Since the characteristic time of diffusion is $L^2/D$, where $L$ is the characteristic length and $D$ is the diffusivity (here, the thermal diffusivity), the characteristic speed is $D/L$, or $10^{-7}\,\mathrm{m/s}$ for a 1 km deep probe and very thermally conductive copper. We can pump liquid at speed v 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. Again, we find that when L is large (or even longer than millimeters), that 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.)

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.)

Since the characteristic time of diffusion is $L^2/D$, where $L$ is the characteristic length and $D$ is the diffusivity (here, the thermal diffusivity), the characteristic speed is $D/L$, or $10^{-7}\,\mathrm{m/s}$ 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. Again, we find that when L is large (or even longer than millimeters), that 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.)

Since the characteristic time of diffusion is $L^2/D$, where $L$ is the characteristic length and $D$ is the diffusivity (here, the thermal diffusivity), the characteristic speed is $D/L$, or $10^{-7}\,\mathrm{m/s}$ for a 1 km deep probe and very thermally conductive copper. We can pump liquid at speed v 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. Again, we find that when L is large (or even longer than millimeters), that 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.)