generally unreliable below normalized frequency contents of 0.3 to 0.2 when Filon’s (3) method is used for data reduction. At normalized frequency contents above 0.2 to 0.3, pulse testing results are in excellent agreement with those obtained by direct frequency forcing and mathematical model solution. Excessive experimental pulse time, due to pulse tailing, was the major cause of the inability to extend the dynamic frequency range using pulse testing techniques. Pulsing temperature limits the high frequency results obtainable because the thermal capacity of the physical system results in energy storage and therefore excessive pulse tailing or time duration. For a given pulse shape, the normalized frequency content does not appear to be affected by pulse amplitude or pulse strength. For pulses of long time duration, the time duration rather than rise or decay times was found to be the best criterion for predicting the relative normalized frequency content of the pulses. Further Work
Although many investigators have done an excellent job in advancing the methodology of pulse testing to its present state, there is still a need for further refinement and improvement. Although the method appears to be adequate for the dynamic analysis of most physical systems, difficulties might be encountered in special cases such as the one discussed. Further studies on in-plant pulse-generating techniques along with studies concerning pulse types (one-sided, two-sided, multiple, etc.) and their characteristics with respect to normalized frequency content could be helpful in the practical application of pulse-testing techniques. Also there is a need for further study of existing data reduction schemes with the intent of improving present methods or developing new ones. Studies of the above nature are at present in progress a t Clemson University.
Acknowledgment
The authors are grateful to the National Science Foundation for support of this work. Nomenclature
G’ h
=
MR R r s t
= = = = =
T,
=
mass velocity of air, lb./(hr.)(sq. ft.)
= convective conductance, B.t.u./(hr.) (sq. ft.) (”
u
= = S(w), =
i(w) x
=
X
=
F.)
magnitude ratio tube radius, ft. variable radius, ft. humid heat capacity, B.t.u./(lb.)(” F.) dynamic temperature, ” F. time duration of pulse, sec. local velocity, ft./hr. frequency content normalized frequency content axial distance, ft. total wetted section length, ft.
GREEKLETTERS = thermal diffusivity, sq. ft./hr. 6 = time, hr. p = density, lb./cu. ft. 4 = phase shift, deg. w = frequency, rad./sec. or cycles/hr. CY
literature Cited (1) Bruley, D. F., Prados, J. W., A.Z.Ch.E. J. 10, No. 5, 612
(1964). (2) Clements, W.C., Jr., Schnelle, K. B., Jr., IND.ENG.CHEM. DEVELOP. 2,94 (1963). PROCESS DESIGN (3) Filon, L. G.N., Proc. Roy. SOC.Edinburg 49, 38-47 (1928-29). (4) Hougen, J. O., “Experiences and Experiment with Process Dynamics,” Chem. Eng. Progr. Mono. 60, No. 4 (1964). (5) Middleton, R. C., “Pulse Circuit Technology,” pp. 11-13, Bobbs-Merrill, New York, 1964. RECEIVED for review April 4, 1966 ACCEPTED January 9, 1967 59th National Meeting, American Institute of Chemical Engineers,
Columbus, Ohio, May 1966. Study carried out under NSF Contract GP-3027.
REACTION KINETICS OF CYCLOHEXANOLACETIC ACID ESTERIFICATION D. J.
McCRACKEN AND P. F. D I C K S O N
Department of Chemical and Petroleum ReJining Engineering, Colorado!School of Mines, Golden, Colo.
kinetic studies of alcohol-acid esterifications are cited in the literature. Goldschmidt (6) developed a n equation which related the rate constant to the initial reactant acid concentration, ester concentration, catalyst concentration, protonated alcohol concentration, and time. He tested the equation for a number of esterification reactions a t 25’ C. and found it successful in obtaining the rate constant. Smith (74, 75) confirmed Goldschmidt’s equation for normal aliphatic acids in methanol catalyzed by hydrochloric acid over the range 20” to 50” C. I n these and other studies (7, 8, 9, 73, 76, 77, ZO), either sealed glass ampoules or a ground-glass-stoppered flask was used as the reactor. The studies were also similar in that the alcohol was always in excess, and the organic acid concentration was initially constant. A high alcohol-to-acid ratio was found to result in a UMEROUS
286
I&EC PROCESS DESIGN A N D DEVELOPMENT
desirable high yield. previous studies:
A number of conclusions developed from
Organic acid-alcohol esterification is catalyzed by a strong mineral acid. The rate constant is a function of catalyst concentration, temperature, and alcohol-acid molar ratio. The equilibrium constant is strongly influenced by the alcohol-acid molar ratio, weakly influenced by temperature, and not influenced by the catalyst concentration. The reaction is second-order with respect to the organic acid concentration. I n previous experiments involving equimolar cyclohexanol and acetic acid, this author found the rate constant to be a function of the initial reactant concentration as well as temperature and catalyst concentration, and the reaction to be third order. T h e present work involved the study of the
Equimolar cyclohexanol and acetic acid, catalyzed by sulfuric acid with dioxane as the diluent, were esterified in a stirred batch reactor. The reverse reaction, cyclohexyl acetate and water forming cyclohexanol and acetic acid, was also studied. An empirical equation relating the rate constant to initial reactant Concentration, catalyst concentration, and temperature was developed. Range of these variables was 1.25 to 5.00M, 0.0930 to 0.1245M, and 104' F. (40' C.) to 176" F. (80' C.),respectively. Experimental forward and reverse reaction rate constants are tabulated for 29 combinations of variables. Nine duplicate esterification runs had a mean deviation of 0.9%. Equilibrium constants are presented for 28 forward runs and seven reverse runs. The empirical equation which was developed predicted rate constants within an accuracy of about 4% in the range studied.
