Correlation of Drag Reduction with Modified Deborah Number for Dilute Polymer Solutions

Correlation of Drag Reduction With Modified Deborah Number For Dilute Polymer Solutions

J. M. RODRIGUEZ U. OF MISSOURI A T ROLLA J. L. ZAKIN ROLLA, MO. G. K. PATTERSON Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 ABSTRACT factor ratio, defined as the ratio of the observed pressure drop to that predicted for a solution of Correlation has been obtained between drag the same viscosity characteristics and density_ reducing characteristics for turbulent flow in a at equal flow rates using the Dodge-Metzner pipe and measurable properties of several polymer friction factor equation 4 solutions. Several concentrations of high molecular weight polymethyl methacrylate in toluene, high _I = molecular weight polyisobutylene in both toluene and cyclohexane, medium molecular weight poly .fi isobutylene in cyclohexane and benzene and low molecular weight polystyrene in toluene were OAO (1) studied. Data obtained in these nonpolar solvents and literature data for more polar solvents were (n ') I 2 successfully correlated as the ratio of measured friction factor to purely viscous friction factor vs Viscous solutions with drag ratios greater than the modified Deborah number VT1/DO.2, where Tl 1.0 can have friction factor ratios less than 1.0. is the first-mode relaxation time of the solution F or practical applications, it is drag reduction estimated by the Zimm theory. A shift factor which which is of interest. However, for correlation the is a function of intrinsic viscosity 11(4[7]] - 1) fundamental ratio is the friction factor ratio. allowed all the data obtained with nonpolar solvents In recen t years, drag reduction has been studied to be correlated as a single function. For these extensively. Recent studies have shown that systems, most of the data fit a single curve to reasonable predictions of the incipience of drag within ± 5 percent of the average friction factor reduction in polymer solutions can be made from ratio. The shift factor did not give a unique the properties of the solutions and the flow function of the data for the more polar systems. variables.S However, it has not been possible to predict accurately the amount of drag reduction to INTRODUCTION be expected for a given polymer solution without The phenomenon of drag reduction in polymer any drag-reducing turbulent flow data on the same solutions was first studied by Toms 1 in dilute solution. 6 solutions of polymethyl methacrylate in mono For flow through circular tubes it is well chlorobenzene. The drag ratio for flow through established that there is a concentration effect circular tubes has been defined2 as the ratio of and a diameter effect on drag reduction. Toms 1 the pressure _drop of the solution to the pressure observed that drag reduction increases with an drop of the solvent at the same flow rate. The increase in the concentration of the solution up drag ratio is less than 1.0 for a drag-reducing to an optimum concentration beyond which, due fluid. Practical use of drag reduction is being to the increased viscosity of the solutions, the made in fracturing operations in the petroleum drag ratio increased with an increase in concen industry.3 tration. Hershey 7 observed that for dilute Newtonian A more fundamental quantity is the friction polymer solutions in nonpolar hydrocarbon solvents, a normal transition region may be obtained followed by a departure from the von Karman equation Original manuscript received in Society of Petroleum Engineers office Oct. 19, 1966. Revised manuscript received (on a friction factor-generalized Reynolds number June 11, 1967. Paper (SPE 1678) was presented at SPE Symposium on Mechanics of Rheologically Complex Fluids chart). The point of departure is a function of held in Houston, Tex., Dec. 15-16, 1966. © Copyright 1967 diameter: the smaller tube s depart at the lower American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Reynolds numbers. Concentrated solutions may lReferences given at end of paper. show no abrupt transition reglOn, but merely an

S EPTEM RER, 1967 325 extension of the laminar regime followed by of the ratio of specific viscosity to concentration gradual departure from the laminar flow line, again favor drag reduction. This was observed experi showing a diameter effect. 4 mentally by Hershey .5, 7 Thus, intrinsic viscosity Several different theories have been offered to [TJ] defined as [TJJ = lim (TJsplc) and related to explain drag reduction such as the slip at the c->o wall theory,8 the anisotropic viscosity theory9 molecular we ight by the Mark-Houwink equation 17 and the various viscoelasticity theories.5 ,10-12 [TJ] = KMva , can be used as an indication of the Presently, viscoelasticity is widely accepted as tendency of a polymer-solvent system to be drag the cause of friction factor reduction, but the reducing. mechanism is not completely understood and it Park 18 was the first experimenter to try to is possible to explain some of the experimentally correlate drag-reducing data with the viscoelastic observed facts about friction factor reduction in properties of the fluids studied. The viscoelasticity terms of the other theories. term appeared in the Weissenberg number, a In a viscoelastic fluid, the stress is dependent dimensionless group relating the first normal stress on both the amount of strain (elastic response) difference to the shear stress, both evaluated at and the rate of strain (viscous response). A measure the wall shear rate. Park correlated data for only

