Journal of Food Engineering 75 (2006) 297–307
www.elsevier.com/locate/jfoodeng
Predicting the effective thermal conductivity
of unfrozen, porous foods
James K. Carson
a,*
, Simon J. Lovatt a, David J. Tanner b, Andrew C. Cleland
c
a
b
AgResearch Ltd., Private Bag 3123 Hamilton, New Zealand
Food Science Australia, P.O. Box 52, North Ryde, NSW 1670 Sydney, Australia
c
IPENZ, P.O. Box 12-241 Wellington, New Zealand
Received 20 February 2005; accepted 18 April 2005
Available online 15 June 2005
Abstract
In this study, it was shown that effective thermal conductivity models that are functions only of the componentsÕ thermal conductivities and volume fractions could not be accurate for both granular-type porous foods (‘‘external porosity’’) and foam-type
porous foods (‘‘internal porosity’’). An extra parameter is needed to make the model sufficiently flexible to allow it to be applied
to porous foods with a range of different structures. A number of effective thermal conductivity models contain the required extra
parameter, and of these, KrischerÕs model appears to have received the greatest use in the food engineering literature; however, for
isotropic materials it is recommended that a modified Maxwell model be used instead, because it assumes an isotropic physical structure, unlike KrischerÕs model, and because the numerical value of the extra parameter may be estimated based on whether the food
has internal or external porosity. A new procedure for predicting the effective thermal conductivity of non-frozen porous foods is
presented as a flowchart.
Ó 2005 Elsevier Ltd. All rights reserved.
Keywords: Effective thermal conductivity; Porous foods
1. Introduction
A large proportion of food engineering practice is
concerned in some way with thermal processing
(Fellows, 2000), and it is often vital to have accurate
thermal property data for food products for modelling
and design purposes. The literature contains a substantial number of thermal conductivity data for minimally
processed foods such as fruits, vegetables, dairy, and
meat products (ASHRAE, 2002; Rahman, 1995; Willix,
Lovatt, & Amos, 1998), but for more highly processed
food products it is often necessary to predict their ther-
*
Corresponding author. Tel.: +64 7 838 5372; fax: +64 7 838 5625.
E-mail address: james.carson@agresearch.co.nz (J.K. Carson).
0260-8774/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfoodeng.2005.04.021
mal conductivities using effective thermal conductivity
models.
There are numerous generally applicable thermal
conductivity models to be found within the food engineering and heat transfer literature, and it is not always
clear which model is appropriate for a given food product. Fig. 1 shows a plot of the effective thermal conductivity predicted by 10 different models which have all
been applied to food products (see Appendix A for
model equations). It is clear from Fig. 1 that there are
large discrepancies between the thermal conductivities
predicted by the different models, and hence a poor
choice of model could result in highly inaccurate predictions. The aim of this paper is to clarify the situation by
providing general guidelines for selecting suitable effective thermal conductivity models for different types of
un-frozen porous foods.
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J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
Nomenclature
distribution factor/empirical parameter
uncertainty in effective thermal conductivity
prediction (Eq. (11)) (W m1 K1)
empirical parameter (Eq. (7))
thermal conductivity (W m1 K1)
temperature (°C, K)
volume fraction of food component
porosity
f
e
j
k
T
v
e
Subscripts
1
component 1
2
component 2
a
ash
c
carbo
e
fat
gas
i
ice
prot
s
water
property of air
property of ash
condensed phase
property of carbohydrate
effective property
property of fat
property of gaseous phase
ith component
property of ice
property of protein
property of food solids
property of water
other than conduction, measured data may lie outside
the bounds calculated from Eqs. (1) and (2)).
1
0.8
ke ¼
1
½ð1 v2 Þ=k 1 þ v2 =k 2
k e ¼ ð1 v2 Þk 1 þ v2 k 2
ke /k1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
v2
Series
Parallel
Geometric
Maxwell (gas phase dispersed)
Maxwell (gas phase continuous)
Levy
Effective Medium Theory
Kopelman Series
Kopelman Isotropic
Hill
Fig. 1. Plots of effective thermal conductivity predicted by 10 models
that have been used in food engineering applications (k1/k2 = 20,
model equations in Appendix A).
