Thermal conductivity correlations of open-cell foams: Extension of Hashin–Shtrikman model and introduction of effective solid phase tortuosity

International Journal of Heat and Mass Transfer 92 (2016) 539–549

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Thermal conductivity correlations of open-cell foams: Extension of Hashin–Shtrikman model and introduction of effective solid phase tortuosity Prashant Kumar ⇑, Frédéric Topin IUSTI, CNRS UMR 7343, Aix-Marseille Université, Marseille, France

a r t i c l e

i n f o

Article history: Received 2 March 2015 Received in revised form 20 July 2015 Accepted 25 August 2015

Keywords: Hashin–Shtrikman model Effective solid phase tortuosity Intrinsic solid phase thermal conductivity Strut morphology

a b s t r a c t Open-cell foams have emerged as promising materials for use in heat sink and heat exchanger applications. The thermal behavior of open-cell foams depends on their microscopic structure. The effective thermal conductivity of open-cell porous foams can be measured using experimental techniques, predicted from the 3-D direct numerical simulations on reconstructed foam structures obtained from micro-computed tomography images or derived from idealized structure thermal analysis. Based on the tetrakaidecahedron unit cell and different strut morphologies, three dependent and interdependent empirical correlations for effective thermal conductivity were derived. They encompass all morphological parameters and ratios of constituent phases of foams of different materials. In this process, the Hashin–Shtrikman (HS) bounds model was first extended and applied to the resistor model. A correlation term, X was introduced to take into account the thermal conductivities of constituent phases and the morphological parameters of the foam structure. Secondly, a more complex effective model that is a combination of series and parallel models was derived by introducing effective solid phase tortuosity. Lastly, a simple model (KT-model) was derived that can be used to predict either effective thermal conductivity or intrinsic solid phase conductivity depending upon which one of these quantities is known. The present study clearly demonstrates that the proposed empirical correlations yield extremely accurate estimates of the effective thermal conductivity for all the experimental and numerical data of different foam materials reported in the literature. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, high porosity foam materials have attracted attention due to their low density and peculiar transport properties, which make them attractive for enhancing the thermal performances of heat transfer devices, while allowing the use of smaller and lighter equipment [1,2]. Accurate knowledge of thermal transport properties of opencell foams is required in order to effectively utilize them in heat transfer applications. From a practical point of view, it is common to assimilate the foam to a homogeneous medium having an effective thermal conductivity (keff ), thereby effectively neglecting the detailed micro-scale effects of the porous structure [3].

⇑ Corresponding author at: 5, Rue Enrico Fermi, Technopole de Château Gombert, 13453 Marseille Cedex 13, France. Tel.: +33 (0) 04 91 10 68 36. E-mail addresses: [email protected] (P. Kumar), frederic.topin@ univ-amu.fr (F. Topin). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.085 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

The effective thermal conductivity of open-cell foams can be obtained either by measuring it directly with experimental techniques (e.g. [4–9]) or numerically calculating the value from simulations, considering the detailed morphology of porous media. With the development of 3-D image processing that can reconstruct the same structure as the original foam sample, many researchers nowadays have performed numerical simulations using the Lattice Boltzmann method (e.g. [10]), on commercial software or in-house code based on finite volume and finite element methods (e.g. [11–23]). However, experiments or numerical approaches can be time consuming and depend on various factors like the intrinsic solid and fluid phase conductivities, morphology of foam structures, verification of local thermal equilibrium condition, boundary conditions, etc. Therefore, an alternative approach in the form of empirical correlations provides fairly straightforward expressions for quick and reasonably accurate estimations of the effective thermal conductivity thus presenting a wide range of applicability.

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Nomenclature Latin symbols A=R side length of strut shape/radius of strut shape (mm) Lc node-to-node length (mm) Ls strut length (mm) M adjustable parameter (Eq. (10), –) N adjustable parameter (Eq. (10), –) Req equivalent circular strut radius, mm S constant (Eq. (8), –) S0 constant (Eq. (14), –) ST effective solid tortuosity (Eq. (12), –) Greek symbols intrinsic solid phase conductivity, W m1 K1 ks kBs parent or bulk material conductivity, W m1 K1 kf fluid phase conductivity, W m1 K1 keff effective thermal conductivity, W m1 K1 kparallel parallel thermal conductivity, W m1 K1 kseries series thermal conductivity, W m1 K1 kHS;Upper HS upper bound thermal conductivity, W m1 K1 (Eq. (1)) kHS;Lower HS lower bound thermal conductivity, W m1 K1 (Eq. (2)) w dimensionless morphological parameter (Eq. (4), –)

