1 Introduction

The thermal conductivity of porous materials, more particularly open-cell foams, plays an important role in many industrial processes. Heat conduction takes place through a solid skeleton and a fluid passing through the 3-D interconnected network of open-cells. In order to effectively utilize open-cell foams in heat transfer applications e.g. heat exchangers, volumetric solar receiver etc., accurate knowledge of their thermal transport properties are needed.

Many heat transfer mechanisms at pore scale could be included in equivalent thermal conductivity. The effective thermal conductivity refers to the case when only diffusion plays a role. The energy transport is then controlled by the effective thermal conductivity (e.g. [1–5]). However, at high temperatures or for foam materials of extremely low conductivity, an equivalent thermal conductivity depends also on radiative heat transfer (e.g. [6–8]) for which the expression or formulation is completely different.

The ETC of open-cell foams is not only dependent on the constituent phase properties and porosity, but also on the structure of the materials (e.g. [4, 9–12]). Moreover, the intrinsic solid phase conductivity may be different from the parent/bulk material conductivity. Miettinen et al. [9] showed experimentally that there exists no simple relationship between intrinsic solid phase heat conductivity and porosity. Dietrich et al. [4] measured the thermal conductivity of strut materials and obtained a decrease in their intrinsic solid phase conductivity compared to the conductivity of pure/bulk material. Kumar et al. [10] showed a decrease of approximately 20–25% in the solid phase thermal conductivity of bulk material when transformed in foams (see also [11, 12]) from their calculated ETC data.

Apart from experiments, substantial efforts have been made in the recent years to estimate the effective thermal conductivity based on: (1) numerical simulations using X-ray µCT images of actual foam samples (e.g. [13–20]) (2) numerical simulations on idealized representation of open-cell foams (e.g. [10, 11, 21, 22]) and, (3) empirical correlations by fitting experimental data (e.g. [1, 2, 4, 5, 12, 23, 24]).

Despite accuracy and precision of ETC results obtained from numerical simulations (or experiments), they are considerably time and resource consuming. Moreover, ETC of a specific structure for a given working fluid has to be obtained individually on a case to case basis. Thus, an empirical correlation is a reasonable compromise between measurements and computational time for quick and accurate evaluation of ETC. It presents general applicability for a wide range of materials, porosities, and ratios of solid and fluid phase conductivities.

Various simplified bounds and models exist in literature to predict the ETC values for different porous media. The most common inputs for these models are material properties (e.g. solid and fluid conductivities) and material morphology (e.g. porosity). Nevertheless, prediction of ETC values for different class of materials i.e. foam like structures is not straightforward. Consequently, various authors proposed empirical correlations using these bounds/models or combination between them to predict reasonable ETC values with or without an adjustable parameter depending on the ratio of constituent phases.

In open-cell foams, the empirical value of adjustable parameter in ETC correlations varies between employed models and used type of materials. Variations in adjustable parameter are due to the preferred choice of model/bound by authors and depend usually on the type of foam material used (metallic or ceramic) and thus, resulting in a massive gap to have access to complete range of conductivity ratios. Consequently, there is no simple model that can be used for all existing types of open-cell foams.

In this work, our intent is to develop generalized ETC correlations in accessible porosity range (0.60 \(< \varepsilon_{o} <\) 0.98) of foams resembling Kelvin-like cell structure and extend the conductivity ratios range (\(\lambda_{s} /\lambda_{f}\) = 10–30,000) in order to fill this gap. It is from this view point, five basic arrangements of simplified models and their combinations including the series and parallel models [25], Hashin–Shtrikman upper and lower bounds [26], effective medium theory (EMT) [27], and, reciprocity model [28] are adapted as generic minimum and maximum bounds of thermal conductivity in both arithmetic and geometric schemes. A functional parameter is obtained by fitting ETC data obtained numerically from our previous works [10, 11]. Obviously, this parameter cannot be obtained beforehand but it has been determined in our present work for the whole range of porosities and thermal conductivity ratios. Thus, it could be used as a function of the traditional input parameters (foam morphology and material property: conductivities) in order to predict ETC value for any Kelvin-like foam structure.

