Solution procedures for nonideal equilibrium stage processes at steady and unsteady state described by algebraic or differential-algebraic equations

Abstract

The Newton-Raphson procedure is used to solve the model equations for a single column and a system of two columns in an extractive distillation process at steady state. The model for the singe column is used as the unit module to solve the extractive system by the Capital Theta method of convergence. The two techniques for solving the system are compared. Broyden's and Schubert's quasi-Newton techniques are compared by application to a distillation problem. An algorithm to apply Broyden's original method in a sparsity preserving manner is presented. The steady state column model is then extended to the unsteady state and used to evaluate the assumption of constant molar holdup in both a distillation column and an adiabatic flash process. Gear's procedures are used to solve the resulting sets of differential and algebraic equations after employing a substitution to put the equations in a form suitable for the procedure. The importance of correct determination of the 0^- and 0^+ values of the independent variables before applying an integration package is demonstrated. The model of the distillation column is then expanded to include tray hydraulic functions and reboiler, condenser, and control system dynamic equations. The resulting closed loop equations are analyzed and a matrix technique proposed to solve the linear equations generated by the convergence procedure in Gear's method. The impact of the control system equations on Jacobian matrix structure and the classification of equation entities into forcing functions and independent variable is emphasized.