Various correlations have also been proposed in the literature based on the experimental and numerical results/values for effective thermal conductivity. Their advantage lies in the fact that empirical correlations are comparatively easy, fast, and inexpensive to use in comparison with individual studies of particular working fluids on a case to case basis on a 3-D reconstructed foam (by X-ray tomography, Voronoi tessellation method, etc.), numerical solutions of transport equations, or computational time. Various simplified models for predicting the effective thermal conductivity of open-cell foams can be found in the literature, and recent reviews have been presented by several researchers (e.g. [24–27]). Empirical correlations are very useful to investigate the influence of the morphological parameters of foams on their apparent thermal conductivity and give insight on how to optimise the thermal performances of various industrial components. The derivation of correlations reported in the literature can be classified in three ways: (1) from experiments [6–9]; (2) from numerical simulations (e.g. [26,18–21]; (3) from congregate data of the literature (e.g. [28–33])). Effective thermal conductivity correlations described in the literature are based either on the asymptotic bound approach (e.g. [29,9,20]) or on the micro-structural approach (e.g. [5,28,7,30,31,18,19,21]). In this context, this work highlights a few critical issues in determining effective thermal conductivity and answers the following questions: (1) What is the physical meaning of intrinsic solid phase thermal conductivity (ks )? (2) Is intrinsic solid phase thermal conductivity (ks ) different than the bulk solid phase conductivity of the parent material (kBs )? (3) Is porosity the only parameter to relate with effective thermal conductivity? (4) Is the combination of series and parallel bounds (also known as Wiener bounds; [34] sufficient to characterize effective thermal conductivity? (5) How should the effective solid phase tortuosity (ST ) be introduced in the correlation? Kumar et al. [18] and Kumar and Topin [19] showed that the ratio of conductivity between phases impacts the effective thermal conductivity under the condition of local thermal equilibrium (LTE). When the fluid phase conductivity has the same order as the solid phase, then the fluid phase starts to play an important

eo en X

aeq b

open porosity (–) nominal porosity (–) correlation factor (Eq. (3), –) ratio of equivalent circular strut radius to node-to-node length (–) ratio of strut length to node-to-node length (–)

Abbreviations LTE local thermal equilibrium HS Hashin–Shtrikman KT Kumar–Topin lCT micro-computed tomography CFD computational fluid dynamics Subscript eq c s det h st rs rh

equivalent circular square diamond (double equilateral triangle) hexagon star (regular hexagram) rotated square rotated hexagon

role in determining effective thermal conductivity. In this situation, the correlations derived by several authors [5,6,28,7,29,11,8,30] do not hold and introduce high error, as also shown in the works of Dietrich et al. [9]. The correlations derived by these authors contain parent material or bulk phase solid thermal conductivity (kBs ) and porosity. Kumar et al. [18] and Kumar and Topin [19] highlighted that since different commercially available foams employ different manufacturing techniques, there are significant changes in intrinsic solid phase thermal conductivity of foams compared to the same parent material one where ks < kBs (a reduction of about 25% in kBs ). Similar observations were demonstrated by Dietrich et al. [9] and Randrianalisoa et al. [17] where these authors measured intrinsic solid phase thermal conductivity of their open-cell foams. Dietrich et al. [9] proposed a parameter b = 0.51 in their correlation that was estimated by an empirical approach based on the conductive heat transfer occurring through the parallel layout without any restriction imposed by the serial part and is related with porosity as the only structural parameter of foam samples. In principle, the parameter should depend on the foam structure under consideration. Edouard [31] used morphological parameters of a foam matrix to derive an effective thermal conductivity correlation based on the cubic lattice unit cell, which does not correspond to the real foam structure and thus, could not be used as periodic unit cell. Kumar and Topin [19] showed that the shape and size of the cells as well as the shape of struts have negligible effects on modeled predictions in the case of isotropic foams. However, in the case of anisotropic foams, the size of the cells as well as the strut shapes strongly impact the effective thermal conductivity over porosity [35]. Nevertheless, porosity was the only structural parameter considered by most of the correlations. Therefore, the direct influence of the morphology of foam structures on the magnitude of the effective thermal conductivity was neglected by numerous correlations, although its influence on the conductive heat transfer is already known to be relevant. Mendes et al. [20] presented an analysis for the determination of effective thermal conductivity of open-cell foam-like

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structures for any working fluid under vacuum condition. Their study clearly demonstrated that the effective thermal conductivity of porous medium under vacuum condition (i.e. ks =kf ! 1) contains the necessary quantitative structural information. These authors studied various simplified models and concluded that using the upper and lower Hashin–Shtrikman bounds [36] as minimum and maximum possible values of effective thermal conductivity, respectively, performs better and yields more accurate estimations for porous structures. Recently, Mendes et al. [21] merged the theoretical descriptions proposed by Krischer [37] and Dietrich et al. [9] and presented a new formulation that contains two adjustable parameters a and b. However, their formulation is based on numerical simulations performed on solid phase alone that give access to determine effective thermal conductivity of solid phase. The adjustable parameter a, is found to be between 0.005 and 0.1 approximately, while the parameter b is found to be a function of effective tortuosity of porous media. In the literature, empirical correlations have been developed for foams of different materials (metallic and ceramic). However, the majority of correlations were focussed on metal foam samples with high porosity (0:88 < eo < 0:97). A few authors have proposed the correlations for ceramic foams (e.g. [9]) for moderate porosity (0:75 < eo < 0:97). The validity and applicability of the correlations were tested against experimental effective thermal conductivity data of ceramic foam samples reported by Dietrich et al. [9]. Table 1 gives an overview of the analytical effective thermal conductivity values obtained from correlations proposed by different authors. The value of the maximum error ranges from 26% to 176%, and the value of the mean error (calculated by the arithmetic mean of all errors from each material) ranges from 18% to 101%. The reasons for such high errors are primarily due to: (1) derivation of correlations based on metal foams, (2) different values of adjustable parameters based on different materials, (3) taking bulk/parent conductivity of the solid phase into account, and, (4) the majority of the expressions were derived using only high porosity data. Thus, there is a need for empirical models that consider simultaneously the whole porosity range (0.60–0.97) as well as open-cell foam samples of different materials/alloys obtained from different manufacturing processes.

