Ind. Eng. Chem. Res. 2001, 40, 1541-1547
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SEPARATIONS Crystal Properties and Nucleation Kinetics from Aqueous Solutions of Na2CO3 and Na2SO4 Bing Shi and Ronald W. Rousseau* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100
Sodium carbonate and sodium sulfate are the major inorganic components in black liquor produced from alkali pulping, and these two compounds have been shown to be the major contributors to the formation of encrustations on the heat-transfer surface of black-liquor evaporators. Therefore, a determination of the variables affecting the nucleation and growth of these species is an important step in elucidating mechanisms by which such encrustations are formed. The metastable zone width for crystallization in the Na2CO3-Na2SO4-H2O system was determined as a function of the total solute concentration and the ratio of sodium carbonate to sodium sulfate. The polythermal method was used in the measurements, and supersaturation was generated by heating the system contents to increase the system temperature at a constant rate. Classical nucleation theory was combined with the metastable limit to correlate the effects of temperature and solute concentration on primary nucleation kinetics. Introduction Chemical recovery is a crucial operation in modern pulp mills because it regenerates the pulping chemicals and provides a substantial amount of thermal energy. The first step of chemical recovery is black-liquor evaporation. Usually, black liquor is concentrated in multiple-effect evaporators to over 60 wt % solids, so that it can be burned without the use of additional fuel in a subsequent boiler. Black-liquor evaporators have long been the cause of reduced productivity and energy efficiency resulting from encrustations at heat-transfer surfaces. These encrustations must be removed periodically by washing (boiling out) with low solids-content black liquor or water. According to a survey on the fouling problems of black-liquor evaporators throughout North America,1 the severity of encrustations at kraft pulping mills differed significantly from one mill to another, and the boil-out frequency ranged from 0.08 to 12.2 times/month. The black-liquor composition, the types of evaporators used, the solids content of the product liquor, and the operating conditions have been identified as some, but not all, of the factors in determining the required boil-out frequency. More must be learned about the mechanisms of fouling before a reliable model can be developed to guide evaporator operation. Sodium carbonate and sodium sulfate are the major inorganic species in black liquor produced by alkali pulping of wood, and these two compounds have been shown to be the major contributors to the formation of encrustations. Several properties of these salts are unusual and may have contributed to the formation of encrustations: both have the abnormal behavior of reduced solubility with increased temperature (over the operating temperature range), they can form the double salt Burkeite (Na2CO3‚2Na2SO4) when they cocrystal-
lize, and Burkeite can form solid solutions with both sodium carbonate and sodium sulfate. Evaporation of high-solids-content black liquor continuously removes solvent and produces crystals of sodium salts. Encrustations result when crystallization occurs on or crystals formed elsewhere adhere to the heating surface. An understanding of the nucleation mechanisms will enhance finding a solution to the formation of these encrustations. The present work examines the nucleation and corresponding widths of metastable zones of sodium carbonate and Burkeite at known operating conditions. These two salts were chosen for study because they are commonly observed in black-liquor evaporators, but they also may be useful surrogates for species that cause fouling in other systems. The metastable limits were determined by measuring the temperatures at which nucleation occurred at specific constant heating rates. This procedure was chosen based on the abnormal solubility behavior of the salts, and it differed accordingly from the procedure in the more frequently used constant cooling rates. The purpose of this work was to use the metastable limits for nucleation to estimate nucleation parameters that could be used in providing a guide to economical operation of the evaporators. Theory Mullin2 reviews the classical theory of nucleation and expresses the rate of nucleation J (the number of nuclei formed per unit time per unit volume) as
(
J ) A exp -
)
∆Gcrit kT
(1)
where ∆Gcrit is the free energy change to form a stable nucleus. In the case of homogeneous nucleation,
10.1021/ie0006559 CCC: $20.00 © 2001 American Chemical Society Published on Web 02/20/2001
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∆Gcrit,hom )
16πγ3v2 3(kT)3(ln S)2
