A Bubble size distribution model has been developed for the numerical simulation of cryogenic high-speed cavitating flow of the turbo-pumps in the liquid fuel rocket engine. The new model is based on the previous one proposed by the authors, in which the bubble number density was solved as a function of bubble size at each grid point of the calculation domain by means of Eulerian framework with respect to the bubble size coordinate. In the previous model, the growth/decay of bubbles due to pressure difference between bubble and liquid was solved exactly based on Rayleigh-Plesset equation. However, the unsteady heat transfer between liquid and bubble, which controls the evaporation/condensation rate, was approximated by a theoretical solution of unsteady heat conduction under a constant temperature difference. In the present study, the unsteady temperature field in the liquid around a bubble is also solved exactly in order to establish an accurate and efficient numerical simulation code for cavitating flows. The growth/decay of a single bubble and growth of bubbles with nucleation were successfully simulated by the proposed model.