The one-dimensional model

The form of the radial temperature profile in a nonadiabatic fixed-bed reactor has been observed experimentally to have a parabolic shape. Data for the oxidation of sulfur dioxide with a platinum catalyst on x -in. cylindrical pellets in a 2-in.-ID reactor are illustrated in Fig. 13-9. Results are shown for several catalyst-bed depths. The reactor wall was maintained at 197°C by a jacket of boiling glycol. This is an extreme case. The low wall temperature resulted in severe radial temperature gradients, more so than would exist in a commercial reactor, where the wall temperature would be higher. The longitudinal profiles are shown in Fig. 13-10 for the same experiment. These curves show the typical hot spots, or maxima, characteristic of exothermic reactions in a nonadiabatic reactor. The greatest increase above the reactants temperature entering the bed is at the center, 

A completely satisfactory design method for nonadiabatic reactors entails ljredictinr-the f5dial and lon fudmal yariations in temperature, such as fho in Figs. 13-9 and analogous concentration 

This one-dimensional approach is useful as a rapid procedure for 

T emperalufe can be Ftermined by m ns of this simplified procedure. Problem 13-6 illustrates calculations for a phthalic anhydride reactor. The oxidation of naphthalene has a high heat of reaction, so that the size of the catalyst tubes is a critical point in the design. 

Examples 13-3 to 13-5 illustrate the simplified design method for different cases. The first is for the endothermic styrene reaction, where the temperature decreases continually with catalyst-bed depth. Example 13-4 is for an exothermic reaction carried out under conditions where radial temperature gradients are not large. Example 13-5 is also for an exothermic case, but here the gradients are severe, and the simplified solution is not satisfactory. 

We model stretched helium configurations using two-electron product states built from the extremal Stark states 5l ) defined in (7.2.7). As shown in Fig. 7.4 these states are quasi-one-dimensional states highly elongated in the direction. But a trivial rotation of the coordinate system results in a coordinate system E in which the states and 

151 ) point into the positive and negative x directions, respectively. On the basis of the extremal Stark states we construct stretched 

The nuclear charge in Fig. 10.2 is not specified in order to be able to describe any other member of the two-electron iso-electronic family, such as H , Li , etc. For helium, Z — 2. Even when focussing on a specific two-electron atom or ion we would like to keep the nuclear charge Z variable in order to study the sensitivity of bound states and resonances to small changes of Z. This topic is covered in Section 10.5.2. 

The positions of the electrons in Fig. 10.2 are denoted by —xi and X2, respectively. Electron number 1 is assumed to be located to the left of the nucleus, electron number 2 to the right. We do not allow the electrons to cross the potential singularity at x = 0. Therefore, we have xi 0, X2 0 for all times. In analogy to the surface state electrons discussed in Chapter 6 we assume perfectly elastic refiection at x = 0. Thus, in the absence of the e - e interaction, and because xi, X2 0, both electrons are governed by the same single-particle Hamiltonian 

Hamiltonian, taking the mutual e - e repulsion into account, reads 

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Although the inadequacy of the one-dimensional model [Goldanskii 1959, 1979] is well by now understood, we start with discussing the simplest one-dimensional version of the theory, since it... 

The Arrhenius plot of k(T) for H and D transfer is presented in fig. 15. Qualitatively, the conclusions about the isotope effect drawn here on the basis of the one-dimensional model remain correct for more dimensions, but turns out to depend more weakly on m than In k This... 

A straightforward calculation of A has been performed by Fraser [1989] within the framework of the one-dimensional model of concerted interconversion. The diabatic terms were taken in the form... 

Krumbhaar. Solidification in the one-dimensional model for a disordered binary alloy under diffusion. Eur Phys J. B 5 663, 1998. 

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. 

Commenge et al. extended the one-dimensional model of reacting flows to include Taylor-Aris dispersion, i.e. they considered an equation of the form... 

Chemical substitution in a pattern that breaks the 2v symmetry of the molecular frame introduces a threefold component to the one-dimensional model torsional potential ... 

