Three dimensional transient heat conduction and thermal stresses in a homogenous sphere via dynamic relaxation

A number of investigations have been made into heat conduction and thermal stresses in pebble bed reactor spherical fuel elements, but these are either two dimensional or steady state analyses. The present study deals with a more general transient three dimensional model, with points of constraint where the fuel sphere is in contact with adjacent spheres. Beryllium oxide (BeO) is the material of the sphere, and all heat transfer and elastic properties are assumed to remain constant under changing temperatures. The sphere is initially at a uniform temperature equal to that of the coolant fluid in the core. In the transient process, which approximates to a sudden increase in power demand on the reactor, the sphere suddenly begins and maintains internal heat generation while the temperature of the coolant remains constant. The resultant change in temperatures and stresses in the sphere with respect to time and space are then the object of the present study. Dynamic Relaxation is used in the present analysis, which is numerical in nature. The three dimensional transient heat conduction equation is transformed, by the introduction of dummy variables, into a set of first order equations, which are then solved by Dynamic Relaxation. Similarly, the set of three dimensional equilibrium equations are solved also by Dynamic Relaxation. It is assumed that stress distribution is only a function of instantaneous temperature distribution and the physical constraints present, but is independent of the time history of the stresses. Computer programs are utilized in applying the Dynamic Relaxation method, and experiments are performed on these programs to find the natural frequency and hence optimum damping required for quickest convergence of the iterative process. ..