esterification of equimolar cyclohexanol and acetic acid catalyzed by sulfuric acid with dioxane as the diluent. Reactions were carried out in a stirred batch reactor. The main objective was to develop an empirical equation which would relate the forward reaction rate constant to initial reactant concentration, catalyst concentration, and temperature. Range of these variables was 1.25 to 5.00M, 0.0930 to 0.1245M, and 104'F. (40' C.) to 176' F. (80' C.),respectively. T h e reverse reaction, cyclohexyl acetate and water forming cyclohexanol and acetic acid, was also studied. Experimental Work
Apparatus a n d Equipment. A 300-ml., 3-necked, roundbottomed, glass boiling flask was constructed for use as the reactor. The center neck was 7 34/45, and the two side necks were both 7 24/40, angled to avoid contact with the mercury-seal stirrer in the center neck. The temperature bath was a MagniWhirl visibility j a r bath, Model MW 1152, manufactured by the Blue M Electric Co. It was an on-off controlled bath with a room temperature to 100' C. temperature range. Commercial mineral oil was the heat transfer medium. Saybolt viscosity thermometers with 0.2' F. increments were used in the reactor. An all-glass, 1-cc. syringe calibrated in 0.01-cc. increments was used to inject the catalyst into the reactor. Equipment used for analytical work included 1-ml. transfer pipets with 0.006-ml. tolerance, 250-ml. Erlenmeyer flasks, a 5-ml. buret inscribed with 0.01-ml. increments, and a Voland and Sons analytical balance. Materials a n d Reagents. Eastman EK-703 cyclohexanol was used with no further purification. Infrared spectral data indicated that the cyclohexanol contained less than 1.O% water. The boiling range was 136.8-37.2' F. at 10 mm. of mercury. Baker and Adamson 20% fuming sulfuric acid was mixed with B and A sulfuric acid (assay 95.5 to 96.5%), and the concentration was adjusted by titration to yield about 50 ml. of 1 0 0 ~ osulfuric acid. This catalyst was stored in a 50-ml. ground-glass volumetric flask. Cyclohexyl acetate was extracted from the reaction products and purified in the following manner. One liter of the reaction products was put in a 2-liter boiling flask. T o this were added 50 ml. of commercial glacial acetic acid, 150 ml. of distilled benzene, and 1 ml. of concentrated sulfuric acid. The solution was refluxed until all of the water had been collected out of a moisture test receiver, then washed in a separatory funnel with tap water, which was slightly basic, to remove the dioxane, unreacted acetic acid, and sulfuric acid. Cyclohexyl acetate is insoluble in water, so as much as 5 liters of tap water were used. T h e solution was then washed with 1 liter of distilled water. Benzene was added to the washed solution, and the solution was refluxed as before until all the water from the washing step was collected. Then the temperature was allowed to rise to about 140' C. by taking off benzene. The resulting solution of approximately 98% ester-2% benzene was vacuum-distilled at room temperature and 5 mm. of mercury until all the benzene was removed. Then the temperature was raised and the ester collected. The first and last 5yo was discarded, even though the refractive index remained constant over the entire distillation. The boiling point was
68.8' C. a t 20 mm. of mercury, and the refractive index was 1.44070 at 20' C. Density us. temperature data were correlated in the 20' to 80' C. range by the empirical equation: Density (grams per ml.) = 0.9893
-
0.000966 T (" C.)
Commercial 1,4-dioxane was purified and dried ( I ) , but the resulting dioxane was more impure than the original. The dioxane was finally purified by drying it over Drierite, passing it through activated alumina, and then distilling it. T h e purified dioxane boiled at 95.2' C. at 622 mm. of mercury, and did not react with calcium hydride. It was stored over calcium hydride in pint bottles which were flushed with nitrogen and sealed with paraffin. Commercial carbonate-free, 1 N sodium hydroxide solution was used to titrate for the acetic and sulfuric acids in water with phenolphthalein as the indicator. Procedure. PREPARATION OF REACTANT SOLUTIONS FOR FORWARD RUNS. The densities of cyclohexanol and acetic acid were determined by weighing 100 ml. of the reagent in a volumetric flask immediately preceding the preparation of the solution. Appropriate amounts of the acid and alcohol were put in a 1- or 2-liter volumetric flask and diluted with dry dioxane. After the solution reached room temperature, the solution was diluted to the volumetric mark and thoroughly mixed. The solution was then poured in 160-ml. amounts C. Each into '/*-pint bottles which were stored a t -20' bottle was labeled as to initial reactant concentration, and each contained enough solution for one experimental run. Preparation was done in this manner to ensure that the initial reactant concentration for each run was identical. PREPARATION OF REACTOR. After each run was completed, the apparatus was disassembled, cleaned with Alconox, washed with acetone, and dried with air to remove any possible catalyst. The equipment was then assembled. INTRODUCTION OF REACTANTS INTO REACTOR.The bath was heated to the desired temperature, and a reactant bottle was taken from the freezer and allowed to warm to room temperature. One milliliter of the solution was titrated with standard base, and the value was recorded as CaoTa, the initial concentration of acid at room temperature. Pipetted into the reactor were 150 ml. of the reactant solution; the reactor was sealed, and the stirring was begun. When the solution reached the desired temperature, another 1-mi. sample was withdrawn and titrated. This value was recorded as Caor, the initial acid concentration a t the operating temperature. INJECTION OF CATALYST INTO REACTOR.An appropriate amount of catalyst was drawn into the syringe. The syringe was weighed before and after injection, and the difference in weight was recorded as "grams of HzS04," the grams of catalyst. The catalyst was injected into the reactor and a stopwatch was started. A 1-ml. sample was taken with a pipet after 0.5 minute (0.0083 hour) and thereafter a t convenient intervals. Each sample was immediately titrated in 150 ml. of distilled water in a 250-ml. Erlenmeyer flask with standard base and phenolphthalein. The amount of base necessary to neutralize the acids was measured to 0.005 ml. and recorded as "ml. of NaOH." Generally 20 to 30 samples were titrated before equilibrium was attained. PROCEDURE FOR REVERSE RUNS. The procedure for reverse runs duplicated the previous procedure, except that each reactant solution of ester, water, and dioxane was mixed separately. VOL. 6
