Mann U. Principles of Chemical Reactor Analysis and Design: New Tools for Industrial Chemical Reactor Operations
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To reduce Eq. 5.2.52 to dimensionless form, we use dimensionless temperature,
defined by Eq. 3.3.4, u ¼ T/T
0
, the dimensionless extent, defined by Eq. 2.7.2,
Z
m
¼ X
˙
m
/(F
tot
)
0
, and the dimensionless space time, defined by Eq. 4.4.8, t ¼
V
R
/v
0
t
cr
. Dividing both sides of Eq. 5.2.52 by T
0
and (F
tot
)
0
and multiplying
both sides by the characteristic reaction time, t
cr
, we obtain
du
dt
¼
Uv
0
t
cr
P
J
j
(F
j
^
c
p
j
)
S
V
(u
F
u)
(F
tot
)
0
T
0
X
n
I
m
DH
R
m
(T
0
)
P
J
j
(F
j
^
c
p
j
)
dZ
m
dt
(5:2:53)
Equation 5.2.53 is the dimensionless, differential energy balance equation for
steady-flow reactors, relating the temperature, u, to the extents of the independent
reactions, Z
m
, as functions of dimensionless space time t. Note that individual dZ
m
/
dt’s are expressed in terms of u(t) and Z
m
(t) by the reaction-based design
equations, as described in Chapter 4.
As in the case of batch reactors, dimensionless energy balance Eq. 5.2.53 is not
conveniently used because the heat capacity of the reacting fluid,
P
J
j
(F
j
^
c
p
j
), is a
function of the temperature and reaction extents and, consequently, varies along
the reactor. To simplify the equation and obtain dimensionless quantities for heat
transfer, we define the heat capacity of the reference stream and relate the heat
capacity at any point in the reactor to it by
Heat capacity
of the
reacting fluid
0
@
1
A
;
Heat capacity
of the
reference stream
0
@
1
A
Correction
factor
or, in mathematical symbols
X
J
j
(F
j
^
c
p
j
) ; (F
tot
)
0
^
c
p
0
CF(Z
m
u)(5:2:54)
where
^
c
p
0
is the specific molar heat capacity of the reference stream, and CF(Z
m
,
u) is a correction factor that adjusts the value of the heat capacity as Z
m
and u
vary. Substituting Eq. 5.2.54 into Eq. 5.2.53 and noting that (F
tot
)
0
¼ C
0
v
0
,we
obtain
du
dt
¼
1
CF(Z
m
, u)
HTN(u
F
u)
X
n
I
m
DHR
m
dZ
m
dt
"#
(5:2:55)
where HTN is the heat-transfer number, defined by Eq. 5.2.22, and DHR
m
is the
dimensionless heat of reaction of the mth independent reaction, defined by
Eq. 5.2.23. Here, C
0
, T
0
, and
^
c
p
0
represent the reference concentration, reference
temperature, and the specific heat capacity of the reference stream.
150 ENERGY BALANCES
Equation 5.2.55 is a simplified dimensionless differential energy balance
equation of steady-flow reactors, where each term is divided by [(F
tot
)
0
^
c
p
0
T
0
],
the reference thermal energy rate. The first term in the bracket of Eq. 5.2.55 rep-
resents the dimensionless heat-transfer rate and its relationship to the dimensionless
driving force (u
F
2 u), where
d
dt
_
Q
(F
tot
)
0
^
c
p
0
T
0
¼ HTN(u
F
u)(5:2:56)
The definition of the specific molar heat capacity of the reference state,
^
c
p
0
,
and the determination of the correction factor, CF( Z
m
, u), deserve a closer
examination.
The specific molar-based heat capacity of the reference stream,
^
c
p
0
, is defined by
Specific molar
heat capacity of the
reference stream
0
@
1
A
;
Heat capacity of the reference stream
Total molar flow rate of the reference stream
and it is expressed mathematically by
^
c
p
0
;
P
J
j
(F
j
^
c
p
j
)
0
(F
tot
)
0
(5:2:57)
As discussed in Section 2.7, for gas-phase reactions, (F
tot
)
0
, is taken as the total
molar flow rate of the reference stream, including inerts, and
X
J
j
(F
j
^
c
p
j
)
0
¼ (F
tot
)
0
X
J
j
y
j
0
^
c
p
j
(T
0
)
where y
j
0
is the molar fraction of species j in the reference stream. Hence, Eq.
