3860
J . Phys. Chem. 1989, 93, 3860-3864
Viscosity of Liquid Hydrocarbons, Mixtures and Petroleum Cuts, as a Function of Pressure and Temperature M. Kanti, H. Zhou, S. Ye, C. Boned,* B. Lagourette, H. Saint-Guirons, P. Xans, and F. Montelt Laboratoire de Physique des MatPriaux Industriels. UniversitP de Pau et des Pays de I'Adour. Centre Universitaire de Recherche Scientifique, Avenue de I'UniversitP, 64000 Pau, France (Received: January 4 , 1988; In Final Form: September 20, 1988)
The viscosity and density of petroleum fractions obtained from "Arabian light" has been measured as a function of pressure ( 1-1 000 bar) and temperature. Different models previously tested on simpler systems (pure alkanes, pure alkylbenzenes, and mixtures of alkanes) have been used, and the experimental data have been fitted to a mean deviation of 2.1% and a maximum deviation of 7.3% for 216 experimental values. The model selected here needs only one viscosity measurement at atmospheric pressure and at one temperature, for a given fraction.
Introduction Viscosity is one of the main physical properties involved in the simulation of the behavior of fluids from natural petroleum reservoirs, allowing differentiation between the oils produced. Viscosity is a measure of the ability of the fluid to circulate within the reservoir and through the production lines and depends strongly on pressure and temperature: so, it is very important to know the viscosity in a wide range of temperature and pressure and to be able to model that behavior. It is very difficult to determine in detail the chemical composition of all the components of a crude oil, owing to its complexity. A crude oil may be fractionated by distillation, so that the composition of the resulting fractions is much simpler than that of the original sample. Little information relative to the behavior of petroleum cuts as a function of temperature and pressure may be found in the literature. In this paper, we intend (a) to present new data that we got in reservoir conditions (temperatures from 20 to 100 "C and pressures from 0.1 to 100 MPa) for different fractions of "Arabian light" and (b) to improve a model previously described on earlier results' obtained for pure alkanes (from nC, to nC18),pure alkylbenzenes (butyl, hexyl, octyl), and mixtures of pure alkanes. Finally, we will analyze and compare various empirical equations giving the viscosity as a function of P and T of the pure components, their mixtures, and the petroleum fractions studied.
Sample Characterization The information on the pure samples as well as the composition
of the mixtures has already been reported in ref 1. The 25 fractions were extracted from Arabian light crude supplied by SNEA(P), at temperatures ranging from 65.0 to 435.9 OC. The composition of each fraction was obtained by gas chromatography. As an example, Table I gives the composition of one fraction for an operating range from 150 to 162.5 "C. The distribution of components for the other fractions may be found in ref 2.
--1
P(P,
1 -Aln(l+?) n - PO(^ bar, 7")
One should mention that an error of 0.01 g/cm3 on the determination of pliqcorresponds to an approximate relative error of for viscosity. The extrapolation up to 1000 bar of paq,from measurements up to 400 bar, with use of the Tait equation (which has an accuracy of 0.0005 g/cm3 between 0 and 400 bar) certainly does not give an error greater than 0.01 g/cm3. As a consequence, the main uncertainty on the viscosity values results essentially from the viscometer and not from the used values of ph. Repeated tests showed that the falling time in the viscometer remains constant to within I%, the temperature can be determined to f0.5 OC, and the pressure can be determined to f l bar. One can estimate then that the viscosity values are determined with an accuracy of 2%. The apparatus is less suited for the study of the lighter cuts because of their very low viscosity, resulting in very short falling times. In addition, owing to the constraints of the initial filling of the measuring cell, the heaviest fractions could not be studied at room temperature and pressure; as a consequence, a complete study as a function of P and T has been limited to the following eight fractions: 150-162.5, 162.5-185, 185-206.1, 206.1-225.9, 225.9-244.5, 244.5-262.2, 262.2-278.1, and 278.7-294.5 "C. The viscosity of all cuts has however been studied, at atmospheric pressure, as a function of temperature with a Lauda ( S / l ) capillary viscometer and a couette rheometer (Contraves Rheomat 30) in steady shearing conditions.
Experimental Results ( a ) Rheological Behavior. The rheological behavior of the different fractions was studied in the 20-100 "C temperature range, with a couette viscometer, in the shear rate range 0.5-1800 SKI.For each sample, the shear stress T scales linearly with the shear rate; as an example, we have plotted on Figure 1 for the 278.7-294.5 oc fraction. a consethe 7(4) quence, the 'I =f(+)curves (Figure 2) are horizontal lines that exmess Newtonian behavior for all the Detroleum fractions defined above, their dynamic viscosity (measured at a given temperature and at atmospheric pressure) being independent of the shear rate.
