Ind. Eng. C h e m . Res. 1988,27, 1714-1721
1714
Registry No. Si, 7440-21-3; S O z , 7631-86-9.
Literature Cited Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables; Applied Mathematics Series 55; National Bureau of Standards: Washington, DC, 1964; p 299. Barin, I.; Knacke, 0. Thermochemical Properties of Inorganic Substances; Springer-Verlag: Berlin, 1973; p 689. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960; pp 352-353. Blanc, J. Appl. Phys. Lett. 1978,33, 424-426. Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids; Clarendon: Oxford, 1959; pp 62-63. Crank, J. Q. J . Mech. Applied Math. 1957,X , 220-231. Crank, J. The Mathematics of Diffusion; Clarendon: Oxford, 1975; pp 32-34. Deal, B. E.; Grove, A. S. J . Appl. Phys. 1965,36, 3770-3778. Dutton, R. W.; Antoniadis, D. A. In Moving Boundary Problems; Wilson, D. G., Solomon, A. D., Boggs, P. T., Eds.; Academic: New York, 1978; pp 233-248. Ghandhi, S. K. VLSZ Fabrication Principles; Wiley: New York, 1983; pp 370-400. Hopper, M. A.; Clarke, R. A.; Young, L. J.Electrochem. SOC.1975, 122, 1216-1222.
Katz, L. E. In VLSI Technology; Sze, S. M., Ed.; McGraw-Hilk New York, 1983; pp 131-167. Kofstad, P. Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides: Wilev-Interscience: New York, 1972; pp 348-355. Lane. C. H. ZEEE Tram. Electronic Devices 1968.ED-15,998-1003. Norton, F. J. Nature (London) 1961,191, 701. ’ Rubinstein, L. I. The Stefan Problem; American Mathematical Society: Providence, RI, 1971; pp 52-55. Singh, S. K.; Fan, L. T. Biotechnol. Prog. 1986,2, 145-156. Talman, Y.; Davis, H. T.; Scriven, L. E. AZChE J. 1981,27,928-937. Tao, L.C. AZChE J. 1967,13, 165-169. Tiller, W. A. J . Electrochem. SOC.1980a,127,619-624. Tiller, W. A. J . Electrochem. SOC.1980b,127,625-632. Tiller, W. A. J . Electrochem. SOC.1981,128, 689-697. Wen, C. Y. Ind. Eng. Chem. 1968,60,34-54. Williams, E. L. J. Amer. Ceramic SOC.1965,48, 190-194. Wilmsen, C. W.; Thompson, E. G.; Meissner, G. H. ZEEE Trans. Electronic Devices 1972,ED-19, 122. Yariv, A. A n Zntroduction to Theory and Applications of Quantum Mechanics; Wiley: New York, 1982; pp 253-259.
Received for review October 23, 1987 Revised manuscript received April 29, 1988 Accepted May 17, 1988
Correlation and Prediction of Physical Properties of Hydrocarbons with the Modified Peng-Robinson Equation of State. 1. Low and Medium Vapor Pressures B. C a r r i e r , M. Rogalski, a n d A. Pbneloux* Laboratoire de Chimie-Physique, Facultk des Sciences de Luminy, 13228 Marseille, Cedex 9, France
A modified version of the Peng-Robinson equation of state is proposed for representing vapor pressure data on compounds currently encountered in petroleum fractions. The proposed equation is valid from the triple point up to 2-3 bar. When one parameter is adjusted using experimental data, a very accurate representation of vapor pressures is obtained, with an overall average absolute deviation of only 0.19% in the case of the 128 substances studied. When this parameter was determined by using a group contribution method, the corresponding deviation was 0.43%. Vapor pressures of pure compounds can be calculated either from some integrated forms of the Clausius-Clapeyron equation or from an equation of state. The first method is currently used to precisly represent experimental vapor pressure data. Numberous expressions proposed in the literature represent experimental data with a good degree of accuracy, but they are often disappointing when used for extrapolation purposes (especially in the very low pressure range). By using the corresponding states principle with expressions of this type, it is possible to obtain general, predictive equations for calculating the vapor pressure of a given compound on the basis of a set of characteristic parameters (usually critical constants and the acentric factor). Results obtained are satisfactory in the high-pressure range and usually only roughly correct at medium (0.06-3 bar) and low pressures (less than 0.06 bar). As examples, the papers by Gomez-Nieto and Thodos (1977), Riedel (1954), Thek and Stiel (1966), and Willman and Teja (1985) can be quoted. Equations of state are usually used as a means of calculating volumetric and thermal properites of pure compounds and mixtures as well as phase equilibria. Thus, it is necessary that they represent the vapor pressures with a good degree of accuracy. Their parameters are usually scaled in terms of the corresponding states principle, i.e., using critical parameters and the acentric factor.
