[FMDenaro]

Originally Posted by Usama1034

Well I actually just want to develop a mathematical model for the evaporation phenomena for a simplistic cooling tower in which I have the freedom to change input air Temperature and input water Temperature.
My methodology is to solve Convection Diffusion equation for the water using FDM, this would simply give me a temperature profile of the water along the length of the tower. I will then use this temperature profile in a formula for convection that is designed to give output temperature of air flowing inside a pipe, if you have the temperature of the internal walls of the pipe and the temperature of the air inlet. I intend to use the Convection Diffusion equation to get the temperature of internal walls.

Finally, I should be left with a mathematical model that gives you the temperature profile of air inside a cooling tower assuming that all the water thrown from the top gets evaporated just when it reaches the bottom.

Why would I need to solve the N-S equations? Is there a mistake or shortcoming in my methodology?

I'm a total beginner so my questions may sound naive. Sorry :/

The convective part of the temperature equation requires the velocity field of the water. Are you considering that the velocity field is already known? Generally, temperature can be solved after the velocityt field in case of incompressible flows but some case requires to take into account the buoyancy effect.