Radial dispersion flow reactors

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R > 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

The solution of Eq. (173) poses a rather formidable task in general. Thus the dispersed plug-flow model has not been as extensively studied as the axial-dispersed plug-flow model. Actually, if there are no initial radial gradients in C, the radial terms will be identically zero, and Eq. (173) will reduce to the simpler Eq. (167). Thus for a simple isothermal reactor, the dispersed plug flow model is not useful. Its greatest use is for either nonisothermal reactions with radial temperature gradients or tube wall catalysed reactions. Of course, if the reactants were not introduced uniformly across a plane the model could be used, but this would not be a common practice. Paneth and Herzfeld (P2) have used this model for a first order wall catalysed reaction. The boundary conditions used were the same as those discussed for tracer measurements for radial dispersion coefficients in Section II,C,3,b, except that at the wall. 

In an adiabatic fixed bed, heat is not exchanged with the environment through the reactor wall. Note that for the derivation of eq. (5.226), it has been assumed that the flow is ideal plug flow and thus the radial dispersion term is eliminated in an adiabatic fixed bed, the assumption of perfect radial mixing is not necessary since no radial gradients exist. 

Radial dispersion Large-diameter reactors with low flow rates (Usually ignored in preliminary models)... 

The space velocity for a given conversion is often used as a ready measure of the performance of a reactor. The use of equation 1.25 to calculate reaction time, as if for a batch reactor, is not to be recommended as normal practice it can be equated to VJv only if there is no change in volume. Further, the method of using reaction time is a blind alley in the sense that it has to be abandoned when the theory of tubular reactors is extended to take into account longitudinal and radial dispersion and other departures from the plug flow hypothesis which are important in the design of catalytic tubular reactors (Chapter 3, Section 3.6.1)... 

The importance of dispersion and its influence on flow pattern and conversion in homogeneous reactors has already been studied in Chapter 2. The role of dispersion, both axial and radial, in packed bed reactors will now be considered. A general account of the nature of dispersion in packed beds, together with details of experimental results and their correlation, has already been given in Volume 2, Chapter 4. Those features which have a significant effect on the behaviour of packed bed reactors will now be summarised. The equation for the material balance in a reactor will then be obtained for the case where plug flow conditions are modified by the effects of axial dispersion. Following this, the effect of simultaneous axial and radial dispersion on the non-isothermal operation of a packed bed reactor will be discussed. 

The third and fourth condition are fulfilled by Tarhan [25]. Axial dispersion is fundamentally local backmixing of reactants and products in the axial, or longitudinal direction in the small interstices of the packed bed, which is due to molecular diffusion, convection, and turbulence. Axial dispersion has been shown to be negligible in fixed-bed gas reactors. The fourth condition (no radial dispersion) can be met if the flow pattern through the bed already meets the second condition. If the flow velocity in the axial direction is constant through the entire cross section and if the reactor is well insulated (first condition), there can be no radial dispersion to speak of in gas reactors. Thus, the one-dimensional adiabatic reactor model may be actualized without great difficulties. ... 

A plug-flow reactor models the conventional plug-flow behavior, assuming radial mixing but no axial dispersion. The reaction kinetics must be specified, and the model has the same limitations as the CSTR model. 

In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

Cao = 0,5 mol/dm. Let s assume the radial dispersion coefficient is equal to the molecular diffusivity. Keeping everything else constant, the average outlet conversion is 52.3%. However, because the flow inside the reactor is modeled as plug flow the concentration profiles are still flat, as shown in Figure E14-3.2. 

The simplest heterogeneous model is that with plug flow in the fluid phase and only external mass and heat transfer resistances between the bulk fluid and the catalyst surface. More complex fluid phase behaviour can be accommodated by including axial and radial dispersion mechanisms into the mode). If tJie reactor is non-adiabatic, radial dispersion is usually more important. 

The above discussion can be illustrated for an isothermal packed bed tubular reactor with negligible diffusional resistances ( /= 1.0) and negligible axial dispersion (plug flow) and instantaneous radial dispersion (one-dimensional model) ... 

The reactor is in plug flow with negligible axial and infinite radial dispersion. 

Figure 5 shows some impellers typically used for low-viscosity mixtures. The six-blade Rushton turbine [8] is commonly used for gas dispersion and to promote gas-liquid mass transfer and heat transfer. It is, however, not often used for the suspension of solids. This stirrer provides mainly radial fluid flow within the reactor. 

In these equations D represents the corresponding diffusion coefficients, and Q the permeate flow rate. The first term of each equation gives the radial dispersion and the second one corresponds to the radial convection. The authors [5.103] used in their model, a biological kinetic rate expression (cp), which was obtained by independent experiments and analysis of a batch reactor, and also made an effort to account for and correlate the permeate flow decrease with the amount of produced biomass. The simulation curves obtained matched well the experimental results in terms of permeate flow rate evolution and product concentration. One of the important aspects of the model is its ability to theoretically determine the biomass concentration profiles, and the relation between the permeate flow rate and the calculated biomass concentration in the annular volume (Fig. 5.24). Such information is important since the biomass evolution cannot be determined by any experimental methodology. 

Quantitative treatment of plug flow reactors is somewhat cumbersome, therefore several assumptions are usually made. The fluid composition is considered to be unform along the reactor cross section (i.e. there is no radial dispersion). This is valid only when... 

The design of such gas-solid catalytic reactors can be approximated by a pseudo-homogeneous model with gas phase in plug flow. In the case of very exothermic reactions accounting for radial dispersion of heat and mass might be useful to prevent excessive particle overheating. The reaction time must find a compromise with the hydrodynamic design, namely the maximum gas velocity and pressure drop. 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

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