3162
Ind. Eng. Chem. Res. 1993,32, 3162-3169
Calculations of Compressed Carbon Dioxide Viscosities Helena SOVOVB.and Jaroslav Prochhzka Institute of Chemical Process Fundamentals, Academy 165 02 Prague 6,Czech Republic
of
Sciences of the Czech Republic,
Compressed carbon dioxide is widely used as a solvent in supercritical and liquefied gas extraction. The knowledge of the fluid-phase viscosity as a function of temperature and pressure is indispensable for predicting the hydrodynamics and mass-transfer rates in these processes. Various equations, both generally valid and especially derived for carbon dioxide, are compared with published experimental data covering gas, liquid, and supercritical regions. The equations of Vesovic et al. and of Altunin and Sakhabetdinov are recommended as most reliable for predicting the viscosity of dense carbon dioxide.
Introduction This paper surveys available methods of predicting the carbon dioxide fluid-phase viscosity as a function of its temperature and pressure and compares them with experimental data. The purpose is torecommend the most reliable methods for engineering calculations of this property. Carbon dioxide is increasingly exploited as a supercritical fluid extraction solvent, e.g., in the food industry, pharmacy, biotechnology, and enhanced oil recovery. The knowledge of its viscosity and other transport properties is necessary for prediction of masstransfer rates of these processes. The most recent equations for the viscosity of carbon dioxide have been developed by Vesovic et al. (1990).They are based upon a great number of experimental data from the literature that have been critically assessed for internal consistency, and their authors declare them as the best equations that can be produced from the available data. The viscosity of a fluid may be expressed as a sum of three independent contributions, (1) p(P,T) = po(T) + &(P,T) + A@(P,T) The first term, po(T), is the dilute gas viscosity at the given temperature, which is insensitive to pressure. The second term, Ap(P,T), is the excess viscosity which represents the effects at elevated densities. Instead of pressure, density can be used as an independent variable. The excess viscosity may then be approximated as a function of density alone, because the effect of temperature on its value at constant density is almost negligible. The critical enhancement, A,+(P,T), arises from the long-range fluctuations that occur in a fluid near its critical point. Viscosity of carbon dioxide exhibits a weak critical enhancement restricted to a rather small region around the critical point. Vesovic et al. analyzed the data of Iwasaki and Takahashi (1981) on the COz viscosity near critical point, fitted them by a relatively complex equation with theoretical background, and concluded that the ratio 4 p I p is smaller than 1%at densities and temperatures outside a range bounded approximately by 300 K < T < 310 K and 300 kg m-3 < p < 600 kg m-3. The critical enhancement is therefore not taken into account in our paper and, for the sake of simplicity, the term 4 p ( P , T ) is omitted in the quoted correlations developed by Vesovic et al. (1990). Both semiempirical and empirical equations contain parameters adjusted according to the experimental data.
* Author to whom correspondence should be addressed. E-mail:
[email protected];
[email protected]. 0888-5885/93/2632-3162$04.00l0
In this respect they can be divided into the general correlations based on the experimental viscosities of a larger group of gases and the special correlations based exclusively on the COZ data. The properties of carbon dioxide used in the relations discussed in this paper are M = 44.01 g mol-', T, = 304.2 K, P, = 73.8 bar, V, = 94.04 cm3 mol-', pc = 468 kg ma, and w = 0.239.
