Ind. Eng. Chem. Res. 1999, 38, 2129-2136
2129
Measurement and Modeling of Viscosity of Supercritical Carbon Dioxide/Biomaterial(s) Mixtures Dang Quoc Tuan,† John A. Zollweg,‡ Peter Harriott,‡ and Syed S. H. Rizvi*,† Institute of Food Science, Stocking Hall, and School of Chemical Engineering, Cornell University, Ithaca, New York 14853
The viscosities of a binary, supercritical carbon dioxide/methyl oleate (SC-CO2/MO, and a multicomponent, SC-CO2/anhydrous milkfat (AMF) system at 40 °C and over a pressure range of 10.6-25.0 MPa were measured in a high-pressure capillary viscometer. The experimental data, and data from the literature, were utilized in viscosity modeling with Ely and Hanley’s corresponding states model for SC-CO2/biomaterial(s) systems. The modified Benedict-WebbRubin equation of state and a viscosity correlation with propane as the reference material were used. An adjustable parameter was added to the energy shape factor equation. This approach worked well for both the fluid and liquid phases of the SC-CO2/biomaterial(s) systems with an average absolute deviation of 3-6%. Compared to the purely correlative Grunberg and Nissan model, this method has better predictive capability, while maintaining comparable accuracy. Introduction The use of supercritical fluids (SCF) as solvents in processing of low-volatility and hard-to-separate biomaterials has received considerable attention by researchers over the last 2 decades. The relatively high and reversibly adjustable density makes SCFs ideally suited for simultaneous extraction and fractionation of various solutes, while their relatively low viscosity provides appreciable penetrating power into the solute matrix. These properties enhance the rates of mass transfer of solutes into SCFs than into liquids and one of the most commonly used solvents for food and biological applications is supercritical carbon dioxide (SC-CO2). Its mild critical temperature and pressure (31 °C and 7.3 MPa) make it highly desirable for dealing with heat-sensitive biomolecules. Furthermore, its unique properties as a nontoxic, nonflammable, and low-cost material have also made it especially attractive to bioprocessing industries.22 The thermophysical property data of biomaterials/ SCF systems are needed for accurate, economical design and safe scale-up of new processing systems. Besides information on equilibrium properties, transport properties such as viscosity and diffusion coefficient are required. Fluid mixture viscosities are necessary for determining the extent of mixing and mass transfer, for calculating other properties such as diffusion coefficients, and for determining pressure drops in piping systems.3 In general, for example, the viscosities of SCCO2 and biomaterials are very different. The dissolution of a small amount of such solutes into a SCF may lead to a dramatic change in viscosity of the system14 which, in turn, will affect the mass-transfer rate of the solute into SCF during extraction. Availability of such data is, however, very limited. Viscosity data for SCF/biomaterials systems are even more scarce since there have been only a few studies * To whom correspondence should be addressed. Telephone: 607-255-7913. Fax: 607-254-4868. E-mail: ssr3@ cornell.edu. † Institute of Food Science. ‡ School of Chemical Engineering.
on the subject. Llave et al.14 measured the viscosity of SC-CO2 containing various dissolved medium molecular weight (100) organic liquids. Peter and Jacob19,20 measured the viscosity and studied the rheological behavior of coexisting phases of the fatty acid/SC-CO2 and fatty acid/SC-ethane systems. Kashulines et al.12 and Yener25 studied the viscosity behavior of high-molecular-weight liquids saturated with CO2 (liquid phase) as well as SCCO2 partially saturated (fluid phase) with fatty acids and fatty acid derivatives. The investigators indicated that the lack of fundamental data is due to many experimental challenges. For example, the presence of low viscosity/high density makes accurate viscosity measurements at high pressure difficult. In addition to the measurement of system viscosity, there is also a need for the development of theoretical models to reliably predict viscosity behavior under various operating conditions. Viscosity Modeling The viscosity of a fluid may be expressed as the sum of three independent contributions:3
η(T,F) ) η0(T) + ∆η(T,F) + ∆ηc(T,F)
(1)
The first term, η0(T), represents the dilute gas viscosity at a given temperature, which is independent of pressure. The second term, ∆η(T,F), is the excess viscosity which represents the effects of elevated pressure. The last term, the critical enhancement, ∆ηc(T,F), which is caused by long-range fluctuations that occur in a fluid near its critical point, can be neglected for most practical engineering applications away from critical points. Various correlations have been proposed to estimate the dilute gas viscosity term, η0(T), as well as the excess viscosity term, ∆η(T,F).17,21 One of the widely used approaches is based on the corresponding states principle (CSP). As the CSP is applied to transport properties, the viscosity of a studied fluid can be calculated from the viscosity of a reference fluid at the same reduced temperature and density. The CSP technique is usually not valid for fluid mixtures, or even for pure
10.1021/ie980363w CCC: $18.00 © 1999 American Chemical Society Published on Web 04/08/1999
2130 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
Figure 1. Schematic diagram of the system for measurement of viscosity of SC-CO2/lipid mixtures.
