2. Conversion and Reactor Sizing

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

aA + bB → cC + dD (2-1)

The uppercase letters represent chemical species, and the lowercase letters represent stoichiometric coefficients. We shall choose species A as our limiting reactant and, thus, our basis of calculation. The limiting reactant is the reactant that will be completely consumed first after the reactants have been mixed. Next, we divide the reaction expression through by the stoichiometric coefficient of species A, in order to arrange the reaction expression in the form

A+baB→caC+daD(2-2)

to put every quantity on a “per mole of A” basis, our limiting reactant.

Now we ask such questions as “How can we quantify how far a reaction [e.g., Equation (2-2)] proceeds to the right?” or “How many moles of C are formed for every mole of A consumed?” A convenient way to answer these questions is to define a parameter called conversion. The conversion X_A is the number of moles of A that have reacted per mole of A fed to the system:

XA=MolesofAreactedMo1esofAfed

Because we are defining conversion with respect to our basis of calculation [A in Equation (2-2)], we eliminate the subscript A for the sake of brevity and let X = X_A. For irreversible reactions, the maximum conversion is 1.0, i.e., complete conversion. For reversible reactions, the maximum conversion is the equilibrium conversion X_e (i.e., X_max = X_e). We will take a closer look at equilibrium conversion in Section 4.3.

[Moles⁢of⁢A⁢reacted(consumed)]=[Moles⁢of⁢A⁢fed]⋅[Moles⁢of⁢A⁢reactedMoles⁢of⁢A⁢fed]

[Moles⁢of⁢A⁢reacted(consumed)]=[NA0]⋅[X]⁢(2−3)

Now, the number of moles of A that remain in the reactor after a time t, N_A, can be expressed in terms of N_A0 and X:

[Moles of A in reactor at time t] = [Moles of A initially fed to reactor at t = 0] - [Moles of A that have been consumed by chemical reaction]

[NA]=[NA0]−[NA0X]

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)]:

dNAdt=rAV (2-5)

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

−dNAdt=(−rA)V

The rate of disappearance of A, —r_A, in this reaction might be given by a rate law similar to Equation (1-2), such as — r_A = kC_AC_B.

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 N_A0 is the number of moles of A initially present in the reactor and is therefore a constant with respect to time.

dNAdt=0−NA0dXdt

Combining the above with Equation (2-5) yields

−NA0dXdt=rAV

For a batch reactor, the design equation in differential form is

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.

To determine the time to achieve a specified conversion X, we first separate the variables in Equation (2-6) as follows:

dt=NA0dX−rAV

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.

[FA0]⋅[X]=Moles⁢of⁢A⁢fedtime⋅Moles⁢⁢of⁢A⁢⁢reactedMoles⁢of⁢A⁢fed

[FA0⋅X]=Moles⁢of⁢A⁢reactedtime

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 F_A. The preceding sentence can be expressed mathematically as

Rearranging gives

The entering molar flow rate of species A, F_A0 (mol/s), is just the product of the entering concentration, C_A0 (mol/dm³), and the entering volumetric flow rate, υ₀ (dm³/s).

For liquid systems, the volumetric flow rate, υ, is constant and equal to υ₀, and C_A0 is commonly given in terms of molarity, for example, C_A0 = 2 mol/dm³.

For gas systems, C_A0 can be calculated from the entering mole fraction, v_A0, the temperature, T₀, and pressure, P₀, using the ideal gas law or some other gas law. For an ideal gas (see Appendix B):

CA0=PA0RT0=yA0p0RT0⁢(2−10)

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.

A+baB→caC+daD⁢(2−2)

the CSTR mole balance Equation (1-7) can be arranged to

V=FA0−FA−rA⁢(2−11)

We now substitute for F_A in terms of F_A0 and X

FA=FA0−FA0X⁢(2−12)

and then substitute Equation (2-12) into (2-11)

V=FA0−(FA0−FA0X)−rA⁢

Simplifying, we see that the CSTR volume necessary to achieve a specified conversion X is

Because the reactor is perfectly mixed, the exit composition from the reactor is the CSTR exit identical to the composition inside the reactor, and, therefore, the rate of reaction, -r_A, is evaluated at the exit conditions.

