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Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams

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Abstract

Various ideal periodic isotropic structures of foams (tetrakaidecahedron) with constant ligament cross section are studied. Different strut shapes namely circular, square, diamond, hexagon, star, and their various orientations are modeled using CAD. We performed direct numerical simulations at pore scale, solving Navier–Stokes equation in the fluid space to obtain various flow properties namely permeability and inertia coefficient for all shapes in the porosity range, \(0.60<\varepsilon <0.95\) for wide range of Reynolds numbers, \(10^{-6}<Re<3000\). We proposed an analytical model to obtain pressure drop in metallic foams in order to correlate the resulting macroscopic pressure and velocity gradients with the Ergun-like approach. The analytical results are fully compared with the available numerical data, and an excellent agreement is observed.

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Abbreviations

\(\Delta P\) :

Pressure drop (\(\hbox {Pa}\))

\(\Delta x\) :

Length of fluid medium (\(\hbox {m}\))

\(K\) :

Forchheimer permeability (\(\hbox {m}^{2}\))

\(K^{\prime }\) :

Darcian permeability (\(\hbox {m}^{2}\)) (Eq. 7)

\(\beta \) :

Inertia coefficient (polynomial curve) (\(\hbox {m}^{-1}\))

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

Forchheimer inertia coefficient (\(\hbox {m}^{-1}\)) (Eq. 13)

\(V_\mathrm{s}\) :

Solid volume of foam (\(\hbox {m}^{3}\))

\(V_\mathrm{T}\) :

Volume of octahedron (\(\hbox {m}^{3}\))

\(V\) :

Superficial fluid velocity (\(\hbox {m}\,\hbox {s}^{-1}\))

\(A\) :

Ergun parameter (dimensionless)

\(B\) :

Ergun parameter (dimensionless)

\(f\) :

Universal inertial coefficient (dimensionless)

\(Re_\mathrm{c}\) :

Critical Reynolds number (dimensionless)

\(D_\mathrm{p}\) :

Particle diameter (\(\mathrm{m}\))

\(Re\) :

Reynolds number (dimensionless)

\(a_\mathrm{c}\) :

Specific surface area (\(\mathrm{m}^{-1}\))

\(d_\mathrm{p}\) :

Pore diameter (\(\hbox {m}\))

\(d_\mathrm{cell} \) :

Cell diameter (\(\hbox {m}\))

\(\nabla P\) :

Pressure gradient (\(\mathrm{Pa\,m^{-1}}\))

\(\nabla \langle P \rangle \) :

Average Pressure gradient (\(\mathrm{Pa}\, \mathrm{m}^{-1}\))

\(\langle V\rangle \) :

average velocity (\(\hbox {m}\,\hbox {s}^{-1}\))

\(P\) :

Fluid force (\(\mathrm{N}\))

\(S\) :

Solid-fluid interface area (\(\hbox {m}^{2}\))

\(d_\mathrm{p}^\mathrm{eq}\) :

Equivalent sphere diameter (\(\hbox {m}\))

\(d_\mathrm{s}\) :

Strut diameter (\(\hbox {m}\))

\(A_\mathrm{side}\) :

Side length of strut shape (\(\hbox {m}\))

\(\varepsilon \) :

Porosity, dimensionless

\(\mu \) :

Fluid viscosity (\(\hbox {kg m}^{-1}\,\hbox {s}^{-1}\))

\(\rho \) :

Fluid density (\(\hbox {kg m}^{-3}\))

\(\varepsilon _\mathrm{sur} \) :

Surface Porosity (dimensionless)

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Acknowledgments

The authors express their gratitude to 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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Authors and Affiliations

  1. IUSTI, CNRS UMR 7343, Aix-Marseille University, 5, Rue Enrico Fermi, Technopole de Château Gombert, 13453 , Marseille Cedex 13, France

    Prashant Kumar & Frederic Topin

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  1. Prashant Kumar
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  2. Frederic Topin
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Correspondence to Prashant Kumar.

