Ind. Eng. Chem. Fundam., Vol. 17, No. i, 1978 Thodos. G., AlChEJ.. 1, 165 (1955). Thodos, G.. AlChEJ., 1, 168 (1955). Thodos, G., AlChEJ., 2, 508 (1956). Thodos, G., AlChE J., 3, 428 (1957). Thomas, G. L., Young, S., Trans. Chem. Soc., 67, 1071 (1895). Travers, M. W., Senter, G.. Jaquerod, A,, 2. Phys. Chem. (Frankfurtam Main), 45, 416 (1903). Tsuruta. K.. Phys., Z., 1, 417 (1900). Van Itterbeek, A . , De Boelpaep, J., Verbeke, O., Theewes, F., Staes, K., Physica. 30, 2119 (1964). Van Itterbeek, A . , Verbeke, O., Staes, K., Physica, 29, 742 (1963). Van Itterbeek, A.. Verbeke, O., Theewes, F., Staes, K., De Boelpaep, J., Physica, 30, 1238 (1964). Vaughan, W. E., Graves, N. R., lnd. Eng. Chem., 32, 1252 (1940). Volova. L. M., J. Phys. Chem. (USSR), 14, 268 (1940). Von Siemens, S., Ann. Phys., 42, 871 (1913). Waddington. G., Smith, J. C., Williamson, K. D., Scott, D. W., J. Phys. Chem., 66, 1074 (1962).
51
(143) Willingham, C . B.. Taylor, W. J., Pignocco, J. M., Rossini, F. D., J. Res. Natl. Bur. Stand., 35, 219 (1945). (144) Woolley, H. W., Scott, R. B., Brickwedde. F. G., J. Res. Natl. Bur. Stand., 41, 374 (1948). 145) Wright, R., J. Chem. Soc., 109, 1134 (1916). 146) Young, S., J. Chem. Soc., 55, 486 (1889). 147) Young, S., J. Chem. Soc., 59, 911 (1891). 148) Young, S., Trans. Chem. Soc., 73, 675 (1898). 149) Young, S., Trans. Chem. SOC.,77, 1145 (1900). 150) Young, S., Sci. Proc. Roy. Dublin Soc., 12, 374 (1910). 151) Young, S., Fortrey, E. C., J. Chem. Soc.. 83, 45 (1903). 152) Young, S., Thomas, G. L.. J. Chem. SOC.,71, 448 (1897) 153) Zmackynski, A.. J. Chem. Phys., 27, 503 (1930).
Reteii)ed f o r rei,ierc July 13, 1977 Accepted October 18. 1977
A Mixing Model for a Continuous Flow Stirred Tank Reactor Thomas R. Hanley"' and Roland A. Mischke Department of Chemical Engineering, Virginia Polytechnic lnstitute and State University, Blacksburg, Virginia 2406 7
The segregated and nonsegregated flow models, using exit age distributions and reaction kinetics, predict steady-state exit stream concentrations for a continuous flow stirred tank reactor. For all reactions other than first-order reactions, the two models predict different results if operating conditions are properly set. The use of the variance of the concentration in the exit age distributions, in conjunction with the mean value of this concentration and the reaction kinetics, was studied in an attempt to accurately predict the exit stream concentration for all conditions.
