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Review of effective thermal conductivity models for foods
International Journal of Refrigeration 29 (2006) 958e967 www.elsevier.com/locate/ijrefrig
Review Article
Review of effective thermal conductivity models for foods James K. Carson* Department of Materials and Process Engineering, University of Waikato, Private Bag 3105, Hamilton 2020, New Zealand Received 16 February 2006; received in revised form 9 March 2006; accepted 13 March 2006 Available online 10 July 2006
Abstract The literature associated with modelling and predicting the thermal conductivities of food products has been reviewed. The uncertainty involved in thermal conductivity prediction increases as the differences between the food components’ thermal conductivities increase, which means that there is greater uncertainty involved with predicting the thermal conductivity of foods which are porous and/or frozen, than with unfrozen, non-porous foods. For unfrozen, non-porous foods, a number of simple effective thermal conductivity models that are functions only of the components’ thermal conductivities and volume fractions may be used to provide predictions to within 10%. For frozen and/or porous foods, the prediction procedure is more complicated, and usually requires the prediction of porosity and/or ice fraction, which introduces another source of error. The effective thermal conductivity model for these foods may require an extra parameter (in addition to the components’ thermal conductivities and volume fractions) whose value must often be determined empirically. Recommendations for selecting models for different classes of foods are provided. There is scope for more research to be done in this area. Ó 2006 Elsevier Ltd and IIR. All rights reserved. Keywords: Food; Survey; Modelling; Thermal conductivity
Etude sur l’efficacite´ des mode`les de la conductivite´ thermique utilise´s pour les produits alimentaires Mots cle´s : Produit alimentaire ; Enqueˆte ; Mode´lisation ; Conductivite´ thermique
1. Introduction Historically, the design of thermal processing units for food products (e.g. refrigerators, ovens, dryers, etc.) has largely been based on experience, but in more recent times analytical approaches have been employed, in particular the implementation of design methods that have been used
* Tel.: þ64 7 838 4206; fax: þ64 7 838 4835. E-mail address: [email protected] 0140-7007/$35.00 Ó 2006 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2006.03.016
by chemical processing engineers [1]. These methods are largely dependent on mathematical models that are derived from the physical laws that govern the process (e.g. mass and energy balances, reaction kinetics, thermodynamics, etc.). The accuracy of any model of a thermal process is limited by, amongst other factors, the accuracy of data for physical properties [2], which may vary during the thermal process. One of the most influential physical properties in thermal processing is thermal conductivity. Thermal conductivity measurement is a relatively complex task, and there are many potential sources of error
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Nomenclature a C, F, G D f j k L n P T v x Z
activity intermediate variables defined within the text diffusivity of water vapour in air empirical weighting factor (sometimes referred to as the ‘‘distribution factor’’) relative weighting factor thermal conductivity of a component (W m1 K1) enthalpy of vaporisation weighting parameter pressure temperature ( C) volume fraction mass fraction weighting parameter
Greek variables 3 porosity r density (kg m3)
[24]. Measurement methods for food products have been reviewed by Rahman [4], Murakami and Okos [14] and Nesvadba [24]. The line-source method (thermal conductivity probe) appears to be the most widely used method for food products, although the guarded hot plate, Fitch apparatus, and several comparative methods have also been employed. A probable explanation for the popularity of the thermal conductivity probe is its relative simplicity coupled with relatively short measurement times; however, since the measurements are localised, it will not be suitable for some foods which have highly non-uniform distribution of phases. The food to be measured needs to be considered when a measurement device is selected since not all methods are suitable for all foods. Databases of thermal conductivities of fresh and minimally processed foods such as fruits, vegetables, grains, cereals, meat and dairy products may be found in the literature [3e8]. The data are sometimes compiled as empirical correlations of temperature and/or composition, and a computer program called COSTHERM has been developed which has condensed much of these data into a predictive thermal properties database [9]. However, it would be impossible to collate databases with measured properties of every single food product, especially since new food products are continually being developed. In the absence of measured physical property data, the best estimate of thermal conductivity may be obtained from effective thermal conductivity models. 2. Thermal conductivity models The literature contains a large number of models for predicting the thermal conductivities of composite or heterogeneous materials based on composition. Some reviews have
Subscripts 1 component 1 2 component 2 ap apparent density b bound water c condensed phase e effective thermal conductivity evap evaporationecondensation effect f initial freezing temperature gas gaseous phase i ith component ice ice prot protein s saturation w total water (ice þ unfrozen free water þ bound water)
provided lists of such models [10e18], although none of these is by any means exhaustive. Many of the models that have been proposed are highly specific to a particular material and contain material-specific parameters. Other models have more general applicability, but may still contain parameters whose values must be determined empirically. Several researchers have proposed generic models by deriving a set of equations, usually based on a conceptual ‘parent’ model that is modified to account for variations in composition and structure [19e23], although many of these still include empirical parameters. However, since new models continue to appear in the literature it seems that, to-date, no single model or prediction procedure has been found with universal applicability. As with most modelling exercises, the prediction of effective thermal conductivity usually involves a trade-off between simplicity/convenience and accuracy. Due to the inherent biological variation of food products it is highly unusual to find measured thermal conductivity data having reported uncertainties of less than 2%, with 3% to 5% being typical figures. Hence, it is unreasonable to expect the accuracy of predicted thermal conductivities to be better than 5%. For design purposes, accuracies to within 10% are usually sufficient for thermal conductivity data [24], which, depending on the food in question, can often be achieved with relatively simple thermal conductivity models. Table 1 shows the thermal conductivities of the major food components at 0 C (correlations for the temperature dependencies of these components may be found in [6] and [71]). Other than ice and air, the thermal conductivities of food components are of similar magnitude. Several studies have shown that the uncertainty involved in the prediction of thermal conductivity increases as the difference between the conductivities of the components increases [25,26]. The
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where
Table 1 Thermal conductivities of food components at 0 C [6,71] Food component
k/W m1 K1
Protein Fat Soluble carbohydrate Fibre Ash Liquid water Ice Air
0.18 0.18 0.20 0.18 0.33 0.57 2.2 0.024
G¼
Unfrozen, non-porous foods (kwater/ksolids z 3) Frozen, non-porous foods (kice/ksolids z 12) Unfrozen, porous foods (kwater/kgas z 25) Frozen, porous foods (kice/kgas z 100)
Pham and Willix [27] performed effective thermal conductivity measurements between 40 C and þ30 C for various meat products and compared the results to the predictions of six simple models (Eqs. (1)e(5), (11)). Series model: ð1Þ
i
Parallel model: X ki vi ke ¼
ð2Þ
i
Kopelman isotropic model [6,28]: # " 1 v2=3 2 ð1 ðk2 =k1 ÞÞ ke ¼ k1 1=3 1 v2=3 2 ð1 ðk2 =k1 ÞÞð1 v2 Þ
ð3Þ
MaxwelleEucken model [29,30]: ke ¼ k1
2k1 þ k2 2ðk1 k2 Þv2 2k1 þ k2 þ ðk1 k2 Þv2
ð4Þ
Levy’s model [31] ke ¼ k1
2k1 þ k2 2ðk1 k2 ÞF 2k1 þ k2 þ ðk1 k2 ÞF
ðk1 k2 Þ2 ðk1 þ k2 Þ2 þk1 k2 =2
X
vi
ki ke ¼0 ki þ 2ke
ð6Þ
which may be rewritten to be explicit in terms of ke for two components:
3. Class I: unfrozen, non-porous food products
1 vi =ki
2
They found that for the meat products in their unfrozen state, all the models other than the Series model provided sufficiently accurate predictions. Mattea et al. [32] found that the effective medium theory (EMT) equation provided predictions of sufficient accuracy for thermal conductivity data of fruits and vegetables at 20 C: EMT model: [32e34]
i
The selection of thermal conductivity models is discussed separately for each of these classes.