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. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

In the formulation of the boundary conditions, it is presumed that there is no dispersion in the feed line and that the entering fluid is uniform in temperature and composition. In addition to the above boundary conditions, it is also necessary to formulate appropriate equations to express the energy transfer constraints imposed on the system (e.g., adiabatic, isothermal, or nonisothermal-nonadiabatic operation). For the one-dimensional models, boundary conditions 12.7.34 and 12.7.35 hold for all R, and not just at R = 0. 

Nevertheless, one feature of the one-dimensional model containing dispersion terms is of... 

The following illustrations indicate the manner in which the one-dimensional model is employed in reactor design analyses. 

The first-stage effluent temperature has been limited to 560 °C in order to prevent excessive catalyst activity losses. The heat of reaction data is slightly inconsistent with the reported activation energies, but use of this expression demonstrates the ease with which temperature dependent properties may be incorporated in the one-dimensional model. 

Equation (43) assumes a specific value for Sq- A more general result follows from the fact that the Fourier transforms in (41) will be small, except for small values of o)p so that, as expected, to obtain high probabilities of charge transfer, Eq should either be in the solid band or, at least, not lie very far from the band. An interesting result, showing the dependence of P, on Eq, can be obtained for the one-dimensional model of section 3.3 with a half-filled band, V(t) of exponential form, P — co) = 0 and V% XB. This is... 

The strict generalization of the one-dimensional model treated in Sec. III,A leads to a crystal with its surface completely covered by adsorbed atoms. For this system, the tight-binding approximation gives a difference equation and boundary conditions which can be solved directly. The results show an important new feature. For a simple cubic lattice, the energy levels are given by the usual equation... 

The interaction parameters z, z, and Ji are defined in the usual way, and t) = /S"//8, where /3" is the resonance integral between nearest neighbors in the adsorbed layer. If rj = 1, the eigenvalue condition. Equation (19), is the same as for the one-dimensional model. The only change is that the discrete localized states (CP and 91) of the one-dimensional model now appear as bands of surface states (CP or 91 bands) associated with the adsorbed layer and the crystal surface. At most, two such bands may be formed, and each band contains levels. This is the number of atoms in the adsorbed layer. Depending on the values of the interaction parameters z and z, these bands may or may not overlap the normal band of crystal states. All this was to be expected, and Fig. 2 gives the occurrence of (P and 91 surface bands when = 1. It is when tj 7 1 (and this will be the usual situation) that a new feature arises. In this case, the second term in the second bracket in Equation (19) does not vanish, and the eigenvalue condition is not the same as in the one-dimensional model. In fact we have z - - 2(1 — jj )(cos 02 - - cos 03) in place of z, and this varies between z - - 4(1 — ij ) and z — 4(1 — tj ). We can still use Fig. 2 if we remember that z varies between these two limits. Then if, for example, this variation... 

The occurrence of homopoiar localized states for the one-dimensional model has been discussed in some detail because the possibility of such states is a natural consequence of the molecular orbital approach to the problem of the surface bond. It is clear, however, that the conditions for their existence are rather stringent. Most sets of interaction parameters do not give a purely homopoiar state but give states with ionic character to a greater or lesser degree. 

To get some idea of the magnitude of the energy change involved, we again consider the one-dimensional model of Sec. III,A. We assume that there is one electron per atom in the chain, so that the band is half-filled and that /3 is negative. We remove the foreign atom to infinity and calculate the total electronic energy of the chain. If no surface states are occupied, the result is (IS). 

It must be mentioned, however, that the one-dimensional model gives only a qualitative explanation of thermal quenching. A quantitatively valid explanation can by obtained only by a multidimensional model. 

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. 4. Local enhancement as a function of radial positions from the particle center using the one-dimensional model presented (D = 1.24 x 10 m s Dr = 0.56,H = 103,dp = 3x 10 m, M = 0)...

Each set of quantum numbers Tig, tit, tic will give rise to an energy level. However, in three dimensions, many combinations of tig, tit, and tic exist which will give the same energy, whereas for the one-dimensional model there were only two levels of each energy n and -n). For example, the following sets of numbers all give +nb/lf +Tic/c )= 108... 

In the following, the one-dimensional model will be presented. The basic ideal models assume that concentration and temperature gradients occur in the axial direction (Froment and Bischoff, 1990). The model for a fixed-bed reactor consists of three equations, which will be presented in the following sections and are... 

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