2 Literature study of ETC correlations

We discuss here ETC as an intrinsic property of conductive heat transfer and not an apparent property representing many other unspecified heat transfer phenomena (e.g. radiation, micro-convection, dispersion, evaporation–condensation etc.). Mixing heat transfer mode in a single apparent coefficient leads to a confusing situation and prohibit comparison and use of obtained data. Nevertheless in several cases, the diffusive heat transfer is not the principal one for a given situation. This contribution could be calculated separately and added eventually (e.g. [6]).

Usually, commercially available real foams are close to periodic in nature. Pieper and Klein [29] demonstrated that the periodic homogenization for real structures close to periodic gives very accurate results. This led researchers to derive empirical correlations for quick estimation of ETC values that are generally based either on asymptotic approaches or on a micro-structural approach in case of open-cell foams. It is widely known that the ETC strongly depends on porosity and the ratio of thermal conductivities of the constituent phases but also on a lesser extent to the distributions of the solid phase (e.g. between struts and lumps) that depends on the manufacturing process. Using analytical modelling, some authors (e.g. [1, 2, 5, 14, 23, 30]) have considered lumps at the node junctions while others (e.g. [4, 10–12, 24]) did not assume the presence of lumps of matter at the node junctions while deriving their empirical correlations. Majority of these works concerns very high conductive metal foams (e.g. [1, 2, 5, 24]) in high porosity range (0.88 \(<\upvarepsilon_{o} <\) 0.97). On the other hand, a few works are dedicated to low conductive ceramic foams (e.g. [4]) for moderate porosity range (0.75 \(<\upvarepsilon_{o} <\) 0.97).

For the validation of empirical correlations against ETC data, it is henceforth of major importance to measure the intrinsic solid phase thermal conductivity of the foam sample because different commercially available foams employ different manufacturing techniques; that lead to significant changes in the intrinsic solid phase thermal conductivity of foams, compared to the same parent/bulk material [4, 9–11]. The correlations reported in the literature can be classified into three major groups that use mainly the same preferences of building correlations. Most commonly, one or two adjustable parameters that contain information about morphology are obtained by fitting experimental or numerical ETC data to derive correlations.

The first group of authors used only porosity and derived correlations based on series and bounds arranged in either geometric (e.g. [24]) or arithmetic (e.g. [4]) scheme. Note that these models do not introduce additional morphological parameters to describe ETC; although this influence has already been shown to be significant (see [10–12, 17, 18]).

The second group of authors tried to introduce other morphological parameters in their correlation based on 2-D (e.g. [1, 2]) or 3-D foam geometry (e.g. [1, 2, 5, 12, 23, 29–31]) and deduced an adjustable parameter from experimental data. Most of these correlations are limited to very high conductive materials (e.g. Al, Cu) while neglecting the influence of low conductive fluid such as air or water (\(\lambda_{s}^{B} /\lambda_{f} >\) 400). These correlations give poor estimates of ETC values for foams of different materials. Constant and variable values of adjustable parameters were obtained by fitting experimental data as a function of morphology and conductivity ratios. Contrary to this, Qu et al. [30] described a so-called ‘morphological parameter’ that was obtained by fitting the experimental ETC data is rather material dependent (thermal conductivity ratio) instead of foam morphology and is only valid for a few types of foam materials. Kumar and Topin [12] derived ETC correlation for low conductive ceramic foam materials (150 \(< \lambda_{s} /\lambda_{f} <\) 900) and modified the Lemlich approach. Their models could be used together to predict the intrinsic solid phase conductivity of foams of different materials irrespective of fluid phase and porosity when ETC is known.

