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Shaft temperature calculation

Q: If one pump is handling oil at temp 320ºC and the bearing housing temperature is around 82ºC. I want to calculate the shaft temperature that is rotating at bearing support. How do I calculate the heat transfer if the shaft is made up of steel 4140?

A: I am not familiar with the geometry. However, it is clear that you must first determine an approximate film coefficient for heat transfer from the oil to the wetted parts of the pump. The overall heat transfer and energy balance around the pump parts, oil and ambient air must be performed to determine shaft temperature. It may be best to approach this problem with finite element analysis after calculating boundary conditions at the surfaces.