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Different arrangements of simplified models to predict effective thermal conductivity of open-cell foams

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Abstract

It is often desirable to predict the effective thermal conductivity (ETC) of a homogenous material like open-cell foams based on its composition, particularly when variations in composition are expected. A combination of five fundamental simplified thermal conductivity bounds and models (series, parallel, Hashin–Shtrikman, effective medium theory, and reciprocity models) is proposed to predict ETC of open-cell foams. Usually, these models use a parameter as the weighted mean to account the proportion of each bound arranged in arithmetic and geometric schemes. Based on ETC data obtained on numerous virtual Kelvin-like foam samples, the dependence of this parameter has been deduced as a function of morphology and phase thermal conductivity ratio. Various effective thermal conductivity correlations are derived based on material properties and foam structure. This is valid for open-cell foams filled with any arbitrary working fluid over a solid conductivity of materials range (\(\lambda_{s} /\lambda_{f}\) = 10–30,000) and over a wide range of porosity (0.60 \(< \varepsilon_{o} <\) 0.95). Arrangement of series and parallel models together using the simplest models for both, arithmetic and geometric schemes, is found to predict excellent results among all the generic combinations.

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Abbreviations

µCT:

Micro-computed tomography

ETC:

Effective thermal conductivity

EMT:

Effective medium theory

HS:

Hashin–Shtrikman

LBM:

Lattice Boltzmann method

RT:

Reciprocity theorem

\(A\,or\,R\) :

Side length of strut shape or radius of strut shape (mm)

\(L_{c}\) :

Node-to-node length (mm)

\(L_{s}\) :

Strut length (mm)

\(F\) :

Correlation factor (Eq. 30)

\(R_{eq}\) :

Equivalent circular strut radius (mm)

\(\varepsilon_{o}\) :

Open porosity

\(\varepsilon_{t}\) :

Total porosity

\(\alpha_{eq}\) :

Ratio of equivalent circular strut radius to node-to-node length

\(\beta\) :

Ratio of strut length to node-to-node length

\(\delta\) :

Functional parameter in arithmetic scheme (Eq. 7)

\(\delta^{\prime }\) :

Functional parameter in geometric scheme (Eq. 8)

\(\psi\) :

Dimensionless geometrical parameter (Eq. 23)

\(\eta\) :

Dimensionless fitting parameter (Eqs. 27 and 29)

\(\eta^{\prime }\) :

Dimensionless fitting parameter (Eq. 28)

\(\lambda_{s}\) :

Intrinsic solid phase conductivity of foam (W m−1 K−1)

\(\lambda_{s}^{B}\) :

Solid/Bulk phase conductivity of foam material (W m−1 K−1)

\(\lambda_{f}\) :

Fluid phase conductivity (W m−1 K−1)

\(\lambda_{eff}\) :

Effective thermal conductivity (W m−1 K−1)

\(\lambda_{parallel}\) :

Effective parallel thermal conductivity (Eq. 1) (W m−1 K−1)

\(\lambda_{series}\) :

Effective series thermal conductivity (Eq. 2) (W m−1 K−1)

\(\lambda_{HS, Upper}\) :

HS upper bound thermal conductivity (Eq. 3) (W m−1 K−1)

\(\lambda_{HS, Lower}\) :

HS lower bound thermal conductivity (Eq. 4) (W m−1 K−1)

\(\lambda_{EMT}\) :

Effective medium theory thermal conductivity (Eq. 5) (W m−1 K−1)

\(\lambda_{RM}\) :

Reciprocity model (Eq. 6) (W m−1 K−1)

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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.

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  1. IUSTI, CNRS UMR 7343, Aix-Marseille Université, Marseille, France

    Prashant Kumar & Frédéric Topin

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  1. Prashant Kumar
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Correspondence to Prashant Kumar.

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Kumar, P., Topin, F. Different arrangements of simplified models to predict effective thermal conductivity of open-cell foams. Heat Mass Transfer 53, 2473–2486 (2017). https://doi.org/10.1007/s00231-017-1993-8

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