Solution of Nonlinear Algebraic Equations

Nonlinear algebraic equations turn up quite frequently in chemical engineering and may appear in several different forms. For example, in thermodynamics, pressure-volume-temperature relatimiships of real gases are often described by equations of state, such as 

In multicomponent distillation, it may be necessary to estimate the minimum reflux ratio using classical methods [1,2]. This estimate usually requires the solution of a polynomial in of degree n, such as the equation 

Fanning friction factor,/, for turbulent flow of an incompressible fluid in a smooth pipe is 

The quantities A, B, and k are constants. is the Reynolds number. Equation 9.3 is not in polynomial form, but can be rearranged to have all nonzero terms on one side of the equation. In fact, all three of the equations mentioned can be represented by the general form 

A standard approach rearranges Equation 9.4 to develop a convergence criterion [3]. 

A binary search is a robust and easily implemented method for finding a root T of a single equation F T) = 0. It is necessary to know boimds T m T within which the root exists. If F(Tmm) and F Tra id differ in sign, there will be an odd number of roots within the bounds and a binary search will find one of them to a specified level of accuracy. It does so by calculating F at the midpoint of the interval, that is, at r = + T - )I2. The sign of F will be the same as at one of the 

Use the binary-search technique to find a zero of a function. 

SOLUTION Code for Example 4.14 works for any arbitrary function, Func T), of the single variable T provided a range on T can be specified in which there is a single zero. The Func (T) used as an example in the code finds a root of an energy balance used in Example 5.7  

Tmax = 450 User supplied value Tmin = 350 User supplied value er = 0.0000005 User supplied value 

If Func(Tmax) Func(Tmin) = 0 Then Stop No root or even number of roots 

The general features of the Newton method are very well known. Nevertheless, it is perhaps worthwhile to offer a very brief review for the scalar case, which is finding a solution to F(y) = 0. The algorithm is stated as 

As illustrated in Fig. 15.5, the initial iterate (point 0) is within the domain of convergence of Newton s method. As a result the iteration converges rapidly. However, imagine the behavior of the algorithm if the starting iterate (initial guess at the solution) were just 

If the system of equations is written in the general vector form 

The iteration procedure continues until the correction becomes negligibly small. The ad- 

One method of improving the convergence properties of Newton s method is to implement a damping strategy [95]. In a damped Newton method, Eq. 15.46 becomes 

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One unresolved question concerns whether it is possible to use volatility parameters as iteration variables in a nonlinear programming algorithm, with an approach similar to that introduced by Boston and Britt (13) for solution of nonlinear algebraic equations involving K-values. Our conclusion is that volatility parameters apply where K-values are used, and would be awkward to use in minimization of Gibbs free energy. 

Appendix A.2 Iterative Solution of Nonlinear Algebraic Equations 611... 

Appendix A.2 Iteradve Solution of Nonlinear Algebraic Equations 621... 

Bain, R. S., Solution of nonlinear algebraic equation systems and single and multiresponse nonlinear parameter estimation problems, Ph. D. thesis, University of Wisconsin-Madison (1993). 

Buzzi-Ferraris, G. and Tronconi, E. (1993) An improved convergence criterion in the solution of nonlinear algebraic equations. Comput Chem. Eng, 17,419. 

Numerical solution of nonlinear algebraic equations with discontinuities. Computers Chemical Engineering, 26,1449. 

R. J. Spiteri and Thian-Peng Ter, pythNon A PSE for the Numerical Solution of Nonlinear Algebraic Equations, JNAIAM J. Numer. Anal. Indust. Appl. Math, 2008, 3, 123-137. 

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