Variable-Volume Operation of a Stirred Tank Reactor Monty M. Lundl and Richard C. Seagrave2 Department of Chemical Engineering, Engineering Research Institute, Iowa State University, Ames, Iowa 60010
Generalized results for relative yield and relative throughput are presented for the semibatch operation
of a stirred tank reactor for both isothermal and adiabatic operation. A special case of continuous variablevolume operation is examined. Reaction mechanisms investigated include irreversible first- and second-order, reversible first-order, and consecutive and competing reactions. It is shown that under certain conditions semibatch operation can produce yield, throughput, and selectivity improvement over continuous operation and, in some instances, over batch or plug flow operation.
I n recent years interest in t,he periodic operation of chemical reactors as a means of improving yield over optimum st,eadystate operation has been increasing rapidly. Periodic operation in this cont'ext refers to the situabion in which some of the output' or input variables of a reactor are periodic functions of time. The variable-volume STR is a special case of the periodic operat'ion of a chemical reactor. A chemical reactor system that is described by a nonlinear system of differential equations or a linear system of differential equations with variable coefficients N-ill, in general, give different time average values for periodic output variables when forced by periodic inputs than the steady-state values obtained by using constant inputs equal t'o t.he average of the oscillating inputs. I n a previous paper, Lund and Seagrave (1971) introduced a special case of periodic Operation called semibatch operation and showed that the average yield for semibatch operation relative to the st,eady-st.ate yield for t,he same average residence time may be increased in the isothermal CSTR with first-order kinet'ics. They extended t,his to show yield increases relative to t'he steady-state yield for adiabatic semibatch operation with a first-order exothermic reaction. Fang and Engel (1967) also showed an average yield increase by semibatch operat.ion relative to the steady-state CSTR for isothermal first- and second-order react.ions. I n another paper, Codell and Engel (1971) showed that when the reaction rate passes through a maximum, semibatch operation can be more efficient than either a backmix or plug flow reactor on a residence time basis. I n this work, generalized resu1t.s are presented for isothermal and adiababic semicontinuous and semibatch operation of a liquid phase stirred tank reactor for reaction schemes of various order. Extension to a special case of continuous variable-volume operation is presented. Finally, a reaction scheme involving successive and competiiig reactions (Van de Vusse, 1964) is examined. Analysis
The configuration and nomenclature used in this paper are similar to those used earlier (Lund and Seagrave, 1971). Figure 1 depicts the reactor configuration and indicates some 1
Present address, Dow Chemical Company, Midland, Mich. To whom correspondence should be addressed.
494 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
of the nomenclature employed. The descriptive operating equations for the reactor may be stated in dimensionless form by introducing the following dimensionless variables.
The subscript R refers to the reference reactor. The reference reactor is chosen as the CSTR with the same maximum volume, feed composition, and feed temperature as the variable-volume STR. The dimensionless concentration is determined with respect to the concentration of a reference reactant A in the feed stream. The reference reactor is isothermal for the case of the isothermal variable-volume STR and adiabatic for the case of the adiabatic variable-volume STR. When the dimensionless variables are substituted into the appropriate material and energy balance equations, the following dimensionless equations result
V* dCj* -=
de
=
Vo*
+
1;
(&I* - &*)de
&r* (CJj*- C j * ) + P V*
2
(1)
Lyi3Ti*
i=l
(i
=
1, 2, . . , , n independent reactions)
(2)
2 (-
(3)
dT* = &I* (1 - T*) + R P dB V*
AHi)*r$*
i=l
or (5)
The dimensionless parameters which appear in the dimensionless equations are defined below. relative rate constant relative thermal energy linear reaction rate parameter Arrhenius reaction rate parnme ter
P R
= (-
L
=
L'
=
=
(see Table I)
AHa)Caf/pCpTf
bTj/Kj E/R,T,
Table 1. Rate Expressions for Reaction Systems Studied Dimensionless rate expression,
n Reaction system
Luijri*
=
rj*
i= 1
AK'B
-C**
isothermal
A
AB
-K*CA*
adiabatic
* A B
5T
isothermal
5
2A B isothermal
5
2A B adiabatic
-K * C A * ~
Van d e Vusse reactions
AK~BK~c 2A
--
3D
Isothermal
