Abstract: In this work, we propose a gradient projected Lagrangian algorithm to address the inadequacies of current free-energy minimization-based solution method for calculating the composition of the ideal gas phase equilibrium. In this algorithm, a weak convergence criterion was used as the convergence criterion of the gradient projection method. Mathematical proofs were then provided for the rationality of the weak convergence conditions used in the gradient projection method, as well as the convergence of this algorithm. On this basis, the result calculated by the gradient projection method was used as the initial value in Newton's method with Lagrange multipliers, and improvements were also made on the iteration step used in Newton's method. We have thus resolved the difficulty of selecting an initial value for Newton's method, and improved the stability and computational speed of the algorithm. The results of example calculations demonstrated that the algorithm proposed in this work converges very quickly and exhibits a high level of computational accuracy.

Key words: free-energy minimization methods, ideal gas phase reactions, Claus reaction, Lagrange multiplier, gradient projection method