Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations
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the reactor at time t, N
j
(t), is related to the number of moles of species j formed by
each of the individual chemical reactions
N
j
(t) N
j
(0) ¼ (n
j
n
j
0
)
1
þþ(n
j
n
j
0
)
i
þþ(n
j
n
j
0
)
n
R
(2:3:2)
where (n
j
2 n
j
0
)
i
is the number of moles of species j formed by the ith chemical
reaction in time t. Note that species j may be formed in some reactions and con-
sumed in others. As will be discussed in Section 2.4, to determine the species com-
position (and, in general, all other state quantities), only a set of independent
reactions should be considered, and not all the chemical reactions that take place.
Hence, using Eq. 2.3.1, Eq. 2.3.2 reduces to
N
j
(t) ¼ N
j
(0) þ
X
n
I
m
(s
j
)
m
X
m
(t) j ¼ A, B, ... (2:3:3)
where m is an index for independent reactions, (s
j
)
m
is the stoichiometric coefficient
of species j in the mth independent reaction, X
m
(t) is the extent of the mth indepen-
dent reaction at time t, and n
I
is the number of independent reactions. Equation
2.3.3 relates the species composition in a batch reactor to the extents of the inde-
pendent chemical reactions.
To relate the total number of moles in the reactor at time t, N
tot
(t) to the extents,
we write Eq. 2.2.3 for each species in the reactor and sum the relations, and then
collect terms by the individual reaction extents,
N
tot
(t) ¼ N
tot
(0) þ
X
n
I
m
D
m
X
m
(t)(2:3:4)
where D
m
is the change in the number of moles per unit extent of the mth indepen-
dent chemical reaction, defined by Eq. 2.2.5, and N
tot
(0) is the total number of
moles initially in the reactor.
When a single chemical reaction takes place, Eq. 2.3.3 reduces to
N
j
(t) ¼ N
j
(0) þ s
j
X(t) j ¼ A, B, ... (2:3:5)
Writing Eq. 2.3.5 for any two species, say A and j,
N
j
(t) ¼ N
j
(0) þ
s
j
s
A
[N
A
(t) N
A
(0)] (2:3:6)
Equation 2.3.6 provides an algebraic relation between the number of moles of any
two species in the reactor at time t without calculating the extent itself. To deter-
mine the total number of moles in a batch reactor at time t, N
tot
(t), use Eq. 2.3.4,
30 STOICHIOMETRY
which reduces to
N
tot
(t) ¼ N
tot
(0) þ D X(t)(2:3:7)
To determine N
tot
(t) without calculating the extent explicitly, take the summation of
Eq. 2.3.6 over all species,
N
tot
(t) ¼ N
tot
(0) þ
D
s
A
[N
A
(t) N
A
(0)] (2:3:8)
Consider now a steady-flow reactor and conduct a species balance over the reac-
tor. At steady state, the molar flow rate of species j at the reactor outlet is equal to
the molar flow rate of species j at the reactor inlet plus the rate species j is being
generated inside the reactor by the reaction, G
j
,
F
j
out
¼ F
j
in
þ G
j
j ¼ A, B, ... (2:3:9)
The generation term, G
j
, is discussed in detail in Chapter 4; here, we focus on its
relation to the extents of the chemical reactions. The rate species j is being gener-
ated by each reaction is expressed in a relation similar to Eq. 2.3.2. Differentiating
Eq. 2.3.3 with respect to time, the rate of generation of species j is
G
j
¼
X
n
I
m
(s
j
)
m
_
X
m
j ¼ A, B, ... (2:3:10)
where X
˙
m
is the extent of the mth independent reaction converted per unit time in
the reactor. Substituting Eq. 2.3.10 into Eq. 2.3.9,
F
j
out
¼ F
j
in
þ
X
n
I
m
(s
j
)
m
_
X
m
j ¼ A, B, ... (2:3:11)
Summing Eq. 2.3.11 over all species, the total molar flow rate at the reactor outlet
relates to the extents of the reactions by
F
tot
out
¼ F
tot
in
þ
X
n
I
m
D
m
_
X
m
(2:3:12)
where D
m
is the change in the number of moles per unit extent of the mth indepen-
dent chemical reaction, defined by Eq. 2.2.5.