2. Ratio of conductivities of components
The Series (Eq. (1)) and Parallel (Eq. (2)) models define the upper and lower bounds (sometimes referred to
as the Wiener bounds) for the effective thermal conductivity of any heterogeneous material for which the componentsÕ volume fractions and thermal conductivities
are known accurately, provided conduction is the only
mechanism of heat transfer involved. (Note that due
to the uncertainty usually involved in determining the
componentsÕ volume fractions and thermal conductivities and the fact that many measurements of effective
thermal conductivity include heat transfer by modes
ð1Þ
ð2Þ
Fig. 2 shows plots of the predictions of the Series and
Parallel models for four different thermal conductivity
ratios (k1/k2). The region between the Series and Parallel
models increases with increasing in k1/k2, and, since this
region contains all the possible effective thermal conductivities (provided ki and vi are accurate, and only conductions is involved), it follows that the uncertainty
involved in thermal conductivity prediction also increases as k1/k2 increases. Based on the ratio of the maximum and minimum thermal conductivity components,
the problem of thermal conductivity prediction for
foods can be divided into four classes:
I. Unfrozen, non-porous foods (kwater/ksolids 3) (cf.
Fig. 2a)
II. Frozen, non-porous foods (kice/ksolids 12) (cf.
Fig. 2b)
III. Unfrozen, porous foods (kwater/kgas 25) (cf. Fig.
2c)
IV. Frozen, porous foods (kice/kgas 100) (cf. Fig. 2d)
Fig. 3 shows experimental data (ASHRAE, 2002; Willix
et al., 1998) for selected Class I foods at 20 °C, along
with the thermal conductivities predicted by six commonly used effective thermal conductivity models:
Series, Parallel, Maxwell–Eucken (water as continuous
phase), Krischer (f = 0.35), Kopelman isotropic and
the effective medium theory (EMT). In plotting the effective thermal conductivity equations it was assumed that
the foods were binary materials made up of water and a
food solids phase. Within the experimental uncertainty
of the measured data (±4%), all of the models, with
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J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
1
Parallel
0.8
0.6
Parallel
0.6
0.4
0.2
0.4
0.2
0
0
0
0.2
0.4
a
0.6
0.8
1
v2
0
0.2
0.4
0.6
0.8
Series
1
Series
Parallel
0.8
Parallel
0.8
ke/k1
0.6
0.4
0.2
1
v2
b
1
ke/k1
Series
0.8
ke/k1
ke/k1
1
Series
0.6
0.4
0.2
0
0
0
0.2
0.4
0.6
0.8
1
v2
c
0
d
0.2
0.4
0.6
0.8
1
v2
Fig. 2. Plots of Series and Parallel models: (a) k1/k2 = 3, (b) k1/k2 = 12, (c) k1/k2 = 25, and (d) k1/k2 = 100.
1
data
egg white
Parallel
Maxwell
0.9
EMT
ke/kwater
Series
fish
0.8
Krischer f=0.35
Kopelman
beef,chicken
0.7
0.6
sausage meat
0.5
cheese
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
vs
Fig. 3. Thermal conductivity data for unfrozen, non-porous foods
with a range of water contents, along with predicted values from six
effective thermal conductivity models (kwater/ks = 3%, 4% measurement
error).
the exception of the Series model, gave reasonable predictions. Fig. 3 shows that, since kwater/ksolids is relatively
low, the choice of effective thermal conductivity model is
not critical for Class I foods.
Pham and Willix (1989) performed measurements on
the effective thermal conductivity of meat, fat and offal
between 40 °C and +30 °C, and compared the results
to the predictions of six of the effective thermal conductivity models plotted in Fig. 1. They found that for the
unfrozen materials (Class I foods) any of the six models
provided satisfactory predictions, consistent with the
previous discussion. However, for the materials in the
frozen state (Class II foods) there were significant discrepancies between the predictions of different models.
The prediction of effective thermal conductivity of frozen or partially frozen foods is complicated by the need
to account for the formation of ice, and while reliable
models for predicting the volumetric ice fraction as a
function of temperature are available (Rahman, 2001),
it is less clear how the growth of the ice crystal structure
affects heat conduction pathways, and hence which
effective conductivity model(s) are most applicable.
Effective thermal conductivity models for Class II materials are beyond the scope of this work.
Unfrozen, porous foods (Class III) which are the subject of this paper, pose a challenge equal to, if not greater than Class II foods, however; firstly because, like the
Class II foods, the structure of the food is much more
influential than for Class I foods, and secondly because
the term ÔporousÕ covers such a wide range of foods (and
hence a wide range of structures). Class IV foods combine the complications of Class II and Class III foods.
3. ‘‘Internal’’ and ‘‘external’’ porosity foods
A diverse range of food products contain significant
amounts of gas (usually air or carbon dioxide) on a
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J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
volume basis. In some instances within the literature the
word ‘‘porous’’ is used to describe materials in which
void spaces exist in the interstices between loose or compacted grains or particles. In other instances the word is
used to describe materials in which a solid, liquid, or solid/liquid matrix contains gas bubbles or pores which
may be either completely isolated or interconnected with
other bubbles (e.g., foams, sponges, honeycombs). However, due to the differences in structure, a foam and a
particulate material may not have the same effective
thermal conductivities, even if they have identical void
fractions and component thermal conductivities (Carson, Lovatt, Tanner, & Cleland, 2005). Hence problems
may arise when a model that has been shown to work
well for one type of porous material is assumed to be
applicable to another type, simply because both materials have been described as ÔporousÕ.