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In this work, we aim at developing three generalized correlations based on: (1) extension of upper and lower HS bounds (known keff narrowest bounds for macroscopically homogeneous, isotropic media), (2) the ‘effective parallel model’ as a combination of parallel and series models with an introduction of effective solid phase tortuosity, ST (introduction of the morphological parameters of foam matrix), (3) the ‘‘Kumar–Topin” (KT) model that serves to predict intrinsic solid phase conductivity of foam materials when effective thermal conductivity is known or vice versa. These correlations use the morphological parameters of foam structure and were validated in the wide porosity range, 0:60 6 eo 6 0:95, for a large range of solid to fluid phase conductivity ratios (ks =kf ¼ 10—30; 000). We have limited the current work to the foams that could be represented by Kelvin-like microstructure with a constant cross section of the ligament along its axis. The empirical corrections were derived for pure diffusion heat transfer conditions under local thermal equilibrium condition (i.e. neither radiative nor convection heat transfer was included). 2. Effective thermal conductivity models The simplified models and correlations provide easy expressions for a quick estimation of the effective thermal conductivity; their range of applicability is generally expected to be limited. Thus, the development of new or modified models and correlations, based on a large dataset for any type of open-cell foams, may be considered an efficient alternative approach in order to achieve a good compromise between the accuracy and the computational effort. Three empirical correlations based on extension of upper and lower HS bounds, the effective parallel model and the KT model are developed in the Sections 2.1–2.3, respectively. The effective thermal conductivity mainly depends on the porosity, the morphology of the structure and the thermal conductivities of the constituent phases. The keff and ks were gathered from the numerical database of 2000 values reported by Kumar and Topin [19] to derive the correlations in order to take into account different strut shapes, morphological properties, variable porosities and pore sizes of different materials (see Fig. 1).

Table 1 Literature models evaluation with reference to experimentally determined effective thermal conductivity (keff ) of Al2O3, Mullite and OBSiC ceramic foams (data taken from [9] of different pore sizes and porosities with air (kf = 0.03 W m1 K1) as an interstitial fluid.

⁄

Grey blocks represent the experimental data while the white blocks represent the calculated/analytical data.

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Fig. 1. Left: representation of an ideal isotropic foam structure. Right: representation of different 3-D strut and ligament shapes (a) Circular (b) Square (c) Diamond (d) Hexagon (e) Star (regular hexagram) (f) Rotated Square (g) Rotated Hexagon. The characteristic and equivalent dimensions (aeq ) corresponding to different strut shapes are also presented (for analytical solutions, see [19]).

2.1. Development of extended HS model

kHS;Lower ¼ kf

It is important to note that the series and parallel models considered by Dietrich et al. [9] are the absolute bounds for the effective thermal conductivity of any heterogeneous two-phase medium. Nevertheless, it is also known that Hashin–Shtrikman (HS) bounds are the narrowest effective thermal conductivity bounds for macroscopically homogeneous, isotropic media (see [20]). These bounds are given as:




kHS;Upper

2ks þ kf  2ðks  kf Þeo ¼ ks 2ks þ kf þ ðks  kf Þeo



ð1Þ




kHS;Lower

 2kf þ ks  2ðkf  ks Þð1  eo Þ ¼ kf 2kf þ ks þ ðkf  ks Þð1  eo Þ

ð2Þ

The approach proposed by Kumar and Topin [19] was applied to the foam structure (based on tetrakaidecahedron geometry) to derive the correlation. HS bounds were utilized in order to incorporate varying individual geometries and impact of the ratios of thermal conductivities of the constituent phases. The effective thermal conductivity lies between these bounds of a two phase system and can be found by incorporating a correlation factor X. This relationship is given by Eq. (3) as: X keff ¼ kXHS;Upper  k1 HS;Lower

06X61


 2kf þ ks  2ðkf  ks Þw 2kf þ ks þ ðkf  ks Þw

ð6Þ

where aeq ¼ Req =Lc (ratio of equivalent circular strut radius to nodeto-node length) and b ¼ Ls =Lc (ratio of strut length to node-to-node length) respectively (see [19]). Eq. (3) is solved for X in terms of kHS;Upper , kHS;Lower and keff which contains the morphological function w. The solution is:

X¼

lnðkeff Þ  lnðkHS;Lower Þ lnðkHS;Upper Þ  lnðkHS;Lower Þ

ð7Þ

For a known eo , one can have only one kHS;Upper and kHS;Lower which leads to only one value of keff for a given value of X. Thus, using this approach implies that the influence of solid foam geometry as well as thermal properties will be taken into account only through the X value. From the database of numerically obtained keff generated by Kumar and Topin [19], values of X using Eq. (7) were extracted

ð3Þ

where keff is simply taken as a geometric average between upper and lower bounds and X is the fraction of the upper bound conductivity material while 1  X is the fraction of the lower bound conductivity material. The elementary structure of foam is idealized as an isotropic Kelvin cell (see Fig. 1-left) and different strut cross sections are considered (see Fig. 1-right). The porosity (eo ) can be expressed as a function of dimensionless morphological parameters (w) of foams of any strut shape by Eq. (4). The HS upper and lower bounds models (see Eqs. (1) and (2)) are rewritten by substituting w over eo as given by Eqs. (5) and (6):

1  eo ¼ w ¼ 12pa2eq b þ
kHS;Upper ¼ ks

32 3 paeq 3

2ks þ kf  2ðks  kf Þð1  wÞ 2ks þ kf þ ðks  kf Þð1  wÞ

ð4Þ  ð5Þ

Fig. 2. Plot of X (dimensionless) and S (dimensionless). A small algorithm is also presented to determine keff from three known input parameters.

