Energy Fuels 2010, 24, 1762–1770 Published on Web 02/26/2010
: DOI:10.1021/ef9011565
Prediction of Waxy Oil Rheology by a New Model Ehsan Ghanaei and Dariush Mowla* School of Chemical and Petroleum Engineering, Shiraz University, Shiraz, Iran Received October 26, 2009. Revised Manuscript Received January 10, 2010
The formation of wax particles influences the rheological behavior of crude oils. Different studies show that crude oils have a Newtonian behavior above the wax appearance temperature (WAT) and shear thinning non-Newtonian behavior below it. In this work, a model based on a combination of the Richardson and Herschel-Bulkley models, together with the simplified Eyring equation, has been presented. The parameters of the model have been obtained as a function of wax content (precipitated wax as a function of temperature), API, and temperature, which is a priority for the new model. To validate the model, some experimental data related to North Sea oils has been applied. The experimental data points utilized to calculate model parameters (162) show an average absolute deviation (AAD%) of 18.45% from the experimental data. Also, to show the reliability of the new model, it has been evaluated by another experimental data including 118 data points related to kinematic and apparent viscosity of 14 oils. Totally, the new model has an AAD% of 19.47% for the 280 experimental data points applied in this work. Also, the new model can predict the trend of viscosity variation against temperature, shear rate, and wax content, in comparison with the experimental data.
thickness of wax layer deposition in petroleum pipelines.7-9 The studies justify the belief that the formation of wax particles changes the rheological behavior of crude oils from Newtonian to non-Newtonian. With regard to the investigation of the influence of solid wax formation on the rheological behavior of oil and thermodynamic modeling of wax precipitation, Roe nningsen et al.,10 Hansen et al.,11 and Pedersen et al.12-14 have performed some valuable studies. In 1991, Roenningsen et al.10 and Pedersen et al.12 measured the WAT and wax content for some samples of North Sea oils experimentally. In 2000, Pedersen and Roe nningsen performed viscosity measurement for some samples of North Sea oils in the Newtonian and shear-rate-dependent region and presented a model to predict the viscosity of oil at different temperatures and shear rates.14 Their model includes application of a combination of Casson15,16 and Richardson17 models, as follows: " # Hφ Iφ4 ð1Þ η ¼ ηliq expðGφÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðdVx =dyÞ ðdVx =dyÞ
1. Introduction Some crude oils contain substantial amounts of heavy components that form wax. These components are more n-alkanes and the lower amounts of iso-alkanes and cyclic alkanes with high molecular weight. The wax crystals formed from n-alkanes have needlelike and platelike morphologies, but the wax formed from iso-alkanes and cyclic alkanes is termed microcrystalline or amorphous wax. Wax precipitation may occur in cold regions of petroleum pipelines and may cause a variety of well-known problems.1-3 At a specified pressure, the wax particles deposit while the temperature decreases from the wax appearance temperature (WAT) or wax disappearance temperature (WDT). Another definition regarding the wax precipitation phenomenon is the wax content, which is the fraction of wax precipitated from the original fluid sample under specified temperature and pressure conditions. The amount of wax content is increased while the temperature is decreased. The prediction of WAT and wax content are performed based on thermodynamic models. Some of these studies performed recently have been reported in the literature.4-6 Another field of study of wax precipitation is the prediction of the rheological behavior of waxy crude oils. The investigation of the rheological behavior of oils is essential to obtain a velocity profile that is eventually applied in the other applications, such as prediction of the temperature profile and the
In the above equation, η is the apparent viscosity, ηliq the dynamic viscosity of liquid, dVx/dy the shear rate, and φ the (7) Mehrotra, A. K.; Bhat, N. V. Energy Fuels 2007, 21, 1277–1286. (8) Correra, S.; Fasano, A.; Fusi, L.; Merino-Garcia, D. Meccanica 2007, 42, 149–165. (9) Azevedo, L. F. A.; Teixeira, A. M. Pet. Sci. Technol. 2003, 21 (3), 393–408. (10) Ro e nningsen, H. P.; Bjorndal, B.; Hansen, A. B.; Pedersen, W. B. Energy Fuels 1991, 5, 895–908. (11) Hansen, A. B.; Larsen, E.; Pedersen, W. B.; Nielsen, A. B.; Roe nningsen, H. P. Energy Fuels 1991, 5, 914–923. (12) Pedersen, W. B.; Hansen, A. B.; Larsen, E.; Nielsen, A. B.; Roe nningsen, H. P. Energy Fuels 1991, 5, 908–913. (13) Pedersen, K. S.; Skovborg, P.; Ro e nningsen, H. P. Energy Fuels 1991, 5, 924–932. (14) Pedersen, K. S.; Ronningsen, H. P. Energy Fuels 2000, 14, 43–51. (15) Barry, E. G. J. Inst. Pet. 1971, 57, 74–85. (16) Ro e nningsen, H. P. J. Pet. Sci. Eng. 1992, 7, 177–213. (17) Richardson, E. G. Kolloid-Z. 1933, 65 (1), 32–37.
