298
Ind. Eng. Chem. Fundam. 1982, 21, 298-305
L = Lagrangian function, $/yr Q = heat duty, Btu/h t = inlet temperature, O F Ts, = steam temperature for the ith exchanger, O F T = outlet temperature, O F U = overall heat transfer coefficient, Btu/h ft2 OF W, = steam flow rate, lb/h w = cooling water flow rate, lb/h x, = arbitrary input variable X , = arbitrary output variable
c L
I
t2 9 5 r
I2 90c
Greek Letters A, = Lagrange multiplier, $ / O F yr 4 = Lagrange multiplier, $ / O F yr 8, = extra temperature variables introduced in problem reformulations 12I 1 I20
I25
I30
I
I35
*
Subscript BC = base-case
X2
Figure 9. Projection of primal and dual functions onto the C vs. x 2 plane. Primal function is given by C(x2)= 14x2 - 36x2 + 36 and the dual function is given by h(x2)= 2/3rz (30 - 3 5 / 2 x 2 ) .
Superscripts + = optimum value * = value satisfying the necessary conditions for optimally but not necessarily the constraints
where xt solves the primal over E,nS. That is, the maximum of the dual function lies below the minimum of the primal and a dual gap is said to occur. (2) The conditions d L / d x i = 0 cannot be used to formulate the dual function because they do not correspond to minimizing L(x,X) at each A. Nomenclature A = area, ft2 C = cost function, $/yr CAi = cost term in linearized cost expression for the ith exchanger, $/ft2 yr CBC= base-case cost, $/yr Coi = cost term in linearized cost expression for the ith exchanger, $/yr C, = heat capacity, Btu/lb O F Cs = steam cost (value), $/(lb/h) yr Cw = cooling water cost, $/(lb/h) yr F = flow rate, lb/h AH = latent heat of vaporization, Btu/lb h = dual function, $/yr
Literature Cited Avery, c. J.; FOSS, A. s. AICM
J . 1971, 17, 998-999. Brosilow, C.; Lasdon. L. AIChE-ICM Symp. Ser. No. 4 1965, 75-83. Brosiiow, C.; Nunez, E. Can. J . Chem. Eng. 1968, 46, 205-212. Guthrie, K. M. Chem. Eng. 1969, 76(6),114. Hancock, M. “Theory of Maxima and Minima”; Dover: New Ywk, 1980. Happel, J.; Jordan, D. G. “Chemical Process Economics”, 2nd ed.; Marcel Dekker: New York, 1975. Lasdon, L. S. “Optimization Theory for Large Systems”; Macmillan: New York, 1970. McGailIrd, R. L.; Westerberg, A. W. Chem. Eng. J . 1972, 4 , 127-138. Stephanopoulos, 0.; Westerberg, A. W. A I C M J . 1973, 19, 1269-1271. Stephanopouios, G.; Westerberg, A. W. Can. J . Chem. Eng. 1975, 53, 551-555. Stephanopouios, G.; Westerberg, A. W. Chem. Eng. Sci. 1976, 3 1 . 195-204. Westerberg, A. W.; Stephanopouios, G. Chem. Eng. Sci. 1975, 3 0 , 963-972. Wismer, D. A., Ed. “Optimization Methods for Large-Scale Systems...with Applications”; Chapter 1, “Static Multilevel Systems”, by Schoeffler, J. D.; McGraw-Hill: New York, 1971.
Receiued for reuiew February 16, 1981 Accepted February 25, 1982
Precipitation Kinetics of Magnesium Hydroxide in a Scaling System Bahram Dablr,’ Robert W. Peters,‘* and John D. Stevens3 Department of Chemical Engineering and The Engineering Research Institute, Iowa State Universitv. Ames,
Iowa 500 1 7
The crystallization of sparingly soluble salts such as magnesium hydroxide and calcium carbonate in the lime-soda ash water softening process presents several obstacles to kinetic analysis. Both systems deposit scale and both usually exist with nonstoichiometric ratios of ions. The v a l i i i of the population balance analysis and a working definition for supersaturation are discussed. Data for magnesium hydroxide and a correlation of kinetic order with hydroxide alkalinity are presented.
