Ind. Eng. Chem. Process Des. Dev. 1980, 79, 509-514
509
REVIEW Crystallization Kinetics from MSMPR Crystallizers John Garside' and Mukund B. Shah Department of Chemical and Biochemical Engineering, University Co//egeLondon, London WC 1E 7JE, United Kingdom
Published crystallizationkinetics obtained in MSMPR crystallizers are reviewed, and measured nucleation and growth rates are compared on a common basis. In general, the range of variables studied (supersaturation, magma density, stirrer speed, etc.) for any given system is extremely limited. Experimental conditions, particularly those related to crystallizer hydrodynamics, are usually poorly defined and scale-up of kinetics on the basis of such results would not be possible. Conditions under which future laboratory studies should be made are suggested.
Introduction The concept of a continuous mixed suspension mixed product removal (MS MPR) crystallizer (Randolph and Larson, 1971) has led bo experimental techniques whereby crystallization kinetics (i.e., nucleation and growth rates) can be determined under conditions where both of these kinetic processes are occurring simultaneously. Such kinetics are usually correlated by semiempirical equations of the form
Table I. Units Used To Compare Kinetic Data
Bo = KRM$G' (1) where KR = f (temperature, hydrodynamics, and impurity concentration) and the relative kinetic order i is the ratio of the nucleation and growth orders. In the period since the mid-1960's results of many such experimental studies have been published for a wide range of systems. Simultaneously with such experimental studies, development of the population balance approach to crystallizer design has enabled design equations to be derived for many crystallizer configurations operating under both steadyand unsteady-state conditions. Before such equations can be used in a quantitative way the kinetic parameters KR, j , and i need to be known. In this paper the results of MSMPR experimental studies are reviewed and the experimental evidence justifying the use of eq 1 is considered. Data published up t o mid-1979 have been included. In addition, the possibility of using published kinetic data for crystallizer design purposes is assessed. The results considered here are restricted to those obtained in MSMPR crystallizers. A number of published studies contain insufficient detail to enable quantitative kinetic data to be deduced and results from these have not been included in this survey. Results of Kinetic Studies The literature contains a wide variety of units which have been employed to record values of the parameters involved in crystallization studies. In order to compare published data a common system of units is used throughout this paper as indicated in Table I. Table I1 lists the systems that have been studied in MSMPR crystallizers. The table is arranged according to the method employed to produce supersaturation and the ranges of temperature, residence time, and magma density
over which measurements were made are indicated. Continuous laboratory crystallizers are generally not designed to run at a predetermined level of supersaturation and so this is not an independent variable under the control of the investigator. Further, supersaturation levels are often very low, making them difficult to measure. Hence only a limited number of studies report a supersaturation range. The growth rate is a measure of the supersaturation and so the range of G over which kinetics were determined is also included in Table 11. Published data were converted to the units shown in Table I and the resulting correlating equations for Bo are given in Table 11. In a number of studies the steady-state crystal size distribution has been measured down to a few micrometers by using a Coulter Counter (e.g., Youngquist and Randolph, 1972; Randolph and Sikdar, 1976; Garside and JanEiE, 1979). A rapid increase in population density is then sometimes observed in these small size ranges and consequently the derived nucleation rates are very much higher than when the upward curvature is not present. Results from such measurements are noted in Table 11. One further complication arises in comparing kinetics, that of size-dependent growth. This produces curvature in the population density plot and a size-dependent growth rate equation has then to be incorporated into the population balance. Two such equations have been most commonly used, those proposed by Canning and Randolph (1967) G = Goc(1 + YCL) (2) and by Abegg et al. (1968)
0196-4305/80/1119-0509$01.00/0
variable nucleation rate growth rate magma density
symbo1
unit
Bo
no./L s
G
m/s
conversion factor
1 no./L s = 0.06 no./cm3 min lo-* m / s = 0.6 ' pm/min = 36 pm/h
MT g / L = kg/m3
G = GoA(1 0 1980 American Chemical Society
+ YAL)~
(3)
510
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
Table 11. Published Crystallization Kinetics from MSMPR Crystallizers
la) 1.
