Why the air moves: the stack effect
Warm air is lighter than cold air. The pressure difference between a column of outside air and the same height of cave air is Δp = (ρout − ρin) g H, with ρ = p / (R T). Here H is the height between the low entrance at the foot of the slope and the chimney tops on the plateau (44 m plus any stack). When the outside is colder than the cave the draught rises through the shafts; on a hot afternoon it reverses, and cool cave air drains out of the low entrance. Around the crossover the draught is almost zero, which is when CO₂ builds up. Wind adds ½ ρ v² ΔCp with ΔCp = 0.5 (up-slope wind helps, the opposite wind opposes).
How much air flows
The network is in series: entrance passage, then at each level a shaft in parallel with the stairway, then the chimney. Each element has a loss Δp = ½ ρ K Q² / A² where K adds entry/exit losses and friction f L / D (f = 0.05 for rough rock-cut walls). Parallel paths add their conductances C = A/√K, series paths add 1/C², so Q = √(Δp / R) with R = ½ρ Σ 1/C². For a single opening this reduces to Q = Cd A √(2Δp/ρ). A rolled-shut stone leaves a 1 cm gap (0.015 m²).
The air budget of one person and one lamp
Oxygen use is V̇O₂ = 3.5 mL/kg/min × MET × 70 kg: 0.25 L/min at rest (1 MET), 0.32 L/min at 1.3 MET. CO₂ output is the respiratory quotient 0.85 times that (0.27 L/min), and the body releases 20.1 kJ per litre of O₂, about 107 W at 1.3 MET. An olive-oil lamp (C₅₇H₁₀₄O₆ + 80 O₂ → 57 CO₂ + 52 H₂O) burning 8 g/h uses 0.27 L/min of O₂, as much as a resting person, makes 0.19 L/min of CO₂ and 82 W of heat; its flame fades as O₂ falls below about 19% and goes out near 15.5%.
Gas balance
Each level is one well-mixed volume: V dC/dt = Q (Cup − C) + exchange + G. In steady state the outlet is Cout = C0 + ΣG / Q (C0 = 420 ppm), which gives the largest population N = Q (Clim − C0) / g, where g is the CO₂ per person plus lamps. Buoyant overturning between levels (stronger when a lower level is warmer than the one above) and a weak background mixing are included; stable layers mix slowly.
Why the rock stays at 13 °C
Deep rock holds the annual mean temperature. The walls are modelled by transient radial heat conduction into the tuff, ρc ∂T/∂t = (1/r²) ∂(r² k ∂T/∂r)/∂r, from a cavity whose area equals each level's wall area. At first the rock absorbs heat quickly (flux ∝ e ΔT/√(π t), thermal effusivity e = √(kρc)), then more slowly as the wall warms. Soft tuff is a poor conductor (k ≈ 0.8 W/m/K), so a crowded, lamp-lit city really does warm up.
What is simplified
Levels are well-mixed boxes with representative volumes and wall areas, not a 3-D flow solution; humidity, latent heat and condensation are ignored (all body heat is treated as sensible); the draught is quasi-steady; the real Derinkuyu has some 50 shafts, a nearly flat site and many more passages, so numbers here show orders of magnitude and trends, not a survey. The city is drawn as a hillside cutaway so that the low entrance and the high chimneys give the stack effect a height. Educational model.