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

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either integers or fractions and should be determined experimentally. Chemical kin-

etics provides a theoretical basis for the values of the orders for some reactions. For

elementary reactions (reactions that take place in a single step without formation of

intermediates), the orders of the species are identical to the absolute value of their

respective stoichiometric coefﬁcients when the reaction is written in its simplest

form. However, in general, we consider Eq. 3.3.7 to be an empirical expression,

and the orders are parameters that should be determined experimentally.

Combining Eqs. 3.3.1 and 3.3.7 and using Eq. 3.3.6, the rate expression for

many homogeneous reactions with two reactants, A and B, is

r ¼ k(T)C

¼ k(T

g(u1)=u

(3:3:8)

Other forms of composition function h(C

’s) are used for different types

of homogenous and heterogeneous reactions. A written chemical reaction is

merely a reaction pathway presentation of many elementary reaction steps that

usually involve unstable intermediate species (e.g., free radicals). Hence, the rate

of a chemical reaction depends on the rates of the individual steps and results in

different forms. For example, the rate of many biological and enzymatically

catalyzed reactions is described by an expression of the form (known as the

Michaelis–Menten expression),

r ¼ k(T)

þ C

(3:3:9)

where k(T ) is the reaction rate constant, and K

is a parameter to be determined

experimentally. Another example is the rate expression of the reaction between

hydrogen and bromine to form hydrogen bromide:

r ¼ k(T)

0:5

1 þ K

HBr

(3:3:10)

For most gas–solid heterogeneous catalytic reactions, rate expressions with the

following general forms are usually used:

r ¼ k(T)

1 þ K

(3:3:11a)

r ¼ k(T)

1 þ K

þ K

(3:3:11b)

r ¼ k(T)

(1 þ K

þ K

)

(3:3:11c)

90 CHEMICAL KINETICS

where K

and K

are parameters that should be determined experimentally. In many

instances, the species concentrations are expressed in terms of the partial pressures.

The dimensions and units of the reaction rate constant, k(T ), depend on the form

of function h(C

’s). For chemical reactions whose rate expressions are power func-

tions of the species concentrations, one can determine the units of the rate constant.

From Eq. 3.3.8, the rate can be written as

r ¼ k(T)C

where n ¼ a þ b þ

...

is the overall order of the reaction. In terms of units,

mole

volume time



[¼] k

mole

volume



Thus, the dimensions of the rate constant are

k [¼]

mole

volume



1n

time



(3:3:12)

3.4 EFFECTS OF TRANSPORT PHENOMENA

The rate expressions derived above describe the dependence of the reaction rate

expressions on kinetic parameters related to the chemical reactions. These rate

expressions are commonly called the intrinsic rate expressions of the chemical reac-

tions. However, as discussed in Chapter 1, in many instances, the local species con-

centrations depend also on the rate that the species are transported in the reaction

medium. Hence, the actual reaction rates are affected by the transport rates of reac-

tants and products. This is manifested in two general cases: (i) gas–solid hetero-

geneous reactions, where species diffusion through the pore plays an important

role, and (ii) gas– liquid reactions, where interfacial species mass-transfer rate as

well as solubility and diffusion play an important role. Considering the effect of

transport phenomena on the global rates of the chemical reactions represents a

very difﬁcult task in the design of many chemical reactors. These topics are

beyond the scope of this text, but the reader should remember to take them

into consideration.

3.5 CHARACTERISTIC REACTION TIME

In the preceding sections, we characterized the rates of chemical reactions in terms

of their variation with changes in temperature and their dependency on species

compositions. The progress of chemical reactions is a rate process that can be

characterized by a time constant. Reactions with the same rate expression (say,

3.5 CHARACTERISTIC REACTION TIME 91

ﬁrst order, second order, etc.) exhibit a similar behavior with respect to concen-

tration variations—the difference between fast and slow reactions is merely the

time scale over which the reaction takes place. To describe the generic behavior

of chemical reactions and to derive dimensionless design equations for chemical

reactors, we deﬁne the characteristic reaction time. The characteristic reaction

time is a measure of the time scale over which the reaction takes place (second,

minute, etc.).