Introduction In a continuous stirred tank reactor the degree of mixing combines with the reaction kinetics t o determine the extent of conversion in a given reaction. The exit concentration from the reactor is therefore a function of a t least two distinct variables, the mixing and the kinetics. The mixing, represented in part by the exit age distribution of the reactor combined with the kinetics, should give an estimate of the exit stream concentration. A general model for predicting the exit stream concentration of a backmix flow reactor, applicable to all variations of reactor conditions, is difficult t o produce. Two models, the maximum-mixedness model of Zwietering (1959) and the segregated flow model of Levenspiel (1965) offer means of estimating the exit stream concentration of a backmix flow reactor for a reaction of known kinetics and a given exit age distribution. Past work indicates, however, that these models provide limits within which the actual exit stream concentration exists. The only times these models accurately predict the actual conversion of the reactor is when segregated or nonsegregated flow exists in the reactor. At least one additional variable is needed in order t o predict the exit stream concentration of the reactor. One variable with distinct possibilities of providing the needed third component of the estimating equation is the fluctuation of concentration in the exit stream for the exit age distribution test. This fluctuation in concentration can be measured as the root mean square, variance, standard de. Address correspondence t o this author a t t h e Department of Chemical Engineering, Tulane University, New Orleans, La. 70118.
viation, or any similar measure of fluctuation from the mean.
The Reaction The saponification of ethyl acetate (reactant B) with sodium hydroxide (reactant A) was chosen as the reaction for this investigation. The rate expression was assumed to be that of an elementary, bimolecular, second-order reaction, yielding
Moelwyn-Hughes (1953) presented several approximations for the rate constant, h . We used that of Warder (1881). 11 200 In k = 16.47 - RT
Exit Age Distributions Exit age distributions, or E curves, are the measures of the distribution of ages of all elements of the fluid stream leaving a vessel. The E curve is a measure of the distribution of residence times of the fluid within the vessel. For steady-state flow of fluid through a vessel and a pulse tracer input, Levenspiel (1965), by means of a material balance, indicates that E = C* (3) Therefore, for the steady-state flow of fluid through a vessel and a pulse tracer input, the exit age distribution can be found experimentally by determining a concentration-time (e*) curve. C 1978 American Chemical Society
52
Ind. Eng. Chern. Fundarn.,Vol. 17, No. 1, 1978
,-Conductivity
I
Probe
I
/ ,
AC Amplifier
Demodulator
Analyzer
1 - L ' E ( 8 ) d8
In solving this equation numerically, it is sufficient to choose an estimated value o f ? = 1 for a value of 8 which is three or four times the mean residence time, and then integrate with respect to 8 (decreasing 8 by small steps). The solution of occurs a t 8 = 0. The two previous models have been shown by Hanley (1970) to provide limits for conversion for a given reactor system. Micromixing, unlike macromixing, depends on the history of the molecules in the reactor (i.e., timeliness of mixing) as well as the residence time distribution. In order to describe micromixing, two-environment models and coalescence-dispersion models have been proposed. Goto (1975) and Nishimura (1970) proposed two-environment models which separate the reactor into two regions, one segregated and one maximally mixed, in an effort to handle the timeliness of mixing concept. Chen and Fan (1971) reversed the order in an effort to characterize polymerization viscosity increases. Spielman and Levenspiel (1965), Kattan and Adler (19671, Rao (1970), and Ziegler (1971) characterize micromixing by the frequency of the coalescence and dispersion of the fluid elements in the reactor. The model results usually depend on generalizing assumptions and methods of calculation. Both two-environment models and coalescence-dispersion models are difficult to prove from experimental data. This is due to the difficulty in data production and the system upset that accompanies such measurements.
c
1
Figure 1. Block diagram of components for reduction of electrical signal resulting from concentration fluctuations.
Mixing Models Once the kinetics and the exit age distribution of the reactor are known, they must be combined into a mixing model in order to predict the conversion and the exit stream concentration. Two models have been proposed for estimating the exit concentration from a reactor having a given exit age distribution. These models are known as the segregated flow, or no-mix, model, proposed by Levenspiel (1965) and the nonsegregated, or maximum-mixedness, model, proposed by Zwietering (1959). Levenspiel (1965) proposes the following model for determining the exit concentration from a backmix reactor processing a macrofluid.
This model for estimating reactor performance is known as the segregated flow model and simulates reaction of a macrofluid. Reaction kinetics supply CBATCH, tracer experiments give E(B),and therefore the exit concentration may be estimated. Zwietering (1959) proposed the following model to estimate the exit stream concentration for bimolecular, second-order reaction of a microfluid.