Appendices

Appendix 1

Specific surface area of a square strut shape is given as

$$\begin{aligned} a_\mathrm{c} =\frac{\left\{ {96A_\mathrm{s} L_\mathrm{s} +24.\frac{3}{4}\left( {\frac{5}{4}A_\mathrm{s} ^{2}} \right) } \right\} }{2\left( {8\sqrt{2}L^{3}} \right) }=\frac{1}{\sqrt{2}L}\left( {6\alpha _\mathrm{s} \chi +\frac{45}{32}\alpha _\mathrm{s} ^{2}} \right) \end{aligned}$$
(25)

where \(\alpha _\mathrm{s} =\frac{A_\mathrm{s} }{L}\) and \(\beta =\frac{L_\mathrm{s} }{L}\).

Specific surface area of a rotated square strut shape is given as

$$\begin{aligned} a_\mathrm{c} =\frac{\left\{ {96A_\mathrm{rs} L_\mathrm{s} +24.\frac{3}{4}\left( {\frac{5}{4}A_\mathrm{rs} ^{2}} \right) } \right\} }{2\left( {8\sqrt{2}L^{3}} \right) }=\frac{1}{\sqrt{2}L}\left( {6\alpha _\mathrm{rs} \chi +\frac{45}{32}\alpha _\mathrm{rs} ^{2}} \right) \end{aligned}$$
(26)

where \(\alpha _\mathrm{rs} =\frac{A_\mathrm{rs} }{L}\) and \(\beta =\frac{L_\mathrm{s} }{L}\).

Specific surface area of a diamond strut shape is given as

$$\begin{aligned} a_\mathrm{c} =\frac{\left\{ {96A_\mathrm{det} L_\mathrm{s} +24.\frac{3}{4}\left( {\frac{5}{4}\left( {\frac{\sqrt{3}}{2}A_\mathrm{det} ^{2}} \right) } \right) } \right\} }{2\left( {8\sqrt{2}L^{3}} \right) }=\frac{1}{\sqrt{2}L}\left( {6\alpha _\mathrm{det} \chi +\frac{45\sqrt{3}}{64}\alpha _\mathrm{det} ^{2}} \right) \end{aligned}$$
(27)

where \(\alpha _\mathrm{det} =\frac{A_\mathrm{rs} }{L}\) and \(\beta =\frac{L_\mathrm{s} }{L}\).

Specific surface area of a hexagon strut shape is given as

$$\begin{aligned} a_\mathrm{c} =\frac{\left\{ {144A_\mathrm{h} L_\mathrm{s} +24.\frac{3}{4}\left( {\frac{5}{4}\left( {\frac{3\sqrt{3}}{2}A_\mathrm{h} ^{2}} \right) } \right) } \right\} }{2\left( {8\sqrt{2}L^{3}} \right) }=\frac{1}{\sqrt{2}L}\left( {9\alpha _\mathrm{h} \chi +\frac{135\sqrt{3}}{64}\alpha _\mathrm{h} ^{2}} \right) \end{aligned}$$
(28)

where \(\alpha _\mathrm{h} =\frac{A_\mathrm{h} }{L}\) and \(\beta =\frac{L_\mathrm{s} }{L}\).

Specific surface area of a star strut shape is given as

$$\begin{aligned} a_\mathrm{c} =\frac{\left\{ {288A_\mathrm{st} L_\mathrm{s} +24.\frac{3}{4}\left( {\frac{5}{4}\left( {3\sqrt{3}A_\mathrm{st} ^{2}} \right) } \right) } \right\} }{2\left( {8\sqrt{2}L^{3}} \right) }=\frac{1}{\sqrt{2}L}\left( {18\alpha _\mathrm{st} \chi +\frac{135\sqrt{3}}{32}\alpha _\mathrm{st} ^{2}} \right) \end{aligned}$$
(29)

where \(\alpha _\mathrm{st} =\frac{A_\mathrm{st} }{L}\) and \(\beta =\frac{L_\mathrm{s} }{L}\).

