An improved model for the effective thermal conductivity of open-cell porous foams

International Journal of Heat and Mass Transfer 75 (2014) 224–230

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

An improved model for the effective thermal conductivity of open-cell porous foams Miguel A.A. Mendes ⇑, Subhashis Ray ⇑, Dimosthenis Trimis 1 Institute of Thermal Engineering, Technische Universität Bergakademie Freiberg, Gustav-Zeuner-Strasse 7, D-09596 Freiberg, Germany

a r t i c l e

i n f o

Article history: Received 20 September 2013 Received in revised form 17 January 2014 Accepted 26 February 2014 Available online 20 April 2014 Keywords: Effective thermal conductivity Porous media Open-cell porous foams Simplified modeling approach Improved model

a b s t r a c t An extension of a simplified model for the effective thermal conductivity (ETC) of open-cell porous foams, that originally contained only one adjustable parameter, is proposed in this article. The present improved model consists of two adjustable parameters those can be calculated directly from detailed numerical predictions of the ETC under vacuum condition and in presence of a reference fluid with known thermal conductivity ratio (of fluid and solid phases). The performance of present simplified model is evaluated by comparing its estimations with results obtained from detailed numerical simulations, those take into account the true morphology of porous media, as well as predictions obtained from the original simplified model, for the complete range of thermal conductivity ratios, lower than unity (0 6 kf =ks 6 1). Geometries of porous media are either synthetically generated for idealized regular structures or obtained from high resolution three-dimensional computed tomography-scan images for real foams. The present study clearly demonstrates that the proposed simplified model yields extremely accurate estimation of the ETC for all the investigated structures, as long as the reference thermal conductivity ratio of fluid and solid phases is judiciously chosen. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The homogenization approach, which may be regarded as a trade-off between modeling accuracy and complexity, is generally used in order to model practical heat and mass transfer applications involving porous media, where the transport processes are analyzed using macroscopic models [1]. In order to obtain accurate predictions, these macroscopic models, however, require reliable information about the effective thermo-physical properties of the porous media those essentially contain the micro-scale effects in an implicit manner. In this respect, the effective thermal conductivity (ETC) of porous media can be regarded as an essential macroscopic parameter in presence of conductive heat transfer. The ETC of porous media can be either measured [2], or can be predicted from detailed numerical simulations, considering the finely resolved morphology of porous media, which nowadays, can be obtained from high resolution, three-dimensional (3D) computed tomography (CT)-scan images [3–5]. However, such approaches could be time consuming, since the ETC of a given foam ⇑ Corresponding authors. Tel.: +49 3731393946 (M.A.A. Mendes), +49 3731393 947 (S. Ray). E-mail addresses: [email protected] (M.A.A. Mendes), [email protected] (S. Ray), trimis@ iwtt.tu-freiberg.de (D. Trimis). 1 Tel.: +49 3731393940. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.02.076 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

for a particular working fluid is obtained individually on a case to case basis. Therefore, in order to achieve a good compromise between the accuracy and the computational (or experimental) effort, an alternative approach could be to consider simplified models those provide fairly straightforward expressions for a quick but reasonably accurate estimation of the ETC, ideally presenting a wide range of applicability. It is expected that these models should be valid for any combination of working fluid and material of the solid matrix of porous media. Various simplified models for predicting the ETC of porous media can be found in the literature and recent reviews on these models have been presented by Coquard et al. [6] and Coquard and Baillis [7]. Most of these models provide closed form correlations or analytical expressions for the ETC, based on either experimental data or theoretical considerations, those grossly simplify either the material morphology or the solution method. Recently, Mendes et al. [5] proposed a simplified model with one adjustable parameter for estimating the ETC of open-cell porous foams using an extremely simplified approach. Their methodology relies upon a single numerical prediction of the dimensionless ETC (defined with respect to the thermal conductivity of solid phase) under vacuum condition, based on the detailed geometry of porous media. The ETC of the same porous medium for any working fluid can be then evaluated from the proposed model that

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225

Nomenclature a; b D d dp k Lr L l q_ Tc Th W w

model parameters characteristic dimension of lumps (m) characteristic dimension of struts cross-section (m) characteristic pore dimension (m) thermal conductivity (W/m K) representative size of porous medium domain (m) representative length of idealized porous medium domain (m) representative length of solid strut of idealized porous medium (m) steady-state heat flow (W) cold temperature (K) hot temperature (K) representative width of idealized porous medium domain (m) representative width of solid strut of idealized porous medium (m)