of the relative amounts of viscous and elastic one system. Since normal stress data on dilute Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 response is the relaxation time. If the time scale polymer solutions in which turbulent pressure drop of the experiment is of the order of or shorter measurements can be made are difficult to obtain, than its relaxation time, any fluid will exhibit his correlation has not been tested on other systems. elastic as well a's viscous properties.13 Hershey,S The correlation presented in this paper relates the Astarita,12 Fabula, Lumley and Taylorll and amount of friction factor reduction for six different Elata, Lehrer and Kahanovitz 10 have suggested polymer solvent systems to variables which can be that friction reduction occurs when the relaxation obtained from simple measurements of fluid time of the polymer molecules in solution is equal properties and flow conditions. to or larger than a certain "characteristic flow time". In general, the characteristic flow time EXPERIMENT AL has been taken as the reciprocal of the shear rate Pumping experiments were carried out in two at the wall (evaluated at the wall using the 'power recirculating systems'? The large system consisted law or the Newtonian viscosity equation relating of a 100-gal reservoir, two pumps (0 to 35 and 0 to shear stress and shear rate), and the relaxation 200 gal/min) in parallel (each equipped with a time has been estimated using the Zimm14 or variable-speed drive), two turbine meters (1.5 to 15 Rouse 15 theory. and 10 to 140 gal/min) in paralle I and Yz-, 1- and The model for the Zimm theory assumes 2-in. ID smooth bore steel tubing test sections in conditions such as monodispersity (molecules parallel. Appropriate manometer systems permitted having all the same length) and Gaussian distribu pressure drops from 0.01 to 50 psi to be measured. tion of the segments of the polymer molecules which Test section and entrance region dimensions are are generally not met by a real polymer. However, listed in Table 1. Friction factor data could be the predicted values of relaxation times are useful obtained in the Reynolds number range of 15,000 for comparisons. The following equation is used to 207,000 with toluene. for predicting relaxation times. 16 The small system consisted of a reservoir, one of two gear pumps (0 to 50 or 0 to 550 ml/min) equipped with a variable-speed drive and stainless steel capillary tubing test sections. The capillary ...... (2) = stream was discharged horizontally into a specially 0586 RT Ak designed funnel and refed to the reservoir. Pressure drops across the capillaries were measured by manometers or Bourdon gauges and covered the where M v is the viscosity average molecular range of 0.005 to 300 psi. Capillary tubing dimen weight, TJ sp is the specific viscosity defined as sions are listed in Table 1. The minimum length in 11- s - 11-0 11- ' 11-0 is solvent viscosity, 11- s is solution 1 0 diameters was 385, and appropriate entrance viscosity, c is concentration, T is absolute corrections were applied to all data. Friction temperature, R is the gas constant and Ak is the factors could be obtained in the laminar region and eigenvalue corresponding to the kth mode of relaxation. The eigenvalues for the different modes have been calculated, the smallest eigen TABLE 1 - TEST SECTION DIMENSIONS Tube Diameter Entrance Length Test Section Length values (longest T corresponding to the first W (in.) in Diameters in Diameters mode. The Zimm theory predicts only discrete .0325 values of relaxation times, while in actual polymer 744 .0463 524 solutions a continuous spectrum of relaxation .0629 385 times is observed. .509 100 300 Si.nce drag reduction is favored by long relaxation .999 75 200 times, increased molecular weight and high values 1.998 50 100

