transport properties

[Simon]

Hello,

I would like to know how the transport properties (diffusion coefficient, thermal conductivity coeffcient) are calculated. Does Cantera use the kinetic theory expressions from Chapman and Enskoog?

sincerly,

Simon

[Joseph Grcar]

Hello Andrew and Simon,

Recently Simon and Andrew asked about transport properties. Andrew and Simon, I am curious if you are working together or perhaps are taking the same university class somewhere?

I was a developer of chemkin on which cantera is based. Cantera appears to have copied the chemkin transport, and I am very familiar with that. I also studied the scientific literature on transport properties. So I may be able to answer your questions. I will refer to cantera/chemkin together because I believe they are the same for neutral gas species. I assume you only care about neutral gases.

The main transport properties are (1) shear viscosity and (2) diffusivity.

(1) For viscosity cantera/chemkin uses a simple approximation based on the pure species viscosities. There are many formulas of this kind. The ones in cantera/chemkin were chosen by Jurgen Warnatz from his knowledge of the literature up to the time that this part of cantera/chemkin was written by Bob Kee in the early 1980s. These formulas can be off by quite a bit. You would have to look in the old chemkin documentation or perhaps in the Kee, Coltrin, Glarborg textbook for the precise sources of the viscosity approximation.

(2) For diffusivity cantera/chemkin supplies mixture-averaged and multicomponent formulas. For mixture-averaged see the above comments about viscosity.

The multicomponent diffusivities are supposed to be the exact values. They are not; instead, they are just the first level of approximations in the Enskog theory. Unfortunately the ultimate source for the cantera/chemkin formulas is a gas kinetics book by Curtiss and Hirschfelder (1949) and a book by Dixon-Lewis (1948). CH and DL messed up the clean Enskog formulas by trying to simply them for calculation; I believe that CH copied from DL. It is this messed-up version that is the source for the cantera/chemkin formulas. When I say they are messed-up, I mean the formulas are overly complicated and limited to the first level of the Enskog theory. Note mechanical engineering textbooks often for this material cite Hirschfelder, Curtiss and Bird (1954) or maybe Bird, Stewart and Lightfoot (1960, 2002). Anything in these books is based on the earlier book by CH.

The cleanest formulas for the multicomponent case were assembled by Waldmann based on his understanding of the Enskog theory with subsequent modifications. Ern and Giovangigli made an attempt to present the Waldmann formulas, but I think it is fair to say that most people find the EG books are very hard to read. At a certain point the formulas from the higher-order Enskog theory are not usable because they require information about collisions for which there is no data. Most of the formulas have to be based on the lowest order approximation. I believe that thermal diffusion is an exception in that it always requires a higher-order formulation.

By the way, I can't help but correct the history of the subject. Enskog was an obscure physicist who only wrote papers in Swedish. Chapman was at a famous English university and was working on the same problem by an approach from Maxwell that ultimately was not developed. Instead Chapman wrote a book in English about Enskog's work in Swedish. This gave us the so-called Chapman and Enskog theory but it is really the Enskog theory.

And finally, why should anyone care? Why did all these books appear in the 1940s and 1950s about transport? During the world war and shortly after transport theory was very important in gaseous diffusion for separating uranium isotopes, which today we hear in the news that Iranians are doing. It turned out that Enskog's theory was used by the Americans. Some people think Enskog was in line for a Nobel prize but then he died unexpectedly.

Best, -- Joe

Joseph Grcar
6059 Castlebrook Drive
Castro Valley, CA 94552 USA
email jfg...@gmail.com
phone 1-510-581-1353

On Aug 21, 2014, at 6:53 AM, Simon wrote:

> Hello,
>

> I would like to know how the transport properties (diffusion coefficient, thermal conductivity coefficient) are calculated. Does Cantera use the kinetic theory expressions from Chapman and Enskog?
>
> sincerely,
>
> Simon
>
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[Andrew]

Hi Joseph,

thank you very much for your support! I don't know who Simon is. I am a student at UIUC and I just started becoming familiar with molecular gasdynamics and Cantera. 

My question about diffusion is: does Cantera (in the source codes TransportFactory/MixTransport) use "Fick's laws of diffusion" or "Maxwell–Stefan diffusion"?

