Ind. Eng. Chem. Res. 1994,33, 681-688
681
Viscosity Correlations for Binary Supercritical Fluids Kevin D. Tilly, Neil R. Foster,' Stuart J. Macnaughton, and David L. Tomaskot School of Chemical Engineering and Industrial Chemistry, University of New South Wales, P.O. Box I, Kensington, NSW, 2033,Australia The viscosities and densities of supercritical mixtures of methanol, ethanol, n-propanol, isopropanol, n-pentane, n-hexane, n-heptane, and acetone in carbon dioxide, a t concentrations between 1 and 5 mol %, were determined using a falling weight viscometer at pressures u p t o 240 bar and a t temperatures between 313 and 328 K. The effects of pressure, temperature, cosolvent concentration, and the physical properties of the cosolvents on the mixture viscosity and density were examined. The viscosities and the densities of the mixtures were found to increase with the size, polarity, and concentration of the cosolvent molecule. The mixture viscosity was correlated with several empirical dense gas viscosity correlations. The best correlation was the Ely and Hanley technique modified with a density-dependent noncorrespondence factor. The Peng-Robinson equation of state was used to correlate the mixture densities.
Introduction The addition of small quantities of an organic cosolvent to supercritical carbon dioxide has been shown to provide a means of increasing the solubility and/or selectivity of a solute in a supercritical fluid. Equilibrium solubilities of various solids in mixtures of cosolvents and supercritical fluids have been determined in a number of recent investigations (Dobbs et al., 1987;Schmitt and Reid, 1986; Dobbs and Johnston, 1987;Gurdial and Foster, 1991),and theoretical frameworks for the correlation of solubility in these mixed solvents have been postulated (Walsh et al., 1987). It has been reported that the addition of a cosolvent to a supercritical fluid can substantially increase the critical point of the mixture (Gurdial et al., 1993). The mixture critical point must, therefore, be known before any experimental investigations are undertaken with these systems to ensure operation in the supercritical region. The use of cosolvent-supercritical fluid systems to improve solubility and/or selectivity can improve the economics of a supercritical fluid extraction process by reducing the so1vent:solute ratio or by enabling operation at a significantly lower pressure than would be practicable using a pure supercritical fluid. However, the commercialization of a supercritical process requires physical data other than equilibrium solubilities. In particular diffusion coefficients,heat-transfer coefficients,heat capacities, and viscosities are required to enable process equipment to be designed reliably. Unfortunately, there have been few reported studies in which these properties have been reported for cosolvent-supercritical fluid systems. The aim of this study was to determine the viscosities of mixtures of cosolvents in supercritical carbon dioxide, and to investigate the effects of the physical properties and the concentrations of the cosolvents on the viscosity of the resulting mixtures. The influence of these properties on the mixture density was also investigated. The viscosity of these fluid mixtures is of particular importance for the design of process equipment, where the viscosity is an important term in a number of dimensionless groups. Increased accuracy in the prediction of the dimensionless groups required should also result in capital expenditure savings by enabling the overdesign factors to be reduced,
* To whom correspondence should be addressed.
t Present Address: Departmentof Chemical Engineering,Ohio State University, Columbus, OH 43210-1180.
0888-5885/94/2633-0681$04.50/0
which when dealing with high-pressure systems can be substantial.
Experimental Section Theory. A falling cylinder technique has been used by Hawkins et al. (1935)to measure the viscosity of superheated steam, Swift et al. (1958)for liquid n-propane, and later (Swift et al., 1960)for methane, ethane, n-propane, and n-butane. Iezzi et al. (1988)used a falling weight viscometer to investigate the effect on viscosities of additives to carbon dioxide, in order to evaluate viscosity enhancers for enhanced oil recovery. The falling weight viscometer consists of a cylindrical weight falling through a close-fitting tube which is closed at one end so that the fluid is forced to flow through the annulus between the weight and the tube. The terminal velocity of the weight is related to the viscosity of the fluid, the size of the annulus, and the density difference between the weight and the fluid. Assuming that the fluid flow through the annulus may be approximated by the flow between parallel plates and that the head resistance can be neglected, the viscosity of the fluid (7) may be related to the fall time ( t )through a fixed distance ( 5 ) by t=
t a3d p -P) 3(d + 26)2 - (26)2
where u = the density of the weight, p = the density of the fluid, and 6 = the annular space between the weight and the tube. The theoretical expression for the viscometer constant was derived by Lohrenz et al. (1960)without using the approximation of parallel plate geometry. In practice it is uncommon to use a viscometer constant calculated from its theoretical definition due to the uncertainty in assuming perfect laminar flow and the difficulty in measuring the required dimensions of the viscometer. Consequently a calibration constant is usually introduced so that 7 = C(u-p)t (2) The calibration of the viscometer is performed by measuring the fall time of the weight for a fluid of known density and viscosity. A plot of the viscosity as a function of the product of the density difference and the fall time yields the viscometer constant as the slope of the calibration curve. In cases where the viscosity is low (i.e., less than 5 mPa-s) Sen and Kiran (1990)reported that the calibration constant becomes a function of the fall time
0 1994 American Chemical Society
682 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
b
a
T-
Led#
chart
F e d Lim
I
Figure 1. Experimental apparatus.
