Validation: Howell Catalog

RadiativeViewFactor.jl is validated against published configuration factors from J. R. Howell, A Catalog of Radiation Heat Transfer Configuration Factors, 3rd ed., https://www.thermalradiation.net, Section C (factors from finite areas to finite areas). The suite lives in benchmarks/howell/; this page summarises its recorded results. benchmarks/howell/RESULTS.md has the full discussion, including runtimes.

Summary

The suite covers 22 of the catalog's 162 Section C entries at 115 parameter points. Every point is run with deterministic quadrature and pair-area Monte Carlo on the same mesh; the 3D points are also run with ray-shooting Monte Carlo. Error is 100 × |F_computed − F_catalog| / F_catalog.

A dash means the kernel was not run: ray-shooting supports only 3D surface meshes, so the 2D cross-section cases (C-1 to C-4, C-8, C-63, C-68, C-69, C-72, C-73) are skipped for that kernel.

Method

Each case builds its geometry with Gmsh (second-order elements, OpenCASCADE kernel), loads it with load_mesh, computes the matrix with compute_view_factors, and compares one entry of the group-to-group matrix against the catalog value at the same parameters. Fifteen cases use a closed-form catalog equation (transcribed in analytic.jl); seven (C-8, C-10, C-33, C-34, C-72, C-73, C-137) use the catalog's published tables, extracted from the page HTML into tables.jl and checked against closure identities. Cases with infinitely long geometry are meshed as 2D cross-sections.

julia --project=benchmarks -e 'using Pkg; Pkg.develop(path="."); Pkg.instantiate()'
julia --project=benchmarks --threads=auto benchmarks/howell/run.jl            # quadrature + pair-area MC
julia --project=benchmarks --threads=auto benchmarks/howell/run_raytrace.jl   # ray-shooting MC

The figures below show each geometry at one representative parameter set; the tables list every parameter point run. Colours identify the surfaces named in each legend. Points with error of 1% or more are marked ⚠.

Cases

C-1: Infinite parallel plates, equal width

Shown with w = 1, h = 1 (cross-section).

Two infinitely long, directly opposed plates of width w separated by h. Reference: closed-form equation, H = h/w. Meshed as a 2D cross-section (per infinite depth).

C-2: Infinite parallel plates, unequal width

As C-1, but with plate widths b and c separated by a, centrelines on a common perpendicular. Reference: closed-form equation. Same geometry builder and figure as C-1.

C-3: Infinite perpendicular plates, common edge

Shown with equal widths and α = 90° (cross-section).

Two infinitely long plates of widths w (A1) and h (A2) sharing an edge at 90°; F is A1 → A2. Reference: closed-form equation. 2D cross-section.

C-4: Infinite equal plates, common edge, angle α

Two infinitely long plates of equal width sharing an edge at included angle α. Reference: closed-form equation, F = 1 − sin(α/2). Same builder and figure as C-3.

C-8: Infinite plane to two rows of tubes

Shown with R = 3 and 9 tubes per row; the benchmark mesh uses 31 tubes per row.

An infinite plane facing two staggered rows of unit-diameter tubes in an equilateral triangular array, R = p/d. F is from a one-pitch central segment of the plane to all tubes of the front row, and to all tubes of the second row. Reference: published table. Every tube is its own obstruction group (obstruction_groups never applies a group to a pair involving itself, so grouping a row together would stop its tubes shadowing one another).

C-10: Rectangle to semi-infinite rectangle at angle

Shown with a = b = 1, φ = 45°, and A2 truncated to width 3 for illustration; the benchmark mesh uses width 80.

A finite rectangle A1 sharing an edge of length b with a second rectangle A2 at included angle φ, A2 extending to infinity away from the edge; A = a/b. Reference: published table. A2 is a finite rectangle of width 80 with a graded mesh (fine near the shared edge). Run with use_duffy=true.

C-11: Identical parallel opposed rectangles

Shown with a = 2, b = 1, c = 1.

Two identical a × b rectangles in parallel planes separated by c. Reference: closed-form equation.

C-14: Perpendicular rectangles, common edge

Shown with l = w = h = 1.

Two rectangles of common length l sharing an edge at 90°: A1 is l × w, A2 is l × h; F is A1 → A2. Reference: closed-form equation. Run with use_duffy=true.

C-33: Hexagonal prism

Shown with l = h = 1.

Interior of a hexagonal prism with side l and height h, L = l/h. Six factors per point: wall to adjacent, next, and opposite wall, end to wall, wall to end, and end to end. Reference: published table. Run with reverse_normals=true, use_duffy=true, and radiating_groups set to the groups each factor reads.

C-34: Parallel regular polygons

Shown with n = 6, l = h = 1.

