Before proceeding to Chapter 3, some terms and equations commonly used in reaction engineering need to be defined. We also consider the special case of the plug-flow design equation when the volumetric flow rate is constant.
2.6.1 Space Time
The space time tau, τ, is obtained by dividing the reactor volume by the volumetric flow rate entering the reactor
τ is an important quantity!
The space time is the time necessary to process one reactor volume of fluid based on entrance conditions. For example, consider the tubular reactor shown in Figure 2-12, which is 20 m long and 0.2 m3 in volume. The dashed line in Figure 2-12 represents 0.2 m3 of fluid directly upstream of the reactor. The time it takes for this upstream fluid volume to enter the reactor completely is called the space time tau. It is also called the holding time or mean residence time.
Space time or mean residence time
τ = V/υ0
For example, if the reactor volume is 0.2 m3 and the inlet volumetric flow rate is 0.01 m3/s, it would take the upstream equivalent reactor volume (V = 0.2 m3), shown by the dashed lines, a time τ equal to
to enter the reactor (V = 0.2 m3). In other words, it would take 20 s for the fluid molecules at point a to move to point b, which corresponds to a space time of 20 s. We can substitute for FA0 = υ0CA0 in Equations (2-13) and (2-16) and then divide both sides by υ0 to write our mole balance in the following forms:
For a PFR
and
For a CSTR
For plug flow, the space time is equal to the mean residence time in the reactor, tm (see Web PDF Chapter 16). This time is the average time that the molecules spend in the reactor. A range of typical processing times in terms of the space time (residence time) for industrial reactors is shown in Table 2-4.
TABLE 2-4 TYPICAL SPACE TIME FOR INDUSTRIAL REACTORS2
Reactor Type
Mean Residence Time Range
Production Capacity
Batch
15 min to 20 h
Few kg/day to 100,000 tons/year
CSTR
10 min to 4 h
10 to 3,000,000 tons/year
Tubular
0.5 s to 1 h
50 to 5,000,000 tons/year
2 Trambouze, Landeghem, and Wauquier, Chemical Reactors (Paris: Editions Technip, 1988; Houston: Gulf Publishing Company, 1988), p. 154.
Practical guidelines
Table 2-5 shows an order of magnitude of the space times for six industrial reactions and associated reactors.
† The reactor is tubular but the flow may or may not be ideal plug flow.
3 Walas, S. M. Chemical Reactor Data, Chemical Engineering, 79 (October 14, 1985).
Typical industrial reaction space times
Table 2-6 gives typical sizes for batch and CSTR reactors (along with the comparable size of a familiar object) and the costs associated with those sizes. All reactors are glass lined and the prices include heating/cooling jacket, motor, mixer, and baffles. The reactors can be operated at temperatures between 20 and 450°F, and at pressures up to 100 psi.
TABLE 2-6 REPRESENTATIVE PFAUDLER CSTR/BATCH REACTOR SIZES AND PRICES†
might be regarded at first sight as the reciprocal of the space time. However, there can be a difference in the two quantities’ definitions. For the space time, the entering volumetric flow rate is measured at the entrance conditions, but for the space velocity, other conditions are often used. The two space velocities commonly used in industry are the liquid-hourly and gas-hourly space velocities, LHSV and GHSV, respectively. The entering volumetric flow rate, υ0, in the LHSV is frequently measured as that of a liquid feed rate at 60°F or 75°F, even though the feed to the reactor may be a vapor at some higher temperature. Strange but true. The gas volumetric flow rate, υ0, in the GHSV is normally reported at standard temperature and pressure (STP).
Example 2-6 Reactor Space Times and Space Velocities
Calculate the space time, τ, and space velocities for the reactor in Examples 2-1 and 2-3 for an entering volumetric flow rate of 2 dm3/s.
Solution
The entering volumetric flow rate is 2 dm3/s (0.002 m3/s).
From Example 2-1, the CSTR volume was 6.4 m3 and the corresponding space time, τ, and space velocity, SV are
It takes 0.89 hours to put 6.4 m3 into the reactor.
