Viscosity Modeling of Light Gases at Supercritical Conditions Using

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Ind. Eng. Chem. Res. 2001, 40, 3848-3854

CORRELATIONS Viscosity Modeling of Light Gases at Supercritical Conditions Using the Friction Theory Claus K. Ze´ berg-Mikkelsen, Sergio E. Quin ˜ ones-Cisneros,* and Erling H. Stenby Center for Phase Equilibria and Separation Processes (IVCSEP), Department of Chemical Engineering, Technical University of Denmark, Building 229, 2800 Lyngby, Denmark

The viscosities of argon, helium, hydrogen, krypton, methane, neon, nitrogen, and oxygen have been modeled using the friction theory (f-theory) for viscosity modeling in conjunction with the Peng-Robinson (PR) equation of state (EOS), Soave-Redlich-Kwong EOS, Stryjek-Vera modification of the PR EOS, and Mathias modification to the SRK EOS. The viscosity modeling has been performed at supercritical conditions and up to 1000 bar, covering most common conditions in industrial processes. The overall average absolute deviation for each fluid ranges from 0.4 to 1.4% and is in excellent agreement with the reported uncertainty of the recommended literature values. These results are obtained using only three friction constants and without any knowledge of the density. Further it has been found that the dilute gas viscosity can also be modeled using only three constants for each fluid, from the critical temperature up to 2000 K, within or close to the uncertainty of the recommended dilute gas viscosity data. The obtained results further show the application of the f-theory for viscosity modeling and its potential for applications to industrial processes. Introduction The industrial use of hydrogen, nitrogen, and oxygen at supercritical conditions is widespread. For instance, ammonia is manufactured by letting nitrogen and hydrogen react over a catalyst at temperatures between 700 and 900 K and at pressures between 200 and 600 bar. Nitrogen is now being used at a large scale in the Gulf of Mexico as an injection gas for oil recovery. Hydrogen is also used in petroleum refinery processes such as hydrocracking or hydrotreating. Oxygen is used in a large number of applications, particularly in combustion processes. These gases are supplied not only in compressed gas cylinders or storage tanks but also through pipelines in the case of industries that require large amounts of nitrogen, hydrogen, or oxygen. In fact, at the homepage of the gas company Air Liquide, it is mentioned that they have a network of approximately 7000 km of pipelines in Europe and USA, primarily for hydrogen, nitrogen, and oxygen. Also the noble gases, such as argon and helium, have found significant applications in industries such as the semiconductor and metallurgical industry in order to avoid oxidation. Therefore, because of the importance that light gases have in diverse industrial processes, modeling of their physical properties over wide ranges of temperature and pressure is important. One of these properties is the viscosity, which is a key property in the design and operation of transport equipments. Thus, reliable and accurate models, which can be applied to wide ranges of temperature and pressure, are required. Recently, the * Corresponding author. E-mail: [email protected]. Phone: (+45) 45 25 29 17. Fax: (+45) 45 88 22 58.

friction theory (f-theory) for viscosity modeling1 has been introduced. Under the f-theory the Amontons-Coulomb friction law is linked to the van der Waals repulsive and attractive pressure terms. This allows for accurate viscosity estimations using simple cubic equations of state (EOSs). Because cubic EOSs are optimized for good pressure-temperature performance, good viscositypressure performance can also be obtainedsregardless of the accuracy of the estimated density. Thus, in this work the concepts of the f-theory are applied to the accurate viscosity modeling of light gases at supercritical conditions. The studied gases are argon, helium, hydrogen, krypton, methane, neon, nitrogen, and oxygen. The Friction Theory According to the f-theory,1 the total viscosity η is separated into a dilute gas viscosity term η0 and a residual friction term ηf

η ) η0 + ηf

(1)

The dilute gas viscosity η0 is defined as the viscosity at the zero density limit, while the friction term ηf is related to friction concepts of classical mechanics. According to the f-theory, the friction term can be expressed as follows:

