Tubular with axial mixing

THE TUBULAR REACTOR WITH AXIAL MIXING THE DISPERSION MODEL... 

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

Compute the profiles in tubular reactors with axial mixing as a function of total length and compare with the plug flow profile. 

The simplest tubular reactor with axial mixing and heat transfer is the TRAM again this has radially uniform composition. It is of particular significance in combustion, being the analt e of one-dimensional laminar-flame propagation. In this Report we shall principally discuss the work on the CSTR, although some mention of tubular reactions is included. 

The simplest realistic model is the TRAM (tubular reactor with axial mixing), in which average temperatures and concentrations are used across the reactor but the finite rates of heat conduction and mass diffusion along the reactor are admitted. 

Figure 15 Steady-state operating curves for the tubular rcMior with axial mixing. Effluent concentration c/c, plotted against feed parameter f= paTtIctQ (= Tr/Ar. ) for various values of the parameter v = qLID. The system has unit Lewis number and the heat-loss parameter xIP ( r /Tn) is chosen as 0.1. Curve v = oo corresponds to the PFTR (D = 0) and v = 0 to the CSTR iD — oo)...

Note that not perfectly mixed reactors give rise to two additional problems the rate of micro-mixings that is treated in sections 4.2 and 5.2, and die effects of non-ideal JloWs that are dealt with in sections 7.2.1 and 7.2.3. Tubular reactors with axial mixing are treated in section 7.2.2. 

Figure 75. The relative concentration of reactant A (one minus the degree of conversion) as a function of the Damkbhler number for first order reactions in a tubular reactor with axial mixing, for different values of the Piclet number for axial mixing see eqs. (721) - (7.23). The Damkbhler number is k X.

Axial mixing in reactors with predominant axial flow can also be modelled by a cascade of a large number of perfectly mixed CSTR s, as described in section 7.1.3. It appears that the residence time distribution (RTD) of a tubular reactor with axial mixing can approach the RTD of a cascade of N perfectly mixed CSTR s when the following condition applies... 

Agrawal, S. and C.D. Han. Analysis of the High Pressure Polyethylene Tubular Reactor with Axial Mixing, AIChE J., 21 (1975), 449-465. 

Steady-state reactors with ideal flow pattern. In an ideal isothermal tubular pZi/g-yZovv reactor (PFR) there is no axial mixing and there are no radial concentration or velocity gradients (see also Section 5.4.3). The tubular PFR can be operated as an integral reactor or as a differential reactor. The terms integral and differential concern the observed conversions and yields. The differential mode of reactor operation can be achieved by using a shallow bed of catalyst particles. The mass-balance equation (see Table 5.4-3) can then be replaced with finite differences ... 

In the next two chapters of this book we turn to the chemical reactor that is probably the most challenging the tubular or plug flow reactor. The inherent distributed nature of the unit (variables change with axial and radial position) gives rise to complex behavior, which is often counterintuitive and difficult to explain. The increase in the number of independent variables makes the development and solution of mathematical models more complex compared to the perfectly mixed CSTR and batch reactor. 

Due to the lack of published data on the special flow field generated in the LDPE tubular reactor by the end pulsing valve, the development of the mathematical model was preceded by a fluiddynamic study, with the aim of evidencing the influence, if any, of the pulsed motion on the axial mixing, the heat transfer coefficient and the pressure drop in the reactor. 

A cooled plug flow reactor when axial dispersion is negligible, a usual assumption for tubular reactors with no mixing tr, tco, C [Pg.2999]

In a plug flow reactor all fluid elements move along parallel streamlines with equal velocity. The plug flow is the only mechanism for mass transport and there is no mixing between fluid elements. The reaction therefore only leads to a concentration gradient in the axial flow direction. For steady-state conditions, for which the term IV is zero the continuity equation is a first-order, ordinary differential equation with the axial coordinate as variable. For non-steady-state conditions the continuity equation is a partial differential equation with axial coordinate and time as variables. Narrow and long tubular reactors closely satisfy the conditions for plug flow when the viscosity of the fluid is-low. 

The reactor is either a concentrated system like a well-mixed stirred tank reactor, that is, parameters such as Tand c do not vary within the whole reactor, or a distributed system, that is, the conditions depend on the local position, for example, a tubular reactor with axial gradients of c and often also of T. 

The concept of the plug flow for piston flow) reactor denotes an ideal tubular reactor, in which all fluid elements travel in one direction with exactly the same speed. They move as a plug would, or as a liquid does when propelled by a piston. That means that radial velocity gradients are negligible, and that there is no axial mixing. Even axial diffusion is neglected. 

To go from volume element models to reactor models the macro flow patterns in the reactor need to be considered. For stirred tank reactors this can be quite simple, in those cases where volume elements in the stirred tank can be described in terms of average conditions. This is not so when macro mixing or residence time distribution are scale dependent, see Chapter 7. When the reactor is tubular, with two countercurrent or parallel flows, the volume element models have to be combined with reactor flow models, including axial mixing. Also this is treated in Chapter 7, for various cases. 

In another land of ideal flow reactor, all portions of the feed stream have the same residence time that is, there is no mixing in the axial direction but complete mixing radially. It is called a.plugflow reactor (PFR), or a tubular flow reactor (TFR), because this flow pattern is characteristic of tubes and pipes. As the reaction proceeds, the concentration falls off with distance. 

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. 

Nicolas has also considered batch and tubular reactors and a cascade of stirred vessels he finds, in agreement with other workers, that a batch process produces narrower distributions than perfectly-stirred continuous reactors, but he finds that tubular reactors (assuming no axial or radial mixing) always produce material of infinite Pw, i.e. gel. 

Thirty years later, Gerhard Damkohler (1937) in his historic paper, summarized various reactor models and formulated the two-dimensional CDR model for tubular reactors in complete generality, allowing for finite mixing both in the radial and axial directions. In this paper, Damkohler used the flux-type boundary condition at the inlet and also replaced the assumption of plug flow with parabolic velocity profile, which is typical of laminar flow in tubes. 

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