Integrating differential equations Runge-Kutta method

Starting with an initial value of cA and given c,(t), Eq. (8-4) can be solved for cA(t + At). Once cA(t + At) is known, the solution process can be repeated to calculate cA(t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. As discussed in Sec. 3, more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations ... 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

In chapter 3, the construction of exponential fitting formulae is presented. In chapter 4, applications of exponential fitting to differentiation, to integration and to interpolation are presented. In chapter 5, application of exponential fitting to multistep methods for the solution of differential equations is presented. Finally, in chapter 6, application of exponential fitting to Runge-Kutta methods for the solution of differential equations is presented. 

Schema for the Runge-Kutta Method of Integrating a Pair of First Order Differential Equations...

Webb and Sardesai (1981) used a fourth-order Runge-Kutta method to integrate the differential equations modeling the condenser. Furno (1986) divided the condenser tube into 40 sections. Both investigators used the same method to calculate physical properties. The Fanning friction was calculated from a correlation obtained by Webb and Sardesai... 

EROS handles concurrent reactions with a kinetic modeling approach, where the fastest reaction has the highest probability to occur in a mixture. The data for the kinetic model are derived from relative or sometimes absolute reaction rate constants. Rates of different reaction paths are obtained by evaluation mechanisms included in the rule base that lead to partial differential equations for the reaction rate. Three methods are available that cover the integration of the differential equations the GEAR algorithm, the Runge-Kutta method, and the Runge-Kutta-Merson method [120,121], The estimation of a reaction rate is not always possible. In this case, probabilities for the different reaction pathways are calculated based on probabilities for individual reaction steps. 

The performance of the EMR may be calculated by means of the measured kinetics and the simultaneous calculation of mass balances of each reactant. The steady-state parameters of the reactor can be estimated by numerical integration of the differential mass-balance equations by means of the Runge-Kutta method. 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

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