4
Experimental Techniques The viscosity of the samples was determined with use Of a body viscometer described in previous technical papers.'14 The dynamic viscosity of the liquid at a given Pressure p and ternperature T is given by q(P3
values of pliq up to 1000 bar were extrapolated with Tait's modified equation?
'0 = K(ps01- Pliq)At
where Ar represents the falling time, K is a parameter dependent on the shape and nature of the falling body, psol is the density of the falling body (here, psol= 8.700 g/cm3), and pliqis the density of the liquid under test. The density pliqwas measured for temperatures ranging from 25 to 100 OC, up to 400 bar, with an Anton-Paar densimeter (DMA 45) equipped with a pressure cell (DMA 512). Temperature was controlled within 0.1 OC. The 'Present address: Sociitt Nationale Elf-Aquitaine, 64000 Pau, France.
0022-3654/89/2093-3860$0l.50/0
( 1 ) Ducoulombier, D.; Zhou, H.; Boned, C.; Peyrelasse, J.; Saint-Guirons, H.; xans, p. J , phys, Chem, 1986, 90, 1692-1700. (2) Kanti, M . These de 3c cycle, Universite de PAU, 1988. (3) Ducoulombier, D.; Lazarre, F.; Saint-Guirons, H.; Xans, P. Rev. Phys.
App'. 1985* 20* 735-740.
(4) Lazarre, F.; Ducoulombier, D. Fr. Patent No. 8211550, 1982. (5) Hogenboom, D. L.; Webb, W.; Dixon, J. A. J . Chem. Phys. 1967, 4 6 ( 7 ) , 2586-2598.
0 1989 American Chemical Society
Viscosity of Liquid Hydrocarbons
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3861
TABLE I: Detailed Composition of the Fraction 150-162.5 OC components %M % wt % vol ~~~
3-ethylpentane methylcyclohexane 2,s-dimethylhexane 2,4-dimethylhexane ethylcyclopentane 1,t-2,~-4-trimethyIcyclopentane toluene 2,3-dimethylhexane 2-methyl-3-ethylpentane 2-methylheptane 4-methylheptane 3-methylheptane 3,4-dimethylhexane 1,~-2,~-4-trimethylcyclopentane 1,~-3-dimethylcyclohexane 1,t-4-dimethylcyclohexane 1,l -dimethylcyclohexane 1-methyl-t-3-ethylcyclopentane n-octane 1,t-2-dimethylcyclohexane 1,t-3-dimethylcyclohexane 1,~-4-dimethylcyclohexane 2,3,5-trimethylhexane 2,2-dimethylheptane 2,4-dimethylheptane 1-methyl-4-ethylcyclopentane 2,6-dimethylheptane 2,s-dimethylheptane 3,s-dimethylheptane ethylbenzene ethylcyclohexane I , 1,3-trimathylcyclohexane
1,1,4-trimethylcyclohexane 1,~-3,c-5-trimethylcyclohexane various C9 naphthenes p-xylene m-xylene 2,3-dimethylheptane 3,4-dimethylheptane 1,~-3,t-5-trimethylcyclohexane 4-methyloctane 3-ethylheptane 3-methyloctane o-xylene
l,f-2,~-3-trimethylcyclohexane 1,f-2,~-4-trimethylcyclohexane isopropylcyclohexane 1,~-2,~-4-trimethylcyclohexane 1-methyl-t-4-ethylcyclohexane n-nonane 1-methyl-c-2-ethylcyclohexane 1-methyl-r-2-ethylcyclohexane
3,3,5-trimethylheptane 4,4-dimethyloctane 2,s-dimethyloctane propylbenzene 2,6-dimethyloctane 2,3-dimethybctane 4,s-dimethyloctane 3-ethyltoluene 4-ethyltoluene tetramethylcyclohexane 2-ethyltoluene 2-methylnonane 1,3,5-trimethylbenzene 3-methylnonane 1,2,4-trimethylbenzene n-decane various Clo naphthenes Cl,+
