Analytical integrals

Two-dimensional models can be used to provide effective approximations in the modelling of polymer processes if the flow field variations in the remaining (third) direction are small. In particular, in axisymraetric domains it may be possible to ignore the circumferential variations of the field unlaiowns and analytically integrate the flow equations in that direction to reduce the numerical model to a two-dimensional form. 

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

For batch operation, the equation for [I] can be derived from an analytical integration of Equation 1. 

Q = Qi (i=l or 2), and W° becomes the total amount of material in the reactor at the start of finishing step. Since the amount of initiator added is insignificant compared to the total reactor charge, it is assumed that the total amount is constant during the finishing process. With this assumption, the analytical integration of Equation 10 is straight forward. 

If the focus is on a wide variety of additives in polyolefins only the OSM/TSM procedure of Marcato and Vianello [210] appears to be far reaching, despite the fact that this total design approach must have required considerable effort in terms of method development. On the other hand, in comparison considerably more method development and optimisation appears to be necessary for SFE of a single polymer/additive matrix, but even then still without the demonstrated benefit of a total approach [322]. Where analyte integrity is of greater concern recourse should be taken to less... 

Note in Table 5.10 that many of the integrals are common to different kinetic models. This is specific to this reaction where all the stoichiometric coefficients are unity and the initial reaction mixture was equimolar. In other words, the change in the number of moles is the same for all components. Rather than determine the integrals analytically, they could have been determined numerically. Analytical integrals are simply more convenient if they can be obtained, especially if the model is to be fitted in a spreadsheet, rather than purpose-written software. The least squares fit varies the reaction rate constants to minimize the objective function ... 

Each centroid potential wfiqf0) as a function of 2 is readily obtained using the analytical expressions of KP1/P20 or KP2/P20. Note that the path integrals for these polynomials have been analytically integrated. 

Since we have seen that the integral cannot be evaluated analytically, there are several alternatives to analytic integration of equation 43-63 we can perform the integration numerically, we can investigate the behavior of equation 43-63 using a Monte-Carlo simulation, or we can expand equation 43-63 into a power series. 

By virtue of the conditions xi+X2 = 1>Xi+X2 = 1, only one of two equations (Eq. 98) (e.g. the first one) is independent. Analytical integration of this equation results in explicit expression connecting monomer composition jc with conversion p. This expression in conjunction with formula (Eq. 99) describes the dependence of the instantaneous copolymer composition X on conversion. The analysis of the results achieved revealed [74] that the mode of the drift with conversion of compositions x and X differs from that occurring in the processes of homophase copolymerization. It was found that at any values of parameters p, p2 and initial monomer composition x° both vectors, x and X, will tend with the growth of p to common limit x = X. In traditional copolymerization, systems also exist in which the instantaneous composition of a copolymer coincides with that of the monomer mixture. Such a composition, x =X, is known as the azeotrop . Its values, controlled by parameters of the model, are defined for homophase (a) [1,86] and interphase (b) copolymerization as follows... 

This value of /B may also be obtained, by means of the E-Z Solve software, by simultaneous solution of equation 22.2-17 and numerical integration of 22.2-13 (with user-defined function fbcr(t, f,), for cylindrical particles with reaction control see file ex22-3.msp). This avoids the need for analytical integration leading to equation 22.2-18. 

Omega Method Model for Compressible Flows The factored momentum balance, Eq. (23-42), can be analytically integrated after first relating the dimensionless specific volume 8 to the dimensionless pressure ratio r. A method to do this, designated the omega method, was suggested by Leung (1986) ... 

When integrating between numerical limits it may be more convenient to proceed directly with a numerical integration rather than through an analytical integration, particularly when the coefficients and integration limits are long numbers, for example, the integral... 

An analytical integration is possible after expansion of the exponential in a... 

The variables are separable, but analytical integration is not possible. A numerical integration will provide the solution B = ftt)... 

This can be substituted into Eq (3), but analytical integrations of the differential equations stop at this point. 

Analytical integration cannot proceed past this stage, so the first three equations are solved numerically by ODE. B displays the characteristic peak of consecutive reactions. 

Analytical integrals are known, but when k is known numerical integration is convenient. 

Although analytical integration is possible, the result is found numerically, f = C/C0 = 0.556 (8)... 

Kramer, M. A., Calo, J. M Rabitz, H., and Kee, R. J., AIM The Analytically Integrated Magnus Method for Linear and Second Order Sensitivity Coefficients. SAND82-8231, Sandia National Laboratories, Livermore, August, 1982b. 

It is possible to construct a modified multivariate density which overcomes this problem. However, the parameters of the model may not be algebraically related to input moments (because the PDF may not be analytically integrated) thus the model cannot be recommended for practical applications [49]. The other... 

Solution of the entire pseudo-steady state problem (commonly referred to as the shrinking core model) is achieved by analytical integration of eqn. (53) and substitution of the result into eqn. (55), subsequently eliminating the unknown Ca by the use of eqn. (54). Substitution into eqn. (56) then gives the overall reaction rate in terms of CAg, and r. This result is not particuleirly useful, however, until the shrinking core radius, r, is related to time. Recalling the chemical stoichiometric relationship [eqn. (50)] the rate of consumption of A in terms of the core radius is... 

Figure 5.6 displays these performance equations and shows that the space-time needed for any particular duty can always be found by numerical or graphical integration. However, for certain simple kinetic forms analytic integration is possible—and convenient. To do this, insert the kinetic expression for in Eq. 

In most biological cases/(C) is nonlinear and analytical integration is difficult or impossible. Numerical integration again allows calculation of concentration at the end of each experiment. Differences between simulated and experimental data are then minimized using, e.g., the least squares criterion. Both experimental approaches are compared in Table 1, especially with respect to their suitability for kinetic screening. 

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