| Author | Message | |||
     Kyle D. Frischknecht (kyle) New member Username: kyle Post Number: 1 Registered: 08-2006 |
Using the 1-D diffusion equation in FlexPDE notation: div(D_H2O*grad(c)) = dt(c), where D_H2O is the diffusion coefficient for the species water, and c is the concentration having units of mass of diffusing species per volume of material, how can FlexPDE handle a boundary change from diffusion through a solid to diffusion through a gas (say nitrogen at 1 atm.)? It is not sufficient to assign a different value of D_H2O in "Region 2" of the script because the concentration definition also changes from mass per volume solid to mass per volume gas. The flux must be continuous over the boundary (diffusing material is not created or destroyed), but the concentration will be discontinuous (step change) according the different solubility limits in the different Regions. The relationship between the two regions could be locked through Henry’s law: [Concentration in solid] = [Solubility Coefficient]*[Concentration in the gas]. How can this be addressed in FlexPDE? | |||
| Bouke Tuinstra (bouke) Member Username: bouke Post Number: 6 Registered: 04-2006 |
Kyle, I don't think you can calculate "c" directly if there is a step in the concentration, because the gradient will be infinite at the interface. I guess you could define a solubility coefficient "K" that has some value in the solid region and the value 1 in the gas region and then you substitute K*c for c in your equation. (On hindsight, you should probably change your diffusion coefficients too). In this way, K*c will be continuous and there will be a step in c at the interface. I attached a simple script to demonstrate this. Does this help?
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| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 676 Registered: 06-2003 |
Have you considered using a "Contact Resistance" boundary condition? This allows a flux determination based on a value jump at the surface. See "Contact" in the Help Index. | |||
| Bouke Tuinstra (bouke) Member Username: bouke Post Number: 7 Registered: 04-2006 |
That is what I thought at first, too. But there is a difference between the contact resistance case and what Kyle descibes. In the contact resistance case, the size of the jump will depend on the flux, i.e., if there is no flux the concentration field will be uniform. In Kyle's case, the concentration in the solid and in the gas will be different also if there is no flux. In fact this is more or less caused by the definition of the concentration variable, since the chemical potential field *will* be uniform (i.e., solid-gas equilibrium so the chemical potential of the compound in the solid will be equal to the chemical potential in the gas). I think my "K*c" variable is in fact a simple representation of the chemical potential. Maybe it would be possible to use a CONTACT boundary with a flux-independent jump? | |||
| Kyle (kyle) New member Username: kyle Post Number: 2 Registered: 08-2006 |
Hi Bouke and Robert: Thank you very much for responding to my post. You guys are right about the "Contact Resistance" boundary condition - it won't work in this case because the "jump" remains even when the flux goes to zero (equilibrium), which is not the case for "Contact Resistance." I think the "K*c" approach is a really great idea. I will try out the script with the "K*c" term and report back later. What do you think about another idea of trying to define unique variables for the concentration in the solid and gas - "c_solid", "c_gas" - setting up two different diffusion equations, one for each variable, and trying to couple them at the interface through some sort of conservation of mass? Does this have any merit? Best Regards, Kyle | |||
| Kyle (kyle) Junior Member Username: kyle Post Number: 3 Registered: 08-2006 |
Hi All: I have now tested out the proposed "K*c" approach for diffusion across a gas-solid interface for both steady-state and time dependent cases. The results are this approach works great for all the steady-state cases I tried, but the time dependent cases are very sensitive - some work and most don't. I manually refined the mesh at regions where the "K" values jump by adding thin "Dummy Regions" that specify a finer mesh_spacing with some success, but this does not always solve the problem. Unfortunately, I have some VALUE boundary conditions that are a central part of my diffusion modeling and the SAWGE function doesn't seem to eliminate the problem. I think I will need to express the initial condition in a better way to reduce startup sensitivity, but in 2D this rather difficult for my geometry. The SWAGE function would have been my choice for startup if it had worked. Overall though, I think this is a generally suitable approach for treating diffusion across gas-solid interfaces. Thank you and Best Regards, Kyle | |||
| Kyle (kyle) Member Username: kyle Post Number: 4 Registered: 08-2006 |
Hi All: I have now tested out the proposed "K*c" approach for diffusion across a gas-solid interface for both steady-state and time dependent cases. The results are this approach works great for all the steady-state cases I tried, but the time dependent cases are very sensitive - some work and most don't. I manually refined the mesh at regions where the "K" values jump by adding thin "Dummy Regions" that specify a finer mesh_spacing with some success, but this does not always solve the problem. Unfortunately, I have some VALUE boundary conditions that are a central part of my diffusion modeling and the SAWGE function doesn't seem to eliminate the problem. I think I will need to express the initial condition in a better way to reduce startup sensitivity, but in 2D this rather difficult for my geometry. The SWAGE function would have been my choice for startup if it had worked. Overall though, I think this is a generally suitable approach for treating diffusion across gas-solid interfaces. Thank you and Best Regards, Kyle |
simple script