due to the onset of nonlaminar flow. In this region the calibration constant may be determined by plotting the quotient of the viscosity and the density difference for various fall times. The equation of the resulting straight line results in the calibration constant as a function of fall time as follows:
C=at+b (3) The fluid density is also required and may be determined experimentally or calculated from an equation of state. The effect of eccentricity of the fall body and the viscometer tube has been examined by Chen et al. (1968). It was found that, in the case where the weight does not fall concentrically with the tube, the shear stress at the wall varies with angular position. This variation in shear stress creates a force couple which tends to rotate the weight end over end, while at the same time other forces tend to realign the cylinder. The force couple and the recentering forces cause the weight to oscillate within the viscometer tube and result in substantial changes to the terminal velocity. In order to minimize the effects of the eccentricity, it has been recommended that the ratio of the fall weight diameter to the tube diameter be kept above 0.93 (Chen et al., 1968; Sen and Kiran, 1990). I t has also been reported that a tube deviation of 1"from the vertical can result in a 2% reduction in fall times (Irving, 1972). Apparatus. The viscometer body was constructed from a thick-walled stainless steel tube of 12.5-mm 0.d. and approximately 600 mm in length. The internal surface of the tube was polished to a mirror finish, and the resulting internal diameter was 9.0 mm. An aluminum fall body was constructed with an outside diameter of 8.90 mm resulting in a wall clearance of 0.05 mm and a ratio of the weight diameter to the tube diameter of 0.989-substantially above the value of 0.93 suggested by Chen et al. (1968) to minimize eccentricity effects. The leading end of the fall weight was hemispherical, and the trailing edge was slightly chamfered to reduce wake turbulence as was suggested by Sen and Kiran (1990). A neodymium alloy magnet (6-mm diameter X 6 mm) was embedded in the trailing end of the fall weight to provide a means of lifting
the fall weight through the tube and to enable the fall time to be measured using the current generated by the magnet as it passed through two coilswound on the exterior of the tube. The coils consisted of approximately 20 turns of 0.2-mm plastic coated copper wire and were connected in series to a chart recorder. The output signal generated by the coils was of the order of 0.3 V. The magnet mounted in the fall weight was also used to lift the weight to its release position with a larger magnet that was lifted manually along the exterior wall of the tube. The construction of the fall weight is illustrated in Figure la. The bottom end of the fall tube was sealed with a 1/2-in. stainless steel Swagelok cap. A Teflon cylinder located inside the cap prevented deformation of the fall weight on contact with the bottom of the tube. The viscometer was located in a Perspex water bath provided with facilities for the accurate alignment of the viscometer tube to the vertical. The bath was heated using a Grant Instruments Type ZD bath heater which also provided sufficient circulation to maintain the desired temperature to within f O . l "C. The entire experimental apparatus is presented schematically in Figure lb. The internal volume of the viscometer was calibrated by bleeding pure carbon dioxide at a known temperature and pressure from the vessel through a wet-gas meter. The volume of gas and the calculated density of carbon dioxide (Pitzer and Schreiber, 1988) at the system conditions allowed the calculation of the internal volume of the viscometer. Both the total volume and the release volume were determined in this way. The calibration of the viscometer was performed by measuring the fall time of the weight for pure carbon dioxide. The density and viscosity of carbon dioxide as functions of temperature and pressure were obtained from Altunin and Sakhabetinou (1972)and Pitzer and Schreiber (1988). The relationship between the calibration constant and fall time (eq 3) was obtained by plotting the ratio of pure COz viscosity to the difference in density between the weight and the COz as a function of experimental fall
Ind. Eng. Chem. Res., Vol. 33, No. 3,1994 683 time. The calibration curves all exhibited linear behavior at the four experimental temperatures a t which calibration took place. The cosolvent-carbon dioxide mixtures were prepared by injecting a known mass of the liquid cqsolvent into the body of the viscometer. The viscometer was subsequently filled, through valve V1, with carbon dioxide and pressurized to the required starting pressure with a Waters M45 HPLC pump. The contents of the viscometer were agitated by raising and lowering the fall weight through the viscometer tube and then allowed to equilibrate for at least 8 h before any measurements were performed. No changes in system pressure or in the measured fall times were observed after this period of time. The concentration of the mixture was determined from the weight of cosolvent added and the amount of carbon dioxide, the latter being measured at the end of the experimental determinations by bleeding the gas through a wet-gas meter. Densities were determined from the volume of the system and the quantities of cosolvent and carbon