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A fin design employing an inverse approach using simplex search method

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Abstract

This paper is aimed at estimating unknown parameters in a rectangular fin satisfying a predefined temperature. The differential transformation along with simplex method is used. The study has been done for different initial guess, random errors and measurement points. It is observed that, there is unique value of the convection-conduction parameter, but different conductivity and radiative parameters exist which will be useful in adjusting the parameters amongst various alternatives.

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Abbreviations

A c :

Cross sectional area of the fin (m2)

a :

Integral constant representing the fin tip temperature

b :

Length of the fin (m)

C :

Variable thermal conductivity parameter (=λT b )

C e :

Estimated value of C calculated from the inverse method

R :

Radiative parameter \(\left( { = \frac{{\varepsilon \sigma T_{b}^{3} Pb^{2} }}{{k_{a} A_{c} }}} \right)\)

R e :

Estimated value of R calculated from the inverse method

e :

Error in calculating the integral constant

e r :

Random measurement error

F :

Objective function

h :

Convective heat transfer coefficient (W/m2 K)

k :

Thermal conductivity (W/m K)

N :

Convection-conduction parameter \(\left( { = \sqrt {\frac{{hPb^{2} }}{{k_{a} A_{c} }}} } \right)\)

n :

Set containing the unknowns to be estimated in the inverse method

P :

Perimeter of the fin (m)

Q :

Heat transfer rate (W)

T :

Temperature (K)

t :

Thickness of the fin (m)

w :

Width of the fin (m)

X :

Non-dimensional location along the fin length (=x/b)

x :

Distance measured from the fin surface (m)

a :

Ambient condition

b :

Fin base

e :

Estimated values

~:

Exact value

\(\chi\) :

Expansion coefficient

ε :

Surface emissivity

\(\gamma\) :

Contraction coefficient

η :

Fin efficiency \(\left( { = \frac{Q}{{Q_{ideal} }}} \right)\)

λ :

Variable thermal conductivity coefficient (K−1)

θ :

Non-dimensional temperature \(\left( { = \frac{T}{{T_{b} }}} \right)\)

\(\rho\) :

Reflection coefficient

σ :

Stefan–Boltzmann constant (=5.67 × 10−8 W/m2 K4)

\(\tau\) :

Shrinkage coefficient

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Authors and Affiliations

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

    Ranjan Das

  2. Department of Mechanical Engineering, Tezpur University, Naapam, 784 028, India

    Ashis Mallick

  3. School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50-Nanyang Avenue, Singapore, 639 798, Singapore

    K. T. Ooi

Authors
  1. Ranjan Das
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  2. Ashis Mallick
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  3. K. T. Ooi
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Correspondence to Ranjan Das.

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Das, R., Mallick, A. & Ooi, K.T. A fin design employing an inverse approach using simplex search method. Heat Mass Transfer 49, 1029–1038 (2013). https://doi.org/10.1007/s00231-013-1146-7

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  • DOI: https://doi.org/10.1007/s00231-013-1146-7

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