4.2.57 reduces to
^
c
p
0
¼
X
J
j
y
j
0
^
c
p
j
(T
0
)(5:2:58)
For liquid-phase reactions, the heat capacity of the reacting stream is usually
defined on a mass basis. Hence,
X
J
j
(F
j
^
c
p
j
)
0
¼
_
m
0
c
p
(5:2:59)
5.2 ENERGY BALANCES 151
where m
˙
0
is the mass flow rate of the reference stream,
c
p
is its mass-based specific
heat capacity. Substituting in Eq. 5.2.58,
^
c
p
0
;
_
m
0
c
p
(F
tot
)
0
(5:2:60)
Note that Eq. 5.2.60 converts the mass-based specific heat capacity to a molar-
based specific heat capacity.
To determine the correction factor of the heat capacity, CF(Z
m
, u), we use Eq.
5.2.54 and the appropriate definition of the specific molar heat capacity of the refer-
ence stream,
^
c
p
0
. For gas-phase reactions,
F
j
¼ (F
tot
)
0
(F
tot
)
in
(F
tot
)
0
y
j
in
þ
X
n
I
m
(s
j
)
m
Z
m
"#
and the heat capacity of the reacting fluid is
X
J
j
(F
j
^
c
p
) ¼ (F
tot
)
0
(F
tot
)
in
(F
tot
)
0
X
J
j
y
j
in
^
c
p
j
(u) þ
X
J
j
^
c
p
j
(u)
X
n
I
m
(s
j
)
m
Z
m
"#
Using Eq. 5.2.54, the correction factor for gas-phase reactions is
CF(Z
m
, u) ¼
1
^
c
p
0
(F
tot
)
in
(F
tot
)
0
X
J
j
y
j
in
^
c
p
j
(u) þ
X
J
j
^
c
p
j
(u)
X
n
I
m
(s
j
)
m
Z
m
"#
(5:2:61)
For liquid-phase reactions, the heat capacity of the reacting fluid is
X
J
j
(F
j
^
c
p
j
) ¼
_
m
c
p
where m
˙
is the mass flow rate of the reacting fluid. Using Eq. 5.2.59, the correction
factor is
CF(Z
m
, u) ¼
_
m
c
p
_
m
0
c
p
0
(5:2:62)
Note that when the inlet stream is taken as the reference stream, and assuming con-
stant mass-based specific heat capacity, CF(Z
m
, u) ¼ 1.
For isothermal operations, du/dt ¼ 0, for plug-flow reactors, Eq. 5.2.55
reduces to
d
dt
_
Q
(F
tot
)
0
^
c
p
0
T
0
¼
X
n
I
m
DHR
m
dZ
m
dt
(5:2:63)
152 ENERGY BALANCES
which can be further simplified to
d
_
Q ¼ (F
tot
)
0
X
n
I
m
DH
R
m
(T
0
) dZ
m
(5:2:64)
Equation 5.2.64 provides the local heating (or cooling) rate along the reactor needed
to maintain isothermal conditions.
For adiabatic operations, HTN ¼ 0 (no heat-transfer area), and for plug-flow
reactors Eq. 5.2.55 reduces to
du
dt
¼
1
CF(Z
m
, u)
X
n
I
m
DHR
m
dZ
m
dt
(5:2:65)
which can be further simplified to
du ¼
1
CF(Z
m
, u)
X
n
I
m
DHR
m
dZ
m
(5:2:66)
Equation 5.2.66 relates changes in the temperature to changes in the extents of the
independent reactions along the reactor. The application of these equations in the
design formulation of plug-flow reactors are discussed in Chapter 7.
To derive the dimensionless energy balance equation for a CSTR, the integral
form of the energy balance equation, Eq. 5.2.48 is used. Since the temperature
in the reactor is uniform, T ¼ T
out
, the rate heat is transferred to the reactor is
_
Q ¼ U
S
V
V
R
(T
F
T
out
)(5:2:67)
where (S/V) is the heat-transfer area per unit volume of reactor. Substitute Eq.
