~~
~
CRYSTALLIZATION EQUIPMENT
THEORY Crystallization is a two-step process
-
nucleation and growth
6y H. M. SCHOEN, Radiation Applications Inc., Long Island City, N.Y.
IT
WOULD BE impossible in a limited space to cover all the aspects of crystallization theory reported in the literature. This paper will present a review and discussion of the more important fundamentals of crystallization upon which the design of crystallizers can be based. Emphasis is placed upon crystal size and crystal size distribution in batch and continuous operation. I n the design of any crystallizer it is possible to list a number of factors of greater or lesser importance, depending upon the specific product requirements. One possible list of important considerations is as follows:
e Crystalquality: soundness, shape, agglomeration, attrition
e Crystal purity e Crystal size a n d size distribution A large number of known (and possibly unknown) factors come into play during crystallization. Any or all of these may play an important role. The pH of the solution, presence of impurities, degree of agitation, rate of production of supersaturation (cooling rate, evaporation rate), and amount and size of seeds present are a few of the factors considered important. The interrelation of these factors is quite complex, therefore crystallizer design and operation is to a great extent based upon experience and know-how. Although there are valid scientific principles which serve as guide lines upon which a rational design can be based, in the final analysis crystallization is not only science but also art. Solubility and Supersolubility As in any mass transfer operation a driving force or state of unbalance is necessary for the process to operate. I n crystallization, this driving force is a concentration difference, which is termed supersolubility-or supersaturation. Ostwald ( 7 7) is generally accepted as having put forth the concept of supersaturation and amplification of the term by means of “meta-stable” and ‘‘labile” fields. According to Miers ( 8 ) , the supersolubility curve is approximately
parallel to the solubility curve, except that it corresponds to higher concentrations (Figure 1). The significance of the diagram is Neither growth nor nucleation can take place in the unsaturated region Growth can take place in the metastable and labile regions Nucleation can take place in the metastable region only if seeds are already present Spontaneous nucleation-without seeds-can take place in the labile region The studies of Ting and McCabe (76) pointed out that the true picture is exceedingly more complex. Supersolubility curves were found, but their position is not only a function of the material being considered but also dependent on a number of other factors such as weight and size of seeds present, cooling rate, and degree of agitation. A definite supersolubility curve, as postulated by Miers, does not exist. For a given solute and solvent, we can consider that there are families of supersolubility curves, their location being dependent upon a t least three factors: specific seed surface, rate of supersaturation production-e.g., cooling rate, evaporation rate-and mechanical dis-
turbances-e.g., agitation. Anothef way to look at the supersolubility curve is that the probability of spontaneous nucleation greatly increases along this line. Spontaneous nucleation will eventually take place anywhere withia the so called meta-stable region. Many investigations have borne out the fact that the concept of “regions” is qualitatively correct, but is far from being a quantitative picture.
Crystal Growth and Nucleation Crystal growth theory dates back to the work of Noyes and Whitney (70) which dealt mainly with the dissolution of crystals. They considered dissolution to be practically a diffusion phenomenon. I n 1904, Nernst (9) modified the equation proposed by Noyes and Whitney, introducing the diffusivity and film thickness. Still later, Berthoud ( 2 ) investigated the possibility that crystal growth is not, as previously assumed, purely a diffusion phenomenon. H e developed a theory based on the assumptions that the diffusional process is followed by a first order interfacial reaction. He obtained the following equation:
dw -- S(C - Co) dt 1 6 ‘3
SPECIAL NOTICE This is the concluding part of the 1961 series on theory, design, and application of process equipment for the chemical industry. The group covers centrifugation, filtration, and crystallization equipment. Reprints of the complete series, 66 pages, may be obtained for $2.50 per copy.
ADDRESS Reprint Department, American Chemical Society, 1155 Sixteenth St., N.W., Washington 6, D. C.
