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Viscosity Prediction of Hydrogen + Natural Gas Mixtures (Hythane) 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 Kgs. Lyngby, Denmark
A scheme based on the friction theory (f-theory) is introduced for the viscosity prediction of mixtures composed of hydrogen and natural gas (hythane). In the f-theory the viscosity is separated into a dilute gas viscosity and a friction contribution term. The mixture friction coefficients are estimated with mixing rules based on pure-component friction coefficients. Because hythane mainly contains hydrogen and methane, the pure friction coefficients of these components are obtained with f-theory models directly fitted to these two components, while the friction coefficients of the other components are obtained with a general f-theory model. For the dilute gas viscosity term, the simple mixing rule of Wilke is capable of an accurate estimation of the dilute gas viscosity. Using this f-theory scheme in conjunction with the Peng-Robinson and Soave-Redlich-Kwong EOSs, the viscosities of four hythane mixtures have been predicted within or close to experimental uncertainty ((1.0%), which is satisfactory for most industrial applications. This scheme is of a predictive character, because only properties and parameters of the pure compounds are required. This work further shows the application of the f-theory for viscosity predictions and its application to industrial processes. Introduction In the 19th century coal gas, a mixture composed mainly of hydrogen (50%) and methane (26%) was used in Great Britain for lighting. Today, mixtures of hydrogen and natural gas, called hythane, are being investigated as alternative fuels in combustion engines. The aim is to reduce the emission of greenhouse gases, such as COx. Because hythane may in the future become an alternative fuel, knowledge of its physical properties is important. One of these properties is the viscosity, which is a key variable in the design of transport and injection equipment. It is, therefore, important to have reliable and accurate models for predicting the viscosity of hythane over wide ranges of temperature, pressure and composition. Recently, a scheme for predicting the viscosity of natural gases1 based on the concepts of the friction theory (f-theory) for viscosity modeling2,3 has been introduced. In this scheme, the viscosities of natural gases are accurately and efficiently predicted by combining a seven-constant f-theory model2 for methane with the general one-parameter f-theory model3 for the other components. In this work, this f-theory scheme is extended to viscosity predictions of hythane mixtures by including an f-theory model for hydrogen,4 because the main components of hythane are hydrogen and methane. Friction Theory In the f-theory, 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 residual term ηf is * Corresponding author. E-mail:
[email protected]. Phone: (+45) 45 25 29 17. Fax: (+45) 45 88 22 58.
related to friction concepts of classical mechanics. According to the f-theory, the residual friction term of an n-component mixture can be expressed as
ηf ) κrpr + κapa + κrrpr2
(2)
where pa and pr are the van der Waals attractive and repulsive pressure terms of the mixture. These repulsive and attractive pressure terms can be obtained from simple cubic equations of state (EOSs), such as the Soave-Redlich-Kwong (SRK) EOS5 or the PengRobinson (PR) EOS.6 For light gas mixtures, the friction coefficients κr, κa, and κrr can be estimated using the linear mixing rules n
κr )
xiκr,i ∑ i)1 n
κa )
xiκa,i ∑ i)1 n
κrr )
xiκrr,i ∑ i)1
(3)
where xi is the mole fraction of component i, while κr,i, κa,i, and κrr,i are the friction coefficients of the pure components. In the case of light gases, such as hythane, these simple mixing rules can deliver satisfactory results for industrial applications. However, if further accuracy enhancement is required, this can be achieved using the mass-weighted mixing rules originally derived for the f-theory.2,3 Estimation of the Pure Friction Coefficients Because the mixture friction coefficients in eq 3 are estimated based on the temperature-dependent friction coefficients of the pure components (κr,i, κa,i, and κrr,i), the mixture friction coefficients can be directly obtained