−dFAdV=−rA⁢(2−14)

¹ This constraint can be removed when we extend our analysis to nonideal (industrial) reactors in the PDF Chapters 16 through 18 on the CRE Web site.

For a flow system, F_A has previously been given in terms of the entering molar flow rate F_A0 and the conversion X

FA=FA0−FA0X⁢(2−12)

Differentiating

dFA=−FA0dX⁢

and substituting into (2-14) gives the differential form of the design equation for a plug-flow reactor (PFR)

We now separate the variables and integrate with the limits V = 0 when X = 0 to obtain the plug-flow reactor volume necessary to achieve a specified conversion X

To carry out the integrations in the batch and plug-flow reactor design equations (2-7) and (2-16), as well as to evaluate the CSTR design equation (2-13), we need to know how the reaction rate — r_A varies with the concentration (hence conversion) of the reacting species. This relationship between reaction rate and concentration is developed in Chapter 3.

The differential form of the design equation [i.e., Equation (2-17)] must be used when analyzing reactors that have a pressure drop along the length of the reactor. We discuss pressure drop in packed-bed reactors in Chapter 5.

In the absence of pressure drop, i.e., ΔΡ = 0, we can integrate (2-17) with limits X = 0 at W = 0, and when W = W then X = X to obtain

Equation (2-18) can be used to determine the catalyst weight W (i.e., mass) necessary to achieve a conversion X when the total pressure remains constant.

A particularly simple functional dependence, yet one that occurs often, is the first-order dependence

−rA=kCA=kCA0(1−X)

Here, k is the specific reaction rate and is a function only of temperature, and C_A0 is the entering concentration of A. We note in Equations (2-13) and (2-16) that the reactor volume is a function of the reciprocal of -r_A. For this first-order dependence, a plot of the reciprocal rate of reaction (l/-r_A) as a function of conversion yields a curve similar to the one shown in Figure 2-1, where

1−rA=1kCA0(11−X)

We can use Figure 2-1 to size CSTRs and PFRs for different entering flow rates. By sizing we mean either determine the reactor volume for a specified conversion or determine the conversion for a specified reactor volume. Before sizing flow reactors, let’s consider some insights. If a reaction is carried out isothermally, the rate is usually greatest at the start of the reaction when the concentration of reactant is greatest (i.e., when there is negligible conversion [X = 0]). Hence, the reciprocal rate (l/-r_A) will be small. Near the end of the reaction, when the reactant has been mostly used up and thus the concentration of A is small (i.e., the conversion is large), the reaction rate will be small. Consequently, the reciprocal rate (l/-r_A) is large.

For all irreversible reactions of greater than zero order (see Chapter 3 for zero-order reactions), as we approach complete conversion where all the limiting reactant is used up, i.e., X = 1, the reciprocal rate approaches infinity as does the reactor volume, i.e.

As X → 1, -r_A → 0 , thus, 1−rA→∞ and therefore V → ∞

Consequently, we see that an infinite reactor volume is necessary to reach complete conversion, X = 1.0.

For reversible reactions (e.g., A ⇄ B), the maximum conversion is the equilibrium conversion X_e. At equilibrium, the reaction rate is zero (r_A = 0). Therefore,

as X → X_e, -r_A → 0 , thus, 1−rA→∞ and therefore V → ∞

and we see that an infinite reactor volume would also be necessary to obtain the exact equilibrium conversion, X = X_e. We will discuss X_e further in Chapter 4.

Examples of Reactor Design and Staging Given — r_A =f(X)

To illustrate the design of continuous-flow reactors (i.e., CSTRs and PFRs), we consider the irreversible isothermal gas-phase isomerization

A → B

We are going to the laboratory to determine the rate of chemical reaction as a function of the conversion of reactant A. The laboratory measurements given in Table 2-1 show the chemical reaction rate as a function of conversion. The temperature was 500 K (440°F), the total pressure was 830 kPa (8.2 atm), and the initial charge to the reactor was pure A. The entering molar flow of A rate is F_A0 = 0.4 mol/s.

