Abstract
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.
Highlights
Shape factors for steady heat conduction have been tabulated in a number of publications [1,2], and most undergraduate heat transfer textbooks derive and use shape factors [3]
We have considered conduction shape factors for twodimensional, connected objects that have two isothermal boundaries, each at different temperature and separated by two adiabatic boundaries
Shape factors for conduction inside an object are equal to those for conduction through the material outside the object, if the only heat sources and sinks are the isothermal segments of the boundary and there is no net heat transfer to the exterior region at great distance from the object
Summary
Shape factors for steady heat conduction have been tabulated in a number of publications [1,2], and most undergraduate heat transfer textbooks derive and use shape factors [3]. Where k is the thermal conductivity and S is the shape factor, which is dimensionless For simple configurations, such as a rectangle or an annular sector with opposing isothermal edges, S is found; but for more complicated shapes, finding S rapidly becomes analytically difficult. This difficulty is especially apparent when the aim is to find S for material exterior to a closed curve. The solution for heat conduction from one side of the interior of a square of height a to the other side is trivial to find analytically: Q_ 1⁄4 kaDT=a, so S 1⁄4 1. We have made the empirical observation that, for some specific cases including this one, the exterior shape factor is the same as the interior shape factor
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