polyatomic fluids, because of nonconformality between the fluid of interest and the reference fluid. Ely and Hanley7,8 developed a method and a computer program (TRAPP) to predict the viscosity of compressed natural gases, using the concept of extended CSP, in which the shape factors were used to correct for nonrandom mixtures, making nonconformal fluids follow the CSP. Critical constants, Pitzer’s acentric factor, and the molecular weight for each component were required as inputs to the model. Considered purely predictive, the method worked well for nonpolar fluids (hydrocarbon mixtures). In general, it fails for mixtures containing polar components or mixtures with large size differences; however, it still works well for SC-CO2 containing less than 5 mol % of a polar cosolvent.24 Hwang and Whiting11 modified Ely and Hanley’s model for polar fluids by employing a viscosity acentric factor and an association parameter to characterize the effects of molecular shape, orientation, and intermolecular forces. Also, mixing rules were proposed on the basis of a density-dependent local-composition model. As usual, methane was chosen as the reference fluid. The method worked well for more than 30 pure fluids including highly branched alkanes and naphthalene, and for polar and hydrogen-bonding compounds, over the entire range of fluid states, from dilute gas to dense liquid. The long-term goal of our research program is experimental measurements and modeling of transport properties and phase equilibrium of biomaterial/SCF systems. The solutes of special interest are long-chain fatty acids found in anhydrous milkfat (AMF), their methyl esters and triglycerides, as well as AMF itself as a
unique multicomponent system. The data would be useful in the design of the AMF fractionation with SCCO2.2 The specific objectives of the study were (1) to measure the viscosity of SC-CO2 containing different levels of methyl oleate (MO) and AMF; (2) to evaluate the applicability of Ely and Hanley’s corresponding states model for viscosity prediction of the biomaterial/ SC-CO2 mixtures; (3) to modify the above model by adding an adjustable parameter in the energy shape factor equation and use propane as the reference fluid instead of methane. Materials and Methods Materials. Methyl oleate (CAS 112-62-9) of 99% purity was purchased from Sigma Chemical Co. (St. Louis, MO). Anhydrous milkfat (AMF) was prepared from commercial grade, unsalted butter (O-AT-KA Milk Prod. Co., Inc., Batavia, NY) by melting it at 60 °C for 2 h. The top layer (protein + water + nonfat components) was then decanted, and the AMF was filtered through filter paper (Whatman no. 1). The AMF was frozen and stored at -15 °C until used. Carbon dioxide (99.9% pure) was purchased from Empire Airgas Inc. (Elmira, NY). Experimental Procedure. A high-pressure capillary viscometer was designed and built previously13 to measure the viscosities of SC-CO2 containing several types of dissolved liquid solutes (Figure 1). It consists of a capillary viscometer, a nonmagnetic cylinder with floating piston (model TOC1-60-w/piston, HIP Inc., Erie, PA), a gear pump (model L-1409, Micropump Inc.,
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2131
Concord, CA), and a sampling loop. The viscometer has a tube diameter of 0.2966 ( 0.0006 cm, a length of 6.753 ( 0.002 m, and a coil diameter of 1.08 m with calming sections of 20 cm, oriented horizontally. The sampling loop for measuring the concentration of the mixtures has a volume of 16.83 mL. More details on the system are available elsewhere.13 The mixture to be measured was prepared by pumping SC-CO2 out of the bottom of the nonmagnetic cylinder (Figure 1) with the gear pump through the view cell where it floated on top of the liquid solute, thereby absorbing some solute. The SC-CO2 with dissolved solute was then returned to the top of the cylinder where it caused the floating piston inside the nonmagnetic cylinder to move. By pumping the mixture from the bottom to the top (and vice versa) of the cylinder (but not through the view cell) repeatedly (at least seven times), the SC-CO2 containing dissolved solute was then thoroughly mixed. The pressure drop in the capillary viscometer section was measured with a differential pressure transducer (model TH-D, T-Hydronics Inc., Westerville, OH) while pumping the test fluid through it. The volume of pumped fluid was determined by measuring the displacement of the sealed free floating