A Generalized Viscosity Equation for Liquid Hydrocarbons - Application to Oil Sand Bitumens

Fluid Phase Equilibria, 75 (1992) 257-268 Elsevier Science Publishers B.V., Amsterdam

257

A generalized viscosity equation for liquid hydrocarbons: Application to oil-sand bitumens

Anil K. Mehrotra

Department of Chemical and Petroleum Engineering, The University of Calgary, Calgary, Alberta T2N IN4 (Canada)

Keywords: viscosity- temperature equation, liquid hydrocarbon viscosity, gas viscosity, bitumen - diluent mixture viscosity

ABSTRACT

A new method is presented for the correlation and prediction of the effect of temperature on viscosity of liquid hydrocarbons and their mixtures. A one- parameter viscosity- temperature equation is described that has been validated with data for approximately 360 pure liquid hydrocarbons, including paraffins, olefins, cyclopentanes, cyclohexanes, naphthenes and aromatics. The viscosity data for all hydrocarbons are represented well by: log (II + 0.8) = 0( a?“‘)*, where lo is in mPa s, T in kelvin, and 8 = 100; Q = 0.01 for pure hydrocarbons. For most compounds, parameter b follows systematic trends with several properties, such as molar mass, boiling point, critical temperature and acentric factor.

The usefulness of the one-parameter equation is extended to the viscosity of oil-sand bitumens diluted with light gases or liquid diluents. This is accom- plished by combining the viscosity equation with simple mixing rules. The predictions for mixture viscosity, spanning several orders of magnitude, are in excellent agreement with data over wide ranges of temperature and composition.

INTRODUCTION

Numerous calculation methods and correlations for the effect of temperature on liquid viscosity are available in literature (Perry et al., 1984; Reid et al., 1986). Amongst the viscosity prediction methods, the approaches given by Ely and Hanley (1981), Teja and Rice (1981), and Pedersen et al. (1984) have been applied successfully for some liquid mixtures. In contrast, a relatively simple predictive approach is presented here.

0378-3312/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved

268

Equation (1) is a one-parameter viscosity-temperature equation that was obtained by Mehrotra (1991a,b) by showing a linear correlation between the two parameters in Walther’s (1931) viscosity correlation.

log (p + 0.8) = e(O7’)” (I)

For approximately 360 pure light, medium and heavy hydrocarbons (including branched and unbranched paraffins and olefins; n-alkyl-branched cyclopentanes, cyclohexanes and aromatics; nonfused and fused-ring naphthenes and aroma- tics), values of 100 and 0.01 were found as optimum (generalized) for constants 8 and 9, respectively (Mehrotra, 1991a,b). For each family of hydrocarbons, parameter b was shown to follow systematic trends with other properties, such as molar mass, boiling point, critical temperature and acentric factor.

Recently, Equation (1) was shown to be applicable also for the viscosity of bitumens (from Canadian oil-sands); with the generalized constants for bitumens and bitumen fractions as 8 = 160 and Q = 0.008 (Mehrotra, 1992). In addition, for the low viscosity of light gases at atmospheric pressure, Equation (1) was modified by inserting a negative sign (Mehrotra, 1992):

log (p + 0.8) = -0.1(0.015T)b (2)

The objectives of this paper are: (i) to document the effectiveness of Equation (1) as a predictive equation for the viscosity of liquid hydrocarbons and their mixtures, and (ii) to describe a method, based on Equation (l), for predicting the viscosity of bitumens (from Canadian oil-sands) diluted with liquid and/or gaseous diluents over a wide range of conditions.

Generalized mixing rules for viscosity prediction

Viscosity of bitumens from the vast deposits of Alberta’s oil-sands is an important property that is needed in reservoir and process simulation, and pipeline calculations. The available calculation methods for the viscosity of bitumens range from empirical correlations to those based on the corresponding states principle (Johnson et al., 1987; Mehrotra and Svrcek, 1987; Svrcek and Mehrotra, 1988). In contrast, a mixing rule approach for calculating the viscosity of CO,-saturated or toluene-diluted bitumens is presented here.

Equation (3) is obtained from a commonly used additive viscosity formula, f(r() = XX;: f( pi), by selecting f( p) = (A4)” log (p + 0.8) and Xi as mole fraction Xi (Mehrotra et al., 1989):

log (~ + 0.8) = C ~i~~~ log (pi + 0.8) P C Ui log (pi + 0.8) (3)

- where Vi 3 xi(Mi/lM)” denotes the geometric mean of mass and mole fractions.