If we consider the family of cubic equations of state, the general form is given as P = R T / ( u - b ) - a / ( u 2 + ubu + W b 2 ) (1) Matching an equation of state of this type with vapor pressures means fitting the parameter a of eq 1 to experimental data. The quality of the fitting depends on the expression chosen to represent variations in parameter a with temperature. An expression currently used was proposed by Soave (1972): u ( T )= a,(l + m,(l - T,”2))2 (2) By means of eq 2 or its modifications proposed in the literature together with one of the forms of eq 1, it is possible to calculate the vapor pressures of hydrocarbons quite correctly in the high-pressure range. A t medium pressures (0.06-3 bar), the representation is still fairly correct (Rauzy, 1982) but is far from being as accurate as experimental data. The main advantage of an equation of state of this type when applied to vapor pressure calculations is its predictive character. It is possible to improve the restitution of vapor pressures by defining the parameter a ( T ) of eq 1with more characteristic parameters per compound than in the case of eq 2. This approach was developed recently by Gibbons and Laughton (1984)
0888-5885/88/2627-1714~01.50/0 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1715 and by Stryjek and Vera (1986). In this paper we are mainly concerned with representing the vapor pressures of the hydrocarbons (including highly boiling ones) encountered in petroleum fractions. For this class of compounds, we intend to propose a form of eq 1 which is able to yield values of vapor pressures from the triple point up to the pressure of 2 bar with an accuracy close to that of experimental data. Limiting the validity of the proposed equation to the range of low pressures, we can generalize its parameters in terms of the normal boiling temperature without using critical constants which are not known or have not been properly established in the case of heavy compounds. The choice of the expression giving parameter a of eq 1as a function of temperature is the crucial point of the present method. Since high-precision vapor pressure representation and predictive possibilities were the main objectives, it was necessary to base the form of this expression on the behavior of thermodynamic properties related to parameter a. The enthalpy of vaporization, L, was thought to be a promising choice. In the case of the van der Waals equation, which is the simplest cubic equation, L is given as follows:
L
- P(u, - u,) =
u, - u, = -d(a/T)/d(l/T) ( l / u g - l / ~ (3) )
In the low-pressure range, this equation can be approximated by the following form:
u, - Ul = ( L - P(u, - U 1 ) ) U l = d ( a / T ) / d ( l / T )
(4)
I t was shown previously (Rogalski, 1987) that below the normal boiling temperature the product of the liquid volume and the enthalpy of vaporization can be represented with a simple expression:
Lul = (LUJT, + A(1 - (T/Tb)")
(5)
where the exponent x is less than 1 (usually between 0.2 and 0.6) and A is a constant. This relation is also valid for the product of the liquid volume and the energy of vaporization. Replacing the left side of eq 4 by eq 5 and integrating, we establish the relationship between parameter a and the temperature:
+
a = aTb(l (1- TX/Tbx)ml+ (1- T/Tb)mz) (6)
This expression obtained with the van der Waals equation can be used with any cubic EOS in the range of low and medium vapor pressures. Parameter aT, is determined by matching an EOS with equilibrium conditions at the boiling point. An iterative procedure for solving the EOS at each step is needed. A rapid convergence is usually obtained with the starting value of U T , equal to 6Tb. The form of eq 6 is identical with that proposed by Gibbons and Laughton (1984) on the basis of different arguments. Development of the Method As the basis of the present method, the following cubic equation of state was used: with y = 4.82843 P = RT/(5 - 6) - a / 5 ( 5 + 76) (7) The pseudovolume 5 and the pseudocovolume 6 should be corrected, according to Rauzy (1982), in the following way: u=jj-c
b=g-c
(8)
When eq 7 is used with eq 8 and c = -0.03222RTc/P,, the
resulting equation is identical with that proposed by Peng and Robinson (1976):
P = RT/(u - b) - u/(u'
+2
b -~ b2)
(9)
