Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations
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Example 4.2 Acrolein is being produced by catalytic oxidation of propylene on a
bismuth molybdate catalyst. The following reactions are taking place in the reactor:
Reaction 1: C
3
H
6
þ O
2
! C
3
H
4
O þ H
2
O
Reaction 2: C
3
H
4
O þ 3:5O
2
! 3CO
2
þ 2H
2
O
Reaction 3: C
3
H
6
þ 4:5O
2
! 3CO
2
þ 3H
2
O
Adams et al. (J. Catalysis 3, 379, 1964) investigated these reactions and
expressed the rate of each as second order (first order with respect to each reac-
tant). Formulate the dimensionless, reaction-based design equations for an ideal
batch reactor, plug-flow reactor, and a CSTR.
Solution The purpose of this example is to illustrate the design formulations
for chemical reactions where the dependency between dependent reactions
and independent reactions is not due to reversible reactions. First, we determine
the number of independent reactions and select a set of independent reactions.
We construct a matrix of stoichiometric coefficients for the given reactions:
C
3
H
6
O
2
C
3
H
4
OH
2
OCO
2
1 1 110
0 3:5 123
1 4:5 033
2
6
4
3
7
5
(a)
Applying Gaussian elimination procedure, Matrix (a) reduces to
1 1 110
0 3:5 123
0 0 000
2
4
3
5
(b)
Matrix (b) is a reduced matrix since all the elements below the diagonal are
zeros, and, because it has two nonzero rows, we have here two independent
chemical reactions. We select the first two reactions as a set of independent reac-
tions; hence, for this case, m ¼ 1, 2, and k ¼ 3. To determine the multipliers
a
km
’s of the dependent reaction (Reaction 3) and the two independent reactions
(Reactions 1 and 2), we write Eq. 2.4.9 for propylene and CO
2
:
a
31
(1) þ a
32
(0) ¼1 (c)
a
31
(0) þ a
32
(3) ¼ 3 (d)
Solving (c) and (d), we obtain a
31
¼ 1 and a
32
¼ 1. (Indeed, Reaction 3 is the
sum of Reactions 1 and 2.) Now that the a
km
factors are known, we can
formulate the design equations. For an ideal batch reactor, we write Eq. 4.4.4
120 SPECIES BALANCES AND DESIGN EQUATIONS
for each of the two independent reactions. Hence, for this case, the design
equations are
dZ
1
dt
¼ (r
1
þ r
3
)
V
R
V
R
0
t
cr
C
0
(e)
dZ
2
dt
¼ (r
2
þ r
3
)
V
R
V
R
0
t
cr
C
0
(f)
To solve these design equations, we have to express r
1
, r
2
, r
3
, and V
R
/V
R
(0) in
terms of Z
1
and Z
2
. For a plug-flow reactor, we write Eq. 4.4.11 for each of the
two independent reactions. Hence, for this case, the design equations are
dZ
1
dt
¼ (r
1
þ r
3
)
t
cr
C
0
(g)
dZ
2
dt
¼ (r
2
þ r
3
)
t
cr
C
0
(h)
To solve these design equations, we have to express r
1
, r
2
, and r
3
in terms of Z
1
and Z
2
. For a CSTR, we write Eq. 4.4.12 for each of the two independent
reactions. Hence, for this case, the design equations are
Z
1
out
Z
1
in
(r
1
out
þ r
3
out
)t
t
cr
C
0
¼ 0 (i)
Z
2
out
Z
2
in
(r
2
out
þ r
3
out
)t
t
cr
C
0
¼ 0(j)
To solve these design equations, we have to express r
1
out
, r
2
out
, and r
3
out
in terms
of Z
1
out
and Z
2
out
and, then, for different values of t, we solve for Z
1
out
and Z
2
out
.