326 SOCIETY OF PETROLEUM ENGINEERS JOURNAL up to Reynolds numbers of 15,000 for toluene. analysis may be performed upon the equation of Temperature control to ± O.IC was maintained on motion using a viscoelastic model as the equation both systems. Friction factors measured with pure of state. For simplicity, the Maxwell model was solvents in the large unit deviate by less than 2 chosen. percent from the von Karman equation, which compares favorably with previous investigations. 7 AU·I OPji 1 Six polymer solvent systems were studied: ::: ::: - +-p .. for i;tJ polyisobutylene (PIB) L-80 in cyclohexane and in axj G at f-L JI benzene,7 polymethyl methacrylate (PMMA) G in toluene,7 polyisobutylene L-200 in toluene and in ...... (3) cyclohexane 19 and polystyrene E in toluene.7 The Lagrangian equation of motion without body Molecular weights and intrinsic viscosities are forces is expressed as given in Table 2. Four concentrations of PIB L-80 in cyclohexane, four of PMMA G in toluene, three of PIB L-200 in toluene, three of PIB L-200 in OPji I + Opij cyclohexane, two of PIB L-80 in benzene and one OXj jjli OXI of polystyrene E in toluene were used in the Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 correlation. All but two of the solutions studied ...... (4) were Newtonian. Viscosities and relaxation times are listed in Table 3. If the Maxwell model is differentiated with respect to Xj. the equation of motion may be substituted for DISCUSSION apj/aXi, To obtain the correlating groups, a dimensional 02Ui P 02Uj I 02p.. II P au·I ~ _._- - _._-+_.- 2 TABLE 2 - INTRINSIC VISCOSITY AND OX~ G Ot G ~tOt J-L MOLECULAR WEIGHT OF SYSTEMS STUDIED J at Intrinsic Temperature Viscosity Molecular op·· Polymer Solvent (OCl (dl/g) Weight' __ . _" , ... (5) PIB L-80 Cyclohexane 25 3.43 860,000 0)1.'! PIB L·80 Benzene 24 0.82 590,000 PMMA.G 30 1.70 1,500,000 Toluene where Pii includes both isotropic pressure and any PIB L.200 Toluene 30 4.20 3,700,000 normal stress terms. PIB L.200 Cyclohexane 25 6.64 2,200,000 Xj = DX , Polystyrene Toluene 30 1.03 240,000 To obtain dimensionless groups let j E ui =vUi , (Opii1axi)= C'J.pIL). (aPiilaXi ), t= (Dlv)T. * Viscosity overage molecular weight, determined from intrinsic The equation of motion for a Maxwell fluid then viscosity measurements. becomes

TABL E 3 - SOLUTIONS STUDI ED fj.pv O~Pij Concentra tions Relaxation Studied Viscosity Time GLD OXjOT Polymer Solvent (percent) (cp) (sec x 103 ) 2 Polyisobutylene Cyclohexane .00 .8892 pv OUj fj.p 0 Pjj L·80 25C .01 .9130 .0435 +_.--_.- . (6) .05 .9981 .0398 o aT J-LL OX .10 1.120 .0431 j .30 1.673 .0481 Polymethyl Toluene .00 .5179 methacrylate G 30C .40 .8660 .0213 Dividing by v1D2, the four groups obtained are 2 .55 .9566 .0224 (1) pv , (2) I'1pD , (3) pvD and (4) /'t"pD2 . Since .70 1.187 .0268 .90 1.313 .0250 C CL fl flvL there are SIX independent variables and three Polyi sobutylene Tol uene .05 .606 .123 L·200 30C .10 .710 .144 fundamental dimensions will represent them, the .42 1.70' .198 Buckingham Pi theorem states that three groups Polyi sobutyl ene Cyclohexane .05 1.17 .278 will adequately represent the variables. Dividing L.200 25C .20 2.20 .310 3 Groups 1 and 2 by Group 4 (1) pv flL (2) 1!3:!... .38 3.80' .346 , I'1pD2C ' CD Polystyrene E Toluene .94 1.056 .0026 and (3) pvD are obtained. 30C fl Polyisobutylene Benzene .00 .6105 Group 1 may be altered by inversion and multi- L·80 24C .50 .856 .0057 plication by Group 2 to obtain a friction factor. .90 1.203 .0077 Group 2 IS a Deborah number, and Group 3 IS a 'Non.Newtonian fluid.average apparent viscosity used for relaxation time calculations. Reynolds number. The three dimensionless groups