The dimensionless reaction rate, Ti*, is defined for several reaction mechanisms in Table I. For this work, two performance criteria, namely, relative yield and relative throughput, are defined. These two criteria represent different ways of evaluating the production rate of the variable-volume S T R with respect to the reference CSTR. The two criteria are not independent and a method for determining relative throughput when the relative yield is known is given. Definition of Relative Yield. Relative yield is defined as the ratio of the flow-average product concentration from the variable-volume S T R to the constant product concentration from the reference CSTR. The variable-volume S T R and t h e reference C S T R have the same maximum volume, the same feed composition and temperature, and t h e same time-average throughput of process material. T h e relative yield of product j is expressed mathematically by
Figure 1 . Schematic diagram of the stirred tank reactor
the variable-volume S T R t o the constant throughput in the reference CSTR. The variable-volume S T R and the reference C S T R haye the same maximum volume, the same feed composition and temperature, and the same flow-average concentration of product from the reactor. T h e relative throughput is defined mathematically by
where
(7)
and ep - ep--l is the time interval for one variable-volume period for which the concentration is a steady periodic function of time. Definition of Relative Throughput. Relative throughput is defined as the ratio of the time-average throughput in
where ep - ep-l is the length of one variable-volume period. Consider the relative yield of product j to be known as a function of the relative rate constant, P . The concentration of component j in the reference CSTR i s determined by solution of the steady-state form of the general dimensionless equations. The flow-average concentration of component j in Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
495
Table II. Equations for the Semibatch Cycle Operation/ fraction of cycle
Time,
Filling Batch Emptying Down
< 6 < OF < 6 < OB 6B < 0 < @E BE < e < O D
e
Feed rate,
Discharge rote,
Qf*
Q*
60
0 0
Q ~ F *
0 0 0
OF
&E*
0
CSTR is determined by solution of the steady-state form of the general dimensionless equations with P replaced by P+. Relative throughput with respect to the second reference CSTR is then given by
+
After has been determined in eq 12, the corresponding conversion of component A is given by
Equations 9, 10, 12, and 13 determine relative throughput as a function of flow average conversion of component A from a knowledge of relative yield vs. relative rate constant and the steady-state solutions to the general dimensionless equations.
*u
I
Semicontinuous Variable-Volume Operation
The cycle for general semibatch operation of the STR is shown in Figure 2 while the equations which describe the semibatch cycle are given in Table 11. These cycles are divided into fractions in which a different residence time function, V*/Ql*, is used for each fraction. The following parameters are defined, by referring to Figure 2, as I
I
b
OF
I
I
I
Bs
bE
'D
DIMENSIONLESS TIME, 8
Figure 2. The semibotch cycle
the reference CSTR and in the variable-volume STR is then given, respectively, by (Cj*)R
=
cj/* + P
(9)
CYij T i *
)fi
The relative throughput, I+, may be determined with respect to a second reference CSTR which has a concentration of component j given by (C,*).,
=
cj"
(11)
where the subscript R' denotes the second reference CSTR. The relative throughput with respect to the second reference 496
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
The relative yield and relative throughput by semibatch operation have been studied in detail. Analytical expressions have been determined for relative yield and relative throughput for semibatch operation of the STR by solving the system of equations consisting of the dimensionless general equation for concentration in the STR given by eq 2, the equations describing the semibatch cycle given in Table 11, and the definitions of relative yield and relative throughput given by eq 6 and 8, respectively. Irreversible First-Order Reactions. The solutions to these systems of equations for relative yield and relative throughput for an irreversible first-order reaction scheme in a n isothermal reactor are given, respectively, by
Relative yield and relative throughput in these relations are given with respect to the same reference CSTR in which the relative rate constant, P , and the flow average conversion of A, SA,are related by x.4
=
P/(1
+ P)
(17)
When the relative yield is known as a function of the relative rate constant in eq 5, then the relative throughput for the same semibatch cycle is also determined as a function of the flow average conversion of A given by
X* = B
relative throughput is greater than unity and semibatch operation of the S T R is preferred to the CSTR. The values of relative throughput a t zero and 100% conversion are given by lim $ = m (23) Z.4-1
1.41
(+p)
The relative throughput a t the conversion of A given by eq 18 is determined as a function of relative yield by the application of eq 10 and 12. The result is $ =
9
1 - P(9 - 1)
(19)