For steady-flow reactors with a single chemical reaction, Eq. 2.3.11 reduces to
F
j
out
¼ F
j
in
þ s
j
_
Xj¼ A, B, ... (2:3:13)
2.3 EXTENT OF A CHEMICAL REACTION 31
and Eq. 2.3.12 reduces to
F
tot
out
¼ F
tot
in
þ D
_
X (2:3:14)
We can write Eq. 2.3.11 for any two species, say A and j, to obtain
F
j
out
¼ F
j
in
þ
s
j
s
A
(F
A
out
F
A
in
) j ¼ B, C, ... (2:3:15)
Taking the summation of Eq. 2.3.15 over all species,
F
tot
out
¼ F
tot
in
þ
D
s
A
(F
A
out
F
A
in
)(2:3:16)
Equations 2.3.15 and 2.3.16 enable us to determine, respectively, the molar
flow rate of any species and the total molar flow rate at the reactor outlet in terms
of the molar flow rate of any other species without calculating the extent itself.
Example 2.1 A batch reactor contains 4 mol of CO and 1 mol of O
2
. Calculate
the extent of the reaction and the composition of the reactor when the reaction
goes to completion.
a. Use Reaction 2.2.1 and determine the species compositions from the extent.
b. Use Reaction 2.2.2 and determine the species compositions without calculat-
ing the extent.
Solution
a. We select the chemical formula
CO þ
1
2
O
2
! CO
2
and its stoichiometric coefficients are
s
CO
¼1 s
O
2
¼
1
2
s
CO
2
¼ 1 D ¼
1
2
The reaction reaches completion when one of the reactants is depleted.
Assuming that CO is depleted—hence, N
CO
(t) ¼ 0, and we write Eq. 2.3.5
to determine the extent of the reaction,
X(t) ¼
N
CO
(t) N
CO
(0)
s
CO
¼
0 4:0
1
¼ 4 mol extent
32 STOICHIOMETRY
Now that we know the extent of the chemical reaction, we can calculate the
moles of O
2
and CO
2
in the reactor at the end of the operation by Eq. 2.3.5:
N
O
2
(t) ¼ N
O
2
(0) þ s
O
2
X(t) ¼ 1 þ (0:5)(4) ¼1 mol
N
CO
2
(t) ¼ N
CO
2
(0) þ s
CO
2
X(t) ¼ 0 þ (1)(4) ¼ 4 mol
Since the number of moles of any species cannot be negative, obtaining such
values is an indication that there is an error in the calculation, or that it is
based on incorrect information. The error here is, of course, the assumption
that the CO is depleted before the oxygen, while actually oxygen is depleted
first. Hence, to determine the extent at completion, we write Eq. 2.3.5 for
oxygen with N
O
2
(t) ¼ 0,
X(t) ¼
N
O
2
(t) N
O
2
(0)
s
O
2
¼
0 1:0
0:5
¼ 2 mol extent
We calculate the moles of CO and CO
2
in the reactor at the end of the oper-
ation by Eq. 2.3.5:
N
CO
(t) ¼ N
CO
(0) þ s
CO
X(t) ¼ 4 þ (1)(2) ¼ 2 mol
N
CO
2
(t) ¼ N
CO
2
(0) þ s
CO
2
X(t) ¼ 0 þ (1)(2) ¼ 2 mol
The total number of moles in the reactor at time t is
N
tot
(t) ¼ N
O
2
(t) þ N
CO
(t) þ N
CO
2
(t) ¼ 0 þ 2 þ 2 ¼ 4 mol
We can also calculate N
tot
(t) by Eq. 2.3.7:
N
tot
(t) ¼ N
tot
(0) þ D X(t) ¼ 5 þ
1
2
(2) ¼ 4 mol
b. In this case, we select the chemical formula
2COþ O
2
! 2CO
2
and its stoichiometric coefficients are
s
CO
¼2 s
O
2
¼1 s
CO
2
¼ 2 D ¼1
Using the given initial composition, N
O
2
(0) ¼ 1 mol, and that at completion
N
O
2
(t) ¼ 0, we write Eq. 2.3.2 for oxygen with N
O
2
(t) ¼ 0,
X(t) ¼
N
O
2
(t) N
O
2
(0)
s
O
2
¼
0 1:0
1
¼ 1 mol extent
2.3 EXTENT OF A CHEMICAL REACTION 33
To determine the moles of CO and CO
2
without calculating the extent,
Eq. 2.3.6 is used:
N
CO
(t) ¼ N
CO
(0)þ
s
CO
s
O
2
[N
O
2
(t) N
O
2
(0)] ¼ 4 þ
2
1
(0 1) ¼ 2 mol
N
CO
2
(t) ¼ N
CO
2
(0) þ
s
CO
2
s
O
2
[N
O
2
(t) N
O
2
(0)] ¼ 0 þ
2
1
(0 1) ¼ 2 mol
Note that the final reactor composition does not depend on the specific
chemical formula selected. Also, note that the value of the extent in part (b)
is half the value of that in part (a).