Introduction Present federal water pollution control standards and those becoming effective in the future require economical
methods for removal of impurities from liquid waste streams. Since the concentration of contaminants in such waste streams is often very low, it is an engineering challenge to find economical ways to remove the impurities. This research effort concerns the use of crystallization as an impurity removal technique. Specifically, our research program has three primary objectives: (1) to evaluate the feasibility of Using crystallization as an economical removal technique for sparingly soluble impurities; (2) to determine the nucleation
Department of Chemical Engineering, Michigan State University, East Lansing, MI 48824. Address correspondence to this author at Environmental Engineem, &hml of Civil Engineering, Purdue University, West Lafayette, IN 47901. Deceased. 0196-4313/82/ 1021-0298$01.25/0
0
1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 299
l Impeller
the crystallizer results in chemical reaction to produce magnesium hydroxide which, because of its low solubility mol/L), precipitates out. Our objective was to measure the rate of this precipitation as a function of various operating parameters and chemical conditions and at the same time to characterize the size distribution. Use of the population balance analysis approach along with the common assumption of size independent growth yields (Randolph and Larson, 1971) n = no exp(-L/GT)
Figure 1. Schematic diagram of crystallizer.
and growth kinetics in the lime-soda ash water softening process for the sake of better design and operation of softening plants; (3) to learn more about fundamental mechanisms in all crystallization processes through the study of specific behavior exhibited by dilute systems but not by more concentrated systems. The discussion herein concerns experimental studies performed to determine the precipitation kinetics of magnesium hydroxide. A brief review of the water chemistry and thermodynamic fundamentals involved is presented. Background Water Chemistry. Hard waters are those containing objectionable amounts of calcium and magnesium salts, usually present as bicarbonates, chlorides, or sulfates. If not removed, these salts form insoluble precipitates with soap. Calcium sulfate, carbonate, and silicate form clogging scales in industrial equipment. The lime softening process involves the addition of slaked lime, Ca(OH)2,to waters containing calcium and magnesium. The hydroxide raises the pH which shifts the carbon dioxide-bicarbonatecarbonate equilibrium in the direction of increased carbonate and thus promotes the precipitation of calcium carbonate. The hydroxide also reacts with magnesium ions to yield magnesium hydroxide precipitate at high pH (- 11). Mg2++ 20H- s Mg(OH),J
(1)
This paper reports on a study of magnesium hydroxide crystallization kinetics in a calcium-free environment. It is interesting to note that most industrial and municipal softening plants are designed and operated with only consideration of equilibrium reactions. The dynamics of the precipitation step are not generally considered, due to lack of data and kinetic models. It is standard practice in the softening industry to refer to concentrations on the basis of CaC03 equivalents expressed as mg/L (or ppm for water systems) of calcium carbonate. Water containing x ppm hardness has the same number of calcium and magnesium equivalents as water containing x milligrams of calcium carbonate per liter. Other ions are expressed as ppm calcium carbonate by equating the number of equivalents of these ions to the number of equivalents of calcium in calcium carbonate. This convention is used here, and all concentrations are expressed as equivalent CaCO,. Steady-StateCrystallization. Figure 1is a schematic diagram of the continuous reactor-crystallizer used in this study. Inlet streams 1and 2 represent feed solutions of magnesium chloride and sodium hydroxide, respectively. Lime was not used as the source of hydroxide because of the tendency of calcium to react with carbonates in the water to form CaC03. The mixing of the feed streams in
(2)
where n = population density, no./mL-pm, n = nuclei population density, no./mL-pm, L = characteristic crystal size, pm, G = growth rate, pm/min, and 7 = drawdown time, min. Equation 2 provides a functional relationship between the size, L, and the population density, n, and thus characterizes the size distribution. Equation 2 may be used to generate expressions for the zeroth, first, second, and thiid moments of the size distribution which represent total numbers, length, area, and mass, respectively (Randolph and Larson, 1971). Nucleation and Growth Kinetics. The actual crystal sizes obtained in a crystallizer are dependent upon the parameters in eq 2. The drawdown time, T , is arbitrary, but the nuclei population density and the growth rate are related to the fundamental crystallization kinetics. If eq 2 successfully models the size distribution data, then semilog plots of n vs. L yield the growth rate, G, and nuclei population density, no. The birth rate, Bo, is available from
Bo = noG (3) The relationship of the supersaturation driving force to nucleation and growth rates is of considerable importance. At constant temperature, the nucleation and growth rates can often be modeled with simple power law models which, when combined, yield Bo = kNG' (4) where k N = constant and i = kinetic order. The values of k N and i may be experimentally determined from a series of runs at different supersaturation levels. This yields a series of Bo and G values which can be fit to eq 4 to yield the model constants. Application to the Softening Process The application of the above described analysis procedure to the lime-soda ash softening process requires that two potential obstacles be dealt with. First, the lime softening process operates at pH levels such that deposition of precipitates on interior surfaces is unavoidable. Therefore, the effect of this scale on the kinetic determinations must be considered. Second, in the more common concentrated crystallization systems the concept of supersaturation is straightforward, since the solvated ionic species are usually present in stoichiometric ratios. However, in the water systems being considered here the ions are not usually present in any particular ratio. Thus, the concept of supersaturation as a difference in concentrations of molecular species must be reexamined. The following discussion concerns these matters. Wall Scale. A persistent problem encountered during the study of magnesium hydroxide and calcium carbonate precipitation in the softening process is the tendency of these materials to cling to internal surfaces of vessels and conduits. An appreciable scale layer develops during the course of a run and is similar to the scale layer observed in commercial practice. The scale appears to be independent of the materials of construction and may be
300
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
1
I
Case I : Suspension with wall deposition.