Figure 3 shows plots of relative yield us. the relative rate constant for different semibatch cycles as determined by eq 15. Figure 4 shows plots of relative throughput us. flow-average conversion of reactant A as determined by eq 18 and 19. Effect of Relative Rate Constant. The relative rate constant is a n independent parameter, in eq 15, for relative yield. It is important to note t h a t the parameters for the semibatch cycle alone do not solely determine how the semibatch operated S T R will perform with respect t o the reference CSTR. At low values of the relative rate constant, which correspond t o a small conversion of reactant, the relative yield is less than unity and steady-state operation is preferred to semibatch operation of the S T R . As the relative rate constant is increased, the relative yield goes through a maximum value greater than unity as shown in Figure 3. The maximum value of relative yield and the corresponding reactant conversions in Figure 3 for the semibatch cycle with U B = 1.0 and Vo* = 0 are q = 1.3 a t P = 1.8; T A= 0.644 in the reference CSTR; T A= 0.834 in the semibatch operated STR. From eq 15 it can be shown that lim q P-
=
I 0.5
0.41 0.2
I
I 5 .O
I 1.0
20.o
I
RELATIVE RATE CONSTANT, P
Figure 3. Effect of relative rate constant and batch fraction on relative yield by semibatch operation of the isothermal stirred tank reactor
1.0
m
o'2
t
0.11
om
Effect of Flow Average Reactant Conversion. The relative throughput is a monotonic increasing function of the flow average conversion of reactant as shown in Figure 4. At high values of flow average reactant conversion, the
I
0.02
I 0.05
I
0.10
I
I
0.20
0.50
DIMENSIONLESS FLOW AVERAGE CONCENTRATION, (1
1
- ZA)
Figure 4. Effect of flow average concentration and batch fraction on relative throughput b y semibotch operation of the isothermal stirred tank reactor Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
497
throughput as a function of flow-average conversion of reactant are
(24) For the semibatch cycles shown in Figure 2 , eq 24 reduces to -lim & =
x,+o
UB
+
=
(+)' {
1-
UP
P
{
1-
[l
- exp(-
Pup)I2 exp(P%p2
1
PUB)
(25)
(1 - Vo*)expi- P(l 1 - Vo* exp[- P(l - vo*)l} V,*)I (26)
Effect of Minimum Volume. Differentiation of eq 15 shows that ~ ( V O=* 1) = 1 is either a maximum or minimum, since the equation
is satisfied only a t Vo*= 1 in the interval 0 5 VO*_< 1. Therefore, relative yield is either a monotonic increasing or monotonic decreasing function of the minimum volume in the interval 0 5 Vo* 5 1. This leads to the following two results: (1) if 7(Vo*)> 1 for some semibatch operation, then the relative yield will be a maximum with respect to minimum volume a t Vo* = 0; and ( 2 ) if ~ ( V O< * )1 for some semibatch operation, then the relative yield will have the maximum value of unity a t Vo* = 1 for which the semibatch cycle has disappeared and the performance of the CSTR is realized. Effect of Unequal Filling and Emptying Times. If the sum of the filling and emptying fractions of the semibatch cycle, U F U E , is held constant, then it is readily seen in eq 15 that the relative yield is symmetric with respect to the filling and emptying fractions. Furthermore, the relative yield is a maximum with respect to the filling and emptying fractions of the semibatch cycle when the filling fraction is equal to the emptying fraction. The Optimal Semibatch Cycle. The optimal semibatch cycle is defined a s the semibatch cycle which affords a maximum relative yield or relative throughput with respect t o the semibatch cycle parameters. The above discussion on the effects of the semibatch cycle parameters on relative yield and relative throughput leads to the following specifications for the optimal semibatch cycle: (1) the minimum volume, Vo*, is equal to zero; ( 2 ) the batch fraction, U B , is as large as possible. A batch fraction equal to unity requires instantaneous filling and emptying and cannot be achieved in any real reactor; (3) the filling fraction, U P , is equal to the emptying fraction, U E ; and (4)the down time is zero. The respective equations for the optimal relative yield as a function of relative rate constant and the optimal relative
+
498
=
(28)
Effect of Batch Fraction. Figure 3 shows that the upper bound of relative yield with respect to the batch fraction, U B , is achieved when u B is equal to unity. This corresponds to instantaneous filling and emptying of the reactor, and the semibatch-operated S T R becomes simply a batch reactor for each semibatch cycle. The initial concentration in the batch reactor is equal to the volumetric average of the fresh feed charged to the reactor and the concentration of the material remaining in the reactor from the previous cycle. This is due t o the assumption of perfect mixing. An expression for the upper bound of relative yield with respect to the batch fraction is obtained by taking the limit in eq 15 as shown below. lim 7
07)optimal
Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
