
db s wb z
dry bulb surface wet bulb maximum
value of the time
Superscripts radius of cylinder or
sphere
T
~0 T
and ex-
heat transfer coefficient latent heat of vaporization
r r0
temperature
temperature product initial temperature
x
isobaric specific heat of water vapour
experimental
t t.
1
Ps H H
e
temperature
* + '
production
I Introduction
dimensionless variable dimensionless independent parameter properties at unsaturated conditions
to centres of processing
and preservation,
is carried out to retain their quality over short duraPrecooling water,
of food products
using refrigerated
prior to their transportation
from
air or
centres of
tions. An a-priori knowledge histories of the product,
of the time-temperature
cooling rate and time during
262
W~rme-
such a cooling process evaluation
is essential
of the refrigeration
sing equipment
and control
duct. Theoretical
thermophysical
perimental
load, design
properties
for an assessment
reliable
diffusivity for the food products or if available,
a
h
Fig. 1. Physical precooling
are often either not
o
*Cooling oir dry bulb temper~lure=tdb Cooling oir wet bulb ternperoture=twb
Thermo-
and thermal
incomplete,
Air flow
Air flow
Such
of the ex-
and methods. conductivity
ir flow
character-
only when
of it are available.
data like thermal
available
of proces-
of the cooling
plant, equipment
physical
for the proper
is possible
data are also necessary
12 (1979)
of the quality of the pro-
prediction
istics of a food product
und Stoffiibertragung
model
and coordinate
system
for air-
fragmentary
or inaccessible. tical utility. Even The
measurement
product
normally
the material are moist in them
of thermal requires
bodies
is always
dient of water moisture
a temperature
to be tested.
porous
resulting
the product
Food
gradient
products
in general,
energy
transfer
conduction
in the liquid within the pores.
tional energy
transfer,
if not properly
accounted
conductivity.
In other words,
any mathematical
which
to estimate
is used
should
consider
transfer.
data for food products which
accounts
estimating
are mostly
the transient
of Heissler
results
of food materials
ponds
as these charts
[10-12]
during
pressure
face temperature
ture. However,
transfer
paper,
of some
to determine
these experimental
Even
tories is described. for any porous
to of food
like slabs,
cylinders
is presented of moisture
consiat the
determined
time-temperature
food models
and actual food
the thermal
and analytical This method
moist
model
characteristics
like fruits and vegetables
a theory
of the cooling air.
air precooling
Experimentally
whereas,
a mathematical
temperature
during
characteristics
A method for
cooled
the effects of evaporation
products
are also presented. properties temperature
from his-
is valid in general
body.
like those
will yield er-
pressure
to model gradients
be a futile mathematical
assume
surface
2 The Analytical
Model
that
at the sur-
is made
to evaluate
migration
of mois-
of reliable values
for
foi~ food products,
the interaction
lem
model
is shown
and coordinate
in Figure
heat conduction
equation
I r
b (rnbt) br b-r
lbt = ~ b~
where n = 0
for slabs
n = 1
for cylinders
of temperature
within the product exercise
The physical
I. One
system
dimensonal
for the model
of the probFourier
is written as
corres-
of water
of internal
in the absence
for
characteristics
air precooling
diffusion coefficients
and moisture
dering
being
of different geometries
surface.
data
that are available
and no attempt
the relative importance
any attempt
products
and spheres
are valid only for pure
at the product
to the saturation
moisture
etc. [7-9]
In the present predict the time
or cooling rates charts
of the product
situations.
of heat and mass
the vapour
[1-6].
use of conventional
The few analyses prediction
ality, it is the wet bulb temperature
model
upon
is not true in practice.
[12] the dry bulb temperature
in re-
property
based
temperatures
or Gurnie-Lurie
heat transfer
thermal
which
example air is in
in
conductivity
for only heat transfer
of food products,
roneous
available
For
tem-
the effects of both heat and mass
The presently
condition, in reference
coarse.
that the cooling
perature
for,
temperature
are rather assumed
in
of the cooling air is taken to be the asymptotic
of thermal
the thermal measured
references
presented
of
This addi-
estimate
the experimentally
Similarly
and free
will only lead to an erroneous
from
gra-
then, the approaches
in [I0] it has been a saturated
gradient
with a partial pressure
in an additional
the above
in
This gives rise to migration
along with molecular
convection
for any
and any temperature
coupled
vapour.
conductivity
will only
without any prac-
and n = 2
for spheres.
