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Optimal effectiveness and efficiency of a fin in steady-state: multiobjective approach

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Abstract

A fin serves as an extended surface to enhance the heat transfer from a larger heated mass to which it is attached. We use a multiobjective optimization (MO) approach for a fin in steady-state to analyze the dynamics of effectiveness and efficiency simultaneously. Our approach is based on a piecewise linear approximation of the design. We use the \(\epsilon\)-constraint method of MO to identify Pareto points for effectiveness and efficiency and hence the Pareto design of the fin. We conduct several numerical experiments using the fmincon routine from the MATLAB. The numerical experiments show that our approach provides a unique interaction between effectiveness and efficiency independent from the initial profiles within the user-defined tolerance range of the fmincon routine.

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Abbreviations

x :

Coordinate along the fin, \(0\le x\le L,\) m

L :

Length of the fin, m

r(x):

Radius of cross section, m

\(\rho (x)\) :

Dimensionless radius of cross section

\(^d\) :

Index that denotes discrete approximation

A(x):

Area of cross section of the fin, \({\mathrm{m}}^2\)

\(A_s(x)\) :

Surface area of the fin on the interval [0, x],  \({\mathrm{m}}^2\)

\(A_s^0(x)\) :

Dimensionless area

\(dA_s(x)\) :

Differential surface area, \({\mathrm{m}}^2\)

\(A_b=A(0)\) :

Base area, \({\mathrm{m}}^2\)

\(q_f\) :

Heat transfer rate, W

\(Bi=hL/k\) :

Biot number, dimensionless

n :

Number of intervals in discretization algorithm

\(\varDelta =L/n,\) :

m

h :

The convective heat transfer coefficient, \({\mathrm{W}}/({\mathrm{m}}^2\,{\mathrm{K}})\)

k :

The conduction coefficient, \({\mathrm{W}}/({\mathrm{m}}\,{\mathrm{K}})\)

T(x):

Temperature of fin, \(^\circ {\mathrm{C}}\)

\(T_b\) :

Fin base temperature, \(^\circ {\mathrm{C}}\)

\(T_{\infty }\) :

Surrounding fluid temperature, \(^\circ {\mathrm{C}}\)

V :

Volume of the fin, \({\mathrm{m}}^3\)

\(m_j=\frac{r_{j+1}-r_j}{\varDelta }\) :

Slope, dimensionless

\(\epsilon _f\) :

Fin effectiveness, dimensionless

\(\tau (x)\) :

Normalized temperature, dimensionless

\(I_s(u)\) :

And \(K_s(u),\,s=1,2\) the modified Bessel functions of order s of the first and second kind

\(\kappa\) :

Parameter characterizing the initial \(\kappa\) profile, dimensionless

\({\mathcal {R}}^p\) :

\(p-\)dimensional Euclidean vector space

\(f=(f_1,\dots ,f_p)^T\) :

Vector valued-function

X :

Set of all feasible solutions of the MOP

\(Y:=f(X)\) :

Set of all outcome vectors of the MOP

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Acknowledgements

The authors would like to thank the anonymous reviewers for their invaluable comments, which allowed the authors to eliminate some mistakes. Most importantly, the reviewers kindly indicated some directions in the heat transfer research that were new to the authors.

The research of the first author was supported by a Faculty Development Grant at the University of Tennessee at Chattanooga.

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Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN, 37403-2598, USA

    Lakmali Weerasena & Boris P. Belinskiy

  2. College of Engineering, The University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN, 37403-2598, USA

    James W. Hiestand

Authors
  1. Lakmali Weerasena
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  2. Boris P. Belinskiy
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  3. James W. Hiestand
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Correspondence to Lakmali Weerasena.

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Weerasena, L., Belinskiy, B.P. & Hiestand, J.W. Optimal effectiveness and efficiency of a fin in steady-state: multiobjective approach. Optim Eng 22, 1157–1180 (2021). https://doi.org/10.1007/s11081-020-09497-9

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  • DOI: https://doi.org/10.1007/s11081-020-09497-9

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