Example 2.2 Ethylene oxide is produced by catalytic oxidation on a silver
catalyst according to the following chemical reaction:
C
2
H
4
þ
1
2
O
2
! C
2
H
4
O
A gaseous stream consisting of 60% C
2
H
4
, 30% O
2
, and 10% N
2
(by mole) is
fed at a rate of 40 mol/min into a flow reactor operating at steady state. If the
mole fraction of oxygen in the reactor effluent stream is 0.08, calculate the pro-
duction rate of ethylene oxide.
Solution We select the written chemical reaction as the chemical formula;
hence, the stoichiometric coefficients are
s
C
2
H
4
¼1 s
O
2
¼
1
2
s
C
2
H
4
O
¼ 1 s
N
2
¼ 0 D ¼
1
2
To determine the extent of the chemical reaction, we use the given composition
of the oxygen in the exit stream. First, we use Eq. 2.3.13 to express the oxygen
molar flow rate at the reactor exit:
(F
O
2
)
out
¼ (F
O
2
)
in
þ s
O
2
_
X ¼ (0:3)(40) þ
1
2
_
X (a)
Next, we use Eq. 2.3.14 to express the total molar flow rate of the effluent
stream:
(F
tot
)
out
¼ (F
tot
)
in
þ D
_
X ¼ 40 þ
1
2
_
X (b)
Thus, the oxygen mole fraction in the exit stream is
( y
O
2
)
out
¼
(F
O
2
)
out
(F
tot
)
out
¼
(0:3)(40) þ
1
2
_
X
40 þ
1
2
_
X
¼ 0:08 (c)
34 STOICHIOMETRY
Solving (c), X
˙
¼ 19.13 mol/min. Now that the value of X
˙
is known, Eq. 2.3.13 is
used to calculate the production rate of ethylene oxide:
(F
C
2
H
4
O
)
out
¼ (F
C
2
H
4
O
)
in
þ s
C
2
H
4
O
_
X ¼ 0 þ (1)(19:13) ¼ 19:13 mol=min
Example 2.3 An aqueous solution with a concentration of 0.8 mol/L of reac-
tant A is fed into a flow reactor at a rate of 150 L/min. The following chemical
reaction takes place in the reactor:
2A ! B
Conductivity cells are used to determine the compositions of the inlet and outlet
streams. The conductivity reading at the inlet is 140 units and at the outlet is 90
units. If the conductivity is proportional to the sum of the concentrations of A
and B, determine:
a. The extent of the reaction
b. The production rate of B
Solution This example illustrates how the reaction extent can be determined
from a measurable quantity. The stoichiometric coefficients are
s
A
¼2 s
B
¼ 1 D ¼1
a. To determine the extent, we have to derive a relationship between the extent
and the conductivity. Since the conductivity is proportional to (C
A
þC
B
), we
can write
l
out
l
in
¼
C
A
out
þ C
B
out
C
A
in
þ C
B
in
¼
90
140
(a)
The species concentrations are related to the molar flow rates and the volu-
metric flow rates by
C
A
¼
F
A
v
C
B
¼
F
B
v
(b)
For liquid-phase reactions, the density is constant. Using Eq. 2.3.12 and
noting that C
B
in
¼ 0, the molar flow rates of species A and B at the reactor
outlet are
F
A
out
¼ F
A
in
þ s
A
_
X ¼ vC
A
in
þ s
A
_
X ¼ 120 2
_
X (c)
F
B
out
¼ F
B
in
þ s
B
_
X ¼ vC
B
in
þ s
B
_
X ¼
_
X (d)
2.3 EXTENT OF A CHEMICAL REACTION 35
Substituting (c) and (d) into (b) and the latter into (a)
l
out
l
in
¼
F
A
in
_
X
F
A
in
¼
120
_
X
120
¼
90
140
(e)
or
_
X ¼ 120 1
90
140
¼ 42:86 mol=min ( f)
b. Using (d), the production rate of product B is 42.86 mol/min.