(I)
K. Badari Narayana and M.V. Krishna Murthy: Determination of Thermal Properties of Food Materials The initial and boundary
t
=ti;
bt
= 0, r = 0, v > 0
b-~
0O .
difference Eq. (I0) to of
for the follow-
[14] : (i) dry bulb temper(tdb = 0 to 3.2),
of cooling air (corresponding and relative humidity
(iii) product initial temperature
3 Numerical
boundary
have been calculated
of parameters
1.6 to 4.8) and (iv) Blot number
Eq. (9) is rewritten by expressing the enthalpy of air as a second degree polynomial in dry bulb temperature (H = a + btdb + ct2b ). Thus
Backward
L15]. The cooling characteristics
moist food products
tdb considered
Ps
on an
using Crank-Nicolson
is used for the nonlinear
avoid oscillations
bulb temperature
with the non-
condition is solved numerically
ature of cooling air 0 - 25r
bT
equation
digital computer
implicit finite difference
ing ranges
~-~ =
(llb)
(ii) wet to
value of 0 to I),
of 20 to 60~
(t~ =
of 0.01 to I0.0.
Results
Figure 2 shows a typical air precooling characteris-
DT ( C 1 T2s + C 2 T s + C 3 ) ; R = 1, r b--~=-Bi
(10)
tics where the present theory is compared with the conventional Gurnie-Lurie chart [8]. It is seen that
where
the rate of cooling as predicted by the present analysis is m u c h higher than that predicted by GurnieLurie theories, because of the additional mechanism of heat transfer namely latent heat transfer that is
C2 =
b + 2t + ) Cp+ WCps wb
(10b)
included in the analysis. The limiting temperature of the product can be observed to be the wet bulb
264
W~rme-
und Stoff~bertragung
12 (1979)
1.0 0.8 0.6 ~- 0.4 0.Z
Fig. 2. Comparison of present theory for pre-cooling with that of Gurnie-Lurie (cylindrical product)
F i g . 4. The a i r cooling tunnel 1. I n s u l a t e d t u n n e l , 2. O r i f i c e p l a t e , 3. Betz m a n o m e t e r , 4. C e n t r i f u g a l b l o w e r , 5. F l e x i b l e c o n n e c t i o n , 6. D i s c h a r g e t h r o t t l e v a I v e , 7. Cooling c o i l s , 8. T h e r m o s t a t bath (low t e m p ) , 9. V a n e , 10. H e a t i n g / c o o l i n g c o i l s , 11. H e a t i n g / c o o l i n g b a t h , 12. Baffle p l a t e s , 13. T h e r m i s t o r s e n s o r , 14. D . B . & W . B . t h e r m o c o u p l e s , 15. 1 2 - p o i n t r e c o r d e r , 16. P r e c i s i o n b a l a n c e , 17. S u s p e n s i o n s y s t e m , 18. Food product
temperature
ysis predicts lesser cooling time (9.23 hr) than the
Presenl Theory
0 0.01
" .
0.1
t.0
6.0
of the cooling air. Gurnie-Lurie
ysis predicts a higher temperature
anal-
for the product
conventional
charts.
This is due to the fact that even
with the limiting value being the dry bulb temperature
with saturated
of the cooling air. The problem
due to the partial vapour
in reference
[163 is reworked
of cooling a meat using the present
slab anal-
between
air, the surface evaporation
the product
that the cooling time for the
increasing
meat
0.0508
to air at
unaffected by changes
an initial
other parameters
slab of thickness
dry bulb temperature
uniform
temperature
m exposed to drop from
of 26.7~
to 4.4~
duct centre is 6.03 hr as compared dicted by the Gurnie'lurie cent relative humidity the present
analysis,
conventional
charts.
saturated
analysis.
at the pro-
to 9.6 hours preA value of 75 per
for cooling air is assumed which is not considered However,
in
in the
even for the case of
air flow over the product,
the present anal-
the overall heat transfer.
and thus
Cooling rate is
in dry bulb temperature
remaining
enthalpy potential remains bulb temperature
can occur
gradient existing
surface and the ambient,
ysis and it is observed
1.7~
pressure
constant.
constant for a fixed wet
of air and product initial temper-
ature irrespective relative humidity
of the dry bulb temperature
is found to increase
with increase
It is observed
from
Figure
rate
is lower at higher air wet bulb temper-
(dis/dr)
in Blot number.