ηf ) κr pr + κapa + κrrpr2

(2)

where κr , κa, and κrr are the friction coefficients and pa and pr the van der Waals attractive and repulsive pressure terms. The friction coefficients in eq 2 are

10.1021/ie000987d CCC: $20.00 © 2001 American Chemical Society Published on Web 07/31/2001

Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001 3849 Table 1. Dilute Gas Viscosity Constants for Eq 5

argon helium hydrogen krypton methane neon nitrogen oxygen

Table 2. Comparison of Deviations between Estimated Dilute Gas Viscosities with Eq 5 and Recommended Values

d1

d2

d3

28.2638 3.65477 -1.55199 37.0292 13.3919 36.6876 19.1275 23.7298

-80.5002 1.80913 2.92788 -101.369 -47.9429 -49.5702 -53.0591 -67.7604

0.206 762 0.758 601 0.645 731 0.232 700 0.160 913 0.325 255 0.184 743 0.192 271

related to the Amontons-Coulomb friction law, and a further discussion on their physical meaning can be found in the original f-theory manuscript.1 For many substances, the repulsive and attractive pressure terms may be obtained from simple cubic EOSs, such as the Soave-Redlich-Kwong2 (SRK) EOS or the PengRobinson3 (PR) EOS. Data Sources Because tabulations of recommended viscosities provide a more uniform weight distribution for a fitting procedure, they can be better used in the development of accurate viscosity models. Thus, recommended viscosities for argon,4 helium,5 hydrogen,5 krypton,5 methane,6 neon,5 nitrogen,7 and oxygen8 are used in order to derive f-theory models for each fluid at supercritical conditions. The main aim of this work is to model the viscosity for temperature and pressure conditions commonly found in industrial processes. Therefore, in spite of the fact that for some gases recommended viscosities are reported at higher pressures, only data up to 1000 bar will be used. In the case of hydrogen, it should be mentioned that for temperatures above 200 K hydrogen (normal hydrogen) is composed of 25% para (p-hydrogen) and 75% ortho (o-hydrogen) in equilibrium (see Figure 1 in work by Vargaftik9). When the temperature is decreased, the equilibrium is shifted toward p-hydrogen, and below 20 K, hydrogen is totally in its para form. Further, according to Vargaftik9 (p 38), there should be no difference between the viscosity of p-hydrogen and hydrogen at supercritical conditions, but Stephan and Lucas5 report differences of 5-10% between their recommended viscosities for p-hydrogen and hydrogen. Because of this, only viscosity data above 200 K will be used in the f-theory viscosity modeling of hydrogen performed in this work. Dilute Gas Limit Because the total viscosity is separated into a dilute gas viscosity term and a residual friction term, any dilute gas viscosity model can be used. However, in this work dilute gas viscosity values were obtained by extrapolating the recommended viscosities of each fluid to zero pressure. On the basis of these values, it is found that the following simple empirical expression can accurately model the dilute gas viscosity limit:

η0 ) d1xT + d2Td3

(3)

The estimated di parameters are given in Table 1; the units of η0 are micropoise, while the temperature is given in Kelvin. The accuracy of this dilute gas viscosity model, eq 3, has been tested by comparing the model results with

argon10 helium10 hydrogen11 krypton10 methane12 methane13 neon10 nitrogen14 oxygen14

no. of points

T range [K]

AAD %

MxD %

57 65 181 51 117 21 73 225 219

160-2000 80-2000 200-2000 220-2000 195-1050 200-400 50-2000 130-2000 160-2000

0.43 1.49 1.19 0.96 0.33 0.15 3.30 1.62 0.48

1.01 at 2000 K 3.56 at 2000 K 2.30 at 2000 K 1.68 at 250 K 0.57 at 910 K 0.47 at 200 K 7.52 at 2000 K 2.42 at 2000 K 1.33 at 180 K

