View Factor Reciprocity

A view factor F₁₂ is the fraction of the radiation leaving surface 1 that lands on surface 2 — pure geometry, no temperatures or materials involved. Reciprocity says that the product of area and view factor is the same in both directions: A₁F₁₂ = A₂F₂₁. It follows directly from the double integral that defines the factor, which is symmetric in the two surfaces, and it is the single most useful labour-saving rule in radiation analysis because view factor integrals are miserable and this one turns a known result into a second free result.

Worked example: a 2 m² heater panel radiating with F₁₂ = 0.3 onto a 1.5 m² absorber. Reciprocity gives F₂₁ = 2 × 0.3/1.5 = 0.4 — the smaller surface sees more of the larger one than the other way round, which is always the direction the arithmetic runs. Pair it with the summation rule, that all factors from any surface in a closed enclosure add to 1, and most textbook enclosures fall out with no integration at all. A small object entirely surrounded by a big one has F₁₂ = 1 by inspection, so F₂₁ = A₁/A₂ immediately. The trap: view factors are geometry only. Two surfaces can have F₁₂ = 1 and exchange almost nothing if one of them is polished, because emissivity enters the heat rate, not the factor.