The third of group of authors calculated ETC from numerical solution of heat equation on idealized or reconstructed open and closed foam structures (e.g. [7, 10–20, 22]). Pore scale numerical simulations (using finite element, finite volume and Lattice-Boltzmann methods) allow solving heat equation within both solid and fluid phases on such foam structures while varying porosity, morphology and conductivity ratios. Druma et al. [22] concluded that some simplified models could not accurately predict the ETC of foams (approximated by spherical pores, homogeneously dispersed within a solid matrix) over the complete range of porosities. Coquard and Baillis [14] deduced that the realistic representations of foam structures (e.g. Cubic, Tetrakaidecahedron and Weaire-Phelan unit cells) did not account for the commercial foam irregularities and imperfections. Mendes et al. [17] observed that using HS bounds to build an ETC correlation give the best results for most of the investigated foam structures. They also proposed an empirical correlation using a complex arrangement of Series and Parallel models [18]. In both cases, the adjustable parameters can only be obtained from the complete set of ETC data. Kumar et al. [10] performed numerical simulations on Kelvin-like cell structure with convex triangular cross section ligaments to determine ETC over a wide range of solid to fluid conductivity ratios. The weighted factor appearing in their correlation is deduced from the morphological parameters and the ratio of solid to fluid phase thermal conductivities. This work was further extended to different strut cross-sections [11]. These authors found that the cell size and the strut cross section shape have negligible effects on ETC.

3 Simplified ETC models of porous media

The effective thermal conductivity of open-cell foams is much higher than those of the granular porous media. Since the solid has a higher conductivity, the interconnected microstructure virtually increases the pathway of thermal conduction (or energy flux) in the homogenous medium compared with the discrete granular microstructure, and therefore enhances the effective thermal conductivity of the foam material.

Two models of a two-phase material, which are not themselves realistic for a foam but provide useful results are the “parallel” and “series” models illustrated in Fig. 1a, b, respectively and are often used as benchmarks for new model validations. The physical structures assumed in the derivations of the Series and Parallel models are layers of the components aligned either perpendicular or parallel to the heat flux as presented in Eqs. 1 and 2.

Fig. 1
figure 1

Five fundamental effective thermal conductivity structural models for two-component (solid and fluid phases) open-cell foam materials (assuming the heat flow is in the vertical direction)

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$$\lambda_{parallel} = (1 - \varepsilon_{o} )\lambda_{s} + \varepsilon_{o} \lambda_{f}$$
(1)
$$\lambda_{series} = \frac{{\lambda_{s} \cdot \lambda_{f} }}{{(1 - \varepsilon_{o} )\lambda_{f} + \varepsilon_{o} \lambda_{s} }}$$
(2)

Hashin and Shtrikman [26] derived effective conductivity bounds on the basis of a variational approach that were the best (i.e. narrowest) possible bounds for macroscopically homogeneous, isotropic, two-phase materials that could be derived from the components’ volume fractions and conductivities. The bounds state that the ETC of any isotropic mixture of several isotropic conducting materials satisfies certain inequalities independently of the structure of a porous medium (see Fig. 1c). The Hashin–Shtrikman (HS) bounds always lie within the Series–Parallel bounds, regardless of the components volume fractions or thermal conductivities and are given by Eqs. 3 and 4.

$$\lambda_{HS, Upper} = \lambda_{s} \left[ {\frac{{2 \lambda_{s} + \lambda_{f} - 2\left( { \lambda_{s} - \lambda_{f} } \right)\varepsilon_{o} }}{{2 \lambda_{s} + \lambda_{f} + \left( { \lambda_{s} - \lambda_{f} } \right)\varepsilon_{o} }}} \right]$$
(3)
$$\lambda_{HS, Lower} = \lambda_{f} \left[ {\frac{{2 \lambda_{f} + \lambda_{s} - 2\left( { \lambda_{f} - \lambda_{s} } \right)\left( {1 - \varepsilon_{o} } \right)}}{{2 \lambda_{f} + \lambda_{s} + \left( { \lambda_{f} - \lambda_{s} } \right)\left( {1 - \varepsilon_{o} } \right)}}} \right]$$
(4)

The problem of electrical conduction in an inhomogeneous medium could be solved by the effective medium theory (EMT). Laudauer’s EMT model [27] consists of a two-component medium composed of two different materials, with neither phase being necessarily continuous or dispersed. The shape of each material element is assumed to be spherical. The main assumption in Landauer’s theory is that the medium surrounding an element is considered homogenous, which has an effective conductivity that characterizes the overall properties of the mixture (see Fig. 1d). Either component may form continuous heat conduction pathways, depending on the relative amounts of the components, making this structure unbiased towards its components. Moreover, for porous media in which the solid phase forms continuous pathways, such as open-cell foams, the minimum ETC bound is expected to be given by the effective medium theory (EMT) as given in Eq. 5.

