Heat transfer in boundary layers

When the fluid and the immersed surface are at different temperatures, heat transfer will take place. If the heat transfer rate is small in relation to the thermal capacity of the flowing stream, its temperature will remain substantially constant. The surface may be maintained at a constant temperature, or the heat flux at the surface may be maintained constant or smface conditions may be intermediate between these two limits. Because the temperature gradient will be highest in the vicinity of the smface and the temperature of the fluid stream will be approached asymptotically, a thermal boundary layer may therefore be postulated which covers the region close to the surface and in which the whole of the temperature gradient is assmned to lie. 

In general, the thermal and momentum boimdaiy layers will not correspond. In the ensuing treatment, the simplest non-interacting case will be eonsidered 

At steady state and with no somce or sink present in the control volume, the heat balance for the control volume ABCD can be stated as follows  

The sum of heat convected in through planes AB and BD and that added by conduction at wall AC must be equal to the heat convected out at plane CD (7.35) 

The rate at which heat enters the control volume at plane AB is  

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