Example 4.3
a. The following gaseous chemical reactions take place in the reactor:
Reaction 1: 4NH
3
þ 5O
2
! 4NO þ 6H
2
O
Reaction 2: 4NH
3
þ 3O
2
! 2N
2
þ 6H
2
O
Reaction 3: 2NO þ O
2
! 2NO
2
Reaction 4: 4NH
3
þ 6NO ! 5N
2
þ 6H
2
O
Formulate the dimensionless, reaction-based design equations for ideal batch
reactor, plug-flow reactor, and CSTR using the heuristic rule.
4.4 DIMENSIONLESS DESIGN EQUATIONS AND OPERATING CURVES 121
b. Formulate the dimensionless, reaction-based design equations for an ideal
batch reactor using a set of independent reactions obtained from the reduced
matrix.
Solution First, we determine the number of independent reactions and select a
set of independent reactions. The matrix of stoichiometric coefficients for these
reactions was constructed and reduced to a reduced matrix in Example 2.12.
Since there are three nonzero rows in the reduced matrix, there are three indepen-
dent reactions. The nonzero rows in the reduced matrix represent the following
set of independent reactions:
Reaction 1: 4NH
3
þ 5O
2
! 4NO þ 6H
2
O
Reaction 5: 2NO ! O
2
þ N
2
Reaction 6: 4NO ! N
2
þ 2NO
2
Note that this set of independent reactions includes two reactions (Reactions 5
and 6) that are not among the given reactions.
a. Applying the heuristic rule of selecting a set of independent reactions whose
rate expressions are known, we should select a set of three independent reac-
tions from the given chemical reactions. Reactions 1, 2, and 3, represent such
a set; hence, m ¼ 1, 2, 3 and k ¼ 4, and we express the design equations in
terms of Z
1
, Z
2
, and Z
3
. The stoichiometric coefficients of the selected inde-
pendent reactions are
(s
NH
3
)
1
¼4(s
O
2
)
1
¼5(s
NO
)
1
¼4(s
H
2
O
)
1
¼6(s
N
2
)
1
¼0(s
NO
2
)
1
¼0
(s
NH
3
)
2
¼4(s
O
2
)
2
¼3(s
NO
)
2
¼0(s
H
2
O
)
2
¼6(s
N
2
)
2
¼2(s
NO
2
)
2
¼0
(s
NH
3
)
3
¼0(s
O
2
)
3
¼1(s
NO
)
3
¼2(s
H
2
O
)
3
¼0(s
N
2
)
3
¼0(s
NO
2
)
3
¼2
D
1
¼1 D
2
¼1 D
3
¼1
To determine the multipliers a
km
’s of the dependent reaction (Reaction 4) and
the three independent reactions, we write stoichiometric relation Eq. 2.4.9
for NO
2
,
a
41
(0) þ a
42
(0) þ a
43
(2) ¼ 0 (a)
and obtain a
43
¼ 0. We write Eq. 2.4.9 for N
2
,
a
41
(0) þ a
42
(2) þ a
43
(0) ¼ 5 (b)
122 SPECIES BALANCES AND DESIGN EQUATIONS
and obtain a
42
¼ 2.5. We write Eq. 2.4.9 for NH
3
:
a
41
(4) þ a
42
(4) þ a
43
(0) ¼4 (c)
and obtain a
41
¼ 21.5. Now that the a
km
factors are known, we can formu-
late the design equations. For an ideal batch reactor, we write Eq. 4.4.4 for
each of the three independent reactions. Hence, the design equations are
dZ
1
dt
¼ (r
1
1: 5r
4
)
V
R
V
R
0
t
cr
C
0
(d)
dZ
2
dt
¼ (r
2
þ 2: 5r
4
)
V
R
V
R
0
t
cr
C
0
(e)
dZ
3
dt
¼ r
3
V
R
V
R
0
t
cr
C
0
(f)
For a plug-flow reactor, we write Eq. 4.4.11 for each of the three independent
reactions. Hence, the design equations are
dZ
1
dt
¼ (r
1
1:5r
4
)
t
cr
C
0
(g)
dZ
2
dt
¼ (r
2
þ 2:5r
4
)
t
cr
C
0
(h)
dZ
3
dt
¼ r
3
t
cr
C
0
(i)
For a CSTR, we write Eq. 4.4.12 for each of the three independent reactions.