piston located inside the nonmagnetic cylinder. The mixture concentration was measured by trapping a sample in the sampling loop. The weight of the sample was determined after venting the CO2 in the sampling loop into the atmosphere through hexane. Fresh hexane (450 mL) was then pumped through the sampling loop to flush out any precipitated lipid. The portions of hexane were then combined and hexane was evaporated from the mixture by a vacuum evaporator. Each treatment was repeated five times and the average pressure drop was calculated. With the apparatus used the experimental standard deviation was 6.0-14.0%. The sampling loop was prepared for another measurement by vacuum evacuation followed by filling with pure SC-CO2 to the desired system pressure using the high-pressure pump. The experimental run time was also recorded for flow rate calculation. Knowing the diameter and length of the viscometer tube and the flow rate, the fluid viscosity was then determined from the Hagen-Poiseuille equation. After the viscosity of a mixture at a given concentration was measured, the SC-CO2/solute mixture was reused to obtain the next higher concentration mixture. Ely and Hanley Extended Corresponding States Model. If the corresponding states method is applicable,7 one can calculate viscosity of a fluid from the viscosity of a reference fluid (usually methane):
( )
η ) ηR
MW MWR
1/2
f1/2h-2/3
(2)
Parameters f and h are the viscosity scale factors and are calculated as follows:
h ) (FcR/Fc)Φ
(3)
f ) (Tc/TcR)Θ
(4)
The size shape factor, Φ, and the energy shape factor, Θ, are density-dependent:
Φ ) [1 + (ω - ωR)G]ZcR/Zc
(5)
Θ ) 1 + (ω - ωR)F
(6)
Variables G and F are functions of reduced temperature and molar volume.8 The pressure and temperature of the reference fluid are calculated from the pressure and temperature of the studied fluid using scale factors:
TR ) T/f
(7)
PR ) Ph/f
(8)
The density of the reference fluid is calculated from a 32-term modified Benedict-Webb-Rubin equation of state (MBWR-EOS), knowing its temperature and pressure. The viscosity of the reference fluid is calculated from the regression provided from its temperature and density.8 The density of the studied fluid, needed for the shape factor calculations, is determined from the density of the reference fluid:
F ) FR/h
(9)
Since the scale factor, in turn, is density-dependent, the procedure for calculation involves iteration. For a fluid mixture, the one-fluid van der Waals mixing rules are applicable. Ely6 indicated three possible areas for improvement of the Ely and Hanley extended corresponding states method: the reference fluid, the shape factor correlations, and the mixing rules. Methane had been used as the reference fluid in the TRAPP program.8,11 One of the problems with using methane is that its normal reduced freezing temperature is 0.48, which is higher than that of many other fluids. Thus, he used propane, which has a lower reduced triple-point temperature and larger molecular weight (hence size) than methane. It leads to reduction of noncorrespondence of the reference fluid to other fluids to a certain degree.16 Ely6 also proposed new, density-independent shape factor correlations, the parameters of which have been reported by Huber and Hanley.10 The method worked well for hydrocarbons. However, our preliminary calculations showed large deviations between calculated and measured viscosity data for the SC-CO2/lipid systems when using Ely’s shape factor correlations. Thus, an approach by Chen and Leland4 was applied in this study: propane was used as the reference fluid and the correlation for the shape factors remained the same as that for methane (eqs 5 and 6). The Pitzer acentric factor of propane was used instead of methane.4 The constants for the propane MBWR-EOS and equations for the propane viscosity correlation are given elsewhere.6 Furthermore, to modify the shape factor correlations, the concept of an association parameter11 was used in this study. The equation for the size shape factor remained unchanged (eq 5). Only a parameter “k” was added to the equation for the energy shape factor, without regression for a “viscosity acentric factor”:
Θ ) 1 + (ω - ωR)F + kTc/T
(10)
The parameter “k” played the role of an optimization parameter and was determined by minimizing the average absolute deviation (AAD) between calculated and measured viscosity values. When the parameter k was set equal to zero, the method became purely predictive.