dioxide. The valves labeled as V2 and V3 enabled a constant volume of the fluid to be isolated from the viscometer body. This isolated volume of fluid could then be vented and the residual cosolvent flushed from its internal surface. The remaining fluid mixture could then be expanded into the total volume by closing V3 and opening V2. In this manner the pressure of the system could be reduced while maintaining a constant cosolvent composition within the viscometer. The density at any given pressure could then be determined from the final density and the quantity of fluid released at each pressure reduction. The measurement of the fall time was performed by raising the weight to its release position using the external magnet. The weight was held in this position for a short period to allow eddy currents created by its movement to dissipate. The external magnet was then carefully moved away from the surface of the viscometer tube, allowing the weight to fall through the fluid. The fall time was measured using a stopwatch, accurate to 0.001 s, to determine the period between the zero voltage positions of the two coils. Five measurements of the fall time were performed at each pressure, and the resulting values were reproducible to less than 0.5% with an average of about 0.25 % The addition of a liquid organic cosolventto supercritical carbon dioxide can result in substantial increases in both the critical temperature and pressure of the resulting mixture. To ensure that a single supercritical phase was present in the viscometer, the concentration of cosolvent was limited to a value below that which would result in the formation of two phases at the operating temperatures. Concentrations of 1.0, 2.0, and 3.0 mol % cosolvent in carbon dioxide were chosen to provide consistency within the systems examined, with the exception of methanol and ethanol, for which concentrations up to 5 and 4 mol % respectively were examined. The critical loci data obtained by Gurdial et al. (1993) were used to determine the maximum cosolventconcentrations, with the exception of isopropanol which was determined by the authors using the technique described by Gurdial et al. (1993). The viscosities of all the cosolvent-CO2 systems considered were measured at temperatures of 318.15 and 328.15 K. In addition, the viscosity of 2 mol % methanol, n-hexane, and acetone in C02 was determined a t 313.15 and 323.15 K to provide information on the temperature dependence of the viscosity of these systems. The system pressure was varied between 93 and 240 bar for all the
.
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Pressure (bar1
1
3
19655%
29645%
6
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59655%
Pvre co2 5 5 %
3.59645%
54b45.6
-P"reC01 4 5 %
Figure 2. Viscosity of methanol-C02 mixtures as a function of pressure. Table 1. Source, Purity, and Some Physical Properties of Chemicals ~
cosolvent methanol ethanol n-propanol isopropanol pentane hexane heptane acetone carbon dioxide
formula CHsOH CzHkOH CSHiOH CsH70H C&12 CeHl4 C7Hle C3HeO
co2
solvent purity mol supplier (%) wt BDH 99.8 32.04 BDH 99.8 46.07 BDH 99.5 60.09 99.5 60.09 BDH 99.5 72.15 BDH 99.5 86.18 BDH 99.5 100.2 BDH 99.5 58.08 BDH 44.01 99.8 Liquid Air
~
dipole bp moment ("C) (D)
64.6 78.3 97.2 82.2 36.0 68.7 98.4 96.3
1.70 1.69 1.68 1.66 0.0 0.0 0.0 2.88 0.0
systems examined. Food-grade carbon dioxide was used, and the purity and some of the physical properties of the cosolvents are presented in Table 1.
Results and Discussion Tabulated experimental viscosity and density data for the systems examined in this study are available as supplementary material (see paragraph at end of paper regarding supplementary material). Effect of Pressure and Temperature. The effect of pressure on the mixture viscosity is primarily related to the density of the mixture. The viscosity of the fluid increases with its density due to the increased intermolecular forces which occur as the molecules in the fluid become more closely packed. The reduced molecular spacing results in greater intermolecular forces, and the viscosity increases because the increased forces between molecules require greater applied force to enable the molecules to flow past one another. The viscosity isotherms for mixtures of methanol in carbon dioxide are presented as a function of pressure at temperatures of 45 and 55 "C and cosolvent concentrations of 2.0,3.5, and 5.0 mol % in Figure 2. The behavior of the methanol-C0z viscosity isotherms is representative of the general trends exhibited by all the systems examined. The viscosity of pure carbon dioxide is also presented in this figure to enable comparison between the pure primary supercritical fluid (SCF) and the primary SCF-cosolvent systems. The addition of the cosolvents to supercritical carbon dioxide increases the viscosity, while the shapes of the isotherms are similar to that of pure carbon dioxide. In the vicinity of the critical point the viscosities of the mixtures increase rapidly with pressure corresponding to the rapid increase in density near the critical point. The pressure dependency of the viscosity decreases with increasing pressure over
684 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 0.0009 I
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A
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HepDnelSDC
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- - - PureCO2 .
