On the Effectiveness of Nature-Inspired Metaheuristic Algorithms for Performing Phase Equilibrium Thermodynamic Calculations

Table 1

Description of thermodynamic functions and optimization problems for phase stability analysis and equilibrium calculations in reactive and nonreactive systems.

Calculation	Description	Thermodynamic function	Optimization problem
Phase stability analysis	It involves the determination of whether a system will remain in one phase at the given conditions or split into two or more phases	Tangent plane distance function; TPDF=∑^𝑐𝑖=1𝑦𝑖𝜇𝑖𝑦−𝜇𝑖𝑧; where 𝑐 is the number of components of the mixture and 𝜇𝑖𝑦 and 𝜇𝑖𝑧 are the chemical potentials calculated at trial composition 𝑦 and feed composition 𝑧	min𝛽TPDF0≤𝛽𝑖≤1𝑖=1,…,𝑐.; The decision variables are 𝛽𝑖∈(0,1) using the following relationships: 𝑛𝑖𝑦=𝛽𝑖𝑧𝑖𝑛𝐹𝑖=1,…,𝑐; 𝑦𝑖=𝑛𝑖𝑦∑^𝑐𝑗=1𝑛𝑖𝑦𝑖=1,…,𝑐; where 𝑛𝑖𝑦 are the mole numbers of component 𝑖 in phase 𝑦 and 𝑛𝐹 is the total moles in the mixture under analysis
Phase equilibrium calculation	It involves the determination of the number, type, and composition of the phases at equilibrium at the given operating conditions	Gibbs free energy of mixing (𝑔); 𝑔=∑^𝜋𝑐𝑗=1∑𝑖=1𝑛𝑖𝑗ln(𝑥𝑖𝑗𝛾𝑖𝑗)=∑^𝜋𝑐𝑗=1∑𝑖=1𝑛𝑖𝑗𝑥ln^𝑖𝑗𝜑𝑖𝑗𝜑𝑖; where 𝜋 is the number of phases at equilibrium and 𝜃𝑖𝑗 denotes the composition (i.e., 𝑥 or 𝑛) or thermodynamic property of component 𝑖 in phase 𝑗	min𝛽𝑔0≤𝛽𝑖𝑗≤1𝑖=1,…,𝑐𝑗=1,…,𝜋−1.; The decision variables are 𝛽𝑖𝑗∈(0,1) using the following relationships: 𝑛𝑖1=𝛽𝑖1𝑧𝑖𝑛𝐹𝑖=1,...,𝑐; 𝑛𝑖𝑗=𝛽𝑖𝑗𝑧𝑖𝑛𝐹−∑^𝑗−1𝑚=1𝑛𝑖𝑚𝑖=1,…,𝑐;𝑗=2,…,𝜋−1; 𝑛𝑖𝜋=𝑧𝑖𝑛𝐹−∑^𝜋−1𝑚=1𝑛𝑖𝑚𝑖=1,...,𝑐
Reactive phase equilibrium Calculations	It involves the determination of the number, type and composition of the phases at equilibrium at the given operating conditions and subject to element/mass balances and chemical equilibrium constraints.	Gibbs free energy of mixing defined using reaction equilibrium constants [2]; 𝐺𝐾=𝑔−∑^𝜋𝑗=1ln𝐊eq𝐍^−1𝐧ref,𝑗; where 𝑔 is the Gibbs free energy of mixing, ln𝐊eq is a row vector of logarithms of chemical equilibrium constants for 𝑟 independent reactions, 𝐍 is an invertible, square matrix formed from the stoichiometric coefficients of a set of reference components chosen from the 𝑟 reactions, and 𝐧ref is a column vector of moles of each of the reference components	min𝑛𝑖𝑗𝐺𝐾; subject to; ^𝜋∑𝑗=1𝑛𝑖𝑗−𝐯𝑖𝐍^−1𝐧ref,𝑗=𝑛𝑖𝐹−𝐯𝑖𝐍^−1𝐧ref,𝐹; 𝑖=1,…,𝑐−𝑟; 𝑛𝑖𝑗>0𝑖=1,…,𝑐𝑗=1,…,𝜋; where 𝑛𝑖,𝐹 is the initial moles of component 𝑖 in the feed, 𝐯𝑖 is the row vector (of dimension 𝑟) of stoichiometric coefficients of component 𝑖 in 𝑟 reactions, and 𝑛𝑖𝑗 is the number of moles of component 𝑖 in phase 𝑗. The constrained global optimization problem can be solved by minimizing 𝐺𝐾 with respect to 𝑐(𝜋−1)+𝑟 decision variables 𝑛𝑖𝑗. In this formulation, the mass balance equations are rearranged to reduce the number of decision variables of the optimization problem and to eliminate equality constraints