Two identical, aligned regular n-gons of side l in parallel planes separated by h, L = l/h, for n = 3, 4, 5, 6, 8. Reference: published table.

C-40: Coaxial parallel disks, equal radius

Shown with r1 = 0.5, r2 = 1, separation 1.

Two coaxial parallel disks of equal radius r separated by a. Reference: closed-form equation (C-41 with r1 = r2).

C-41: Coaxial parallel disks, unequal radius

Coaxial parallel disks of radii r1 and r2 separated by a; F is disk 1 → disk 2. Reference: closed-form equation. Same builder and figure as C-40.

C-63: Concentric infinite cylinders

Shown with r1 = 0.5, r2 = 1 (cross-section).

Infinitely long concentric cylinders; F is outer → inner, F = D1/D2. Reference: closed-form equation. 2D cross-section.

C-68: Infinite parallel cylinders, equal diameter

Shown with r1 = 0.5, r2 = 0.75, s = 1 (cross-section).

Two infinitely long parallel cylinders of radius r with surface-to-surface gap s (not axis-to-axis). Reference: closed-form equation. 2D cross-section, reverse_normals=true.

C-69: Infinite parallel cylinders, unequal radius

As C-68 with radii r1 and r2; s is again the surface-to-surface gap. F is cylinder 1 → 2. Reference: closed-form equation. Same builder and figure as C-68.

C-72: Cylinder in square array

Square array shown with P = 1.5 (cross-section).

A centre cylinder in an infinite square array of cylinders, P = p/d; F is from the centre cylinder to its nearest and diagonal neighbours. Reference: published table. Modelled as a 5 × 5 array, each cylinder its own obstruction group, with radiating_groups=[1,2,3].

C-73: Cylinder in triangular array

As C-72 for an equilateral triangular array; F is to the nearest and next neighbour. Reference: published table. Same builder as C-72 (the figure shows the square variant).

C-79: Cylinder base to inside lateral surface

Shown with r = 0.5, h = 1.

Base of a right circular cylinder of radius r and height h to its inside lateral surface. Reference: closed-form equation. Run with reverse_normals=true, use_duffy=true.

C-109: Cone interior to base

Shown with r = 0.5, h = 1.

Interior lateral surface of a right circular cone to its base, F = 1/√(1 + (h/r)²). Reference: closed-form equation. Run with reverse_normals=true, use_duffy=true. Must be meshed with quads (see Limitations found by the suite).

C-125: Sphere to coaxial disk

Shown with rs = 0.3, r = 1, a = 1.

Sphere of radius rs to a coaxial disk of radius r whose plane lies a from the sphere centre. Reference: closed-form equation (independent of rs).

C-135: Concentric spheres

Shown with r1 = 0.5, r2 = 1.

Concentric spheres; F is outer → inner, F = (r1/r2)². Reference: closed-form equation. Only the outer sphere's normals are reversed (reverse_group_normals), since the boolean cut already leaves the inner sphere outward-facing.

C-137: Two spheres of unequal radius

Shown with r1 = 0.4, r2 = 1, s = 0.6.

Two spheres of radii r1 and r2 with surface-to-surface gap s, R = r1/r2, S = s/r2; F is sphere 1 → 2. Reference: published table (3 significant figures).

Limitations found by the suite

• Shared-edge singularity. Surfaces that share an edge (C-3, C-4, C-10, C-14, C-33, C-79, C-109) are the least accurate. use_duffy=true is what makes them usable: it takes C-14 from 3.5% to 0.023% error. C-10 at A = 0.4 converges, but slowly, and the benchmark uses a moderate mesh.
• Duffy is quad-only. It silently skips triangles. C-109 meshed with quads gives 0.2% error; the same geometry meshed with triangles gives 3.6–9.2%. Mesh edge-sharing geometries with quads.
• Obstruction accuracy is set by facet count. Obstruction soups use only element corner nodes, so a curved blocker is an inscribed polygon. This is the source of C-8's second-row error (0.6–3.7%), which converges as tube facets are refined.
• Ray-shooting bias on coarse curved meshes. Its worst points are on spheres (C-125, C-135, C-137). The error does not shrink with n_rays but does shrink with mesh refinement (C-135: 2.8% → 1.0% → 0.4% as element size halves twice), so it is a faceting effect, not sampling noise.

Cases not included

C-35 and C-154 were attempted and excluded: their published figures are ambiguous about which surface is meant, and no reading tried reproduced the table (errors of 39–83%, and two readings gave a self view factor above 1). C-13c (no geometry definition on the catalog page) and C-101 (needs raster equations as well as a table) were set aside. Most of the remaining entries carry an equation that must be transcribed by hand and a bespoke geometry; C-156 to C-162 (people and animals) have no meaningful mesh.

Recorded run

Raw results, one row per point per kernel, are in benchmarks/howell/results.csv.