ke ¼ P
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=G 1 þ 2v2 Þ2 8v2 =G
and
difference between components’ thermal conductivities is therefore a useful method of classifying materials from an effective thermal conductivity perspective. Carson et al. [26] proposed the following classifications for predicting the effective thermal conductivity of food products: I II III IV
F¼
2=G 1 þ 2v2
ð5Þ
ke ¼
1 ð3v2 1Þk2 þ ½3ð1 v2 Þ 1k1 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½ð3v2 1Þk2 þ ð3f1 v2 g 1Þk1 2 þ8k1 k2
Murakami and Okos [14] recommended the use of the Parallel model for unfrozen, non-porous foods in general (other than meat products), which had been the recommendation of the COST 90 group [35,36]. Hsu and Heldman [37] compared the thermal conductivities of aqueous starch solutions at temperatures between 5 C and 45 C to the predictions of six effective thermal conductivity models including the Kopelman (Eq. (3)), Maxwell (Eq. (4)), and EMT (Eq. (6)) models, and found that while the Maxwell model provided the best predictions overall none of the prediction errors from any of the models were greater than 10%. Fig. 1 shows a plot of Eqs. (1)e(6) where component 1 is water and component 2 is a food solids phase having a thermal conductivity of 0.2 W m1 K1. Although the physical structures assumed in the derivations of each of Eqs. (1)e(6) are very different, other than the Series model the model predictions are all very similar. The inference to be drawn from this observation is that because the thermal conductivities of these food components are similar, the influence of material structure on effective thermal conductivity is minimal. Evidence supporting this conclusion is easily seen in the studies of Pham and Willix [27] and Hsu and Heldman [37]. The choice of effective thermal conductivity model for non-frozen, non-porous foods is therefore relatively straightforward since any of the models listed above, other than the Series model, may be used with sufficient accuracy for most
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For their study on the thermal conductivity of meat products, Pham and Willix [27] related the bound (i.e. unfreezable) water to the protein mass fraction [43]:
1
0.8
xb ¼ 0:4xprot
ð8Þ
ke /k1
0.6
Tchigeov (cited in Ref. [17]), recommended the following empirical ice fraction models for use with meat, fish, milk, eggs, fruits and vegetables:
0.4
xice 1:105 ¼ xw 1 þ 0:70138=ln Tf T þ 1
0 0
0.2
0.4
0.6
0.8
1
v2 Series Maxwell-Eucken Effective Medium Theory
Parallel Levy Kopelman Isotropic
Fig. 1. Plots of Eqs. (1)e(6) for k1/k2 ¼ 3 (equivalent to an unfrozen, non-porous food).
purposes (10%). It should be noted: firstly that while the Parallel model is the simplest of all these models it is most likely to over-predict (Fig. 1); secondly, the Maxwelle Eucken model requires identification of a continuous phase (component 1 in Eq. (4)) and a dispersed phase (component 2 in Eq. (4)). Unless there is clear indication to the contrary, it should be assumed that water, rather than the solids phase, forms the continuous phase of the food product. 4. Class II: frozen, non-porous food products Thermal conductivity predictions involving frozen food require knowledge of the food’s ice content (xice) which in turn requires knowledge of the initial freezing temperature (Tf). These data may be determined experimentally from measured enthalpyetemperature data, or alternatively may be predicted. The initial freezing temperature for a number of foods may be found in the literature [6,7]. For high moisture-content foods such as fresh fruits and vegetables, and meat and seafood products, the initial freezing temperature is typically between 0.5 C and 3 C. For lower moisture content foods such as cheese and egg yolks the initial freezing temperature can be much lower. Models for predicting the initial freezing temperature based on the food’s composition may be found in the literature [17,38e40]. Models for predicting ice fractions have been reviewed by Rahman [17]. One of the most widely used models (including [27,41,42]) is based on Raoult’s law and the ClausiuseClapeyron equation:
ð10Þ Fig. 2 shows Eqs. (7), (9) and (10) plotted assuming an initial freezing temperature of 1 C, a protein mass fraction of 0.2 and a total water mass fraction of 0.65. It is clear from the discrepancies between the predictions of the different models shown in Fig. 2, that the selection of an ice fraction model introduces an extra source of uncertainty. For example, the discrepancy between the predictions of Levy’s model based on ice fractions calculated firstly from Eq. (7) and secondly from Eq. (10) may be as high as 15%, depending on the ice fraction. This error occurs independently of any further error that may result from the selection of an unsuitable thermal conductivity model. Ideally, a thermal conductivity model should therefore be tested independently of an ice fraction model, since if the ice fraction model overpredicts and the conductivity model under-predicts (or vice versa), thermal conductivity predictions may appear to be accurate for a given set of data, but may only be so by coincidence. However, the most common approach when thermal conductivity models are being compared
0.7
Eq. (7) Eq. (9)
0.6
Eq. (10)
0.5 0.4 0.3 0.2 0.1 0 -15
Tf xice ¼ ðxw xb Þ 1 T
ð9Þ
expð 9:703 103 Tf þ 4:794 106 Tf2 Þ 1 xice ¼1 xw exp 9:703 103 T þ 4:794 106 T 2 1
x ice
0.2
-12
-6
-9
-3
0
T (°C)
ð7Þ
Fig. 2. Plots of three different ice fraction prediction models (Eqs. (7), (9) and (10)), with Tf ¼ 1, xprot ¼ 0.2, xb ¼ 0.08, xw ¼ 0.65.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
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appears simply to have been to base all thermal conductivity models on a single ice fraction model. The majority of the effective thermal conductivity studies for this class of food have been concerned with meat products. Hill et al. [44] performed thermal conductivity measurements on fresh and frozen meat products between 0 F and 150 F (18 C to þ66 C), and derived an equation to model the effective thermal conductivity as a function of temperature, fat and total moisture content, but not explicitly as a function of ice fraction. The model assumed that heat was conducted through the meat by three parallel pathways: through the meat fibre, through the aqueous phase, and through the meat fibre and aqueous phase in series (see Figs. 2 and 3 of Ref. [44]): Hill’s model [44]: 8k1 k2 C C2 2 2 ke ¼ k2 2C C þ k1 1 4C þ 3C þ k1 C þ k2 ð4 CÞ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C ¼ 2 4 2v2 and component 1 is the aqueous phase, component 2 is the meat fibre phase. Pham and Willix [27] found that Hill’s model was not as accurate as others such as Levy’s, and it does not appear to have received widespread use. Mascheroni et al. [45] developed a model for frozen and unfrozen meats in which the meat was modelled as a bundle of partially dehydrated meat fibres surrounded by ice. The meat fibres in turn were assumed to be a meat solids’ matrix containing unfrozen water as a dispersed phase. Above the initial freezing temperature, all the water was assumed to be contained within the fibre; below the initial freezing temperature it was assumed that ice progressively formed in the extra-fibril region at the expense of water within the fibres. Heat flows both parallel to, and perpendicular with, the fibre were
1
0.8
ke /k1
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
v2 Series Maxwell-Eucken Effective Medium Theory
Parallel Levy Kopelman Isotropic
Fig. 3. Plots of Eqs. (1)e(6) for k1/k2 ¼ 20 (equivalent to an unfrozen, porous, high water-content food).
considered. The model was relatively complex, and subsequent workers [27,42] have found that a simpler model such as Levy’s provides predictions that have comparable accuracy. Pham and Willix [27] found that Levy’s model, Eq. (5), based on ice fractions calculated from Eqs. (7) and (8) provided good predictions for both the frozen and unfrozen meat products, and was substantially better for frozen meats than the other models they considered in the study. They were, however, reluctant to place too much confidence in Levy’s model due to its ‘‘. lack of physical justification, since it was based on mathematical rather than physical arguments.’’ Wang et al. [23] have subsequently shown that Levy’s model can have a physical interpretation, which should allay any further reservations about its use on that basis. Renaud et al. [46] studied the thermal conductivity of frozen model foods between 40 C and þ20 C and compared the predictions of the Series, Parallel, MaxwelleEucken models and their own model (Eq. (12)): Renaud model: 1 v2 v2 ke ¼ f ½ð1 v2 Þk1 þ v2 k2 þ ð1 f Þ ð12Þ þ k1 k2 They found that when Eq. (12) was fitted to their experimental data, the empirically determined weighting factor, f, was strongly dependent on the specific food in question, and hence they concluded that as a predictive tool the Maxwelle Eucken model was better, even though it produced prediction errors of up to 28%. Tarnawski et al. [42] used the data of Pham and Willix to compare the predictions of several models including those that had been used previously for meats (such as the Mascheroni and Levy Models) and some models that had been used for soils, such as the models of De Vries [47] and Gori [48]. Some of the more complex models provided slightly more accurate predictions than Levy’s (based on the root mean square error); however, the differences were marginal, and some, such as De Vries’ model, contained empirically determined parameters, which is undesirable. Recently, Wang et al. [49] compared the predictions of 31 models (including the Series, Parallel, MaxwelleEucken and Levy models) to the thermal conductivity data for 22 meat and seafood products between 40 C and þ40 C [7,27], based on ice fractions calculated from Eqs. (7) and (8). Levy’s model was again found to be only marginally less accurate than some more complex models. It is difficult to give comprehensive, generic guidelines for predicting the thermal conductivity of frozen, non-porous foods. For meat products, Levy’s model based on ice fractions predicted by Eqs. (9) and (10) appears to be tried and trusted; however, for other Class II products fewer studies appear to have been performed and there remains a significant level of uncertainty, both in the selection of thermal conductivity models and in the selection of ice fraction models. There is scope for more work to be done in this area.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