Thanks a lot,

Andrew

[Joseph Grcar]

Hi Andrew,

I guess you can say all the diffusion formulas are Fickian because by depend on the first derivative of concentration, but of what? The mixture-averaged case has the derivative of mass fraction, and the multicomponent case has the derivative of mole fraction.

Stefan-Maxwell is just a clever way to calculate fluxes (which is what is really needed in conservation equations) from diffusion coefficients. The SM fluxes should be equivalent to writing out the fluxes explicitly, provided you define the diffusion coefficients properly in the SM equations. The SM equations by themselves are not a deeper or ultimate theory.

-- Joe

Joseph Grcar
6059 Castlebrook Drive
Castro Valley, CA 94552 USA

please correspond via
email jfg...@gmail.com
phone 1-510-581-1353

[Simon]

Hi Joseph,

thank you very much for your detailed answer! Fortunately, I have an example of Hirschfelders book at my disposal.

Their formula of binary diffuison coefficients in practical units is: 0.002628*sqrt(T^3/M_reduced)/(p*sigma^2*Omega22). As far as I know this is also the formular of the cassical first order Enskog approach (besides the prefactor, which differs in every paper or book I have read so far).

So am I right when I assume that cantera/chemkin uses this formular together with the mixing laws of Kee, Coltrin for the mixture average case? And for the multicomponent casechemkin/cantera uses  the model of Hirschfelder 1954 in capter 8.2?

Do you know by any chance if the coefficient of  thermal conductivity is also calculated according to hirschfelders book?

I would like to thank you in advance for your help!

sincerly,
Simon

[Joseph Grcar]

Hi Simon,

I have a question: where are you and why do you care so much about diffusion?

According to my notes, the formula given by HCB (1954) for the binary diffusion coefficient is

D_j,k = (3/16) Sqrt ( 2 pi k_B^3 T^3 / m_j,k ) / ( P pi sigma_j,k^2 Omega(1,1)* )

This was the formula used by chemkin and I believe is the formula used by cantera (you would have to look at the cantera code).

The trick is how all the intermediate quantities are defined: m_j,k, sigma_j,k, and Omega(1,1)*

For example, if one species is polar and the other not, then there is a different formula for sigma. It turns out that chemkin coded the formula wrong (and probably cantera if they just copied the chemkin code).

And exactly where does Omega(1,1) come from, and don't forget the *?

I expanded the formulas out in my notes, but it is hard to reproduce it here.

Best, -- Joe

Joseph Grcar
6059 Castlebrook Drive
Castro Valley, CA 94552 USA

please correspond via
email jfg...@gmail.com
phone 1-510-581-1353

[Simon]

Hi Joseph,

I am a student at the Technical University of Darmstadt and I'm working on transport properties of gases in the frame of my bachelor thesis.
I believe we mean the same equation for D12, just that in my equation Hirschfelder summed the constants like Pi and kb together to the number in front of m equation.
(You are right, it is of course the reduced collison integral Omega*).


For polar-polar interactions i use the Stockmayer potential.

For the interaction of polar and nonpolar gases I used the mixing laws  and the Lennard Jones Potential that are described by Kee and also in the book of Hirschfelder. They multiply sigma and Epsilon of the mixture of two gases with Zeta^2 or Zeta^-1/6 respectively . Zata is calculated by constants like polarisability and dipolmoment (described in Hirschfelder). However, I noticed, that Zeta in Kee and so I guess also in Cantera is calculated by

Zeta=1-alpha_reduced*µ_reduced*sqrt(epsilon1/epsilon2)

with Epsilon= depth of the potential of both components

alpha reduced=reduced dpolarisability of the unpolar component
µ_reduced= reduced polarisability.



So that the mixture Epsilon and sigma is:

Epsilon12 = ( Epsilon(i) .* Epsilon(j) ) .^ ( 0.5 ) .* Zeta.^2;
   
  sigma12   = ( sigma(i) + sigma(j) ) ./ 2 .* zeta^(-1/6);



Hirschfelder uses a similar formula, with the only difference that it contains µ_reduced^2 instead of only µ_reduced.

Is this the mistake of Cantera/Chemkin you were talking about or is there a further mistake ?

Thank you very much for your help, would it be possible for you to send the answer to my mailadress, I think it is easier to commucate this way.
I send

sincerly,
Simon