Nonlinear equations numerical solutions

These equations look innocuous, but they are highly nonlinear equations whose solution is almost always obtainable only numerically. The nonlinear terms are in the rate r (Ca, T), which contains polynomials in Ca, especially the very nonlinear temperature dependence of the rate coefficient k(T). For first-order kinetics this is... 

It is proper to emphasize the fact that there is no mathematical way of determining how correct is the numerical solution of a nonlinear differential equation, just by looking at what an algorithm predicts for linear forms of the same problem. Even if a computer program generates accurate numerical solutions which are in agreement with analytical solutions of the linear forms of a nonlinear equation, the solutions in the nonlinear domain may not be correct. 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

An analytical solution to Equation 15.48 can also be obtained for a first-order reaction. The solution is Equation 15.43. Beyond these cases, analytical solutions are difficult since the 5 is usually nonlinear. For numerical solutions. Equation 15.48 can be treated as though it were an initial-value problem. Guess a value for dout = (0). Integrate Equation 15.48. If a k) remains finite at large X, the correct d(0) has been... 

In general we need to solve sets of nonlinear equations numerically, but again the special structure of this problem allows an analytical solution. Notice that we can subtract the material balance for B from the materia) balance for A to obtain... 

For reactions other than first order, the reaction-diffusion equation is nonlinear and numerical solution is required. We will see, however, that many of the conclusions from the analysis of the first-order reaction case still apply for other reaction orders. We consider nth-order, irreversible reaction kinetics... 

The Wegstein method is a secant method applied to g x) — x — F x). Numerical Solution of Simultaneous Nonlinear Equations... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

As mentioned above, the numerical solution of exact equations breaks down for low flame speeds, where the strength of the leading shock approaches zero. To complete the entire range of flame speeds, Kuhl et al. (1973) suggested using the acoustic solutions by Taylor (1946) as presented earlier in this section. Taylor (1946) already noted that his acoustic approach is not fully compatible with the exact solution, in the sense that they do not shade into one another smoothly. In particular, the near-piston and the near-shock areas in the flow field, where nonlinear effects play a part, are poorly described by acoustic methods. In addition to these imperfections, the numerical character of Kuhl etal. (1973) method inspired various authors to design approximate solutions. These solutions are briefly reviewed. 

Only numerical solutions are possible when Equation (9.24) is solved simultaneously with Equation (9.14). This is true even for first-order reactions because of the intractable nonlinearity of the Arrhenius temperature dependence. 

The combined fiuld fiow, heat transfer, mass transfer and reaction problem, described by Equations 2-7, lead to three-dimensional, nonlinear, time dependent partial differential equations. The general numerical solution of these goes beyond the memory and speed capabilities of the current generation of supercomputers. Therefore, we consider appropriate physical assumptions to reduce the computations. 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

Because the term r(CA T) is exponentially dependent on T and can be nonlinear as well, a numerical solution or piecewise linearization must be used. To simplify the numerical manipulations, equations in Table IX are normalized by = z/L, r = ut/L, and jc = 1 - C,/(C,)0, where i is normally S02. y also is a normalized quantity. The Peclet numbers for mass and heat are written PeM = 2Rpu/D) and PeH = 2Rpcpul t for a spherical particle. They are also written in terms of bed length as Bodenstein numbers. It is... 

Model-fitting procedures are usually based on analytical solutions of the model however, model parameters may be estimated by fitting the differential equations describing the model. Since the numerical solution of the differential equations introduces another source of error, fitting of differential equations is usually limited to cases where nonlinearities are present. 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Consequently, modeling of a two-phase flow system is subject to both the constraints of the hydrodynamic equations and the constraint of minimizing N. Such modeling is a nonlinear optimization problem. Numerical solution on a computer of this mathematical system yields the eight parameters ... 

We pose the problem for the remaining equations by specifying the total mole numbers Mw, Mi, and of the basis entries. Our task in this case is to solve the equations for the values of nw, mt, and - The solution is more difficult now because the unknown values appear raised to their reaction coefficients and multiplied by each other in the mass action Equation 4.7. In the next two sections we discuss how such nonlinear equations can be solved numerically. 

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. 

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