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and a best fit was obtained that includes morphological parameters and the ratio of constituent phases. A plot of X as function of S ¼ lnðw2:25 :ks =kf Þ is shown in Fig. 2. It is observed that X increases roughly with increase in S, following an approximately quadratic polynomial regime, and all the values of factor X for different porosities collapse on a single curve. There is no physical reason to choose a quadratic polynomial function and we do not claim any physical meaning to the curve fitting. The quadratic polynomial function is the simplest function that gives a good approximation of effective thermal conductivity data. From Fig. 2, it is found that:

and parallel bounds respectively. The physical interpretation of the new proposed model in Eq. (10) is presented in Fig. 3

"

keff ¼ N 

X ¼ 0:0028S þ 0:0395S þ 0:8226

ð8Þ

Due to the scattering of experimental data in Fig. 2, the root mean square deviation (RMSD) of the fitting relationship was calculated, which is given by Eq. (9):

RMSD ¼ 10RMSðELOGÞ  1with ELOG ¼ log

    keff keff  log kf calc kf exp ð9Þ

An RMSD value of 0.0455 (or 4.55%) was obtained for calculated values of the effective thermal conductivity. From Eq. (8) and Fig. 2, it is evident that X is a function of morphological parameters and the ratios of the thermal conductivities of constituent phases and is applicable to a wide range of thermal conductivity ratios ðks =kf Þ. Note that the parameters X and S obtained in the present work using HS bounds are different than those reported in our previous works (obtained using Wiener bounds). To build our correlation, it is better to use the closest possible bounds to obtain accurate value of X. The HS bounds always lie within the Series–Parallel bounds. This is mainly interesting for low porosity or high phase conductivity ratio as classical bounds can be several orders of magnitude apart. HS bounds provide closer estimates of effective thermal conductivity values even at low porosity as well as low thermal conductivity ratios.

In this section, the theoretical model proposed by Mendes et al. [21] was extended to predict effective thermal conductivity. The new ‘effective parallel model’ is based on combination of series

1M kparallel

1 #

kparallel ¼ wðks  kf Þ þ kf 1

¼

    1 1 1 þ w  ks kf kf

þ ð1  NÞ  kparallel

ð10Þ

ð11aÞ ð11bÞ

where, w is a function of morphological parameters (see also Eqs. (5) and (6)). Eq. (10) is an effective parallel model that makes more sense for a material with a continuous solid phase, because heat transfer can occur in the parallel part without restriction from an effective serial part for the conditions when ks  kf . For the conditions when ks P kf , the adjustable parameters M and N ensure that they are explicit functions of morphological parameters of foam structure and thermal conductivities of constituent phases, thus allowing for accurate predictions of effective thermal conductivity values. In open-cell foams, effective solid phase tortuosity (ST ) is a characteristic path responsible for heat career and has been known to describe heat transfer characteristics for decades. Let us consider an idealized 3-D porous structure inside a cubic volume of length L and surface area W 2 , in which the solid phase, immersed in a fluid phase, forms a continuous path, as schematically shown in Fig. 4 (left). 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 area w2 . ST can be evaluated using the morphological characteristics of open-cell foams. From basic definitions, the volume fraction of the solid phase (1  eo ) and the effective solid tortuosity ST , for this idealized porous medium can be expressed by Eq. (12):

1  eo ¼ 2.2. Effective parallel model: Introduction of effective solid phase tortuosity (ST )

kseries

þ

where M and N are two adjustable parameters in the range [0,1]. kparallel and kseries of Eq. (10) can be written as (see [19]):

kseries 2

M

ST ¼

lw

2

LW 2

8ð1  eo Þ

pa2eq

ð12aÞ

ð12bÞ

pffiffiffi where, ST ¼ Ll , W ¼ 2 2Lc and Lc is node-to-node length of the foam structure respectively (see [19]).

Fig. 3. Schematic presentation of ‘effective parallel model’ (Eq. (10)) as a combination of (a) series and parallel connections arranged in series and (b) parallel combination of: the resultant of (a) and parallel connection (see also [21]).

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Fig. 4. Left: schematic representation of an idealized three-dimensional foam-like structure inside a cubic unit cell. Right: plot of M (dimensionless) and S0 (dimensionless).

Mendes et al. [21] presented a relation to relate N and ST by the following expression. The effective solid phase tortuosity (ST ) appearing in Eq. (13) depends on the effective thermal conductivity of the solid phase in vacuum and porosity, which implies that for each foam sample, a unique numerical simulation or experiment is required to obtain ST . These authors obtained this relation only by performing numerical simulations and no correlation was provided to predict analytically beforehand the value of parameter, N.

N ¼1

1 S2T

ð13Þ

However, Eq. (12b) contains the information of purely morphological parameters of foam structure of any strut shape, porosity and size. This equation can be used to calculate easily the values of parameter, N. Values of M using Eq. (10) have been determined by substituting N obtained directly from solid tortuosity, ST (Eq. (12)) and keff (from the database of [19]. A better fit was obtained to describe M that includes morphological parameters and ratio of constituent phases. The representation of M as function of S0 ¼ lnðw:ks =kf Þ is shown in Fig. 4 (right) and is given by the Eq. (14). RMSD for the calculated values of M is 10%.

