The Two-dimensional Model

The mass balance of reactant for these conditions is similar to Eq. (13-1). The term for longitudinal diffusion is-omitted, and the mass velocity 

The objective is to solve Eqs. (13-42) and (13-20) for the temperature and conversion at any point in the catalyst bed. As boundary conditions at the entrance we need the temperature and conversion profile across the diameter of the reactor. Further boundary conditions applicable at any axial location are that the conversion is flat (dx/dr = 0) at both the centerline and the wall of the tube. The temperature gradient at the centerline is zero, but the condition at the wall is determined by the heat-transfer character- 

Equations (13-42) and (13-20) are solved by a stepwise numerical procedure, starting at the entrance to the reactor. The equations are first written in difference form. Let n and L represent the number of increments in the radial and axial directions, respectively, and Ar and Az their magnitude, so that 

The temperature at any point in the bed can be represented by that is, the temperature at r = n Ar and z = L Az. Note that r is measured from the center of the bed and z from the feed entrance. 

With these definitions, the approximate difference form of Eq. (13-20) is 

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In two-dimensional solids theory, the size of the solid in a fixed direction is assumed to be small as compared to the other ones. Therefore, all characteristics of the thin solid are referred to a so-called mid-surface, and one obtains the two-dimensional model. Let us give the construction of plate and shell models (Donnell, 1976 Vol mir, 1972 Lukasiewicz, 1979 Mikhailov, 1980). 

A very recent application of the two-dimensional model has been to the crystallization of a random copolymer [171]. The units trying to attach to the growth face are either crystallizable A s or non-crystallizable B s with a Poisson probability based on the comonomer concentration in the melt. This means that the on rate becomes thickness dependent with the effect of a depletion of crystallizable material with increasing thickness. This leads to a maximum lamellar thickness and further to a melting point depression much larger than that obtained by the Flory [172] equilibrium treatment. 

In contrast with the one-dimensional model, the two-dimensional model allows to determine the actual parameter distribution in flow fields of the working fluid and its vapor. It also allows one to calculate the drag and heat transfer coefficients by the solution of a fundamental system of equations, which describes the flow of viscous fluid in a heated capillary. 

The rate equation for the two-dimensional model is then given by (12), where A = pT ptlT and /3 is a constant related to the individual reaction, and Topt represents the temperature where the reaction occurs most efficiently. 

The various energy transfer constraints enter into the analysis primarily as boundary conditions on the difference equations, and we now turn to the generation of the differential equations on which the difference equations are based. Since the equations for the one-dimensional model are readily obtained by omitting or modifying terms in the expressions for the two-dimensional model, we begin by deriving the material balance equations for the latter. For purposes of simplification, it is assumed that only one independent reaction occurs within the system of interest. In cases where multiple reactions are present, one merely adds an appropriate term for each additional independent reaction. 

Lin et al. [44] simulated the enhancement using the two-dimensional model, as a function of all important parameters. It is clear from this work that the role of lateral diffusion depends mainly on the particle capacity, that is, on the value of partition coefficient, H. This conclusion was drawn by Brilman et al. [42,54], as well. The one-dimensional model does not contain the lateral effect and, as a consequence, the absorption rate would be underestimated by this model and the difference of the results obtained by the one- and two-dimensional models would be increased with increasing partition coefficient. The question is in which parameter range this error can be neglected. 

Fig. 3. Comparison the one- (Eq. 24) and the two dimensional model (Lin et al. [44]) results as a function of dispersed phase holdup at different values of particle distance from the interface...

However, the total dissociation wavefunction is useful in order to visualize the overall dissociation path in the upper electronic state as illustrated in Figure 2.3(a) for the two-dimensional model system. The variation of the center of the wavefunction with r intriguingly illustrates the substantial vibrational excitation of the product in this case. As we will demonstrate in Chapter 5, I tot closely resembles a swarm of classical trajectories launched in the vicinity of the ground-state equilibrium. Furthermore, we will prove in Chapter 4 that the total dissociation function is the Fourier transform of the evolving wavepacket in the time-dependent formulation of photodissociation. The evolving wavepacket, the swarm of classical trajectories, and the total dissociation wavefunction all lead to the same general picture of the dissociation process. 

Figure 4.2 depicts snapshots of the evolving wavepacket for the two-dimensional model system that we discussed in Section 2.5 to illustrate vibrational excitation (see Figure 2.3). Because the PES is quite steep in the proximity of the ground-state equilibrium, the molecule dissociates at once. Immediately after the wavepacket is released it slides down the potential slope and disappears into the asymptotic channel for good. Since 4>/(0) starts significantly displaced from the minimum energy path of the PES, the evolving wavepacket oscillates along the r-direction, which in-...

Fig. 10.14. Absorption spectrum for the two-dimensional model of the photodissociation of FNO via the Si state. E is the energy in the excited electronic state relative to F+NO(re). The resonances are assigned to excitation of the bending degree of freedom with quantum numbers k = 0,..., 3 (see Figure 10.13). Adapted from Ogai et al. (1992).

Melendez-Martinez, J J., Gomez-Garcla, D., and Dominguez-Rodriguez, A., Acritical analysis and a recent improvement of the two-dimensional model for solution-precipitation creep application to silicon nitride ceramics , Phil. Mag., 2004, 84, 2305-16. 

