thermal conductivity models for porous baked foods
TRANSCRIPT
THERMAL CONDUCTIVITY MODELS FOR POROUSBAKED FOODS
C. REYES1, S.A. BARRINGER1,3, R. UCHUMMAL-CHEMMINIAN2 andG. KALETUNC2
1Department of Food Science and TechnologyThe Ohio State University
2015 Fyffe Road, Columbus, OH
2Department of Food, Agricultural and Biological EngineeringThe Ohio State University
Columbus, OH
Accepted for Publication March 1, 2006
ABSTRACT
Models for the thermal conductivity of porous food were developed bymodifying the parallel model to include structural data such as pore diameter,pore wall thickness and shortest path ratio. The predictions from these modelsand 12 literature models were compared to values determined for 12 porousfoods. The models from the literature considered the air volume fraction but noother structural information. The proposed parallel, Kopelman random (KR)and Kopelman fibrous parallel models gave predictions that were not signifi-cantly different from the experimental thermal conductivity. All acceptablemodels except the KR model are variations of the parallel model, indicatingthat heat flows in parallel through the air and solid portions of porous foods.Regression analysis determined that only the thermal conductivity of the solidand shortest path ratio significantly correlate to the final thermal conductivity,indicating that the configuration of the pores is important in porous foods.
INTRODUCTION
Many foods are thermally processed to preserve them, such as canning,drying or freezing. These processes involve heat transfer; therefore, the heattransfer properties of foods are essential to predict rates for these processes.Thermal conductivity is needed to determine the temperature profile during aprocess and to select the appropriate equipment.
3 Corresponding author. TEL: (614) 688-3642; FAX: (614) 292-0218; EMAIL: [email protected]
Journal of Food Processing and Preservation 30 (2006) 381–392. All Rights Reserved.© 2006, The Author(s)Journal compilation © 2006, Blackwell Publishing
381
Porous foods are complex structural systems, and their thermal conduc-tivity is difficult to predict (Sweat 1995). Models that consider only compo-sition are not accurate; therefore, structurally based models must be used topredict the thermal conductivity of porous foods (Sweat 1995). Several modelsto predict thermal conductivity have been proposed. The Maxwell, series,parallel, random, Mattea–Urbicain–Rotstein (MUR) (Maroulis et al. 1990)and several Kopelman models (Kopelman 1966; Marschoun et al. 2001) aregeneral thermal conductivity models used for porous foods. These structuralmodels consider the air volume fraction of each food sample and the thermalconductivity of the solid and air portions.
Articles disagree on the accurateness of these structural models. At 20C,the Maxwell model closely predicted the thermal conductivities of fresh andfrozen foods (van den Berg and Lentz 1975). Van den Berg and Lentz (1975)predict that the Maxwell model will effectively estimate the effect of air andtherefore, the Maxwell model could predict the thermal conductivity of porousfoods. However, models that predict the thermal conductivity for nonporousfoods frequently do not for porous foods. For example, for granular starches,considered to behave like porous foods, a comparison of the parallel, series,random, MUR and Maxwell models determined that the parallel model givesthe smallest error, and the series model the largest. The parallel model had thelowest error because heat flow occurs simultaneously through adjoining starchgranules and through the pores (Maroulis et al. 1990). When equivalentstarches were gelatinized (nonporous), the series model best predicted thermalconductivity while the parallel model gave the largest error (Maroulis et al.1991).
On the basis of the effective medium theory, the MUR model accuratelypredicts the thermal conductivity of granular starch in a wide porosity andtemperature range (Shah 1996). Another model that accurately predictsthermal conductivity is the Kopelman parallel parallel (KPP) model, whichassumes that the heat flow is parallel to layers within the food. The Kopelmanrandom (KR) model was tested for starch and casein at selected temperaturesand gave acceptable standard errors at 25C (Sabilov 1998).
The effect of air on thermal conductivity is of great importance. Moistureexhibits a major effect on thermal conductivity, but for porous material, theeffect of air on thermal conductivity is greater than that of water (Goedkenet al. 1998). Coefficients of variation of predicted thermal conductivity valuesfor porous systems are large and increase in proportion to porosity (Sherifet al. 1976; Polezhaev 1997). The reason for the large coefficients of variationhas been attributed to the different pore sizes, shapes and orientations in thesystems (Sherif et al. 1976; Polezhaev 1997). In general, it is difficult topredict the thermal conductivity of porous foods, and it has been suggested thatthermal conductivity predictions for systems with less than 48% solids are not
382 C. REYES ET AL.
possible (Sherif et al. 1976). The difficulty in modeling thermal conductivityfor porous foods is due to their structural complexity (Sweat 1995).