† Proprietary coded data courtesy of Jofostan Central Research Laboratory, Çolow, Jofostan, and published in Jofostan Journal of Chemical Engineering Research, Volume 21, page 73 (2019).

Recalling the CSTR and PFR design Equations, (2-13) and (2-16), we see that the reactor volume varies directly with the molar flow rate F_A0 and

with the reciprocal of −rA,(1−rA), e.g., V=(FA0−rA)X. Consequently, to size reactors, we first convert the raw data in Table 2-1, which gives -r_A as a function of X first to (1−rA) as a function of X. Next, we multiply by the entering molar flow rate, F_A0, to obtain (FA0−rA) as a function of X as shown in Table 2-2 of the processed data for F_A0 = 0.4 mol/s.

To size reactors for different entering molar flow rates, F_A0, we would use rows 1 and 3 in Table 2-2 to construct the following figure:

However, for a given F_A0, rather than use Figure 2-2A to size reactors, it is often more advantageous to plot (FA0−rA) as a function of X, which is called a

Levenspiel plot. We are now going to carry out a number of examples where we have specified the flow rate F_A0 at 0.4 mol A/s.

Plotting (FA0−rA) as a function of X using the data in Table 2-2 we obtain the plot shown in Figure 2-2B.

We are now going to use the Levenspiel plot of the processed data (Figure 2-2B) to size a CSTR and a PFR.

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.

F_Ai = F_A0 – F_A0X_i

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.

For the first reactor, the rate of disappearance of A is -r_A1 at conversion X₁.

A mole balance on reactor 1 gives

Reactor 1:

In – Out + Generation = 0

FA0−FA1+rA1V1=0⁢(2−19)

The molar flow rate of A at point 1 is

FA1=FA0−FA0X1⁢(2−20)

Combining Equations (2-19) and (2-20), or rearranging

In the second reactor, the rate of disappearance of A, -r_A2, is evaluated at the conversion of the exit stream of reactor 2, X₂. A steady-state mole balance on the second reactor is

Reactor 2:

In – Out + Generation = 0

FA1−FA2+rA2V2=0⁢(2−22)

The molar flow rate of A at point 2 is

FA2=FA0−FA0X2⁢(2−23)

Combining and rearranging

V2=FA1−FA2−rA2=(FA0−FA0X1)−(FA0−FA0X2)−rA2

For the second CSTR, recall that -r_A2 is evaluated at X₂ and then use (X₂-X₁) to calculate V₂.

In the examples that follow, we shall again use the molar flow rate of A used in Example 2-1 (i.e., F_A0 = 0.4 mol A/s) and the reaction conditions given in Table 2-3.

Solution

For Reactor 1, we observe from either Table 2-3 or Figure 2-2B that when X = 0.4, then

(FA0-rA1)X=0.4=2.05m3

Then, using Equation (2-13)

V1=(AA0-rA1)41X1=(FA0-rA1)0.4X1=(2.05)(0.4)=0.82m3=820dm3

For Reactor 2, when X₂ = 0.8, then (FA0-rA)X=0.8=8.0m3

using Equation (2-24)

V2=(FA0-rA2)(X2-X1) (2-24)

V2=(8.0m3)(0.8-0.4)=3.2m3=3200dm3

V2=3200dm3 (liters)

The shaded areas in Figure E2-4.1 can also be used to determine volumes of CSTR 1 and CSTR 2.

Note again that for CSTRs in series, the rate — r_A1 is evaluated at a conversion of 0.4 and rate —r_A2 is evaluated at a conversion of 0.8. The total volume for these two reactors in series is

V=V1+V2=0.82m3+3.2m3=4.02m3=4020dm3

By comparison, the volume necessary to achieve 80% conversion in one CSTR is

y=(FA0-rA1)X=(8.0)(0.8)=6.4m3=6400dm3

Notice in Example 2-5 that the sum of the two CSTR reactor volumes (4.02 m³) in series is less than the volume of one CSTR (6.4 m³) to achieve the same overall con-

Analysis: When we have reactors in series, we can speed our analysis and calculations by defining an overall conversion at a point in the series, rather than the conversion of each individual reactor. In this example, we saw that 40% was achieved at point 1, the exit to the first reactor, and that a total of 80% conversion was achieved by the time we exit the second reactor.