NO. 3
JULY 1967
287
Calculation Procedures
Alcohol Reacted and Water Formed by Catalyst-Alcohol Reaction. To obtain the acetic acid present, the free acidity must be corrected for the presence of the catalyst. I t was found in this investigation that sulfuric acid reacts with cyclohexanol as it does with many other alcohols (78). Cyclohexanol was diluted with dry dioxane to the required concentration and made to react with a volume of 100% sulfuric acid until equilibrium was reached, within 10 minutes in all runs. Therefore, the same correction was applied to the free acidity throughout the entire esterification run. Table I shows the corrections that were applied. Sulfuric acid reacts with a n alcohol to form the corresponding monoalkyl hydrogen sulfate and water (78). The amount of water and cyclohexyl hydrogen sulfate formed, #, will equal the difference in the initial equivalents of sulfuric acid added and the equivalents of acid after equilibrium is reached. The following equation was used : Cao $J = 2 6 -2 Caoro
- y NaOH
0.500 0.450 0.400
I / CAT 0.300 0.250 0.200
~
TIME (HR)
where 4 = concentration of catalyst (moles per liter, M ) calculated from grams of catalyst added
Figure 1. Relation between reactant concentration and time for runs at 2.5M initial reactant concentration
caor
-= alcohol-dioxane volume correction from room to Cao~o operating temperature y = ml. of N a O H necessary to neutralize a 1-ml. sample N a O H = normality of base
The equation for the amount of cyclohexyl hydrogen sulfate formed will appear in the equilibrium constant equation, because the amount of sulfate and water formed equals the amount of alcohol depleted by the reaction with the catalyst. Order of Esterification Reaction. The order of the esterification reaction was determined graphically. Figure 1 is a graph of f ( C u ) us. time for a variety of runs. For each run in this investigation, 1/Car2us. time gave a straight line, indicating the reaction was third order (77). Figure 2 shows the results of a run in which the alcohol was in excess. The plot of l/CaT us. time is a straight line, indicating that the reaction was second-order with respect to the acid. This result is compatible with the work done by Leyes and Othmer (72) in their study of butanol-acetic acid esterification. Bennett ( 2 ) suggests that acetic acid exists as a double molecule, (CzH402)2, in the present range of temperature, which suggests a secondorder reaction with respect to the acid. From Figures 1 and 2 it was concluded that the equimolar cyclohexanol-acetic acid esterification was second order with respect to the acid, first order with respect to the alcohol, and therefore third order for the equimolar reaction. Therefore, all rate constants have units of (liter)2 (moles) -2 (hr.) Equilibrium Constant. The equilibrium constant is defined as the ratio of the equilibrium concentration of products
Table 1.
104' F .
0.25 0.75 1 .oo
0.045 0.125 0.165
288
122' F. 0,045 0,120 0.160
140" F. 0,045 0.120 0.160
158' F. 0.040 0.115 0.155
Forward reaction
raised to the reaction order with respect to each product divided by the concentration of reactants raised to the reaction order with respect to each reactant.
K = [ (ester)2 (water)
1
(alcohol) (acid)2
eq
The number of moles of water and ester formed in the esterification reaction are equal, and each equals the number of moles of acid reacted. The equilibrium constant can then be written as a function of known values as follows:
K =
(CaoT
- CafTY (CaoT - cafT [ c a f T - ($11 cafT2
+ $1
776'F. 0.040 0.115 0.155
l & E C PROCESS DESIGN A N D DEVELOPMENT
(2)
where CaoT = initial acetic acid concentration at operating temperature, T CafT = final acetic acid concentration at operating temperature, T The values of the final acetic acid concentration were obtained from the graph of Car us. time, where Car is the concentration of acetic acid during the reaction. The following change was made in the equilibrium constant equation to calculate values for reverse reactions. The initial ester and water concentration, Go, was mixed to equal, for example, 1.25M, but could not be checked by titration. Equation 2 with Caor replaced by Co was used to calculate the equilibrium constant. Experimental Rate Constant. The reversible reaction under consideration can be represented by A 4-B C D, where A and B are equimolar, and C and D are equimolar. If A and B react to form C and D, it can be assumed that the reaction is irreversible and can be written A B +C D. After about 70 to 80% of the reaction had taken place, the graph of 1/CaT2us. time deviated from a straight line. At this point the reverse reaction began to exert a noticeable effect. U p to the break-off point on the graph, however, the forward reaction can be used by itself to obtain the rate constant, k,. For the forward reaction the defining third-order differential equation becomes :
+
+
Free Acidity Corrections
(MI. basejml. sample) MI. of HzS04/ 150 MI.