5.2.67 into Eq. 5.2.48, divide both sides of the equation by T
0
and (F
tot
)
0
, and mul-
tiply both sides by t
cr
. Using Eq. 5.2.54, the energy balance equation reduces to
HTNt(u
F
u
out
)
_
W
sh
(F
tot
)
0
^
c
p
0
T
0
¼
X
n
I
m
DHR
m
(Z
m
out
Z
m
in
) þCF(Z
m
, u)
out
(u
out
1)
CF(Z
m
, u)
in
(u
in
1) (5:2:68)
This is the dimensionless energy balance equation for a CSTR, relating the outlet
temperature, u
out
, to the inlet temperature, u
in
, and the extents of the independent
reactions in the reactor, (Z
m
out
Z
m
in
)’s.
For isothermal CSTRs with negligible shaft work, and selecting the inlet stream
as the reference stream, u
out
¼ u
in
¼ 1, and, using Eq. 5.2.56, Eq. 5.2.68 reduces to
_
Q
(F
tot
)
0
^
c
p
0
T
0
¼
X
n
I
m
DHR
m
(Z
m
out
Z
m
in
)(5:2:69)
5.2 ENERGY BALANCES 153
which can be further simplified to
_
Q ¼ (F
tot
)
0
X
n
I
m
DH
R
m
(T
0
)(Z
m
out
Z
m
in
)(5:2:70)
Equation 5.2.70 provides the heating (or cooling) rate needed to maintain isother-
mal conditions. For adiabatic CSTRs, HTN ¼ 0 (no heat-transfer area), and assum-
ing negligible shaft work, Eq. 5.2.68 reduces to
X
n
I
m
DHR
m
(Z
m
out
Z
m
in
) ¼ CF(Z
m
, u)
out
(u
out
1)
CF(Z
m
, u)
in
(u
in
1) (5:2:71)
Equation 5.2.71 relates the temperature at the reactor outlet to the extents of the
independent chemical reactions. The applications of these equations are discussed
in Chapter 8.
Example 5.4 The following gas-phase chemical reactions take place in a cat-
alytic reactor producing ethylene oxide:
Reaction 1: C
2
H
4
þ 0: 5O
2
! C
2
H
4
O
Reaction 2: C
2
H
4
þ 3O
2
! 2CO
2
þ 2H
2
O
A feed stream at 600 K and 2 atm, consisting of 67.7% C
2
H
4
and 33.3% O
2
(by
mole), is introduced into the reactor. Based on the data below, determine
CF(Z
m
, u).
Data: The specific molar heat capacities of the species (in cal /mol K) are
^
c
p
C
2
H
4
¼ 2:83 þ 28:6 10
3
T þ 8:73 10
6
T
2
^
c
p
O
2
¼ 7:83 þ 1:005 10
3
T 0:451 10
5
T
2
^
c
p
C
2
H
4
O
¼0:765 þ 46: 62 10
3
T þ 18:47 10
6
T
2
^
c
p
CO
2
¼ 10:84 þ 2:076 10
3
T 2:30 10
5
T
2
^
c
p
H
2
O
¼ 6:895 þ 2:881 10
3
T þ 0:240 10
5
T
2
Solution This example illustrates how to determine the correction factor
CF(Z
m
, u) that depends on both Z
m
and u. The two chemical reactions are inde-
pendent, and their stoichiometric coefficients are
(s
C
2
H
4
)
1
¼1(s
O
2
)
1
¼0:5(s
C
2
H
4
O
)
1
¼ 1(s
CO
2
)
1
¼ 0(s
H
2
O
)
1
¼ 0
(s
C
2
H
4
)
2
¼1(s
O
2
)
2
¼3(s
C
2
H
4
O
)
2
¼ 0(s
CO
2
)
2
¼ 2(s
H
2
O
)
2
¼ 2
154 ENERGY BALANCES
We select the inlet stream as the reference stream; hence T
0
¼ 600 K,
(N
tot
)
0
¼ ( N
tot
)
in
and y
j
in
¼ y
j
0
. In this case, ( y
C
2
H
4
)
0
¼ 0:667, ( y
O
2
)
0
¼ 0:333,
( y
CO
2
)
0
¼ 0, ( y
CO
2
)
0
¼ 0, and ( y
H
2
O
)
0
¼ 0. Using T ¼ T
0
