Integration step size

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. 

Results of calculations for Sc = 1000 are given in Table 5.4-23. The total stream of B added was divided into 20 portions (N = 20). Each CPU time was less than 5 s at an integration step size of lO ". The formation of the bisazo product is promoted by poor mixing (low values of E, and, consequently, high values of Da) and high volume ratio of reagent solutions Va/Vb-... 

Effect of integration step size with Euler integration. 

Figure 5.3 shows results for a step change in the disturbance of 0.2 at time equal zero. An integration step size of 0.1 min is used. We will return to this simple system later in this book to discuss the selection of values for and r, that is, how we tune the controller. . 

The explicit Euler integration technique involves specifying the integration step size, At ... 

This method often requires very small integration step sizes to obtain a desired level of accuracy. Runge-Kutta integration has a higher level of accuracy than Euler. It is also an explicit integration technique, since the state values at the next time step are only a function of the previous time step. Implicit methods have state variable values that are a function of both the beginning and end of the current... 

Most commonly used ordinary differential equation (ODE) solvers provide options of several different integration techniques. Most solvers also automatically vary the integration step size during the simulation to allow the best trade-off between accuracy and solution time, based on user-specified numerical tolerances. There is no single best integration technique—different methods work better for various problems. 

To assure sufficient accuracy of the numerical integration, we repeatedly halved the integration step size until no significant difference (0.1% ) in solutions occurred. For the fourth-order Runge-Kutta method this required a step size of. 04 sec (43 steps). An upper bound of the total error introduced by the numerical integration procedure can be obtained for the Runge-Kutta method (29). At a step size of 0.04 sec, the error estimate calculated is 0.004% (14). 

Here r = hk Jhk, where hj = xJ+1 —Xj is the integration step size. Two variants are presented, as well as a simple Euler formula suitable for starting an integration series. 

Figure 8.43 (a) Setting integrator step size, (b) Effect of integrator step. 

Set integration step size Go to Run in the toolbar. In the Run menu click on solver options, which opens the Solver Properties window. In the Integrator tab, pick the hiplicit Euler method, and click on the Fixed radio button. Typically use a step size that is approximately (run time)/200. Might try 0.025 as a first try. For now, ignore the other tabs. Click OK. 

This involves the knowledge of the order of the discontinuity. Numerical approaches to determine this order work only reliable for orders 0 and 1 [G084]. We therefore suggest to restart the integration method for safety reasons. When restarting a multistep method, much time is spent for regaining the appropriate integration step size and order. The best one can do is to use the information available from the last time interval before the discontinuity in the case of the BDF method for... 

Figure C.l shows the Simulink diagram for Eq. C-1 (transport delay 1 = 5 for both models). To generate a transient response, the simulation menu is selected to allow parameters for the simulation to be specified (start time, stop time, integration routine, maximum integration step size). Numerical values of time t are entered into the input-output data set via a clock block. After the simulation has been completed, the resulting data can be plotted (see Fig. C.2), manipulated, and analyzed from the MATLAB command window.

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