Example 2.4 The following two simultaneous chemical reactions take place in
a batch reactor:
Reaction 1: CO þ
1
2
O
2
! CO
2
(a)
Reaction 2: C þ
1
2
O
2
! CO (b)
Initially, the reactor contains 4 mol of CO, 4 mol of O
2
, and 2 mol of C. At the
end of the operation, the reactor contains 2 mol of CO and 2 mol of O
2
.
Determine:
a. The composition of the reactor at the end of the operation
b. The portion of O
2
that reacts in each reaction
Solution Since each chemical reaction has a species that does not participate in
the other, the two reactions are independent. The stoichiometric coefficients of
the species in the respective reactions are
(s
CO
)
1
¼1(s
O
2
)
1
¼
1
2
(s
CO
2
)
1
¼ 1(s
C
)
1
¼ 0 D
1
¼
1
2
(s
CO
)
2
¼ 1(s
O
2
)
2
¼
1
2
(s
CO
2
)
2
¼ 0(s
C
)
2
¼1 D
2
¼
1
2
a. To determine the extents of the chemical reactions, we first write Eq. 2.3.3
for CO:
N
CO
(t) ¼ N
CO
(0) þ (s
CO
)
1
X
1
(t) þ (s
CO
)
2
X
2
(t) ¼ 2 mol
Substituting the numerical values for the stoichiometric coefficients and the
initial composition, we obtain
X
1
(t) þ X
2
(t) ¼2 (c)
36 STOICHIOMETRY
Next we write Eq. 2.3.3 for O
2
:
N
O
2
(t) ¼ N
O
2
(0) þ (s
O
2
)
1
X
1
(t) þ (s
O
2
)
2
X
2
(t) ¼ 2 mol
and obtain
X
1
(t) þ X
2
(t) ¼ 4 (d)
Solving (c) and (d), X
1
(t) ¼ 3 mol and X
2
(t) ¼ 1 mol. Now that the extents of
the two chemical reactions are known, we can calculate the amount of C and
CO in the reactor. Using Eq. 2.3.3,
N
C
(t) ¼ N
C
(0) þ (s
C
)
1
X
1
(t) þ (s
C
)
2
X
2
(t) ¼ 1 mol
N
CO
2
(t) ¼ N
CO
2
(0) þ (s
CO
2
)
1
X
1
(t) þ (s
CO
2
)
2
X
2
(t) ¼ 3 mol
The total number of moles at time t is
N
tot
(t) ¼ N
CO
(t) þ N
O
2
(t) þ N
C
(t) þ N
CO
2
(t) ¼ 8 mol
It can be calculated also by using Eq. 2.3.4,
N
tot
(t) ¼ 10:0 þ
1
2
(3:0) þ
1
2
(1:0) ¼ 8:0 mol
b. To determine the number of moles of oxygen reacted by each chemical reac-
tion we use Eq. 2.3.1:
(n
O
2
n
O
2
0
)
1
¼ (s
O
2
)
1
X
1
(t) ¼
1
2
(3:0) ¼1:5 mol
(n
O
2
n
O
2
0
)
2
¼ (s
O
2
)
2
X
2
(t) ¼
1
2
(1:0) ¼0:5 mol
The minus sign indicates that oxygen is being consumed in the chemical
reactions. Thus, 75% of the oxygen is consumed by Reaction 1 and 25% by
Reaction 2.