The mathematical model for the PBE reactor should in general be a two-dimensional model describing the potential and concentration distributions within the packed bed electrode. However, the model can be simplified in certain cases. Under a recycle flow operation, for example, the conversation per pass through the packed bed is small, so that the PBER can be treated as a differential reactor, the potential distribution only in the lateral direction is considered. In this case which is similar to the case in 5.4, the two-dimensional model can be written in a onedimensional Poisson equation form as... 

The two-dimensional model can be used to develop an equivalent one-dimensional model with a bed-side heat-transfer coefficient defined as [see, e.g., Froment, Chem. Eng. Sci. 7 29 (1962)]... 

The remainder of this paper is organized as follows In Sect. 5.2, we present the basic theory of the present control scheme. The validity of the theoretical method and the choice of optimal pulse parameters are discussed in Sect. 5.3. In Sect. 5.4 we provide several numerical examples i) complete electronic excitation of the wavepacket from a nonequilibrium displaced position, taking LiH and NaK as examples ii) pump-dump and creation of localized target wavepackets on the ground electronic state potential, using NaK as an example, and iii) bond-selective photodissociation in the two-dimensional model of H2O. A localized wavepacket is made to jump to the excited-state potential in a desirable force-selective region so that it can be dissociated into the desirable channel. Future perspectives from the author s point of view are summarized in Sect. 5.5. 

In order to demonstrate the efficiency and accuracy of the semiclassical formulation of optimal control theory, let us consider the control of two elementary types of motion (a) a shift of the position of the ground-state wavepacket in the two-dimensional model system of H20 and (b) an acceleration of the ground-state wavepacket at the same position in the same model. 

Finally, the question rises whether an accurate simulation of the reformer tubes does not have to include consideration of radial gradients. The two dimensional model developed for this purpose in this work neglects interfacial gradients, for reasons explained already above, but maintains the mass transfer limitations inside the catalyst, of course. 

The outside tubeskin temperature was taken to be identical to that generated in the previous simulation. The input data were also identical. Radial process temperature profiles are given in Figure 7. The ATg between the bed centerline and the wall amounts to 33°C, which is not excessive and permits the radially averaged temperature to be accurately simulated by means of the one dimensional model with "equivalent" heat transfer parameters, as discussed above. The methane conversion at the wall never differed more them 2% absolute from that in the centerline of the bed. The more detailed description which is possible by the two dimensional model would only be required if thermodynamic s predict possible carbon formation, and therefore catalyst deactivation, at locations different from those simulated by the one dimensional model. 

The chain surface charge density oa used in Eq. (19) to obtain the same force as the one calculated from the two-dimensional model... 

By equating the force provided by the two-dimensional model to that obtained from Eq. (19), the values listed in Table 1 have been obtained for era as a function of pHo. 

Figure 2. Grid system used in the two-dimensional model.

Berend and Benson [49] have compared experimental rotational relaxation rates for / -H2-/ -H2, />-H2-He, and o-D2-He with their classical theory utilizing the two-dimensional model of Figure 3.5, with atom-centered Morse potentials. Their conclusion that H2-H2 collisions are more effective than are H2-He collisions differs from the predictions of Roberts and is not confirmed experimentally however, there is fair agreement with regards to both magnitude and temperature dependence, particularly for the H2-H2 case. 

The steady-state heat and mass balance equations of the different models were numerically integrated using a fourth-order Runge-Kutta-Gill method for the one-dimensional models, while the Crank-Nicholson finite differences method was used to solve the two-dimensional models. 

Differences in the Responses of the Different Types of Models. The basic differences that exist in the heat and mass balances for the different types of models determine deviations of the responses of types I and II with respect to type III. In a previous work (1) a method was developed to predict these deviations but for conditions of no increase in the radial mean temperature of the reactor (T0 >> Tw). In this work,the method is generalized for any values of T0 and Tw and for any kinetic equation. The proposed method allows the estimation of the error in the radial mean conversions of models I and II with respect to models III. Its validity is verified by comparing the predicted deviations with those calculated from the numerical solution of the two-dimensional models. A similar comparison could have been made with the numerical solution of the one-dimensional models. 

At present there is no systematic work on simulation and design of packed bed nonadiabatic reactors of industrial size where a deactivation process occurs. The purpose of this work is to analyze the operation of a nonadiabatic deactivating catalyst bed and to develop simple techniques for simulation. Based on hydrogenation of benzene,full-scale reactor behavior is calcu lated for a number of different operational conditions. Radial transport processes are incorporated in the model, and it is shown that the two-dimensional model is necessary in some cases. 

The computer time for simulating 50 hours of deactivation was 13 seconds for the one-dimensional model and 15 seconds for the two-dimensional model on the CDC Cyber 174 (13). 

Table 14.2 shows the annihilation and reaction parameters of die positron-molecule states PsA of 0.01 M aqueous solutions of NaF, NaCl, NaBr, and Nal, as well as for pure water. The two-dimensional model functions based on equations (6) and (7) were shown to describe the AMOC... 

Figure 14.8 Lineshape functions (t) of solid rare gases. Data points with error bars are calculated from the AMOC data, solid lines from the two-dimensional model... 

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