+
roach and - infinity. From the triple partial product rule applied to P,, v,, T , space either (bT/bP,),, or (bP,/bvJT, or both must pass through zero at the cusp point in order to create an indeterminate form for the triple product such t h a t the possibility is created for the product to achieve the required - 1 value. Imposing the stability condition that ( b P , / b ~ , )5~ ~ 0 requires (bT,/bP,),, to change sign a t the cusp point. If the isothermal slope is finite then (bT,/bPJOr must be zero in order t h a t the triple partial product be -1. If,however, the isothermal slope is zero, then (bT,/bP,) may change sign a t zero or at infinity to satisfy the triple partial product rule. Thus (bT,/bP,) or must have either a continuous or a discontinuous extremum corresponding to bhe isobaric cusp in the 2, T , plot. This is rejected on intuitive grounds. This conclusion can also be reached from another point of view. With (dv,/bTJp, changing sign and becoming infinite a t the cusp point the product p c P mustjhave similar behavior. For positive c,, p must change sign a t the cusp point, eit’her by passing through zero or through infinity. The former case is rejected because i t again gives a p = 0 point not associated with the usual inversion curve. This latter is also rejected because i t requires a n iseiithalpic curve t o have an infinite slope in the T,, P, plane. This of course does not exhaust the discontinuous possibilities. It is still possible for the isobar to have a cusp in t’he2, T , plane such that the common tangent line t o the two segnieiits of the isobar a t the cusp is not oriented parallel to the 2 axis. Then, if 2 and T, are both increasing on one branch of the isobar in the vicinity of the slanted cusp, they must decrease together 011 the other branch of the isobar-again in the neighborhood of the cusp-as the isobar is followed from one side of the locus to the other. This means that T, is not moiiatonic on the isobar joining the crossover points in the T,, P, plane. I n going from one branch of the p = 0 curve a t the low temperature to the other branch along this isobar which is less than P,,,,the temperature increases, then decreases before again increasing. Thus a fold in the properties
occurs in this T,, P, plane and this kind of behavior of the properties is not ordinarily associated with the T,, P, plane; therefore the slanted cusp possibility is also rejected. Conclusions
The loop in the three-constant inversion curve shown in Miller’s Figure 4 cannot exist. Instead it should be compressed into a cusp a t the maximum pressure point in order to satisfy the fact that the inversion curve envelopes the isotherms in the 2, P, plane. As previously noted, hliller comments that the inversion curve closely follows the envelope of the constant-T, lilies for reduced pressures less than or equal to 10, that is, for reduced pressures less than or equal t o t h a t of the inversion curve’s crossover point in the 2, P, plane. T o repeat, the inversion curve must do better than closely follow the envelope of the isotherms; it is the envelope of such curves, not just for reduced pressures 5 10 but for all reduced pressures associated with the inversion curve. It is this envelope property of the inversion curve that becomes a n extremely sensitive test of an equation of state, especially with respect to data correlated according to the corresponding states principle. literature Cited
Bursik, J. W., Onuf, B. O., Final Report, USARTC Grant DA TC 44 177, June 1963. Lydersen, A. L., Greenkorn, R. A., Hougen, 0. A,, “Generalized Thermodynamic Properties of Pure Fluids,” University of Wisconsin, College of Engineering, Rept. 4, Oct 1955. Miller, 11. G., IND.ENG.CHEM.,FENDAM. 9, 58.5 (1970). Van Wylen, G. T., Sonntag, R. E., “Fundamentals of Classical Thermodynamics,” p 354, Wiley, New York, N. Y., 1965. RECEIVED for review December 14, 1970 ACCEPTEDAugust 20, 1971
JOSEPH W. BURSIK Rensselaer Polytechnic Institute Troy, N . Y . 12181
Drag Reduction in Two-Phase Flows The addition of small amounts of high molecular weight polymers to the liquid during two-phase (gasliquid) flows sizably reduced the pressute drop compared to flows without the polymer present. Data for aqueous and hydrocarbon systems containing 50 ppm of viscoelastic polymers indicate that pressure drop reductions as high as 50% are possible. The usefulness of adding soluble polymers to the liquid during twophase flow for mathematical modeling purposes i s also discussed.
