Heat and mass transfer within a vertical pipe with a surface heating element of variable size

Heat and mass transfer due to upstream fluid flow in a vertical pipe which is heated in some region due to an external heating element on the surface of the pipe is considered. Unlike most studied in the literature which consider heating uniformly over the entire pipe, we allow for the heater to act over a smaller sub-region of the pipe surface. We first derive a heat and mass transfer model to describe the velocity, pressure, and temperature evolution in a vertical pipe under the assumption of cylindrical symmetry. Using a finite element method we are able to obtain numerical simulations to this model. We compare solutions under a variety of different heater configurations, in order to understand how the size and placement of the heating element on the surface of the pipe will modify the thermal properties of the fluid. We find that a smaller heating element placed near the top of the pipe can still deliver sufficient heat so that the temperature of fluid exiting the top of the pipe has desirable thermal properties for a specific application, and in such cases it is not necessary to heat the entire length of the pipe. Such a configuration could be more efficient, as it requires less material for the heating element, while also requiring less energy for the heating. On the other hand, if the heating element is too small, or poorly placed along the pipe, then it may not be possible to obtain desirable thermal properties in the fluid that would have been possible with a heating element covering the entire pipe length.