Since the vapor pressure calculations were the main objective of the present study, we use eq 7 without the volumetric correction. With this equation we prppose a new method for determining parameters a and b. Choice of Experimental Data. The vapor pressure data base was selected to cover available information about compounds representative of petroleum mixtures and mixtures boiling above the temperature T = 282.68 K, which corresponds to the normal boiling temperature of neopentane. The set of 2800 data points for 128 compounds in the range of pressures from the triple to the normal boiling point was used in all calculations. For each set of data, the range of pressure and the reference are given in Table I. Values of the normal boiling temperatures accepted in the literature are listed in the same table. All available very low pressure data were included in the data base to extend the validity of the present method to this little known region. When several discordant data sets existed on a given compound, they were all included in the data base. Data on pressures of up to several bars were also included. Determination of the Pseudocovolume, 6 . The pseudocovolume of a cubic EOS is usually calculated using critical parameters. An attempt has been made to express the pseudocovolume in terms of a group contribution method. A regression was performed for all the hydrocarbons having known critical parameters. The resulting equation is as follows:
6
13
= -2.40086
+ j=l CBjG,
(10)
where the pseudocovolume of a given compound is calculated by summing the product of group parameters Bj and the number of corresponding groups Gj in this compound. The specification of groups and the values of parameters B, are given in Table 11. It should be observed that the pseudocovolumes calculated with critical parameters cannot be represented precisely by a linear group contribution method such as that given by eq 10. This difference does not affect the results of vapor pressure correlation, however. Pseudocovolumes resulting from eq 10 are listed in Table I. Determination of Parameter a . In all the calculations that follow, eq 7 will be used with a ( T )expressed by eq 6. The pseudocovolume was calculated according to eq 10. In an initial stage, preliminary calculations were performed to fix the value of the exponent x in eq 6 and to study the possibility of simplifying eq 6. The value of x = 0.5 was found to be an appropriate choice. It turned out that with hydrocarbons the parameters ml and m2 are highly correlated and that only one characteristic parameter per compound is needed (parameter m in eq 11). Thus, the final form of eq 6 selected for further calculation was as follows:
m, = 7.1562m - 1.9829 m2 = 2.5780m - 0.9914 To calculate the vapor pressures of one compound, the pseudocovolume, the normal boiling temperature, and parameter m must be known. The results obtained by
1716 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 Table I. Pure Compound Parameters and Average Absolute Pressure Deviations
compd neopentane isopentane isopentane pentane pentane cyclopentane cyclopentane 2,2-dimethylbutane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylpentane 3-methylpentane hexane hexane methylcyclopentane 2,2-dimethylpentane benzene benzene 2,4-dimethylpentane cyclohexane cyclohexane cyclohexane 2,2,3-trimethylbutane 2,2,3-trimethylbutane 3,3-dimethylpentane 1,l-dimethylcyclopentane 2,3-dimethylpentane 2-methylhexane 2-methylhexane cis-1,3-dimethylcyclopentane 3-methylhexane trans-1,2-dimethylcyclopentane 3-ethylpentane 3-ethylpentane heptane isooctane cis-1,2-dimethylcyclopentane methylcyclohexane methylcyclohexane ethylcyclopentane 1,1,3-trimethylcyclopentane 2,2-dimethylhexane 2,5-dimethylhexane cis,trans,cis-1,2,4-trimethylcyclopentane 2,4-dimethylhexane toluene toluene toluene 3,3-dimethylhexane 2,3,4-trimethylpentane 1,1,2-trimethylcyclopentane 2,3-dimethylhexane 2-methyl-3-ethylpentane cis,cis,trans-1,2,4-trimethylcyclopentane 2-methylheptane 4-methylheptane 4-methylheptane 3,4-dimethylhexane 3-methyl-3-ethylpentane 3-ethylhexane 3-methylheptane trans-1,4-dimethylcyclohexane 1,l-dimethylcyclohexane trans-1,3-dimethylcyclohexane 1-methyl-1-ethylcyclopentane 1-methyl-1-ethylcyclopentane 2,2,4,4-tetramethylpentane 2,2,4,4-tetramethylpentane frans-l,2-dimethylcyclohexane 2,2,5-trimethylhexane cis-1,4-dimethylcyclohexane cis-1,3-dimethylcyclohexane octane isopropylcyclopentane 2,2,4-trimethylhexane cis-2-methyl-1-ethylcyclopentane
I",,. K
b, cm3
282.68 301.00
49.93 52.37
m 0.491 82 0.521 36
309.19
55.90
0.553 28
322.41
40.89
0.479 33
322.89
60.97
0.528 88
331.14 333.42 336.43 341.89
59.88 63.41 63.41 66.95
0.537 77 0.574 67 0.570 38 0.604 06
344.96 352.35 353.25
53.55 72.02 41.30
0.527 84 0.588 30 0.502 95
14 5 38 10 14 34
353.65 353.86
70.93 49.98
0.597 11 0.511 54
354.03
68.49
0.547 54
14 14 14
359.21 361.00 362.93 363.20
72.02 67.65 70.93 74.46
0.570 37 0.568 29 0.592 11 0.625 84
363.92 365.00 365.02 366.62
66.21 74.46 66.21 74.46
0.565 94 0.622 41 0.56846 0.619 26
371.57 372.39 372.68 374.08
77.99 79.54 66.21 64.89
0.652 58 0.597 33 0.568 24 0.544 56
376.62 378.04 379.95 382.25 382.44 382.55 383.77
64.60 80.31 83.07 81.98 78.86 81.98 54.12
0.567 47 0.594 76 0.641 53 0.646 16 0.599 85 0.641 45 0.549 52
385.15 386.62 386.88 388.75 388.75 389.88 390.80 390.85
83.07 78.45 80.31 81.98 81.98 78.86 85.51 85.51
0.622 79 0.607 00 0.594 44 0.641 47 0.633 31 0.595 77 0.67194 0.671 37
390.85 391.35 391.65 392.07 392.45 392.69 393.25 394.67
81.98 78.01 85.51 85.51 79.80 74.74 79.80 78.70
0.638 37 0.593 29 0.666 47 0.669 93 0.584 17 0.563 36 0.592 94 0.59969
395.43
88.15
0.610 67