T h e addition of minute amounts of suitable high molecular weight polymers to liquids yields sizable reduction of drag during turbulent flow-.This effect, described by Toms (1949) has been extensively investigated by a number of researchers as indicated in a recent literature review by Patterson, et al. (1969). The majority of the research carried out in this area can essentially be subdivided into two broad categories: fuIldamelita1 characterizatioll of turbulent flolv of dilute solutions of high molecular lveight viscoelastic materials, and data collection in the search for the most suitable poly646 Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
mers with respect to drag reducing properties, solvent compatibility, and polymer degradation. The object of the present’ work was to evaluate the drag flows. phenomelloll ill tTvo-phase, hlthough a voluminous amount of lit’erature information existed for liquid flow, the concept of drag reduction in gasliquid systems, to our knonledge, has not’ been previously investigated. I n order t o evaluate this phenomenon, tests were carried out in our laboratory for both aqueous and
Table 1. Pressure Drop Reduction in Two-Phase Gas-liquid Flows
Air-Water Flow jn a 1.5-in. Pipe Mixture Froude no., N F ~
2 2 4 4 20 20 60 100 20 60 60 100
Ait rate, scfm
Water rate, gpm
0.77 1.64 0.77 1.20 3.28 4.92 8.74 11.68 5.46 9.40 10.05 13.10
9.92 3.36 16.30 13.05 24.80 12.58 20.01 23.00 8.55 15.20 10.30 24.40
liquid fraction
Pressure drop for pure fluids, psi/lOO ft
Pressure dropn for 5 0 ppm Polyox in the water, psi/lOO ft
Pressure drop reduction,
%
Flow regime
0.63 0.22 0.74 0.59 0.50 0.26 0.24 0.21 0.17 0.18 0.12 0.11
1.05 0.40 1.65 1.45 4.90 2.45 6.05 8.35 1.90 5.00 3.75 5 40
0.83 0.20 1.25 0.95 2.90 1.65 3.75 4.80 1.05 3.10 2.10 3.30
19 50 24 34 41 44 38 43 45 38 44 39
Plug Plug Plug-slug Plug-slug Slug Slug Slug Slug Slug Slug Slug Slug
Nitrogen-Kerosene Flow in a 1.5411. Pipe Mixture Froude m., N F ~
60
Nitrogen rate, scfm
9.40
Kerosene rate, gPm
liquid fraction
Pressure drop for pure fluids, psi/I 00 ft
15.20
0.18
4.80
Pressure drop* for 50 ppm Vistanex Pressure drop 1-200in the kerosene, reduction, psi/lOO ft %
3.40-3.60
Polyox-polyethylene oxide -7 X 1 0 6 mol wt. * Vistanex L-200-polyisobutylene -4-6
hydrocarbon systems, and it is the purpose of this paper to present some of these results. Apparatus and Procedures
The apparatus consisted of a 1.50-in., 80-ft long acrylic pipe with pressure taps conveniently placed along the length. Prior studies of horizontal two-phase flow involving this apparatus have been previously reported by Greskovich, et al. (1969, 1971). F r o m these studies, it was found t h a t a t least 20-30 ft is necessary t o “establish” two-phase flow especially in t,he slug flow regime. Therefore, the pressure drops report,ed here were t’aken over a 20-f t interval approximately 40 ft from the pipe inlet. T h e operating procedure used in our study was to inject concentrated polymer solutions (1000-2500 wppm; wppm = parts per million b y weight) into the liquid before gas inject,ion. Approximately 20 ft of pipe plus a 20-ft section of flexible hose was used between the polymer injection point and the gas injection tee. The .polymer concentration of the resulting liquid reported here was maintained at 50 wppm. Good mixing of the dilute polymer solution and gas was observed at the gas injection tee, and for the purpose of this paper, it is assumed that the resulting liquid in the gasliquid mixture was 50 wppm as it flowed through the test section. Since the pressure taps in the test section were located approximately 400 pipe diameters downstream from the gas injection tee, this assumptioil seems justified. Fluids and Polymers Tested
Slug
106 mol wt.
polymers utilized were polyethl+leneoxide coagulant (Polyox) as supplied by the Union Carbide Co. and polyisobutylene (Vistanex L-200) as supplied b y Enjay Chemical Co. Polyox is water soluble and has a molecular weight of approximately 7 million, while the Vistanex L-200 is soluble in kerosene and has a molecular \\-eight of approximately 4-6 million. As mentioned earlier, the concentration of polymer in the liquid was maintained a t 50 wppm. Results
The results of our tests are summarized in Table I. For air-water flows, i t was found that the pressure drop reduction was as high as 50% when the flow regime was mainly slug
0 I
For the studies reported, the two-phase systems were airwater and nitrogen-kerosene. The system was operated at room temperature and near atmospheric pressure. The
x
25-29
Flow Regime
10
100
MIXTURE FROUDE NUMBER.