Concentration Detection The electrical conductivity detection system of Lamb et al. (1960) was chosen for this investigation. This system, presented in Figure 1, was capable of detecting concentration variations in volume elements of the order of 3 X cm3 over a concentration range of six orders of magnitude. Two Pace TR-10 analog computers were used to analyze, square, and average the signal. This circuitry is shown in Figure 2. The following changes in tho schematic, shown in Figure 3, were made in order to reduce noise in the output and to provide a wide range of conductivity.
SIGNAL TO
TEN
Y
OF
MEAN
MICROFARAD
. ..
+ y X-SQUARE
s
0.5
1.0 L
TO V
s
OF
VA RlA N C E X - V
RECORDEQ
0.1
Figure 2. Analog schematic for mean and variance calculation.
o
0.05
J
X - V
RECORDER
CAPACITOR
Ind. Eng. Chem. Fundam., Vol. 17, No. 12AX7
212AU7
CALIBRATING RESISTOR
.
6SN 7
bF5
i,1978
53
8 + - 2 7 O v
.01
Figure 3. Circuit for electrical conductivity probe
-6. 0 25"
BATH THERMOREGULATOR
0 121"
PROBES
~
EPOXY
~
STIRRING
HOLDER
F BAFFLE
RING
> BAT1
-
-
SHRINK
STEEL TUBlNG
0.2 5 "
0.417"
BAFFLES
0.125'
-
RES~N
VIEW
STAINLESS
COVER
7
- _ END
ROD
FLUID
[EXIT
LINE
LEVEL
b-
TUBING SHIELD
REACTOR
13.0"
EPOXY RES,N
SIDE
C R O S S - S E C T I O N b L VIEW
THREE
CONDUCTOR
AGITATOR
LEVEL
ri
/ GLASS TUBING
,
O.b
LEbDWIRE
'
TYP
Figure 4. Conductivity cell construction.
BAT*
AQlTlTOR
THERHOREQULITOR
1 TO
DRAIN VALVE
LEFT-SIDE
RIGUT-SIDE
INPUT
lNPdT
Figure 6. Reactor details (front elevation).
Figure 5. Schematic diagram of experimental apparatus.
(1) The output of the 15-kHz sine-wave generator was passed through a one to five voltage divider to reduce the output of this signal t o f0.400 V. (2) T h e 500 K potentiometer leading to tube 6F5 was replaced with two 200K resistors and a lOOK potentiometer in series in order to set the gain of the potentiometer more accurately. (3) The calibrating resistor was a 27 000-ohm resistor. The conductivity probe consisted of two platinum wires insulated with 2-mm glass tubing and contained in a 2-in.
section of %-in. stainless steel tubing. The outside of the stainless steel shield was covered with shrink tubing. The two probe wires were soldered to two leads of a three-conductor shielded cable. T h e cable shield was soldered to the stainless steel tubing. The cell was then sealed a t both ends with epoxy resin. For a more complete description of the cell, see Figure 4. A solution of sodium chloride was chosen for the tracer. The relationship between concentration and electrical conductivity was found to be nearly linear in ranges of concentration of 0.0000 to 0.0300 M, the range suitable for the tracer investigations. Experimental Apparatus The design of the reactor followed that of Javinsky and Kadlec (1968). According to these authors, their reactor was perfectly mixed. However, modification of the mixing condi-
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
54
pulse and injecting the pulse through the diaphragm into the flow line.
-
>.00*Dll
Figure 7. T o p cross-sectional view of reactor taken 9 in. above base.