(2)
where g
In the case of heterogeneous nucleation, the presence of a foreign surface can lower the overall free energy change associated with the formation of a critical nucleus,
F)
n(i) ∑ i)1
(12)
and g
∆Gcrit,het ) φ∆Gcrit,hom
The quantity φ is less than 1, and it takes into account the reduction in ∆G due to participation of foreign surfaces in the nucleation process. Furthermore,
φ ) [(2 + cos R)(1 - cos R)2]/4
(4)
where R, the contact angle between the crystalline deposit and the foreign solid surface, is a measure of the affinity between the nuclei and the foreign solid surface. From eqs 1-4, the heterogeneous nucleation rate can be expressed as
(
Jhet ) A exp -
φ16πγ3v2 3(kT)3(ln S)2
)
(5)
The molecular volume v ) M/FSNA may be approximated as a constant because it is a weak function of temperature. As for the interfacial energies of anhydrous salts in aqueous solution, γ, Mersmann3 proposed the following predictive relationship:
γ ) 0.414kT
( ) ( ) FSNA M
CS CL
2/3
ln
(6)
where
FS CS ) CL FLw
(7)
Jhet ) A exp[-1.19φ ln3(FS/FLw) ln-2 S]
(8)
From eqs 5-7,
The nucleation rate also may be expressed in terms of the supersaturation created by heating a solution of salts exhibiting an inverse solubility behavior:
J)
[
]
dn(g) d(∆w) dn(g) ) dt d(∆w) dT
Tnuc
dT dt
(9)
where n(g) is the number of nuclei in a unit volume of solution having a number of molecules larger that the number in a critical nucleus, g. The equation can be simplified by letting b ) dT/dt, the heating rate, and
dn(g) d(∆w)
f)
(10)
This expression is an approximation to the change in number of nuclei with supersaturation, where supersaturation is proportional to ∆w. Using the approach of Frenkel,4 the critical-cluster concentration n(g) can be estimated as
n(g) )
∫g∞F(NF )
x
(
exp -
N)
(3)
)
∆G(x) dx kT
(11)
in(i) ∑ i)1
(13)
Although the ratio of F to N (N/F equals the average number of molecules in a cluster) is close to unity at low supersaturation,5 this term cannot be neglected because g . 1. So for the time being, it is not possible to estimate f. However, combining eqs 8-10 and taking the natural logarithm of both sides
ln
{[
] } ()
d(∆w) dT
b ) ln
Tnuc
( )
FS A - 1.19φ ln3 ln-2 S f FL w (14)
it is now possible to treat A/f and φ as parameters to be estimated from experimental data. Experimental Section An experimental crystallizer was constructed from a Jerguson gauge (a liquid-level gauge for high-pressure applications) that provided for circulation of liquid, visual observation of the system contents, and operation at pressures up to 1000 kPa and temperatures over the range from 50 to 170 °C. The crystallizer held up to 110 cm3 of solution, which was pumped through the whole system by a Ruska magnetic pump at a rate of 30 cm3/ min. The entire system was placed in a mechanicalconvection oven, whose temperature is controlled to (1 °C. Metastable zone widths were determined by the polythermal (as opposed to the constant-temperature) method in the following procedure. An aqueous solution of sodium carbonate or a mixture of sodium carbonate and sodium sulfate was filtered through 2-µm stainless steel solvent filters and then charged to the crystallizer. The solution was then circulated through the preheated system at 50 °C and sampled after 20 min. Subsequently, the system was subjected to a constant heating rate and pressurized to suppress boiling. Because the wall of the crystallizer was very thick, primary heating of the solution was on its circulation pathway. The first crystalline particles were detected by visual observation, and the corresponding temperature T1 (see Figure 1) was taken as nucleation temperature Tnuc. After the crystals had settled at constant temperature, the solution was sampled with a Whitey sample cylinder. The residue in the crystallizer was then washed with distilled water and dissolved for analysis of the contents. For comparison, metastable limits also were determined for aqueous solutions of Na2CO3 with a differential scanning calorimeter (DSC; Setaram TG-DSC 111). In that method, about 100 µL of a solution of known concentration was sealed tightly in a stainless steel crucible of volume 0.15 cm3 that could withstand more than 1000 kPa pressure. The crucible and solution were heated rapidly to a known temperature that was well below the saturation temperature of the solute(s) and then at a constant rate until nucleation was detected. Figure 2 shows a typical DSC curve used to
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Figure 1. Experimental apparatus for metastable zone measurement. Figure 3. Solubility of Burkeite interpolated from the original data obtained by various researchers: Schroeder et al. (SBG);8 Dawkins (D) as cited in Seidell and Linke;6 Green and Frattali (GF);7 Makarov and Krasinkov (MK) as cited in Seidell and Linke;6 DeMartini (DM);9 the present work (SR).