Hence, for this case, the design equations are
Z
1
out
Z
1
in
( r
1
out
1: 5r
4
out
)t
t
cr
C
0
¼ 0(j)
Z
2
out
Z
2
in
( r
2
out
þ 2: 5r
4
out
)t
t
cr
C
0
¼ 0 (k)
Z
3
out
Z
3
in
r
3
out
t
t
cr
C
0
¼ 0 (l)
To solve the design equations, we have to express the rates of the individual
reactions in terms of the dimensionless extents of the independent reactions,
Z
1
, Z
2
, and Z
3
.
b. Taking the reactions of the reduced matrix as a set of independent reactions,
m ¼ 1, 5, 6, and k ¼ 2, 3, 4. Hence, we express the design equations in terms
4.4 DIMENSIONLESS DESIGN EQUATIONS AND OPERATING CURVES 123
of Z
1
, Z
5
, and Z
6
. The stoichiometric coefficients of the three independent
reactions are
(s
NH
3
)
1
¼4(s
O
2
)
1
¼5(s
NO
)
1
¼4(s
H
2
O
)
1
¼6(s
N
2
)
1
¼0(s
NO
2
)
1
¼0
(s
NH
3
)
5
¼0(s
O
2
)
5
¼1(s
NO
)
5
¼2(s
H
2
O
)
5
¼0(s
N
2
)
5
¼1(s
NO
2
)
5
¼0
(s
NH
3
)
6
¼0(s
O
2
)
6
¼0(s
NO
)
6
¼4(s
H
2
O
)
6
¼0(s
N
2
)
6
¼1(s
NO
2
)
6
¼2
D
1
¼1 D
5
¼1 D
6
¼1
To determine the multipliers a
km
’s of the three dependent reactions
(Reactions 2, 3, and 4) and the three independent reactions, we write stoichio-
metric relation Eq. 2.4.9. For k ¼ 2, we write Eq. 2.4.9 for NO
2
,
a
21
(0) þ a
25
(0) þ a
26
(2) ¼ 0 (m)
and obtain a
26
¼ 0. We write Eq. 2.4.9 for N
2
,
a
21
(0) þ a
25
(1) þ a
26
(1) ¼ 2 (n)
and obtain a
25
¼ 2. We write Eq. 2.4.9 for NH
3
,
a
21
(4) þ a
25
(0) þ a
26
(0) ¼4 (o)
and obtain a
21
¼ 1. For k ¼ 3, we write Eq. 2.4.9 for NO
2
,
a
31
(0) þ a
35
(0) þ a
36
(2) ¼ 2(p)
and obtain a
36
¼ 1. We write Eq. 2.4.9 for N
2
,
a
31
(0) þ a
35
(1) þ a
36
(1) ¼ 0 (q)
and obtain a
35
¼ 2 1. We write Eq. 2.4.9 for NH
3
,
a
31
(4) þ a
35
(0) þ a
36
(0) ¼ 0(r)
and obtain a
31
¼ 0. For k ¼ 4, we write Eq. 2.4.9 for NO
2
,
a
41
(0) þ a
45
(0) þ a
46
(2) ¼ 0 (s)
and obtain a
46
¼ 0. We write Eq. 2.4.9 for N
2
,
a
41
(0) þ a
45
(1) þ a
46
(1) ¼ 5 (t)
124 SPECIES BALANCES AND DESIGN EQUATIONS
and obtain a
45
¼ 5. We write Eq. 2.4.9 for NH
3
,
a
41
(4) þ a
45
(0) þ a
46
(0) ¼4 (u)
and obtain a
41
¼ 1. Hence, the a
km
factors for the dependent reactions are
a
21
¼ 1 a
25
¼ 2 a
26
¼ 0
a
31
¼ 0 a
35
¼1 a
36
¼ 1
a
41
¼ 1 a
45
¼ 5 a
46
¼ 0
Now that the a
km
factors are known, we can formulate the design equations.
