Concepts

Unsteady Transport Across a Membrane

The concentration in the membrane satisfies the equation

with initial conditions

C(x,0) = F(x)

and boundary conditions

C(0,t) = C0
C(L,t) = C1

The solution can be written as its Steady-State, Css(x), plus a Transient C'(x,t) where the steady-state is

Css(x) = C0 + (C1 - C0)x / L

and the transient satisfies

with initial condition

(NOTE: the prime does NOT denote derivative, instead, it denotes the perturbation from the steady state) and boundary conditions

Solution: The solution is given by

where

Approach to Steady State: As t increases, each mode decays proportional to . For larger n, the decay is faster. Thus, the n = 1 mode decays the slowest. Even so, afer t = L2 / 2D, this mode will be 1/e of its original value. (Jon, can you supply values (L,D) here for an example?)


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