Growth and Dissolution Kinetics of Potassium Sulfate Crystals in an Agitated Vessel John Garside,* John W. Mullin, and Sibendu N. Das Chemical Engineering Department University College London, London WC 1E 7JE, England
Growth and dissolution rates of potassium sulfate crystals have been measured in an agitated vessel. T h e effect of temperature (over the range 20-50°C),agitation, and crystals size (over the range 0.571.18 m m ) w e r e investigated. From t h e results t h e surface integration growth kinetics were deduced a n d , for 0.71-mm crystals over t h e above temperature range, were given by an equation of t h e form RG = k , ’ ~ ’ . ~where , the value of k r ’ varies with temperature. T h e results seem to indicate that the surface integration kinetics of potassium sulfate and a number of other salts are size dependent over t h e range 0.4-1.6 m m . T h e paper also discusses the problems involved in eliminating volume diffusion effects in order to determine t h e surface integration kinetics, and t h e reproducibility of results obtained in crystal growth experiments.
Introduction This paper presents new experimental data on the growth and dissolution of potassium sulfate crystals in an agitated vessel. Previous work on the growth of potassium sulfate crystals carried out in this laboratory has been performed in a fluidized bed (Mullin and Gaska, 1969)in a draft-tube agitated vessel (Jones and Mullin, 1973) and on single crystals (Mullin and Gaska, 1973). Dissolution rates in an agitated vessel have been measured by Nienow (1969). One aspect of the present work is therefore concerned with a comparison of the results obtained using these different experimental conditions. Since different types of industrial crystallizer operate under very different hydrodynamic conditions it is important to know in advance how changes in environment are likely to change the crystal growth characteristics. A second objective of this work was to investigate the different methods that have been proposed for deriving the surface integration rate from overall growth rate measurements. Measurements in which growth rate is measured as a function of supersaturation are necessary if theories of crystal growth are to be tested against experimental data. The classical chemical engineering approach to crystallization theory has been to consider the overall growth rate as being determined by the relative magnitudes of two resistances in series, namely, bulk or volume diffusion through the mass transfer boundary layer, followed by an “integration” step in which growth units are incorporated into the crystal lattice. These processes are conventionally represented by the equations b u l k diffusion: R, = k,(c - ci) surface integration: R , = k,(c, - c*)’
(1) (2)
The equation representing the surface integration step is essentially an empirical equation that allows for all processes, other than bulk diffusion, that are involved in the growth process and so enables growth data to be correlated without having to postulate any specific physical mechanism for this “surface integration” step. Recently more explicit theories of crystal growth from solution have been developed. Bennema (1969), in particular, has shown how the Burton-Cabrera-Frank (BCF) theory (Burton, et al., 1951) can be applied to growth from solution. Physically this theory describes the integration step as a process in which surface diffusion and integration of growth units into the crystal lattice are coupled and predicts that this combined process should obey an equation of the form
a2 v = C- t a n h oc
where u is the linear rate of advance of a face, u is the supersaturation, and C and g c are constants related to the structure of the surface and the energetics of the growth process. Computer simulation models of crystal growth have recently been developed by a number of workers, including Gilmer and Bennema (1972). Whereas the BCF theory envisages “ad-atoms” diffusing over the crystal surface and being responsible for growth, these simulation models take account of the possibility that the crystal surface will be rough if it has a “low effective bond strength” due, for example, to strong interaction with the solution. With such surfaces, both clusters of ad-atoms and surface vacancies will be involved in the growth process in addition to the motion of single ad-atoms. By using Monte Carlo simulation techniques Gilmer and Bennema showed that such a “nuclei above nuclei” model gave rise to equations of the form
v = Aa5I6 exp( -B/a)
(4)
A and B are constants related to the surface structure and energetics. Measurement of the surface integration rate as a function of supersaturation is required in order to test the validity of growth models such as those represented by eq 3 and 4. Experimentally the difficulty involved in these measurements is one of eliminating the bulk diffusion contribution to the total resistance. The diffusion step is influenced by the hydrodynamics of the growth environment and the resistance offered by the bulk diffusion process will decrease as the relative velocity between crystal and solution is increased. Measurements have therefore frequently been made a t increasing relative velocity and the growth rate obtained a t high velocities taken as the surface integration rate. It has recently been shown, however, that there are several unforeseen difficulties in this procedure and many previously published results may be suspect (Garside, 1971). A second method of deducing the surface integration rate is to perform dissolution experiments under identical hydrodynamic conditions to similar growth measurements. If the dissolution rate is now assumed to be determined solely by bulk diffusion and if the kinetics of bulk diffusion are assumed identical for growth and dissolution taking place in identical hydrodynamic situations (i.e . , that the mass transfer coefficients, k d , are identical) then the surface concentration, cL,can be calculated from eq 1 Ind. Eng. Chem., Fundam., Vol. 13, No. 4 , 1974
299
24
t'
22t
Figure 1 . Agitated vessel used for growth and dissolution experiments (all dimensions in m m ) .
