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Entropy Generation Rate Due to Radiative Transfer Within a Vertical Channel Filled with a Semi-Transparent Porous Medium

  • Research Article - Mechanical Engineering
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Abstract

The present work deals with entropy generation rate due to flow and heat transfer by unsteady natural convection coupled with thermal radiation within a vertical channel open at both ends and filled with a semi-transparent porous medium. The radiative transfer equation is solved by the finite volume method. The governing equations for this problem and the relevant boundary conditions are non-linear differential equations depending on different dimensionless numbers. Various results are obtained for the average dimensionless conductive, convective, radiative and total Nusselt numbers for different combinations of the Planck number N, optical thickness τ, walls’ emissivity \({\varepsilon}\), single scattering albedo ω and temperature ratio R. The present work mainly investigates the rate of local as well as total entropy generation enhancement due to radiation. Profiles of spatial variations of local entropy generation rate and time variations of the total entropy generation rate are presented for different values of \({10^{-1}\le N \le 10^{-3},0.1\le \tau\le 5, 0\le \varepsilon\le 1,0.1~\le \omega\le 0.9}\) and 0.1 ≤ R ≤ 0.9.

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Abbreviations

A :

Aspect ratio of the channel, \({\frac{H}{D}}\)

Bi i,o :

Modified inlet (respectively, outlet) Biot numbers, \({\frac{h_{\rm i,o}D}{\lambda}}\)

Br :

Brinkman number, \({\frac{\mu v^{2}_{\rm ref}}{\lambda\Delta^T}}\)

Br m :

Modified Brinkman number, \({Br = \frac{T_\infty}{\Delta T}}\)

d p :

Particle diameter, m

D :

Channel’s width, m

Da :

Darcy number, \({\frac{k}{D^{2}}}\)

E gen :

Total dimensionless entropy generation number

f :

Binary factor (Eqs. 10–11)

g :

Acceleration due to gravity, m s−2

H :

Channel’s height, m

h i,o :

Heat transfer coefficient at the inlet (respectively at the outlet) of the channel, W m−2 K−1

I :

Dimensionless intensity

I b :

Dimensionless blackbody intensity

k :

Permeability of the porous medium, m2

n :

Refractive index

N :

Planck number, \({\frac{\lambda \beta \Delta T}{4n^2\sigma T_{\rm w}^4}}\)

Nu :

Nusselt number

P :

Dimensionless motorize pressure

R :

Temperature ratio, \({\frac{T_\infty}{T_{\rm w}}}\)

Ra :

Modified Rayleigh number, \({\frac{kg\beta _{\rm f} D\Delta T}{\alpha v_{\rm f}}}\)

R d :

Viscous dissipation term, \({\frac{Br}{Da}}\)

s :

Dimensionless distance traveled by a beam

\({S^{\prime\prime\prime}_{\rm gen}}\) :

Local entropy generation rate per unit volume, W m−3 K−1

S gen :

Dimensionless local entropy generation rate

t :

Dimensionless time

T w :

Wall temperature, K

T :

Dimensionless temperature

ΔT :

Temperature difference, T w − T ∞

T ∞ :

Environmental temperature, K

u, v :

Dimensionless transverse and axial velocity components, respectively

x, z :

Dimensionless transverse and axial coordinates, respectively

α :

Thermal diffusivity, \({\frac{\lambda}{(\rho c_p)_{\rm f}},\; {\rm m}^{2}~{\rm s}^{-1}}\)

β f :

Fluid coefficient of volume expansion, k−1

β :

Extinction coefficient, m−1

ε :

Emissivity

δ :

Average porosity

\({\phi}\) :

Viscous dissipation function

\({\kappa }\) :

Absorbing coefficient, m−1

λ :

Thermal conductivity, W m−1 K−1

γ :

Volumetric specific heat ratio, \({\frac{(\rho c_{\rm p})}{(\rho c_{\rm p})_{\rm f}}}\)

μ f :

Fluid’s dynamic viscosity, kg m−1 s−1

v f :

Fluid’s kinematic viscosity, m2 s−1

τ :

Optical thickness, βD

ρ :

Reflectivity

ρ f :

Fluid’s density, kg m−3

c p :

Specific heat capacity at constant pressure, J kg−1 K−1

σ :

Stefan–Boltzmann constant

σ s :

Scattering coefficient, m−1

ω :

Single scattering albedo, \({\frac{\sigma_{\rm s}}{\beta}}\)

f:

Fluid

p:

Particle

ref:

Reference

w:

Along a domain boundary

\({\varpropto}\) :

Surrounding quantity

′:

Dimensional quantity

cd:

Conductive

cv:

Convective

ra:

Radiative

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Authors and Affiliations

  1. Higher Institute of Transport and Logistics, Sousse University, P.O. Box 4023, Riadh City, Sousse, Tunisia

    K. Slimi

  2. Department of Mechanical Engineering, Umm Al-Kura University, Makkah, 21955, Saudi Arabia

    A. A. Saati

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  1. K. Slimi
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Correspondence to K. Slimi.

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Slimi, K., Saati, A.A. Entropy Generation Rate Due to Radiative Transfer Within a Vertical Channel Filled with a Semi-Transparent Porous Medium. Arab J Sci Eng 37, 803–820 (2012). https://doi.org/10.1007/s13369-012-0205-6

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  • DOI: https://doi.org/10.1007/s13369-012-0205-6

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