Figure 2. Typical DSC curve obtained in the measurement of a metastable limit.
determine metastable limits. The nucleation temperature, which is taken to be the local maximum just before the sharp decrease on the curve, was defined as the metastable limit Tnuc. The effect of water evaporation on the solution concentration in the crucible was estimated to be negligible. The solubilities of mixtures of sodium carbonate and sodium sulfate (in the composition range of Burkeite) in water were also determined in an attempt to resolve the disagreements among data from the literature.6-9 Most of the earlier research used the method of bringing solid and solution together to equilibrate (dissolving method), but in the present work, equilibrium was approached from the opposite direction, i.e., by causing the solution to crystallize and then equilibrate (crystallization method). The apparatus and procedure were basically the same as those described in determining metastable limits except that the crystals were kept in equilibrium with the solution for 48 h. Nucleation was induced by depressurizing the crystallizer and evaporating the supersaturated solution for a second or so at the designated temperature, if necessary. During the equilibrating period, the temperature did not deviate by more than 1 °C from the designated temperature. Solution concentrations were determined by a MettlerToledo DL58 titrator. The relative error of the concentration was less than 1%.
higher than those obtained in the present work. The literature values were obtained by interpolating data on the Na2CO3-Na2 SO4-H2O system6-9 to estimate the solubility of Burkeite in water. The differences between the present and literature values are thought to be due to the methods of measurement. As described above, the literature results were obtained using the dissolving method, but the present results were obtained using the crystallization method. We also obtained a measurement of solubility at 120 °C using the dissolving method and found that it resulted in a higher estimate of the solubility than was obtained by the crystallization method. The difference was about 0.01 g of Burkeite/g of solution (a difference of ∼4%). It is probable that the differences in solubilities measured by the two techniques are due to variations in the rates at which equilibrium is attained. When Burkeite is one of the equilibrated solid phases, the dissolving method requires solids (Na2CO3 and Na2SO4) placed in the solvent to dissolve and then recrystallize as Burkeite; on the other hand, the crystallization method results in the immediate formation of Burkeite, and a solution-mediated transformation to the equilibrated solid phase is unnecessary. Furthermore, the solids placed in the solvent in the dissolving method were relatively large crystals, with a low surface-tovolume ratio, requiring even longer for the solutionmediated transformation to approach equilibrium. The temperatures at which nucleation occurred in either sodium carbonate and Burkeite solutions were determined at different heating rates over the range 100-170 °C and are shown in Figures 4 and 5. The discontinuity in the data for sodium carbonate (Figure 4) is caused by a phase change of the solid from monohydrate to anhydrous Na2CO3. Metastable zone widths can be determined at a specific temperature as the difference between the solution composition w and the equilibrium composition w*. All of the solubility data (except those of Makarov and Krasinkov, cited in Seidell and Linke6) were correlated with the following approximation to ideal solution theory:
Results and Discussion Solubilities and Nucleation. As shown in Figure 3, literature values for the solubility of Burkeite are
ln
(
) ( )
∆H ˆ fus(T - Tref) wFL C ≈ ln ) wrefFL,ref Cref RTTref
(15)
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Figure 4. Solubility and metastable limits of Na2CO3 in aqueous solutions at different heating rates.