2132 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999
Figure 3. Increase in viscosity of SC-CO2 containing different mass fractions of solute (MO) at various temperatures and pressures.
Figure 2. Molecular weight of AMF fractions soluble in SC-CO2 at 40 °C as a function of pressure.
Estimation of Corresponding State Properties. The critical properties and acentric factors of pure substances, which are necessary for computation with the corresponding states model, are available in the literature.5 The data for methyl linoleate (MLO), which are not available in the Daubert and Danner database, were estimated in this study by Ambrose’s method.21 The data for AMF and AMF fractions were estimated from their average molecular weights as follows. The triglyceride compositions of AMF and the AMF fractions in SC-CO2 at 40 °C and various pressures have been reported.26 From these data, the average molecular weight of the AMF fractions in the SC-CO2 phase at 40 °C was calculated as presented in Figure 2. As a result, the molecular weight of the AMF fraction in SC-CO2 at 25.0 MPa and 40 °C was estimated to equal 673.3. The average molecular weight of whole AMF was estimated to be 713.1. Yu26 has estimated the critical pressure, temperature, and acentric factor for AMF triglycerides of various acyl carbon numbers, namely, C24, C30, C36, C42, C48, and C54 by Ambrose’s method. Additionally, in this study the critical volume of those substances was estimated with Ambrose’s method. The physical properties of those triglycerides were correlated with their molecular weights (MW) and the results are as follows:
For critical temperature (in K): Tc ) 349.144151 + 1.229957 × MW 0.000669 × MW2 (11) For critical pressure (in bar): Pc ) 18.431493e-0.001296×MW
(12)
For critical volume (in cm3/mol): Vc ) 1.693246 × MW+1.11323
(13)
For acentric factor: ω ) 7.444072 × 10-01 + 1.202760 × 10-03 × MW 1.365326 × 10-07 × MW2 (14)
The physical properties of AMF and one of its fractions and of several pure substances are given in Table 1. Results and Discussion Viscosity Measurements. The viscosity of the SCCO2/MO (methyl oleate) system was measured at 40 °C and 11.5 and 10.6 MPa and the SC-CO2/AMF system at 40 °C and 25.0 MPa. Viscosity data for fluid and liquid phases of several lipid/SC-CO2 systems have been reported in the literature. Regarding the fluid phase (CO2-rich), the viscosity of the SC-CO2/oleic acid (OA) system was measured at 60 °C/30.0 MPa, 40 °C/30.0 MPa, and 40 °C/20.5 MPa.25 The SC-CO2/MO system was measured at lower pressures: 60 °C/15.5 MPa, 50 °C/13.7 MPa,25 and 40 °C/11.5 MPa.13 As for the liquid phase, OA and linoleic acid (LOA), and their methyl esters, saturated with CO2, were measured at 40 and 60 °C, while AMF saturated with CO2 was measured at 40 °C.13 The numerical experimental data are given elsewhere (Tuan, D. Q. Measurement and modeling of viscosity and diffusion coefficients of lipids and supercritical carbon dioxide mixtures. Ph.D. Dissertation, Cornell University, Ithaca, NY, 1998). Figures 3 and 4 show the percentage increase in mixture viscosity (compared to that of pure CO2) versus the solute mass fraction for the SC-CO2/MO systems, and the SC-CO2/OA and SC-CO2/AMF systems, respectively, for several pressures and temperatures. The dependence of the viscosity increase on the solute mass fraction appeared to be linear. From the best-fit lines and the information on saturated concentrations,27 the viscosity of SC-CO2 saturated with solute can be estimated and is shown in Table 2. The highest increase in viscosity was 35% for the SC-CO2/MO system at 11.5 MPa and 40 °C where the solute mass fraction at saturation was as high as 9%. At this level, the viscosity increase is no longer trivial. Thus, by using the viscosity of pure CO2 instead of a mixture, as usually practiced in engineering calculations, misleading results are likely be obtained. At a mass fraction of 4%, MO contributed a 14-17% increase in mixture viscosity. Because of their lower solubility in SC-CO2, oleic acid and AMF were measured at much smaller concentrations than methyl oleate. At a mass fraction of only 1.5%, OA contributed a 5.0-6.0% increase in viscosity, while the AMF contribution was
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2133 Table 1. Physical Properties of AMF and AMF Fractions and Several Pure Substancesa component
MW
Pc (MPa)
Tc (K)
Tb (K)
Vc (cm3/mol)
Zc
ω
methane CO2 propane OA LOA MO MLOb AMF-250b AMF (whole)b n-decanol isooctane 2-ethylhexanol
16.04 44.01 44.10 282.47 280.45 296.49 294.48 673.30 713.10 158.29 114.23 130.23
