Runge-Kutta Equation

Figure 3-30. Excel spreadsheet for the numerical integration of the rate law for the reaction 2A, f s B using 4th order Runge-Kutta equations.

The Rimge-Kutta methods for numerical solution of the differential equation dy/dx = F(x, y) involve, in effect, the evaluation of the differential function at intermediate points between xi and Xj+i. The value of yi+ is obtained by appropriate summation of the intermediate terms in a single equation. The most widely used Runge-Kutta formula involves terms evaluated at X(, Xj + Ax/2 and X + Ax. The fourth-order Runge-Kutta equations for dy/dx = F(x, y) are... 

The radial stress (7 in Equation 7 can be found by numerical integration techniques, e.g. the Runge-Kutta equation as follows ... 

Equation 16 can also be written as the ordinary form in Equation 9. Accordingly, the value at r < Rp can be calculated from the Runge-Kutta equation in Equation 10. 

The reaction rate equations give differential equations that can be solved with methods such as the Runge-Kutta [14] integration or the Gear algorithm [15]. 

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. 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

Therefore, the slope of the linear plot Cg versus gives the ratio kj/kj. Knowing kj -i- kj and kj/kj, the values of kj and kj ean be determined as shown in Figure 3-10. Coneentration profiles of eom-ponents A, B, and C in a bateh system using the differential Equations 3-95, 3-96, 3-97 and the Runge-Kutta fourth order numerieal method for the ease when Cgg =Cco = 0 nd kj > kj are reviewed in Chapter 5. 

Coneentration profiles ean he developed witli time using tlie differential Equations 3-180, 3-181, and 3-182, respeetively, with the Runge-Kutta fouith order mediod at known values of kj and kj for a hateh system. 

Cbo = Cco = 0- For known values of kj and kj, simulate die eoneen-tradons of A, B, and C for 10 minutes at a dme interval of t = 0.5 min. A eomputer program has been developed using die Runge-Kutta fourdi order mediod to determine die eoneentrations of A, B, and C. The differential Equations 5-64, 5-65, and 5-66 are expressed, respeetively, in die form of X-arrays and funetions in die eomputer program as... 

Equations 5-81, 5-82 and 5-83 are first order differential equations that ean be solved simultaneously using the Runge-Kutta fourth order method. Consider two eases ... 

Equations 5-88, 5-89, and 5-90 are first order differential equations and the Runge-Kutta fourth order method with the boundary eonditions is used to determine the eoneentrations versus time of the eomponents. 

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. 

Equations 5-118, 5-120, 5-121, and 5-122 are first order differential equations. A simulation exereise on the above equations using the Runge-Kutta fourth order method, ean determine the numher of moles with time inerement h = At = 0.2 hr for 2 hours. Computer program BATCH58 evaluates the numher of moles of eaeh eomponent as a funetion of time. Table 5-7 gives the results of the simulation, and Eigure 5-17 shows the plots of the eoneentrations versus time. 

The differential Equations 5-127 and 5-133 are solved using the Runge-Kutta fourth order while eomponents C and D are ealeulated from the mass balanee of Equations 5-130 and 5-132, respeetively, at a time inerement of h = At = 0.5 min. The eomputer program BATCH59 ealeulates the eoneentrations of A, B, C, and D as funetion... 

Equations 5-146, 5-149, and 5-152 are first order differential equations. The eoneentration profiles of A, B, C, and the volume V of the bateh using Equation 5-137 is simulated with respeet to time using the Runge-Kutta fourth order numerieal method. 

The eomputer program BATCH510 was developed ineorporating Equations 5-156, 5-157, 5-158, and 5-159 in the subprogram of the Runge-Kutta fourth order program. The results of the simulation are... 

The computer program PLUG51 employing die Runge-Kutta fourdi order numerical mediod was used to determine die conversions and the compositions of die components. Applying die Runge-Kutta mediod, Equations 5-328 and 5-329 in differential forms are... 

Equations 6-66 and 6-67, respectively, are two coupled first order differential equations. This is because dC /dt is a function of T and while dT/dt is also a function of and T. The Runge-Kutta fourth... 

Equations 6-94 and 6-97 are first order differential equations, and it is possible to solve for both the eonversion and temperature of hydrogenation of nitrobenzene relative to the reaetor length of 25 em. A eomputer program PLUG61 has been developed employing the Runge-Kutta fourth order method to determine the temperature and eonversion using a eatalyst bed step size of 0.5 em. Table 6-6 shows... 

Complex reactions require the solution of simultaneous differential equations, and the Runge-Kutta procedure is applicable to these problems. To illustrate the method, Scheme XIV will be used. The rate equations are, in incremental form. 

The error in Runge-Kutta calculations depends on h, the step size. In systems of differential equations that are said to be stiff, the value of h must be quite small to attain acceptable accuracy. This slows the calculation intolerably. Stiffness in a set of differential equations arises in general when the time constants vary widely in magnitude for different steps. The complications of stiffness for problems in chemical kinetics were first recognized by Curtiss and Hirschfelder.27... 

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... 

Runge-Kutta. The mixed first-order and second-order equation... 

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. 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

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