I n d . Eng. Chem. Res 1990,29, 1574-1578
1574
E, = order of error in calculating f due to truncation Ey = order of error in calculating due to truncation f, = nth expansion coefficient g(x) = function defined in eq 6 H,,H,, H2 = polynomial coefficients, defined in eq 18 I ( k n...) = integrals defined by eq 22 K = Langmuir constant q = solid-phase concentration r = position vector R = reference length U = total uptake x = fluid-phase concentration fraction, c/co x ' = approximate solution, fluid phase f = fluid-phase uptake f ' = approximate fluid-phase uptake y = solid-phase concentration fraction, q / q o y' = approximate solution, solid phase 7 = solid-phase uptake 7' = approximate solid-phase uptake z = dimensionless distance, z = r / R 8 = isotherm parameter, defined in eqs 11 and 12 bkn = Kronecker delta c = constant void fraction of medium A, = nth eigenvalue of the solution of the linear problem A = capacity ratio, defined in eq 6 7 = dimensionless time, T = D t n 2 / t R 2 h 4, = nth eigenfunctionof the solution of the linear problem Subscripts
Literature Cited Crank J. The Mathematics of Diffusion, 2nd ed.; Oxford University Press: Oxford, 1975. Gottlieb, D.; Orszag, S. Numerical Analysis of Spectral Methods: Theory and Applications; Society for Industrial and Applied Mathematics: Philadelphia, PA, 1977. Hermans, J. J. Diffusion with Discontinuous Boundary (1). J . Colloid Sci. 1947, 2, 387. Ott, R. J.; Rys, P. Sorption-Diffusion in Heterogeneous Systems Part 5-A General Diffusion/Immobilization Model. J. Chem. Soc., Faraday Trans. 1 1973a, 69, 1694. Ott, R. J.; Rys, P. Sorption-diffusion in Heterogeneous Systems Part 6-General Behavior of Diffusion Controlled Competitive Sorption Process. J . Chem. Soc., Faraday Trans. I 197313, 69, 1705. Tsang, T.; An Approximate Solution of Ficks Diffusion Equation. J. Appl. Phys. 1961,32, 1518. Tsang, T.; Hammarstrom, C. A. Nonlinear Diffusion in the Solid Phase. Ind. Eng. Chem. Res. 1987,26, 855. Vieth, W. R.; Howell, J. M.; Hsieh, J. H. Duel Sorption Theory. J . Membrane Sei. 1976, I , 177. Weisz, P. B.; Hicks, J. S. Sorption-diffusion in Heterogeneous Systems Part 2-Quantitative Solutions for Uptake Rates. Trans. Faraday SOC.1967, 63, 1807.
Stuart R. MacPherson, James E. Maneval Ben J. McCoy* Department of Chemical Engineering University of California Davis, California 95616 Received for review September 14, 1989 Revised manuscript received April 3, 1990 Accepted May 1, 1990
max = maximum 0 = reference
Modeling the Effects of Temperature, Pressure, and Composition on the Viscosity of Crude Oil Mixtures A model is presented for the temperature, pressure, and composition effects on the viscosity of crude oils and their mixtures. The proposed three-parameter model uses Walther's (or ASTM) viscosity-temperature correlation and its modification t o include the effect of pressure on viscosity. The stepwise modeling approach is illustrated using published viscosity data for three middle-East crude oils and their binary and ternary mixtures. One of the three parameters in the viscosity model is found t o be a constant, and simple correlations are provided for the remaining two parameters in terms of crude oil mixture densitv at 25 O C . The validitv of the viscositv model is demonstrated by a good match between the c a h l a t e d and experimental crude oil viscosities.
