I chose to use the defined reaction method for the chemical equilibrium as opposed to the free energy minimization method. This choice provides more control over which reactions are in equilibrium. The compounds and reactions selected are shown below.
The formula line shows the order of the species. Each of the columns in the stoichiometric matrix below corresponds to one specie. The above reaction network was also used by Carberry.
The equilibrium coefficients.
Using methods shown in earlier posts, the equilibrium coefficients are obtained. The plot below shows their values in the temperature range of interest.
The high values for the NO oxidation reaction confirms that this reaction may be treated as irreversible.
Comparison with Hüpen et al.
Hüpen et al. provided the formulas they used for computing the equilibrium coefficients as a function of temperature in their Table 1. Their R2 and R5 reactions correspond to my R1 and R2 reactions. Their equilibrium coefficients have dimensions of 1/kPa for these reactions indicating that the standard state fugacity product was not included. This is a common practice when using the equilibrium coefficients in a rate expression.
To obtain the true equilibrium constant, I multiply their constant by 1 atm. which is the net standard state fugacity after cancellation. The figure below compares the Hüpen coefficients with my values for the two equilibrium reactions.
The results for my R1 and Hüpen R2 show a perfect match. The other reaction shows a large discrepancy.
I suspect that the Hüpen formula is for the log base 10 instead of the natural log as reported. After making that modification, the comparison shows great improvement.
The two curves are nearly identical in shape. This indicates that the heat capacities for the participating compounds are the same for both sources for the equilibrium coefficients I suspect that the offset is due to different sets of standard states. I will show later in this post that the Hüpen form for the equilibrium coefficients isn't useful for an equilibrium model without knowledge of the standard states.
A test of the equilibrium coefficient
Carberry provided enough information to test the equilibrium coefficients. First, he states that the liquid from the condenser contains 20.9 mol/hr HNO3 which is 40 wt% of the stream. This specifies the liquid stream composition and rate. He also gives the composition and rate of the absorber gas feed stream, which is also the gas stream from the condenser. By adding these two streams and then performing a reactive flash at 35 C and 7 atm, I should get a liquid stream that contains 40 wt% HNO3. I performed the test and got the required result, thereby validating my equilibrium coefficients.
Similarly, Kenig and Seferlis provided the compositions for a similar example. The reactive flash of those streams also validated my equilibrium coefficients.
The composition form for K
I'm going to call the equilibrium coefficient, K, provided thus far as the "thermodynamic equilibrium coefficient." At equilibrium, the "composition equilibrium coefficient", KX, defined below, will equal the thermodynamic value.
Because activities are dimensionless, KX is also.
In the definition for lnKX, I have chosen to use the liquid composition, m, and the liquid state fugacities. Instead, I could have chosen the gas composition and fugacities and achieved the same result because the fugacities of the vapor and liquid are equal at the equilibrium condition. In this sense, it doesn't matter in which phase an equilibrium reaction occurs.
The fugacity of a specie is independent of the standard state. However, the activity depends upon the standard state chosen. Since we can chose the standard state for the activity, how is it possible that KX can be equated to K? The answer is that K also depends upon the standard state chosen for the free energies and enthalpies. Therefore, the free energies, enthalpies and the activities need to be based on the same standard state. The standard state used by Prode, and probably most process simulation programs, is ideal gas at 1 atm.
The chemical equilibrium error function
As the flash routine changes the liquid composition, the error between K and KX will be computed as fK, as defined below. To achieve chemical equilibrium, this error needs to be minimized. Note that I have excluded the NO oxidation reaction from this comparison because it is not equilibrium limited.
Hüpen, Bernhard, and Eugeny Y. Kenig. “Rigorous Modelling of NOx Absorption in Tray and Packed Columns.” Chemical Engineering Science 60, no. 22 (2005): 6462–71. doi:10.1016/j.ces.2005.04.060.
Carberry, James J., Chemical and Catalytic Reaction Engineering, Dover (2001)
Kenig, E. and P. Seferlis, "Modeling Reactive Absorption", Chem. Eng. Prog., pg 65-73, Jan. 2009.