SEPTEMBER. 1967 327 D!'J.p system. The curves obtained are positioned in 4L the same order as the intrinsic viscosity [7]] are (1) (2) ~ and (3) pvD pv2 ' GD fl of the systems (Table 2). A shift factor of the form 1/(4[7]]-1) was found to fit the data for the 2 five drag-reducing systems and a single curve The Deborah number NDe is defined as the fluid was obtained. Fig. 4 shows the data for the five relaxation time fllG divided by a characteristic flow time. It is the reciprocal of that proposed by systems plotting Illpu vs (T1 vIDO.2) (1/(4[7]]-1). Reiner and the same as that used by Metzner. 20 Nondrag-reducing polystyrene in toluene data are The relaxation time used for this investigation was also shown, and the correlation accurately predicts no drag reduction (that is, low values of the the first mode '1 calculated from Eq. 2. The attempt abscissa). to use Dlv, the pipe diameter divided by bulk mean velocity, as the flow time was not successful. The proposed correlation is good to ± 5 percent As will be shown later, the Deborah number in the value of I I I pu for most of the data. Scatter in the data for PIB L-200 in cyclohexane can be successfully used was one involving a length term attributed to degradation effects. These solutions more representative of turbulent shear stress fluctuations. This length term was a weak function showed the largest change in intrinsic viscosity Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 of diameter and was represented by the diameter to after pumping. The values of [7]] and Mu for this system were obtained with a partially degraded a small power. (A Deborah number defined as NDe 0.38 percent solution. Points that deviate the most in = '1 ('wi fls), where the characteristic flow time Figs. 3 and 4 correspond to a 0.20 percent was the reciprocal shear rate at the wall was tried but it failed to account for both concentration effects and diameter effects in the drag-reduction data.) The drag-reducing data used to test this correla tion yield parallel curves when plotted as log friction factor vs log ~eynolds number at constant

Deborah number. Because of this, the friction /0 factor ratio is a unique function of the Deborah number. The friction factor ratio used by Metzner and Park21 (I - 11)1(1 pv - 11)' where I is the friction factor of drag reauction, 11 is the extension of the laminar friction factor line 161NRe and I pu ... 4\ is the friction factor for no drag reduction, was not 0 0.80 \ a unique function of Deborah number for this data. ~ e That ratio was also very sensitive to small errors t B at low Reynolds number. fplI ~ eo e Fig. 1 is a plot of Illpu vs '1 vlD for PIB L-80 070 J o 0 ~ in cyclohexane at concentrations of 0.01, 0.05, 6. 0.10 and 0.3 percent. There is no consistent " x concentration effect in this correlation, but a 060 " diameter effect is apparent. To eliminate the ~ xL 6. diameter effect it was necessary to use DO.2 x in the Deborah number instead of D. As noted above, x 0;:J,Y this means that the proper length term is actually proportional to DO.2 since the Deborah number as modified was T1 vIDO. 2. The weak diameter effect in the modified flow o 2 ~ 4 5 6 '7 /3 time is in accord with energy spectrum measure ~ 7/.10 ments in both solvents and polymer solutions by Patterson.22 Little effect. of diameter was found PIB L-80 in Cyc10hexane on the spectra at a given velocity. Since frequency of velocity fluctuation iii closely related to a Tube/Concentration characteristic flow time for the viscoelastic fluid, (in) the flow time should be a weak function of 0.01% 0.05% 0.1% 0.3% diameter. 0.063 Figs. 2 and 3 show the data for all systems .& V studied as III pu vs the modified Deborah number. 0.046 0 e A unique curve is obtained for each polymer • solvent system. The natures of the polymer 0.033 6 X fJ• solvent systems affect the location of the corre lation curves, but the correlation accounts for 0.51 ~ 0 concentration, diameter and flow rate in each FIG. 1 - CORRELATION USING FLOW TIME OF• DIV.

328 SOCIETY OF PETROLEUM ENGINEERS JOURNAL solution that was very degraded. To mllllmize time of each pressure drop measurement. Degradation scatter, Mv and [7]] should be measured at the in the other system was not as significant 23, 24 and the amount of scatter in data of the other systems was considerably less. The correlation was established using data obtained in one laboratory with solvents that were not considerably different in their properties. The literature was surveyed to obtain data that could be treated in a similar manner. Three references were found that gave values of intrinsic viscosity: Toms: Fabula15 and Elata, Lehrer andKahanovitz.10 Although Toms reported a value of [7]] = 3.9 dUg for PMMA in monochlorobenzene, recalculation of his data gave a value of 1.2 dUg. From this a Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021

(>3

o 2 s G 7 B .'L r;.l0· Do.~ PMMA-G in Toluene - Da ta B Tube/Concentration .- . (in) 0.9% 0.7% 0.55% 0.4% ,,' 0.063 \l 0.046

