Questions on Gibbs vs CSTR-eq

[nrgeng]

I compared several simulations of the Gibbs Reactor Unit (Gibbs) to simulations of the CSTR Unit in equilibrium mode (CSTR-eq). Please refer to the chart. I expected the CSTR-eq to show changes in Extent of Reaction (Extent, §) for changes in operating pressure (p) given a single value of Equilibrium Constant (Keq). This is true as shown in Row 4 for 325 C, 1 atm and 300 atm.

What I did not expect was different § for the same Keq at constant p. See Extent for 1 atm and 300 atm. For example; @ 325 C & 1 atm, § = 3.75E-04 for the Gibbs, but § = 1.53E-04 for the CSTR-eq. Another example is @ 325 C & 300 atm, § = 5.36E-04 for the Gibbs, but § = 2.19E+00 for the CSTR-eq.

Since Keq = f(T) only, pressure should have no effect on it, but the CSTR-eq should correct § for pressure.

Any suggestions as to the deviations would likely add to my learning.

Chart of Reaction Extents

Questions on Gibbs vs CSTR-eq.GIF (15.07 KiB) Viewed 25443 times

[jasper]

Check the density? The equilibrium constant expressed in molar concentrations for which the sum of molecules of reactants does not match the sum of molecules of reaction products is density dependent. And density (especially for the vapor phase) depends on pressure.

[nrgeng]

Thanks for your suggestion regarding density.

I compared Gibbs (1), CSTR-eq (2), and I added Equilibrium Reactor Unit (Equil) (3). The inlet and outlet densities were all identical, ρ_hat =20.37 mol/m^3. The Ideal Gas Law is valid at low pressures so I used 1 atm for the simulations. For identical inlet conditions, T1, p1, F(i)1 = T2, p2, F(i)2 = T3, p3, F(i)3, then ρ_hat1 = ρ_hat2 = ρ_hat3, which is the case for all of my simulations. For Keq = 2.99E-07, the question remains, why is § = 3.75E-04 for the Gibbs, but § = 1.53E-04 for the CSTR-eq and Equil?

Any additional suggestions are solicited.

[jasper]

Based on a textual description it is hard for me to pinpoint the source of differences. Is it possible to send me an fsd that includes the comparison for both reactors?

[nrgeng]

I have sent the file: Gibbs Unit (Forum) 13 May 06.fsd by e-mail. I hope an explanation is possible.

[jasper]

You sent only the Gibbs reactor. This contains no reaction package and does not use a specified equilibruim constant.

The reaction it uses is

0.408248 Carbon monoxide + 0.816497 Hydrogen = 0.408248 Methanol

considering you state that

Keq = 2.99E-07 m^6/mol^2

I image that you formulate

CO + 2 H2 = CH3OH

so that

kEq = [CH3OH]/([H2]^2*[CO]) = X[CH3OH]*rho/((X[H2]*rho)^2*rho*X[CO]) = X[CH3OH]/(X[H2]^2*rho^2*X[CO]) = V^2*X[CH3OH]/(X[H2]^2*X[CO]) = 2.98989e-007 m^6/mol^2

using, from the product stream:

X[CO]=0.33332721 mol/mol
X[CH3OH]=1.8374913e-05 mol/mol
X[H2]=0.66665442 mol/mol
V=0.049096525 m³/mol

So the results are the same you state, which is nice. The conversions are therefore the same, so the extents relate as the stroichiometries, as the amount converted for a particular compound equals the reaction extent multiplied by its stoichiometric coefficient.

Hence, the stoichiometric ratios between

0.408248 CO + 0.816497 H2 = 0.408248 CH3OH (Gibbs)

and

CO + 2 H2 = CH3OH (Equilibrium reactor)

are 0.408248 for all compounds. This should be the ratio between the extents as well then.

why is § = 3.75E-04 for the Gibbs, but § = 1.53E-04 for the CSTR-eq and Equil?

Their ratio is 1.53E-04/3.75E-04 = 0.408, corresponding to the ratios of the stoichiometric coefficients in the reactions used by the various reactors.

[nrgeng]

Thanks for locating my error with the Gibbs reaction extent.

I am wondering if you have considered normalizing the reaction stoichiometry to the conventional coefficient of one for the first compound of the reaction as written? This could be a tripping point for users as it was for me.

[jasper]

It is normalized. The vector of stoichiometries has a length of unity. Or: sqrt(0.816497^2+2*0.408248^2)=1. This way all reaction count equally heavy. Multiple reactions are also orthogonal. So the matrix of stroichiometries is an orthonormal basis of the reaction space. A result of an SVD composition. Details are here:

van Baten, J.M., Szczepanski, R., A thermodynamic equilibrium reactor model as a CAPE-OPEN unit operation, Computers and Chemical Engineering 35 (2011) 1251-1256

[jasper]

Another way to look at it:

CO + 2 H2 = CH3OH

has a stroichiometry vector of [-1 -2 1] which has a length of sqrt(-1^2+(-2)^2+1^2) = sqrt(6); this is where the 0.408248 ratio comes from, it is 1/sqrt(6).

[nrgeng]

That is good information to know. Thanks.

I guess that I am lazy. I would like to look at the Gibbs Reactor Unit (Gibbs) Report and the Equilibrium Reactor Unit (Equil) Report and see §(Gibbs) = §(Equil), but instead I see §(Equil) = §(Gibbs) * (6)^(-0.5). If the Gibbs is normalized, then what can be said about the Equil?

Example : §(Gibbs) = 3.75E-04, then §(Equil) = 3.75E-04 * (6)^(-0.5) = 1.53E-04

It is confusing to me to have to convert §'s while reviewing the two Unit Reports for the same reaction. Am I asking too much?