3640
Ind. Eng. Chem. Res. 2009, 48, 3640–3648
Expanded Fluid-Based Viscosity Correlation for Hydrocarbons H. W. Yarranton* and M. A. Satyro Department of Chemical and Petroleum Engineering, UniVersity of Calgary
The viscosity of pure hydrocarbons was correlated using a simple function based on the fluid density. The correlation has three adjustable parameters, a compressed state density, Fs°, an empirical parameter, c2, that scales the viscosity response to fluid expansion, and another empirical parameter, c3, used to tune at pressures above 10 MPa. The inputs to the correlation are the fluid density, pressure, and low-pressure gas viscosity. The correlation fit experimental viscosities for 39 pure hydrocarbons including n-alkanes, branched alkanes, alkenes, cyclics, and aromatics, within experimental error over a broad range of temperatures and pressures. Heavy hydrocarbons such as mineral oils were also fit with an AAPRD of 2.7%. Binary mixture viscosities were predicted to within 10% using simple volumetric mixing rules. The method provides a single framework for liquid and vapor phases, is simple to implement, and is very fast computationally, making it ideal for incorporation into process and reservoir simulators. Introduction The calculation of viscosities is an important part of process and reservoir simulation where the correct calculation of pressure drops and heat transfer coefficients is paramount. When applied to computer simulation problems, correlations should have the following properties: 1) a small number of adjustable parameters to ensure a maximum of physical significance and predictable extrapolation behavior, 2) continuity of vapor and liquid values across the critical point of the solution, 3) easily determined parameters from incomplete or estimated data, 4) speed. There are several methods available for the calculation of viscosities, some applicable only to some specific phases,1 some applicable to the vapor and liquid phases using a corresponding states principle,2-6 empirical residuals,7,8 or newer approaches based on factoring a contribution to viscosity coming from a dilute gas term and a contribution from a friction term.9 The empirical residuals method is popular in reservoir simulation due to its simple formulation and speed, as well as its ability to model pure components and mixtures using the same basic formulation. The method is based on the accuracy of density predictions and when coupled with different equations of state different results are produced, and empirical fine-tuning is frequently necessary. Corresponding states methods are used by several process simulators and are useful for the prediction of mixtures of relatively simple hydrocarbons. With appropriate empirical modifications to the calculation of the density and temperature shape parameters, the method can also be used to compute the viscosity of complex fluids.10 The method can be extended to mixtures via the definition of mixing rules for the shape parameters, including the possibility of introducing interaction parameters for the modeling of complex viscosity behavior that cannot be estimated using simple mixing rules. The F-Theory of Quinones-Cisneros and co-workers9 is an interesting new development, and it is conceptually easier to use than a corresponding states method where, depending on the formulation, an iterative procedure for the shape factors may be necessary. Unfortunately, the F-Theory formulation uses a significant number of parameters for the characterization of a fluid, equal to 5 in the original method. Therefore, a significant amount of * To whom correspondence should be addressed. E-mail: hyarrant@ ucalgary.ca.
experimental data is necessary for the proper determination of the parameters necessary for the use of the model. No meaningful study was available for the behavior of the model with incomplete data such as normally found in process and reservoir simulation. Corresponding states methods or the F-Theory model involve complex calculations for the determination of its adjustable parameters, normally coupled with the computation of volumetric roots of equations of state. In addition, the calculation of actual viscosities is also dependent on the calculation of volumetric roots of equations of state and complex equations for the corresponding states or friction term. As a result, these methods can be computationally intensive for process and reservoir simulation. Many of the above methods were developed and tested on pure hydrocarbons and light petroleum fluids. However, with the increasing development of heavy oil and bitumen resources, a reliable viscosity correlation for heavy petroleum is also required. Both thermal- and solvent-based processes have been used or are being considered to recover, process, and transport heavy oils.11 Hence, a viscosity correlation is required that can extend to light hydrocarbons, conventional crude oil, heavy oils, and their mixtures as a function of temperature and pressure with a minimal amount of experimental data. Our goal in the course of this study was to develop a simple correlation based on the fact that viscosities correspond to densities and use this idea for the correlation of a large amount of experimental data. The constraints were: 1) to develop model parameters that have a simple physical interpretation, 2) to have simple mixing rules that provide reasonable predictive capabilities even without adjustable interaction parameters, and 3) to be computationally efficient and directly applicable to commercial simulators. A second goal was to ensure that the correlation could be used to estimate the viscosities of heavy oils, mixtures of heavy oils, and mixtures of heavy oils and solvents. Development of the Correlation. The general principle behind the correlation is very simple: as a fluid expands there is greater distance between molecules and its fluidity increases. Eyring12 attempted to relate this change in fluidity to “holes” in the fluid that increase in size as the fluid expands. Whereas his results were more theoretically based, they are difficult to generalize and are known to be incorrect from an actual fluid structure point of view.13 Corresponding states methods provide an empirical relationship between fluid expansion and viscosity