3 that the actual cooling
is higher at higher wet bulb temperature. clear from crease
_.
and
of the cooling air. The cooling rate
ature, even though the relative cooling rate
i 1.0~ ~ ~ ~ , _ ~ _ _ ~
with
This is because
the definition of T. Because
in both temperature
ference between
and vapour
(dTs/dV~)
This fact is of the inpressure
dif-
the product and the surrounding
at higher product temperatures, cooling rate also increases
it is observed
air that
with t.. 1
4 Experiments
- ---F-/ &0O5
0.01
R=1.0 R =0.5 RI =~
0:~ 213 l q 1.121 s l 6 qq 1:50
I lg
0.1
I 1.0
The experimental
setup shown
in Figure
4 consists of
a closed circuit air tunnel with a test section size of 20 cm
square.
humidity
The required
temperatures,
relative
and velocity of air in the duct are main-
Fig.3. Time-temperature characteristics of m o i s t materials during air-precooling ( e f f e c t of wet bulb
tained and controlled
temperature
- cooling coils and throttle valve. Experiments
)
in the test section using heating are
K. Badari
Narayana
='-C~"" ~
and M.V.
Krishna
Murthy:
Determination
Composition : 100 water;30 sugar;3 agor Size :50.0 mm OIA.sphere Cooling medium : Air \ Dry bulb temperature : 2g.0~ k Wet bulb temperature :23.5 ~ ') Velocity :1A75 m/s ~'\\ \ Product initial \ ~ temperature : 32.0oC, 49.0 C
9 4.s 4.6 44
o
'~ k\
42
\
X
....
~
-
R= 0.75
1 9
4.O
o
Properties
of Food
Materials
265
~x~, ~ . Composition:~ 1 0 0 water; 25 sugar:3 ogor ~ \~ - - - - 1 0 0 woteq 10 sugar;3ogor _ \"~ x~. / Shape :slob 40 \~'~ ~x%.\ 11 Eooling medium :Air % ' ~ \HI Dry bulb temperature :19.5~
%,, " ~ .
l
i
Welbulb~emperoture:16.0~
35
R = 0.25,
R=O
%,\
"xx,~ ',,
of Thermal
30
" 4 ~
- a (o.so),,~-
-'--,
~
%.
x'N
Nurnbegi/nthe///bmc//k/efsh//ow the////posffbn of the thermocoupbin cm) 1
~. 25
4
0
8
. . . . . .
12
16 rain
L F--
20
Time Fig. 6. Experimental time-temperature histories during air-preeooling (effect of variation of sugar content)
32 30 28
from
Z
4.
5
8 10 tZ ]ime ,-..-----
Fig.5. Air cooling of model duct initial-temperature)
1/+
16 rnin t0
Figure
crease.
food gel (effect of pro-
5 at higher product initial temperatures,
when both evaporation Figure
and sensible heat transfer in-
6 shows
that the effect of decreasing
the sugar content is to increase is due to the increase
the cooling rate. This
in the initial moisture
of the product at lower sugar contents. conducted
on (i) food models
agar-agar,
of sugar,
and water in different proportions
(ii) fruits and vegetables.
The temperature
ious depths in the product dia copper-constantan the product. recorder
consisting
is measured
history of the product.
in
as shown
7 that lower sugar content products
are
by higher moisture
loss
is found to increase
Figures
by suspending in Figure
0.03
the product
4. A Sartorius
the time-temperature
istics of some
products
It is observed
from
ratio and hence
in wet bulb
3
is used for this purpose.
5 and 6 show
of the product
in air velocity,
and decrease
temperature.
Results
bulb temperatures
with increase
product initial temperature
0.02 5 Experimental
loss. Moisture
the time-temperature
The weight loss of the product
during cooling is measured on to a balance
embedded
Figure
accompanied
12 point potentiometric
is used to record
precision balance
using 0.1 mm
thermocouples
Hartman-Braun
and at var-
from
content
It is observed
character-
obtained during air cooling.
the experiments
that at lower wet
of the cooling air, the cooling rate
increases.
