Batch and Piston Flow Reactors

Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is 

Example 7.4 The following data have been obtained in a constant-volume, isothermal reactor for a reaction with known stoichiometry A B - - C. The initial concentration of component A was 2200 mol/m. No B or C was charged to the reactor. 

Solution A suitable rate expression is = —fea . Equation (7.14) can be integrated analytically or numerically. Equation (7.8) takes the following form for H / 1  

Reaction order n Rate constant Standard deviation cr 

The fit with H= 1.53 is quite good. The results for the fits with n = 1 andn = 2 show systematic deviations between the data and the fitted model. The reaction order is approximately 1.5, and this value could be used instead of n= 1.53 with nearly the same goodness of fit, a = 0.00654 versus 0.00646. This result should motivate a search for a mechanism that predicts an order of 1.5. Absent such a mechanism, the best-fit value of 1.53 may as well be retained. 

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TABLE 1.1 Relationships between Batch and Piston Flow Reactors... 

In the usual case, t and ain will be known. Equation (1.49) is an algebraic equation that can be solved for aout. If the reaction rate depends on the concentration of more than one component, versions of Equation (1.49) are written for each component and the resulting set of equations is solved simultaneously for the various outlet concentrations. Concentrations of components that do not alfect the reaction rate can be found by writing versions of Equation (1.49) for them. As for batch and piston flow reactors, stoichiometry is used to relate the rate of formation of a component, say Sl-c, to the rate of the reaction SI, using the stoichiometric coefficient vc, and Equation (1.13). After doing this, the stoichiometry takes care of itself. 

Table 1.1 Relationships between batch and piston flow reactors...

If several reactive components are involved, a version of Equation (8.12) should be written for each component. Thus, for complex reactions involving N components, it is necessary to solve N simultaneous PDEs (partial differential equations). For batch and piston flow reactors, the task is to solve N simultaneous ODEs. Stoichiometric relationships and the reaction coordinate method can be used to eliminate one or more of the ODEs, but this elimination is not generally possible for PDEs. Except for the special case where all the diffusion coefficients are equal, = , stoichiometric relationships should not... 

When kinetic measurements are made in batch or piston flow reactors, the reaction rate is not determined directly. Instead, an integral of the rate is measured, and the rate itself must be inferred. The general approach is as follows ... 

One other type of reactor allows this in principle. Dijferential reactors are so short that concentrations and temperatures do not change appreciably from their inlet values. However, the small change in concentration makes it very hard to determine an accurate rate. The use of dilferential reactors is not recommended. If a CSTR cannot be used, a batch or piston flow reactor is preferred over a dilferential reactor even though the reaction rate is not measured directly but must be inferred from measured outlet concentrations. 

When kinetic measnrements are made in batch or piston flow reactors, the reaction rate is not directly determined. Instead, an integral of the rate is measured, and the rate itself mnst be inferred. All the parameters of the model must be specified, for example, 5 odei(, m, n,r, s,ko, Tact), but this will be done by the optimization rontine. The integration can be done analytically in simple cases or numerically in more complicated cases. For a batch reactor. 

There are two important types of ideal, continuous-flow reactors the piston flow reactor or PFR, and the continuous-flow stirred tank reactor or CSTR. They behave very diflerently with respect to conversion and selectivity. The piston flow reactor behaves exactly like a batch reactor. It is usually visualized as a long tube as illustrated in Figure 1.3. Suppose a small clump of material enters the reactor at time t = 0 and flows from the inlet to the outlet. We suppose that there is no mixing between this particular clump and other clumps that entered at different times. The clump stays together and ages and reacts as it flows down the tube. After it has been in the piston flow reactor for t seconds, the clump will have the same composition as if it had been in a batch reactor for t seconds. The composition of a batch reactor varies with time. The composition of a small clump flowing through a piston flow reactor varies with time in the same way. It also varies with position down the tube. The relationship between time and position is... 

The circumflex over a and b allows for spatial variations. It can be ignored when the contents are perfectly mixed. Equation (2.36) is the form normally used for batch reactors where d = a t). It can be applied to piston flow reactors by setting ao = Ui and d = a z), and to CSTRs by setting ao = and d = Uout-... 

Chapter 2 developed a methodology for treating multiple and complex reactions in batch reactors. The methodology is now applied to piston flow reactors. Chapter 3 also generalizes the design equations for piston flow beyond the simple case of constant density and constant velocity. The key assumption of piston flow remains intact there must be complete mixing in the direction perpendicular to flow and no mixing in the direction of flow. The fluid density and reactor cross section are allowed to vary. The pressure drop in the reactor is calculated. Transpiration is briefly considered. Scaleup and scaledown techniques for tubular reactors are developed in some detail. 

Chapter 1 treated the simplest type of piston flow reactor, one with constant density and constant reactor cross section. The reactor design equations for this type of piston flow reactor are directly analogous to the design equations for a constant-density batch reactor. What happens in time in the batch reactor happens in space in the piston flow reactor, and the transformation t = z/u converts one design equation to the other. For component A,... 

All the results obtained for isothermal, constant-density batch reactors apply to isothermal, constant-density (and constant cross-section) piston flow reactors. Just replace t with z/u, and evaluate the outlet concentration at z = L. Equivalently, leave the result in the time domain and evaluate the outlet composition t = L/u. For example, the solution for component B in the competitive reaction sequence of... 

The emphasis in this chapter is on the generalization of piston flow to situations other than constant velocity down the tube. Real reactors can closely approximate piston flow reactors, yet they show many complications compared with the constant-density and constant-cross-section case considered in Chapter 1. Gas-phase tubular reactors may have appreciable density differences between the inlet and outlet. The mass density and thus the velocity down the tube can vary at constant pressure if there is a change in the number of moles upon reaction, but the pressure drop due to skin friction usually causes a larger change in the density and velocity of the gas. Reactors are sometimes designed to have variable cross sections, and this too will change the density and velocity. Despite these complications, piston flow reactors remain closely akin to batch reactors. There is a one-to-one correspondence between time in a batch and position in a tube, but the relationship is no longer as simple as z = ut. 

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