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Closed form solution for a conductive–convective–radiative annular fin with multiple nonlinearities and its inverse analysis

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An Erratum to this article was published on 07 September 2016

Abstract

The performance characteristics and temperature field of conducting–convecting–radiating annular fin are investigated. The nonlinear variation of thermal conductivity, power law dependency of heat transfer coefficient, linear variation of surface emissivity, and heat generation with the temperature are considered in the analysis. A semi-analytical approach, homotopy perturbation method is employed to solve the nonlinear differential equation of heat transfer. The analysis is presented in non-dimensional form, and the effect of various non-dimensional thermal parameters such as conduction–convection parameter, conduction–radiation parameter, linear and nonlinear variable thermal conductivity parameter, emissivity parameter, heat generation number and variable heat generation parameter are studied. For the correctness of the present analytical solution, the results are compared with the results available in the literature. In addition to forward problem, an inverse approach namely differential evolution method is employed for estimating the unknown thermal parameters for a given temperature field. The temperature fields are reconstructed using the inverse parameters and found to be in good agreement with the forward solution.

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Abbreviations

r i , r o , t :

Inner radius, outer radius and thickness of the fin

h(T):

Coefficient of convective heat transfer

k(T):

Thermal conductivity

k o , h b , q o , ε s :

Parameters describing the coefficients of thermal conductivity, convective heat transfer, internal heat generation and surface emissivity at ambient temperature

κ, γ :

Parameters describing the linear and non-linear variation of thermal conductivity

λ, e :

Parameters describing the variation of surface emissivity and internal heat generation

β, β 1 :

Non-dimensional parameters describing the variation of linear and non-linear thermal conductivity parameter

n :

Exponent of variable convective heat transfer coefficient

N :

Non-dimensional thermo-geometric parameter, \(\left( {2hr_{i}^{2} /k_{o} t} \right)^{0.5}\)

M :

Non-dimensional conduction-radiation parameter, \((2r_{i}^{2} \sigma \varepsilon_{s} T_{b}^{3} /k_{o} t)\)

G :

Non-dimensional heat generation parameter, \(G = q_{o} r_{i}^{2} /k_{o} T_{b}\)

E G :

Non-dimensional parameter describing the variation of heat generation

T b :

Base temperature of fin

T a :

Ambient temperature

c 1, c 2, C 1, C 2 :

Constants of integration

\(\xi\) :

Dimensionless radius of fin, \(\xi = \, \left( {r - r_{i} } \right)/r_{i}\)

R :

Dimensionless outer radius ratio, R = r o /r i

θ :

Dimensionless temperature, θ = (T − T a)/(T b  − T a)

p :

Imbedding parameter

η :

Fin efficiency

Q f :

Actual heat transfer

Q max :

Maximum possible heat transfer

F(ξ):

Objective function

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian School of Mines, Dhanbad, 826 004, India

    Rajiv Ranjan & Ashis Mallick

  2. Computational Intelligence Graduate Lab, Nanyang Technological University, Singapore, 639798, Singapore

    Dilip K. Prasad

Authors
  1. Rajiv Ranjan
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  2. Ashis Mallick
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  3. Dilip K. Prasad
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Corresponding author

Correspondence to Ashis Mallick.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00231-016-1913-3.

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Ranjan, R., Mallick, A. & Prasad, D.K. Closed form solution for a conductive–convective–radiative annular fin with multiple nonlinearities and its inverse analysis. Heat Mass Transfer 53, 1037–1049 (2017). https://doi.org/10.1007/s00231-016-1872-8

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