Ind. Eng. Chem. Res. 1989,28, 122-127
122
Effects of Surface Active Agents on Minimum Impeller Speeds for Liquid-Liquid Dispersion in Baffled Vessels The effects have been examined of nonionic, anionic, and cationic surface active agents (S.A.A.)on the minimum impeller speed for complete dispersion in agitated vessels. Variables included size and form of impeller, fluid properties in two equal-volume liquid-liquid systems, and surfactant concentration. The Skelland and Ramsay correlation (eq 6) served to predict the minimum impeller speed, when used with diminished interfacial tensions due to the presence of S.A.A., with an overall absolute deviation of 11.67%. This contrasts with some droplet phenomena, where more elaborate allowance for contamination effects is required. The difference between deviations among the surfactant systems was accounted for by experimental error and verified by two methods of statistical analysis. Dispersion in liquid-liquid systems is one of the most common forms of industrial contacting, as in mixer-settlers in extraction operations and in emulsion polymerization. In such processes, impellers are used to disperse one immiscible liquid in the other, with agitator speeds that are sufficiently high to achieve complete dispersion. In industrial operations, trace amounts of surface active contaminants are widely encountered, and the effect of these materials in distorting performance from that predicted by correlations for pure systems requires accommodation. This is the object of this study. Surface active agents (S.A.A.) may modify the hydrodynamics by reducing the rate of internal circulation in drops (Gamer and Skelland, 1955), lowering the interfacial tension (Davies and Rideal, 19631, and retarding the rate of droplet coalescence (Lang and Wilke, 1971). Previous work on the minimum impeller speed needed for complete dispersion began with Nagata's (1950) study. Using an unbaffled, flat-bottomed vessel, with centrally mounted, four-bladed, flabblade turbine agitator with TID equal to 3 and blade width of 0.06T, Nagata developed the empirical relationship
He found no effect of interfacial tension on the minimum impeller speed. The study by van Heuven and Beek (1971) examined liquid-liquid dispersion in a baffled vessel using a sixbladed disk turbine. From a combination of theory and experiment, they found 3,28g0.38~p0.38P 0.08,0.08( 1 + 2.5$)0.90 C "in
=
Zwietering (1958) employed turbines, propellers, paddles, and vaned disks in baffled vessels for producing solid-liquid suspensions and obtained
cy'
Nmin=
CDDaoPc1/g~d-l/9,,0.3Ap0.25
(5)
where H I T = 1, $ = 0.5, and T was constant. Skelland and Ramsay (1987) developed their correlation as an extension of the earlier work by Skelland and Seksaria (1978). They combined 251 runs from their own study with 35 runs compiled by van Heuven and Beek (1971) and 195 runs reported by Skelland and Seksaria (1978). Using a total of 481 data points on 5 impeller types (3 radial and 2 axial flow), 4 impeller locations, and 11 systems, they present the following more general correlation based on dimensional analysis and an assumed exponential form:
where the densities of the two phases were accommodated using a mean density, defined as PM = 4Pd
+ (1 - 4)Pc
(7)
The mean viscosity is that due to Vermeulen et al. (1955) and is PM
=
(l+Z)
(2)
D0.77PM0.54
where the ratio of impeller to tank diameter was fixed at 0.3. Pavlushenko et al. (1957) studied a related but different phenomenon, namely the suspension of sand and iron particles in liquids agitated by three-bladed, squarepitched propellers. They correlated their results by
where C'and
location. The conflicting directional effect of the continuous-phase viscosity, P ~ indicated , by eq 3 and 4 should be noted. Skelland and Seksaria (1978) studied minimum impeller speed using propellers and pitched-, flat-, and curved-blade turbines with liquid-liquid systems in a baffled vessel. Their correlation is given by
are constants based on impeller type and 0SS8-5SS5/S9/262S-0122$01.50/0
The third group on the right-hand side of eq 6 is the product of the reciprocals of the Galileo and the Bond numbers. The former ( N G a ) is proportional to the ratio of inertial times gravitational to viscous force squared in the system. The Bond number is proportional to the ratio of gravitational to interfacial tension forces prevailing in the system. Skelland and Ramsay (1987) therefore expressed eq 6 in abbreviated form as
which expands to