10.0
+
2.0 I/ CAT 1.6 2.2
1.4
/’
0.5
0
ORDER/
IET
- *LN- 03 C*r CAO,
-
- 0.2 -0.1 I
I
2.0
410
3.0
510°‘0
TIME (HR)
Figure 2.
Determination of order
8 M alcohol, 1 M mid, 104’ F., 0.75 ml. of catalyst/l50 ml.
where CbT = concentration of cyclohexanol. If the reaction is equimolar in acid and alcohol, Equation 3 may then be written:
I
I
1.62
l/CUT2 = 2 k f ( t )
+z
(5)
where I is the constant of integration. If this is the governing equation, the graph of l/CaTz should yield a straight line with a slope of 2kf. It was assumed in Equation 4 that the concentration of the alcohol equaled the concentration of the acid. There was a reaction, however, between the alcohol and the catalyst which depleted the alcohol by an amount #. Since the reaction of catalyst and alcohol was complete in less than 10 minutes for all runs, $ was considered to be a constant. For this assumption the defining third-order differential equation becomes:
- (dCaT/dt) = k f C a T 2(Car - #)
(6)
Equation 4 may be written in finite difference form:
t)/CaT3 = k f
(7)
Equation 6 may be rewritten in the same manner:
-
I
Figure 3. Relation between rate constant, k,, and temperature at 0.75 ml. of catalyst per 150 ml.
and integrated to yield :
(A
1.72
I / T X IO’ OABS
(4)
-
I
1.67
(A CaT/A t)/CaTz ( C U T
- #)
=
kf
(8)
When the rate constant, k f , was calculated from Equations 7 and 8 by means of original data, the range of per cent deviation of Equation 8 from Equation 7 was 0.4 to 7.270, the majority of the runs having a deviation of about 3y0. After consideration of the error involved, the stability of the solution, and the relative ease of calculation, the rate constant was calculated by means of Equation 5 by the graphical method. Only the values of Car in the range 20 to 80% reaction were used to construct the graph of 1/CaT2us. time. The slope was then calculated to be 2 k f . Theoretically, the same type of analysis could have been used to determine the governing equation for the reverse reaction. I t was found experimentally, however, that even a n ester-water concentration of 2.5M was heterogeneous upon addition of the catalyst. Because samples could not be taken, the reverse rate constant, k,, could not be calculated for the majority of possible combinations of variables. Other aspects of the reverse reaction are discussed later.
Goldschmidt’s equation (6) is applicable only to a secondorder reaction or one in which the alcohol is in excess. Reaction Rate Constant Equation. T h e following explains how the various relationships among the variables, initial reactant concentration, catalyst concentration, and temperature were combined to obtain an equation which could be used to predict the reaction rate constant. T h e rate expression for many reactions can be written as a function of temperature times a function of composition, or: r = f(temperature)
. g(composition)
where f(temperature) equals the rate constant, k . Arrhenius-type dependency is assumed, then :
k = k, exp [ - E / R ( I / T
- l/T,)] (70)
If an
(9)
If the graph of -In k f us. 1/T’ abs. yields a straight line, then the slope of that line equals E/R. Figure 3 is that graph at constant catalyst concentration and various values of initial reactant concentration. I t was found that E/R was a function of initial concentration, but not catalyst concentration. Figure 4, a plot of E/R us. CaoTo,was assumed to yield a straight line. This line had the equation: E/R = 9860 788 CaoT,. A base temperature of 618’ R. (158’ F.) was chosen, and the exponential term was written:
+
exp[-E/R(l/T
- l/T,)]
exp[(9860
=
+ 788 Caoro) (0.0016181 - 1 / T ) ]
(10)
C’ was then defined as grams of H2S04 per gram of cyclohexanol times 100. Figure 5 is a plot of kf/C‘ us. CaoT, a t the base temperature and constant catalyst concentration. The equation of the straight line obtained was k f = C’ (0.0425
- 0.0034 C a o T o )
(11)
Given the volume of 0.1 5 liter, the weight of the cyclohexanol VOL. 6
NO. 3
JULY 1 9 6 7
289
1.4
*I
Run
2 1.3 x
1 2 3 4 5 6 7 8 9 10 11 12
1.2 Id
1.1
'"!).5
115
315
e15
415
5!5
CAOTO
Figure 4. Relation between activation energy and initial reactant concentration with 0.75 rnl. of catalyst per 150 rnl.
13 14
15
I
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 41 42
I
3.75
5.00
C%O
Figure 5. Relation between rate constant and initial reactant concentration at 158" F. and 0.75 rnl. of catalyst per 150 rnl.
