may be found from the result (Tanner and Simmons, 1967)
pa,
cc
(
1 - C anC1 -
S T -
n
(1
+
~ X P(-B/Xn./)I
(4)
The curve for B = 12 is fairly satisfactory (Figure 1). Although the errors might be reduced by letting B vary with n, this is not attempted here because the lack of information a t very low shear rates and frequencies leads to some uncertainties. From the knowledge of A,, an, and B already given, the elongational viscosity may be predicted under certain assumptions concerning the rupture of the network. The simplest assumption (Tanner and Simmons, 1967) gives the result p =
C aflrn-l
from a t least one point of view (Tanner, 1966), the tensile and shearing flows are the most divergent steady homogeneous flows possible in an incompressible medium. Nomenclature
6 a,
-
B, B = rupture strain magnitudes G = extensional rate, d u / d z , see.-’ =
n
= integer
q
= integer
V
r,
+
+
This may be solved numerically for a given value of B. Calculation using B = 12 and Equations 5 and 6 gives the predicted Trouton viscosity (Figure 1). Depending on the purpose of the exercise, the prediction may or may not be considered satisfactory. If one considers the limited amount of experimental information usually available in practice and the need to make reasonable predictions of moderate accuracy, the rupture theory seems to be helpful for engineering purposes. The theory can be expected to work equally well for any steady homogeneous flow, since
E
t, t’ = times, sec. u, v, w = components of velocity vector, cm./sec. y
where r = AnG. To relate the value of B to B it is assumed that the trace of the Finger strain measure (Lodge, 1964) governs rupture of the network. The strain matrix is computed relative to the present time, t, as reference; a t a time t’ in the past it may readily be shown (Lodge, 1964) that in simple shearing the trace of the strain matrix is y 2 ( t- t ’ ) ; when this quantity reaches a value of BZ,rupture occurs. Similarly, in elongational flow the quantity G(t - t ’ ) occurs in the calculation. The trace of the strain matrix may be computed and equated to B2; one finds the equation for the rupture parameter B, which defines the rupture time, t‘, as 3 B2 = exp 2 B 2 exp - B Hence, (6 ) B-lnBforB>>l
47
i
5 , y,
n
= amplitude of sinusoidal shear rate, sec.-l = strength of nth time constant, poises
pd ps pt p* Xn
i;
w
= velocity vector, cm./sec. z = coordinates, cm. = shear rate, see.-’ = = X,G = dynamic viscosity, poises = shearing viscosity, poises = tensile viscosity, poises = complex viscosity, poises = nth time constant, sec. = complex amplitude of shear stress, dynes/sq. cm. = frequency, rad./sec.
literature Cited
Ballman, R. L., Rheol. Acta 4, 137 (1965). Lodge, A. S., “Elastic Liquids,” Academic Press, New York, 1964.
Tanner, R. I., IND.E m . CHEM.FUXDAMENTAL~ 6, 55 (1966). Tanner, R. I., J. A p p l . Polymer Sei. 12, 1649 (1968). Tanner, R. I., Simmons, J. M., Chem. Eng. Sci. 22, 1803 (1967). Trouton, F. T., Proc. Roy. SOC.A77, 426 (1906). ROGER I. TANNER Brown University Providence, R.I. 0.9912 RICHARD L. BALLMAN Momanto Co. Springjield, Mass. RECEIVED for review February 15, 1968 ACCEPTEDMarch 3, 1969 Investigation supported in part by the National Aeronautics and Space Administration under the hlultidisciplinary SpaceRelated Research Program (Grant NGR40-002-009) at Brown University.
CORRELATION OF DIFFUSION COEFFICIENTS FOR PARAFFIN, AROMATIC, AND CYCLOPARAFFIN HYDROCARBONS IN WATER Diffusion coefficients for all hydrocarbons investigated can be correlated over a temperature range of 2’ to using the Wilke-Chang empirical equation.
60’ C.
UNTIL recently, very few data on the diffusion of hydrocarbons in water have been available. Kartsev et al. (1959) reported work credited to Antonov for the diffusion of methane, ethane, propane, and *hexane in water. Recently, diffusion coefficients were reported for methane at 25’, 45’, and 65’ C. (Gubbins et al., 1966). With regard to other hydrocarbons, diffusion data have been reported
only for acetylene in water (Tammann and Jensen, 1929), propylene in water (Vivian and King, 1964), and ethylene, propylene, and butylene in water (Unver and Himmelbau, 1964). A project has therefore been under way for some time in this laboratory to study the diffusion through water of the various hydrocarbons found in petroleum. Results have been VOL.