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by combining different kinds of f-theory models, as shown in the f-theory scheme for viscosity predictions of natural gases.1 Thus, because hythane is composed of hydrogen and natural gas, the f-theory natural gas scheme1 can be further combined with the threeconstant f-theory model derived for hydrogen.4 In this way an accurate and efficient procedure for predicting the viscosity of hythane can be achieved. The hydrogen f-theory model4 has been derived for supercritical conditions above 200 K, because hydrogen (normal hydrogen) at these conditions consists of 25% para hydrogen and 75% ortho hydrogen in equilibrium.7 Further, because the application of the introduced f-theory scheme for viscosity predictions of hythane is for temperatures above 200 K, which correspond to temperatures above the critical temperature of methane, the derived threeconstant f-theory model for methane4 is also used in this work. Therefore, both the hydrogen and the methane friction coefficients are obtained with the expressions4
Table 1. Friction Constants for Methane4 and Hydrogen4
κr ) kr
The PR and SRK EOSs will be used in this work for viscosity predictions of hythane mixtures, along with the original van der Waals mixing rules and without any binary interaction parameters. Thus, the three constants required in eq 4 for hydrogen4 and for methane4 are reported in Table 1 for the PR and the SRK EOSs, respectively. The PR and SRK universal constants in eqs 6 and 7 together with the required characteristic critical viscosities are given by Quin˜onesCisneros et al.3 All of the other required compound constants in the EOSs are taken from the DIPPR Data Compilation.8
κa ) ka κrr ) krrΓ2
(4)
while the friction coefficients of the other components in the natural gas are estimated with the following expressions derived from the general one-parameter f-theory model3
κr,i )
ηc,iκˆ r,i Pc,i
ηc,iκˆ rr,i Pc,i2
SRK
kr [µP/bar] ka [µP/bar] krr [µP/bar2]
Methane 0.073 179 6 -0.382 909 0.000 066 361 5
0.080 306 -0.422 054 0.000 094 862 9
kr [µP/bar] ka [µP/bar] krr [µP/bar2]
Hydrogen -0.001 853 08 -0.332 575 0.000 135 146
0.002 564 07 -0.436 199 0.000 229 206
with
ψ ) RTc,i/Pc,i
(5)
where ηc,i and Pc,i are a characteristic critical viscosity and the critical pressure of the pure components. The κˆ r,i, κˆ a,i, and κˆ rr,i parameters are separated into a critical isotherm contribution and a residual temperaturedependent contribution
where R is the gas constant and Tc,i the critical temperature of the pure component. In both eqs 4 and 7, Γ is defined as
Γ ) Tc,i/T
κˆ a,i )
xMwT η0 ) 40.785 2/3 Fc vc Ω*
+ ∆κˆ a,i
c κˆ rr,i ) κˆ rr,i + ∆κˆ rr,i
(6)
For the temperature-dependent residual terms, the following empirical functions have been derived:3
(κr,2,0 + κr,2,1ψ + κr,2,2ψ2)(exp(2Γ - 2) - 1)
Ω* )
1.16145 0.52487 + + T* exp(0.77320T*) 2.16178 - 6.435 × 10-4T*0.14874 × exp(2.43787T*) sin(18.0323T*-0.76830 - 7.27371) (11)
with
T* ) 1.2593T/Tc
∆κˆ a,i ) κa,0,0(Γ - 1) + (κa,1,0 + κa,1,1ψ)(exp(Γ - 1) - 1) + (κa,2,0 + κa,2,1ψ + κa,2,2ψ2)(exp(2Γ - 2) - 1) (7)
(10)
where the reduced collision integral Ω* corresponds to
∆κˆ r,i ) κr,0,0(Γ - 1) + (κr,1,0 + κr,1,1ψ)(exp(Γ - 1) - 1) +
∆κˆ rr,i ) κrr,2,1ψ(exp(2Γ) - 1)(Γ - 1)2
(9)
For fluids in the vapor phase or at supercritical conditions, the contribution from the dilute gas viscosity to the total viscosity is important. The dilute gas viscosity model proposed by Chung et al.9 can be applied for the prediction of the dilute gas viscosities of several polar and nonpolar pure fluids over wide ranges of temperature within an uncertainty of (1.5%. This model is an empirical correlation derived from the Chapman-Enskog theory,10 and the expression for the reduced collision integral is obtained from the LennardJones 12-6 potential (Neufeld et al.).11 The Chung et al. model9 is given by
c + ∆κˆ r,i κˆ r,i ) κˆ r,i c κˆ a,i