$$\lambda_{EMT} = \frac{1}{4}\left[ {\left( {3\varepsilon_{o} - 1} \right) \lambda_{f} + \left\{ {3\left( {1 - \varepsilon_{o} } \right) - 1} \right\} \lambda_{s} + \sqrt {\left[ {\left( {3\varepsilon_{o} - 1} \right) \lambda_{f} + \left\{ {3\left( {1 - \varepsilon_{o} } \right) - 1} \right\} \lambda_{s} } \right]^{2} } + 8 \lambda_{f} \lambda_{s} } \right]$$
(5)

The reciprocity model was based on the reciprocity theorem [28] which assumes that a microstructure of two-component remains statistically equivalent when exchanging the volume fractions of the components (see Fig. 1e) and is given by Eq. 6. Reciprocity model is quite different than other models and predictions agreed well with many granular materials of spherical nature [32].

$$\frac{{\lambda_{RM} }}{{ \lambda_{f} }} = \frac{{1 + \left( {\sqrt { \lambda_{f} / \lambda_{s} } - 1} \right)\varepsilon_{o} }}{{1 + \left( {\sqrt { \lambda_{s} / \lambda_{f} } - 1} \right)\varepsilon_{o} }}$$
(6)

4 Comparison between experimental and numerical ETC

Most of the ETC data were reported in the literature for the cases where \(\lambda_{s}\) (or \(\lambda_{s}^{B}\)) \(\gg \lambda_{f}\) (usually metal foams). In order to perform numerical simulations to calculate ETC of foam samples, it is necessary to provide solid and fluid conductivities as an input parameters on foam structures. As discussed in Sects. 1 and 2, intrinsic solid phase thermal conductivity of foam materials (\(\lambda_{s}\)) has not been generally measured in majority of the works. It is from this view point, we chose to perform numerical simulations using the measured intrinsic solid phase conductivities of \({\text{Al}}_{2} {\text{O}}_{3}\), Mullite and \({\text{OBSiC}}\) ceramic foams (\(\lambda_{s}\) = 26, 4.4, 15 W m−1 K−1 respectively) by Dietrich et al. [4]. The fluid conductivity (\(\lambda_{f}\)) of air used was 0.03 W m−1 K−1.

As ETC is mainly porosity driven, virtual open-cell foams were generated based on total porosity (\(\varepsilon_{t}\) ) reported by Dietrich et al. [4]. 3-D numerical simulations at pore scale were performed on these virtual open-cell foams with circular strut cross-section in LTE condition (for detailed description, see our previous work [11]). In the Fig. 2, total porosity as well as ETC data obtained numerically were compared against experimental data and are in excellent agreement. However, there are very minimal differences which could be attributed to the measurements uncertainties, numerical errors and morphological disparity. This agreement confirms quantitatively and qualitatively the previously published numerical dataset of ETC, which lends confidence to develop different ETC configurations based on different simplified models.

Fig. 2
figure 2

Comparison and validation of experimental ETC data (taken from the works of Dietrich et al. [4]) against numerically calculated ETC data of ceramic foams

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5 Empirical modelling of effective thermal conductivity

Depending on the availability of foam material (ceramic, alloys and pure metals), authors found out different values of adjustable parameters based on different arrangement of bounds. These adjustable parameters possessed fixed (e.g. [4, 23, 31]) and variable (e.g. [10–12, 24]) values in the most of the reported works. On the other hand, a few authors did not obtain any adjustable parameter in ETC determination for very high conductive material. The most common belief in this case is that heat conduction is mainly due to solid conductivity occurring mainly through the parallel layout while neglecting the influence of working fluid conductivity like air or water.

The present work investigates in predicting ETC values by determining a functional parameter in terms of the traditional input parameters valid for the complete range of porosities and accessible thermal conductivity ratios of Kelvin-like foams. This has been carried out by arranging various combinations of simplified bounds and models (see Eqs. 7 and 8).

