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.