A rapid and robust method for solving the Rachford–Rice equation using convex transformations

The Rachford–Rice equation must be solved in an inner loop of flash calculations. It consists in a sum of hyperbolas, and the desired root is located between two asymptotes which define the negative flash window. First, a simple procedure to remove the adjacent poles by changing the variable is proposed. Then, a class of convex transformations is identified, and several elementary functions that respect the general convexity condition are recognized. A new extremely simple solution window is proposed (depending only on compositions of the lightest and heaviest components), as well as an initialization procedure. Based on several convex formulations, two solution algorithms are proposed, based on the selection of the appropriate convex function (such that the monotonic convexity condition is fulfilled), and several Newton methods (including higher order) can be used without any control (it is guaranteed that an overshoot of the solution cannot occur). The transformations to make the functions convex within the search region, enable the use of Newton-type updates without the need for bounds checking and interval reduction lead to significant increase in speed of solution. The proposed algorithms are simple, very easy to implement, robust and rapid; as shown by various test examples (for selected very difficult cases and for a very large number of randomly generated flashes), only few iterations are required for convergence and the proposed method compares favorably to previous solution methods.