10.1021/ie801698h CCC: $40.75 2009 American Chemical Society Published on Web 03/06/2009
Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3641
but use shape factors to develop a correspondence between a reference fluid and the desired fluid.2 In the proposed correlation, the compressed state is used as a reference instead. The fluidity was assumed to be an exponential function of the expansion of a fluid from a near-solid state, based on an original idea put forth by Hildebrand:14
[ {( ) } ]
1 ) f exp µ
Fs* F
n
-1 -1
(1)
where µ is the viscosity (inverse of fluidity) and F is the density. Fs* is the density beyond which the fluid cannot be compressed without incurring a solid-liquid phase transition, and n is an empirical exponent that was found to improve the fit near the critical region. This form of correlating parameter was chosen so that, as the density approaches the compressed state, the fluidity approaches zero, and, as the density approached zero, the fluidity approaches infinity. The correlation is recast as a viscosity departure-like function as follows:
[
1
µ - µG ) f
{( ) }
exp
Fs* F
n
-1 -1
]
Table 1. Model Parameters and Correlating Properties for n-Alkanes component
Fs° (kg/m3)
c2
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-decane n-dodecane n-tetradecane n-hexadecane n-eicosane n-tetracosane
540 724 778 813 837 849.1 857.8 862.7 868.1 871.4 875.5 878.6 885.5 900
0.100 0.156 0.174 0.190 0.198 0.205 0.213 0.221 0.236 0.249 0.265 0.278 0.306 0.307
c3 (×10 kPa-1) 6
0 0 0.10 0.15 0.18 0.18 0.17 0.17 0.20 0.22 0.24 0.28
viscosity at 25 °C (mPa · s)
Fs°/Ftp
0.011 0.035 0.083 0.16 0.22 0.29 0.39 0.51 0.85 1.3 2.1 3 6.2 6.5
1.20 1.10 1.05 1.10 1.10 1.11 1.11 1.13 1.13 1.13 1.13 1.13 1.14 1.13
(2)
where µG is the gas viscosity. Now, when the density approaches zero, the viscosity approaches the gas low-pressure viscosity. When the density approaches the compressed state density, the viscosity goes to infinity. To determine the form of the function, viscosity data for n-alkanes were plotted versus preliminary estimates of the correlating parameter. The following function was observed to fit the form of the data: µ - µG ) c1(exp{c2β} - 1)
(3)
where c1 and c2 are the fit parameters and β is the correlating parameter given by: β)
1
{( ) }
exp
Fs* F
n
(4)
-1 -1
Fs* )
exp(-c3P)
The parameters used to fit this data set are provided in Table 1. The following two parameters were fixed for all hydrocarbon components: n ) 0.65
(dimensionless)
c1 ) 0.165
This form of correlation was found to be adequate at atmospheric pressures over a moderate range of temperatures. The correlation was improved by introducing a pressure dependence into the compressed state density as follows: Fso
Figure 1. Relationship between parameter c2 and viscosity at 25 °C.
(5)
where Fs° is the compressed state density in a vacuum and c3 is a fitting constant. The correlation has three required inputs, the fluid density, the pressure, and the gas viscosity, plus five parameters. Fs°, n, c1, c2, and c3. The next step in developing the correlation is to fix as many parameters as possible. Values of Parameters Based on n-Alkane Data. The correlation was fitted to n-alkane data from the NIST database.15 The data set includes densities and viscosities for the saturated and compressed liquid. Because the densities and viscosities were not measured at the same conditions, the saturated liquid density data was curve fit to compute viscosities at the same conditions as the measured data. It was not practical to do so for the compressed liquid data, and therefore an error analysis was only performed for the saturated liquid data.
(dimensionless)
(6) (7)
Parameter c2 was found to correlate to the viscosity at 25 °C as follows: c2 ) 0.241µ0.13 25C
(dimensionless)
(8)
where µ25C is the viscosity at 25 °C in mPa · s. The correlation held for all hydrocarbons investigated here except complex aromatics, as shown in Figure 1. The final parameter roughly correlated to molar mass as follows: c3 ) 1.68·10-8M0.5
(1/kPa)
(9)
where M is the molar mass (g/mol). Note that eq 9 did not predict the c3 parameter very accurately for many hydrocarbons, as shown in Figure 2. This parameter accounts for the pressure effect, and therefore caution is advised when using eq 9 at pressures above 10 MPa. The compressed state densities were approximately 10 to 15% greater than the triple point densities, as shown in Figure 3 and Table 1. However, the deviations were too significant for use in the correlation. Hence, the compressed state density must be fitted for each pure component, and at
3642 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 Table 2. Summary of Correlation Error for Saturated Liquid Viscosity of n-Alkanes
Figure 2. Relationship between parameter c3 and molar mass.