Figure 3. Population density with and without classified product removal.
Case I I : Suspension with no deposition.
Figure 2. Effects of wall scale.
contributed to by the irregular habit of the crystals. The effect of this scale on the crystallization kinetics and crystal size distribution is of prime importance to this research. For purposes of discussion, consider the two cases illustrated in Figure 2. Case I is a schematic of a process in which the fraction a of the solids production (QCi- QCJ remains in suspension while the fraction (1- a)of the total solids production deposits on the walls and other surfaces. Case I1 represents a similar system except that since no wall deposition occurs the feed concentration of solute need only be some fraction a of the case I inlet concentration in order to produce an effluent stream with the same suspension density as in case I. Since the same suspension density, MT,is produced in both cases for the same volumetric flow rate, Q, case I may be thought of as having a second effluent stream in which the fraction (1- a) of the total solids produced leave. The fundamental question is whether the crystal size distribution (CSD) in the crystal suspension is the same in both cases. If the two CSD’s are identical, it may be concluded that the scale has no effect on the kinetic behavior of the crystallizing system. The measured crystal size distribution of the suspension is not affected if the wall deposits are a representative sample of the crystal suspension. If this is not the case and the smaller particles preferentially adhered to the walls, the crystal size distribution would appear similar to that of the classified product removal system illustrated as line A in Figure 3. LPIis the size below which smaller crystals would preferentially adhere to the walls. Similarly, if crystals larger than size L,, preferentially adhered to the walls, then the expected population density distribution (solid line in Figure 3) would exhibit a break as illustrated by line B in the figure. In practice one would not expect the breaks in the population density curve to be as sharp as illustrated in Figure 3, but rather to generally resemble an inverted concave curve. For all experiments conducted during the course of our studies on
the lime-soda ash softening process we have not observed behavior resembling that of a crystallizer with classified product removal (see Figure 5, for example). This observation is based on Coulter Counter measurements over the size range of 2.8-56 pm. Since the crystal size distribution is apparently unaffected by the wall deposition, the nucleation rate, growth rate, and kinetic order i in eq 4 are similarly unaffected. It appears that one may think of the wall deposition as layers of crystals, one upon another. The underlayers are insulated from fresh mother liquor and may be thought of as “dead” crystals. The crystals at the surface probably exhibit some growth but at a much slower rate than crystals in suspension. This growth of wall crystals, although inhibited, nevertheless may cause the supersaturation level in the crystallizer to be slightly lower than in the case with no wall deposition. In the conduct of the experiments reported here, an attempt was made to maximize the volume to surface area ratio in the crystallizer in order to minimize the wall deposition effect. Some runs were also made in which the volume to surface area ratio was changed, but no changes in kinetic behavior were observed in those runs. Previous Studies on Magnesium Hydroxide Precipitation. Magnesium hydroxide precipitates from highly supersaturated solutions as a gelatinous colloid. Many investigators [ (Shirasaki (1961); Murotani and Shirasaki (1961); Chauhan et al. (1968);and Tagai and Saito (1968)l have been concerned with the development of a method for the preparation of well-characterized crystallites of the salt. Few studies have been concerned with the crystallization kinetics of magnesium hydroxide. Published values of the solubility of magnesium hydroxide cover a wide range of values [Liu and Nancollas (1973); Nasanen (1941); Gjaldback (1925); Phillips et al. (1977); Wiechers (1977); Travers and Nouvel (1929); and Ryznar et al. (1946)j. Gjaldback (1925) claimed that magnesium hydroxide existed in two well-defined modifications with a more soluble labile form and a less soluble stable form. More recently, Klein et al. (1967) reported the results of a homogeneous nucleation study of magnesium hydroxide. They suggested that the number of magnesium hydroxide stoichiometric units in a critical nucleus is 33. From their experiments, an initial critical concentration product, log [M$*] [OH-I2 = -9.2, was required to initiate homogeneous nucleation of magnesium hydroxide. After an initial surge of growth, Liu and Nancollas (1973) found that an equation first order with respect to supersaturation could be used to explain their data. A surface diffusion process was
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 301