M ¼ Expð0:0147S02  0:824S0  0:253Þ

ð14Þ

In short, the present new model in Eq. (10) requires ks , kf and w as primary inputs. Based on these inputs, solid phase tortuosity (ST ), followed by determination of parameters, M and N can be uniquely obtained. Using these parameters, keff can be precisely calculated. 2.3. Development of KT model In the literature, intrinsic solid phase thermal conductivity of foam materials has not been generally measured and thus, most of the authors have used bulk or parent material thermal conductivity to derive empirical correlations. Moreover, the majority of these correlations were derived for samples made of metal or reticulated vitreous carbon with porosities larger than 0.89 and air as an interstitial fluid. In such cases, heat conduction is mainly through the solid phase. Dietrich et al. [9] experimentally measured the intrinsic solid phase conductivity for Al2 O3, Mullite and OBSiC ceramic foams (ks ¼ 26; 4:4; 15 W m1 K1 respectively). The bulk phase thermal conductivities of these ceramic foams are kBs ¼ 35; 6:1;

90 W m1 K1 respectively. Similarly, Kumar et al. [18] predicted

ks ¼ 150 W m1 K1 and ks ¼ 128 W m1 K1 for Al T-6201 and Al T-6101 ERG foams respectively. The bulk phase thermal conductivities of these metal-alloys foam samples are kBs ¼ 218—237 W m1 K1 and kBs ¼ 160—180 W m1 K1 . Manufacturing processes greatly impact the solid phase thermal conductivity of the parent material when transformed into foams and thus, there must be significant difference in the intrinsic solid phase thermal conductivity of foam compared to the same parent material one. Moreover, it will be cumbersome and time consuming to measure experimentally intrinsic solid phase conductivity of foam (or strut) for different samples. Thus, there is a need to find an empirical correlation that incorporates the intrinsic solid phase thermal conductivity of foams, fluid phase and morphological characteristics of foam structure. On one hand, the correlations developed in Eqs. (3) and (10) can be used to determine intrinsic solid phase conductivity when effective thermal conductivity is known. However, the formulation would be rather complex as these involve many dimensionless morphological parameters as well as the ratio of constituent phases. Using this method, the intrinsic solid phase conductivity could be determined by several iterative processes. On the other hand, a better solution might be the derivation of a simplified yet efficient correlation that can yield an accurate estimate of intrinsic solid phase conductivity for a given value of effective thermal conductivity. For this purpose, several combinations of keff , ks ; kf , w and X (from Section 2.1) have been tested. Using the database of keff values for all porosities and strut shapes in the wide range of solid to fluid conductivity ratios (ks =kf ¼ 10—30; 000), the best fit has been obtained and is presented in Fig. 5. From Fig. 5, the relation is given by Eq. (15) and RMSD for calculated values of keff is 5.02%.

keff ¼

2  k  ðwÞ1:2  ðXÞ1:5 3 s

ð15Þ

where, keff ¼ kkeff and ks ¼ kks . f

f

A simple method to calculate the intrinsic solid phase conductivity (ks ) for a given w, keff and kf has been described below:Step I: Rewrite Eq. (15) as:

"   2 keff 2 ks ks 1:2 ¼  ðwÞ 0:0028 ln w2:25 3 kf kf kf    1:5 ks þ 0:8226 þ 0:0395 ln w2:25 kf

ð16Þ

Step II: Assign ks =kf ¼ K s and keff =kf ¼ K e and apply natural log functions on both sides:

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545

3.1. Comparison and validation of extended HS model and effective parallel model

Fig. 5. Presentation of KT model. Plot of keff vs. 23 ks :ðwÞ1:2 :ðXÞ1:5 (dimensionless).

h i 2 lnðK s Þ  1:5 ln n1 flnðw2:25 :K s Þg þ n2 flnðw2:25 :K s Þg þ n3 þ 1:2lnðwÞ ¼ lnð1:5:K e Þ

ð17aÞ

h i 2 lnðK s Þ  1:5 ln n1 flnðw2:25 :K s Þg þ n2 flnðw2:25 :K s Þg þ n3 þ 1:2 lnðwÞ  lnð1:5:K e Þ ¼ 0

ð17bÞ

where n1 = 0.0028, n2 = 0.0395, n3 = 0.8226. Step III: Using iterative process, solve for K s .

3. Comparison and validation The effective thermal conductivity correlations (developed in the Sections 2.1–2.3) are compared and validated against experimental and numerical data of open-cell foams of different materials.