Dilute Gas Viscosity General Equations. The Chapman-Enskog theory (Chapman and Cowling, 1970) for the dilute gas viscosity is written as = 0.02669(MT)1/2/a2Qz2(T), T* = kT/t (2) where elk and u are the energy and length scaling parameters and Q 2 2 ( T * ) is the reduced collision integral, which was evaluated using the Lennard-Jones (12-6) potential model and correlated by Neufeld et al. (1972) as po
fizz = 1.16145/A* + 0.52487/exp(0.77320T) + 2.16178/exp(2.43787T) 0.0006435A* ~ i n ( 1 8 . 0 3 2 3 / ( T )-~7.27371) *~~~ A* = (T*)o.14874
(3)
Equation 2 is accurate for nonpolar monatomic gases. To extend to polyatomic molecular gases, Chung et al. (1988) multiplied eq 2 by a factor F,,
+
F, = 1- 0.2756~+ 0.059035p: K (4) where w is the acentric factor, pris the dimensionless dipole moment, and K is a correction factor for the hydrogenbonding effect of associated substances. They also employed the following relations for the scaling parameters: u = 0.809V,'/3,
tlk = TJ1.2593
(5) Substitution of the carbon dioxide properties including pr = 0 and K = 0 into eqs 4 and 5 and into eq 2 with the factor F, yields po =
1.222T1'2/Qzz(T*), T* = Tl241.56 (6) Stiel and Thodos (1961) have correlated the viscosities of 52 nonpolar gases with reduced temperature TR= TIT,: pot = 0.34TR0.94for T R 5 1.5 pot
= 0.1778[4.58TR- 1.67]0.626for TR> 1.5
Viscosity was multiplied by a factor E, 0 1993 American Chemical Society
(7)
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3163 Table I. Low Pressure Data Sets no. reference 1 Touloukian et al. (1975) 2 Harris et al. (1979) 3 Iwaeakiand Takahashi (1981) 4 Haepp (1976) 5 Vogel and Barkow (1986) 6 Kestin et al. (1977) 7 Kestin et al. (1972)
-1
I
T,K
claimed accuraw 170-2000 LT, MT i 2 % , H T i 5 % 203-310 LT >1%,MT, H T *l% 298-323 i0.3% 298-475 298-623 298-773 298-973
iO.9% LT *0.1%, HT i0.3% LT *0.1%, H T *0.3% LT *0.1%, HT +0.3%
0 LT, MT, H T low, middle, and high temperatures in the given range.
5 = 1.0088T~/6/M'.t2P:/3
(8)
where the critical pressure is in atmospheres. The value of the factor for carbon dioxide is [ = 0.0224. Similarly, Lucas (Reid et al., 1987) has used a viscosity factor X = 0.176T,'/6/M'/2P:/3
-44
1200
800
400
0
1600
2000
TIK
Figure 1. Deviations of low pressure viscosity correlations from Reichenberg's correlation. (-) Chung; (-+-) Stiel; (-) Lucas.
(9)
and described the viscosity of nonpolar gases by
+ 0.340 exp(-4.058TR) + 0.0181
p o x = 0.1[0.807T~0~61a - 0.357 eXp(-0.449T~)
(10)
The factor for carbon dioxide is X = 0.00391. A simple equation based on the viscosity data of 79 organic compounds has been developed by Reichenberg (1975) in the form po = aT/[1
+ 0.36(1 + 4/Tc)TR(TR- 1)11"
(11)
The authors of this paper have determined the value of the adjustable parameter in eq 11for COZ as a = 0.050. The range of temperatures where the general correlations are valid is wide and includes the conditions of all data sets from Table I. Special Equations for Carbon Dioxide. Two equations for the dilute carbon dioxide viscosity were published in the same year. Kestin et al. (1972) used eq 2 to evaluate the empirical function for the collision integral
+
OZz= exp[0.46461- 0.56612(1n P ) 0.19565(1n PI2-
0.0303(ln P I 3 ] (12) with the scaling parameters u = 0.3703 nm and elk = 266.13 K from viscositiesmeasured in the temperature range 298973 K. Altunin and Sakhabetdinov (1972) correlated the carbon dioxide viscosity in the temperature range 220 K 5 T I1300 K with reduced temperature: po
= TR0.5(27.2246461 - 16.6346068lTR + 4.669205561T;)
(13)
Vogel and Barkow (1986) have represented the results of their measurements in the range 295-647 K by the empirical equation po =
E e a
14.918 exp[0.619420 (In TA) - 0.483713/TA+ 0.0652807/TA2+ 0.418465)
Vesovic et al. (1990) evaluated 18 data sets measured by 5 groups of workers and obtained an equation for the collision integral.