The existing models have been tested on granular foods such as starchesor casein rather than continuous systems such as bread. Researchers typicallyconsider granular foods as porous foods. Granular and porous foods are bothtwo-media systems composed of air and solids; therefore, models developedfor granular foods are also used for porous foods. Also, no thermal conduc-tivity models the authors are aware of consider structural data such as porediameter or pore wall thickness. In this study, structural data is taken intoaccount to predict the thermal conductivity of selected porous foods such asbread, cakes and muffins. The models using structural data are compared withmodels from the literature to determine which models are most accurate inpredicting the thermal conductivity of porous foods.
MATERIALS AND METHODS
Thermal conductivity was determined using the probe method describedby Sweat and Haugh (1974). Food samples 5-cm long by 3-cm wide weretested at 35C. The solid thermal conductivity was determined on compactedsamples. The air volume fraction or porosity was calculated by the volumedifference of a sample before and after compression. The samples used werecorn bread, three types of wheat bread, two different rye breads, hamburgerbuns, chocolate muffins, oat bran muffins, angel food cake, pound cake andsponge cake.
The samples were sliced 1-mm thick so the light from a microscope willpass through and illuminate the structure. Five slices were used for eachsample. After slicing the samples, they were painted with a mixture of brownand black ink using a brush, placed under a stereomicroscope (Stemi DV4;Zeiss, Jena, Germany) and a digital picture was taken through the stereomi-croscope with a digital camera (DC 290; Eastman Kodak, Rochester, NY). Animage of a 5-mm reference slide was also taken as a reference image. Photo-Shop (Adobe Systems Inc., San Jose, CA) was used to turn the images into ablack and white image where the air-filled pores were well differentiated fromthe solids. Scion Image (Scion Corp., Frederick, MD) software was used toconvert the images into measurements. The reference image was used todetermine the pixel to millimeter ratio for each image. The black and whiteimages were converted into binary images, and the Scion software (Scion)determined pore diameters and areas. The line selection tool was used todetermine pore wall thickness and shortest path ratio. The cell wall thicknesswas obtained by calculating the mean of the shortest distance between adjacentpores. The diameter of each pore was calculated by determining its total area
383CONDUCTIVITY MODELS FOR BAKED FOODS
and by calculating the diameter of a circle with the same area. The diametersobtained from the areas were used to calculate the mean pore diameter. Theunit length was calculated by adding the mean cell wall thickness to the meanpore diameter. The shortest path ratio was obtained by dividing the length ofthe shortest path that heat could follow through the solid portion of the imageby the length of the image. These values are presented in Table 1.
Twelve literature models were used to calculate the thermal conductivityof the samples. The selected models were the parallel, series, Maxwell,random, MUR (for z = 4, 5 and 6) (Maroulis et al. 1990), KR, Kopelmanfibrous parallel (KFP), Kopelman fibrous (KFPe) perpendicular, KPP andKopelman parallel perpendicular (KPPe) (Kopelman 1966; Marschoun et al.2001) models.
Analysis of variance was used to determine if predicted values weresignificantly different from experimental values. The c2 value and mean devia-tion from the experimental values were calculated to determine which modelsgave the most accurate predictions. A least squares linear regression analysis inJMPIN (SAS Institute Inc., Cary, NC) statistical software determined whichfactors were significant. A P value � 0.05 was considered significant.
RESULTS AND DISCUSSION
Five models gave predictions not significantly different from the experi-mental thermal conductivity of 12 porous foods. These are Eqs. (1) and (2), theKFP, the KR and the parallel model.