We found only one set of experimental data containing simultaneously morphological data, intrinsic solid phase thermal conductivity and effective thermal conductivity [9]. To our knowledge, we did not find any experimental measurements of intrinsic solid phase conductivity for commonly used metal/metal-alloy (Al/Cu/ Ni) foams. In this latter case, only effective thermal conductivity values are available while intrinsic solid phase conductivity differs largely from bulk/parent material conductivity [19]. Extended HS model and effective parallel model are first compared and validated against the numerical dataset of keff values outlined in the works of Kumar and Topin [19] and subsequently, against the experimental data reported by Dietrich et al. [9]. From Fig. 6 (left), it is found that the Eq. (8) underestimates and overestimates the correlation factor X, but the error lies within ±2%. This justifies the present choice of X to assimilate the impact of very low and high solid to fluid conductivity ratios with morphological parameters. Further, analytical values of X obtained from Eq. (8) are substituted in Eq. (3). Fig. 6 (right) demonstrates an excellent agreement. The errors are minimal. However, an error of ±5% is observed for circular strut shape that could be attributed to the approximation of complex node shape at very low porosities (eo < 0:70). Thus, the present study states that effective thermal conductivity can be precisely calculated using extended HS bounds approach if the morphological parameters or the relations between them are known. This validation confirms that the extended HS model gives precise results for the whole range of porosity (0:60 < eo < 0:95) as it can encompass all the strut shapes; even the complex cases depict the model’s robustness. The effective parallel model developed in Eq. (10) introduced the importance of effective solid phase tortuosity of the foam structure. Values of adjustable parameter M obtained from numerical values are compared and validated against those obtained using Eq. (14) and presented in Fig. 7 (left). It is observed that Eq. (14) underestimates and overestimates the parameter M in the error range of ±10% for all strut shapes. Effective thermal conductivity values were then calculated using Eq. (10) from the analytically obtained values of M. The validation is presented in Fig. 7

Fig. 6. Comparison and validation of analytical and experimental X (left) and keff (right). An excellent agreement is observed for extended HS model. The error is mainly due to the node approximation for strut shapes at low porosity (eo < 0:70).

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Fig. 7. Comparison and validation of analytical and experimental M (left) and keff (right). Effective parallel model predicts accurate results by introducing effective solid phase tortuosity and morphological parameters in the current approach.

Table 2 Validation of analytically determined effective thermal conductivity (keff ) and intrinsic solid phase conductivity (ks ) of Al2O3, Mullite and OBSiC ceramic foams (data taken from [9] of different pore sizes and porosities with air as an interstitial fluid.

⁄

Grey blocks represent the experimental data while the white blocks represent the calculated/analytical data.

(right) and errors for different strut shapes are in the range of ±7%. This correlation agrees qualitatively against the numerical dataset of effective thermal conductivity, which lends confidence to the current approach of introducing effective solid phase tortuosity and morphological parameters. The errors in the effective parallel model are a bit higher compared to those in the HS model. This was expected due to the fact that the effective parallel model takes series and parallel bounds models into account. The accuracy and applicability of these models are expected to decrease drastically for porous media where the distribution of solid and fluid phases considerably departs from the parallel arrangement; where the difference between solid and fluid phase is of the same order. The correlation factor X and adjustable parameter M that is obtained from experimental data of Dietrich et al. [9] and analytically calculated from Eqs. (8) and (14) are compared and validated

(see Table 2), which suggests that the empirical correlations (Eqs. (8) and (14)) are good predictions of X and M obtained from experiments. Further, calculated effective thermal conductivity values obtained from the extended HS model (Eq. (3)) and effective parallel model (Eq. (10)) are validated against experimental effective thermal conductivity data of ceramic foams (Table 2). The errors could be associated to morphological approximation at node junction and uncertainties in the estimated experimental data and thus, are comparable.

3.2. Comparison and validation of KT model The analytically calculated intrinsic solid phase conductivity values (ks ) obtained from the KT model using Eq. (17a) (or (17b)) are compared and validated against the experimental ks values of

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547

Table 3 Validation of analytical effective thermal conductivity (keff ) against experiments for foam samples of different pore sizes and porosities with air as interstitial fluid.

⁄

Grey blocks represent the experimental data while the white blocks represent the calculated/analytical data.

ceramic foams reported by Dietrich et al. [9] as presented in Table 2 as it is actually the only available full set of experimental data. The intrinsic ks values for each effective thermal conductivity value and its averaged value for the same type of material are calculated and presented. The comparison shows an excellent validation of the KT model’s prediction of the intrinsic value of solid phase conductivity from experimental keff values. All of these add to the confidence of the proposed correlation as given by Eq. (15). Further, analytical intrinsic ks values are calculated using KT model for various metal foams and are substituted in the extended HS model and effective parallel model to predict effective thermal conductivity values, which is presented in Table 3. An excellent agreement between predicted and measured keff values is observed for different authors (e.g. [4,6–8,16,22,23] and different foam materials. This comparison shows that the three models are consistent with each other and their uses can be combined.

metallic or ceramic). They were validated against experimental data obtained from the reduced set of foam samples of the same material. They may or may not work for foams of different materials as discussed in Section 1. In the present work, we tried to obtain ‘‘reliable” correlations by using an extended set of input morphological and thermo-physical data. The obtained correlations were then tested against numerical and experimental data of foam samples of different materials reported in the literature. We have focussed on open-cell (Kelvinlike) foams where only heat conduction phenomena were modeled. Other heat transfer modes can be added using specific descriptions e.g. dispersion, radiative heat transfer, etc. The correlations were primarily developed based on effective thermal conductivity values obtained from 3-D pore scale numerical simulations on virtual Kelvin-like foam samples of different strut cross sections. The present correlations should be applicable to any open-cell foams meeting the following constraints:

4. Limitations and validity range of empirical correlations Many correlations were proposed in the literature for predicting effective thermal conductivity of foams in a generic way. Nevertheless, they were derived for a specific set of foam samples (either

   

Can reasonably be idealized as an isotropic Kelvin-cell structure. Porosity ranging from 0.60 to 0.95. Phase thermal conductivity ratio ks =kf P 10. Constant strut cross section along the ligament axis.