and the surface integration kinetics can be established. The nature and implications of these assumptions are discussed in detail in this paper.
4OoC
2
4
8
6
/30q~
IO
I2
1
114
Concentration Driving Force, Ac, lkg &SO, I k g d n . 1 n IO'
Figure 2. Growth rates of pota_ssium sulfate crystals in an agitated vessel (mean crystal size, L = 0.71 mm; speed of stirring = 8.33 rev/sec).
Experimental Technique Growth and dissolution rates of potassium sulfate crystals (molecular weight = 174.3) were measured in an agitated vessel using a procedure similar to that used previously for measurements made in fluidized beds (Mullin and Garside, 1967; Mullin and Gaska, 1969; Mullin, et al., 1966). The size of the vessel, made to conform to a standard geometry, is indicated in Figure 1 (Rushton, et al., 1950). Solution temperature was controlled to &0.05"C and solution concentration measurements were made by evaporating a sample to dryness giving an estimated accuracy of kg of K2S04/kg of solution. Seed crystals were prepared by successive growth and dissolution until single crystals of regular shape were obtained. The shape factor for these seed crystals was determined as previously described (Garside, et al., 1973). A growth rate measurement was made as follows: 5 g of closely sized seed crystals were introduced into the supersaturated solution contained in the growth vessel. After growing for sufficient time to increase their weight to about 8 g, the crystals were removed, dried, and sized using perforated-plate round-hole sieves. The growth rate was calculated as described previously (Mullin and Garside, 1967; Mullin and Gaska, 1969). The solution concentration was measured before and after the run. Changes in ICduring the run were always less than about 1.5 x kg of K2S04/kg of solution. The effect of supersaturation, temperature, crystal size, and stirrer speed of rotation were investigated for both growth and dissolution. Results A . Effect of Temperature. Figure 2 shows the variation of overall growth rate with concentration driving force, Ac, for a given set of hydrodynamic conditions (mean crystal size 0.71 mm, speed of stirring 8.33 rev/sec). The effect of temperature is shown. Growth rate increases with both supersaturation and temperature, the dependence on supersaturation being approximately second order. The overall growth rate may thus be correlated by the equation
R, 300
= K,(c
- c*)2
Ind. Eng. Chem., Fundam., Vol. 13, No. 4, 1974
(5)
Temperature
50
("CI 30
40
20
Surface integration rate coefficient I
a
-
L
0.1
* 0.08
-; x
T 0.3
Overall dissolution mass transfer coefficient
a 0.1
Overall growth rate coefficient
L1
3.1
3.2 T
(K-II
3.3
3.4
xi03
Figure 3. Effect of temperature on rate coefficients
The temperature dependence of Kc can be represented by an Arrhenius type relationship as shown in Figure 3, the activation energy for the overall growth process being 30 kJ/mol. This compares with a value of 18 kJ/mol obtained previously by Mullin and Gaska (1969) for growth in a fluidized bed. These two values are not strictly comparable, however, since the bulk diffusion contribution to the overall growth process is probably different in the two cases. The effect of temperature on the dissolution rate of similar crystals (mean crystal size = 0.57 mm) under similar hydrodynamic conditions (speed of stirring = 8.33 rev/ sec) was also measured. Dissolution rates were always first order with respect to undersaturation and the variation of
Table I. Mass Transfer Coefficients for the Dissolution of Potassium Sulfate Crystals in an Agitated ____Vessel _
-
_
-
______
~