396.55 397.23 397.45 397.60 398.81 399.57 399.70 401.20
79.80 90.59 79.80 79.80 89.04 73.49 90.59 77.25
0.581 15 0.660 94 0.596 17 0.600 31 0.697 32 0.58183 0.638 71 0.606 09
ref' 17 33 38 25 38 3 38 28 38 38 38 38 20 38 38 38 38 1
18 14
14 14 14 14
34 38 38 14 35 38 14 14 38 38 14 38 38 4 32 38 38 14
38 38 14 38 38 38 38 38 38 38 38 14 38 14
30 14
34 38 14 38 38 38 14 30 14
pressure, bar min max 0.36 1.45 0.01 1.04 0.06 1.04 0.004 0.68 0.06 1.04 0.008 0.27 0.06 1.04 0.06 1.04 0.106 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.22 0.50 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.11 2.06 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 1.04 0.06 0.12 1.99 0.06 1.04 0.06 1.04 0.06 1.04 0.02 0.21 0.06 1.04 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 0.12 1.99 1.04 0.06 1.04 0.06 1.04 0.06 0.06 0.90 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 0.38 0.09 1.23 0.09 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.67 1.04 1.04 0.06 1.04 0.06 1.04 0.06 0.06 1.04 0.06 1.04 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 0.10 2.70 0.06 1.04 0.12 1.99 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.06 1.04 0.0003 0.03 0.06 1.04
ap% (1) 0.10
0.20 0.03 0.19 0.11 0.32 0.16 0.17 0.04 0.02 0.02 0.01 0.05 0.02 0.06 0.03 0.22 0.25 0.03 0.12 0.11 0.18 0.05 0.07 0.01 0.07 0.02 0.05 0.02 0.06 0.01 0.05 0.03 0.03 0.05 0.04 0.06 0.07 0.08 0.05 0.05 0.05 0.01 0.01 0.04 0.23 0.22 0.52 0.04 0.01 0.03 0.02 0.08 0.06 0.05 0.04 0.05 0.06 0.10 0.08 0.03 0.06 0.07 0.02 0.03 0.03 0.03 0.09 0.03 0.01 0.03 0.03 0.07 0.19 0.02 0.05
(2) 0.11 1.18 0.08 1.17 0.13 0.39 0.20 0.50 0.17 0.33 0.08 0.27 0.21 0.12 0.35 0.36 0.22 0.26 0.34 0.10 0.11 0.21 0.33 0.34 0.05 1.03 0.02 0.69 0.29 0.33 0.05 0.44 0.21 0.10 0.14 0.07 0.49 0.14 0.15 0.06 0.07 1.08 0.70 0.07 0.38 0.28 0.24 0.45 0.39 0.03 0.02 0.36 0.28 0.40 0.49 0.45 0.38 0.16 0.27 0.11 0.33 0.60 0.15 0.06 0.06 0.09 0.48 0.36 0.86 1.32 0.31 0.61 0.24 0.56 1.58 0.11
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1717 Table I (Continued)
a m
pressure, bar compd cis-2-methyl-1-ethylcyclopentane cis-1,2-dimethylcyclohexane 2,4,4-trimethylhexane propylcyclopentane ethylcyclohexane 2,2,3,4-tetramethylpentane 2,3,3-trimethylhexane ethylbenzene ethylbenzene 1,1,3-trimethylcyclohexane p-xylene p-xylene m-xylene m-xylene 2,2,3,3-tetramethylpentane 2,3,3,4-tetramethylpentane o-xylene o-xylene 3,3-diethylpentane cis-1-methyl-3-ethylcyclohexane nonane isopropylbenzene isopropylbenzene isopropylcyclohexane propylcyclohexane propylbenzene 1-methyl-3-ethylbenzene 1-methyl-4-ethylbenzene 1,3,5-trimethylbenzene 1-methyl-2-ethylbenzene tert-butylbenzene 1,2,4-trimethylbenzene isobutylcyclohexane tert-butylcyclohexane isobutylbenzene sec-butylbenzene decane decane 1-methyl-3-isopropylbenzene 1,2,3-trimethylbenzene 1-methyl-4-isopropylbenzene indan 1-methyl-2-isopropylbenzene sec-butylcyclohexane butylcyclohexane 1,3-diethylbenzene butylbenzene butylbenzene 1,2-diethylbenzene 1,4-diethylbenzene trans-decaline 2-methyldecane cis-decaline undecane tetraline 1,4-diisopropylbenzene 1,3,5-trimethyl-2-ethylbenzene dodecane dodecane naphthalene naphthalene tridecane 2-methylnaphthalene 2-methylnaphthalene 1-methylnaphthalene 1-methylnaphthalene tetradecane tetradecane biphenyl diphenylmethane diphenylmethane 2,3-dimethylnaphthalene pentadecane pentadecane acenaphthene acenaphthene decylcyclopentane
rep 30 38 14 14 38 14 30 9 38 14 38 29 32 38 14 14 32 38 14 29 14 12 38 14 14 14 14 14 14 14 14 14 14 14 14 14 24 38 23 14 23 2 23 14 14 14 14 21 14 14 8 30 8 8 16 23 6 24 38 15 8 8 8 37 8 37 24 8 11 19 36 29 24 8 29 26 7
Tb,K
b, cm3
m
402.88 403.80 404.10 404.94 406.17 407.60 409.34
79.80 90.59 75.65 75.94 87.06 90.59 65.17
0.591 43 0.637 94 0.615 78 0.591 85 0.612 22 0.641 26 0.591 49
409.78 411.50
89.64 66.94
0.612 54 0.596 09
412.25
66.94
0.601 09
413.42 474.79 417.56
88.15 87.06 66.94
0.61001 0.609 18 0.601 57
419.32 421.62 423.97 425.54
94.12 90.85 100.09 73.45
0.633 82 0.642 42 0.73889 0.620 33
427.71 429.87 432.37 434.45 435.14 437.87 438.30 422.27 442.50 444.47 444.74 445.91 446.45 447.27
85.99 86.99 76.22 77.98 77.98 79.75 77.98 84.05 79.75 94.50 96.58 83.73 83.73 111.14
0.623 70 0.637 26 0.628 15 0.648 60 0.639 15 0.666 67 0.644 83 0.647 47 0.654 10 0.655 88 0.641 51 0.640 89 0.643 53 0.780 50
448.08 449.23 450.26 451.12 451.47 452.48 454.10 454.25 456.42
86.27 79.75 86.27 67.76 86.27 94.50 98.04 89.03 87.27
0.67198 0.651 53 0.667 61 0.590 37 0.659 16 0.657 80 0.681 74 0.691 90 0.673 08
456.57 456.90 460.42 462.37 468.93 469.04 480.51 483.33 483.57 489.43
89.03 89.03 89.57 118.66 89.57 122.19 76.71 105.60 103.62 113.24
0.684 62 0.680 74 0.600 49 0.797 57 0.610 73 0.81965 0.609 05 0.726 94 0.743 91 0.855 88
491.11
70.17
0.580 44
508.58 514.20
144.29 82.98
0.893 22 0.632 98
517.84
82.98
0.634 23
526.67
155.33
0.926 09
528.36 537.51
85.49 96.54
0.66802 0.713 45
541.15 543.76
95.80 166.38
0.696 53 0.963 88
550.38
87.96
0.650 85
552.53
152.99