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The correlations are expected to predict accurate effective thermal conductivity values for variable strut cross section provided all the struts vary the same way. These models will probably hold true in case of a smooth variation of morphological properties (e.g. gradient of pore size). In fact, the available commercial foam samples (e.g. ERG) are slightly anisotropic in nature. However, such foams are nearly isotropic in terms of physical properties and this is one of the reasons that the present correlations based on an isotropic unit cell provide excellent predictions of commercial foam properties (see Section 3). Obviously, these correlations would not be applicable for more anisotropic, deformed foam structures, and thus specific models should be developed.

5. Concluding remarks The empirical correlations reported in the literature for the estimation of effective thermal conductivity are based on the choice of different resistor networking models (series, parallel, upper and lower HS models, etc.). In this view, most of the correlations are related only to the porosity (mainly for high porosity foam samples) and an adjustable parameter that has been determined from experiments. Most commonly, the intrinsic solid phase thermal conductivities of foam materials are not measured. Nowadays, for a variety of applications, many researchers have directly estimated the effective thermal conductivity values of different open-cell foams using the lCT-CFD approach. However, this method consumes a considerable amount of times and resources and therefore, its application is justified only when higher accuracy is required or when validating or calibrating simpler models. Based on the findings from the literature and present work, it can be suggested that:  Most of the correlations presented in the literature were derived by taking the bulk/parent conductivity of the solid phase into account. Manufacturing process impacts greatly on the thermal conductivity of foam material and thus, intrinsic solid phase conductivity should be measured/known as a priori.  Different morphological parameters such as variation of strut cross section along ligament axis could impact strongly on effective thermal conductivity value for the same porosity of foam samples.  The simplest series–parallel models are sufficient to characterize effective thermal conductivity. However, choosing the bounds that are narrower than series–parallel bounds provide more accurate results.  Effective solid tortuosity estimates the main heat flux path tortuosity. However, when phase thermal conductivity ratio decreases, some corrections should be included. In the perspective of their greater simplicity, empirical models are sufficient and efficient to obtain satisfactory estimates of the effective thermal conductivity of open-cell foams of different materials for most applications. Three dependent/interdependent empirical correlations have been derived: the extended HS model, the effective parallel model, and the KT model. These correlations account for a large range of solid to fluid phase conductivity ratios, variable porosities, morphological parameters and different strut shapes together. It has been demonstrated that morphological parameters, effective solid phase tortuosity and intrinsic solid phase thermal conductivity are indeed the salient functions to determine effective thermal conductivity. The correlations presented here are validated against the numerical and experimental values reported in the literature and are found to predict accurately the effective thermal conductivity for different metallic and ceramic foams. These

