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Unsteady thermal response of a porous fin under the influence of natural convection and radiation

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Abstract

In this study, the effects of transient thermal performance of a rectangular porous fin in the presence of radiation and natural convection heat transfer are considered. The porous fin allows the flow to infiltrate through it and solid–fluid interaction takes place. This study is performed using Darcy’s model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered, namely, long fin, finite length fin with insulated tip and finite length fin with tip exposed. The effects of the porosity parameter Sh, radiation parameter G and the temperature ratio CT on the dimensionless transient temperature distribution and heat transfer rate are discussed.

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Abbreviations

Bi:

Biot number \(\left( {\frac{h L}{k}} \right)\)

Cp :

Specific heat (J kg−1 K−1)

CT :

Temperature ratio \(\left( {\frac{{T_{\infty } }}{{T_{b} - T_{\infty } }}} \right)\)

Da:

Darcy number (K/t 2)

g:

Gravity constant (m s−2)

G:

Radiation parameter

Gr:

Grashof number

h:

Heat transfer coefficient (W m−2 K−1)

k:

Thermal conductivity (W m−1 K−1)

Kr:

Thermal conductivity ratio (k eff /k f )

K:

Permeability of porous fin

L:

Length (m)

Pr:

Prandtl number \(\left( {\frac{\nu }{\alpha }} \right)\)

q:

Heat transfer rate (W m−2)

q r :

Radiation heat transfer rate (W m−2)

Ra:

Rayleigh number (Gr Pr)

Sh :

Porous parameter

T(x):

Temperature at any point (K)

T b :

Temperature at fin base (K)

t:

Thickness of the fin (m)

t*:

Time (s)

v w (x):

Velocity of fluid passing through the fin any point (m s−1)

W:

Width of the fin (m)

x:

Axial coordinate (m)

X:

Dimensionless axial coordinate (x/L)

α :

Thermal diffusivity (m2 s−1)

β :

Coefficient of volumetric thermal expansion (K−1)

ε :

Porosity or void ratio

σ :

Stephen–Boltzmann constant [σ = 5.6703 × 10−8 (W/m2 K4)]

ϵ :

Emissivity of porous fin

θ :

Dimensionless temperature \(\left( {\theta = \frac{{T_{\left( x \right)} - T_{\infty } }}{{T_{b} - T_{\infty } }}} \right)\)

θ b :

Base temperature difference, \(\left( {T_{b} - T_{\infty } } \right)\) (K)

v :

Kinematic viscosity (m2 s−1)

ρ :

Density (kg m−3)

μ :

Absolute viscosity (kg m−1 s−1)

τ :

Dimensionless time

b:

Base of fin conditions

eff:

Porous properties

f:

Fluid properties

s:

Solid properties

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Authors and Affiliations

  1. Department of Mechanical Engineering, Cleveland State University, Cleveland, OH, USA

    Rama Subba Reddy Gorla

  2. Department of Mathematics, Razi University, 67149, Kermanshah, Iran

    M. T. Darvishi & F. Khani

Authors
  1. M. T. Darvishi
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  2. Rama Subba Reddy Gorla
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  3. F. Khani
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Correspondence to Rama Subba Reddy Gorla.

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Darvishi, M.T., Gorla, R.S.R. & Khani, F. Unsteady thermal response of a porous fin under the influence of natural convection and radiation. Heat Mass Transfer 50, 1311–1317 (2014). https://doi.org/10.1007/s00231-014-1341-1

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  • DOI: https://doi.org/10.1007/s00231-014-1341-1

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