(29) General Characteristics of Semibatch Operation. The general characteristics of the semibatch operation of the S T R which may be used to explain why it is possible t o obtain a relative yield and a relative throughput greater than unity are determined by considering the mixing characteristics of the CSTR and the plug flow reactor, which are combined in semibatch operation of the S T R . Semibatch operation of the S T R may be visualized as simulating a level of mixing between the mixing levels in the plug flow reactor and the CSTR. Since the average reaction rate for a single reaction is a maximum in the plug flow reactor and a minimum in the C S T R for the isothermal case, the decrease in mixing has the effect of increasing the average reaction rate for semibatch operation of the S T R relative t o the CSTR and a relative yield and relative throughput greater than unity may be obtained. However, when the decrease in flow average residence time more than compensates for the increase in average reaction rate by semibatch operation, the relative yield and relative throughp u t obtained by semibatch operation of the S T R are less than unity. Reversible First-Order Reactions. The isothermal reversible reaction can be treated as a n irreversible reaction if concentrabion is measured in excess of the equilibrium concentration. Relative yield and relative throughp u t were determined analytically for this case for semibatch operation of the S T R . The system of equations is the same as for the case of the first-order irreversible reaction except t h a t the reaction rate term and the relative rate constant are modified as given by Table I. The resulting expression for relative yield for the product B is given by eq 15, with P as defined in Table I. The resulting expression for relative throughput is given by eq 16 with replaced by X A / X Awhere ~ X A e is the equilibrium conversion of reactant A. Thus the form of the analytical expressions for relative yield and relative thoughput is the same for the reversible and irreversible case of the firstorder reaction in the semibatch operated S T R . Figure 3 for relative yield with P defined in Table I and Figure 4 for relative throughput with x A replaced by ~ , / X A are , correct for the reversible first-order reaction. It follows that the same semibatch cycle is optimal for both the reversible and irreversible first-order reaction in the isothermal STR. Second-Order Reactions. Larger values of relative yield and relative throughput are attained when the reaction is second order than when the reaction is first order. The expressions for the upper bounds of relative yield and relative throughput for the second-order reaction of Table I are given, respectively, by 9 P2
and
An analytical expression for relative yield and relative throughput for semibatch operation for second-order reactions was not obtained, since the general equation for concentration in the STR is nonlinear and probably cannot be solved analytically. Adiabatic Operation
I n a n earlier paper (Lund and Seagrave, 1971) the steadystate optimal adiabatic parameters were summarized. Some general comments regarding semibatch operation for this case are offered here. The adiabatic case in general is more complicated and the relationships between the semibatch cycle parameters, namely Vo*, u F , uE, and u g , for relative yield or relative throughput for a given adiabatic system are not the same as for the isothermal case. dris (1962) has shown that for adiabatic steady-state operation, the reactor system which maximizes the average reaction rate will require the least residence time to obtain a given conversion for fixed feed conditions. Since the plug flow reactor, in which the average reaction rate is a maximum, gives the upper bounds of relative yield and relative throughput for semibatch operation in the isothermal case, it is reasonable to assume that the steadystate reactor system which maximizes the average reaction rate will provide the upper bounds for relative throughput for semibatch operation of the adiabatic STR. Based on this assumption, the following two conclusions may be reached. (1) When the adiabatic derivative (the derivative of reaction rate with respect to conversion) is positive a t the conversion in the reference adiabatic CSTR, the relative yield and relative throughput will always be less than unity. This is due to the fact that the average reaction rate by semibatch operation must be less than the reaction rate in the reference adiabatic CSTR. (2) When the adiabatic derivative is negative in the reference adiabatic CSTR, there exists an optimal adiabatic CSTR and semibatch operated STR combination. I n this optimal combination the adiabatic CSTR is operated a t the maximum reaction rate and is followed in series by the semibatch operated adiabatic STR. As the batch fraction, uB, approaches unity and the minimum volume, Vo*,approaches zero for the semibatch cycle, the relative yield and relative throughput approach their upper bounds, which are obtained in the optimal adiabatic CSTR and plug flow reactor combination. The upper bounds of relative yield and relative throughput obtained by the optimal adiabatic CSTR and plug flow reactor combination may be expressed analytically. The expressions are given below for the case where the dimensionless rate constant, K *, is a linear function of temperature.