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Figure 5. Viscosity of 2 mol % pentane, hexane, and heptane at 45 and 55 O C as a function of pressure. *
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Figure 4. Viscosity of 2 mol % methanol, acetone, and hexane in carbon dioxide a t 120 and 200 bar as a function of temperature.
the pressure range examined. To illustrate this point, the density and the viscosity of the 2 mol % n-hexane-carbon dioxide system are presented as a function of pressure in Figure 3. The viscosity data presented in Figure 2 show that as the temperature increases the viscosities of the mixtures decrease, and that at the higher temperatures the data exhibit a decreased dependency on the system pressure. In order to quantify the effect of temperature on the viscosity of cosolvent-COz mixtures, the viscosities of 2 mol % methanol, n-hexane, and acetone in carbon dioxide were determined at temperatures of 40,45,50, and 55 "C. The viscosity of these systems are presented as a function of temperature in Figure 4, at pressures of 120 and 200 bar. The corresponding pure carbon dioxide isobars are included for comparative purposes. The viscosity of pure carbon dioxide was observed to be a linear function of temperature over the temperature range examined, and the slope of the function varies with the system pressure. The mixture viscosity is observed to deviate from the linear function observed for pure carbon dioxide to a small degree at 200 bar and is essentially linear at the lower pressure. The slope of the curves is substantially different from that of the pure carbon dioxide system. Effect of Molecular Size. In order to examine the effects of cosolvent molecular size, the viscosity of each of the nonpolar, noninteracting, straight-chain n-alkane402 systems is presented in Figure 5 at the same concentration of 2 mol % and at 45 and 55 "C. The molecular weights of the cosolvents are 72.15,86.18,and 100.2,for n-pentane, n-hexane, and n-heptane, respectively. The viscosity of the mixtures increased with increasing molecular weight of the cosolvent molecule. The increase in the mixture viscosity is primarily due to increasing intermolecular attraction as the cosolvent molecule increases in size. The
.MeOH
.Pent
5.40E.04 ~20
30
40
so
60
70
no
go
inn
~ I O
Molecular Weight
Figure 6. Viscosity of 2 mol % cosolvent-COz systems at 45 "C and 120 bar.
polarizability increases with molecular size, resulting in greater attraction between cosolvent and COz for the larger cosolvent molecules. Greater intermolecular attraction results in greater fluid viscosity as the fluid is bound together more tightly, thus impeding flow. The same increase in intermolecular attraction also accounts for the increase in the mixture densities. The viscosities of the 2 mol % cosolvent-carbon dioxide mixtures at 45 "C and 120 bar are presented as a function of molecular weight in Figure 6. An approximately linear relationship was observed for the straight-chain primary alcohols and the straight-chain n-alkanes. The viscosity of isopropanol was significantly lower than that of n-propan01although the two cosolventshave identical molecular weights. This difference in viscosity can be related to the mixture critical properties of each system (as discussed under Modeling of the Experimental Density Data). Effect of Cosolvent Polarity. In order to evaluate the effect of the cosolvent polarity, a n-alkane, a n-alcohol, and acetone were compared at constant cosolvent concentration. The individual cosolvents for the comparison were chosen as those with molecular weights closest to that of acetone to reduce the effects of molecular weight. The viscosities of 2 mol % cosolvent-carbon dioxide systems containing n-propanol, n-pentane, and acetone, are presented as a function of pressure in Figure 7 at 45 "C. n-Pentane, n-propanol, and acetone have dipole moments of 0.0, 1.68, and 2.88 D, respectively, and the mixture viscosity for these systems increases in this order. However, the relationship between the mixture viscosity and the cosolvent polarity exhibits nonlinear behavior. The viscosities of the acetone and propanol mixtures are quite similar and converge at higher pressures. The
9.IME-04
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Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 685
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160
180
200
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80
100
I20
140
160
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Pressure (bar)
Pressure (bar)
Figure 7. Viscosity of mixtures of cosolventa of differing polarity in carbon dioxide at 45O C as a function of pressure.
Figure 8. Viscosity of 2 mol % isopropanol and n-propanol at 45 OC.
viscosity of the n-pentane mixture, however, is only marginally greater than the viscosity of pure carbon dioxide and approaches the pure carbon dioxide isotherm at higher pressures. The behavior of these mixtures may be explained by the identical polarity of the primary SCF and cosolvent for the n-pentane-carbon dioxide system. The cosolvent and primary SCF in this system are both nonpolar, no dipole interactions exist in the mixture, and no hydrogen bonding is possible. Therefore, the only influences on the viscosity are due to the polarizability of the cosolvent molecule, and possibly solvent clustering in the vicinity of the critical point. The absence of cosolvent polarity is also reflected in the normal boiling points of the cosolventsexamined-higher polarity results in greater intermolecular attractions and higher boiling points. The normal boiling points of n-pentane, n-propanol, and acetone are 36.0, 97.2, and 96.3 "C, respectively. The boiling point of n-pentane is significantly lower than those of the other cosolvents, due to the absence of polar interactions and hydrogen bonding effects. The presence of the hydrogen bonding group may also explain the large increase in viscosity exhibited by mixtures of liquid alcohols in supercritical carbon dioxide. Although carbon dioxide has no overall dipole moment, the individual carbon-oxygen bonds have strong dipoles and the oxygen molecules have two pairs of unshared electrons which can act as proton-accepting sites for the hydrogen bonding hydroxyl group on an alcohol. The association of methanol in supercritical carbon dioxide has been examined by Fulton et al. (1991), and the degree of association of methanol was found to increase rapidly above concentrations of 2 mol % They also postulated a specific interaction between carbon dioxide and methanol. It has been reported that carbon dioxide is similar in hydrogen bond accepting ability to acetone (Walsh et al., 1987).An alcohol-C0z system may, therefore, possess hydrogen bonds acting between the hydroxyl group of the alcohol and the oxygen molecules of the carbon dioxide. The existence of hydrogen bonding would result in increased mixture viscosity due to the attractive forces between solvent and cosolvent molecules reducing the mobility of the fluid.