4.604 7.382 4.248 1.390 1.410 1.280 1.310 0.770 0.731 2.370 2.568 2.730
190.58 304.19 369.83 781.00 775.00 764.00 777.07 874.00 886.03 684.40 540.33 640.25
111.66 194.60 231.11 633.00 628.00 617.00 599.30 755.00 767.60 504.07 369.41 475.75
99.3 94.0 200.0 1000.0 990.0 1060.0 1034.1 2383.2 2540.6 600.0 466.3 485.0
0.288 0.274 0.276 0.214 0.217 0.214 0.214 0.253 0.252 0.250 0.266 0.249
0.0108 0.2276 0.1523 1.1872 1.1762 1.0494 0.9530 1.4928 1.5308 0.6612 0.3117 0.5490
a T : normal boiling point. T : critical temperature. P : critical pressure. V : critical volume. ω: acentric factor. MW: molecular weight. b c c c Zc: critical compressibility factor. OA: oleic acid. LOA: linoleic acid. MO: methyl oleate. MLO: methyl linoleate. AMF-250: AMF fraction in SC-CO2 at 25.0 MPa and 40 °C. b Estimated by Ambrose’s method21; other data from Daubert and Danner.5
Table 2. Estimated Viscosity of SC-CO2 Saturated with a Solute at Various Temperatures and Pressures pres. (MPa)
temp. (K)
CO2 densitya (kg/m3)
CO2 viscosityb (E+07 Pa‚s)
solute mass fraction at saturationc (E+02)
increase in viscosityd (%)
MO
10.6 11.5 13.7 15.5
313.15 313.15 323.15 333.15
666.0 704.8 665.4 624.5
513.2 566.7 512.6 487.3
6.07 8.88 6.24 4.39
25.8 35.3 24.7 14.8
OA
20.5 30.0 30.0
313.15 313.15 333.15
846.5 911.8 831.3
778.8 930.2 773.7
1.90 3.68 3.83
7.9 15.2 14.2
25.0
313.15
881.5
850.5
2.32
12.0
solute
AMF a
Angus et al.1
b
Stephan and Lucas.23
c
Yu.26
d
Viscosity increase over that of pure CO2 at the same pressure and temperature.
Figure 4. Increase in viscosity SC-CO2 containing different mass fractions of solute (OA or AMF) at various temperatures and pressures.
larger, about 8.0%. Tilly et al.24 found that the viscosity of a CO2-rich mixture increased with increasing cosolvent molecular weight at the same molar concentration. Viscosity Modeling. (1) Fluid Phase. Table 3 summarizes the results of the viscosity of SC-CO2/solute systems calculated by the corresponding states model, using methane or propane as the reference fluid, with and without the optimized parameter. The AAD from the propane-based model was lower than that from the methane-based model. By using the optimized param-
eter, the AAD was reduced from 15.8 to 5.6%. If the data for alcohols and isooctane were excluded, the AAD was just 3.3%. The values of the optimized parameter for all fluid systems were negative. It is worthy to note that the association parameters for hydrogen-bonding fluids had negative values in Hwang and Whiting’s study.11 Without the optimized parameter, the method (TRAPP and PROP programs) overpredicted the viscosity of all SCCO2/lipid mixtures. Using the optimized parameter, the AAD for mixtures of SC-CO2 with OA and AMF were less than 1% (Figure 5). For SC-CO2/MO systems the AAD were higher, at 4-8%. The viscosity of the SCCO2/n-decanol system, where the mixture density was greater than 700 kg/m3, was overpredicted by the method without the optimized parameter. However, the viscosity of CO2 containing isooctane, where the system density was less than 610 kg/m3 was underpredicted with the AAD as high as 33%. The calculation errors were higher at lower mixture density, showing that prediction accuracy is sensitive to fluid density. For the SC-CO2/2-ethylhexanol system at 40 °C and 13.9 MPa, the method overpredicted the mixture viscosity at a lower cosolvent mole fraction but underpredicted the mixture viscosity when the cosolvent mole fraction was greater than 0.1. (2) Liquid Phase. Table 4 presents results of viscosity calculations for lipid systems saturated with SC-CO2 at 40 and 60 °C and various pressures. The AADs from the TRAPP and PROP programs were 28 and 29%, respectively, and the AAD was reduced to 6% using the optimized parameter. Without the optimized parameter, both programs underpredicted the viscosity of the systems containing fatty acids or AMF, with AADs ranging from 40 to 53%. Viscosity for the fatty acid methyl ester systems (except for MO/CO2 at 60 °C) was overpredicted at a higher and underpredicted at a lower
2134 Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 Table 3. Summary of Results of Modeling for Viscosity of SC-CO2 Partially Saturated with Organic Solute (Fluid Phase) by the Corresponding States Methoda PR-OPT
material
temp. (K)
pres. (MPa)
no. of points
meth AAD
prop AAD
AAD
k
data source
MO MO MO MO OA OA OA AMF n-decanol n-decanol isooctane 2-ethylhexanol
313.15 313.15 323.15 333.15 313.15 313.15 333.15 313.15 313.15 323.15 323.15 313.15
10.6 11.5 13.7 15.5 20.5 30.0 30.0 25.0 var.