Superscript  dimensionless variable Subscript k parallel arrangement bound for ETC ? serial arrangement bound for ETC a serial arrangement of minimum and maximum bounds dependent on parameter a eff effective f fluid max maximum bound for ETC; maximum value min minimum bound for ETC model result from simplified model ref reference s solid simulated result from detailed simulation

Greek symbols ss effective tortuosity of conduction / macroscopic porosity

porous medium for

heat

combines minimum and maximum ETC bounds, where the adjustable parameter in the model can be directly as well as uniquely calculated from the dimensionless ETC under vacuum condition. A particular form of this simplified model, that uses serial and parallel limits of the ETC as minimum and maximum bounds, respectively, was previously proposed by Dietrich et al. [8]. However, they estimated the adjustable parameter on the basis of an empirical approach, where measurements of ETC for different samples of open-cell ceramic foams were best fitted in order to suit the model. Therefore, the only structural parameter of porous media that could directly influence the modeled prediction remained as the porosity [8]. Two main advantages of the approach, proposed by Mendes et al. [5], are the following. First, it offers significantly reduced computational time for calculating the dimensionless ETC under vacuum condition, which scales with the order of porosity of the medium and is also independent of the thermal conductivity of solid phase. Secondly, the dimensionless ETC under vacuum condition implicitly includes the morphological information of porous media that is relevant for heat conduction. From their results, Mendes et al. [5] concluded that the simplified model, with lower and upper Hashin–Shtrikman bounds as minimum and maximum limits for the ETC, performs better than alternative choices for most of the investigated structures. However, the model that uses the Effective Medium Theory (EMT) and the upper Hashin–Shtrikman bound, apparently yields more accurate estimations for porous structures with higher porosity, especially when the fraction of solid phase in lumps is also higher than that in struts. The main objective of the present investigation is to extend the simplified ETC model of Mendes et al. [5] by introducing one additional adjustable parameter. The suggested improvement of the model is expected to increase its accuracy in the ~ ¼ k =ks ) by intermediate range of thermal conductivity ratios (k f f performing an additional numerical prediction of the ETC corresponding to a reference fluid. In the present article, the performance of this simplified model is evaluated by comparing its estimations with results obtained from the detailed numerical model.

2. Theoretical considerations Let us consider an open-cell porous foam formed by two distinct phases with thermal conductivities ks and kf , for solid and fluid phases, respectively, where kf 6 ks . The simplified model for ETC, proposed by Mendes et al. [5], is given as:

~ ¼ bk ~ þ ð1  bÞk ~ k min max eff

ð1Þ

~ and k ~max are the generic minimum and maximum dimenwhere k min sionless bounds for the ETC, respectively, and b is the adjustable parameter in the model. All thermal conductivities are made dimensionless with respect to the thermal conductivity of solid phase, i.e., ~ ¼ k=ks . An extended version of this model is proposed in the presk ent study as follows:

~ ¼ bk ~a þ ð1  bÞk ~max k eff " ~a ¼ k

a

~ k min

þ

ð1  aÞ ~ k

ð2aÞ

#1 ð2bÞ

max

where a and b are two adjustable parameters in the range ½0; 1 and ~a represents the ETC as a function of a due to serial arrangement of k ~ ~ materials having equivalent thermal conductivities k min and kmax . The physical interpretation of the proposed model in Eq. (2) is shown in Fig. 1. Nevertheless, the original model of Mendes et al. [5] in Eq. (1) can be readily recovered by setting a ¼ 1 in Eq. (2). Although different ETC bounds, proposed by Mendes et al. [5], ~ ~ could be used for k min and kmax , in the present investigation they are taken according to serial and the parallel arrangement models, respectively, in order to guaranty a 2 ½0; 1, independent of the porous medium under consideration. Justification for such a choice is provided in Appendix A. Serial and parallel ETC bounds, are given by Eqs. (3a) and (3b), respectively, as functions of porosity of the ~: medium / and k f

" #1 / ~ ~ kmin ¼ k? ¼ þ ð1  /Þ ~ k f

ð3aÞ

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Fig. 1. Physical interpretation of the proposed model in Eq. (2) for ETC of porous media.

