Runge-Kutta technique

Solution of Batch-Mill Equations In general, the grinding equation can be solved by numerical methods—for example, the Luler technique (Austin and Gardner, l.st Furopean Symposium on Size Reduction, 1962) or the Runge-Kutta technique. The matrix method is a particiilarly convenient fornmlation of the Euler technique. 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

The computer model consists of the numerical integration of a set of differential equations which conceptualizes the high-pressure polyethylene reactor. A Runge-Kutta technique is used for integration with the use of an automatically adjusted integration step size. The equations used for the computer model are shown in Appendix A. 

We also use a restricted form of Equation 19 for the kinetics studies described previously. Smog chamber analyses uses just the first and last terms so that they depend on ordinary differential equations. These are solutions which describe the time-dependent behavior of a homogeneous gas mixture. We used standard Runge-Kutta techniques to solve them at the outset of the work, but as will be shown here, adaptations of Fade approximants have been used to improve computational efficiency. 

Equations 24.11-24.16 can either be solved as such or can be nondimensiona-lized and then solved. Solutions can be obtained by the IMSL subroutine DBVFD along with a third-order Runge-Kutta technique. Experimental data on the dehydrogenation of ethane (Champagnie et al., 1992) reasonably uphold the predicted ethane profiles in both the tube and shell sides. 

The Runge-Kutta technique is different from the predictor-corrector method in the sense that, instead of using a number of points from backward it uses a number of functional evaluations between the current point t and the desired point f + i. 

This differential equation was integrated using the Runge-Kutta technique with initial conditions /=0 at r=0. This returned a set of data points for r, / that were then curve-fitted to produce a microgravity growth rate equation, I = -2.60 10 + 5.01 10" t + 6.56 10", shown graphically in Figure 4. 

The numerical solution of simultaneous, ordinary differential equations can be accomplished with a number a standard mathematical packages. Appendix 7-A.2 illustrates the use of a spreadsheet to solve these equations, via a fourth-order Runge-Kutta technique. The result is z - 0.0246 gcat-min/1. 

The following spreadsheet shows how the fourth-order Runge-Kutta technique can he used to solve Example 7-2. After entering the values of the known parameters (Cao, Cbo, k, andk2) into the spreadsheet, a step size h = Ax was selected. The value of h was chosen arbitrarily to be 4 min, so that the interval between the reactor inlet (t = 0) and the reactor outlet (t = 40) was divided into 10 slices. The corresponding values of x were entered in the first column of the spreadsheet. The second column was set up to contain values of Cb. The initial value of Cb (0.10 mol/1) was entered into the first row of this column. The other values were calculated as described below. 

The following spreadsheet shows how the Runge-Kutta technique for simultaneous differential equations can be used to solve Example 7-4. 

Equation (8-26) now can be solved numerically. It could be solved directly as a differential equation, using any available program, including the techniques described in Chapter 7. Use of the fourth-order Runge-Kutta technique in EXCEL to solve Eqn. (8-26) is illustrated in Appendix 8-A. The numerical solution gives XA,f = 0.70 whenr = 1.5 h. Direct solution ofthe differential equation is straightforward because the final value of the independent variable, time, is known. 

Equations (4.4.1) and (4.4.4) can be solved numerically using the Runge-Kutta technique when n is known. In order to do so, the mass transfer problem in the film is first solved. This is discussed in Appendix 4.1, where an analytical solution is developed using a similarity transformation. From these results, it is possible to prepare a computer program that gives n . 

The results of simulation have been plotted in Figure 6.4 for [M] and T, Figure 6.5 for Apo and and Figure 6.6 for and A, 2- We have also calculated results using the Runge-Kutta technique and compared the computations in these figures. We find that the results fi om the semianalytical technique arc inherently stable, whereas those from Runge-Kutta require excessive computational time. For certain choices of At, there is a numerical overflow. Results from both of these techniques are identical until the thermal runaway conditions are encountered. 

In this chapter, we have considered the reaction engineering of chain-growth polymerization. In order to manufacture polymers of desired physical and mechanical properties, the performance of the reactors must be closely controlled. To do this, various transport equations governing their performance must be established, which, in principle, can be solved numerically. The usual Runge-Kutta technique takes considerable computational time and, at times, gives numerical instability. To overcome aU of these problems, a semianalytical approach can be used. 

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