5. Class III: unfrozen, porous food products Many, if not most of the effective thermal conductivity models that may be found in the literature are concerned with porous materials. The term ‘porous’ may refer to granular or particulate materials in which the void volume may be occupied by either liquid or gaseous components, or alternatively, it may refer to a material having a continuous solid matrix that contains pores/bubbles which may be isolated (sometimes referred to as ‘closed pores’) or interconnected (sometimes referred to as ‘open pores’) [50]. The measurement and prediction of porosity and pore formation is a field of study in itself, particularly for drying foods [51]. An unfrozen porous food product may be thought of as a binary mixture of a ‘condensed phase’ containing immobile water and food solids (which, for the purposes of effective thermal conductivity prediction, may be treated as a Class I food), and a gaseous phase such as air or carbon dioxide. The simplest method for determining a food’s porosity is from its apparent density and the density of the solid and gaseous components: 3¼
rc rap rap z1 rc rgas rc
ð13Þ
Fig. 3 shows a plot of Eqs. (1)e(6) for a theoretical porous, high water-content food in which the thermal conductivity of the condensed phase is 0.48 W m1 K1. By contrast with Fig. 1, the predictions of the different models vary significantly, and the effect of the physical structure that each model is based on becomes significant. Since thermal conductivity data for porous foods may lie anywhere between the predictions of the Series and Parallel models (the so-called ‘‘Wiener Bounds’’ [52]), no single model which is a function solely of the components’ thermal conductivities and volume fractions (including Eqs. (1)e(6)) will be suitable for all types of porous foods [26,53]. Due to differences in their structures, a foam and a particulate material may have different effective thermal conductivities, even if they have identical void fractions and component thermal conductivities [53]. 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 ‘porous’. Carson et al. [53] 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. Upper and lower bounds were proposed for the effective thermal conductivity of two types of isotropic porous materials: for isotropic external porosity materials it was proposed that the effective thermal conductivity is bounded above by the EMT model (Eq. (6)), and below by the MaxwelleEucken model (Eq. (4)) with air as the continuous phase; for isotropic internal porosity materials, the upper bound is provided by the
963
MaxwelleEucken model with air as the dispersed phase and the lower bound is provided by the EMT model (refer to Fig. 6 of Ref. [53]). The merit of these bounds is the significant reduction in the range of possible thermal conductivity values for the two types of porous foods, compared with the range constrained by the Wiener bounds. However, since data will most likely lie between the bounds, they are not in themselves a complete solution to the problem. The literature contains some correlations of thermal conductivity as functions of porosity and moisture content for selected fruits and vegetables [54e56]; however, the applicability of these models is clearly limited to the dataset from which the correlation was calculated. Neural network techniques have also been employed [57,58], but are considerably more complex to implement than the simple algebraic models that are available. A semi-empirical approach to the prediction of the thermal conductivity of porous materials was introduced by Krischer [59]. He reasoned that since the thermal conductivity of any two-component material must lie between the Wiener bounds, its structure could be modelled as a mixture of the Series and Parallel structures. He proposed that the effective thermal conductivity of the combined structure should be the weighted harmonic mean of the Series and Parallel conductivities (Eq. (14)): Krischer model [59]: 1 1f 1 v2 v2 þf þ ð1 v2 Þk1 þ v2 k2 k1 k2
ke ¼
ð14Þ
The value of the parameter f was determined empirically. By suitable adjustment of the f parameter between 0 and 1, Krischer’s model may predict a thermal conductivity anywhere within the Wiener Bounds. A number of studies of thermal properties of foods have used Krischer’s model, and hence values for f (often referred to as the ‘‘distribution factor’’) may be found in the literature for some foods, including granular foods between 20 C and 70 C [14,60], dried fruits and vegetables between 5 C and 100 C [61,62], and partially baked French bread between 35 C and þ25 C [63,64]. Several other models (including Eq. (12)) are similar to Krischer’s in that they may predict thermal conductivities anywhere between the Series and Parallel models by suitable adjustment of the f, Z, n and j parameters: ChaudharyeBhandari model [65]: ke ¼ ½ð1 v2 Þk1 þ v2 k2 f
ð1f Þ 1 v2 v2 þ k1 k2
ð15Þ
Kirkpatrick model (modification of Eq. (6)) [66]: X i
vi
ki ke ¼0 ki þ ðZ=2 1Þke
ð16Þ
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
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ð17Þ
As for Class II foods, it is difficult to give comprehensive, generic guidelines for model selection. If the f-value for the food in question is known, then Krischer’s model should be used, otherwise the algorithms of Maroulis et al. [62] or Carson et al. [26] should be consulted.
ð18Þ
6. Class IV: frozen, porous food products
Hamilton model (modification of Eq. (4)) [67]: ke ¼ k1
ðn 1Þk1 þ k2 ðn 1Þðk1 k2 Þv2 ðn 1Þk1 þ k2 þ ðk1 k2 Þv2
Carson’s modified Maxwell model [18,26]: 2 j = 1 j2 ks þ ka j2 = 1 j2 ðks ka Þ3 ke ¼ ks j2 = 1 j2 ks þ ka þ ðks ka Þ3 Carson’s modified EMT model [18,26]: X i
vi
ki ke ¼0 ki þ ðj=ð1 jÞÞke
ð19Þ
Of these models (Eqs. (12), (14)e(19)), Krischer’s model appears to have been employed most frequently. However, the fact that the f-value of Krischer’s model (and equivalent parameters of other models) cannot be determined mechanistically is a significant shortcoming, since its determination may require a thermal conductivity measurement e which might defeat the purpose of the prediction (although the j parameters of Carson’s models may be related to the thermal conductivity bounds for internal and external porosity materials [26]). Ideally all parameters in the model should be related to some physical property. Carson et al. [68] concluded that after the food components’ thermal conductivities and volume fractions the next most important variable was the extent (or quality) of thermal contact between particles (in the case of particulate materials) or pores (in the case of sponge or foam-like materials). However, as yet there does not appear to be any suitable method for predicting or measuring this property other than for regular arrangements of regularly shaped objects. Unfortunately, there remains an inevitable degree of empiricism involved in the thermal conductivity prediction of porous foods, particularly for those with external porosity [26], and there is scope for more work in this area. A separate issue to the influence of structure on the thermal conductivity of a porous food is the potential increase in the apparent thermal conductivity of the gaseous phase due to the evaporation and condensation of moisture across the pores. While this phenomenon is more likely to be an issue above temperatures likely to be encountered in refrigeration applications, Hamdami et al. [63] observed this behaviour even at sub-freezing temperatures. They employed a model introduced by Sakiyama et al. [70] to account for the increase in thermal conductivity: kap ¼ kgas þ kevap kevap ¼
D P dPs Law dT RT P aw Ps
ð20Þ ð21Þ
Similarly, radiation in the gaseous phase may also alter the apparent thermal conductivity of the gaseous phase [69], although it is unlikely to be significant in the range of temperatures typically encountered in refrigeration.
Class IV foods combine the difficulties associated with Class II and Class III foods, but, conveniently, the effects of ice formation and porosity may be dealt with sequentially rather than simultaneously, as demonstrated in the studies performed by Cogne´ et al. [41] with ice cream and Hamdami et al. [63] with partially baked French bread. The study by Cogne´ et al. was very thorough, complete with a micrograph of ice cream which allowed the selection of thermal conductivity models to be based on the food’s microstructure. Fig. 7 of [41] showed that ice cream is composed of a continuous aqueous phase in which are suspended discrete air bubbles, ice crystals and fat globules, and since the physical basis of the Maxwell-type models is a structure comprised of distinct continuous and dispersed phases, this family of models was an obvious choice for this application. Cogne´ et al. calculated the ice cream’s thermal conductivity in three steps: firstly the Parallel model was used to combine the thermal conductivities of the liquid water and food solids’ components (consistent with the discussion on Class I foods); secondly, the resulting thermal conductivity from the first step was combined with the thermal conductivity of ice using a modification of Maxwell’s model (De Vries model) to produce the thermal conductivity of the condensed phase; thirdly, the thermal conductivity of the condensed phase was combined with the thermal conductivity of air using the MaxwelleEucken model (Eq. (4)) to give the overall thermal conductivity. Hence, in the terms used in this review, the Class IV food (ice cream) was treated successively as a Class I food, a Class II food and a Class III food, as outlined diagrammatically in Fig. 9 of [41]. Using this technique they were able to predict the thermal conductivity of ice cream to within 8% mean relative uncertainty. Hamdami et al. [63] used successive application of the MaxwelleEucken model, as outlined schematically in their Fig. 2b. They also considered another approach in which they combined the thermal conductivities of all the components in one step using Krischer’s model, which, unlike the MaxwelleEucken model, is capable of handling multicomponent materials. The predictions of both approaches were compared to experimental data, and it was found that Krischer’s model could provide more accurate predictions; however, it was acknowledged that this was because the f-value for Krischer’s model was determined from their own experimental data, and hence the model had essentially been fitted to the data, whereas the MaxwelleEucken model had provided genuine predictions.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
such as ice cream, it may be apparent from a visual inspection of the food’s structure not only which effective thermal conductivity models are appropriate, but also the order in which the components should be combined. However, there may be many situations where very little may be inferred from examining the food’s structure. The literature does not appear to contain any definitive guidelines on this issue, but, consistent with previous discussion, unless there is clear indication from the material’s structure that a different order should be used, as a general rule components with the most similar thermal conductivities should be combined first, followed by the component with the next most similar thermal conductivities and so on.
Start
Class I foods unfrozen, non-porous
Sufficient to treat food as binary mixture of water and solids. Use any of Eq. (2) to (6) to determine ke.