(a) 10

(b)10

0

0

D/d=1 - simulation D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - simulation D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

10

D/d=1 - simulation D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - simulation D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

-1

10

-1

φ 10-2 -3 10

10-2

10-1

(c) 10

φ 10-2 -3 10

100

10-2

10-1

100

0

D/d=1 - simulation D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - simulation D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

10-1

φ 10-2 -3 10

10-2

10-1

100

~ for cubic cell Fig. 2. Comparison between detailed numerical predictions and estimations from different simplified models for the dimensionless ETC as functions of k f structures with different porosity and D=d ratio: (a) / ¼ 0:95; (b) / ¼ 0:8; (c) / ¼ 0:7.

~max ¼ k ~k ¼ k ~ / þ ð1  /Þ k f

ð3bÞ

~ ! 0, for which By applying Eqs. (2) and (3) and in the limit as k f ~ ~ ~ kmin ¼ 0; kmax ¼ 1  / and ka ¼ 0, an expression for b can be obtained as [5]:

b¼1

~ k eff;s ð1  /Þ

ð4Þ

~ where k eff;s is the dimensionless ETC under vacuum condition [5]. Physical interpretation of b in relation to the tortuosity of the foam is provided in Appendix A. The parameter a can be expressed from Eqs. (2) and (3) as:

a¼

!1 !1 ~k k b 1 1 ~? ~ ~ k 1k eff;ref =kk

ð5Þ

~ where k eff;ref is the ETC in presence of a reference fluid with ~ . ~ ¼k k f f;ref ~ ~ In short, the present model in Eq. (2) requires /; k eff;s and keff;ref as primary inputs, those can be either experimentally measured or numerically predicted by resolving the detailed structure of porous ~min and k ~max are calculated from foam. Based on these inputs, k Eq. (3) and hence parameters b and a are obtained uniquely using Eqs. (4) and (5), respectively. 3. Results and discussions The performance of present simplified model for ETC is evaluated by comparing its predictions with results obtained using the detailed numerical model, explained by Mendes et al. [5]. These comparisons are carried out for ideal cellular structures as well

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as for real open-cell porous foams. For both cases, the ETC is predicted for the complete range of kf =ks ratios lower than unity i.e., ~ 6 1. for 0 < k f 3.1. Results for ideal cellular structures Cubic cells with porosities 0.95, 0.8 and 0.7 are considered as ideal cellular structures. They consist of struts of square cross-section with a characteristic dimension d and cubic lumps with two characteristic dimensions: D ¼ d and D ¼ 4d [4–6]. Other than detailed numerical predictions, by taking into account contributions of both solid and fluid phases, the ETC is also ~ evaluated under vacuum condition. Table 1 presents k eff;s and ~ ~ k k (assuming ¼ 0:02) as well as values of a and b, required eff;ref f;ref in Eq. (2), for different cubic cell structures. Such a specific choice ~ of k f;ref will be shortly apparent. ~ as functions of k ~ for cubic cells with two Predicted values of k eff f different D=d ratios are presented in Fig. 2(a)–(c) for porosities 0.95, 0.8 and 0.7, respectively. Additionally, these figures also show Table 1 Geometric properties of different cubic cell structures, along with respective adjustable parameters required in Eq. (2). /

0.95

0.8

D=d

1

4

4

1

4

2 ~ k eff;s  10 2 ~ k eff;ref  10 a, Eq. (5) b, Eq. (4)

1.88

0.87

9.36

3.12

15.31

5.14

3.97

3.10

11.62

6.17

17.63

8.90

0.93 0.62

0.88 0.83

0.86 0.53

0.72 0.84

0.84 0.48

0.62 0.82

1

0.7

Remarks

~ ¼ 0) (k f ~ (k f;ref ¼ 0:02)

~ with estimations obthe comparison of detailed predictions for k eff tained from proposed simplified ETC models, given by Eqs. (1) and (2). It can be easily observed from Fig. 2 that in general, estimations ~ obtained with Eq. (2) appear to be in excellent agreement of k eff with simulated (i.e., ideal) values. Estimations from Eq. (1), on the other hand, show higher deviations at intermediate values of ~ , especially for cubic cells with low porosity and high D=d ratio. k f This is clearly demonstrated in Fig. 3, which presents the relative ~ , obtained with both models for different error for the estimated k eff ideal structures. The relative error in this article is defined as:

error ¼

~ ! 1, both k ~ ~ According to Eq. (3), in the limit as k min and kmax f assume a value equal to unity and their derivatives with respect ~ are equal to /. Using this simple knowledge and by differento k f ~ , it can be easily demonstrated that: tiating Eq. (2) with respect to k f

