Mass Transfer Value

Question

I'm trying to model the following boundary condition using MassTransferValue.

$$ D_A \frac{\partial[A]} {\partial x} + D_B \frac{\partial[B]} {\partial x}+ D_C \frac{\partial[C]} {\partial x}=0 $$

The help file does have an example where it handles two species and I can add a third one.

MassTransferValue[
 x >= 0, {{Subscript[c, 1][x], Subscript[c, 2][x]}, {x}}, <|
  "AmbientConcentration" -> {Subscript[c, ext1][t, x, y], 
    Subscript[c, ext2][t, x, y]}, 
  "MassTransferCoefficient" -> {Subscript[k, 1], Subscript[k, 2]}|>]

However the output to this consists of two separate Neumann conditions (rather than one). Maybe I can link the two together using the MassTransferCoefficient but is there a cleaner way to do it?

Answer 1

What you can do is:

vars = {{Subscript[c, 1][x], Subscript[c, 2][x]}, {x}};
pars = <|"DiffusionCoefficient" -> {{1, 0}, {0, 1}}, 
   "AmbientConcentration" -> {Subscript[c, ext1][t, x, y], 
     Subscript[c, ext2][t, x, y]}, 
   "MassTransferCoefficient" -> {Subscript[k, 1], Subscript[k, 2]}|>;

Total[MassTransferValue[x >= 0, vars, pars]]

However, I am not quite sure why you would want to do that. The corresponding PDE is a system of 2 PDEs:

MassTransportPDEComponent[vars, pars]
{Inactive[Div][{{-1}} . Inactive[Grad][Subscript[c, 1][x], {x}], {x}], 
 Inactive[Div][{{-1}} . Inactive[Grad][Subscript[c, 2][x], {x}], {x}]}

And this fits nicely with the boundary condition that also returns two equations:

MassTransportPDEComponent[vars, pars] == 
 MassTransferValue[x >= 0, vars, pars]

{Inactive[Div][{{-1}} . Inactive[Grad][Subscript[c, 1][x], {x}], {x}], 
  Inactive[Div][{{-1}} . Inactive[Grad][Subscript[c, 2][x], {x}], {x}]} == 
 {NeumannValue[Subscript[k, 1]*(-Subscript[c, 1][x] + 
     Subscript[c, ext1][t, x, y]), x >= 0], 
  NeumannValue[Subscript[k, 2]*(-Subscript[c, 2][x] + 
     Subscript[c, ext2][t, x, y]), x >= 0]}

If for some reason (and I'd be curious to know) you want the boundary condition to be associated with a specific equation you can use:

MassTransportPDEComponent[vars, pars] == {0, 
  Total[MassTransferValue[x >= 0, vars, pars]]}

If you wanted to associate it with the second equation. Leave me a comment why this is important to be able to do.