Carson et al. (2005) proposed that porous materials
(including foods) should be divided into ‘‘external
porosity’’ materials (i.e., grains and particulates) and
‘‘internal porosity’’ materials (i.e., foams and sponges),
because the mechanism for heat conduction in a granular material is different from that in a foam. (Note that it
is possible for a food product to contain both internal
and external porosity, e.g., a box of popcorn). Upper
and lower bounds were proposed for the effective thermal conductivity of the two types of porous materials,
based on the optimal heat conduction pathways in each
type of material. Assuming that the food may be treated
as a binary mixture of a gaseous phase and a condensed
phase (i.e., solid and/or immobilised liquid), it was proposed that the effective thermal conductivity of isotropic
external porosity materials is bounded above by the
effective medium theory (EMT) model (Eq. (3)):
1
ke ¼
ð3e 1Þk a þ ½3ð1 eÞ 1k c
4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
þ ½ð3e 1Þk a þ ð3f1 eg 1Þk c 2 þ 8k a k c
ð3Þ
and below by the Maxwell–Eucken model, with air as
the continuous phase (Eq. (4)):
ke ¼ ka
2k a þ k c 2ðk a k c Þð1 eÞ
2k a þ k c þ ðk a k c Þð1 eÞ
ð4Þ
For isotropic internal porosity materials, the upper
bound is provided by the Maxwell–Eucken model, with
air as the continuous phase (Eq. (5)):
ke ¼ ks
2k c þ k a 2ðk c k a Þe
2k c þ k a þ ðk c k a Þe
ð5Þ
and the lower bound is provided by the EMT model
(Eq. (3)). Fig. 4 shows a plot of the internal and external
porosity thermal conductivity bounds with schematic
representations of the two types of porous material.
The merit of these bounds is the significant reduction
Fig. 4. Effective thermal conductivity bounds for ‘‘external’’ and
‘‘internal’’ porosity materials (k1/k2 = 20).
in the range of possible thermal conductivity values
for the two types of porous foods, by comparison with
the range constrained by the Wiener bounds.
Fig. 5a and b shows published thermal conductivity
data for granular starch (Maroulis, Drouzas, & Saravacos, 1990), and spray dried milk powder (MacCarthy,
1985) respectively, along with the proposed thermal conductivity bounds. Consistent with the discussion above,
these data for granular-type materials lay within the
‘‘external porosity region’’, within the experimental
uncertainty. Fig. 5c and d shows the effective thermal
conductivity measurements for a porous food analogue
(guar-gel/polystyrene beads, Carson, Lovatt, Tanner,
& Cleland, 2004) and cake (Carson & Lovatt, 2003),
along with the proposed thermal conductivity bounds.
Consistent with the discussion above, these data for
foam-type materials lay within the ‘‘internal porosity
region’’, within the experimental uncertainty (see also
Figs. 7a to 8d of Carson et al., 2005).
Fig. 4 shows that the external porosity region is larger
than the internal porosity region, and this is true for any
value of k1/k2. The implication of this observation is that
there is inherently more uncertainty in the prediction of
the thermal conductivities of external porosity materials
than there is for internal porosity materials, if they have
the same k1/k2. This may be explained by an analysis of
heat conduction pathways. For internal porosity materials, the majority of the heat flow is through the condensed phase, and is not influenced greatly by the
extent of thermal contact between the pores. However,
for external porosity materials the optimal heat transfer
pathways are strongly dependent on the extent/quality
of thermal contact between particles. Since thermal
contact is affected by the shape and packing arrange-
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J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
1
1
Eq. (3)
0.8
0.8
0.6
0.6
ke/kc
ke/kc
Eq. (3)
0.4
0.2
0.4
0.2
Eq. (4)
0
0
0
a
0.2
0.4
0.8
1
b
0
1
0.8
0.8
Eq. (5)
0.4
Eq. (3)
0.2
0.2
0.4
0.6
0.8
1
Eq. (5)
0.6
ke/kc
ke/kc
0.6
1
0.6
0.4
Eq. (3)
0.2
0
0
c
Eq. (4)
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
d
Fig. 5. Thermal conductivity data for selected foods plotted with porosity bounds: (a) granular starch, (kc/ka = 9%, 10% measurement error),
(b) whole milk powder, (kc/ka = 7%, 10% measurement error), (c) guar-gel/polystyrene, (kc/ka = 19%, 3% measurement error), and (d) cake,
(kc/ka = 14%, 10% measurement error).
ment of the inclusions, there is a much greater level of
randomness involved, and hence more uncertainty in
predicting the thermal conductivity.
