Axial Dispersion as Elastic Diffusion: Micromixing in Tubular Reactors

The axial dispersion model of a tubular reactor is closed by Danckwerts’ boundary conditions, argued over as physics since 1953. Read probabilistically they are a boundary behaviour: the closed vessel is the exit law of a drifted Brownian particle reflected at the inlet and removed at the outlet at rate $\mathrm{Pe}$ per unit of its local time there, Feller’s elastic boundary. First-order conversion is then a Feynman–Kac expectation, Wehner–Wilhelm’s formula follows, and an absorbing outlet is another vessel, off by up to 0.64 in outlet concentration. For a second-order reaction, the dispersion model lies between Zwietering’s complete segregation and maximum mixedness for the same residence-time distribution. The three agree through $\mathrm{Da}^2$; at $\mathrm{Da}^3$ they differ in one integral, the mean square of the age averaged over nothing, over position or over life expectancy. Near plug flow the dispersion model’s position in the band tends to $\xi_\infty(\mathrm{Da}) = (1+\mathrm{Da})(\mathrm{Da} – \ln(1+\mathrm{Da}))/\mathrm{Da}^2$, from one half toward one (0.58 at $\mathrm{Pe} = 1000$, $\mathrm{Da} = 1$, against 0.61); at low conversion it rises from 0 near the stirred tank to 0.23 at $\mathrm{Pe} = 10$ and 0.46 at $\mathrm{Pe} = 300$. The method of lines, collocation, Laplace inversion and a Monte Carlo compute every number at least twice. Above $\mathrm{Pe} \approx 20$ the average of the two limits gives the dispersion model’s conversion to within 0.1 percentage points for $\mathrm{Da} \le 10$.