1.800 0.088 0.017 0.017 0.027 0.017 0.350 0.043 0.044 0.305 0.103 0.297 0.043 0.014 0.156 0.156 0.119 0.205 2.278 0.024 0.199 0.080 0.029 0.032 0.190 0.070 0.040 0.683 0.266 0.920 1.727 0.088 0.239 0.866 2.720 1.550 2.362 0.013 0.013 0.439 2.755 0.393 3.654 0.139 0.462 0.462 0.470 0.490 1.464 26.221 0.058 0.518 1.857 0.838 0.568 1.871 1.817 0.040 1.223 7.587 5.132 0.227 8.266 3.155 0.202 2.388 0.202 0.04 1 8.867 0.000
1.428 0.069 0.016 0.016 0.021 0.015 0.256 0.039 0.039 0.276 0.093 0.268 0.039 0.012 0.138 0.138 0.106 0.182 2.060 0.022 0.177 0.071 0.030 0.033 0.193 0.062 0.041 0.694 0.270 0.773 1.534 0.088 0.239 0.866 2.7 18 1.303 1.985 0.0 14 0.0 14 0.439 2.797 0.399 3.710 0.117 0.461 0.461 0.470 0.490 1.463 26.620 0.058 0.518 2.092 0.944 0.640 1.780 2.046 0.045 1.378 7,.218 4.882 0.252 7.865 3.553 0.192 2.690 0.192 0.047 9.846 0.000
~
1.575 0.069 0.018 0.017 0.021 0.0 16 0.227 0.042 0.042 0.305 0.102 0.293 0.042 0.012 0.139 0.140 0.105 0.183 2.259 0.022 0.174 0.070 0.032 0.035 0.209 0.061 0.044 0.743 0.290 0.687 1.502 0.086 0.231 0.852 2.635 1.166 1.771 0.014 0.014 0.432 2.996 0.423 3.974 0.102 0.456 0.452 0.474 0.474 1.421 28.591 0.056 0.503 2.228 0.992 0.681 1.601 2.173 0.047 1.425 6.441 4.372 0.245 6.892 3.769 0.171 2.830 0.169 0.049 9.311 0.000
( b ) Density and Dynamic Viscosity versus Temperature and Pressure. T a b l e I1 gives t h e density values (g/cm3) measured for t h e eight fractions for which viscosity was studied a s a function of pressure. T h e pressure values a r e given in bars a n d t h e temp e r a t u r e values in degrees centigrade. T a b l e 111 indicates t h e experimental values of t h e dynamic viscosity (centipoise). Figure
/
1
1
1
500
1000
1500
y
Figure 1. Rheological curves for the fraction 278.7-294.5 temperatures.
OC
-
(5-9
at various
T = 25'C
-
T 40'C
.:
_ _ _ .-
T = 60'C
T = 8O'C
0 500 1060 1500 f (s-'I Figure 2. Dynamic viscosity as a function of shear rate (278.7-294.5 OC cut).
3 shows the variation of dynamic viscosity 9 for various fractions. Data Analysis ( a ) W e have already given in a previous paper' t h e viscosity values of pure alkanes, p u r e alkylbenzenes, a n d alkane mixtures
Kanti et al.
3862 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 TABLE 11: Density (g/cm3) of the Cuts (P,bar; T , "C) T P 25 40 60 80
100
TABLE 111: Dynamic Viscosity (cP) of the Cuts (P, bar; T , "C) T P 25 40 60 80 100
0.69 0.85 1.02 1.19 1.385 1.585
Cut: 150-162.5 "C 0.58 0.475 0.71 0.57 0.845 0.685 0.995 0.805 1.16 0.925 1.33 1.065
0.84 1.06 1.28 1.55 1.83 2.21
Cut: 162.5-185 "C 0.69 0.55 0.85 0.67 1.04 0.79 1.25 0.94 1.44 1.11 1.65 1.27
0.40 0.54 0.65 0.77 0.895 1.03
1.1 1.35 1.64 2.00 2.36 2.86
Cut: 185-206.1 OC 0.905 0.73 1.13 0.84 1.33 1.05 1.57 1.22 1.86 1.42 2.20 1.67
0.535 0.685 0.8 15 0.96 1.13 1.29
200 400 600 800 1000
1.43 1.87 2.27 2.79 3.33 4.04
Cut: 206.1-225.9 OC 1.1 1 0.84 1.42 1.06 1.73 1.28 2.1 1 1.54 2.53 1.84 3.01 2.14
0.635 0.81 0.99 1.19 1.39 1.60
0.755 0.762 0.768 0.778 0.790 0.797
1 200 400 600 800 1000
1.86 2.30 2.89 3.60 4.40 5.37
Cut: 225.9-244.5 O 1.42 1.03 1.78 1.30 2.18 1.59 2.69 1.92 3.25 2.29 3.89 2.70
0.795 0.99 1.21 1.45 1.71 1.99