(8)
Dilute Gas Viscosity
ηc,iκˆ a,i κa,i ) Pc,i κrr,i )
PR
(12)
For nonpolar fluids the empirical expression for the Fc factor is
Fc ) 1 - 0.2756ω
(13)
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Table 2. Composition (mol %) of the Natural Gas and the Hythane Mixtures natural gas13 mixture 113 mixture 213 mixture 313 mixture 413
hydrogen
methane
ethane
nitrogen
4.95 15.4 29.9 74.9
94.67 89.98 80.09 66.364 23.762
3.50 3.33 2.96 2.454 0.879
1.83 1.74 1.55 1.283 0.459
where ω is the acentric factor. The dilute gas viscosity obtained by eq 10 has units of µP, when the temperature is in K and the critical volume vc is in cm3/mol. For vc, the real critical volume of the fluid must be used. A comparison of this model with reported dilute gas viscosities for pure compounds found in natural gas1 showed that the predicted dilute gas viscosities are within the accuracy reported in the literature. For hydrogen, however, the following empirical expression has been found to model better the dilute gas viscosity limit4 with an uncertainty of 1.2%
η0 ) -1.55199xT + 2.92788T0.645731
η0 )
∑ i)1
xiη0,i
φi,j )
[ ( )( ) ] [ ] Mw,j 0.25 Mw,i 0.5 M w,i 4 1+ Mw,j x2 0.5
299-399 299-399 299-399 300-400 296-399 299-399
1-67 1-63 1-68 1-71 1-58 1-67
0.90 0.75 0.41 1.42 0.86 1.14
1.84 1.55 1.46 1.97 2.06 2.07
0.82 0.67 0.35 1.51 0.82 1.06
1.68 1.40 1.29 2.12 2.06 1.93
NP is the number of points.
Figure 1. Performance of the PR f-theory model for viscosity prediction of hythane mixture 1 (s) along with the experimental points13 (b).
evaluate the performance of the investigated viscosity models, the following quantities are defined:
Deviationi )
xjφi,j
where
η0,i 1+ η0,j
a
56 53 56 33 76 59
PR f-theory SRK f-theory P range AAD MxD AAD MxD [bar] % % % %
(15)
n
∑ j)1
mixture 113 mixture 213 mixture 313 mixture 413 hydrogen13 natural gas13
NPa
T range [K]
(14)
where η0 is in µP and the temperature in K. At low pressure and constant temperature, gas mixtures composed of hydrogen and hydrocarbons show a viscosity maximum as a function of the composition. This kind of behavior is normal in the case of gas mixtures composed of compounds with large differences in size and shape. For the dilute gas viscosity limit, the mixing rule based on a simplification of the kinetic gas theory proposed by Wilke12 is capable of describing this kind of behavior. The Wilke mixing rule for the dilute gas viscosity is written as n
Table 3. Performance of the f-theory Model for Viscosity Predictions of the Hythane Mixtures, Hydrogen, and Natural Gas
(16)
This mixing rule is of a predictive nature because no information about the mixture properties is required; only the dilute gas viscosity, the molecular weight, and the mole fraction of the pure components are needed. Viscosity Prediction of Hythane Recently, Nabizadeh and Mayinger13 measured the viscosity of four hythane mixtures in the temperature range 298-400 K and from 1 to 71 bar using an oscillating-disk viscometer with an experimental uncertainty of (1.0%. The viscosities of hydrogen and the natural gas, used to prepare the hythane mixtures, were also measured within the same temperature and pressure conditions. The compositions of these four hythane mixtures and the natural gas are given in Table 2. Thus, the viscosities of the four hythane mixtures, as well as the related hydrogen and natural gas, are predicted using the procedure outlined above in conjunction with the PR and SRK EOSs, respectively. To
(17)
NP
∑|Deviationi| NP i)1
(18)
MxD ) Max |Deviationi|
(19)
AAD )
2
1
ηcalc,i - η exp,i η exp,i