The values of C and cy as determined by Skelland and Ramsay for various impellers and locations are given in Table I. It is interesting to note the similarity between Q 1989 American Chemical
Society
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 123 Table I. Skelland and Ramsay (1987) Correlations and Average Deviation between Nawm and N,,. Equation lon 959i confidence a interval set HIT imDeller location C 9i av dev 1 1 square pitch, 4.38 0.67 f0.44 16.9 ~ 1 4 downthrusting 2 1 3H/4 2.76 0.95 f0.44 14.4 1 3 4.33 0.79 f0.44 17.7 prop e11er HI2 1 1.46 1.33 f0.44 13.7 4 Hf4, 3Hf4 (3 blades) 1 1.95 1.44 f0.47 13.9 5 dowthrusting ~ 1 4 1 6 3H/4 pitched-blade 1.96 1.17 f0.47 14.2 1 7 0.84 1.97 f0.47 18.6 turbine HI2 H/4, 3H/4 1 0.94 1.27 h0.47 (6 blades) 10.0 8 1 0.91 2.02 flat-blade 9.9 f0.45 9 ~ 1 4 * * * * 1 3H/4 10 turbine 11 1 0.95 1.38 f0.13 (6 blades) 11.5 HI2 * * * * H/4, 3H/4 12 1 0.70 1.24 10.2 18 f0.28 HI2 14.7 1.10 1.70 19 f0.28 Hi2 31/ 2 13 f0.46 10.2 1.03 1.86 curved- blade ~ 1 4 * * * * 1 14 3H/4 turbine 1 1.34 1.20 f0.46 8.9 (6 blades) 15 HI2 Hf4,3H/4 1.20 1 0.94 f0.46 10.7 16 1.70 15.8 1 17 0.53 disk turbine (6 blades) HI2 Asterisks indicate insufficient data due to splashing. Table 11. Fluid Properties at 23 O C dynamic viscosity, fluid densitv. ke/m3 (Nd / m 2 chlorobenzene 1107 0.0010 benzaldehyde 1041 0.0014 water 998 0.0010
Experimental Apparatus and Procedure Fluids Used. Distilled water formed one phase in all runs; the other phase consisted of either chlorobenzene or benzaldehyde. All runs were performed at 23 OC, and the fluid properties appear in Table 11. The chlorobenzene and benzaldehyde were Purified and Certified grades,
* 1.27 *
0.85 2.23 2.71
*
0.73 -0.05 2.23
't
interfacial tension with water. N/m 0.0352 0.0154
eq 10 and the expression given by Zwietering (1958) for solid-liquid suspensions. This is at least consistent with the suggestion by van Heuven and Beek (1971) that similar mechanisms exist between liquid-liquid dispersion and solid-liquid suspension. Further studies on impeller speeds required for dispersion in agitated liquid-liquid systems have been made by Esch et al. (1971), Godfrey et al. (1984), Grilc (19791, and DeMaerteleireand Heyndrickx (1986). However, none of the previous work considered the influence of surface active contamination on Nmi,,which is the object of the present study. Questions to be addressed include whether eq 6 will be rendered inadequate in the presence of such common surfactant contamination. This could arise, for instance, from modification of the disperse-phase settling rate or of phase separation because of the reduction in drop size, internal circulation, and terminal velocity of the drops on the one hand, or the retardation of droplet coalescence on the other, as induced by traces of surface active impurities. Such influences are of course not necessarily measured by u, the interfacial tension, so it is of interest to test eq 6 under such previously unexplored circumstances. In any event, the exponent on u is very low; namely 0.04 (see eq 6 and 10). This is midway between the exponent of zero found by Nagata (1950) and 0.08 found by van Heuven and Beek (1971) in eq 2; it was determined statistically by Skelland and Ramsay (1987) using data from 481 runs, drawn from 3 published sources, as 0.5 (0.084 f 0.017) at the 95% confidence level. Evidently only slight correction would be possible in eq 6 through interfacial tension alone.
3a - 2.87 -0.86 -0.02 -0.50 1.12 1.45 0.64 3.04 0.94 3.19
30
I
O
2
I
!
4
6
B
I
I
l
O
1
2
C (gram/Iiter) Figure 1. Chlorobenzene-water interfacial tension versus surfactant concentration.
n
E
s C
z
-0
v
b
-
0 0
2
4
6
8
1
0
1
2
C (gram/iiier) Figure 2. Benzaldehyde-water interfacial tension versus surfactant concentration.
respectively, as supplied by Fisher Scientific Co. Interfacial tensions were measured by using a 1 scher surface tensiometer, Model 21. The force necessary to pull a platinum-iridium duNuoy ring through the liquid-liquid
124 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 Table 111. ADDaratus SDecifications 0.2135 m internal diameter of vessel 0.2135 m liquid height in vessel 0.2500 m height of vessel 0.0130 m diameter of shaft 0.2300 m baffle length 0.0190 m baffle width 0.0030 m baffle thickness length of baffle in liquid from air-liquid interface 0.1930 m 0.50 volume fraction of organic liquid R ope1 1e r
Pit ched-bhde turbine
? l e t bottom Denser l l p u l d
S i d e Y!eu
Flat-blede turbine
Curved-blade turbine
Figure 4. Mixing impellers used.