Figure 3. Relationship between compressed state density and triple point density.
least one liquid viscosity data point is required for any given fluid. A least-squares method was used to find the value of the compressed state density that minimized the following objective function: OF )
∑
[ { }] ln
µcorr µmeas
2
(10)
where subscript corr indicates the predicted value from the correlation and subscript meas is the measured value. Note, a minimum of one viscosity is required to determine a value for the compressed state density for a given component.
component
number of data points
AAD mPa · s
MAD mPa · s
AAPRD
MAPRD
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-decane n-dodecane n-tetradecane n-hexadecane n-eicosane n-tetracosane ALL
22 54 73 47 70 235 191 132 136 151 51 64 34 13 1273
0.015 0.020 0.056 0.017 0.036 0.009 0.015 0.015 0.010 0.012 0.023 0.024 0.035 0.099 0.019
0.032 0.11 0.29 0.17 0.37 0.16 0.54 0.21 0.064 0.097 0.097 0.097 0.12 0.23 0.54
16 5.9 4.4 3.1 3.0 2.9 2.7 1.7 1.5 1.2 1.7 1.5 2.4 5.4 2.7
39 22 20 9.3 13 12 13 7.9 9.9 7.9 11 5.0 13 13 39
To illustrate the performance of the correlation, the correlated viscosities are compared with the data for propane and n-decane in Figures 4 and 5, respectively. Note, measured densities (and therefore correlated viscosities) were only available up to 40 MPa for propane even though viscosity data were available up to 80 MPa. The average absolute deviation (AAD), maximum absolute deviation (MAD), average absolute percent relative deviation (AAPRD), and maximum absolute percent relative deviation (MAPRD) are provided in Table 2. In all cases, the correlation fit the measured viscosities to within the accuracy of the data both for saturated and compressed liquid data. Note that the apparent jumps and discontinuities in the correlated viscosities in these and later figures are a result of scatter in the density data, which was not smoothed prior to applying the correlation. Also note that the units used here are mPa · s for viscosity, kg/m3 for density, Kelvin for temperature, and kPa for pressure. Testing the Correlation on Pure Hydrocarbons. The next step was to test the correlation on pure hydrocarbons. Common hydrocarbons and hydrocarbons for which mixture data were available were selected including branched alkanes, alkenes, aromatics, and cyclics. All of the data were obtained from the NIST database.15 In all cases unless otherwise noted, parameters n, c1, and c2 were determined from eqs 6 to 8, respectively. For each component, the compressed state density, Fs°, was used to fit the saturated liquid viscosity. Parameter c3 was used to fit the compressed liquid viscosities at pressures above 10 MPa. The correlation parameters for branched alkanes, alkenes, cyclics, and simple aromatics are given in Table 3. The AAD,
Figure 4. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) propane. Symbols are data;15 lines are the correlation. Note that the correlation was only applied where density data was available.
Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3643
Figure 5. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) n-decane. Symbols are data;15 lines are the correlation. Note that the jumps in the correlated viscosities in this and other figures result from the scatter in the density data, which was not smoothed prior to applying the correlation. Table 4. Summary of Correlation Error for Saturated Liquid Viscosity of Branched Alkanes, Alkenes, Cyclics, and Aromatics
Table 3. Model Parameters and Correlating Properties for Branched Alkanes, Alkenes, Cyclics, and Aromatics component 3-methylpentane 2,2-dimethylpentane 2-methylhexane 2,2-dimethylhexane isooctane 2,6,10,15,19,23hexamethyltetracosane propylene 1-hexene 1-octene cyclopentane cyclohexane methylcyclohexane cyclooctane decalin benzene toluene o-xylene p-xylene ethylbenzene