As a first step, a generalization of the simplified model for ETC similar to the works of Mendes et al. [17] is proposed in the arithmetic scheme as presented in Eq. 7. This approach is extended and applied to a geometric scheme (also proposed by a few authors e.g. [10–12, 24]) as described in Eq. 8. The two schemes are chosen in order to identify how simple models can be best arranged based on generic minimum and maximum bounds and could predict accurate ETC values of open-cell foams.

$$\lambda_{eff} = \delta \lambda_{min} + \left( {1 - \delta } \right)\lambda_{max}$$
(7)
$$\lambda_{eff} = \lambda_{max}^{\delta \prime } \cdot \lambda_{min}^{1 - \delta \prime }$$
(8)

where, \(\lambda_{max}\) and \(\lambda_{min}\) are generic maximum and minimum bounds for the ETC respectively, and \(\delta\) and \(\delta^{\prime }\) are the functional parameters to be determined from the fitting of various experimental/numerical data.

Mendes et al. [17] used an adjustable parameter \(\delta\) that was an explicit function of the structure of porous media and obtained by calculating ETC of solid phase alone (i.e. under vacuum condition). \(\delta\) values were obtained for each case from numerical simulations by giving \(\lambda_{eff,s}\) and respective values for \(\lambda_{min}\) and \(\lambda_{max}\) under vacuum conditions.

A different strategy is applied in the current work to obtain the parameters, \(\delta\) and \(\delta^{\prime }\) by solving Eqs. 7 and 8 in terms of \(\lambda_{eff}\), \(\lambda_{min}\) and \(\lambda_{max}\) as:

$$\delta = \frac{{\lambda_{max} - \lambda_{eff} }}{{\lambda_{max} - \lambda_{min} }}$$
(9)
$$\delta^{\prime} = \frac{{ln\left( {\lambda_{eff} } \right) - ln\left( {\lambda_{min} } \right)}}{{ln\left( {\lambda_{max} } \right) - ln\left( {\lambda_{min} } \right)}}$$
(10)

where, 0 \(< \delta \left( {{\text{or }}\delta^{\prime } } \right) <\) 1.

Based on the proposed generic ETC model, given by Eqs. 9 and 10, and considering different possible minimum (\(\lambda_{min} = \lambda_{series} , \lambda_{HS, Lower} , \lambda_{EMT}\) and \(\lambda_{RM}\)) and maximum (\(\lambda_{max} = \lambda_{parallel}\) and \(\lambda_{HS, Upper}\)) bounds, twelve different arrangements of these models are formed (Models 1–12, six for each scheme, Eqs. 11–22 in Table 1) by selecting different expressions for \(\lambda_{max}\) and \(\lambda_{min}\). In the following sections, they are called as “model” for the sake of simplicity.

Table 1 Simplified models for effective thermal conductivity, based on the generic model given by Eqs. 9 and 10, obtained by selecting different thermal arrangements for \(\lambda_{min}\) and \(\lambda_{max}\)
Full size table

These models are presented in Table 1 where explicit expressions of \(\delta\) and \(\delta^{\prime }\) are provided. The dataset of 2000 numerically obtained values of ETC obtained on virtual and ideal isotropic foam structures was gathered from our previous work [11] to determine values of \(\delta\) and \(\delta^{\prime }\) in order to account for a wide range of porosities, different morphological parameters of foam structures as well as low to high ratios of solid to fluid conductivity ratios. Previously, porosity (\(\varepsilon_{o}\)) of an idealized isotropic Kelvin-like cell was expressed as a function of dimensionless morphological parameters (\(\psi\)) of foams of any strut cross section [10–12] (Eq. 23).

Same procedure was followed and \(\delta\) values for minimum and maximum bounds arranged in arithmetic scheme were plotted against \(\psi\) for Models 1–6 (see Eqs. 11–16) based on Eq. 9. It can be seen in Fig. 3a, c corresponding to the Model 1 (\(\lambda_{series}\) and \(\lambda_{parallel}\) bounds) and Model 3 (\(\lambda_{EMT}\) and \(\lambda_{parallel}\) bounds), \(\delta\) values (obtained from Eq. 9 using numerical ETC values [11]) collapse very well and a unique curve is obtained as presented in Eqs. 24–25.

$$1 - \varepsilon_{o} = \psi = 12\pi \alpha_{eq}^{2} \beta + \frac{32}{3}\pi \alpha_{eq}^{3}$$
(23)
$$\delta_{1} = 0.6249\left( {1 - \psi } \right)$$
(24)
$$\delta_{3} = 0.6213\left( {1 - \psi } \right)$$
(25)

where \(\alpha_{eq} = R_{eq} /L_{c}\) (ratio of equivalent circular strut radius to node-to-node length) and \(\beta = L_{s} /L_{c}\) (ratio of strut length to node-to-node length) respectively. This formulation (Eq. 23) is presented in [11].

