EXISTENCEOFTWOTYPESOFDRAG REDUCTION IN PIPE FLOW OFDILUTE POLYMERSOLUTIONS H A R R Y C. H E R S H E Y ’ A N D JACQUES L . Z A K l N
Department of Chemical Engineering, Unioersity of Missouri at Rolla, Rolla, Mo.
Drag reduction in the pipe flow of polymer solutions is shown to b e of two types which apparently occur by two separate mechanisms. In turbulent flow, drag reduction is probably caused by viscoelastic effects. The critical solvent Reynolds number a t the onset of drag reduction is proportional to about the first power of the diameter. Thus, the critical velocity is independent of tube diameter. Polymers dissolved in good solvents show more drag reduction than in poor solvents. The other type of drag reduction occurs when the laminar region is extended to high Reynolds numbers. It is followed by a transition region and a turbulent region in which the drag is not affected.
HE
reduction in pressure drop caused by the introduction
Tof a small amount of additive to a solvent in turbulent flow was first noted during World War I1 ( 7 ) . Savins defined this phenomenon as drag reduction (32). His drag ratio, D E , is the ratio of the pressure gradient for the solution in question to the pressure gradient for the solvent a t the same flow rate in the same tube:
Any fluid whose DR is less than unity is a drag-reducing fluid. ‘There is much confusion in the literature regarding the term “turbulence suppression.” For the purpose of this discussion, turbulence suppression describes the phenomenon of the friction factors falling below the values expected of “ordinary Newtonian” fluids. T h u s a friction factor of a polymer solution may be lower than for a Kewtonian fluid of the same viscosity (turbulence suppressing) but not as low as that of the solvent a t the same flow rate (not drag-reducing). Many excellent reviews of non-Newtonian fluid technology (4. 6 , 8. 9, 76, 79, 20, 27. 38) are available, and the material is not repeated here. I n view of recent discoveries of elastic behavior in apparently Newtonian fluids (27), the classical definition of a Newtonian fluid ( 7 6 ) no longer suffices. A fluid whose viscosity is independent of shear rate or shear stress over the range of interest will be referred to as apparently Newtonian, regardless of its behavior under extreme conditions or of the magnitude of elastic properties which have a negligible effect on the flow curve under such conditions. Previous Work
Poly(methy1 methacrylate) in monochlorobenzene was studied by Toms (37) in 0.050- and 0.159-inch tubes and found to be drag-reducing. For many years after Toms’ work investigators were primarily concerned with correlation of friction factors with various non-Newtonian viscosity parameters. Shaver (33, 34) measured friction factors in five tube sizes from 0.182 to 0.709 inch in six solutions of sodium carboxymethylcellulose (CMC), :seven of ammonium alginate, and one each of poly(viny1 alcohol) and Carbopol, all in water, as \vel1 as 0.5% PIB B-100 in cyclohexane. However, the results in Present address, Depai-trnent of Chemical Engineering, Ohio State University, Columbus, Ohio.
the 0.1 82-inch tube Lvere discarded because the water friction factors were inconsistent and not in agreement with Nikuradse’s measurements on lvater as expressed by Equation 2 :
dG
= 4.0
log
(SRefi)
- 0.4
(2)
Shaver derived a Reynolds number, now Lvidely referred to as the pow-er-law Reynolds number, from dimensional analysis considerations and the integration of the velocity profile of a power-law fluid in laminar tube floiv. With this Reynolds number and the poiver-law constants determined from a rotational viscometer, he obtained a correlation based on an extension of the Blasius equation. Dodge, whose thesis was completed some months after Shaver’s, also measured friction factors in pseudoplastic solutions (8,9 ) . He studied aqueous solutions of Carbopol 934 (four concentrations), Attagel (Attasol) clay suspensions (three concentrations), and C M C (two concentrations) in tubes from I,/? to 2 inches in diameter. H e used the generalized non-Sewtonian Reynolds number of Metzner and Reed (23):
(3) in a theoretical analysis for turbulent floiv of non-Se\vtonian fluids. His resultant friction factor equation was a generalization of Equation 2, and reduced to Equation 2 when n’ = 1 :
Dodge’s Carbopol 934 and Attagel clay were extremely well represented by Equation 4, whereas his C M C solutions \vere nowhere close. As can be seen in Dodge’s plots, in the laminar region the C M C friction factor data ivere on the theoretical line f = 16: .VRe’. Holyever, Lvhen .Vne’ exceeded 2500, the friction factors \yere lolver than those predicted by Equation 4. Also there \vas a significant diameter effect, u i t h the smallest tube friction factors loivest and the largest tube highest for a particular value. None of Shaver’s friction factors really fits Equation 4 well; in fact, most of his data fall below the predictions of Equation 4. Later investigators ( 7 7 > 20, 22, 25) shoived that the earlier correlations in the literature ivere not generally applicable. Thus the present state of the art requires some turbulent pressure drop measurements ( 6 ) on the fluid of interest for prediction of turbulent flow friction factors. Simple measurements such as viscosity-shear rate data in the laminar region are insufficient. VOL. 6
NO. 3
AUGUST
1967
381
Theories of Drag Reduction