A m m o n i u n alum
Charnblisr 11966)
22
15-45
50-220
3.5-8.3
2.
Ammonium s u l p h a t e
Chambliss (1566)
22
15-45
30- 7 5
4.3-12.2
I.
Ammonium s u l p h a t e
Y a v n g q u i i f and R a n d o l p h ( 1 9 7 2 )
34
5-16
4.
h o n i u n sulphate
Larran and M u l l i n 1 1 9 7 3 )
18
8-20
5.
h o n i v m sulphate
L a r s a n and Klekar ( 1 9 7 3 )
15
9-32
6.
S a g n e i ~ u ms u l p h a t e
S l k d a r and R a n d o l p h (19761
7.
P a r a s s i u n alum
O r f e n s and d e Jong ( 1 9 7 1 )
8. P o r a r E i u m alum
C a r r i d e and J a n E i ;
(1579)
9.
PaLarsium c h l c r i d e
G e n c k and hson 0972)
10.
P o t d l s i ~ mc h l o r i d e
Randolph e t a l .
11.
P o t a s s ~ u nd i c h r o m a t e
T i m and C o o p e r ( 1 9 7 1 )
12.
Potassium dichromare
Desai
11.
Porarsium dichromate
Janse 1 1 9 7 7 )
14.
Polasslum n i t r a t e
S h o r and Larsan (1971)
15.
Potassium n i t r a t e
G e n c k and Larson 119721
16.
P o f a ~ s i u mn i t r a t e
J u i a r n e k and Larsen ( 1 9 7 7 )
17.
Patarslum Illtrace
H e l t and L a r s o n ( 1 9 7 7 )
18.
Potassium su:phafe
R a n d o l p h and Rajugapa:
et
5 -200
IO
10-60
15-120
12-30
15-43
a l . (1974)
(1970)
5.2-11.7
20-100
2.6-10.0
21
15-45
25-87
4 . 8-1 3 . 1
11-40
15-45
25-45
4.3-15.7
20
14-23
10-40
7.7-13.2
..- - I 6
10
19-88
21.
Sodium c h l o r i d e
Bennecr e t a,.
Sodium chloride
Asselbergs (1975)
23,
Citric a c i d
S i k d a r and R a n d o l p h 0 9 : 6 )
24.
Lrea
Bennett and v a n B u r e n ( 1 9 6 9 )
-55
25.
Lrea
Ladaya e t SI.
1-16
,-.
3.7-12.6
3.5-11
5-28
0.3-2.5
0.6-6.1
23-108
1.2-9.5
2.8-7. 1
5
29
Rose> and H u l b u r c ( 1 5 7 1 )
30
2-10
2
bO-liC
70-190
0.*8-2.0
50
10-54
25-200
3.5-11
150-4OC
320-110
0.4-1.2
-.
1-7
1.7-1:.8
16-24
1.1-3.7
30-130
Reaction p r e c i p i f a i i o n
26,
Barium n i t r a t e
Blumenfndl e : a l . (1911)
25-45
5-20
27.
Barium soap
Chiiare e t a l . (1576)
15-60
7.5-15
28.
Calcium c a r b a n a r e
S c h i e r h a l z and Stevens ( 1 9 7 5 )
-23
10-1:
29.
Cal'lum
D r a c h e t 8 1 . 119781
10.
Calciua s u l p h a t e !H20
3;.
Calcium sulphate jH20
S i k d a r e l a!.
12.
S i l v e r bromide
Key e t 8 1 .
33.
h o n ~ u alunlH,CiEtOH r
Yvrray
31.
h o n l u m alun/H,01EfOH
T i m 8nc L e r r a n 11968)
27
35.
Annnar.um
Tim a n d L d r s o n ( 1 9 6 8 )
7
36.
Soclum rhlor.CeIH,O!ttOH
TIT
37.
