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Diffusion Coeff: dependence on local composition

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Old   August 2, 2005, 09:07
Default Diffusion Coeff: dependence on local composition
  #1
Christine
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I am looking for a physical reason of why the diffusion coefficient D has a very slight dependence on local mass fraction in a binary gaseous mixture, according to what Hirschfelder et al. says in his book "Molecular Theory of Liquid and Gases".

I have modelled a binary mixture flow, and all what I have found is the formula for D given in that book.

Thanks in advance.
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Old   August 2, 2005, 20:22
Default Re: Diffusion Coeff: dependence on local compositi
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Runge_Kutta
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Christine,

You might look over the book by Maitland, Rigby, Smith, and Wakeham entitles "Intermolecular Forces." Also, the 1972 book by Ferziger and Kaper.

I can't give you a great answer off the top of my head but in a dilute monatomic gas mixture, all 4 transport coeffs are functions of temperature and concentration but not pressure. Once the pressure gets high enough then three body collisions become nonnegligible in the transfer of energy and momentum then you have a dense gas and the transport coeffs are also a function of pressure. Also, I'm not sure why you say "a very slight dependence." Look at the form of the reduced temperature and the reduced collision integral. They are very much influenced by the molecular mass. There is also a potentially big effect caused by different intermolecular potentials.

On the off chance you're considering using true multicomponent diffusion coefficients rather than the binary ones in either a fickian approximation or with the Stefan-Maxwell relations, don't use the one in Hirschfelder, Curtiss, and Bird. Use the one that is described by Ferziger/Kaper and Maitland ... Giovangigli's book talks about it but the book is couched is so much "mathematics for mathematics sake" that it's nearly unreadable.

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Old   August 3, 2005, 10:35
Default Re: Diffusion Coeff: dependence on local compositi
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Christine
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Thanks Runge.

The fact is I don't have the book you mentioned at hand. I think you have not understood my point or I have not explained it right. I am modelling a binary gaseous mixture flow. In order to calculate the viscosity coefficient I have used the Wilke's law, in which the molar fraction X appears inside. On the contrary, when calculating the binary coefficient of diffusion D, it does not depend on local molar fraction X. I mentioned the word "slight" because there is a function for correcting D taking into account the molar fraction, but as Hirsch says, it is always near 1.

Thank you.
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Old   August 3, 2005, 15:29
Default Re: Diffusion Coeff: dependence on local compositi
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Runge_Kutta
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Christine,

I went over to my shelf and looked at Maitland et al. (chapter 5 and the appendix). You are right that the first-order Chapman-Enskog approximation to the binary diffusion coeff. contains no mole fraction. And, as you implied, the second-order does. I suspect the reason for this is related to the H^{ij} and L^{ij} matrices used to compute the transport coeffs (again, see Maitland. Transport coeffs. are a determinant over a determinant. each determinant is that of a supermatrix composed of smaller ones.The upper rightmost column and lower row can make the first approximation vanish. This happens with the thermal conductivity using the L^{ij} matrices and also happens with thermal diffusion. Ordinary diffusion does not have a vanishing first approximation with the L^{ij}'s. Viscosity is based on the H^{ij} family of matrices and has a nonvanishing first approximation.So, I suspect that what you are seeing is the particular structure of the first order matrices.Some have mole fractions in the first terms of the lower row and upper right column.

If you want to see the structure of the matrices and see why the mole fractions do or do not show up, I could email them to you ...
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Old   August 3, 2005, 17:25
Default Re: Diffusion Coeff: dependence on local compositi
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Christine
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Thanks again Runge.!!

Well, to be honest I am not looking for a mathematical explanation of the event, I am looking for a physical explanation. All the maths you described me has to have a translation into physics, for anyone who is not an specialist in calculating from the bottom this stuff. I'm looking for a "visual" explanation, not one explanation full of determinants.
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Old   August 3, 2005, 18:36
Default Re: Diffusion Coeff: dependence on local compositi
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Runge_Kutta
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Christine,

A visual or physical explanation is very unlikely in this context. People set about to derive approximate solutions to the Boltzmann equation in certain very specific situations. The Chapman-Enskog procedure gives you approximate solutions to the Boltzmann equation in the limit of small Knudsen numbers (and other restrictions) subject to the solution having a particular form - one where the constitutive relations are functions of the gradients in the hydrodynamic variables only.With all of these assumptions and more, one then sets about to solve the very complicated math. That's when you get into determinants.Your question may have a simple explanation but it is at a level of nuanced understanding that is beyond me.

By the way, look at Hirschfelder, Curtiss, and Bird equation 7.4-61 and notice the mole fractions on the rightmost column.
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Old   August 4, 2005, 12:50
Default Re: Diffusion Coeff: dependence on local compositi
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Angen
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There is a difference between calculation of viscosity or thermal conductivity and diffusivity from Chapman-Enscog expansion. The former can be done for a single component systems. The later make sense only for multicomponent systems. Evaluation of Knudsen number for a single component system is strightforward. For multi-component systems it is not obvious and any reasonable estimate of Knudsen number should depend on the composition.

Angen
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