*Author to whom correspondence should be addressed. Tel.: þ98711-234-3833. Fax: þ98-711-628-7294. E-mail:
[email protected]. (1) Musser, B. J.; Kilpatrick, P. K. Energy Fuels 1998, 12, 715–725. (2) Dirand, M.; Chevallier, V.; Provost, E.; Bouroukba, M.; Petitjean, D. Fuel 1998, 77, 1253–1260. (3) Martos, C.; Baudilio, C.; Espada, J. J.; Robustillo, M. D.; Gomez, S.; Pena, J. L. Energy Fuels 2008, 22, 708–714. (4) Esmaeilzadeh, F.; Fathi Kaljahi, J.; Ghanaei, E. Fluid Phase Equilib. 2006, 248, 7–18. (5) Ghanaei, E.; Esmaeilzadeh, F.; Fathi Kaljahi, J. Fluid Phase Equilib. 2007, 254, 126–137. (6) Ghanaei, E.; Esmaeilzadeh, F.; Fathi Kaljahi, J. World Acad. Sci., Eng., Technol. 2007, 23 (August), 129-134. r 2010 American Chemical Society
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: DOI:10.1021/ef9011565
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wax content (volume fraction of precipitated wax). The parameters G, H, and I are constants (in the literature, the original letters are D, E, and F), and their values have been reported in the literature.14 Pedersen and Roe nningsen evaluated their model for 18 samples of North Sea oils.14 With consideration of the experimental data reported in the literature,14 the rheological model should be able to predict the viscosity above and below WAT; that is, the viscosity is shear-rate-dependent. Thus, in this work, a new model based on the Richardson model17 and the Herschel-Bulkley model18 has been presented. The Herschel-Bulkley model has the following form:18 dVx n τxy ¼ τ0 þ k ð2Þ dy
Table 1. Density and API Values for Oils 1-32
In this equation, τ0 is the yield stress, k the consistency coefficient, and n the power index. In the case of τ0 = 0, this model changes to the power law equation. Also, Newtonian model is generated when τ0 = 0 and n equals to unity. The Richardson model17 has been presented in eq 3: η ¼ ηliq expðMj0 Þ ð3Þ This equation is applied to predict the viscosity of water-oil emulsion in which j0 is the volume fraction of the dispersed phase and M is a constant. Thus, a combination of eqs 2 and 3 has been applied to present a new model for the prediction of rheological behavior of waxy oils. The new model can predict the viscosity at different values of temperature, shear rate, and wax content. A superiority of this model over the Pedersen and Ronningsen model14 is in the dependency of the new model parameters to the wax content and oil API.
