What is the final temperature of the air in a tube

[roldy]

Homework Statement

I have a copper tube with outer radius r₂ and inner radius of r₁. Half the tube is exposed to the surrounding air while the other half is embedded into the ground. The outside air temperature is T₂ and the ground temperature is T₃. What is the air temperature inside the tube , T₁, after equilibrium is reached. This is not a homework assignment but more of a thought experiment.

Homework Equations

$$\dot{Q} \frac {T_1 - T_2}{R_{Total}}$$
$$\Delta T_{tube} = \dot{Q}R_{pipe}$$

3. The Attempt at a Solution [/B]

I was following along using the method described here https://www.engineersedge.com/heat_transfer/heat_loss_insulated_pipe_13865.htm. I then realized that the resistance of the ground needs to come into play. I'm not sure how to account for this since the problem is no longer symmetric about the axis. One possible way to solve this is to split the tube in half and solve for the temperature change in the top half and the temperature change in the bottom half. Then do another heat transfer analysis for the mixing of the two air "regions" to find the equilibrium temperature. Is this thinking correct? I can't recall from college heat transfer courses if I had such an example as this.

 

[BvU]

This is not a homework assignment but more of a thought experiment

You still want a complete problem statement: I suppose the lower half of the tube is filled with air, not with Earth ?
What makes you write the problem is no longer symmetric about the axis ? There is rotational symmetry as far as I can follow the description ?

What is the air temperature inside the tube , T1, after equilibrium is reached.

What makes you think there is one single temperature and not a temperature profile ?

 

[roldy]

You still want a complete problem statement: I suppose the lower half of the tube is filled with air, not with Earth ?
What makes you write the problem is no longer symmetric about the axis ? There is rotational symmetry as far as I can follow the description ?
What makes you think there is one single temperature and not a temperature profile ?

The whole tube is filled with air. The reason why I don't consider this problem symmetric is because of the 2 different temperature boundary conditions. After a long time, the inside of the tube should reach an equilibrium temperature.

 

[dRic2]

Look at "figure" A and tell me If I understood the problem correctly.

Assuming that the temperature of the fluid inside the pipe is greater than the temperature of the surroundings, figure B should represent the temperature profile in the simpest case where outside the pipe there is only air with homogeneus properties. (The one I sketched is a radial temperature profile because cylindrical coordinates are easier to use in this case).
Case B is Simpler.

Figure C is What I think the Solution should look Like for your problem (Assuming the air is colder than the ground). I don't think there is a Simple Solution here...

Sent from my ATU-L21 using Physics Forums mobile app

 

[BvU]

That's one way of having half a tube in the ground. And here's me thinking it is standing straight up !

 

[dRic2]

I'm also guessing