Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations

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c. To determine the reactor volume, we use either the reaction operating curve

or the integral form of the design equation:

t ¼

0:675

2:99  1:21Z

0:9  Z

dZ ¼ 3:458 (i)

Using Eq. 7.1.3, the volume of the reactor is

¼ tv

¼ 4584 L ( j)

d. The mass of the catalyst bed is

bed

¼ r

bed

¼ 5959 kg

We conclude the discussion with two comments. First, note that the design

equation provides us with the reactor volume needed to obtain a given extent (or

conversion). However, it does not indicate whether we should use a long reactor

with a small diameter or a short reactor with a large diameter (provided, of

course, that the plug-ﬂow assumption is valid). The reactor diameter is determined

by other considerations such as the heat-transfer area needed to provide (or remove)

heat to the reactor, and the pressure drop (pumping cost). The effect of heat-transfer

on the performance of plug-ﬂow reactors is discussed in Section 7.4. The effect of

pressure drop on the performance of gaseous plug-ﬂow reactors is discussed in

Section 7.5. Second, in the examples above, we designed the reactor for a given

speciﬁed extent. However, the question of what is the most suitable reaction

extent for the operation has not been discussed. The cost of the reactants, the

value of the products, the cost of the equipment, and the operating expenses

(including separation costs) affect the optimal level of the extent. These points

are covered in Chapter 10.

Figure E7.4.2 Species operating curves.

260 PLUG-FLOW REACTOR

7.2.2 Determination of Reaction Rate Expression

In the preceding section, we described how to apply the design equation when the

reaction rate expression is given. Now, we will discuss how to determine the rate

expression from data obtained in plug-ﬂow operations. The method is based on

measuring the exit composition of the reactor at different space times, and then,

by differentiating the data, we obtain the reaction rate at the exit conditions.

Different space times are obtained by either withdrawing samples at different

points along the reactor (different reactor volumes) or by varying the feed ﬂow

rate. The approach is similar to the differential method applied to batch reactors.

To derive a relation between the reaction rate and the extent, we rearrange Eq.

7.2.2 and use Eq. 7.1.3 to obtain

r ¼ C

(7:2:10)

where t

¼ V

is the space time deﬁned by Eq. 4.4.6. For the reactor outlet,

out

¼ C

out

(7:2:11)

Hence, by plotting C

out

versus t

and taking the derivatives, we can determine

out

as shown schematically in Figure 7.4. Note that r

out

occurs at the outlet

concentration.

The main difﬁculty in using Eq. 7.2.11 is that the extent is not a measurable

quantity. Therefore, we have to derive a relationship between Z

out

and an appropri-

ate measured quantity. We do so by using the design equation and relevant stoichio-

metric relations. In most applications, we measure the concentration of a species at

the reactor outlet and calculate the extent by either Eq. 7.2.5 for liquid-phase reac-

tions or Eq. 7.2.6 for gas-phase reactions. We can then determine the orders of the

individual species for power rate expressions.

Figure 7.4 Determination of reaction rate from PFR operating data.

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS 261

Example 7.5 A stream of gaseous reactant A at 3 atm and 308C

¼ 120 mmol=L) is fed into a 500-L plug-ﬂow reactor where it decomposes

according to the reaction:

A ! 2B þ C

The concentration of reactant A is measured at the outlet of the reactor for differ-

ent feed ﬂow rates. Based on the data below, determine:

a. The order of the reaction

b. The rate constant at 308C

Data:

(L=min) 5:010:012:516:725:050:0 125 250 1

out

(mmol=L) 7:113:516:220:026:840:662:679:5 120

Solution The stoichiometric coefﬁcients are:

¼1 s

¼ 2 s

¼ 1 D ¼ 2

We select the inlet stream as the reference stream; hence,

¼ 1, y

¼ y

¼ 0, and C

¼ C

¼ 120 mol=L. Using Eq. 7.2.6, we cal-

culate Z

out

for each outlet concentration:

out

 C

out

D C

out

 s

(a)

The reactor space time is

(b)

For each run, we calculate Z

out

using (a), and t

using (b), and then calculate

out

a. To determine the reaction rate at the reactor outlet for each run (using Eq.

7.2.11), we differentiate the data. Since the data points are not equally

(L/min) 5.0 10.0 12.5 16.7 25.0 50.0 125 250 1

out

(mmol=L) 7.1 13.5 16.2 20 26.8 40.6 62.6 79.5 120

out

0.841 0.724 0.681 0.622 0.537 0.395 0.234 0.145 0

(min) 100 50 40 30 20 10 4 2 0

out

(mmol/L) 101 86.9 81.7 74.6 64.4 47.4 28.1 17.4 0

262 PLUG-FLOW REACTOR

spaced, we calculate the derivative at the midpoints of each of two adjacent

points. The calculated values are given in the table below:

Assuming the rate expression is of the form r ¼ kC

, substituting

Eq. 7.2.11, and taking the log,

ln C

out



¼ ln (k) þ a ln (C

out

) (c)