S o d l v i c h l ~ - r i d eH 0 l E I O H
Yarns e l 3 1 .
OXdldCe
Salting
38.
1.4-5.2
9-30
R a n d o l p h and S i k d a r 1 1 5 7 6 )
11977)
2-12 1.4-11.0
26-40
10
2-10
11.7-20,i
1.2-3.1
Patassiurr s u l p h a t e
(1971)
0.9-1.5
14-42
10-2s
1,6-I. 3 27.6-67.6
5-1L
18-60
Pacassium sulphate
(11)
14-25 60-250
10-45
19.
'
2.6-7.1 5-50
20.
22.
18-23 9.7-21
18-26
12
(1577)
38 51-61
bmir a n d L a c i o n
(19681
(1978)
11978)
20-120
2.8-18.9
6.00 x I0l6
15-23
5 , 3-11, i
1.72
-0.3
38
5-20
45-75
12-45
0.0:5-0.06 15-18
0 07-0.12
1.6-5.6
70-90
0.20-0.11
80
0.019-0.13
a t 25OC
- ~ , ~ ~
1 . 4 2 x 1021 G 2
0.4-1.1
0.33-1. 7
b$
1 0 - l ~G
2 . 2 8 x 10'' 5 . 3 2 x IOz4 G2" r e a g e n t g r a d e . 7OoC 2.16 x IOz2 G 2 ' 6 p l a n t g r a d e , 70°C 8 . 5 9 x 10"
2.2
a c 800C
6 . 3 7 x I O L 9 '6
out
iulphdtelH OIYe0k 2
Sodium i ' h l u r l d a l H O l F f O H
and L a i i o r ( 1 5 6 8 )
64
2.5-8.1
15-45
110
1.4-3.1
15-&5
33
1.7-2.8
I4 11
2 8-9.1
15-:
5
1.0-5.8
l5-"5
(15721
L i u and B o t s a r i i : I 9 7 3 1
2.9-8.9
15-.5
a n d Larson ( 1 5 6 5 )
27
15-45 c
35.
S o d i u m c h l n r i d c r H OrtlDH
40.
C > c l o l L C i tl\O
,H I: > 2
5-20
S o n e and D n u p l a s ( . 9 7 5 l
Branrorr e t a l .
11549)
b7
3.1"
2.3-3.9
("I 32.5-53 1
3.5-!4.1
a A dash indicates that the data are not recorded in the original reference. Size analysis down to 1.26 p m with curvature in population density in this size range; N = 545-675 rpm. Size analysis down to 8 Effect of Cr3+investigated. pm, linear population density plot t o this size. N o effect of N o n Bo. e E = 0.1-11.3 W/kg, G o = growth rate of crystals below 150 pm. f N = 850-2100 rpm. G o= growth rate of crystals below 100 km. Results also for size analysis down t o 5 pm-different kinetics then obtained. Correlations also given for e = 1 2 and 30 "C. Solutions near-saturated with Effect of several impurities investigated. NaCl and containing 0.75 mg of MgS0,/100 g of H,O. N = 800-1250 rpm. Correlation also given for e = 11, 31, and 40 "C. Size analysis in range Correlation also given for e = 1 0 and 2 5 "C. 10-30 pm, linear population density plot in this range. Size analysis down t o 7 pm, nonlinear population density plot, G o = growth rate in equation of Abegg at al. (1968),(eq 3). TIPS = impeller tip speed (ft/min), TO = s/turnover, conditions probably not MSMPR. P N = 414-804 rpm. Q N = 600-900 rpm. For stoichiometric CaSO, and Na,CO,. Results also for excess Na,CO,. From phosphoric acid. t Precipitation in gelatin. Size analysis in submicrometer range. Ci = concn PbCl, (ppm). Range of Ci = 1-30 ppm.
In both these cases Go represents the growth rate of "nuclei" and it is this growth rate that is correlated with the nucleation rate to obtain the relative kinetic order. Potassium alum and potassium sulfate are systems which appear to exhibit size-dependent growth.