Effect of Cosolvent Molecular Shape. The influence of the molecular shape on the viscosity of a supercritical mixture is examined by comparing mixtures of n-propanol and isopropanol with carbon dioxide. These two alcohols have identical molecular weights and very similar dipole moments, yet the normal boiling points differ by 15 "C. The difference in the boiling points is a direct consequence of the shape of the two molecules. In the liquid state the straight-chain alcohol can align its molecules and achieve closer intermolecular spacing than can the secondary alcohol. The closer molecular spacing allows the van der Waals and dipole forces to act over a shorter distance increasing the overall attractive forces, resulting in a higher boiling point of the primary alcohol. The viscosities of these two systems, at 2 mol % alcohol and 45 "C, are plotted as a function of pressure in Figure 8. The viscosity of the primary alcohol is greater than that of the secondary alcohol, following the trend in the boiling points. The increased viscosity of the n-propanol system compared to the isopropanol system is due to the greater intermolecular attraction between the solvent and the straight-chain cosolvent. Modeling of the Mixture Viscosity. The mixture viscosity was correlated using five dense gas correlations described in the literature (Lucas, 1974;Chung et al., 1988; Pedersen et al., 1984, 1987; Ely and Hanley, 1981; Dean and Stiel, 1965). The data required and the fitting parameters involved, and the resulting deviations between the experimental and calculated data for the correlations examined, are presented in Table 2. The correlations have been used in a semipredictive and purely correlative manner, and the resulting deviations between calculated and experimental viscosity are also presented. In the semipredictive method, experimental mixture densities and suggested parameter values were used. In general, the correlation of the data in the absence of fitting parameters resulted in a poor representation of the experimental viscosity, with average absolute deviations ranging from 4% ' to 10%. The inadequate representation of the experimental data results from the fact that the correlations were not developed for the case where one of the components of the mixture is a liquid, and they do not
.
Table 2. Results of the Viscosity Correlations Using Ootimized Parameters
correlation
data required
optimized params
% AARD using
% AARD without
fitting params
fitting params 10.00 4.65 5.46 10.41 4.19
686
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994
adequately describe the increased molecular interactions as the critical point is approached. Most of the correlations examined include adjustable parameters which were used to fit the equations to the experimental data. Optimizing the fitting parameters improved the correlation between the experimental and calculated data substantially. The most successful of the viscosity correlations examined was that proposed by Ely and Hanley (1981),with the modifications outlined below. This correlation relates the viscosity of the dense fluid to that of methane and requires only the component critical constants, acentric factors, and molecular weights. The noncorrespondence correction X , was set equal to unity for all systems but one, methane-n-decane, in the development of the correlation. Ely and Hanley (1981) found that systems in which there were large differences between the critical volumes of the mixture components required values of X, other than unity. The ratios of the critical volumes of the cosolvents to the solvent molecules in the present study range from 3.24 for the n-pentane-carbon dioxide system to 4.65 for the n-heptane-carbon dioxide system. The large differences in critical volume indicate that a value of unity for the noncorrespondence correction will not be adequate to describe the viscosity behavior of these systems. The following empirical equation for the correction was proposed to account for the differing sizes of the molecules.