->-
ROTAMETERS
-
~
i
A-
---HEATING
TAPE
F L O W LINES THERMOREGULATORS
Procedure Exit age distributions were determined for total flow rates of 213.6 and 397.0 ml/min for five different agitation conditions. The five agitation conditions were as follows. (1) Use of Agitator No. 1. The agitator blades were bent a t 20" angles to produce a downflow in the reactor. (2) Use of Agitator No. 2. This agitator was smaller than the first agitator with no bending of the blades. This agitator was designed to create a condition of lesser agitation. (3) Use of Agitator No. 3. This agitator was the same size as agitator no. 1, but the blades were bent a t 30" angles to produce an upflow in the reactor. This agitator was designated to produce a high degree of mixing in the reactor. (4) Use of No Agitation. This condition was designed to produce a reactor with no agitation. ( 5 ) Use of Agitator No. 4. Agitator no. 4 was merely a smooth rod. This agitator was designed to produce a mixing condition only slightly better than no mixing. The test was allowed to proceed for a period of four residence times a t the conditions of operations. Then, the flow to the reactor was stopped and the fluid in the vessel was agitated with either agitator no. 1 or agitator no. 3 for 5 min. After this time flow was reintroduced to the reactor a t a low rate, and the final concentration was quite different from the reading a t four residence times for conditions of poor agitation. This final concentration was needed to complete a material balance on the reactor system. Fifty-one exit age distributions were produced with a minimum of five tests for each condition. Reactions of 0.0401 M sodium hydroxide and 0.0393 M ethyl acetate were performed a t the same conditions of flow and agitation a t 55 "C. Samples of the reactor effluent were quenched with 0.0780 M hydrochloric acid and back-titrated. This process continued until stable results were obtained for each set of operating conditions.
OUTSIDE BATH
Figure 8. Flow diagram of constant temperature system.
tions could be easily obtained by modification of the agitator or by changing the flow rate to the reactor, thereby offering a perfectly mixed reactor in addition to relatively worse conditions of mixing. A schematic flow diagram of the system used in this investigation appears in Figure 5. The reactor itself was made of copper. The two feed streams entered annularly in the center of the base of the reactor. The overflow exit line drained the reactor on the side. The reactor was surrounded by a constant-temperature bath constructed from a &gal acid drum. The bath was used during the reaction studies, whereas the exit age distributions were performed a t room temperature. The reactor is shown in Figures 6 and 7 and the constant temperature system appears in Figure 8. The inlet streams to the reactor were gravity fed from two 10-gal reactant tanks through l/z-in. copper tubing. The flow rate of each stream was controlled by a needle valve and was monitored by a rotameter. The flow line was wrapped with heating tape from the rotameter to a point 6 in. from the base of the reactor. This heating tape was used to heat the fluid in the line during the reaction exit stream studies. The flow line was insulated to prevent heat loss. In order to introduce a pulse into the inlet line, a pulsing port constructed of a Y 2 - h tee was installed in the flow line 4 in. from the base of the reactor. One side of the tee was equipped with a circle of chromatograph rubber diaphragm held in place by an open threaded cap. The pulse was injected by filling a hypodermic syringe with the desired amount of
Results Figures 9 and 10 illustrate exit age distributions for two of the conditions tested. Note that the better mixed test yields results closer to the perfect mixing plot with the distribution varying with the amount of mixing. The values of VARV, the variance of the concentration fluctuations in the exit stream, exhibit a high degree of fluctuation during the exit age distribution. The method of analysis for these data is the use of the area of the variance readings over a given range of reduced time in which the data were taken. All variance data were taken starting at reduced time equal to 0.25. This was done in order to keep the computer from overloading during the first part of the exit age testing when the concentration of the exit stream changed from a zero reading to a peak reading due to the injection of the pulse tracer. I t should be noted here that because the signal contained some 60-Hz noise, the signal was passed through an averaging integrator to eliminate this noise. This integrator also averaged the signal to some degree, therefore reducing both the signal and the variance. Because of this fact, variance data were correlated in the form of a ratio, using some value of variance as a reference point in order to place all data on the same basis. In the search for a method of analyzing the variance of data, VOMSQ was defined as VOMSQ(0) = VARV(0)/AMEANV(0)2
(6)
This dimensionless quantity was integrated by means of a computer starting a t 8 = 0.75 in order to find the area of the
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978 1
55
.0
J
0
Q
0.9
- - R a n g e Of E V a l u e s A t A G l v e n V a l u e Of R e d u c e d T i m e Exit Age DiStributlons O n e Throuh Six.