Figure 7. Correlating metastable limits of Burkeite in aqueous solutions at different heating rates.
Figure 5. Solubility and metastable limits of Burkeite in aqueous solutions at different heating rates.
Figure 8. Comparison between the metastable limits obtained using the crystallizer and the DSC at a heating rate of 0.6 K/min.
an expression of the same form as eq 16 to relate w and Tnuc; i.e.,
ln w ) C1/Tnuc + C2
(17)
The correlations using eqs 16 and 17 are shown in Figures 6 and 7. A comparison of the metastable limits obtained using the circulating crystallizer with those obtained using the DSC is shown in Figure 8. The remarkable similarity of these provides a basis for DSC applications in future experiments with black liquor in which opaqueness limits the ability to observe crystal formation. Equations 16 and 17 were used to estimate the supersaturation ratio S at nucleation, and the following simplifications were made to eq 14 in order to interpret the experimental results: Figure 6. Correlating metastable limits of anhydrous Na2CO3 in aqueous solutions at different heating rates.
When the effect of temperature on the density of the solution is neglected, the correlation can be reduced further to
ln w* ) C1/T + C2
(16)
The fits to eq 16 are shown in Figures 6 and 7. The similarity of solubility and metastable-limit (nucleation) curves in Figures 4 and 5 suggests using
( )
FS ln-2 S FLw
(18)
y ) ln{[d(∆w)/dT]Tnucb}
(19)
x ) 1.19 ln3 and
where Tnuc is the nucleation temperature. This means that eq 14 becomes
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y ) ln(A/f) - φx
(20)
The experimental nucleation data for Na2CO3 and Burkeite are plotted in Figures 9 and 10, respectively. They show linearity between x and y; the slopes are -φ and the intercepts are ln(A/f). Estimated values of the parameters are given in Table 1. The dependence of the nucleation temperature on the heating rate may be attributed to three possibilities: (1) the time lag between the changing of the solution state and subsequent cluster-size redistribution, (2) the induction period of the nuclei, and/or (3) the time required for the nuclei to grow to detectable sizes. All of these factors mean the nucleation parameters obtained should be closer to their true value when the heating rate decreases. Influence of Additives. Sodium hydroxide is an intrinsic component of black liquor, and it generally decreases the solubility of sodium carbonate and sodium sulfate in this system without changing the composition of the crystals. The effects of sodium hydroxide on the solubility and nucleation of Burkeite solutions at a heating rate of 0.6 K/min are reflected in the data of Figure 11. As shown, sodium hydroxide, present at 5 wt %, significantly lowers the metastable limits of Burkeite and reduces the metastable zone width by about 30%. The influence of organic material from black liquor on the metastable limits was studied by adding 1 wt % black liquor (∼45 wt % dissolved solids) into the aqueous solution of sodium sulfate and sodium carbonate. A higher organic content could not be used because the added black liquor darkened the solution and made it impossible to observe crystals formed. Figure 12 compares the metastable limits obtained at a heating rate of 0.6 K/min with and without black liquor added. The decrease of the metastable zone width was expected because the surface-active agents contained in black liquor tend to decrease the free energy change associated with the formation of an embryo. Critical Cluster Size Estimation. From the classical nucleation equations,2 the critical cluster size dc can be estimated from the equation
dc )
4γv kT ln S
(21)
The metastable-limit data for a heating rate of 0.3 K/min were used to estimate the sizes of sodium carbonate nuclei at different temperatures. The results (Figure 13) show that the sizes of critical nuclei increase as the temperature increases, which is consistent with the analysis by Mullin,2 and range from 8 to 10 nm. These results compare favorably with those of others: the average cluster sizes in various systems estimated based on the concentration gradient were 0.78-4.32 nm by Ginde and Myerson5 and 3.1-9.8 nm by Larson and Garside;10 the average cluster size obtained by Buyanov et al.11 was 2-4 nm based on electron microscopy. Crystal Composition. The crystals were sampled soon after nucleation from a Burkeite solution (one in which sodium carbonate and sodium sulfate were present at a molar ratio of 1:2) and examined later by X-ray powder diffraction. The comparison (Figure 14) with the standard pattern of orthorhombic Burkeite crystals shows the same position of all major reflection lines but with a change in the relative intensities. The changes could be explained by ion replacement, which
Figure 9. Correlating the kinetics of Na2CO3 nucleation from aqueous solutions at a heating rate of 0.6 K/min.
