Heating and cooling of thin walled vessels

The relationships for steady heat flow can also be applied to the solution of a transient heat transfer problem, namely to the calculation of the temperature change with time during the heating and cooling of a thin walled vessel filled with a liquid. Two simplifications have to be made  

The temperature of the liquid inside the vessel is the same throughout, it only changes with time i9F = i9F (t). 

The heat stored in the vessel wall, or more precisely the change in its internal energy, can be neglected. 

The first assumption is true for most cases, as free or forced convection due to an agitator in the vessel, lead to almost the same temperature throughout the liquid. The second is only correct when the heat capacity of the contents is much larger than the heat capacity of the vessel wall. This happens in the heating and cooling of liquids in thin walled vessels, but may not be applied to vessels containing gases, which have either thick or well insulated walls. 

When both these assumptions are valid, at every point in time the temperature of the fluid is spatially uniform, and the wall temperature will be predicted by the equations valid for steady state. In a flat vessel wall the temperature changes linearly, however the straight line moves according to time. 

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