Figures 1 and 2 depict the crystallization kinetics tabulated in Table I1 as plots of Bo against G. Many of the studies employing cooling or evaporative crystallization have investigated the influence of magma density. Where the effect of this variable has been included in the corre-
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 511 ,
1os7
-
105
-
VI
L L
a3
G 'm
10'.
d
e
e
-C
0
Q 0
U
z' i o 3 -
10'
10-9
, 10-8
!7,,:/4
, 10-7
1u 9
1o
1 o.8 Growth r a t e , G
-~
1 o-6
Irnisi
Figure 2. Crystallization kinetics for precipitation and salting out crystallization. Kinetics for a given system are represented by lines of a given form. Numbers indicate reference number in Table 11. Notes: curve 38, Ci = 1 ppm.
lation the curves in Figure 1 have been drawn for MT = 50 g/L if this value is within the range investigated. Such a magma density is at the lower end of those generally used commercially. If the investigation did not extend to this value, the experimental magma density closest to 50 g/L was used to calculate a kinetic curve. In cases where the effect of hydrodynamics has been explored a value of stirrer speed or power input near the middle of the measured range has been used to evaluate the curve, actual values being given in the figure key. Effect of Variables; on Crystallization Kinetics 1. Systems Studied. Choice of systems for use in academic studies is generally made on the basis of ease of working and familiarity with the material. In consequence, many materials of industrial importance are not adequately represented. For example, no data appear to be available for ammonium nitrate, the ammonium phosphates, sodium sulfate, and alumina, while potassium chloride, sodium chloride, urea, and calcium sulfate have only been the subjects of a small number of studies. Yet, the annual U.S.A. production of each of these materials is in excess of lo6 t (van Damme, 1973). Organic systems are very poorly represented. Precipitation processes are being increasingly studied in MSMPR crystallizers and most of the systems listed in Table I1 are encountered on the industrial scale. Much of the early experimental work using the MSMPR technique was carried out using salting-out since this has many experimental advantages. There appears to be increasing industrial interest in such processes as a means of increasing yields and conserving energy. 2. S u p e r s a t u r a t i o n , Supersaturation within a MSMPR crystallizer is forced to different levels by changing the residence time. As a consequence, the growth rate changes and this is generally taken as a measure of the supersaturation as implied by the use of eq 1 to cor-
relate kinetics. The range of supersaturation, and hence growth rates, that can be achieved by a given change in residence time depends on the value of i. It can be shown (Randolph and Larson, 1971) that for constant magma density G
a 7-4/(i+3)
(4)
If growth rate is related to supersaturation by the expression G
0:
sg
(5)
the corresponding change in supersaturation is s
a ?-4/g(i+3)
(6)
Assuming that the maximum ratio of residence times that can be achieved in a given crystallizer is 8 (e.g., from say 7.5 to 60 min) the resulting ratio of G is 8 and 5.3 for i = 1 and 2, respectively. If g = 1 the corresponding ratio of s is the same as that for G, while if g = 2 the ratio of s is 2.8 and 2.3 for i = 1 and 2, respectively. With the exception of the data for barium nitrate (Blumenthal et al., 1974, ref 26) for which G varies by a factor of 13.9, the maximum ratio for G is about 8 but most sets of data cover a considerably smaller range. The range of driving force over which kinetics may be obtained is thus rather limited in practice and this sets a limit on the accuracy with which kinetic parameters may be determined. With few exceptions, values of G measured in published studies are within the range 2 X to 2 X lo-' m/s (- 1-10 pm/min) and are remarkably insensitive to the particular systems. Precipitation conditions tend to produce growth rates that are somewhat lower. Nucleation rates cover a much wider range of values. In part, this must be due to the large number of variables that affect nucleation rate and that are not standardized in Figures 1 and 2; magma density and hydrodynamics are the most important of these. For cooling and evaporative conditions Bo appears to cover about three orders of
