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Estimation of critical dimensions for a trapezoidal-shaped steel fin using hybrid differential evolution algorithm

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Abstract

This paper deals with the inverse prediction of parameters in a trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient. Three critical dimensions along with the relevant heat transfer coefficient at the fin base have been simultaneously predicted for satisfying a given temperature distribution on the surface of the trapezoidal fin. The inverse problem is solved by a hybrid differential evolution-nonlinear programming (DE-NLP) optimization method. For a given fin material which is considered to be stainless steel, it is found from the present study that many feasible dimensions exist which satisfy a given temperature distribution, thereby providing flexibility in selecting any dimensions from the available alternatives by appropriately regulating the base heat transfer coefficient. A very good estimation of the unknown parameters has been obtained even for temperature distribution involving random measurement errors which is confirmed by the comparisons of the reconstructed distributions. It is concluded that for a given fin material, the hybrid DE-NLP algorithm satisfactorily estimates feasible dimensions of a trapezoidal fin even with random measurement error of 11 %.

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Abbreviations

a 1 , b 1 , c 1 , d 1 , e 1 :

Vectors in DE algorithm

b :

Semi-thickness at the base (m)

e r :

Non-dimensional measurement error

e r(dim.) :

Dimensional measurement error (K)

F :

Objective function

f :

Scaling factor

Gr:

Grashof number

g :

Acceleration due to gravity (m/s2)

h :

Local heat transfer coefficient W/(m2·K)

h b :

Base heat transfer coefficient W/(m2·K)

j :

Set containing the constraints

k :

Thermal conductivity W/(m·K)

k a :

Thermal conductivity at ambient condition W/(m·K)

L :

Length of the fin (m)

M :

Number of temperature measurement points

N :

Non-dimensional fin parameter

Nu :

Nusselt number at any location

Nu b :

Nusselt number at the base of the fin

n :

Exponent for variable heat transfer coefficient

p :

Number of constraints

Pr:

Prandtl number

Ra:

Rayleigh number

T :

Temperature (K)

T b :

Base temperature (K)

T a :

Ambient temperature (K)

W :

Width of the fin (m)

x :

Any location along fin length (m)

Y :

Mutant vector in DE algorithm

Z :

Parent vector in DE algorithm

z :

Child vector in DE algorithm

α f :

Thermal diffusivity of fluid medium (m2/s)

β :

Variable conductivity coefficient (K−1)

χ :

Crossover probability in DE algorithm

δ :

Semi fin offset (m)

ɛ 1 :

Non-dimensional thermal conductivity parameter

ɛ 2 :

Taper ratio

ɛ 3 :

Non-dimensional heat transfer coefficient parameter

λ :

Set containing the unknowns

ν f :

Kinematic viscosity of fluid (m2/s)

θ :

Non-dimensional temperature = \({\raise0.7ex\hbox{${\left( {T - T_{a} } \right)}$} \!\mathord{\left/ {\vphantom {{\left( {T - T_{a} } \right)} {\left( {T_{b} - T_{a} } \right)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\left( {T_{b} - T_{a} } \right)}$}}\)

\(\tilde{\theta }\) :

Exact value of non-dimensional temperature

ψ :

Lagrange multiplier in NLP algorithm

ψ f :

Volume expansion coefficient (K−1)

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Authors and Affiliations

  1. School of Mechanical, Materials and Energy Engineering, Indian Institute of Technology, Ropar, Punjab, 140001, India

    Ranjan Das & Kuljeet Singh

  2. Department of Mechanical Engineering, Tezpur University, Napaam, Assam, 784028, India

    Tapan K. Gogoi

Authors
  1. Ranjan Das
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  2. Kuljeet Singh
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  3. Tapan K. Gogoi
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Correspondence to Ranjan Das.

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Das, R., Singh, K. & Gogoi, T.K. Estimation of critical dimensions for a trapezoidal-shaped steel fin using hybrid differential evolution algorithm. Neural Comput & Applic 28, 1683–1693 (2017). https://doi.org/10.1007/s00521-015-2155-x

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