8
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Table 1.
Experimental Results for Diffusion Coefficients of Paraffin Hydrocarbons in Water D X 105, Sq. Cm./Second
Hydrocarbon Methane Ethane Propane n-Butane n-Pentane a
40
0.85 0.69 0.55 0.50 0.46
c.
200
f 0.02 f 0.01 f 0.03 f 0.02 f 0.02a
c.
60' C. 3.55 f 0.15 2.94 f 0.12 2.71 f 0.05 2.51 f 0.05 2.24 f 0.10
40' C.
1.49 f 0.04 1.20 f 0.06 0.97 f 0.02 0.89 f 0.04 0.84 f 0.04
2.38 f 0.07 1.94 f 0.04 1.77 f 0.04 1.59 f 0.04 1.49 f 0.03
New result not previously published.
Table 11.
Experimental Results for Diffusion Coefficients of Aromatic and Cycloparaffin Hydrocarbons in Water D X 105, Sq. Cm./Second
--
100 c.
c.
c.
Hydrocarbon Benzene Toluene Ethylbeneene
0.58 f 0.02 0.45 f 0.02 0.44 f 0.04
0.75 f 0.02 0.62 f 0.02 0.61 f 0.05
1.02 f 0.03 0.85 f 0.03 0.81 f 0.03
40' C. 1.60 f 0.05 1.34 f 0.05 1.30 f 0.05
2.55 f 0.05 2.15 f 0.06 1.95 f 0.05
Cyclopentane Methylcyclopentane Cyclohexane
0.56 f 0.02 0.48 f 0.08 0.46 f 0.04
0.64 f 0.03 0.59 f 0.06 0.57 f 0.04
0.93 f 0.02 0.85 f 0.10 0.84 f 0.05
1.41 f 0.09 1.32 f 0.12 1.31 f 0.07
2.18 f 0.07 1.92 f 0.15 1.93 f 0.12
20
reported for the diffusion coefficients of five paraffins (methane, ethane, propane, n-butane, and n-pentane) ; three aromatics (benzene, toluene, and ethylbenzene); and three cycloparaffins (cyclopentane, methylcyclopentane, and cyclohexane) at temperatures ranging from 2" to 80" C. (Bonoli and Witherspoon, 1968; Sahores and Witherspoon, 1969; Saraf et al., 1963; Witherspoon et al., 1968; Witherspoon and Saraf, 1965). The capillary-cell method of measuring diffusion coefficients was used throughout these studies (Saraf et al., 1963; Witherspoon et al., 1968; Witherspoon and Saraf, 1965). Experimental results obtained in these investigations are summarized in Table I for the paraffin hydrocarbons and in Table 11 for the aromatic and cycloparaffin hydrocarbons. The data represent the averages of 10 or more repetitive runs, and the precision shown is the standard deviation of the arithmetic mean. Since diffusion coefficients in liquids are increasingly important in many theoretical and engineering calculations in-
1
\
03 *ASSOCIATED
LIOUIDS, X.26
UNASSOCIATED LIQUIDS. x . 1.0
UNASS0;I
\
-I
b&
I-
r
\\\\
2 1 :V
+
BUTANE
P PENTANE
'
\ I
%*
-
P
\
1.0
LIQUIDS,
' \\\
x
f;,',"
\ \
co0 ' O F
A ETHANE V PROPANE
60° C.
I
I
1
1
1
1
I
I
I
I
1 \ 1 1
Y10
1.0 .~
0.3
Figure 2. Correlation of diffusion Coefficients in water for aromatic and cycloparaffin hydrocarbons 590
l&EC
FUNDAMENTALS
Figure 2 shows the same correlation for aromatic and cycloparaffin hydrocarbons. These results are in agreement with the correlation line for associated liquids, and indicate that the Wilke-Chang equation will predict diffusion coefficients for the aromatic and cycloparaffin hydrocarbons with a maximum error of 10%. Therefore, the Wilke-Chang equation provides a reliable basis for predicting the diffusion of hydrocarbons in pure water. Apparently, this equation will be somewhat more reliable when applied to aromatic and cycloparaffin hydrocarbons than when applied to paraffin hydrocarbons. However, V does not vary as much for the aromatic and cycloparaffin hydrocarbons as for the paraffins of this investigation. Nomenclature
D= AI = T= V=
diffusion coefficient, sq. cm./sec. molecular weight of solvent absolute temperature, O K . liquid molal volume of solute a t normal boiling point, cc./ (9. mole) z = association parameter equal to 2.6 for water and 1.0 for unassociated liquids, dimensionless 7 = viscosity of solvent a t temperature T, cp.
literature Cited
Bonoli, L., Witherspoon, P. A., J . Phys. Chem., 72, 2532 (1968). Gubbins, K. E., Bhatia, K. K., Walker, R. D., A.Z.Ch.E.J. 12, 548 (1966).