 ~  dk eff  ~ ~ dk f

0

-5

-5

-10

-10

-15

D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

φ

10-2

10-1

100

ð7Þ

This result, although remarkably simple, clearly shows that in the ~ ! 1, the ETC of any porous medium turns out to be a limit as k f function only of the porosity / and hence the structure of porous medium has no direct influence. This is clearly demonstrate in Fig. 2. Consequently, it is expected that in the vicinity of this limit, ~ from Eqs. (1) and (2) are also weakly dependent estimations of k eff on a and b. This explains lower relative errors yielded by simplified ~ ! 1, shown in Fig. 3. models as k f

0

-20 -3 10

¼/

kf ¼1

(b)

D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

ð6Þ

eff;simulated

(a)

-15

~ ~ k eff;model  keff;simulated ~ k

-20 -3 10

10-2

φ 10-1

100

(c) 0

D/d=1 - model, Eq.(1) D/d=1 - model, Eq.(2) D/d=4 - model, Eq.(1) D/d=4 - model, Eq.(2)

-5

-10

-15

φ -20 -3 10

10-2

10-1

100

~ for cubic cell structures with different porosity and D=d ratio: (a) / ¼ 0:95; (b) Fig. 3. Relative errors for the ETC, obtained with different simplified models, as functions of k f / ¼ 0:8; (c) / ¼ 0:7.

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M.A.A. Mendes et al. / International Journal of Heat and Mass Transfer 75 (2014) 224–230

(a)

Table 2 Geometric properties of different samples of real open-cell porous foams, along with respective adjustable parameters required in Eq. (2).

1

0.8

0.6

0.4

0.2

0 -3 10

D/d=1, D/d=1, D/d=1, D/d=4, D/d=4, D/d=4,

10

Foam 1

Foam 2

Foam 3

Foam 4

Pores per inch (ppi) / dp (mm) Lr (mm) CT resolution (lm) ~  102 k

10 0.57 3 18 65 23.98

0.74 4 18 65 11.04

30 0.79 2.8 18 65 9.43

0.88 2.8 9 17 3.15

2 ~ k eff;ref  10 a, Eq. (5) b, Eq. (4)

27.05

14.32

12.89

6.48

0.8 0.43

0.74 0.58

0.63 0.56

0.7 0.74

eff;s

=0.95 =0.8 =0.7 =0.95 =0.8 =0.7

-2

Sample

10

-1

10

Remarks

~ ¼ 0) (k f ~ (k f;ref ¼ 0:025)

0

10

0

(b)10 8

D/d=1, D/d=1, D/d=1, D/d=4, D/d=4, D/d=4,

=0.95 =0.8 =0.7 =0.95 =0.8 =0.7

10

-1

Foam 1 - simulation Foam 1 - model, Eq.(1) Foam 1 - model, Eq.(2) Foam 2 - simulation Foam 2 - model, Eq.(1) Foam 2 - model, Eq.(2) Foam 3 - simulation Foam 3 - model, Eq.(1) Foam 3 - model, Eq.(2) Foam 4 - simulation Foam 4 - model, Eq.(1) Foam 4 - model, Eq.(2)

6

4

2

10 0 -3 10

10

-2

10

-1

10

-2

10-3

10-2

10-1

100

0

~ Fig. 4. Influence of reference condition k f;ref for cubic cell structures with different porosity and D=d ratio: (a) on the parameter a in Eq. (2); (b) on the maximum absolute relative error.

