Modeling continuous aqueous two-phase systems for control purposes

A mathematical model was proposed to allow the analysis of steady-state and transient behaviors of single-stage continuous aqueous two-phase systems. Since the complete system of simultaneous equations contains more equations than unknown variables, a program based on the method of least squares was developed to solve the problem. The methodology was tested using a system composed of thaumatin, sodium chloride, and a contaminant protein. A poly(ethylene-glycol)/phosphate salt/water system was selected to isolate the thaumatin. For the steady-state and transient operations, a constrained optimization procedure - from the Matlab Optimization Toolbox (MathWork, Inc.)—was implemented after recasting the system of equations as a minimization problem. Euler’s method was used in the transient case to discretize the differential equations. The steady-state concentrations agreed with published data. An input–output model based on a 4% step change decrease in the inlet stream flow rate showed that output variables such as concentrations of sodium chloride and phosphate salt settled to their final values in different time periods. The proposed analysis may be helpful in the dynamic control of large-scale commercial extractor units using advanced control schemes.