V-CPL-04 — Frank-Kamenetskii slab: subcritical steady state + criticality bracket

Tags: reaction_conduction, bifurcation, exothermic, coupled_jacobian

References:

• D.A. Frank-Kamenetskii, Diffusion and Heat Exchange in Chemical Kinetics
• Verification Suite Plan 2026-07-06, V-CPL-04

Problem statement

Zero-order exothermic Arrhenius source in a slab with Dirichlet walls (Frank-Kamenetskii). The only case where reaction heat release feeds back on the temperature field with a sharply known stability boundary δ_c = 0.878 (slab), δ = (E/RT₀²)·S₀L²/k, L = half-width, S₀ = |h|·A·e^(−E/RT₀).

Realization: the volumetric heat source is a SELF-REGENERATING reaction A → A (order n = 0, h < 0) — product index = reactant index nets ξ̇ ≡ 0 exactly (verified: drift 0.0) while depositing |h|·A·e^(−E/RT) per unit volume, i.e. a pure zero-order Arrhenius source with no reactant-depletion clock. The depletion limiter's tanh(ξ/1) factor is exactly 1.0 at ξ = 1000 kg/m³, and the default gates (Tmin = 0) are exactly open, so the kinetics are pure Arrhenius. A Tmax gate at T₀ + 100 K caps the SUPERCRITICAL branch so the runaway saturates instead of chasing e^θ to integrator death; the subcritical branch (rise ≤ 15 K) never feels it.

References (exact/frank_kamenetskii.jl, self-tested): the FK closed-form profile at E/RT₀ → ∞, AND a full-Arrhenius RK4-shooting steady solution at the case's finite E/RT₀ = 40 — the primary comparison, since the FK-limit truncation error (~2–5% at E/RT₀ = 40) is physics, not discretization.

Assert-mode protocol: main solve δ = 0.44 ≈ 0.50·δc → steady; center/interior T vs shooting bracket δ = 0.70 ≈ 0.80·δc → must stay bounded (near-critical, stable) δ = 1.32 ≈ 1.50·δc → must run away past the detection rise The δc bisection itself (turning-point hunt) is report-mode work.

Quantities of interest (n = 160)

Convergence

Observed order 2.474 (norm), expected 2.0.

Solver configuration