0.175 O.I5O;
I
I I /)__,
I
I
-
0.150
0.125
k, 0.075 0.050
0 .0250,0 0.025i0
/@ 1 0.5
1
I
1.5
2.0
1.0
; I
/I
62
GRAMS Has04
I
2.5
Figure 6. Relation between rate constant and catalyst weight at 2.5M and 158' F.
was calculated to be 15.024CaoT,. Therefore, Equation 11 could be simplified to:
k,
= grams of HzSOa
(E - 0g .02263)
(12)
COOTO
An initial reactant concentration of 2.5Mwas chosen as the base concentration. The rate constant was found to be a function of catalyst concentration at constant initial reactant concentration and constant temperature. Figure 6 is a plot of k, us. grams of catalyst a t 2.5Mand 158' F. The equation for the straight line through the points was
+ 0.0815 . (grams of catalyst)
k, = 0.01152
(13)
Equation 13 was substituted for grams of H2SOa in Equation 12,and the result was multiplied by Equation 10 to yield 290
l & E C PROCESS DESIGN A N D DEVELOPMENT
Table It. Reaction Rate Constant Results G. HzSOa T , ' F. CaoT, keslcd. kexptl. 1.3637 104 2.498 0.0197 0.0203 1 .8247 104 2.503 0.0256 0.0255 1.3741 122 2.498 0.0378 0.0344 1 ,8227 122 2.498 0.0490 0,0443 1 .8297 122 2.503 0.0491 0.9435 0.4785 140 2.503 0.0283 0.0257 1 .3780 140 2.498 0.0697 0.0705 1 .3728 140 2.518 0.0687 0.0713 1.8311 140 2.518 0.0896 0.0908 0.4593 158 2.503 0.0488 0.0460 0.4631 158 2.402 0.0497 0.0479 1,3725 158 2.503 0.1232 0.1236 1 ,8266 158 2.539 0.1573 0.1574 1,8323 158 2.487 0.1618 0.1590 0.4668 176 2.498 0.0852 0.0780 1.3723 176 2.498 0.2124 0.2170 1 .3761 176 2.507 0.2119 0.2190 1.3750 122 5.014 0.0115 0.0103 1.3750 122 5.033 0.0114 0.0108 1.8313 122 5.006 0.0150 0.0144 1.3782 140 5.079 0.0232 0.0232 1.3815 140 5.089 0.0217 0.0239 0.4599 140 5.021 0.0093 0.0084 1 .8297 158 5.000 0.0603 0.0605 1.3783 158 4.998 0.0464 0.0455 1.3658 176 5.033 0.0857 0.0825 1.3658 176 5.033 0.0857 0.0881 0.4640 140 3.757 0.0153 0.0143 1.3718 140 3.757 0.0385 0.0401 1.3819 158 3.719 0.0733 0.0739 1 .3803 176 3.762 0.1296 0.1320 0,4624 140 1.264 0.0644 0.0610 1 .3766 140 1.264 0.1619 0.1563 1,3746 176 1.264 0.4511 0.4258 1.3779 176 1.259 0.4544 0.4258 0,4584 140 1.250 0.0387 1 .3820 140 1.250 0.1185
70Dev.
-
3.0
f 0.4
+4-10.6 9.9 $12.9 $10.1 - 1.1 3.6 - 1.3 6.1 f 3.8 0.3 0.0 1.8 9.2 - 2.1 3.2
-
+ -
++ -
f11.7 f 5.6 f 4.2
0.0
- 9.2 +10.7 - 0.3
+ 2.0 f 3.9
- 2.1 f 7.0 4.0
- 0.8 - 1.8
f 5.6
4- 3.6 +4- 6.7 5.9
+
k, = [0.01152 0.0815(grams of catalyst)] X
-
(0.2829/CaoT, 0.02263) . exp [(9860 4- 788 CaoT,) X (0.0016181
- 1/T)]
(14)
At the base concentration and base temperature, Equation 14 should become only a function of catalyst concentration. The exponential term becomes exp (0) at the base temperature. A value of 2.5Mwas substituted into the second term of Equation 14,and the term was calculated to be 0.09053. The first term of Equation 14 was divided by 0.09053to obtain the term: 0.1272 0.900 (grams of catalyst). Equation 14 for k , did not contain a volume term; therefore (gram of catalyst) was converted to moles per liter, 9, given the volume 0.15 liter. The term thus became equal to (0.1272 13.24084). Therefore the equation for the prediction of the reaction rate constant became
+
+
k,
+ 13.24084) [0*282g - - 0.022631 x CaOTo exp [(9860 + 788 Caoro) (0.0016181- 1/T)] (15)
= (0.1272
Calculated and experimental rate constants, per cent deviations, and data leading to the results are tabulated in Table 11. Equilibrium constants of the forward and reverse runs and the results of the steps leading to their calculation are tabulated in Table 111.
Interpretation of Results
Activation Energy. The activation energy was calculated from a plot of -In k, us. 1/T' abs. At an initial reactant concentration of 2.5M; 0.25,0.75,and 1.00 ml. of catalyst gave values of E / R equal to 11,800', 11,830', and 11,800' abs.,
Table 111.
Run
1 2 3 4 5 6 7 9 10 12 14 15 16 18 19 20 21 22 23 24 26 27 28 29 31 32 33 34
T,
O F .
