Open Access
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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