There are so many thermodynamic models commonly used in chemical process simulators that it would be overwhelming to cover all of them in great detail. This is why the discussion in the text focuses on a few representative models. Nevertheless, students interested in process engineering will often face the need to choose the most appropriate thermodynamic model, and the most appropriate model may not be one of those that we have covered in detail. Fortunately, the differences between many of the thermodynamic models and the ones that we have studied are generally quite small. In this appendix, we review some of most common thermodynamic models and put them into context with others that we have studied. This should help students to feel a bit more comfortable wading through the wealth of models from which to choose.
Students interested in becoming process simulation experts will be interested in reading the recent articles reviewing the selection of thermodynamic models. Schad1 and Carlson2 provide some significant examples and cites several relevant articles. A common thread throughout these articles is the emphasis on accurate application of thermodynamic principles. It is interesting to see the large number of examples in which practical engineering applications were so deeply affected by the fundamentals of thermodynamics.
We have covered the Peng-Robinson and virial equations in fair detail, but there are many others. Some that we have mentioned but not treated in detail are the Redlich-Kwong equation (homework problem 7.9),3 the Lee-Kesler equation (Eqn. 7.11 on page 260),4 and a popular form of its extension to mixtures, the Lee-Kesler-Plocker equation,5 the Soave equation (Eqn. 7.65 on page 294),6 also known as the Soave-Redlich-Kwong or SRK equation), the ESD equation,7 and the SAFT equation.8 A slight variation on the Soave equation is the API equation;9 it changes only the value of κ as a function of acentric factor in order to obtain a slight improvement in the predicted vapor pressures of hydrocarbons. A specific implementation of the virial equation useful for associating systems is the Hayden-O’Connell method.10 The Soave equation, Peng-Robinson equation, Lee-Kesler-Plocker equation, and API equation are all very similar in their predictions of VLE behavior of hydrocarbon mixtures. They are accurate to within ~5% in correlations of bubble-point pressures of hydrocarbons and gases (CO, CO2, N2, O2, H2S) and about ~15% for predictions based on estimated binary interaction parameters. The Lee-Kesler-Plocker equation can be slightly more accurate for enthalpy and liquid density for some hydrocarbon mixtures, but the advantage is generally slight with regard to enthalpy and there are better alternatives to equations of state for liquid densities if you want accurate values. The cubic equations have some convergence advantages for VLE near critical points and their relative simplicity makes them more popular choices for adaptations of semi-empirical mixing rules to tune in an accurate fit to the thermodynamics of a specific system of interest. The best choice among these is generally the one for which the binary interaction parameters have been determined with the greatest reliability. (Accurate reproduction of the most experimental data at the conditions of your specific interest wins.)
The primary role of equations of state is that they can predict thermodynamic properties at any conditions of temperature and pressure, including the critical region. The disadvantage is that they tend to be inaccurate for strongly hydrogen-bonding mixtures. This disadvantage is diminishing in importance with the development of hydrogen-bonding equations of state (like the SAFT and ESD equations), but it is not clear at this time whether these newer equations of state will displace any of the long-standing cubic equations of state with their semi-empirical modifications.
We have covered many solution models in fair detail: the Margules equation, the Redlich-Kister expansion, the van Laar equation, the Scatchard-Hildebrand theory (the most common implementation of regular solution theory), the Flory-Huggins equation, the Wilson equation, NRTL, UNIQUAC, and UNIFAC. Once again, the best choice will most often depend on the availability of binary interaction parameters which are relevant to the specific conditions of interest.
The primary role of solution models is to provide semi-empirical models which have a greater degree of flexibility than equation of state models, owing to the greater number of adjustable parameters and their judicious choice such that both magnitude and skewness of the free energy curves can be accurately tuned.
Another set of models that have been developed relatively recently can be referred to as “hybrid” models in the sense that they combine equation of state models with solution models. The two most prevalent of these are the Modified Huron-Vidal (MHV) method and the Wong-Sandler mixing rules.11 The basic idea is to apply a solution model at high density or pressure to characterize the mixing rules of the equation of state and then interpolate from this result to the virial equation at low density. These methods tend to compete with the hydrogen bonding models in the sense that they enhance accuracy for nonideal solutions at high temperatures and pressures. They are more empirical, but they tend to leverage the well-developed solution models (like UNIFAC) more directly. They also tend to be more efficient computationally than the hydrogen bonding equations of state.
When faced with choosing a thermodynamic model, it is helpful to at least have a logical procedure for deciding which model to try first. A decision tree is included in Fig. D.1. For nonpolar fluids, an equation of state may suffice. For polar fluids, a fitted activity coefficient model is preferred, possibly in combination with the Hayden-O’Connell method or in combination with some other equation of state for the vapor phase (like the Peng-Robinson equation). This approach can often provide satisfactory predictions as long as the pressures are 10 bar or less. Predictions by this approach should be checked against literature data to the greatest extent possible. If there are no experimental data for one of the binary systems in this event, then UNIFAC can be used to generate “pseudo-data” that can be used to predict the Gibbs excess energy for that binary, and these pseudo-data can be used to regress UNIQUAC or NRTL parameters if desired (homework problem 13.16). Above 10 bars, the choices are not so obvious. The most obvious method to try if you are satisfied with the correlations below 10 bars is to apply the MHV or Wong-Sandler approach. If you need to predict phase behavior over a broad range of conditions based on few data in a narrow range of conditions, a hydrogen bonding equation might provide more reliable leverage in light of its clearer connection with the physical chemistry in the solution. If you are dealing with compounds which dissociate electrolytically or associate strongly and specifically in solution, then it will probably be necessary to apply a simultaneous reaction and phase equilibrium approach. These kinds of systems are common in gas strippers for compounds like CO2, H2S, and amines. For these systems, it is especially important to check your correlations against experimental data near the conditions of your specific interest.