# Thermal nonequilibrium, non-darcian forced convection in a channel filled with a fluid saturated porous medium—A perturbation solution

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## Abstract

This paper concentrates on the analysis of the thermal nonequilibrium effects during forced convection in a parallel-plate channel filled with a fluid saturated porous medium. The flow in a channel is described by the Brinkman-Forchheimer-extended Darcy equation and the thermal nonequilibrium effects are accounted for by utilizing the two energy equations model. Applying the perturbation technique, an analytical solution of the problem is obtained. It is established that the temperature difference between the fluid and solid phases for the steady fully developed flow is proportional to the ratio of the flow velocity to the mean velocity. This results in a local thermal equilibrium at the walls of the channel if the Brinkman term which allows for the no-slip boundary condition at the walls is included into the momentum equation.

### Key words

forced convection Brinkman-Forchheimer-extended Darcy equation thermal nonequilibrium flow### Nomenclature

*a*_{s}*f*specific surface area common to solid and fluid phases, m

^{2}/m^{3}*c*_{p}specific heat at constant pressure, J kg

^{−1}K^{−1}- Da
Darcy number,

*K/H*^{2}- F
Forchheimer coefficient

- \(\tilde F\)
scaled Forchheimer coefficient, −F(d〈

*p*_{ f }〉^{ f }/d*x*)(〈ω_{ f }〉^{ f }*H*^{4}/*K*^{1/2}μ_{ f }^{2})*h*_{w}heat transfer coefficient at the porous bed walls,

*W*m^{−2}K^{−1}*h*_{sf}fluid-to-particle heat transfer coefficient between solid and fluid phases,

*W*m^{−2}K^{−1}*H*one half of channel width, m

*K*permeability of the porous medium, m

^{2}- <
*pf*>^{f} intrinsic average pressure, Pa

*q*_{w}heat flux at the walls of the channel,

*W*m^{−2}*T*_{ref1}and*T*_{ref2}reference temperatures, K

- <
*T*_{f}>^{f} intrinsic average temperature of the fluid phase, K

- <
*T*_{s}>^{s} intrinsic average temperature of the solid phase, K

- \(\bar T_f \)
dimensionless fluid phase temperature, (

*T*_{ref1}−⊂*T*_{ f }⊃^{ f })/(*T*_{ref1}−*T*_{ref2})- \(\bar T_s \)
dimensionless solid phase temperature, (

*T*_{ref1}−⊂*T*_{ s }⊃^{ s })/(*T*_{ref1}−*T*_{ref2})- ⊂
*u*_{f}⊃ superficial average velocity, m s

^{−1}*u*_{m}mean velocity, (1/

*H*) ∫_{0}^{ H }̩*u*_{ f }〉d*y*, m s^{−1}*u*_{∞}superficial average velocity outside the momentum boundary layer, m s

^{−1}- \(\tilde u\)
dimensionless velocity, −(

*u*_{ f }〈*u*_{ f }〉)/(*H*^{2}d〈*p*_{ f }〉^{ f }/d*x*)*U*dimensionless velocity, ⊂

*u*_{ f }⊃/*u*_{∞}*x*streamwise coordinate, m

- \(\bar x\)
dimensionless streamwise coordinate,

*x*(〈*ρ*_{ f }〉^{ f }(*c*_{ p })_{ f }*u*_{∞})/(λ_{ feff}+λ_{ seff})*y*transverse coordinate, m

- \(\tilde y\)
dimensionless transverse coordinate,

*y*(〈*ρ*_{ f }〉^{ f }(*c*_{ p })*u*_{∞})/(λ_{ feff}+λ_{ seff})- \(\tilde y\)
dimensionless transverse coordinate,

*y/H*

### Greek letters

- δ
dimensionless small parameter (1/

*h*_{ sf }*a*_{ sf })([〈*ρ*_{ f }〉^{ f }(*c*_{ p })_{ f }]^{2}*u*_{∞}^{2})/(λ_{ feff}+λ_{ seff})- \(\Delta \bar T\)
dimensionless temperature difference between the fluid and solid phases

- ΔΘ
dimensionless complex characterizing temprature difference between the fluid and solid phases, [⊂

*T*_{ s }⊃^{ s }−⊂*T*_{ f }⊃^{ f }](*h*_{ sf }*a*_{ sf }*H/q*_{ w })- ε
porosity

- λ
thermal conductivity,

*W*m^{−1}K^{−1}- A
_{1} inertial parameter, ɛ

^{3/2}F(*u*_{∞}*H/v*_{ f })*μ*_{f}fluid dynamic viscosity, kg m

^{−1}s^{−1}*v*_{f}fluid kinematic viscosity, m

^{2}s^{−1}*μ*_{eff}effective viscosity in the Brinkman term, kg m

^{−1}s^{−1}- ρ
density, kg m

^{−3}

### Subscripts/Superscripts

- eff
effective property

*f*fluid phase

*s*solid phase

*Other symbols ⊂...⊃*volume average

- 〈...〉
^{f}and 〈...〉^{s} intrinsic volume average

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