Equation 1 is a modification of the parallel model where ks is the thermalconductivity of the solid portion; ka is the thermal conductivity of air; w is themean cell wall thickness, and q is the mean pore diameter.
k kw
wk
w=
+( )+
+( )s aθθ
θ (1)
In Equation 1, the unit structure is assumed to be a pore and its surroundingwall (w + q). The entire sample is a series of randomly placed units. Becausethermal conductivity is independent of sample size, the thermal conductivity ofthe unit cell is equivalent to the thermal conductivity of the entire sample, aslong as the unit cell is representative of the sample. The heat flows by conductionthrough the solid portion and simultaneously flows by convection through theair in the pores. The thermal conductivity is then the sum of the individualthermal conductivities of each of the components in the system, air and solids,multiplied by the proportion of each component on the basis of structuralmeasurements. Instead of volume fraction, Eq. (1) uses the linear fraction.
384 C. REYES ET AL.
TAB
LE
1.ST
RU
CT
UR
AL
DA
TAFO
RPO
RO
US
FOO
DS
Sam
ple
Pore
diam
eter
(mm
)D
iam
eter
rang
e(m
m)
Wal
lth
ickn
ess
(mm
)U
nit
leng
th(m
m)
Solid
k(W
/mK
)A
irvo
lum
efr
actio
nSh
orte
stpa
thra
tio
Spon
geca
ke1.
583
17.5
500.
666
2.24
90.
164
0.74
41.
216
Whe
atbr
ead
21.
324
14.7
700.
601
1.92
50.
278
0.73
41.
347
Ang
elfo
odca
ke0.
542
8.07
00.
311
0.85
30.
287
0.83
01.
213
Whe
atbr
ead
10.
689
22.9
200.
284
0.97
30.
286
0.56
01.
270
Cor
nbr
ead
1.37
34.
600
0.90
42.
277
0.33
40.
720
1.30
0B
urge
rbu
ns0.
483
1.29
00.
129
0.61
20.
379
0.75
31.
203
Cho
cola
tem
uffin
s0.
561
5.07
00.
866
1.42
70.
272
0.50
01.
077
Poun
dca
ke0.
657
4.75
00.
429
1.08
60.
321
0.70
81.
182
Bar
ley
muf
fins
1.16
516
.530
0.61
41.
779
0.38
70.
710
1.08
3R
yebr
ead
20.
845
11.9
000.
651
1.49
60.
468
0.46
41.
121
Rye
brea
d1
2.22
476
.390
1.67
43.
898
0.38
80.
629
1.19
7W
heat
brea
d3
0.65
54.
370
0.44
41.
099
0.46
50.
759
1.12
4
385CONDUCTIVITY MODELS FOR BAKED FOODS
Thermal conductivity predicted with Eq. (1) was not significantly differ-ent from the experimental thermal conductivities (Table 2). This suggests thatthe heat flow in a porous sample is mostly governed by the proportion of solidmaterial to air in the food (Wallapapan et al. 1983). Equation 1 assumes thatheat flows through the solid and air simultaneously, similar to the parallelmodel, except that the parallel model uses total porosity as the structuralcomponent of the model. Size and distribution of pores in a porous mediaaffect thermal conductivity (Sherif et al. 1976). Therefore, in Eq. (1), moreattention is given to the size of the pores filled with air.
Equation 2 was developed that considers the pore diameter, cell wallthickness and also the shortest path ratio, where s is the shortest path ratio.
k kw
s wk
w=
+( )+
+( )s aθθ
θ (2)
Equation 2 also considers a structure of randomly distributed cell wallpore units. In this case, though, a shortest path ratio was included to modify thelength of the path that heat takes through the solid phase. The shortest pathratio is a structurally determined correction factor that accounts for the randomdistribution of pores in the solid phase (Fig. 1). The structurally determinedcorrection factor accounts for the fact that heat travels a longer distancethrough the solid fraction than a straight line will predict, because heat goesaround the pores following a curved path. The term in the model for the solidcomponent is divided by the ratio of the shortest distance that heat can travelby conduction to the width of the sample.
The KFP (Eq. 3) and the KR (Eq. 4) models gave thermal conductivitypredictions that were not significantly different from the experimental values(Table 2). The KFP model produced the best fit to the experimental valuesamong the literature models. This model was created for a fibrous systemwhere heat flows parallel to the fibers. This model is also a modification of theparallel model. The volume fraction of air is ea.
k kk
kc= − −⎛⎝⎜
⎞⎠⎟
⎡⎣⎢
⎤⎦⎥
1 12εaa
s
(3)
The KR model (Eq. 4) gives acceptable standard errors when predictingthermal conductivity for casein and starch powders at 25C (Sabilov 1998). TheKR model was developed for two phase systems in which the discontinuousphase, air in this case, is randomly distributed in the continuous phase. Thefoods used in this study were a solid matrix with pores randomly distributedthroughout the samples. Therefore, the samples used match the system forwhich the Kopelman ransom model was developed.