proposed as the controlling step in the crystallization process. Several investigators have studied the kinetics of magnesium hydroxide in continuous flow systems. Munk (1976) studied the removal of magnesium chloride with varying lime dosages at different residence times in a continuous well-mixed reactor. He used an initial magnesium hardness of 150 and 250 ppm as CaC03. These hardness levels are higher than those expected in most water softening operations. The softening results are probably in error because he assumed the lime dosage remained constant throughout the run, after measuring the concentration a t 2 to 3 residence times. Carbon dioxide, however, causes a side reaction with the lime, according to the reaction C02 + Ca(OH)2 F? CaCO31 + H 2 0 This lowers the lime concentration with time. This has the effect of not having the same lime dosage for the hardness during runs of differing residence times, due to the fact that the lime concentration has lowered more at 10 residence times in the 40-min residence time runs than for the 20-min residence time runs. The kinetic exponents are therefore somewhat in question. Hartwick and Schierholz (1976) studied the precipitation of magnesium hydroxide using magnesium chloride and sodium hydroxide feed streams. The concentrations of both feed streams were varied and studied using different reactor residence times. To prevent crystal buildup in the system, the reactor was scraped every 5 min with a rubber spatula. While this has the advantage of allowing the system to level out at a study state level for the feed concentrations and crystal size distribution (CSD), several of the basic assumptions in the development of the MSMF’R model are no longer valid, i.e., no crystals in the feed (by mixing the crystals back into solution) and no attrition or agglomeration (as crystals may be broken or agglomerated with the action of the spatula). The particles scraped back into solution may serve as sites for nucleation. Allowing the accumulation of solids on the wall would be a better technique to not invalidate the basic assumptions used. Though steady state may not be truly achieved, this can be incorporated into the model. A pseudo steady state might be achieved where the CSD and feed concentrations tend to level out. The particles scraped back into solution serve to increase the area for deposition, thereby lowering the supersaturation. Since the surface area is higher, the growth rate will be smaller. With the nuclei density (no) being a function of supersaturation, it will likewise decrease. The number density n = no exp(-L/GT) will decrease due to the decrease in both no and G. The kinetics of the distribution are such that the nucleation more nearly resembles growth kinetics, with the result being the kinetic exponent i will be lower than would otherwise be expected for the nucleation kinetics. The kinetics thus tend to be more growth oriented. Driving Forces for Birth and Growth. The fundamental driving force for crystallization is the difference between the chemical potential of a substance in solution (state 1) and in the crystal form (state 2). This may be written as (5) AP = P2 - P1 For a crystallizing system, Ap C 0. To avoid negative driving forces it is convenient to define r#J as (6) 4 = AP = 1.11 - CLZ Chemical potential may be defined as p = po + R T l n a (7)
where po is the standard state and a is activity. By choosing the same standard state for crystal and solute we may combine eq 6 and 7 to give
r#J/RT= In (a,/a,*)
(8)
where a, and a,* are the activities of solute at the actual and equilibrium concentrations, respectively. It can be shown for dilute systems near equilibrium (Mullin, 1977) that
a,
( g ) v
a,*
where C is concentration and v is the number of moles of ions formed from one mole of solute. The ratio (C/C*) is called relative supersaturation. Substitution of eq 9 into eq 8 yields r#J/RT= v In (C/C*) (10)
For values of (C/C*) near 1a Taylor expansion of the In term yields
r#J/RT= v(C/C* - 1) = v(C - C*)/C*
(11)
Since RT, v, and C* are constant at a given temperature, we may write = k(C -
C*) (12) where in this work we refer to (C- C*) as supersaturation, r#J
s. Equation 12 shows the driving force in crystallization
systems is proportional to supersaturation. The above derivation applies to electrolytes if the concentrations and activities are thought of as mean ion concentrations and mean ion activities, respectively. That is
C = 12,”;a = atY
(13)
where C,” = (C+.+)(C-lr),etc., and v+, v- are stoichiometric Coefficients of the cation and anion, respectively. In this case, eq 12 becomes I$
= kT(C,’+)(C_‘) - (C,*”+)(C*l-)]
(14)
Equation 14 implies that the driving force for crystallization is proportional to the difference between the ion product of the solute in solution and its value at equilibrium (i.e., the solubility product). As written, eq 14 applies for concentration solubility products. For the concentration levels encountered, these values are close to those of the activity solubility products, as indicated in eq 9. The effect of the activity can be included in eq 14 by including the appropriate activity coefficients in that equation. Liu and Nancollas (1973) have addressed this topic. Thus, since the systems under study here do not usually possess ionic species in proportion to the stoichiometric coefficients of dissociation, eq 14 provides a means of calculating the supersaturation driving force from knowledge of ion concentrations alone. The use of ion products to follow supersaturation has experienced considerable success in sparingly soluble salt systems [Nielsen (1955,1964,1979,1981);Davies and Jones (1949,1955);van Leeuwen (1979); Swinney (1979); Peters (1980), Peters and Stevens (1981); Peters et al. (1981) and Klein et al. (1959)l. Davies and Jones (1949, 1955) found that silver chloride crystals in nonequivalent solutions grew with a rate that was a function of the ion product ?r = [Ag+][Cl-]. The rate was: r = K , ( T ~-/K,,1/”)2. ~ Nielsen (1979, 1981) showed that many electrolytes crystallizing from aqueous solution follow a parabolic rate law with the linear growth rate proportional to ( d I ” - Ksp1/y)2 where v
302
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
Table i. Chemical Analysis of the Reactor Effluent and Measured Growth and Nucleation Rates with Feed Containing 75 ppm NIgz Hardness +
run no.
