Transient Heat transfer from water through pipe to ground

[AtypicalEngineer]

Hi I am trying to do a transient heat transfer calculation.

The water flowing through the pipe is laminar (1 m/s, id = 0.0127 m, Re_d = 0.01) transferring heat to the pipe via convection, then the pipe (od = 0.015875 m, l = 144 m) transfers heat to the ground again via conduction.

I found this thread here, that details some similar concepts, but just stops short of what I need.
https://www.physicsforums.com/threads/heat-transfer-from-pipe.395855/

I am wondering if anyone can help with this problem. I am not quite sure how to tackle this one.
I guess I am looking for equations for convection through the pipe for laminar flow, conducting in the pipe to the ground as an infinite medium. I would be able to manipulate it to suit, but I guess I need some starting points.

Can anyone help?
Thanks for anything that anyone can provide.

 

[Vincenzo]

Yes, look for the following:

For the convection inside the pipe: Internal Forced Convection with Laminal Flow.
For the conduction through the solid walls look for Conduction for cylindrical bodies
And in general, resistance analogies for heat transfer.

Also if you wan to find the heat flow from the outer surface of the pipe to the surronding you need to have the thermal conductivity.

If you can find a copy of Heat Transfer from Yunes Cengel in a Library you should check it out, you will find all what you need there.

 

[Chestermiller]

Hi I am trying to do a transient heat transfer calculation.

The water flowing through the pipe is laminar (1 m/s, id = 0.0127 m, Re_d = 0.01) transferring heat to the pipe via convection, then the pipe (od = 0.015875 m, l = 144 m) transfers heat to the ground again via conduction.

I found this thread here, that details some similar concepts, but just stops short of what I need.
https://www.physicsforums.com/threads/heat-transfer-from-pipe.395855/

I am wondering if anyone can help with this problem. I am not quite sure how to tackle this one.
I guess I am looking for equations for convection through the pipe for laminar flow, conducting in the pipe to the ground as an infinite medium. I would be able to manipulate it to suit, but I guess I need some starting points.

Can anyone help?
Thanks for anything that anyone can provide.

Can you give more details such as the length of the pipe and the temperatures involved. The hard part to model is going to be the conductive heat transfer from the pipe to the ground. Is the ground considered an infinite medium surrounding the pipe, or is the pipe close to some other boundary? Is it really transient heat transfer or is it steady velocity and temperature profiles?

 

[AtypicalEngineer]

Thanks Vincenzo & Chestermiller.

I don't have Yunes Cengel, but I am using Incropera. I am looking these concepts up, but I guess I am struggling with the questions Chester has for me.
I guess I should have described the system more completely. The ultimate goal is to find the steady state temperature of the water in the tank.

It is a closed loop.
a) 1.5 m3 water tank at 50C (guess), with 15 kW power source
b) water is piped to a ground loop (144 m long) that goes into the ground (~ 2 m depth), where heat is dumped (plus more, but I haven't calculated this yet)
c) then the water loops back to the tank to start over again

Loop = 144 m
pipe id = 0.0127 m
pipe od = 0.01587 m
mass flow = 0.1267 kg/s
Re_d = 23200 (turbulent)
The ground is infinite medium. I guess east coast is the closest body of water (assume New Jersey)

My methodology is to solve for the overall heat flux then use the general convection equation to solve for the final temperature at the end of the loop (going back into the tank), then use this as the starting point to repeat the calculation until the error is small.

The problem is that I cannot complete the heat flux calculation.

Heres what I have found. Please tell me where I am wrong.
Convection through pipe (turbulent, not laminar like I said)
Resistance is 1 / (2*pi*r1*L*h_w)

Conduction through pipe
Resistance is ln(r2/r1) / (2*pi*L*k_p)

Conduction through ground
Resistance is where I am not sure what to do next.

I would then add these resistances and solve for the total resistance, and subsequently the overall heat flux.
From there, I would calculate the outlet temperature (T2) of the pipe using
q_conv = m_dot*C_p*(T2-T1)

If anyone can comment if I am doing things correctly and help me with the heat transfer into the ground that would be great!

 

[Chestermiller]

This doesn't seem like too difficult a problem. If you neglect the convective heat transfer resistance at the ground surface, then the surface of the ground can be taken as a constant temperature surface. The trick is to determine the heat flux from the pipe as a function of the temperature difference between the outer surface of the pipe and the ground surface (2 meters above). Assume for the moment that the temperature of the outer surface of the pipe is constant along its entire length at $T_p$ while the ground surface is at temperature $T_g$. If we could solve the steady state heat conduction in the ground with these boundary conditions, then we could determine the heat flux.

There is a simple trick to solving this. We extend the ground upward to infinity, and we put a heat sink pipe 2 meters above the ground, immediately above our heat source pipe at z = -2 meters. So we have a linear heat source at z = - 2, and a heat sink at z = +2. This will result in a uniform temperature at z = 0, which is our required boundary condition. The method I'm describing is called the "method of images."

Since the heat conduction equation is linear, the temperature profile in the ground will be the linear superposition of the temperature profile from the heat source plus the temperature profile from the heat sink.

The solution to this problem is given by: $$q=\frac{k}{r_0\ln{\frac{2H}{r_0}}}(T_p-T_g)$$where q is the heat flux at the outer surface of the pipe, k is the thermal conductivity of the ground, H is the depth of the pipe below the ground, and $r_0$ is the outer radius of the pipe.