\
For
\
\P
0.8
The Broken Line Represents The Exit A g e Distribution For
(-e)
Perfect Mlxlng, Exp.
\
\ \
0.7
\ \
'\"
0.6
\
0.5
\o
0.4
0.3
. --
0.2
-0.
rn
0. I
0.0 0
1 .o
0.5
1.5
2.5
2.0
3.0
3.5
A.0
@ I Reduced Time(Dimena~onleSa)
Figure 9. Exit age distributions for tests using agitation condition one and a flow rate of 397.0 mL/min.
1.0
0.9
--
\
0.8
R a n g e O f E V a l u e s A t A Given V a l u e Of R e d u c e d T i m e F a r
Exit A g e Distribut8ons Forty-Two Through F o r t y
-
Sir.
The B r o k e n Line Represents T h e E x i t A g e Distribution For Perfect M ~ ~ , ~ ~ .
(-e)
\
0.7
0. b
0.5
0.4
0.3 \
0
0.2
c
0.1
1
0.0 0
0
\ '\ \
...
1 .
O
Q
O
._ --
o
c
----_._
--
'
--
3 .. 1
0.5
1.0
;.5
2.0
@ ' R e d u c e d Time
3.0
2.5
3.5
4 .0
(Dimrnslonlcss)
Figure 10. Exit age distributions for tests using agitation condition 5 and a flow rate of 397 0 mL/min.
curve. Because the poor mixing present in exit age distributions 32-41 allowed the reactor to operate for some time before any of the tracer left the system, values of VOMSQ between R = 0.25 and 0 = 0.75 for these tests approached infinity. Therefore all VOMSQ areas were started a t 0 - 0.75. The values of VOMSQ will be plotted as a function of S , the fraction of mixing, defined as
S=
Cexp
Cmax
- Cmin - Cmin
(7)
The values of the area of the VOMSQ curves starting a t 0 = 0.75 were averaged a t each condition of flow rate and agi-
Am,
tation. This area, was plotted as a function of S and a least-squares regression analysis yielded
+
In = - 11.0730s 5.8168 (8) In order to place the equation in the form of a ratio, the value
-
of In A075 a t S = 0.50 was substracted from both sides of the equation. In A075 (at S = 0.50) = 0.2803. Substituting this value into eq 8 In R , = 11.0730s -
+ 5.5365
__
(9)
where R A is the A075 value minus the A075 value of the regression line a t S = 0.50. This model, henceforth referred to as the Hanley-Mischke model, allows the prediction of the actual conversion from a backmix reactor. This model employs the mean and variance of the concentration fluctuations in the exit stream for an exit age distribution and the kinetics of the reaction. This model is presented graphically in Figure 11. The exit stream concentration estimations for the segregated and non-segregated flow models were calculated using batch reactor kinetics, the results of the exit age distribution tests, and the temperature of the reaction test. The reactor performance for the segregated flow model was estimated
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
b t 0.5
A
--Segregated
0
--Non-Segregated
X
--Experimental
Flow
Model Est,mat#on Flow M o d e l E s t i m a t i o n
Value
?.