+
(11
where
dw -
= mass growth rate
dt S
=
C
= bulk solution concentration
surface area of the crystal
CO = saturation concentration
K = rate constant for interfacial reaction
6
= film thickness
D = diffusivity The assumption of a first order interfacial reaction is not a generally valid one; however, the concept of the two reactions in series is still accepted. I n the case of sucrose growth from unagitated solutions (77), the ratio of diffusion rate to surface reaction rate is 25.7 a t 20’ C., 2.1 a t 40’ C., and 0.03 a t 70” C. McCabe and Stevens (7) developed VOL. 53, NO. 8
AUGUST 1961
607
p
an empirical approach to the correlation of the diffusional portion of the growth rate reaction. They derived the following equation :
2G C
The growth rates were dependent upon both the mass transfer coefficient. which varied with the fluid velocity, and the surface reaction rate coefficient. which varied with the temperature. Either rate process may be controlling depending upon the material being crystallized and the conditions under which the crystallization is carried out. For magnesium sulfate crystals, the rate of the surface reaction varied directly with the driving foice (concentration difference), and for copper sulfate, a second order dependence was observed. Crystallization is a two-step process involving first nucleation and then the groivth of the nucleus to a macro size. I n practice. both steps are raking place a t the same time within the crystallizer. Supersaturation is necessary for both nucleation and growth, although the effect is different in the two steps. Nucleation was described by LaMer (6) to be the process of generating within a meta-stable mother phase the initial fragments of a new and more stable phase capable of developing spontaneously into gross fragments of the stable phase. Nucleation involves the activation of smaller unstable particles, called embryos. The critical rate-determining embryo becomes a nucleus, which differs from an equal-numbered aggregate of ordinary molecules in solution by possessing sufficient excess surface energy to form a new phase. I n supersaturated solutions, nucleation may be brought about by means of
where the over-all growth rate coefficient = interfacial growth rate 7 ” = growth rate at zero velocity 0 = a constant i o=
7i
ZL
solution velocity
=
T h e equation may be an oversimpiified model of crystal growth. Interpretation of the equation is as follows: The mass transfer to the surface (ro p u ) consists of two parallel processes, a diffusion effect? independent of velocity, and a flow effect dependent on velocity. The mass transfer is followed in series by the interfacial reaction. The rate of growth of crystals depends upon the rate of transfer of solute to the crystal surface and the rate of orientation of these molecules at the surface. Hixon and Knox (5), using a fundamental theoretical approach to the problem, related the mass transfer coefficient to the properties of the s o h tion.
+
mass transfer coefficient, grammole/(cm.z)(sec.)(mole fraction) D e = diameter of a sphere having same surface area as the crystal, cm. D,*= molal diffusivity, gram-mole/
k,
=
(cm.)(sec.)
=
solution
velocity
past
solution , , viscosity, gram/(cm.) (sec.) = average molecular weight of solution = constant
=
crystal,
cm./sec. = solution density, g r a m / ~ m . ~
CURVE
SUPERSATURATED REGION (LABILE)
/
/
dV
-
= nucleation
rate
(number/unit
dt vol./time) A = isothermal work of formation of a
nucleus Boltzman constant T = absolute temperature
k
=
The work of nucleation, A , is a function of the temperature and is infinitely large at the saturation curve. It decreases with increasing temperaturee.g., increasing supersaturation. The greater the energy required to form such a nucleus, the less probable is its formation. Although much has been written on the subject of nucleation. it is difficult to make direct use of these findings in an industrial crystallizer. I t is important, however: to design a crystallizer so that the supersaturation levels for nucleation are such that unwanted nuclei do not form. Particular attention must often be given to the temperature and supersaturation levels in recirculating lines. Cold spots can lead to the formation of a large number of nuclei and/or the clogging of lines.
Crystal Size and Crystal Size Distribution U N SAT UR ATE D
, I .-,
REGION
I
TEMPERATURE
~
+
Figure 1. This classical picture of supersaturation, although an oversimplification of the true state of affairs, is a powerful guide in explaining the driving force
608
mechanical shock or by the presence of impurities. Silver iodide is used to crystallize supercooled clouds in the rain making experiments. The closer the lattice parameters of the “impurity” to those of the substance being crystallized, the less the degree of super cooling required to cause nucleation. Where several hydrates are possible in a system, seeding with the desired form of hydrate can cause selective crystallization of that form. The critical nucleus size varies with the system being considered. For water, the critical nucleus is 80 molecules, while for barium sulfate, it is of the order of the unit cell. A basis for the understanding of the factors which influence nucleation is provided by the theoretical work of Becker ( I ) , Stranski (75), and Volmer (78). Becker postulated that the rate of nucleus formation is proportional to the isorhermal work of formation of a nucleus.