Fig. 3
figure 3

Plot of \(\delta\) (dimensionless) for different arrangements of simplified models (Models 1–6) in arithmetic scheme. The dotted line represents the best fitting of \(\delta\) values calculated using Eq. 9 from numerical ETC values of Kumar and Topin [11] corresponding to different Models (1–6). Different colours of cubic data points represent \(\delta\) values obtained for different porosities (colour figure online)

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The RMSD (root-mean-square-difference) values (see Eq. 26) are 0.77 and 1.93% for \(\delta\) values of Models 1 and 3.

$$RMSD = 10^{{RMS\left( {ELOG} \right)}} - 1 with\,\, ELOG = \log \left( \delta \right)_{calc} - log\left( \delta \right)_{exp}$$
(26)

In case of Model 2 (\(\lambda_{HS, Lower}\) and \(\lambda_{HS, Upper}\) bounds), data of \(\delta\) values fit very well (see Fig. 3b). An intercept has been introduced in order to obtain the best fit of ETC data. However, no explanation is yet found for the meaning of the intercept. The models 4–6 (see Fig. 3d, f) clearly indicate that they follow same trend with porosity but do not form unique characteristics. However, these fittings are not coherent over the entire range of porosity and thus, they are not suitable to be arranged in arithmetic scheme to predict accurate values of ETC.

In the case of arrangements of simplified models in geometric scheme, the derivation of an empirical correlation is more difficult to obtain an excellent combination of dimensionless morphological parameters and ratio of constituent phases. The geometric scheme of series and parallel models using a weighted parameter (Model 7, Eq. 17) has already been presented [10–12] and thus, this scheme is not shown here. Following their procedure, \(\delta^{\prime }\) values for Models 8–12 (see Eqs. 18–22) are calculated (obtained from Eq. 10 using numerical ETC values [11]) and further efforts have been made to obtain an accurate relation between \(\delta^{{^{\prime } }}\) and a combination of \(\psi\) and \(\lambda_{s} /\lambda_{f}\) as presented in Fig. 4. However, we could not advance to obtain any relationship between \(\delta^{{^{\prime } }}\), \(\psi\) and \(\lambda_{s} /\lambda_{f}\) for Models 9 and 10. The plots of \(\updelta_{2}^{{\prime }}\), \(\updelta_{5}^{{^{{^{\prime } }} }}\) and \(\updelta_{6}^{\prime }\) are represented against \(\eta = ln(\psi^{2.25} \lambda_{s} /\lambda_{f} )\), \(\eta^{\prime } = ln(\psi^{1.5} \lambda_{s} /\lambda_{f} )\) and \(\eta = ln(\psi^{2.25} \lambda_{s} /\lambda_{f} )\) in Fig. 4a–c respectively. The dimensionless parameters \(\eta\) (and \(\eta^{\prime }\)) are the best fitting parameters to estimate effective thermal conductivity (using Eq. 8). It can be easily observed that all the values of \(\delta^{\prime }\) in relation with \(\eta\) (and \(\eta^{\prime }\)) collapsed on a single curve for all the different strut shapes. The RMSD values are 1.04, 1.82 and 1.6% for Models 8, 11, and 12 respectively, obtained just by replacing \(\delta\) with \(\delta^{\prime }\) in Eq. 26. From Fig. 4, numerical approximation of \(\delta^{\prime }\) for Models 8, 11, and 12 are given by Eqs. 27–29 as:

Fig. 4
figure 4

Plot of \(\delta '\) (dimensionless) versus fitting parameters \(\eta\) and \(\eta '\) (dimensionless) of different strut shapes. a Model 8–plot of \(\delta_{2}^{ '}\) versus \(\eta = ln(\psi^{2.25} \lambda_{s} /\lambda_{f} )\). b Model 11—plot of \(\delta_{5}^{ '}\) versus \(\eta '= ln(\psi^{1.5} \lambda_{s} /\lambda_{f} )\). c Model 12—plot of \(\delta_{6}^{ '}\) versus \(\eta = ln(\psi^{2.25} \lambda_{s} /\lambda_{f} ) .\) The errors in these models are evenly distributed. Different colours of cubic data points represent \(\delta'\) values obtained for different porosities (colour figure online)