0.775 0.780 0.785 0.794 0.803 0.811
0.760 0.766 0.771 0.781 0.790 0.799
1 200 400 600 800 1000
2.37 3.09 3.78 4.78 5.98 7.39
Cut: 244.5-262.2 OC 1.75 1.26 2.28 1.64 2.84 2.03 3.53 2.47 4.32 2.95 5.21 3.54
0.95 1.24 1.51 1.83 2.17 2.56
0.75 0.98 1.19 1.42 1.68 1.96
0.794 0.799 0.804 0.813 0.821 0.829
0.780 0.786 0.792 0.801 0.810 0.818
1 200 400 600 800
3.00 3.92 5.06 6.42 8.06 9.93
Cut: 262.2-278.7 OC 2.15 1.50 2.86 2.01 3.61 2.49 4.49 3.08 5.57 3.75 6.71 4.49
1.13 1.51 1.87 2.26 2.73 3.26
0.93 1.16 1.42 1.72 2.05 2.41
Cut: 278.7-294.5 O C 0.810 0.796 0.824 0.802 0.828 0.815 0.819 0.806 0.832 0.827 0.814 0.839 0.834 0.822 0.847 0.830 0.853 0.842
0.781 0.787 0.792 0.801 0.809 0.8 18
3.84 5.32 6.86 8.79 11.21 13.86
Cut: 278.7-294.5 OC 2.67 1.81 3.67 2.49 4.70 3.10 6.26 3.86 7.41 4.78 9.23 5.77
1.32 1.82 2.25 2.72 3.31 3.91
1.10 1.38 1.70 2.06 2.43 2.89
I 50 100 200 300 400
0.763 0.772 0.776 0.785 0.793 0.800
Cut: 150-162.5 "C 0.751 0.736 0.759 0.743 0.764 0.748 0.7735 0.759 0.7815 0.768 0.7895 0.777
1
50 100 200 300 400
0.769 0.773 0.778 0.786 0.795 0.800
Cut: 162.5-185 "C 0.758 0.742 0.762 0.748 0.767 0.753 0.776 0.763 0.785 0.772 0.791 0.779
0.727 0.733 0.739 0.749 0.758 0.767
0.71 1 0.7 18 0.725 0.736 0.746 0.755
1 50 100 200 300 400
0.781 0.785 0.789 0.797 0.805 0.811
Cut: 185.0-206.1 OC 0.755 0.770 0.775 0.760 0.779 0.765 0.788 0.775 0.783 0.796 0.803 0.790
0.741 0.747 0.752 0.762 0.772 0.780
0.726 0.733 0.738 0.750 0.759 0.768
I 50 100 200 300 400
0.788 0.791 0.797 0.804 0.8 12 0.8 18
Cut: 206.1-225.9 OC 0.777 0.764 0.783 0.769 0.787 0.775 0.783 0.795 0.791 0.803 0.810 0.799
0.750 0.755 0.762 0.771 0.779 0.788
0.736 0.742 0.749 0.758 0.767 0.777
1 50 100 200 300 400
0.813 0.8 17 0.821 0.828 0.834 0.842
Cut: 225.9-244.5 OC 0.786 0.802 0.806 0.792 0.797 0.810 0.818 0.805 0.813 0.825 0.832 0.821
0.771 0.777 0.782 0.792 0.802 0.809
1 50 100 200 300 400
0.814 0.817 0.822 0.828 0.835 0.841
Cut: 244.5-262.2 "C 0.803 0.788 0.793 0.807 0.812 0.798 0.806 0.819 0.826 0.813 0.821 0.833
1
50 100 200 300 400
0.832 0.836 0.840 0.846 0.853 0.860
Cut: 262.2-278.7 OC 0.808 0.822 0.813 0.827 0.817 0.830 0.838 0.826 0.833 0.845 0.840 0.852
1 50 100 200 300 400
0.835 0.839 0.842 0.849 0.856 0.862
0.720 0.725 0.732 0.743 0.753 0.762
0.705 0.708 0.715 0.728 0.739 0.749
(binary and quaternary), measured at different temperatures and pressures. The analysis of these results showed that the following equation, based on the principle of corresponding states, gives a reasonable representation of the data:
1
200 400 600 800 1000 1
200 400 600 800 1000 1
200 400 600 800 1000 1
1000
1
200 400 600 800 1000
0.52 0.65 0.80 0.94 1.10 1.27
C
homologous quantities T,, Pd, and M , correspond to a reference of known viscosity to The A , B, and C coefficients are obtained' by fitting of the experimental data. A mean deviaton of 6.9% and a maximum deviation of 52.2% are obtained when eq I is applied to the 266 experimental points relative to the pure alkanes (from C, to CI8),with nC14as a reference. The deviation is defined as texpt!
