 1.1 The Rate of Reaction, rA
 1.2 The General Mole Balance Equation
 1.3 Batch Reactors (BRs)
 1.4 ContinuousFlow Reactors
 1.5 Industrial Reactors
1.4 ContinuousFlow Reactors
Continuous flow reactors are almost always operated at steady state. We will consider three types: the continuousstirred tank reactor (CSTR), the plug flow reactor (PFR), and the packedbed reactor (PBR). Detailed physical descriptions of these reactors can be found in both the Professional Reference Shelf (PRS) for Chapter 1 and in the Visual Encyclopedia of Equipment on the DVDROM.
1.4.1 ContinuousStirred Tank Reactor (CSTR)
A type of reactor used commonly in industrial processing is the stirred tank operated continuously (Figure 17). It is referred to as the continuousstirred tank reactor (CSTR) or vat, or backmix reactor, and is used primarily for liquid phase reactions. It is normally operated at steady state and is assumed to be perfectly mixed; consequently, there is no time dependence or position dependence of the temperature, concentration, or reaction rate inside the CSTR. That is, every variable is the same at every point inside the reactor. Because the temperature and concentration are identical everywhere within the reaction vessel, they are the same at the exit point as they are elsewhere in the tank. Thus, the temperature and concentration in the exit stream are modeled as being the same as those inside the reactor. In systems where mixing is highly nonideal, the wellmixed model is inadequate, and we must resort to other modeling techniques, such as residencetime distributions, to obtain meaningful results. This topic of nonideal mixing is discussed in DVDROM Chapters DVD13 and DVD14, on the DVDROM included with this text, and in Chapters 13 and 14 in the fourth edition of The Elements of Chemical Reaction Engineering (ECRE).
Figure 17(a) CSTR/batch reactor. [Courtesy of Pfaudler, Inc.]
Figure 17(b) CSTR mixing patterns. Also see the Visual Encyclopedia of Equipment on the DVDROM.
When the general mole balance equation
Equation 14
is applied to a CSTR operated at steady state (i.e., conditions do not change with time),
in which there are no spatial variations in the rate of reaction (i.e., perfect mixing),
it takes the familiar form known as the design equation for a CSTR:
Equation 17
The CSTR design equation gives the reactor volume V necessary to reduce the entering flow rate of species j from F_{j} _{0} to the exit flow rate F_{j} , when species j is disappearing at a rate of –r_{j} . We note that the CSTR is modeled such that the conditions in the exit stream (e.g., concentration, and temperature) are identical to those in the tank. The molar flow rate F_{j} is just the product of the concentration of species j and the volumetric flow rate u:
Equation 18
Similarly, for the entrance molar flow rate we have F_{j}_{0} = C_{j}_{0} · v_{0}. Consequently, we can substitute for F_{j} _{0} and F_{j} into Equation (17) to write a balance on species A as
Equation 19
The ideal CSTR mole balance equation is an algebraic equation, not a differential equation.
1.4.2 Tubular Reactor
In addition to the CSTR and batch reactors, another type of reactor commonly used in industry is the tubular reactor. It consists of a cylindrical pipe and is normally operated at steady state, as is the CSTR. Tubular reactors are used most often for gasphase reactions. A schematic and a photograph of industrial tubular reactors are shown in Figure 18.
In the tubular reactor, the reactants are continually consumed as they flow down the length of the reactor. In modeling the tubular reactor, we assume that the concentration varies continuously in the axial direction through the reactor. Consequently, the reaction rate, which is a function of concentration for all but zeroorder reactions, will also vary axially. For the purposes of the material presented here, we consider systems in which the flow field may be modeled by that of a plug flow profile (e.g., uniform velocity as in turbulent flow), as shown in Figure 19. That is, there is no radial variation in reaction rate, and the reactor is referred to as a plugflow reactor (PFR). (The laminar flow reactor is discussed on the DVDROM in Chapter DVD13 and in Chapter 13 of the fourth edition of ECRE.)
Figure 18(a) Tubular reactor schematic. Longitudinal tubular reactor. [Excerpted by special permission from Chem. Eng., 63 (10), 211 (Oct. 1956). Copyright 1956 by McGrawHill, Inc., New York, NY 10020.]