NaOH exces
series
min
effl. PH
21 22 8 13 17 18 19 14 24 25 27 23
20% 20% 20% 20% 0% 0% 0% 0% -20% -20% -20% -20%
I I I I I1 I1 I1 I1 I11 I11 I11 I11
20 20 30 40 20 30 40 40 20 30 30 40
10.47 10.60 10.65 10.62 10.40 10.23 10.43 10.62 10.25 10.18 10.20 10.25
7,
series avpH 10.59
10.41
10.22
is the number of ions per formula unit. This can be explained by the Burton-Cabrera-Frank theory adapted for electrolytes, assuming that the ions are adsorbed in equivalent amounts on the crystal surface. Klein and Gordon (1958) performed nucleation studies on barium sulfate using the supersaturation ratio S = (Kimpmt/Ksp)l/z to control supersaturation allowing the precipitate crystal growth to be studied. Nielsen (1979) used the same type of analysis involving the supersaturation ratio to study the growth rate of calcium carbonate, barium sulfate, and silver chloride from aqueous solutions. Liu and Nancollas (1973) studied the growth of magnesium hydroxide seed crystals in supersaturated solutions. They found that the expression dt
effl. Mg2+,
series avMg2+, P P ~ ppm 33 33 38 46 48 48 47 46 58 56 58 55
37.5
47.2
56.8
G,
fim/ min 0.10 0.11 0.08 0.06 0.30 0.22 0.1 7 0.16 0.26 0.17 0.17 0.14
no x
lo-",
36.0 37.5 23.0 13.4 1.80 1.47 1.25 1.28 0.130 0.125 0.140 0.130
vL
CONTROLLER
___. T M S
Bo X
no./rm-mL no./mL-min 37.0 42.0 18.0 8.1 5.4 3.3 2.1 2.0 0.34 0.21 0.23 0.19 RPM ,COKlffOLLER
% TEW. M T H
Figure 4. Experimental equipment flow diagram.
([Mg2+][OH-]2)1/3 - ($)1'3]n
adequately described the data. After a rapid initial surge of growth, an equation first order with respect to supersaturation described the growth of magnesium hydroxide crystals. Peters (1980), Swinney (1979), Peters and Stevens (1981), and Peters et al. (1981) used the supersaturation ratio to determine the phase transformation from aragonite to calcite in calcium carbonate precipitation. The use of ion product ratios or linearized differences is a conventional definition of supersaturation in a multi-ion system (Randolph, 1981). The problem in more complicated systems lies in the calculation or measurement of the species concentrations. This often necessitates equilibria computer programs or specific ion electrodes. For example, in this magnesium hydroxide system, hardness titrations, alkalinity titrations, conductivity measurements, and temperature measuremeats are all required to determine the supersaturation. The hardness titrations involve titration for calcium and total hardness with EDTA (Diehl, 1974),with magnesium hardness assumed to be the difference between total and calcium hardness. The hydroxide concentration is determined from the pH measurement, with the bicarbonate and carbonate alkalinity determined from the initial pH and total alkalinity using equilibria considerations (Sawyer and McCarty, 1978). The conductivity measurements are required to calculate the activity coefficients for the temperature involved. All the measurements cause some degree of error (up to about 10%) in the calculated ion product value, so that the ion product definition is not without its problems in experimentally determining the various parameters involved in the definition of supersaturation.