oil
reference
oil in reference
density at 15 °C [kg/m3]
API
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 10 10 10 10 10 10 10 10 10 10 10 10 10 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 7 8 9 10 12 13 14 15 16 17
866.0 856.0 858.0 860.0 891.0 911.0 841.0 851.0 811.0 907.0 879.0 857.0 841.0 878.0 790.0 798.0 847.0 849.0 883.3 842.4 848.7 824.7 852.2 840.9 874.9 805.6 795.5 776.5 861.9 805.2 870.6 857.0
31.9 33.8 33.4 33.0 27.3 23.8 36.8 34.8 43.0 24.5 29.5 33.6 36.8 29.7 47.6 45.8 35.6 35.2 28.5a 36.3a 35.1a 39.9a 34.4a 36.6a 30.1a 44.0a 46.2a 50.5a 32.5a 44.1a 30.9a 33.4a
a
Calculated from the oil density at ∼15 °C.
Table 2. Experimental Data of Wax Content at Different Temperatures for Oils 1 and 2
2. Experimental Data The experimental data used in this work consist of 32 samples of North Sea crude oils for which the amount of dynamic viscosity above the WAT and the shear-rate-dependent viscosity below the WAT at different temperatures and shear rates have been reported in the literature.10,12,14 The studied oils have API from 23.8 to 50.5, as reported in Table 1. As Table 1 shows, for oils 19-32, the API values have not been reported in the literature, the values of API have been calculated by the density at 15 °C. Also, the wax content values in a temperature range of 303.15-363.15 K for oils 1 and 2 used to obtain the new model parameters have been presented in Table 2. Another experimental data of the amount of wax content related to eight oils from Table 1 in the number range of 19-32 applied to evaluate the new model has been reported in the literature.12 The total number of data points utilized to obtain the parameters and evaluate the new model, including dynamic viscosity and apparent viscosity (shear-rate-dependent viscosity) data, are 280 data points. More details of the experimental data, such as the composition and WAT of oils, have been reported in the literature.10,12,14
Wax Content (wt %)
Wax Content (wt %)
temperature [K]
oil 1
oil 2
temperature [K]
oil 1
oil 2
318.15 313.15 308.15 303.15 298.15 293.15 288.15
0.3 0.5 0.4 1.1 2.1 4.4 5.9
0.2 0.4 0.1 0.5 0.9 2.3 3.2
283.15 278.15 273.15 268.15 263.15 258.15 253.15
8.2 8.6 9.8 10.0 10.4 9.9 11.8
4.9 4.9 5.5 5.7 5.8 5.8 6.9
non-Newtonian behavior is observed. Therefore, the rheological model should predict the viscosity in both states including Newtonian and non-Newtonian behavior. In the other words, the model should be dependent on temperature, shear rate, and wax content. In this work, a new model, which is a combination of the Richardson model17 and Herschel-Bulkley model18 has been presented as follows: dVx C dVx þ ηliq expðDÞ τxy ¼ A þ B ð4Þ dy dy
3. New Model As mentioned in the previous section, the formation of wax particles changes the behavior of oil from the Newtonian model to the non-Newtonian model. Above the WAT, there are no wax particles; therefore, the oil shows Newtonian behavior.14 In this case, the decreasing of temperature increases the dynamic viscosity, according to an Arrhenius-type equation.19 Below the WAT, wax precipitation occurs and
In this equation, τxy is the shear stress, (dVx/dy) the shear rate, and ηliq the dynamic viscosity of the oil. By considering the definition of apparent viscosity, which is shear stress divided by shear rate, this equation changes to the following equation: A dVx C -1 þB þ ηliq expðDÞ ð5Þ η ¼ ðdVx =dyÞ dy
(18) Herschel, H.; Bulkley, R. Proc. Am. Soc. Test. Mater. 1926, 26 (II), 621–633. (19) Ro e nningsen, H. P. Energy Fuels 1993, 7, 565–573.