Figure 18(b) Tubular reactor photo. Tubular reactor for production of Dimersol G. [Photo Courtesy of Editions Techniq Institut français du pétrole].
Figure 19 Plugflow tubular reactor.
The general mole balance equation is given by Equation (14):
Equation 14
The equation we will use to design PFRs at steady state can be developed in two ways: (1) directly from Equation (14) by differentiating with respect to volume V, and then rearranging the result or (2) from a mole balance on species j in a differential segment of the reactor volume ΔV. Let's choose the second way to arrive at the differential form of the PFR mole balance. The differential volume, ΔV shown in Figure 110, will be chosen sufficiently small such that there are no spatial variations in reaction rate within this volume. Thus the generation term, ΔG_{j}, is
Figure 110 Mole balance on species j in volume Δ V.
Equation 110
Dividing by ΔV and rearranging
the term in brackets resembles the definition of a derivative
Taking the limit as ΔV approaches zero, we obtain the differential form of steady state mole balance on a PFR.
Equation 111
We could have made the cylindrical reactor on which we carried out our mole balance an irregular shape reactor, such as the one shown in Figure 111 for reactant species A.
Figure 111 Pablo Picasso's reactor.
However, we see that by applying Equation (110), the result would yield the same equation (i.e., Equation [111]). For species A, the mole balance is
Equation 112
Consequently, we see that Equation (111) applies equally well to our model of tubular reactors of variable and constant crosssectional area, although it is doubtful that one would find a reactor of the shape shown in Figure 111 unless it were designed by Pablo Picasso.
The conclusion drawn from the application of the design equation to Picasso's reactor is an important one: the degree of completion of a reaction achieved in an ideal plugflow reactor (PFR) does not depend on its shape, only on its total volume.
Again consider the isomerization A B, this time in a PFR. As the reactants proceed down the reactor, A is consumed by chemical reaction and B is produced. Consequently, the molar flow rate F _{A} decreases, while F _{B} increases as the reactor volume V increases, as shown in Figure 112.
Figure 112 Profiles of molar flow rates in a PFR.
We now ask what is the reactor volume V _{1} necessary to reduce the entering molar flow rate of A from F _{A0} to F _{A1}. Rearranging Equation (112) in the form
and integrating with limits at V = 0, then F _{A} = F _{A0}, and at V = V _{1}, then F _{A} = F _{A1}.
Equation 113
V_{1} is the volume necessary to reduce the entering molar flow rate F_{A0} to some specified value F_{A1} and also the volume necessary to produce a molar flow rate of B of F_{B1}.
1.4.3 PackedBed Reactor (PBR)
The principal difference between reactor design calculations involving homogeneous reactions and those involving fluidsolid heterogeneous reactions is that for the latter, the reaction takes place on the surface of the catalyst (see Chapter 10). Consequently, the reaction rate is based on mass of solid catalyst, W, rather than on reactor volume, V. For a fluid–solid heterogeneous system, the rate of reaction of a species A is defined as
= mol A reacted/(time x mass of catalyst)
The mass of solid catalyst is used because the amount of catalyst is what is important to the rate of product formation. The reactor volume that contains the catalyst is of secondary significance. Figure 113 shows a schematic of an industrial catalytic reactor with vertical tubes packed with solid catalyst.
Figure 113 Longitudinal catalytic packedbed reactor. [From Cropley, American Institute of Chemical Engineers, 86(2), 34 (1990). Reproduced with permission of the American Institute of Chemical Engineers, Copyright © 1990 AIChE. All rights reserved.]
In the three idealized types of reactors just discussed (the perfectly mixed batch reactor, the plugflow tubular reactor [PFR]), and the perfectly mixed continuousstirred tank reactor [CSTR]), the design equations (i.e., mole balances) were developed based on reactor volume. The derivation of the design equation for a packedbed catalytic reactor (PBR) will be carried out in a manner analogous to the development of the tubular design equation. To accomplish this derivation, we simply replace the volume coordinate in Equation (110) with the catalyst mass (i.e., weight) coordinate W (Figure 114).