4. ‘‘Rigid’’ and ‘‘flexible’’ effective thermal conductivity
models
4.1. Suitability of effective thermal conductivity models
for porous materials
Fig. 5a shows that Eq. (3) would give acceptable predictions of the effective thermal conductivity of the
starch granules data; similarly Eq. (4) would be acceptable for predicting the thermal conductivity of the milk
powder (Fig. 5b), and Eq. (5) would provide reasonable
predictions for the thermal conductivity of the guar-gel/
polystyrene mixture (Fig. 5c). However, it is clear that
none of these equations could provide accurate predictions for all four of the porous foods in Fig. 5a–d; for
that matter neither would any of the models plotted in
Fig. 1. This is because these models are functions of
the componentsÕ thermal conductivities and volume
fractions alone, and therefore plots of ke = ke(ki, vi) for
these equations lie in fixed positions within the Wiener
Bounds. In order for a single equation to be able to provide accurate predictions for all the data in Fig. 5a–d, it
must contain at least one parameter in addition to ki and
vi in order to account for differences in structure between
different types of porous materials.
Carson, Lovatt, Tanner, and Cleland (2003) concluded that after the componentsÕ thermal conductivities
and volume fractions the next most influential parameter was the extent of thermal contact between neighbouring inclusions of the dispersed phase (i.e., the
extent of particle-particle contact in external porosity
materials, or the extent of interconnectivity between
neighbouring bubbles in internal porosity materials).
However, this parameter is material specific (Kunii
& Smith, 1960) and is difficult to predict (Nozad,
Carbonell, & Whitaker, 1985), and there do not appear
to be any methods within the literature that describe
how to determine its value other than by empirical
means.
4.2. ‘‘Flexible’’ models
For convenience, models that are functions of ki and
vi alone will be referred to as ‘‘rigid’’ models, while those
that contain an extra parameter will be referred to as
‘‘flexible’’ models. Several ‘‘flexible’’ effective thermal
conductivity models may be found in the food engineering literature, of which KrischerÕs model (Eq. (6))
appears to be the most widely used. Inspection of
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J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
KrischerÕs model can be fitted to any thermal conductivity data point by suitable adjustment of f, it often
performs better than the other model(s). Hamilton
(Hamilton & Crosser, 1962), Chaudhary and Bhandari
(1968), Kirkpatrick (1973), and Renaud, Briery,
Andrieu, and Laurent (1992) also derived ‘‘flexible’’
models that could lie anywhere between the Wiener
bounds by suitable selection of the extra parameter f
(see Appendix A for equations) but they do not appear
to be as widely used in the food engineering literature.
Like KrischerÕs model, the f-parameters in the Chaudhary–Bhandari, and Renaud models were simply an
empirical weighting parameter. HamiltonÕs model was
a modification of the Maxwell–Eucken model, and the
f-parameter was intended to be related to the sphericity
of the inclusions of the dispersed phase, while KirkpatrickÕs model was a variation of the EMT, and the
f-parameter was related to the number of dimensions
of the system being considered; however, there is no reason why these parameters could not have an equivalent
role to the empirical weighting factor in the Krischer,
Chaudhary–Bhandari, and Renaud models.
Fig. 6a–d shows plots of ke = ke(ki, vi, f) for a range of
values of f for each of the Krischer, Chaudhary–
Bhandari, Hamilton and Kirkpatrick models respectively. Two important observations can be made from
these plots: firstly each of these models has a distinct
characteristic shape, secondly the position of the plot
of ke relative to the Wiener bounds has a highly ÔnonlinearÕ dependence on the value of f.
KrischerÕs model reveals that it is a weighted harmonic
mean of the predictions of the Series and Parallel
models, where the weighting parameter (f) is often referred to as the ‘‘distribution factor’’.
ke ¼
1
1f
1 v2 v2
þf
þ
ð1 v2 Þk 1 þ v2 k 2
k1
k2
ð6Þ
When f is zero, KrischerÕs model is reduced to the
Parallel model, and when f is unity KrischerÕs model is
reduced to the Series model; hence by modifying f
between 0 and 1.0 the plot of ke = ke(ki, vi, f) may lie anywhere in the region bounded by the Wiener bounds (Fig.
6a), and therefore KrischerÕs model could be expected to
provide acceptable predictions of the thermal conductivity of each of the porous foods shown in Fig. 5a–d, provided a suitable value of f was chosen. A number of
studies have correlated the distribution factor as a function of temperature and composition for different foods
(e.g., Chen, Xie, & Rahman, 1998; Murakami & Okos,
1989); however, these correlations of f are dependent
on additional empirical constants, and, because f is
determined empirically, the correlations cannot be used
for food products that were not included in the dataset
that was used to derive the correlation.