Equilibrium Constant Results CaoTo Caor cafT 9
104 104 122 122 122 140 140 140 158 158 158 176 176 122 122 122 140 140 140 158 176 176 140 140 176 140 140 176
2.498 2.503 2.498 2.498 2.503 2.503 2.498 2.518 2.503 2.503 2.487 2.498 2.498 5.014 5.033 5.006 5,079 5.089 5.021 5.000 5.033 5.033 3.757 3.757 3.762 1.264 1.264 1.264
104 122 140 158 176 140 140
CO 2.500 2.500 2.500 2.500 2.500 1.250 1.250
FORWARD RUNS 2.482 1.113 2.492 1.113 2.477 1.123 2.482 1.102 2.498 1.123 2.482 1.082 2.482 1.118 2.482 1,123 2.471 1.061 1.102 2.461 1.139 2.456 2.461 1.081 2,461 1.113 4.956 2.187 4.980 2.187 4.936 2.111 5.006 2.116 4.995 2.116 4.938 2.064 4.917 2.090 4.898 2.173 4.917 2.173 3.715 1.589 3.705 1.625 3.658 1.609 1.249 0.575 1.249 0.596 1.245 0.601
K
0.0536 0.0746 0.0598 0.0790 0.0812 0.0174 0.0608 0.0782 0.0198 0.0632 0.0838 0.0206 0.0636 0.0493 0.0495 0.0907 0.0604 0.0590 0.0146 0.0826 0.0693 0.0701 0.0152 0.0584 0.0622 0.0150 0.0596 0.0652
2.03 2.15 1.93 2.24 2.10 2.23 2.01 2.01 2.43 2.08 1.77 2.15 1.97 2.11 2.17 2.58 2.68 2.64 2.73 2.65 2.09 2.13 2.44 2.24 2.21 1.69 1.60 1.52
0.0544 0.0596 0.0616 0.0638 0.0634 0.0147 0.0602
2.29 2.36 2.09 2.27 2.15 2.18 1.79
REVERSE RUNS 36 37 38 39 40 41 42
1.097 1.092 1.118 1.102 1.113 0.549 0.585
respectively. This shows the activation energy, E, not to be a function of catalyst concentration. As is shown in Figure 3, activation energy is a function of initial reactant concentration. Values of E/R ranged from 10,610O abs. a t 1.25M to 13,690' abs. a t 5.00M. Since a small error in the rate constant would cause a large error in --In k , the fact that good straight lines were obtained in all cases indicated that the values of k, were correct, and E / R was a function of initial reactant Concentration. EquiIibrium Constant. After 20 to 25 hours, the acetic acid concentration tended to drift with an approximately constant negative slope with time. The corks in the reaction vessel were examined after 40 hours in one run and found to contain water and dioxane. This would explain the drift in end point, since, as the water was removed and held in the corks, the reaction could proceed further. This drift was about 0.0016M per hour. A slight amount of evaporation may also have taken place over a 30-hour period because the reactor was opened to take 20 to 30 samples in that period. I n the reverse runs, 36 to 42, the end point could be determined more accurately. As the reaction proceeded, the concentration of acetic acid increased to a maximum and then in some cases decreased. When the concentration began to decrease, the drift in the end point had begun. Therefore, the maximum acetic acid concentration was taken as the equilibrium value. T h e results from the reverse reactions were therefore assumed to be more accurate than those of the forward runs. Table I11 shows the calculations and results of the equilibrium constant, K. From the results of runs 36 to 40, it was surmised that temperature had at most only a slight effect on the equilibrium constant. This effect is compatible with the
fact that the maximum E / R from Figure 3 was 13,690' abs. T h e results in Table I11 were examined to determine if initial concentration had an effect on the equilibrium constant. T h e average K was 1.60 for 1.25M (runs 32 to 34), 2.08 for 2.5M (runs 1 to 16), 2.30 for 3.75M (runs 28 to 31), and 2.42 for 5.00M (runs 18 to 27). These results indicate that the equilibrium constant might be a function of initial reactant concentration. I n order to justify or refute this result, it was first assumed that the catalyst concentration had no effect on the equilibrium constant. No generalization could be drawn from the data and results, but at most it could have had only a slight effect. I t was also concluded from previous papers on esterification that the catalyst had no effect on the equilibrium constant. Since it has been stated that temperature did not affect K to any degree, the only variable that could have caused a change in K with initial concentration was the initial concentration itself. If initial concentration had no effect on K , then an error in titration or a faulty end point, or both, might have caused the differences in the average values of K . R u n 33 gave an equilibrium constant which was the average for the 1.25M group. Errors of 0.005M in titration and 0.02M in end point were assumed to be maximum. Using these errors the equilibrium constant for run 33 was calculated to be 2.08. This value is as great as the average value of the 2.5M group but about 0.35 less than the average of the 5M group. R u n 12 gave an equilibrium constant of 2.08, the average value of the 2.5M group. The equilibrium constant for run 12 using the previous maximum errors gave K equal to 2.38. This value is nearly as large as the average value of the 5.00M group. The average K of all the runs in Table I11 is 2.16. Runs 2 and 1 5 have K equal to 2.15. Assuming the previous errors were made in these runs, the value of K in each run became 2.46, which is greater than the average a t 5.00M, but still not as large as several runs a t 5.00M. I t was concluded that the scattering in each initial concentration group could be explained by errors, probably in the determination of the correct end point, but that the four groups were too different to be entirely explained by errors. An all-glass reaction system in which no drift was possible would have given more exact data. Reaction Rate Constant. The results of Figures 1 and 2 indicated that the reaction was first order with respect to the cyclohexanol and second order with respect to the acetic acid. A graph of 1/CaT2us. time was drawn for each run, and the slope was calculated to be 2 k , the forward rate constant. Rate constants for 35 combinations of variables were determined. Nine duplicate runs are also given. T h e average deviation ranged from 0.0 to 3.3y0 with the median at 0.9%. Therefore, a good degree of precision was obtained with the stirred batch reactor and the procedure that was used. The drift in acetic acid concentration discussed previously did not affect the rate constant results, because most of the runs reached 80% reaction in less than 4 hours, the maximum being only 10 hours for run 32. An increase in either temperature or catalyst concentration caused an increase in the rate constant. However, an increase in the initial reactant concentration caused a decrease in the rate constant. The rate constant was determined from Figures 5 and 6 to be a linear function of both catalyst and initial reactant concentration. Equation 1 5 related the rate constant to the three variables. Rate constants calculated from Equation 1 5 , experimental rate constants, and the per cent deviations are given in Table 11. The per cent deviations ranged from 0.0 to 12.9y0,with a 3.8% mean deviation and a 4.8% average deviation. VOL.
6
NO. 3
JULY 1 9 6 7
291
i
'Ob6
RUN
T°F
0.400
1
104
36 37 38 39 40
122 140 158
1 1
176
0.300
0.200 0.100
o.ooo~,o
I
I
I
I
10.0
20.0
30.0
40.0
I
I
50.0 55.0
TIME (HRI
Figure 7. Acetic acid concentration vs. time for a series of temperatures at 2.5M and 0.75 ml. of catalyst per 150 ml. Reverse reaction
0.800
Reverse Reaction. I t was desired to calculate the rate constant for the reverse reaction. However, an ester and water concentration of 2.5M or greater was heterogeneous after the catalyst was added. Figure 7 is a graph of Cur us. time for runs 36 to 40, each at an initial reactant concentration of 2.5M. The only variable in these runs was temperature. The first point on each graph was taken as close as possible to the time when the reaction mixture became homogeneous. Reverse rate constants could not be calculated for these runs, but the runs were useful in determining the effect of temperature on the equilibrium constant. Solutions at an ester-water concentration of 1.25M with either 0.25 or 0.75 ml. of catalyst per 150 ml. of solution were homogeneous throughout the reaction. These were the only two combinations of variables covered in this study that gave homogeneous solutions at all times. I t was determined from Figure 8, a plot off(&) us. time, where Cer is the ester concentration, that the reverse reaction was third order. Rate constants were calculated by the graphical method discussed earlier and are tabulated in Table 11. Reactions in Dimethyl Sulfoxide. The discussion thus far has pertained to reactions carried out in 1,4-dioxane. Esterification reaction rates should increase as the polarity of the solvent is lowered and so dioxane was chosen because of its low polarity ( p = 0.3 Debye above 30' C.) (3) and low dielectric constant [ E = 2.252 - 0.00717'(' C.)] (79). T o examine the effect of solvent polarity on the rate constant three runs were carried out in dimethyl sulfoxide with a dipole moment of 3.96 D (5) and a dielectric constant of [e = 48.5 - 0.075T (' C.)] (4). Runs were also made in dioxane with the same initial reactant concentration, catalyst concentration, and temperature (Table IV). The esterification in dimethyl sulfoxide was third order over-all. Therefore, the units on the rate constant in both solvents are (liter)2 (moles) -* (hr.) -l. At 122' F. (50' C.), the highest temperature for which the dielectric constant of dioxane is applicable, the ratio eDMBO/ ~ D I O X is 20.65. The ratio, P D J T ~ O / ~ D I Ois~ 13.2. Although these two ratios are within 25% of the ratio of the rate constants, there exists no theoretical justification for this.
1.100 /
0
I . 500
42
Conclusions 1.300
0.900 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 TIME (HR)
Figure 8. and time
Relation between initial reactant concentration Reverse reactions
Table IV.
The esterification reaction was first order with respect to cyclohexanol and second order with respect to acetic acid. The reverse reaction was third order. Esterification reaction rate constants can be predicted within 5y0 by an empirical equation. The rate constant was a function of initial reactant concentration, catalyst concentration, and temperature. The activation energy was not a function of catalyst concentration, but was a function of initial reactant concentration. The equilibrium constant was at most weakly influenced by temperature and not influenced by catalyst concentration. I t could not be determined from the experimental data if the equilibrium constant was influenced by initial reactant concentration.
Reactions of 2.5M Cyclohexanol and Acetic Acid in Dimethyl Sulfoxide Acknowledgment
140 158 176
292
0.75 1 .oo 1.oo
0.0710 0.1555 0.2475
0.0041 0.0090
0,0144
17.30 17.29 17.15
IBEC PROCESS DESIGN A N D DEVELOPMEN1
The authors express their appreciation for financial aid received from a National Aeronautics and Space Administration graduate fellowship (D.J.M.) granted to the Colorado School of Mines.
literature Cited ( 1 ) Bartlett, P. D., Dauben, H. J., J . A m . Chem. SOG.62, 1344 ( 1940). (2) Bennett, G. M., J . Chem. SOG.(London) 107, 357-8 (1915). ( 3 ) Bogomolov, N. A., Stepanenko, N. N., Z h . Fir. Khim. 26, 1664 (1952). ( 4 j Doucet, Y., Calmes-Perrault, F., Durand, M., Compt. Rend. 260(7), 1878-81 (1965). ( 5 ) Dreizler, H., Dendl, G., Z . Naturforsch. 19a, 512-14 (1964). ( 6 ) Goldschmidt, H., Udby, O., Z. Physik. Chem. 60,728 (1907). ( 7 ) Hartman, R. J., Borders, A. M., J . A m . Chem. Soc. 59, 2107-12 11937). ( 8 ) Hartman, R. J., Gassmann, A. G., Ibid., 62, 1559-60 (1940). ( 9 ) Hartman, R. J., Hoogsteen, H. M., Moede, J. A., Zbid., 66,1714-18 (1944). ( 10) Levanspiel, Octave, “Chemical Reaction Engineering,” p. 23, Wiley, New York, 1962. \ - - - - , -
\ - - -
I -
( 1 1 ) Zbid., p. 51. (12) Leyes, C. E., Othmer, D. F., Znd. Eng. Chem. 37, 968-77 (1945). (13) Rolfe, A. C., Hinshelwood, C. N., Trans. Faraday Soc. 30, 935-44 (1934). (14) Smith, H. A.,J . A m . Chem.Soc. 61,254-60 (1939). ( 1 5 ) Zbid., 62, 1136-40 (1940). ( 1 6 ) Smith, H. A., Burn, James, Zbid., 66, 1494-97 (1944). (17) Smith, H. A., Levenson, H. S., Zbid., 62, 2733-5 (1940). ( 1 8 ) Suter, C. M., Oberg, Elmer, Zbid., 56, 677-9 (1934). 119) Weast. R. C.. ed.. “Handbook of Chemistrv and Phvsics.” ’
46th ed.,’ p. E-50, Chemical Rubber Co., Cieveland, ’Ohib,
1965. (20) Williamson, A. T., Hinshelwood, C. N., Trans. Faraday Soc. 30,1145-9 (1934).
RECEIVED for review May 23, 1966 ACCEPTEDFebruary 4, 1967
MOMENTUM TRANSFER STUDIES IN EJECTORS Correlationsfor Single-Phase and Two-Phase @stems G.
s.
D A V I E S , ~A . K . M I T R A , A N D A . N . R O Y
Department of Chemical Engineering, Indian Institute of Technology, Kharagpur, India The performance of an ejector in single-phase (air-air) and two-phase (air-liquid) systems has been studied with air as the motive fluid, and air and various liquids as the entrained fluid. Data for air-air system have been analyzed, using energy and momentum equations; the values of mass entrained calculated from the theoretical expression agreed with experimental results only in the limited range of small area ratios. Correlations based on the method of dimensional analysis have been developed for both air-air and air-liquid systems relating the mass ratio of entrained fluid to motive fluid in terms of Reynolds number of motive fluid, geometry of the ejector, and the physical properties of the fluid system. HE mechanism of jet flow has largely been exploited in Tejectors or jet pumps in which the momentum and kinetic energy of a high velocity fluid stream are used to entrain and pump a second fluid stream. Ejectors with steam o r compressed air as the motive fluid have found application in industrial operations for the creation of vacuum, exhausting corrosive fumes, pneumatic conveyor feeding, etc. Its incorporation especially in slurry-type chemical reactors holds considerable promise. By this technique the kinetic energy of the reactant motive gas can be utilized to maintain the solid catalyst particles in suspension, cause intense mixing between gas and fluid, and circulate the catalyst slurry through an external side tube, thereby obviating the difficulties normally encountered in conventional mechanically stirred reactors. Satisfactory operation of such a type of reactor employing an ejector as a pump for hydrocarbon synthesis has been reported ( 9 ) . The present investigation on the momentum transfer in ejectors has been carried out to obtain necessary data and correlation, with the ultimate object of designing and incorporating ejectors in slurry-type chemical reactors and allied process operations. In a multiphase system consisting of gas, liquid, and solid a large number of variables are involved which usually give rise
Present address, Indian Institute of Technology, Madras, India.
to problems of great complexity. Apart from the importance of fluid physical properties and solid particle characteristics, there are interrelated problems such as solubility, holdup, and slip. Hence, it is considered logical to carry out this investigation on momentum transfer in stages-first, using a singlephase system, where the motive and entrained fluids are gases, then a two-phase system where the motive fluid is gas and the entrained fluid is liquid, and finally a three-phase system where the motive fluid is gas and the entrained fluid is solid-liquid slurry. Studies carried out in single-phase systems with air as the motive and entrained fluid in the motive pressure range of 25 to 100 p.s.i.g. have been reported (8). This paper presents studies carried out in a single-phase (air-air) system in the low motive pressure range and in twophase (air-liquid) systems. Single-phase (Air-Air) System
The many analyses (7,4-7, 76, 78) that have been attempted in the design of ejectors, notably those of Keenan, Neumann, and Lustwerk (5, 6), Kastner and Spooner ( 4 ) , Smith (76), and Van der Lingen (78), have all been made using the equations of continuity, momentum, energy, and state. Most of them deal with constant area and constant pressure mixing. Nearly all the experimental investigations reported in the literature on the performance of ejectors relate to relatively high motive pressure, the only exception being the work of Kastner and Spooner. VOL. 6
NO. 3 J U L Y 1 9 6 7
293
