For the axial dispersion model

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

Solution The first step in the solution is to find a residence time function for the axial dispersion model. Either W t) or f(t) would do. The function has Pe as a parameter. The methods of Section 15.1.2 could then be used to determine which will give the desired relationship between Pe and... 

For a relatively small amount of dispersion, what value of Pei would result in a 10% increase in volume (V) relative to that of a PFR (Vpf) for the same conversion (/a) and throughput (q) Assume the reaction, A - products, is first-order, and isothermal, steady-state, constant-density operation and the reaction number, Mai = at, is 2.5. For this purpose, first show, using equation 20.2-10, for the axial-dispersion model with relatively large Per, that the % increase s 100(V - V pfWpf = 100MAi/Pei. 

Besides, if SHa — oo, then xa w=i.o = xAb This also corresponds to a negligible external mass transfer resistance. In both cases, that of a finite Sherwood number SHa or for SHa — oo, we get a two-point boundary value differential equation. For the nonlinear case this has to be solved numerically. However, as for the axial dispersion model, we will start out with the linear case that can be solved analytically. 

For the nonlinear case, the nonlinear two-point boundary value differential equation(s) for the catalyst pellet can be solved using the same method as used for the axial dispersion model in Section 5.1, i.e., by the orthogonal collocation technique of MATLAB s bvp4c. m boundary value solver. 

The H-Oil reactor (Fig. 21) is rather unique and is called an ebullated bed catalytic reactor. A recycle pump, located either internally or externally, circulates the reactor fluids down through a central downcomer and then upward through a distributor plate and into the ebullated catalyst bed. The reactor is usually well insulated and operated adiabatically. Frequently, the reactor-mixing pattern is defined as backmixed, but this is not strictly true. A better description of the flow pattern is dispersed plug flow with recycle. Thus, the reactor equations for the axial dispersion model are modified appropriately to account for recycle conditions. 

Reference was made earlier to some specific numerical studies reported for the axial dispersion model employing rate expressions other than first order. Some results given by Fan and Balie for half-, second-, and third-order kinetics are illustrated in Figure 5.21. In these results the parameter R is defined as... 

A transient solution for step-change in the inlet concentration for the axial dispersion model, using Type A boundary conditions, has been given by Fan and Ahn [L-T. Fan and Y-R. Ahn, Chem. Eng. Progr. Symp. Ser., 46(59), 91 (1963)]. For... 

A comparison of these transient forms for first-order kinetics, equation (5-115) for the CSTR sequence and (5-116) for the axial dispersion model, on the basis of their steady-state equivalence in terms of n and Npe, is suggested in the exercises. The basic equation to be answered is whether the equivalence of the two models in the... 

In many ways, the two models are rather similar, although the mathematical details for the tanks in series model is much simpler than for the axial dispersion model On the other band, no theoretical justification such as Taylor diffusion is possible in general nor are theoretical estimates of the model parameter, n that is, n is strictly empirical The only exceptions to this are the finite stage models for packed bed interstices as briefly discu ed in Chapter 11. 

Next, let us discuss the various techniques that have been used for the actual parameter estimation, describing their strong and weak points. For the axial dispersion model, this has been done by Boxkcs and Hofmann [96], and illustrates the typical problems involved. 

C(g, 0) represents the solution to the Danckwarts model Equation 3.329 with the boundary conditions (3.331) and (3.334), then the RTD function for the axial dispersion model is... 

Formulation and Solution Strategy for the Axial Dispersion Model... 

For both cases of limited Sh and Sh oo, we get a two-point boundary-value dilferential equation. For the nonlinear cases, it has to be solved iteratively (we can use Fox s method, as explained for the axial dispersion model in Chapter 4 or orthogonal collocation techniques as explained in Appendix E). 

Axial dispersion model Here, let us work through the RTD for the axial dispersion model. The equation to solve is... 

By equating the variance of the model to that for the axial dispersion model one obtains... 

Figure 4. Independence of k, with bed depth for the axial dispersion model...

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