Exact Solutions for Heat Conduction
This work involves computing exact solutions for conductive heat transfer problems, primarily to verify finite element codes so that a determination can be made as to the accuracy of the finite element solutions. This work is being conducted with James V. Beck, Professor Emeritus at Michigan State University and president of Beck Engineering Consultants Company. For more information on his work, you can visit his web site at www.BeckEng.com. Much of this work is documented in a paper entitled “Precise Solutions for Verification in Transient Three-Dimensional Cartesian Heat Conduction” in the Journal of Numerical Heat Transfer by McMasters, Dowding Beck and Yen. The tables giving all of the Green's functions for this work can be viewed at Green's Function Tables.
One particular aspect of this work is solving conduction problems which include solid body motion. Below is the development of a solution for a one dimensional case with known heat flux on both boundaries. An interesting point associated with this solution is that a non-zero eigenvalue appears for the zeroth term in the infinite series solution. This is something which does not occur in solutions involving non-moving solids.
We now consider the determination of these eigenfunctions and eigenvalues so that the Green’s functions may be given. The problem can be described by
(1)
(2)
(3)
The temperature problem can be transformed to remove the solid body motion in eq. (1) by using
(4)
which yields the W problem of
(5)
(6)
(7)
Using the product solution in the separation of variables for W(x,t) = X(x)Q (t) gives
(8)
Notice the positive sign before the 
term; this is an anomalous term and does not usually occur but may be present depending upon the effective heat transfer coefficients. The presence of this term causes the solid body flow problem to be unique. The solution for the X problem can be written as
(9)
The boundary conditions for the X problem are
(10)
Using the boundary condition at x = 0 gives
(11)
Using the condition at x = L gives the eigencondition,
(12)
where
(13)
(14)
Using these definitions, eq. (12) can be written as
(15)
Using this eigencondition for the XU22 case (which has B0 = BL = 0) gives the eigenvalue of
(16)
The eigencondition given by eq. (15) written in a form similar to the trigonometric relation is
(17)
The eigenfunction can be written as
(18)
The solution of the time problem in eq. (8) is
(19)
The zeroth term for the Green's function can then be written as
(20)
where the norm is given by
(21)
For the special XU22 case, this reduces to
(22)
Also for this special case, the eigenfunction simplifies to
(23)
Using these two equations then gives the zeroth term for the XU22 Green's function of
(24)
The XU22 Green’s function for the other eigenvalues is found from the X33 case with B0 = -P/2 and BL = P/2 and multiplied by the group indicated by eq. (4), with t replaced by u. The result is
(26)
(27)