{
2RLP ( R L P - P - 1) - [ ( R L P
+P +
- 4RLP]'jz
where y =
(RL
- 1) + l ) P - 2(RL RL + 1
0.m
0.10
0.20
0.50
1.00
2.00
RELATIVE RATE CONSTANT, P
5.00
10.00 20.00
Figure 5. Limiting cases of relative yield by semibatch operation of the adiabatic stirred tank reactor. The rate constant, K*, used in determining these curves is a linear function of temperature: curve A, isothermal plug flow reactor: RL = 0.0; curve 8, optimal adiabatic continuous stirred tank reactor and plug flow reactor combination: RL = 2.0; curve C, optimal adiabatic continuous stirred tank reactor and plug flow reactor combination: RL = 40.0; curve D, adiabatic plug flow reactor: RL = 40.0; curve E, semibatch operation of the adiabatic stirred tank reactor: RL = 40.0, Vo* = 0.1 , U B = 0.0, U F = UE = 0.5
and
where
The optimal adiabatic CSTR and plug flow reactor combination may also be thought of as the optimal adiabatic CSTR and batch reactor combination. When the continuous reactor is followed by the batch reactor or by the semibatch operated STR, an adiabatic accumulator vessel in which no reaction occurs would be required between the two reactors. For the limiting case where Tio*approaches the volume of the batch reactor, the resulting expressions correspond to the relative yield and relative throughput of two adiabatic CSTR's in series. The first adiabatic CSTR operates a t the conversion corresponding to the maximum reaction rate, and the second adiabatic CSTR brings the conversion to the desired level. Curves A, B, and C of Figure 5 show plots of eq 32 for the relative yield in systems with different amounts of heat sensitivity as measured by the parameter RL. By comparing curves A, B, and C it is seen that the maximum value of the relative yield decreases as the heat sensitivity of the reaction system increases. However, for highly heat sensitive reactions there is a range of values of the relative rate constant for which the relative yield in the adiabatic reactor is greater than in the isothermal reactor. Further, the maximum value of relative yield occurs a t smaller values of the relative rate Ind. Eng. Chern. Fundarn., Vol. 10, No. 3, 1971
499
exist values of the semibatch cycle parameters for which semibatch operation of a single adiabatic STR will result in a relative yield between the relative yields obtained in the adiabatic plug flow reactor and in the optimal adiabatic CSTR and plug flow reactor combination, Figure 6 shows the relationship between relative throughput and flow-average concentration for the same situations as in Figure 5. The results shown in Figures 5 and 6 are valid in a quantitative sense only for reactor systems in which the rate constant may be approximated as a linear function of temperature. The dependency of the rate constant on temperature is more accurately given by the Arrhenius equation (eq 5). For reactions with a greater heat sensitivity, the plots for relative yield us. relative rate constant and relative throughput us. conversion obtained when the rate constant is given by the Arrhenius equation will be less accurately aproximated by plots obtained when the rate constant is given as a linear function of temperature. However, the discussion on the upper bounds of relative yield and relative throughput and the optimal semibatch cycle is equally valid for both the linear and the Arrhenius relations. Continuous Variable-Volume Operation
0.11 1 0.01 0.10 DIMENSIONLESS FLOW AVERAGE CONCENTRATION, (1
-
1.( XA)
Figure 6. Limiting cases of relative throughput b y semibatch operation of the adiabatic stirred tank reactor. The rate constant, K*, used in determining the curves in this figure was a linear function of temperature: curve A, isothermal plug flow reactor: RL = 0.0; curve B, optimal adiabatic continuous stirred tank reactor and plug flow reactor combination: RL = 2.0; curve C, optimal adiabatic continuous stirred tank reactor and plug flow reactor combination: RL = 40.0; curve D, adiabatic plug flow reactor: RL = 40.0; curve E, semibatch operation of the adiabatic stirred tank reactor: RL = 40.0, Vo* = 0.1, U B = 0 . 0 , ~=~U E = 0.5
constant for highly heat sensitive reactions than for reactions of lesser heat sensitivity. Curves D and E in Figure 5 show the relative yield obtained by a single adiabatic plug flow reactor and by a particular semibatch operation of a single adiabatic STR, respectively. Curve D shows that the yield may be greater in an adiabatic CSTR than in an adiabatic plug flow reactor with the same residence time and feed conditions. The point where the relative yield is equal to unity on curve D corresponds to the conversion for which the average reaction rate in the adiabatic plug flow reactor is the same as the reaction rate in the adiabatic CSTR. The optimal adiabatic CSTR and plug flow reactor combination may be thought of as simulating a level of mixing between complete mixing in the CSTR and no mixing in the plug flow reactor. Thus, it may be concluded that an intermediate level of mixing will maximize the yield of product in an adiabatic reactor system with fixed feed conditions and a residence time greater than that corresponding to an adiabatic derivative equal to zero. Since semibatch operation of the STR simulates an intermediate mixing level, it may be expected that for sufficiently heat-sensitive reactions there 500 Ind. Eng. Chem. Fundom., Vol. 10, No. 3 1971
