Here is a Mickey Mouse example that shows the what the Dumux diffusion formulation can produce:
[cid:[email protected]] I just plotted an imaginary molfraction gradient and 2 gradient of molar density over some spatial domain. For an ideal gas, the molar density is propotional to the gas pressure (ideal gas law). So the molar density gradient corresponds to a pressure gradient which is again proportional to 1/K (the permeability). I also plotted the molar concentration which is rho*x. If only a gradient of the molfraction drives the diffusion, transport will be from right to left. However, it is obvious that for case 2 (high pressure gradient), the concentration has its maximum not at the right boundary but inside the domain. I think this illustrates that Fickian diffusion breaks down even for a binary system in porous media if the permeability is low. A better approach would probably be the dusty gas model (Stefan-Maxwell + Knudsen diffusion) but I need to check how it is derived. Maybe the same problem can occur when the dusty gas model is used. Best regards Georg Von: Dumux [mailto:[email protected]] Im Auftrag von Alexander Kissinger Gesendet: Donnerstag, 3. Dezember 2015 16:06 An: DuMuX User Forum Betreff: Re: [DuMuX] A fundamental question concerning diffusion in Dumux sorry one more clarification to the last post: I wrote: Mass or molar gradients are only valid if I meant: Mass or molar concentration gradients [mol_comp/m3] are only valid if On 12/03/2015 03:56 PM, Alexander Kissinger wrote: Dear Dumux, one clarification to the last post: The driving force for Fickian diffusion is a gradient in the mole fractions x [mol_comp/mol_total] as implemented in the Dumux models: Diffusive flux: J_D = -rho_molar [mol_total/m3] * D * grad x Mass or molar gradients are only valid if - the volume of the solute is much smaller than the total concentration or molar density [mol_total/m3] and - isothermal conditions prevail, i.e. no volume change due to changes in temperature A detailed explanation can be found in: Taylor, Ross, and Rajamani Krishna. Multicomponent mass transfer. Vol. 2. John Wiley & Sons, 1993. Chapter 3.1 and 3.1.1 Best regards Alex On 12/03/2015 01:40 PM, Bernd Flemisch wrote: Hi, I discussed this a bit here with Alex, Holger and Rainer. The main point is that Fickian diffusion is described by _molar_ concentrations [mol/m3], not _mass_ concentrations [kg/m3], https://en.wikipedia.org/wiki/Fick's_laws_of_diffusion<https://en.wikipedia.org/wiki/Fick%27s_laws_of_diffusion> The multiplication by molar/mass density comes then from the fact that we balance moles/mass. But it happens outside of the gradient. This indeed allows diffusion to occur against the mass concentration gradient, if that differs from the molar concentration gradient like in the setup that you prescribe. Kind regards Bernd On 12/02/2015 10:45 AM, [email protected]<mailto:[email protected]> wrote: I forgot the attached file... Von: Dumux [mailto:[email protected]] Im Auftrag von [email protected]<mailto:[email protected]> Gesendet: Mittwoch, 2. Dezember 2015 10:44 An: [email protected]<mailto:[email protected]> Betreff: Re: [DuMuX] A fundamental question concerning diffusion in Dumux Hello Dumux, I am back with the same question and some more infos. The modeling approach for diffusion in Dumux is (in my opinion) wrong and gives unphysical results. This is most pronounced for diffusion in a gas phase and when the intrinsic permeability is low. Consider the model setup depicted in Figure1.jpg. In this setup the only transport mechanism for H2O from the right boundary is diffusion because the sink of N2 is high and the advective flow is from left to right. In this setup, the pressure will drop to the right while the gradient of x_g^H2O is vice versa. However, the concentration of H2O will be lower at the right boundary and diffusion will occur from a lower to a higher concentration! This is completely unphysical. There is no reason why the component should flow against its concentration gradient. The reason for this lies in the formulation of the diffusive fluxes: [cid:[email protected]] Where [cid:[email protected]] In the Dumux formulation, the second (pressure dependent) term on the very right is neglected resulting in the possibility that species diffuse against their concentration gradient. The density gradient is proportional to the pressure gradient from left to right while the molfraction gradient is vice versa. I hope this made things more clear. I would recommend to use grad(rho*x) instead of grad(x) for the calculation of the diffusive fluxes. I am always open for discussion. Kind regards Georg Von: Dumux [mailto:[email protected]] Im Auftrag von [email protected]<mailto:[email protected]> Gesendet: Mittwoch, 25. November 2015 14:30 An: [email protected]<mailto:[email protected]> Betreff: [DuMuX] A fundamental question concerning diffusion in Dumux Hello Dumuxers, I was wondering why the diffusive fluxes in Dumux are defined as D*rho*grad(mol-or-massfraction). Typically one would use D*grad(c) (e.g. Ficks law) where c=rho*mol-or-massfraction. Using the Dumux equation means that local differences in the density are neglected for diffusive fluxes. Is there any reason/justification for this? Or is my thinking just wrong? Best regards Georg Futter -------------------------- German Aerospace Center (DLR) Institute of Engineering Thermodynamics | Computational Electrochemistry | Pfaffenwaldring 38-40 | 70569 Stuttgart Dipl.-Ing. Georg Futter | Ph.D. student Telefon 0711/6862-8135 | [email protected]<mailto:[email protected]> www.DLR.de<http://www.DLR.de> _______________________________________________ Dumux mailing list [email protected]<mailto:[email protected]> https://listserv.uni-stuttgart.de/mailman/listinfo/dumux -- _______________________________________________________________ Bernd Flemisch phone: +49 711 685 69162 IWS, Universität Stuttgart fax: +49 711 685 60430 Pfaffenwaldring 61 email: [email protected]<mailto:[email protected]> D-70569 Stuttgart url: www.hydrosys.uni-stuttgart.de<http://www.hydrosys.uni-stuttgart.de> _______________________________________________________________ _______________________________________________ Dumux mailing list [email protected]<mailto:[email protected]> https://listserv.uni-stuttgart.de/mailman/listinfo/dumux -- Alexander Kissinger Institut für Wasser- und Umweltsystemmodellierung Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Pfaffenwaldring 61 D-70569 Stuttgart Telefon: +49 (0) 711 685-64729 E-Mail: [email protected]<mailto:[email protected]> _______________________________________________ Dumux mailing list [email protected]<mailto:[email protected]> https://listserv.uni-stuttgart.de/mailman/listinfo/dumux -- Alexander Kissinger Institut für Wasser- und Umweltsystemmodellierung Lehrstuhl für Hydromechanik und Hydrosystemmodellierung Pfaffenwaldring 61 D-70569 Stuttgart Telefon: +49 (0) 711 685-64729 E-Mail: [email protected]<mailto:[email protected]>
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