From Example 2-3, the PFR volume was 2.165 m3, and the corresponding space time and space velocity are
Analysis: This example gives an important industrial concept. These space times are the times for each of the reactors to take the volume of fluid equivalent to one reactor volume and put it into the reactor.
Many times, reactors are connected in series so that the exit stream of one reactor is the feed stream for another reactor. When this arrangement is used, it is often possible to speed calculations by defining conversion in terms of location at a point downstream rather than with respect to any single reactor. That is, the conversion X is the total number of moles of A that have reacted up to that point per mole of A fed to the first reactor.
Only valid for NO side streams!!
For reactors in series
However, this definition can only be used when the feed stream only enters the first reactor in the series and there are no side streams either fed or withdrawn. The molar flow rate of A at point i is equal to the moles of A fed to the first reactor, minus all the moles of A reacted up to point i.
FAi = FA0 – FA0Xi
For the reactors shown in Figure 2-3, X1 at point i = 1 is the conversion achieved in the PFR, X2 at point i = 2 is the total conversion achieved at this point in the PFR and the CSTR, and X3 is the total conversion achieved by all three reactors.
To demonstrate these ideas, let us consider three different schemes of reactors in series: two CSTRs, two PFRs, and then a combination of PFRs and CSTRs in series. To size these reactors, we shall use laboratory data that give the reaction rate at different conversions.
2.5.1 CSTRs in Series
The first scheme to be considered is the two CSTRs in series shown in Figure 2-4.
For the first reactor, the rate of disappearance of A is −rA1 at conversion X1.
For a batch reactor, we saw that conversion increases with time spent in the reactor. For continuous-flow systems, this time usually increases with increasing reactor volume, e.g., the bigger/longer the reactor, the more time it will take the reactants to flow completely through the reactor and thus, the more time to react. Consequently, the conversion X is a function of reactor volume V. If FA0 is the molar flow rate of species A fed to a system operated at steady state, the molar rate at which species A is reacting within the entire system will be FA0X.
The molar feed rate of A to the system minus the rate of reaction of A within the system equals the molar flow rate of A leaving the system FA. The preceding sentence can be expressed mathematically as
Rearranging gives
The entering molar flow rate of species A, FA0 (mol/s), is just the product of the entering concentration, CA0 (mol/dm3), and the entering volumetric flow rate, υ0 (dm3/s).
Liquid phase
For liquid systems, the volumetric flow rate, υ, is constant and equal to υ0, and CA0 is commonly given in terms of molarity, for example, CA0 = 2 mol/dm3.
For gas systems, CA0 can be calculated from the entering mole fraction, yA0, the temperature, T0, and pressure, P0, using the ideal gas law or some other gas law. For an ideal gas (see Appendix B):
Gas phase
Now that we have a relationship [Equation (2-8)] between the molar flow rate and conversion, it is possible to express the design equations (i.e., mole balances) in terms of conversion for the flow reactors examined in Chapter 1.
2.3.1 CSTR (Also Known as a Backmix Reactor or a Vat)
Recall that the CSTR is modeled as being well mixed such that there are no spatial variations in the reactor. For the general reaction
Simplifying, we see that the CSTR volume necessary to achieve a specified conversion X is
Perfect mixing
Because the reactor is perfectly mixed, the exit composition from the reactor is identical to the composition inside the reactor, and, therefore, the rate of reaction, −rA, is evaluated at the exit conditions.
In most batch reactors, the longer a reactant stays in the reactor, the more the reactant is converted to product until either equilibrium is reached or the reactant is exhausted. Consequently, in batch systems the conversion X is a function of the time the reactants spend in the reactor. If NA0 is the number of moles of A initially present in the reactor (i.e., t = 0), then the total number of moles of A that have reacted (i.e., have been consumed) after a time t is [NA0X].