From Figure 2-6, we note a very important observation! The total volume to achieve 80% conversion for five CSTRs of equal volume in series is “roughly” the same as the volume of a PFR. As we make the volume of each CSTR smaller and increase the number of CSTRs, the total volume of the CSTRs in series and the volume of the PFR will become identical. That is, we can model a PFR with a large number of CSTRs in series. This concept of using many CSTRs in series to model a PFR will be used later in a number of situations, such as modeling catalyst decay in packed-bed reactors or transient heat effects in PFRs.

We can see from Figure 2-8 and from the following equation

∫0X2FA0dX-rA≡∫0X1FA0-X-rA+∫X1X2FA0dX-rA

that it is immaterial whether you place two plug-flow reactors in series or have one continuous plug-flow reactor; the total reactor volume required to achieve the same conversion is identical!

A schematic of the industrial reactor system in Figure 2-9 is shown in Figure 2-10.

For the sake of illustration, let’s assume that the reaction carried out in the reactors in Figure 2-10 follows the same (FA0-rA)vs.X vs. X curve given by Table 2-3.

The volumes of the first two CSTRs in series (see Example 2-5) are:

Reactor 1: V1=FA0X1-rA1   (2-13)

Reactor 2: V2=FA0(X2-X1)-rA2   (2-14)

Starting with the differential form of the PFR design equation

FA0dXdV=-rA (2-15)

rearranging and integrating between limits, when V = 0, then X = X₂, and when V = V₃, then X = X₃ we obtain

Reactor 3: V3=∫X2X3FA0-rAdX    (2-25)

The corresponding reactor volumes for each of the three reactors can be found from the shaded areas in Figure 2-11.

The (F_A0/-r_A) versus X curves we have been using in the previous examples are typical of those found in isothermal reaction systems. We will now consider a real reaction system that is carried out adiabatically. Isothermal reaction systems are discussed in Chapter 5 and adiabatic systems in Chapter 11.

If we look at Figure E2-5.2, the area under the curve (PFR volume) between X = 0 and X = 0.2, we see that the PFR area is greater than the rectangular area corresponding to the CSTR volume, i.e., V_PFR > V_CSTR- However, if we compare the areas under the curve between X = 0.6 and X = 0.65, we see that the area under the curve (PFR volume) is smaller than the rectangular area corresponding to the CSTR volume, i.e., V_Cstr ^> V_F¥R. This result often occurs when the reaction is carried out adiabatically, which is discussed when we look at heat effects in Chapter 11.

In the sequencing of reactors, one is often asked, “Which reactor should go first to give the highest overall conversion? Should it be a PFR followed by a CSTR, or two CSTRs, then a PFR, or ...?” The answer is "It depends." It depends not only on the shape of the Levenspiel plot (F_A0/-r_A) versus X, but also on the relative reactor sizes. As an exercise, examine Figure E2-5.2 to learn if there is a better way to arrange the two CSTRs and one PFR. Suppose you were given a Levenspiel plot of (F_A0/-r_A) vs. X for three reactors in series, along with their reactor volumes V_CSTR1 = 3 m³, V_CSTR2 = 2 m³, and V_PFR = 1.2 m³, and were asked to find the highest possible conversion X. What would you do? The methods we used to calculate reactor volumes all apply, except the procedure is reversed and a trial-and-enor solution is needed to find the exit overall conversion from each reactor (see Problem P2-5_B).

The previous examples show that if we know the molar flow rate to the reactor and the reaction rate as a function of conversion, then we can calculate the reactor volume necessary to achieve a specified conversion. The reaction rate does not depend on conversion alone, however. It is also affected by the initial concentrations of the reactants, the temperature, and the pressure. Consequently, the experimental data obtained in the laboratory and presented in Table 2-1 as -r_A as a function of X are useful only in the design of full-scale reactors that are to be operated at the identical conditions as the laboratory experiments (temperature, pressure, and initial reactant concentrations). However, such circumstances are seldom encountered and we must revert to the methods we describe in Chapters 3 and 4 to obtain –r_A as a function of X.

