Effect of radius on transient heat xfer from pipe in semi-infinite soil

[Sandi]

TL;DR Summary

Hypothesis: heat loss increases with radius but at a less than 1:1 rate

1. My first hypothesis is that the thermal conditions of a ground source heat pump (GSHP) ground loop never reach steady state because the thermal mass of the surrounding soil is large enough that the time constant for temperature gradients moving away from the pipe before reaching boundary conditions is comparable to the time between the change of seasons

2. My second hypothesis is that a larger pipe diameter will transfer more heat to the surrounding soil, but that the increase will be less than 1:1, meaning that a doubling of the pipe diameter will not result in twice as much heat moving into the soil.

To simplify the calculation, I am assuming a single cylindrical ground-loop pipe rather than the typical pair of pipes joined at the end with a u-fitting.

Has anyone derived a formula showing the correlation between pipe radius and heat transfer per unit length assuming all other conditions remain fixed? If a mathematical correlation cannot be expressed, has anyone used thermal modeling software to derive an approximate expression for the correlation?

 

[Baluncore]

Welcome to PF.

TL;DR Summary: Hypothesis: heat loss increases with radius but at a less than 1:1 rate

... the time constant for temperature gradients moving away from the pipe before reaching boundary conditions is comparable to the time between the change of seasons

You assume a tube in a trench near the surface. Depending on the soil, seasonal changes are not noticed below a metre or two. With a vertical hole, seasons will be irrelevant.

TL;DR Summary: Hypothesis: heat loss increases with radius but at a less than 1:1 rate

2. My second hypothesis is that a larger pipe diameter will transfer more heat to the surrounding soil, but that the increase will be less than 1:1, meaning that a doubling of the pipe diameter will not result in twice as much heat moving into the soil.

It should double the heat flow, for the same temperature difference. Twice the diameter is twice the surface area. If the soil thermal conductivity is less than the thermal conductivity of the fluid in the pipe, then it will transfer more than twice. The heat will get a head start in the double radius pipe, as it moves away from the axis.

 

[Chestermiller]

See the book Heat Conduction by Max Jacob.

In your system, for constant heat flux, the heat transfer is always transient.

 

[Sandi]

It should double the heat flow, for the same temperature difference. Twice the diameter is twice the surface area. If the soil thermal conductivity is less than the thermal conductivity of the fluid in the pipe, then it will transfer more than twice. The heat will get a head start in the double radius pipe, as it moves away from the axis.

Thanks. I am assuming that the thermal conductivity of the pipe is higher than that of the soil so the limiting interface is the pipe to soil boundary. I agree that the emitting surface has doubled, but I am thinking that sets an upper bound on the increase in heat transfer. The reason I was intuitively thinking the actual increase would be somewhat less is because of the decreased edge effect as the radius increases. If the radius is very large, then the conduction of each unit of perimeter area begins to match that of a planar boundary whereas at a small radius, the same unit of perimeter area has much more curvature, making it easier for the pipe to push heat into the surrounding soil wedge resulting in a higher heat flux. Is there a flaw in my visualization?

 

[Baluncore]

Is there a flaw in my visualization?

I view it in polar coordinates.
All heat flow is radial, so I believe the curvature is not relevant to the analysis.

Take the analysis to the extreme where the pipe is a filament.
The problem with radial flow from a filament, is the high flux density close to the filament axis, with the higher temperature difference that causes. The temperature difference available can be better utilised by increasing the radius of the better conducting pipe and fluid flow.

 

[Baluncore]

See the book Heat Conduction by Max Jacob.

I can only find: "Heat Transfer", by Max Jakob. 1949. Two volumes.
It is very rare, and only second hand.

Is there a more available text on the subject ?

 

[Sandi]

I view it in polar coordinates.
All heat flow is radial, so I believe the curvature is not relevant to the analysis.

Take the analysis to the extreme where the pipe is a filament.
The problem with radial flow from a filament, is the high flux density close to the filament axis, with the higher temperature difference that causes. The temperature difference available can be better utilised by increasing the radius of the better conducting pipe and fluid flow.

Interesting. So my takeaway is you think that each doubling of the OD of the pipe could result in more than a doubling of total heat transfer. It will be interesting to see if anyone else has a different opinion. Thanks for your input!

 

[marcusl]

Carslaw and Jaeger, Conduction of Heat in Solids. Lots of inexpensive used copies. I have the 2nd edition.