The development and compilation of formal solutions to heat transfer problems which occur in reactor design is an important phase of reactor engineering. Formal analytical solutions are useful both for making first approximations and as a check of more detailed work. Three solutions to three different cases of transient heat transfer in a conduit cooled on the inside by a flowing coolant are presented. The heat transfer mechanism is described by a pair of coupled partial differential equations applicable to nuclear reactor design and analysis.The first solution is for the case of coolant flowing at constant velocity through a conduit with internal heat generation a function of distance. The heat transfer coefficient from conduit to coolant is infinite for transfer so that the conduit and coolant temperatures are always equal. The coolant inlet temperature varies with time. All physical properties of the coolant and conduit are taken as constant. Four specific sets of conditions are considered.In the second case the coolant inlet temperature is constant, the heat transfer coefficient is infinite, the internal heat generation is a function of distance, and the coolant velocity decreases with time, as on loss of pumping power. Three specific sets of conditions are considered.The third case is the same problem as case one except that the heat transfer coefficient between the conduit and coolant is finite.

The thermal stresses in the nozzles of heavy-walled reactor vessels, caused by sudden changes in coolant temperature, are of great interest to the reactor designer. An analysis has been made of this problem for both high and low flows of cold water. The results show that the nozzle investigated can endure a large number of cycles of the worst case analyzed.