For an ideal batch reactor, we write Eq. 4.4.4 for each of the three indepen-
dent reactions. Noting that independent Reactions 5 and 6 do not actually
take place r
5
¼ r
6
¼ 0, and the design equations are
dZ
1
dt
¼ (r
1
þ r
2
þ r
4
)
V
R
V
R
0
t
cr
C
0
(v)
dZ
5
dt
¼ (2r
2
r
3
þ 5r
4
)
V
R
V
R
0
t
cr
C
0
(w)
dZ
6
dt
¼ r
3
V
R
V
R
0
t
cr
C
0
(x)
Note that, in this case, the design equations have seven terms (seven rate
expressions), whereas in the formulation in part (a), design equations (d),
(e), and (f), have only five terms. This illustrates that, by adopting the heur-
istic rule, we minimize the number of terms in the design equations. To solve
the design equations, we have to express the rates of the individual reactions
in terms of the dimensionless extents of the independent reactions, Z
1
, Z
5
,
and Z
6
.
4.5 SUMMARY
In this chapter, we discussed the application of the conservation of mass principle
to derive design equations for chemical reactors. The following topics were
covered:
†
We derived (in Appendix B) the species continuity equations, describing the
variations in species concentrations at any point in the reactor.
†
We integrated the continuity equation over the reactor volume and obtained
the general, species-based design equation.
4.5 SUMMARY 125
†
We noticed the difficulty of solving the general form of the design equation.
†
We derived the species-based design equations for three ideal reactor models:
ideal batch reactor, plug-flow reactor, and CSTR.
†
We derived the reaction-based design equations for three ideal reactors.
†
We converted the reaction-based design equations to dimensionless forms
that, upon solution, provide the dimensionless operating curves.
PROBLEMS*
4.1
2
A 200 L/min stream of freshwater is fed to and withdrawn from a well-
mixed tank with a volume of 1000 L. Initially, the tank contains pure
water. At time t ¼ 0, the feed is switched to a brine stream with a concen-
tration of 180 g/L that is also fed at a rate of 200 L/min.
a. Derive a differential equation for the salt concentration in the tank.
b. Separate the variables and integrate to obtain an expression for the salt
concentration as a function of time.
c. At what time will the salt concentration in the tank reach a level of 100 g/L?
4.2
2
Many specialty chemicals are produced in semibatch reactors where a reac-
tant is added gradually into a batch reactor. This problem concerns the gov-
erning equations of such operations without considering chemical reactions.
A well-mixed batch reactor initially contains 200 L of pure water. At time
t ¼ 0, we start feeding a brine stream with a salt concentration of 180 g/L
into the tank at a constant rate of 50 L/min. Calculate:
a. The time the salt concentration in the tank is 60 g/L
b. The volume of the tank at that time
c. The time and salt concentration in the tank when the volume of the tank is
600 L
Assume the density of the brine is the same as the density of the water.
4.3
3
Solve Problem 4.2 when the feed rate of the brine is not constant. Consider the
case where the feed rate is a function of time given by
n
in
(t) ¼ 100–10t (t is in
minute and
n
is in liter/minute), and the exit flow rate is constant at 20 L/min.
a. Plot a graph of the salt concentration as a function of time.
b. What is the salt concentration when the amount of solution in the tank is
the maximum?
c. What is the salt concentration of the last drop in the tank?
*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 modifi-
cations of given equations. Subscript 3 indicates problems whose solutions require more comprehen-
sive 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.
126
SPECIES BALANCES AND DESIGN EQUATIONS
Hint: You may resort to numerical solution, but you have to show the
equations you solve and the initial conditions.
