Time dependent PDEs - mathematical modelling - diffusion equation

  1. 1. The problem statement, all variables and given/known data

    Porous membranes are used to separate mixtures in industry, because smaller
    compounds permeate through them more easily than larger ones. KnoGas Pty
    Ltd are trialling an experimental separation process, using a membrane to sep-
    arate compounds A and B: compound A permeates the membrane, while com-
    pound B cannot. The ratio of A to B in the original mixture is held constant,
    and any compound A that has di used through the membrane is instantly re-
    moved for storage when it emerges at the other side.
    It is discovered, however, that compound A interacts with the membrane:
    during this interaction, some (but not necessarily all) of compound A turns
    into a useless by-product. This reaction doesn't a ect the rate of di usion of
    compound A, but the permeation process is no longer described by di usion
    alone. Importantly for KnoGas, the production rate of compound A { the rate
    at which it leaves the membrane { is changed as a result of this reaction.
    You have been asked by KnoGas to develop and solve a model for this
    process, in order to advise them on
    1. how the production rate of compound A changes with time; and
    2. how the interaction rate between compound A and the membrane aff ects
    this change in the production rate over time.

    2. Relevant equations

    Diffusion equation: dc/dt=D(d^2c/dx^2) these should be partial derivatives

    3. The attempt at a solution

    dA/dt = D(d^2A/dx^2) + R(A)

    R(A) is the interaction term.

    Don't know if i'm going about this the wrong way?
    kind of confused by the wording. Please help!
     
  2. jcsd
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