It may be that one of the reasons why the Krischer
model is popular is that it is often compared against
‘‘rigid’’ models, such as the Maxwell–Eucken model, in
food engineering studies (e.g., Hamdami, Monteau, &
Le Bail, 2004; Murakami & Okos, 1989), and since
1
1
f = 0.04
f = 0.9
f=1
f=0
0.8
0.8
f = 0.78
f = 0.2
0.4
f = 0.35
0.2
0.6
f = 0.45
0.4
f = 0.365
0.2
f = 0.6
f=1
0
f = 0.25
f=0
0
0
a
k e /k 1
k e /k 1
f = 0.1
0.6
1
0.2
0.4
v2
0.6
0.8
1
0
b
1
f=2
0.2
0. 4
v2
f = ∞
f=6
f = 1.7
0.6
k e /k 1
k e /k 1
1
0.8
0.8
f = 1.3
0.4
f = 1.15
00
.2
0.6
f=4
0.4
f=3
0.2
f = 1.05
f=1
f = 2.4
f=2
0
0
c
0.8
f = 12
f=∞
0.2
0.6
0.4
v2
0.6
0.8
1
0
d
0.2
0.4
v2
0.6
0.8
1
Fig. 6. Plots of effective thermal conductivity predicted ‘‘flexible’’ models for different values of f (k1/k2 = 20, model equations in Appendix A):
(a) KrischerÕs model, (b) Chaudhary–Bhandari model, (c) HamiltonÕs model, and (d) KirkpatrickÕs model.
J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
The different characteristic shapes of the models can
be related to the different physical models on which
the effective thermal conductivity models are based:
the Krischer and Chaudhary–Bhandari models are
based on the Series and Parallel models, which have
highly anisotropic physical structures, while the Kirkpatrick and Hamilton models are based on macroscopically isotropic, random dispersions of components.
Carson (2002) showed that if f was constant over a range
of compositions, the Kirkpatrick and Hamilton models
consistently provided better predictions for isotropic
structures than the Krischer and Chaudhary–Bhandari
models. Hence it was concluded that even though these
‘‘flexible’’ models are semi-empirical, it was still important to use effective conductivity models based on isotropic physical structures for isotropic materials, and (it
was assumed) vice versa.
Carson (2002) proposed a modified version of
MaxwellÕs model (Eq. (7)) in which the position of the
plot of ke relative to the Wiener bounds had a ÔlinearÕ
dependence on the value of the weighting parameter
j (i.e., a j-value of 0.5 meant that the plot of ke lay
approximately midway between the Series and Parallel
bounds, as shown in Fig. 7).
2
2
j
j
þ
k
k
ðk c k a Þe
c
a
2
1j
1j2
ke ¼ kc 2
ð7Þ
j
þ
k
þ
ðk
k
Þe
k
c
a
c
a
2
1j
Although j in Eq. (7) is essentially an empirical parameter, intuitively it can be thought of as a measure of
the Ôquality of heat conduction pathwaysÕ within the
material; the greater the value of j, the higher the quality
of the heat conduction pathways. In other words, since
there is significant contact resistance (to heat transfer)
between neighbouring particles in granular materials,
the j-value would be lower for granular materials than
1
j = 0.4
0.8
j = 0.5
j = 0.6
0.6
ke /kc
j = 0.7
j = 0.8
0.4
j = 0.9
j = 0.9999
j = 0.3
0.2
j = 0.2
j = 0.1
j =0
0
0
0.2
0.4
0.6
0.8
Fig. 7. Plots of Eq. (7) for different values of j (k1/k2 = 20).
1
303
for materials with a continuous solid-phase matrix, all
other factors being equal. Similarly, due to the obstruction to heat flow caused by gaseous components, interconnecting bubbles within a continuous matrix would
cause the material to have a lower j-value than isolated
bubbles.
Similar to KrischerÕs models, Eq. (7) is reduced to the
Series model when j is zero, and to the Parallel model
when j is unity. The thermal conductivity bounds for
‘‘internal’’ and ‘‘external’’ porosity materials can also
be expressed in terms of the j-values of Eq. (7). Since
Eq. (7) is based on MaxwellÕs model, Eqs. (4) and (5)
can be obtained by inserting j-values from Eqs. (8) and
(9) respectively into Eq. (7):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k a =k c
j¼
ð8Þ
1 þ 2k a =k c
pffiffiffiffiffiffiffiffi
j ¼ 2=3
ð9Þ
Eq. (3) is not based on MaxwellÕs model and so an exact
fit by Eq. (7) is not possible; however, based on a leastsquares fitting exercise, j-values between 0.68 and 0.7
for Eq. (7) predicted similar values of ke to Eq. (3) over
the range of composition. Hence, based on the proposed
bounds for isotropic porous materials, a foam or a
sponge (internal porosity) would be expected to have a
j-value between 0.68 and 0.82, and a particulate material
(external porosity) with ks/ka = 10, would be expected to
have a j-value between 0.4 and 0.7. When Eq. (7) was fitted to the thermal conductivity data displayed in Fig. 5a–
d, the best-fit j-values were 0.61, 0.48, 0.82, and 0.71
respectively, all of which fell within the expected ranges.