B o t t m Vlcv
Figure 3. Schematic diagram of the experimental apparatus.
interface is converted directly into interfacial tension and read from a calibrated dial. The interfacial tensions of chlorobenzene and benzaldehyde with distilled water containing surfactant are shown in Figures 1 and 2, respectively. Surface Active Agents Used. The surfactants, their grades, and their sources were as follows: nonionic, octylphenoxypolyethoxyethanol(Triton X-loo), laboratory grade, supplied by the Sigma Chemical Co., St. Louis, MO; anionic, dodecyl sodium sulfate, Certified grade, from Fisher Scientific Co.; cationic, dodecyl pyridinium chloride, research grade, provided by Pfaltz and Bauer, Inc., Stamford, CN. Apparatus. The experimental agitator Model ELB manufactured by the Bench Scale Equipment Company was used to study the liquid-liquid dispersions. The unit included a llq-hpdrive motor, which provided an infinitely variable output speed of 0 to 20 rps. The speed control dial was calibrated directly into rpm by a tachometer. A 0.01-m3cylindrical, flat-bottomed glass vessel was used to enable visual observations. Four baffles were placed radially at 90' intervals to prevent vortex formation. Specifications of the apparatus are given in Table 111, and a diagram is shown in Figure 3. Agitation was effected by attaching an impeller to a centrally mounted vertical shaft. The shaft, impellers, and baffles were all made of 316 stainless steel. Two copper wire electrodes, placed 0.01 m apart, were attached to a conductivity meter to identify the continuous phase. Four types of impellers, available in two different sizes, were used, as shown in Figure 4. The characteristics of
the impellers are described below, based on their flow patterns. Axial Flow. Propellers and pitched-blade turbines belong to this group. They have suction and discharge in a direction parallel to the shaft axis. The propellers used were three-bladed with square pitch and had diameters of 0.10 and 0.06 m. The pitched-blade turbines had six blades at 4 5 O from the vertical, with diameters of 0.10 and 0.062 m. Radial Flow. The flat- and curved-blade turbines are in this category. They have suction in a direction parallel to the shaft axis and discharge normal to the shaft. The curved-bladeor "backswept" turbines curve away from the direction of rotation. Both the flat- and curved-blade turbines were six-bladed and had diameters of 0.10 and 0.065 m. Operational Procedure. Before filling with liquids, all glassware was washed with chromic acid, and then all equipment was washed with hot water containing the appropriate detergent, thoroughly rinsed with distilled water, and air-dried. Two immiscible liquids were poured into the vessel, to a height equal to its diameter, and in equal volumetric proportions for all runs. The impeller was centrally located at H / 2 throughout, with its speed being gradually increased in small increments until complete dispersion was visually observed, as described by Skelland and Seksaria (1978). Techniques for Identifying Continuous and Disperse Phases. The identification of the continuous phase was usually determined visually in the pure systems, as described in detail by Quinn and Sigloh (1963) and by Selker and Sleicher (1965). However, in the presence of surfactants, the most reliable means of identification was provided by the conductivity electrodes. In the anionic and cationic systems, the cloudy emulsion-like dispersions made the conductivity probe the only reliable method to determine the continuous phase. The conductivity of the organic liquids was much lower than that of the aqueous phase, making identification of the continuous phase a simple matter. Thus, a nearly full-scale deflection of 0.01 mh/m was observed on the meter when water was con-
Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 125
20 10
i 1
5
10
20
10
20
Nexp (rps)
Nexp (rps)
Figure 5. Nd from eq 10 versus Ne,, for the propeller.
Figure 7. Nd from eq 10 versus Ne,, for the curved-blade turbine.
20
L
I 4
A
0 0
l:i
z
1
5
10
20
5
1
10
20
Nexp (rps)
Nexp (rps)
Figure 6. Nd from eq 10 versus Nm for the pitched-blade turbine.
