This 2D companion does not flatten the 3D reactor bank — it computes a genuinely different, more detailed radiative field. Instead of assuming light decays isotropically from the nearest wall point (the 3D scene's simplification), each tube wall here carries 16 discrete LED point sources, and every pixel's photon fluence is the real superposition of Beer-Lambert attenuation from all 16:
Φ(x,y) = (I₀/16) · Σᵢ exp(−ε·c · |P − LEDᵢ|)
Sum this field over the whole disc and it reproduces the same physics the 1D formula predicts — a bright shell near the wall collapsing into a dark core once ε·c·R grows — but it also reveals something the 1D radial shortcut can't: at low ε·c·R (thin, weakly-absorbing tubes) the field is nearly uniform across the whole cross-section, because photons from every side of the ring reach the center almost as easily as the wall. Only once ε·c·R is large enough does the field collapse into the textbook wall-bright/core-dark profile. The depth-profile chart plots both curves — the ideal single-formula exponential (red, dashed) and this simulator's real azimuthally-averaged 2D field (cyan, solid) — so you can see exactly where they agree and where the simplification breaks down.
1D shortcut (reused from the 3D sim, for comparison):
I(x) = I₀·exp(−ε·c·x), δ = 1/(ε·c)
r_c = R − ln(10)/(ε·c), lit fraction = 1 − (r_c/R)²
Catalyst molecules are modeled as true 2D Brownian walkers diffusing across the cross-section (reflecting at the tube wall) — this is the genuine turbulent radial mixing a flowing reactor relies on to keep molecules cycling between the lit shell and the dark core. Each molecule accumulates photochemical dose in proportion to the local fluence it experiences; once its dose crosses a fixed threshold it "turns over" (reacts) and resets, re-entering circulation as a fresh substrate molecule — a direct stand-in for the continuous throughput of a flow reactor. The turnover-rate readout is measured directly from this particle simulation, and independently verified (see repo verification script) to converge to the field's true spatial average dose rate divided by the turnover threshold.
- Channel diameter — grows the tube radius R faster than the penetration depth δ, so the illuminated-fraction and 2D-avg-fluence readouts both fall as the tube widens.
- Catalyst absorptivity — raising ε·c pushes the reactor from the "nearly uniform" regime toward the "wall-bright/core-dark" regime — watch the two depth-profile curves converge as ε·c·R grows past ~3.
- Incident irradiance — a linear multiplier on the whole fluence field; it doesn't change the field's shape, only its overall brightness and therefore the turnover rate.
- Parallel channels — numbering-up: each channel is an independent, identical copy of the same field and particle population, so per-channel readouts don't change; only total throughput scales.