Figure 10. Correlating the kinetics of Burkeite nucleation from aqueous solutions at a heating rate of 0.6 K/min.
is not an uncommon phenomenon among minerals. Further elemental analysis showed a higher sulfate to carbonate molar ratio (2.2:1) in crystals from the present study than in the standard (2:1). This result partially confirmed the conclusions of other authors7,8 that Burkeite can form solid solutions with both sodium carbonate and sodium sulfate. An additional set of experiments was performed to determine the effects of aging on crystal composition. In each run, the initial solution had a ratio of 2 mol of Na2SO4 to 1 mol of Na2CO3; the crystallizations were conducted with the same heating rate, and the resulting solid-liquid mixtures were allowed to equilibrate at 120 °C for different periods of time. The results are shown in Figure 15. In each of these experiments, the final solution composition was about 1.97 ( 0.02 mol of Na2SO4/mol of Na2CO3. Although estimates of the compositions of the crystals were not highly accurate because of the presence of residual liquid, the data show (1) that the molar ratios are consistently higher in the crystal than in the solution and (2) that these ratios change over time toward the ideal composition of Burkeite (2 mol of Na2SO4/mol of Na2CO3). These observations can be explained as follows: because of the rapid kinetics of crystallization, the crystals initially formed were not in equilibrium with the solution but were transformed over time. As the crystals remained in contact with the solution, transformations were toward the equilibrium composition, probably by a
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Figure 11. Influence of sodium hydroxide on the metastable limits of Burkeite solutions.
Figure 13. Size of sodium carbonate nuclei estimated from metastable limit data.
Figure 12. Influence of the organic material from black liquor on the metastable limits of Burkeite solutions (heating rate of 0.6 K/min).
solution-mediated process of recrystallization. The small sizes of the crystals and the relatively high temperature of the solution facilitated such a transformation. Summary Nucleation conditions were determined for aqueous solutions of both sodium carbonate and Burkeite (2 mol of Na2SO4/mol of Na2CO3) at temperatures above 100 °C. Classical nucleation theory was used to correlate the metastable zone width to parameters associated with nucleation kinetics. Sizes of critical clusters (nuclei) were estimated based on the metastable limits, and the values were consistent with results others have found for different systems. Adding sodium hydroxide or organic materials from black liquor to the solution reduced the metastable zone width. Finally, the crystals have been identified by X-ray powder diffraction as a
Figure 14. Comparison between (a) the X-ray powder diffraction pattern obtained by the present experiments and (b) the standard pattern of orthorhombic Burkeite (Powder Diffraction File, Inorganic, #24-1134, JCDPS, International Center for Diffraction Data, Swarthmore, PA).
Burkeite phase in which some carbonate ions may have been replaced by sulfate ions. The composition of these crystals changes when held in contact with the mother liquor.