Example 2.5 Wafers for integrated circuits are made of pure silicon that is pro-
duced by reacting raw silicon with HCl to form silicon trichloride, SiHCl
3
. The
silicon trichloride is reduced later with hydrogen to provide pure silicon. At the
reactor operating conditions, silicon tetrachloride, SiCl
4
, is also formed. Molten
raw silicon is fed into a flow reactor at a rate of 80 lbmol/h, and gaseous HCl is
fed in proportion of 4 mol HCl per mole silicon. The reactor operates at 12508C
and the following reactions take place:
Reaction 1: Si þ 3HCl ! SiHCl
3
þ H
2
(a)
Reaction 2: Si þ 4HCl ! SiCl
4
þ 2H
2
(b)
2.3 EXTENT OF A CHEMICAL REACTION 37
If all the silicon is consumed, and the effluent stream contains 40% H
2
(by
mole), determine:
a. The production rate of SiHCl
3
b. The amount of HCl reacted
Solution Since each chemical reaction has a species that does not appear in the
other, the two reactions are independent, and their stoichiometric coefficients are
(s
Si
)
1
¼1(s
HCl
)
1
¼3(s
SiHCl
3
)
1
¼ 1(s
SiCl
4
)
1
¼ 0(s
H
2
)
1
¼ 1 D
1
¼2
(s
Si
)
2
¼1(s
HCl
)
2
¼4(s
SiHCl
3
)
2
¼ 0(s
SiCl
4
)
2
¼ 1(s
H
2
)
2
¼ 2 D
2
¼2
a. We select the inlet stream as a basis for the calculation; thus, (F
Si
)
in
¼ 80 lbmol/
hand(F
HCl
)
in
¼ 320 lbmol/h, and (F
tot
)
in
¼ 400 lbmol/h. Using Eq. 2.3.11,
the production r ate of SiHCl
3
is
(F
SiHCl
3
)
out
¼ 0 þ (s
SiHCl
3
)
1
_
X
1
þ ( s
SiHCl
3
)
2
_
X
2
¼
_
X
1
(c)
To obtain the extents of the r ea ctions, we first use the fa ct that all the silicon is
consumed in the r ea ctor; hence, (F
Si
)
out
¼ 0. Using Eq. 2.3.11,
(F
Si
)
out
¼ 80 þ (1)
_
X
1
þ (1)
_
X
2
or
_
X
1
þ
_
X
2
¼ 80 lbmol=h (d)
Next we use the given composition of H
2
in the outlet s tream:
( y
H
2
)
out
¼
(F
H
2
)
out
(F
tot
)
out
¼ 0:4 (e)
Using Eq. 2.3.11,
(F
H
2
)
out
¼ 0 þ
_
X
1
þ 2
_
X
2
(f)
and using Eq. 2.3.12,
(F
tot
)
out
¼ 400 2
_
X
1
2
_
X
2
(g)
Substituting (f) and (g) into (e),
(1:8)
_
X
1
þ (2 :8)
_
X
2
¼ 160 lbmol=min (h)
Solving (h) and (d), X
˙
1
¼ 64 lbmol/h, X
˙
2
¼ 16 lbmol/h, and from (c), the
production rate of SiHCl
3
is (F
SiHCl
3
)
out
¼ 64 lbmol=h.
38 STOICHIOMETRY
b. Using Eq. 2.3.11, the flow rate of HCl at the reactor exit is
(F
HCl
)
out
¼ 320 þ (3)(64) þ (4)(16) ¼ 64 lbmol=h (i)
The amount of HCl reacted in the reactor is
(F
HCl
)
in
(F
HCl
)
out
¼ 320 64 ¼ 256 lbmol=h(j)
2.4 INDEPENDENT AND DEPENDENT CHEMICAL REACTIONS
In the preceding section we noted that when multiple chemical reactions take place,
only a set of independent chemical reactions should be considered to determine the
species composition. As indicated, the summations in Eq. 2.3.3 and Eq. 2.3.11 are
taken over a set of independent chemical reactions and not over all the reactions that
take place. This point deserves a closer examination.
The concept of independent reactions, or, more accurately, independent stoichio-
metric relations, is an important concept in stoichiometry and reactor analysis. The
number of independent reactions indicates the smallest number of stoichiometric
relations needed to describe the chemical transformations that take place and to
determine all the state quantities of a chemical reactor (species composition, temp-
erature, enthalpy, etc.). As will be seen later, the number of independent reactions
also indicates the smallest number of design equations needed to describe the reac-
tor operation. Since state quantities are independent of the path, we can select
different sets of independent reactions to determine the change from one state to
another. Below, we discuss the roles of independent and dependent reactions in
describing reactor operations. We also describe a procedure to determine the
number of independent reactions and how to identify a set of independent reactions.
To develop insight into the concept of independent reactions, consider the fol-
lowing reversible chemical reaction:
CO þ H
2
O
CO
2
þ H
2
(2:4:1)
According to the stoichiometric methodology adopted here, reversible reactions are
represented as two distinct reactions, a forward and a reverse. Hence, Reaction 2.4.1
is described by two chemical reactions: a forward reaction,
CO þ H
2
O ! CO
2
þ H
2
(2:4:1a)
whose stoichiometric coefficients are
s
CO
¼1 s
H
2
O
¼1 s
CO
2
¼ 1 s
H
2
¼ 1
and a backward reaction,
CO
2
þ H
2
! CO þ H
2
O(2:4:1b)
2.4 INDEPENDENT AND DEPENDENT CHEMICAL REACTIONS 39