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

Подождите немного. Документ загружается.

CHEMICAL KINETICS

This chapter covers the second fundamental concept used in chemical reaction

engineering—chemical kinetics. The kinetic relationships used in the analysis

and design of chemical reactors are derived and discussed. In Section 3.1, we dis-

cuss the various deﬁnitions of the species formation rates. In Section 3.2, we deﬁne

the rates of chemical reactions and discuss how they relate to the formation (or

depletion) rates of individual species. In Section 3.3, we discuss the rate expression

that provides the relationship between the reaction rate, the temperature, and species

concentrations. Without going into the theory of chemical kinetics, we review the

common forms of the rate expressions for homogeneous and heterogeneous reac-

tions. In the last section, we introduce and deﬁne a measure of the reaction

rate—the characteristic reaction time. In Chapter 4 we use the characteristic reaction

time to reduce the reactor design equations to dimensionless forms.

3.1 SPECIES FORMATION RATES

Consider a batch reactor with volume V where species j is formed (or consumed) by

chemical reactions. We denote the total moles of species j in the reactor by N

;

hence, the rate species j being formed in the reactor is dN

/dt. The magnitude of

/dt depends, of course, on the size of the reactor. To deﬁne an intensive quantity

for the formation rate of species j, we consider the nature of the chemical reactions

that are taking place. For homogeneous chemical reactions (reactions that take

Principles of Chemical Reactor Analysis and Design, Second Edition. By Uzi Mann

place throughout a given phase), the volume-based species formation rate, (r

), is

deﬁned by

) ¼ (r

)

;

(3:1:1a)

Thus, (r

) is the formation rate of species j expressed in terms of moles of species j

formed per unit time per unit volume. For heterogeneous chemical reactions (reac-

tions that take place on the surface of a solid catalyst or at the interface of the two

phases), the surface-based formation rate of species j,(r

)

, is deﬁned by

)

;

(3:1:1b)

where (r

)

is expressed in terms of moles of species j formed per unit time per unit

surface area. In some cases, it is convenient to deﬁne the mass-based formation rate

of species j (on the basis of the mass of solid catalyst):

)

;

(3:1:1c)

Here, (r

)

is the mass-based formation rate of species j expressed in moles of

species j formed per unit time per unit mass of catalyst.

From Eqs. 3.1.1a–3.1.1c, it is easy to show that

) ¼



)

(3:1:2)

) ¼



)

(3:1:3)

Hence, when any one of these rates is known, the other two can be determined if the

properties of the reactor (mass of solid catalyst per unit volume or catalyst surface

per unit volume) are provided.

Note that all these deﬁnitions do not relate to any speciﬁc chemical reaction but

rather to the net formation of species j, which may be formed simultaneously by

some reactions and consumed by others. Also, when species j is consumed, (r

)

is negative.

3.2 RATES OF CHEMICAL REACTIONS

Consider a speciﬁc chemical reaction and focus on the rate that it progresses. As

discussed in Section 2.2, we ﬁrst have to select the chemical formula for this reac-

tion (select a basis for the calculation) and then express the progress of the reaction

82 CHEMICAL KINETICS

in terms of the extent of that formula. For a homogeneous chemical reaction, the

volume-based rate of a reaction is deﬁned by

r ;

(3:2:1a)

where r denotes the reaction rate expressed in terms of moles extent per unit time

per unit volume. For heterogeneous reactions, the surface-based rate of the chemi-

cal reaction is deﬁned by

;

(3:2:1b)

where r

is expressed in terms of moles extent per unit time per unit surface area.

Similarly, the mass-based rate of the chemical reaction is deﬁned by

;

(3:2:1c)

where r

is expressed in terms of moles extent per unit time per unit mass. As

before, relationships among these three reaction rate deﬁnitions can be easily

obtained:

r ¼



(3:2:2)

r ¼



(3:2:3)

Next, we consider the relationship between the formation rates of individual

species and the rates of the chemical reactions. When a single chemical reaction

takes place, we can easily relate the rate of the chemical reaction, r, to the formation

rate of a speciﬁc species, say species j,(r

). Differentiating the deﬁnition of the reac-

tion extent (Eq. 2.3.1) with time,

(3:2:4)

and by substituting this into Eq. 3.2.1a,

) ¼ s

r (3:2:5)

Equation 3.2.5 provides a relationship between the reaction rate and the rate of

formation (or depletion) of any species j by that chemical reaction.