It is important to remember that if the rate of reaction is available or can be obtained solely as a function of conversion, –r_A =f(X), or if it can be generated by some intermediate calculations, one can design a variety of reactors and combinations of reactors.

Ordinarily, laboratory data are used to formulate a rate law, and then the reaction rate-conversion functional dependence is determined using the rate law. The preceding sections show that with the reaction rate-conversion relationship, i.e., –r_A = f(X), different reactor schemes can readily be sized. In Chapters 3 and 4, we show how we obtain this relationship between reaction rate and conversion from rate law and reaction stoichiometry

The space time tau, τ, is obtained by dividing the reactor volume by the volumetric flow rate entering the reactor

τ≡Vv0 (2-26)

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.

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

τ=0.2m30.01m3/s=20s

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 F_A0 = v₀C_A0 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 τp=rv0VB-)=CA0∫0XdX-rA

and

For a CSTR τ=(Vv0)=CA0-rAX

For plug flow, the space time is equal to the mean residence time in the reactor, t_m (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 2-5 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).

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.

SV≡v0VSV=1τ (2-27)

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).

LHSV=υ0|liquidV (2-28)

GHSV=υ0|STPV (2-29)

For batch systems: X=NA0-NANA0 (S2-1)

For flow sytems: X=FA0-FAFA0 (S2-2)

For reactors in series with no side streams, the conversion at point i is

Xi=Total moles of A reacted up to Point iMoles A fed to the first reactor (S2-3)

2. In terms of the conversion, the differential and integral forms of the reactor design equations become:

3. If the rate of disappearance of A is given as a function of conversion, the following graphical techniques can be used to size a CSTR and a plug-flow reactor.

A. Graphical Integration Using Levenspiel Plots

The PFR integral could also be evaluated by

B. Numerical Integration

See Appendix A.4 for quadrature formulas such as the five-point quadrature formula with ΔΧ = 0.8/4 of five equally spaced points, X_l = 0, X₂ = 0.2, X₃ = 0.4, X₄ = 0.6, and X₅ = 0.8.

4. Space time, τ, and space velocity, SV, are given by

τ=Vv0 (S2-4)

SV=υ0V (at STP for gas-phase reactions) (S2-5)

1. Web P2-1_A Reactor Sizing for Reversible Reactions (http://umich.edu/~elements/5e/02chap/expanded_ch02_homeproblem.pdf)

2. Web P2-2_A Puzzle Problem, “What’s Wrong with this Solution?” (http://umich.edu/~elements/5e/02chap/expanded_ch02_puzzk.pdf)

• Learning Resources (http://umich.edu/~elements/5e/02chap/learn.html)

1. Summary Notes (http://umich.edu/~elements/5e/02chap/summary.html)

2. Self-Tests

A. Exercises (http://www.umich.edu/~elements/5e/01chap/summary-selfiest.html)

B. i>clicker Questions (http://www.umich.edu/~elements/5e/02chap/iclicker_ch2_ql.html)

3. Web Module Hippopotamus Digestive System
(http://umich.edu/-elements/’5e/web_mod/hippo/index.htm)

4. Interactive Computer Games

Reactor Staging (http://umich.edu/~elements/5e/icm/staging.html)

5. Solved Problems

A. CDP2-A_B More CSTR and PFR Calculations—No Memorization (http://umich.edu/~elements/5e/02chap/learn-cdp2.html)

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A = ● B = ◼ C = ◆ D = ◆◆

Q2-2 (a) Without referring back, make a list of the most important items you learned in this chapter.

(b) What do you believe was the overall purpose of the chapter?

Q2-3 Go to the Web site www.engr.ncsu.edu/learningstyles/ilsweb.html. Take the Inventory of Learning Style test, and record your learning style according to the Solomon/Felder inventory and then use Appendix 1.2 to suggest two ways to facilitate your learning style in each of the four categories.

