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Entropy generation in moving exponential porous fins with natural convection, radiation and internal heat generation

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Abstract

In this article, convection, radiation and internal heat generation for entropy generation in the moving exponential porous fin are investigated. We have considered fins of fixed dimension with insulated tip and convective coefficient at fin’s tip. The second law of thermodynamics for one-dimensional heat transfer is applied to study the entropy generation in the exponential porous fins. The shooting method is applied to solve the system numericaly. The amount of entropy generation depends on the porosity parameter, temperature distribution, and temperature ratio of the exponential porous fin which is observed. It is also observed that the highest generation of entropy exists at the fin’s base when the temperature difference is increased, at particular values of temperature ratio and porosity parameter. The data also reveal that the effect of porosity parameter on entropy generation number is substantially stronger than the influence of temperature ratio, with an increase in Porosity from 1 to 80 resulting in a 15-time increase in average entropy generation number. However, an increase in temperature ratio from 1.1 to 1.9 results an average of 30% increase in entropy generation number.

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Abbreviations

A :

Area of fin’s surface m\(^2\)

h :

Convection heat transfer coefficient W/m\(^2\) K

L :

Fins length m

q :

Internal heat generation W

t :

Fin’s thickness m

\(T_{\gamma }\) :

Temperature ratio

U :

Speed of moving fin m/s

x :

Direction along x-axis m

\(C_{p}\) :

Specific heat of the material J/kg K

\(k_\mathrm{eff}\) :

Effective thermal conductivity W/m K

\(k_{f}\) :

Air thermal conductivity W/m K

\(N_{r}\) :

Dimensionless radiation parameter

\(R_{a}\) :

Rayleigh number \(\beta \,t^3\,(T-T_{a})/\lambda \,\nu \)

\(T_{a}\) :

Ambient temperature K

\(\alpha \) :

Fin shape parameter

\(\beta \) :

Volume expansion coefficient K\(^{-1}\)

\(\varepsilon \) :

Surface emissivity

\(\lambda \) :

Thermal diffusivity of air m\(^2\)/s

\(\varphi \) :

Porosity of the fin

\(\theta _{a}\) :

Dimensionless ambient temperature

g :

Gravitational acceleration m/s\(^{2}\)

K :

Porous fins permeability Darcy

P :

Fins perimeter m

Q :

Dimensionless internal heat generation

T :

Dimensional fin temperature K

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

Mass flow rate kg/s

W :

Width of the fin m

X :

Dimensionless coordinate

\(D_{a}\) :

Darcy number K/t\(^2\)

\(K_{s}\) :

Solid thermal conductivity W/m K

Bi :

Biot number

\(P_{e}\) :

Peclet number

\(S_{h}\) :

Porosity parameter

\(T_{b}\) :

Base temperature K

\(\alpha ^*\) :

Dimensionless fin shape parameter

\(\sigma \) :

Stefan–Boltzmann constant W/m\(^2\) K\(^4\)

\(\nu \) :

Kinematic viscosity of air m\(^2\)/s

\(\rho \) :

Density of material kg/m\(^3\)

\(\theta \) :

Dimensionless local temperature

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Authors and Affiliations

  1. Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan

    Zia Ud Din, Amir Ali & Gul Zaman

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  1. Zia Ud Din
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  2. Amir Ali
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Din, Z.U., Ali, A. & Zaman, G. Entropy generation in moving exponential porous fins with natural convection, radiation and internal heat generation. Arch Appl Mech 92, 933–944 (2022). https://doi.org/10.1007/s00419-021-02081-2

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