3.2 RATES OF CHEMICAL REACTIONS 83

When multiple chemical reactions take place simultaneously, species j may par-

ticipate in several reactions, formed by some and consumed by others. The change

in the molar content of species j in the reactor relates to its formation by all chemi-

cal reactions. Using the deﬁnition of the reaction extent (Eq. 2.3.1), the molar con-

tent relates to the reaction extents by

(t)  N

(0) ¼

i¼1

)

(t)

where n

is the number of reactions (reaction pathways). Differentiating this

expression with respect to time and substituting into Eq. 3.1.1a, the formation rate

of species j is

) ¼

i¼1

)

(3:2:6)

where (s

)

is the stoichiometric coefﬁcient of species j in the ith chemical reaction,

and r

is the rate of the ith reaction. Equation 3.2.6 relates the formation rate of any

species to the rates of the chemical reactions. It is important to note that the sum-

mation in Eq. 3.2.6 is over all the chemical reactions that actually take place in the

reactor (both dependent and independent). Also, note that when a single chemical

reaction takes place, Eq. 3.2.6 reduces to Eq. 3.2.5.

Example 3.1 The heterogeneous catalytic chemical reaction

þ O

! 2C

is investigated in a packed-bed reactor. The reactor is ﬁlled with catalytic pellets

whose surface area is 7 m

/g, and the density of the bed is 1.4 kg/L. The

measured consumption rate of ethylene is 0.35 mol/h g catalyst. Determine:

a. The volume-based and surface-based formation rates of ethylene

b. The volume-based rate of the chemical reaction

c. The volume-based formation rate of oxygen

d. The volume-based formation rate of ethylene oxide

Solution The stoichiometric coefﬁcients of the chemical reaction are

¼2 s

¼1 s

¼ 2

The given mass-based formation rate of ethylene is 20.35 mol/h g (the minus

sign indicates that ethylene is consumed).

84 CHEMICAL KINETICS

a. Using Eq. 3.1.3, the volume-based formation rate of ethylene is

) ¼



)

¼ r

bed

)

¼ 1:4



0:35

mol Et



1000 g



¼490

mol Et

(a)

To determine the surface-based formation rate of ethylene, we ﬁrst calculate

the speciﬁc surface in the reactor:



¼ r

bed



¼ (1:4kg=L)(7:0m

=kg)(1000 g=kg) ¼ 9800 m

=L (b)

Using Eq. 3.1.2, (a), and (b), the surface-based formation rate of ethylene is

)

490 mol Et =hL

9:80  10

¼5:00  10

2

mol Et=hm

(c)

b. The volume-based rate of the reaction is determined by using Eq. 3.2.5

and (a):

r ¼

)

490 mol Et=hL

2 mol Et=mol extent

¼ 245 mol extent=h L (d)

c. The volume-based formation rate of oxygen is determined by using Eq. 3.2.5

and (d):

¼1

mol oxygen

mol extent



245

mol extent



¼245 mol oxygen=h L (e)

The negative sign indicates that oxygen is consumed.

d. The volume-based formation rate of ethylene oxide is determined by using

Eq. 3.2.5 and (d):

¼ 2

mol oxide

mol extent



245

mol extent



¼ 490 mol oxide=hL (f)

3.2 RATES OF CHEMICAL REACTIONS 85

3.3 RATE EXPRESSIONS OF CHEMICAL REACTIONS

The rate of a chemical reaction is a function of the temperature, the composition

of the reacting mixture, and, if present, the catalyst. The relationship between the

reaction rate and these parameters is commonly called the rate expression or,

sometimes, the rate law. Chemical kinetics is the branch of chemistry that deals

with reaction mechanisms and provides a theoretical basis for the rate expression.

When such information is available, we use it to obtain the rate expression. In many

instances, the reaction rate expression is not available and should be determined

experimentally.

For most chemical reactions, the rate expression is a product of two functions,

one of temperature, k(T ), and the second of species concentrations, h(C

’s):

r ¼ k(T)h(C

s) (3:3:1)

The function k(T ) is commonly called the reaction rate constant. However, note

that the reaction rate depends on the temperature. The term rate constant comes

about because k(T ) is independent of the composition and is constant at isothermal

operations.

For most chemical reactions, k(T ) relates to the temperature by the Arrhenius

equation:

k(T) ¼ k

E

=RT

(3:3:2)

where E

is a parameter called the activation energy, k

is a parameter called the

frequency factor or the preexponential coefﬁcient, and R is the universal gas con-

stant. Both parameters are characteristic of the chemical reaction and the presence

(or absence) of a catalyst. The value of E

indicates the sensitivity of the reaction

rate to changes in temperature. When the activation energy is relatively large (200–

400 kJ/mol), the reaction rate is sensitive to temperature. Such values are typical of

combustion and gasiﬁcation reactions that take place at high temperatures and are

very slow at room temperature. On the other hand, when the value of E

is relatively

low (20– 40 kJ/mol), the reaction rate is not insensitive to temperature. These

values are typical of biological and enzymatic reactions that take place at room

temperature. Figure 3.1 shows the relation between k and T for large and small acti-

vation energies.

