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The unknown fraction

A grain bin is the easiest volume in agriculture to instrument: rigid, bounded, and already monitored. Give it the industry-standard sensors and ask a question the industry does not report — what fraction of the grain can they actually see?

The bin, and what its sensors reach

Heat moves through grain by diffusion, so a temperature probe senses a sphere whose radius grows as √(α·t). That single fact sets everything below.

reached by a sensor unknown a spoilage hot spot
99.7% of the grain is unknown — outside the reach of every sensor in the bin
0.26 m diffusion length √(α·t) — how far a probe sees in the time you allowed
10,900 temperature sensors it would take to reach every cubic metre

Enough for what?

There is a good objection to everything above: sensor-placement studies find that five well-sited soil probes get estimation error near 2%, and ten near 1%, with little gained after that. Both things are true, because they answer different questions. Interpolation covers the gaps between sensors when the quantity varies smoothly. A spoilage pocket, a disease focus and an insect colony are not smooth. They are localized events, and there is nothing to interpolate them from.

If the quantity varies smoothly ±1.8% estimation error for a field-wide average, falling as 1/√N. The sensors you have are close to sufficient, and more buy little.
If it is a localized event 99.7% chance it starts where nothing can reach it. Interpolation cannot help, because the event is not a sample of a field — it is its own thing, somewhere. The unknown fraction is the probability you miss it.

What a longer reach buys

Sensor count to fill a volume scales as 1/r³, so reach is worth far more than count. Heat and CO₂ both move through grain by diffusion, so both reaches follow √(kt) and differ only by their transport coefficients: grain's thermal diffusivity is near 1.1×10⁻⁷ m²/s, while measured effective CO₂ diffusivity through bulk corn is 3.10–3.93×10⁻⁶ m²/s.

Every reach in this table is computed from a measured transport coefficient, not assumed. Acoustic carries no radius because this review did not locate a published detection distance in metres for grain.
ModalityReachNodes to resolve this binLocates?Where the reach comes from

The energy the answer costs

Each wireless node spends roughly 50 mJ per measurement cycle (LoRa uplink at +20 dBm plus MCU wake). Sensing density is capped by joules long before it is capped by sensor price.

—joules for the season, as configured above
—joules if you tried to resolve the bin with temperature alone
—joules to resolve it with CO\u2082 instead, at its own diffusion reach

What this is. A first-order model, not a bin simulator. Reach is taken as the diffusion length L=√(α·t) with grain thermal diffusivity α near 1×10⁻⁷ m²/s, and reached volume as the union of spheres of radius L, assumed non-overlapping at these spacings. Real bins have convection currents, non-uniform moisture and wall effects, all of which move the number — none of which make it large. What it is for. The unknown fraction is a quantity every sensing system could report and, in the record this review searched, none does. Part of the Glass Lab × Charlot Lab track on Physical AI and integrated sensors for agriculture surveillance.