For volume-based reaction rate expressions, the characteristic reaction time, t

is deﬁned by

;

Reference concentration

Reference reaction rate

(3:5:1)

where C

is a characteristic concentration and r

is a characteristic reaction rate,

both conveniently selected. We select them in a way so that they can be applied

for all forms of rate expressions. For batch reactors, the reference concentration,

, is deﬁned by

;

tot

)

(3:5:2a)

where (N

tot

)

and V

are, respectively, the total number of moles and the volume of

a conveniently-selected reference state. For flow reactors, C

is deﬁned by

;

tot

)

(3:5:2b)

where (F

tot

)

and v

are, respectively, the total molar ﬂow rate and the volumetric

ﬂow rate of a conveniently selected reference stream. Note that ( N

tot

)

and (F

tot

)

were used to deﬁne the dimensionless extents in Eqs. 2.7.1 and 2.7.2, respectively.

The selection of the reference reaction rate, r

, should be done in a way that can

be used for all forms of rate expressions. Further, since the reaction rate depends on

the initial (or inlet) composition, r

should be selected such that it is independent

of the speciﬁc composition of the reference state (or stream). To deﬁne r

,we

express the reaction rate at any instance by

Reaction

rate



Reference

rate, r



Dimensionless

correction

due to

temperature

Dimensionless

correction

due to

composition

(3:5:3)

The dimensionless correction term due to temperature variation is given by Eq.

3.3.6. The dimensionless correction term due to changes in composition is readily

derived by expressing the species compositions in terms of dimensionless species

concentrations. The reference reaction rate, r

, is deﬁned as the value for which the

two dimensionless correction factors are equal to 1.

92 CHEMICAL KINETICS

For convenience, the following procedure is adopted in calculating the character-

istic reaction time:

1. Write the reaction rate expression, Eq. 3.3.1.

2. Express the reaction rate constant, k(T ), in terms of Eq. 3.3.6.

3. Express the species composition in terms of a dimensionless relation. Either

substitute

;



or express C

in term of the dimensionless extents.

4. Combine all the dimensionless terms and equate them to 1.

5. The remaining term, the dimensional term, is r

. Determine its value.

6. Determine the characteristic reaction time using Eq. 3.5.1.

For an nth order reaction (overall), the characteristic reaction time is

k(T

n1

(3:5:4)

The four examples below illustrate how the characteristic reaction time is deter-

mined.

Example 3.3 The homogeneous chemical reaction

A þ B ! C

is carried out in a batch reactor. The reaction is ﬁrst order with respect to reactant

A, and the order of reactant B is 1.5. Initially, the reactor contains 2 mol of A,

3 mol of B, and 0.5 mol of C, and its initial volume is 2 L. The reaction rate

constant at the initial temperature is 0.1 (L/mol)

1.5

min

. Determine the charac-

teristic reaction time.

Solution First, calculate the reference concentration. Selecting the initial state

as the reference state,

tot

)

¼ N

tot

(0) ¼ N

(0) þ N

(0) ¼ 5:50 mol

and V

¼ V

(0) ¼ 2 L. Hence, using Eq. 3.5.2a,

tot

)

(0)

5:5

2:0

¼ 2:75 mol=L (a)

3.5 CHARACTERISTIC REACTION TIME 93

The rate expression of the chemical reaction is

r ¼ k(T

g(u1)=u

1:5

(b)

Following the procedure, we express the species concentrations in dimensionless

form:

¼ C



¼ C



and substitute in (b)

r ¼ k(T

2:5

g(u1)=u



1:5

(c)

The dimensional portion of the rate expression is

¼ k(T

)=C

2:5

Using Eq. 3.5.1, the characteristic reaction time is

k(T

2:5

(0:1)(2:75)

1:5

¼ 2:19 min

Example 3.4 A biological waste, A, is decomposed by an enzymatic reaction

A ! B þ C

in aqueous solution. The rate expression of the reaction is

r ¼

þ C

An aqueous solution with a concentration of 2 mol/L of A is charged into a

batch reactor. For the enzyme type and concentration used, k ¼ 0.1 mol/L

min and K

¼ 4 mol/L. Determine the characteristic reaction time.