Mass t r a n s f e r coefficient, K,, kg/m2 s e c
Z = 0.57 m m T e m p , "C
= 1.06 m m
__
8.33 r e v / s e c
8.33 r e v / s e c
11.7 r e v / s e c
13.3 r e v / s e c -
20 30 40 50
0.099 0.127 0.155 0.183
0.110 0.139 0.170
0.144 0.170 0.185
0.178
overall mass transfer coefficient, K D ,defined by the equation
RD = K,-,(C* -
(6 ) with temperature is shown in Table I and Figure 3. The activation energy for dissolution is 17 kJ/mol, which compares with a value of 14 kJ/mol obtained previously (Mullin and Gaska, 1969), and a value of 15 kJ/mol obtained using a standard mass transfer correlation given by Fbwe and Claxton (1965) to evaluate the overall mass transfer coefficient from measured physical properties (Mullin and Nienow, 1964). In order to calculate the surface integration kinetics it was assumed that the mass transfer coefficient measured for dissolution applies for the volume diffusion step occurring during growth. For any given growth rate occurring a t a solution concentration c, the interfacial concentration c z can thus be estimated from eq 1 and the relation between the growth rate and interfacial driving force, (c, - c * ) , or interfacial supersaturation, gz ( = pz - p * / p * ) can be established (note that the units of concentration used to evaluate uz are kg of KzS04/m3 of solution). Figure 4 shows the overall growth rate plotted against the interfacial supersaturation for the temperature range 20-50°C. Potassium sulfate growth rates obtained in a fluidized bed by Mullin and Gaska (1969) were also recalculated in a similar manner using their measured dissolution rates and the correlating lines for these results are also included in Figure 4. The present data refer to crystals with a mean size of 0.71 m m while those of Mullin and Gaska refer to a mean size of 0.85 mm. Although the measurements were made by different workers using different seed crystals and under very different hydrodynamic conditions, the agreement between the two sets of results is excellent. The maximum deviation of any of the present points from the lines representing the data of Mullin and Gaska is about 25%. The relationships shown in Figure 4 can therefore be taken to represent the surface integration kinetics of potassium sulfate, the fact that consistent results were obtained with different hydrodynamics indicating that the effect of system environment has been effectively eliminated. The combined results for the surface integration rate of crystals -0.8 mm in size are well represented by an equation of the form C)
where the magnitude of h,.' varies with temperature (see Table 11). The exponent of ut is essentially constant with changing temperature, linear regression on the combined data giving values of the exponent varying between 2.28 and 2.48 for different temperatures but with too great a standard deviation to justify using a varying exponent. The variation of k,' with temperature is plotted in Figure
, 1 0.01
(19691
0,030.05 0.1
1 0.2
Supersaturation a t Crystal /Solution Interface. U,
Figure 4. Surface integration kinetics of potassium sulfate 0.71 mm).
(t=
-
Table 11. Variation of k,' with Temperature (Crystal Size 0.8 mm) Temp, "C
k r ' , kg/m2 s e c
20 30 40 50
0.0537 0.100 0.210 0.377
3 and indicates that the activation energy of the surface integration process is 52 kJ/mol, considerably higher than the value for the overall growth process and similar to the value of 43 kJ/mol obtained previously for the surface integration step in potash alum (Garside and Mullin, 1968). The surface integration kinetics are not well represented by an equation of the form indicated by the BCF theory (eq 3). For uL