Table 3. Critical Constants and Acentric Factors Used in This Work


Tc [K]

Pc [bar]

ω

150.86 5.20 33.18 209.35 190.58 44.40 126.1 154.58

48.981 2.275 13.130 55.020 46.043 26.53 33.944 50.43

0.0 -0.3900 -0.2150 0.0 0.0108 -0.0414 0.0403 0.0218

recommended dilute gas viscosity values found in the literature. To also test the performance of the model under extrapolation, the comparison has been performed for temperatures up to 2000 K. The average absolute deviation (AAD) and the absolute maximum deviation (MxD) for the comparisons are presented in Table 2. The found deviations are in agreement with the reported uncertainties for the tabulated dilute gas viscosity values, except for neon, where the largest deviations are obtained for temperatures below 100 K and above 1000 K. However, the obtained AAD for neon in the temperature range 100-1000 K is only 1.03% with an MxD % of 3.03% at 1000 K. It should also be mentioned that the only way to obtain “experimental” dilute gas viscosities is by extrapolating viscosity measurements performed at low densities to the zero density limit. Further, because of the low value of the dilute gas viscosity, a difference of 1-1.5 µP can easily correspond to a deviation of 1-2%. Use of a Cubic Equation of State in the Supercritical Region In general, cubic EOSs require the fluid acentric factor (ω), critical temperature (Tc), and critical pressure (Pc). Thus, all of the EOS-required compound properties have been taken from the DIPPR Data Compilation.15 However, because these work models are based on cubic EOSs, the f-theory models are also sensitive to the used values of the compound properties. Therefore, the DIPPR constants used in this work are also reported in Table 3. Before proceeding with the viscosity modeling procedure, it is important to make some remarks regarding the use of cubic EOSs for supercritical fluids. Practically all of the cubic EOSs are based on different modifications of the attractive term of the van der Waals EOS.16 In general, they have the form

P)

a(T) RT v-b f(v)

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Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001

where a(T) is, in most cases, based on the attractive function proposed by Soave2 for the Redlich-Kwong17 (RK) EOS and f(v) is some sort of quadratic function of volume. The clever parametrization of the attractive term in the RK EOS introduced by Soave has led to an enormous success of the cubic EOSs due to their capability in representing the vapor/liquid equilibria for a large number of fluids of industrial interest. Further improvements of the attractive Soave function have extended the application of cubic EOSs to many polar and nonpolar fluids, as is the case with the Mathias18 modification of the SRK EOS (SRKM) or the Stryjek and Vera19 modification of the PR EOS (PRSV). However, because the Soave attractive function was originally intended for vapor/liquid equilibria, some remarks are necessary when such a function is applied at supercritical conditions. The original Soave attractive function has the form

a(T) ) acR(T)

(4)

R0.5 ) 1 + m(1 - Tr0.5)

(5)

Figure 1. Behavior of the Soave R function as a function of the reduced temperature Tr and the acentric factor ω.

where

Here, Tr is the reduced temperature (T/Tc) and

m ) 0.480 + 1.574ω - 0.176ω2

(6)

For fluids with small positive acentric factors and critical temperatures above 100 K, the Soave attractive function has an adequate performance for the temperature ranges that are relevant to industrial applications. However, in some cases the Soave R function, eq 5, may have problems because of the fact that it is an unbounded function of temperature. Figure 1 shows the behavior of the Soave R function for different acentric factors. Clearly, for a fluid such as methane, with an acentric factor close to zero and a critical temperature close to 200 K, the Soave R function is a monotonically decreasing function (up to around 2000 K) for all possible temperature ranges of industrial interest. However, in some other cases, such as helium, hydrogen, or neon, the behavior of the Soave R function at high reduced temperatures may not be adequate. One way to correct this problem is the use of a different R function for the supercritical region. For example, for the supercritical region of the SRKM EOS, Mathias18 has used an expression suggested by Boston and Mathias20 (BM) of the form