7 12 12 14 5 10 8 6 4 4 4 11
26.0 24.9 26.4 19.4 15.0 14.3 13.4 12.4 20.9 27.9 32.4 17.9
20.4 19.2 20.7 14.6 9.1 7.3 6.4 7.1 14.4 19.7 32.8 17.3
7.1 7.7 5.5 4.0 0.4 0.2 0.6 0.5 3.1 1.2 20.7 16.6
-0.0227 -0.0252 -0.0351 -0.0306 -0.0235 -0.0235 -0.0235 -0.0235 -0.0305 -0.0453 -0.0381 -0.0143
b b, c d d d d d b e e e e
20.9
15.8
5.6
13.9
col. avg. a
METH: methane as the reference fluid (TRAPP program). PROP: propane as the reference fluid; PR-OPT: propane as the reference fluid with an optimized parameter (Hwang and Whiting).11 k: optimized parameter. b This work. c Kashulines.13 d Yener.25 e Llave et al.14 Table 4. Summary of Results of Modeling for Viscosity of Lipids Saturated with SC-CO2 (Liquid Phase) by the Corresponding States Method material
temp. (K)
OA OA LOA LOA MO MO MLO MLO AMF
313.15 333.15 313.15 333.15 313.15 333.15 313.15 333.15 313.15
col. avg. a
PR-OPT no. of meth prop points AAD AAD AAD k 8 7 9 8 7 7 7 7 5
39.5 35.7 36.6 30.2 9.6 9.6 20.7 21.6 49.9
43.6 40.3 40.4 35.3 8.5 8.5 18.2 15.2 52.7
13.1 3.0 3.3 2.0 8.2 8.1 11.5 6.1 2.0
28.2
29.2
6.4
0.1230 0.1411 0.1283 0.1219 -0.0033 -0.0042 -0.0389 -0.0357 0.1979
data source a a a a a a a a a
Kashulines.13
Figure 5. Comparison of calculated and measured viscosity by TRAPP and PR-OPT programs for SC-CO2/OA or SC-CO2/AMF system.
lipid mole fraction, with a threshold value of around 0.11. The AAD for these systems were lower than that for fatty acids and AMF, ranging from 9 to 18%. Viscosity of the AMF saturated with CO2 is about 3 times higher than that of MO saturated with CO2, and about 10-27 times higher than that of SC-CO2 at the same conditions. Pedersen et al.18 indicated that the TRAPP program, while satisfactory for less viscous fluids, is unable to represent highly viscous fluids. Using the optimized parameter, the method correlated viscosity data for the systems of OA, LOA, and AMF saturated with CO2 in sufficient agreement, except for the OA/ CO2 system at 40 °C (Figure 6), with an AAD of 2-3%. For the fatty acid methyl ester systems, the AADs were higher than that for fatty acids and AMF (6-12%). The
Figure 6. Measured and calculated viscosities of OA, LOA, and AMF saturated with CO2 at 40 and 60 °C.