200
600
1000
1400
1
TIK
Figure 2. Deviations of low-pressure viscosity correlations from Reichenberg's correlation. (-) Kestin; (+-.+) Altunin, (-+-) Vogel; (-) Vesovic.
O,, = 1.25 expI0.235156 - 0.491266(1n P )+ 0.05211155(1n P)2 + 0.05347906(1n P ) 30.01537102(1n P I 4 ] (15) with the scaling parameters u = 0.3751 nm and elk = 251.196 K valid in the range 200-1500 K. Differences in the course of the individual correlations are depicted in Figures 1and 2 in terms of the deviation from Reichenberg's correlation, Acor(T) = ~ O O ~ [ ~ O , ~ , ( T ) / ~ ~ ~ ~1)~ ~(16) ~~~(T)I The maximum deviations are smaller than 4 % Except for the correlations of Chung and Lucas, the curves intersect close to the room temperature where the viscosity measurement is most accurate. Comparison with Experimental Data. A'list of experimental data used in this work for comparison with the equations for dilute carbon dioxide viscosity is presented in Table I. The data from the monograph by Touloukian et al. (1975) are recommended values based on 28 primary data sets, while the other rows of the table refer to the individual more recent data sets. The results of the comparison are presented in Table I1 as average deviations of data sets from the equations,
.
The calculations with special correlations for COz were
3164 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 Table 11. Fit of Low Pressure Viscosity Models, d(w) no. Chum Stiel Lucas Reichenbera 1.84 1.30 3.19 1.15 1 2.01 0.29 1.12 0.15 2 1.24 0.12 1.48 0.12 3 0.29 1.20 2.46 0.82 4 1.76 0.44 0.99 0.91 5 2.64 1.68 0.37 1.99 6+7 av 1.63 0.84 1.60 0.86
:j -4
0
,
~
400
,
,
,
,
1200
800
Kestin 2.60 0.15 0.19 1.84 0.63 0.13 0.92
Altunin 2.09 0.30 0.45 1.88 0.58 0.33 0.94
Voael 1.61 0.13 0.13 1.30 0.00 0.62 0.63
Vesovic 2.36 0.48 0.16 1.40 0.15 0.43 0.83
;J 1600
2
I
-54 0
400
1200
800
Figure 3. Deviations of low pressure viscosity data from ReichenTouloukian; (+) Harris; (*) I w d , (m) Haepp; berg'scorrelation. (0) (X) Vogel; (A)Kestin.
limited to the respective temperature ranges. The correlation of Vogel and Barkow, which covers the smallest range of temperatures in the region of minimum experimental errors, has therefore the minimum total average deviation (= averaged deviations from the six data sets). The total average deviations given in the last row of Table I1 range from 0.6 to 1.6 % Both experimental data and correlations may be divided into two groups according to their course above the room temperature. The general equations of Chung et al., Stiel and Thodos, and Reichenberg are parallel to the recommended data of Touloukian et al. (1975) and to the data of Haepp (1976) and Harris et al. (1979). The correlation of Reichenberg describes best Touloukian's and Harris' data, and we recommend it as a representative for this group (see Figure 3). The special equations for carbon dioxide and, to some extent, also the general correlation of Lucas follow the data of Kestin et al. (1972,1977) and Vogel and Barkow (1986). They predict a value of viscosity at 800 K by 2-3% higher than the equations of the first group; this deviation diminishes at higher temperatures. The proper representative of this group is the equation of Vesovic et al. (see Figure 4). We are not able to decide on the basis of the compared data sets whether the equation of Vesovic et al. is more reliable than the equation of Reichenberg. With regard to the developments in the viscosity measurement technique, the equation of Vesovic et al., which is based on the more recent data, should be preferred.
.