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$$\delta_{2}^{\prime } = - 0.0028 \eta^{2} + 0.0395 \eta + 0.8226$$
(27)
$$\delta_{5}^{\prime } = - 0.005 \eta '^{2} + 0.0876 \eta ' + 0.4914$$
(28)
$$\delta_{6}^{\prime } = - 0.0029 \eta^{2} + 0.0458 \eta + 0.759$$
(29)

There is no physical reason to choose a quadratic polynomial function in Eqs. 27–29 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 ETC data.

The development of functional parameter, \(\delta\) (or \(\delta^{\prime }\)) is clearly a function of foam morphology (\(\psi\)) and constituent conductivities (\(\lambda_{s} /\lambda_{f}\)) depending the combination of bounds and models employed. For each combination, the functional parameter is unique. Figure 5 presents a simple algorithm to predict ETC from input parameters for foam structures. Moreover, this algorithm can also be used to predict the solid phase conductivity and morphology from a known ETC value that could help in optimizing foam structure. The applicability of these functional parameters to predict ETC is validated in the next section.

Fig. 5
figure 5

Algorithm to predict effective thermal conductivity (ETC) by morphological and material properties characteristics of a foam matrix. This algorithm can be used in reciprocal way-from input to output and vice versa

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6 Comparison and validation

ETC correlations developed in the Sect. 5 for different arrangements and schemes are compared and validated against the experimental and numerical data reported in the literature data for foams of different materials (ceramic and metal).

6.1 With ceramic open-cell foams

Predicted ETC results are firstly compared and validated against the experimental data reported by Dietrich et al. [4] for ceramic foams. Values of \(\delta\) and \(\delta^{\prime }\) of different models are calculated using empirical correlations according to different arrangements and schemes (Models 1, 3, 8, 11 and 12) and further applied them to estimate ETC. The analytical results are presented in Table 2. From Table 2, it can be clearly observed that all these models predict excellent ETC results (see also Fig. 6-left).

Table 2 Validation of analytically determined effective thermal conductivity (\(\lambda_{eff}\)) of Al2O3, Mullite and OBSiC ceramic foams (data taken from Dietrich et al. [4]) of different pore sizes and porosities with air as an interstitial fluid
Full size table
Fig. 6
figure 6

Comparison and validation of experimentally and analytically obtained effective thermal conductivity (\(\lambda_{eff}\)) data by using different models (Models 1, 3, 8, 11, 12) for ceramic foams (left) and metal foams (right). Note that, Ana. analytical, Exp. experimental. The fitting comparison between experimental and analytical ETC values is also presented (right). The ETC data of ceramic foams were taken from the works of Dietrich et al. [4] while ETC data of metal foams were taken from Bhattacharya et al. [2], Solórzano et al. [3], Bodla et al. [15], Ranut et al. [19], Wulf et al. [20], Takegoshi et al. [33] and Paek et al. [34]

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6.2 With metal open-cell foams

A few authors (e.g. [9–11]) have already highlighted the problem of non-reporting of intrinsic solid phase conductivity of most of the foams. Most commonly, the correlations that have been reported in the literature were based on parent/bulk solid phase conductivity. Different metal foam materials (or alloys) and their associated ETC values of various authors are presented in Table 3.

Table 3 Determination of intrinsic solid phase conductivity (\(\lambda_{s}\)) and validation of analytical effective thermal conductivity (\(\lambda_{eff}\)) against experiments for foam samples of different pore sizes and porosities with air as interstitial fluid
Full size table

Consequently, the methodology described as modified Lemlich model [12] has been used to determine intrinsic solid phase conductivity while using the experimental ETC data. These authors proposed the following formulation of modified Lemlich model that is valid for any arbitrary fluid as:

$$\frac{{\lambda_{eff} }}{{\lambda_{f} }} = \frac{2}{3} \cdot \frac{{\lambda_{s} }}{{\lambda_{f} }} \cdot \left( \psi \right)^{1/F}$$
(30)

where, \(F = - 0.004\left( {ln(\psi^{2} \cdot \lambda_{s} /\lambda_{f} )} \right)^{2} + 0.0593 ln(\psi^{2} \cdot \lambda_{s} /\lambda_{f} ) + 0.7144\).