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
512
Table 111. Effect of Temperature on Kinetics ref (see
Table 111
9
system potassium
range of e, “C
i = const., K R
chloride
13 15 17 23
comments
12-30
increases with e
potassium dichromate potassium nitrate potassium nitrate
26-40
i = const., B o a
11-40
i = const., K R
10-25
i = const., negative
citric acid
16-24
0 -0.57
decreases with e
activation
energy for B o i = const., negative
activation
26 30
barium nitrate calcium
sulfate-water 31
calcium
sulfate-water
energy for B o 25-45 45-75
B o decreases with
70-90
increasing G i = const., positive
activation
energy for B o
magnitude, from about 2 x 102 to 2 X 105 no./L s ( 10-10~ no./cm3 min). Results of several studies are not included in Figure 1. Youngquist and Randolph (1972, ref 3) and Randolph and Sikdar (1976, ref 20) determined steadystate crystal size distributions down to a few micrometers with a Coulter Counter. Sharp upward curvature was observed and so the resulting kinetics are not strictly comparable. Lodaya et al. (1977, ref 25) provide insufficient details of experimental conditions to enable a curve to be constructed. For precipitating systems values are much more variable, covering a least four orders of magnitude. The results of two studies are not included in Figure 2, Wey et al. (1978, ref 32) with silver bromide for which nucleation rates are of the order of 10’O to l O I 4 no./L s, and Koros et al. (1972, ref 37) where nucleation rates appear very low, -lo2 no./L N
S.
The relative kinetic order for cooling and evaporative conditions is generally between 1 and 2. The only recorded values significantly higher than 2 are for KCl where i is reported as 2.55 (Genck and Larson, 1972, ref 9) and 4.99 (Randolph et al., 1977, ref 10). For potassium dichromate and citric acid, i appears to be between 0 and 1. Negative values of i are probably suspect since it is difficult to envisage kinetic mechanisms giving rise to such values. As pointed out by Keight (1978), operation of a high yield MSMPR crystallizer a t constant residence time and magma density forces the nucleation and growth rates to follow the relation Bo a G-3. The resulting apparent value of i = -3 is then unrelated to the kinetics of the system but arises since too small a range of residence time and magma density was covered. Nevertheless, there is a suggestion that potassium sulfate and urea exhibit unusual kinetic behavior. With precipitation and salting-out i tends to be higher. In these systems primary as opposed to secondary nucleation is likely to be the dominating mechanism. Primary nucleation rate expressions are highly nonlinear in supersaturation and so a high value of i may be expected. 3. Temperature. Few studies have investigated the effect of temperature on crystallization kinetics. Table I11 summarizes those for which sufficient data are available to draw tentative conclusions. The range of temperature over which measurements have been made is very limited, particularly if attempts are made to determine activation energies. Apart from studies with CaS04J/2H20from phosphoric acid, all work
has been conducted close to ambient conditions. Over these narrow temperature ranges the evidence suggests that i can be considered constant. In many cases the apparent activation energy for the nucleation process appears to be negative, and Wey and Terwilliger (1980) discuss why such results may arise. 