Axial diffusion reactors

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The thoughtful reader may wonder about a real reactor with a high level of radial diffusion. Won t there necessarily be a high level of axial diffusion as well and won t the limit of oo really correspond to a CSTR rather than... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

Fig. 2.4p shows three types of post-column reactor. In the open tubular reactor, after the solutes have been separated on the column, reagent is pumped into the column effluent via a suitable mixing tee. The reactor, which may be a coil of stainless steel or ptfe tube, provides the desired holdup time for the reaction. Finally, the combined streams are passed through the detector. This type of reactor is commonly used in cases where the derivatisation reaction is fairly fast. For slower reactions, segmented stream tubular reactors can be used. With this type, gas bubbles are introduced into the stream at fixed time intervals. The object of this is to reduce axial diffusion of solute zones, and thus to reduce extra-column dispersion. For intermediate reactions, packed bed reactors have been used, in which the reactor may be a column packed with small glass beads. 

A plug flow or tubular flow reactor is tubular in shape with a high length/diameter (1/d) ratio. In an ideal case (as in the case of an ideal gas, this only approached reality) flow is orderly with no axial diffusion and no difference in velocity of any members in the tube. Thus, the time a particular material remains within the tube is the same as that for any other material. We can derive relationships for such an ideal situation for a first-order reaction. One that relates extent of conversion with mean residence time, t, for free radical polymerizations is ... 

Much work has been focused on the significance of dispersion terms in the transient material and energy equations for packed bed reactors. In general, axial diffusion of mass and energy and radial diffusion of mass have been... 

Although this assumption is often not too restrictive, additional assumptions must be introduced to reduce the mathematical model to the pseudohomogeneous form for reactors that are nonadiabatic or require radial temperature considerations. For the full mathematical model, the following assumptions would be necessary in addition to Eq. (68) first negligible energy accumulation in the gas phase second, no axial diffusion in the gas phase or... 

In this section we have presented the first example of two-point boundary value problems that occur in chemical/biological engineering. The axial dispersion model for tubular reactors is a generalization of the plug flow model for tubular reactors which removes some of the limiting assumptions of plug flow. Our model includes additional axial diffusion terms that are based on the simple physics laws of Fick for mass and of Fourier for heat dispersion. 

The flow patterns, composition profiles, and temperature profiles in a real tubular reactor can often be quite complex. Temperature and composition gradients can exist in both the axial and radial dimensions. Flow can be laminar or turbulent. Axial diffusion and conduction can occur. All of these potential complexities are eliminated when the plug flow assumption is made. A plug flow tubular reactor (PFR) assumes that the process fluid moves with a uniform velocity profile over the entire cross-sectional area of the reactor and no radial gradients exist. This assumption is fairly reasonable for adiabatic reactors. But for nonadiabatic reactors, radial temperature gradients are inherent features. If tube diameters are kept small, the plug flow assumption in more correct. Nevertheless the PFR can be used for many systems, and this idealized tubular reactor will be assumed in the examples considered in this book. We also assume that there is no axial conduction or diffusion. 

A maximum reactor temperature of 500 K is used in this study. This maximum temperature occurs at the exit of the adiabatic reactor under steady-state conditions. Plug flow is assumed with no radial gradients in concentrations or temperatures and no axial diffusion or conduction. 

To achieve desired conversions predicted by ideal design equations, plug flow is required. This implies turbulent flow and higher energy costs if packing is used. Mass transfer can also be a problem. Axial diffusion or dispersion tends to decrease residence time in the reactor. High values of the length-to-diameter ratios (L/D > 100) tend to minimize this problem and also help heat transfer. 

Figure 1 shows the schematic of a tubular reactor, of radius a and length L, where a — a/Lis the aspect ratio. Clearly, ifa>S> 1, or a <3C 1, a physical length scale separation exists in the reactor. This length scale separation could also be interpreted in terms of time scales. For example, a 1 implies that the time scale for radial diffusion is much smaller than that of either convection and axial diffusion, and concentration gradients in the transverse direction are small compared to that in the axial direction. 

Neglecting axial diffusion (or assuming L/a > 1), the three-dimensional CDR equation of a laminar flow loop reactor is given by... 

The boundary conditions are dx/dz = Px at z =0 and dx/dz = 0 at z = 1. Here, P is the axial Peclet number, defined as the product of velocity and reactor length, divided by the effective axial diffusivity. 

Backmixing by axial diffusion, and stability of adiabatic reactors, 344... 

From the point of view of practical computation, the equations without axial diffusion are solved by starting at the inlet to the reactor and computing the conditions stepwise downstream, making no adjustment for what happens further downstream. When there is axial diffusion, the conditions at all points in the reactor have to be adjusted to meet boundary conditions at both ends, and the difficulty of the computation is multiplied by a large factor. 

This equation is a generalization of the Aris (1968) result given in Eq. (98) The apparent overall order is (a + l)/a times the intrinsic order of 2. Scaramella et al. (1991) have also analyzed a perturbation scheme around this basic solution, which, incidentally, lends itself to a solution via reduction to an integral Volterra equation of the same type as discussed in Section IV,C with regard to a plug flow reactor with axial diffusion. Scaramella et al. have also shown that if b x,y) can be expressed as the sum of M products of the type b x)B y) + B(x)b(y), which is... 

Aris (1991a), in addition to the case of M CSTRs in series, has also analyzed two other homotopies the plug flow reactor with recycle ratio R, and a PFR with axial diffusivity and Peclet number P, but only for first-order intrinsic kinetics. The values M = 1(< ), R = >(0), and P = 0( o) yield the CSTR (PFR). The M CSTRs in series were discussed earlier in Section IV,C,1. The solutions are expressed in terms of the Lerch function for the PFR with recycle, and in terms of the Niemand function for the PFR with dispersion. The latter case is the only one that has been attacked for the case of nonlinear intrinsic kinetics, as discussed below in Section IV,C,7,b. Guida et al. (1994a) have recently discussed a different homotopy, which is in some sense a basically different one no work has been done on multicomponent mixture systems in such a homotopy. 

Diffusion with a convection and simultaneous first order reaction in a rectangular plate can be simulated using the program described above by using minor modifications. Consider the composition profile in a packed tube reactor undergoing isothermal linear kinetics with axial diffusion. The governing equation is... 

Axial diffusion and conduction in an adiabatic tubular reactor can be described by [6]... 

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