Equilibrium reaction rate

[Juan]

I am trying to understand how Cantera calculates the net reaction
rates for an Ideal Gas Mixture. Specifically I need to know how it
models the equilibrium constant for calculating the reverse reaction
rate.

Is it using the Gibb's free energy? Which components is it using?

Translational free?
Rotational free?
Vibrational free?
others?

Thank you,

Juan

[Steven DeCaluwe]

Hi Juan,

Yes, Cantera uses the Gibbs free energy to calculate the equilibrium
constant, and hence the reverse rates.

The specific method of calculating delta_G for the reaction depends
on the species thermodynamic inputs in your cti file. In most cases,
I would assume that species thermodynamics are calculated using the
NASA polynomials, though other models (such as constant Cp) are also
allowed.

The attached pdf can be downloaded over at the old sourceforge
hosting site, and gives a pretty good overview of how cantera works.
Some of it may be outdated, but it is still a good introductory read.

Cheers,
Steven

[j s]

It looks like it is the NASA9 model in the file I was given.

Is there a description anywhere of how the equilibrium constant is calculated?  It is not apparent from the source code as there are a lot of function calls to cached model parameters so it is not clear how things are calculated.

There seems to be several Gibbs Energy models in this modeling document I was given, and I want to make sure I am using the right ones to match up with Cantera.  For example, I have a description for

Translational Free Gibbs Energy
Rotational Free Gibbs Energy
Vibrational Free Gibbs Energy
Electronic Free Gibbs Energy

and they seem to be based on physical constants and temperature not related to the model.

Regards,

Juan

-- You received this message because you are subscribed to the Google Groups "Cantera User's Group" group.
To post to this group, send email to canter...@googlegroups.com.
To unsubscribe from this group, send email to cantera-user...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/cantera-users?hl=en.

--
You received this message because you are subscribed to the Google Groups "Cantera User's Group" group.
To post to this group, send email to canter...@googlegroups.com.
To unsubscribe from this group, send email to cantera-user...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/cantera-users?hl=en.

[Steven DeCaluwe]

Hi Juan,

I'm not sure specifically what you're asking - whether you want specific info about what the NASA polynomials represent, or whether you want info about the Cantera calculations - but the go-to reference for the NASA polynomials is the report by McBride, Zehe, and Gordon, which you can find a link to here (click on the link that says 'View PDF FIle').  Hopefully this can answer some of your questions.

Regards,

Steven

[j s]

Thanks for your help.

What I am trying to understand is how the Gibbs Free Energy is calculated in Cantera, and how, from that, it calculates the Equilibrium Reaction Rate.

Does the equation for the Gibbs energy and equilibrium constant depend on the thermo model being used?  What are their equations in Cantera?

I don't really understand how a polynomial fit to a bunch of coefficients can control how the reverse reaction is being evaluated, but only the coefficient values?  Is it always based on Gibbs Free Energy?  What is the Cantera equation for Gibbs Free Energy?

Juan

[Steven DeCaluwe]

Juan, 

The relation between delta_G and the reaction rates comes from two equations:

1) At equilibrium, the forward and reverse rates of production for a given reaction are equal.  If you're using mass-action kinetics, you can bring the k's to one side -  k_fwd/k_rev -  and the product of the species concentrations (raised to their net stoichiometric coeffiicients) on the other side.

2) Also, at equilibrium, the change in Gibbs free energy for the reaction equals zero.  For your system, the change in Gibbs free energy is composed of two parts - the standard state change in free energy dG_0, where G_k for all speices k is calculated at standard T and with X_k = 1, and the nonstandard part, which is -RT*ln( __ ) where the inside of the log is, as above, the product of the species concentrations raised to their net stoichiometric coefficients.  

With a little algebra, we make the right-hand side of equation 2 the same as in equation 1 (the product of species concentrations raised to their net stoichiometric coefficients), and we set the two left-hand-sides equal to each other, with the following result: 

k_fwd/k_rev = exp(-dG_0/RT)

Thus if we know k_fwd, T, and dG_0, we can find k_rev.

