Piston Flow Reactors

The piston flow reactor has an advantage over a stirred tank reactor when the kinetics is of positive order, but the reverse is true when the... 

In Fig. 28, the abscissa kt is the product of the reaction rate constant and the reactor residence time, which is proportional to the reciprocal of the space velocity. The parameter k co is the product of the CO inhibition parameter and inlet concentration. Since k is approximately 5 at 600°F these three curves represent c = 1, 2, and 4%. The conversion for a first-order kinetics is independent of the inlet concentration, but the conversion for the kinetics of Eq. (48) is highly dependent on inlet concentration. As the space velocity increases, kt decreases in a reciprocal manner and the conversion for a first-order reaction gradually declines. For the kinetics of Eq. (48), the conversion is 100% at low space velocities, and does not vary as the space velocity is increased until a threshold is reached with precipitous conversion decline. The conversion for the same kinetics in a stirred tank reactor is shown in Fig. 29. For the kinetics of Eq. (48), multiple solutions may be encountered when the inlet concentration is sufficiently high. Given two reactors of the same volume, and given the same kinetics and inlet concentrations, the conversions are compared in Fig. 30. The piston flow reactor has an advantage over the stirred tank... 

Kinetics c/c0 in piston flow reactor Si, Sc o c/c0 in stirred tank reactor St,... 

These two parameters describe the change in fraction unconverted with a percentage change in kt or in c0. The first sensitivity is also the slope of the curves in Fig. 28. The values of these sensitivities are given in Table IX. In a piston flow reactor where the conversion level is c/c0 = 0.1, the value of Stt is —0.23 for the first-order kinetics, —0.90 for the zero-order kinetics, and —4.95 for the negative first-order kinetics. In the stirred tank reactor, the value of the sensitivities Skt is —0.09 for the first-order kinetics, — 0.90 for the zero-order kinetics, and +0.11 for the negative first-order kinetics. A positive sensitivity means that as kt is increased, the fraction unconverted also increases, clearly an unstable situation. 

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... 

Example 1.3 Find the outlet concentration of component A from a piston flow reactor assuming that A is consumed by a first-order reaction. 

We now formalize the definition of piston flow. Denote position in the reactor using a cylindrical coordinate system (r, 6, z) so that the concentration at a point is denoted as a(r, 9, z) For the reactor to be a piston flow reactor (also called plug flow reactor, slug flow reactor, or ideal tubular reactor), three conditions must be satisfied ... 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

FIGURE 1.4 Differential element in a piston flow reactor. 

Example 1.4 Determine the reactor design equations for the various elementary reactions in a piston flow reactor. Assume constant temperature, constant density, and constant reactor cross section. (Whether or not all these assumptions are needed will be explored in subsequent chapters.)... 

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

Piston flow reactors and most other flow reactors have spatial variations in concentration such as a = a(z). Such systems are called distributed. Their... 

Figures 1.6 and 1.7 display the conversion behavior for flrst-and second-order reactions in a CSTR and contrast the behavior to that of a piston flow reactor. It is apparent that piston flow is substantially better than the CSTR for obtaining high conversions. The comparison is even more dramatic when made in terms of the volume needed to achieve a given conversion see Figure 1.8. The generalization that...

Equation (1.45) gives the spatial distribution of concentration, u(z), in a piston flow reactor for a component that is consumed by a first-order reaction. The local concentration can be used to determine the local reaction rate, S Aiz)-... 

Autocatalytic reactions often show higher conversions in a stirred tank than in either a batch flow reactor or a piston flow reactor with the same holding time, tjjatch = i. Since d = agut in a CSTR, the catalyst, B, is present at the... 

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. 

FIGURE 3.1 Differential volume elements in piston flow reactors (a) variable cross section (b) constant cross section. 

Example 3.2 Consider the reaction 2A B. Derive an analytical expression for the fraction unreacted in a gas-phase, isothermal, piston flow reactor of length L. The pressure drop in the reactor is negligible. 

Computational Scheme for Gas-Phase PFRs. A general procedure for solving the reactor design equations for a piston flow reactor using the marching-ahead technique (Euler s method) has seven steps ... 

Solution of the design equations for liquid-phase piston flow reactors is usually easier than for gas-phase reactors because pressure t5q)icaUy has no effect on the fluid density or the reaction kinetics. Extreme pressures are an exception that theoretically can be handled by the same methods used for gas-phase systems. The difficulty will be finding an equation of state. For ordinary pressures, the... 

If the pilot reactor is turbulent and closely approximates piston flow, the larger unit will as well. In isothermal piston flow, reactor performance is determined by the feed composition, feed temperature, and the mean residence time in the reactor. Even when piston flow is a poor approximation, these parameters are rarely, if ever, varied in the scaleup of a tubular reactor. The scaleup factor for throughput is S. To keep t constant, the inventory of mass in the system must also scale as S. When the fluid is incompressible, the volume scales with S. The general case allows the number of tubes, the tube radius, and the tube length to be changed upon scaleup ... 

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