Radiation View Factors

Radiation View Factors

View Factors and the Grey-Body Network

Thermal radiation leaves a surface in every direction, and only part of it lands on any one other surface. The view factor (configuration factor, shape factor, angle factor) F₁₂ is the fraction of the radiation leaving surface 1 that arrives at surface 2, for a surface that emits and reflects diffusely (Lambert's cosine law). It depends only on the geometry: F₁₂ = (1/A₁) ∫_A1∫_A2 cos θ₁ cos θ₂ / (π r²) dA₂ dA₁, where r joins the two elements of area and θ₁, θ₂ are the angles between r and the surface normals. Two identities follow at once and the tool prints both as checks: reciprocity, A₁F₁₂ = A₂F₂₁, and the summation rule, Σ_j F_ij = 1 over every surface an enclosure contains, surface i itself included when it is concave (F_ii > 0 for the inside of a cylinder or sphere, 0 for a plane or convex body).

The Hamilton–Morgan closed forms

Hamilton and Morgan (NACA TN 2836, 1952) integrated the defining equation for the configurations an engineer meets most often and tabulated the results; their letters (P for a point source dA₁, L for a line source, A for a finite area) are used on the form. The tool evaluates their Appendix C equations with the dimensionless parameters of the report:

• A-1, aligned parallel rectangles a × b at distance c, x = b/c, y = a/c: F₁₂ = 2/(πxy){ln[(1 + x²)(1 + y²)/(1 + x² + y²)]^½ + y√(1 + x²) tan⁻¹[y/√(1 + x²)] + x√(1 + y²) tan⁻¹[x/√(1 + y²)] − y tan⁻¹y − x tan⁻¹x}. Table 12 of the report.
• A-2, perpendicular rectangles with a common edge b; surface 1 is b × c, surface 2 is b × a, N = a/b, L = c/b: F₁₂ = (1/πL){L tan⁻¹(1/L) + N tan⁻¹(1/N) − √(N² + L²) tan⁻¹[1/√(N² + L²)] + ¼ ln[…]} with the logarithm's three factors raised to the powers 1, L² and N² as printed on the results page. Table 13 (φ = 90° block); the report notes those values may be in error by 0.0005 because they came from a numerical integration of the line-source case L-2.
• A-5, coaxial parallel discs, source radius r₁, receiver radius r₂, spacing c, E = r₂/c, D = c/r₁: F₁₂ = ½[X − √(X² − 4E²D²)], X = 1 + (1 + E²)D². Table 14.
• A-7 and A-8, concentric cylinders of finite length l with open ends. The report tabulated these from a numerical integration (Tables 10 and 11, three decimals); the tool uses the closed forms of Leuenberger and Person (1956) for the outer cylinder's view of the inner one, F₂₁, and of itself, F₂₂, with R = r₂/r₁ and L = l/r₁, and reciprocity for F₁₂ = R F₂₁. Whatever is left, 1 − F₂₁ − F₂₂, leaves through the two annular ends, which the tool treats as the surroundings.
• P-1, element to a parallel rectangle a × b whose corner lies on the element's normal at distance c, x = a/c, y = b/c: F = (1/2π){x/√(1 + x²) tan⁻¹[y/√(1 + x²)] + y/√(1 + y²) tan⁻¹[x/√(1 + y²)]}. Table 1. A rectangle not cornered on the normal is handled by adding and subtracting cornered rectangles (view-factor algebra).
• P-2, element to a rectangle at an angle. The rectangle's edge b lies on the line where its plane meets the element's plane at the included angle φ; the element sits at distance c from that line, level with one end of the edge, N = a/b, L = c/b: F = (1/2π){tan⁻¹(1/L) + V(N cos φ − L) tan⁻¹V + (cos φ/W)[tan⁻¹((N − L cos φ)/W) + tan⁻¹(L cos φ/W)]}, V = 1/√(N² + L² − 2NL cos φ), W = √(1 + L² sin²φ). Table 2 (φ = 30°, 60°, 90°, 120°, 150°). As c → 0 the factor tends to ¼(1 + cos φ), the element's view of a quarter-infinite plane.
• P-6, element parallel to a disc of radius r₂ at distance c, offset s from the axis, E = r₂/c, D = c/s: F = ½[1 − (X − 2E²D²)/√(X² − 4E²D²)] with the same X. On the axis the factor is r₂²/(c² + r₂²). Table 3.
• Limiting cases: two infinite parallel plates, F₁₂ = F₂₁ = 1; long concentric cylinders (A-9), F₁₂ = 1, F₂₁ = r₁/r₂; concentric spheres (A-10), F₁₂ = 1, F₂₁ = (r₁/r₂)²; a sphere over an infinite plane, F₁₂ = ½ exactly, because the plane fills half of every point's hemisphere once the whole sphere is averaged.

Every form was checked against the report's tables and against direct quadrature of the defining integral (scipy, agreement to 10⁻¹⁰); the few table cells that disagree with the closed forms are misprints of the 1952 printing and are listed in the project's source notes. The chart on the results page sweeps one geometric ratio with the others fixed and marks the case, so the sensitivity of F to the spacing or the aspect ratio can be read at a glance.