4. Magma Density. Secondary nucleation rates which are “removal limited” (Evans et al., 1974) will be influenced by the amount of crystals in suspension. A first-order dependence on magma density is predicted if crystal-agitator or crystal-crystallizer collisions are responsible for secondary nuclei production (Ottens and de Jong, 1973). Most results quoted in Table I1 show such a relation, although it should be noted that in a number of cases the relation was assumed rather than determined from experimental results. Several correlations show an exponent on magma density that is close to 0.5 (Desai et al., 1974, ref 12; Juzaszek and Larson, 1977, ref 16; and two studies with potassium sulfate, Randolph and Rajugopal, 1970, ref 18, and Randolph and Sikdar, 1976, ref 20) while the study of Randolph et al. (1977) with potassium chloride gives the very low exponent 0.14. These low values may arise since the nucleation is in part “regeneration limited” (Evans et al., 1974) or “survival limited” (Randolph and Sikdar, 1976). Precipitation studies have not included measurements on the influence of MT. This reflects the assumption that primary, rather than secondary, nucleation dominates the behavior of such processes. The range of magma density over which measurements have been made is frequently rather limited and in many cases is far below typical industrial values. Since crystal-crystal collisions are predicted to result in a secondorder dependence of nucleation rate on magma density, their relative importance would be expected to increase at high magma densities. Little experimental work has been undertaken to test this prediction. 5. Crystallizer Hydrodynamics. Crystallizer hydrodynamics are likely to exert a profound effect on both primary and secondary nucleation rates. Lack of standardization and poor specification of hydrodynamic conditions are the most likely causes of differences between kinetics measured by different workers for the same system. The influence of hydrodynamics on secondary nucleation was first discussed by Ottens et al. (1972). They showed that for crystallizers of similar geometry operating in the removal limited regime Bo 0: 6 which for a given vessel size implies Bo 0: IP,As shown in Table IV this relation is approximated in a number of studies. More recent modifications to this theory as summarized by Garside and Davey (1980) do not give rise to substantially different conclusions. Several studies show nucleation rates to be significantly less sensitive to stirrer speed which may again reflect regeneration or survival limitation, while the very high order dependence for ammonium sulfate (Youngquist and Randolph, 1972, ref 3) may be related to the method of determining the nucleation rate in the micrometer size range. The range of stirrer speeds covered in any single investigation is generally very narrow. The minimum stirrer speed is set by the necessity to suspend all the crystals and maintain a mixed suspension while the upper limit is set by vortex formation and air entrainment. This imposes a limit on the precision with which the exponent of N can be determined. Since primary nucleation rates are very sensitive to supersaturation, mixing of the reacting components in pre-
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 513
Table IV.
Effect of Civstallizer Hydrodynamics o n Nucleation Kinetics
ref (see Table 11) 3 6 7 8 13 20 21 22 23 25 31
system ammonium sulfate magnesium sulfate potash alum potash alum
correlating relation for B o
potassium dichromate potassium sulfate sodium chloride sodium chloride citric acid urea calcium sulfate
NO
system
4 ammonium sulfate 1 4 potassium nitrate
29
calcium oxalate 38 sodium chloridewater-ethanol - borax (Randolph and Koontz, 1976) - potash alum (Rousseau and Woo, 1978)
N 3 ;results for 1 0 and 40 L for B o determined with Coulter Counter, B o a N";results for 1, 5, and 3 0 L
E a h1.8
N3 N2.5
TIPS2/T0 N2
TIPSZ/TO a N 3 55-L vessel
NO 1V 2.3
NO
Table V. Effect of Impurities o n Kinetics ref (see Table 111
comments
N7.84
impurity and concentration range Cr3+,0-20 ppm Co2+,0-4 g/100 mL soln Cr3+,0-500 ppm methylamine hydrochloride, 0-1000 ppm dodecylamine hydrochloride, 0-50 ppm fluorocarbon, 0-30 ppm pyrophosphate methylene blue Pb", 1-30 ppm sodium oleate dodecylbenzene sulfonate, 0-18 ppm quinoline yellow, 0-300 ppm
