scholarly journals Exterior Shape Factors From Interior Shape Factors

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
John H. Lienhard
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Conformal Mapping ◽  
Analytical Determination ◽  
Two Dimensional ◽  
Shape Factors ◽  
Transfer Rates ◽  
Steady Heat Conduction ◽  
Steady Heat

Shape factors for steady heat conduction enable quick and highly simplified calculations of heat transfer rates within bodies having a combination of isothermal and adiabatic boundary conditions. Many shape factors have been tabulated, and most undergraduate heat transfer books cover their derivation and use. However, the analytical determination of shape factors for any but the simplest configurations can quickly come to involve complicated mathematics, and, for that reason, it is desirable to extend the available results as far as possible. In this paper, we show that known shape factors for the interior of two-dimensional objects are identical to the corresponding shape factors for the exterior of those objects. The canonical case of the interior and exterior of a disk is examined first. Then, conformal mapping is used to relate known configurations for squares and rectangles to the solutions for the disk. Both a geometrical and a mathematical argument are introduced to show that shape factors are invariant under conformal mapping. Finally, the general case is demonstrated using Green's functions. In addition, the “Yin-Yang” phenomenon for conduction shape factors is explained as a rotation of the unit disk prior to conformal mapping.

Heat transfer at small Grashof numbers

1957 ◽  
Vol 238 (1214) ◽  
pp. 412-423 ◽  
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Ambient Temperature ◽  
Heat Conduction Equation ◽  
Body Shape ◽  
The Body ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Transfer Rates ◽  
Theory And Experiment

Rough physical arguments suggest that the heat transfer from a body, immersed in a fluid, should be determined by the heat-conduction equation alone whenever the Grashof number, G , associated with the problem is small. However, heat-transfer rates predicted in this fashion are not always in accordance with the experimentally determined values. It is shown that, while convection is negligible in comparison with conduction near the body, it becomes as important at distances from the body of the order ( G ) -n , where n varies between 1/3 and ½ with the body shape. Whenever this distance is large in comparison with all the dimensions of the body the use of the conduction equation yields correct heat-transfer rates. If, however, this distance is small in comparison with the body length, the heat transfer may be calculated from the two-dimensional convection solution. An examination of the solutions in these two extreme cases reveals that the heat loss is the same as that by conduction to a certain surrounding surface maintained at ambient temperature. This interpretation enables certain qualitative deductions to be made for the case when the ratio of the lengths is neither large nor small. The agreement between theory and experiment is satisfactory.

Inverse Determination of Boundary Conditions and Sources in Steady Heat Conduction With Heat Generation

1996 ◽  
Vol 118 (3) ◽  
pp. 546-554 ◽  
Author(s):  
T. J. Martin ◽  
G. S. Dulikravich
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Conduction Problem ◽  
Analytic Solutions ◽  
Heat Sources ◽  
Two Dimensional ◽  
Simple Modification ◽  
Ill Posed ◽  
Steady Heat Conduction ◽  
Steady Heat

A Boundary Element Method (BEM) implementation for the solution of inverse or ill-posed two-dimensional Poisson problems of steady heat conduction with heat sources and sinks is proposed. The procedure is noniterative and cost effective, involving only a simple modification to any existing BEM algorithm. Thermal boundary conditions can be prescribed on only part of the boundary of the solid object while the heat sources can be partially or entirely unknown. Overspecified boundary conditions or internal temperature measurements are required in order to compensate for the unknown conditions. The weighted residual statement, inherent in the BEM formulation, replaces the more common iterative least-squares (L2) approach, which is typically used in this type of ill-posed problem. An ill-conditioned matrix results from the BEM formulation, which must be properly inverted to obtain the solution to the ill-posed steady heat conduction problem. A singular value decomposition (SVD) matrix solver was found to be more effective than Tikhonov regularization for inverting the matrix. Accurate results have been obtained for several steady two-dimensional heat conduction problems with arbitrary distributions of heat sources where the analytic solutions were available.

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