Numerical Solution of the Model Equations

The differential equations (7.164), (7.165), (7.166), and (7.168) form a pseudohomogene-ous model of the fixed-bed catalytic reactor. More accurately, in this pseudohomogeneous model, the effectiveness factors rji are assumed to be constantly equal to 1 and thus they can be included within the rates of reaction ki. Such a model is not very rigorous. Because it includes the effects of diffusion and conduction empirically in the catalyst pellet, it cannot be used reliably for other units. 

We need to solve the I VP from / = 0 to / = Lt, the total length of the bed. This can be achieved readily by any of the MATLAB IVP solvers ode.., once the initial conditions 

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Despite the assumptions and simplifications we have made in arriving at a model we feel that the physical basis we have adopted is sufficiently realistic to give good predictions, certainly as far as our present experimental results eneible us to make tests. The numerical solution of the model equations we have used presented no difficulties using a fast computer ( v 5 secs per solution). ... 

Algorithm for the numerical solution of the model equations Modelling for the different configurations of the ammonia converters... 

With todays computers and the state of the art regarding numerical techniques, it does not seem that the numerical solution of the model equations presents any serious problems. With the fast development of computer hardware and software, this problem will become almost trivial in the near future. 

Theoretical breakthrough curves for nonlinear systems may be calculated by numerical solution of the model equations using standard finite difference or collocation methods. Such solutions have been obtained by many authors and a brief summary is given in Table 8.4. In all cases plug flow was assumed and the equilibrium relationship was taken to be of cither Langmuir or Freundlich form. As linearity is approached ( ->1.0) the linearized rate models approach the Anzelius model (Table 8.1, model la) while the diffusion models approach the Rosen model (Table 8.1, model la). The conformity of the numerical solution to the exact analytic solution in the linear limit was confirmed by Garg and Ruthven. ... 

The model is completely dynamic, accounting for the accumulation of mass in the bulk phases of gas and liquid as well as in the pores of the catalyst particles. Our previous experience [3] has demonstrated that dynamic (or pseudodynamic) models should be preferred not only because of the prediction of transient operation periods, but also to ensure an improved robustness in the numerical solution of the model equations, particularly for countercurrent operations. 

Concentration and temperature profiles inside the catalyst pellet were obtained by using orthogonal collocation for the numerical solution of the model equations.Let us briefly recall how the technique works.The variables f and T are approximated by polynomials of degree N+l,i.e.,... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Model-fitting procedures are usually based on analytical solutions of the model however, model parameters may be estimated by fitting the differential equations describing the model. Since the numerical solution of the differential equations introduces another source of error, fitting of differential equations is usually limited to cases where nonlinearities are present. 

Davidson et al. (13) developed numerical solutions of the differential equation for solute transport for a model that... 

This example based on the reactor described by Murase et al. (1970) shows one way to mesh the numerical solution of the differential equations in the process model with an optimization code. The reactor, illustrated in Figure E14.2a, is based on the Haber process. 

Results for two types of model systems are shown here, each at the two different inverse temperatures of P = 1 and P = 8. For each model system, the approximate correlation functions were compared with an exact quantum correlation function obtained by numerical solution of the Schrodinger equation on a grid and with classical MD. As noted earlier, testing the CMD method against exact results for simple one-dimensional non-dissipative systems is problematical, but the results are still useful to help us to better imderstand the limitations of the method imder certain circumstances. 

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. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

Figure 3.31 As (due to orientational response of aqueous solvent) versus e, calculated for ET in a large binuclear transition metal complex (D (Ru2+/3+) and A (Co2+/3+) sites bridged by a tetraproline moiety) molecular-level results obtained from a nonlocal polarization response theory (NRFT, solid lines) continuum results are given by dashed lines, referring to numerical solution of the Poisson equation with vdW (cont./vdW) and SAS (cont./SAS) cavities, or as the limit of the NRFT results when the full k-dependent structure factor (5(k)) is replaced by 5(0) 5(k) for bulk water was obtained from a fluid model based on polarizable dipolar spheres (s = 1.8 refers to ambient water (square)). For an alternative model based on TIP3 water (where, nominally, 6 = ), ambient water corresponds to the diamond. (Reprinted from A. A. Milishuk and D. V. Matyushov, Chem Phys., 324, 172. Copyright (2006), with permission from Elsevier).

Certain crude approaches are available to predict overall results, that is, nonequilibrium compositions. More refined techniques are available for the analysis of simplified models. Solution of the reaction kinetics of homogeneous gas phase combustion is possible through numerical solution of the rate equations. With the exception of the role of an overall highly exothermic reaction, the procedures are similar to those described in the preceding section on nozzle processes. The solution of the droplet burning problem including the role of chemical reaction rates, while not particularly tractable, is feasible. 

The steady-state permeation model of in situ coal gasification is presented in an expanded formulation which includes the following reactions combustion, water-gas, water-gas shift, Boudouard, methanation and devolatilization. The model predicts that substantial quantities of unconsumed char will be left in the wake of the burn front under certain conditions, and this result is in qualitative agreement with postburn studies of the Hanna UCG tests. The problems encountered in the numerical solution of the system equations are discussed. 

Fu and Zhang (1991) developed a general method for calculating burning rate of coal and char particles, and a chart was made through numerical solution for the model equations, as shown in Fig. 5. The dimensionless... 

The above phenomena me physically miomalous and can be remedied through the introduction of a hyperbolic equation based on a relaxation model for heat conduction, which accounts for a finite thermal propagation speed. Recently, considerable interest has been generated toward the hyperbolic heat conduction (HHC) equation and its potential applications in engineering and technology. A comprehensive survey of the relevant literature is available in reference [6]. Some researchers dealt with wave characteristics and finite propagation speed in transient heat transfer conduction [3], [7], [8], [9] and [10]. Several analytical and numerical solutions of the HHC equation have been presented in the literature. 

The measured uptake kinetics are modeled by a numerical solution of the difiSiaon equation... 

When a significant change in the substrate concentration in the column is expected, numerical integration of the model equations (26) and (27) should be applied to the experimental data. The solution was simplified by combining Eq. (27) with Eq. (26) in order to obtain the following equation, where the term (-AHr)... 

During the 1970s the ECD became firmly established as the most sensitive gas chromatographic detector for some compounds. The kinetic model was described in terms of a numerical solution of the differential equations. This was assisted by the development of the constant current mode of measuring the response and the development of Ni-63 sources for the detector. The purification of the carrier gas and the further development of capillary columns improved the operation of the ECD. In addition, chemical reactions were used to make derivatives with a greater sensitivity in the ECD. Other ion molecule reactions were used to improve the sensitivity of... 

At this stage, two approaches are possible. The first one calculates solutions of the mass balance equation (Eq. 10.60) and uses finite-difference schemes that give a numerical error of the second order. The second approach calculates solutions of the mass balance equation of the ideal model (Eq. 10.72) and uses finite-differences schemes that give an error of the first order. The parameters of the numerical integration are then selected in such a way that the numerical error introduced by the calculation is equivalent to the dispersion term, so the approximate numerical solution of the approximate equation and the exact solution of the correct equation are equal to the first order. 

If DJuL, and of course kd, are known, Eq. (6-45) gives the conversion predicted by the dispersion model, provided the reaction is first order. For most other kinetics numerical solution of the differential equation is necessary. ... 

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