Class II foods frozen, non-porous
Select ice fraction model from Eqs. (7) to (10), and ascertain Tf from predictive model (e.g. [39]).
Use Levy's model (Eq. 5) to predict ke.
Class III foods unfrozen, porous
Treat food as binary mixture of condensed phase and air. Determine ke of condensed phase as for Class I foods.
7. Conclusion The uncertainty involved in thermal conductivity prediction increases as the differences between the food components’ thermal conductivities increase, which means that foods which are frozen and/or porous require greater consideration than unfrozen, non-porous foods for which thermal conductivity prediction is a relatively straightforward exercise. Recommendations for selecting models for different classes of foods are summarised in Fig. 4, although it should be noted that they are only basic guidelines. Greater understanding and characterisation of the effects of ice and/or porosity on food structures would help to improve the prediction accuracy for frozen and/or porous foods.
Estimate porosity. Use Eq. (13); alternatively refer to [62]
If f-value available in literature, use Krischer's model (Eq. 14) to determine ke; alternatively refer to [26] for model selection guidelines
Class IV foods frozen, porous
965
Deal with ice content (as for Class II food) and porosity (as for Class III food) sequentially. Follow example of [41].
Acknowledgments
Finish
The author would like to acknowledge the assistance of Dr. Milan Housˇka of the Food Research Institute Prague and Dr. Paul Nesvadba of Rubislaw Consulting Ltd in compiling this review. The majority of the work was performed while the author was at AgResearch Ltd, New Zealand.
Fig. 4. Summary of recommendations for selecting models for different classes of foods.
References Models such as the Series, Parallel and EMT models and their derivatives (e.g. Eqs. (8), (14)e(16) and (19)) are capable of handling multi-component materials and hence the thermal conductivity prediction may be achieved in one step. However, in these models all components are assumed to have identical structures and spatial distributions, and so it is questionable whether this approach is as sound as the sequential approach which allows for different structural models to be applied to the different phases of the mixture. It should be noted that when thermal conductivity is predicted by sequential application of one or more models, the order in which the food’s components are included is significant; i.e. for a food comprised of components A, B and C, the result of combining A and B first to give AB and then combining AB with C, will not necessarily be the same as the result of combining A and C first, etc. For some foods,
[1] S. Thorne, Mathematical Modelling of Food Processing Operations, Elsevier Applied Science, London, 1992. [2] Proceedings of a Conference e Workshop e Practical Instruction Course on Modelling of Thermal Properties and Behaviour of Foods during Production, Storage and Distribution, Food Research Institute Prague, Prague (Czech Republic), 1997. [3] M. Housˇka, et al., Thermophysical and Rheological Properties of Foods: Milk, Milk Products and Semiproducts, Food Research Institute Prague, Prague, 1994. [4] M.S. Rahman, Food Properties Handbook, CRC Press, Boca Raton, 1995 (Chapter 5). [5] M. Housˇka, et al., Thermophysical and Rheological Properties of Foods: Meat, Meat Products and Semiproducts, Food Research Institute Prague, Prague, 1997. [6] ASHRAE Handbook of Refrigeration, American Society of Heating, Refrigeration and Air-conditioning Engineers, Atlanta, GA, 2002.
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[7] J. Willix, S.J. Lovatt, N.D. Amos, Additional thermal conductivity values of foods measured by a guarded hot plate, J. Food Eng. 37 (2) (1998) 159e174. [8] Z. Mayer, M. Housˇka, M. Ciga´nı´k, Luong Thi Cam Tu, J. Sˇesta´k, P. Nesvadba, Thermophysical Properties of Food: Selected Fruits and Vegetables, Food Research Institute Prague, Prague, 2004. [9] C.A. Miles, M.J. Morley, Estimation of the thermal properties of foods: a revision of some of the equations used in COSTHERM, in: Proceedings of a Conference e Workshop e Practical Instruction Course on Modelling of Thermal Properties and Behaviour of Foods during Production, Storage and Distribution, Prague (Czech Republic), 1997, pp. 135e146. [10] P. Harriott, Thermal conductivity of catalyst pellets and other porous particles e 1. Review of models and published results, Chem. Eng. J. 10 (1) (1975) 65. [11] R.C. Progelhof, J.L. Throne, R.R. Reutsch, Methods for predicting the thermal conductivity of composite systems: a review, Polym. Eng. Sci. 16 (1976) 615e625. [12] O.T. Farouki, Thermal properties of soils, in: Series on Rock and Soil Mechanics, vol. II, Trans Tech Publications, 1986. [13] E. Tsotsas, H. Martin, Thermal conductivity of packed beds: a review, Chem. Eng. Process. 22 (1987) 19e37. [14] E.G. Murakami, M.R. Okos, Measurement and prediction of thermal properties of foods, in: R.P. Singh, A.G. Medina (Eds.), Food Properties and Computer Aided Engineering of food Processing Systems, Kluwer Academic Publishers, New York, 1989, pp. 3e48. [15] A.G. Leach, Thermal conductivity of foams. I. Models for heat conduction, J. Phys. D: Appl. Phys. 26 (5) (1993) 733e739. [16] P. Cheng, C.-T. Hsu, Heat conduction, in: D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Pergamon Press, 1998, pp. 57e76. [17] M.S. Rahman, Thermophysical properties of foods, in: D.-W. Sun (Ed.), Advances in Food Refrigeration, Leatherhead Publishing, Surrey, England, 2001. [18] J.K. Carson, Prediction of the thermal conductivity of porous foods, PhD thesis, Massey University, New Zealand, 2002. [19] D.A.G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen: I. Dielektrizita¨tskonstanten und Leitfa¨higkeiten der Mischko¨rper aus isotropen Substanzen, Ann. Phys. 24 (1935) 636e664. [20] E. Behrens, Thermal conductivities of composite materials, J. Compos. Mater. 2 (1968) 2e17. [21] T.H. Bauer, A general analytical approach toward the thermal conductivity of porous media, Int. J. Heat Mass Transfer 36 (1993) 4181e4191. [22] E.E. Gonzo, Estimating correlations for the effective thermal conductivity of granular materials, Chem. Eng. J. 90 (2002) 299e302. [23] J.F. Wang, J.K. Carson, M.F. North, D.J. Cleland, A new approach to the modelling of the effective thermal conductivity of heterogeneous materials, Int. J. Heat Mass Transfer 49 (17e18) (2006) 3075e3083. [24] P. Nesvadba, Methods for the measurement of thermal conductivity and diffusivity of foodstuffs, J. Food Eng. 1 (2) (1982) 93e113. [25] S.C. Cheng, R.I. Vachon, Prediction of thermal conductivity of two and three phase solid heterogeneous mixtures, Int. J. Heat Mass Transfer 12 (1969) 249e264.
