# Thermodynamically Consistent Limiting Forced Convection Heat Transfer in a Asymmetrically Heated Porous Channel: An Analytical Study

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## Abstract

Fluid transport and the associated heat transfer through porous media is of immense importance because of its numerous practical applications. In view of the widespread applications of porous media flow, the present study attempts to investigate the forced convective heat transfer in the limiting condition for the flow through porous channel. There could be many areas, where heat transfer through porous channel attain some limiting conditions, thus, the analysis of limiting convective heat transfer is far reaching. The primary aim of the present study is focused on the limiting forced convection analysis considering the flow of Newtonian fluid between two asymmetrically heated parallel plates filled with saturated porous media. Utilizing a few assumptions, which are usually employed in the literature, an analytical methodology is executed to obtain the closed-form expression of the temperature profile, and in the following the expression of the limiting Nusselt numbers. The parametric variations of the temperature profile and the Nusselt numbers in different cases have been shown highlighting the influential role of different performance indexing parameters, like Darcy number, porosity of the media, and Brinkman number of forced convective heat transfer in porous channel. In doing so, the underlying physics of the transport characteristics of heat has been delineated in a comprehensive way. Moreover, a discussion has been made regarding an important feature like the onset of point of singularity as appeared on the variation of the Nusselt number from the consideration of energy balance in the flow field, and in view of second law of thermodynamics.

### Keywords

Porous channel Asymmetrically heated Nusselt number Dracy number Porosity Point of singularity### List of Symbols

### Variables

- \(A\)
Degree of asymmetrical wall heating

- Br
Brinkman number

- \(C_p\)
Specific heat at constant pressure

- \(C_1\)–\(C_2 \)
Integration constants

- Da
Darcy number

- \(h_1 \)
Heat transfer coefficient in limiting condition at lower wall

- \(h_2 \)
Heat transfer coefficient in limiting condition at upper wall

- \(H\)
Half channel height

- \(I_1\)–\(I_{11}\)
Constants

- \(k\)
Thermal conductivity

- \(K_\mathrm{p}\)
Permeability of the media

- Nu
Nusselt number

- \(\mathrm{Nu}_1\)
Nusselt number in the conduction limit at the lower plate

- \(\mathrm{Nu}_2\)
Nusselt number in the conduction limit at the upper plate

- \(p\)
Pressure

- \(\bar{{Q}}\)
Volumetric heat generation

- \(Q\)
Dimensionless volumetric heat generation

- \(R_1\)–\(R_2\)
Constants

- \(S\)
Parameter to characterize \({({1-A})}/{({1+A})}\)

- \(T\)
Temperature

- \(\bar{T}\)
Average temperature (K)

- \(T_1\)
Lower plate temperature (K)

- \(T_2\)
Upper plate temperature (K)

- \(T_\mathrm{in}\)
Uniform fluid inlet temperature (K)

- \(T_\mathrm{m}\)
Mean temperature in the conduction limit

- \(\bar{{u}}\)
Dimensional velocity

- \(\hat{{u}}\)
\(\tilde{u}={\mu _\mathrm{f} \bar{{u}}}/{\gamma H^{2}}\)

- \(\tilde{u}\)
Dimensionless velocity

- \(u_\mathrm{m}\)
Mean velocity (m/s)

- \(\bar{{x}}\)
Axial coordinate (m)

- \(\bar{{y}}\)
Coordinate perpendicular to flow (m)

- \(\hat{{y}}\)
Dimensionless coordinate perpendicular to flow

- \(w\)
Width of the channel

### Greek Symbols

- \(\theta \)
Dimensionless temperature

- \(\theta _\mathrm{m}\)
Dimensionless mean temperature in the conduction limit

- \(\beta \)
Shape factor to characterize \(\sqrt{\varepsilon /\mathrm{Da}}\)

- \(\lambda \)
Negative of applied pressure gradient

- \(\varepsilon \)
Porosity of the media \({\mu _\mathrm{f} }/{\mu _\mathrm{eff} }\)

- \(\mu _\mathrm{f} \)
Viscosity of fluid

- \(\mu _\mathrm{eff} \)
Effective viscosity of the media

- \(\rho \)
Density

### Subscripts

- DR
Denominator

- NR
Numerator

- c
Conduction limit

- eff
Effective

- in
Incoming fluid

- f
Fluid

- m
Mean in the conduction limit

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