# Nonsimilar Boundary Layer Analysis of Free Convection Heat Transfer Over a Vertical Cylinder in Bidisperse Porous Media

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## Abstract

This work presents a boundary layer analysis for the free convection heat transfer from a vertical cylinder in bidisperse porous media with constant wall temperature. A boundary layer analysis and the two-velocity two-temperature formulation are used to derive the nonsimilar governing equations. The transformed governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The effects of inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Results show that an increase in the modified thermal conductivity ratio and the permeability ratio can effectively enhance the free convection heat transfer of the vertical cylinder in a bidisperse porous medium. Moreover, the thermal nonequilibrium effects are strong for low values of the inter-phase heat transfer parameter.

### Keywords

Nonsimilar solutions Free convection Heat transfer Bidisperse porous medium Vertical cylinder### List of Symbols

### Variables

- \(c\)
Specific heat at constant pressure

- \(f\)
Dimensionless stream function for the f-phase

- \(g\)
Dimensionless stream function for the p-phase

- \(g^{{*}}\)
Acceleration due to gravity

- \(h\)
Inter-phase heat transfer coefficient

- \(H\)
Inter-phase heat transfer parameter

- \(k\)
Thermal conductivity

- \(K\)
Permeability

- \(K_\mathrm{r}\)
Permeability ratio

- \(Nu\)
Local Nusselt number

- \(r_0\)
Radius of the vertical cylinder

- \(Ra\)
Rayleigh number

- \(T_{\infty }\)
Ambient temperature

- \(T_\mathrm{w}\)
Wall temperature

- \(u, v\)
Dimensional velocity components along \(x\) and \(r\) axes

- \(x, r\)
Dimensional Cartesian coordinates

### Greek Symbols

- \(\beta \)
Modified thermal capacity ratio

- \(\beta _\mathrm{T}\)
Volumetric thermal expansion of the fluid

- \(\gamma \)
Modified thermal conductivity ratio

- \(\varepsilon \)
Porosity within the p-phase

- \(\varsigma \)
Coefficient for momentum heat transfer between the two phases

- \(\eta ,\xi \)
Dimensionless coordinates

- \(\mu \)
Viscosity of the fluid

- \(\rho _\mathrm{F}\)
Density of the fluid

- \(\sigma _\mathrm{f}\)
f-phase momentum transfer parameter

- \(\tau \)
Porosity parameter

- \(\theta \)
Dimensionless temperature

- \(\varphi \)
Volume fraction of the f-phase

- \(\psi \)
Stream function

### Subscripts

- f
Fracture phase

- p
Porous phase

## Notes

### Acknowledgments

This work was supported by the National Science Council of Republic of China under Grant No. NSC 100-2221-E-218-045.

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