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The Effect of MHD on a Porous Fin Attached to a Vertical Isothermal Surface

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Abstract

The effect of MHD on the total heat transfer from a porous fin attached to a vertical isothermal surface has been investigated. The Maxwell equations have been used, and also Rosseland approximation for radiation heat transfer and Darcy model for simulating the flow in porous medium have been adapted. The governing equations are reduced to a nonlinear ODE. The fin is supposed to be an infinite fin, which is exposed to a magnetic field. The dimensionless temperature profile, and the average Nusselt number profiles have been obtained for different Rayleigh numbers and porosities. Validation is carried out by comparing the results obtained in this study with those predicted by Darcy–Brinkman–Forchheimer model.

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Abbreviations

A :

Cross sectional area (m2)

B 0 :

Magnetic field intensity (T)

c p :

Specific heat (J kg−1 K−1)

g :

Gravitational acceleration (m s−2)

h :

Heat transfer coefficient (J m−2 K−1)

J :

Total current intensity (A)

J c :

Conduction current intensity (A)

k :

Thermal conductivity (W m−1 K−1)

k r :

Thermal conductivity ratio, k eff/k f

K :

Permeability of the porous fin (m2)

\({\dot{m}}\) :

Mass flow rate (kg s−1)

P :

Fin perimeter (m)

Rd :

Radiation–conduction parameter, \({Rd=(4\sigma T_\infty^3)/(3\beta_{\rm R}k_{\rm eff})}\)

R 2 :

Surface-ambient radiation parameter, \({R_2 =(4\sigma T_\infty^3 b)/k_{\rm eff}}\)

Ra*:

Modified Rayleigh number, Ra* = g β (T b−T ∞)Kb/α vk r

T :

Temperature (K)

T b :

Temperature at the fin base (K)

u :

Axial velocity (m s−1)

v :

Normal velocity (m s−1)

V :

Macroscopic velocity of electrons (m s−1)

\({\overline{{\vartheta}_{\rm w}}}\) :

Average velocity of the fluid passing through the fin at any point (m s−1)

x :

Axial coordinate

y :

Transverse coordinate

α :

Thermal diffusivity (m2 s−1)

β R :

Rosseland extinction coefficient

ε :

Emissivity

\({\tilde{\varepsilon}}\) :

Porosity

ζ :

Magnetic interaction parameter, \({{\sigma B_0^2 x}/{\rho u_\infty}}\)

η :

Pseudo similarity variable, y(u ∞/vx)1/2

θ :

Dimensionless temperature

θ b :

Surface temperature parameter, θ b = T b/T ∞

μ :

Dynamic viscosity (kg m−1 s−1)

ν :

Kinematic viscosity (m2 s−1)

σ :

Electric conductivity (Ω−1 m−1)

σ st :

Stefan–Boltzmann constant (W m2 K4)

ρ :

Density of the fluid (kg m−3)

ρ ε :

Electrical density (A m−3)

eff:

Effective properties

f:

Fluid

b:

Conditions at the fin base

s:

Solid

∞:

Ambient conditions

1:

Clear fluid domain

2:

Porous domain

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Tehran, Tehran, Iran

    A. Taklifi

  2. Department of Mechanical Engineering, K.N.T. University of Technology, Tehran, Iran

    C. Aghanajafi & H. Akrami

Authors
  1. A. Taklifi
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  2. C. Aghanajafi
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  3. H. Akrami
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Correspondence to A. Taklifi.

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Taklifi, A., Aghanajafi, C. & Akrami, H. The Effect of MHD on a Porous Fin Attached to a Vertical Isothermal Surface. Transp Porous Med 85, 215–231 (2010). https://doi.org/10.1007/s11242-010-9556-1

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  • DOI: https://doi.org/10.1007/s11242-010-9556-1

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