Solution Selecting the initial state as the reference state, and since A is the only

species charged into the reactor, C

¼ C

(0). To express C

in dimensionless

form,

¼ C



(a)

and substituting (a) into the rate expression,

r ¼ k

ðÞþC

ðÞ



(b)

94 CHEMICAL KINETICS

The dimensional term (the characteristic reaction rate) is

¼ k (c)

Using Eq. 3.5.1, the characteristic reaction time is

2 mol=L

0:1 mol=L min

¼ 20 min (d)

Example 3.5 The heterogeneous, gas-phase catalytic chemical reaction

A þ B ! C

is carried out in a ﬂow reactor. The volume-based rate expression of the chemical

reaction is

r ¼ k(T)

(1 þ K

þ K

)

A gas stream at 2 atm and 2308C is fed into a reactor. At this temperature,

k ¼ 80 L/mol min. Determine the characteristic reaction time.

Solution Note that the units of K

and K

are (mol/L)

. We select the inlet

stream as the reference stream. For gas-phase reactions, and assuming ideal gas

behavior,

2 atm

(0:08206 L atm=mol K)(503 K)

¼ 4:85  10

2

mol=L (a)

To express the species concentrations in dimensionless form

¼ C



¼ C



(b)

Substituting (b) into the rate expression,

r ¼ k(T

g(u1)=u

)(C

)

[1 þ K

) þ K

)]



(c)

The dimensional portion of (c) (the characteristic reaction rate) is

¼ k(T

3.5 CHARACTERISTIC REACTION TIME 95

Using Eq. 3.5.1, the characteristic reaction time is

k(T

(80 L=mol min)(4:85 10

2

mol=L)

¼ 0:258 min (d)

Example 3.6 Allyl chloride is produced at 4008F by a homogeneous gas-

phase reaction between chlorine and propylene:

þ C

! C

Cl þ HCl

Based on experimental runs, the chlorine depletion rate is

(r

) ¼ k(T)P

where (r

) is in lbmol/hft

, and P is in psia. A gas stream at 30 psia and

4008F, consisting of 50% propylene and 50% chlorine, is fed into the reactor.

The reaction rate constant at 4008F is 0.02 lbmol/hft

psia

. Determine the

characteristic reaction time.

Solution Since the chlorine depletion rate is provided (rather than the reaction

rate), ﬁrst we convert the species rate to a reaction rate. Using Eq. 3.2.5, the reac-

tion rate is

r ¼

r

¼r

(a)

To determine the characteristic reaction time, ﬁrst the reference concentration,

, is calculated. Selecting the inlet stream as the reference stream, for gas-

phase reactions, assuming ideal-gas behavior

30 psia

(10:73 psia ft

=lbmol8R)(8608R)

¼ 3:25  10

3

lbmol= ft

(b)

To express the species partial pressure in dimensionless form

¼ P



¼ P



(c)

where P

is the total reference pressure. Substituting these into the rate

expression,

r ¼ k(T

g(u1)=u



(d)

96 CHEMICAL KINETICS

The dimensional portion of the rate expression is

¼ k(T

(e)

Substituting (b) and (e) into Eq. 3.5.1, the characteristic reaction time is

k(T

)RT

¼ 1:81  10

4

h(f)

3.6 SUMMARY

In this chapter, the main concepts and relations of chemical kinetics that are used in

reactor design were discussed. We covered the following topics:

1. Deﬁnition of species formation rates on the basis of volume, mass, and sur-

face and the relations among them

2. Deﬁnition of reaction rates and their relation to the species formation rates

3. The general forms of the reaction rate expressions for most homogeneous

reactions (power expression)

4. The reaction activation energy and how to determine it

5. The orders of the reaction

6. Different forms of the rate expressions for biological and heterogeneous

catalytic reactions

7. Deﬁnition and determination of the characteristic reaction time as a measure

of the reaction time scale

For convenience, Table A.2 in the Appendix provides a summary of the kinetic

deﬁnitions and relations.

PROBLEMS*

3.1

The rate of a heterogeneous reaction was measured in a rotating basket

reactor. The volume of the basket was 100 mL and it was ﬁlled with

240 g of catalytic particles whose speciﬁc surface area was 9.5 m

/gof

Subscript 1 indicates simple problems that require application of equations provided in the text.