R0.5 ) exp[c(1 - Trd)]

(7)

where

c)1+

m + 0.3p 2

(8)

c-1 c

(9)

and

d)

The m in eq 8 is similar to the m proposed by Soave, eq 6, but with slightly different numerical values because of the fact that the Mathias correlation in the SRKM EOS was regressed from a larger data set. The Mathias subcritical R function is essentially the same

Figure 2. Behavior of the BM R function correction as a function of the reduced temperature Tr and the acentric factor ω.

as the Soave function except for the addition of another empirical parameter, q, intended for a better vapor/ liquid representation of polar substances. The transition between the classical subcritical Soave R function and a BM type of supercritical modification is first-order smooth, i.e., a continuous function up to the first derivative. As illustrated in Figure 2, for reasonable values of the acentric factor, the BM correction ensures that the supercritical term of the R function will either vanish or remain bounded. However, although probably outside of realistic application ranges, it is important to mention that the mathematical structure of the BM correction for the supercritical R function also allows for the divergence of the R function for large positive or negative acentric factor values. Thus, in this work four different viscosity models for light gases are presented. Two of the models are based on the unmodified SRK EOS and PR EOS, i.e., the unmodified original R function is extrapolated to the supercritical region. The other two viscosity models are based on the SRKM and PRSV EOSs, where the BM correction of the R function has been used for the supercritical region. In the case of the SRKM and PRSV EOSs, the additional empirical parameter used in the R function has been neglected because, in addition of not being tabulated in standard compilations, the main purpose of this parameter is to improve the vapor/liquid equilibrium performance of the EOS. f-Theory Modeling of Light Gases The empirical observation of the viscosity behavior of supercritical fluids indicates that, as high reduced temperatures are approached, the viscosity turns into an almost linear function in pressure and also appears

Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001 3851 Table 4. Friction Constants for the PR and SRK EOSs kr [µP/bar]

ka [µP/bar]

krr [µP/bar2]

Table 5. Friction Constants for the PRSV and SRKM EOSs kr [µP/bar]


PR EOS 0.102 756 -0.727 451 -0.042 703 5 -0.507 319 -0.001 853 08 -0.332 575 0.152 164 -0.941 704 0.073 179 6 -0.382 909 0.051 744 4 -0.794 717 0.080 672 0 -0.675 406 0.063 389 3 -0.643 424

1.508 31 × 0.027 288 8 1.351 46 × 10-4 1.437 47 × 10-4 6.636 15 × 10-5 7.446 34 × 10-4 2.810 86 × 10-4 1.172 49 × 10-4


SRK EOS 0.115 788 -0.835 290 -0.036 778 8 -0.831 727 0.002 564 07 -0.436 199 0.173 573 -0.992 514 0.080 306 0 -0.422 054 0.060 631 5 -0.846 085 0.090 114 5 -0.760 370 0.072 435 4 -0.714 059

1.783 45 × 10-4 2.646 80 × 10-2 2.292 06 × 10-4 2.366 28 × 10-4 9.486 29 × 10-5 1.002 09 × 10-3 3.508 77 × 10-4 1.577 48 × 10-4

10-4

to increase linearly with temperature. On the other hand, at low reduced supercritical temperatures, close to the critical temperature, the viscosity is clearly not a linear function of pressure and for an accurate viscosity modeling a quadratic repulsive term is necessary.1 Thus, the following simple empirical expressions for the friction coefficients are found to deliver a good performance:

κa ) ka

(10)

κr ) kr

(11)

