9.54. The combination of nonuniform heat-source distribution and ir­regular geometries, which occur in many reactor components, complicates the calculation of thermal-conduction rates and temperature distributions. The problem is an important one because high thermal stresses frequently result from large temperature gradients. A study of the temperature stresses in reactor components, which frequently have irregular shapes, depends upon a knowledge of the temperature pattern within the materials. A number of relatively simple codes suitable for microcomputers are available to solve multidimensional conduction problems under both steady-state and transient conditions. A stepwise, finite-difference approach, as de­scribed in the following sections for transient heat conduction, is effective. An example of such a situation is found in graphite-moderated, gas-cooled reactors. These generally consist of massive graphite blocks pierced by long holes with cylindrical fuel elements and passages for coolant. Pressure vessels containing the reactor and thermal and biological shields frequently present similar difficulties in the determination of temperature distribution and thermal-stress analysis. For such cases, a rigorous solution of the basic heat-conduction equation (9.16) is usually impossible. However, approx­imate methods have been developed in some instances; thus, a circular harmonics series solution has been proposed for the case of a massive graphite block treated as equivalent to a number of adjacent cylinders [3].

9.55. Temperature patterns for nonregular boundary conditions may be determined using numerical “relaxation” methods. The procedure is first to divide the irregular volume, so far as is possible, into a number of regular subvolumes. The heat-conduction equation may now be applied to
these subvolumes in a systematic manner based on an assumed temperature distribution compatible with the boundary conditions. By carrying out a heat-balance calculation for each of the sub volumes, the inaccuracy of the assumed temperature pattern is determined and expressed as a so-called residual for each subvolume. On the basis of these residuals, the assumed temperatures are changed in a systematic manner to reduce (or “relax”) the residuals from point to point. The heat-balance procedure and tem­perature adjustment are repeated until the residuals have been reduced to zero; the corresponding temperature distribution is then taken to be the correct one.