The value of the rate constant at a given temperature is readily calculated when

both parameters in the Arrhenius equation, k

and E

, are known. However, it is

convenient to calculate the value of the rate constant at one temperature on the

basis of its value at a different temperature, using only the activation energy. To

obtain a relationship between the values of the reaction rate constant at two temp-

eratures, T

and T

, we take the log of Eq. 3.3.2 for each and combine the two

equations to obtain

k(T

)

k(T

)

¼





(3:3:3)

86 CHEMICAL KINETICS

The values of the parameters in the Arrhenius equation, k

and E

, are deter-

mined from experimental measurements of the reaction rate constant, k(T ), at

different temperatures. Using Eq. 3.3.3, the plot ln k(T ) versus 1/T gives a straight

line with a slope of 2E

/R, as shown schematically in Figure 3.2. The value of k

can be determined from the intercept. However, this procedure is not accurate

because it involves an extrapolation far beyond the temperature range of the exper-

imental data. Instead, we ﬁrst determine the slope and then use Figure 3.2 to obtain

the value of the rate constant at the average temperature, T

. We then calculate k

from Eq. 3.3.2 using the value of E

obtained from the slope (see Fig. 3.2).

Figure 3.1 Reaction rate constant as a function of temperature.

Figure 3.2 Determination of activation energy.

3.3 RATE EXPRESSIONS OF CHEMICAL REACTIONS 87

Physically, the activation energy represents an energy barrier that should be

overcome as the reaction proceeds. This barrier and its relationship to the heat of

reaction is shown schematically in Figure 3.3 for exothermic and endothermic reac-

tions. Note that for reversible chemical reactions, the heat of reaction is the differ-

ence between the activation energy of the forward reaction and the activation

energy of the backward reaction, DH

¼ (E

)

for

 ( E

)

back

As will be discussed later, we formulate the design equations of chemical reac-

tors in terms of dimensionless quantities and would like to express the reaction rate

constants in terms of them. We deﬁne dimensionless temperature

u ;

(3:3:4)

where T

is a conveniently selected reference temperature. We also deﬁne a dimen-

sionless activation energy, g,by

g ;

(3:3:5)

which is a characteristic of the chemical reaction and the reference temperature.

Using Eqs. 3.3.4 and 3.3.5, the Arrhenius equation reduces to

k(u) ¼ k(T

g(u1)=u

(3:3:6)

where k(T

) is the reaction rate constant at the reference temperature T

Figure 3.3 Activation energy for (a ) endothermic reactions and (b) exothermic reactio ns.

88 CHEMICAL KINETICS

Example 3.2 The rate constant of a chemical reaction is determined exper-

imentally at two temperatures. Based on the data below, determine:

a. The activation energy

b. The dimensionless activation energy if the reference temperature is 308C

c. The preexponential coefﬁcient

Data: T (8C) 30 50

k (min

1

)0:25 1:4

Solution

a. To determine the activation energy, write Eq. 3.3.3 for the two given temp-

eratures, T

¼ 30 þ 273.15 ¼ 303.15 K and T

¼ 50 þ 273.15 ¼ 323.15 K,

1:4

0:25



¼

323:15



303:15



(a)

to obtain E

/R ¼ 8438 K. Hence,

¼ (8438 K)(1:987 cal=mol K) ¼ 16:767 kcal=mol (b)

b. The reference temperature is T

¼ 303.15 K, and, using Eq. 3.3.5, the dimen-

sionless activation energy is

g ¼

16,767

1:987  303:15

¼ 27:83

c. To determine the preexponential coefﬁcient, we write Eq. 3.3.2 for one of the

given temperatures. For T

¼ 303.15 K,

(0:25 min

1

) ¼ k

8438=303:15

to obtain k

¼ 3.06  10

min

Next, we consider the dependence of the rate expression on the species compo-

sition–function h(C

’s) in Eq. 3.3.1. For many homogeneous chemical reactions,

h(C

’s) is expressed as a power relation of the species concentrations:

h(C

s) ¼ C

(3:3:7)

where C

, C

, ... are the concentrations of the different species. The powers in

Eq. 3.3.7 are called the orders of the reaction: a is the order of the reaction with

respect to species A, b is the order of the reaction with respect to species B, and

so on, and a þ bþ

...

is the overall order of the reaction. The orders can be

3.3 RATE EXPRESSIONS OF CHEMICAL REACTIONS 89