- #1
Hypatio
- 151
- 1
Ignoring cross-diffusion, diffusive mass fluxes down chemical potential gradients can be described by the equation (I am working from de Groot and Mazur's 1984 text on non-equilibrium thermodynamics):
[itex]\frac{\partial C_k}{\partial t} = L_{kk}\frac{\partial (\mu_k-\mu_n)}{\partial x}[/itex]
where [itex]C_k[/itex] is the concentration of species [itex]k[/itex], [itex]L_{kk}[/itex] is a phenomenological coefficient, [itex]\mu_k[/itex] is the chemical potential of species k, [itex]\mu_n[/itex] is the chemical potential of a different species, and [itex]x[/itex] is distance.
However, if the chemical potential of a species is high (relative to others) when its concentration approaches zero, this equation will predict that a flux will continue to remove the component, resulting in negative concentrations. Mass conservation is "preserved" because the concentration of a different component will then be greater than 1, but this is obviously still incorrect.
How can this behavior be corrected? Must you simply enforce non-negative concentrations ad hoc, or is there a more obvious method.
[itex]\frac{\partial C_k}{\partial t} = L_{kk}\frac{\partial (\mu_k-\mu_n)}{\partial x}[/itex]
where [itex]C_k[/itex] is the concentration of species [itex]k[/itex], [itex]L_{kk}[/itex] is a phenomenological coefficient, [itex]\mu_k[/itex] is the chemical potential of species k, [itex]\mu_n[/itex] is the chemical potential of a different species, and [itex]x[/itex] is distance.
However, if the chemical potential of a species is high (relative to others) when its concentration approaches zero, this equation will predict that a flux will continue to remove the component, resulting in negative concentrations. Mass conservation is "preserved" because the concentration of a different component will then be greater than 1, but this is obviously still incorrect.
How can this behavior be corrected? Must you simply enforce non-negative concentrations ad hoc, or is there a more obvious method.