Fs° (kg/m3)
c2
858.5 850.1 861.2 856.8 855.4 893.0
0.206 0.214 0.210 0.221 0.218 0.372
815 880 885 930 922.1 926 957 1010 1066.4 1056.0 1052.9 1045.5 1052.0
0.176 0.203 0.219 0.215 0.237 0.229 0.269 0.270 0.226 0.223 0.232 0.226 0.227
c3 ( × 106 kPa-1)
viscosity at 25 °C (mPa · s)
1.04 1.07 1.10
0.15 0.26
0.3 0.4 0.35 0.52 0.47 28
0.18 0.165 0.155 0.09 0.135 0.14 0.14 0.14 0.14
0.09 0.27 0.48 0.42 0.89 0.67 2.3 2.4 0.60 0.55 0.75 0.61 0.63
1.07 1.09 1.08 1.11 1.12 1.04 1.14 1.19 1.09 1.15 1.19 1.08
Fs°/Ftp
1.06
MAD, AAPRD, and MAPRD are summarized in Table 4. In all cases, the correlation fit the data within experimental error. To illustrate, the correlated viscosities for isooctane, cyclohexane, and toluene are compared with the data in Figures 6, 7, and 8, respectively. The correlation was also tested on more complex aromatics including styrene, naphthalene, 1-methylnapthalene (Figure 9), 2-methylnaphthalene, phenanthrene, and tetralin. The correlation parameters for these hydrocarbons are given in Table 5. The AAD, MAD, AAPRD, and MAPRD are summarized in Table 6. The values of parameter c2 from eq 8 did not provide the best fit to the data. Instead, the following correlation provided a better fit to the data:
component 3-methylpentane 2,2-dimethylpentane 2-methylhexane 2,2-dimethylhexane isooctane 2,6,10,15,19,23hexamethyltetracosane propylene 1-hexene 1-octene cyclopentane cyclohexane methylcyclohexane cyclooctane decalin benzene toluene o-xylene p-xylene ethylbenzene ALL
number of AAD MAD AAPRD MAPRD data points mPa · s mPa · s 6 9 25 7 74 9
0.011 0.014 0.002 0.028 0.012 1.12
0.017 0.041 0.018 0.044 0.036 2.54
3.5 3.5 6.6 4.6 2.4 4.9
5.2 10 0.7 7.4 6.8 11
51 7 7 80 188 80 13 11 310 347 86 108 159 1577
0.14 0.002 0.006 0.004 0.012 0.015 0.028 0.041 0.010 0.016 0.016 0.009 0.011 0.023
2.0 0.002 0.011 0.015 0.11 0.25 0.055 0.11 0.066 0.29 0.077 0.050 0.046 2.54
4.6 0.6 1.5 0.9 1.7 2.1 1.7 2.2 2.0 2.3 2.6 1.8 2.1 2.2
21 1.0 2.2 3.3 17 8.9 4.9 4.6 8.9 11 7.0 10 7.0 21
Mixtures of Hydrocarbons. Volumetric mixing rules are proposed to extend the correlation to mixtures as follows: m
o ) Fs,mix
∑φF
o i si
m
c2,mix )
∑φc
i 2i
(dimensionless)
(11)
The previous correlation was developed for components where dispersion forces dominate the interaction between the molecules. With more complex aromatics, π-π bonding may also be present. Because the parameter c2 sets the proportionality of viscosity to fluid expansion, it is expected to be sensitive to the type of intermolecular forces in the hydrocarbon.
(13)
i)1
c3,mix )
1
(14)
m
∑ i)1
c2 ) 0.20µ0.115 25C
(12)
i)1
φi
/c3i
where i is the pure component, m is the number of components, and φ is the volume fraction. Recall that n and c1 are identical for all pure hydrocarbon components and therefore no mixing rule is required. The model was also tested on binary mixtures of hydrocarbons. The data was obtained from Chevalier et al. 16 and includes n-alkanes, branched alkanes, cyclics, and aromatics. The binary mixtures considered in this study are summarized in Table 7.
3644 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009
Figure 6. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) isooctane. Symbols are data;15 lines are the correlation.
Figure 7. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) cyclohexane. Symbols are data;15 lines are the correlation. Note that the correlation does not extend to the full range of the viscosity data when density data is not available.
Figure 8. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) toluene. Symbols are data;15 lines are the correlation.