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The parameters of eq 5 are obtained as a function of wax content of crude oil at different temperatures for two samples of North sea crude oils (oils 1 and 2), for which the amount of wax content at different temperatures has been measured experimentally.10,12,14 The following equations have been obtained for the model parameters: A ¼ 0:04j expð1:37849jÞ ð6Þ B ¼ 43:54937j expð0:19348jÞ
ð7Þ
C ¼ 0:7
ð8Þ
D ¼ 0:225j0:93446
ð9Þ
Table 3. Calculated AAD% Values in the Prediction of Dynamic Viscosity by the New Model for Oils 1-18
In the above equations, j is the wax content (weight percent of precipitated wax). Also, a component similar to the Eyring equation20 has been utilized to predict the temperature dependence of the dynamic viscosity which is represented as F ηliq ¼ E exp ð10Þ T
oil
temperature range [K]
number of data points
AAD%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
313.15-353.15 308.15-363.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 303.15-353.15 313.15-353.15 303.15-353.15 308.15-353.15 313.15-353.15 308.15-353.15 313.15-353.15 323.15-353.15 303.15-353.15
5 7 5 5 5 5 5 5 5 6 5 6 6 5 6 5 4 6
9.11 6.54 19.73 23.83 58.57 10.24 16.80 14.71 4.26 51.90 7.94 4.28 1.87 5.08 6.23 0.86 4.13 11.59
96
14.30
total
5. Results and Discussion in which E and F are physical parameters that can be obtained using experimental data. The calculations performed in this work show that these parameters are dependent on the API of oil. Therefore, the dependency of these parameters on the API for oils 1-1810,12,14 has been obtained as follows: E ¼ 3:59459 10 -5 API2 -0:00179API þ 0:0225
ð11Þ
F ¼ 1:58895API2 -210:05596API þ 7646:4541
ð12Þ
As mentioned previously, the experimental data of oils 1-18 has been considered to obtain the parameters in eq 10 and their dependency on the API. Table 3 shows the AAD% values calculated using the following equation in the prediction of dynamic viscosity by the new model for oils 1-18: N P ðjηCal -ηExp j=ηExp Þ 100 ð13Þ AAD% ¼ i ¼1 N
It should be noted that the coefficients of eqs 6-12 have been obtained with consideration of the units of mPa s for the viscosity, wt % for the wax content, and K for the temperature.
In this equation, ηCal and ηExp are calculated and experimental value of viscosity, respectively, and N is the number of points for which the viscosity has been calculated. Among oils 1-18, oils 13, 16, and 17, which have the lowest AAD% value (according to Table 3), have been chosen to present their results in schematic form in Figures 1-3; for other oils, the results are available in the Supporting Information. Also, to evaluate the new model, more data have been used, which were named oils 19-32 in this work, as reported in Table 1. By knowing the density of oils 19-32 (from Table 1), the kinematic viscosity has been calculated and compared with the experimental data. Table 4 presents the AAD% values, in comparison to the experimental data in the prediction of kinematic viscosity for oils 19-32 by the new model. The results of the new model for oils 19 and 25, which have less deviation from the experimental data, according to this table, have been shown in Figures 4 and 5, and the others have been reported in the Supporting Information. By comparing the total AAD% values in Table 3 with those in Table 4, it is inferred that the new model has total AAD% values of 14.30% and 17.34%, relative to 96 and 78 experimental data points, respectively, in the prediction of dynamic viscosity (or kinematic viscosity) for 32 oil samples investigated in this work. On the other hand, although oils 19-32 have not been considered to obtain the new model parameters, the results reported in Table 4 show an acceptable error in comparison with that reported in Table 3. Therefore, the reliability of the new model in the prediction of viscosity in the Newtonian region for the oils studied in this work is justified.
