I'm currently designing a single pass shell and tube heat exchanger. In the Nusselt number equation I have chosen to use, it requires me to calculate the Prandtl number at the SURFACE of the tube, shell side. I'm trying to find what the temperature at the surface of the tube will be. I've tried calculating this using constant heat flux however I do not know what the convective heat transfer coefficient is yet so I cannot use this method. Any suggestions?
You must take into account the conduction heat through the wall of the tube. If the flow inside the tube is turbulent, the temperature of the inner tube wall is very close to the fluid temperature, so the heat flux through the tube is the conduction heat that depends on the thermal conductivity of the tube material and inner and outer radius of the tube.
Try to use a numerical method such as the finite volume method to find the surface temperature by knowing the heat flow of the fluid and check your answer by the concept that says the quantity of heat in the series connecting are equals.
Try numerical approach first (i.e. FVM, FEM), it will help solving what you really want to do. In your case, the heat flux through the tube seems like conduction heat that depends on the thermal conductivity of the tube material and diameter of specific tube you're using. About surface temp, the numerical solution will help you in that case.
You can used a fixed surface temperature value that can be calculated based on your inlet temperature and the expected outlet temperature using the principles of conduction and convection.
My guess is that your Nu-correlation contains a correction term, e.g. (Pr_wall/Pr_bulk)^n. This is to adjust for variable properties in the boundary layer, and is typically in the order of unity. So, my suggestion is: Use bulk properties in calculating Nu, then check the impact of the Pr_w term (since you can now calculate T_w) and reiterate if you feel that the influence is significant. This procedure converges rapidly and only 1-2 iterations are required (if any).
From Newton's laws of Convectional Cooling
QS = h(Ts-Ta)
Where Qs = Heat loss by convection to Ambient
h = heat transfer coefficient
Ts = Temperature of the Surface
Ta = Temperature of the Ambient
You can use correlation from process heat transfer book by donald q. Kern page 101 but you must do iteration first, you use surface temperature assumption and then find heat transfer coefficient shell side hs and tube side hi so you will get wall/surface temperature
Either do a numerical simulation or calculate it from the convection process.
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