~ , for evaluating a The choice of optimal reference condition k f;ref in Eq. (2b), requires a careful look. Fig. 4(a) presents the variation ~ of a, calculated from Eq. (5), as function of k f;ref for different cubic cell structures. One can readily observe from the figure that the value of a is strongly influenced by the porosity and the D=d ratio of cubic cells in a sense that it decreases with the decrease in porosity and with the increase in D=d ratio (see also Table 1). Nevertheless, for each of the cellular structures, a remains almost constant for ~ ~ lower values of k f;ref i.e., kf;ref 6 0:05. ~ The influence of kf;ref on the maximum relative error for the estimation of ETC, obtained with the proposed model, given by Eq. (2), is shown in Fig. 4(b) for different cubic cell structures. One can clearly conclude from the figure that lower values of ~ , for which a is less sensitive, yield lower relative errors, k f;ref ~ although marginal variations could be detected for k f;ref 6 0:05. 3.2. Results for real open-cell foam structures The accuracy of simplified model for ETC in Eq. (2), as well as its ~ of real open-cell original version, given by Eq. (1), in estimating k eff porous foams is also investigated by considering four different foam samples (named as Foam 1, Foam 2, Foam 3 and Foam 4). Structures of these foam samples and their detailed numerical simulations were already presented by Mendes et al. [5] and hence they are not shown here for the sake of brevity. Table 2, however, summarizes their main geometric properties, along with the ~ ~ ~ predicted k eff;s and keff;ref (assuming kf;ref ¼ 0:025), as well as values of a and b.

Fig. 5. Comparison between detailed numerical predictions and estimations from ~ for opendifferent simplified models for the dimensionless ETC as functions of k f cell porous foam samples described in Table 2.

~ and Fig. 5 presents the comparison between the predicted k eff ~ for all the four estimations from Eqs. (1) and (2) as functions of k f samples of open-cell porous foams. Respective relative errors, obtained with both simplified models, are presented in Fig. 6. It can be clearly observed from these figures that the present model with two adjustable parameters, given by Eq. (2), yields more accurate ~ than the original model, given by Eq. (1), for predictions for k eff all the four tested samples. Nevertheless, although the maximum relative error, obtained with Eq. (2), is found to be consistently ~ lower than 1% (considering k f;ref ¼ 0:025), its accuracy consider~ ably depends on the choice of reference condition k f;ref for evaluating the parameter a. Results, shown in Fig. 5, once again clearly demonstrate the validity of argument extracted from Eq. (7) that ~ ! 1, the only structural parameter of porous medin the limit as k f ium influencing the ETC is the porosity. Values of a, calculated according to Eq. (5), are shown in ~ Fig. 7(a) as functions of k f;ref for all the four samples of real foams. ~ It is evident from the figure that for lower values of k f;ref i.e., ~ kf;ref 6 0:1; a remains nearly constant and depends only on the foam structure, as it is observed for ideal cellular structures in ~ Fig. 4. Fig. 7(b) presents the influence of k f;ref on the maximum absolute relative error yielded by the present model. The figure ~ , estimations from Eq. (2) shows that for lower values of k f;ref produce a lower relative error. However, for the case of Foams 3 and 4, the relative error marginally increases (by about 1%) for ~ . Consequently, based on present reextremely small values of k f;ref ~ sults, the optimal value of k f;ref for evaluating the new adjustable ~ parameter a is found to be in the intermediate range, i.e., k f;ref lying approximately between 0.02–0.1 for real foams and 0.005–0.05 for

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M.A.A. Mendes et al. / International Journal of Heat and Mass Transfer 75 (2014) 224–230

(a)

(b)

0

0

-5

-5

-10

-10

model, Eq.(1) model, Eq.(2)

-15 -3 10

10

-2

model, Eq.(1) model, Eq.(2)

10

-1

10

0

(c)

-15 -3 10

-2

10

-1

10

0

(d)

0

0

-5

-5

-10

-10

model, Eq.(1) model, Eq.(2)

-15 -3 10

10

10-2

model, Eq.(1) model, Eq.(2)

10-1

100

-15 -3 10

10-2

10-1

100

~ for open-cell porous foam samples described in Table 2: (a) Foam 1; (b) Foam 2; Fig. 6. Relative errors for the ETC, obtained with different simplified models, as functions of k f (c) Foam 3; (d) Foam 4.

