Tritium permeation

Solve a 1D permeation transient and read the breakthrough curve and time lag.

A 1D slab $0 \le x \le L$ holds mobile tritium $c_m(x, t)$ with Fickian diffusion at a caller-supplied diffusivity $D$ (T1, trap-free here):

\frac{\partial c_m}{\partial t} = D\,\frac{\partial^2 c_m}{\partial x^2}. \tag{T1}

The demo below solves the permeation setup $c(0, t) = c_0$ , $c(L, t) = 0$ , $c(x, 0) = 0$ with the crate’s implicit finite-volume stepper and plots the normalized outlet flux $J(L, t)/J_{ss}$ (G2):

\frac{J(L,t)}{J_{ss}} = 1 + 2\sum_{n=1}^{\infty}(-1)^n e^{-D n^2 \pi^2 t / L^2}. \tag{G2}

The dashed marker sits at the time lag $t_{\mathrm{lag}} = L^2/6D$ (G2-lag), the intercept of the cumulative-permeation asymptote. The default inputs replay fixtures/tritium/g2_timelag.json ( $D = 10^{-9}\ \mathrm{m}^2/\mathrm{s}$ , $L = 10^{-3}\ \mathrm{m}$ , $c_0 = 1.0\ \mathrm{mol}/\mathrm{m}^3$ , $t_{\mathrm{lag}} \approx 166.67\ \mathrm{s}$ ). All inputs are validated inline (finite, with $L > 0$ , $D > 0$ , $c_0 \ge 0$ , $t_{\mathrm{end}} > 0$ ) before any solve runs.

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