5. Thermal conductivity prediction
The prediction of the thermal conductivity of a food
product usually involves more than simply selecting an
appropriate effective thermal conductivity model, since
the composition of the product may need to be predicted. Maroulis, Krokida, and Rahman (2002) developed a step-by-step method for estimating the thermal
conductivity of fruits and vegetables during drying,
which included procedures for calculating the componentsÕ thermal conductivities and volume fractions, with
the effective thermal conductivity model involved in the
final step. The following procedure is recommended for
selecting appropriate thermal conductivity models for
unfrozen, porous foods, assuming that the componentsÕ
thermal conductivities and volume fractions have been
determined using the method of Maroulis et al. (2002):
1. Combine kprot, kfat, kcarbo, and kash using Eq. (2) to
produce ks (the thermal conductivities of the basic
food components may be found as functions of temperature in ASHRAE, 2002).
J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
Assuming the proposed thermal conductivity bounds for
isotropic porous materials are valid, the maximum
uncertainty (e) introduced by assuming a mid-range
value would be:
e ¼ Maxbk e ðjmid-range Þ k e ðjbound Þc
ð11Þ
0.12
0.1
e (W m-1 K-1)
The use of Eq. (2) is recommended for Step 1 because it
is the simplest of all the models, but since the thermal
conductivities of protein, fat and carbohydrate are similar, any of the models listed in the Appendix A would
be suitable. The use of Eq. (3) in Step 2 is not critical;
however, Eq. (2) may over-predict kc slightly, and without knowledge of whether water forms the continuous
or dispersed phase in order to determine whether Eq.
(4) or Eq. (5) is applicable, Eq. (3) is a reasonable compromise, since it assumes a completely random arrangement of components. [Steps 1 and 2 could be combined
in one step by using Eq. (3) to determine kc from kprot,
kfat, kcarbo, kash, and kwater, since it may applied to a
multi-component mixture (see Eq. (A1), Appendix A);
however, this is not recommended because Eq. (3) is
implicit with respect to ki for mixtures with more than
two components, requiring more complex calculations
than would otherwise be necessary.]
If KrischerÕs distribution factor is not available in the
literature, the weighting parameter of the effective thermal conductivity model could be determined empirically
(Maroulis et al., 2002). However, for isotropic foods, a
reasonable estimate may be determined without the need
for experimentation by using Eq. (7) with a j-value midway between the j-values corresponding to the bounds
for thermal conductivity of internal and external porosity materials. For internal porosity materials this j-value
would be 0.75 (the average of 0.68 and 0.82), while for
external porosity materials the mid-range value would
need to be determined from Eq. (10):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2k a =k c
0.7 þ
1 þ 2k a =k c
jmid-range ¼
ð10Þ
2
Figs. 8 and 9 show plots of e for external and internal
porosity foods respectively, over a range of component
thermal conductivity ratios and porosities. The majority
of granular porous foods (external porosity foods) have
relatively low moisture contents, hence it is probable
that kc/ka < 12, and, based on the Fig. 8, the thermal
conductivity predicted using Eq. (7) with a j-value determined from Eq. (10) would have an uncertainty of less
than 0.02 W m1 K1. Fig. 9 shows that the thermal
conductivity predicted using Eq. (7) with a j-value of
0.75 would be expected to have an uncertainty of less
than 0.02 W m1 K1 other than for internal porosity
0.08
0.06
0.04
0.02
0
85
0.
65
0.
45
0.
ε
25
0.
9
05
0.
13
21
17
25
ka
k c/
5
Fig. 8. Plots of maximum uncertainty e (as defined by Eq. (11))
expected when using Eq. (7) to predict the effective thermal conductivity of isotropic, external porosity foods with a j-value calculated
from Eq. (10).
0.06
0.05
-1
2. Combine ks and kwater using Eq. (3) to produce
kc (effective thermal conductivity of condensed
phase).
3. If the distribution factor of KrischerÕs model (Eq. (6))
is available (Chen et al., 1998; Murakami & Okos,
1989) or if the food is fibrous/anisotropic, use
KrischerÕs model to calculate ke from ka and kc;
otherwise go to Step 4.
4. If the food may be considered isotropic on a macroscopic scale, use Eq. (7) to calculate ke from ka and kc:
(a) If food has internal porosity, select a j-value
such that 0.68 < j < 0.82.
(b) If food has
external porosity, select a j-value
qffiffiffiffiffiffiffiffiffiffiffiffiffi
2k a =k c
such that 1þ2k
< j < 0:7:
a =k c
e (W m-1 K )
304
0.04
0.03
0.02
0.01
0
85
0.
65
0.
45
0.
ε
25
0.
9
05
0.
5
13
17
k c/
21
25
ka
Fig. 9. Plots of maximum uncertainty e (as defined by Eq. (11))
expected when using Eq. (7) with j = 0.75 to predict the effective
thermal conductivity of isotropic, internal porosity foods.