1600
2 '0
TIK
TIK
Figure 4. Deviations of low pressure viscosity data from Vesovic's Touloukian; (+) Harris; (*) Iwaeaki; (m) Haepp; (X) correlation. (0) Vogel; (A)Kestin.
density:
G2 = (Al[l - exp(-A&l/y
+ A2G1exp(A9) + A.&'1J/(A1A4
p**
= @A7y2G2 exp[A,
+ A2 + A31
+ A,(T*)-' + Alo(T*)-2]
Ai = ai+ biw + tip: + d i ~ ,i = 1-10; @ = 3.6344(MT~'/~/ V:I3 (19) Substitution of carbon dioxide properties for w, pr and K into eq 19 yields A1 = 18.372 46, A2 = 9.34490 X 1W, A3 = 66.03941, A4 = 15.72750, A6 = 21.56905, A6 = -4.896 191, A7 = 25.09892, As = 1.064276, Ag = -0.221 980 9, Ai0 = 0.151 784 3, @ = 20.335. Jossi, Stiel and Thodos (1962) correlated the excess viscosity with reduced density in the range 0.1 IPR < 3: A
Values of the coefficients are a0 = 1.0230, a1 = 0.233 64, a2 = 0.585 33, a3 = -0.407 58, and a4 = 0.093 324.
Compressed Fluid Viscosity Generation Equations. The equations for dilute gas viscosity listed above were extended to the conditions with increased pressure. Chung et al. (1988) developed the expression for the dense fluid viscosity
Recently Lee and Thodos (1990) have derived a method for determining the excess viscosity especially suited for liquids and gases at temperatures below their normal boiling points. The quantities in this correlation are related to the triple point characterized by the temperature Tt and the molar volume ult. The excess viscosity is calculated from the equation
p = pi$*(PR) + p**(T*,PR) (18) containing two functions of reduced temperature and
Apy = exp(2.9328.ifSW + 4.5424g0.922e)- 1 (21) where the viscosity factor y and the expansion factorg are
Ind. Eng. Chem. Res., Vol. 32, No. 12,1993 3165 y = u,:I3/ (MTt)'I2; g = ~ ( 0 . 9 7 6 t ) - ~ . ~ 5 6 6 /(22) ~~'~'~
The variable x is a function of temperature and density:
An equation for the excess viscosity as a function of temperature and density has been developed by Ulybin and Makarushkin (1976):
= g7-0.079*~'3. 0 = P/Plt;
T/Tt (23) and t = ult/ust is the ratio of molar volumes of the liquid and solid phases at the triple point. For C02 the following values have been recommended by Lee and Thodos: Tt = 216.55 K; ult = 37.37 cm3 mol-'; y = 0.1145; e = 1.206 27. Lucas (Reid et al., 1987) proposed an equation for the viscosity of compressed gases in the range of conditions 1 < TR < 40,O < PR < 100:
The correlation factors Fp and FQ = 1 for CO2 and the parameters a to f are functions of reduced temperature:
Ap
= S(1+ k S / p ) ; S = x a i ( p X lo3)'
with coefficients a1 = 20.861 869 11, a2 = -226.831 846 5, a3 = 1845.657 533, a4 = -5091.665 242, a5 = 6872.877 227, a6 = -4460.627 564, a7 = 1140.901 439, and k = 62. The
correlation was based on a new set of data measured with liquid COZand 11 literature data sets for dense carbon dioxide, and it is valid in the temperature range 220 IT I 1300 K and for pressures up to 3000 bar. On the basis of their own measurements at pressures up to 200 bar, Herreman et al. (1978) developed a correlation for the viscosity of liquid C02 as a function of the density using a free volume model: p = exp13.3882
a = (0.001245/TR)e~p(5.1726/T$~'~); b = a(1.6553T~- 1.2723); c = (0.4489/TR) X
(30)
i=l
+ 1.3423p1/(1.909- pl)l;
p1 = 0.001p