Figure 114 Packedbed reactor schematic.
As with the PFR, the PBR is assumed to have no radial gradients in concentration, temperature, or reaction rate. The generalized mole balance on species A over catalyst weight ΔW results in the equation
Equation 114
The dimensions of the generation term in Equation (114) are
which are, as expected, the same dimensions of the molar flow rate F _{A}. After dividing by ΔW and taking the limit as ΔW 0, we arrive at the differential form of the mole balance for a packedbed reactor:
Equation 115
When pressure drop through the reactor (see Section 5.5) and catalyst decay (see Section 10.7 in DVDROM Chapter 10) are neglected, the integral form of the packedcatalystbed design equation can be used to calculate the catalyst weight.
Equation 116
W is the catalyst weight necessary to reduce the entering molar flow rate of species A, F _{A0}, down to a flow rate F _{A}.
For some insight into things to come, consider the following example of how one can use the tubular reactor design in Equation (111).
Example 1–2. How Large Is It?

Consider the liquid phase cis – trans isomerization of 2–butene
which we will write symbolically as
The reaction is first order in A (–r _{A} = kC _{A}) and is carried out in a tubular reactor in which the volumetric flow rate, v, is constant, i.e., v= v_{0}.
 Sketch the concentration profile.
 Derive an equation relating the reactor volume to the entering and exiting concentrations of A, the rate constant k, and the volumetric flow rate v _{0}.
 Determine the reactor volume necessary to reduce the exiting concentration to 10% of the entering concentration when the volumetric flow rate is 10 dm^{3}/min (i.e., liters/min) and the specific reaction rate, k, is 0.23 min^{–1}.
Solution

Sketch C _{A} as a function of V.
Species A is consumed as we move down the reactor, and as a result, both the molar flow rate of A and the concentration of A will decrease as we move. Because the volumetric flow rate is constant, v = v_{0}, one can use Equation (18) to obtain the concentration of A, C _{A} = F _{A}/v_{0}, and then by comparison with Figure 112 plot, the concentration of A as a function of reactor volume, as shown in Figure E12.1.
Figure E12.1 Concentration profile.

Derive an equation relating V, v_{0}, k, C _{A0}, and C _{A}.
For a tubular reactor, the mole balance on species A (j = A) was shown to be given by Equation (111). Then for species A (j = A)
Equation 112
For a firstorder reaction, the rate law (discussed in Chapter 3) is
Equation E12.1
Because the volumetric flow rate, v, is constant (v = v0), as it is for most all liquidphase reactions,
Equation E12.2
Multiplying both sides of Equation (E12.2) by minus one and then substituting Equation (E12.1) yields
Equation E12.3
Separating the variables and rearranging gives
Using the conditions at the entrance of the reactor that when V = 0, then C _{A} = C _{A0},
Equation E12.4
Carrying out the integration of Equation (E12.4) gives
Equation E12.5
We can also rearrange Equation (E12.5) to solve for the concentration of A as a function of reactor volume to obtain
C = C _{Ao}exp(–kV/v_{0})

Calculate V. We want to find the volume, V_{1}, at which for k = 0.23 min^{–1} and v_{0} = 10 dm^{3}/min.
Substituting C _{A0}, C _{A}, v_{0}, and k in Equation (E12.5), we have
Let's calculate the volume to reduce the entering concentration to C _{A} = 0.01 C _{A0}. Again using equation (E12.5)
Note: We see that a larger reactor (200 dm^{3}) is needed to reduce the exit concentration to a smaller fraction of the entering concentration (e.g., C _{A} = 0.01 C _{A0}).
We see that a reactor volume of 0.1 m^{3} is necessary to convert 90% of species A entering into product B for the parameters given.
 Analysis: For this irreversible liquidphase first order reaction (i.e., –r_{A} = kC_{A}) being carried out in a PFR, the concentration of the reactant decreases exponentially down the length (i.e., volume V) of the reactor. The more species A consumed and converted to product B, the larger must be the reactor volume V. The purpose of the example was to give a vision of the types of calculations we will be carrying out as we study chemical reaction engineering (CRE).