Experimental Section The flow diagram for the experimental equipment is shown in Figure 4. Feed solutions of magnesium chloride
and sodium hydroxide were pumped through 0.45-pm filters to remove foreign particles and then passed through a constant temperature bath to maintain the crystallizer temperature at 25 f 0.05 "C. The feed stream volumetric flow rates were adjusted in order to maintain 16 L of magma in the crystallizer at the desired drawdown time. The levels of drawdown time used in this work were 20, 30, and 40 min. Steady-state operation was generally achieved after 9 to 10 residence times and most of the data were collected between 14 and 18 residence times. A Coulter Counter Model TAII with a 140-pm micron aperture tube was used to determine the crystal size distribution. Details of specific chemical analysis performed, and other experimental details are available elsewhere (Dabir, 1978).
Results and Interpretation Nucleation and Growth Rates. Because the population densities are very low for precipitating systems of calcium carbonate and magnesium hydroxide, it is more accurate to plot In N vs. size L, where N is the total number of particles per unit volume in the size range L to m. It is easy to show that integration of J-L- ndL yields N(L,m) = nOGT exp(-L/GT)
(15)
Thus, a plot of In N vs. L will have a slope of ( - ~ / G Tand ) an intercept of In (nOGr). Figure 5 is an example of the sue distribution plots obtained for each experimental run. The data were fit using a linear regression and, in the particular case shown in Figure 5, a growth rate of 0.16 Km/min and nuclei density of 1.3 X lo4 were computed. The birth rate, B O , is computed from n"G and equals 2.1 x IO3 no./mL-min. Several runs were made at each of three different NaOH/Mg2+feed ratios. The ratios used were 20% excess, stoichiometric, and 20% deficiency of the theoretical
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 303 12 -
Growth Rate 0 ,
pm/min
Figure 6. Determination of i for series I, 11, and 111. HYDROXIDE A L K A L I N I T Y (as ppm COCO,)
I
0
I
I
2
I
I
I
4
[OH-] x 1 0 4 , moles/I
Figure 7. Change of kinetic exponent i with hydroxide concentration.
OH- concentration and because softening plants are operated using pH or hydroxide alkalinity as a control variable, an attempt was made to correlate i with these control variables. This effort results in the correlation shown in Figure 7,which shows that it is approximately a linear function of hydroxide ion concentration. Since in industrial practice it is common to describe hydroxide concentration in terms of hydroxide alkalinity, defined as hydroxide alk (as CaC03) = 50000
X 10(pH-pKw)
the figure also shows an hydroxide alkalinity scale. Using linear regression analysis, the kinetic order i was correlated
304
Ind. Eng. Chem. Fundam.. Vol. 21. NO. 3, 1982 Table 111. Percent Change in G and L D as T is Doubled and M T Remains Constant
7
1.0
-!
-, -,
1.6 2.7
-
1.7
Table IV. Percent Change in G and LD with 10-Fold Increase in MT and No Change in T
i 1.0 Figure 8. Photomicrograph of magnesium hydroxide from run 18 (3000x1.
with hydroxide concentration and hydroxide alkalinity to give i = 8100(OH-) - 0.43
(16)
and
1.6 2.7 7.7
G
Ln -
+ 78%
+ 78% + 65%
+65% + 50% +24%
+ 50% + 24%
where k , = a shape factor and p = crystal density as mass per unit volume. If MTisassumed constant for a series of runs at a given pH, eq 18 may be used to show that for crystallizations 1and 2, the following relations are true
(G,/G,) = ( T ~ / T J ~ / ' ~ + ~ )
i = 0.1621[OH-I
- 0.43
(17)
where parentheses indicate molar concentrations and brackets indicate the alkalinity expressed in mg/L as the calcium carbonate equivalent. The correlation coefficients r for both correlations were 0.9971. Industrial processes commonly remove magnesium hardness a t a pH of approximately 11(hydroxide alkalinity = 50). Extrapolation of the line in Figure 7 suggests that the kinetic order in a process operating at pH 11is approximately 7.7. Kinetic orders of this magnitude are uncommon and thus this suggests that extrapolation of the data in Figure 7 may be invalid. Further studies are being performed to substantiate the behavior at high NaOH/Mg*+ ratios. The authors do not believe the variation in the kinetic order i was due to an entrance region mixing effect. Preliminary studies (Munk, 1976; Dabir, 1978) showed locating the feed stream inlets at various points in the crystallizer had little effect on the crystal size distribution. No noticeable difference was observed in the resulting