Borehole Ground Loop: Heat Exchanger Resistance

Operating Mode

Fluid enters colder than the ground and warms up on its way through the loop — heat flows from the ground into the fluid.

Ground Loop

Flow rate 18 L/min

Undisturbed ground temp 11 °C

Grout thermal conductivity 1.4 W/m·K

Borehole depth fixed at 100 m, single U-tube (32 mm HDPE), 150 mm borehole. Real ground-loop designers size exactly these three levers.

How it works

Heat travels from the ground to the loop fluid (or back) through three resistances in series, per metre of pipe:

R' = R_conv + R_pipe + R_grout   [K·m/W]

R_conv  = 1 / (h · π · d_in)         (fluid → pipe wall)
R_pipe  = ln(d_out/d_in) / (2π k_pipe)
R_grout = ln(r_b/r_pipe) / (2π k_grout)

h from Dittus–Boelter: Nu = 0.023 Re^0.8 Pr^0.4

Along the pipe the fluid temperature relaxes toward the ground temperature exponentially — the same "fin equation" solution used for a pipe losing heat to its surroundings:

T(z) = T_ground + (T_in − T_ground)·exp(−z / L_c)
L_c = ṁ·c_p·R'   (characteristic decay length)

A slower flow (small ṁ) or a poor grout (small k_grout, big R_grout) shrinks L_c, so the fluid equilibrates faster — it leaves closer to the ground temperature, but moves less total heat because ṁ is small. A faster flow moves more heat per trip but each parcel spends less time in contact, so it approaches the ground temperature less closely. This trade-off is exactly why ground-loop design optimizes flow rate rather than just maximizing it.

Mode — heating pulls heat out of the ground (fluid enters cold); cooling rejects heat into the ground (fluid enters warm), as in a real reversible heat pump.
Flow rate — sets both the convective resistance R_conv and the residence time, so it moves the exponential decay length L_c.
Grout conductivity — a poured grout with more solids (e.g. thermally-enhanced grout) lowers R_grout and raises heat transfer for the same pipe.

Simplification: this model treats each U-tube leg as a single pipe centered in the borehole and ignores "thermal short-circuiting" (heat leaking directly between the down-leg and up-leg through the grout) — real design software adds a correction for that, but the resistance stack and exponential approach shown here are the same physics used in first-pass borehole sizing (Kavanaugh & Rafferty method).

Cold fluid Warm fluid

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