Channel side view — cake wedge & flowAxial profile: C(x) & J(x)
Feed particles (crossflow)
Cake layer (deposited foulant)
A crossflow nanofiltration membrane doesn't foul uniformly along its length. Feed entering at the inlet still carries its full load of rejected foulant, so cake deposits fastest right where the stream first meets the membrane; by the time that same stream reaches the outlet, upstream deposition has already stripped much of the foulant out of it, so less is left to deposit there. This simulator solves that axial mass balance directly as a continuum model: a bulk concentration profile C(x) that depletes along the channel length exactly by however much cake mass is being laid down at each position, coupled through the real resistance-in-series Darcy law to a position-dependent permeate flux J(x). Watch the cake resistance form a decaying wedge — high at the inlet, low at the outlet — and see how raising crossflow velocity flattens that wedge by lowering the capture probability everywhere at once, while a backwash pulse resets it, never quite back to a bare membrane.