Figure 8. Nd from eq 10 versus Nexpfor the flat-blade turbine
tinuous, whereas the deflection was negligible when the continuous phase was organic. Results for each run are reported by Moeti (1984).
Table IV. Average Absolute Deviation between Ne=*and Noelfrom Equation 10 no. of runs propeller 9.978% 72 overall 24 nonionic 6.075% 24 anionic 10.413% 24 13.446% cationic pitched blade 14.302% overall 72 9.333% nonionic 24 15.454% anionic 24 18.119% cationic 24 curved blade 11.718% 72 overall 7.265% 24 nonionic 24 anionic 13.380% 24 14.508% cationic flat blade overall 10.671% 72 nonionic 8.266% 24 anionic 11.310% 24 12.435% 24 cationic overall deviation 288 11.67%
Results and Discussion Type of Mixing. The type of mixing occurring with impellers at H/2in this study is qualitatively similar to that described by Skelland and Seksaria (1978) for Type 5 mixing, in which dispersion begins in the vicinity of the impeller and moves through the liquid mass until a well-mixed system is attained at higher speeds. Minimum Mixing Speed and Reproducibility. The minimum mixing speed, as described by Skelland and Seksaria, is visually determined as the speed just sufficient to completely disperse one liquid in the other, such that no clear liquid is present at either the top or bottom of the agitator vessel, with only relatively small nonstationary pockets remaining in the bulk dispersion. This is not necessarily the speed required for a homogeneous or uniform dispersion. In the case of the nonionic system, the visual determination of minimum impeller speed was easily ascertained. In such systems, there was minimal cloudiness and the large droplets made the reproducibility of the minimum impeller speed good. In the anionic and cationic systems, however, visual determination of the minimum impeller speed was more difficult. The much lower interfacial tensions, compared to the nonionic system, led to smaller droplets in a cloudy emulsion and made it necessary to repeat runs and take the average minimum impeller speed as representative of the true value. Comparison between Experimental and Calculated Results. The experimentally determined minimum impeller speeds were compared to the values calculated from
the Skelland and Ramsay (1987) correlation, using diminished interfacial tensions due to the S.A.A. Figures 5-8 show plots of NCdversus Nexp for the different types of impellers. Table IV gives the deviation between the experimental and calculated values of N- based on impeller type and surfactant system. The absolute deviation is defined as deviation =
INcd - N e x p l II
x 100%
(11)
JVCd
From Table IV it is observed that the Skelland and Ramsay correlation applied in this manner is a good pre-
126 Ind. Eng. Chem. Res., Vol. 28, No. 1, 1989 Table V. Constants Evaluated from Regression Analysis propeller overall nonionic anionic cationic pitched blade overall nonionic anionic cationic curved blade overall nonionic anionic cationic flat blade overall nonionic anionic cationic
c
01
4.165 f 0.317 4.342 4.142 4.017
0.791 & 0.071 0.804 0.787 0.782
0.655 f 0.063 0.739 0.606 0.628
2.123 f 0.089 2.072 2.182 2.114
0.988 f 0.100 1.033 0.982 0.952
1.456 f 0.097 1.494 1.423 1.450
0.847 f 0.085 0.840 0.816 0.888
1.502 f 0.096 1.610 1.493 1.403
dictor of Nminfor the surfactant systems studied, giving an overall absolute deviation of 11.67%, compared to a 12.7% overall absolute deviation reported by Skelland and Ramsay in their work. Techniques Used for Analyzing Data. It is perhaps noteworthy to observe the apparent trend in deviation indicated by the surfactant systems. The nonionic surfactant systems produce smaller deviations than those observed in the anionic and cationic systems. It was of interest to determine whether such trends were due solely to experimental error or were based on some inherent characteristic of the surfactant system. Two methods were used to evaluate the directional effects of the surfactants, namely, regression analysis and one-way analysis of variance (one way ANOVA). These are outlined below. Regression Analysis. Although the Skelland and Ramsay (1987) correlation seemed adequate for all impellers and surfactant systems, it was desirable to determine whether there was any significant difference between the effects of the three surface active agents used in this study. Forcing the exponents on the physical properties as given by eq 10, the constants C and cy were evaluated for the impellers and surfactant systems. Table V shows the constants and the attendant 95% confidence intervals. It was observed that the constants for the surfactant systems fell mostly within the confidence intervals indicated. In addition, hypothesis testing on the equality