Is Covection-Diffusion the right equation for my problem?
 

I have a simple internal pipe flow. The walls are made of water (if that makes sense) and I know the walls's temperature at the top and at the bottom. I also know the inlet temperature of the air, but I want to know the outlet temperature of the air - I also want the temperature profile of the whole wall, knowing temperatures at every point.

Which equation should I be solving, I have a feeling it should be Convection-Diffusion equation.

Here, I've made a simple illustration to explain my problem.
(PLEASE HELP :confused:)
https://drive.google.com/open?id=1hy...On8bnMn9JEi8sb

I have doubts on the configuration, as even in deep space without gravity you wouldn't be able to keep "water walls" still with the air flowing along, not to mention how you could blow air between them and not move the water.

However, it can be done simply by treating the "walls" as solid (as in more regular scenarios) but using the water properties.

But I'm not sure if this actually represents what you have in mind... and, in any case, you have to solve the NS equations, not a simple convection diffusion one

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.

Yes, I'm considering constant velocity of both water and air in the cooling tower and I've confirmed that experimentally, it's a very reasonable assumption.

But does the temperature equation take into account the changes in latent heat as water evaporates? and by temperature equation, you mean the energy governing (N-S) equation, right?

Thanks

Yes, I'm considering constant velocity of both water and air in the cooling tower and I've confirmed that experimentally, it's a very reasonable assumption.

But does the temperature equation take into account the changes in latent heat as water evaporates? and by temperature equation, you mean the energy governing (N-S) equation, right?

Thanks

Yes, the internal energy equation is that for solving the temperature field.
Be aware that a uniform velocity field is a vere very rude assumption.

In the energy equation, how do I incorporate temperature change due to phase change process? In my case, water temperature drops as it loses heat of vaporization and water vapors leave the bulk water surface. But air also loses temperature as a temperature difference is created between water and air due to phase change.

you have to look for multiphase flow problems, a very different and more complex NSE problem than the simple linear convection-diffusion equation