By plotting ln(C

out

/dt

) versus ln(C

out

), we should get a straight line

whose slope is a. The plot is shown in Figure E7.5.1. The slope is

Slope ¼

ln 10  ln 0:35

ln 100  ln 10

¼ 1:48 (d)

b. Now that the order of the reaction is known ( a ¼ 1.5), we use the integral

form of the design equation (Eq. 7.2.4) to determine the value of k.For

(min)

out

(mmol/L)

out

(mmol=L)

DðC

out

Þ=D t

ðmmol=L minÞ

out

)

ave

(mmol=L)

0 0.0 120

2 17.4 79.5 8.70 99.7

4 28.1 62.6 5.35 71.0

10 47.4 40.6 3.21 51.6

20 64.4 26.8 1.70 33.7

30 74.7 20.2 1.03 23.5

40 81.7 16.2 0.70 18.2

50 86.9 13.5 0.52 14.9

100 101.0 7.1 0.28 10.3

Figure E7.5.1 Determination of reaction order.

7.2 ISOTHERMAL OPERATIONS WITH SINGLE REACTIONS 263

a ¼ 1.5, Eq. 7.2.2 reduces to

1  Z

1 þ 2Z



1:5

(e)

Using Eq. 3.5.4, the characteristic reaction time is

0:5

(f)

Separating the variables and integrating (e),

t ;

out

(1 þ 2Z)

1:5

(1  Z)

1:5

dZ ¼ G(Z

out

) (g)

The right-hand side of (g) is a function of Z

out

, G(Z

out

). Hence, by plotting

G(Z

out

) versus the space time, t

, we obtain a line whose slope is 1/t

The table below provides the calculated data, and the plot is shown in

Figure E7.5.2.

The slope is

Slope ¼

¼ kC

0:5

¼ 0:1 min

1

(h)

and the rate constant at 308Cisk ¼ 9.13  10

(L/mmol)

0.5

min

Figure E7.5.2 Determination of reaction rate constant.

(min) 100 50 40 30 20 10 4 2 0

out

0.841 0.724 0.681 0.622 0.537 0.395 0.234 0.145 0

G(Z

out

) 10.01 5.00 4.00 3.00 2.00 1.00 0.40 0.20 0

264 PLUG-FLOW REACTOR

7.3 ISOTHERMAL OPERATIONS WITH MULTIPLE REACTIONS

When more than one chemical reaction takes place in a plug-ﬂow reactor, we

have to address several issues before we start the design. First, we have to deter-

mine how many independent reactions there are among the given reactions and

then select a set of independent reactions. Next, we have to identify all the reac-

tions that actually take place in the reactor (including dependent reactions) and

express their rates. As discussed in Chapter 4, we have to write the design

equation for each independent chemical reaction. To solve the design equations

(obtain relationships between Z

’s and t), we have to express the rates of the

individual chemical reactions, r

’s and r

’s, in terms of the Z

’s and t. The

procedure for designing plug-ﬂow reactors with multiple chemical reactions

goes as follows:

1. Identify all the chemical reactions that take place in the reactor and deﬁne

the stoichiometric coefﬁcients of each species in each reaction.

2. Determine the number of independent chemical reactions.

3. Select a set of independent reactions from among the reactions whose rate

expressions are given.

4. For each dependent reaction, determine its a

multipliers with each of the

independent reactions using Eq. 2.4.9.

5. Select a reference stream [determine (F

tot

)

, C

, v

] and its species compo-

sitions, y

’s.

6. Write Eq. 7.1.1 for each independent chemical reaction.

7. Select a leading (or desirable) chemical reaction and determine the

expression and the value of its characteristic reaction time, t

8. Express the reaction rates in terms of the extents of the independent

reactions, Z

’s.

9. Specify the inlet conditions.

10. Solve the design equations for Z

’s as functions of the dimensionless space

time, t, and obtain the reaction operating curves.

11. Determine the species operating curves using Eq. 2.7.8.

12. Determine the reactor volume using Eq. 7.1.3.

Below, we describe the design formulation of isothermal plug-ﬂow reactors with

multiple reactions for various types of chemical reactions (reversible, series, paral-

lel, etc.). In most cases, we solve the design equations numerically by applying a

numerical technique such as the Runge-Kutta method or using commercial math-

ematical software such as HiQ, Mathcad, Maple, and Mathematica. In some

simple cases, we can obtain analytical solutions. Note that, for isothermal oper-

ations, du ¼ 0, and we do not have to solve the energy balance equation simul-

taneously with the design equations.

7.3 ISOTHERMAL OPERATIONS WITH MULTIPLE REACTIONS 265

Example 7.6 Product B is produced in an isothermal tubular reactor where the

following gas-phase, ﬁrst-order chemical reactions take place:

Reaction 1: A ! 2B

Reaction 2: B ! C þ D

A gaseous stream of reactant A (C

¼ 0:04 mol=L) is fed into a 200 L reactor at

a rate of 100 L/min. At the reactor operating temperature, k

¼ 2 min

and

¼ 1 min

. The pressure drop along the reactor is negligible.

a. Derive the design equations and plot the reaction and species curves.

b. Derive and plot the reaction, yield, and selectivity curves.

c. Determine the conversion of reactant A and the production rate of products B,

C, and D for the given reactor.

d. Determine the optimal reactor volume to maximize the production rate of

product B.

e. Determine the conversion of A, the production rate of products B and C, and

the yield of product B in the optimal reactor.

