Runge-Kutta routine

The trapezoidal program of CHAPRA CANALE and the Runge-Kutta routine of CONTANTINIDES are compared for the integral... 

The kinetic profiles displayed in Figure 3-32 have been integrated numerically with Matlab s stiff solver odel5s using the rate constants /ci=1000 M 1s 1, /c2=100 s 1 for the initial concentrations [A]o=l M, [Cat]o=10 4 M and [B]o=[C]o=0 M. For this model the standard Runge-Kutta routine is far too slow and thus useless. 

This method generally gives accurate results without requiring an excessively small stepsize At. Of course, some problems are nastier, and may require small steps in certain time intervals, while permitting very large steps elsewhere. In such cases, you may want to use a Runge-Kutta routine with an automatic stepsize control see Press et al. (1986) for details. 

These two non-linear differential equations were solved using a fourth order Runge-Kutta routine with variable step size to ensure accuracy. 

Then T is calculated from Eq. (m) and subsequently r T). Next the integral equation is solved, starting with the assumed Pm as first approximation for the profile inside the particle. The (pm ), obtained from Eqs. (o) is compared with the assumed value. If they do not correspond the procedure described here is repeated. If they do, pco, and T/ are recalculated so that everything is known at z = 0. Then using the Runge-Kutta routine, the values of Pm, pco2> and T at the end of the first increment Az are calculated and from this F,. Then the values of p, Po)t , and so on have to be computed in the way outlined above. Cappelli et aL performed the simulation on a Univac 1108 computer. One such computation... 

Runge-Kutta routine, DiffEq-3D, with stepsize 0.05, available through Ref. 18. 

Figure 24 Period-doubling bifurcations in the Rossier model, Eqs. [106]. For these simulations (computed using the Runge-Kutta routine DiffEq-3D, with stepsize = 0.1, available through Ref. 18), the following parameters were used a = 0.1 and b = 0.2 with c variable. In (a) c = 2.5 ind the initial values were Xq = -2.98, =... 

The set of Equations 25.42-25.48 can be solved provided the following information is available vapor-liquid equilibrium data, for example, the ternary equilibrium data for a typical esterification reaction mass and enthalpy balances around the feed point, reflux inlet, and reboiler to account for the flow rates, compositions, and thermal conditions of the external streams mass transfer coefficients in the absence of reaction (either by experimental determination or estimation from available correlations) liquid holdup (usually from available correlations) and an expression for the reaction rate. Then the equations can be solved by any convenient method, preferably the Runge-Kutta routine, to get the mole fraction of each component as a function of height. 

Equations (6) are solved with a forth-order Runge-Kutta routine combined widt a shooting procedure to End die proper (Lb/Lo)eff. When the potential is low (ty < 5), (L /Lo)e t is obtained in the following way for a solution wiA given ionic strength, we guess a starting value of (L /Lo)eff and obtain initial values of y and diy/9r from (9) and... 

All the equations are thus available to calculate G, L, y. and x. in plane h - Ah using, e.g. the Runge-Kutta routine. In this procedure, the derivatives of y. and x- are calculated, however, without taking into consideration the lvalue of, or change in, other mole fractions. For this reason the mole fractions must be normalized at each step of the calculation. Hence... 

Runge-Kutta routine with automatic step size (to ensure accuracy) to obtain the values of Pc and Sc for the next time step. The differential Eqs. (6.147), (6.148), and (6.149) are integrated using the same routine to obtain the values of Pb, Sj, and X for the next time step. 

Such a set of equations could be integrated by means of Runge-Kutta routines, were it not that with the elementary step and Single Event Kinetics approach described in Chapter 1, it becomes extremely stiff because of the orders of magnitude difference between the concentrations of molecular and radical species. The Gear routine [1971] was developed to cope with such problems. 

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