(31) Diller and Ball (1985) correlated the viscosities of liquid exp(3.0578/TR37'7332); f = 0.9425 e ~ p ( - O . l 8 5 3 T $ ~ ~ ~ ) ; carbon dioxide at pressures up to 300 bar with a modified Hildebrand equation: d = (1.7368/TR)e ~ p ( 2 . 2 3 1 0 / T 2 . ~ ~e~= ' ) ;1.3088 (25) For the region of gas at temperatures below critical and p = (10.537 - exp(6.726 - 0.0433111 [(Mlp)- Val]-' (32) pressures below saturated the following relation has been where VO= 0.0295 L mol-' is the estimated specific volume recommended by Lucas: at zero fluidity. Vesovic et al. (1990) treated the gas phase and liquid pclx/FpFQ = 0.060 0.076PRa + (0.699PR'- 0.06)(1- T R ) phase separately. They presumed the excess viscosity of gaseous carbon dioxide to be temperature independent a = 3.262 14.98PR5.508; = 1.390 5.746PR (26) and correlated it with density,
+
+
+
The equation of Lucas (eq 24) was derived from a similar equation for the viscosity of dense gases above the critical temperature which was proposed by Reichenberg (Reid et al., 1987):
The parameters A, B, C, and D are functions of reduced temperature:
A = (O.O019824/TR) e~p(5.2683/T,"~~~~)
D = (2.9496/TR)
(28)
For CO2 as a nonpolar compound Q = 1. Special Equations for Carbon Dioxide. Altunin and Sakhabetdinov (1972) proposed an empirical equation for the viscosity ratio as function of reduced temperature and density:
where po is calculated from eq 13. Coefficients a10 = 0.248 566 120, all = 0.004 894 942, azo = -0.373 300 660, a21 = 1.227 534 88,am = 0.363 854 523,a31= -0.774 229 021, a40 = -0.0639070755, and a41 = 0.142507049 were evaluated from nine data sets for carbon dioxide viscosity covering the temperature range 220 IT I 1300 K and pressures up to 1200 bar.
+
Ap = 3.6350734~' 72.09997~:
+ 30.O306pl7; p1
= 0.001p (33)
The estimated uncertainty is between 1 and 5% in dependence on the pressure and temperature region. The liquid carbon dioxide viscosity was represented by the equation p =
[(18.56
+ O.O14T)((l/p)- 7.41 X lo4 + 3.3 x 10-~ 111-l (34)
The liquid phase viscosity data from the literature differed significantly among themselves, and there was no strong evidence on which to base a judgment for a preferred data set. All data were therefore taken into account, and eq 34 reproduces almost all of them within rt7 5%. Equations 33 and 34 together cover the temperature range 200 K IT I1500 K and pressures up to 1000 bar. Carbon dioxide viscosities calculated with the abovementioned equations for the pressure range up to 3000 bar are compared in Figures 5-7 for the gas phase and Figures 8 and 9 for the liquid phase. Mutual deviations of the equations in the gas phase do not exceed 10% except for the pressures above 1000 bar and also for the correlations of Reichenberg and Lucas in the critical region (see Figure 5). These correlations express the excess viscosities as functions Ap(P,n with a relatively low number of coefficients which cannot describe the extreme changes of properties close to the critical point with the same accuracy as the equations correlating the viscosity with density. The special equations of Ulybin and Makarushkin (1976) and Altunin and Sakhabetdinov (1972) deviate from the equation of Vesovic et al. by less than 3 % Deviations of general correlations which were not adjusted to the carbon dioxide viscosities are larger according to expectations.
.
3166 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 10
7”
a30620-
z
4-
10
5
2-
Q
o--
e
L
e a
0
-2-10
-4-
X -20
0
400
p
800 ikg m-3
-64
1200
Figure S. Deviations of high pressure viscosity correlations at 305.15 K from Vesovic’s correlation. (-+-) Jossi; (--) Ulybin; (--) Altunin; (-k) Chung; (-X-) Reichenberg; (-A-) Lucas.