Now, the number of moles of A that remain in the reactor after a time t, NA, can be expressed in terms of NA0 and X:
The number of moles of A in the reactor after a conversion X has been achieved is
When no spatial variations in reaction rate exist, the mole balance on species A for a batch system is given by the following equation [cf. Equation (1-5)]:
Moles of A in the reactor at a time t
This equation is valid whether or not the reactor volume is constant. In the general reaction, Equation (2-2), reactant A is disappearing; therefore, we multiply both sides of Equation (2-5) by -1 to obtain the mole balance for the batch reactor in the form
The rate of disappearance of A, −rA, in this reaction might be given by a rate law similar to Equation (1-2), such as −rA = kCACB.
For batch reactors, we are interested in determining how long to leave the reactants in the reactor to achieve a certain conversion X. To determine this length of time, we write the mole balance, Equation (2-5), in terms of conversion by differentiating Equation (2-4) with respect to time, remembering that NA0 is the number of moles of A initially present in the reactor and is therefore a constant with respect to time.
For a batch reactor, the design equation in differential form is
Batch reactor (BR) design equation
We call Equation (2-6) the differential form of the design equation for a batch reactor because we have written the mole balance in terms of conversion. The differential forms of the batch reactor mole balances, Equations (2-5) and (2-6), are often used in the interpretation of reaction rate data (Chapter 7) and for reactors with heat effects (Chapters 11–13), respectively. Batch reactors are frequently used in industry for both gas-phase and liquid-phase reactions. The laboratory bomb calorimeter reactor is widely used for obtaining reaction rate data. Liquid-phase reactions are frequently carried out in batch reactors when small-scale production is desired or operating difficulties rule out the use of continuous-flow systems.
Uses of a BR
To determine the time to achieve a specified conversion X, we first separate the variables in Equation (2-6) as follows:
Batch time t to achieve a conversion X
This equation is now integrated with the limits that the reaction begins at time equals zero where there is no conversion initially (when t = 0, X = 0) and ends at time t when a conversion X is achieved (i.e., when t = t, then X = X). Carrying out the integration, we obtain the time t necessary to achieve a conversion X in a batch reactor
The longer the reactants are left in the reactor, the greater the conversion will be. Equation (2-6) is the differential form of the design equation, and Equation (2-7) is the integral form of the design equation for a batch reactor.
Be more concerned with your character than with your reputation, because character is what you really are while reputation is merely what others think you are.
—John Wooden, former head coach, UCLA Bruins
Overview. In the first chapter, the general mole balance equation was derived and then applied to the four most common types of industrial reactors. A balance equation was developed for each reactor type and these equations are summarized in Table S1-1 in Chapter 1. In Chapter 2, we will show how to size and arrange these reactors conceptually, so that the reader may see the structure of CRE design and will not get lost in the mathematical details.
In this chapter, we
• Define conversion
• Rewrite all balance equations for the four types of industrial reactors in Chapter 1 in terms of the conversion, X
• Show how to size (i.e., determine the reactor volume) these reactors once the relationship between the reaction rate and conversion is known—i.e., given −rA = f(X)
• Show how to compare CSTR and PFR sizes
• Show how to decide the best arrangements for reactors in series, a most important principle
In addition to being able to determine CSTR and PFR sizes given the rate of reaction as a function of conversion, you will be able to calculate the overall conversion and reactor volumes for reactors arranged in series.
A batch reactor is used for small-scale operation, for testing new processes that have not been fully developed, for the manufacture of expensive products, and for processes that are difficult to convert to continuous operations. The reactor can be charged (i.e., filled) through the holes at the top (see Figure l-5(a)). The batch reactor has the advantage of high conversions that can be obtained by leaving the reactant in the reactor for long periods of time, but it also has the disadvantages of high labor costs per batch, the variability of products from batch to batch, and the difficulty of large-scale production (see Industrial Reactor Photos in Professional Reference Shelf [PRS] on the CRE Web sites, www.umich.edu/~elements/5e/index.html). Also see http://encyclopedia.che.engin.umich.edu/Pages/Reactors/menu.html.