0.897 58
min 0.0003 0.06 0.06 0.06 0.06 0.06 0.0002 0.01 0.06 0.06 0.06 0.31 0.002 0.06 0.06 0.06 0.002 0.06 0.06 0.25 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.002 0.06 0.06 0.06 0.06 0.04 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.0002 0.06 0.01 0.06 0.06 0.06 0.09 0.06 0.08 0.0007 0.06 0.001 0.32 0.04 0.004 0.0002 0.06 0.002 0.03 0.06
max 0.01 1.04 1.04 1.04 1.04 1.04 0.02 0.27 1.04 1.04 1.04 2.70 0.07 1.04 1.04 1.04 0.03 1.04 1.04 2.70 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 0.03 1.04 1.04 1.04 1.04 2.06 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 0.04 1.04 0.39 1.04 1.04 1.04 8.21 1.04 4.15 0.01 1.04 1.42 1.47 6.70 0.02 0.01 1.04 0.02 1.26 1.04
(1)
0.31 0.05 0.04 0.02 0.04 0.01 0.54 0.11 0.04 0.04 0.02 0.04 0.85 0.18 0.01 0.11 0.76 0.15 0.09 0.08 0.05 0.41 0.07 0.02 0.02 0.01 0.10 0.08 0.12 0.08 0.03 0.11 0.02 0.01 0.01 0.06 0.23 0.07 0.64 0.09 0.09 0.07 0.72 0.02 0.02 0.07 0.08 0.27 0.06 0.07 0.03 0.07 0.05 0.06 0.34 0.12 0.83 1.20 0.10 0.24 0.49 0.11 0.13 0.49 0.16 0.69 0.75 0.17 0.55 0.74 1.09 0.31 0.61 0.17 0.63 1.23 0.05
(2) 0.47 0.10 0.35 0.18 0.04 0.08 0.67 0.31 0.08 0.15 0.32 0.10 0.99 0.25 0.46 0.39 0.94 0.06 0.15 0.21 0.23 0.58 0.21 0.67 0.05 0.49 0.33 0.39 0.94 0.09 0.16 0.10 0.32 0.14 0.64 0.43 1.16 0.28 0.68 0.20 0.15 0.23 0.89 0.47 0.13 0.31 0.42 0.48 0.22 0.48 0.58 0.55 0.26 0.24 0.77 0.63 1.13 1.13 0.17 0.39 0.47 0.13 0.23 0.45 0.31 0.74 0.97 0.15 0.96 0.78 1.44 2.23 0.67 0.14 0.87 1.20 0.05
1718 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 Table I (Continued)
ap%
pressure, bar compd decylcyclopentane non ylbenzene hexadecane hexadecane decylbenzene octadecane octadecane phenanthrene anthracene anthracene eicosane
ref" 31 24 13 7
7 22 24 27 27 26 22
Tb,
K
b, cm3
m
553.25 559.94
142.51 177.43
0.875 14 0.997 43
571.04 589.87
153.56 199.53
0.899 94 1.057 99
613.37 614.96
99.03 99.03
0.622 33 0.622 89
616.93
221.62
1.12843
average
min
max
0.06 0.00003 0.000 03 0.06 0.06 0.00001 0.00003 0.06 0.06 0.06 0.00001
1.04 0.01 0.01 1.04 1.04 0.0001 0.01 1.04 1.04 1.04 0.0001
(1) 0.27 1.80 2.81 0.16 0.09 1.92 2.67 0.31 0.43 0.74 2.34
(2) 0.27 2.19 2.80 0.14 0.95 3.00 2.82 0.28 0.48 0.77 4.75
0.19
0.43
"(1) Ambrose, D. J. Chem. Thermodyn. 1981,13, 1161. (2) Ambrose, D.; Sprake, C. H. S. J. Chem. Thermodyn. 1975,8,601. (3) Aston, J. G.; Fink, H. L.; Schumann, S. C. J . Am. Chem. SOC. 1943,65,341. (4) Besley, L. M.; Bottomley, G. A. J. Chem. Thermodyn. 1974,6,577. (5) Boublik, T. Thesis, Vscht Prague, 1960. (6) Buck, F. R.; Coles, K. F.; Kennedy, G. T.; Morton, F. J. Chem. SOC. 1949, 2377. (7) Camin, D. L.; Forziati, A. F.; Rossini, F. D. J. Phys. Chem. 1954, 58, 440. (8) Camin, D. L.; Rossini, F. D. J . Phys. Chem. 1955, 59, 1173. (9) Chaiyavech, P.; Van Winkle, M. J. Chem. Eng. Data 1959, 4, 53. (10) Cruichshank, A. J. B.; Cutler, A. J. B. J. Chem. Eng. Data 1966,12, 326. (11) Cunningham, G. B. Power 1930, 72, 374. (12) Dreyer, R.; Martin, W.; Von Weber, U. J. Prakt. Chem. 1955, 273, 324. (13) Eggersten, F. T.; Seibert, E. E.; Stross, F. H. Anal. Chem. 1969,41, 1177. (14) Forziati, A. F.; Norris, W. R.; Rossini, F. D. J. Res. Natl. Bur. Stand. 1949,43, 555. (15) Fowler, L.; Trump, N. W.; Vogler, C. E. J. Chem. Eng. Data 1968, 13, 209. (16) Hertz, W.; Schuftan, P. Z. Phys. Chem. 1922, 101, 269. (17) Hoepfner, A.; Parekh, N.; Hoerner, Ch.; Adbel-Hamid, A. Ber. Bunsenges. Phys. Chem. 1975, 79, 216. (18) Huffman, H. M.; Gross, M. E.; Scott, D. W.; Mc Cullough, J. P. J. Phys. Chem. 1961, 65, 495. (19) Krafts, J. M. Ber. 1915, 13, 105. (20) Letcher, T. M.; Marsicano, F. J. Chem. Thermodyn. 1974,6,509. (21) Linek, J.; Fried, V.; Pick, J. Collect. Czech. Chem. Commun. 1965,30, 1358. (22) Macknick, A. B.; Prausnitz, J. M. J. Chem. Eng. Data 19798 24, 175. (23) McDonald, R. A.; Shrader, S. A.; Stull, D. R. J. Chem. Eng. Data 1959,4, 311. (24) Merlin, J. C.; Allemand, N.; Jose, J. 2nd Codata Symposium, Paris, 1985. (25) Messerly, G. H.; Kennedy, R. H. J . Am. Chem. SOC. 1940,62, 2988. (26) Mortimer, F. S.; Murphy, R. V. Ind. Eng. Chem. 1923, 15, 1140. (27) Nelson, C. A.; Senseman, C. E. Ind. Eng. Chem. 1922, 14, 58. (28) Nicolini, E.; Laffite, P. C. R. Seances Acad. Sci., Ser A 1949, 229, 757. (29) Osborn, A. G.; Douslin, D. R. J. Chem. Eng. Data 1966,20,229. (30) Osborn, A. G.; Douslin, D. R. J. Chem. Eng. Data 1974,19, 114. (31) Pasek, G. J.;Thodos, G. J. Chem. Eng. Data 1962, 7, 21. (32) Pitzer, K. S.; Scott, D. W. J. Am. Chem. SOC.1943, 65, 803. (33) Schumann, S. C.; Aston, J. G.; Sagenkahn, M. J . Am. Chem. SOC. 1942, 64, 1039. (34) Smith, E. R. J. Res. Natl. Bur. Stand 1941, 26, 129. (35) Varushchenko, R. M.; Belikova, N. A.; Skuratov, S. M.; Plate, A. F. Zh. Phys. Khim. 1970,44,3022. (36) Wieczorek, S. A.; Kobayashi, R. J. Chem. Eng. Data 1980, 25, 302. (37) Wieczorek, S. A.; Kobayashi, R. J . Chem. Eng. Data 1981,26, 8. (38) Willingham, C. J.; Taylor, W. J.; Pignocco, J. M.; Rossini, F. D. J . Res. Natl. Bur. Stand 1945, 35, 219.