[26] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, Predicting the effective thermal conductivity of unfrozen, porous foods, J. Food Eng. 75 (3) (2006) 297e307. [27] Q.T. Pham, J. Willix, Thermal conductivity of fresh lamb meat, offals and fat in the range 40 to þ30 C: measurements and correlations, J. Food Sci. 54 (3) (1989) 508e515. [28] I.J. Kopelman, Transient heat transfer and thermal properties in food systems, PhD thesis, Michigan State University, USA. [29] J.C. Maxwell, A Treatise on Electricity and Magnetism, third ed. Dover Publications Inc., New York, 1954 (reprinted). [30] A. Eucken, Allgemeine Gesetzma¨ßigkeiten fu¨r das Wa¨rmeleitvermo¨gen verschiedener Stoffarten und Aggregatzusta¨nde, Forschung Gabiete Ingenieur 11 (1) (1940) 6e20. [31] F.L. Levy, A modified MaxwelleEucken equation for calculating the thermal conductivity of two-component solutions or mixtures, Int. J. Refrigeration 4 (4) (1981) 223e225. [32] M. Mattea, M.J. Urbicain, E. Rotstein, Prediction of thermal conductivity of vegetable foods by the effective medium theory, J. Food Sci. 51 (1) (1986) 113e115 (134). [33] R. Landauer, The electrical resistance of binary metallic mixtures, J. Appl. Phys. 23 (1952) 779e784. [34] H.T. Davis, L.R. Valencourt, C.E. Johnson, Transport processes in composite media, J. Am. Ceram. Soc. 58 (1975) 446e452. [35] C.A. Miles, G. van Beek, C.H. Veerkamp, Calculation of thermophysical properties of foods, in: R. Jowitt, F. Escher, B. Hallstro¨m, H.F.T. Meffert, W.E.L. Spiess, G. Vos (Eds.), Physical Properties of Foods, Applied Science Publishers, London, 1983. [36] H.F.T. Meffert, History, aims, results and future of thermophysical properties within COST 90, in: R. Jowitt, F. Escher, B. Hallstro¨m, H.F.T. Meffert, W.E.L. Spiess, G. Vos (Eds.), Physical Properties of Foods, Applied Science Publishers, London, 1983. [37] C.-L. Hsu, D.R. Heldman, Prediction of the thermal conductivity of aqueous starch, Int. J. Food Sci. Tech. 39 (7) (2004) 737e743. [38] E.G. Murakami, M.R. Okos, Calculation of initial freezing point, effective molecular weight and unfreezable water of food materials from composition and thermal conductivity data, J. Food Proc. Eng. 19 (3) (1996) 301e320. [39] C.A. Miles, Z. Mayer, M.J. Morley, M. Housˇka, Estimating the initial freezing point of foods from composition data, Int. J. Food Sci. Tech. 32 (5) (1997) 389e400. [40] R.G.M. van der Sman, E. Boer, Predicting the initial freezing point and water activity of meat products from composition data, J. Food Eng. 66 (4) (2005) 469e475. [41] C. Cogne´, J. Andrieu, P. Laurent, A. Besson, J. Nocquet, Experimental data and modelling of thermal properties of ice creams, J. Food Eng. 58 (4) (2003) 331e341. [42] V.R. Tarnawski, D.J. Cleland, S. Corasaniti, F. Gori, R.H. Mascheroni, Extension of soil thermal conductivity models to frozen meats with low and high fat content, Int. J. Refrigeration 28 (6) (2005) 840e850. [43] Q.T. Pham, Calculation of bound water in food, J. Food Sci. 54 (3) (1989) 508e515. [44] J.E. Hill, J.D. Leitman, J.E. Sunderland, Thermal conductivity of various meats, Food Tech. 21 (1967) 1143e1148. [45] R.H. Mascheroni, J. Ottino, A. Calvelo, A model for the thermal conductivity of frozen meat, Meat Sci. 1 (1977) 235e243. [46] T. Renaud, P. Briery, J. Andrieu, M. Laurent, Thermal properties of model foods in the frozen state, J. Food Eng. 15 (2) (1992) 83e97.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967 [47] D.A. DeVries, Thermal properties of soils, in: W.R. van Wijk (Ed.), Physics of Plant Environment, Wiley, New York, 1963, pp. 210e235. [48] F. Gori, A theoretical model for predicting the effective thermal conductivity of unsaturated frozen soils, in: Proceedings of the Fourth International Conference on Permafrost, Fairbanks (Alaska), National Academy Press, Washington, DC, 1983, pp. 363e368. [49] J.F. Wang, J.K. Carson, D.J. Cleland, M.F. North, A comparison of effective thermal conductivity models for frozen foods, in: Proceedings: Innovative Equipment and Systems for Comfort and Food Preservation, Auckland, 2006, pp. 336e343. [50] M.S. Rahman, Mechanism of pore formation in foods during drying: present status, in: J. Welti-Chanes, G.V. BarbosaCanovas, J.M. Aguilera (Eds.), Proceedings of the Eighth International Congress on Engineering and Food, Technomic Publishing Co., Lancaster, PA, 2001, pp. 1111e1116. [51] M.A. Hussain, M.S. Rahman, C.W. Ng, Prediction of pores formation (porosity) in foods during drying: generic models by the use of hybrid neural network, J. Food Eng. 51 (3) (2002) 239e248. [52] O. Wiener, Die theorie des Mischkorpers fu¨r das Feld der Statona¨ren Stromu¨ng I. Die Mittelwertsatze fu¨r Kraft, Polarisation und Energie, Der Abhandlungen der MathematischPhysischen Klasse der Ko¨nigl, Sachsischen Gesellschaft der Wissenschaften 32 (1912) 509e604. [53] J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, Thermal conductivity bounds for isotropic, porous materials, Int. J. Heat Mass Transfer 48 (11) (2005) 2150e2158. [54] M.S. Rahman, Thermal conductivity of four food materials as a single function of porosity and water content, J. Food Eng. 15 (1992) 261e268. [55] M.S. Rahman, X.D. Chen, A general form of thermal conductivity equation as applied to an apple: effects of moisture, temperature and porosity, Drying Technol. 13 (1995) 1e18. [56] M.S. Rahman, X.D. Chen, C.O. Perera, An improved thermal conductivity prediction model for fruits and vegetables as a function of temperature, water content and porosity, J. Food Eng. 31 (2) (1997) 163e170. [57] S.S. Sablani, O.-D. Baik, M. Marcotte, Neural networks for predicting thermal conductivity of bakery products, J. Food Eng. 52 (3) (2002) 299e304. [58] S.S. Sablani, M.S. Rahman, Using neural networks to predict thermal conductivity of food as a function of moisture
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content, temperature and apparent porosity, Food Res. Int. 36 (6) (2003) 617e623. O. Krischer, Die wissenschaftlichen Grundlagen der Trocknungstechnik. (The Scientific Fundamentals of Drying Technology), Springer-Verlag, Berlin, 1963 e cited in English in: R.B. Keey, Drying of Loose and Particulate Materials, Hemisphere Publishing Corporation, New York, 1992 (Chapter 7). M.K. Krokida, Z.B. Maroulis, M.S. Rahman, A structural generic model to predict the effective thermal conductivity of granular materials, Drying Tech. 19 (9) (2001) 2277e2290. X.D. Chen, G.Z. Xie, M.S. Rahman, Application of the distribution factor concept in correlating thermal conductivity data for fruits and vegetables, Int. J. Food Prop. 1 (1) (1998) 35e44. Z.B. Maroulis, M.K. Krokida, M.S. Rahman, A structural generic model to predict the effective thermal conductivity of fruits and vegetables during drying, J. Food Eng. 51 (1) (2002) 47e52. N. Hamdami, J.-Y. Monteau, A. Le Bail, Effective thermal conductivity of a high porosity model food at above and sub-freezing, Int. J. Refrigeration 26 (7) (2003) 809e816. N. Hamdami, J.-Y. Monteau, A. Le Bail, Thermophysical properties evolution of French partly baked bread during freezing, Food Res. Int. 37 (7) (2004) 703e713. D.R. Chaudhary, R.C. Bhandari, Heat transfer through a three-phase porous medium, J. Phys. D (Brit. J. Appl. Phys.) 1 (1968) 815e817. S. Kirkpatrick, Percolation and conduction, Rev. Mod. Phys. 45 (1973) 574e588. R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous two-component systems, Ind. Eng. Chem. Fundam. 1 (3) (1962) 187e191. J.K. Carson, S.J. Lovatt, D.J. Tanner, A.C. Cleland, An analysis of the influence of material structure on the effective thermal conductivity of porous materials using finite element simulations, Int. J. Refrigeration 26 (8) (2003) 873e880. A.J. Loeb, Thermal conductivity: VIII, a theory of thermal conductivity of porous materials, J. Am. Ceramic Soc. 37 (2) (1954) 96e99. T. Sakiyama, M. Akutsu, O. Miyawaki, T. Yano, Effective thermal diffusivity of food gels impregnated with air bubbles, J. Food Eng. 39 (3) (1999) 323e328. J.P. Holman, Heat Transfer, McGraw-Hill Book Company, Singapore, 1992.