Subscript 2 indicates problems whose solutions require some more in-depth analysis and modiﬁcation

of given equations. Subscript 3 indicates problems whose solutions require more comprehensive

analysis and involve application of several concepts. Subscript 4 indicates problems that require the

use of a mathematical software or the writing of a computer code to obtain numerical solutions.

PROBLEMS 97

catalyst. At 1008C, the measured volume-based reaction rate constant was

2.5 s

. Determine:

a. The order of the reaction, based on the units of the reaction rate constant

b. The mass-based rate constant

c. The surface-based rate constant

3.2

For a ﬁrst-order homogeneous reaction, the following experimental data

were recorded:

T (8C) 020406080

k  10

(min

1

)0:36:18 86:4 879 6900

Determine:

a. The activation energy

b. The rate constant at 258C

c. The frequency factor

d. The dimensionless activation energy if the reference temperature is 258C

e. The characteristic reaction time at 408C

3.3

A common rule-of-thumb for many chemical reactions states that, at normal

conditions, the reaction rate doubles for each increase of 10 8C. Assuming a

“normal” temperature of 208C, estimate the activation energy of a “typical”

chemical reaction. If we select 208C as the reference temperature, what is a

“typical” dimensionless activation energy?

3.4

Fine and Beall ( Chemistry for Engineers and Scientists, 1990, p. 854) pro-

vide the kinetic data for a particular proton transfer reaction:

T (K) 273 283 293 303

k (L mol

1

) 35 150 500 1800

Determine:

a. The overall order of the reaction from the units of k

b. The activation energy

c. If we select 300 K as the reference temperature, what is the dimensionless

activation energy?

d. If C

¼ 2 mol/L, what is the characteristic reaction time at 300 K?

3.5

The use of a pressure cooker is a common method to increase the reaction

rate and to shorten the cooking time. A typical pressure cooker operates

at 15 psig, thus raising the cooking temperature from 212 to 2498F.

98 CHEMICAL KINETICS

Neglecting the cooking that occurs during the heating-up period and assum-

ing the rate constant is inversely proportional to the cooking time,

a. Estimate the cooking activation energy of the foods listed below.

b. If we select 2128F as the reference temperature, what is the dimensionless

activation energy of each food?

c. Assuming ﬁrst-order reaction, what are the characteristic reaction times

at 2128F?

Normal Cooking

Time

Cooking Time in a

Pressure Cooker

Green beans 12 min 3 min

Corn beef 4 h 60 min

Chicken stew 2.5 h 20 min

BIBLIOGRAPHY

More detailed treatments of chemical kinetics, reaction mechanisms, catalysis, and

the theoretical basis for the rate expression can be found in:

S. W. Benson, The Foundation of Chemical Kinetics, McGraw-Hill, New York, 1960.

M. Boudart, Kinetics of Chemical Processes, Prentice-Hall, Englewood Cliffs, NJ, 1968.

J. H. Espenson, Chemical Kinetics and Reaction Mechani sms, McGraw-Hill, New York,

1981.

W. C. Gardiner, Rates and Mechanisms of Chemical Reactions, Benjamin, New York,

1969.

O. A. Hougen and K. M. Watson, Chemical Process Principles III; Kinetics and Catalysis,

Wiley, New York, 1947.

K. J. Laidler, Chemical Kinetics, 3rd, Harper & Row, New York, 1987.

R. I. Masel, Principles of Adsorption and Reaction on Solid Surfaces, Wiley Interscience,

New York, 1996.

R. I. Masel, Chemical Kinetics and Catalysis, Wiley Interscience, Hoboken, NJ, 2001.

Compilations of experimental and theoretical reaction rate data can be found in:

C. H. Bamford, and C. F. H. Tipper, eds., Comprehensive Chemical Kinetics, Elsevier,

Amsterdam, 1989.

J. A. Kerr, and M. J. Parsonage, Evaluated Kinetic Data of Gas-Phase Addition Reactions,

Butterworth, London, 1972.

V. N. Kondratiev, Rate Constants of Gas-Phase Reactions—Reference Book, Trans. L.

J. Holtschlag, R. M. Fristrom, ed., National Bureau of Standards, NITI Service,

Springﬁeld, VA, 1972.

NIST, NIST Chemical Kinetic Database on Diskette, NIST, Gaithersburg, MD, 1993.

BIBLIOGRAPHY 99