The science
In constant-pressure cake filtration the filtrate flux is Darcy's law across the growing cake plus the septum:
dV/dt = A dP / ( mu ( alpha c V / A + R_m ) ), t/V = a V + b, a = mu alpha c / (2 A^2 dP), b = mu R_m / (A dP)
The specific cake resistance comes from Kozeny-Carman: alpha = 180 (1 - eps) / (rho_s d^2 eps^3). V grows as sqrt(t) once the cake dominates, and the cake thickness is c V / (A rho_s (1 - eps)).
Honest simplifications: the cake is incompressible with uniform porosity (real diatomite cakes compress slightly), the particle size is a single mean diameter, particle bridging and pore blocking in the septum are ignored, and the cake thickness is capped at 30 mm to define one time-lapse cycle of 15 s. Checks: RK4 versus the closed-form root, the least-squares slope of t/V versus V, and Kozeny-Carman through permeability.