## Ideal Reactors for a Single Reaction

**Ideal Reactors for a Single Reaction**

In this chapter we develop the performance equations for a single fluid reacting in the three ideal reactors shown in Fig.6. We call these homogeneous reactions. Applications and extensions of these equations to various isothermal and nonisothermal operations are considered in the following four chapters.

In the batch reactor, or BR, of Fig.6a the reactants are initially charged into a container, are well mixed, and are left to react for a certain period. The resultant mixture is then discharged. This is an unsteady-state operation where composition changes with time; however, at any instant the composition throughout the reactor is uniform.

The first of the two ideal steady-state flow reactors is variously known as the plug flow, slug flow, piston flow, ideal tubular, and unmixed flow reactor, and it is shown in Fig.6b. We refer to it as the plugpow reactor, or PFR, and to this pattern of flow as plugpow. It is characterized by the fact that the flow of fluid through the reactor is orderly with no element of fluid overtaking or mixing with any other element ahead or behind. Actually, there may be lateral mixing of fluid in a plug flow reactor; however, there must be no mixing or diffusion along the flow path. The necessary and sufficient condition for plug flow is for the residence time in the reactor to be the same for all elements of fluid.

**Figure **6 The three types of ideal reactors: (a) batch reactor, or BR; (b) plug flow reactor, or PFR; and *(c) *mixed flow reactor, or MFR.

The other ideal steady-state flow reactor is called the mixed reactor, the backmix reactor, the ideal stirred tank reactor, the *C* *(meaning C-star), CSTR, or the CFSTR (constant flow stirred tank reactor), and, as its names suggest, it is a reactor in which the contents are well stirred and uniform throughout. Thus, the exit stream from this reactor has the same composition as the fluid within the reactor. We refer to this type of flow as mixed pow, and the corresponding reactor the mixed flow reactor, or MFR. These three ideals are relatively easy to treat. In addition, one or other usually represents the best way of contacting the reactants-no matter what the operation. For these reasons, we often try to design real reactors so that their flows approach these ideals, and much of the development in this book centers about them

In the treatment to follow it should be understood that the term *V, *called the reactor volume, really refers to the volume of fluid in the reactor. When this differs from the internal volume of reactor, then *V, *designates the internal volume of reactor while V designates the volume of reacting fluid. For example, in solid catalyzed reactors with voidage * *we have

**1. ****IDEAL BATCH REACTOR**

Make a material balance for any component A. For such an accounting we usually select the limiting component. In a batch reactor, since the composition is uniform throughout at any instant of time, we may make the accounting about the whole reactor. Noting that no fluid enters or leaves the reaction mixture during reaction, Eq. 1, which was written for component A, becomes

Evaluating the terms of Eq. 7, we find

This is the general equation showing the time required to achieve a conversion **X**_{A} for either isothermal or nonisothermal operation. The volume of reacting fluid and the reaction rate remain under the integral sign, for in general they both change as reaction proceeds. This equation may be simplified for a number of situations. If the density of the fluid remains constant, we obtain

For all reactions in which the volume of reacting mixture changes proportionately with conversion, such as in single gas-phase reactions with significant density changes, Eq. **9 **becomes

Eqs. 8 to 11 are applicable to both isothermal and nonisothermal operations. For the latter the variation of rate with temperature, and the variation of temperature with conversion, must be known before solution is possible. Figure 7 is a graphical representation of two of these equations.

**Figure **7 Graphical representation of the performance equations for batch reactors, isothermal or nonisothermal

**Space-Time and Space-Velocity**

Just as the reaction time *t *is the natural performance measure for a batch reactor, so are the space-time and space-velocity the proper performance measures of flow reactors. These terms are defined as follows:

Space-time:

Space-velocity:

Thus, a space-velocity of 5 hr-l means that five reactor volumes of feed at specified conditions are being fed into the reactor per hour. A space-time of 2 min means that every 2 min one reactor volume of feed at specified conditions is being treated by the reactor.

Now we may arbitrarily select the temperature, pressure, and state (gas, liquid,or solid) at which we choose to measure the volume of material being fed to the reactor. Certainly, then, the value for space-velocity or space-time depends on the conditions selected. If they are of the stream entering the reactor, the relation between **s **and *r *and the other pertinent variables is

It may be more convenient to measure the volumetric feed rate at some standard state, especially when the reactor is to operate at a number of temperatures. If, for example, the material is gaseous when fed to the reactor at high temperature but is liquid at the standard state, care must be taken to specify precisely what state has been chosen. The relation between the space-velocity and space-time for actual feed conditions (unprimed symbols) and at

standard conditions (designated by primes) is given by In most of what follows, we deal with the space-velocity and space-time based on feed at actual entering conditions; however, the change to any other basis is easily made

**2 STEADY-STATE MIXED FLOW REACTOR**

The performance equation for the mixed flow reactor is obtained from Eq. 1, which makes an accounting of a given component within an element of volume of the system. But since the composition is uniform throughout, the accounting may be made about the reactor as a whole. By selecting reactant A for consideration, Eq. 1 becomes

input = output disappearance by reaction accumulation (15)