of the constants using t-tests was performed. The hypotheses tested were of the type that for a given impeller the constants were equal, e.g., for the propellers, Canionic = Cationic, Canionic = Cnonionic, anonionic - cyc,tionic,and SO on. Within a 95% confidence interval, the hypothesis of equality of constants among impeller groups could not be rejected. One-way Analysis of Variance. The second method used to evaluate the apparent trend indicated by the surfactant systems employed the one-way analysis of variance and covariance which tests the equality of coefficients between groups. In this application, the groups were taken as the surfactant systems for each impeller. Using F-tests, the equality of coefficients could not be rejected a t 95% confidence levels. All statistical analysis was performed with the aid of BMDP Statistical Software (1983). In this work, programs iR and 9R were used for regression analysis and program iv was used for analysis of variance. In the present study, it was found that the Skelland and Ramsay (1987) correlation, used with the diminished in-
terfacial tension due to the presence of nonionic, anionic, or cationic surfactants, was adequate for predicting minimum impeller speed for complete dispersion. This contrasts with other droplet phenomena, such as the eccentricity and oscillation frequency of drops in free fall. Correlations for these latter quantities in contaminant-free systems have been provided by Wellek et al. (1968) and by Lamb (1945), respectively. However, Mekasut et al. (1979) showed that use of the diminished values of u in surface actively contaminated systems did not bring these correlations into line with measured values. It should, of course, be understood that the purpose of the present work is not to recorrelate Nmkin terms of the particular species of surface active contaminants used here. Rather, the surfactants are intended to simulate some of the impurities encountered industrially, which are usually unknown with regard to number of species, structure, and concentration. The object, then, is to ascertain whether characterization of their effects by measurement of a single property, u, is adequate for purposes of prediction of Nmh via eq 10 without further modification. The indications from this study are in the affirmative. For developments regarding scale-up, the reader is referred to Skelland and Ramsay (1987).
Acknowledgment Partial financial support from National Science Foundation Grants CPE 82-03872 and CPE 82-03872101 is gratefully acknowledged.
Nomenclature B = baffle width, m C, C,, C’ = constants, “shape factors‘! D = impeller diameter, m D, = particle diameter, m g = acceleration due to gravity, m/s2 H = height of liquid in the vessel, m NBo= Bond number, D2gAp/a NFr= Froude number, DN2pM/gAp NG, = Galileo number, D3p&Ap/pM2 Nmin= minimum rotational speed of impeller for complete liquid-liquid dispersion in agitated, baffled vessels, without regard to uniformity, rev/s P = power input to the system, W R = weight fraction of solids T = tank diameter, m V = volume of total liquid, m3 W = width of impeller blade, m Greek Symbols a,ao, a’ = constants p c , & = viscosities of continuous and disperse phases, (Ns)/m2 WM = given in eq 8, (N s)/m2 po Pd = densities of continuous and disperse phases, kg/m3 pM = given in eq 7 , kg/m3 AP = I P C - P& kg/m3 0 = interfacial tension, N/m
4 = volume fraction of disperse phase
Literature Cited BMDP Statistical Software; University of California Press: Berkley, 1983.
Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic Press: New York, 1963. DeMaerteleire, E.; Heyndrickx, G. V. Rijksuniu, Gent 1986, 51 (4), 1397-1406. Esch, D. D.; d’Angelo, P. J.; Pike, R. W. Can. J . Chem. Eng. 1971, 49, 872-875. Garner, F. H.; Skelland, A. H.P. Chem. Eng. Sci. 1955, 4, 149. Godfrey, J. C.; Reeve, R. N.; Grilc, V. Inst. Chem. Eng., Symp. Ser. 1984, 89, 107-126.
Ind. Eng. Chem. Res. 1989,28, 127-130 Grilc, V. Vestn. Slov. Kem. Drus. 1979, 26, 123-135. Lamb, H. Hydrodynamics, 6th ed.; Cambridge University Press: New York, 1945. Lang, S. B.; Wilke, C. R. Znd. Eng. Chem. Fundam. 1971,10, 341. Mekasut, L.; Molinier, J.; Angelino, H. Can. J. Chem. Eng. 1979,57, 688. Moeti, L. T. M.S. Thesis in Chemical Engineering, Georgia Institute of Technology, 1984. Nagata, S. Trans. SOC.Chem. Eng. Jpn. 1950,8, 43. Pavlushenko, I. S.; Kostin, N. M.; Matveev, S. F. Zh. Prikl. Khim. 1957, 30, 1160. Quinn, J. A.; Sigloh, D. B. Can. J. Chem. Eng. 1963, 41,15. Selker, A. H.; Sleicher, C. A., Jr. Can. J. Chem. Eng. 1965,43, 298. Skelland, A. H.P.; Ramsay, G. G. Znd. Eng. Chem. Res. 1987,26,77. Skelland, A. H. P.; Seksaria, R. Ind. Eng. Chem. Process Des. Deu. 1978, 17, 56.