Review Article
Review of effective thermal conductivity models for foods James K. Carson* Department of Materials and Process Engineering, University of Waikato, Private Bag 3105, Hamilton 2020, New Zealand Received 16 February 2006; received in revised form 9 March 2006; accepted 13 March 2006 Available online 10 July 2006
Abstract The literature associated with modelling and predicting the thermal conductivities of food products has been reviewed. The uncertainty involved in thermal conductivity prediction increases as the differences between the food components’ thermal conductivities increase, which means that there is greater uncertainty involved with predicting the thermal conductivity of foods which are porous and/or frozen, than with unfrozen, non-porous foods. For unfrozen, non-porous foods, a number of simple effective thermal conductivity models that are functions only of the components’ thermal conductivities and volume fractions may be used to provide predictions to within 10%. For frozen and/or porous foods, the prediction procedure is more complicated, and usually requires the prediction of porosity and/or ice fraction, which introduces another source of error. The effective thermal conductivity model for these foods may require an extra parameter (in addition to the components’ thermal conductivities and volume fractions) whose value must often be determined empirically. Recommendations for selecting models for different classes of foods are provided. There is scope for more research to be done in this area. Ó 2006 Elsevier Ltd and IIR. All rights reserved. Keywords: Food; Survey; Modelling; Thermal conductivity
Etude sur l’efficacite´ des mode`les de la conductivite´ thermique utilise´s pour les produits alimentaires Mots cle´s : Produit alimentaire ; Enqueˆte ; Mode´lisation ; Conductivite´ thermique
1. Introduction Historically, the design of thermal processing units for food products (e.g. refrigerators, ovens, dryers, etc.) has largely been based on experience, but in more recent times analytical approaches have been employed, in particular the implementation of design methods that have been used
* Tel.: þ64 7 838 4206; fax: þ64 7 838 4835. E-mail address: [email protected] 0140-7007/$35.00 Ó 2006 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2006.03.016
by chemical processing engineers [1]. These methods are largely dependent on mathematical models that are derived from the physical laws that govern the process (e.g. mass and energy balances, reaction kinetics, thermodynamics, etc.). The accuracy of any model of a thermal process is limited by, amongst other factors, the accuracy of data for physical properties [2], which may vary during the thermal process. One of the most influential physical properties in thermal processing is thermal conductivity. Thermal conductivity measurement is a relatively complex task, and there are many potential sources of error
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Nomenclature a C, F, G D f j k L n P T v x Z
activity intermediate variables defined within the text diffusivity of water vapour in air empirical weighting factor (sometimes referred to as the ‘‘distribution factor’’) relative weighting factor thermal conductivity of a component (W m1 K1) enthalpy of vaporisation weighting parameter pressure temperature ( C) volume fraction mass fraction weighting parameter
Greek variables 3 porosity r density (kg m3)
[24]. Measurement methods for food products have been reviewed by Rahman [4], Murakami and Okos [14] and Nesvadba [24]. The line-source method (thermal conductivity probe) appears to be the most widely used method for food products, although the guarded hot plate, Fitch apparatus, and several comparative methods have also been employed. A probable explanation for the popularity of the thermal conductivity probe is its relative simplicity coupled with relatively short measurement times; however, since the measurements are localised, it will not be suitable for some foods which have highly non-uniform distribution of phases. The food to be measured needs to be considered when a measurement device is selected since not all methods are suitable for all foods. Databases of thermal conductivities of fresh and minimally processed foods such as fruits, vegetables, grains, cereals, meat and dairy products may be found in the literature [3e8]. The data are sometimes compiled as empirical correlations of temperature and/or composition, and a computer program called COSTHERM has been developed which has condensed much of these data into a predictive thermal properties database [9]. However, it would be impossible to collate databases with measured properties of every single food product, especially since new food products are continually being developed. In the absence of measured physical property data, the best estimate of thermal conductivity may be obtained from effective thermal conductivity models. 2. Thermal conductivity models The literature contains a large number of models for predicting the thermal conductivities of composite or heterogeneous materials based on composition. Some reviews have
Subscripts 1 component 1 2 component 2 ap apparent density b bound water c condensed phase e effective thermal conductivity evap evaporationecondensation effect f initial freezing temperature gas gaseous phase i ith component ice ice prot protein s saturation w total water (ice þ unfrozen free water þ bound water)
provided lists of such models [10e18], although none of these is by any means exhaustive. Many of the models that have been proposed are highly specific to a particular material and contain material-specific parameters. Other models have more general applicability, but may still contain parameters whose values must be determined empirically. Several researchers have proposed generic models by deriving a set of equations, usually based on a conceptual ‘parent’ model that is modified to account for variations in composition and structure [19e23], although many of these still include empirical parameters. However, since new models continue to appear in the literature it seems that, to-date, no single model or prediction procedure has been found with universal applicability. As with most modelling exercises, the prediction of effective thermal conductivity usually involves a trade-off between simplicity/convenience and accuracy. Due to the inherent biological variation of food products it is highly unusual to find measured thermal conductivity data having reported uncertainties of less than 2%, with 3% to 5% being typical figures. Hence, it is unreasonable to expect the accuracy of predicted thermal conductivities to be better than 5%. For design purposes, accuracies to within 10% are usually sufficient for thermal conductivity data [24], which, depending on the food in question, can often be achieved with relatively simple thermal conductivity models. Table 1 shows the thermal conductivities of the major food components at 0 C (correlations for the temperature dependencies of these components may be found in [6] and [71]). Other than ice and air, the thermal conductivities of food components are of similar magnitude. Several studies have shown that the uncertainty involved in the prediction of thermal conductivity increases as the difference between the conductivities of the components increases [25,26]. The
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where
Table 1 Thermal conductivities of food components at 0 C [6,71] Food component
k/W m1 K1
Protein Fat Soluble carbohydrate Fibre Ash Liquid water Ice Air
0.18 0.18 0.20 0.18 0.33 0.57 2.2 0.024
G¼
Unfrozen, non-porous foods (kwater/ksolids z 3) Frozen, non-porous foods (kice/ksolids z 12) Unfrozen, porous foods (kwater/kgas z 25) Frozen, porous foods (kice/kgas z 100)
Pham and Willix [27] performed effective thermal conductivity measurements between 40 C and þ30 C for various meat products and compared the results to the predictions of six simple models (Eqs. (1)e(5), (11)). Series model: ð1Þ
i
Parallel model: X ki vi ke ¼
ð2Þ
i
Kopelman isotropic model [6,28]: # " 1 v2=3 2 ð1 ðk2 =k1 ÞÞ ke ¼ k1 1=3 1 v2=3 2 ð1 ðk2 =k1 ÞÞð1 v2 Þ
ð3Þ
MaxwelleEucken model [29,30]: ke ¼ k1
2k1 þ k2 2ðk1 k2 Þv2 2k1 þ k2 þ ðk1 k2 Þv2
ð4Þ
Levy’s model [31] ke ¼ k1
2k1 þ k2 2ðk1 k2 ÞF 2k1 þ k2 þ ðk1 k2 ÞF
ðk1 k2 Þ2 ðk1 þ k2 Þ2 þk1 k2 =2
X
vi
ki ke ¼0 ki þ 2ke
ð6Þ
which may be rewritten to be explicit in terms of ke for two components:
3. Class I: unfrozen, non-porous food products
1 vi =ki
2
They found that for the meat products in their unfrozen state, all the models other than the Series model provided sufficiently accurate predictions. Mattea et al. [32] found that the effective medium theory (EMT) equation provided predictions of sufficient accuracy for thermal conductivity data of fruits and vegetables at 20 C: EMT model: [32e34]
i
The selection of thermal conductivity models is discussed separately for each of these classes.
ke ¼ P
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2=G 1 þ 2v2 Þ2 8v2 =G
and
difference between components’ thermal conductivities is therefore a useful method of classifying materials from an effective thermal conductivity perspective. Carson et al. [26] proposed the following classifications for predicting the effective thermal conductivity of food products: I II III IV
F¼
2=G 1 þ 2v2
ð5Þ
ke ¼
1 ð3v2 1Þk2 þ ½3ð1 v2 Þ 1k1 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½ð3v2 1Þk2 þ ð3f1 v2 g 1Þk1 2 þ8k1 k2
Murakami and Okos [14] recommended the use of the Parallel model for unfrozen, non-porous foods in general (other than meat products), which had been the recommendation of the COST 90 group [35,36]. Hsu and Heldman [37] compared the thermal conductivities of aqueous starch solutions at temperatures between 5 C and 45 C to the predictions of six effective thermal conductivity models including the Kopelman (Eq. (3)), Maxwell (Eq. (4)), and EMT (Eq. (6)) models, and found that while the Maxwell model provided the best predictions overall none of the prediction errors from any of the models were greater than 10%. Fig. 1 shows a plot of Eqs. (1)e(6) where component 1 is water and component 2 is a food solids phase having a thermal conductivity of 0.2 W m1 K1. Although the physical structures assumed in the derivations of each of Eqs. (1)e(6) are very different, other than the Series model the model predictions are all very similar. The inference to be drawn from this observation is that because the thermal conductivities of these food components are similar, the influence of material structure on effective thermal conductivity is minimal. Evidence supporting this conclusion is easily seen in the studies of Pham and Willix [27] and Hsu and Heldman [37]. The choice of effective thermal conductivity model for non-frozen, non-porous foods is therefore relatively straightforward since any of the models listed above, other than the Series model, may be used with sufficient accuracy for most
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For their study on the thermal conductivity of meat products, Pham and Willix [27] related the bound (i.e. unfreezable) water to the protein mass fraction [43]:
1
0.8
xb ¼ 0:4xprot
ð8Þ
ke /k1
0.6
Tchigeov (cited in Ref. [17]), recommended the following empirical ice fraction models for use with meat, fish, milk, eggs, fruits and vegetables:
0.4
xice 1:105 ¼ xw 1 þ 0:70138=ln Tf T þ 1
0 0
0.2
0.4
0.6
0.8
1
v2 Series Maxwell-Eucken Effective Medium Theory
Parallel Levy Kopelman Isotropic
Fig. 1. Plots of Eqs. (1)e(6) for k1/k2 ¼ 3 (equivalent to an unfrozen, non-porous food).