.. 2 • o 2 0.033 0 rJ 0.51 b. X .. PIB L-200 in Toluene - Data A 1.00 'Q • o • Tube/Concentration -- (in) PIB L-80 in Cyc10hexane - Data A 0.05% 0.10% 0.42% Tube/Concentration (in) 0.063 ~ 0.01% 0.05% 0.1% 0.3/0 .. 0.046 (5 0 o 0.063 .. \l 0.046 • 0 e • 0.033 l::::. X • 0.033 b. X f1 PIB L-200 in Cyc10hexane - Data B 0.51 .. 0 D... Tube/Concentration PIB L-80 in Benzene - Data C (in) 0.20% 0.38% Tube/Concentration (in) 0.5% 0.9% 0.51 • 0.51 0 1.00 <)• FIG. 2 - CORRELATION OF PMMA-G AND PIB L-BO DATA. FIG. 3 - CORRELATION OF PIB L-200 DATA.

329 SEPTEMBER, 1967 molecular weight of 240,000 was estimated. This where k'was taken as 0.40. The maximum possible value was used to calculate relaxation times from error in Tjsplc, due to the estimated value of k', is Toms' data. less than 10 percent. Toms' and Elata's data Fabula's data for polyethylene oxide in water gave curves of the same general shape as above. showed increasing friction factor ratios as the The correlation accounted for variations in diameter Reynolds number increased for a given concentra and concentration as shown in Fig. 5. Toms' data tion. This was probably due to a severe degradation were for five concentrations and two tubes, while effect Fabula mixed his solutions gently, and as Elata's were for four tubes and four concentrations. his system was of the one - pass eype, shear Toms' data fall to the left of PIB L-80 in benzene degradation in the tubes was probably considerable and Elata's data superimpose on the PIB L-80 in because fresh polymer solutions are extremely cyclohexane. The shift factor 1/( 4[Tj]-1) fails with senSItIve to degradation. Values of molecular these two sets of data, the curves being shifted to weight and intrinsic viscosity on the exit solutions the left of the correlation curve obtained above. were not available, so it was not possible to There is a qualitative difference in the polarity recalculate the data. of the solvents used by these other investigators Elata gave values of intrinsic viscosity and molecular weight for guar gum in water, but no Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 solution ViSCOSItIeS were reporred. They were ~ ./04 (1'.1 r; estimated from the Huggin's equation. 10 0 /0 z.o 2.5 .so 3.5 "'.

0 9 A 0 ' e • 1.0 08 ~

Vt! 0 0 ~.) ~l::, e 0.' yOlk t .... o .... fpV(11S 'il 0 .... 0 ... 06 oro .6- • 0,6 TOMS "" l~ ... V Q:8. v~ ·v l::, OOy ;"-.6- 0 • 07 04 • OV V rio ELATA 0 ~ Oy • .6- ~. l::, 05 V 01S O~~ .... ~ • l::, \l)!':II 0.5 0 o • l::,l::, • l::, 0.4 • • Elata - Guar Gum in Water Tube/Concentration (cm) 0.5 0.047. 0.087. 0.15% 0.3% 5.07 e 0 \l o 2 3 ~ ., 9101112 .. V'" I 4 3.22 D'.• ".~.'O • 0 PIB L-80 in Cyclohexane 2.22 6. 1. 22 A PIB L-80 in Benzene • Toms - PMHA in Monochlorobenzene PIB L-200 in Toluene Tube/Concentration (g/l) • (cm) PIB L-200 in Cyclohexane ~ 2.5 1.0 0.50 0.25 0.10 \l PMMA-G in Toluene 0.404 o • Polystyrene in Toluene 0.128 o • FIG. 5 - CORRELATION OF THE DATA OF ELATA FIG. 4 - CORRELATION WITH SHIFT FACTOR. AND TOMS.

330 SOCIETY OF PETROLEUM ENGINEERS JOURNAL and the nonpolar hydrocarbon solvents studied time here. It is not clear whether the general correlation T absolute temperature obtained with six systems is fortuitous or whether T dimensionless quantity the mechanisms for drag reduction are different in ui velocity in the ith direction polar and nonpolar solvents. Also, the applicability of the Zimm theory to polymer solutions in polar U i dimensionless quantity solvents is uncertain. v average velocity, ft/sec Only two of the solutions on which the correlation Xj length in the direction of the jth axis was based had n 'values which differed from unity, Xj dimensionless quantity with the lowest value of n 'being 0.85. Since Tom's fl viscosity data appear to include both Newtonian and non Newtonian solutions, and since they fit the same flo solvent viscosity correlation curve, variations in n' seem to have no fls solution viscosity effect on the corre la tion. 7]sp specific viscosity CONCLUSIONS [7]] intrinsic viscosity, dl/g Ak eigenvalue corresponding to the kth mode