4. Modified Pedersen and Roe nningsen Model In the Pedersen and Roe nningsen model, the liquid viscosity is calculated by the corresponding states model.14 In this work, in addition to the new model, a modification of the Pedersen and Roe nningsen model has been presented. This modification includes the determination of the liquid viscosity in the Pedersen and Roe nningsen model14 by the simplified Eyring equation and its functions presented in eqs 10-12 for the new model. Therefore, above the WAT, the new model and the modified Pedersen and Roe nningsen model have similar results. In the other words, the difference between the new model and the modified Pedersen and Roe nningsen model appears in the prediction of viscosity below the WAT. Therefore, the comparison of the results has been only presented in the apparent viscosity calculations. The reason for this modification of the Pedersen and Roe nningsen model is to evaluate the main structure of their model (eq 1) by comparing the results of the modified model with the new model. Also, it shows the reliability of the simplified Eyring equation and its functions presented in this work, which are dependent on the API, in comparison with the corresponding states model applied in the Pedersen and Roe nningsen model. (20) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena, 2nd Ed.; John Wiley & Sons: New York, 2002; pp 29-31.
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Figure 1. Comparison of the calculated dynamic viscosity, as a function of temperature, by the new model with the experimental data for oil 13.
the modified Pedersen and Roe nningsen model for oils 1 and 2 have been reported in Tables 5 and 6. These tables indicate that the new model has total AAD% values of 20.38% and 29.42%, relative to 36 and 30 experimental data points, respectively, which is less than that in the modified Pedersen and Roe nningsen model. Also, to show the prediction of apparent viscosity, variations, as a function of shear rate by the new model and modified Pedersen and Roe nningsen model, in comparison with the experimental data shown in Figures 6 and 7, related to oil 1 at a temperature of 303.15 K and oil 2 at a temperature of 288.15 K have been presented, respectively. As reported in the literature, oil has a shear thinning behavior at the temperature below the WAT. It seems that, because of the geometrical form of wax crystals (needle form), the wax particles get more alignment in the direction of flow while the shear rate is increased. Also, to better represent the reliability and accuracy of the new model, in Figures 8 and 9, the prediction of viscosity, as a function of wax content (wax content is a function of temperature) by the new model and modified Pedersen and Roe nningsen model, at shear rates of 100 and 500 s-1 for oils 1 and 2 have been illustrated, respectively. Other figures related to prediction of the apparent viscosity can be found in the Supporting Information. To show the reliability of the new model with regard to predicting the apparent viscosity, similar to the evaluation procedure for the viscosity of oil above the WAT, eight oils from Table 1 in the oil number range of 19-31 have been considered to evaluate the new model below the WAT, with regard to estimating the apparent viscosity at a shear rate of 100 s-1. The results of the prediction of the apparent viscosity for these eight oils have been reported in Table 7. This table indicates that the new model has a total AAD% value of 27.72% for 40 experimental data points, which is less than that in the modified Pedersen and Roe nningsen model. Also, this table indicates that the modified Pedersen and Roenningsen model has less deviation from the experimental data for oils 19 and 24 in comparison with the new model. Among these eight oils, for oils 19 and 25, for which the new model has less deviation from the experimental data, Figure 10 is used to illustrate the better comparison of the new model and the modified Pedersen and Roe nningsen model with the experimental data in the prediction of apparent viscosity, as a function of temperature. Because of the existence of wax
Figure 2. Comparison of the calculated dynamic viscosity, as a function of temperature, by the new model with the experimental data for oil 16.
Table 4. Calculated AAD% Values in the Prediction of Kinematic Viscosity by the New Model for Oils 19-32 oil
temperature range [K]
number of data points
AAD%
19 20 21 22 23 24 25 26 27 28 29 30 31 32
313.15-353.15 308.15-363.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 313.15-353.15 303.15-353.15 313.15-353.15 303.15-353.15 308.15-353.15 313.15-353.15
6 6 6 6 3 3 6 6 6 6 6 6 6 6
3.39 16.56 7.91 41.55 25.48 11.63 3.37 17.39 18.41 24.36 24.56 6.28 30.21 12.85
78
17.34
total
Below the WAT, the wax particles are formed and influences the rheological nature of oil. As mentioned in the literature, the formation of wax crystals changes the behavior of oil from Newtonian to non-Newtonian.10,14 Therefore, the parameters of the new model, related to the presence of wax particles, have been computed as a function of wax content by the experimental data of oils 1 and 2. The results of the prediction of the apparent viscosity by the new model and 1765
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Figure 3. Comparison of the calculated dynamic viscosity, as a function of temperature, by the new model with the experimental data for oil 17.
