# Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

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## Abstract

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.

## Keywords

Porous medium Partially filled porous media Channel flow Fully developed.## List of Symbols

- \(\text{ Da}\)
Darcy number

- \(f\)
Friction coefficient

- \(G\)
Pressure gradient in \(x\)- direction

- \(h\)
Convective heat transfer coefficient, \(\text{ W/m}^{2}\) K

- \(H\)
Half height of channel

- \(k_m \)
The overall thermal conductivity, W/m K

- \(K\)
Permeability, \(\text{ m}^{2}\)

- \(K^{*}\)
The ratio of effective conductivity over fluid conductivity

- \(L\)
Length of the channel, m

- \(M\)
Dimensionless viscosity ratio parameter

- \(Nu_m \)
Overall Nusselt number

- Re
Reynolds number

- \(q^{{\prime }{\prime }}\)
heat flux, \(\text{ W/m}^{2}\)

- \(S\)
Porous media shape parameter

- \(T\)
Temperature, \(\text{ K}\)

- \(T_i \)
Initial temperature, \(\text{ K}\)

- \(u\)
Velocity component along \(x\)- direction, m/s

- \(U\)
Dimensionless velocity component along dimensionless \(X\)- direction, m/s

- \(U_{i}\)
Dimensionless interface velocity component along dimensionless \(X\)- direction

- \({\hat{u}}\)
Dimensionless normalized velocity

- \(x\)
Coordinate along the axis of the channel, m

- \(y\)
Coordinate normal to the surfaces of the channel, m

- \(X,\;Y\)
Dimensionless coordinates

## Greek letters

- \(\beta \)
Coefficient for the walls

- \(\varepsilon \)
Porosity

- \(\mu \)
Dynamic viscosity, kg/m s

- \(v\)
Kinematic viscosity of fluid, \(\text{ m}^{2}\)/s

- \(\theta \)
Dimensionless temperature

- \(\theta _i \)
The dimensionless interface temperature

- \(\rho \)
Density, \(\text{ kg/m}^{3}\)

- \(\xi \)
Porous thickness

## Subscripts

- \(c\)
Clear fluid region

*eff*Effective

- \(f\)
Fluid

- \(m\)
Mean

- \(p\)
Porous

- \(r\)
Ratio

- \(s\)
Solid

- \(t\)
Total

- \(w\)
Wall

*wu*Upper wall

*wl*Lower wall

- \(w\,\mu \)
Average wall

- \(u\)
Belongs to the upper wall

- \(l\)
Belongs to the lower wall

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