386 C. REYES ET AL.
TAB
LE
2.C
OM
PAR
ISO
NO
FE
XPE
RIM
EN
TAL
AN
DPR
ED
ICT
ED
TH
ER
MA
LC
ON
DU
CT
IVIT
Y(k
),W
ITH
c2V
AL
UE
SA
ND
ME
AN
DE
VIA
TIO
NS
BE
TW
EE
NE
XPE
RIM
EN
TA
ND
PRE
DIC
TIO
N
Sam
ple
Exp
erim
enta
lk
valu
e(W
/mK
)
Stru
ctur
alm
odel
sL
itera
ture
mod
els
Eq.
(2)
Eq.
(1)
KFP
KR
Para
llel
KFP
eK
PPM
axw
ell
Seri
esK
PPe
MU
RM
UR
MU
RR
ando
m
**
**
*(z
=6)
(z=
5)(z
=4)
Spon
geca
ke0.
058
0.05
90.
068
0.05
20.
055
0.06
20.
043
0.04
00.
043
0.03
50.
030
0.02
60.
023
0.01
90.
698
Whe
atbr
ead
20.
081
0.08
30.
106
0.07
40.
080
0.09
40.
056
0.05
20.
048
0.03
60.
030
0.02
60.
019
0.01
10.
782
Ang
elfo
odca
ke0.
083
0.10
40.
122
0.05
80.
060
0.07
10.
045
0.04
30.
039
0.03
20.
029
0.02
60.
019
0.01
00.
859
Whe
atbr
ead
10.
084
0.08
50.
103
0.11
00.
124
0.14
10.
084
0.07
30.
068
0.04
50.
032
0.02
60.
019
0.01
00.
709
Cor
nbr
ead
0.09
80.
118
0.14
90.
088
0.09
50.
113
0.06
50.
059
0.05
00.
037
0.03
00.
025
0.01
60.
006
0.81
0
Ham
burg
erbu
ns0.
101
0.08
80.
101
0.08
80.
094
0.11
40.
064
0.05
90.
048
0.03
50.
030
0.02
40.
014
0.00
20.
853
Cho
cola
tem
uffin
s0.
104
0.16
40.
176
0.11
80.
133
0.15
00.
091
0.07
80.
076
0.04
90.
033
0.02
60.
019
0.01
10.
686
Poun
dca
ke0.
110
0.12
40.
143
0.08
80.
095
0.11
30.
065
0.05
90.
051
0.03
70.
030
0.02
50.
017
0.00
70.
796
Bar
ley
muf
fins
0.17
00.
141
0.15
10.
101
0.10
90.
132
0.07
30.
066
0.05
20.
037
0.03
00.
024
0.01
30.
001
0.83
7
Rye
brea
d2
0.17
40.
197
0.21
90.
204
0.23
30.
263
0.15
20.
127
0.09
50.
055
0.03
50.
021
0.00
8-0
.006
0.85
3
Rye
brea
d1
0.18
00.
155
0.18
20.
123
0.13
60.
161
0.08
90.
079
0.06
30.
042
0.03
10.
024
0.01
30.
001
0.80
7
Whe
atbr
ead
30.
201
0.18
30.
204
0.10
10.
108
0.13
30.
071
0.06
60.
048
0.03
50.
030
0.02
10.
008
-0.0
060.
896
c20.
060
0.13
10.
128
0.13
10.
146
0.26
30.
325
0.44
40.
671
0.81
20.
930
1.10
31.
325
53.2
09
Ave
rage
devi
atio
nfr
omex
peri
men
tal
valu
e0.
019
0.02
60.
032
0.03
10.
031
0.04
60.
054
0.06
40.
081
0.09
00.
096
0.10
50.