models can be easily coupled with one another if all the information about the morphological and material characteristics is not known. Conflict of interest None declared. Acknowledgement The authors would like to thank the ANR (Agence Nationale de la Recherche) for financial support in the framework of FOAM project and all project partners for their assistance. References [1] T.J. Lu, H.A. Stone, M.F. Ashby, Heat transfer in open-cell metal foams, Acta Mater. 46 (10) (1998) 3619–3635. [2] S.Y. Kim, J.W. Paek, B.H. Kang, Flow and heat transfer correlations for porous fin in a plate-fin heat exchanger, ASME J. Heat Transfer 122 (3) (2000) 572–578. [3] M. Kaviany, Principles of heat transfer in porous media, in: Mechanical Engineering Series, Springer-Verlag, 1995. [4] E. Takegoshi, Y. Hirasawa, J. Matsuo, K. Okui, A study on effective thermal conductivity of porous metals, Trans. Japan Soc. Mech. Eng. B 58 (1992) 879– 884. [5] V.V. Calmidi, R.L. Mahajan, The effective thermal conductivity of high porosity fibrous metal foams, ASME J. Heat Transfer 121 (2) (1999) 466–471. [6] J.W. Paek, B.H. Kang, S.Y. Kim, J.M. Hyun, Effective thermal conductivity and permeability of aluminium foam materials, Int. J. Thermophys. 21 (2) (2000) 453–464. [7] A. Bhattacharya, V.V. Calmidi, R.L. Mahajan, Thermophysical properties of high porosity metal foams, Int. J. Heat Mass Transfer 45 (5) (2002) 1017–1031. [8] E. Solórzano, J.A. Reglero, M.A. Rodríguez-Pérez, D. Lehmhus, M. Wichmann, J. A. de Saja, An experimental study on the thermal conductivity of aluminium foams by using the transient plane source method, Int. J. Heat Mass Transfer 51 (25–26) (2008) 6259–6267. [9] B. Dietrich, G. Schell, E.C. Bucharsky, R. Oberacker, M.J. Hoffmann, W. Schabel, M. Kind, H. Martin, Determination of the thermal properties of ceramic sponges, Int. J. Heat Mass Transfer 53 (1) (2010) 198–205. [10] M. Wang, N. Pan, Modeling and prediction of the effective thermal conductivity of random open-cell porous foams, Int. J. Heat Mass Transfer 51 (5–6) (2008) 1325–1331. [11] A. Druma, M. Kharil Alam, C. Druma, Surface area and conductivity of open-cell carbon foams, J. Miner. Mater. Charact. Eng. 5 (2006) 73–86. [12] J. Vicente, F. Topin, J.V. Daurelle, F. Rigollet, Thermal conductivity of metallic foam: simulation on real X-ray tomographied porous medium and photothermal experiments, in: Proceedings of IHTC 13, Sydney, 2006. [13] E.N. Schmierer, A. Razani, Self-consistent open-celled metal foam model for thermal applications, J. Heat Transfer 128 (11) (2006) 1194–1203. [14] S. Krishnan, S. Garimella, J.Y. Murthy, Simulation of thermal transport in opencell metal foams: effects of periodic unit-cell structure, J. Heat Transfer 130 (2) (2008) 24503–24507. [15] R. Coquard, D. Baillis, Numerical investigation of conductive heat transfer in high-porosity foams, Acta Mater. 57 (18) (2009) 5466–5479. [16] K.K. Bodla, J.Y. Murthy, S.V. Garimella, Resistance network-based thermal conductivity model for metal foams, Comput. Mater. Sci. 50 (2010) 622–632. [17] J. Randrianalisoa, R. Coquard, D. Baillis, Microscale direct calculation of solid phase conductivity of Voronoi’s foams, J. Porous Media 16 (5) (2013) 411–426. [18] P. Kumar, F. Topin, J. Vicente, Determination of effective thermal conductivity from geometrical properties: application to open cell foams, Int. J. Therm. Sci. 81 (2014) 13–28. [19] P. Kumar, F. Topin, Simultaneous determination of intrinsic solid phase conductivity and effective thermal conductivity of Kelvin like foams, Appl. Therm. Eng. 71 (1) (2014) 536–547. [20] A.A.M. Mendes, S. Ray, D. Trimis, A simple and efficient method for the evaluation of effective thermal conductivity of open-cell foam-like structures, Int. J. Heat Mass Transfer 66 (2013) 412–422. [21] A.A.M. Mendes, S. Ray, D. Trimis, An improved model for the effective thermal conductivity of open-cell porous foams, Int. J. Heat Mass Transfer 75 (2014) 224–230. [22] P. Ranut, E. Nobile, L. Mancini, High resolution X-ray microtomography-based CFD simulation for the characterization of flow permeability and effective thermal conductivity of aluminum metal foams, Exp. Therm. Fluid. Sci. (2014), http://dx.doi.org/10.1016/j.expthermflusci.2014.10.018. [23] R. Wulf, A.A.M. Mendes, V. Skibina, A. Al-Zoubi, D. Trimis, S. Ray, U. Gross, Experimental and numerical determination of effective thermal conductivity of open cell FeCrAl-alloy metal foams, Int. J. Therm. Sci. 86 (2014) 95–103. [24] R. Coquard, D. Baillis, Radiative and conductive thermal properties of foams, in: A. Öchsner, G. Murch, M.D. Lemos (Eds.), Cellular and Porous Materials: Thermal Properties Simulation and Prediction, John Wiley & Sons, 2008, pp. 343–384.

P. Kumar, F. Topin / International Journal of Heat and Mass Transfer 92 (2016) 539–549 [25] M. Wang, N. Pan, Predictions of effective physical properties of complex multiphase materials, Mater. Sci. Eng. R 63 (1) (2008) 1–30. [26] R. Coquard, D. Rochais, D. Baillis, Conductive and radiative heat transfer in ceramic and metal foams at fire temperatures, Fire Technol. 48 (2012) 699–732. [27] J. Randrianalisoa, D. Baillis, Thermal conductive and radiative properties of solid foams: traditional and recent advanced modelling approaches, C.R. Phys. 15 (8–9) (2014) 683–695. [28] K. Boomsma, D. Poulikakos, On the effective thermal conductivity of a threedimensionally structured fluid-saturated metal foam, Int. J. Heat Mass Transfer 44 (2001) 827–836. [29] R. Singh, H.S. Kasana, Computational aspects of effective thermal conductivity of highly porous metal foams, Appl. Therm. Eng. 24 (2004) 1841–1849. [30] Z. Dai, K. Nawaz, Y. Park, J. Bock, A. Jacobi, Correcting and extending the Boomsma–Poulikakos effective thermal conductivity model for threedimensional, fluid-saturated metal foams, Int. Commun. Heat Mass Transfer 37 (6) (2010) 575–580.

549

[31] D. Edouard, The effective thermal conductivity for ‘‘slim” and ‘‘fat” foams, AIChE 57 (6) (2011) 1646–1651. [32] P. Kumar, F. Topin, The geometric and thermohydraulic characterization of ceramic foams: an analytical approach, Acta Mater. 75 (2014) 273–286. [33] H. Yang, M. Zhao, Z.L. Gu, L.W. Jin, J.C. Chai, A further discussion on the effective thermal conductivity of metal foam: an improved model, Int. J. Heat Mass Transf. 86 (2015) 207–211. [34] O. Wiener, Die theorie des mischkörpes für das feld der stationären strömung, Abh. Math. Phys. Kl. Königl. Ges. 32 (1912) 509–604. [35] J.M. Hugo, Heat transfer in cellular porous medium at high porosity: application to structural optimization of fin exchangers (Ph.D. thesis), AixMarseille University, 2012. [36] Z. Hashin, S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33 (1962) 3125– 3131. [37] O. Krischer, W. Kast, Die wissenschaftlichen Grundlagen der Trocknungstechnik, 1er Band, third ed., Springer-Verlag, Berlin, Germany, 1978.