The increase
the enhancement
driving force is the reason
in the humidity
of the mass
for this. Also,
in the cooling rate of the product
transfer an increase
could be observed
~ O.Ol
Z
J
oY 0
ti= 40~
12 Time ,
=lg~176
1
100
2 3 4
100 100 100
5
100
6
100 18
10 15 20 25 30 40 2~4
--
3 3 3 3 3 3
_
rain
P
F i g . 7. D e h y d r a t i o n c h a r a c t e r i s t i c s of food m o d e l s during a i r - p r e c o o l i n g (cylindrical product)
30
266
6
W~rme-
of
Determination
Thermal
Properties
12 (1979)
und Stoff~bertragung
+ Tz
x
[Tn(~)- T e , n ( ~ ) ] 2 d ~ Thermal
properties
evaluated
of the products
by matching
time-temperature
method
procedure
The non-linear
adopted
The method
using
governing
estimation
upon
on the theory
parameters
and arriving
is nor-
values
limits.
on the parameter
A good
estimation
non-linear
in the estimation
estimation
method
In the present
case
duct is a function of Bi, R between
the measured
temperature
meter
and theoretical
with respect
in least squares
sense
of the pro-
~*, the deviation
at different locations
to be minimized
of
in [19-22].
and
values
F
of the
in the product
to the physical
It could be observed
from
limit of integration
becomes
Eq. (14) that the upper a constant
The derivative
once
c~0 and
of Eq. (14) with re-
spect to Bi and
is given in [18]
as the temperature
c/0T/r02 9
~z are specified.
The use of the
of food products
and for those of solid materials
9 ~=
at the final values
[17].
properties
and
of the
is given by Pfal and Mitchell
thermal
2
+ = C~O TZ /r 0 ~'Z
the process.
of least squares
within specified
state of the art paper
where
pro-
consuming
a set of initial guess
of these parameters
obtained
method
to hasten
(14)
n=l 0
a trial and error
and time
in such cases
is based
for iterating
are
with the theoretical
will be a combersome
process. mally
the experimentally
curves
files. This matching
considered
e* are then equated to zero and the z correction ABi and A~* to be applied to the paraz meters are evaluated [14]. A computer programme is developed
to evaluate
tions [14].
It is observed
the above
scheme
are sufficient to satisfy the required given by the following
of calcula-
that about 6-8 iterations accuracy
criteria
expressions:
is
I ABi/Bi[ ~ 0.00001
para-
and is written as follows:
and x
@ Tz
(12)
f [Tn(~*)-Te,n(~*)]2dT* n=l 0
F(Bi,~*) =z. ~
IA T@/ z/~zl@ 40.00001 if the initial estimates ues.
where
Thermal
conductivity
are evaluated
Tz@ = ~ T z / r ~ "
from
and thermal
the final corrected
of the Eq. (12) with respect
Bi and
~*z when equated
taneous
equations
to zero
to be solved
to
will yield two simul-
for obtaining
to be applied to the initial guess
the cor-
values
of Bi
and
diffusivity values
temperature Figure
data are used
8 shows
in the same
the sequential
matching
process
It is observed
from
Table
I that for food models
for a given increment
is 70 to 88.5 per cent. These trends followthe
difficulty the following
value of ~. dimensionless
To overcome
this
ratio is intro-
[14] :
(13)
conductivity
with increase
in sugar
tern as observed who
have
ce0 is any reference
diffusivity value.
of Eq. (13) in (12) yields
Intro-
diffusivity decrease
content. content ( wet basis) considered
by Bakal
[23 ] and Keppler
determined
the thermal
uated for some
fresh fruits and vegetables
mentioned
properties
of the earlier transient
in the literature
fects are either minimized
eval~
are given experi-
the convection
or neglected
pat-
and Boose
solutions.
ments
The thermal
same
diffusivity val-
ues of sucrose
in Table 2. In most where
and thermal
The range of moisture
[24],
+ = -~0 -
for
a food model.
thermal
duction
of Bi
calculation.