J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
foods with kc/ka > 12 and e > 0.5. Measured thermal
conductivity data for porous foods typically range
between 0.05 and 0.5 W m1 K1, so if a ke of less than
0.1 W m1 K1 is predicted, an uncertainty of 0.02
W m1 K1 may be unacceptable, in which case j should
be determined empirically, for greater accuracy. This
may be achieved by fitting Eq. (7) to measured ke data
by suitable adjustment of j, which is a common method
for determining f when using KrischerÕs model. Fig. 10
shows the recommended procedure for selecting effective
thermal conductivity models for porous foods summarised as a flowchart.
This discussion has only considered heat transfer by
conduction; however, for temperatures above approximately 40 °C it may be necessary to account for the increased apparent thermal conductivity of air due to
radiation and evaporation, or, if there is significant fluid
flow, convection. In addition, for many cooking processes heat is generated or consumed as certain chemical
and physical processes occur (e.g., protein denaturation,
kprot (T), kcarbo (T), kfat (T), kash (T)
kwater (T)
Eq. (2)
Eq. (3)
305
gelation of starch etc.), which may require separate consideration, since the models recommended above may
not be adequate.
Effective thermal conductivity prediction can be a
reasonably involved process, as indicated by the flow
charts in Fig. 10 and Maroulis et al. (2002), and since
the process may require experimental measurements of
other physical properties, it may be simpler and more
reliable to measure the thermal conductivity directly,
provided a thermal conductivity measurement device is
available that is suitable for the food in question
(although thermal conductivity measurement itself, is
by no means a trivial task). However, if a thermal conductivity measurement device is unavailable, or is
unsuitable for the food product in consideration, then
an effective thermal conductivity model may be the only
alternative.
Another attractive feature of thermal conductivity
models is that they are capable of dealing with changes
in composition which may occur during a process, such
as the increase in porosity of bakery products during
baking. In this case, even if one thermal conductivity
measurement would be required to determine an empirical parameter (such as j or f), the use of a model could
circumvent the need for a series of thermal conductivity
measurements that would otherwise be required in order
to cover the range of compositions.
ks
6. Conclusions
kc
f-value from
literature
f-value
known?
Y
empirical
f-value
N
anisotropic
food?
Y
Eq. (6)
N
particulate
food?
ka (T)
Eq. (6)
j-value from
Eq. (10)
Y
Eq. (7)
j = 0.75
N
Eq. (7)
ke
Fig. 10. Flowchart for predicting the effective thermal conductivity of
unfrozen, porous foods once the componentsÕ thermal conductivities
and volume fractions have been determined.
The degree of difficulty involved in predicting a porous foodÕs thermal conductivity is significantly greater
than for an unfrozen non-porous food, due to the difference in thermal conductivities between the gaseous and
condensed phase components of food products. ‘‘Rigid’’
effective thermal conductivity models (i.e., those that are
functions only of ki and vi) are not versatile enough to be
used for both ‘‘external porosity’’ and ‘‘internal porosity’’ foods. An extra parameter is needed to make the
model more flexible and to allow it to be used on foods
with a range of different structures. Ideally this parameter should be related to the extent of thermal contact between neighbouring inclusions of the dispersed phase;
however, at this stage it appears to be more practical
to treat it as an empirical parameter. Of the ‘‘flexible’’type models, KrischerÕs model appears to have received
the greatest use in the food engineering literature; however, for isotropic materials it is recommended that a
modified Maxwell model (Eq. (7)) be used instead,
because it assumes an isotropic physical structure, unlike KrischerÕs model. The range of values that the
empirical j-parameter of Eq. (7) can have is limited
depending on whether the material has internal or external porosity.
306
J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
Acknowledgement
This work was funded by the Foundation for Research, Science & Technology (New Zealand) as part of
Objective 2 of Contract MRI 801.
Appendix A
‘‘Rigid’’ models: ke = ke(ki, vi).
EMT/‘‘Random Mixture’’ (Brailsford & Major, 1964;
Landauer, 1952):
X ki ke
vi
¼0
ðA1Þ
k i þ 2k e
i
For two components:
k e ¼ 1=4 ð3v2 1Þk 2 þ ½3ð1 v2 Þ 1k 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
þ ½ð3v2 1Þk 2 þ ð3f1 v2 g 1Þk 1 þ 8k 1 k 2
ðA2Þ
Geometric:
ð1v2 Þ v2
k2
ke ¼ k1
ðA3Þ
Hill (Hill, Leitman, & Sunderland, 1967):
8k 1 k 2 ðU U2 Þ
k e ¼ k 2 ð2U U2 Þ þ k 1 ð1 4U þ 3U2 Þ þ
k 1 U þ k 2 ð4 UÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
U ¼ 2 4 2v2
ðA4Þ
Kopleman Isotropic (cited in Rahman, 2001):
"
#
2=3
1 v2 ð1 k 2 =k 1 Þ
ke ¼ k1
ðA5Þ
2=3
1=3
1 v2 ð1 k 2 =k 1 Þð1 v2 Þ
Kopelman Series (cited in Rahman, 2001):
!