INDUSTRIAL AND ENGINEERING CHEMISTRY
One of the most important considerations in many crystallization processes is the crystal size distribution (CSD). Specifications as to size and size distribution are often imposed by the consumer of a crystalline product. The basis for such specifications may hinge upon sound technical reasons or perhaps upon sales requirements such as product appearance. Sometimes limits are placed on both extremes of the CSD-i.e., no more than a certain weight per cent greater than a
CE/THLORY given size and no more than a certain weight per cent finer than a given size. Such specifications can readily be met by passing the product through a sieving operation or by increased crystallizer complexity. Only under the most fortuitous circumstances can one expect the specifications imposed on both ends of the CSD to be easily met. For any given mode of crystallizer operation, the upper and lower limits of the CSD are intimately related and cannot be arbitrarily varied with respect to each other. There are certain important conclusions that can be drawn from theory which permits "prediction" and "control" of the final product size. As was previously pointed out, both nucleation and growth rates depend upon the same driving force, supersaturation. Nucleation is of a higher order dependence on supersaturation than is growth. A finer size product is obtained a t high supersaturations-Le., short residence time, high feed concentrations, high evaporation or cooling rates. Growth rate = .f (supersaturation)z, length/ time Nucleation rate = f (supersaturation)y, number/time since y > x , the crystal size index decreases with increasing supersaturation growth rate nucleation rate
-
length number
-
HIGH
LOW
SUPERSATURATION
Figure 2. finer
1.0
0.9
) .
As the supersaturation driving force increases, product will become
I
I
I
I
1
I
I
I
I
I
I
I
I
I
t
an index of crystal size
This same information is presented graphically in Figure 2. I n batch crystallization processes, the level of supersaturation continually changes, making it extremely difficult to "tailor make" a product size except within very broad ranges. Continuous crystallizers operate within relatively narrow supersaturation ranges permitting much better control of the CSD to be exercised. Any analysis of the CSD suffers from a number of more or less important deviations from ideal conditions. At high agitation rates, or when crystals are fragile (because of shape or structure) attrition may become a significant source of fines. Under conditions of poor mixing or high supersaturation agglomeration may become an important factor. Imperfect mixing can also lead to deviations from the ideal situation in a number of ways. Imperfect mixing can lead to local cold or hot spots where the nucleation and growth are not what is designed for. Representative product withdrawal in a continuous operation is also made difficult when the slurry is not perfectly mixed. If a perfectly mixed qituation is not obtained within a crystallizer, then variations in growth rate, as a function of crystal size, will be observed because of differences in settling velocity.
GROWTH OF IO "UNITS"
0
I
2
3
4
5
7
6 SIZE
8
9
IO
I/
12
13
1 4 1 5
"I("
Figure 3. Size distribution of the product crystals depends on size distribution of the seeds. These curves result when seed size i s normally distributed
Crystal Size DistributionBatch Operation Case I. Uniform Seed Size. The simplest case for studying crystal size distribution is in a seeded batch process. Even in this situation a number of simplifying assumptions are necessaryif these assumptions are not made the analysis becomes quite complex. All the seeds are of uniform size Spontaneous nucleation does not occur 0 Attrition is negligible
0 0
Agglomeration is negligible Crystals are invariant (shape is not a function of size) 0 Growth rate is independent of size (all crystals grow at the same linear rate) 0
0
If all of these assumptions are valid the analysis of the CSD is trivial. A study of this nature was conducted by Palermo (72) using a rotating drum crystallizer developed by Grove (4). The following equation was derived and experimentally verified for potassium alum. VOL. 53, NO. 8
AUGUST 1961
609
0.8
where
weight of product crystals weight of seed crystals number of crystals growth constant A C = supersaturation 6 = time
W, = Wa = n = K = C
0.6
T
If the supersaturation is held constant throughout the run, a very simple expression results :
z 0 c
2
0.4
WP1!3- Ws1!3 = K.AC.6.n'/3
LL
(6)
LL
since
$
w
'3
= =
0.2
P
1, - /, 0
8
12
16
MESH
20 SIZE
30
40
50
70
____f
t C R Y S T A L .SIZE
"I"
Figure 4. Both proper weight and proper size of seeds are required to produce the required product size distribution A.