~*. However, the numerical evaluation of the z derivative with respect to ~* offers some difficulty z as the upper limit of integration in Eq. (12) varies
duced
val-
and ~* values respectively. The non-linear estimaz tion method is quite accurate as all the transient
The partial derivative
rections
are close to the correct
ef-
and the mass
K. Badari Narayana
and M.V.
Krishna Murthy: Determination
1.0
No.
x,,,, !1 -'~_
".
of F o o d M a t e r i a l s
T a b l e 1. E f f e c t i v e t h e r m a l v a l u e s of f o o d m o d e l s
Composition: 100 water: 10sugoQ 3 agor Shape: 60.60.10 mm slab 9 or - - - surface o or - - - - centre
0.8
of T h e r m a l P r o p e r t i e s
1 2 3 4 5 6 7
Experimental points
0.6
267
conductivity and diffusivity
Sugar content (grams)
k (W/Km)
10 15 20 25 30 35 40
0.820 0.812 0.810 0.789 0.768 0.765 0.756
~ x
10 7
(m2/s) 1.119 1.067 1.040 1.010 0.980 0.972 0.952
Size: 60• 10 m m , ti = 4 3 . 5 ~ td~= 1 9 . 5 ~ twb = 1 6 . 7 5 ~ cooling air velocity = 2.01 m/s, comp o s i t i o n : 100 g w a t e r , 3 g A g a r - a g a r , the sugar contents are tabulated
0.4 '5 Numeruls denote the iteration number l I 1 I I 02 0.4 E6 0.8 1.0
0.2 0
1.2
1.4
1.6
lowance
for mass
transfer at the product exposed
face during air precooling Fig. 8. Matching time-temperature
of the theoretical and experimental curves using least squares theory
ing equations
are presented.
with the associated
sur-
The govern-
nonlinear equations
are solved numerically. ii. Surface evaporation transfer effects are neglected. striction imposed mass
in the present
transfer effects at the product
cluded.
Thermal
periments
properties
which simulate
effect on the total heat transfer.
There is no such remethod
iii. Faster
and also the
from
actual processing
cooling rates for food products
dicted by the present
surface is in-
are evaluated
is found to have significant
model
tional forced convection
ex-
iv. Thermal
conditions.
properties
fruits and vegetables
compared
boundary
the non-linear
to the conven-
condition analysis.
of some
food models,
are evaluated by matching
theoretical and experimental estimation
are pre-
thermal
the
histories using
method.
7 Conclusions References i.
Mathematical
sient heat transfer
models
for the evaluation
in moist food products
of t r a n -
in the shape
of i n f i n i t e s l a b , i n f i n i t e c y l i n d e r a n d s p h e r e
with al-
Table 2. Effective thermal conductivity vegetables estimated during precooling
No.
1 2 3 4 5 6 7 8 9 I0 II
Hayakawa, K.I. : Estimating Temperatures of Foods during various Heating or Cooling Treatments. New York: ASHRAE Journal 1972
and diffusivity values of fresh fruits and
Moisture content (per cent)
k
c~• I07
(~
Cooling air velocity (m/s)
(W/Km)
(m2/s)
19.0 18.4 20.3 20.0 19.5 19.5 19.5 19.1 19.0 18.5 17.0
1.40 1.38 1.38 1.55 1.50 1.38 1.38 1.38 1.50 1.38 1.38
86.0 75.8 86.5 85.8 93.3 82.2 92.1 89.7 84.5 94.2 90.4
1.376 0.890 1.837 0.958 1.615 1.074 1.450 2.130 1.878 1.390 0.860
1.037 0.891 1.271 1.836 1.103 0.725 1.134 1.586 1.182 0.958 1.796
ti
tdb
twb
(~
(~
43.75 49.00 46.00 45.75 47.50 43.00 45.00 46.25 48.50 43.50 48.00
22.0 21.4 22.5 23.0 22.2 21.5 21.4 21.4 21.5 20.2 19.6
Material
Apple Banana Beetroot Carrot Cucumber Mango Melon Papaya Potato Pumpkin Raddish
1.
268
2.
3.
4.
5.
6.
7.