cipitation and salting out systems may be of importance. Local regions of high supersaturation near the mixing point could produce extremely high nucleation rates which would be very sensitive to thLe degree of mixing and hence stirrer speed. None of the studies reported in sections (ii) and (iii) of Table I1 adequately investigated these problems. 6. Impurities. The dramatic effect of trace quantities of impurities on crystallization kinetics is well known. Little attention has been paid, however, to quantitative investigations of their influence in MSMPR crystallizers. In large part, this is 11-10doubt due to the lack of an adequate theoretical framework on which to base experiment planning and interpretation. Table V lists those studies that have been reported and includes two references where insufficient data are available to determine kinetics of the pure system and so are not included in Table I1 (Randolph and Koontz, 1976; Rousseau and Woo, 1978). Increasing impurity concentration generally decreases both nucleation and growth rates for a given supersaturation. In an MSMPR crystallizer EL decrease in, say, nucleation rate will result in a decrease iin surface area and so a rise in supersaturation. The imam balance constraint, therefore, forces the nucleation and growth rates to move in opposite directions, and the separate responses of nucleation and growth kinetics are difficult to disentangle. The evidence available at present suggests that i is relatively independent of impurity concentration although KR can be very sensitive to small changes in impurities. Resulting crystal size distributions can hence be changed
significantly by impurity addition. Discussion and Conclusions Although it was not the objective of many of the studies reported here to obtain crystallization kinetics that could be related to crystallizer design and operation, such an aim is important if the power of the population balance to explain crystallizer behavior is to be exploited. The relative kinetic order, i, is probably the single most important parameter and for several systems the values determined by different workers are encouragingly similar: e.g., ammonium sulfate, i = 1.03 to 1.5; potash alum, i = 1.58 to 1.8; potassium dichromate, i = 0.48 to 0.53, and CaS04l/zHzO,i = 2.8 to 3.2. For some other systems there are large differences: e.g., potassium chloride, i = 2.55 to 4.99; potassium sulfate, i = -1 to 0.54 and sodium chloride/ H,O/EtOH, i = 1.72 to 9.0. The absolute value of the nucleation rate for a given growth rate frequently varies by several orders of magnitude as with, for example, potassium dichromate, potassium nitrate (see Figure l ) , CaS04J/2H20and some of the salting out systems (see Figure 2). Some possible reasons for this have already been discussed. It should also be noted that the volumetric basis on which nucleation rates are reported is not always clear in the original references. The basic form of the kinetic equation (eq 1) seems adequate to represent experimental data but the ranges of magma density and hydrodynamics over which a given study have been made are in general very limited. The frequently accepted value of j = 1 needs more extensive experimental support, particularly at high magma densities and for large-scale crystallizers, and the effect of mixing in precipitating and salting-out systems needs further investigation. Hydrodynamics of crystallizers used for kinetic studies need to be more clearly defined by the investigators. A useful step would be to record power numbers for the agitator/crystallizer system used in kinetic investigations since these could then be related directly to the specific power input as suggested by the nucleation model of Ottens et al. (1972). Standardization of vessel and stirrer geometry would be of the greatest value. It is probably true to say that few of the data recorded in Table I1 could be used with any conficence for crystallizer design purposes. Finally, comparison between different sets of data would be facilitated if a common system of units were used for the important crystallization parameters. Nomenclature b = exponent in size-dependent growth equation (eq 3)
Bo = nucleation rate, no./(L s)
g = growth order (eq 5) G = growth rate, m / s GOA= nuclei growth rate (eq 3) (m/s)
514
Ind. Eng. Chern. Process Des. Dev.