When necessary, the model was tuned to exactly fit the pure component viscosities reported in the source data so that the only deviation from the predicted mixture viscosity would result from the mixing rules. In general, very good predictions were obtained for similar components such as a pair of n-alkanes or a pair of aromatics, as shown in Figure 10. Not surprisingly, less satisfactory predictions were obtained for dissimilar components such as an n-alkane and a branched alkane or an n-alkane and an aromatic, as shown in Figure 11. Nonetheless, the predictions are all within 10% of the measured value, as shown in Table 8 and Figure 12, and are in most cases superior
to an ideal volumetric mixing rule given below and shown on Figures 10 and 11: m
ln(µmix) )
∑ φ ln(µ ) i
i
(15)
i)1
It was possible to better fit the data for binary mixtures of dissimilar components using a nonideality term of the following form: ln(µmix) ) [ln(µmix)]corr - φiφjkij
(16)
Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3645
Figure 9. Comparison of proposed correlation with viscosity of saturated (left) and compressed (right) 1-methylnaphthalene. Symbols are data;15 lines are the correlation. Table 5. Model Parameters and Correlating Properties for Complex Aromatics viscosity c3 at 25 °C Fs°/Ftp 6 -1 (×10 kPa ) (mPa · s)
component
Fs° (kg/m3)
c2
styrene naphthalene 1-methylnaphthalene 2-methylnaphthalene phenanthrene tetralin
1058 1141 1138 1130 1228 1087
0.191 0.211 0.225 0.222 0.256 0.217
0.14 0.16
0.66 1.6 2.8 2.5 8.8 2.0
1.11 1.17 1.08 1.14 1.15 1.07
Table 7. Binary Mixtures Evaluated in This Study component 1 n-hexane
n-octane
Table 6. Summary of Correlation Error for Saturated Liquid Viscosity of Complex Aromatics component styrene naphthalene 1-methylnaphthalene 2-methylnaphthalene phenanthrene tetralin ALL
number of AAD MAD AAPRD MAPRD data points mPa · s mPa · s 37 18 108 28 4 69 264
0.008 0.035 0.054 0.033 0.014 0.030 0.037
0.017 0.066 0.41 0.096 0.017 0.013 0.41
1.5 15 2.8 2.4 2.7 3.2 3.5
3.5 26 12 6.3 5.1 7.0 26
where i and j are the components in the binary mixture, [ln(µmix)]corr is the viscosity predicted from the correlation and kij is a binary interaction parameter for viscosity. However, the correlation must then be fitted to each binary mixture and the predictive value is lost. A larger data set is required to determine if the viscosity interaction parameters can be predicted for different components. Heavy Hydrocarbons and Crude Oils. Heavy Hydrocarbons. The viscosities and densities of two series of viscosity standards were obtained from Cannon Instruments.17 The data set included 39 standards with viscosities ranging from 0.3 to 100 000 mPa · s at temperatures from 20 to 100 °C, all at atmospheric pressure. The viscosity standards are mixed hydrocarbons (low viscosity), petroleum based mineral oils, synthetic base oils, and hydrocarbon polymers (high viscosity), but no other information was available on their chemical composition. The correlation was fitted to the data by adjusting both the compressed state density and the c2 parameter. Examples of the fitted viscosity data are shown in Figure 13. The c3 parameter was set to zero because all of the data was at low pressure. Parameters n and c1 were not adjusted. The values of Fs° and c2 used to fit each standard are provided in Table 9. A dispersion plot is shown in Figure 14. The AAPRD and MAPRD for the 263 data points are 2.7% and 17%, respectively.
n-decane
component 2
component 1
n-heptane n-decane n-dodecane n-tetradecane n-hexadecane isooctane methylcyclohexane o-xylene n-decane n-tetradecane n-hexadecane
benzene
cyclohexane o-xylene p-xylene
toluene
methylcyclohexane o-xylene
n-tetradecane n-hexadecane 3-methylepentane 2,2-dimethylpentane 2,2-dimethylhexane isooctane methylcyclohexane benzene toluene o-xylene n-tetradecane n-hexadecane 2-methylhexane isooctane toluene o-xylene n-hexadecane 3-methylepentane 2,2-dimethylpentane 2,2-dimethylhexane cyclohexane methylcyclohexane
o-xylene
component 2
p-xylene cyclohexane p-xylene
The fitted values of parameter c2 are compared with eqs 8 and 11 in Figure 15. The data for the mixed hydrocarbons (HC) and the synthetic base oils (SBO) are consistent with eq 8 whereas, the data for mineral oils (MO) and hydrocarbon polymers (HCP) are consistent with eq 11. The trends suggest that the mixed hydrocarbons and synthetic base oils are dispersion force dominated, whereas additional intermolecular forces may be at play in mineral oils and hydrocarbon polymers. Without more information about the chemistry of the viscosity standards, it is difficult to draw further conclusions. Heavy Oil and Bitumen. We conclude with two examples of diluted heavy crude oils. The first example is a heavy oil diluted with n-decane investigated by Barrufet and Setiadarma.18 The viscosity of the heavy oil was 2150 mPa · s at 22.5 °C and atmospheric pressure. Viscosities were measured from 22 to
3646 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009
Figure 10. Example of a good prediction of mixture viscosity; hexane and hexadecane at 25 °C and atmospheric pressure. Data from Chevalier et al.16
Figure 13. Example of measured and correlated viscosity of standards. Data from Cannon Instruments.17 Table 9. Model Parameters and Viscosity at 25 °C for Viscosity Standards17
Figure 11. Example of a poor prediction of mixture viscosity; decane and toluene at 25 °C and atmospheric pressure. Data from Chevalier et al.16 Table 8. Summary of Correlation Error for Binary Mixtures mixture n-alkane/n-alkane n-alkane/branched alkane n-alkane/cyclic n-alkane/aromatic aromatic/aromatic aromatic/cyclic ALL
number of AAD MAD AAPRD MAPRD data points mPa · s mPa · s 60 50 20 34 25 15 204
0.007 0.037 0.036 0.024 0.004 0.038 0.022
0.044 0.13 0.10 0.039 0.008 0.061 0.13
0.8 3.2 2.7 3.5 0.6 5.9 2.4
3.8 8.7 5.3 6.5 1.3 9.8 9.8
175 °C for the heavy oil, n-decane, and mixtures of 10.2, 26.2 and 52.9 wt % n-decane in the heavy oil, as shown in Figure 16.