0:E x p e r i m e n t a l
0--Hanley-
YI VI
Data
-u E
Perfect
0
7 u
Mischke M o d e l Estlmatlon
Segregated
0.4
Mixing
Flow M o d e l E s t i m a t i o n
N o n - S e g r e g a t e d Flow Model
E
Estimation
v
C
2 0.3
L
-! W
C U 0
u
0.2
u
"
A
2
0
u
a
'0 0.1
0.0
3
I
0.2
O
0.4
0.8
0.6
4
5
kcAO T S =
%
Figure 11. Plot of In R vs. S: the Hanley-Mischke model. A - -Segregated 0.5
a--HanIey-
A
F l o w Model Estimation
Value
12
( ~ ~ m e n s ~ o n l e ~ s )
0.5
Miochke Model Estimation
F l o w M o d e l Estimation
Non-Segregated
--Segregated
Flow M o d e l Estimation
--Non-Segregated
Perfect Mixing S e g r e g a t e d F i o w M o d e l Estimation
\
0.4
11
Model Estlmatlon
Flow
X--Experlmentai
IO
9
Figure 13. Experimental and estimated values of ?, reduced concentration, as a function of kCAor for reactions using agitation condition 2.
M ~ x i n g (Dimensionless)
C--Non-Segrcgated
8
7
b
I .0
,.
X
Flow Model EstImatlOn
- - E x p e r ~ m e n t a lValue
0 --Hanley-
Mischke
M o d e l Estimation
s
-0c 0
d
c
__
0.4
\ -
Perfect Segregated
Mixing
Flow Model Estimation
Non- S e g r e g a t e d Flow M o d e l Estimation
01
! w
0.3
0
0.3
8
uv
0.2
0
0.2 V
"3 U
V u
0.1
a 10
0.1
0.0 3
4
5
6
7
8
9
I O
11
I2
0.0 kCAOT
~ D ~ m e n 5 ~ o n I e s ~ l
Figure 12. Experimental and estimated valves of c, reduced concentration, as a function of kC.~.or for reactions using agitation condition 1.
using eq 4.The value of ??BATCH was calculated using eq 1.The values of E ( 0 ) were calculated from the AMEANV data. Equation 4 was integrated by means of a computer using Simpson's rule between values of 8 = 0 and 8 = 4. The reactor performance for the nonsegregated flow model was calculated using eq 5. The values of E ( 8 )were calculated in the previously mentioned manner. The area of the E ( 8 ) curve between 8 = and d = 4 was calculated using Simpson's rule. F o r e q 5, the concentration a t the exit of th_ereactor is given by C evaluated a t 0 = 0. By assuming that C = 1at 8 = 4,the equation was solved by a fourth-order Runge-Kutta
k Cn
1
( DI m e nSI o n I e
s s)
Figure 14. Experimental and estimated values of %, reduced concentration, as a function of kCAor for reactions using agitation condition 3.
method using a time increment of A8 = -0.10 between 8 = 4 and 8 = 0.20 and A0 = 0.05 between 8 = 0.20 and 0 = 0.0. The values of the exit stream concentration estimated by the segregated flow model, the nonsegregated flow model, and the Hanley-Mischke model are plotted with the experimental determinations of the exit stream concentration in Figures 12 through 16. The values of the exit stream concentration estimated by the segregated flow model and the nonsegregated flow model for a perfect mixing exit age distribution are also
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
A -0 -X
Flow Model Estlmatlon
Segregated Non-
Flow M o d e l EstlmatlOn
Segregated
--Experimental Value
0 --Uanley-Mtschhe c 0
Perfect Segregated
Model Estimation
M Ia , n g
Flow Model Estimation
Non-Segregated
Flow M o d e l E s t i m a t i o n
L
C
'.
D U c
0
v uW
0.2
-\
V W
[L
0
&
(Dimensionless)
kCAo
Figure 15. Experimental and estimated values of ?, reduced concentration, as a function of k C A o i for reactions using agitation condition 4.
A
,.
0.5
e E 0
VI
c 0.4 W
\
Flaw M o d e l E s t i m a t i o n
--Segregated
0 --
NO"
-
segregated
FIOW M
X
- - ~ x p e r , m e n t a i value
c
--Hanley
O
~
~~ s It t m a t ~ o n
- Mischhe Model Estimation Perfect Mlxlnp
--
Segregated Non
!