9.WE.04
-
1.00E44
-
I.WE.04
-
% W E04
-
X,, = Eqn 4 & 6 2WE04
no
(5)
and a, b, and c are empirical constants, Vmhis the smaller of the critical volumes of the component molecules, and the noncorrespondence factor 9 = 1.5. The term g, is a function of the component critical temperature and a thermodynamic shape factor. The average absolute relative deviation (AARD) over all the systems examined improves from a value of 10.45% when X, = 1to a value of 4.9% when eq 4 is used for the noncorrespondence correction. While the deviations encountered when X, = 1 increase substantially with the molecular weight of the cosolvent, they do not when the noncorrespondence correction is used. This indicates that the size difference is adequately compensated for by employing eq 4 for the noncorrespondence correction. The noncorrespondence factor, $, was given a value of 1.5 by Ely and Hanley (1981). However, this value does not satisfactorily describe the departure from corresponding states theory in the present case where a liquid cosolvent is dissolved in a supercritical fluid. Comparison of the calculated viscosity with the experimental data revealed that 9 could be correlated with the fluid density. A relationship between the noncorrespondence factor and density was considered meaningful for the systems examined as it has been shown earlier that the interactions which affect the viscosity also affect the density in the same manner. Examination of a number of relationships for 9 as a function of density revealed the best fit of the data could be obtained using an inverse relationship of 9 with density. The equation used for the noncorrespondence factor was as follows: + ( P ) = A / P+ B (6) The results of correlating the data using the above
100
IZO
140
160 Pressure (bar1
180
zw
120
140
Figure 9. Experimental and predicted viscosity of the 2 mol % hexane-carbon dioxide system at 45 "C using the Hanley and Ely technique. Table 3. Correlation Results for the Hanley and Ely Technique Using a Density-Based Expression for the Noncorrespondence Correction cosolvent methanol ethanol n-propanol isopropanol pentane hexane heptane acetone overall
where
- - Xq =Equation 4
I
x. as a function of density A B % AARD -1.5058 3.4857 2.83 -1.9838 4.5434 1.63 -1.8403 4.6436 3.06 -1.6869 4.2295 1.86 -1.3168 3.5760 2.11 -1.0250 3.2210 2.53 -1.0466 3.8288 2.88 -1.2061 3.9075 3.74 2.62
expression for 9 are shown in Table 3. The values of the coefficients A and B were determined by the minimization of the AARD. The AARD between experimental and calculated data improves to a value of 2.6 % through the use of eq 6, and the calculated values of the constants A andB vary little between the cosolvent systems examined. The experimental and predicted viscosity using the Ely and Hanley technique with eq 6 and the fitted parameters A and B are presented as a function of temperature in Figure 9 for the 2 mol 5% n-hexane-COz system at 45 "C. The use of corresponding states theory (CST) to correlate the mixture viscosity has provided reasonable accuracy for these systems. Since critical properties are the reducing parameters used in CST, it appears reasonable that the mixture viscosities could be ranked in order of increasing viscosity according to the critical constants of the mixture. The critical temperatures and pressures of the systems studied have been determined previously (Gurdial et al., 1993), and were used to examine the relationship between the mixture viscosity and the mixture critical properties. The relationship between mixture viscosity and P c , m i r is presented in Figure 10. The viscosities of the mixtures increase with increasing mixture critical property within the alcohol and n-alkane homologous groups. It was also observed that isopropanol and n-propanol in mixtures with carbon dioxide exhibit increasing viscosity with increasing T c , m i r and Pc,mix, whereas this was not the case when the viscosities were related to any of the pure cosolvent properties. This behavior is due to the fact that the mixture critical properties are directly related to the intermolecular interactions between the solvent and cosolvent in the mixture. Modeling of the Experimental Density Data. The density of cosolvent-supercritical fluid systems is often predicted using the Peng-Robinson equation of state (EOS). However, insufficient experimental data exist for
Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 687 Table 4. Correlation Results for the Calculation of Mixture Densities Using Peng-Robinson EOS with ku Temperature Dependent kij
cosolvent methanol
45OC 9.733-3 -1.223-4 -5.OOE-3 -8.063-3 -9.143-3 -1.5OE-2 -1.083-2 -1.323-2
40 O C -1.563-2'
ethanol n-propanol isopropanol pentane hexane heptane acetone
-3.203-2 -3.093-2
50°C 8.583-3
'
-1.91E-2 -1.953-2
6.10E.04
-
6.00E.04
-
S.9OE.04
-
5.8OE.04
-
% AARD
4.17 3.50 3.66 2.92 3.71 4.20 4.14 2.95
max dev ( % ) 19.75 12.65 11.53 10.89 11.78 15.55 13.10 9.49
3.66
overall a -1.563-2
55 "C 2.573-2 1.763-2 9.893-3 -2.723-3 -7.893-3 -2.383-3 O.OOE+O -3.463-3
represents -1.56
etc.
X
.AcetQM
on.PrOH =Heptane
0i.PrOH
the phenomena of solvent clustering (or increased solventsolute interactions) in the critical region cannot be accounted for with cubic EOS's. When the Peng-Robinson EOS is used to predict the molar volumes of these systems, accurate representation of the molar volume was only obtained at reduced pressures above 1.4. At reduced pressures below this value the accuracy of the correlation declines rapidly.