The "Cantera equation" depends entirely on what equation of state you provide in the input file.  Since you specify an ideal gas, then the standard-state Gibbs free energy is a function of temperature only.  The polynomial coefficients you provide in the input file are used to calculate H_0 and S_0 as a function of temperature for each species, which leads to G_0  = H_0 - T*S_0.  This is used to calculate dG_0 for the reaction.  Look to the reference I provided earlier to see the exact form of the polynomials, but they are fit to the thermodynamics with high accuracy over a wide range of temperatures (with the temperature limits provided along with the polynomials).

Hope this helps,

Steven

[j s]

Hello Steven,

Thanks.  That actually explains a lot.  The models I had been given for Gibbs energy had a much different formulation then the one you mention.

Regards,

Juan

[Steven DeCaluwe]

Juan,

You're quite welcome.  By way of (possible) explanation, the NASA polynomial is not a physically-based model for the thermodynamic properties, but is an empirical fit, which is probably why it looks different from the models you've been looking at.  Section 4.2.3 of the 'definingphases.pdf' file that I sent earlier describes the NASA parameters a little further, as well as two other formulations available in Cantera (the Shomate parameters and constant Cp), and is worth looking over.

Also, in the interest of full disclosure, there is an errant minus sign in my explanation below; the nonstandard part of the Gibbs free energy is just RT*ln( __ ), with no negative sign.

Best of luck,

Steven

[andr1976]

Your desciption based on a partition into Translational Free Gibbs
Energy, Rotational Free Gibbs Energy,
Vibrational Free Gibbs Energy, Electronic Free Gibbs Energy, has it's
origon in statistical thermodynamics (see e.g. F. Reif, Fundamentals
of Statistical and Thermal Physics, McGraw-Hill, 1985).
In my thesis I made a comparison between gas phase thermodynamics
calculated by JANAF and statistical thermodynamics for various
molecules (methanol, co2, h2o etc.) and the agreement between the two
methods was very good. Usually polynomial coefficients e.g. NASA and
the like are available for many gases. However, in some cases such
coefficients are not available and an approach may be to calculate
e.g. Cp, gibbs free energy, entropy by statistical thermodynamics and
then fit e.g. NASA polynomials in order to provide input to cantera.
For instance the thermodynamics of molecular species adsorbed on a
substrate is usually not known. In those cases vibrational energy
levels measured with spectroscopy e.g. IR, EELS, Raman can be used as
input for a statistical thermodynamics approach. One also needs to
account for the lost translational and rotational energy levels which
is tranformaed into so-called "frustrated" translational (2D like) and
rotational energy levels. Those can be calculated assuming e.g. a
simple model of the potentional well. The last parameter is the ground
state energy (accounting for the electronic contribution). Often
measured heat of adsorption or quatum mechanics e.g. DFT methods can
be used.

best reagards

Anders Andreasen

On 29 Mar., 21:04, j s <j.s4...@gmail.com> wrote:
> It looks like it is the NASA9 model in the file I was given.
>
> Is there a description anywhere of how the equilibrium constant is
> calculated?  It is not apparent from the source code as there are a lot of
> function calls to cached model parameters so it is not clear how things are
> calculated.
>
> There seems to be several Gibbs Energy models in this modeling document I
> was given, and I want to make sure I am using the right ones to match up
> with Cantera.  For example, I have a description for
>
> Translational Free Gibbs Energy
> Rotational Free Gibbs Energy
> Vibrational Free Gibbs Energy
> Electronic Free Gibbs Energy
>
> and they seem to be based on physical constants and temperature not related
> to the model.
>
> Regards,
>
> Juan
>
> On Mon, Mar 29, 2010 at 1:25 PM, Steven DeCaluwe

> <steven.decal...@gmail.com>wrote:

> >> cantera-user...@googlegroups.com<cantera-users%2Bunsu...@googlegroups.com>

> >> .
> >> For more options, visit this group at
> >>http://groups.google.com/group/cantera-users?hl=en.
>
> > --
> > You received this message because you are subscribed to the Google Groups
> > "Cantera User's Group" group.
> > To post to this group, send email to canter...@googlegroups.com.
> > To unsubscribe from this group, send email to

> > cantera-user...@googlegroups.com<cantera-users%2Bunsu...@googlegroups.com>