Conduction in rectangular plates with boundary temperatures specified: Fid-Bau Portal

Conduction in rectangular plates with boundary temperatures specified

Beck, James V. / Wright, Neil T. / Haji-Sheikh, A. / Cole, Kevin D. / Amos, Donald E.

The exact steady-state heat conduction problem for a rectangle with temperature boundary conditions is treated. It is associated with one or more transient problems. The series solutions available in some advanced heat conduction books for the temperature, and particularly the heat flux, converge slowly at the surface where a non-zero temperature is prescribed. Solutions with improved convergence are available. Even with these latter solutions, further improvement is possible based on a series identity given by Morse and Feshbach. This identity is used in this paper and extended. The solution is particularly effective for large aspect ratios in the nonhomogeneous direction, which is the x-direction for the X11B10Y11B00 problem. (When the body is semi-infinite in the x-direction and T(0) = T₀, it is denoted as the X10B1Y11B00 problem since L = L/W tends to infinite.) Another algebraic form for a series present in the semi-infinite in the y-direction problem is derived. This problem is denoted X11B10Y10B0 and is for the aspect ratio L = L/W tends to 0. These two complementary semi-infinite heat conduction problems are examined in some detail. Contrary to common usage, Cole, Yen, Beck, and coworkers have shown that convergence in steady-state problems is frequently improved by using the eigenvalues in direction of the nonhomogeneous boundary conditions, rather than in the homogeneous direction. Numerical values are given for some cases. An analysis is given to determine the required number of terms in the series to obtain a desired accuracy; the number of terms can be specified in 'for' loops in Matlab or 'DO' loops in Fortran, providing better computer programming practice.