purposes (10%). It should be noted: firstly that while the Parallel model is the simplest of all these models it is most likely to over-predict (Fig. 1); secondly, the Maxwelle Eucken model requires identification of a continuous phase (component 1 in Eq. (4)) and a dispersed phase (component 2 in Eq. (4)). Unless there is clear indication to the contrary, it should be assumed that water, rather than the solids phase, forms the continuous phase of the food product. 4. Class II: frozen, non-porous food products Thermal conductivity predictions involving frozen food require knowledge of the food’s ice content (xice) which in turn requires knowledge of the initial freezing temperature (Tf). These data may be determined experimentally from measured enthalpyetemperature data, or alternatively may be predicted. The initial freezing temperature for a number of foods may be found in the literature [6,7]. For high moisture-content foods such as fresh fruits and vegetables, and meat and seafood products, the initial freezing temperature is typically between 0.5 C and 3 C. For lower moisture content foods such as cheese and egg yolks the initial freezing temperature can be much lower. Models for predicting the initial freezing temperature based on the food’s composition may be found in the literature [17,38e40]. Models for predicting ice fractions have been reviewed by Rahman [17]. One of the most widely used models (including [27,41,42]) is based on Raoult’s law and the ClausiuseClapeyron equation:
ð10Þ Fig. 2 shows Eqs. (7), (9) and (10) plotted assuming an initial freezing temperature of 1 C, a protein mass fraction of 0.2 and a total water mass fraction of 0.65. It is clear from the discrepancies between the predictions of the different models shown in Fig. 2, that the selection of an ice fraction model introduces an extra source of uncertainty. For example, the discrepancy between the predictions of Levy’s model based on ice fractions calculated firstly from Eq. (7) and secondly from Eq. (10) may be as high as 15%, depending on the ice fraction. This error occurs independently of any further error that may result from the selection of an unsuitable thermal conductivity model. Ideally, a thermal conductivity model should therefore be tested independently of an ice fraction model, since if the ice fraction model overpredicts and the conductivity model under-predicts (or vice versa), thermal conductivity predictions may appear to be accurate for a given set of data, but may only be so by coincidence. However, the most common approach when thermal conductivity models are being compared
0.7
Eq. (7) Eq. (9)
0.6
Eq. (10)
0.5 0.4 0.3 0.2 0.1 0 -15
Tf xice ¼ ðxw xb Þ 1 T
ð9Þ
expð 9:703 103 Tf þ 4:794 106 Tf2 Þ 1 xice ¼1 xw exp 9:703 103 T þ 4:794 106 T 2 1
x ice
0.2
-12
-6
-9
-3
0
T (°C)
ð7Þ
Fig. 2. Plots of three different ice fraction prediction models (Eqs. (7), (9) and (10)), with Tf ¼ 1, xprot ¼ 0.2, xb ¼ 0.08, xw ¼ 0.65.
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appears simply to have been to base all thermal conductivity models on a single ice fraction model. The majority of the effective thermal conductivity studies for this class of food have been concerned with meat products. Hill et al. [44] performed thermal conductivity measurements on fresh and frozen meat products between 0 F and 150 F (18 C to þ66 C), and derived an equation to model the effective thermal conductivity as a function of temperature, fat and total moisture content, but not explicitly as a function of ice fraction. The model assumed that heat was conducted through the meat by three parallel pathways: through the meat fibre, through the aqueous phase, and through the meat fibre and aqueous phase in series (see Figs. 2 and 3 of Ref. [44]): Hill’s model [44]: 8k1 k2 C C2 2 2 ke ¼ k2 2C C þ k1 1 4C þ 3C þ k1 C þ k2 ð4 CÞ ð11Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where C ¼ 2 4 2v2 and component 1 is the aqueous phase, component 2 is the meat fibre phase. Pham and Willix [27] found that Hill’s model was not as accurate as others such as Levy’s, and it does not appear to have received widespread use. Mascheroni et al. [45] developed a model for frozen and unfrozen meats in which the meat was modelled as a bundle of partially dehydrated meat fibres surrounded by ice. The meat fibres in turn were assumed to be a meat solids’ matrix containing unfrozen water as a dispersed phase. Above the initial freezing temperature, all the water was assumed to be contained within the fibre; below the initial freezing temperature it was assumed that ice progressively formed in the extra-fibril region at the expense of water within the fibres. Heat flows both parallel to, and perpendicular with, the fibre were
1
0.8
ke /k1
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
v2 Series Maxwell-Eucken Effective Medium Theory
Parallel Levy Kopelman Isotropic
Fig. 3. Plots of Eqs. (1)e(6) for k1/k2 ¼ 20 (equivalent to an unfrozen, porous, high water-content food).
considered. The model was relatively complex, and subsequent workers [27,42] have found that a simpler model such as Levy’s provides predictions that have comparable accuracy. Pham and Willix [27] found that Levy’s model, Eq. (5), based on ice fractions calculated from Eqs. (7) and (8) provided good predictions for both the frozen and unfrozen meat products, and was substantially better for frozen meats than the other models they considered in the study. They were, however, reluctant to place too much confidence in Levy’s model due to its ‘‘. lack of physical justification, since it was based on mathematical rather than physical arguments.’’ Wang et al. [23] have subsequently shown that Levy’s model can have a physical interpretation, which should allay any further reservations about its use on that basis. Renaud et al. [46] studied the thermal conductivity of frozen model foods between 40 C and þ20 C and compared the predictions of the Series, Parallel, MaxwelleEucken models and their own model (Eq. (12)): Renaud model: 1 v2 v2 ke ¼ f ½ð1 v2 Þk1 þ v2 k2 þ ð1 f Þ ð12Þ þ k1 k2 They found that when Eq. (12) was fitted to their experimental data, the empirically determined weighting factor, f, was strongly dependent on the specific food in question, and hence they concluded that as a predictive tool the Maxwelle Eucken model was better, even though it produced prediction errors of up to 28%. Tarnawski et al. [42] used the data of Pham and Willix to compare the predictions of several models including those that had been used previously for meats (such as the Mascheroni and Levy Models) and some models that had been used for soils, such as the models of De Vries [47] and Gori [48]. Some of the more complex models provided slightly more accurate predictions than Levy’s (based on the root mean square error); however, the differences were marginal, and some, such as De Vries’ model, contained empirically determined parameters, which is undesirable. Recently, Wang et al. [49] compared the predictions of 31 models (including the Series, Parallel, MaxwelleEucken and Levy models) to the thermal conductivity data for 22 meat and seafood products between 40 C and þ40 C [7,27], based on ice fractions calculated from Eqs. (7) and (8). Levy’s model was again found to be only marginally less accurate than some more complex models. It is difficult to give comprehensive, generic guidelines for predicting the thermal conductivity of frozen, non-porous foods. For meat products, Levy’s model based on ice fractions predicted by Eqs. (9) and (10) appears to be tried and trusted; however, for other Class II products fewer studies appear to have been performed and there remains a significant level of uncertainty, both in the selection of thermal conductivity models and in the selection of ice fraction models. There is scope for more work to be done in this area.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
5. Class III: unfrozen, porous food products Many, if not most of the effective thermal conductivity models that may be found in the literature are concerned with porous materials. The term ‘porous’ may refer to granular or particulate materials in which the void volume may be occupied by either liquid or gaseous components, or alternatively, it may refer to a material having a continuous solid matrix that contains pores/bubbles which may be isolated (sometimes referred to as ‘closed pores’) or interconnected (sometimes referred to as ‘open pores’) [50]. The measurement and prediction of porosity and pore formation is a field of study in itself, particularly for drying foods [51]. An unfrozen porous food product may be thought of as a binary mixture of a ‘condensed phase’ containing immobile water and food solids (which, for the purposes of effective thermal conductivity prediction, may be treated as a Class I food), and a gaseous phase such as air or carbon dioxide. The simplest method for determining a food’s porosity is from its apparent density and the density of the solid and gaseous components: 3¼
rc rap rap z1 rc rgas rc
ð13Þ
Fig. 3 shows a plot of Eqs. (1)e(6) for a theoretical porous, high water-content food in which the thermal conductivity of the condensed phase is 0.48 W m1 K1. By contrast with Fig. 1, the predictions of the different models vary significantly, and the effect of the physical structure that each model is based on becomes significant. Since thermal conductivity data for porous foods may lie anywhere between the predictions of the Series and Parallel models (the so-called ‘‘Wiener Bounds’’ [52]), no single model which is a function solely of the components’ thermal conductivities and volume fractions (including Eqs. (1)e(6)) will be suitable for all types of porous foods [26,53]. Due to differences in their structures, a foam and a particulate material may have different effective thermal conductivities, even if they have identical void fractions and component thermal conductivities [53]. 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 ‘porous’. Carson et al. [53] 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. Upper and lower bounds were proposed for the effective thermal conductivity of two types of isotropic porous materials: for isotropic external porosity materials it was proposed that the effective thermal conductivity is bounded above by the EMT model (Eq. (6)), and below by the MaxwelleEucken model (Eq. (4)) with air as the continuous phase; for isotropic internal porosity materials, the upper bound is provided by the
963
MaxwelleEucken model with air as the dispersed phase and the lower bound is provided by the EMT model (refer to Fig. 6 of Ref. [53]). The merit of these bounds is the significant reduction in the range of possible thermal conductivity values for the two types of porous foods, compared with the range constrained by the Wiener bounds. However, since data will most likely lie between the bounds, they are not in themselves a complete solution to the problem. The literature contains some correlations of thermal conductivity as functions of porosity and moisture content for selected fruits and vegetables [54e56]; however, the applicability of these models is clearly limited to the dataset from which the correlation was calculated. Neural network techniques have also been employed [57,58], but are considerably more complex to implement than the simple algebraic models that are available. A semi-empirical approach to the prediction of the thermal conductivity of porous materials was introduced by Krischer [59]. He reasoned that since the thermal conductivity of any two-component material must lie between the Wiener bounds, its structure could be modelled as a mixture of the Series and Parallel structures. He proposed that the effective thermal conductivity of the combined structure should be the weighted harmonic mean of the Series and Parallel conductivities (Eq. (14)): Krischer model [59]: 1 1f 1 v2 v2 þf þ ð1 v2 Þk1 þ v2 k2 k1 k2
ke ¼
ð14Þ
The value of the parameter f was determined empirically. By suitable adjustment of the f parameter between 0 and 1, Krischer’s model may predict a thermal conductivity anywhere within the Wiener Bounds. A number of studies of thermal properties of foods have used Krischer’s model, and hence values for f (often referred to as the ‘‘distribution factor’’) may be found in the literature for some foods, including granular foods between 20 C and 70 C [14,60], dried fruits and vegetables between 5 C and 100 C [61,62], and partially baked French bread between 35 C and þ25 C [63,64]. Several other models (including Eq. (12)) are similar to Krischer’s in that they may predict thermal conductivities anywhere between the Series and Parallel models by suitable adjustment of the f, Z, n and j parameters: ChaudharyeBhandari model [65]: ke ¼ ½ð1 v2 Þk1 þ v2 k2 f
ð1f Þ 1 v2 v2 þ k1 k2
ð15Þ
Kirkpatrick model (modification of Eq. (6)) [66]: X i
vi
ki ke ¼0 ki þ ðZ=2 1Þke
ð16Þ
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
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ð17Þ
As for Class II foods, it is difficult to give comprehensive, generic guidelines for model selection. If the f-value for the food in question is known, then Krischer’s model should be used, otherwise the algorithms of Maroulis et al. [62] or Carson et al. [26] should be consulted.