Flow M o d e l E s t # m a t t o n
- S e g r e g a t e d Flow Model E s t i m a t i o n
v
c 0
2
0.3
C
" U
\
0 c
u
'. '.--1
0.2
"
0
U
B a
U
L I
10
0.1
0.0
kCAOT
(Dimensionless)
Figure 16. Experimental and estimated values of ?, reduced concentration, as a function of k C ~ ofor i reactions using agitation condition 5. plotted in these figures to illustrate the deviation from perfect mixing in the experimental data.
Discussion The exit age distribution testing for the five conditions of agitation yielded five different exit age distributions, with the differences detectable using the shape of the distribution. For agitation condition no. 1, the exit age distributions were very close to that of perfect mixing with the slope of the line indi-
57
cating a small amount of deadwater region present. Agitation condition no. 2 provides an exit age distribution similar to that of agitation condition no. 1 with a slightly smaller slope, indicating a larger deadwater region. Exit age distributions for agitation condition no. 3 indicate that a small amount of bypassing occurs as the slope of the exit age distribution is greater than that for perfect mixing. Agitation condition no. 4 yields exit age distributions with a large time delay before any tracer leaves the reactor, indicating poor mixing, and a very small slope after the exit age distribution peaks, indicating a large deadwater region. Exit age distributions for agitation condition no. 5 also indicate a large deadwater region. However, as the distribution shows no significant deadtime and a slope greater than that of agitation condition no. 4, the deadwater region for this test is smaller than that of agitation condition no. 4. The lower flow rate a t a given agitation condition produced an exit age distribution which indicated better mixing. This fact can be deduced from the fact that the slopes of the exit age distributions obtained a t the lower flow rates were larger than those obtained a t the higher flow rates. The variance of the concentration fluctuations in the exit stream fluctuated through a series of peaks and drops during the course of the test. This fluctuation can be explained by the fact that during the greater portion of the test, no steady-state condition is obtained. Therefore, globs of fluid exiting the reactor at random times would cause this peak and drop effect. The area of the variance-time relationship obtained during a test provided a reliable fit for the data. Each exit age distribution test provided fairly reproducible areas of the variance, no matter which point of reduced time was chosen to start the area. The determination of the reduced time used as a starting point for the area calculations was influenced heavily by the no mixing tests. A point of A = 0.75 was chosen as a starting point because this point eliminated the rather unstable conditions during which the reactor exit concentration rose from a zero value to a peak value, yet still included a significant portion of the variance curve. For all exit age distribution tests, the segregated and nonsegregated flow models provided bounds for the experimentally determined value of the exit stream concentration. This was to be expected as the two models are based on extreme conditions of agitation and should therefore produce estimates of the limits of conversion. As these models depend only on the experimentally determined exit age distributions and the reaction kinetics, the estimations of the exit stream conversion for the models should only be as good as the exit age distribution. As the exit age distributions were quite reproducible, the model estimations varied approximately 1%from the average for the exit age distribution tests performed using the same conditions of agitation and flow rate. As the Hanley-Mischke model was formulated using the experimental data determined during this investigation, the model will obviously fit these data. The validity of this model must be determined considering the following circumstances. (1)Is the model feasible in considering the actual conditions of mixing present in the backmix flow reactor? In considering completely segregated flow, one would expect that the globs of fluid would maintain a constant size and number, therefore producing a finite value of the variance of the concentration fluctuations a t this condition. The model predicts a finite value by extrapolation of the curve to S = 0. However, as no tests yielded results closer to S = 0 than the test where S = 0.104, this assumption cannot be considered true, even though it appears to be true. In considering maximum mixedness, one would expect the variance of the concentration fluctuations to approach some finite value as maximum mixedness by