The important variables In drag reduction Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 p density correlation in dilute polymer solutions are v, Mv' [7]1 fls' flo, c and D. A unique curve suitable for Tw shear stress at the wall scale-up in diameter, velocity and concentration Tk relaxation time corresponding to the kth was obtained for each polymer solvent system mode, second tested by plotting flfpv vs (T1 v)/DO.20. This T1 relaxation time corresponding to the first treatment was also successful with data on two mode, second systems from the literature. In nonpolar hydrocarbon solvents, the curves ACKNOWLEDGMENTS obtained for five drag-reducing systems could be shifted to a single curve by an abscissa shift Acknowledgment is made to the donors of the factor of the form 1/(4[7]]-1). This shift factor did Petroleum Research Fund, administered by the not apply to the data for the polar solvents. American Chemical Society, for support of this research. Some of the data used in this correlation NOMENCLATURE were obtained by Harry C. Hershey.

a constant in the Mark-Houwink equation REFERENCES c concentration, gldl 1. Toms, B. A.: "Some Observations on the Flow of D pipe diameter, ft Linear Polymer Solutions Through Straight Tubes f friction factor at Large Reynolds Numbers", Proc., International Congress on Rheology, North Holland Publishing fpv friction factor calculated from the Dodge- Co., Amsterdam (1949) 11-135. Metzner equation 2. Savins, J. G.: J. Inst. Pet. (1961) Vol. 47, 329. It = laminar friction factor 3. Ousterhout, R. S. and Hall, C. D.: "Reduction of G = shear rigidity modulus Friction Loss in Fracturing Operations", J. Pet. Tech. (March, 1961) 217-222. k' = constant in the Huggin's equation 4. Dodge, D. W. and Metzner, A. B.: "Turbulent Flow K = coefficient in the Mark-Houwink equation of Non-Newtonian Systems", AIChE J. (1959) Vol. K' = consistency index 5, 189. L pipe length 5. Hershey, H. C. and Zakin, J. L.: "A Study of the Turbulent Drag Reduction of Solutions of High viscosity average molecular weight Polymers in Organic Solvents", Paper presented flow behavior index; ,exponent in the equa- at the 58th AIChE meeting, Philadelphia (1965). . K,(sv)n 6. Bowen, R. L.: "Designing Turbulent Flow Systems", tl0n Tw = D Chern. Eng. (1961) Vol. 68, 15-143. generalized Reynolds number, defined by 7. Hershey, H. C.: "Drag Reduction in Newtonian Polymer Solutions", PhD dissertation, U. of Missouri Dn' v2-n' p at Rolla (1965). K'sn'-l 8. Oldroyd, J. G.: "A Suggested Method of Detecting Wall Effects in Turbulent Flow Through Tubes", N Re Reynolds number Proc., International Congress on Rheology, North NDe Deborah number Holland Publishing Co., Amsterdam (1949) II-130. 9. Shin, H.: "Reduction of Drag in Turbulence by P ji stress tensor component Dilute Polymer Solutions", PhD dissertation, MIT Pii normal stress (1965). p .. II dimensionless quantity 10. Elata, C., Lehrer, J. and Kahanovitz, A.: "Reduction /).P pressure drop of Friction in Flow of Liquids by Minute Polymer Additives", Israel J. Tech. (1966) Vol. 4, 87. R gas constant 11. Fabula, A. G., Lumley, J. L. and Taylor, W. D.: s deformation "Some Interpretations of the Toms Effect", Paper

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17. Tompa, H.: Polymer Solutions, Academic Press, 25. Fabula, A. G.: "The Toms Phenomena in the Downloaded from http://onepetro.org/spejournal/article-pdf/7/03/325/2152848/spe-1678-pa.pdf by guest on 26 September 2021 New York (1956). Turbulent Flow of Very Dllute Polymer Solutions", 18. Park, M. G.: "Turbulent Flow and Normal Stresses Proc., Fourth Int. Congress of Rheology (1963) Part of Viscoelastic Fluids", Masters thesis, U. of 3, 455; Interscience Publishers, New York (1965). Delaware, Newark, Dela. (1964). • ••

332 SOCIETY OF PETROLEUM ENGINEERS JOURNAL