dev = 1001
- tcaic
71exptl
In this equation, Tc,Pc, and M a r e the critical temperature, critical pressure, and molar mass of the component, respectively, and the
0.295 0.43 0.535 0.63 0.735 0.825
1
The mean deviation is 9.7% and the maximum deviation is 21.4% for 90 experimental points of the alkylbenzene series.
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3863
, 0
15-
0
0
I -
0
.
0
0
0 0l I
. /O
-0-e*-
/ /
I
200
0-
.o
I
0'
--
/t
. I
I
I
0
0 0
. I
4.
I
.
.
,.'
.. ...
,
0 0 0 .
1, relation 2 requires neither a reference component nor critical values P, and T,. In addition, numerical fitting7 of the parameters a-g leads to a better fitting of the data. For example, maximum deviation is decreased to 12.5% for all the n-alkanes (nCI-nCI8, including the data of nC14,Le., 294 experimental points), and the mean deviation is decreased to 3.5%. The agreement is still good when applied to the alkane mixtures studied in ref 1, with the same coefficients as for the pure alkanes: mean deviation is 5.6% and maximum deviation is 12.4%. However, the deviations obtained for a different chemical family (alkylbenzenesl) are much less satisfactory (mean deviation 21.7%, maximum deviation 32.7% for 90 experimental points). This is probably related to the fact that, in eq 2, the only characteristic parameter of the sample is its molar mass M . So, the chemical family is not taken into account as far as the coefficients of the equation are concerned. We introduced the acentric factorg w in order to improve the fitting of the samples' viscosity: this quantity is easy to derive for the pure components and for mixtures containing a few components (the blending law is then w = Cluixiwhere xi is the mole fraction). Equation 2 was then modified7 as
/ /
-t-/
/
I
400
WT 2 0 6 J
- 221.9.C
.'
/
0
=/ ,
,
a M + bw
[+-
+ c + d-1 M- w
jP+ g + ( h P +
+e-)
w
1-w
x \
1 i) exp
k- M
1-w
+ 1-1 - w
where a-1 are fitting parameters. This last expression gives a mean deviation of 4.1% and a maximum deviation of 13.3% with the experimental data for all pure components (alkanes alkylbenzenes, 384 experimental points). The mean and maximum deviations are 3.8 and 13.3% for the whole experimental data set reported in ref 1 (468 points). When the experimental viscosity data found in the literature for other pure components (in particular, naphthenes and olefinslO) at P = 1 bar were used, with the alkane and alkylbenzene fitting parameters, deviations were found to scatter from an average of 2-60%, depending on the components. This demonstrates, in fact, that the fitting parameters a-1 depend on the two reference families from which they were derived. The addition of the acentric factor w improved the model but is not adequate to extend the use of eq 3 to other chemical families. Furthermore, it is difficult to determine correctly M and w in the case of petroleum fractions. (b) Kashiwagi and Makitall suggested a different approach, namely, representing the dynamic viscosity of a liquid at a given temperature T, as a function of pressure, by measuring only the viscosity at atmospheric pressure and temperature T. They consider that this viscosity value contains specific information on the sample studied. They propose the following equation
+
0
, ,,
/ -
I
0
I
200
I
400
I
I
600
000
I
-
KKKI P(W11
Figure 3. Comparison between experimental data (D,0,e, A, 0 ) and calculated values from eq 5 (- -).
In the case of the mixtures (four binaries, one quaternary; 84 experimental points), the classical equation used to define the critical values P, and T, is indicated in ref 6. In that case, the mean and maximum deviations are 9.2% and 18.9%. The very large number of chemical species involved in the composition of the petroleum fractions (see Table I) makes illusory attempts to define significant critical quantities in this way. So we have been using7 the following modified Van Velzen8 theory to fit our experimental data cP + d
where E and D are fitting parameters. This equation turns out to be reasonable when applied to each individual component. However, the values of D and E vary widely from one temperature to another and from one component to another. Furthermore, another disadvantage is to have to make a measurement (at Po = 1 bar) at each temperature. To correct this, we propose the following expression
1
+ (eP + d ) exp(gM)
where a-g are adjustable fitting parameters. Compared with eq (6) Pedersen, K. S.; Fredenslund, A,; Christensen, P. L.; Thomassen, P. Chem. Eng. Sci. 1984, 39(6), 1011-1016. (7) Zhou, H. Thke de Doctorat, Universite de PAU, 1988.