As shown in Fig. 8, if the molar feed rate of component A to the reactor, then considering the reactor as a whole we have

input of A, moles/time = F_{A0}(l – X_{A0}) = F_{A0}

output of A, moles/time = F_{A} = F_{A0}(l – X_{A})

disappearance of A moles A reacting volume of by reaction, = (-r_{A})V =

Introducing these three terms into Eq. 15, we obtain

F_{A0}X_{A} = (-r_{A})V

which on rearrangement becomes

**Figure **8 Notation for a mixed reactor

where X_{A} and r_{A} are measured at exit stream conditions, which are the same as the conditions within the reactor. More generally, if the feed on which conversion is based, subscript 0, enters the reactor partially converted, subscript i, and leaves at conditions given by subscript *f, *we have

For the special case of constant-density systems X_{A} = 1 - C_{A}/C_{A0in}, which case the performance equation for mixed reactors can also be written in terms of concentrations or

These expressions relate in a simple way the four terms X_{A}, -r_{A}, *V, *F_{AO}; thus, knowing any three allows the fourth to be found directly. In design, then, the size of reactor needed for a given duty or the extent of conversion in a reactor of given size is found directly. In kinetic studies each steady-state run gives, without integration, the reaction rate for the conditions within the reactor. The ease of interpretation of data from a mixed flow reactor makes its use very attractive in kinetic studies, in particular with messy reactions (e.g., multiple reactions and solid catalyzed reactions). Figure 9 is a graphical representation of these mixed flow performance equations. For any specific kinetic form the equations can be written out directly.

**Figure **9 Graphical representation of the design equations for mixed flow reactor

As an example, for constant density systems *C*_{A}/C_{A0}, = 1 - *X*_{A} thus the performance expression *for first-order reaction *becomes

On the other hand, for linear expansion

thus *for first-order reaction *the performance expression of Eq. 16 becomes

For *second-order reaction, *A ------> products, *-r*_{A} = *k C*_{A }^{2}, ^{ }*ε*_{A} = 0, the performance equation of Eq. 16 becomes

Similar expressions can be written for any other form of rate equation. These expressions can be written either in terms of concentrations or conversions. Using conversions is simpler for systems of changing density, while either form can be used for systems of constant density.

**3. ****STEADY-STATE PLUG FLOW REACTOR**

In a plug flow reactor the composition of the fluid varies from point to point along a flow path; consequently, the material balance for a reaction component must be made for a differential element of volume dV. Thus for reactant **A, **Eq. 1 becomes

input = output disappearance by reaction accumulation (22)

Referring to Fig. 10, we see for volume dV that

input of **A, **moles/time = *F*_{A}

output of **A, **moles/time = *F*_{A} *dF*_{A}

disappearance of **A **by

reaction, moles/time = (-r_{A})dV

**Figure 10** Notation for a plug flow reactor.

This, then, is the equation which accounts for A in the differential section of reactor of volume dV. For the reactor as a whole the expression must be integrated. Now F_{A0}, the feed rate, is constant, but r_{A}, is certainly dependent on the concentration or conversion of materials. Grouping the terms accordingly, we obtain

Equation 23 allows the determination of reactor size for a given feed rate and required conversion. Compare Eqs. 17 and 23. The difference is that in plug flow r_{A} varies, whereas in mixed flow r_{A} is constant. As a more general expression for plug flow reactors, if the feed on which conversion is based, subscript 0, enters the reactor partially converted, subscript i, and leaves at a conversion designated by subscript *f, *we have

in which case the performance equation can be expressed in terms of concentrations, or

**Figure **11 Graphical representation of the performance equations for plug flow reactors.

These performance equations, Eqs. 23 to 25, can be written either in terms of concentrations or conversions. For systems of changing density it is more convenient to use conversions; however, there is no particular preference for constant density systems. Whatever its form, the performance equations interrelate the rate of reaction, the extent of reaction, the reactor volume, and the feed rate, and if any one of these quantities is unknown it can be found from the other three. Figure 11 displays these performance equations and shows that the spacetime needed for any particular duty can always be found by numerical or graphical integration. However, for certain simple kinetic forms analytic integration is possible-and convenient. To do this, insert the kinetic expression for r, in Eq.23 and integrate. Some of the simpler integrated forms for plug flow are as follows: Zero-order homogeneous reaction, any constant *ε*_{A}

First-order irreversible reaction,*, *A -------> products, any constant *ε*_{A},

Where the density is constant, put *ε*_{A} = 0 to obtain the simplified performance equation.

By comparing the batch expressions with these plug flow expressions we find:

(1)For *systems of constant density *(constant-volume batch and constant-density plug flow) the performance equations are identical, *T *for plug flow is equivalent to *t *for the batch reactor, and the equations can be used interchangeably.

(2) For *systems of changing density *there is no direct correspondence between the batch and the plug flow equations and the correct equation must be used for each particular situation. In this case the performance equations cannot be used interchangeably.