A simple method has been described below to calculate the intrinsic solid phase conductivity (\(\lambda_{s}\)) for a given \(\psi\), \(\lambda_{eff}\) and \(\lambda_{f}\).

Step I: Eq. 30 can be rewritten as:

$$\frac{{\lambda_{eff} }}{{\lambda_{f} }} = \frac{2}{3} \cdot \frac{{\lambda_{s} }}{{\lambda_{f} }} \cdot \left( \psi \right)^{{1/\left[ { - 0.004\left( {ln(\psi^{2} \cdot \lambda_{s} /\lambda_{f} )} \right)^{2} + 0.0593 ln(\psi^{2} \cdot \lambda_{s} /\lambda_{f} ) + 0.7144} \right]}}$$
(31)

Step II: Assign \(\lambda_{s} /\lambda_{f} = K_{s}\) and \(\lambda_{eff} /\lambda_{f} = K_{e}\) and apply natural log functions to both sides:

$$\ln \left( {1.5 \cdot K_{e} } \right) = ln\left( {K_{s} } \right) + \frac{1}{{\left[ {n_{1} \left\{ {ln\left( {\psi^{2} \cdot K_{s} } \right)} \right\}^{2} + n_{2} \left\{ {ln\left( {\psi^{2} \cdot K_{s} } \right)} \right\} + n_{3} } \right]}} \cdot ln\left( \psi \right)$$
(32a)
$$\ln \left( {1.5 \cdot K_{e} } \right) - ln\left( {K_{s} } \right) - \frac{1}{{\left[ {n_{1} \left\{ {ln\left( {\psi^{2} \cdot K_{s} } \right)} \right\}^{2} + n_{2} \left\{ {ln\left( {\psi^{2} \cdot K_{s} } \right)} \right\} + n_{3} } \right]}} \cdot ln\left( \psi \right) = 0$$
(32b)

where, \(n_{1}\) = −0.004, \(n_{2}\) = 0.0593,\(n_{3}\) = 0.7144.

Step III: Using iterative process, solve for \(K_{s}\).

Intrinsic solid phase conductivities were calculated using Eqs. 31, 32a for each foam material and these values were subsequently substituted in different models (Models 1, 3, 8, 11 and 12) to predict analytical ETC values (see Table 3). From Table 3, it can be prompted that the different models are consistent with each other and their uses can be combined to predict accurate ETC values (see also Fig. 6-right). However, the predicted values underestimate the experimental ETC data which can be attributed to the formulation [12] by which the intrinsic solid phase conductivity of a different material is under-evaluated.

7 Conclusion

Different arrangements of simplified models in arithmetic and geometric schemes were tried and tested to determine effective thermal conductivity of foam samples of different materials. It has been demonstrated that arrangements of different simplified models may work for one scheme but may not work for another.

The highlight of the present work is to significantly improve the notion of fixed value of adjustable parameter and thus, determining a functional parameter of coupled thermal bounds and models as a function of foam material properties and morphology. Their validity and adaptability has been shown to predict accurate ETC values. The predicted values are compared and validated against experimental data for foams of different materials in a wide porosity range.

It is from this view point, arrangement of Parallel and Series models in both arithmetic and geometric schemes predicts the most accurate effective thermal conductivity results. Moreover, the combination of Parallel and EMT models in the arithmetic scheme as well as Parallel and Reciprocity models, and HS upper bound and Reciprocity models arranged in geometric scheme also predict accurate results.

Depending on the availability of morphological resources, any of these models can be easily used to determine effective thermal conductivity from morphology and thermal conductivity ratio. From the present results, it can be safely concluded that the proposed correlations are most suitable for evaluating the functional parameters of both schemes. Finally, it may be emphasized that the most remarkable feature of the proposed models lies in the fact that they allow one to quite accurately predict the effective thermal conductivity of open-cell foam structures for any working fluid.