.Hexane
5.60E.04 -
5.5OE.04
OEtOH
oMeOH
.Pl"lP"e
5.40E.04 74
75
16
77
78
79
BO
81
81
Critiepl Mixture Pressure (bar)
Figure 10. Viscosity of 2 mol % cosolvent-C02 systems at 120 bar as a function of critical mixture pressure.
the accuracy of this correlation to be evaluated. The PengRobinson EOS was therefore used to correlate the density data obtained in the present study in order to evaluate the ability of this EOS to predict mixture densities. The density data obtained as a function of temperature, pressure, and composition were modeled using the PengRobinson equation of state with the mixing rules suggested by Reid et al. (1987). A binary interaction parameter, kij, was used as an adjustable parameter to fit the equation of state to the experimental data. The interaction parameter was determined in two ways: firstly, for each cosolvent the parameter was optimized independently of temperature. Secondly, it was assumed that kij was a function of temperature, and the parameter was optimized for each cosolvent and for each individual temperature as shown in Table 4. The effect of making kij temperature dependent is a reduction in the overall average absolute deviation from 4.3% to 3.7% and a slight reduction in the maximum deviations. As was the case when kij was independent of temperature, the average absolute relative deviation indicates that while the prediction of the molar volume is reasonably accurate for most of the pressure range studied the prediction breaks down as the critical point is approached, resulting in the large maximum deviations. Although the Peng-Robinson equation of state predicted the molar volume to within about 4% AARD, the maximum deviations from the experimental data are substantially higher than this. In most of the systems examined the maximum absolute deviation occurred at the lowest pressure examined, indicating an inability of the equation of state to accurately predict the molar volume as the critical point is approached. The region in the vicinity of the critical point exhibits large negative partial molar volumes for systems of this type, resulting in large increases in density per mole for cosolvent. In addition
Conclusions The viscosity of supercritical mixtures of COZwith three n-alkanes, four alcohols, and acetone was determined and found to increase, relative to pure carbon dioxide, with the addition of liquid cosolvent. The magnitude of the increase in the viscosity is dependent upon the size of the cosolvent molecules and upon the concentration. The magnitude of the viscosities could be ranked with the critical properties of the mixtures within a homologous group. Examination of the available correlations for the calculation of dense gas mixture viscosity revealed that all of these could reproduce the experimental data to less than 10% without modification. The most accurate description of the viscosity of these supercritical fluid mixtures was obtained using the Ely and Hanley technique with modifications. The inability of the Peng-Robinson equation of state to correlate the molar volume of cosolvent-C0z mixtures as the critical point is approached may be attributed to the increased molecular interaction which occurs in this region. Although the overall deviations between experimental and predicted densities are found to be within about 4 % AARD, individual deviations of up to 20 5% occur in the vicinity of the critical point. Acknowledgment This work was funded in part by the Australian Research Council under Grant No. A89131800. D.L.T. gratefully acknowledgesfinancial support from the National Science Foundation through Grant No. INT-9203312. Nomenclature 9 = fluid viscosity (g/cm2s) u = density of the weight (g/cm3) p = density of the fluid (g/cm3) 6 = annular space between the weight and the tube (cm2) d = diameter of the fall weight (cm) s = distance through which the weight falls (cm) t = time taken to fall through the distance s (s) C = viscometer constant a, b = parameters in the viscometer constant equation 3 X , = noncorrespondence correction used in the Hanley and Ely correlation $ = noncorrespondence factor
688 Ind. Eng. Chem. Res., Vol. 33, No. 3, 1994 T = Temperature (K) g, = thermodynamic shape factor Ziix = critical compressibility of the mixture Z:ef = critical compressibility of the reference compound,
methane a, b, and c = constants in the noncorrespondence correction
equation (eq 4) R = correction for molecular size difference in the noncorrespondence correction equation x i = mole fraction of component i = critical volume of component i Vmin = smaller of the component critical volumes $ ( p ) = density-dependent correction to the noncorrespondence correction A and B = parameters in $ ( p ) Pc,mix = mixture critical pressure (bar) Tc,mix= mixture critical temperature (K) kij = binary interaction parameter
Supplementary Material Available: Tabulated data listing the experimental viscosities and densities for the following systems: methanol-COz (2, 3, 5 mol 5%) at 45 and 55 OC, also 2 mol % at 40 and 50 "C; ethanol-COz (2, 3,4 mol % ) at 45 and 55 OC; n-propanol-C02 (1,2,3 mol 7%) at 45 and 55 O C ; 2-propanol-COz (1,2,3 mol % ) at 45 and 55 "C; n-pentane-COz (1,2, 3 mol % ) at 45 and 55 OC; n-hexane-C0z (1,2, 3 mol % ) a t 45 and 55 OC, also 2 mol % at 40 and 50 OC; n-heptane-C0z (1,2,3 mol % ) at 45 and 55 "C; acetone-C0z (1,2 , 3 mol % ) at 45 and 55 "C, also 2 mol % a t 40 and 50 "C (9 pages). Ordering information is given on any current masthead page. Literature Cited Altunin, V. U.; Sakhhabetinou, M. A. Teploenergetica 1972,8,85. Chen, M. C. S.; Lescarboura, J. A.; Swift, G. W. The Effect of Eccentricity on the Terminal Velocity of the Cylinder in a Falling Cylinder Viscometer: AZChE J. 1968,14 (l),123-127. Chung, T.-H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Generalized Multiparameter Correlation for Non-polar and Polar Fluid Transport Properties. Znd. Eng. Chem. Res. 1988,27,671-679. Dean, D. E.; Stiel, L. I. The Viscosity of Nonpolar Gas Mixtures at Moderate and High Pressures. AZChE J. 1965,ll (3),526-532. Dobbs, J. M.; Johnston, K. P. Selectivities in Pure and Mixed Supercritical Fluid Solvents. Ind. Eng. Chem.Res. 1987,26,14761482. Dobbs, J. M.; Wong, J. M.;Lahiere,R. J.;Johnston,K.P. Modification of Supercritical Fluid Phase Behaviour Using Polar Cosolvents. Znd. Eng. Chem. Res. 1987,26,56. Ely, J. F.; Hanley, H. J. M. Prediction of Transport Properties. I. Viscosity of Fluids and Mixtures. Znd. Eng. Chem.Fundam. 1981, 20,323-332. Fulton, J. L.; Yee, G. G.; Smith, R. D. Hydrogen Bonding of Methyl Alcohol-d in Supercritical Carbon Dioxide and Supercritical Ethane Solutions. J. Am. Chem. SOC.1991,113,8327-8334.
