# Numerical Simulation of Gas/Solid Heat Transfer in Metallic Foams: A General Correlation for Different Porosities and Pore Sizes

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## Abstract

In the present research work, numerical simulations were performed to investigate the effects of structural parameters on fluid flow and heat transfer under unsteady state conditions in aluminium foams, with various physical specifications such as different porosities (76–96%), pores diameter (100–500 μm) and tortuosity (1.024–1.14), by meshing computed micro-tomography images. In all the simulated cases, the fluid was considered as air with a temperature of 500 K and different superficial velocities (1–6 m/s) entered the foam with a temperature of 300 K. Calculation of the pressure gradient based on a generic formula Δ*P/L *=* αv *+* βv*^{2} shows that by increasing porosity and pore diameter, coefficients *α* and *β* decrease. Moreover, heat transfer analysis shows that the average convection heat transfer coefficient (*h*_{ave}) depends on the geometrical parameters of the foam and also on the superficial velocity of the fluid. In fact, the minor changes in the pore diameter can greatly affect *h*_{ave} (e.g. the variation of *h*_{ave} for samples with 86% porosity at inlet velocity of 5 m/s and different pore diameters from 500 to 100 μm: 250 to 600 J/m^{2} s K). However, the porosity variations do not have significant effects on *h*_{ave}. On the other hand, by using the nonlinear least square fitting technique and also including the structural factor (*F*_{s}, function of the foam geometrical parameters) to the *Nu* correlation, the equation *Nu* = 0.0305 *Re*^{0.77} *F*_{s} [where *F*_{s} = ((1 − *ε*)/*τ*)^{−0.27} (*d*_{p}*/d*_{t})^{−5.108}] for determining the *Nu* in the different foams has been proposed. The equation and simulated results are agreed with each other very well and additionally are similar to the previous studies. Therefore, it’s expected that this equation can be used in design and performance evaluation of porous heat exchangers and porous catalysts.

## Keywords

Fluid flow Convection heat transfer Foam structural factor Micro-tomography Nusselt number## List of symbols

*C*_{f}Inertia coefficient

*C*_{p}Specific heat (J kg

^{−1}K^{−1})*d*_{p}Pore diameter (µm)

*d*_{s}Strut diameter (µm)

*d*_{t}Total diameter (pore diameter + strut diameter) (µm)

*f*Friction factor

*h*_{l}Local heat transfer coefficient (W m

^{−2}K^{−1})- \( \bar{h}_{\text{l}} \)
Average local heat transfer coefficient (W m

^{−2}K^{−1})*h*_{ave}Average heat transfer coefficient (W m

^{−2}K^{−1})*K*Permeability (m

^{2})*k*Thermal conductivity(W m

^{−1}K^{−1})*Nu*Nusselt number

*P*Pressure (Pa)

- Δ
*P* Pressure drop (Pa)

*Pr*Prandtl number

*q*Heat rate (W)

*Re*Reynolds number

*T*Temperature (K)

- \( T_{\text{f}}^{0} \)
Initial temperature of fluid (K)

- \( T_{\text{s}}^{0} \)
Initial temperature of foam (K)

*T*_{sl}Local temperature of foam (K)

*T*_{eq}Equilibrium temperature

*t*Time

- \( t_{\text{eq}} \)
Equilibrium time

*v*Velocity (m s

^{−1})*x*,*y*,*z*Cartesian coordinates (m)

*L*Foam length (mm)

- \( s_{0} \)
Specific surface area (m

^{−1})

## Greek symbols

*α*Darcian coefficient (kg m

^{−3}s^{−1})*β*Non-Darcian coefficient (kg m

^{−4})*θ*Non-dimensional temperature

*ρ*Density (kg m

^{−3})*µ*Dynamic viscosity (kg m

^{−1}s^{−1})*ε*Porosity

*τ*Tortuosity

## Subscripts

- air
Air, in the pore region

- ave
Average

- f
Fluid

- l
Local

- s
Solid

## Notes

### Acknowledgements

This study was conducted in the Particulate Materials Research Group (Department of Materials Engineering, Isfahan University of Technology). The authors would like to express great appreciation to Dr. Andreas Wiegmann for the GeoDict (trial version) software.

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