It is well-known that the viscosity of crude oils, like that of other liquids, decreases with an increase in temperature. The rate of change of viscosity with temperature depends largely on the nature of the crude oil; it can be quite dramatic for viscous crude oils such as the bitumens from Alberta's oil sands. With an increase in pressure, crude oil viscosity increases due to the compressibility effects (Al-Besharah et al., 1989; Mehrotra and Svrcek, 1986, 1987). Numerous correlations are available in the literature on the effect of temperature on liquid viscosity. The following viscosity-temperature correlation, proposed originally by Walther (1931), is particularly relevant to this study: log log (U + 0.7) = bl + b2 log T (1) The ASTM D341 viscosity-temperature chart for highviscosity liquid petroleum products is based on the above correlation (ASTM, 1982). Equation 1 has been found to be applicable for a variety of liquid hydrocarbons, from light to very heavy crude oils, of paraffinic as well as naphthenic and/or aromatic nature. It was shown to be suitable for the dynamic viscosity of bitumens and their fractions, with viscosities ranging from 1to 800000 mPa-s 0888-5885/90/2629-1574$02.50/0
(Khan et al., 1984; Mehrotra, 1990; Mehrotra et al., 1989; Svrcek and Mehrotra, 1988). Al-Besharah et al. (1989) presented kinematic viscosity data for three (light, medium, and heavy) middle-East crude oils at temperatures ranging from 10 to 50 "C. Also provided were viscosity data for 12 binary and 5 ternary mixtures of the three crude oils. As summarized in Table I, the viscosity-temperature-pressure-composition data were obtained at four temperatures (25-50 OC) and five pressures (14.7-8 000 psi or 0.1-55 MPa). Al-Besharah et al. (1989) correlated the compressed crude oil viscosity to the atmospheric pressure viscosity ( y o ) and density as U P = uOeR(P-14.7) (2) where u denotes the kinematic viscosity,at pressure P (in psi). $arameter R in eq 2 was expressed as a linear function of the crude oil density. For crude oil blends (assumed to be simple mixtures), Al-Besharah et al. (1989) used a mass-averaged density. However, the mixture density should be calculated as Pmir = [E~i/~il-' (3) where x i denotes the mass (weight) fraction. More im-
0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1575 Table I. Summary of Data by Al-Beeharah et al. (1989) for the Effects of Pressure and Temperature on the Viscosity of Middle-East Crude Oils and Mixtures r T - P data sample
light
composition, wt % medium
light medium heavy
100 0 0
0 100 0
90 60 40 10 90 60 40 10 0
0
10 40 60 90 0 0 0 0 90 60 40 10
30 30 55 20 10
30 50 40 30 70
7 at 25-50 "C,
Pmix
at 25 "C, g/cm3
no. of pts
0.8376a 0.8992" 0.9601"
22 22 22
4.850-17.81 24.98-189.1 130.0-2470.0
0.8434 0.8612 0.8735 0.8926 0.8484 0.8826 0.9070 0.9463 0.9049 0.9226 0.9348 0.9536
22 22 22 22 18 22 22 18 22 18 22 22
5.625-21.14 8.192-38.92 11.95-66.68 17.80-139.21 5.909-24.44 11.98-71.81 20.70-190.29 75.02-1110.60 25.10-226.70 38.06-448.21 60.81-795.50 109.02-1880.60
Ternary Mixtures 40 0.9022 20 0.8908 5 0.8669 50 0.9148 20 0.9040
22 22 22 22 22
18.20-135.63 13.30-96.03 7.651-38.52 26.02-229.20 21.71-179.90
heavy Crude Oils 0 0 100
0.1-55 MPa, mm2/s
Binary Mixtures 1
2 3 4 5 6 7 8 9 10 11 12
0 0
0 0 0 0 10 40 60 90 10 40 60 90
Measured; pmixcalculated from eq 3.
portantly, Al-Besharah et al. (1989) did not give any correlation for vo or up with temperature. For calculating the viscosity of compressed crude oils at any temperature from eq 2, the atmospheric pressure viscosity at the same temperature must be known. Hence, eq 2 is useful only at temperatures for which the viscosity data were provided. In this study, a viscosity-temperature-pressure equation is developed from the data of Al-Besharah et al. (1989). Firstly, the atmospheric pressure-viscosity-temperature data are fitted to the Walther (1931) correlation. Secondly, the effect of temperature on the viscosity of all three middle-East crude oils, as well as their binary and ternary mixtures, is shown to be represented with a single value of parameter bz in eq 1. Thirdly, the compressibility effect on crude oil viscosity is incorporated by using a functional form given by Mehrotra and Svrcek (1986). The two parameters in the viscosity model are expressed in terms of crude oil density. Finally, the calculation procedure is demonstrated with an example that highlights the effect of mixture composition on viscosity.
Correlation of the Effect of Temperature on Crude Oil Viscosity By using qo to denote either the dynamic (p,, in m P a d or the kinematic ( y o in mmz/s or cSt) viscosity a t atmospheric pressure, eq 1 is written as log log ( 9 0 + 0.7) = b1 + bz log T (4) The atmospheric pressure viscosity data of Al-Besharah et al. (1989) were regressed by using eq 4. The results are summarized in Table 11. As noted, correlation of each set of data for the three crude oils and their binary and ternary mixtures is very good. Average absolute deviations (AADs) are less than 4 % , in most cases. One-Parameter Viscosity Correlation. The values of parameter bz, Table 11, are noted to be scattered in a narrow range, from -3.3 to -3.7, for most samples. This indicates that a one-parameter viscosity correlation, similar to that for bitumens by Svrcek and Mehrotra (1988), is
9.2
crude oils 0 binary 1-4
Q
A
binaN 5-8
0 binaG9-12
v ternary 1-5 8.6 0.82
0.86
0.90
Density a t 25
OC,
0.94
0.98
g/cm3
Figure 1. Variation of parameter bl with the middle-East crude oil density at 25 "C.
