Time dependent PDEs - mathematical modelling - diffusion equation

[Anabelle37]

Homework Statement

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 diused 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 aect the rate of diusion of
compound A, but the permeation process is no longer described by diusion
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 affects
this change in the production rate over time.

Homework Equations

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

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!

 

[mighty2000]

Hello there,

I can understand your confusion with the wording of this problem. However, I believe you are on the right track with your attempt at a solution.

To address the first question, we can use the diffusion equation to model the change in concentration of compound A over time. This will give us the production rate of compound A at any given time. However, as stated in the problem, the interaction between compound A and the membrane affects the production rate. So, we need to also consider the interaction term in our model.

To answer the second question, we need to determine how the interaction rate between compound A and the membrane affects the production rate over time. This can be done by analyzing the relationship between the interaction term and the production rate. Is the interaction rate directly proportional to the production rate, or is it inversely proportional? Does it have a linear or nonlinear relationship with the production rate?

Based on your knowledge of diffusion and reactions, you can make assumptions about the form of the interaction term and use it to solve the model. I would recommend looking into reaction-diffusion equations and how they are used to model similar processes.

I hope this helps! Let me know if you have any further questions or if you need additional clarification. Good luck with your model!