3
2
1=2
v2
1
7
6
1 k 1 =k 2
7
6
7
6
!
ke ¼ k16
7
1=2
5
4
v
2
12
1 ð1 v2 Þ
1 k 1 =k 2
ðA6Þ
ke ¼ k1
f¼
K¼
2
ðk 1 k 2 Þ
2
ðk 1 þ k 2 Þ2 þ k 1 k 2 =2
ke ¼ k1
ðA7Þ
2k 1 þ k 2 2ðk 1 k 2 Þv2
2k 1 þ k 2 þ ðk 1 k 2 Þv2
ðcomponent 1 continuousÞ
2k 2 þ k 1 þ 2ðk 1 k 2 Þð1 v2 Þ
ke ¼ k2
2k 2 þ k 1 ðk 1 k 2 Þð1 v2 Þ
ðA8Þ
ðcomponent 2 continuousÞ
ðA9Þ
Note: the ‘‘Maxwell’’, ‘‘Maxwell–Eucken’’, ‘‘solid
continuous’’ and ‘‘fluid-continuous’’ models may all be
represented by equations (A12) and (A13); however,
they are often listed as different models. Algebraic
manipulation of the model for heterogen mischkörper
(heterogeneous mixtures) in Eucken (1940) will produce
Eq. (A14). Mathematically the Maxwell and Maxwell–
Eucken models are identical; the only difference is that
MaxwellÕs model was applied to electrical conductivity,
whereas Eucken applied it to thermal conductivity.
The terms ‘‘solid continuous’’ and ‘‘fluid-continuous’’
appear to have been introduced first by Brailsford and
Major (1964). They simply referred to the Maxwell
model with either the solid component as the continuous
phase, or the fluid component as the continuous phase.
Eqs. (8) and (9) in Brailsford and Major (1964) may be
manipulated algebraically into Eqs. (A13) and (A14).
Hence if a study comparing the predictions of different
models includes the ‘‘fluid continuous’’ and ‘‘solid continuous’’ models, it should not list the Maxwell or
Maxwell–Eucken model(s) separately (as has happened
in some well-known studies), since there would be
redundancy.
Parallel:
k e ¼ ð1 v2 Þk 1 þ v2 k 2
ðA10Þ
Series (also known as the Perpendicular model):
ke ¼
Levy (Levy, 1981):
2k 1 þ k 2 2ðk 1 k 2 Þf
2k 1 þ k 2 þ ðk 1 k 2 Þf
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2=K 1 þ 2v ð2=K 1 þ 2vÞ 8v=K
Maxwell/Maxwell–Eucken/‘‘Solid Continuous’’/‘‘Fluid
Continuous’’ (Brailsford & Major, 1964; Eucken, 1940;
Maxwell, 1954):
1
½ð1 v2 Þ=k 1 þ v2 =k 2
ðA11Þ
‘‘Flexible’’ models: ke = ke(ki, vi, f).
Chaudhary–Bhandari (Chaudhary–Bhandari, 1968):
ð1f Þ
v2
f 1 v2
k e ¼ ½ð1 v2 Þk 1 þ v2 k 2
ðA12Þ
þ
k1
k2
Kirkpatrick (Kirkpatrick, 1973). Note modification
of EMT model:
X
ki ke
vi
¼0
ðA13Þ
k
þ
ðf
=2 1Þk e
i
i
Krischer (Krischer, 1963):
ke ¼ h
1f
ð1v2 Þk 1 þv2 k 2
1
þf
1v2
k1
þ kv22
i
ðA14Þ
J.K. Carson et al. / Journal of Food Engineering 75 (2006) 297–307
Hamilton (Hamilton & Crosser, 1962). Note: modification of MaxwellÕs model:
ke ¼ k1
ðf 1Þk 1 þ k 2 ðf 1Þðk 1 k 2 Þv2
ðf 1Þk 1 þ k 2 þ ðk 1 k 2 Þv2
ðA15Þ
Renaud (Renaud et al., 1992):
k e ¼ f ½ð1 v2 Þk 1 þ v2 k 2 þ ð1 f Þ
1 v2 v2
þ
k1
k2
Modified Maxwell (Carson, 2002):
2
2
j
j
ks þ ka
ðk s k a Þe
2
1j
1 j2
2
ke ¼ ks
j
k s þ k a þ ðk s k a Þe
1 j2
Modified EMT (Carson, 2002):
X
ki ke
¼0
vi
f
i
ki þ
ke
1f
ðA16Þ
ðA17Þ
ðA18Þ
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