Figure 4. Comparison of the calculated kinematic viscosity, as a function of temperature, by the new model with the experimental data for oil 19.
Figure 5. Comparison of the calculated kinematic viscosity, as a function of temperature, by the new model with the experimental data for oil 25.
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Table 5. Calculated Apparent Viscosity and AAD% Values at Different Shear Rates by the New Model and the Modified Pedersen and Roe nningsen Model for Oil 1 Calculated Apparent Viscosity [mPa s] temperature [K]
shear rate = 20 s-1
shear rate = 30 s-1
shear rate = 70 s-1
313.15 303.15 293.15 288.15 283.15 278.15
17.73 36.00 216.93 414.70 2124.79 3333.04
16.61 33.23 194.74 363.79 1599.99 2433.65
14.67 28.43 156.97 283.38 952.03 1351.16
shear rate = 100 s-1
shear rate = 300 s-1
shear rate = 500 s-1
number of data points
AAD%
12.30 22.57 111.75 195.00 488.05 625.15
11.68 21.05 100.12 173.28 407.78 509.82
6 6 6 6 6 6
11.83 3.43 30.12 34.21 27.36 15.30
36
20.38
6 6 6 6 6 6
36.74 47.53 63.88 60.48 47.87 48.14
36
50.77
New Model 13.99 26.75 143.91 257.11 788.50 1087.50
total 313.15 303.15 293.15 288.15 283.15 278.15
9.11 15.66 86.40 197.11 615.50 850.68
Modified Pedersen and Ro e nningsen Model 8.82 8.76 8.64 14.77 14.61 14.26 70.22 67.97 63.95 146.20 139.61 128.56 412.62 387.49 347.10 564.03 528.67 472.05
9.00 15.31 79.18 173.86 521.66 717.94
8.61 14.15 62.95 126.04 338.52 460.10
total
Table 6. Calculated Apparent Viscosity and AAD% Values at Different Shear Rates by the New Model and the Modified Pedersen and Roe nningsen Model for Oil 2 Calculated Apparent Viscosity [mPa s] temperature [K]
shear rate = 30 s-1
shear rate = 70 s-1
303.15 293.15 288.15 283.15 278.15 274.15
16.98 72.20 115.32 238.81 244.43 317.37
15.04 59.51 94.17 191.05 196.67 252.98
shear rate = 100 s-1 New Model 14.36 55.06 86.79 174.71 180.32 231.43
shear rate = 300 s-1
shear rate = 500 s-1
number of data points
AAD%
12.67 44.03 68.46 134.75 140.36 179.59
12.05 40.01 61.80 120.39 126.00 161.20
5 5 5 5 5 5
30.47 45.28 7.20 15.89 35.76 41.89
30
29.42
5 5 5 5 5 5
15.23 35.65 56.77 48.25 61.30 59.79
30
46.17
total 303.15 293.15 288.15 283.15 278.15 274.15
9.39 25.73 42.47 99.16 115.20 173.04
9.21 24.28 39.20 86.55 100.55 147.86
Modified Pedersen and Ro e nningsen Model 9.15 9.02 23.86 22.99 38.32 36.61 83.45 78.03 96.95 90.65 141.79 131.44
total
content experimental data over a wide range of temperatures of these oils in the literature,12 prediction of the apparent viscosity has been performed in a wider range than that of available experimental data about the apparent viscosity. Figure 10 shows that the new model has an acceptable trend, in comparison with the experimental data and modified Pedersen and Roenningsen model. Although the experimental data of these eight oils have not been considered to obtain the new model parameters, comparing the AAD% value reported in Table 7 with those given in Tables 5 and 6 justifies the acceptable accuracy of the new model in the prediction of apparent viscosity, in addition to the viscosity prediction above the WAT. Also, it seems that the modified Pedersen and Roe nningsen model has less error than the Pedersen and Roe nningsen model, which has a value of ∼47%, as reported in the literature.14 By considering the difference between the Pedersen and Roe nningsen model and the modified version of it, it is inferred
