Underground Heat and Ventilation

What is modelled

Heat flows up through the crust at a surface rate q₀. Rock that contains radioactive elements produces extra heat, so the temperature rises with depth more slowly than a straight line. A tunnel cut in that rock is cooler than the rock around it, so heat flows from the rock into the tunnel air; machines add their own heat, and the ventilation carries it away. The air warms toward the rock temperature with a characteristic length that grows with airflow.

Equations

• Rock temperature T(z) = T0 + q0 z / k - A z² / (2 k) with uniform radiogenic heat production A, from d²T/dz² = -A / k.
• Heat flow from the rock to a tunnel of radius r, across the rock out to Ro = 2 √(α t) after t years: q' = 2 π k (Trock - Tair) / ln(Ro / r) per metre.
• Air heating m cp dT/dx = q', solved by T(x) = Trock - (Trock - Tstart) e-x / Lc with Lc = m cp ln(Ro / r) / (2 π k).
• Air mass rate m = ρ Q, machine heat P (1 - 0.8 coolant) added at the start.

Checks run in the page

• The second derivative of the rock temperature and the integral of the heat-flow gradient against the closed form.
• Runge-Kutta integration of the air heating against the exponential solution.
• Heat from the rock summed along the tunnel against the air enthalpy gain.
• Equal radial heat flow at three radii around the tunnel against the formula.

Honest simplifications

Uniform rock, uniform radiogenic heating, no groundwater, no moisture or latent heat in the air, no autocompression of the air with depth and a steady-state conduction approximation with a fixed outer radius instead of the full transient solution. The 3D lab's sliders are percentages without units; here they scale to at most 500 kW of machine heat and 150 m3/s of air, which are assumptions of this page.

• Choose Weak ventilation, hot machines: the air at the end climbs past 30 C.
• Step through the four stratum presets and compare the rock temperatures with depth.
• Raise the coolant slider and see the machine heat reaching the air fall.