a
Figure 12. Dispersion plot for the correlated viscosities of the 41 binary mixtures from Table 7. Data from Chevalier et al.16
standard
Fs° (kg/m3)
c2
viscosity at 25 °C (mPa · s)
N 0.4 N 0.8 N2 N4 N7.5 N10 N14 N26 N35 N44 N75 N100 N140 N250 N350 N415 N750 N1000 n1400 N2500 N4000 N5100 N10200 N15000 N18000 S3 S6 S20a S60 S200 S200a S600 S2000a S2000 S8000 S30000a
861.0 1046.4 870.8 874.8 886.7 956.7 891.0 899.1 935.9 906.3 916.8 951.2 921.8 924.8 919.9 931.1 936.6 938.1 946.0 951.9 947.5 951.9 962.0 967.2 970.7 964.6 959.2 935.7 944.6 915.6 950.3 932.5 959.8 941.8 960.1 970.7
0.248 0.220 0.255 0.260 0.300 0.260 0.320 0.360 0.320 0.395 0.460 0.385 0.505 0.540 0.515 0.590 0.650 0.670 0.750 0.850 0.560 0.590 0.670 0.700 0.740 0.235 0.250 0.305 0.350 0.475 0.400 0.600 0.470 0.525 0.630 0.740
0.4 0.6 2 4.5 9.5 16 20 37 56 71 105 200 250 480 600 830 1600 2000 3000 5100 11000 16000 32000 41000 57000 3.3 8 30 105 330 418 1100 1434 4700 20000 75320
Old standards (laboratory samples).
The correlation was fit to the heavy oil data using the parameters given in Table 10. Note that this is a discrepancy between the correlation and the data for the heavy oil at 22.5 °C. It is possible that the correlation is not adequate for such as large change of viscosity with temperature. However, the correlation worked well for other heavy oils over similar conditions, as shown in the next example. Also, the correlation performed well for the diluted oil at 22.5 °C. Therefore, it was concluded that the heavy oil viscosity at 22.5 °C was an outlier and it was not included in the fitting. The AAPRD and MAPRD of the fitted correlation were 9.0% and 17%, respectively. The
Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009 3647
Figure 14. Dispersion plot for the correlated viscosities of 39 viscosity standards. Data from Cannon Instruments.17
Figure 15. Fitted parameter c2 for viscosity standards (HC ) mixed hydrocarbons, MO ) mineral oil, SBO ) synthetic base oil, HCP ) hydrocarbon polymer) compared with values predicted for simple hydrocarbons (eq 8) and for complex aromatics (eq 11).
Figure 17. Viscosity of a Cold Lake bitumen diluted with toluene; data from Eastick19 and Mehrotra.20 The oil viscosities were fitted and the mixture viscosities were predicted.
The viscosities of the mixtures of heavy oil and n-decane were then predicted using the mixing rules given in eqs 12 to 14. The AAPRD and MAPRD were 18% and 45%, respectively. Whereas these errors seem high, the predictions appear to be within the accuracy of the data, which had scatter in the viscosity measurements as high as 30%. The measured and predicted viscosities are compared in Figure 16. The second example is a Cold Lake bitumen diluted with toluene investigated by Eastick19 and Mehrotra.20 The viscosity of the bitumen was 54 500 mPa · s at 23.9 °C and atmospheric pressure. Viscosities were measured from 23 to 121 °C for the bitumen, toluene, and mixtures of 1.61, 4.71 and 5.29 and 9.55 wt % toluene in the heavy oil, as shown in Figure 17. Note that the data at 5.29 wt % toluene was very close to data at 4.71 wt % and is not included on the plot. Density was only reported at 23.9 °C and was determined at other temperatures based on data from other heavy oils and bitumens21 using the following relationship: FCL ) 1010 - 0.63T
Figure 16. Viscosity of a heavy oil diluted with n-decane; data from Barrufet and Setiadarma.18 The oil and the n-decane viscosities were fitted and the mixture viscosities were predicted. Table 10. Model Parameters and Viscosity at 25°C for a Heavy Oil18 and Cold Lake Bitumen19,20 standard
Fs° (kg/m3)
c2
measured viscosity at 23 °C (mPa · s)
predicted viscosity at 25 °C (mPa · s)
heavy oil Cold Lake bitumen
984.0 1055.9
0.32 0.51
2150 55 400
4700 45 800
densities and viscosities of n-decane reported by the authors deviated from those in the NIST database. The data was used as reported and the value of Fs° was adjusted from 868.1 to 852 kg/m3 to fit the data.