900
-20-
-30
0
400
800
e /kg m4
1200
Figure 6. Deviations of high pressure viscosity correlations at 323.15 (-+-) Jossi; (.-) Ulybin; (-) Altunin; (-k) Chung; (-X-) Reichenberg; (-A-) Lucas.
K from Vesovic’s correlation.
1300
(-e)
-25
-25-
1200
1100 9 ikg m-3
Figure 8. Deviations of liquid COz viscosity correlations at 273.15 K from Vesovic’s correlation. (-+-) Diller; Ulybin; (-) Altunin; (+) Chung; (-W Lee; ( 4 - 1 Herreman.
-21
-1 5-
1000
-30 1150
,
,
1200
1250
,
1300
\ 1350
1 00
/kg m-3
Figure 9. Deviations of liquid COz viscosity correlations at 220 K from Vesovic’s correlation. (-+-) Diller; (--) Ulybin; (-) Altunin; (-k) Chung; (-X-) Lee; (-k) Herreman. Table 111. High Pressure Data Sets
10
no.
reference Diller and Ball (1985) Haepp (1976) Iwasaki and Takahashi (1981) Michele et al. (1957) Ulybin and Makarushkin (1976) Herreman et al. (1970) Kestin et al. (1964) Kurin and Golubev (1974)
8 -5-
T,K
P,bar
220-320 298-475 298-323 273-348 223-293 219-303 304-323 293-423
7-311 1-150 1-145 9-2097 62-547 10-196 38-118 99-3561
4
-10-
-151 0
200
400
c /kg m-’
600
Figure 7. Deviationsof high pressureviscositycorrelationsat 1273.15 K from Vesovic’s correlation. (-+-) Jossi; (-.) Ulybin; (-) Altunin; (+) Chung; (-X-) Reichenberg; (-A-) Lucas.
In the liquid-phase region, the correlation of Chung et al. deviates from others. The remaining six correlations of viscosity with density show a similar course. The equations of Herreman and Diller form one group, the equations of Altunin, Ulybin, and Lee form another group shifted to the lower viscosity values, and the equation of
Vesovic runs mostly between these groups. The discrepancy among the groups increases as the temperature decreases; at 273 K the difference is 4% and at 220 K it reaches about 10’3%. Comparison with Experimental Data. A list of data sets compared with the equations for dense fluid viscosity in this work is presented in Table 111. As the density was not quoted in the data sets of Ulybin and Makarushkin and of Kurin and Golubev, it was calculated with Altunin and Gadetskii’s equation (Angus et al., 1976) from the experimental pressures. The original densities were used in calculations with the other data. Experimental points measured close to the critical point have been omitted from three data seta: the data of Iwasaki and Takahashi in the region bounded by 303.25 K IT I 304.95 K and 326 kg m-3 I p I 528 kg m-3, the data of Kestin et al. in the region 304.25 K I T 5 305.15 K and 403 kg m-3 Ip
Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 3167 Table IV. Fit of High Pressure Viscosity Models,. d(p) no. Vesovic Altunin Ulybin Chung 1 2 3 4 7 8 av
4.9 1.4 1.1 1.0 1.3 1.2 1.8
3.6 1.5 1.4 0.9 2.1 1.3 1.8
4.5 1.4 1.6 1.3 2.0 2.1 2.2
2.7 3.3 4.5 3.5 7.3 2.6 4.0
Reichenberg Gas Phase 1.9 3.1 7.9 4.5 8.2 13.8
6.6 Liquid Phase
Lucas
Jossi
2.8 4.8 9.3 4.7 8.8 8.4 6.5
3.0 3.3 4.0 4.9 5.8 5.7 4.5
Lee
Herreman
Diller
1 1.7 4.4 5.6 7.5 17.7 5.2 2.4 4 1.8 1.2 1.7 1.3 6.6 2.2 11.2 5 4.1 1.7 0.8 3.5 13.6 0.8 6.8 6 3.3 5.5 6.7 6.9 16.6 6.3 1.9 8 1.7 1.0 3.4 1.2 6.6 2.4 7.2 av 2.5 2.8 3.6 4.1 12.2 3.4 5.9 av 4,5,8 2.5 1.3 2.0 2.0 8.9 1.8 8.4 a Maximum pressures for the special correlations: Herreman, 200 bar; Diller, 300 bar; Vesovic, lo00 bar; Altunin, 1200 bar.