κrr ) krr/Tr2

(12)

and

For all of the considered EOSs, even for the SRK and PR EOSs, eqs 10 and 11 are enough to model the viscosity in the linear regions, i.e., in the high-temperature region and in the low-pressure region close to the critical temperature region. On the other hand, because the relative contribution of the second-order repulsive term should decrease as the temperature increases away from the critical temperature, the mathematical structure of eq 12 should be such that the quadratic temperature dependency implicit in the pr2 term of eq 2 is canceled out. Thus, eq 12 represents the simplest mathematical expression that would achieve this purpose. A more detailed discussion on the behavior and contribution of the different friction viscosity terms has already been presented by Quinõnes-Cisneros et al.1 Finally, based on a least-squares fit, the friction constants in eqs 10-12 have been estimated for each fluid and the results are given in Table 4 for the f-theory PR and SRK models and in Table 5 for the f-theory PRSV and SRKM models. Table 6 contains the AAD and MxD obtained by the f-theory viscosity modeling together with the temperature and pressure ranges. Figures 3 and 4 show the performance of the PR f-theory models for argon and oxygen, respectively. The reported uncertainty of the recommended data for argon4 is within (2.0%. For oxygen8 the reported uncertainty depends on the temperature and pressure conditions, but primarily it is within (5%, except in the critical region and at high pressures, where the uncertainty of the recommended data can go up to (12% and (8%, respectively. Thus, the obtained AAD and MxD in this

ka [µP/bar]

krr [µP/bar2]


PRSV EOS 0.065 286 7 -0.761 658 -0.021 374 5 -1.145 14 -0.002 600 14 -0.337 98 0.122 511 -0.957 231 0.054 285 4 -0.399 96 0.037 802 2 -0.784 327 0.049 835 7 -0.684 294 0.037 893 1 -0.662 772

1.673 69 × 10-4 0.014 801 3 1.384 23 × 10-4 1.588 80 × 10-4 7.525 00 × 10-5 7.951 84 × 10-4 3.030 99 × 10-4 1.269 28 × 10-4


SRKM EOS 0.068 398 8 -0.877 401 -0.020 283 3 -1.349 00 -0.001 364 68 -0.400 596 0.136 152 -1.012 27 0.056 294 -0.444 138 0.043 168 6 -0.825 002 0.054 065 1 -0.765 597 0.042 232 1 -0.735 723

2.034 42 × 10-4 0.018 398 4 2.221 97 × 10-4 2.562 69 × 10-4 1.078 92 × 10-4 1.080 49 × 10-3 3.802 82 × 10-4 1.712 42 × 10-4

work for argon and oxygen are in good agreement with the reported uncertainties. This is also the case for helium, hydrogen, and krypton, where the uncertainty of the recommended data ranges from (1% to (2%.5 For neon the largest deviations are obtained below 100 K, primarily in the critical region, as shown in Figure 5 for the PR f-theory model. This is in excellent agreement with the reported uncertainty of (2% for neon5 above 200 K and which increases with decreasing temperatures. If the viscosity of neon is calculated above 100 K, using the PR f-theory model, an AAD of 0.64% with an MxD of 2.40 at 100 K and 180 bar is obtained. For nitrogen the largest deviations are found at high temperatures and pressures and close to the critical region, primarily because of the fact that the recommended viscosity data7 at high temperature and high pressure are extrapolated values. To evaluate the PR f-theory model for methane, a comparison with recommended viscosities,13 which were not used in the derivation of this work f-theory models, in the temperature range 200-400 K and from 1 to 500 bar has been performed. The results of this comparison are shown in Figure 6, indicating an AAD of 1.01% with a MxD of 4.56% at 200 K and 100 bar, which are in agreement with the deviations obtained for the modeled methane viscosity data. Recently, Nabizadeh and Mayinger21 measured the viscosity of hydrogen using an oscillating disk viscometer in the temperature range 296-399 K and from 1 to 58 bar with an uncertainty of (1.0%. A comparison of the Nabizadeh and Mayinger hydrogen data with this work f-theory models for hydrogen gives an AAD of 0.86% for the PR, PRSV, and SRKM f-theory models and 0.82% for the SRK f-theory model with a MxD of 2.06% at 399 K and 1 bar for all models. The obtained deviations are within the uncertainty of the f-theory models and the experimental data. Using a vibrating-wire viscometer, Wilhelm and Vogel22 recently measured the viscosity of argon up to 200 bar at temperatures between 298 and 423 K and of krypton up to 160 bar at temperatures between 298 and 348 K. The claimed uncertainty for these measurements is (0.2%. Based on an EOS, the reported viscosity measurements are tabulated against density instead of pressure. Despite this, the direct substitution of the reported densities in this work f-theory models also gives good results. For argon, the obtained AADs are