possible for the middle-East crude oils. A single value of b, = -3.5 was found to be optimum for all 20 sets (i.e., 3 crudes and 12 binary and 5 ternary mixtures) of the atmospheric pressure-viscosity data. As given in Table 11, the AADs are again low and compare well with those for the two-parameter case. The resulting one-parameter viscosity-temperature correlation for the middle-East crudes and their mixtures is log log (uO + 0.7) = b1 - 3.5 log T (5)
Dependence of Parameter b l on the Oil Density. Figure 1 suggests a relationship between parameter bl in eq 5, Table 11, and the corresponding densities a t 25 "C listed in Table I. However, a similar trend was not found between the bl values and the density for the two-parameter case (eq 4). There is some scatter in Figure 1, particularly in the results for binary mixtures 6-8 and all ternary mixtures. The deviation may be attributable to possible discrepancies in the reported composition (or proportions) of the binary and ternary mixtures. Additional evidence of inconsistencies in the results for binary mixtures 6-8 and ternary mixtures is given below. Mixing Rule for Parameter b 1. Several methods for estimating the liquid mixture viscosity are available in the
1576 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 Table 11. Parameters in Viscosity-Temperature Correlations for Middle-East Crude Oils and Mixtures [Data of Al-Besharah et al. (1989)] one-parameter correlation (eq 5) two-parameter correlation (eq 4) correlated b, and b, correlated b, predicted b, b, AAD,” % bi AAD,” % bi Crude Oils -3.3797 3.5 8.6466 3.7 8.3482 4.1 4.1 8.9245 -3.3845 8.6379 1.3 1.2 9.1062 -3.4525 8.9883
samde light medium heavy
6, (eq 9) AAD,” 7%
8 9 10 11 12
8.0545 8.6059 8.3944 9.3473 8.4859 8.7840 9.2562 9.1371 9.4058 9.2964 8.7196 9.0025
-3.2502 -3.4385 -3.3308 -3.6831 -3.4174 -3.4832 -3.6374 -3.5315 -3.6885 -3.6228 -3.3736 -3.4637
Binary Mixtures 4.7 8.6742 2.5 8.7586 3.8 8.8142 2.0 8.8929 6.1 8.6909 5.2 8.8257 3.3 8.9152 1.5 9.0589 1.8 8.9382 3.3 8.9918 3.6 9.0333 1.8 9.0927
4.7 2.6 3.8 3.5 6.1 5.2 3.4 1.8 3.8 3.4 4.9 2.0
8.6839 8.7797 8.8335 8.9035 8.7215 8.8903 8.9748 9.0769 8.9465 9.0065 9.0424 9.0911
6.7 15.7 16.5 9.4 20.7 71.1 80.4 26.3 8.0 17.5 12.5 2.3
1 2 3 4 5
8.9689 9.8056 8.9714 9.3427 9.4061
-3.5324 -3.8841 -3.5900 -3.6615 -3.6945
Ternary Mixtures 8.8882 2.1 8.8491 0.6 8.7472 1.0 8.9406 0.3 8.9169 0.2
2.1 3.4 1.2 1.6 2.1
8.9523 8.9078 8.8088 8.9909 8.9485
74.8 58.9 48.9 61.9 32.3
1 2 3 4 5 6 n
‘AAD =
tl/”),X(Iuca~
- uexpl/Yexp).
literature (Irving, 1977). Most simple viscosity blending formulas do not involve any viscous interaction term and can be expressed as (Reid et al., 1977)
f ( $ = Cx,f(7,1
(6)
In eq 6, x, may be the mass (weight), volume, or mole fraction, while many forms of the viscosity function f ( q ) have been suggested in the literature (Reid et al., 1977). The use of 9 in eq 6 again implies that the available liquid mixture viscosity formulas are for either p or u. Mehrotra et al. (1989) evaluated several viscosity functions for the blending of bitumen fractions. If we chose f ( 7 ) E log (7 + 0.71, then eq 6 becomes log ( q + 0.7) = EX,log
(vi
+ 0.7)
(7)
Next, eq 5 is written as
+ 0.7) = 10bl-3.51WT = 10b1T-3.5 Combining eq 7 (with 9 = v ) with eq 8 gives log
(uo
1061~-3.5=
(8)
(C~~JO(bl)t)T-3,5
or
6,
= log
[,&lO(*l)l]
(9)
Equation 9 provides a simple mixing rule for the single parameter bl in eq 5 or 8. Note that a similar temperature-independent mixing rule cannot be obtained for either of the two parameters in eq 4. As mentioned previously, three possible choices for x i in eq 6,7, or 9 have been suggested in the literature. Molar masses of the middle-East crude oils are not available; hence, the choice of x i E mole fraction cannot be verified. The choice of xi-= mass (weight) fraction was tested, using eq 9 to predict bl followed by viscosities calculated from eq 8. The results are summarized in the last column of Table 11. For binary mixtures 6-8 and all ternary mixtures, the AADs are large (>20%). Note that the results for these mixtures were scattered in Figure 1. The AADs are reasonable (