where NP is the number of experimental points, ηexp the experimental viscosity, and ηcalc the calculated viscosity. The absolute average deviation (AAD) and the maximum absolute deviation (MxD) indicate how close the calculated values are to the experimental values. The obtained AAD and MxD for the viscosity predictions of the four hythane mixtures, hydrogen, and the natural gas are given in Table 3. The performance of the PR f-theory procedure is shown in Figures 1-4 for the four hythane mixtures. It should be remarked that, in the cases studied in this work, the dilute gas viscosity is the main contribution to the total viscosity. The maximum contribution of the viscosity in excess of the dilute gas limit is on the order of 10% compared to the total viscosity. Overall, as indicated by the results reported in Table 3, the obtained AADs with the PR and SRK f-theory models are within or close to the uncertainty reported for the experimental values. The slightly higher deviation that mixture 4 shows is related to a small overprediction of the dilute gas limit. However, if the dilute gas viscosity of mixture 4 is reduced by 1%, an AAD of 0.53% and an MxD of 0.96% are obtained
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Figure 5. Deviation between PR f-theory predicted viscosities of hythane and reported values at 1 bar: (O) mixture 1, (+) mixture 2, (-) mixture 3, (×) mixture 4. Figure 2. Performance of the PR f-theory model for viscosity prediction of hythane mixture 2 (s) along with the experimental points13 (b).
Figure 3. Performance of the PR f-theory model for viscosity prediction of hythane mixture 3 (s) along with the experimental points13 (b).
of 1.39%, and the highest deviation is found for mixture 4, or at the highest temperatures. These results demonstrate that the simple mixing rule proposed by Wilke12 can be used to estimate the dilute gas viscosity limit of hythane, because the obtained AAD % is within the uncertainty of the pure compound dilute gas viscosity estimations. It should also be stressed that, because of the low viscosity values of hythane, a difference of only 1 µP can easily correspond to a deviation of 1-1.5%. As a comparison, on the basis of the assumption that the hythane mixtures correspond to binary mixtures composed of hydrogen and natural gas, Nabizadeh and Mayinger13 modeled their measured hythane viscosities at 1 bar using the procedure proposed by Hirschfelder et al. (p 530).14 After fitting the characteristic binary parameters in the Chapman-Enskog theory, σ12 and 12, to the experimental data, Nabizadeh and Mayinger obtained a mean deviation of (0.3% with an MxD of 0.5%. However, such an approach cannot be considered predictive because the fitted binary parameters correspond to hythane mixtures prepared from the specific used natural gas and can only be applied to these specific mixtures. On the other hand, although better results for the mixture dilute gas viscosity can be obtained by directly modeling the experimental data at low pressure, the results obtained in this work are satisfactory given their predictive nature. Conclusion
Figure 4. Performance of the PR f-theory model for viscosity prediction of hythane mixture 4 (s) along with the experimental points13 (b).
with the PR f-theory, while the SRK f-theory gives an AAD of 0.61% and an MxD of 1.12%. In Figure 5, the deviations between the predicted viscosities using the PR f-theory model and the reported values at 1 bar are shown. At this pressure the friction viscosity contribution in excess of the dilute gas viscosity is less than 0.5%. Therefore, Figure 5 mainly shows the performance of the Wilke mixing rule12 used to predict the dilute gas viscosity of the studied hythane mixtures. The overall AAD for all mixtures is 0.78% with an MxD
In this work the friction theory approach for viscosity modeling2-4 has been extended to viscosity prediction of hythane. A scheme has been introduced by combining the simple mixing rule of Wilke12 for the dilute gas viscosity with three different f-theory models for the friction viscosity term. This can be done because the mixture friction coefficients are linked to mixing rules based on the friction coefficients of the pure components, provided that the same EOS is used. Because the main components in hythane are hydrogen and methane, an efficient and accurate scheme for predicting the viscosity of hythane is achieved using a three-constant f-theory model for both hydrogen and methane together with the general one-parameter f-theory model for the remaining components. On the basis of this scheme, the viscosities of four hythane mixtures have been predicted with the SRK and PR EOSs. The obtained AAD ranges from 0.4% to 1.5% and is within or close to the experimental uncertainty. Although the use of more complex models may deliver better results, the mixing rule proposed by Wilke,12 in conjunction with simple models for the pure dilute gas viscosities, and the linear mixing rules for