Also see http://encyclopedia.che.engin.umich.edu/Pages/Reactors/Batch/Batch.html.
A batch reactor has neither inflow nor outflow of reactants or products while the reaction is being carried out: Fj0 = Fj = 0. The resulting general mole balance on species j is
If the reaction mixture is perfectly mixed (Figure l-5(b)) so that there is no variation in the rate of reaction throughout the reactor volume, we can take rj out of the integral, integrate, and write the mole balance in the form
Perfect mixing
Let’s consider the isomerization of species A in a batch reactor
As the reaction proceeds, the number of moles of A decreases and the number of moles of B increases, as shown in Figure 1-6.
We might ask what time, t1, is necessary to reduce the initial number of moles from NA0 to a final desired number NA1. Applying Equation (1-5) to the isomerization
rearranging,
and integrating with limits that at t = 0, then NA = NA0, and at t = t1, then NA = NA1, we obtain
This equation is the integral form of the mole balance on a batch reactor. It gives the time, t1, necessary to reduce the number of moles from NA0 to NA1 and also to form NB1 moles of B.
1.4 Continuous-Flow Reactors
Continuous-flow reactors are almost always operated at steady state. We will consider three types: the continuous-stirred tank reactor (CSTR), the plug-flow reactor (PFR), and the packed-bed 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, http://encyclopedia.che.engin.umich.edu/Pages/Reactors/CSTR/CSTR.html, and on the CRE Web site.
1.4.1 Continuous-Stirred Tank Reactor (CSTR)
A type of reactor commonly used in industrial processing is the stirred tank operated continuously (Figure 1-7). It is referred to as the continuous-stirred tank reactor (CSTR) or vat, or backmix reactor, and is primarily used 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 well-mixed model is inadequate, and we must resort to other modeling techniques, such as residence time distributions, to obtain meaningful results. This topic of nonideal mixing is discussed on the Web site in PDF Chapters 16, 17, and 18 on nonideal reactors.
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),
The ideal CSTR is assumed to be perfectly mixed.
it takes the familiar form known as the design equation for a CSTR
The CSTR design equation gives the reactor volume V necessary to reduce the entering molar flow rate of species j from Fj0 to the exit molar flow rate Fj, when species j is disappearing at a rate −rj. 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 Fj is just the product of the concentration of species j and the volumetric flow rate υ
Similarly, for the entrance molar flow rate we have Fj0 = Cj0 · υ0. Consequently, we can substitute for Fj0 and Fj into Equation (1-7) to write a balance on species A as
The ideal CSTR mole balance equation is an algebraic equation, not a differential equation.
The first step to knowledge is to know that we are ignorant.
—Socrates (470-399 B.C.)
How is a chemical engineer different from other engineers?
The Wide Wild World of Chemical Reaction Engineering
Chemical kinetics is the study of chemical reaction rates and reaction mechanisms. The study of chemical reaction engineering (CRE) combines the study of chemical kinetics with the reactors in which the reactions occur. Chemical kinetics and reactor design are at the heart of producing almost all industrial chemicals, such as the manufacture of phthalic anhydride shown in Figure 1-1. It is primarily a knowledge of chemical kinetics and reactor design that distinguishes the chemical engineer from other engineers. The selection of a reaction system that operates in the safest and most efficient manner can be the key to the economic success or failure of a chemical plant. For example, if a reaction system produces a large amount of undesirable product, subsequent purification and separation of the desired product could make the entire process economically unfeasible.