58
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
definition is not perfect mixing. This value of variance should be very small, however, and the model, which predicts the exponential approach to this small value, seems to follow this reasoning. Again, as no points were determined near S = 1, this cannot be considered a proven conclusion. (2) Is the model limited to the geometry and set conditions of the reactor system? As the model uses exit stream data, the timeliness of mixing (early or late mixing), a function of reactor geometry and mixing conditions, cannot be detected. I t is reasonable to assume, however, that a given reactor will tend to promote early or late mixing. This reactor setup promotes early mixing, so the model should be generally valid for early mixing reactors; a similar relationship should exist for other time of mixing configuration. (3) Would the exponential decay of A075 vs. S illustrated mixing increases by the model be the condition expected asin the reactor? The exponential decay of A075 vs. S for the model was not expected by this author, but in turn, it was considered a possibility during the actual formulation of the model. As no data other than those taken during this investigation are available, it is impossible to say what shape of curve was expected or even which shape is valid. All that can be said is that an exponential decay was a possible shape of the investigated curve, and that the choice of this shape was made only after the data indicated that it was so. (4) Would the model be applicable to any reaction, or is it only applicable to the second-order reaction used? As the segregated and nonsegregated flow models are valid for all types of reaction kinetics, this model, which is a function of the two models, should be valid for all reaction situations.
Conclusions The following conclusions are valid for the reactor system used in this investigation using flow rates to the reactor of 397.0 and 213.6 mL/min. The exit age distributions are dependent on the position of the pulse injection point, therefore making all estimations of the exit stream concentration using the exit age distribution dependent on this condition. The results are also valid for a bimolecular, second-order reaction of ethyl acetate and sodium hydroxide operated a t 55 "C. With the limitations noted above, the conclusions of this investigation are as follows. (1)The exit age distributions for the ten conditions of flow rate and agitation were determined to be reproducible to within f 2 % . (2) The values of the exit stream Concentration for the reaction tests were determined to be reproducible to within f1%. (3) The values of the exit stream concentration estimated by the segregated and nonsegregated flow models provided limits for the experimentally determined exit stream concentration. (4) The values of the exit stream concentration estimated by the segregated and nonsegregated flow models were determined to be reproducible to within f l % . (5) The Hanley-Mischke model was determined capable of predicting the exit stream concentration from the backmix reactor to within fl% for values between S = 0.104 and S = 0.862. Nomenclature AMEANV = the mean of the concentration fluctuations of t h e exit stream, V A075 = the area of the VOMSQ curve from 0 = 0.75 to 0 = 4.00, dimensionless C = exit concentration, M
-
C = fraction of reactant unreacted = CA/CAO, dimensionless CAO = initial concentration of reactant A, M CBO = initial concentration of reactant B, M CA = concentration of reactant A a t time t , M CB = concentration of reactant B a t time t , M CBATCH = fraction of reactant remaining in an aggregate of age between t and t dt = (CA/CAO)BATCH dimensionless Co = concentration of tracer if the injected pulse were per- fectly mixed throughout the vessel, M Cexp= measured exit stream concentration, dimensionless C,,, = exit stream concentration predicted by the segregated - flow model, dimensionless ,,C , = exit stream concentration predicted by the maximum-mixedness flow model, dimensionless C* = dimensionless concentration function = C/Co E = exit age distribution k = the reaction rate constant, L/(s mol) M = CBO/CAO,dimensionless R = eas constant. cal/(mol K) ~ ( c ) reaction rate, mol/(L min) R a = A075 - A075 &t S = 0.50, dimensionless
+
t = time, min VARV = the variance in the concentration fluctuations of the exit stream, V2 VOMSQ = VARV/AMEANV*, dimensionless u = incoming flow rate, mL/min V = volume of the reactor, mL
Greek Letters 6' = ut/V = reduced time = t / r , dimensionless 7 = space time = V / u ,min
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Received f o r review August 2, 1976 Accepted October 31,1977