Grey-body exchange: the radiosity network

For opaque, grey, diffuse surfaces at uniform temperature the radiation leaving a surface per unit area, its radiosity J, is the emitted part εE_b plus the reflected part (1 − ε)G, with E_b = σT⁴ the blackbody emissive power (σ = 5.670 × 10⁻⁸ W/(m²·K⁴)) and G the irradiation. Eliminating G gives the surface resistance form q_i = (E_bi − J_i) / [(1 − ε_i)/(ε_iA_i)] for the net heat leaving surface i, and the exchange between two surfaces is q_ij = (J_i − J_j) / [1/(A_iF_ij)], with the space resistance 1/(A_iF_ij) = 1/(A_jF_ji). Oppenheim's network (1956) draws these as an electrical circuit; Hottel's treatment in McAdams is the same algebra. A node balance at each surface gives one linear equation per unknown radiosity:

• surface with known temperature: ε_iA_i/(1 − ε_i) (E_bi − J_i) = Σ_j≠i A_iF_ij (J_i − J_j);
• black surface (ε = 1) or very large surroundings: J_i = E_bi (zero surface resistance);
• re-radiating (adiabatic, insulated) surface: Σ_j≠i A_iF_ij (J_i − J_j) = 0, so its net heat is zero and its radiosity, hence its temperature T = (J/σ)^¼, is a result.

A surface's view of itself drops out of its own balance (the F_ii term multiplies J_i − J_i), which is why the inside of a cylinder or sphere needs no special treatment. The tool prints the coefficient matrix and right-hand side it solves, the radiosities, and the net rate of each surface, positive when the surface loses heat; the rates sum to zero. In USCS mode the coefficients are in ft², the rates in Btu/h and the radiosities in Btu/(h·ft²).

The surroundings

Only the infinite plates, the long cylinders and the concentric spheres enclose each other; for every finite configuration part of each surface's view, F₁₃ = 1 − F₁₁ − F₁₂ and F₂₃ = 1 − F₂₂ − F₂₁, misses the other surface and reaches whatever lies beyond. The form offers four treatments of that third surface. Re-radiating closes the enclosure with an insulated surface, the usual idealisation of refractory furnace walls or of a well-insulated housing; the heat the hot surface loses then all reaches the cold one, by direct exchange and by the detour over surface 3. Large black surroundings at T₃ fix J₃ = σT₃⁴: a room, the sky, a large vacuum chamber. A grey third surface with its own emissivity and area has a surface resistance of its own; its view factors follow from reciprocity, which is why its area cannot be smaller than A₁F₁₃ + A₂F₂₃. None applies the textbook two-surface formula q₁₂ = (E_b1 − E_b2) / [(1 − ε₁)/(ε₁A₁) + 1/(A₁F₁₂) + (1 − ε₂)/(ε₂A₂)]; it is exact for the enclosing pairs and is always printed for comparison, but for finite surfaces it silently assumes that nothing leaves to, or arrives from, the rest of the world. For the default case (1 × 2 m plates 1 m apart, 500 and 300 K, ε 0.8 and 0.5) the two-surface formula gives 1,299 W while re-radiating surroundings carry 2,199 W between the plates and large black surroundings at 300 K draw 4,773 W from the hot plate.

A differential element

The three P configurations give the factor from an element dA₁, which has no area to put in a network. The results page reports the factor, the irradiation the element receives from a black surface 2, G = F σT₂⁴, and the element's net flux ε₁F σ(T₁⁴ − T₂⁴) with surface 2 taken black and the element grey (absorptivity equal to emissivity). Element factors are the building blocks for a surface whose temperature varies: the factor from a finite surface is the area average of the element factor over it, which is how the coaxial-disc formula was checked.

Radiation shields

A thin shield of low emissivity placed between two surfaces adds two surface resistances and one space resistance to the chain. For infinite parallel plates with N shields of emissivity ε_s on both faces the exchange falls to q_N/q₀ = (1/ε₁ + 1/ε₂ − 1) / [1/ε₁ + 1/ε₂ − 1 + N(2/ε_s − 1)]; with every emissivity equal this is 1/(N + 1). For long cylinders and spheres the tool places the shields at equal radial spacing between r₁ and r₂ and sums the chain 1/(ε₁r₁ⁿ) + Σ_k(2/ε_s − 1)/r_kⁿ + (1 − ε₂)/(ε₂r₂ⁿ) with n = 1 for cylinders and 2 for spheres. Shields between finite surfaces change the view factors themselves and are not modelled.

Conventions and limits

Temperatures may be typed as absolute (K or °R, the natural radiation scale and the one handbook examples use) or as °C / °F; the tool converts to kelvin before raising to the fourth power. Lengths go through the unit dictionary; the view factor is dimensionless and independent of the unit. Surface 1 is the source of the report's tabulated factor: the lower plate, the source disc, the inner cylinder or sphere, or the element. The analysis assumes grey, diffuse, opaque surfaces of uniform temperature and radiosity, a non-participating medium between them, and the configuration's idealised geometry (no edge effects, exact alignment); specular reflection, spectral properties, gases that absorb, and non-uniform surface temperature need the methods of Howell, Mengüç and Siegel.