crystal habit. The conclusion reached was that the kinetic order was not appreciably affected by entrance effects. The kinetic order was affected by the hydroxide concentration. This is due in part to the system being enhanced primary nucleation (Randolph, 1981). Strictly speaking, the system is self-induced heterogeneous nucleation, due to the presence of the other magnesium hydroxide crystals. This system is usually considered primary or homogeneous nucleation, due to the low concentrations involved. Crystal Morphology. Figure 8 is a photomicrograph of the Mg(OH), crystals mmt commonly observed in this research. The crystals shown in this figure were dried before being studied under a scanning electron microscope. Shirasaki (1961) and Munk (1976) reported similar blossom-petal patterns in magnesium hydroxide crystals. A similar habit for Mg(OH), crystalswas o k r v e d in system with simultaneous crystallization of Mg(OH), and CaC03 and from samples obtained from the Ames,Iowa, softening plant. Application of Results If MT represents the suspension density in units of crystal mass per unit volume of slurry, the suspension density may be calculated (Randolph and Larson, 1971) as
MT = 6kvpno(Gr)'
(18)
(L~,/LD,)= ( T 2 / T 1 ) ( i - 1 ) / ( i + 3 ) where LD = dominant crystal size = ~ G T . For a 100% increase in drawdown time (state 2), Table III summarizes the changes that occur in G and LDfor the case of i equals 1.0, 1.6, and 2.7. Also shown is the case of i equal to 7.7, the projected value a t pH 11.0. The results clearly show the implications of doubling the drawdown time. If the process is one operating so that i is approximately unity, no advantage is gained from the change since the dominant crystal size remains unchanged. However, if the system is operating so that the kinetic order is greater than 1,then increasing the drawdown time increases the dominant size, thus producing a crystal size distribution with larger crystals and improved settling properties. The disadvantage, of course, is the increased capacity required to permit the increased drawdown time. Another change commonly made in such softening processes is to hold the drawdown time constant but increase the suspension density by means of seeding with recycled sludge For this case we may show (G2/G1) = (MTJMT,)1/(H3) l/(i+3)
(LnJLD,) = (MTJMTJ
The changes that occur in G and LDfor a 10-fold increase in MTare illustrated in Table lV. In this case the systems with lower kinetic orders exhibit greater change, but in all cases G and Ln increase, thus yielding more favorable size distributions. It is interesting to observe that for a 1Wfold increase in MTthe increase in LD for i = 7.7 is only 54% above that for no seeding. Comparison of Tables I11 and IV shows that designers and operators have several options available to accomplish given objectives. For example, increasing the dominant crystal size may be achieved by either increasing T or MT. However, knowledge of the kinetics is critical for a correct decision. For the case of i equal 1, the choice is clearly to increase the suspension density, MT. On the other hand, for very large i the greater response of LD to an increase in T mav make increasing- T a viable economic alternative to sludge recycle. The above discussion illustrates the need for the value of t h e type of data reported herein. The kinetic data should permit much more intelligent decisions and judg-
Ind. Eng.
ments with respect to the design and operation of softening plants.
Summary Several ramifications of having cationlanion ratios of the precipitating salt in nonstoichiometric ratios have been considered. A definition of supersaturation for such cases has been proposed and the effects of scale have been examined. It has been experimentally observed that the kinetics of magnesium hydroxide are a sensitive function of pH. The growth rate, nucleation rate, and kinetic order were measured for three levels of hydroxide feed. The kinetic order i was correlated with crystallizer hydroxide concentration. The practical use of the kinetics determined in this research has been illustrated. Although the crystallization of Mg(OH)2occurs simultaneously with the crystallization of CaC03 in the commercial process, the authors have observed in other studies that the crystallizations of these two substances appear to be independent. Thus,the results presented here should be applicable to commercial processes. Acknowledgment The authors wish to acknowledge the support of the Iowa State University Engineering Research