CHf
CGf
C Af
C Af
CGn
CHn C An
Feature Report Engineering Practice
C Af = C An
C An
X n =
CSTR Design for Reversible Reactions
C An–1
C A1
C A0
X f f =
C A0 – C A1 C A0
C A0
C A0 – C Af
> C A0 = M >
1.0
Product: CGf and CHf
C A0
Multiple backmixed reactors
FIGURE 1. Conversion in plug-flow reactors and CSTRs for second order reactions is shown here, with conversion per stage shown for the CSTR case
M
ultiple CSTRs (continuous stirred-tank reactors) are advantageous in situations where the reaction is slow; two immiscible liquids are present and require higher agitation rates; or viscous liquids are present that require high agitation rates. Unlike in plugflow reactors, agitation is easily available in CSTRs. In this article, batch and plugflow reactors are analyzed and compared to multiple CSTRs. The number of reactors required in a CSTR system is based on the con version for each stage. stage. When When the final stage obtains the fraction of uncon verted reactant that is equal to the desired final value from the plug-flow case, the CSTR system is complete. The volumetric efficiency of multiple CSTRs is expressed as a function of conversion per stage and gives the total conversion required. In this article, we will apply this to reversible second-order reactions.
stage, the number of stages of equal volume, as well as the volumetric efficiency of the CSTR stages and the plugflow reactor. The reactor design is developed by selecting a conversion in the first stage. Then, the second-stage conversion is equal to that of the first stage, since it requires an equal volume. This procedure is continued until the fraction of reactant exiting each reactor stage reaches the desired value in the last stage, or slightly less than the plugflow case, as illustrated in Figure 1. The kinetic rate conversion of a re versible bi-molecular reaction at constant temperature and flowrate is represented by Equation (1). The reaction is illustrated below (nomenclature is defined on p. 49. 2 A G H
r
−
The first case presented here is a kinetic process requiring a double component (2 A) to be fed to a reactor, and producing two products ( G and H ). ). The design may be calculated for both CSTR and plug-flow reactors, determining the conversion in the first
=
k F C A
2
−
kR CG CH
( 1) k R
2nd-order, reversible reactions
CHEMICAL ENGINEERING
C A1
C Ao
Plug-flow reactor
Ralph Levine
46
C A1 – C A2
C A1
X 1 =
Here, a design approach for continuous stirred-tank reactors is outlined for three cases of second-order reactions
C A(n–1)
C An–1
X 2 =
C Ao
C A(n–1) – C An
kF K
(2)
2
G0
f
Gf
A0
C Hf
=
C
A 0
C A 0 X f
(2b)
C A0 X f
Xf
2
(2c)
C
A0
C Af
2
H0
Hf
SEPTEMBER 2009
CGf
C Hf
(2d) The ratio may be different, as, for example, the concentration of G in the product may be five times that of H . However, we assume that the products pr oducts are equal ( CGf = C Hf ). Expressing reaction rate. The rate equation can be modified to include conversion and equilibrium constant terms. Substituting Equations (1), and (2d) into Equation (1) give an expression for rate. r k F C A2 0 1
Xf
2
K C A0 X f
kF
2
(3a) (2a)
Af
Assume that G and H compounds compounds are not present in the feed. Therefore, CG0
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CGf
2
C X C C C C C C A 0
and C H0 are equal to zero, and the following expressions are true:
1 2 X f X f K 1 K 2
r k F C A
2 0
(3b)
r k F C A K 1 X f 2 X f 1 K 2
V k C 0 X ln 1 X f X e v F A e
2
0
K 1 X 2 X 1 K 2
e
e
(4a)
Volume of each CSTR stage . An expression for the first stage of a CSTR is given in Equation (7). The first stage conversion, X 1, occurs in each of the successive stages ( X 2, X 3, and so on), and each has the same volume and reaction temperature.
Using the quadratic equation, Equation (4a) is simplified to Equation (4c). X e
2
2 4 K 1 2 K 1 K
2
X e
K 1
1
X f
K
(4b)
X f
v
r
=
K 1
1 1
k F CA2 0 ( X e
K
K
(5)
K K
−
X f )
(5b)
−r = k F CA20 Xe ⎡⎣1 − ( X f Xe ) ⎦⎤
(5c)
Stirred reactor in a batch or plug flow reactor . The batch reactor case and the ideal continuous plugflow case are given in Equation (6). The reaction time is t for the batch case, and V / v for the plugflow case. X V = ⌠ f dX vC A 0 ⌡0 −r
(6)
Substituting Equation (5c) into (6) and rearranging, gives Equation (6a). X f V dX k F C A2 0 X e = ∫ 0 vC A 0 1 − ( X f X e )
(
(7)
1
A 2
Ae
A1
Ae
e
A1
Ae
A 0
Ae A
A 2
Ae
A 0
Ae
1
)
(6a)
X
X 1 X
1
Number of stages. Conversion for stage 1 is expressed by equation (8).
