Successive Substitution Technique

Successive substitution technique. In this technique, through an 

Define a = V(F then Eq. (4.11) becomes the Rachford-Rice (1952) expression. 

Step 1. Guess the initial values of at the fixed temperature and pressure. The Wilson (1969) correlation can be used for this purpose. In this correlation, which is based on the direct application of Raoult s law (see Example 1.7, Chapter 1), the ideal value of component i is given by 

Step 2. Solve Eq. (4.12) for oc. This equation is readily solved by Newton s method, to be discussed shortly. 

Step 3. Calculate Xi andy from Eqs. (4.9) and (4.10), then the compressibility factors of the liquid and gas phases from an EOS. 

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The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

The magnitude of the derivative of the successive substitution function evaluated at the solution point must be less than unity for the error to decrease with each iteration. If the derivative is negative, the calculated values will oscillate about the solution point while if it is positive the calculated values will monotonically approach the solution point. As the magnitude of the derivative approaches unity, the convergence of the technique becomes very slow and the successive substitution technique works best when the substitution function is a slowly varying function of X. ... 

Some formulas, such as equation 98 or the van der Waals equation, are not readily linearized. In these cases a nonlinear regression technique, usually computational in nature, must be appHed. For such nonlinear equations it is necessary to use an iterative or trial-and-error computational procedure to obtain roots to the set of resultant equations (96). Most of these techniques are well developed and include methods such as successive substitution (97,98), variations of Newton s rule (99—101), and continuation methods (96,102). 

The bottom-up method uses the same substitution and expansion techniques, except that now, the operation begins at the bottom of the tree and proceeds up. Equations containing only basic failures are successively substituted for higher faults. The bottom-up approach can be more laborious and time-consuming however, the minimal cutsets are now, 1 obtained for every intermediate fault as well as the top event. 

The application of this procedure to a single tear stream variable is tantamount to solving an equation of the form x = f(x), where f(x) is the function that generates a new value of the tear stream variable x by working around the cycle. Techniques described in Appendix A.2—successive substitution and Wegsiein s algorithm—can be used to perform this calculation. 

A compromise between the explicit and implicit methods is the predictor-corrector technique, where the explicit method is used to obtain a first estimate of and this estimated y +i is then used in the RHS of the implicit formula. The result is the corrected y + i, which should be a better estimate to the true y +i than the first estimate. The corrector formula may be applied several times (i.e., successive substitution) until the convergence criterion (7.49) is achieved. Generally, predictor-corrector pairs are chosen such that they have truncation errors of approximately the same degree in h but with a difference in sign, so that the truncation errors compensate one another. 

This appendix presents a number of solution methods to solve systems of algebraic equations. We will start with the basic techniques, such as bisection and successive substitution, and then discuss one of the most widely used Newton-Raphson methods. These methods are useful in solving roots of... 

The main advantage of the implicit techniques is their stability for any given value of the step size. This advantage, however, requires the solution of a set of nonlinear equations via an iterative approach. For this purpose, methods such as successive substitution or Newton-Raphson can be used [1]. 

Example 1.1 Solution of the Colebrook Equation by Successive Substitution, Linear Interpolation, and Newton Raph on Methods. Develop MATLAB functions to solve nonlinear equations by the successive substitution method, the linear interpolation, and the Newton-Raphson root-finding techniques. Use these functions to calculate the friction factor from the Colebrook equation [Eq. (1.4)] for flow of a fluid in a pipe with e/Z> =10 and Njf, = 10. Compare these methods with each other. 

The method of successive substitutions (also called fixed point iteration) is perhaps the simplest method of obtaining a solution to a nonlinear equation. This technique begins by rearranging the basic F(x) = 0 equation so that the variable x is given as some new function of the same variable x. The original equation is thus converted into an equation of the form ... 

Several methods have received considerable research attention as alternatives to salt curing. These include use of sodium bisulfite as a disinfectant to allow preservation with or without decreased salt in a brine cure use of disinfectants such as quatenary amines for temporary preservation in direct shipping to the taimery from the packing plant (see Disinfectants and antiseptics) preservation of hides by radiation sterilization (see Sterilization techniques) and substitution of materials such as potassium chloride for sodium chloride. These methods have found only limited commercial success. 

If cyclic ketones are monosubstituted in the a-position, their rates of reaction decrease as compared to the rate for the parent ketone (9,41). More highly substituted ketones (e.g., diisobutyl ketone, diisopropyl ketone) can be caused to react using newer preparative techniques (39,43,44, see Section VII). Monosubstituted acetones often can give selfcondensation products, but the recent literature (13,39,43) contains reports of the successful formation of the enamines of methyl ketones. 

Substances, 255q density ofj 14 15 properties of, 13-19 solubility of, 15-17 Substitution reaction, 603 Successive approximations A technique... 

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