Difference point equation numerical integration

To demonstrate the superior dynamic controllability of high-conversion and low-temperature designs, the nonlinear differential equations are numerically integrated for the four different design cases. Disturbances in feed flowrate, temperature controller set-point, and overall heat-transfer coefficient are made, and the peak deviations in reactor... 

It is useful at this point to note that the Newton forward difference formula is utilized here for the development of the numerical integration formula, while the Newton backward difference formula was previously used (in Chapter 7) for the integration of ordinary differential equations of the initial value type. 

Under such simplified description of the streamer development, we could model the subsequent evolution of the gas phase in a standard way, using the continuity equations for each chemical species and solving a system of mono-dimensional first-order differential equations easily and quickly tackled by numerical integration (Riccardi, 2000). From a chemical engineering point of view, indeed it means that the model can be formulated as a well-mixed reactor (Benson, 1982). The gas-phase composition in the reactor is determined by the chemical reactions among the reactive species and the transport processes. The time evolution of the concentration of the different N sp>ecies in the gas phase is determined by integrating each balance equation for the density nk of the Id spiecies ... 

The runaway limits determined by Morbidelli and Varma [1982] are based on the occurrence of an inflection point in the temperature profile before the hot spot. They used the method of isoclines, which requires the numerical integration of a differential equation. The method is also applicable to reaction orders different from 1, as shown in Fig. 11.5.3-1. The runaway region becomes more important as the order decreases. Tjahjadi et al. [1987] developed a new approach, applicable to more complex reactions, for example, the radical polymerization of ethylene in a tubular reactor. Hosten and Froment [1986]... 

Other popular methods for numerical solutions of DEs are the Runge-Kutta methods. They again come in forms of different order, depending on the number of selected points on each sub-interval for which the function is evaluated and averaged. The development of these methods includes quite sophisticated analyses of errors (deviations from the true solutions) which occur with functions of different properties. A major problem in the numerical integration of rate equations is stiffness. A differential equation is called stiff if, for instance, different st s in the process occur on widely different time scales. It is very in dent to compute with time intervals suitable for the steepest part of the progress curve (see Press et al., 1986, chapter 16 and commercial programs recommended on p. 36). 

The partial-differential equations (3) are solved numerically through finite difference approximation for the spatial derivatives and the method of line for time advancement. The model medium is represented by a discretized line with a resolution from 50 up to 200 points. The resulting set of ordinary differential equations is integrated with a stiff ODE solver [85]. Care is taken to vary the spatio-temporal resolution in order to check the reliability of the reported phenomena. 

The calculus of finite differences may be characterized as a two-way street that enables the user to take a differential equation and integrate it numerically by calculating the values of the function at a discrete (finite) number of points. Or, conversely, if a set of finite values is available, such as experimental data, these may be differentiated, or integrated, using the calculus of finite differences. It should be pointed out, however, thatnumerical differentiation is inherently less accurate than numerical integration. 

The weak point in the kinetic concept of Morawetz et al. was the assumption that cyclization of oligomers and polymers do not need to be considered. This short-coming was revised in the work of Mandolini et al. [13-15] who demonstrated that the cyclization factor C of the monomer (Mi) depends on the cyclization factors of the oligomers and vice versa. However, the main purpose of their work was different and defined as follows We now describe a more refined approximation treatment, where the formation of both, linear and cyclic oligomers with DP s up to 12 is taken into account. The procedure involves the micro-computer-assisted numerical integration of the proper system of differential rate equations by the simple Euler method [16] . 

The numerical integrations are performed by means of finite-difference techniques using a rectangular grid in the (co, t) plane having (nonuniform) intervals Sco and St, We denote by the vector of dependent variables at grid point n St)J S(o), The equations to be integrated [(4.12) and (4.13)] may then be written as the matrix equation... 

Thus, exact or integrable Pfaffians lead to non-intersecting solution surfaces, which requires that solution curves that lie on different solution surfaces cannot intersect. For a given point p. there will be numerous other points in very close proximity to p that cannot be connected to p by a solution curve to the Pfaffian differential equation. No such condition exists for non-integrable Pfaffians, and, in general, one can construct a solution curve from one point to any other point in space. (However, the process might not be a trivial exercise.)... 

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