ð18Þ
6. Class IV: frozen, porous food products
Hamilton model (modification of Eq. (4)) [67]: ke ¼ k1
ðn 1Þk1 þ k2 ðn 1Þðk1 k2 Þv2 ðn 1Þk1 þ k2 þ ðk1 k2 Þv2
Carson’s modified Maxwell model [18,26]: 2 j = 1 j2 ks þ ka j2 = 1 j2 ðks ka Þ3 ke ¼ ks j2 = 1 j2 ks þ ka þ ðks ka Þ3 Carson’s modified EMT model [18,26]: X i
vi
ki ke ¼0 ki þ ðj=ð1 jÞÞke
ð19Þ
Of these models (Eqs. (12), (14)e(19)), Krischer’s model appears to have been employed most frequently. However, the fact that the f-value of Krischer’s model (and equivalent parameters of other models) cannot be determined mechanistically is a significant shortcoming, since its determination may require a thermal conductivity measurement e which might defeat the purpose of the prediction (although the j parameters of Carson’s models may be related to the thermal conductivity bounds for internal and external porosity materials [26]). Ideally all parameters in the model should be related to some physical property. Carson et al. [68] concluded that after the food components’ thermal conductivities and volume fractions the next most important variable was the extent (or quality) of thermal contact between particles (in the case of particulate materials) or pores (in the case of sponge or foam-like materials). However, as yet there does not appear to be any suitable method for predicting or measuring this property other than for regular arrangements of regularly shaped objects. Unfortunately, there remains an inevitable degree of empiricism involved in the thermal conductivity prediction of porous foods, particularly for those with external porosity [26], and there is scope for more work in this area. A separate issue to the influence of structure on the thermal conductivity of a porous food is the potential increase in the apparent thermal conductivity of the gaseous phase due to the evaporation and condensation of moisture across the pores. While this phenomenon is more likely to be an issue above temperatures likely to be encountered in refrigeration applications, Hamdami et al. [63] observed this behaviour even at sub-freezing temperatures. They employed a model introduced by Sakiyama et al. [70] to account for the increase in thermal conductivity: kap ¼ kgas þ kevap kevap ¼
D P dPs Law dT RT P aw Ps
ð20Þ ð21Þ
Similarly, radiation in the gaseous phase may also alter the apparent thermal conductivity of the gaseous phase [69], although it is unlikely to be significant in the range of temperatures typically encountered in refrigeration.
Class IV foods combine the difficulties associated with Class II and Class III foods, but, conveniently, the effects of ice formation and porosity may be dealt with sequentially rather than simultaneously, as demonstrated in the studies performed by Cogne´ et al. [41] with ice cream and Hamdami et al. [63] with partially baked French bread. The study by Cogne´ et al. was very thorough, complete with a micrograph of ice cream which allowed the selection of thermal conductivity models to be based on the food’s microstructure. Fig. 7 of [41] showed that ice cream is composed of a continuous aqueous phase in which are suspended discrete air bubbles, ice crystals and fat globules, and since the physical basis of the Maxwell-type models is a structure comprised of distinct continuous and dispersed phases, this family of models was an obvious choice for this application. Cogne´ et al. calculated the ice cream’s thermal conductivity in three steps: firstly the Parallel model was used to combine the thermal conductivities of the liquid water and food solids’ components (consistent with the discussion on Class I foods); secondly, the resulting thermal conductivity from the first step was combined with the thermal conductivity of ice using a modification of Maxwell’s model (De Vries model) to produce the thermal conductivity of the condensed phase; thirdly, the thermal conductivity of the condensed phase was combined with the thermal conductivity of air using the MaxwelleEucken model (Eq. (4)) to give the overall thermal conductivity. Hence, in the terms used in this review, the Class IV food (ice cream) was treated successively as a Class I food, a Class II food and a Class III food, as outlined diagrammatically in Fig. 9 of [41]. Using this technique they were able to predict the thermal conductivity of ice cream to within 8% mean relative uncertainty. Hamdami et al. [63] used successive application of the MaxwelleEucken model, as outlined schematically in their Fig. 2b. They also considered another approach in which they combined the thermal conductivities of all the components in one step using Krischer’s model, which, unlike the MaxwelleEucken model, is capable of handling multicomponent materials. The predictions of both approaches were compared to experimental data, and it was found that Krischer’s model could provide more accurate predictions; however, it was acknowledged that this was because the f-value for Krischer’s model was determined from their own experimental data, and hence the model had essentially been fitted to the data, whereas the MaxwelleEucken model had provided genuine predictions.
J.K. Carson / International Journal of Refrigeration 29 (2006) 958e967
such as ice cream, it may be apparent from a visual inspection of the food’s structure not only which effective thermal conductivity models are appropriate, but also the order in which the components should be combined. However, there may be many situations where very little may be inferred from examining the food’s structure. The literature does not appear to contain any definitive guidelines on this issue, but, consistent with previous discussion, unless there is clear indication from the material’s structure that a different order should be used, as a general rule components with the most similar thermal conductivities should be combined first, followed by the component with the next most similar thermal conductivities and so on.
Start
Class I foods unfrozen, non-porous
Sufficient to treat food as binary mixture of water and solids. Use any of Eq. (2) to (6) to determine ke.
Class II foods frozen, non-porous
Select ice fraction model from Eqs. (7) to (10), and ascertain Tf from predictive model (e.g. [39]).
Use Levy's model (Eq. 5) to predict ke.
Class III foods unfrozen, porous
Treat food as binary mixture of condensed phase and air. Determine ke of condensed phase as for Class I foods.
7. Conclusion The uncertainty involved in thermal conductivity prediction increases as the differences between the food components’ thermal conductivities increase, which means that foods which are frozen and/or porous require greater consideration than unfrozen, non-porous foods for which thermal conductivity prediction is a relatively straightforward exercise. Recommendations for selecting models for different classes of foods are summarised in Fig. 4, although it should be noted that they are only basic guidelines. Greater understanding and characterisation of the effects of ice and/or porosity on food structures would help to improve the prediction accuracy for frozen and/or porous foods.
Estimate porosity. Use Eq. (13); alternatively refer to [62]
If f-value available in literature, use Krischer's model (Eq. 14) to determine ke; alternatively refer to [26] for model selection guidelines
Class IV foods frozen, porous
965
Deal with ice content (as for Class II food) and porosity (as for Class III food) sequentially. Follow example of [41].
Acknowledgments
Finish
The author would like to acknowledge the assistance of Dr. Milan Housˇka of the Food Research Institute Prague and Dr. Paul Nesvadba of Rubislaw Consulting Ltd in compiling this review. The majority of the work was performed while the author was at AgResearch Ltd, New Zealand.
Fig. 4. Summary of recommendations for selecting models for different classes of foods.
References Models such as the Series, Parallel and EMT models and their derivatives (e.g. Eqs. (8), (14)e(16) and (19)) are capable of handling multi-component materials and hence the thermal conductivity prediction may be achieved in one step. However, in these models all components are assumed to have identical structures and spatial distributions, and so it is questionable whether this approach is as sound as the sequential approach which allows for different structural models to be applied to the different phases of the mixture. It should be noted that when thermal conductivity is predicted by sequential application of one or more models, the order in which the food’s components are included is significant; i.e. for a food comprised of components A, B and C, the result of combining A and B first to give AB and then combining AB with C, will not necessarily be the same as the result of combining A and C first, etc. For some foods,
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