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. 298 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 m
1 K
1) empirical parameter (Eq. (7)) thermal conductivity (W m
1 K
1) 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 299 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 300 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- 301 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 302 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
 1
f 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 1
j 1
j2 ke ¼ kc  2  ð7Þ j þ k þ ðk
k Þe k c a c a 2 1
j 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 m
1 K
1. 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 m
1 K
1 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 m
1 K
1, so if a ke of less than 0.1 W m
1 K
1 is predicted, an uncertainty of 0.02 W m
1 K
1 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: ð1
v2 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):
ð1
f 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 1
f ð1
v2 Þk 1 þv2 k 2 1 þf  1
v2 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 1
j 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 1
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