fitting eq 7 with eq 11to experimental vapor pressure data are given in Table I. The values of parameter m are those obtained by regression. The corresponding average absolute percent pressure deviations are given in column (1). Determination of Parameter m by the Group Contribution Method. From Table I, it is clear that parameter m changes regularly in the homologous series. When m is plotted against the molar mass (Figure 1)the correlation existing between both quantities is evident. These observations lead us to express parameter m in terms of a group contribution method. Parameter m is calculated as a sum of group contributions according to m = 0.22942 + S - 0.21311S2 (12) with 13
S = EMjG, - 0.015G13(0.5G13 + j=1
0.75(Gll + GI2 - G13 - 6)) where S is a sum calculated for all groups forming a compound. Mj and Gj are the characteristic parameter of the group j and the number of groups of this type in the molecule, respectively. Since eq 12 is quadratic in S , it can satisfactorily represent the nonlinear character of the relationship between m and the molar mass (Figure 1). The group parameters were obtained by consecutively fitting eq 11 with m expressed by eq 1 2 to the values of parameter a calculated directly from the vapor pressure data on various classes of hydrocarbons. The resulting parameters are given in Table 11. The 13 groups given in this table were the minimum number of groups with which accurate representation of the vapor pressures of hydro-
carbons was possible. In the case of polyaromatic compounds, the value of the group parameter of the carbon joining two cycles depends on the molecular structure of a compound. It was found that the term completing the expression for S in eq 1 2 makes it possible to represent the influence of structural factors on the vapor pressures of polyaromatics. The scarceness of the available experimental data makes this difficult to check, however. As an example, the structure of several compounds is presented in Table I11 in terms of the proposed group contribution method. It was observed that when the isopropyl or tert-butyl group is attached to a cycle (isopropylbenzene, tert-butylcyclohexane, etc.), good results are obtained if the substituted carbon of the cycle is taken to be one and a half a group and the quaternary (or tertiary) carbon to be half a group. The distribution of the number of groups is given in Table I11 in the case of isopropylbenzene. Table IV gives a list of compounds which should be treated individually due to the particularity of their structure. This list is rather short and includes benzene, toluene, cyclohexane, and two disubstituted pentanes. With these compounds the value of parameter m calculated by using eq 1 2 should be modified by adding a corresponding correction Am. Column (2) of Table I gives the average absolute percent deviations of vapor pressures predicted by using the group contribution method.
Results From the average absolute percent pressure deviations given in Table I, it can be seen that the present form of the cubic equation of state is a very satisfactory means of
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1719 am
+
t + *
I
+
OA
+
0
+
+
0
+
+
D
Q
0
+
O
0
0
u
0 0
0.5 A I
100
200
*
I l/T,
Figure 1. Relationship between the parameter m of eq 8 and the molar mass for several families of hydrocarbons: (+) n-alkanes, (0) alkylcyclopentanes, (0) alkylcyclohexanes, (A)alkylbenzenes, (*) bicycles. Table 11. Group Contribution Parameters for Calculating Parameter m and the Pseudocovolume group parameters i B; M; 1 12.576 0.0608 2 11.049 0.0522 alkanes 5.989 0.0091 CH 3 4 2.022 -0.0448 ( C quaternary 0.0378 5 8.658 cyclopentanes 6 8.739 0.0219 7 10.266 -0.0016 C quaternary 8 8.730 0.0348 11.063 0.0265 cyclohexanes 9 C quaternary 10 8.332 -0.0130 11 0.0325 7.284 aromatics C substituted 12 7.524 0.0357 C polycyclic 13 7.147 0.0357
{E? {E?
representing the vapor pressures of hydrocarbons from the triple point up to 2-3 bar. Only two characterizing parameters (the boiling temperature and the parameter m) are needed. In many cases, the accuracy is similar to that of experimental data. There exist, however, cases where the average pressure deviation is higher than the estimated dispersion of experimental data, and the reasons for this seem to be inherent to the model. This is the case with benzene, toluene, and several methyl-substituted benzenes. The same applies to cyclopentane and cyclohexane. In all the above-mentioned cases, the results were nevertheless still very acceptable. It can be noted that, in the few cases where experimental vapor pressure data exceeded 1 bar, the representation is still correct and that the model is valid up to pressures of about 2-3 bar. One of the most interesting features of the present equation is its ability to represent low and medium vapor pressures with the same parameters. This can be seen very clearly from the results presented in Table I. It is therefore to be hoped that the low-pressure extrapolation will be successful. In Figure 2, the results of the vapor pressure extrapolation Table 111. Examde of the Structure of Several Hydrocarbons in Terms of the Present Group Contribution Method no. of groups of the jth type for jth group positions compd 1 2 3 4 5 6 7 8 9 1 0 1 1 12 2,2-dimethylpentane 4 3 0 1 0 0 0 0 0 0 0 0 1-methyl-1-ethylcyclopentane 2 1 0 0 4 0 1 0 0 0 0 0 0 0 1,3,5-trimethylcyclohexane 3 0 0 0 0 0 0 3 3 0 isobutylbenzene 2 1 1 0 0 0 0 0 0 0 5 1 2-methylnaphthalene 1 0 0 0 0 0 0 0 0 0 7 1 1.5 isopropylbenzene 2 0 0 . 5 0 0 0 0 0 0 0 5
13 0 0 0 0
2 0
1720 Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 Table IV. List of Compounds Requiring a Correction of the Value of Parameter m Calculated by Using Equation 11; m = m ( e q 11) Am compound correction term, Am 0.0170 benzene cyclohexane 0.0130 3,3-dimethylpentane -0.0130 toluene 0.0060 3,3-diethylpentane -0.0390
+
Acknowledgment The a u t h o r s t h a n k eng. G. Auxiette for helpful discussions during t h e work o n this paper. T h i s work was supp o r t e d b y TOTAL-Compagnie FranGaise d e s PBtroles. W e also t h a n k Dr. J. Blanc for h e l p with t h e English translation.
Nomenclature a, b = cubic equation of state parameters
Table V. Prediction of Vapor Pressures Using the Group Contribution Method pressure no. of data sets ap% 10 1.96 alkanes low 0.33 medium 60 1.08 aromatics low 5 0.37 34 medium cyclanes low 0.23 3 0.31 39 medium polyaromatics low 1.04 3 11 medium 0.69 overall av % dev
low medium
21
144
1.08 0.43
Table VI. Calculation of t h e Normal Boiling Point and Vapor Pressures Using One Experimental Boiling Temperature no. of data av for series exptl vapor pressure range, kPa sets considered, n ATb, K aP% 40-47 144 0.09 0.28 7-11 146 0.18 0.35 44 0.33 1.02 0.1-3 0.001-0.1 17 0.30 1.55 n o r m a l boiling t e m p e r a t u r e was as low as 0.01 kPa. T h e p r e s e n t m e t h o d can therefore also be useful w h e n t h e experimental information is very scarce (one experimental p o i n t i n the pressure range 0.001-100 kPa).