By combining Equation (3) and the Grunberg (1954) formula, the following mixing rule is obtained for the viscosity of a bitumen-diluent mixture:

259

log (i + 0.8) = c’ ui log (p, + 0.8) + c’ ci u, ujBii (4)

where Bii (B, = Bii = 0 and B, = Bji) is a binary viscous interaction term that could be determined from mixture viscosity data.

Next, Equations (3) and (4) are combined with Equation (1) to obtain the following mixing rules based on the generalized one-parameter viscosity equation.

Mixing Rule I: log (cl + 0.8) = 2 I+ @(W)“J (5)

Mixing Rule II: log (G + 0.8) = 2 ui e(WJbl + 2 ci ui uj B, (6)

Data used for validating the viscosity predictions

Data for the viscosity of a Cold Lake bitumen and its five fractions were reported by Mehrotra et al. (1989). These five bitumen fractions (or “cuts”) were obtained by vacuum distillation of a Cold Lake bitumen sample. The boiling point ranges for Cuts I and 2 are bp < 343°C and 343 c bp < SlO”C, respectively. As indicated in Table 1, the molar mass of Cuts I to 5 ranges from 209 to 2500 g/mol.

TABLE 1

Data and viscosity correlation for the five fractions of Cold Lake bitumen ’

Sample Mass Molar Density fraction mass

(g/m00 (g/cm3) Cut1 0.103 209 0.879 Cut2 0.283 310 0.941 Cut3 0.168 667 0.990 Cut4 0.065 800 0.999 Cut5 0.381 2500 1.072

‘Data from Mehrotra et al. (1989)

Correlation of viscosity data with Eq. (1): log (/J + 0.8) = 160(0.008T)b Parameter b AAD %

- 6.0234 14.2 - 4.9632 8.5 -4.0444 4.5 - 3.7853 3.3 -2.9073 14.8

Cuts 1 to 4 of Cold Lake bitumen contained little or no asphaltenes whereas approximately 50 mass % of Cut 5 was asphaltenes. Also, Cuts I and 2 were clear liquids with low viscosities, Cuts 3 and 4 had consistencies and viscosities much like the whole bitumen, and Cut 5 was a glass-like solid with a softening temperature of approximately 100°C (Mehrotra et al., 1989). From viscosity- temperature data for all five bitumen fractions, Figure 1, it is clear that viscosities of bitumen fractions differ by several orders of magnitude.

260

\ -b 0 6. cut Cut 2 1

!&

0 cut 3 v cut 4

10’ 0 cut 5

- Equation (1) i

100 I I I 0 50 100 150 200 250

Temperature, ‘C

Fig. 1. Viscosity of the five fractions of Cold Lake bitumen (data from Mehrotra et al., 1989).

The viscosity- temperature-composition data for 12 reconstituted binary blends of bitumen fractions were presented by Eastick and Mehrotra (1990). These data are used for obtaining the viscous interaction term (B,) for Cut i- Cut j pairs. Viscosity data for mixtures of toluene and Cuts 3, 4 and 5 as well as the whole bitumen were reported by Mehrotra (1990). These data are used for evaluating the Cut i-Cut j binary interactions and for validating the calculation approach.

Data for the viscosity and CO,-solubility of the five bitumen fractions and the whole bitumen were collected at temperatures of 25 - 150°C and at pressures up to 10 MPa (Eastick et al., 1992). The data for the five Cut i-CO, binaries are used to obtain Bii for each pair. The data for CO,-saturated Cold Lake (whole) bitumen are used for further validation of the viscosity calculation approach.

RESULTS AND DISCUSSION

Results are presented first for the correlation of viscosity with temperature, by use of Equation (1) or (2), for the individual components, i.e. pure liquid hydrocarbons, bitumen fractions, and gases. The calculations for the viscosity of CO,-saturated bitumen fractions, to validate Equations (5) and (6) for binary mixtures, are described next. The data for the viscosity of the reconstituted blends of bitumen fractions are used to obtain Cut i-Cut j interaction parameters. Results are also described for the viscosity of toluene-diluted bitumen and bitumen fractions. Finally, the calculation procedure is used for

261

predicting the viscosity of CO,-saturated and toluene-diluted bitumen, and the results are compared to the measured values.

Correlation and prediction of component viscosities

For approximately 360 pure liquid hydrocarbons, the optimum values of parameter b in Equation (1) were given by Mehrotra (1991a,b). Figures 2 and 3, for example, present the variation of parameter b for light, medium and heavy hydrocarbons as a function of boiling temperature. Similar consistent trends for parameter b were also noted with other properties, namely molar mass, critical temperature and acentric factor (Mehrotra, 1991a,b). For toluene (the liquid diluent), the value of parameter b in Equations (1) is -6.0853.