Institute and the National Science Foundation through grant number ENG76-17985. Nomenclature a = activity a, = activity of solute based on concentration a+ = mean ionic activity of solute based on concentration Bo = crystal nucleation rate, number/mL-min C = concentration of solute, mol/L Ci = inlet crystallizer concentration, mol/L C, = outlet crystallizer concentration, mol/L C, = mean molar concentration of solute, mol/L C, = concentration of cations, mol/L C- = concentration of anions, mol/L f = activity coefficient G = crystal growth rate, pm/min i = kinetic exponent relating nucleation rate to growth rate k , = rate constant for crystal growth kN = kinetic rate constant k , = kinetic rate constant k , = volumetric shape factor K, = ion product KSp= thermodynamic solubility product L = crystal size, pm LD = dominant crystal size, pm MSMPR = mixed suspension, mixed product removal MT = suspension density, mg/L n = population density at size L , number/mL-pm no = nuclei density, number/mL-pm N = cumulative number of crystals per mL pH = -log (H+) concentration pK, = -log (dissociation constant for water) Q = volumetric flowrate r = net rate of deposition = -dc/dt R = gas constant
Chem. Fundam., Vol. 21, No. 3, 1982 305
s = supersaturation = K , - K, S = supersaturation ratio = d / K , , t = time, min T = abolute temperature
Greek Letters A = operator meaning change p = chemical potential ?r = ion product p = crystal density, g/L v = number of moles of ions formed from 1mole of electrolyte v+,v- = stoichiometric coefficients of cation and anion, re-
spectively 4 = affinity of reaction (Mullin and Sohnel, 1977) T = drawdown time, min Superscripts * = equilibrium state - = overbar meaning average
Literature Cited Chauhan, D. A.; Shukla. B. K.; Datar, D. S. Trans. SOC. Advan. Electrochem. Sci. Techno/. 1988, 3 , 14. Dabir, B. M.S. Thesis, Iowa State University, Ames, Iowa, 1978. Davies, C. W.; Jones, A. L. Discuss Faraday SOC. 1949, 5 , 103-111. Davies, C. W.; Jones, A. L. Trans. Faraday SOC. 1955, 57, 812-817. Diehi, H. C. “Quantltative Analysis: Elementary Principles and Practice”, 2nd ed.; Oakland Street Science Press: Ames, IA, 1974. Gjaldback, J. K. Z . Anorg. A//g. Chem. 1925, 744, 145-168. Hartwick, P.; Schierholz, P. M. presented in part at the 69th Annual AIChE Meeting, Chicago, IL, Nov 1976. Juzaszek, P.; Larson, M. A. AIChEJ. 1977, 2 3 , 480-468. Klein, D. H.; Gordon, L.; Walnut, T. H. Talanta 1959, 3 , 177-190. Klein, D. H.; Gordon, L. Talanta 1958, 7 , 334-343. Liu, S. T.; Nancollas, G. H. Desallnation 1973, 72, 75-84. Mullin, J. W.; Sohnei, 0. Chem. Eng. Sci. 1977, 3 2 , 883-686. Munk, P. M.S. Thesis, Iowa State University, Ames, IA. 1976. Murotani, H.; Shlrasakl, T. Bull. Tokyo Inst. Technoi. 1981, 42, 15. Nasanen, R. Z . Phys. Chem. 1941, A188. 272-283. Nielsen, A. E. J. CoUoM Interface Sci. 1955, IO, 576-586. Nielsen, A. E. “Kinetics of Precipitatlon”; Pergamon Press: New York, 1964. Nleisen, A. E. I n “Industrial CrystailIratiin 78”; de Jong, E. J.; JanEiE, S. J., Ed.; North Holland Publishing Co.: Amsterdam, The Netherlands, 1979; pp 159- 168. Nielsen, A. E. Presented in part at the Fourth International Conference on Surface and Colloid Science, Jerusalem, Israel, July, 1981. Peters, R. W.; Stevens, J. D. Presented in part at the 74th Annual AIChE Meeting, New Orleans, LA, Nov 1981. Peters, R. W.; Swinney, L. D.; Stevens, J. D. AIChE Symp. Ser, 1981, 77(209), 49-66. Peters, R. W. Ph.D. Dissertation, Iowa State University, Ames, IA, 1980. Phillips, V. A.; Kolbe, J. L.; Opperhauser, H. J . Cryst. Growth 1977, 47, 235-244. Randolph, A. D.; Larson, M. A. “Theory of Particulate Processes“; Academic Press, New York, 1971. Randolph, A. D. University of Arizona, personal communication, 1981. Ryznar, J. W.; Green, J.; Winterstein, M. G. Ind. Eng. Chem. 1946, 3 8 , 1057-1061. Sawyer, C. N.; McCarty, P. L. “Chemistry for Environmental Engineering”, 3rd ed.; McGraw-Hili: New York, 1978. Shirasaki, T. Denki Kagaku 1961, 2 9 , 656. Shirasaki, T. DenkiKagaku 1961, 2 9 , 557; Chem. Abstr. 1965, 62, 7186a. Swinney, L. D. M.S. Thesis, Iowa State University, Ames, IA, 1979. Tagai, H.; Saito, K. Yogyo Kyokai Shi. 1988, 76, 81. Travers, A.; Nouvel, R. Compt. Rend. 1929, 788, 499-501. Van Leeuwen, C. J . Cryst. Growth 1979, 46, 91-95. Wiechers, H. N. S. Prog. Water Technoi. 1977, 9 , 531-545.
Received for review May 14, 1980 Accepted April 16, 1982
This paper was presented at the 72nd Annual Meeting of the American Institute of Chemical Engineers, San Francisco, CA, NOV25-29, 1979.