Conclusions W i t h the proposed modification of t h e Peng-Robinson e q u a t i o n of state, i t is possible t o correctly r e p r e s e n t t h e v a p o r pressures of c o m p o u n d s c u r r e n t l y e n c o u n t e r e d in p e t r o l e u m fractions in the pressure range f r o m t h e triple p o i n t u p t o 2 bar. Three possible applications of t h e p r e s e n t e q u a t i o n a r e illustrated: (1)Representation of vapor pressure data obtained with t h e f i t t e d p a r a m e t e r m is usually close to t h e accuracy of experimental d a t a . (2) When only the boiling t e m p e r a t u r e is k n o w n , parameter m can be determined b y using the g r o u p contrib u t i o n method, and v a p o r pressures are calculated to within about 0.3% in the case of m e d i u m pressures and to within 1%i n that of low pressures. (3) W h e n t h e experimental boiling t e m p e r a t u r e is n o t available, it can be estimated with any experimental vapor pressure data, and the method can be a p p l i e d as i n t h e previous case. T h e accuracy of calculated vapor pressures d e p e n d s o n the accuracy of the experimental data used, but t h e results a r e satisfactory i n m o s t cases. T h e p r o b l e m of integrating t h i s m e t h o d i n t o t h e cubic e q u a t i o n valid u n d e r high-pressure conditions (supercritical region included) will be considered in a forthcoming paper. I n conclusion, t h e p r e s e n t m e t h o d can be said t o b e a useful tool for m a n y petroleum calculations. It can be used whenever an equation of state giving a good representation of vapor pressures is needed. It is also a n efficient method of correlating or predicting t h e vapor pressures of hydrocarbons.
aTb= value of parameter a calculated at normal boiling conditions
B, = j t h group parameter for calculation of b Gj = number of t h e j t h group in a given compound L = enthalpy of vaporization, J mol-' ml, m2,m = parameters of eq 11 M , = j t h group parameter for calculating m m, = parameter of t h e Soave function (eq 2)
P = pressure, bar T = absolute temperature, K Tb = normal boiling absolute temperature, K T , = reduced temperature
U, = gas energy, J mol-' U1 = liquid energy, J mol-l V , = gas molar volume, cm3 mol-' V,=, liquid molar volume, cm3 mol-' 6,b = noncorrected volume a n d pseudocovolume related t o t h e cubic equation of state, cm3 mol-' = average absolute deviation of X calculated from n experimental determinations, X = lXFPt1- XFalcdl/n
ax%
Registry No. Neopentane, 463-82-1; isopentane, 78-78-4; pentane, 109-66-0;cyclopentane, 287-92-3; 2,2-dimethylbutane, 75-83-2; 2,3-dimethylbutane, 79-29-8; 2-methylpentane, 107-83-5; 3-methylpentane,96-14-0; hexane, 110-54-3;methylcyclopentane, 96-37-7; 2,2-dimethylpentane, 590-35-2; benzene, 71-43-2; 2,4dimethylpentane, 108-08-7; cyclohexane, 110-82-7; 2,2,3-trimethylbutane, 464-06-2; 3,3-dimethylpentane, 562-49-2; 1,l-dimethylcyclopentane, 1638-26-2;2,3-dimethylpentane, 565-59-3; 2-methylhexane, 591-76-4; cis-l,3-dimethylcyclopentane, 2532-58-3; 3-methylhexane,589-34-4; trans-1,2-dimethylcyclopentane, 82250-4; 3-ethylpentane, 617-78-7; heptane, 142-82-5; isooctane, 540-84-1;cis-1,2-dimethylcyclopentane, 1192-18-3;methylcyclohexane, 108-87-2;ethylcyclopentane, 1640-89-7;1,1,3-trimethylcyclopentane, 4516-69-2; 2,2-dimethylhexane, 590-73-8; 2,5-dimethylhexane, 592-13-2; cis,trans,cis-1,2,4-trimethylcyclopentane, 16883-48-0; 2,4-dimethylhexane, 589-43-5; toluene, 108-88-3; 3,3-dimethylhexane,563-16-6; 2,3,4-trimethylpentane, 565-75-3; 1,1,2-trimethylcyclopentane,4259-00-1; 2,3-dimethylhexane, 584-94-1; 2-methyl-3-ethylpentane,609-26-7; cis,cis,trans-1,2,4trimethylcyclopentane, 4850-28-6; 2-methylheptane, 592-27-8; 4-methylheptane, 589-53-7; 3,4-dimethylhexane, 583-48-2; 3methyl-3-ethylpentane, 1067-08-9; 3-ethylhexane, 619-99-8; 3methylheptane, 589-81-1; trans-l,4-dimethylcyclohexane, 220704-7; 1,l-dimethylcyclohexane,590-66-9; trans-1,3-dimethylcyclohexane, 2207-03-6; 1-methyl-1-ethylcyclopentane, 16747-50-5; 2,2,4,4-tetramethylpentane, 1070-87-7;trans-1,2-dimethylcyclohexane, 6876-23-9; 2,2,5-trimethylhexane, 3522-94-9; cis-l,4-dimethylcyclohexane, 624-29-3; cis-1,3-dimethylcyclohexane, 63804-0; octane, 111-65-9;isopropylcyclopentane, 3875-51-2; 2,2,4trimethylhexane, 16747-26-5;cis-2-methyl-l-ethylcyclopentane, 930-89-2; cis-1,2-dimethylcyclohexane,2207-01-4; 2,4,4-trimethylhexane, 16747-30-1; propylcyclopentane, 2040-96-2; ethylcyclohexane, 1678-91-7;2,2,3,4-tetramethylpentane,118653-4; 2,2,3-trimethylhexane, 16747-25-4;ethylbenzene, 100-41-4; 1,1,3-trimethylcyclohexane,3073-66-3; p-xylene, 106-42-3; mxylene, 108-38-3;2,2,3,3-tetramethylpentane, 7154-79-2; 2,3,3,4tetramethylpentane, 16747-38-9;o-xylene, 95-47-6; 3,3-diethylpentane, 1067-20-5;cis-l-methyl-3-ethylcyclohexane, 19489-10-2; nonane, 111-84-2;isopropylbenzene,9882-8; isopropylcyclohexane, 696-29-7;propylcyclohexane, 1678-92-8;propylbenzene, 103-65-1; l-methyl-3-ethylbenzene,620-14-4; l-methyl-4-ethylbenzene, 622-96-8; 1,3,5-trimethylbenzene, 108-67-8; 1-methyl-2-ethylbenzene, 611-14-3; tert-butylbenzene, 98-06-6; 1,2,4-trimethylbenzene, 95-63-6; isobutylcyclohexane, 1678-98-4; tert-butyl-
Ind. Eng. Chem. Res. 1988,27, 1721-1728 cyclohexane, 3178-22-1; isobutylbenzene, 538-93-2; sec-butylbenzene, 135-98-8;decane, 124-18-5;l-methyl-3-isopropylbenzene, 535-77-3; 1,2,3-trimethylbenzene, 526-73-8; 1-methyl-4-isopropylbenzene, 99-87-6;indan, 496-11-7; 1-methyl-2-isopropylbenzene, 527-84-4; sec-butylcyclohexane,7058-01-7;butylcyclohexane, 1678-93-9;1,3-diethylbenzene,141-93-5;butylbenzene, 104-51-8; 1,2-diethylbenzene, 135-01-3;1,6diethylbenzene,10505-5; trans-decaline, 493-02-7; 2-methyldecane, 6975-98-0; cisdecaline, 493-01-6;undecane, 1120-21-4; tetraline, 119-64-2;1,4diisopropylbenzene, 100-18-5; 1,3,5-trimethyl-2-ethylbenzene, 3982-67-0;dodecane, 112-40-3;naphthalene, 91-20-3; tridecane, 629-50-5;2-methylnaphthalene,91-57-6; 1-methylnaphthalene, 90-12-0; tetradecane, 629-59-4; biphenyl, 92-52-4; diphenylmethane, 101-81-5;2,3-dimethylnaphthalene,581-40-8;pentadecane, 629-62-9; acenaphthene, 83-32-9; decylcyclopentane, 1795-21-7;nonylbenzene, 1081-77-2;hexadecane, 544-76-3; decylbenzene,104-72-3;octadecane,593-45-3;phenanthrene, 85-01-8; anthracene, 120-12-7;eicosane, 112-95-8.
