Hybrid Newton–Successive Substitution Method for Multiphase Rachford-Rice Equations

In multiphase (≥3) equilibrium calculations, when the Newton method is used to solve the material balance (Rachford-Rice) equations, poorly conditioned Jacobian can lead to false convergence. We present a robust successive substitution method that solves the multiphase Rachford-Rice equations sequentially using the method of bi-section while considering the monotonicity of the equations and the locations of singular hyperplanes. Although this method is slower than Newton solution, as it does not rely on Jacobians that can become poorly conditioned, it can be inserted into Newton iterations upon the detection of a poorly conditioned Jacobian. Testing shows that embedded successive substitution steps effectively improved the robustness. The benefit of the Newton method in the speed of convergence is maintained.