Table 1. Parameter Correlation for Solubility, Metastable Limits, and Nucleation Kinetics system
type of data
C1
C2
Na2CO3-H2O Na2CO3-H2O Na2CO3-H2O Na2CO3-H2O Burkeite-H2O Burkeite-H2O Burkeite-H2O
solubility metastable, 0.3 K/min metastable, 0.6 K/min metastable, 1.2 K/min solubility metastable, 0.6 K/min metastable, 1.2 K/min
499.94 526.95 497.12 479.4 362.68 404.27 838.25
-2.476 -2.4905 -2.4166 -2.3389 -2.2346 -2.1981 -2.1321
A/f
φ
R, deg
8.34 × 10-6 5.87 × 10-5 1.42 × 10-3
2.17 × 10-4 6.57 × 10-4 4.36 × 10-3
10.6 14.0 22.7
1.50 × 10-5 8.76 × 10-4
2.3 × 10-3 3.9 × 10-3
19.2 22.0
Ind. Eng. Chem. Res., Vol. 40, No. 6, 2001 1547 w ) mass fraction Greek Letters R ) contact angle between crystalline deposit and foreign solid surface φ ) factor defined in eq 3 γ ) interfacial tension, J/m2 θ ) time, s F ) density, kg/m3 Superscript * ) equilibrium Subscripts
Figure 15. Evolution of Burkeite crystal composition when contact is maintained with the mother liquid at 120 °C.
L ) liquid S ) solid het ) heterogeneous hom ) homogeneous ref ) reference conditions
Literature Cited Acknowledgment The authors are grateful for the assistance provided by Dr. W. J. Frederick from the Institute of Paper Science and Technology and for the financial support by project member companies (Andritz-Ahlstrom Corp., Mead Corp., Potlatch Corp., and Weyerhaeuser Co.) and the Department of Energy under a subcontract from the Institute of Paper Science and Technology. Nomenclature A ) nucleation parameter in eq 8 b ) heating rate, K/s C ) solute concentration, mol/m3 C1, C2 ) parameters in the solubility and metastability correlation dc ) critical cluster size, nm f ) defined in eq 10 g ) number of molecules in a critical cluster ∆Gcrit ) free energy change to form a nucleus, J/mol ∆H ) enthalpy of dissolving, J/mol J ) nucleation rate, m-3‚s-1 k ) Boltzmann constant, 1.3805 × 10-23 J/K M ) molecular weight, g/mol n ) number of critical nuclei per unit volume NA ) Avogadro’s number, 6.02 × 1023 S ) supersaturation ratio, C/C* R ) gas constant T ) temperature, K Tnuc ) nucleation temperature, K v ) molecular volume, m3
(1) Schmidl, W.; Frederick, W. J. Current Trends in Evaporator Fouling. 1998 International Chemical Recovery Conference; TAPPI Press: Atlanta, GA, 1998; p 367. (2) Mullin, J. W. Crystallization, 3rd ed.; Butterworth-Heinemann: Oxford, U.K., 1992. (3) Mersmann, A. Calculation of Interfacial Tensions. J. Cryst. Growth 1990, 102, 841. (4) Frenkel, J. Kinetic Theory of Liquids; Dover: New York, 1946. (5) Ginde, R. M.; Myerson, A. S. Cluster Size Estimation in Binary Supersaturated Solutions. J. Cryst. Growth 1992, 116, 41. (6) Seidell, A.; Linke, W. Solubility of Inorganic and Metal Organic Compounds, 4th ed.; American Chemical Society: Washington, DC, 1965; Vol. II. (7) Green, S.; Frattali, F. The System Sodium CarbonateSodium Sulfate Hydroxide-Water at 100 °C. J. Am. Chem. Soc. 1946, 68, 1789. (8) Schroeder, A.; Berk, A.; Gabriel, A. Solubility Equilibria of Sodium Sulfate at Temperatures from 150 to 350 °C. 2. Effect of Sodium Hydroxide and Sodium Carbonate. J. Am. Chem. Soc. 1936, 58, 843. (9) DeMartini, N. Internal Report, Institute of Paper Science and Technology, Atlanta, 1998. (10) Larson, M. A.; Garside, J. Solute Clustering in Supersaturated Solutions. Chem. Eng. Sci. 1986, 41, 1285. (11) Buyanov, R. A.; Krivoruchko, O. P.; Ryzhak, I. A. Study of the Mechanism of Generation and Growing of Crystal of the Ferric Hydroxide and Oxides in Mother Solution. Kinet. Catal. 1972, XIII (2), 470.
Received for review July 12, 2000 Revised manuscript received January 9, 2001 Accepted January 12, 2001 IE0006559