Typical product size distribution obtained in a batch crystallizer lunseeded) 5. Distribution obtained b y seeding with a small amount of coarse crystals. In this case, too little surface area was introduced into the system resulting in o bimodal distribution. The peak at the left i s "grown up" seeds, the peak at the right i s due to spontaneously formed crystals. The peak at the right corresponds to even finer crystals than in the unseeded case because some of the solute deposited out on the seeds and was unavailoble for growth on the spontaneously formed crystals C. Corresponds to too many fines used in seeding. This results in a narrower distribution of finer average size
100
80
5
60
w V
LT w Q
c r 2
40
W
3
= =
volume crystal density
(71
K'.AC.6.
Case 11. Nonuniform Seed Size. If all the assumptions stated above are valid, except that a distribution of seed sizes (rather than a single seed size) is used, a somewhat more complicated situation results. Suppose a crystallizer were charged with a normal distribution of seeds-or for that matter any distribution of sizes. Such a distribution is generally represented as weight fraction of a given size. The distribution of product crystals may skew to the right or left, may have a higher or lower mode depending upon the seed size distribution. An example of this is shown in Figure 3. I n the limit, a normal distribution of seeds leads to a narrow distribution skewed to the left. The fact that the weight rate of growth is a cubic function of the size IS the reason for the "distortion" of the distribution as the crystals grow. Virtually any product size distribution may be obtained depending upon the distribution of the seeds. Case 111. Spontaneous Nucleation Does Occur. A more common situation involves seeding in a batch operation in order to produce a more desirable product-quite often seeding is employed to obtain a coarser, more uniform product. Schlichtkrull (74)investigated the crystallization of insulin crystals, in which the product consisted of grown seeds as well as spontaneously formed nuclei. In this study, a uniform seed size was used. We = knld
20
W, n
I, k 0 0
2
3
3.67 4
6
4
RELATIVE SIZE L --
Figure 5.
6 10
a13.p
crystal size = shape factor relating size and
1 a
W
Cumulative and differential distribution for mixed product removal
INDUSTRIAL AND ENGINEERING CHEMISTRY
IO
(8)
weight of seeds number of seeds = size of seeds = proportionality constant which includes the crystal density as well as shape factor = =
The total weight of crystals obtained from the batch is rVP. aWp
=
kn23
(99
CE/TREORY where
1
(1
Distribution Equations
= weight fraction of product
a!
-
grown from the seeds = size of grown up seeds a ) = fraction of spontaneously formed crystals
or
If no crystals are formed spontaneously, a = 1 and the maximum crystal size is obtained.
As a n example of the problems which may be encountered in seeding a batch operation for the purpose of obtaining a coarser product, consider the distributions shown in Figure 4. Proper amounts (weight and size) of seeds are required to produce the required product size distribution. I t is clear from these examples that seeding does not necessarily produce a more uniform, coarser product.
Crystal Size Distribution in a Continuous Operation Continuous crystallizers have long been used commercially. The design and operation of such units has greatly depended upon experience and knowhow with little underlying theory to draw upon. In recent years, several papers have been published (3, 73) which serve as a basis for rational design and operation of crystallizers. T h e following discussion refers to a continuous operation under steady-state conditions. Case IV. Perfectly Mixed CrystalIn the lizer (Mixed Withdrawal). analysis of the perfectly mixed crystallizer the following assumptions are made:
The suspension is completely mixed The system is at steady state The numerical rate of product withdrawal plus fines equals the nucleation rate The weight rate of product withdrawal equals the crystallization rate The shape factor is constant (the crystals are invariant) The following equation is a modified form of the one derived by Saeman (73).
