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RESEARCH PAPERS: Solution Methods

The Simplified-Fredholm Integral Equation Solver and Its Use in Thermal Radiation

[+] Author and Article Information
K. G. Terry Hollands

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

The accuracy of the numerical method is determined by the goodness of fit of the relation between $Nd$ and the value of $u$ computed at that $Nd$ setting.

Textbooks have attached various names to $Ii,j$: interchange factor (15), gray-body shape factor (16), total exchange factor (17) (this after division by $εiεj$), transfer factor (18), and exchange factor (4-5). Several authors (19-20) elected not to give it a name. Thus the choice here of the term exchange factor is somewhat arbitrary.

It should be noted that a simple extension to Eq. 24 applies when the adiabatic surfaces are of prescribed uniform heat flux instead. For simplicity, we do not consider that case here.

We switch to $χ$ as the second variable in our integral equation instead of $x$ because $x$ can also represent the coordinate in the $X-Y$ plane.

J. Heat Transfer 132(2), 023401 (Dec 01, 2009) (6 pages) doi:10.1115/1.4000183 History: Received October 23, 2008; Revised April 13, 2009; Published December 01, 2009; Online December 01, 2009

Abstract

The application of a new Fredholm integral equation solver to problems in thermal radiation is explored. The new method provides a simplified version of Fredholm’s own 1903 solution which, while being highly important from a theoretical point of view, had been considered too complex to provide a practical tool for solving integral equations. The method does not involve solving large arrays of simultaneous equations; rather, the simplified-Fredholm method provides an explicit solution. The solution consists of an infinite series with each term containing multiple integrals. It has been found, however, that the series can be safely truncated after about a dozen terms, and the multiple integrals can be resolved through repeated matrix multiplications, all of this leading to a practical methodology. Implicit in the method and highly useful in radiant analyses is the idea of the resolvent kernel, which permits generalized solutions to be obtained, independent of the forcing function. The method also adapts itself to a simple technique for establishing the possible error in any result. It is illustrated here on some enclosure problems that can be reduced to solving Fredholm’s equation in a single variable.

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Figures

Figure 2

Cross section of two concentric cylinders with a small gap between them, the outer one being at uniform temperature T0 and the inner one open at both ends to an environment at 0K

Figure 1

Plot of resolvent kernel corresponding to a hollow cylinder open at both ends to an environment at 0K and with a prescribed heat flux applied to its interior wall

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