4.4
1
Ethylene oxide is produced by a catalytic reaction of ethylene and oxygen
at controlled conditions. However, side reactions cannot be completely
prevented, and it is believed that the following reactions take place:
2C
2
H
4
þ O
2
! 2C
2
H
4
O
C
2
H
4
þ 3O
2
! 2CO
2
þ 2H
2
O
C
2
H
4
þ 2O
2
! 2CO þ 2H
2
O
2CO þ O
2
! 2CO
2
CO þ H
2
O ! CO
2
þ H
2
Formulate the dimensionless, reaction-based design equations for an ideal
batch reactor, plug-flow reactor, and a CSTR.
4.5
1
Cracking of naphtha to produce olefins is a common process in the petro-
chemical industry. The cracking reactions are represented by the simplified
elementary gas-phase reactions:
C
10
H
22
! C
4
H
10
þ C
6
H
12
C
4
H
10
! C
3
H
6
þ CH
4
C
6
H
12
! C
2
H
4
þ C
4
H
8
C
4
H
8
! 2C
2
H
4
Formulate the dimensionless, reaction-based design equations for a plug-
flow reactor.
4.6
1
The cracking of propane to produce ethylene is represented by the simplified
kinetic model:
C
3
H
8
! C
2
H
4
þ CH
4
C
3
H
8
! C
3
H
6
þ H
2
C
3
H
8
þ C
2
H
4
! C
2
H
6
þ C
3
H
6
2C
3
H
6
! 3C
2
H
4
C
3
H
6
! C
2
H
2
þ CH
4
C
2
H
4
þ C
2
H
2
! C
4
H
6
PROBLEMS 127
Formulate the dimensionless, reaction-based design equations for a
plug-flow reactor.
4.7
1
The following reactions are believed to take place during the direct oxidation
of methane:
2CH
4
þ 3O
2
! 2CO þ 4H
2
O
CH
4
þ 2O
2
! CO
2
þ 2H
2
O
2CH
4
þ O
2
! 2CH
3
OH
CH
4
þ O
2
! HCHO þ H
2
O
CO þ H
2
O ! CO
2
þ H
2
CO þ 2H
2
! CH
3
OH
CO þ H
2
! HCHO
2CH
4
þ O
2
! 2CO þ 4H
2
2CO þ O
2
! 2CO
2
Formulate the dimensionless, reaction-based design equations for a plug-
flow reactor.
4.8
1
In the reforming of methane, the following chemical reactions occur:
CH
4
þ 2H
2
O ! CO
2
þ 4H
2
CH
4
þ 2O
2
! CO
2
þ 2H
2
O
2CH
4
þ O
2
! 2CH
3
OH
2CH
4
þ 3O
2
! 2CO þ 4H
2
O
CO
2
þ 3H
2
! CH
3
OH þ H
2
O
2CH
3
OH þ 3O
2
! 2CO
2
þ 4H
2
O
2CO þ O
2
! 2CO
2
CO þ H
2
O ! CO
2
þ H
2
Formulate the dimensionless, reaction-based design equations for a plug-
flow reactor.
128 SPECIES BALANCES AND DESIGN EQUATIONS
BIBLIOGRAPHY
More comprehensive treatments of the species continuity equations and the molar
fluxes can be found in:
R. B. Bird, The Basic Concepts in Transport Phenomena, Chem. Eng. Ed., Winter, 102,
1994.
R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley,
Hoboken, NJ, 2002.
J. R. Welty, C. E. Wicks, and R. E. Wilson, Fundamentals of Momentum Heat and Mass
Transfer, 2nd ed., Wiley, New York, 1984.
Applications of the continuity equations for turbulent flows can be found in:
J. O. Hinze, Turbulence, 2nd ed., McGraw-Hill, New York, 1975.
H. Tennekes, and J. L. Lumley, A First Course in Turbulence, MIT Press, Cambridge, MA,
1972.
Applications of the species continuity equation in reacting systems is found in:
L. A. Belfiore, Transport Phenomena for Chemical Reactor Design, Wiley-Intersciences,
Hoboken, NJ, 2003.
D. E. Rosner, Transport Processes in Chemically Reacting Flow Systems, Butterworth-
Heineman, Boston, 1986.
BIBLIOGRAPHY 129