(17)
where FCL is the density of the Cold Lake bitumen in kg/m3 and T is temperature in °C. The correlation was fit to the bitumen data using the parameters given in Table 10. The AAPRD and MAPRD of the fitted correlation were 2.1% and 7.9%, respectively. The parameters for toluene were not adjusted. The viscosities of the mixtures of bitumen and toluene were then predicted using the mixing rules given in eqs 12 to 14, as shown in Figure 17. The AAPRD and MAPRD were 17% and 47%, respectively. The poorest predictions were at the lowest temperatures. Fulem at al. 22 has observed changes in heat capacity and viscosity trends at temperatures below 50 °C. The nature of these changes has not yet been established. A solid-liquid phase transition may occur in the heavy components of the bitumen or there may be structure changes in the fluid related to asphaltene self-association. Asphaltene association is known to increase at lower temperatures.23 The correlation was fitted to the bitumen data and may implicitly include these transitions. If these transitions are different when the bitumen is diluted with toluene, the correlation will not account for the difference. In other words, the correlation does not apply when phase or structural changes occur and it should be used with caution for heavy oils at lower temperatures. At this stage, the compressed state density and the c2 parameter cannot be predicted for a heavy hydrocarbon. For example, the c2 parameters used for the heavy oil and the bitumen are compared with the c2 parameters used for the heavy
3648 Ind. Eng. Chem. Res., Vol. 48, No. 7, 2009
hydrocarbons in Figure 15. The crude oil c2 parameters are significantly lower than those of the heavy hydrocarbons. The deviation may result from the low-temperature transitions in heavy oil, discussed previously. Alternatively, heavy oils contain heteroatoms, which may deviate from correlations based on pure hydrocarbon. Whatever the reason, at least some viscosity data is required to tune the correlation to heavy oil data. Once tuned, the correlation can be used to predict the viscosity of diluted heavy oils. The method can be used to extend the usefulness of limited experimental data by providing a simple correlating platform that captures changes in viscosity from temperature, pressure, and composition.
Conclusions Hydrocarbon viscosity was correlated to density based on an expanded fluid concept. The inputs to the correlation are the fluid density, the pressure, and the low-pressure gas viscosity. The correlation has three adjustable parameters. a compressed state density, Fs°, an empirical parameter, c2, that scales the viscosity response to fluid expansion, and another empirical parameter, c3, used to tune at pressures above 10 MPa. For pure hydrocarbons, the c2 parameter correlated to the viscosity at 25 °C. The c3 parameter correlated approximately to the molar mass. The correlated viscosity was very sensitive to the compressed state density, and therefore this parameter could not be correlated with sufficient accuracy and was fitted for each component. If the parameter correlations are used, the model has four inputs: fluid density, pressure, low pressure gas viscosity, and liquid viscosity at 25 °C; and only one adjustable parameter, the compressed state density. The correlation fit experimental viscosities for 39 pure hydrocarbons including n-alkanes, branched alkanes, alkenes, cyclics, and aromatics, within experimental error over a broad range of temperatures and pressures. Heavy hydrocarbons such as mineral oils were also fit with an AAPRD of 2.7%. Binary mixture viscosities were predicted to within 10% using simple volumetric mixing rules. The viscosities of a heavy oil and a bitumen over a range of temperatures were fit with the correlation with an AAPRD of 9% and 2%, respectively. The viscosity of the heavy oil diluted with n-decane was predicted to within the accuracy of the data. The viscosity of the bitumen diluted with toluene was also predicted to within the accuracy of the data except possibly at temperatures below 50 °C. There is evidence of structural or phase transitions in bitumens at low temperature and these would not be accounted for in the correlation. The main advantage of the expanded fluid-based correlation is its simplicity and a relatively small number of adjustable parameters. Its main disadvantage is the need for accurate density data and at least one viscosity measurement. The dependence on density data can potentially be eliminated by retuning the correlation based on densities predicted with an equation of state.
