Numerical solution different parameters

Numerical solutions of the maximum mixedness and segregated flow equations for the Erlang model have been obtained by Novosad and Thyn (Coll Czech. Chem. Comm., 31,3,710-3,720 [1966]). A few comparisons are made in Fig. 23-14. In some ranges of the parameters n or fte ihe differences in conversion or reaclor sizes for the same conversions are substantial. On the basis of only an RTD for the flow pattern, perhaps only an average of the two calculated extreme performances is justifiable. 

The EPR spectra of cell walls saturated with copper has been fitted to the numerical solutions of the spin hamiltonian describing the EPR lineshape of cupric ions. Two simulations have been performed. The first one (Fig. 4.a) considers that all uronic acids of the cell walls are similar the best fit is rather poor. The second one assumes existence of two populations of exchange sites with different parameters. In this case, the optimization is much better and confirms the existence of two different types of uronic acids in the cell wall (Fig. 4.b). 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

The parameter y reflects the sensitivity of the chemical reaction rate to temperature variations. The parameter represents the ratio of the maximum temperature difference that can exist within the particle (equation 12.3.99) to the external surface temperature. For isothermal pellets, / may be regarded as zero (keff = oo). Weisz and Hicks (61) have summarized their numerical solutions for first-order irreversible... 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

This chapter addresses the planning, design and optimization of a network of petrochemical processes under uncertainty and robust considerations. Similar to the previous chapter, robustness is analyzed based on both model robustness and solution robustness. Parameter uncertainty includes process yield, raw material and product prices, and lower product market demand. The expected value of perfect information (EVPI) and the value of the stochastic solution (VSS) are also investigated to illustrate numerically the value of including the randomness of the different model parameters. 

Figure 1 Electron escape probability as a function of the applied electric field. The solid lines are obtained from Eq. (23) for different values of the initial electron cation distance ro- The broken lines are calculated for ro = 1 nm from the numerical solution of Eq. (16) with the sink term given by k r) = A exp[—a(r—

Thus even approximate analytical solutions are often more instructive than the more accurate numerical solutions. However considerable caution must be used in this approach, since some of the approximations, employed to make the equations tractable, can lead to erroneous answers. A number of approximate solution for the hot spot system (Eq 1) are reviewed by Merzhanov and their shortcomings are pointed out (Ref 14). More recently, Friedman (Ref 15) has developed approximate analytical solutions for a planar (semi-infinite slab) hot spot. These were discussed in Sec 4 of Heat Effects on p H39-R of this Vol. To compare Friedman s approximate solutions with the exact numerical solution of Merzhanov we computed r, the hot spot halfwidth, of a planar hot spot by both methods using the same thermal kinetic parameters in both calculations. Over a wide range of input variables, the numerical solution gives values of r which are 33 to 43% greater than the r s of the approximate solution. Thus it appears that the approximate solution, from which the effect of the process variables are much easier to discern than from the numerical solution, gives answers that differ from the exact numerical solution by a nearly constant factor... 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

The term parameters of the lowest two allowed transitions of ethene calculated with different methods and different choices of computational parameters (48,51,98,105) are summarized in Table I. Included in the table are results obtained with four different basis sets. In combination with these basis sets the MCD parameters were obtained in the transition-based approach through solution of Eq. (60) by direct numerical solution (labeled Direct in Table I) and by expansion in a set of transition densities according to Eq. (72) (labeled SOS ). In some cases approximate forms of the A(1) and B(1) matrices were used (labeled Approx, see Eq. (64) and the discussion following it). MCD parameters derived from a fit to a spectrum obtained by calculation of the imaginary part of the Verdet constant are labeled as Im[V]. The parameters obtained from a fit to the spectrum obtained from the approximate form of Im[V] (see Section... 

For the reconstruction of the size distribution, the theoretical signal course of a particle collective is calculated directly by a weighted summation of LII signals from monodisperse size classes, which are computed from the numerical solution of the power balance. The count median particle diameter dp,med, the distribution width a and the ambient temperature T0 are input parameter for this calculation. Fitting of this theoretical signal of a size-distributed particle ensemble in two different time intervals after the laser pulse ((Af)j, (Af)2) provides two characteristic signal decay times (tj = r((Af)j), r2 = t((Af)2)). 

Numerical method. The choice of the proper numerical solution procedure should be left to specialists. Even when a particular CFD package has been chosen, the user can usually choose different solution strategies for the linearized equations (point or line relaxation techniques versus whole field solution techniques) and associated values of the relaxation parameters. The proper choice of relaxation factors to obtain converged solutions (at all) within reasonable CPU constraints is a matter of experience where cooperation between the engineer and the specialist is required. This is especially true for new classes of fluid flow problems where previous experience is nonexistent. 

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