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I am trying to model the flow of steam through a vertical cylinder. Ultimately I would like to find the surface temperature as a function of the length of the pipe. I am assuming the steam cools as it moves upward along the length of the pipe and it is a steady-state process, so T only varies with position (1-dimensional).

Doing a shell balance around a small portion of the pipe I am confused about the convection term entering in from the base and leaving through the top of the shell. I understand the cross sectional area concerning this convection term is the area of a circle. However, I am uncertain about how to determine the gradient for the surface temperature when all of the terms in the energy balance are due to convection.

I have seen the similar process done for a solid pipe (long fin) where there was conduction through the shell and convection out of the side. But I am confused by fluid motion inside the shell.

Any advice on this conceptually, or if you could recommend another approach, would be appreciated.

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# Heat Convection through a Vertical Pipe

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