optimized parameter for AMF and fatty acids had positive values while the values for fatty acid methyl esters were negative. Comparison to the Correlative Models. Kashulines13 evaluated the relatively simple Grunberg and Nissan model for the viscosity of SC-CO2/lipid mixtures. Their method utilized the mixture composition (mass fractions) and the pure component viscosity at the
Ind. Eng. Chem. Res., Vol. 38, No. 5, 1999 2135 Table 5. Comparison of Results for Liquid- and Fluid-Phase Viscosities of SC-CO2/Lipid Systems Obtained by Three Different Methods AAD without optimized parameter
AAD with optimized parameter
method
nature of method
liquid phase
fluid phase
liquid phase
fluid phase
ref
modified Ely and Hanley’s Grunberg and Nissan’s modified Grunberg and Nissan’s
predictive/correlative correlative correlative
29.2 39.5 76.3
15.8 10.0 4.6
6.3 8.5 24.7
3.3 4.0 0.8
this work Kashulines12 Yener25
mixture temperature and pressure. Grunberg and Nissan’s equation is a specific case of the Arrhenius equation for binary systems, with a so-called “interaction parameter”, G12. Kashulines et al.12 mentioned that an attempt to predict the interaction parameter by the group contribution method failed, so G12 was determined by the best-fit method from experimental viscosity data. He found that the method overestimated the fluid-phase viscosity with an AAD less than 4% (only for the SCCO2/MO system at 11.5 MPa and 40 °C). The liquidphase viscosity was estimated and had an AAD of 9% using the optimized G12. The original Arrhenius equation produced a result with an AAD of 40% and 10% for the liquid and fluid phases, respectively. In general, the parameter G12 is system-dependent and sometimes temperature-dependent, making generalization difficult.15 Yener,25 in another attempt, correlated G12 with excess Gibbs free energy, assuming that the fluid mixture followed a regular solution theory. As a result, the liquid phase and fluid phase could be modeled with an AAD of 25 and 1%, respectively, using an adjustable interaction parameter. This is compared to an AAD of 76 and 5% for the liquid and fluid phase, respectively, when the adjustable parameter was not used. Although these relationships are practical for quick estimation, extrapolation of pure component viscosity data to a high-pressure region limits their accuracy. The empirical nature of the methods requires experimental data to determine the model parameters. Furthermore, the need for an EOS to determine the mixture density (in the equation based on excess Gibbs free energy) requires additional experimental work. Therefore, predictive viscosity models for the SC-CO2/biomaterials systems were needed. The comparison of the results from different correlation methods is presented in Table 5. The propane-based corresponding states model predicted the liquid-phase viscosity better than Grunberg-Nissan’s methods with and without the adjustable parameter. For the fluid phase, the AAD from the corresponding states method was 3% for the SC-CO2/lipid systems, comparable with that from the Grunberg-Nissan equation. It is worth noting that the Grunberg-Nissan equation with excess Gibbs free energy,25 while showing improved results for the vapor phase, needs four sets of experimental data (two for viscosity of components and two for mixture molar volume and viscosity) to determine the adjustable parameter. The corresponding states method needs only one data set (mixture viscosity) for the determination of the optimized parameter. Conclusions The viscosity of SC-CO2 partially saturated with lipids was measured with a high-pressure viscometer. A solute at saturation can contribute up to a 35% increase in mixture viscosity compared to that of pure SC-CO2 at studied pressures and temperatures. The applicability
of the Ely and Hanley corresponding states technique was tested for 12 fluid-phase systems and 9 liquid-phase systems. The use of an optimized parameter in the energy shape factor equation increased the accuracy of the model considerably, with an AAD of 3% for the fluid phase and 6% for the liquid phase. It has been shown that the modified Ely and Hanley method was better than the correlative GrunsbergNissan equation in terms of accuracy and applicability to the viscosity prediction of SC-CO2/biomaterial(s) systems. Notations AAD ) average absolute deviation:
AAD )
∑|
1 N
|
(ηcalc. - ηexp. ηexp.
× 100%
AMF ) anhydrous milkfat f, h ) scale factors F, G ) shape factors functions k ) adjustable parameter in the energy shape factor LOA ) linoleic acid MLO ) methyl linoleate MO ) methyl oleate MW ) molecular weight OA ) oleic acid P ) pressure (MPa) T ) temperature (K) V ) molar volume (L/mol) Z ) compressibility factor Greek Letters F ) density (mol/L) Φ ) size shape factor Θ ) energy shape factor η ) viscosity (Pa‚s) ω ) acentric factor Indices c ) critical property R ) reference fluid
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Received for review June 8, 1998 Revised manuscript received March 3, 1999 Accepted March 5, 1999 IE980363W