115
0.67
9
*N
otsi
gnifi
cant
lydi
ffer
ent
from
expe
rim
enta
lva
lues
.K
FP,
Kop
elm
anfib
rous
para
llel;
KR
,K
opel
man
rand
om;
KFP
e,K
opel
man
fibro
uspe
rpen
dicu
lar;
KPP
,K
opel
man
para
llel
para
llel;
KPP
e,K
opel
man
para
llel
perp
endi
cula
r;M
UR
,M
atte
a–U
rbic
ain–
Rot
stei
n.
387CONDUCTIVITY MODELS FOR BAKED FOODS
k
k
k
k
k
=− −⎛
⎝⎜⎞⎠⎟
− −⎛⎝⎜
⎞⎠⎟
−( )
1 1
1 1 1
2
2
ε
ε ε
aa
s
aa
sa
(4)
The final model to produce results not significantly different from theexperimental values was the parallel model (Eq. 5). As expected, the parallelmodel slightly overpredicted the thermal conductivity for most samples. Theparallel model assumes that heat flows through the air and solid in parallel, anduses the air volume fraction to determine the relative quantities of heat flow.The parallel model also accurately predicted thermal conductivity for granularstarches (Maroulis et al. 1990) and defatted soy flour (Wallapapan et al. 1983).
k k k= −( ) +1 ε εa s a a (5)
The remaining models predicted thermal conductivities that were signifi-cantly different from the experimental thermal conductivities. The KPP modelis very similar to the KFP model, and it is expected to accurately predictthermal conductivity. The KFPe and KPPe models are variations of the seriesmodel. These two and the series model assume thermal flow through the poresand the solid portion in series, which means that a large amount of heat transferis assumed to occur through the air. The MUR model assumed that the two
FIG. 1. SHORTEST PATH RATIO (S) = CURVY LINE/STRAIGHT LINE
388 C. REYES ET AL.
phases in a porous system can be mixed together to give one homogenousphase, and that the structural distribution of the phases is not important to heattransfer. Other studies were also unsuccessful in using series-based models orthe MUR model to predict the thermal conductivity of porous foods.
The Maxwell model is reported to predict thermal conductivity almost aswell as the parallel model for granular starch (Maroulis et al. 1990). TheMaxwell model is also reported to yield accurate thermal conductivities forfoods with low moisture content (Goedken et al. 1998). The Maxwell modelconsiders porosity; however, it is assumed that the air cells are discrete and donot interact with any other air cells. None of the components of highly porousfood systems are isolated. There are many structural interactions, and the airportion of a porous system is not an isolated constituent. The Maxwell modelmay accurately predict thermal conductivity for some low porosity foods, butin this study, the Maxwell model gave predictions significantly different fromthe experimental thermal conductivities.
Regression analysis determined that the factors affecting thermal conduc-tivity include the thermal conductivity of the solid component and the shortestpath ratio (Fig. 2). Cell wall thickness, thermal conductivity of air, pore diam-eter, pore size distribution and porosity were not significant factors in predict-ing thermal conductivity (Fig. 3).
The regression analysis suggests that heat flow in porous foods occursonly through the solid portion. Because dry air at room temperature has a verylow thermal conductivity, the majority of heat should be transferred throughthe solid component, while the heat transfer through air will be negligible.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19
Experimental k value (W/mK)
Th
erm
al c
ondu
ctiv
ity
(W/m
K)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Sh
orte
st p
ath
rat
io
k of solid
Shortest path ratio
FIG. 2. RELATIONSHIP OF EXPERIMENTAL THERMAL CONDUCTIVITY (k) TO MEANTHERMAL CONDUCTIVITY OF THE SOLID COMPONENT AND SHORTEST PATH RATIO
389CONDUCTIVITY MODELS FOR BAKED FOODS
Lack of heat transfer through the air is a major violation of the series model,and so series-based models will not give accurate predictions, as reported inother studies. Models based on the parallel model allow the majority of heattransfer to occur through the solid portion and produce accurate results(Table 2).
The shortest path ratio indicates the shortest length that heat can followby conduction through the solid portion. Pore shape and configuration havepreviously been reported as important factors influencing thermal conductiv-ity, as well as the basis for large SDs in predicted thermal conductivity values(Sherif et al. 1976). Thus, including pore configuration will improve predic-tions. Porosity was previously determined as an important factor influencingthermal conductivity (Goedken et al. 1998). Porosity was expected to correlatewell with thermal conductivity, but in this study, it did not. The shortest pathratio may not be practical to determine in most samples, but the fact that itcorrelated significantly with the experimental thermal conductivity indicatesthat some inclusion of tortuosity or pore structure into thermal conductivitymodels will improve thermal conductivity predictions for porous foods.