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Table 6. f-Theory Viscosity Modeling Results no. of points

T range [K]

P range [bar]

argon

921

155-500

1-1000

helium

69

80-1300

1-800

hydrogen

60

200-1000

1-1000

krypton

284

250-600

1-500

methane

459

210-476

1-919

neon

832

50-500

1-200

nitrogen

1452

140-1100

1-1000

oxygen

816

160-1400

1-1000

PR AAD % MxD % (MxD Point)

SRK AAD % MxD % (MxD point)

PRSV AAD % MxD % (MxD point)

SRKM AAD % MxD % (MxD point)

0.95 5.14 (160 K, 60 bar) 1.09 3.25 (150 K, 150 bar) 0.72 2.15 (200 K, 600 bar) 0.44 1.57 (270 K, 1 bar) 0.97 4.22 (324 K, 597 bar) 0.87 6.64 (50 K, 50 bar) 1.06 4.90 (1100 K, 1000 bar) 1.15 5.26 (170 K, 70 bar)

0.89 7.80 (160 K, 60 bar) 1.06 3.76 (80 K, 300 bar) 0.87 2.47 (200 K, 600 bar) 0.40 1.57 (270 K, 1 bar) 0.90 5.04 (229 K, 92 bar) 1.00 7.44 (50 K, 50 bar) 1.29 6.75 (1100 K, 1000 bar) 1.36 7.38 (160 K, 60 bar)

1.33 5.14 (155 K, 70 bar) 1.20 5.52 (80 K, 300 bar) 0.70 2.10 (200 K, 600 bar) 0.66 1.99 (270 K, 54 bar) 1.28 4.50 (324 K, 597 bar) 0.82 5.50 (50 K, 50 bar) 0.84 2.64 (220 K 70 bar) 0.80 4.90 (160 K, 70 bar)

1.28 6.53 (160 K, 60 bar) 1.22 5.66 (80 K, 300 bar) 0.75 2.28 (200 K, 600 bar) 0.62 1.88 (270 K, 54.3 bar) 1.27 4.39 (324 K, 597 bar) 0.89 5.93 (50 K, 50 bar) 0.86 2.74 (140 K, 50 bar) 0.85 6.29 (160 K, 60 bar)

Figure 5. Deviation of modeled viscosities of neon by the PR f-theory model from the recommended viscosities.5 Figure 3. Viscosity of argon with the PR f-theory model (s) along with the recommended values4 (b).

Figure 6. Deviation between predicted methane viscosities with the PR f-theory model and recommended viscosities.13

Conclusion Figure 4. Viscosity of oxygen with the PR f-theory model (s) along with the recommended values8 (b).

0.63%, 0.37%, 1.20%, and 0.98% for the PR, SRK, PRSV, and SRKM f-theory models, respectively. For krypton, the PR, SRK, PRSV, and SRKM f-theory model AADs are 0.66% 0.91%, 0.90%, and 0.69%, respectively. Thus, despite using a simple cubic EOS, the obtained AADs for argon and krypton are in good agreement with the reported uncertainty.

