The literature is rich with advanced analytical equations of state. These equations are usually empirical or semi-empirical with a large number of multi-parameters adjusted to experimental or molecular simulated data. Another type of these equations is derived based on statistical mechanical theories. Examples of advanced analytical equations are:
- Statistical association fluid theory-based on statistical mechanics
[1]
- Equation of state by Nicolas et al. based on fitting molecular simulated data for Lennard-Jones molecules
[2]
- Empirical multiparameter equations of state
[3]
Unlike the commonly used cubic equations of state, these equations have involved mathematical expressions. The use of such involved equations of state is usually acceptable in the thermodynamic community provided that the model is accurate and offers some desirable features that cannot be found in simple models. If one inspects their respective references, he/she will appreciate the excellent comparison with molecular simulated data or experiments. The apparent success of these models might not always be true at some state conditions. However, the failure of a certain model might not been noticed even by the developers. In fact, in the development process, it is unlikely to discover hidden pitfalls in these models. In the literature, unphysical behavior associated with any equation of state usually discovers after using the model in many applications. In our group, we offer a new method that can pinpoint any unphysical behavior exists in any model. We use concepts taken from bifurcation analysis to generate bifurcation diagrams. From the analysis of these diagrams, we can show the limitations and defects in any studied model.
What can bifurcation analysis do for us?
1- To point out any possible hidden pitfalls in thermodynamic models
2- To know how far a thermodynamic model deviates from the correct physical behaviour
3- To pinpoint spurious two-phase separation regions
4- To specify the location of spurious two-phase separation regions
5- To find unphysical branches
6- To evaluate the risk of the presence of unphysical branches
7- To determine the number of volume roots at any temperature
What is bifurcation analysis?
Bifurcation analysis is a big field that is commonly used with dynamic systems. Unfortunately, it is not common in thermodynamic community. We borrow some of its concepts and we utilize them with thermodynamic models. The reader may consult our work to read about the bifurcation analysis and how we utilize it with equation of state. You may want to read
section 2 in [4].
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How the bifurcation diagram is generated?
The following figure shows how we generate the bifurcation diagram.