1721
Literature Cited Gibbons, G.; Laughton, G. J. Chem. SOC., Faraday Trans. 1984,80, 1019.
Gomez-Nieto, M.; Thodos, G. AIChE J. 1977,23,904. Merlin, J. C.; Allemand, N.; Jose, J. 2nd Codata Symposium, Paris, 1985. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Rauzy, E. Thesis, Marseille, 1982. Riedel, L. Chem.-1ng.-Tech.1954, 26, 83. Rogalski, M. Thermochim. Acta 1987, submitted to publication. Soave, G . Chem. Eng. Sci. 1972,27, 1197. Stryjek, R.; Vera, J. H. Can. J. Chem. Eng. 1986, 64, 323. Thek, R. E.; Stiel, L. I. AIChE J. 1966, 12, 599. Willman, B.; Teja, A. S. Ind. Eng. Chem. Process Des. Deu. 1985,24, 1033.
Received for review October 15, 1987 Accepted May 3, 1988
Crystallization and Agglomeration Kinetics in the Batch Precipitation of Strontium Molybdate Otakar Sohnel,?John W.Mullin,* and A l a n G. Jones Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London WClE 7JE, England Kinetic processes were studied during the batch precipitation of SrMo04(mixing equimolar solutions of SrC12and Na2Mo04at 25 "C) over a range of supersaturations and under different stirring modes. Primary heterogeneous nucleation was predominant, with homogeneous nucleation becoming significant at S > 27, followhg which diffusion-controlled growth became dominant. Secondary nucleation was not detected under the conditions studied. Soon after the induction period, the small individual crystals agglomerated orthokinetically and t h e agglomerate size depended on both t h e intensity of stirring and the initial supersaturation. Toward t h e end of t h e precipitation, the agglomerate particle size distribution stabilized and was no longer affected by prolonged agitation. 1. Introduction
The crystallization of sparingly soluble substances, generally referred to as precipitation, has been studied on a scientific basis since the time of von Weimarn.' Despite much work, however, and the vast amount of information gathered as a result, there is little agreement either on the experimental data themselves or on their interpretation, even for the same precipitating system. The majority of reported precipitation data refer to batch experiments since these are the simplest to perform on a laboratory scale, and although the time-varying data can be difficult to interpret, careful analysis can yield much useful information. The precipitation process may proceed through different nucleation and growth mechanisms, depending on the prevailing conditions; e.g., nucleation can be primary (homogeneous or heterogeneous) or secondary and crystal growth can be controlled by diffusion, by surface nucleation, or by screw-dislocation mechanisms. This enormously complicates the analysis. Moreover, the effects of secondary processes (secondary nucleation, agglomeration, Ostwald ripening, polymorphic transformation, crystal habit modification, etc.) are often ignored in determining crystallization kinetics, and their omission can lead to substantial error. Some of the widely contradictory conclusions, inferred from experimental precipitation data, which have been put forward may be illustrated for the case of BaS04,one of Permanent address: Research Institute of Inorganic Chemistry, RevoluEni 86, 400 60 Usti nad Labem, Czechoslovakia.
the most widely studied precipitated substances. For example, nucleation has been determined as primary with an insignificant proportion of secondary: and exclusively secondary;' crystal growth has been reported as being controlled by diffusion? screw-dislocation,8~9and surface nucleationlo mechanisms; the kinetic order of the crystal growth process has been evaluated as 1,11 2,899J2 3,7J3J4and 4;" agglomeration during the early stages of precipitation has been found to be virtually absent2 and substantial;15J6nuclei were observed to grow into individual discrete crystals3J7 and agglomerate soon after their formation to form pseudosingle c r y ~ t a l s . ' ~ J ~ The object of the present work was to study the batch precipitation of a model substance under carefully controlled conditions and, by comparison of the experimental data with models of nucleation, growth, and agglomeration, to deduce which processes play a decisive role during this mode of operation. SrMo04 was chosen because it (i) forms as a result of a simple ionic reaction without the formation of complexes in the solution, (ii) precipitates as compact shaped crystals of reasonable size that do not change their morphology over the range of supersaturation studied, without the formation of precursors, different crystalline modifications, or an amorphous phase, (iii) readily agglomerates, and (iv) is easily removable from the experimental equipment when deposits are formed. 2. Theoretical Section 2.1. Nucleation a n d Growth. Methods for the identification of nucleation and/or growth mechanisms predominating during the early stages of precipitation have
0888-5885/88/ 2627- 1721$01.50/0 0 1988 American Chemical Society