On the basis of the concepts of the f-theory, the viscosities of argon, helium, krypton, methane, neon, nitrogen, normal hydrogen, and oxygen have been modeled at supercritical conditions from 1 to 1000 bar in conjunction with the PR, SRK, PRSV, and SRKM EOSs. Overall, the obtained AAD (0.4-1.4%) is in good agreement with the reported uncertainty of the recommended viscosities and is satisfactory for most industrial applications. The obtained MxDs are primarily found to be close to the critical region, where the

Ind. Eng. Chem. Res., Vol. 40, No. 17, 2001 3853

viscosity-pressure slope tends to diverge, or at high pressures and temperatures. However, because of the fact that cubic EOSs are optimized to exactly match the critical pressure and temperature of pure components and the fact that the f-theory is based on a correlation of stresses rather than density, the f-theory models can also deliver a good viscosity-pressure performance close to the critical point, as illustrated in Figures 3 and 4. It is also important to remark that, despite the anomalous behavior that may be present at high reduced temperatures in the attractive Soave function of the PR and SRK EOSs, in the case of the light gases studied in this work, all models deliver an equivalent good viscosity performance. This work also illustrates how accurate f-theory models for light gases can be obtained using a simple three-friction-constant model. In addition, the viscosity modeling performed in this work further illustrates the f-theory potential for accurate viscosity modeling using simple cubic EOSs. Further, with the use of simple mixing rules for the f-theory friction coefficients, it may be possible to achieve good viscosity predictions for light and dense fluid mixtures that may include the light gases studied in this work. Although the viscosity prediction of fluid mixtures is an ongoing research area, some applications such as normal alkane mixtures,23 oil reservoir fluids,24 carbon dioxide plus hydrocarbon mixtures,25 natural gas,26 hythane,27 among others, have already been carried out. Additionally, the extension of the f-theory approach to polar fluids is another current research area for which a first study on pure alcohols has already been carried out.28 Finally, for the sake of completeness and accuracy, simple three-constant equations have been derived for the dilute gas viscosity term of the studied light gases. The performance of these equations is compared with recommended dilute gas viscosities up to 2000 K, and an uncertainty within or close to the uncertainty of the recommended values is obtained. However, it is important to remark that the friction models presented here are not contingent on the dilute gas models also derived in this work. In fact, any other accurate model for the dilute gas limit can also be used with good results. Acknowledgment This work is accomplished within the European project EVIDENT under the JOULE Program, Contract No. JOF3-CT97-0034. List of Symbols Latin Letters d1, d2, d3 ) dilute gas viscosity constants in eq 5 ka ) attractive friction constant in eq 3 kr ) repulsive friction constant in eq 3 krr ) quadratic repulsive friction constant in eq 3 P ) pressure [bar] pa ) attractive pressure term pr ) repulsive pressure term T ) temperature [K] Tc ) critical temperature [K] Greek Letters η ) viscosity η0 ) dilute gas viscosity ηf ) residual friction term

κa ) linear attractive friction coefficient κr ) linear repulsive friction coefficient κrr ) quadratic repulsive friction coefficient

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(26) Ze´berg-Mikkelsen, C. K.; Quinõnes-Cisneros, S. E.; Stenby, E. H. Viscosity Prediction of Natural Gases Using the Friction Theory. Int. J. Thermophys., in press. (27) Ze´berg-Mikkelsen, C. K.; Quinõnes-Cisneros, S. E.; Stenby, E. H. Viscosity Prediction of Hydrogen + Natural Gas Mixtures (Hythane). Ind. Eng. Chem. Res. 2001, 40, 2966. (28) Ze´berg-Mikkelsen, C. K.; Quinõnes-Cisneros, S. E.; Stenby, E. H. Viscosity Modeling of Associating Fluids Based on the Friction Theory: Pure Alcohols; Ninth International Conference

on Properties and Phase Equilibria for Product and Process Design (PPEPPD 2001), Kurashiki, Japan, May 20-25, 2001.

Received for review November 27, 2000 Revised manuscript received May 17, 2001 Accepted June 1, 2001 IE000987D

Viscosity Modeling of Light Gases at Supercritical Conditions Using

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