-3

-4

-6

Branched Paraffins and Olefins -6

Non-fused Aromatics

D -5 g -5

b u” -6

z a

E

-6

z -3

rz -7 -4

-5

Fused ring Aromatics

Non-fused Nophthenes

Fused ring Naphthenes

2.0 2.5

(l/T/,) X 103, K-l (l/T;')X 103, K-'

Fig. 2. Dependence of parameter b on normal boiling point of light and

Fig. 3. Dependence of parameter b

medium hydrocarbons. on boiling temperature (at 10 mmHg) of heavy hydrocarbons.

-3

-4

-5

-6

Parameter b for bitumen fractions, Table 1, was found to be satisfactorily correlatable with the molar mass (A4 in g/mol) and the density at 25°C (P in g/cm’) :

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With n4ozar ie7ss (M): ( Ir 1 = 0.990)

b = -27.23 + 13.56 [log M] - 1.886 [log iWJ2 (7)

With Density (p at 25°C): ( It-1 = 0.998)

b = -40.95 + 58.74 p - 21.67 p2 (8)

Calculations with the one-parameter equation for the viscosity of gases also yielded an excellent match of the data, as is indicated in Table 2. Note that, in Equation (2) for gases, the values of 8 and Q are -0.1 and 0.015, respectively (Mehrotra, 1992).

TABLE 2

Correlation of viscosity of five light gases at 0-200°C Gas Molar mass Correlation of viscosity data with Equation (2):

log (/J + 0.8) = -0.1(0.015T)b

(g/mol) Parameter 6 AA.u (%)

N2 28.01 - 0.0908 0.8 co 28.01 - 0.0899 0.9 CH‘I 16.04 - 0.0632 0.6 CO, 44.01 -0.0791 1.9 CAH, 30.07 - 0.0568 0.9

Viicosity prediction of bitumen fraction -diluent (binary) mixtures

For a Cut i - CO, binary mixture, Equation (5) is written as:

log (cl + 0.8) = I+ [160(0.008T)bi] + u6 [ -O.l(O.O15l)“6] (9)

where the first term is for a bitumen fraction (i.e. Cut i) and the second for CO,. The predicted viscosities were found to be well within one order of magnitude of the data by Eastick et al. (1992).

Equation (6) for a Cut i - CO2 binary mixture becomes:

log (i + 0.8) = Ui [160(0.0087Jbi] + ug [-0.1(0.0157)b6] + 2 Ui u,5 Bi6 (10)

Calculations to obtain Bi6 for each data point were performed with Equation (10). For each Cut i-CO, pair, the B, values were regressed. linearly with temperature and the results are summarized in Table 3 and plotted in Figure 4. Overall, the results indicated an improvement in viscosity predictions from Equation (10) over those obtained from Equation (9).

263

TABLE 3

Correlations for binary viscous interaction parameters @Iii) in Equation (6) J Bc - T correlation: i

Binary pair Ao A,

Cut ;LCu:, (B& atz cut’

- 1.09 0.0101 Cut 1 -Cut 4 (BJ - 1.09 0.0101 Cut 1 -Cut 5 (B,,) 2.15 -0.0100 Cut 2-Cut 3 (B,) -0.19 0.0037 Cut 2 -Cut 4 (B& -0.19 0.0037 Cut f:Cut 5 (B,,) 0.14 0.0085 Clltz co Cut I-Cd; (Bib) - 1.17 0.0192 Cut 2-co, - 1.81 0.0204 Cut 3-co* - 1.53 0.0155 Cut 4-co, -0.51 0.0065 Cut 5-co, (BM) 9.19 0.0

*- ne Cut 3 - toluene (Bj7) -1.28 0.0136 Cut 4 - toluene (B4,) -1.28 0.0136 Cut 5 - toluene (B& 4.79 - 0.0233

B25- /kdBz4

/Wh

857

NW/B47

Fig. 4. The Bil- T correlations for Cut i-EQ, Cut i-Cut j and Cut i- toluene binary pairs.

