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Table B2

Analytical models.

Researcher Model Remark
Behrang et al. [37] (Parallel-series-Maxwell model) Sw : wetting phase saturation
K: thermal conductivity
Δ: volume fraction
ε: Porosity
Bhattacharya et al. [13] r: half thickness of fiber
L: half-length of fiber
Yao et al. [82] a2: geometrical parameter controlling cross-sectional size of ligament
L: distance between centers of neighboring nodes
Gong et al. [83] In this theory, phases (or components) are treated as small spheres dispersing into an assumed uniform medium with the thermal conductivity km.
m: uniform medium with dispersed spheres.
e: two components system
Kou et al. [38] Fractal dimensions Df , Dt , and Df,w (or Df,g )
ʎ: Pore size
Bhattacharya [84] circular lump corrected Developed for metal foams. 2D array of hexagonal cells, with hexagonal lumps of metal at fibers' intersection.
Four layers in series
Boomsma and Poulikakos [32], revised Developed for metal foams. 3D tetrakaidecahedron, struts of cylindrical cross section, with cubic nodes at fibres' intersection. Four layers in series. The parameter e must be estimated from data fitting.
r: lump thickness
L: strut length
d: strut radius
Dai et al. [34] Developed for metal foams. Improvement of the above
model. Accounts for struts' orientation Φ = 45
Good predictions but with an unfeasible value of e/r
Yang et al. [85] Developed for metal foams. Improvement of the acove model; considers the variation of e with porosity a0,a1,a2,a3 calibrated against data of [32]:
valid for 0.905 < ε < 0.978
Yang et al. [86] Developed for metal foams. 1/16th of a tetrakaidecahedron unit cell. No lumps at ligaments' intersection. The coefficient
1/3 is the reciprocal of thermal tortuosity. The same form of the Lemlich model is obtained
Yang et al. [11] Accounting for the variation of struts'cross section:
1/16 th of a tetrakaidecahedron unit cell. Cubic nodes at ligaments' intersection. From experimental measurements on SEM images for Al foams: e = 0.3 and α = 1.5. α ≥ 1: node to ligament cross-sectional area ratio e ≥ 0: node thickness to strut length ratio
φ: ratio of ligament cross-sectional area in the middle to that at the end
Kumar and Topin [87] Developed for metal foams.
F is fraction of material oriented in heat flow direction
Kumar and Topin [87] PF model Developed for metal foams.
Γ is defined as resistor model.
Valid for 0.6 < ε < 0.95
Dul'nev and Novikov [88]  
Hsu et al. [89] touching square cylinders 2 Dimensional
l: length of one side of solid square cylinder
Hsu et al. [89] touching cubes 3 Dimensional
l: length of one side of cube
Wang et al. [90] Efficient for all ranges of porosities and hollow or solid struts
Singh et al. [91] spherical inclusions n ': fractional volume of the more highly conducting phase
a/b: node to cell size ratio
Singh et al. [91] cubic inclusions n ': fractional volume of the more highly conducting phase and a/b is node to cell size ratio
C1 and C2: constants
Jagjiwanram and Singh [92] ellipsoidal inclusion n : fractional volume of the more highly conducting phase and a/b is node to cell size ratio
Zehner and Schundler, from Ref. [36] n is shape factor and at n = 1 solid becomes sphere
k r  = k f /k s : fluid/solid thermal conductivity ratio
Hsu et al. [36] Area contact model n is shape factor and at n = 1 solid becomes sphere
k r  = k f /k s : fluid/solid thermal conductivity ratio
α: deformation factor
Hsu et al. [36] Phase-symmetry model n is shape factor and at n = 1 solid becomes sphere
k r  = k f /k s : fluid/solid thermal conductivity ratio
α: deformation factor
Russel, from Ref. [58] pores are cubes of the same size with solid walls of uniform thickness
Ahern et al. [93] : the fraction of solid material contained in the windows and β declares strut shapes : β = 1/3 for cylindrical struts and β = 2/3 for the plate-like windows
Schuetz and Glicksmann, from Ref. [93] β = 1/3 for cylindrical struts and β = 2/3 for the plate-like windows
Fiedler et al. [94] Determining ETC based on relative density of solid
Singh et al. [95] In order to take into accouter the random distribution of cells in the continuous matrix as well as the wide difference in the thermal
conductivity of the constituents, a non-linear second-order correction term is correlated
Donea Variational Model from Ref. [96] r: radius of inclusion,
r ':radius of the largest possible spherical shell surrounding the inc!iision for random dispersion of spheres
Bauer [97] Compatible with experimental data obtained from wide range of thermal conductivity, pore shape and distribution k0: thermal conductivity of unperturbed continuous medium
Kuwahara et al. [98] Analytical correlation established for air and aluminium as fluid and sold phases under non-equilibrum thermal condition

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