.i +++
1.2 3.6 6.7 3.2 4.3 3.8 4.9
,.,
+
++++
6l
4
+
+
+
++ I
m
1
I -7 I
0
I
-2
I
I
m
m
-7
400
800
c lkg m-3
1200
Figure 10. Deviations of high pressure viscosity data at 313-324 K Iwasaki; (1) from Altunin's correlation. (+) Diller; (X) Haepp; (0) Michels; (A)Kestin;).( Kurin. S 533 kg m4, and the data of Michels et al. in the region 298 K I T I 313 K and 200 kg m a I p I 700 kg m-3. (The omitted data of Michels et al. are erroneous due to the inappropriate analysis of the experimental resulis in this region, as was stated by Vesovic et al.). Two points measured at the lowest pressure and at 323.15 K were omitted from the data of Kurin and Golubev because they deviated from both analyzed equations and the other experimental data by 12-16%. The average deviations of these data from the individual equations, d ( p ) , are given in Table IV. The viscosity of dense carbon dioxide in the gas phase is best represented by the correlations of Vesovic and Altunin, and also the total average deviation of the Ulybin's correlation is low. The same order of correlations according to their accuracy is valid in the liquid phase. The general correlations are less accurate, except for the correlation of Lee and Thodos for the liquid carbon dioxide. In this group, the correlations of viscosity with the density (Chung et al., Jossi et al.) are more reliable than the correlations with the temperature (Reichenberg, Lucas). In Figures 10-15 we present a comparison of experimental data from Table I11 with the equations of Vesovic and Altunin, including pressures above 1200 bar. Deviations of the measured viscosities of gaseous and supercritical carbon dioxide from the equations in the temperature range 39-50 "C are depicted in Figures 10 and 11. The data of Iwasaki et al., Michels et al., Kurin and Golubev, and Haepp are on the same level. As was stated
Figure 11. Deviations of high pressure viscosity data at 313-324 K from Vesovic's correlation. (+) Diller; (X) Haepp; (a)Iwasaki; (1) Michels; (A)Kestin;).( Kurin.
I
c 2
a
0 -2 -44
0
400
800
c ikg m-3
1200
Figure 12. Deviations of high pressure viscosity data at 346474 K from Altunin's correlation. (X) Haepp; (0)Michels;).( Kurin.
by Vesovic et al., the measurements performed by Diller and Ball are, on average, some 4 % above these data. The measuremenis of Kestin et al. are, on the contrary, approximately 1% below these data. The equation of Vesovic et al. represents rather Kestin's data, while the equation of Altunin and Sakhabetdinov is closer to the other four data sets. In the temperature range 75-200 OC the scatter of the data grows due to the increasing experimental error.
3168 Ind. Eng. Chem. Res., Vol. 32, No. 12, 1993 Table V. Values of p Given at T, P, and p for Checking Computer Code u. uPa.5
T,K 220 300 800 304 220 300 800
f
" -4
I
C =
I
I
I I
0
400
800
9 lkg m-'
1200
I
Figure 13. Deviations of high pressure viscosity data at 348-474 K from Vesovic's correlation. (X) Haepp; (0)Michels; (M) Kurin.
25i
101 15
I
I
T? +*+A. Y
+ 1000 lkg m''
so0
1200
Figure 14. Deviations of liquid COz viscosity data at 273-304 K from Altunin's correlation. (0)Diller; (X) Michels; (+) Ulybin; (1) Herreman; (*) Kurin.
251 20-
E
