Adiabatic reactor, axial dispersion model

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

Compare Equation (11.42) with Equation (9.1). The standard model for a two-phase, packed-bed reactor is a PDE that allows for radial dispersion. Most trickle-bed reactors have large diameters and operate adiabatically so that radial gradients do not arise. They are thus governed by ODEs. If a mixing term is required, the axial dispersion model can be used for one or both of the phases. See Equations (11.33) and (11.34). 

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. 

Vanden Bussche and Proment [13] simulated an adiabatic Bench Scale Reactor using a pseudo-homogeneous one-dimensional model. Using a similar pseudo-homogeneous axial dispersion model Jakobsen et al [5] obtained axial concentration and temperature profiles that were hardly distinguishable from the pseudo-homogeneous one-dimensional model results of Vanden Bussche and Proment [13]. 

Packed-bed reactors can be adiabatic, and Equation 9.3 takes a particularly simple form with no radial gradients in temperature or composition arising when the feed is premixed. If the fluid is uniform in the radial direction when it enters the reactor, it remains uniform. Thus adiabatic packed beds are normally modeled as PERs. This assumption may be overly optimistic in terms of yields and selectivities. The axial dispersion model in Section 9.2 adds a correction term to avoid undue optimism. Unmixed feed streams can also be treated provided the reactants enter the reactor in a manner that preserves radial symmetry. 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

It was pointed out by Young and Finlayson [55] that for reactors with finite wall heat transfer, the ultimate conditions for a large reactor will be determined by the heat exchange, whereas in isothermal and adiabatic situations the concentrations and temperatures are determined only by the reaction, and so for the latter case, the differences between plug flow and axial dispersion model results always diminish with increasing reactor length. But, differences between the two models at the entrance may persist for reactors with wall heat transfer. They provide alternate criteria for the unimportance of axial dispersion (of both mass and heat) effects. Mears [59] pointed out that these new criteria were not general, and provided alternate ones for equal feed and wall temperatures. The new criteria are... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

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. ... 

All simulators provide one-dimensional, plug-flow models that neglect axial dispersion Thus, there are no radial gradients of temperature, composition, or pressure and mass diffusion and heat conduction do not occur in the axial direction. Operation of the reactor can bt adiabatic, isothermal, or nonadiabatic, nonisothermal. For the latter, heat transfer to or fron the reacting mixture occurs along the length of the reactor. 

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. 

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