W~rme-
Olson, F.C.W.~ Schultz, O.T.: Temperatures in solids during Heating and Cooling. Ind. Eng. Chem. 34 (1942) 874 Pflug, l.J.; Blaisdell, J.L.; Kopelman, J.: Developing Temperature-Time curves for Objects that can be Approximated by a Sphere, Infinite Plate or Infinite cylinder. Trans. ASHRAE. 71 (1965) 238 Pflug, I.J.~ Kopelman, I.J.: Correlating and Predicting Transient Heat Transfer Rates in Food Products. Comm. I.I.R. 2 (1966) 89 H ayakawa, K.I. ; Bakal, A. : New Computational Procedure for Determining the Apparent Thermal Diffusivity of a Solid Body Approximated with an Infinite slab. J. Food Science. 38 (1973) 623 Kopelman, I.J. ; Mizrahi, S. ; Kauffman, I. : Thermal Conductivity in Transient Cooling of Oranges. Comm. 2. I.I.R. (1973) 223 Heisler, M.P. : Temperature Charts for Induction and Constant Temperature Heating. Trans. ASME.
69 (1947) 227 8.
9. 10.
II.
12.
13. 14.
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Gurnie, H. P. ~ Lurie, J. : Charts for E stimating Temperature Distribution in Heating and Cooling of solid shapes. Ind. Eng. Chem. 15 (1923) 1170 Schneider, P.J. : Conduction Heat Transfer. Reading, Man. : Addison Wesley 1966 Dyner, H.~ Hesselchwerdt, A.L. : TemperatureTime Characteristics During Food Precooling. Trans. ASHRAE. 70 (1964) 249 Hodgson, T. : The Effect of Environmental Conditions on Chilling Rates of Meat. Bull. I.I.R. Annexe 1966-1 (1966) 635 Srinivasa Murthy, S. ; Krishna Murthy, M.V. ; Ramachandran, A. : Heat Transfer During Aircooling and Storing of Moist Food Products. Trans. ASAE. 17 (1974) 769 Stoecker, W.F. : Refrigeration and Air Conditioning. New York: McGraw Hill 1958 Badari Narayana, K. : Heat and Mass Transfer Studies and Evaluation of Thermal Properties of Food Products. Ph. D. Thesis, Ind. Inst. Technology, Madras 1976 Von Rosenberg, D.U. : Methods for the Numerical Solution of Partial Differential Equations. New York: Elsevier 1969 ASHRAE Guide and Data Book (Applications). New York: ASHRAE 1971
und Stofffibertragung
12 (1979)
17. Pfal, R.C.; Mitchel, B.T.: A General Method for Simultaneous Measurement of Thermal Properties. AIAA PaperNo. 69 (1969) 602 18. Hundtoft, E.B. ; Wu, S.M. : Determining Specific Gravity of Alfalfa Solids by Nonlinear Least Squares Method. Trans. ASAE. 13 (1970)181 19. Beck, J.V. : Calculation of Thermal Diffusivity from Temperature Measurements. Trans. ASME. J. of Heat Transfer. 85 (1963) 181 20. Beck, J.V. : Transient Determination of Thermal Properties. Nuclear Eng. Design. 3 (1966) 373 21. Pfal, R.C. : Nonlinear Least Squares: A Method for Simultaneous Thermal Property Determination in Ablating Polymeric Materials. J. Appl. Polymer Sci. I0 (1966) I111 22. Clark, B.L. : A Parametric Study of the Transient Ablation. Trans. ASME. J. of Heat Transfer. 94 (1972) 347 23. Bakal, A.I. : Conduction Heat Transfer with Phase Change and Its Application to Freezing or Thawing of Foods. Ph. D. Thesis, Rutgers State University, New Brunswick, New Jersey 1970 24. Keppeler, R.A.~ Boose, J.R.: Thermal Properties of Frozen Sucrose Solutions. Trans. ASAE. 13 (1970) 335
Dr. K. Badari Narayana Prof. Dr. M.V. Krishna Murthy ~ Refrigeration and Airconditioning Laboratory Department of Mechanical Engineering Indian Institute of Technology Madras-600 036, India Present
address"
Institut ffir W~irmetechnik und Thermodynamik Univer sittit Stuttgart Pfaffenwaldring b D-7000 Stuttgart-Vaihingen Bundesrepublik Deutschland
Received
February
21~ 1979