Global/Sequential_____

Active/Reflective_____

Visual/Verbal_____

Sensing/Intuitive_____

P2-1_A (a) Revisit Examples 2-1 through 2-3. How would your answers change if the flow rate, F_A0, were cut in half? If it were doubled? What conversion can be achieved in a 4.5 m³ PFR and in a 4.5 m³ CSTR?

(b) Revisit Example 2-2. Being a company about to go bankrupt, you can only afford a 2.5 m³ CSTR. What conversion can you achieve?

(c) Revisit Example 2-3. What conversion could you achieve if you could convince your boss, Dr. Pennypincher, to spend more money to buy a 1.0 m³ PFR to attach to a 2.40 CSTR?

(d) Revisit Example 2-4. How would your answers change if the two CSTRs (one 0.82 m³ and the other 3.2 m³) were placed in parallel with the flow, F_A0, divided equally between the reactors.

(e) Revisit Example 2-5. (1) What is the worst possible way to arrange the two CSTRs and one PFR? (2) What would be the reactor volumes if the two intermediate conversions were changed to 20% and 50%, respectively? (3) What would be the conversions, X_h X₂, and X₃, if all the reactors had the same volume of 100 dm³ and were placed in the same order?

(f) Revisit Example 2-6. If the term CA0∫0XdX-rA is 2 seconds for 80% conversion, how much fluid (m³/min) can you process in a 3 m³ reactor?

Interactive Computer Games

P2-2_A ICG Staging. Download the Interactive Computer Game (ICG) from the CRE Web site, http://www.umich.edu/~elements/5e/icm/staging.html. Play this game and then record your performance number, which indicates your mastery of the material. Your professor has the key to decode your performance number. Note: To play this game you must have Windows 2000 or a later version. ICG Reactor Staging Performance # ___________________________________

Problems

P2-3_B You have two CSTRs and two PFRs, each with a volume of 1.6 m³. Use Figure 2-2B on page 43 to calculate the conversion for each of the reactors in the following arrangements.

(a) Two CSTRs in series. (Ans.: X_l = 0.435, X₂ = 0.66)

(b) Two PFRs in series.

(c) Two CSTRs in parallel with the feed, F_A0, divided equally between the two reactors.

(d) Two PFRs in parallel with the feed divided equally between the two reactors.

(e) Caution: Part (e) is a C level problem. A CSTR and a PFR in parallel with the flow equally divided. Calculate the overall conversion, X_OV

Xov=FA0-FACSTR-FAPFRFA0, with FACSTR=FA02-FA02XCSTR, and FAPFR=FA02(1-XPFR)

(f) A PFR followed by a CSTR.

(g) A CSTR followed by a PFR. (Ans.: X_CSTR = 0.435, X_PFR = 0.71)

(h) A PFR followed by two CSTRs. Is this arrangement a good arrangement or is there a better one?

P2-4_B The exothermic reaction of stillbene (A) to form the economically important trospophene (B) and methane (C), i.e.,

A → B + C

was carried out adiabatically and the following data recorded:

The entering molar flow rate of A was 300 mol /min.

(a) What are the PFR and CSTR volumes necessary to achieve 40% conversion? (V_PFR = 72 dm³, VCSTR = 24 dm³)

(b) Over what range of conversions would the CSTR and PFR reactor volumes be identical?

(c) What is the maximum conversion that can be achieved in a 105-dm³ CSTR?

(d) What conversion can be achieved if a 72-dm³ PFR is followed in series by a 24-dm³ CSTR?

(e) What conversion can be achieved if a 24-dm³ CSTR is followed in a series by a 72-dm³ PFR?

(f) Plot the conversion and rate of reaction as a function of PFR reactor volume up to a volume of 100 dm³.

P2-5_B The financially important reaction to produce the valuable product B (not the real name) was carried out in Jesse Pinkman’s garage. This fly-by-night company, Breaking Bad, is on a shoestring budget and has very little money to purchase equipment. Fortunately, cousin Bernie has a reactor surplus company and you can get reactors from Bernie. The reaction

A → B + C

takes place in the liquid phase. Below is the Levenspiel plot for this reaction. You have up to $10,000 to use to purchase reactors from those given below.