Kartsev, A. A., Tabasaranskii, Z. A., Subotta, M. I., Mogilevskii, G. A., “Geochemical Prospecting for Petroleum,” p. 69, University of California Press, Berkeley, Calif., 1959. Othmer, D. E., Thakar, M. S., Znd. Eng. Chem. 46,589 (1953). Sahores, J. J., Witherspoon, P. A., “Advances in Organic Geochemistry,” Pergamon Press, New York, in press, 1969. Saraf, D. N., Witherspoon, P. A,, Cohen, L. H., Science 142,955 (1963).
Scheibel, E. G., Znd. Eng. Chem. 46, 2007 (1954). Tammann, G., Jensen, V. Z., 2. Anorg. Allgem. Chem. 179, 125 (1929).
Unver, A. A., Himmelbau, D. M., J . Chem. Eng. Data 9, 428 (1 Qfi4). \-- 8 .
Vivian, J. E., King, C. J., A.Z.Ch.E.J. 10, 220 (1964). Wilke, C. R., Chang, P., A.Z.Ch.E.J. 1. 264 (1955). Witherspoon,’P. A.,-Bonoli, L., Sahores, J. J.,’All-Union Conference on Origin of Oil and Gas, Moscow, USSR, January 1968 (to be translated into Russian and published with Proceedings). Witherspoon, P. A,, Saraf, D. N., J.Phys. Chem. 69,3752 (1965). P. A. WITHERSPOON LUCIAN0 BONOLI University of California Berkeley, Calif. 94720 RECEIVED for review April 1, 1968 ACCEPTEDNovember 19, 1968 Research supported by a grant from the National Science Foundation.
TRUE GAS CONTENT FOR HORIZONTAL GAS-LIQUID FLOW The use of the mixture Froude number as a parameter for correlating the true and input gas content for twophase, gas-liquid flows is discussed. The correlation, primarily based on the air-water system but also involving other systems, apparently represents the experimental data better than prior holdup correlations. Furthermore, the transitions from separated to intermittent flows are easily noted by the S-shaped regions in the curves. Holdup data were obtained for air-water flowing in an 80-foot-long, 1.5-inch i.d. acrylic pipe in order to substantiate the correlation.
THEpurpose of this communication is to discuss the use of
the mixture Froude number as a correlating parameter for horizontal two-phase flow holdup data, with specific reference to the data of Mamayev (1965) and Guzhov et al. (1967), which have been compared with limited results obtained in our laboratory. It is important for design purposes to predict the true gas content or holdup accurately for isothermal two-phase (gasliquid) flow in horizontal pipes. This is necessary not only for determining pressure drop but in some instances for accurately determining residence times. However, most published correlations, such as those cited in the extensive compilation of references by Gouse (1963, 1964, 1966), are too unreliable or too limited in their range of applicability for general use. Of particular importance to the petroleum industry are long-distance two-phase (gas-liquid) pipelines mainly involving crude and natural gas systems. In many instances, economic considerations dictate the use of moderate velocities (10 to 20 feet per second), resulting in the predominance of flow in the stratified, bdbble, and intermittent (slug or plug) regimes. For these conditions, the most satisfactory published holdup correlation appears to be that of Hughmark (1962), which involves somewhat lengthy calculation. This
correlation is reasonably accurate a t low gas qualities but yields uncertain values a t higher gas qualities, notably for relatively low flow rates. A simpler but less reliable correlation, developed earlier by Lockhart and Martinelli (1949), is also still in use. More recently, another simple correlation has been proposed by Mamayev (1965) and discussed by Guzhov et al. (1967). This method correlates the actual gas holdup with the input gas content, employing only the mixture Froude number as a parameter, and evidently represents experimental data better than the correlations of Hughmark or of Lockhart and Martinelli. From a similarity analysis of horizontal two-phase flow, based on the appropriate hydrodynamic equations, it can be shown (Mamayev, 1965) that the true gas content is described as follows:
d = f(P,FrmRern,P, P, We) For a horizontal pipe, Mamayev (1965) deduced from experimental data that the Reynolds number, Weber number, density ratio, and viscosity ratio have little effect on the true gas content. These data represented primarily the air-water system but also included systems involving a 20-fold range of liquid viscosity, fivefold range of gas density, and a threefold range of Weber number. Despite some uncertainty VOL.
8
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