264

Viscosity of reconstituted binary blerxis of bitumen fractions

As mentioned previously, viscosity data over a range of temperatures for twelve binary blends of bitumen fractions (prepared by mixing Cuts I or 2 with Cuts 3, 4 or 5) were provided by Eastick and Mehrotra (1990). The viscosity predictions for those data from Mixing Rule I, rewritten as Equation (ll), were found to be well within one order of magnitude.

log (G + 0.8) = ui [160(0.0087’)*i] + uj [160(0.008Y’)bj] (11)

Next, the binary blend data were used to evaluate the Cut i - Cutj interaction parameters from Equation (12).

log (i + 0.8) = ui [160(0.0087’)bi] + uj [160(0.008ZJbj] + 2 ui uj Bq (12)

The resulting Biis were regressed linearly with temperature and the results are presented in Table 3. The calculated viscosities for all binary blends were in good agreement with the data (AAD typically less than 10%). These B,- T correlations are plotted in Figure 4. Note that all BV- T lines, except for B,,, have a positive slope. Also, the Bqs for the blends involving Cuts 3 and 4 are identical, i.e. B13 = B,, and B, = B,.

Prediction of the viscosity of CO,-saturated (whole) bitumen

In the following calculations, the CO,-saturated (whole) bitumen is modeled as a 6-component mixture comprising the five bitumen fractions and CO* as the sixth component. Sixteen data points for the CO,-solubility, density and viscosity of Cold Lake bitumen were reported by Eastick et al. (1992). The temperature and pressure ranges for these data were 23- 100°C and 0- 10.5 MPa, respectively. Obviously, an adequate prediction of these data is an important test of the viscosity calculation procedure outlined here.

Calculations were first performed with Mixing Rule I, Equation (5), which does not involve any binary interaction term. The bitumen composition, in terms of its five fractions (Mehrotra et al., 1989), together with the CO,-solubility data (Eastick et al., 1992) were used to calculate all ui terms. The values of parameter b for the six components were taken from Tables 1 and 2. The predicted viscosities of CO,-saturated bitumen, plotted in Figure 5, are remarkably close to the experimental viscosities with an AAD of 23.6%.

Calculations for the viscosity of CO,-saturated bitumen with Mixing Rule II, Equation (6), would involve 15 binary interaction terms. Of these, values for the five Bg and six B,,s are given in Table 3. The remaining four Cut i-Cut j interaction terms are B,2, Bj4, B,, and Bd5, for which no binary-mixture viscosity data were available. It was reasoned that these four BVs would be small owing to the similar viscosities of both components in each case. That is, it was assumed that B,, = B3, = B,, = B,, = 0.

265

The viscosity predictions from Equation (6) for CO,-saturated (whole) bitumen were higher than the measured data, as is shown in Figure 5. The AAD for the sixteen data points was 61.8%. Of the eleven B+s in the double-summation of Equation (6), it was noted that the term for the Cut 5 -CO, interaction (i.e. &) had a dominant effect on the viscosities calculated from Mixing Rule II. In Table 2, the value of B, is 9.19 which is much larger than all other Bi8.

104, ““..I . “‘.“I . ‘“.?a 1 “.““n “.“..I ..“.‘.I ‘PI

? : o Mixing

cl

t

0 Mixing

>;103: .e

E

Experimental Viscosity, mPa.s Experimental viscosity, mPa.s

Fig. 5. Comparison of predicted and experimental viscosity of CO,-saturated Cold Lake (whole) bitumen.

Fig. 6. Comparison of predicted and experimental viscosity of toluene-diluted Cold Lake (whole) bitumen.

Comparison of predictions j?om Mixing Rules I and II

The information required for using Mixing Rule I consists of (i) parameter b in Equation (1) or (2) for each constituent, and (ii) the gas solubility for each Cut i-CO, mixture. Thus, Mixing Rule I is a truly predictive method for the viscosity of CO,-saturated bitumen. In contrast, the use of Mixing Rule II for another CO,-saturated bitumen may require additional data for the viscosity of CO,-saturated bitumen fractions and Cut i-Cut j blends to obtain the B,s. Hence, Mixing Rule II should be viewed as a semi-predictive method.

In Figure 5, the predictions for the viscosity of CO,-saturated (whole) bitumen from Mixing Rule I are slightly lower than the data, while those from Mixing Rule II are higher. Both sets of results, nevertheless, are within one-half to two-times the data. These predictions are, indeed, acceptable in view of the fact that the calculations involved a mixture of six widely different components, from CO, (a gas) to Cut 5 (a glass-like solid), which have very different viscosities spanning several orders of magnitude.

