Mathematical-thermodynamic solubility model developed by the application of discrete Volterra functional series theory

• For the first time, the appropriate thermodynamic-mathematical development of empirical models is presented.
• The model is two parametric and based on the discrete form of Volterra functional series.
• A substantial number of data points is used for performance analysis.
• Peng–Robinson equation of state (PR-EOS) together with the two adjustable parameters van der Waals mixing rule (vdW2) and a number of recent empirical models were considered.
• Particle swarm optimization (PSO) was used for optimizations.

Based on the discrete form of Volterra functional series in approximation of multivariable functions, a mathematical model proposed for the modeling of solute solubility in supercritical fluids with four independent variables, which can be regarded as the general form of majority of available empirical models in literatures. As far as we know, it is the first time the thermodynamic-mathematical development of empirical models is presented. A substantial number of data points for different solutes and solvents is used to analyze the performance and reliability of the proposed model. Peng–Robinson equation of state together with the two adjustable parameters van der Waals mixing rule, as an example of theoretical models, and a number of recent empirical models, as examples of empirical/statistical models, were considered to demonstrate the superiority of the proposed model. The particle swarm optimization (PSO) was used for determination of the optimum coefficients of all models. The results were obtained and discussed in details.