Cooling of a thick walled pipe transporting water

[Sorade]

TL;DR

How does the outer temperature of the pipe changes as the water flows through it.

Hi everyone,

I'm trying to solve the following problem but can't quite get my head around it.

I have a pipe of length, L (m), diameter, d (m), the pipe walls are thick (multiple meters). I know the inflow temperature of the pipe, say, Tin (K), and the pipe's external wall temperature, Tw.

I found a solution which answers my question by assuming the pipe is infinitely thick. What it does is it determines the radius in which the temperature will change in the pipe based on the amount of water flowing through it and the time it has been flowing for.

I am happy with that assumption as long as the radius of temperature change is lower than that of the pipe. When that happens it doesn't make sense and the pipe's wall external temperature, Tw, should be dropping cooling markedly.

When that happens I am unsure how I can determine, both by how much Tw, cools, and from there Tout. My flow rate through the pipe is constant.

Can I ask if:

A) this is a simple problem that can be solved analytically?
B) If so how would you go about it?

I know all the thermal and material properties of the pipe and water.

Thank you.

 

[anorlunda]

The key thing you are missing is the heat transfer from the outer wall to the surrounding environment.

If the pipe is surrounded by perfect insulation, then the outer wall temperature will eventually match the inner wall temperature. If the pipe is on the ocean floor, the outer temperature will be very close to water temperature. There are infinite intermediate cases.

 

[Sorade]

Hi anorlunda,

Thank you for your response. You are absolutely correct, and in my case the pipe is surrounded by perfect insulation...(in practice I actually have a network of pipes evenly spaced together, so half way between the pipes the heat transfer is equal but in opposite direction).

Does that help to narrow down the cases?

Also, the flow in the pipe is turbulent.

 

[russ_watters]

Wait, is the outer wall temp given or are you trying to find it?

 

[Sorade]

I know the initial outer wall temperature yes. But eventually the cooling will reach the outer wall and then the outer wall's temperature is starting to drop. When that happens I need to figure out both the temperature drop of the outer wall, and the outflow temperature, as function of time. Essentially, I have a finite amount of heat in my thick pipe. I'm not sure that what I'm trying to obtain is possible analytically. It sounds like it should be, but I've been scratching my brain and notepad for hours without any success. Here's a quick sketch to help define the problem: