Ind. Eng. Chem. Res. 1989, 28, 844-850
844
R ;i is the number of 13C atoms in the molecule t = index xi = Components Of x;i represents the ~ t d xofr deuterium atoms in the molecule ri = components of
Functions L ( i , j ) = the larger of i or j S(i,j ) = the smaller of i or j Registry No. D,, 7782-39-0;benzene-d,, 1120-89-4;benzene-d,, 25323-71-1; benzene-4, 25321-35-1; 25321-36-2; benzene-d5,13657-09-5;benzene-d6,1076-43-3;hexane, 110-54-3; hexane-d,, 98821-94-4;hexane-d,, 98821-93-3;hexane-da, 9882192-2; hexane-d,, 120295-95-6;hexane-d,, 120311-04-8;hexane-d6, 120295-96-7: hexane-d,. 120295-97-8:hexane-dn. 120295-98-9: hexane-d9, 120295-99-01 hexane-d,,, 120296-0016; hexane-d,,; 120296-01-7; hexane-d,,, 120296-02-8;hexane-d,,, 120296-03-9; '
hexane-d,,, 120296-04-0.
Literature Cited Dibeler, V. H.; Mohler, F. L.; de Hemptinne, M. Mass Spectra of the Deuteroethylenes. J . Res. Natl. Bur. Stand. 1954, 53(2), 107. Koski, W. S.; Kaufman, J. J.; Friedman, L.; Irsa, A. P. Mass Spectrometric Study of the B,D8-B5H9 Exchange Reaction. J. Chem. Phys. 1956, 24(2), 221-225. Lenz, D. H.; Conner, Wm. C. Computer Analysis of the Cracking Patterns of Deuterated Hydrocarbons. Anal. Chim. Acta 1985, 173, 227-238. Ozaki, A. Isotopic Studies of Heterogeneous Catalysis; Kodonsha Ltd./Academic Press: New York, 1977. Schissler, D,0.; Thompson, S. 0.; Turkevich, J. Behaviour of paraffin Hydrocarbons on Electron Impact: Synthesis and Mass Spectra of some Deuterated Paraffin Hydrocarbons. Discuss. Faraday SOC. 1951, 10, 46.
Received for review March 8, 1988 Revised manuscript received November 28, 1988 Accepted January 24, 1989
Supersaturation and Crystallization Kinetics of Potassium Chloride Ru-Ying Qian,* Xian-Shen Fang, and Zhi-Keng Wang Shanghai Research Institute o f Chemical Industry, Shanghai 200062, People's Republic
of
China
The crystallization kinetics of potassium chloride from its aqueous solution is studied with a novel temperature float method for supersaturation determination. The relative supersaturation is with a precision of 1 X The growth rate is proportional t o the measured to be (1.1-2.9) X supersaturation over a wide range of retention times and tip speeds, while the nucleation rate increases rapidly with respect to supersaturation. The supersaturation, growth rate, and nucleation rate are independent of tip speed up to 3 m/s. Beyond 3 m/s, the power index of the tip speed in the kinetic equation increases to 4. A change in the dominant nucleation mechanism is discussed. A slight impeller vibration does not affect the power indexes in the kinetic equation but considerably increases the nucleation rate. A temperature difference of 4-6 K between cooling water and bulk suspension leads to extraordinarily high nucleation rates but does not yet affect growth rates. The crystallization kinetics of potassium chloride has been studied by many investigators (Genck and Larson, 1972; Ploss et al., 1985; Randolph et al., 1981;Qian et al., 1987). However, the width of the metastable zone for this system is so narrow that the supersaturation in crystallizers has not yet been precisely determined. Currently, the supersaturation in crystallizers can be measured accurately only for systems having wide metastable zones. For the potash alum-water system, with the maximum allowable supercooling of 4 K (Mullin, 1972), the supersaturation data are still not precise enough to be correlated with crystallization kinetic data (Garside and JanEiE, 1979). The conventional kinetic equation expresses nucleation rate as a power function of growth rate and operation factors, such as suspension density, agitation speed, and impeller diameter. This empirical equation does not directly include, however, the driving force in the crystallization process, i.e., supersaturation, and cannot be used to correlate kinetic data of different systems. Without supersaturation data, it is impossible to compare the kinetic data of various investigators. The development of crystallization as a unit operation has long been hindered due to the lack of supersaturation data in crystallizers. In this paper, supersaturation of the potassium chloride-water system in a crystallizer was determined precisely by a new method. The corresponding nucleation rate and growth rate were measured simultaneously and were *Present address: 11 Cameron Rd, Wayland, MA 01778.
well correlated with supersaturation. The crystallizer was operated as an MSMPR crystallizer in a wide range of retention times and tip speeds. Excessive nuclei and incrustation were observed at short retention times due to the low cooling water temperature. The effect of impeller vibration on the kinetic equation was studied. The dominant nucleation mechanisms under different operation conditions were discussed. The crystallization kinetics of the KC1-water system was compared with the KC1-brine system.
Apparatus and Procedure In this work, a 2.5-dm3glass crystallizer with a streamlined bottom and a hollow stainless steel draft tube is used. The details of the crystallizer have been reported in our previous work (Qian et al., 1987). The flow system used in that work has been further improved in this study. The system consists of a crystallizer, reservoirs, a saturator, a metering pump, preheaters, a flowmeter, a filter, a sampling bottle, and thermostats. The temperature control of the system has been improved to meet the requirement of precise supersaturation control and measurement, being achieved by using special fine-temperature regulators in thermostats to minimize their temperature fluctuation within K. The crystallization temperature is maintained within jzO.01 K by fine-temperature control of three thermostats, which are used for adjusting the temperatures of the cooling water in the draft tube, the water bath of the crystallizer, and the feed solution, re-
0888-588518912628-0844$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 845 spectively. Careful regulation of feed flow, withdrawal flow, and level in the crystallizer is also very important. The nucleation rate and growth rate are calculated from the crystal size distribution in the crystallizer a t steady state by population balance (Randolph and Larson, 1988). The correlation coefficient of linear regression of In n versus L is 0.97 or better, usually 0.99. Only when the retention time is very long, i.e., 4286 s, a t a tip speed of 2.58 m/s does the classification of coarse crystals occur in the crystallizer; then the correlation coefficient is 0.95 or less. Photographs of crystal samples on every sieve are taken under the microscope for each run. The potassium chloride crystals are regular cubes except for coarse crystals at a high agitation speed.
Supersaturation Measurement A modified temperature float method is proposed in this study for the precise measurement of supersaturation in crystallizers. The float method has been well established for deuterium analysis in natural water and heavy water (Kirshenbaum, 1951), but it is rarely considered as a precise method for the determination of the density of other liquids. The flotation temperature is highly sensitive to the liquid density, and therefore, the difference in flotation temperatures between two liquids can be used to determine precisely the slight difference in their densities. It is proven that the flotation temperature difference between the supersaturated and corresponding saturated solutions can be correlated very well with the supersaturation of the solution. The sensitivity of this method is fO.OOO1 kg of KC1/100 kg of water. In the deuterium analysis, the purification of the water sample by distillation eliminates almost all impurities and thus ensures the high precision of the float method. This purification method, however, is not applicable to supersaturation measurement. The interference of soluble impurities in supersaturation has to be minimized by keeping the content of impurities in the saturated solution as close as possible to that in the supersaturated solution. The precision of this new method is f0.004 kg of KC1/100 kg of water, which is still much better than that of previous methods and is accurate enough for the correlation of kinetic data. The details of this method will be published elsewhere (Wang et al., 1989). The sampling technique is another important procedure for determining supersaturation in crystallizers. Fine crystals suspended in the sample or a slight fluctuation of impurities in the solutions will lead to serious error in supersaturation. Separation of fine crystals from the sample solution by filtration with filter paper, cotton, or sintered glass has been tested, but the results are disappointing. Satisfactory separation is achieved by using a settling tube and a sampling pipet. The settling tube is 12 mm i.d. and 280 mm long, having a baffle hat at its lower end and a rubber stopper with a vent tube a t its upper end. The sample flow is controlled so that the sample is introduced very slowly into the settling tube from its lower end and the supersaturated solution sample is then taken with the pipet from its upper layer after a settling time of 20 min. The settling tube is preheated to slightly above the crystallization temperature before it is inserted into the crystallizer for sampling. In the period of sampling the supersaturated solution, all of the solution withdrawn from the crystallizer is collected in a bottle and is equilibrated with recrystallized coarse potassium chloride crystals under stirring in a thermostat at the same crystallization temperature. Then, after 20 min of settling, the saturated solution sample is
taken from the upper layer with a pipet.
Experimental Results Two series of experiments were made to study the crystallization kinetics at different retention times and agitation speeds, respectively. In the first series, supersaturation, growth rate, and nucleation rate were determined with a retention time ranging from 16 to 59 min at an agitation speed of 820 rpm. In the second series, these data were measured at an agitation speed range of 600-1276 rpm a t a retention time of 25 min. Another series of experiments was designed to study the vibration effect of the impeller and carried out in the same crystallizer but with a worn agitator bearing. The crystallization kinetics was studied in the ranges 25-62.5 min and 820-1140 rpm. For all runs, the crystallization temperature was 303.11 f 0.01 K, the temperature of the saturator was 313.2 f 0.1 K, and the suspension density was 23 f 3 kg/m3. (1) Effect of Retention Time. In this series, experiments were conducted at five retention times, i.e., 952, 1485,1500,2372, and 3529 s, but a t a constant tip speed of 2.58 m/s. In the crystallizer, the annular space between the draft tube and the vessel is divided by four baffle plates into four equal sections, and the introduction of samples into the sampling device is a t the center of a section. The supersaturation a t this sampling point is taken to be representative of the solution bulk in the crystallizer. The relative supersaturation, s, can be well correlated with the retention time, T, by linear regression of In s versus In T with a correlation coefficient of 0.962; Le., s = 0.431~+.~~
Similarly, the growth rate, G, and nucleation rate, Bo, are also correlated with the retention time, T, by
G = 2.08
X
10-5~4.74
(2)
with a correlation coefficient of 0.972 and by
B o = 4.30
X
1011~-1.89
(3)
with a correlation coefficient of 0.979. From the above equations, supersaturation, growth rate, and, especially, nucleation rate decrease when the retention time increases. The growth rate, G, can also be expressed as a function of relative supersaturation, s, by direct regression of In G versus In s, as shown by the solid line in Figure 1and by the following equation: G = 2.45 x 10-5so.92
(4)
with a correlation coefficient of 0.899. The power index of 0.92 in eq 4 agrees well with the corresponding value of 1.03 obtained from substituting eq 1in eq 2. So the growth rate of potassium chloride in the crystallizer is proportional to supersaturation. This relationship has also been found by Botsaris et al. (1966) and Klein Haneveld (1971) in the study on. growth rate of a potassium chloride crystal in the low-supersaturation range. Thus, the linear relationship between growth rate and supersaturation in continuous crystallization can be used for the calculation of potassium chloride crystallizers. The direct regression of In B o versus In s gives the following equation and is shown by the solid line in Figure 2 BO
= 1.65 x 1 0 ~ 4 9
with a correlation coefficient of 0.962.
(5)
.
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
846
-15 6
-16 0
TIPS=2 58m: s
0
r = 952 - 3529s
TIPS= 1.89mIs
0 TIPS= 3.58m/ s r = 1474-1531s
i
U
-16.41
-
-16.6
r = 9 5 2 - 3529s -E~q(6)
1 -16.8
0
!-
I
-168
-16 4
InG
I
I
I
-16 0 (m: s)
-
6
Figure 3. Relationship between nucleation rate and growth rate.
In 5
Figure 1. Relationship between growth rate and relative supersaturation. I
I
15
I
..--Eq(15)
a
I
of 2.0 published by Ploss et al. (1985). From the population balance in a continuous MSMPR crystallizer, the following equation holds (Randolph and Larson, 1988):
/
TIPS=2,58m/ s
MT = 6pclz,n0(Gr)4
4'I
0 r=952-3529~
(7)
Combining eq 7 with the kinetic equation
B o = KIG'
(8)
B o = noG
(9)
and the equation
2
v
i
131
Ic
we have
and
-6
a
-6 4
-60
-5 6
Ins
Figure 2. Relationship between nucleation rate and relative supersaturation.
The traditional kinetic equation may be given by the direct correlation of the nucleation rate, B o ,with growth rate, G , as follows:
B o = 6.23 X 1OZ1G2.30 (6) with a correlation coefficient of 0.906. The plot of In B o versus In G is shown by the solid line in Figure 3. The power index, i, of 2.30 in eq 6 agrees with the value of i = 2.55 published by Genck and Larson (1972) and that
Thus, the theoretical value of the power index of the retention time in eq 2 at constant suspension density, MT, should be -4/(i + 3). Substituting i for 2.30 in eq 6, we have a theoretical power index value of -0.75 from eq 10, which agrees well with the experimental value of -0.74 from eq 2. Likewise, the theoretical value of the power index of the retention time in eq 11 is -1.74 with i = 2.30 and agrees well with the experimental value of -1.89 in eq 3. From the above agreements, we conclude that our crystallizer was well operated as an MSMF R crystallizer in the retention time range 3529-952 s. (2) Effect of Tip Speed of the Impeller. In this series experiments were made at five tip speeds, i.e., 1.89, 2.20, 2.58, 3.58, and 4.01 m/s but at a constant retention time of 1500 f 30 s. It has been proven in our previous work (Qian et al., 1987) that the scale-up criterion of secondary nucleation is constant tip speed (TIPS) and the charac-
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 847
r = 1474-1531s
t I
06
08
10
12
14
(mi s)
In(TIPS)
Figure 4. Effect of tip speed on kinetics.
teristic group for hydrodynamics in crystallizers is MT(TIPS)3or the dimensionless group x. So it is more reasonable to correlate supersaturation (s), growth rate (G ), and nucleation rate ( B O ) with tip speed (TIPS) or MT(TIPS)3rather than with agitation speed. The correlations with tip speed are expressed by the following equations in the range 1.89-3.58 m/s: s = 2.71 X 10-3(TIPS)-0.047
(12)
G = 1.01 x 10-7(TIPS)-0.035
(13)
B o = 3.78 X 105(TIPS)0.17
(14)
Equations 12-14 show clearly that the effect of tip speed in this range is within the limit of experimental error and can be neglected. The relationship of In B o versis In G of these runs agrees very well with eq 6 for different retention times. The regression equation of all eight runs is
BO = 7.74 X
1021G2.31
(15)
with a correlation coefficient of 0.911 and is shown by the dotted line in Figure 3. But at a tip speed of 4.01 m/s, Le., at an agitation speed of 1276 rpm, the supersaturation (s) and growth rate (G ) decrease; however, the nucleation rate ( B") increases. The effect of agitation on crystallization kinetics of potassium chloride can be summarized more clearly by a In (Bo/Gi) versus In TIPS plot, as Figure 4. In the first hydrodynamic region, where In TIPS is less than 1.22, the value of In ( B"/G 2.30) is independent of In TIPS as indicated above. In the second region, where In TIPS is greater than 1.22, In ( B"/G 2.30) increases sharply with In TIPS and the relationship can be expressed by Bo/G2.30= 4.57 X 1019(TIPS)4
(16)
The power index of 4 in eq 16 agrees well with the results of the third series of experiments, as described in eq 18 below. The broken line in Figure 4 suggests a transition of dominant nucleation mechanism, which we will discuss later. (3) Effect of Vibration of the Impeller. In this work, the agitator was supported by a Teflon bearing. Its advantage is no contamination of the lubricating oil in the crystallizer; however, it may wear out after a long period of operation of the agitator. At first, the slight wear of bearing caused a microvibration of the impeller and a little eccentricity between the impeller and draft tube. An ap-
-17.4
-17.2
-17.0
-16.8
-16.6
-16.4
InG (m/s) Figure 5. Relationship between nucleation rate and growth rate after a long period of operation.
preciable wobbling of the impeller was observed when the eccentricity due to bearing wear was measured to be 1mm. A t that time, the nucleation rate increased by a factor of 9.2, while the growth rate decreased to 45% of its normal value. Photographs showed that most of large crystals were cracked rather than rounded. In fact, even though the vibration of the impeller cannot be sensed manually, changes in nucleation rate and growth rate were measurable, as indicated in the following series of experiments. In this third series, two sets of experiments were designed to study the power indexes of growth rate and tip speed in the kinetic equation, respectively. The comparison of this kinetic equation with that from the two previous series of experiments shows the effect of vibration on the crystallization kinetics. In the first set, 15 runs were performed at a constant tip speed of 2.58 m/s and at retention times of 1500,1764, 2502, and 3750 s. The relationship between nucleation rate ( B") and growth rate (G ) is shown in Figure 5 and can be expressed well by the regressive equation
B o = 8.43 X 1020G2.09
(17)
with a correlation coefficient of 0.926. In the second set, 11runs were conducted at a constant retention time of 1500 s and at tip speeds of 2.58,3.08, and 3.58 m/s. The kinetic equation is obtained by linear regression, as in Figure 6, and is expressed by
Bo/G2.O9= 1.60 X 1019(TIPS)4.08
(18)
with a correlation coefficient of 0.947. The agreement between the power indexes of 2.09 from eq 17 and 2.30 from eq 6 indicates that the operation condition of the agitator does not affect the index of growth rate in the kinetic equation. In our previous work (Qian et al., 1987), we have proven that this power index is independent of hydrodynamic conditions. This study provides another proof on this independence.
848
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
7
I
09
I 10
In(TI P S)
I
I
11
= 1500s
I
I 12
I
L
J
13
(m s )
Figure 6. Effect of tip speed on kinetics after a long period of operation.
The power index of 4.08 for the effect of tip speed in eq 18 agrees well with that of the value of 4 in eq 16 or the slope in the second region in Figure 4. This means that the power index of the tip speed is also independent of the slight vibration of the agitator. The slight vibration due to the wear of the bearing, like the increase of tip speed, will stimulate the secondary nucleation and at the same time reduce the supersaturation and the growth rate. For example, at a retention time of 1500 s and a tip speed of 2.58 m/s, the growth rate decreases from 9.8 X m/s in the two previous series to 7.3 x m/s in this series, and the nucleation rate increases from 3.9 X lo5 no./(m3.s) in the former two to 8.9 X lo5 no./(m3.s) in the third. It means that the coefficient in the kinetic equation, K1 in eq 8 or K in eq 20), increases when the bearing wears out. In this series of experiments, the eccentricity due to the bearing wear is estimated to be less than 1/2 mm. Moreover, during reinstallation of the crystallizer,a specification of an 1/3-mmeccentricity at the clearance of 4 mm between impeller and draft tube has been proven to have good reproducible results in the kinetic data. So the eccentricity does not seem to play an important role in this series of experiments. On the other hand, the vibration of impeller appears to play a more significant role on nucleation. The crystals are cracked by the additional impact energy due to the vibration of the impeller.
Discussion (1) Inhomogeneity of Supersaturation at Short Retention Time. In a cooling crystallizer, the local temperature of suspension in the boundary layer near the cooling surface is lower than that of the bulk suspension. The temperature of the cooling water was controlled at 300.1 K or higher in all of the above-mentioned experiments to minimize the corresponding local supersaturation (slot) near the cooling surface.
At short retention times of 759 and 723 s, the temperature differences between the cooling water and the bulk of suspension at a tip speed of 2.58 m/s were as great as 4.0-4.3 and 5.5-6.0 K, respectively. The supercooling of the suspension near the cooling surface was calculated to be much greater than the maximum allowable supercooling of 1.1 K. The extraordinary high nucleation rates were measured and shown by the dotted line in Figure 2. The nucleation mechanism in these cases should be primary nucleation. An evidence on this local inhomogeneity in supersaturation was the incrustation on cooling surface in these experiments. The supersaturation in bulk solution (s) in these experiments was measured to be 0.00284.0031, independent of 7 in the range 952-723 s. But for MSMPR crystallizer, the supersaturation should increase with the decrease of retention time. This indicated an inhomogeneity of supersaturation in the crystallizer. The high nucleation rates at 759 and 723 s could not be produced by the low bulk supersaturation. This provided more evidence of the existence of the high local supersaturation, sloe which attributed to the extraordinary high nucleation rates. The bulk of the suspension was still homogeneous in these cases. This can be proven by the crystal size distribution and the relationship between the growth rate and supersaturation in the bulk solution. In these two experiments, sieve analysis of seven crystal samples was made. The correlation coefficients of the linear regression on In n versus L were 0.993-0.998. It indicated that the bulk suspension was well mixed. The growth rate ( G ) and bulk supersaturation (s) in the whole range of retention times, 3529-723 s, can be well correlated as follows:
G = 2.32 x 10-5so.91
(19)
with a correlation coefficient of 0.878. This equation agrees well with the corresponding equation in the range 3529-952 s, eq 4,as shown in Figure 1. It implies that in the short retention times of 759 and 723 s, as in the long retention times, the bulk of the suspension is well mixed. In summary, our experimental results indicate that at short retention time there is local inhomogeneity of supersaturation near the cooling surface but supersaturation in bulk still remains homogeneous. (2) Comparison of the KC1-Water System with the KC1-Brine System. Under the same operation conditions, the growth rate for the KC1-water system is greater than that for the KC1-brine system and the nucleation rate for the KC1-water system is much less. For example, a t a retention time of 1500 s, a tip speed of 2.58 m/s, a suspension density of 23 kg/m3, and a crystallization temperature of 303.11 K, the growth rate and nucleation rate are 9.8 X m/s and 3.9 X lo5 no./(m3.s), respectively, for the KC1-water system and are estimated to be 5.5 X m/s and 1.4 X lo6 no./(m3-s),respectively, for theKC1-brine system (Qian et al., 1987). The generalized kinetic equation for the crystallization of potassium chloride from its aqueous solution can be expressed as follows from Figure 4 and eq 16:
B o / G L= K(MT(TIPS)3)J
(20)
where i is 2.30 at 303.11 K. When In (MT(TIPS)3)is less than 6.8 kg/s3, j is zero. Therefore, Bo/G2.30= 6.02
X
loz1
(2Oa)
When In ( ? K ~ ~ ( T I PisSgreater ) ~ ) than 6.8 kg/s3, j is 4/3. Therefore, B 0 / G 2 . 3 0= 7.15 X 1 0 " ( k f ~ ( T I p S ) ~ ) " ~(20b) ~
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 849 Similarly, the kinetic equation under a slight impeller vibration, eq 18, can also be expressed by
B o / G 2.09 = 2.20 X 10'7(hf~(TIpS)3)''36 (20c) for the In (MT(TIPS)~) range 6.0-7.0 kg/s3. The general kinetic equation for the KC1-brine system a t 313.2 K is as follows (Qian et al., 1987): When In (MT(TIPS)3)< 7.1 kg/s3,
B "/G
2.78
= 1.70 X 1024(MT(TIpS)3)0.70 @la)
When 9 > In (MT(TIPS)3)> 7.1 kg/s3,
B "/G
2.78
= 5.24
X
1022(MT(TIpS)3)'.20 (21b)
The power indexes j , in eq 20b, 20c, and 21b agree well. It means that the dominant nucleation mechanisms are the same for both systems in the high MT(TIPS)3range. The shape of the crystals in the crystallizer often reveals a clue to the nucleation mechanism. The shape of the large crystals for the KC1-water system is more rounded at higher tip speeds, when it is greater than 3 m/s. For example, at 3.58 m/s, crystals of 1062-1146 pm are slightly rounded and those larger than 1146 pm are rounded, while at 4.01 m/s some crystals of 639-722 pm and all crystals greater than 722 pm are rounded. Similar results were reported for the KC1-brine system by Qian et al. (1987). Thus, the greater nucleation rate a t higher tip speeds can be well correlated with more rounded crystals, i.e., higher fracture or more macroattrition of crystals. This provides evidence that the secondary nucleation by macroattrition is the dominant mechanism at high tip speeds for both systems. (3) Nucleation Mechanism at Low Tip Speeds. For the KC1-Water system, when the tip speed decreased to lower than 3 m/s at a retention time of 1500 s, the shapes of the large crystals were no longer rounded as they were a t high tip speeds, but instead were almost regular cubic, even for sizes larger than 1146 pm. In addition, the nucleation rate became independent of the tip speed as shown by eq 20a and in Figure 4. These implied a change in dominant nucleation mechanism from the macroattrition of crystals a t high tip speeds. In the following discussion, we postulate the dominant nucleation mechanism at low tip speeds with evidence on the secondary nucleation by microattrition of crystals, probably controlled by surface regeneration. As presented earlier, the crystallizer at low tip speeds was operated as an MSMPR crystallizer at retention times of 3529-952 s with relative supersaturation determined to be (1.1-2.9) X The latter value is only one-third of the limit of the metastable zone. So the nucleation mechanism should be secondary nucleation. Moreover, the regular cubic shape of the crystals a t low tip speeds indicates that the damage of the crystals is very slight under the low contact energy. This implies that in this case the mechanism should be secondary nucleation by microattrition of the crystals. Furthermore, we suggest that the nucleation rate by microattrition is controlled by surface regeneration. In our crystallizer, the turnover times of crystals at low tip speeds of 1.89 and 3 m/s were still as short as 3.6 and 2.2 s, respectively. These contact intervals of the crystals may be shorter than the surface regeneration time. In a related study on the contact nucleation of MgS0,.7H20 by Khambaty and Larson (1978), the surface regeneration time is more than 8 s. In addition, our experimental results a t low tip speeds from eq 14 proved that the nucleation rate was independent of tip speed, further enhancing the assumption that this secondary nucleation was controlled
by the surface regeneration. However, further investigation needs to be performed for the direct evidence on the dominant nucleation mechanism a t low tip speeds. Conclusions Supersaturation in a continuous MSMPR crystallizer for the KC1-water system is determined by the temperature float method with a precision of 0.004 kg of KC1/100 kg of H20,which is much better than by previous methods. Both the growth rate and nucleation rate are correlated well with supersaturation in the bulk solution over a wide range of retention times and tip speeds. The growth rate is proportional to supersaturation, while the nucleation rate increases much more rapidly with the increase of supersaturation. In the kinetic equation at 303.11 K, the power index of growth rate is 2.3. The power index of tip speed is 0 up to tip speeds of 3 m/s and is 4.0 beyond 3 m/s. A change in the dominant nucleation mechanism is proposed. A slight impeller vibration due to bearing wear decreases the growth rate and sharply increases the nucleation rate, but it does not affect both power indexes of the kinetic equation. Acknowledgment The authors thank He-Gen Ni, Han-Ming Shi, and other members of the research group on crystallization a t Shanghai Research Institute of Chemical Industry for their help in this work. Nomenclature B o = nucleation rate, no./(m3.s) G = growth rate, m/s g = acceleration of gravity, m/s2 i = relative kinetic order in eq 8, power index of growth rate in the kinetic equation j = power index of suspension density in the kinetic equation K = coefficient in eq 20, n ~ . / ( m ~ . s ) ( m / s ) ~ ( k g / m ~ ) j ( m / s ) ~ j K1 = coefficient in eq 8, no./(m3-s)(m/s)' k , = volume shape factor of crystals L = crystal size, m MT = suspension density, kg/m3 n = population density, n0./m4 no = population density of nuclei, no./m4 s = relative supersaturation in the bulk solution slw = local relative supersaturation near the cooling surface TIPS = tip speed of the impeller, m/s Greek L e t t e r s Y
= kinetic viscosity of the solution, m2/s
pc = T
density of crystals, kg/m3
= retention time, s
x = dimensionless group, (MT(TIPS)3)/pc~g Registry No. KC1, 7447-40-7.
Literature Cited Botsaris, G. D.; Mason, E. A.; Reid, R. C. Growth of Potassium Chloride Crystals from Aqueous Solutions I. The Effect of Lead Chloride. J. Chem. Phys. 1966,45, 1893-1899. Garside, J.; JanEiE, S. J. Measurement and Scale-up of Secondary Nucleation Kinetics for the Potash Alum-Water System. AZChE J . 1979,25,948-958. Genck, W. J.; Larson, M. A. Temperature Effect on Growth and Nucleation Rates in Mixed Suspension Crystallization. AIChE Symp. Ser. 1972,68 (No. 1211,57. Khambaty, S.; Larson, M. A. Crystal Regeneration of Growth of Small Crystals in Contact Nucleation. Ind. Eng. Chem. Fundam. 1978,17,160-165. Kirshenbaum, I. In Physical Properties and Anaysis of Heavy Water; Urey, H. C., Murphy, G. M., Eds.; McGraw-Hill: New
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I n d . E n g . C h e m . Res. 1989, 28, 850-856
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Received for review July 25, 1988 Accepted January 17, 1989
Fluid Mixing in a 90" Pipeline Elbow Linda M. Sroka and Larry J. Forney* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
An experimental investigation was conducted to determine mixing quality downstream from a 90° pipeline elbow. Mixing quality, as determined by a concentration second moment, was measured for a range of jet-to-pipe momentum ratios a t both 5 and 10 pipeline diameters downstream from the jet injection point. T h e ratio of the centerline radius of curvature of the elbow to the pipeline diameter was varied from a mitered corner of 0.5 to a maximum value of 1.14. The injection point was located such that it entered normal to the pipeline flow from inside, outside, or perpendicular to the plane of the elbow. The jet was also positioned on the outside of the elbow such as t o enter the pipeline flow head-on. I t was found that mixing is significantly improved for all elbow geometries compared to a straight pipeline. Furthermore, optimum mixing in a 90° elbow was obtained a t reduced jet momentum for all injection geometries.
I. Introduction The use of pipeline mixing techniques is common in the chemical industry to promote chemical reactions, heat transfer, mixing, and combustion processes. A conventional configuration for pipeline mixing is a side tee followed by a length of straight pipe as discussed in the recent reviews of Forney (1986) and Gray (1986). Many of these processes do not have a sufficient length of straight pipe after the mixing tee to achieve a desired mixing quality. However, an elbow in the pipeline near the injection point may create sufficient secondary flow and turbulent intensity to significantly shorten the pipe length for a desired degree of mixing. A substantial amount of work has been performed to determine the quality of mixing downstream from a conventional side tee (Ger and Holley, 1976; Forney and Kwon, 1979; Fitzgerald and Holley, 1981; Marauyama et al. 1981,1983; Forney and Lee, 1982; O'Leary and Forney, 1985; Sroka and Forney, 1988; Sroka, 1988). The effects of a 90" pipeline elbow on the mixing quality directly downstream of the tee, however, are not available in the literature. Nevertheless, some work is cited in the literature on related subjects. Hiby (1970) found that a single 90" pipeline elbow placed six pipeline diameters downstream from the feed entry reduced the distance to mix parallel streams of equal flow rate and velocity. The mixing length to obtain an intensity of segregation of was reduced from 98 pipe diameters for the straight pipeline to 62 pipeline diameters with the elbow. Fitzgerald and Holley (1981) examined the effects of secondary flow caused by a three-blade, fixed propeller. The propeller produced a single swirl with a rotation of approximately eight pipeline diameters. The swirl deflected the jet to the side of the pipe, giving an
* To whom correspondence should be addressed.
asymmetric profile, and thus increased the mixing distance. The effects of a single swirl, however, are not a complete picture of the secondary flow downstream from an elbow since backflow and increased turbulent intensity may also be present. It is often the case that rapid mixing is desired within a minimum pipeline distance. While the work of Hiby deals with mixing of a parallel feed at large distances of 30 5 x / D 5 100 pipeline diameters downstream from the elbow, the present work focuses on the effects of elbows with small radii of curvature 0.5 5 R J D 5 1.14 at short distances 5 5 x / D (. 10 downstream from a more conventional side-tee geometry. In contrast to the straight pipe, the asymmetry of a pipeline elbow provides several possible jet orientations. For this study, the side tee was positioned such that the secondary fluid entered the pipe flow from inside, outside, or perpendicular to the plane of the elbow. The side tee was also positioned on the outside of the elbow such as to enter the flow head-on. Shown in Figure 1 are the head-on, inside, and outside tee orientations.
11. Elbow Fluid Flow The turbulence within the pipe and injected fluid cause the two fluids to mix rapidly as they travel downstream. An elbow creates secondary flow and increased turbulent intensity, which changes the pipe flow characteristics and anticipated mixing quality. The elbow-induced secondary flow and pressure gradient contribute to mixing by transporting fluid from the outside of the bend toward the inside wall. Tunstall and Harvey (1968) found that elbows of circular cross section in which separation occurred caused a bistable secondary flow pattern. The secondary flow consisted of a single vortex or helical swirl changing its rotational direction from clockwise to anticlockwise with a period of approximately 1s (see Figure 2). Furthermore, Weske (1948) determined that regions of backflow and flow
8 1989 American Chemical Society 088~-588~/89/2628-085Q$01.~0/0
