In the design of high-temperature equipment, two problems are often encountered: (1) determining the effect of slow changes of the thermal conductivity (which can occur as a result of chemical and/or mechanical factors) on the thermal behaviour of the equipment, and (2) finding the most efficient combination of materials for the design of the equipment when several materials are available. Both of these problems are customarily solved by trial-and-error techniques. In this paper, it is first shown that the temperatures before and after such changes of the thermal conductivity are related by simple linear equations. It is demonstrated that the process of trial-and-error in the solution of the above problems can be entirely eliminated by the use of these simple equations.