NF,,
Figure 1. The effect of volume fraction liquid and mixture Froude number on the pressure drop reduction Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
647
flow. For the only hydrocarbon-nitrogen run reported, the pressure drop reduction was approximately 25 to 30%. It is interesting to note that during two-phase slug flow, it has been proposed by Hubbard (1965) and Hubbard and Dukler (1968) that the total pressure drop is composed of frictional losses (at the wall) and acceleration losses at the noise and tail of the slug. From the limited amount of data collected in this study, it is observed (Figure 1) that when 50 wppm Polyox is present in the liquid, the pressure drop reduction tends t o level off at approximately 40’%. These results suggest that the friction losses tend t o level off and thus support the predictions of Hubbard (1965). The contradiction between Hubbard’s results and the results of Dascher (1968) may be easily explained using the data presented here. Conclusions
The experimental data presented here have conclusively demonstrated the effectiveness of dilute polymer solution in reducing the pressure drop in two-phase flow, presumably by reducing frictional losses. For fundamental studies and modeling of two-phase flows, addition of dilute polymer solutions to the flow can be extremely helpful in assessing the pressure losses due t o friction. I n addition, large scale uses of dilute polymer solutions may exist over relatively short distances where polymer degradation is limited. Rhenever the results of these studies become available, more useful correlations may be obtained from theoretical modeling.
Acknowledgment
The authors wish t o express appreciation t o Jack Martin for the collection of data and Esso Research and Engineering Co. for releasing data contained in this paper. literature Cited
Dascher, R. E., h1.S. Thesis, University of Houston, Houston, Tex., 1968. Greskovich, E. J., Shrier, A. L., Bonnecaze, R. B., IND.ENG. CHEW,FCNDAM. 8, 591 (1969). Greskovich, E. J., Shrier, A. L., A.Z:Ch.E. J . 17, 1214 (1971). Hubbard, h i . G., Ph.D. Thesis, University of Houston, Houston, Tex.. 1966. Patterson, G. K., (1969). Toms, B. A,!, “Proceedings of the 1st International Congress on Rheology, Yol. 11, p 135, North-Holland Publishing Co., Amsterdam, 1949. EUGENE J. GRESKOVICH*
Bucknell University Lewisburg, Pa. 17837
ADAM L. S H R I E R Esso Research and Engineering Co. Linden, iV. J . RECEIVED for review January 27, 1971 ACCEPTEDJuly 9, 1971
A Postwithdrawal Expression for Drainage on Flat Plates A theoretical expression for predicting the film thickness, obtained in a two-step process of withdrawal and drainage, is derived by extending the Denson equation. The theory i s verified by measurements of postwithdrawal film thickness of a 19-Poil in the top 5 to 20y0of the 18-cm film over a 7-85-sec range of withdrawal time. The theory is shown to reduce to the Jeffreys theory of drainage at long drain times.
w h e n a plate is withdrawn from a liquid bath and then motion is stopped, the entrained film thins by drainage. This condition of postwithdrawal drainage occurs in many applications, including dipcoating, measuring contact angle, and rinsing to minimize water pollution. Although drainage has been described by Jeffreys (1930) for times long enough t o neglect the initial conditions, the study of postwithdrawal drainage requires consideration of short time effects. Consider a vertical plate immersed in a quiescent bath of a wetting Sewtonian liquid. The solid is then withdrawn from the liquid bath a t a constant speed (U,) for a distance L and stopped (Figure 1). The resultant thin film is then allowed t o drain under the influence of gravity. The location of the liquid-solid-gas junction remains fixed with respect to the solid support. The problem in question is the prediction of film thickness ( h ) as a function of drain time ( t D ) , distance (2) from the top of the film, and other relevant parameters, or h(2,fD). The Jeffreys (1930) expression for film thickness in drainage is
648
Ind. Eng. Chem. Fundam., Vol. 10, No. 4, 1971
As will be shown, eq 1 is inaccurate for the data obtained herein. The deficiency of eq 1 for this case is that it does not indicate the influence of withdrawal speed or other withdrawal variables. I n order to do this one must consider the entire history of the film. This might be done in a t least b o ways, both starting with the liquid a t rest. One method would be to use partial differential equations and describe flow for both types of solid motion, including the proper discontinuities when withdraival was stopped. The second approach, which is used below, splits the process into two parts and uses the first-stage withdrawal result as a boundary condition (initial profile) for the second stage of drainage. I n developing eq 1, Jeffreys (1930) considered the different case of liquid lowering, which involves a fixed solid support and a moving liquid. H e then supposed a film of constant thickness and concluded that the resultant profile would be given by eq 1 after a short time. The assumption of a constant initial thickness is not suitable for Figure 1 films, however. Many other authors have derived and discussed equivalent forms of eq 1, some of which start from more complex equations. For example, see eq 88 of Rhitaker (1966). However,