127
van Heuven, J. W.; Beek, W. J. Paper No. 51, Proc. Int. Solvnt. Extract. Confer., 1971. Vermeulen, T.; Williams, A. M.; Langlois, G. E. Chem. Eng. h o g . 1955, 51, 85f. Wellek, R. M.; Agrawal, A. K.; Skelland, A. H. P. AZChE J. 1968,2, 854. Zwietering, T. N. Chem. Eng. Sci. 1958,8, 244.
A. H. P. Skelland,* L. T.Moeti School of Chemical Engineering Georgia Institute of Technology Atlanta, Georgia 30332 Received for reuiew January 16, 1987 Reuised manuscript received August 22, 1988 Accepted October 16, 1988
Correlation for the Second Virial Coefficient for Nonpolar Compounds Using Cubic Equation of State A correlation for the prediction of the second virial coefficient of nonpolar compounds has been developed by using a cubic equation of state and requires the availability of the critical pressure, critical temperature, and Pitzer acentric factor a. Predictions are in excellent agreement with the experimental data and compare well with the values obtained by means of Tsonopoulos correlation and four cubic equations of state: Soave-Redlich-Kwong, Peng-Robinson, Kubic, and LielmezsMerriman modification of Peng-Robinson equation. The expanded form of the pressure explicit virial equation of state may be written as follows:
P=
q 1 +v+ V
B -
C
D
v2
v3
- + - + ...
where B is the second, C the third, D the fourth virial coefficient, etc. Often, however, at moderate pressures, density is less than half the critical and the virial equation truncated after the second term provides an excellent estimate of the vapor-phase fugacity coefficient. While there are several methods for predicting the second virial coefficient (Pitzer and Curl, 1957; Tsonopoulos, 1974; Hayden and O’Connell, 1975; Martin, 1967, 1979, 19841, for our use in this work, we have adopted the general form of the Tsonopoulos (1974) modification of the three-parameter corresponding states correlation of Pitzer and Curl (1957):
3=
f‘0’ + ,f(l) (2) R TC where f ( O ) represents the reduced second virial coefficient of simple fluid which has zero acentricity, f ( l )is the correction term for normal fluids, and w is the Pitzer acentric factor. Recent work (Adachi et al., 1983; Yu et al., 1985) discloses that none of the currently popular cubic equations of state give accurate values of second virial coefficients except the equation of Kubic (1982). The Kubic equation uses Tsonopoulos’ correlation to predict one of its constants. However, even the Kubic equation yields not very accurate values for vapor pressures and densities. The noted shortcomings of cubic equations of state to accurately predict the values of second virial coefficients may be better understood if we consider the generalized form of the cubic equation of state (Schmidt and Wenzel, 1980): a(T) p = - -RT (3) V - b V 2+ ubV+ wb2 If eq 3 is expanded in inverse molar volume similar to eq
0888-5885/89/2628-0127$01.50/0
1,the second virial coefficient is given as
(4) where
a ( T ) = a(TC)a (5) The reduced form of the second virial coefficient is
where
F = a/T,
(7)
Shaw and Lielmezs (1985) showed that for most cubic equations of state instead of eq 7 the F function could be written as a power series in inverse reduced temperature. For instance, the F function for Soave-Redlich-Kwong (Soave, 1972,1980) and Peng-Robinson (1976) equations would assume the form of
F
=
c 2 c3 c1+ + T,0.5 Tr
Constants C1-C3 are functions of the acentric factor, w. Figures 1 and 2 show that below T, = 0.8 the slope of the second virial coefficient increases rapidly with decreasing temperature. If we compare the functional dependence of F on reduced temperature T,, we see that the F function developed from the generalized cubic equation of state (eq 7 and 8) and, depending on reduced temperatue T, with the power of -1, does not adequately describe the swift change of the second virial coefficient in the low-temperature region. Replacing the simple a function as given by Soave (1972) or Peng-Robinson (1976) by a more complex function (eq 7 and 8), however, would correct for this inadequacy, permitting us to follow the second virial coefficient change in the low-temperature range with a relatively high degree of accuracy. These observations and the availability of new experimental data (Dymond 0 1989 American Chemical Society
