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A new iterative Broyden Legendre Wavelet Galerkin FEM applied to study heat transfer in two-dimensional elliptic and plate fins

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Abstract

In this paper, a novel hybrid method is applied to study heat transfer in 2-D fins experiencing a range of conditions, including dry, wet, and partially wet, subjected to the most generalized boundaries. This model represents a boundary value problem involving nonlinear heat equations of second order. A new iterative Broyden Legendre Wavelet Galerkin Finite Element approach is used for solving the problem. The process of discretizing the Y coordinate and applying Hadamard, Khatri–Rao, and face-splitting matrices products with the Legendre wavelet Galerkin technique transforms the main problem into a system of nonlinear algebraic equations. The solution for this system is obtained by using the iterative Broyden technique. It has been found that when the values of latent heat and Biot number rise, the temperature in an elliptic fin falls. In a specific case, the present results are compared with exact values and found to be approximately the same. The impacts of various parameters, including Biot number, latent heat, Kirpichev number, fin thickness, Lewis number, \(\mu\), \(\eta\) and \(\xi\) on the temperature profile of a fin are discussed in detail. A comparative analysis of elliptic and plate fin efficiencies for different boundary conditions is provided and highest efficiencies observed in the plate fin. The finding indicates that better fin efficiency requires a lower fin thickness and higher Biot number values. The present method has been successfully applied to linear and nonlinear problems.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors' comment: Data available on the request.]

Abbreviations

A :

Surface area of the fins (mm\(^2)\)

Ar :

Tube axis proportion (a/b)

a :

The elliptic tube’s semi-major axis (mm)

b :

The elliptic tube’s semi-minor axis (mm)

Bi :

Biot number, (hb/k)

Ki :

Kirpichev number

\(c_p\) :

Dry air specific heat (kj kg\(^{-1}\) \(^\circ {\text {C}}^{-1})\)

h :

Convective heat coefficients average (W m\(^{-1}\) \(^\circ {\text {C}}^{-1})\)

\(h_d\) :

Average mass transmission coefficient on humidity ratio variance (kg \({{\text {m}}^{-2}} {{\text {s}}^{-1}})\)

i :

Humid air enthalpy (kj \({\text {kg}}^{-1})\)

\(i_{fg}\) :

Latent heat (kj \({\text {kg}}^{-1})\)

k :

Fins thermal conductivity (Wm\(^{-1}\) \(^\circ C^{-1})\)

Le :

Lewis number, \((Le={(h/{h_d}{c_p})}^{3/2})\)

l :

Height of the fins (mm)

q :

Rate of heat transfer (W)

T :

Temperature \((^\circ {\text {C}})\)

W :

Humidity proportion (kg water vapor/kg dry air)

\(W_a\) :

Ambient air humidity proportion (kg water vapor/kg dry air)

\(\eta _{f}\) :

Efficiency of the fins

\(\delta\) :

Thickness of the fins (mm)

\(l^*\) :

Dimensionless fin height

X :

Dimensionless coordinate

Y :

Dimensionless coordinate

\(\delta ^*\) :

Dimensionless fin thickness

\(\theta\) :

Dimensionless temperature

BLWGFEM:

Broyden Legendre Wavelet Galerkin Finite Element Method

BVP:

Boundary Value Problem

BC Ist:

Boundary Condition of first kind

BC IInd:

Boundary Condition of second kind

BC IIIrd:

Boundary Condition of third kind

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Acknowledgements

Authors are thankful to Eternal University Baru Sahib, India, for providing necessary facilities.

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This research did not receive any grants from public, commercial, or non-profit organizations.

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Authors and Affiliations

  1. Department of Mathematics, Indian Naval Academy, Kannur, Kerala, India

    Subrahamanyam Upadhyay

  2. Department of Mathematics, Eternal University, Baru Sahib, Himachal Pradesh, India

    Priti Sharma & Surjan Singh

  3. Department of Mathematical Sciences, IIT BHU, Varanasi, U.P., India

    K. N. Rai

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  1. Subrahamanyam Upadhyay
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Upadhyay, S., Sharma, P., Singh, S. et al. A new iterative Broyden Legendre Wavelet Galerkin FEM applied to study heat transfer in two-dimensional elliptic and plate fins. Eur. Phys. J. Plus 138, 1154 (2023). https://doi.org/10.1140/epjp/s13360-023-04788-3

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