Appendix 2

For a square strut shape, \(R_\mathrm{eq} =A_\mathrm{s} /\sqrt{\pi }\)

$$\begin{aligned} \varepsilon =\frac{1-\frac{1}{3}\left( {36A_\mathrm{s} ^{2}L_\mathrm{s} +24.\frac{4}{3}A_\mathrm{s} ^{3}/\sqrt{\pi }} \right) }{8\sqrt{2}L^{3}}\Leftrightarrow 12\alpha _\mathrm{s} ^{2}\chi +\frac{32}{3\sqrt{\pi }}\alpha _\mathrm{s} ^{3}=8\sqrt{2}\left( {1-\varepsilon } \right) \end{aligned}$$
(30)

For a rotated square strut shape, \(R_\mathrm{eq} =A_\mathrm{rs} /\sqrt{\pi }\)

$$\begin{aligned} \varepsilon =\frac{1-\frac{1}{3}\left( {36A_\mathrm{rs} ^{2}L_\mathrm{s} +24.\frac{4}{3}A_\mathrm{rs} ^{3}/\sqrt{\pi }} \right) }{8\sqrt{2}L^{3}}\Leftrightarrow 12\alpha _\mathrm{rs} ^{2}\chi +\frac{32}{3\sqrt{\pi }}\alpha _\mathrm{rs} ^{3}=8\sqrt{2}\left( {1-\varepsilon } \right) \end{aligned}$$
(31)

For a diamond strut shape, \(R_\mathrm{eq} =A_\mathrm{det}.\sqrt{\sqrt{3}/2\pi }\)

$$\begin{aligned} \varepsilon&= \frac{1-\frac{1}{3}\left( {36\frac{\sqrt{3}}{2}A_\mathrm{det} ^{2}L_\mathrm{s} +24.\frac{4}{3}.\frac{\sqrt{3}}{2}.\sqrt{\frac{\sqrt{3}}{2\pi }}A_\mathrm{det} ^{3}} \right) }{8\sqrt{2}L^{3}}\Leftrightarrow 6\sqrt{3}\alpha _\mathrm{det} ^{2}\chi +\frac{16}{\sqrt{3}}\sqrt{\frac{\sqrt{3}}{2\pi }}\alpha _\mathrm{det} ^{3}\nonumber \\&= 8\sqrt{2}\left( {1-\varepsilon } \right) \end{aligned}$$
(32)

For a hexagon strut shape, \(R_\mathrm{eq} =A_\mathrm{h}.\sqrt{3\sqrt{3}/2\pi }\)

$$\begin{aligned} \varepsilon&= \frac{1-\frac{1}{3}\left( {36\frac{3\sqrt{3}}{2}A_\mathrm{h} ^{2}L_\mathrm{s} +24.\frac{4}{3}.\frac{3\sqrt{3}}{2}.\sqrt{\frac{3\sqrt{3}}{2\pi }}A_\mathrm{h} ^{3}} \right) }{8\sqrt{2}L^{3}}\Leftrightarrow 18\sqrt{3}\alpha _\mathrm{h} ^{2}\chi +16\sqrt{3}\sqrt{\frac{3\sqrt{3}}{2\pi }}\alpha _\mathrm{h} ^{3}\nonumber \\&= 8\sqrt{2}\left( {1-\varepsilon } \right) \end{aligned}$$
(33)

For a star (regular hexagram) strut shape, \(R_\mathrm{eq} =A_\mathrm{st} .\sqrt{3\sqrt{3}/\pi }\)

$$\begin{aligned} \varepsilon&= \frac{1-\frac{1}{3}\left( {36\sqrt{3}A_\mathrm{st} ^{2}L_\mathrm{s} +24.\frac{4}{3}.3\sqrt{3}.\sqrt{\frac{3\sqrt{3}}{\pi }}A_\mathrm{st} ^{3}} \right) }{8\sqrt{2}L^{3}}\Leftrightarrow 36\sqrt{3}\alpha _\mathrm{st} ^{2}\chi +32\sqrt{3}\sqrt{\frac{3\sqrt{3}}{\pi }}\alpha _\mathrm{st} ^{3}\nonumber \\&= 8\sqrt{2}\left( {1-\varepsilon } \right) \end{aligned}$$
(34)

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Kumar, P., Topin, F. Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams. Transp Porous Med 105, 57–81 (2014). https://doi.org/10.1007/s11242-014-0358-8

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