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

Image111 (1)

Image112 (2)

Image113 (3)

The temperature problem can be transformed to remove the solid body motion in eq. (1) by using


which yields the W problem of

Image115 (5)

Image116 (6)


Using the product solution in the separation of variables for W(x,t) = X(x)Q (t) gives


Notice the positive sign before the Image119 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

Image120 (9)

The boundary conditions for the X problem are

Image121 (10)

Using the boundary condition at x = 0 gives

Image122 (11)

Using the condition at x = L gives the eigencondition,

Image123 (12)


Image124 (13)

Image125 (14)

Using these definitions, eq. (12) can be written as

Image126 (15)

Using this eigencondition for the XU22 case (which has B0 = BL = 0) gives the eigenvalue of

Image127 (16)

The eigencondition given by eq. (15) written in a form similar to the trigonometric relation is

Image128 (17)

The eigenfunction can be written as

Image129 (18)

The solution of the time problem in eq. (8) is

Image130 (19)

The zeroth term for the Green's function can then be written as

Image131 (20)

where the norm is given by

Image132 (21)

For the special XU22 case, this reduces to

Image133 (22)

Also for this special case, the eigenfunction simplifies to

Image134 (23)

Using these two equations then gives the zeroth term for the XU22 Green's function of

Image135 (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

Image137 (26)

Image138 (27)