u, we express the
specific species molar heat capacities of the species in terms of u:
^
c
p
C
2
H
4
O
(u) ¼ 2:83 þ17:16u þ3:143u
2
^
c
p
O
2
(u) ¼ 7:23 þ0:603u 0:125u
2
^
c
p
C
2
H
4
O
(u) ¼0:765 þ27:97u þ6:65u
2
^
c
p
CO
2
(u) ¼ 10:84 þ1:246 u 0:639 u
2
^
c
p
H
2
O
(u) ¼ 6:895 þ1:729 u þ0:0667 u
2
For gas-phase reactions, the reference specific molar heat capacity is determined
using Eq. 5.2.58:
^
c
p
0
¼
X
J
j
y
j
0
^
c
p
j
(T
0
) ¼ (0:667)23:13 þ(0:333)7:71 ¼ 18:0 cal=mol K (a)
For this case Eq. 5.2.61 reduces to
CF(Z
m
, u) ¼
1
^
c
p
0
X
J
j
y
j
0
^
c
p
j
(u) þ
X
J
j
^
c
p
j
(u)
X
J
m
(s
j
)
m
Z
m
"#
(b)
Substituting the
^
c
p
j
(u)’s, the first term in the bracket on the right reduces to
1
18
(0:667)
^
c
p
C
2
H
4
(u)þ(0:333)
^
c
p
O
2
(u)
hi
¼
1
18
(0:667)(2:83þ17:16uþ3:243u
2
)
þ(0:333)(7:23þ0:608u0:125u
2
)
()
¼
4:295þ11:65uþ2:096u
2
0:0416u
2
18
Expanding the second term in the bracket on the right of (b),
¼
1
^
c
p
0
(Z
1
Z
2
)
^
c
p
C
2
H
4
(u)þZ
1
^
c
p
C
2
H
4
O
(u)þ(0:5Z
1
3Z
2
)
^
c
p
O
2
(u)
þ2Z
2
^
c
p
O
2
(u)þ2Z
2
^
c
p
H
2
O
(u)
()
5.2 ENERGY BALANCES 155
¼
1
^
c
p
0
(Z
1
Z
2
)(2:83þ17:16uþ3:143u
2
)
þZ
1
(0:765þ27:97uþ6:65u
2
)
þ(0:5Z
1
3Z
2
)(7:233þ0:603u0:125u
2
)
þ2Z
2
(10:84þ1:246u0:639u
2
)
þ2Z
2
(6:895þ1:729u0:067u
2
)
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
¼
1
18
(7:21Z
1
þ10:95Z
2
)(10:51Z
1
þ13:02Z
2
)u
þ(3:08Z
1
3:143Z
2
)u
2
þ(0:0625Z
1
0:770Z
2
)u
2
Combining the two terms, the correction factor is
CF(Z
m
,u)¼
1
18
(42957:21Z
1
þ10:95Z
2
)
þ(11:65þ10:51Z
1
þ13:02Z
2
)u
þ(2:096þ3:507Z
1
3:143Z
2
)u
2
þ(0:04160:0625Z
1
0:770Z
2
)u
2
8
>
>
<
>
>
:
9
>
>
=
>
>
;
5.3 SUMMARY
In this chapter, we reviewed the thermodynamic relations related to reactor oper-
ations and derived the energy balance equations for batch and flow reactors.
Main topics covered are:
1. Heat of reaction of a chemical reaction and its dependence on temperature
and pressure
2. Equilibrium constant of a chemical reaction and its dependence on
temperature
3. Definition of the specific molar heat capacity of the reference state,
^
c
p
0
, for
gas-phase and liquid-phase reactions
4. Definition of the correction factor of heat capacity, CF(Z
m
, u), for gas-phase
and liquid-phase reactions
5. Definition of the dimensionless heat of reaction, DHR
m
6. Definition of the heat-transfer number, HTN
7. Derivation of the macroscopic energy balance equation of batch reactors and
reducing it to a dimensionless form
8. Derivation of the integral and differential energy balance equations for flow
reactors and reducing these equations to dimensionless forms
For convenience, Table A.4 in Appendix A provides the energy balance equations
and auxiliary relations for the three ideal reactors.
