Numerical method order

The following examples review some complex reactions and determine the concentrations history for a specified period using the Runge-Kutta fourth order numerical method. 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

Equation 5-247 is a polynomial, and the roots (C ) are determined using a numerical method such as the Newton-Raphson as illustrated in Appendix D. For second order kinetics, the positive sign (-r) of the quadratic Equation 5-245 is chosen. Otherwise, the other root would give a negative concentration, which is physically impossible. This would also be the case for the nth order kinetics in an isothermal reactor. Therefore, for the nth order reaction in an isothermal CFSTR, there is only one physically significant root (0 < C < C g) for a given residence time f. 

In practice the finite-field calculation is not so simple because the higher-order terms in the induced dipole and the interaction energy are not negligible. Normally we use a number of applied fields along each axis, typically multiples of 10 " a.u., and use the standard techniques of numerical analysis to extract the required data. Such calculations are not particularly accurate, because they use numerical methods to find differentials. 

Many ab initio packages use the two key equations given above in order to calculate the polarizabilities and hyperpolarizabilities. If analytical gradients are available, as they are for many levels of theory, then the quantities are calculated from the first or second derivative (with respect to the electric field), as appropriate. If analytical formulae do not exist, then numerical methods are used. 

Semi-open formulas are used when the problem exists at only one limit. At the closed end of the integration, the weights from the standard closed-type formulas are used and at the open end, the weights from open formulas are used. (Weights for closed and open formulas of various orders of error may be found in standard numerical methods texts.) Given a closed extended trapezoidal rule of one order higher than the preceding formula, i.e.. 

Norris et al. [1254] discuss the application of several numerical methods to the determination of rate coefficients and of orders of solid state reactions of the contracting interface type. 

In order to see how accurate this perturbation treatment actually is, we have substituted numerical values for the S s directly into the secular equation, and then solved it rigorously by numerical methods. The calculations are not given in detail, since they are quite straightforward and proceed along well-known lines. The results are shown in Table I. 

A solution to Equation (8.12) together with its boundary conditions gives a r, z) at every point in the reactor. An analytical solution is possible for the special case of a first-order reaction, but the resulting infinite series is cumbersome to evaluate. In practice, numerical methods are necessary. 

Unsteady behavior in an isothermal perfect mixer is governed by a maximum of -I- 1 ordinary differential equations. Except for highly complicated reactions such as polymerizations (where N is theoretically infinite), solutions are usually straightforward. Numerical methods for unsteady CSTRs are similar to those used for steady-state PFRs, and analytical solutions are usually possible when the reaction is first order. 

RANS codes were not unsuccessful for the study of piston engines [25-27]. However, it is only with LES [30], for example, that the study of cycle-to-cycle variations becomes possible. For such studies, the solver must have moving-grid capabilities for the piston and the valves, while retaining all the required properties for LES, such as a high-order numerical method. From the point of view of modeling, the combustion model must handle... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

The experimental results imply that the main reaction (eq. 1) is an equilibrium reaction and first order in nitrogen monoxide and iron chelate. The equilibrium constants at various temperatures were determined by modeling the experimental NO absorption profile using the penetration theory for mass transfer. Parameter estimation using well established numerical methods (Newton-Raphson) allowed detrxmination of the equilibrium constant (Fig. 1) as well as the ratio of the diffusion coefficients of Fe"(EDTA) andNO[3]. 

Numerical methods. Computer-intensive numerical methods like quantum mechanics, molecular mechanics, or distance geometry [8] do not normally fall into the scope of automatic model builders. However, some model builders have built-in fast geometry optimization procedures or make use of distance geometry in order to generate fragment conformations. 

Again, these functional relationships should ideally be available in an explicit form in order to ease the numerical method of solution. Two-solute batch extraction is covered in the simulation example TWOEX. 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

One may clearly extend the technique to include as many reactions as desired. The irreversibility of the reactions permits one to solve the rate expressions one at a time in recursive fashion. If the first reaction alone is other than first-order, one may still proceed to solve the system of equations in this fashion once the initial equation has been solved to determine A(t). However, if any reaction other than the first is not first-order, one must generally resort to numerical methods to obtain a solution. 

For the case where all of the series reactions obey first-order irreversible kinetics, equations 5.3.4, 5.3.6, 5.3.9, and 5.3.10 describe the variations of the species concentrations with time in an isothermal well-mixed batch reactor. For series reactions where the kinetics do not obey simple first-order or pseudo first-order kinetics, the rate expressions can seldom be solved in closed form, and it is necessary to resort to numerical methods to determine the time dependence of various species concentrations. Irrespective of the particular reaction rate expressions involved, there will be a specific time... 

For constant-separation factor systems, the /(-I rails formal ion of Helfferich and Klein (gen. refs.) or the method of Rhee et al. [AlChE J., 28, 423 (1982)] can be used [see also Helfferich, Chem. Eng. Sci., 46, 3320 (1991)]. The equations that follow are adapted from Frenz and Horvath [AlChE ]., 31, 400 (1985)] and are based on the h I ransiomialion. They refer to the separation of a mixture of M — 1 components with a displacer (component 1) that is more strongly adsorbed than any of the feed solutes. The multicomponent Langmuir isotherm [Eq. (16-39)] is assumed valid with equal monolayer capacities, and components are ranked numerically in order of decreasing affinity for the stationary phase (i.e., Ki > K2 > Km). 

In order to control the air supply of what is known as an atmospheric appliance, the main challenge lies in finding a low-cost and robust actuator to achieve an optimal air ratio. Numerous methods are mentioned in the relevant literature and in patent specifications. Unfortunately, most of these concepts are far too costly and elaborate to be used in serial production. Despite these difficulties there have been some promising approaches. 

The turbulence models discussed in this chapter attempt to model the flow using low-order moments of the velocity and scalar fields. An alternative approach is to model the one-point joint velocity, composition PDF directly. For reacting flows, this offers the significant advantage of avoiding a closure for the chemical source term. However, the numerical methods needed to solve for the PDF are very different than those used in standard CFD codes. We will thus hold off the discussion of transported PDF methods until Chapters 6 and 7 after discussing closures for the chemical source term in Chapter 5 that can be used with RANS and LES models. 

Fractional time stepping is widely used in reacting-flow simulations (Boris and Oran 2000) in order to isolate terms in the transport equations so that they can be treated with the most efficient numerical methods. For non-premixed reactions, the fractional-time-stepping approach will yield acceptable accuracy if A t r . Note that since the exact solution to the mixing step is known (see (6.248)), the stiff ODE solver is only needed for (6.249), which, because it can be solved independently for each notional particle, is uncoupled. This fact can be exploited to treat the chemical source term efficiently using chemical lookup tables. 

In summary, DQMOM is a numerical method for solving the Eulerian joint PDF transport equation using standard numerical algorithms (e.g., finite-difference or finite-volume codes). The method works by forcing the lower-order moments to agree with the corresponding transport equations. For unbounded joint PDFs, DQMOM can be applied... 

In developing the equations governing the thermal and diffusional processes, Hirschfelder obtained a set of complicated nonlinear equations that could be solved only by numerical methods. In order to solve the set of equations, Hirschfelder had to postulate some heat sink for a boundary condition on the cold side. The need for this sink was dictated by the use of the Arrhenius expressions for the reaction rate. The complexity is that the Arrhenius expression requires a finite reaction rate even at x = —°°, where the temperature is that of the unbumed gas. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

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