266

Prediction of the viscosity of toluene-diluted (whole) bitumen

For calculating the viscosity of toluene-diluted (whole) bitumen, the mixture is modeled as a 6-component mixture; i.e. five bitumen fractions along with toluene (denoted by subscript 7) as the sixth component. The calculation steps with Walther’s (1931) two-parameter correlation have been described elsewhere (Mehrotra, 1990); hence, only the significant results are given,

In Figure 6, the predicted viscosities of Mixtures ll- 14 (Mehrotra, 1990) from Equation (5) are in fair agreement with the data. Calculations for the viscosity of toluene-diluted bitumen with Mixing Rule II, Equation (6), again involved 15 binary interaction terms. Of these, six Bus for Cut i-Cut j interactions are given in Table 3. Three Cut i - toluene interaction terms, i.e. B3, = B0 and B57, were evaluated from the data for Mixtures 1- 10 (Mehrotra, 1990) and are given in Table 3. All remaining interactions were neglected due to a lack of binary-mixture data, i.e. B,, = BN = Bj5 = Bd5 = B,, = B, = 0. The calculated viscosities, plotted in Figure 6, show remarkable agreement with the experimental viscosities, especially for Mixtures 12 and 13.

CONCLUSIONS

A generalized one-parameter equation was presented for predicting the viscosity of pure hydrocarbons, bitumen fractions, and gases. The single parameter in this generalized viscosity equation is correlated well with other hydrocarbon properties. Subsequently, the viscosity equation was used to predict the mixture viscosity of diluted Cold Lake bitumen which was modeled as a 6-component mixture. Of the two mixing rules for the viscosity of CO,-saturated or toluene-diluted bitumens, Mixing Rule I was demonstrated to predict viscosities that are well within one order of magnitude of the data. Mixing Rule II, involving binary viscous interaction terms that are determined empirically, is shown to yield improved viscosity predictions.

ACKNOWLEDGEMENTS

Financial support was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC).

NOMENCLATURE

b Bfi

average absolute deviation, % single viscosity parameter binary viscous interaction term

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M M !I

x

molar mass, g/m01 average (mixture) molar mass, g/m01 correlation coefficient absolute temperature, K mole fraction

Greek symbols dynamic viscosity, mPa s predicted mixture viscosity, mPa s generalized viscosity constant density at 25”C, g/cm’ generalized viscosity constant geometric mean of mass and mole fractions

Subscripts i j l-5 6 7

component i component j Cuts 1 to 5 (bitumen fractions) CO2 (gaseous diluent) toluene (liquid diluent)

REFERENCES

Eastick, R.R. and Mehrotra, A.K., 1990. Viscosity data and correlation for mixtures of bitumen fractions. Fuel Processing Technology, 26: 25-37.

Eastick, R.R., Svrcek, W.Y. and Mehrotra, AK, 1992. Phase behaviour of CO,-bitumen fractions. Can. J. Chem. Eng., 70: 159-164.

Ely, J.F. and Hartley, H.J.M., 1981. A Computer Program for the Prediction of Viscosity and Thermal Conductivity in Hydrocarbon Mixtures, NBS 1039, National Bureau of Standards, Washington, DC.

Gnmberg, L., 1954. The viscosity of regular solutions. Trans. Faraday Sot., 50: 1293-1303.

Johnson, S.E., Svrcek, W.Y. and Mehrotra, AK, 1987. Viscosity prediction of Athabasca bitumen using the extended principle of corresponding states. Ind. Eng. Chem. Res., 26: 2290-2298.

Mehrotra, AK., 1990. Development of mixing rules for predicting the viscosity of bitumen and its fractions blended with toluene. Can. J. Chem. Eng., 68: 839-848.

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Mehrotra, A.K., 1991b. Generalized one-parameter viscosity equation for light and medium hydrocarbons. Ind. Eng. Chem. Res., 30: 1367-1372.

Mehrotra, AK, 1992. Mixing rules for predicting the viscosity of bitumens saturated with pure gases. Can. J. Chem. Eng., 70: 165-172.

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Reid, R.C., Prausnitz, J.M. and Poling, B.E., 1986. The Properties of Gases and Liquids. 4th ed., McGraw-Hill, Chapter 9.

Svrcek, W.Y. and Mehrotra, AK., 1988. One parameter correlation for bitumen viscosity. Chem. Eng. Res. Des., 66: 323-327.

Teja, A.S. and Rice, P., 1981. Generalized corresponding states method for the viscosities of liquid mixtures. Ind. Eng. Chem. Fundam., 20: 77-81.

Walther, C., 1931. The evaluation of viscosity data. Erdiil und Teer, 7: 382-384.