Reactor Type	Number	Volume (dm3)	Cost
CSTR	2	2	$1,000
CSTR	1	4	$2,000
PFR	2	4	$2,000
CSTR	2	6	$4,000
CSTR	3	12	$8,000
PFR	2	12	$6,000

What reactors do you choose, how do you arrange them, and what is the highest conversion you can get for $10,000? Approximately what is the corresponding highest conversion with your arrangement of reactors?

Scheme and sketch your reactor volumes.

P2-6_B Read the Web Module “Chemical Reaction Engineering of Hippopotamus Stomach” on the CRE Web site (http://www.umich.edu/’-elements/’5e/web_mod/hippo/index. htm).

(a) Write five sentences summarizing what you learned from the Web Module.

(b) Work problems (1) and (2) in the Hippo Web Module.

(c) The hippo has picked up a river fungus, and now the effective volume of the CSTR stomach compartment is only 0.2 m³. The hippo needs 30% conversion to survive. Will the hippo survive?

(d) The hippo had to have surgery to remove a blockage. Unfortunately, the surgeon, Dr. No, accidentally reversed the CSTR and the PFR during the operation. Oops!! What will be the conversion with the new digestive arrangement? Can the hippo survive?

P2-7_B The adiabatic exothermic irreversible gas-phase reaction

2A + B → 2C

is to be carried out in a flow reactor for an equimolar feed of A and B. A Levenspiel plot for this reaction is shown in Figure P2-7_B.

(a) What PFR volume is necessary to achieve 50% conversion?

(b) What CSTR volume is necessary to achieve 50% conversion?

(c) What is the volume of a second CSTR added in series to the first CSTR (Part b) necessary to achieve an overall conversion of 80%? (Ans.: V_CSTR = 1.5 X 10⁵m³)

(d) What PFR volume must be added to the first CSTR (Part b) to raise the conversion to 80%?

(e) What conversion can be achieved in a 6 x 10⁴ m³ CSTR? In a 6 x 10⁴ m³ PFR?

(f) Think critically (cf. Preface, Section H, page xxviii) to critique the answers (numbers) to this problem.

P2-8_A Estimate the reactor volumes of the two CSTRs and the PFR shown in the photo in Figure 2-9. [Hint: Use the dimensions of the door as a scale.]

P2-9_D Don’t calculate anything. Just go home and relax.

P2-10_c The curve shown in Figure 2-1 is typical of a reaction carried out isothermally, and the curve shown in Figure P2-10_B (see Example 11-3) is typical of a gas-solid catalytic exothermic reaction carried out adi-abatically

(a) Assuming that you have a fluidized CSTR and a PBR containing equal weights of catalyst, how should they be arranged for this adiabatic reaction? Use the smallest amount of catalyst weight to achieve 80% conversion of A.

(b) What is the catalyst weight necessary to achieve 80% conversion in a fluidized CSTR? (Ans.: W = 23.2 kg of catalyst)

(c) What fluidized CSTR weight is necessary to achieve 40% conversion?

(d) What PBR weight is necessary to achieve 80% conversion?

(e) What PBR weight is necessary to achieve 40% conversion?

(f) Plot the rate of reaction and conversion as a function of PBR catalyst weight, W.

Additional information: F_A0 = 2 mol/s.

• Additional Homework Problems can be found on the CRE Web site under Chapter 2 hot button

1. CDP2-A_A An ethical dilemma as to how to determine the reactor size in a competitor’s chemical plant (http://www.umich.edu/~elements/5e/02chap/add.html).

2. CDP2-E_A Levenspiel Plots CSTR and PFR in Series

3. CDP2-F_A Levenspiel Plots CSTR and PFR in Series

BURGESS, THORNTON W, The Adventures of Poor Mrs. Quack, New York: Dover Publications, Inc., 1917.

KARRASS, CHESTER L., Effective Negotiating: Workbook and Discussion Guide, Beverly Hills, CA: Karrass Ltd., 2004.

LEVENSPIEL, O., Chemical Reaction Engineering, 3rd ed. New York: Wiley, 1999, Chapter 6, pp. 139-156.