The continuous variable-volume operation of the STR employs feed and discharge flow rates that are continuous periodic functions of time. The study of this mode of operation of the STR was undertaken to show that it is possible to obtain a relative yield greater than unity by a variablevolume operation which employs continuous periodic feed and discharge streams. The continuous variable-volume cycle which was studied and with which a relative yield greater than unity was obtained employs feed and discharge flow rates that are sinusoidal functions of time. The kinetic scheme is that of an isothermal first-order irreversible reaction. Douglas and Rippen (1966) studied the periodic operation of an isothermal STR with second-order kinetics in which the feed and discharge streams were sinusoidal functions of time, in phase, and of the same amplitude and frequency; that is, the volume of the STR was constant. They found that the average conversion was decreased by this type of operation for all values of the amplitude and frequency. However, in this work the results obtained by semibatch operation of the STR lead to the conclusion that a relative yield greater than unity may be obtained by a continuous variable-volume operation in which the feed and discharge flow rates are sinusoidal functions of time with the same amplitude and frequency but out of phase. This occurs because the product is withdrawn from the reactor a t a low rate during that portion of the cycle when the product concentration is low and a t a high rate during that portion of the cycle when concentration is high. The feed rate and discharge rate are given in dimensionless form as sinusoidal functions of time by Q,* = 2 sin Q* = 2 sin
(W*e
+1 - 4) + 1
W*e
(34) (35)
where 2 = amplitude, W * = 2 ~ j in * radians/residence time, f* = ~ / ( Q R / V in ~cycles/residence ~ ~ ) time, and 4 = phase lag of output flow behind input flow in radians. The average feed and discharge flow rates are the same as in the reference CSTR. The amplitude, 2, is restricted such that
OlZll
since an amplitude greater than unity would result in a negative flow rate during some portion of the cycle. The volume is determined as a periodic function of time by evaluating the integral in eq 1. The result is c
v* =
2
- [i W*
- COS w*e
+
COS
(w*e - 4) -
COS
61
+ [v*(e- 0)l
E 5m
(37)
y1
The volume in eq 37 is restricted such that 0.961
O ~> K f , some intermediate level of mixing will maximize yield. Semibatch operation of the STR may be visualized as a simulation of a level of mixing between the CSTR and the plug flow reactor. Thus, semibatch operation of an STR may result in a yield of desired intermediate product which is greater than the CSTR yield. Furthermore, the yield by semibatch operation may be greater than the yield of either the CSTR or plug flow reactor for some values of the reaction parameters for systems of consecutive reactions with higher order side reactions. For the Van de Vusse reaction scheme it was therefore inInd. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
501
Table 111. Solutions to Isothermal Constant Volume Equations Reaction scheme
Steady-rtote concn, C*
Tronrient concentration profile,
c*w
A+B K
A Z B K'
2A
5B
Van d e Vurre
0.5t K3CAt/Kl
K2/Kl
-
0.14
0.5
= 1 .O
0.40
0.3C
* -
0.12
-
0.10
-
IV
9
F 0.2(
0.081 -
IU
9 Y,
0.1c
'A
0.06-
b 0 .c
0
I
I
1
I
0.2
0.4
0.6
0.8
0.04 -
FLOW AVERAGE CONVERSION, Figure 9. Yield of intermediate B in different reactor systems with Van de Vusse reactions vs. conversion of feed component: curve A, plug flow reactor; curves B and C, semibatch operation of the stirred tank reactor: Vo* = 0.2; curve B, U B = 0.75, UF = U E = 0.125; curve C, uB = 0.0,uF = uE = 0.5; curve D, continuous stirred tank reactor
0.02
-
I 1
0.00
0.0
1
0.2
I
0.4
1
I
0.6
0.8'
F l O W AVERAGE CONVERSION,
vestigated whether, in fact, the yield (=*) of the desired intermediate could be increased by semibatch operation of the STR over the yield obtained in the CSTR or in the plug flow reactor. Yield of the desired intermediate is defined as the flow average dimensionless concentration of the intermediate product B in the discharge from the reactor, The concentration profiles during the semibatch cycle and yield of intermediate B were determined by numerical solution of the equations using the digital computer and by simulation of the process on the analog computer. Analytical solutions for concentrations in the CSTR are given in Table 111. Van de Vusse (1964) gives analytical solutions for concentrations in the plug flow reactor.
=*.