156 ENERGY BALANCES
PROBLEMS*
5.1
1
The following liquid-phase chemical reactions take place in a CSTR:
A ! 2B
B ! C þ D
Both reactions are first order and the desired product is B. An aqueous sol-
ution whose concentration is C
A
0
¼ 3 mol=L is fed at a rate of 200 L/min
into a 500-L CSTR with a heat-transfer area of 20 m
2
/m
3
. The inlet tempera-
ture is 608C, and the temperature of the jacket is 1008C. The density and heat
capacity of the solution are essentially the same as those of water (r ¼ 1g/
cm
3
; c
¯
p
¼ 1 cal/gC). The heat-transfer coefficient is estimated as 5 cal/scm
2
8C. Based on the data below, determine:
a. The specific molar heat capacity of the reference stream.
b. The dimensionless heat of reaction of each chemical reaction.
c. The dimensionless heat-transfer number.
d. The correction factor of heat capacity.
e. Formulate the dimensionless energy balance equation for the operation.
Data:At608C,
DH
R
1
¼ 20,000 cal=mol extent DH
R
2
¼ 12,000 cal=mol extent
k
1
¼ 0:8 min
1
k
2
¼ 0:2 min
1
5.2
2
A stream consisting of 40% O
2
and 60% CH
4
is fed into a plug-flow reactor
where the following reactions take place:
2CH
4
þ O
2
! 2CH
3
OH
CH
4
þ 2O
2
! CO
2
þ 2H
2
O
The feed temperature is 600 K. Based on the data below, determine:
a. The dimensionless heat of reaction of each chemical reaction at 600 K
b. The specific molar heat capacity of the feed (reference stream)
c. The correction factor of heat capacity, CF(Z
m
, u)
*
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.
PROBLEMS 157
Data: The standard heat of formations of the species at 298 K (in joule/
mole):
Methane: 274,520
Methanol (gas): 2200,600
CO
2
: 2393,509
Water (gas): 2241,818
The species heat capacities in Joule/mol K (T in K)
Methane ¼ 1.702 þ 9.081 10
23
T 2 2.164 10
26
T
2
Methanol ¼ 2.211 þ 12.216 10
23
T 2 3.450 10
26
T
2
Oxygen ¼ 3.639 þ 0.506 10
23
T2 0.227 10
5
T
22
CO
2
¼ 5.457 þ 1.045 10
23
T 2 1.157 10
5
T
22
Water ¼ 3.470 þ 1.450 10
23
T 2 0.121 10
5
T
22
BIBLIOGRAPHY
More detailed treatment of the energy balance equations and deriving them for
microscopic systems can be found in:
R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley,
New York, 2002.
J. R. Welty, C. E. Wicks, and R. E. Wilson, Fundamentals of Momentum Heat and Mass
Transfer, 4th ed., Wiley, New York, 2001.
More detailed treatment of chemical equilibrium and thermodynamic properties
can be found in:
J. M. Smith and H. C. Van Ness, Introduction to Chemical Engineering Thermodynamics,
7th ed., McGraw-Hill, New York, 2005.
158 ENERGY BALANCES
6
IDEAL BATCH REACTOR
A batch reactor is a vessel where chemical reactions take place, and no material is
added or withdrawn during the operation. Reactants are charged into the reactor
before the commencement of the operation, and the content of the reactor is dis-
charged at the end of the operation. An ideal batch reactor, schematically described
in Figure 6.1, is a mathematical model of “idealized” batch reactors. The model is
based on the assumption that, due to good agitation, the same species compositions
and temperature exist everywhere in the reacting fluid. Hence, at any instant, the
same reaction rates prevail throughout the entire reactor volume.
In practice, batch reactors of various sizes are used, ranging from a few cubic
centimeters in a test tube, through a beaker agitated by a magnetic stirrer in a chemi-
cal laboratory, to a few-liters pilot-scale reactor, to several thousand-gallon
production reactors. Achieving good mixing in small reactors is relatively
simple, but it is a difficult task in large reactors. Mixing in very large reactors is
an important practical issue, but it is beyond the scope of this text. In very large
vessels and in reactors with viscous liquids, temperature and concentration gradi-
ents may exist. In those cases, the ideal batch reactor model serves only as an
approximation of the actual reactor operation. In general, the ideal batch reactor
model provides a good representation of the actual operations when the mixing
time is short in comparison to the reaction time.
In this chapter, we analyze the operation of ideal batch reactors. In Section 6.1,
we review how the design equations are utilized and discuss the auxiliary relations
that should be incorporated in order to solve the design equations. In the rest of the
Principles of Chemical Reactor Analysis and Design, Second Edition. By Uzi Mann
Copyright # 2009 John Wiley & Sons, Inc.
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