Nonhomogeneous systems

The two equations, [53] and [55], form a system of coupled ODEs with the variable z playing the role of the independent variable. Given initial conditions at a point Zq these equations can be solved by standard numerical routines such as those discussed in the previous section. Because much computational effort is required to evaluate each p, at each increment of the independent variable z, a method that does not require too many evaluations of the right hand side of the iterative equation is desirable. Usually, a simple forward Euler routine is quite adequate for these purposes. If a multistep algorithm is used, the Adams-Bashforth method has been recommended by Kubicek and Marek the first-order Adams-Bashforth algorithm is, in fact, equivalent to the simple forward Euler algorithm. 

Computer codes for implementation of the continuation method are available in several places. A self-contained program known as AUTO has been developed by Eusebius Doedel and is available from the Applied Mathematics Department, California Institute of Technology, Pasadena. Alternatively, FORTRAN code for a similar routine is given in an appendix of the recent book by Marek and Schreiber.22 

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If these properties vary continuously over the entire mass in the system in some simple way, the resulting problems may be solved analytically, but most often the mathematics are too difficult. However, many complex systems can be conceptually divided into a group of subsystems, each of which is homogeneous. Then for a nonhomogeneous system made up of homogeneous subsystems, we can rewrite Eq. 4.34 as... 

In this next example, we will solve a nonhomogeneous system of two equations. Consider... 

The constant of proportionality D which appears in (5.9) and (5.10) is called the diffusion coefficient. In general it will be a function of the nuclear properties of the medium (and therefore a spatially dependent function in nonhomogeneous systems) and of the neutron speed. If we define the net neutron current in the units neutrons per unit area per unit time, it follows that the diffusion coefficient D has the units of length. This definition differs from the one customarily used in gas-diffusion problems. In the gas problems the coefficient D has the units length squared divided by time." Thus if the current of gas particles is defined analogously to the neutron problem, D = vD. Evidently the particle speed v is absorbed into the definition of the proportionality constant in the gas problem. 

A second general approach to the problem of determining the angular distribution of the neutron flux may be developed by utilizing the methods of integral equations. These methods are generally quite powerful, and formal solutions are readily derived even for nonhomogeneous systems with complex source distributions. Unfortunately, the mathematical techniques required for these applications are of a somewhat sophisticated... 

An overall RTD of a single CSTR requires that a = 3= 0.5. At an activity ratio of p = 9, while still keeping Dai = 9, we get Ab 0.792 and = 0.925. Clearly, for a nonhomogeneous system the knowledge of the RTD is not sufficient in predicting reactor performance even for a first-order process. This has important ramifications in dealing with heterogeneous systems. 

The well-known heterogeneity of amorphous glassy pol5miers plastic deformation [1,2] allows to assume them nonhomogeneous systems. The same affirmation is valid in respect of semicrystalline pol5miers amorphous phase [3,4], As well nevertheless, both models of continues (let us remind, that Ihe known Dugdale model, often used for crazes characteristics, was developed originally for a metals [5]) and molecular concepts, are applied successfully for both classes polymers behavior description. In this cormection ihe question arises about scale, which can be considered as lower boimdary of models of continua applicability. 

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