The chemical reaction engineering (CRE) principles learned here can also be applied in many areas, such as waste treatment, microelectronics, nanoparticles, and living systems, in addition to the more traditional areas of the manufacture of chemicals and pharmaceuticals. Some of the examples that illustrate the wide application of CRE principles in this book are shown in Figure 1-2. These examples include modeling smog in the Los Angeles (L.A.) basin (Chapter 1), the digestive system of a hippopotamus (Chapter 2 on the CRE Web site, www.umich.edu/~elements/5e/index.html), and molecular CRE (Chapter 3). Also shown are the manufacture of ethylene glycol (antifreeze), where three of the most common types of industrial reactors are used (Chapters 5 and 6), and the use of wetlands to degrade toxic chemicals (Chapter 7 on the CRE Web site). Other examples shown are the solid-liquid kinetics of acid-rock interactions to improve oil recovery (Chapter 7); pharmacokinetics of cobra bites (Chapter 8 Web Module); free-radical scavengers used in the design of motor oils (Chapter 9); enzyme kinetics (Chapter 9) and drug delivery pharmacokinetics (Chapter 9 on the CRE Web site); heat effects, runaway reactions, and plant safety (Chapters 11 through 13); and increasing the octane number of gasoline and the manufacture of computer chips (Chapter 10).
Overview. This chapter develops the first building block of chemical reaction engineering, mole balances, which will be used continually throughout the text. After completing this chapter, the reader will be able to:
• Describe and define the rate of reaction
• Derive the general mole balance equation
• Apply the general mole balance equation to the four most common types of industrial reactors
Before entering into discussions of the conditions that affect chemical reaction rate mechanisms and reactor design, it is necessary to account for the various chemical species entering and leaving a reaction system. This accounting process is achieved through overall mole balances on individual species in the reacting system. In this chapter, we develop a general mole balance that can be applied to any species (usually a chemical compound) entering, leaving, and/or remaining within the reaction system volume. After defining the rate of reaction, −rA, we show how the general balance equation may be used to develop a preliminary form of the design equations of the most common industrial reactors (http://encyclopedia.che.engin.umich.edu/Pages/Reactors/menu.html).
• Batch Reactor (BR)
• Continuous-Stirred Tank Reactor (CSTR)
• Plug-Flow Reactor (PFR)
• Packed-Bed Reactor (PBR)
In developing these equations, the assumptions pertaining to the modeling of each type of reactor are delineated. Finally, a brief summary and series of short review questions are given at the end of the chapter.
1.1 The Rate of Reaction, −rA
The rate of reaction tells us how fast a number of moles of one chemical species are being consumed to form another chemical species. The term chemical species refers to any chemical component or element with a given identity. The identity of a chemical species is determined by the kind, number, and configuration of that species’ atoms. For example, the species para-xylene is made up of a fixed number of specific atoms in a definite molecular arrangement or configuration. The structure shown illustrates the kind, number, and configuration of atoms on a molecular level. Even though two chemical compounds have exactly the same kind and number of atoms of each element, they could still be different species because of different configurations. For example, 2-butene has four carbon atoms and eight hydrogen atoms; however, the atoms in this compound can form two different arrangements.
Identify
– Kind
– Number
– Configuration
As a consequence of the different configurations, these two isomers display different chemical and physical properties. Therefore, we consider them as two different species, even though each has the same number of atoms of each element.
When has a chemical reaction taken place?
We say that a chemical reaction has taken place when a detectable number of molecules of one or more species have lost their identity and assumed a new form by a change in the kind or number of atoms in the compound and/or by a change in structure or configuration of these atoms. In this classical approach to chemical change, it is assumed that the total mass is neither created nor destroyed when a chemical reaction occurs. The mass referred to is the total collective mass of all the different species in the system. However, when considering the individual species involved in a particular reaction, we do speak of the rate of disappearance of mass of a particular species. The rate of disappearance of a species, say species A, is the number of A molecules that lose their chemical identity per unit time per unit volume through the breaking and subsequent re-forming of chemical bonds during the course of the reaction. In order for a particular species to “appear” in the system, some prescribed fraction of another species must lose its chemical identity.