Gurdial, G. S.; Foster, N. R. Solubility of o-Hydroxybenzoic Acid in Supercritical Carbon Dioxide. Znd. Eng. Chem. Res. 1991, 30, 575. Gurdial, G. S.;Foster, N. R.; Yun,S. L. J.;Tilly, K. D. Phase Behaviour of Supercritical Fluid-Entrainer Systems; Supercritical Fluid Engineering Science: Fundamentals and Applications; Kiran, E., Brennecke, J. F., Eds.; ACS Symposium Series 514;American Chemical Society: Washington, DC, 1993;pp 34-45. Hawkins, G. A.; Solberg, H. L.; Potter, A. A. The Viscosity of Water and Superheated Steam: Trans. Am. SOC.Mech. Eng. 1935,FSP57-11,395-400. Iezzi, A.; Enick, R.; Brady, J. Direct Viscosity Enhancement of Carbon Dioxide. In Supercritical Fluid Science and Technology; Johnston, K. P., Penninger, J. M. L., Eds.; ACS Symposium Series 406; American Chemical Society: Washington, DC, 1988;pp 122-139. Irving, J. B. The Effect of Nonvertical Alignment on the Performance of a Falling Cylinder Viscometer. J. Phys. D: Appl. Phys. 1972, 5,214-224. Lohrenz, J.; Swift, G. W.; Kurata, F. An Experimentally Verified Theoretical Study of the Falling Cylinder Viscometer. AIChE J . 1960,6 (4),547-550. Lucas, K. A Simple Technique for the Calculation of the Viscosity of Gases and Gas Mixtures. Chem.-1ng.-Tech, 1974,46, 157. Pedersen, K. S.;Fredenslund, A. An Improved Corresponding States Model for the Prediction of Oil and Gas Viscosities and Thermal Conductivities. Chem. Eng. Sci. 1987,42 (l),182-186. Pedersen, K. S.; Fredenslund, A.; Christensen, P. L.; Thomassen, P. Viscosity of Crude Oils. Chem. Eng. Sci. 1984,39(6),1011-1016. Pitzer, K. S.;Schreiber, D. R. Improving Equation-of-State Accuracy in the Critical Region; Equations for Carbon Dioxide and Neopentane as Examples. Fluid Phase Equilib. 1988,41,1. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Schmitt, W. J.; Reid, R. C. Solubility of Monofunctional Organic Solids in Chemically Diverse Supercritical Fluids. J. Chem. Eng. Data 1986,31, 204. Sen, Y. L.;Kiran, E. A New Experimental System to Study the Temperature and Pressure Dependence of Viscosity, Density, and Phase Behavior of Pure Fluids and Solutions. J. Supercrit. Fluids 1990,3 (2),91-99. Swift, G. W.; Christy, J. A.; Heckes, A. A.; Kurata, F. Determining Viscosity of Liquefied Gaseous Hydrocarbons a t Low Temperatures and High Pressures. Chem. Eng. Prog. 1958,54(6),47-50. Swift, G. W.; Lohrenz, J.; Kurata, F. Liquid Viscosities Above the Normal Boiling Point for Methane, Ethane, Propane, and n-Butane. AZChE J. 1960,6 (3),415-419. Walsh, J. M.; Ikonomou, G. D.; Donohue, M. D. Supercritical Phase Behavior: The Entrainer Effect. Fluid Phase Equilib. 1987,33, 295-314. Received for review March 2 , 1993 Revised manuscript received July 29, 1993 Accepted September 24, 1993'
Abstract published in Advance A C S Abstracts, December
1, 1993.