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the friction coefficients used in this work give satisfactory results for hythane. The scheme introduced in this work for the viscosity prediction of hythane is totally predictive, because only properties and parameters of the pure compounds are required. Furthermore, the use of the PR and SRK EOSs for nonpolar light gases is well-known to deliver satisfactory density estimations for most industrial applications. In fact, for the studied mixtures in this work, the density predictions are within an AAD of 1.2%. Therefore, the approach described in this work can deliver viscosity and density estimations within a satisfactory accuracy. Acknowledgment This work is accomplished within the European Project EVIDENT under the JOULE Program, Contract No. JOF3-CT97-0034. List of Symbols Latin Letters Fc ) defined in eq 13 ka ) attractive friction constant in eq 4 kr ) repulsive friction constant in eq 4 krr ) quadratic repulsive friction constant in eq 4 Mw ) molecular weight [g/mol] P ) pressure [bar] Pc ) critical pressure [bar] pa ) attractive pressure term pr ) repulsive pressure term R ) gas constant T ) temperature [K] Tc ) critical temperature [K] T* ) defined in eq 12 vc ) critical molar volume [cm3/mol] xi ) mole fraction of component i Greek Letters Γ ) defined in eq 9 η ) viscosity η0 ) dilute gas viscosity ηc ) characteristic critical viscosity ηf ) residual friction term κa ) linear attractive friction coefficient κr ) linear repulsive friction coefficient κrr ) quadratic repulsive friction coefficient
Ω* ) reduced collision integral, defined in eq 11 ω ) acentric factor ψ ) defined in eq 8
Literature Cited (1) Ze´berg-Mikkelsen, C. K.; Quin˜ones-Cisneros, S. E.; Stenby, E. H. Viscosity Predictions of Natural Gas Using the Friction Theory. Int. J. Thermophys. 2001, in press. (2) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. The Friction Theory (f-theory) for Viscosity Modeling. Fluid Phase Equilib. 2000, 169, 249. (3) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. One Parameter Friction Theory Models for Viscosity. Fluid Phase Equilib. 2001, 178, 1. (4) Ze´berg-Mikkelsen, C. K.; Quin˜ones-Cisneros, S. E.; Stenby, E. H. Viscosity Modeling of Light Gases at Supercritical Conditions Using the Friction Theory. Ind. End. Chem. Res. 2000, submitted for publication. (5) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (6) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (7) Vargaftik, N. B. Tables on the Thermophysical Properties of Liquids and Gases, 2nd ed.; Hemisphere Publishing Corp.: London, 1975. (8) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals Data Compilation; Hemisphere Publishing Corp.: New York, 1989. (9) Chung, T.-H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Generalized Multiparameter Correlation for Nonpolar and Polar Fluid Transport Properties. Ind. Eng. Chem. Res. 1988, 27, 671. (10) Chapman, S.; Cowling, T. G.; Burnett, D. The Mathematical Theory of Nonuniform Gases, 3rd ed.; University Printing House: Cambridge, U.K., 1970. (11) Neufeld, P. D.; Janzen, A. R.; Aziz, R. A. Empirical Equations to Calculate 16 of the Transport Collision Integrals Ω(l,s)* for the Lennard-Jones (12-6) Potential. J. Chem. Phys. 1972, 57, 1100. (12) Wilke, C. R. A Viscosity Equation for Gas Mixtures. J. Chem. Phys. 1950, 18, 517. (13) Nabizadeh, H.; Mayinger, F. Viscosity of Binary Mixtures of Hydrogen and Natural Gas (Hythane) in the Gaseous Phase. High Temp.-High Pressures 1999, 31, 601. (14) Hirschfelder, J. B.; Curtiss, C. F.; Bird, R. B. The Molecular Theory of Gases and Liquids, 4th ed.; John Wiley & Sons Inc.: New York, 1967; Chapter 8.
Received for review December 4, 2000 Revised manuscript received April 2, 2001 Accepted April 10, 2001 IE0010464
