Plotting pressure distribution of fluid flow in a pipe

[shevir1]

Hi all,

I am analysing the flow of water in a subsea pipe, and have been getting stuck in trying to solve the energy equation to plot a temperature distribution across the pipe diameter

I have derived the equation as such seen in my attachement. But my main question is how should/would I determine the wall temperature of the flowline.

Initially I was assuming to take the operating temperature of the flow line to be the temperature of the inner wall. Am I correct in saying this?
Also there is the ambient sea temperature which could well have an effect on the innerwall temperature due to convection.

What would be the best way for me to determine the inner wall temperature of the flowline?

Assuming operating temp of flowline is:
Assuming ambient sea temp of flowline is:
Steady state

Please see attached my derivation of the energy equation which then results in my final equation I am to use.

 

[shevir1]

see pic attached