8.99 22.74 36.14 76.72 89.13 129.03
that the method of viscosity estimation above the WAT presented in this work is probably more reliable than the Pedersen and Roe nningsen model. The obtained results of apparent viscosity calculations demonstrate more accuracy of the new model, as compared with the Pedersen and Roenningsen model14 and the modified Pedersen and Roe nningsen model. To evaluate the new model in the prediction of viscosity in both states, including above the WAT and below it, in addition to the results of apparent viscosity, the AAD% value in the Newtonian region also should be considered. Therefore, according to the AAD% value and the number of data points presented in Tables 3-7, for all 280 data points applied in this work, the new model has an AAD% value of 19.47%, in comparison with the modified Pedersen and Roe nningsen model with an AAD% value of 26.88%. Thus, the new model is more reliable than the Pedersen and Roe nningsen model14 and the modified Pedersen and Roe nningsen model, with 1767
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Figure 6. Comparison of the calculated apparent viscosity, as a function of shear rate, by the new model with the experimental data and the modified Pedersen and Roe nningsen model for oil 1 at T = 303.15 K.
Figure 7. Comparison of the calculated apparent viscosity, as a function of shear rate, by the new model with the experimental data and the modified Pedersen and Roe nningsen model for oil 2 at T = 288.15 K.
Figure 8. Comparison of the calculated apparent viscosity, as a function of wax content, by the new model with the experimental data and the modified Pedersen and Roe nningsen model for oil 1 at a shear rate = 100 s-1.
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Figure 9. Comparison of the calculated apparent viscosity, as a function of wax content, by the new model with the experimental data and the modified Pedersen and Roe nningsen model for oil 2 at a shear rate of 500 s-1. Table 7. Calculated Apparent Viscosity and AAD% Values, Using the New Model and the Modified Pedersen and Roe nningsen Model, for Eight Oils from Oils 19-31 at a Shear Rate of 100 s-1 Apparent Viscosity [mPa s] temperature [K]
oil 19
oil 20
oil 21
303.15 293.15 288.15 283.15 278.15
15.84 19.61 25.78 29.29 31.71
7.78 8.46 12.04 14.50 31.75
13.06 20.40 52.41 69.25 114.56
number of data points AAD%
5 19.75
5 25.54
5 31.64
303.15 293.15 288.15 283.15 278.15
13.11 16.11 20.75 24.70 28.37
5.70 6.41 7.85 9.26 14.95
number of data points AAD%
5 13.80
5 30.54
oil 24 New Model 7.55 10.92 19.50 29.85 47.29 5 36.74
oil 25
oil 27
oil 30
oil 31
16.64 23.60 47.24 64.72 97.38
1.34 2.60 3.92 7.93 9.48
2.91 3.06 3.23 5.90 13.17
19.12 26.25 41.12 65.70 90.70
5 20.52
5 25.41
5 33.05
5 29.11
1.34 1.51 1.69 2.06 2.32
1.82 1.97 2.15 2.54 3.37
13.19 17.28 25.48 40.52 59.88
5 57.81
5 51.62
5 47.54
Modified Pedersen and Ro e nningsen Model 7.71 5.46 12.87 10.22 6.65 16.99 19.70 9.13 29.56 27.03 12.50 42.16 46.48 18.42 67.67 5 51.41
5 33.92
5 31.21
total
40 27.72
40 39.73
Figure 10. Comparison of the calculated apparent viscosity, as a function of temperature, by the new model with the experimental data and the modified Pedersen and Roe nningsen model for oils 19 and 25 at a shear rate of 100 s-1.
