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Investigation of Heat Transfer in Rectangular Porous Fins \((\mathrm{Si}_3 \mathrm{N}_4 )\) with Temperature-Dependent Internal Heat Generation by Galerkin’s Method (GM) and Akbari-Ganji’s Method (AGM)

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Abstract

In this paper, heat transfer in rectangular porous fins with temperature-dependent internal heat generation, is investigated and Darcy’s model for passage velocity is applied. Heat transfer governing equation is solved by two simple and accurate methods, Galerkin’s Method (GM) and Akbari-Ganji’s Method (AGM). AGM is a simple and innovative approach can be applied for many linear and nonlinear problems and in this case by this method can reach solutions faster in comparison with other methods. Results indicate that GM and AGM are very accurate and effective in comparison with the numerical results. Porous fin is made from \(\mathrm{Si}_3 \mathrm{N}_4\). Effects of Darcy number and Rayleigh number on temperature distribution are investigated. As a main outcome, the heat generation rate is very effective in fin temperatures. When heat generation rate increases, fin temperatures increase since more amount of heat is dissipated to the surrounding.

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Abbreviations

a :

Constant

A :

Section area of fin

AGM :

Akbari-Ganji’s method

\(C_p \) :

Specific heat at constant pressure

\(D_a \) :

Darcy number

G :

Generation number,dimensionless

GM :

Galerkin’s method

h :

Convection coefficient

k :

Thermal conductivity

\(K_{r }\) :

Thermal conductivity ratio \(\left[ {\frac{k_{eff} }{k_f }} \right] \)

K :

Permeability

L :

Length of fin

M :

Convective parameter

NUM :

Numerical method

q :

Conducted heat

\(q^{*}\) :

Internal heat generation

Ra :

Rayleigh number

\(S_h \) :

Porous parameter

T :

Temperature

\(T_b \) :

Temperature at fin base

\(T_\infty \) :

Sink temperature for convection

t :

Thickness of the fin

\(V_w \) :

Velocity of fluid passing through the fin

w :

Width of the fin

x :

Axial coordinate

X :

Dimensionless axial coordinate [x / L]

\(\beta \) :

Coefficient of volumetric thermal expansion

\(\Delta \) :

Temperature difference

\(\varepsilon _G \) :

Internal heat generation parameter

\(\theta \) :

Dimensionless temperature

\(\vartheta \) :

Kinematic viscosity

\(\rho \) :

Density

\(\varphi \) :

Porosity variable

f :

Fluid properties

\(e_{ff}\) :

Porous properties

s :

Solid properties

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Authors and Affiliations

  1. Mechanical Engineering Faculty, Babol Noshirvani University of Technology, Babol, Mazandaran, Iran

    Hasan Asadian, Mohsen Zaretabar, D. D. Ganji, Mofid Gorji-Bandpy & Sina Sohrabi

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  1. Hasan Asadian
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  2. Mohsen Zaretabar
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  3. D. D. Ganji
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  4. Mofid Gorji-Bandpy
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  5. Sina Sohrabi
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Correspondence to D. D. Ganji.

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Asadian, H., Zaretabar, M., Ganji, D.D. et al. Investigation of Heat Transfer in Rectangular Porous Fins \((\mathrm{Si}_3 \mathrm{N}_4 )\) with Temperature-Dependent Internal Heat Generation by Galerkin’s Method (GM) and Akbari-Ganji’s Method (AGM). Int. J. Appl. Comput. Math 3, 2987–3000 (2017). https://doi.org/10.1007/s40819-016-0279-z

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