(a)

1

(b) 10

0.8

8

6

0.6

4

0.4

0.2

0 -3 10

Foam 1 Foam 2 Foam 3 Foam 4

Foam 1 Foam 2 Foam 3 Foam 4

10-2

2

10-1

100

0 -3 10

10-2

10-1

100

~ Fig. 7. Influence of reference condition k f;ref for open-cell porous foam samples described in Table 2: (a) on the parameter a in Eq. (2); (b) on the maximum absolute relative error.

ideal structures, which depends on the porosity and the D=d ratio of the considered porous structure. 4. Conclusions In the present study, an extended version of the simplified model for ETC of open-cell porous foams [5] is proposed that contains two adjustable parameters a and b. From the detailed comparison the following conclusions can be drawn:

~ ! 1, the ETC turns out to be a function only of the porosity 1. As k f / and hence the structure of porous medium has only a negligible influence. Both models obey this limit, irrespective of the chosen values of adjustable parameter(s). 2. The proposed simplified model, yields extremely accurate estima~ tion of the ETC for all the investigated structures, as long as k f;ref is properly chosen for evaluating the newly introduced parameter a. ~ , a remains nearly constant and depends 3. For lower values of k f;ref only on the foam structure.

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~ , used for evaluating the adjustable 4. The optimal value of k f;ref parameter a, is found to be in the intermediate range i.e., approximately between 0.005 and 0.1, depending upon / and D=d ratio of the considered porous medium. 5. Once both adjustable parameters a and b are obtained for a particular porous medium, following the method outlined in this article, the proposed model can be used for any combination of fluid and solid, irrespective of their nature.

Th

ks

Appendix A. Physical interpretation of parameter b and ~ ~ appropriate choice of k min and kmax Evaluation of b according to Eq. (4) requires a closer look for its appropriate physical interpretation. Let us consider an idealized two-dimensional porous structure of length L and width W, in which the solid phase, immersed in a fluid phase, forms a continuous path, as schematically shown in Fig. A.8. The path, available for heat transfer through the solid phase, has a total length l and in the highly simplified idealized condition, offers a constant width w. A temperature difference T h  T c (T h > T c ) is applied between boundaries of the domain in x1 direction, inducing a steady-state heat flow q_ through the medium. From basic definitions, the volume fraction of solid phase ð1  /Þ and the effective tortuosity ss for this idealized porous medium can be expressed as:

ss

l ¼ L

ðA:1aÞ

ðA:1bÞ

Th  Tc l

ðA:2Þ

From the macroscopic perspective, however, q_ can also be written in terms of the ETC under vacuum condition keff;s as:

q_ ¼ Wkeff;s

Th  Tc L

ðA:3Þ

Equating Eqs. (A.2) and (A.3), the dimensionless ETC under vacuum condition can be expressed in terms of geometric parameters of the porous structure as:

keff;s w L ~ k ¼ eff;s ¼ W l ks

L

Fig. A.8. Schematic representation of an idealized two-dimensional porous structure.

~ k 1 eff;s ¼ ð1  /Þ s2s

ðA:5Þ

Utilizing the simple consideration presented here and relating Eqs. (4) and (A.5), the parameter b in model given by Eq. (2) can be related to ss as:

b¼1

1

s2s

ðA:6Þ

The analysis for idealized structure clearly shows that for finite val~ , the effective tortuosity of porous media would be smaller ues of k f than that under vacuum condition, since the presence of finite thermal conductivity fluid phase would allow for additional shorter paths for heat flow, which would. Therefore, it is expected that the simplified model of Mendes et al. [5], which could be retrieved by setting a ¼ 1 in Eq. (2), would always underpredict the ETC of ~ ~ ~ ~ porous media for k min ¼ k? and kmax ¼ kk . Consequently, this guaranties that the parameter a, evaluated from Eq. (5) for the reference ~ ¼k ~ , would always lie in the range ½0; 1. It, therefore, condition k f f;ref justifies the present choice of minimum and maximum ETC bounds, given in Eq. (3). References

Assuming that the idealized porous medium is under vacuum condition, the Fourier’s law of heat conduction yields the following _ expression for q:

q_ ¼ wks

x1

~ Using definitions in Eq. (A.1), an alternative expression for k eff;s can be written as:

The authors would like to thank the German Research Foundation (DFG) for supporting the investigation in the subproject B02, which is a part of the Collaborative Research Center CRC 920.

lw LW

w W

kf

x2

Acknowledgments

ð1  /Þ ¼

kf

l

Conflict of interest Myself and the co-authors Subhashis Ray and Dimosthenis Trimis of the manuscript, entitled ‘‘An improved model for the effective thermal conductivity of open-cell porous foams’’, declare that there is no conflict of interest regarding the publication of this manuscript in the International Journal of Heat and Mass Transfer.

Tc

ðA:4Þ

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