Nonlinear algebraic systems complex solutions

Kunii and Levenspiel(1991, pp. 294-298) extend the bubbling-bed model to networks of first-order reactions and generate rather complex algebraic relations for the net reaction rates along various pathways. As an alternative, we focus on the development of the basic design equations, which can also be adapted for nonlinear kinetics, and numerical solution of the resulting system of algebraic and ordinary differential equations (with the E-Z Solve software). This is illustrated in Example 23-4 below. 

Using these methods to describe an aqueous electrolyte system with its associated chemical equilibria involves a unique set of highly nonlinear algebraic equations for each set of interest, even if not incorporated within the framework of a complex fractionation program. To overcome this difficulty, Zemaitis and Rafal (8) developed an automatic system, ECES, for finding accurate solutions to the equilibria of electrolyte systems which combines a unified and thermodynamically consistent treatment of electrolyte solution data and theory with computer software capable of automatic program generation from simple user input. 

Since the CSTR equation is simply a system of nonlinear algebraic equations, it is possible to obtain multiple CSTR steady states for a fixed feed concentration and residence time. For example, if there are polynomial terms in r(C), then more than one concentration will exist as roots to the equation. Even if these roots do not bear any physical significance to the system, they must still be known in order for the appropriate construction of the AR to be carried out. More complex expressions are also valid (and common) in modern-day rate expressions. Be wary of this when attempting to solve for CSTR solutions. The presence of multiple steady states presents... 

Some simple reaction kinetics are amenable to analytical solutions and graphical linearized analysis to calculate the kinetic parameters from rate data. More complex systems require numerical solution of nonlinear systems of differential and algebraic equations coupled with nonlinear parameter estimation or regression methods. 

Nonlinear differential and/or algebraic equations cannot, in general, be solved analytically, and computer-aided numerical solutions are required. Numerical solutions are also preferred for the equations which can be solved analytically, when the analytic solutions are very complex and provide little insight in the behavior of the system. 

Formally, the nonlinear reconciliation problem can be stated in the same manner as in the linear case see Section 10.1. As the examples in the section show the problem can be, however, not well-posed. It can happen that the solution does not exist, or an effective way of finding a solution would require an algebraic pre-treatment of a complex system, not feasible in practice. 

The holdup effects can be neglected in a number of cases where this model approximates the column behavior accmately. This model provides a close approximation to the Rayleigh equation, and for complex systems (e.g., azeotropic systems) the synthesis procedures can be easily derived based on the simple distillation residue curve maps (trajectories of composition). However, note that this model involves an iterative solution of nonlinear plate-to-plate algebraic equations, which can be computationally less efficient than the rigorous model. 

---