Could metal rods conduct geothermal energy to the surface? Currently geothermal heat pumps circulate a working fluid through a loop running in either a deep well or a long series of more shallow trenches.  Boring a well is expensive and digging trenches chews up a large chunk of land.
Could an array of metal rods (rebar connected to a sacrificial anode?) driven into the ground effectively conduct heat to a heat-exchanger at the surface? Would this cost less than the other methods?
 A: Since the characteristic time of diffusion is L²/D, where L is the characteristic length and D is the diffusivity (here, the thermal diffusivity), the characteristic speed is D/L, or 10-7 m/s for a 1 km deep probe and very thermally conductive copper. We can pump liquid far faster than this.
Looking at the conductive power delivery, we have kAΔT/L for the probe, where k is the thermal conductivity, A is the cross-sectional area and ΔT is the temperature difference. Compare this to ρCVΔT/t for advective transfer, where ρ is the material density, C is its specific heat capacity, V is its volume, and t is the turnaround time. Dividing by AΔT, we’re comparing the magnitudes of k/L and ρCv, where v is the liquid pumping speed. We find again that v need only exceed 10-7 m/s for water to best a strongly conductive rod.
The conclusion is that when L is large (or even longer than millimeters), pumping liquid is generally a far more effective way to transfer thermal energy than to rely on conduction.
(I’ll leave it to you to incorporate the relevant cost comparison, which takes this topic into engineering.)