502 Ind. Eng. Chern. Fundorn., Vol. 10, No. 3, 1971
1
FA
Figure 10. Yield of intermediate B in different reactor systems with Van de Vusse reactions vs. conversion of feed component: curve A, plug flow reactor; curve B, semibatch operation of a stirred tank reactor: VO* = 0.2, uB = 0.75, u p = uE = 0.1 25; curve C, continuous stirred tank reactor
Some results for yield of B (c,*)by semibatch operation of the STR are shown in Figures 9 and 10. Figure 9 shows a case where the reaction rate constant ( K 3 C a l ) for the higher order side reaction is small compared to the reaction rate constant ( K z )for the degradation of B. The result is that the yield of B (c,*)may be greater for semibat,ch operation
of the STR, as is seen for the semibatch cycles of Figure 9, than the yield of B in the CSTR. However, the plug flow reactor gives the upper limit of yield of B. Figure 10 shows a case where the reaction rate constant (K3C.47) is large compared to K z . It is seen in Figure 10 that the maximum yield of B is obtained by semibatch operation of the STR. A region in the plane of the dimensionless parameter, KsCar/ K l , vs. the dimensionless parameter, K2/K1, was determined for which the yield of B is greater by a particular semibatch operation than in either the CSTR or the plug flow reactor. The particular semibatch cycle is defined by fixing the semibatch cycle parameters as uE = 0.75; cF = U E ; uD = 0.0; Vo* = 0.2; = 1.0. The result is shown in Figure 11. The upper curve is the locus of points for which the maximum yield of B is the same in the plug flow reactor and in the semibatch operated STR, and the lower curve is the locus of points for which the maximum yield is the same in the CSTR and in the semibatch operated STR. For all points between the upper and lower curves the maximum yield of B is greater in the semibatch operated STR than in either the plug flow reactor or in the CSTR. This region is denoted as region I1 in Figure 11. The middle curve in Figure 11 was obtained by Van de Vusse (1964) and is the locus of points for which the maximum yield of B is equal in the plug flow reactor and in the CSTR. Region I1 in the (KSCA?/Ki, K 2 / K 1 )plane for which the maximum yield is greater in the semibatch operated STR than in the plug flow reactor in the CSTR will be different for each semibatch cycle as defined by thesemi batch cycle parameters. It should be possible to determine the semibatch cycle which maximizes the area of region 11; however, this has not been done. A second problem which has not been solved is the determination of the semibatch cycle which maximizes the yield of B for values of the reaction parameters K B C A I / K Iand K*/K1 for which the maximum yield of B may be obtained by semibatch operation of the STR.
10.
f Y
Y LL
I-
12 Z
Q
+
Acknowledgment
This work was supported by the Engineering Research Institute, Iowa State University, Ames, Iowa.
1.
6
2 R
5 Z
0 VI
0.
5 5 P
DIMENSIONLESS REACTION PARAMETER, K3CAt/KI
Figure 1 1. Region of maximum yield of 6 for Van de Vusse reactions by semibatch operation of the stirred tank reactor in the ( K 3 C ~ , / K l r K 2 / K I ) plane. Semibatch cycle parameters: uE = 0.75, u p = uE = 0.125, VO* = 0.2: region I, maximum yield greatest in the plug flow reactor; region II, maximum yield greatest in the semibatch operated stirred tank reactor; region 111, maximum yield greatest in the continuous stirred tank reactor; region IV, maximum yield greater in the plug flow reactor than in the continuous stirred tank reactor; region VI maximum yield greater in the continuous stirred tank reactor than in the plug flow reactor
&fk*
= dimensionless feed flow rate during k fraction
Nomenclature
R
= dimensionless relative thermal energy
b
r* T
CJ CAt
* c,* cJ
T
slope of reaction velocity curve with temperature = concentration of component j = concentration of component A in the feed = dimensionless concentration of component j , =
CJ /C.Af
dimensionless flow average concentration of component j in the discharge stream = heat capacity of reactor contents CP E = activation energy in Arrhenius rate expression (- AH,) = heat of reaction for i t h reaction (- AH,*) = dimensionless heat of reaction for i t h reaction, (- AH$)/(- AHI) K = reaction rate constant = reaction rate constant a t feed temperature = dimensionless reaction rate constant, K / K , L = dimensionless linear reaction rate parameter L’ = dimensionless Arrhenius reaction rate parameter P = dimensionless relative rate constant = discharge flow rate Q = feed flow rate QJ QR = feed and discharge flow rate for the reference reactor = dimensionless discharge flow rate, Q / Q R Q* = dimensionless feed flow rate, Q ~ / & R Qf* Q k* = dimensionless discharge flow rate during k fraction of the semicontinuous cycle =
Tf T*
t
U V VM
vo
V* X A
XA
2
of the semicontinuous cycle reaction rate dimensionless reaction rate temperature feed temperature dimensionless temperature, T / T j time = heat transfer coefficient = volume = maximum volume = minimum volume = dimensionless volume = conversion of component A = flow average conversion of component A in the discharge stream = amplitude = = = = = =
GREEKLETTERS ff,J
‘I1
e
8k 00
P Uk
stoichiometric coefficient of reactant j in reaction i = relative yield of product j = dimensionless time, t & R / V M = dimensionless time a t end of k fraction of the semicontinuous cycle, ~ ~ & R / V M = dimensionless time a t beginning of semicontinuous cycle, t & ! R / V M = density = the k fraction of the semicontinuous cycle =
Ind. Eng. Chem. Fundom., Vol. 10, No. 3, 1971
503
*
4 w
= phaselag = relative yield =
literature Cited
frequency
SUBSCRIPTS
i j
k F B E
D
=
refers to ith reaction: i
=
1, 2, 3, . . .