Property
Differential
Weight
e-X.X3
Area
e-X-X2
Length Number
e-X.X e-x
Cumulative
a xz - e--x x - +6 2
+X + 1 X2 1 - e-x - + X + 1 2 1 - e - X [ X + 11 1 - e-x 1
where
I = crystal size under consideration Wc = cumulative weight of crystals smaller than “I” Wt = total weight of crystals i = linear growth rate e = residence time (since the crystallizer is well mixed with representative withdrawl; the crystal, clear liquor and magma residence times are equal) L = ie = size of the crystal which has grown for the nominal residence time The use of relative size (1/L) permits analysis of crystal size distribution without detailed knowledge of the mathematical relation between growth rate and supersaturation. The differential CSD for the above case is given by the following equation:
w t= e-x.xa
(14)
These two distributions are shown in Figure 5. With the above expression as a starting point, other equations may be derived, describing the cumulative differential number, length, and area distributions. I n addition, useful properties of these distributions, such as the mean, mode, and median, are readily obtained. The differential weight distribution is f’(X),
= C-X
Mode
x 3
6
Value of X Mean Median
3
3.67
4
2
2.67
3
1 0
1.67 0.67
2 1
Case V. Classified Withdrawal. If the crystallization operation is carried out in the same manner as described above-perfectly mixed operation-except that the product crystals are withdrawn a t full size only, the CSD within the crystallizer is of a different form. In this case exact knowledge of the form of the distribution is not as important as it is in the case of mixed withdrawal. For classified withdrawal, there is only one product size which depends upon the operating parameters and the properties of the system under consideration. I n classified withdrawal, Saeman (73) has shown that the age of the product crystals is four times the nominal crystal residence time (total weight of crystals in suspension/production rate).
literature Cited (1) Becker, J., Doring, W., ilnn. Physik 24, 719 (1935). (2) Berthoud, A., J . Chem. Phys. 10, 624 (19 12). (3) Bransom, S. H., Dunning, W. J., Millard. B.. Disc. Far. Sac. 5. 83 (1949). (4) Grove, C. S., Jr., Ph.D. disseAation, University of Minnesota, Minneapolis, Minn., 1940. (5) Hixon, A. W., Knox, K. L., IND. ENG. CHEM.43, 2144 (1951). (6) LaMer, V. K., Ibid., 44, 1270 (1952). (7) McCabe, W. L., Stevens, R. P., Chem. Eng. Prog. 47, 168 (1951). (8) Miers, H. A., Isaac, F. J., J . Chem. Sac. 89, 413 (1906). (9) Nernst, H. W., Z. Phys. Chem. 47, 52 II 904). - .,. (10) Noyes, A. A., Whitney, W. R., Z. Physzk. Chem. 23, 689 (1897). (11) Ostwald, W., Ibid., 34, 495 (1900). (12) Palermo; J.. A., Ph.D. dissertation, Syracuse University, Syracuse, N. Y., 1952. Saeman, W. C., A.I.Ch.E. Journal 2, 107 (1956). (14) Schlichtkrull, J., Acta Chem. Scand. 11, 3 (1957). Stranski, I. N., 2.physik. Chem. 136, . . \ - -
The mode is found by setting the first derivative of Equation 14 equal to zero and solving for X. X has a value of 3. The median value of X is 3.67-when f ( X ) = 0.5. The mean is calculated using the expression below
Sorn
Xf’(X)dX
The differential number distribution is obtained by dividing the differential weight distribution by A?.
( 1 $ ‘ g 8 h , H., McCabe, W. L., IND. ENG.CHEM.34, 1201 (1934). 117) Van Hook. A , , in “Principles of Sugar ‘ Technology,’; Honig, ed., V d . 11, Chgps. 3 & 4, Elsevier, Amsterdam, 1959. (18) Volmer, M., “Kinetik der Phasenbildung,” Dresden-Leipzig, 1939.
In a like manner, a complete set of
RECEIVED for review March 20. 1961 ACCEPTED March 20, 1961
expressions characterizing all the possible distributions may be determined. These are summarized in the table.
Division of Industrial and Engineering Chemistry, 139th Meeting, ACS, St. Louis, Mo., March 1961. VOL. 53. NO. 8
AUGUST 1961
61 1