Definition of Rate of Reaction
There are three basic ways a species may lose its chemical identity: decomposition, combination, and isomerization. In decomposition, the molecule loses its identity by being broken down into smaller molecules, atoms, or atom fragments. For example, if benzene and propylene are formed from a cumene molecule,
A species can lose its identity by
• Decomposition
• Combination
• Isomerization
the cumene molecule has lost its identity (i.e., disappeared) by breaking its bonds to form these molecules. A second way that a molecule may lose its chemical identity is through combination with another molecule or atom. In the above reaction, the propylene molecule would lose its chemical identity if the reaction were carried out in the reverse direction, so that it combined with benzene to form cumene. The third way a species may lose its chemical identity is through isomerization, such as the reaction
Here, although the molecule neither adds other molecules to itself nor breaks into smaller molecules, it still loses its identity through a change in configuration.
To summarize this point, we say that a given number of molecules (i.e., moles) of a particular chemical species have reacted or disappeared when the molecules have lost their chemical identity.
The rate at which a given chemical reaction proceeds can be expressed in several ways. To illustrate, consider the reaction of chlorobenzene with chloral to produce the banned insecticide DDT (dichlorodiphenyl-trichloroethane) in the presence of fuming sulfuric acid.
CCl3CHO + 2C6H5Cl → (C6H4Cl)2CHCCl3 + H2O
Letting the symbol A represent chloral, B be chlorobenzene, C be DDT, and D be H2O, we obtain
A + 2B → C + D
The numerical value of the rate of disappearance of reactant A, −rA, is a positive number.
What is −rA?
The rate of reaction, −rA, is the number of moles of A (e.g., chloral) reacting (disappearing) per unit time per unit volume (mol/dm3·s).
A + 2B→C + D
The convention
−rA = 10 mol A/m3·s
rA = −10 mol A/m3·s
−rB = 20 mol B/m3·s
rB = −20 mol B/m3·s
rC = 10 mol C/m3·s
Example 1–1 Rates of Disappearance and Formation
Chloral is being consumed at a rate of 10 moles per second per m3 when reacting with chlorobenzene to form DDT and water in the reaction described above. In symbol form, the reaction is written as
A + 2B → C + D
Write the rates of disappearance and formation (i.e., generation) for each species in this reaction.
Solution
(a) Chloral[A]:
The rate of reaction of chloral [A] (−rA) is given as 10 mol/m3·sRate of disappearance of A = −rA = 10 mol/m3·sRate of formation of A = rA = −10 mol/m3·s
(b) Chlorobenzene[B]:
For every mole of chloral that disappears, two moles of chlorobenzene [B] also disappear.Rate of disappearance of B = −rB = 20 mol/m3·sRate of formation of B = rB = −20 mol/m3·s
(c) DDT[C]:
For every mole of chloral that disappears, one mole of DDT[C] appears.Rate of formation of C = rC = 10 mol/m3·sRate of disappearance of C = −rC = −10 mol/m3·s
(d) Water[D]:
Same relationship to chloral as the relationship to DDTRate of formation of D = rD = 10 mol/m3·sRate of disappearance of D = −rD = −10 mol/m3·s
Analysis: The purpose of this example is to better understand the convention for the rate of reaction. The symbol rj is the rate of formation (generation) of species j. If species j is a reactant, the numerical value of rj will be a negative number. If species j is a product, then rj will be a positive number. The rate of reaction, −rA, is the rate of disappearance of reactant A and must be a positive number. A mnemonic relationship to help remember how to obtain relative rates of reaction of A to B, etc., is given by Equation (3-1) on page 73.
In Chapter 3, we will delineate the prescribed relationship between the rate of formation of one species, rj (e.g., DDT [C]), and the rate of disappearance of another species, −ri (e.g., chlorobenzene [B]), in a chemical reaction.
Heterogeneous reactions involve more than one phase. In heterogeneous reaction systems, the rate of reaction is usually expressed in measures other than volume, such as reaction surface area or catalyst weight. For a gas-solid catalytic reaction, the gas molecules must interact with the solid catalyst surface for the reaction to take place, as described in Chapter 10.
The dimensions of this heterogeneous reaction rate, (prime), are the number of moles of A reacting per unit time per unit mass of catalyst (mol/s·g catalyst).
(prime), are the number of moles of A reacting per unit time per unit mass of catalyst (mol/s·g catalyst).