C A 0
C A1
C
A0
C Ae
(8)
C A0
(9)
Subtract Equation (9) from (8) and di vide by (9) to obtain Equations (10a) and (10b). X e
X e
X1
−
C
X 1 =
X e
1
X
1
C Ae
C A1
−
C Ae
C A 0
−
C Ae
X e
A1
A 0
Ae
X 1 1 X
(12)
C C C 0 C
log
An
Ae
A
Ae
X n X C C 1 1 f An Ae X e C A C Ae X e 1
(12b)
0
Total volume of all stages . Substitute Equation (7a) into (13a). VT
V T
nV 1
k F C A0
v
(13a)
n
V 1 v
kF C A0
X X 1 e n 1 X1 X e (13b)
C A0
The equilibrium conversion is based on time to reach a net reaction rate of zero, which may be calculated by Equation (4b) or (9).
Ae
(12a)
V T
k F C A0
v
X e
An
(7a)
Equation (7a) has only one independent variable (V 1). If each stirred reactor stage is to be of equal volume and volumetric flowrate, then the result is a constant conversion per stage. That is, each stage, when at a fixed set of conditions, has the same conversion from each stage, expressed as: X 1 = X 2 = X 3... = X n
X1
n log
n
C C C C
e
X e
1
(11a)
Continue this process for the nth stage to obtain the following equations.
X1 X e
(4c)
(5a) The reaction rate expression can then be expressed as Equations (5b) and (5c). −
r
Substituting Equation (5c) into (7) and rearranging gives Equation (7a).
K
K 1
1
X 1
V 1 C A 0 kF
4 4 1 1
X C C C C 1 X C C C C C C C C
e
1 1 K
2
V 1 vC A 0
The quadratic equation can also be used to simplify Equation (3c), resulting in Equation (5a). 2
tion (11a), based on each stage having the same volume and conditions. 2
(3c) At equilibrium, the net reaction rate equals zero. 0
(6b)
C A0
(10)
(13c) C C C 0 C X1 X log 1 X1 X 1 X1 X log
An
Ae
A
Ae
e
e
e
By definition, C An = C A0(1 – X f ), where X f is the total conversion in the nth stage or the final desired conversion of the plug-flow reactor. By the method used to obtain Equation (10), the following equation is similarly derived. Substitution of Equation (12b) into (13c) gives Equation (13d). V T
(10a)
C A1
C Ae
C A 0
C Ae
k C v F A0 log 1 X f X e
(10b)
The exit concentration, C A1, can be calculated from Equation (10b). Also, the exit concentration from the second stage, C A2, can be calculated from Equa-
(13d)
X1 X e
log 1 X1 X e 1 X1 X e
Volumetric efficiency Since V T / v in Equation (13c) is residence time, as is V / v in Equation (6b), for CSTRs, these terms are equiva-
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TABLE 1.VOLUMETRIC EFFICIENCY FOR EQUATION (14A)
Engineering Practice
lent. The volumetric flowrate is the same in all cases (a batch operation for one complete reaction cycle). Thus, the ratio of comparison should be V for plugflow or batch operation (reaction volume and time only) compared to V T for multiple CSTRs. This ratio ( V / V T ), volumetric efficiency is expressed as Equation (14), and is derived from Equations (6b) and (13d). V k C X v F A0 e V T k F C A0 (14) v 2.303 log 1 X f X e
X1 X e
V V T
2.303 log 1 X1
X e X1 X e
log 1 X1
X e
X e
C B0 X f C pf
CB0
0.1
0.7
0.9
–0.046
0.948
1.355
0.2
0.7
0.8
–0.097
0.893
1.275
0.3
0.7
0.7
–0.155
0.832
1.189
0.4
0.7
0.6
–0.222
0.766
1.095
0.5
0.7
0.5
–0.301
0.693
0.990
0.6
0.7
0.4
–0.398
0.611
0.873
0.7
0.7
0.3
–0.523
0.516
0.737
0.8
0.7
0.2
–0.699
0.402
0.575
0.9
0.7
0.1
–1.000
0.256
0.366
X M 1 X M KC
The volumetric efficiency is calculated as Equation (23).
2
e
e
M 1 4 M KC KC 2
M 1
X e
B0
B0
2
(18b)
Using Equations (17) and (18a), an expression for X f is found.
r k F CB 2
M 1 X f M KC B (19a)
X f 2 0
0
Equation (19a) can be simplified to Equation (19b) using the quadratic equation.
X f
2
M 1 KC B
M 1 4 M (19b) KCB
0
0
kRC p C Bf
r k F CB2 0 X e 1 X f Xe
ln1
V1 k F CB0 X e v
(16b)
At equilibrium, the rate is zero. WWW.CHE.COM
1
(23) X e
Reversible production of a dimer from twin reactants In another alternate but similar case, Equation (15) is modified for double components that are reversibly reacted to form a dimer, as shown in the reaction below. As in this last case, there is only one product. 2 A P
r
−
k F C A 0
=
C Pf
=
2 −
kR CP
(24)
C A0 X f
(25)
X
1
1
X
1
X e
C Bn CBe C B0 C Be
X
1
C B1
C Be
C B0
CBe
X e
X e
(20a)
(20b)
1 X1 X e
SEPTEMBER 2009
k F C A0 2 1
Xf
2
k
F
K C A0 X f
(26a)
r k F C A0
2
X 2 2 KC X 1 A0 f f (26b)
At equilibrium, the rate is zero. 0 X e
2
2
KCA0 X e
1
(27a) Using the quadratic equation, Equation (27a) becomes (27b).
