Adiabatic Reactor with Plug Flow

For simple irreversible reactions, a (semi) analytical solution of the continuity and energy equations is possible. Douglas and Eagleton [1962] published solutions for zero-, first-, and second-order reactions, both with a constant and varying number of moles. For a first-order reaction with constant density, the integration proceeds as follows  

Note the simple relation between the conversion and the temperature variation in adiabatic situations the temperature variation is a measure of conversion, and vice versa. 

After substituting dx by its expression based on (9.3.1-3) and of k by its Arrhenius epression, the following equation is obtained  

For given feed conditions, (9.3.1-5) permits the calculation of V/Fao, which limits the outlet temperature and therefore the outlet conversion to a set value. Obviously for a given V/Fao, one can calculate the corresponding outlet conditions, but the expression is implicit with respect to T. 

For more complicated rate equations, semianalytical integration is no longer possible. 

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Example 93-1 An Adiabatic Reactor with Plug Flow Conditions... 

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 reactor is assumed to be adiabatic with plug flow. Axial dispersion can be ignored. Any effect of limitations of mass or heat transfer inside the catalyst pellet is lumped into the rate constants given in Table 1. The catalyst activity is assumed to be constant. Use the conversion of ethylbenzene or water in the set of continuity equations. Use the Ergun equation to describe the pressure drop. 

EXAMPLE 9J ADIABATIC PLUG FLOW REACTOR WITH RECYCLE... 

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. 

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. 

While the adiabatic batch reactor is important and presents many control issues in its own right, we are concerned here primarily with continuous systems. We consider in detail two distinct reactor types the continuous stirred tank reactor (CSTRj and the plug-flow reactor. They differ fundamentally in the way the reactants and the products... 

We noted earlier in this chapter that many reactions in the chemical industries are exothermic and require heat removal. A simple way of meeting this objective is to design an adiabatic reactor. The reaction heat is then automatically exported with the hot exit stream. No control system is required, making this a preferred way of designing the process. However, adiabatic operation may not always be feasible. In plug-flow systems the exit temperature may be too hot due to a minimum inlet temperature and the adiabatic temperature rise. Systems with baekmixing suffer from other problems in that they face the awkward possibilities of multiplicity and open-loop instability. The net result is that we need external cooling on many industrial reactors. This also carries with it a control system to ensure that the correct amount of heat is removed at all times. 

Figure 4.27 shows another method of controlling plug-flow systems. Instead of cooling the effluents from each adiabatic step in a heat exchanger, we introduce a cold shot of fresh feeds. The cold shot technique increases the concentration of reactant in all the segments. Mixing the cold feed with the reactor effluent lowers the inlet temperature to the next reactor. 

In Chap. 4 we mentioned that the simplest reactor type from a control viewpoint is the adiabatic plug-flow reactor. It does not suffer from output multiplicity, open-loop instability, or hot-spot sensitivity. Furthermore, it is dominated by the inlet temperature that is easy to control for an isolated unit. The only major issue with this reactor type is the risk of achieving high exit temperatures due to a large adiabatic temperature rise. As we recall from Chap. 4, the adiabatic temperature rise is proportional to the inlet concentration of the reactants and inversely proportional to the heat capacity of the feed stream. WTe can therefore limit the temperature rise by diluting the reactants with a heat carrier. 

It now looks as if we have achieved the best of all worlds a thermally efficient process with an easy-to-control reactor Can this be true Not quite. What we forget are the undesirable effects on the reactor that thermal feedback introduces. In Chap. 4 we explained in detail how7 process feedback is responsible for the same issues we tried to avoid in the first place by selecting an adiabatic plug-flow reactor. It is necessary that we take a close look at the steady-state and dynamic characteristics of FEHE systems. 

Just as we approached reactor control in Chap. 4, we will start by exploring the open-loop effects of thermal feedback. Consider Fig. 5.19, which shows an adiabatic plug-flow reactor with an FEHE system. We have also included two manipulated variables that wall later turn out to be useful to control the reactor. One of these manipulated variables is the heat load to the furnace and the other is the bypass around the preheater. It is clear that the reactor feed temperature is affected by the bypass valve position and the furnace heat load but also by the reactor exit temperature through the heat exchanger. This creates the possibility for multiple steady states. We can visualize the different... 

Figure 5.19 Adiabatic plug-flow reactor with feed-effluent heat exchanger and trim heater.

Figure 5.2S Control of packed adiabatic plug-flow reactor with FEHE and bypass control.

A model for an adiabatic HDS reactor (see Fig. 4-7) with a single quench is given by Shah et al.46 Under plug-flow conditions and assuming that there are no external mass-transfer resistances, the governing material and energy-balance equations are... 

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