Acknowledgment The authors thank Mr. Florian Schoeggl for locating the data on viscosity standards and Dr. Anil Mehrotra for providing the data on Cold Lake bitumen. Literature Cited (1) Poling, B. E.; Prausnitz, J. M., O’Connell, J. P. The Properties of Gases and Liquids, 5th Ed.; McGraw-Hill: New York, 2000. (2) Ely, F. J.; Hanley, H. J. M. Prediction of Transport Properties. 1. Viscosity of Fluids and Mixtures. Ind. Eng. Chem. Fundam. 1981, 20, 323. (3) Baltatu, M. Prediction of the Liquid Viscosity for Petroleum Fractions. Ind. Eng. Chem. Process Des. DeV. 1982, 21, 192–195. (4) Twu, C. H. Generalized Method for Predicting Viscosities of Petroleum Fractions. AIChE J. 1986, 32 (12), 2091–2094. (5) Aasberg-Petersen, K.; Knudsen, K.; Fredenslund, A. Prediction of Viscosities of Hydrocarbon Mixtures. Fluid Phase Equilib. 1991, 70, 293– 308. (6) Huber, M. L.; Ely, J. F. Prediction of Viscosity of Refrigerants and Refrigerant Mixtures. Fluid Phase Equilib. 1992, 80 (4), 239–248. (7) Lohrenz, J.; Bray, B. G.; Clark, C. R. Calculating Viscosities of Reservoir Fluids from their Compositions. J. Pet. Technol. 1964, 1171– 1176. (8) Letsou, A.; Stiel, L. I. Viscosity of Saturated Nonpolar Liquids at Elevated Pressures. AIChE J. 1973, 19, 409–411. (9) Quinones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Friction Theory (F-Theory) for Viscosity Modeling. Fluid Phase Equilib. 2000, 169 (2), 249–276. (10) VMGSim Version 4.0; Virtual Materials Group, Inc.: Calgary, Alberta, Canada, 2008. (11) Thomas, S. Enhanced Oil Recovery - An Overview. Oil & Gas Sci. Technol. 2008, 63 (1), 9–19. (12) Eyring, H. Viscosity, Plasticity and Diffusion as Examples of Absolute Reaction Rates. J. Chem. Phys. 1936, 4, 283–291. (13) Jhon, M. S.; Eyring, H. Significant Liquid Structures; John Wiley and Sons: New York, 1969. (14) Hildebrand, J. H. Viscosity and DiffusiVity - A PredictiVe Treatment; John Wiley and Sons: New York, 1977. (15) NIST Standard Reference Database; NIST/TRC Source Database; WinSource, Version 2008.. (16) Chevalier, J. L. E.; Petrino, P. J.; Gaston-Bonhomme, Y. H. Viscosity and Density of some Aliphatic, Cyclic, and Aromatic Hydrocarbons Binary Mixtures. J. Chem. Eng. Data 1990, 35, 206–212. (17) Cannon Instruments website, www.cannoninstrument.com/HighVisc.htm, July 2008. (18) Barrufet, M. A.; Setiadarma. A. Reliable Heavy Oil - Solvent Mixing Rules for Viscosities up to 450 K, Oil-Solvent Viscosity Ratios up to 4 × 105, and Any Solvent Proportion. Fluid Phase Equilib. 2003, 213, 65–79. (19) R. R., Eastick Phase Behavior and Viscosity of Bitumen Fractions Saturated with CO2, M. S. Thesis, University of Calgary, 1989; p 30. (20) Mehrotra, A. Development of Mixing Rules for Predicting the Viscosity of Bitumen and its Fractions Blended with Toluene. Can. J. Chem. Eng. 1990, 68, 839–848. (21) Takamura, K.; Isaacs, E. E. Interfacial Properties. In AOSTRA Technical Handbook on Oil Sands, Bitumens, and HeaVy Oils; Hepler, L. G., His C., Eds.;AOSTRA Technical Publication Series #6: Edmonton, 1989, p 104. (22) Fulem, M.; Becerra, M.; Hasan, A.; Zhao, B.; Shaw, J. M. Phase Behaviour of Maya Crude Oil Based on Calorimetry and Rheometry. Fluid Phase Equilib. 2008, 272, 32–41. (23) Yarranton, H. W.; Alboudwarej, H.; Jakher, R. Investigation of Asphaltene Association with Vapour Pressure Osmometry and Interfacial Tension Measurements. Ind. Eng. Chem. Res. 2000, 39, 2916–2924.
ReceiVed for reView November 7, 2008 ReVised manuscript receiVed January 19, 2009 Accepted January 28, 2009 IE801698H