B , C, D refers to k fraction of the semicontinuous cycle k = F , B , E, D = filling = batch = emptying = down = refers to jth component: j = A , =
Ark, R., Can. J. Chem. Eng. 40, 87 (1962). Codell, R. B., Enael, A. J.. A.I.Ch.E. J. 17. 220 (1971). Douglas, J., Ripp&,'D., Chem. Eng. Sci. 21,'305 (1966): Fang, M., Engel, A. J., paper presented a t 61st National AIChE Meeting. New Orleans. La.. 1967. Gillespie, %., Carberry, J., Chim. Eng. Sci. 21, 472 (1966). Lund, M., Seagrave, R., A.I.Ch.E. J. 17, 30 (1971). Van de Vusse, J., Chem. Eng. Sci. 19, 994 (1964). RECEIVED for review September 21, 1970 ACCEPTED June 4,1971 Presented a t the Division of Industrial and Engineering Chemistry, 159th National Meeting of the American Chemical Society, Houston, Tex., Feb 1970.
Calculation of Isothermal Vapor-Liquid Equilibrium Data for Binary Mixtures from Heats of Mixing Richard W. Hanks,' Avinash C. Gupta, and James J. Christensen Department oj Chemical Engineering and Center for Thermochemical Studies, Brigham Young University, Provo, Utah 84601
The Gibbs-Helmholtz relation is used together with two popular semitheoretical excess free energy relations (Wilson's equation and the NRTL equation) to calculate isothermal vapor-liquid equilibrium (x-y) data for six highly nonideal binary systems. This method is shown to produce reliable x-y curves directly from measured heat of mixing (AhM)data and pure component vapor pressures without the necessity of measuring any x-y data. This method, coupled with titration calorimetry techniques for measuring AhM easily and quickly, has considerable practical potential for phase equilibrium work. Although Wilson's equation and the NRTL equation were used, the calculation method is valid for any excess free energy function. The accuracy of the calculated x-y data appears to be limited only by the accuracy of the Ahw and vapor pressure data and the semitheoretical excess free energy model chosen.
Vapor-liquid equilibrium data are of considerable industrial and academic importance. Although these data can readily be calculated for ideal solutions, most solutions of interest are not ideal in behavior. A great deal of study has been perfumed on nonideal solutions, but a t present no way exists which will permit the accurate theoretical prediction of deviations from ideality. Therefore, one must turn to experimental measurement to determine nonidealities for a given system. I n this paper, we discuss a method of estimating isothermal vapor-liquid equilibrium which does not involve the experimental measurement of the composition of either the vapor or liquid in a vapor-liquid equilibrium mixture, a procedure which is often quite laborious. The method consists of using measured heats of mixing to evaluate the constants in semitheoretical thermodynamic equations which in turn are used to calculate equilibrium data from pure component vapor pressures. The conventional approach to vapor-liquid equilibrium involves a n attempt to predict heats of mixing from measured vapor-liquid equilibrium data and then to compare these values with measured heats of mixing determined by calorimetric techniques. 111general, the agreement between calculated and experimental heats of mixing has been poor. One To whom all correspondence should be addressed. 504 Ind. Eng. Chem. Fundam., Vol. 10, No. 3, 1971
reason for this poor agreement is inherent in the calculation methods employed. All of the theoretical equations used are based upon the concept of the excess free energy, G E , which is directly related to the activity coefficients, yi. The latter are defined by the expression
where P is the total system pressure, yi, G ~ j ,z o L , and x i are, respectively, the vapor phase mole fraction, the vapor phase fugacity coefficient, the pure liquid fugacity, and the liquid phase mole fraction of component i in the equilibrium mixture. Since this expression involves the ratio yi/x2, errors in y t a t a given x L are reflected directly in the G E data derived therefrom, especially a t low values of z, where this ratio is essentially the first derivative of the equilibrium curve. The G E data are used to determine adjustable constants in semitheoretical thermodynamic equations. Finally, the heats of mixing, Ah", are calculated from the G E data by differentiation of the semitheoretical thermodynamic equations using the well known Gibbs-Helmholtz relation
Thus, the error magnification inherent in two differentiations