Goc = nuclei growth rate (eq 2) (m/s)
1980, 19, 514-521
Crystallization from Solution", presented at 66th Annual AIChE Meeting, Philadelphia, 1973. Larson, M. A., Mullin, J. W., J. Cryst. Growth, 20, 183 (1973). Liu, Y-A., Botsaris, G. D., AIChE J., 19, 510 (1973). Lodaya, K. D., Lahti, L. E., Jones, M. L., Ind. Eng. Chem. Process Des. Dev., 16, 294 (1977). Murray, D. C., Larson, M. A,, AIChE J., 11, 728 (1965). Ottens, E. P. K., Janse, A. H., de Jong, E. J., J. Cryst. Growth, 13/14, 500 (1972). Ottens, E. P. K., de Jong, E. J., Ind. Eng. Chem. Fundam., 12, 179 (1973). Randolph, A. D., Rajagopal, K., Ind. Eng. Chem. Fundam., 9, 165 (1970). Randolph, A. D., Larson, M. A., "Theory of Particulate Processes", Academic Press, New York, 1971. Randolph, A. D., Sikdar. S . K., Ind. Eng. Chem. Fundam., 15, 64 (1976). Randolph, A. D., Koontz, S.,"Effects of Habit and Nucleation Modifiers In Crystallization of Sodium Tetraborate Decahydrate", presented at 69th Annual AIChE Meeting, Chicago, 1976. Randolph, A. D., Beckman, J. R., Kraljevich, Z. I., AIChE J., 23, 500 (1977). Rosen, H. N., Hulburt, H. M., Chem. Eng. frog. Symp. Ser. No. 110, 67, 18 (1971). Rousseau, R. W., Woo, R., "Effects of Operating Variables on Potassium Alum Crystal Size Distrlbutions", presented at 84th National AIChE Meeting, Atlanta, 1978. Schierhoiz, P. M., Stevens, J. D., AIChE Symp. Ser. No. 151, 71, 248 (1975). Sikdar, S.K., Randolph, A. D., AIChE J., 22, 110 (1976). Sikdar, S. K., Ore, F., Moore, J. H., "Crystailizatlon of Calcium Sulfate Hemihydrate in Reagent Grade Phosphoric Acid", presented at 84th National AIChE Meeting, Atlanta, 1978. Shor. S. M., Larson, M. A,, Chem. Eng. frog. Symp. Ser. No. 110, 67, 32 (1971). Song, Y. H., Douglas, J. M., AIChE J., 21, 924 (1975). Timm, D. C., Larson, M. A., AIChE J., 14, 452 (1968). Timm, D. C., Cooper, T. R., AIChE J., 17, 285 (1971). van Damme-van Weeie, M. A., in "Crystal Growth: an Introduction", Hartman, P., Ed., North Holland, p 248, 1973. Wey, J. S., Terwilliger, J. P., Glngello, A. D., "Analysis of Silver Bromide Precipitation in a Continuous Suspension Crystallizer", presented at 84th National AIChE Meeting, Atlanta, 1978. Wey, J. S., Terwllliger, J. P., Chem. Eng. Commun., 4, 297 (1980). Youngquist, G. R., Randolph, A. D., AIChE J., 18, 421 (1972).
i = relative kinetic order
j = exponent of magma density (eq 1) KR = nucleation rate constant, no./L s (g/L)I (m/s)' MT = magma density, g/L N = agitator speed, rpm s = supersaturation, g of crystallizing substance/g of water YA, yc = constants in size-dependent growth equations (eq 3 and 2), rn-l E = specific power input, W/kg T = residence time, s 0 = temperature, "C
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Received f o r review January 28, 1980 Accepted April 30, 1980
ARTICLES
Electrochemical Study of Liquid-Solid Mass Transfer in Packed Beds with Upward Cocurrent Gas-Liquid Flow Ghlslalne Delaunay, Alain Storck, Andre Laurent, and Jean-Claude Charpentler' Laboratoire des Sciences du G6nie Chimique, CNRS-ENSIC, 54042 Nancy, France
An electrochemical technique was used for global measurement of overall solid-liquid mass transfer coefficients in upward cocurrent gas-liquid flow through a packed bed under bubble flow and surging flow conditipns. The feasibility of the technique was ascertained and the coefficients were compared with those obtained by other techniques. An energetic correlation has been proposed for both single and gas-liquid flow which extends the range of application of previous works.
Introduction The use of fixed bed reactors operated under cocurrent upflow conditions has widely increased during these past years, especially in the petrochemical industries: coal liquefaction, catalytic hydrodesulfurization, selective hydrogenations, etc. (Shah, 1979). The overall rate of the process may depend either on the chemical reaction kinetics or on the physical gas-liquid and liquid to solid 0196-4305/80/1119-0514$01.00/0
particle mass transfer resistances. However, under certain operating conditions the limit of the process is only located at the diffusional mass transfer resistance near the liquid-solid interface. There are very few data available on the determination of the space_average value of the particle mass transfer coefficient k for upward cocurrent flow in packed beds. The recent works published on the topic together with the 0
1980 American Chemical Society