1
KCA0
X f
1
KCA0
1
KCA0
2
X e 1 KCA0
(21)
1 X f X e n
(19c)
X1 X e
(16a)
(17) CHEMICAL ENGINEERING
V1 k F CB0 X e v
CB0 X f
2.303 1 X 1 X e log 1 X1 X e X 1 X e
r
(15) C pf
V T
2
The volume of each backmixed stage is equal.
V
X r k F C B20 1 X f M X f f KC B0
48
V /V T
(14a)
C DP
k F CBCD
log[1–(X 1/X e )] (V /V T )X e
(18a) Using the quadratic equation, Equation (18a) becomes (18b).
Another case exists, where two components are reversibly reacted to form a single product, a dimer, rather than two products (as shown in the reaction below). This case is similar to the pre vious case, but but with only one product, as shown below in Equation (15).
1–(X 1/X e )
B0
Reversible production of a dimer from two reactants
r
X e
0
Volumetric efficiency is independent of the initial or final concentration and velocity constant at constant temperature, as well as overall conversion. It is dependent on only the ratio of the first stage conversion compared to the equilibrium conversion. Calculations for Equation (14a) are shown in Table 1.
X 1/X e
TABLE 1. For any ratio of conversion per stage to equilibrium conversion, this table provides the corresponding volumetric efficiency, based on Equation (14a)
log 1 X f X e
1 X1 X e log 1 X X 1 e
(14a)
r k F C A0 1 X f X e
2
1
(27b)
1
(28a)
2
(22)
(28b)
For a plugflow reactor, the following expression is true.
SUMMARY OF EQUATIONS [6 ]
B D P S 2
M 1 M 1 4 M 1 K X e 2 K 1 K 2
M 1 M 1 4 M 1 K X f 2 K 1 K G H 1 1 1 K X e K 1 K 2 A
X f
1
1 K K 1 K 1
C D P
X e
X f
2
2
M 1 4 M KCB
3. Levine, R. A New Design Approach for Backmixed Reactors — Part I, Chem. Eng. July 1, 1968, pp. 62–67.
0
4. Levine, R. A New Design Approach for Backmixed Reactors — Part II, Chem. Eng. July 29, 1968, pp. 145–150.
2
P X e 1 KC A 1 2 A
0
X
1
KC A0
V C A0 kF X e v
1
KC A0 KC A0
2
1
5. Levine, R. A New Design Approach for Backmixed Reactors — Part III, Chem. Eng. Aug 12, 1968, pp. 167–171.
2
1
6. Levine, R. CSTRs: Bound for Maximum Conversion, Chem. Eng. Jan. 2009, pp. 30–34.
v
X1 X e
1
Ralph Levine is a retired chemical engineer currently working as a consultant for plants, design or operations and R&D (578 Arbor Meadow Dr., Ballwin, Mo. 63021; Email: ralphle2000@ yahoo.com). Levine earned a B.S.Ch.E. from the City Uni versity of New York, and did graduate work at Louisiana State University and the Uni versity of Delaware. Levine later served as an engineer for the U.S. Army Chemical Corps. He has worked for DuPont, Cities Service Co., and most recently, Columbian Chemical Co. Levine has filed several U.S. patents during his career, and is a published author, with his work featured in Chemical Engineering and Hydrocarbon Processing.
ln 1 X f X e (29)
The expression for multiple CSTRs is given as Equation (30). V1 C A 0 kF
Subscripts Initial conditions First, second and third stages For component A For component C For component D Equilibrium conditions Overall or final conditions Conditions for forward reaction For component G For component H Any stage in the series of reactor stages The nth stage For component P Conditions for reverse reaction The total of all n stages
Author
2. Levine, R., Hydro. Proc., July 1967, pp. 158–160.
2
0
A C D e f F G H j n P R T
1. Levenspiel, O., “Chemical Reaction Engineering,” John Wiley & Sons, Inc., 1962.
0
M 1 KC B
K
C Concentration, moles/unit volume k Reaction rate constant K Equilibrium constant M Initial mole ratio of D/B n Number of stages r Reaction rate t Reaction time V Reactor volume v Volumetric flowrate X Conversion
0 1,2,3
References
M 1 4 M K C B
M 1 KC B0
K
NOMENCLATURE
X
1
X e
(30)
The best way to heat and cool the most corrosive materials.
n
1 X X 1 X X 1 e f e C C Ae An C A0 C Ae
(31)
The volumetric efficiency is found to be Equation (32).
2.303 1 X1 X X X V X 1 log1 X1 X V T
e
e
(32)
e
e
The last two cases presented here are reversible and have only one product. The differences between these cases are the values calculated based on the quadratic equation for both X e and X f . All second order reactions that are reversible and produce one or two products require the quadratic equation for the calculation of X e and X f for each case. A summary of these equations is presented in the box above. ■ Edited by Kate Torzewski
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