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Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

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Abstract

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.

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Abbreviations

Br :

Brinkman number=Ec Pr

Cp :

Specific heat (J kg−1 · K−1)

Pr :

Prandtl number=C p /k

Ec :

Eckert number = \({u_{m}^{2}}/{Cp\Delta T}\)

Da :

Darcy number K/h 2

F :

Empirical coefficient

Fr i :

Froude number \({{u_{0}^{1}}/{hg_i}}\)

h :

Half transverse distance (m)

g i :

Gravity in i-direction

k f :

Thermal conductivity (W · m−1 · K−1)

K :

Permeability (m 2)

ℓ :

Characteristic length in the x-direction (m)

N DB :

Entropy generation number, Darcy–Brinkman dissipation = Bru 2/ΩDa

N F :

Entropy generation number, velocity gradient dissipation = Br(∂u/∂y)2/Ω

N S :

Entropy generation number, total

N Y :

Entropy generation number, conduction = (∂θ/∂ y)2

P :

Dimensionless pressure \({={\tilde{P}}/ \left({\mu u_0}\ell/{h^2}\right)}\)

P 0 :

Dimensionless pressure gradient dP/dx

Re :

Cross-flow Reynolds number = ρu 0 h/μ

S G :

Entropy generation rate (W · m−3 · K−1)

S G,C :

Characteristic entropy transfer rate

t :

Dimensionless time \({={\tilde {t}}/{\left({{h^{2}\rho}/\mu}\right)}}\)

T :

Temperature (K)

T 0 :

Reference temperature (K)

T 1 :

Temperature at the lower plate (K)

T 2 :

Temperature at the upper plate (K)

\({\tilde {u}}\) :

Velocity in x-direction (m · s−1)

u 0 :

Characteristic velocity (m · s−1)

\({\tilde{v}}\) :

Velocity in y-direction (m · s−1)

v 0 :

Characteristic velocity in y-direction (m · s−1)

\({\tilde {w}}\) :

Velocity in z-direction (m · s−1)

u, v, w:

Dimensionless velocities

\({\tilde{x}}\) :

Axial direction

\({\tilde{y}}\) :

Transverse direction

\({\tilde{z}}\) :

Normal to the x-direction

x, y, z:

Dimensionless directions

θ :

Dimensionless temperature

Ω:

Dimensionless temperature difference = ΔT/T 0

\({\Phi}\) :

Total viscous dissipation

μ :

Dynamic viscosity (Pa · s)

ρ :

Density of the fluid (kg · m−3)

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  1. Department of Chemical Engineering, Firat University, 23119, Elazig, Turkey

    Fethi Kamışlı

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  1. Fethi Kamışlı
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Kamışlı, F. Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium. Transp Porous Med 80, 345–371 (2009). https://doi.org/10.1007/s11242-009-9364-7

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