Thermodynamic modelling is a set of different strategies that are used by engineers and scientists to develop models capable of evaluating different thermodynamic properties of a system. At each thermodynamic equilibrium state of a system, the thermodynamic properties of the system are specified. Generally, thermodynamic models are mathematical relations that relate different state properties to each other in order to eliminate the need of measuring all the properties of the system in different states. The easiest thermodynamic models, also known as equations of state, can come from simple correlations that relate different thermodynamic properties using a linear or second-order polynomial function of temperature and pressures. They are generally fitted using experimental data available for that specific properties. This approach can result in limited predictability of the correlation and as a consequence it can be adopted only in a limited operating range. By contrast, more advanced thermodynamic models are built in a way that can predict the thermodynamic behavior of the system, even if the functional form of the model is not based on the real thermodynamic behaviour of the material. These types of models contain different parameters that are gradually developed for each specific model in order to enhance the accuracy of the evaluating thermodynamic properties.
Cubic model development Cubic equations of state refer to the group of thermodynamic models that can evaluate the specific volume of gas and liquid systems as a function of pressure and temperature. To develop a cubic model, first, it is essential to select a cubic functional form. The most famous functional forms of this category are Redlich-Kwong, Soave-Redlich-Kwong and Peng-Robinson. Although their initial form is empirically suggested, they are categorised as semi-empirical models as their parameters can be adjusted to fit the real experimental measurement data of the target system.
Pure component modelling In case the development of a cubic model for a pure component is targeted, the purpose would be to replicate the specific volume behaviour of the fluid in terms of temperature and pressure. At a given temperature, any cubic functional form results in two separate roots which makes us capable of modelling the behaviour of both vapour and liquid phases within a single model. Finding the roots of the cubic function will be done by simulating the vapour-liquid equilibrium condition of the pure component where the fugacity coefficients of the two phases are equal to each other. So, in this case, the main aim can be limited to deriving fugacity coefficients of vapour and liquid phases from the cubic model and refining the adjustable parameters of the model such that they will become equal to each other at different equilibrium pairs of temperature and pressure. As the equilibrium pressure and temperature are related together in the case of a pure component system, the functional form of cubic models are able to evaluate the specific volume of the system in the wide range of temperature and pressure domain.
Multi-component modelling
Cubic model development for mixtures of more than one component is different as, according to the Gibbs phase rule, at each temperature level of a multi-component system, equilibrium states can exist at multiple pressure levels. Because of that, development of the thermodynamic model should be performed following different steps:
Selection of the cubic model: The initial step is the selection of a cubic functional form. Essentially, there exists no specific rule for this step. It can be done based on common practices of the cubic models already developed for the pure components existing in the mixture. Single phase: Although a cubic model for a pure component is capable of predicting the specific volume of the system at both vapour and liquid phases, this is not the case for multi-component systems. Currently cubic models are used for the prediction of specific volume only in the vapour phase, while the liquid phase is modelled with more complex models based on the excess Gibbs energy, such as UNIFAC, UNIQUAQ, etc. Vapour and liquid phases: Cubic models can be expanded to model multi-component systems at both vapour and liquid phases by integrating a proper mixing rule in their structural function.
Mixing rules
Mixing rules refer to different approaches that can be used to modify the cubic model in the case of multi-component mixtures. The simplest mixing rule is proposed by van der Waals and is called the van der Waals one fluid (vdW1f) mixing rule. As it can be understood from its name, this mixing rule is only used in case of modelling of a single phase (vapor phase). As a first step, to combine the model parameters for each binary combination of the mixture, the following equations are suggested:
a i j = a i i a j j ( 1 − k i j ) {\displaystyle a_{ij}={\sqrt {a_{ii}a_{jj}}}(1-k_{ij})}
b i j = b i i + b j j 2 ( 1 − l i j ) {\displaystyle b_{ij}={\frac {b_{ii}+b_{jj}}{2}}(1-l_{ij})}
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