(8) Van Velzen, D.; Lopes Cardozo, R.; Langenkamp, H. Ind. Eng. Chem. Fundam. 1972, 11(1), 20-25.
~ ( bar, 1 7') = ~ ( bar, 1 To)exp
[ ( k- -io)] cy
-
(9) Passut, C. A,; Danner, R. P. Ind. Eng. Chem. Process Des. Dev. 1973, 12(3), 365-368. (10) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: London, 1977; Chapter 9, pp 39 1-469. ( 1 1) Kashiwagi, H.; Makita, T. Int. J . Thermophys. 1982,3(4), 289-305.
3864
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989
TABLE I V List of Optimized Coefficients Intervening in Equation 5 (Alkanes and Alkvlbenzenes) a = 0.275 832 b = 0.533 739 c = 1.838385 d = 40.598 32 e = 236.3475 f = 1610.261 g = 6.729026 h = 481.5716 i = 1278.456
in agreement with van Velzen’s equation. The a coefficient has been developed as a = gxo2+ hxo + i
in which xo = In ( ~ ( 1bar, T o ) )
where Tois a selected reference temperature. Besides, we propose to develop the constants E and D of eq 4 in trinomial form
E = ax2+bx+ c
D + 1 = dx2+ex+f
where x = In [ ~ ( bar, l T)]. At last, substituting these expressions in eq 4, we obtain the following equation with nine fitting parameters a-i:
This equation involves only the reference measurement V ( 1 bar, To)at Po = 1 bar and one temperature To. For obvious practical reasons, it is reasonable to choose a value of To close to room temperature. As a consequence, the nCI and nC4 data will not be considered, within the series of pure components.I This last equation when applied to the other alkanes and alkylbenzenes (3 13 experimental points) gives a mean deviation of 3.0% and a maximum deviation of 9.9%, with the parameter values indicated in Table IV. The same coefficients applied to the alkane mixtures give a 3.3% mean deviation and a 9.6% maximum deviation. These various equations used to fit the data are purely empirical; their importance lies, however, in the need to provide an accurate estimate of the viscosity from a minimum of experimental data.
Kanti et al. TABLE V: List of Optimized Coefficients Intervening in Equation 5 (Fractions) u = -0.483416
d = -727.7665 g = 20.74
b = 1.339004 e = 962.8084 h = 427.365
c = 2.6534
f = 2310.752 i = 1323.012
Comparison of Equation 5 versus Experimental Data on Petroleum Fractions For the 216 experimental points of these eight oil fractions, a mean deviation of 3.0% and a maximum deviation of 12.7% are obtained with the previous values of constants a-i, which is a reasonable result. However, just as for the other methods, the set of coefficients (Table IV) depends on the chemical families (Le., alkanes or alkylbenzenes). As a consequence, we have determined a specific set of coefficients of these systems (Table V) to take into account the diversity of composition of the samples. The numerical analysis from eq 5 along with the coefficients used above provides better fitting of the experimental data, since the mean deviation falls to 2.1% and the maximum deviation to 7.3%. We have plotted in Figure 3 the curves representing the values derived by numerical analysis compared with the experimental points for various fractions. In this type of numerical approach with use of equations involving a relatively large number of coefficients, it frequently occurs that coefficient values obtained depend on the families of samples used as references. Our data show that, among the nine coefficients involved in eq 5, all of the coefficients are modified sharply when changing from Table IV to Table V, except coefficients h and i . This observation clearly shows the limits of this type of empirical formulation in which it seems hopeless to seek any physical meaning for the various parameters. However, the interest of this approach is to provide a very easy means of estimating the fraction viscosity under reservoir conditions from a single reference measurement made at atmospheric pressure and, generally, room temperature. Equation 5 is, however, empirical, and its coefficients are dependent on the chemical nature of the components. The use of Table IV (pure’ alkanes and alkylbenzenes) coefficients gives 3% of mean deviation (1 2.7% maximum), and the use of Table V coefficients gives 2.1% (7.3%). As a consequence, we do believe that eq 5 (Table V) can be applied to various crudes, mainly composed of alkanes and alkylbenzenes. Acknowledgment. We are indebted to the Socittt Nationale Elf-Aquitaine for the financial support of this work.
