Nonlinear Equation Solver Program

The sequence of equations presented below is required to solve the isothermal gas-liquid CSTR problem for the chlorination of benzene in the liquid phase at 55°C. After some simplifying assumptions, the problem reduces to the solution of nine equations with nine unknowns. Some of the equations are nonlinear because the chemical kinetics are second-order in the liquid phase and involve the molar densities of the two reactants, benzene and chlorine. The problem is solved in dimensionless form with the aid of five time constant ratios that are generated by six mass transfer rate processes (1) convective mass transfer through the reactor, (2) molecular transport in the liquid phase across the gas-liquid interface for each of the four components, and (3) second-order chemical reaction in the liquid phase. 

Inlet molar density of liquid benzene as a pure component, g-mol/mL  

Factor by which pure liquid benzene is diluted in the liquid feed stream, 1  

Second-order kinetic rate constant in the liquid phase (mL/g-mol s)  

Diameter of the gas bubbles that rise through the CSTR s liquid phase (mm)  

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NONLINEAR EQUATION SOLVER PROGRAM Liquid-phase diffusivity of benzene at 55°C (cm /s) ... 

Alternatively, all thej model equations may be listed singly, and then solved simultaneously using a standard nonlinear equation solver, such as a spreadsheet program. For the two-component system, the equations include the van Laar equations for both components in each liquid phase ... 

Q Linear Programming, Nonlinear Programming, Nonlinear Equations, and Nonlinear Regression Using Solver... 

In this example, you have two iterations - one because of the circular reference due to the recycle streams and one because of the nonlinear Rachford-Rice equation. Some computer programs cannot handle both of these complications together. Neither Goal Seek nor Solver worked for this example, and you iterated the vapor fraction by hand. 

The Matlab Simulink Model was designed to represent the model stmctuie and mass balance equations for SSF and is shown in Fig. 6. Shaded boxes represent the reaction rates, which have been lumped into subsystems. To solve the system of ordinary differential equations (ODEs) and to estimate unknown parameters in the reaction rate equations, the inter ce parameter estimation was used. This program allows the user to decide which parameters to estimate and which type of ODE solver and optimization technique to use. The user imports observed data as it relates to the input, output, or state data of the SimuUnk model. With the imported data as reference, the user can select options for the ODE solver (fixed step/variable step, stiff/non-stiff, tolerance, step size) as well options for the optimization technique (nonlinear least squares/simplex, maximum number of iterations, and tolerance). With the selected solver and optimization method, the unknown independent, dependent, and/or initial state parameters in the model are determined within set ranges. For this study, nonlinear least squares regression was used with Matlab ode45, which is a Rimge-Kutta [3, 4] formula for non-stiff systems. The steps of nonlinear least squares regression are as follows ... 

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