CONCLUSIONS
The thermal conductivities of porous foods were acceptably predicted inthis study by Eqs. (1) and (2), the KFP, KR and parallel models. Linear
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19
Experimental k value (W/mK)
Por
e si
ze r
ange
(m
m)
0.0
0.5
1.0
1.5
2.0
2.5
Dia
met
er, c
ell w
all t
hick
ness
(m
m)
or p
oros
ityPore size range
Pore diameter
Porosity
Cell wall thickness
FIG. 3. THERMAL CONDUCTIVITY (k) WAS NOT SIGNIFICANTLY CORRELATEDTO MEAN CELL WALL THICKNESS, POROSITY, PORE DIAMETER OR PORE
SIZE DISTRIBUTION
390 C. REYES ET AL.
regression calculations demonstrated that the thermal conductivity of the solidportions of porous foods and the shortest path ratio are the factors that sig-nificantly correlate with thermal conductivity. In porous foods, heat apparentlymoves in parallel through the solid and air portions, but the size and distribu-tion of the pores is also important.
ACKNOWLEDGMENT
Support by the Ohio Agricultural Research and Development CenterInterdisciplinary Team Research Competition is gratefully acknowledged.
REFERENCES
GOEDKEN, D.L., SHAH, K.K. and TONG, C.H. 1998. True thermal conduc-tivity determination of moist porous food materials at elevated temper-atures. J. Food Sci. 61(6), 1062–1065.
KOPELMAN, I.J. 1966. Transient heat transfer and thermal properties infood systems. PhD Thesis, Michigan State University, East Lansing, MI.
MAROULIS, Z.D., DROUZAS, A.E. and SARAVACOS, G.D. 1990. Model-ing of thermal conductivity of granular starches. J. Food Eng. 11(4),255–271.
MAROULIS, Z.D., SHAH, K.K. and SARAVACOS, G.D. 1991. Thermalconductivity of gelatinized starches. J. Food Sci. 56(3), 773–776.
MARSCHOUN, L.T., MUTHUKUMARAPPAN, K. and GUNASEKARAN,S. 2001. Thermal properties of cheddar cheese: Experimental and mod-eling. Int. J. Food Prop. 4(3), 383–403.
POLEZHAEV, Y. 1997. Porous medium. In International Encyclopedia ofHeat and Mass Transfer, (G.F. Hewit, G.L. Shires and Y. Polezhaev, eds.)pp. 867–871, CRC Press, New York, NY.
SABILOV, C.M. 1998. A prediction model for thermal conductivity of porousfoods. Masters Thesis, University of Missouri, Columbia, MI.
SHAH, K.K. 1996. Thermal conductivity of porous, hygroscopic food powdersunder extrusion conditions. PhD Thesis, The State University of NewJersey, New Brunswick, NJ.
SHERIF, I.I., AMMAR, A.S. and EL-MESSIH, S.A. 1976. The effect of poreshape, configuration and orientation on the thermal conductivity ofporous materials. Int. J. Heat Mass Tran. 19(2), 227–229.
SWEAT, V.E. 1995. Thermal properties of foods. In Engineering Properties ofFoods, (M.A. Rao and S.S. Rizvi, eds.) pp. 99–138, Marcel Dekker Inc.,New York, NY.
391CONDUCTIVITY MODELS FOR BAKED FOODS
SWEAT, V.E. and HAUGH, C.G. 1974. A thermal conductivity probe forsmall food samples. T. ASAE 17(1), 56–58.
VAN DEN BERG, L. and LENTZ, C.P. 1975. Effect of composition onthermal conductivity of fresh and frozen foods. Can. Inst. F. Sci. Tec. J.8(2), 79–83.
WALLAPAPAN, K., SWEAT, V.E., DIEHL, K.C. and ENGLER, C.R. 1983.Thermal properties of porous foods. ASAE paper no. 83–6515, Chicago,IL.
392 C. REYES ET AL.