The Windrow Oxygen Bound: How Deep Air Reaches, Why No Turning Interval Keeps a Tall Pile Aerobic, and Why Passive Chimneys Fail the Same Bound

A static compost windrow breathes only through its skin. This paper derives, from diffusion against microbial oxygen consumption, how deep aerobic conditions penetrate a pile, what fraction of a windrow is therefore anaerobic, what turning can and cannot do about it, and whether passive chimney aeration can meet the same demand. SIMULATED, NOT MEASURED. At peak respiration the penetration depth is 0.1949 m; at mean respiration it is 0.3081 m. For a triangular windrow 2.0 m high, the anaerobic core is 0.8146 of the volume at peak respiration and 0.7156 at mean respiration. The oxygen stored in the pore space of the core is consumed in 0.34 hours at peak respiration, so the turning interval that would keep the core continuously aerobic is under an hour — no practical turning interval exists, and turning must be understood as periodic exposure by mixing, requiring about 15 turns to expose 95 per cent of a 2.0 m pile's material to the aerobic shell. A passive chimney 2.5 m tall over a 30 K temperature difference develops only 2.703 Pa of draft and, even through coarse material, supplies 0.1005 of peak oxygen demand and 0.2513 of mean demand: the passive design fails the bound and is printed here as a failure. Under stated central assumptions, the anaerobic core of a 100 m³, 2.0 m pile corresponds to about 733.1 kg of methane over the active phase, or 20,527.9 kg CO₂-equivalent at a 100-year factor of 28. The designs that meet the bound are forced aeration, or piles no taller than about twice the penetration depth.