V-HC-05 — steady Kirchhoff profile with k(T) = k₀(1+βT)

Tags: heat_conduction, temperature_dependent_k, kirchhoff, steady

References:

• Kirchhoff transform (exact steady flux/profile), standard texts

Problem statement

Steady conduction across a slab with k(T) = k₀(1 + βT), fixed T at both faces. The Kirchhoff transform gives the exact flux and implicit profile:

q = (k₀/L)·[(Ttop − Tbot) + β(Ttop² − Tbot²)/2] z(T) = k₀[(T − Tbot) + β(T² − Tbot²)/2] / q (bottom cold face)

The implicit form is compared invert-free: evaluate z(Tnum[i]) and compare to the cell position. k varies 2.25× across 300–800 K with β = 0.01 (the linear form caps the ratio at Ttop/T_bot; 2.25× is char-vs-virgin scale).

What is verified: the T-dependent property path (LinearProperty) and the distance-weighted harmonic-mean face interpolation of cell-centered k(T) under strong nonlinearity. Note k(T) is evaluated at cell centers and harmonic-averaged to faces — NOT the integrated Kirchhoff face conductance — so unlike V-HC-04 this is a genuine discretization-error measurement, with a convergence check. Transient T-dependent {k,c} has no exact solution (that regime is B-02, code-to-code).

Quantities of interest (n = 320)

Convergence

Observed order 2 (L2), expected 2.0.

Solver configuration