Dispersion model second order reactions

FIG. 23-15 Chemical conversion by the dispersion model, (a) First-order reaction, volume relative to plug flow against residual concentration ratio, (h) Second-order reaction, residual concentration ratio against kC t. 

Determine the yield of a second-order reaction in an isothermal tubular reactor governed by the axial dispersion model with Pe = 16 and kt = 2. 

A second order reaction is to be conducted in a vessel whose RTD is an Erlang with n = 3. Find conversion with the dispersion and other models for... 

Direct contact heat transfer, 185 Dispersion model, 560-562 first order reactions, 561 second order reactions, 562 Distillation, 371-457 batch, 390 binary, 379 column assembly, 371 flash, 375... 

Ming-heng et al. (M15) employ a model for liquid-liquid dispersions similar to those mentioned previously. They analyzed droplet mixing on conversion for a second-order reaction occurring in the dispersed phase. The solution was obtained using the method of moment equations. 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

The utility of the model to predict the effects of interdroplet mixing on extent of reaction was demonstrated for the case of a solute diffusing from the dispersed phase and undergoing second-order reaction in the continuous phase. For this comparison the normalized volumetric dispersed-phase concentration distribution is deflned as fv(y) dy equal to the fraction of the total volume of the dispersed phase with dimensionless concentration in the range y to y -i- dy, where y = c/cq and... 

Use the dispersion model to estimate the conversion for a second-order reaction with fc = 0.1 dmVmol s and Qo I mol/dm-. ... 

Determine the yield of a second order reaction, A -F B — prodnct with ai = bm in an isothermal tubular reactor governed by the axial dispersion model. Specifically, plot fraction unreacted versus aiJd for a variety of Pe. Be sure to show the limiting cases that correspond to a PER and a CSTR. 

Under these circumstances, a general stationary heterogeneous dispersion (PD-)model for an irreversible catalytic second order reaction between a gaseous and a liquid reactant in dimensionless form consists of the balance equations shown in Fig. 18. In this model the whole fluiddynamics are lumped into a single parameter, i.e. the Bodenstein number, here based on the reactor length. 

This method of formulation by von Smoluchowski and Fuchs is limited to small concentrations of particles. Then the fixed particle can at most feel the presence of one other particle, and (p is equal to the sum of the van der Waals attraction and the electrical double-layer repulsion poteitial, or, as discussed in previous sections. In this limit it is also legitimate to model the reaction as a second-order reaction (i.e., only two-particle collisions can occur and the higher body collisions are virtually nonexistent). In aerosols, which arc colloidal dispersions in air, there is no significant electrical repulsion betwerai particles. Hence the effect of interparticle forces on the initial coagulation rate is negligible, and we find... 

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

The parameters in the model, which witb-rare exception should no exceed two in number, are obtained from the RTD data. Once the parame ters are evaluated, the conversion in the model, and thus in the real reactoi can be calculated. For typical tank-reactor models, this is the conversion i a series-parallel reactor system. For the dispersion model, the second-orde differential equation must be solved, usually numerically. Analytical soli tions exist for first-order reactions, but as pointed out previously, no mod has to be assumed for the first-order system if the RTD is available. 

For nonlinear reaction kinetics, a numerical solution of the balance Equation 4.121 is carried out. For example, for second-order kinetics, R = kcACB, with an arbitrary stoichiometry, the generation rate expressions, ta = —va CaCb and tb = —vb caCb, are inserted into the mass balance expression, which is solved numerically using, for example, a polynomial approximation (orthogonal collocation method). The performances of the normal dispersion model and its segregated or maximum-mixed variants are compared in Figure 4.34. The symbols are explained in the figure. The comparison reveals that the differences between the segregated, maximum-mixed, and normal axial dispersion models are notable at moderate Damkohler numbers R = Damkohler number). 

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