Solution Since each reaction has a species that does not participate in the

other, the two reactions are independent, and there is no dependent reaction.

The stoichiometric coefﬁcients are

¼1 s

¼ 2 s

¼ 0 s

¼ 0 D

¼ 1

¼ 0 s

¼1 s

¼ 1 s

¼ 1 D

¼ 1

We write Eq. 7.1.1 for each of the independent reactions (m ¼ 1, 2),

¼ r



(a)

¼ r



(b)

We select the inlet stream as the reference stream; hence, Z

¼ Z

¼ 0. Since

only reactant A is fed into the reactor, C

¼ C

¼ 4 mol=L, y

¼ 1, and

¼ y

¼ 0. Also,

tot

)

¼ v

¼ 4 mol=min

a. For gas-phase reactions, isothermal operation (u ¼ 1), and negligible

pressure drop, we use Eq. 7.1.15 to express the species concentrations, and

266 PLUG-FLOW REACTOR

the reaction rates are

¼ k

1  Z

1 þ Z

þ Z

(c)

¼ k

 Z

1 þ Z

þ Z

(d)

We deﬁne the characteristic reaction time on the basis of Reaction 1; hence,

¼ 0:5 min (e)

Substituting (c), (d), and (e) into (a) and (b), the design equations become

1  Z

1 þ Z

þ Z

(f)



 Z

1 þ Z

þ Z

(g)

We solve (f) and (g) numerically subject to the initial condition that at t ¼ 0,

and Z

¼ Z

¼ 0. The reaction curves are shown in the Figure E7.6.1. Once

we have Z

and Z

as a function of the dimensionless space time, we use

Eq. 2.7.8 to determine the species curves:

tot

)

¼ 1  Z

(h)

tot

)

¼ 2Z

 Z

(i)

tot

)

¼ Z

(j)

Figure E7.6.1 Reaction operating curves.

7.3 ISOTHERMAL OPERATIONS WITH MULTIPLE REACTIONS 267

tot

)

¼ Z

(k)

Figure E7.6.2 shows the species curves.

b. Using Eq. 2.6.2,

(t) ¼

 F

out

¼ 1  Z

(t) (l)

The desirable reaction is Reaction 1. Using Eq. 2.6.15, the yield of product B is

(t) ¼

1



[2Z

(t)  Z

(t)] (m)

Using Eq. 2.6.19, the selectivity of product B is

(t) ¼

1



(t)  Z

(t)

(n)

Figure E7.6.3 shows the conversion, yield, and selectivity curves.

Figure E7.6.2 Species operating curves.

Figure E7.6.3 Conversion, yield, and selectivity curves.

268 PLUG-FLOW REACTOR

c. For the given reactor and feed ﬂow rate, using Eq. 7.1.3 and (i), the dimen-

sionless space time is

t ¼

¼ 4

From the reaction curves (or tabulated calculated data), for t ¼ 4, Z

¼ 0.874

and Z

¼ 0.832. Using (c) through (f), the species molar ﬂow rates at

the reactor outlet are F

¼ 0.504 mol/min, F

¼ 3.66 mol/min, F

¼ 3.33 mol/min. Using (l), f

¼ 0.874, using (m), the yield of product

B is 0.458, and using (n) the selectivity is 0.524.

d. To determine the reactor volume that provides the highest production rate of

product B, we use the species curves (or tabulated calculated data) and ﬁnd

that the highest F

is reached at t ¼ 2.30. Using Eq. 7.1.3 and (e), the opti-

mal reactor volume is

¼ 115 L

From the reaction operating curves, at t ¼ 2.30, the solutions of (f) and (g)

are Z

¼ 0.751 and Z

¼ 0.501. Using (h) through (k), the species molar ﬂow

rates at the optimal reactor outlet are: F

¼ 0.996 mol/min, F

¼ 4.00 mol/

min, F

¼ F

¼ 2.00 mol/min. Using (l), the conversion of reactant A is

0.751. Using (m), the yield of product B is 0.500. Using (n), the selectivity

of product B is 0.667. The table below provides a comparison between the

performance of the given reactor and the optimal reactor.

We see that the volume of the optimal reactor is slightly more than half that

of the given reactor. While the feed rate is the same, the conversion of

reactant A is slightly lower, the production rate of product B is about 10%

higher, and the production of the by-products (products C and D) is reduced

by 40%.

Given Reactor Optimal Reactor

Reactor volume (L) 200 115

Volumetric feed rate (L/min) 100 100

Conversion of reactant A 0.874 0.751

Yield of product B 0.458 0.500

Selectivity of product B 0.524 0.667

Production rate of product B (mol/min) 3.66 3.99

Production rate of product C (mol/min) 3.33 2.00

7.3 ISOTHERMAL OPERATIONS WITH MULTIPLE REACTIONS 269