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Energy Fuels 2010, 24, 1762–1770
: DOI:10.1021/ef9011565
Ghanaei and Mowla
regard to calculating the Newtonian and non-Newtonian viscosity for the experimental data used in this work.
Acknowledgment. The authors are grateful to Shiraz University for supporting this research.
6. Conclusions
Note Added after ASAP Publication. This paper was published February 26, 2010 with an error in eq 1 and an error in the descriptive text following eq 4. The revised version was published on March 2, 2010.
The formation of wax particles causes the rheological behavior of crude oils to change from Newtonian to nonNewtonian.10 In this work, a new model based on a combination of the Richardson model17 and the Herschel-Bulkley model18 has been presented. Experimental data have been applied to obtain the model parameters, and another dataset has been used to show the validity of the new model. In addition to the comparison with the experimental data, a modified model of the Pedersen and Roenningsen approach has been applied to present the reliability of the new model. In the Pedersen and Roenningsen model,14 the liquid viscosity is estimated by the corresponding states model; however, in the modified version, the simplified Eyring equation with parameters that are dependent on the oil API presented in this work is used. According to the results, the modified Pedersen and Roenningsen model has less error, in comparison with the Pedersen and Roenningsen model, which shows the accuracy and reliability of the equations presented in this work with regard to determining the simplified Eyring parameters over a wide range of oil API. In the case of Newtonian behavior (without wax precipitation), the model is similar to the simplified Eyring equation.20 The parameters of this part of the model have been obtained as a function of oil API by considering of 18 North Sea oil samples (including 96 experimental data points). Also, to demonstrate the reliability of the new model, another dataset (including 14 oil samples, consisting of 78 experimental data points) has been considered. For temperatures below the WAT, wax particles are formed and the model changes to nonNewtonian, in accordance to the rheological behavior of the fluid. To obtain the new model parameters and their dependency on the oil API and wax content below the WAT, 66 experimental data points have been applied. Also, the new model has been evaluated by applying another experimental data, including 40 data points in the case of calculation of the apparent viscosity. According to the results of the new model, increasing the amount of wax content increases the apparent viscosity at a specified shear rate, which is in agreement with the experimental data. Also, the new model predicts that increasing the shear rate reduces the apparent viscosity, which is also in agreement with the experimental data. Overall, the results show that the new model has an average absolute deviation (AAD%) value of 19.47% for the total 280 experimental data points used in this work that is less than the error of the Pedersen and Roe nningsen model14 and the modified Pedersen and Roe nningsen model.
Supporting Information Available: In the Supporting Information, 10 figures related to the prediction of oil viscosity in the Newtonian and non-Newtonian region have been reported. (PDF) This information is available free of charge via the Internet at http://pubs.acs.org/.
Nomenclature Parameters A = coefficient in eqs 4, 5, and 6 B = coefficient in eqs 4, 5, and 7 C = coefficient in eqs 4, 5, and 8 D = coefficient in eqs 4, 5, and 9 E = coefficient in eqs 10 and 11 F = coefficient in eqs 10 and 12 G, H, I = constants in eq 1 k = consistency coefficient in eq 2 M = constant in eq 3 n = power index (in eq 2) N = number of data points (in eq 13) T = temperature Greek Letters η = viscosity τ = shear stress τ0 = yield stress φ = volume fraction of precipitated wax (in eq 1) j0 = volume fraction of dispersed phase (in eq 3) j = weight percent of precipitated wax (in eqs 6, 7, and 9) Subscripts and Superscripts Cal = calculated Exp = experimental liq = liquid x, y = directions in the xy coordinate Abbreviations AAD% = average absolute deviation percentage WAT = wax appearance temperature WDT = wax disappearance temperature wt% = weight percent
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