Dodge ( 8 , 9 ) suggested viscoelastic effects as a n explanation of the failure of his C M C data to fit Equation 4, and most investigators (2, 5, 70, 73, 75, 20, 22, 25, 37, 40)have preferred his hypothesis. But there is still confusion in the literature regarding earlier theories of drag reduction. The earliest attempt to explain drag reduction was by Oldroyd (24), who offered a wall effect hypothesis for Toms’ data (37). Oldroyd proposed the existence of a n abnormally mobile laminar sublayer whose thickness was comparable to molecular dimensions and which caused apparent slip a t the wall. Toms later showed that slip a t the wall failed to explain his data (36). Savins (37) showed that a slip velocity could be calculated from his data. H e also indicated that the anomalous diameter effects so noticeable in Dodge’s C M C data (8, 9 ) as well as in those of later investigators were qualitatively explained by a slip hypothesis. Astarita’s analysis (2) indicated, however, that wall slip effects were not sufficiently large to explain the literature data quantitatively. Furthermore, Dodge’s C M C data (8) show no diameter effect in a plot of rw us. 8 V / D in laminar flow over a diameter range of 60 to 1. Since Dodge’s laminar capillary data extend to somewhat higher wall shear stresses than his turbulent data, the conclusion is that wall effects dependent on rw are absent. Comparison of velocity profiles in ordinary fluids with velocity profiles in drag-reducing fluids is the most direct method of determining the presence or absence of wall effects. If there were slip a t the wall, the velocity profiles would have to be more blunt. Shaver measured velocity profiles in drag-reducing fluids (33). Although his correlation [velocity deficiency, (U,,, U ) / U * ,us. Y / R with n as a parameter] predicts larger values of velocity deficiency in a non-Newtonian fluid than in water a t any radial position, the conclusion that the profiles are therefore steeper (as has been erroneously stated in numerous literature articles over the past seven years) is not warranted. The term velocity deficiency is proportional to the friction factor to the -0.5 power. Since in Shaver’s drag-reducing solutions the friction factors and hence U* were lower in solutions with low n, the friction factor dependence of velocity deficiency causes higher deficiencies a t low n. Because a velocity deficiency plot is a confusing way of comparing profiles for drag-reducing fluids, Hershey recalculated and compared Shaver’s Newtonian and non-Newus. Y / R a t similar Reynolds numbers tonian profiles as U/UmaX (77) and found that there was no significant difference between the profiles in drag-reducing solutions and in water. Other investigators have reported velocity profiles in dragreducing fluids (73, 40). Comparisons with water or some other Newtonian fluid are not available for their data. Also it must be remembered that the Pitot equation for an elastic fluid contains, in addition to the usual kinetic contributions, two additional terms that may cause a n apparent negative velocity if they are neglected (3, 7 ) . Thus if drag reduction and viscoelasticity occur mutually, it may not be possible to measure velocity profiles directly in a viscoelastic liquid with a Pitot tube. Another conjecture, discussed first by Shaver and Merrill ( 3 4 , is that drag reduction may be a result of the non-Newtonian viscosity gradient. Since the shear rate is maximum a t the tube wall and zero a t the tube center, a turbulent vortex must encounter a n ever-increasing viscosity in a pseudoplastic liquid. However, this theory fails to explain the absence of drag reduction in the highly shear thinning Carbopol and 382
I&EC F U N D A M E N T A L S
Attagel solutions of Dodge (8) or the presence of drag reduction in dilute (and apparently Newtonian) polymer solutions of Toms (37),Fabula ( 7 4 , Hershey (77),and others. Astarita suggested that turbulence in viscoelastic fluids was less dissipative and offered some order of magnitude calculations to support his proposal (2). Hershey and Zakin (77, 78) proposed that turbulence suppression begins a t a critical Reynolds number which is reached when a characteristic time of the flow is of the same order as the longest relaxation time of the polymer solution. Using relaxation times estimated from a modification of the theory of Zimm (47) and reciprocals of the shear rate a t the wall as a measure of the characteristic flow time, they obtained good predictions of the start of turbulence suppression in their experimental pipe flow data. At about the same time Fabula, Lumley, and Taylor (75) offered a similar proposal. Recently, Elata, Lehrer, and Kahanovitz (72) and Ram, Finkelstein, and Elata (26) have used a similar approach. Thus, there is evidence in the literature to discount all of the proposed theories of drag reduction except viscoelasticity. Furthermore, some rough predictions of turbulence suppression can be made from viscoelastic theory. Nevertheless, the relative importance of the various flow variables and the solution properties which affect drag reduction have not been completely defined. This paper reports experimental pipe flow data in the turbulent regime which show the effect of a wide range of variation of the flow variables velocity and diameter, and the effect of solution variables such as polymer type, polymer molecular weight, polymer concentration, and polymer-solvent interactions on drag reduction. In addition, the existence of a second type of drag reduction, not previously recognized, is illustrated. Experimental
Nonpolar solvents were used for three reasons. First, a large body of theoretical and experimental results has been reported on the dilute solution characteristics of nonpolar polymer systems; the behavior of polymer molecules in these systems is well understood. Second, such solvents permit studying polymers in different solvent environments. Third, only a few experimental results on nonpolar systems have been reported (26, 33, 37), so that data on the extent of drag reduction and on the concentration ranges necessary in these systems are sparse. Emphasis was placed on apparently Newtonian fluids (as opposed to earlier studies where pseudoplastic behavior was sought), because analysis and interpretation of the results are not clouded by poorly understood shear thinning effects. Only data that directly bear on this paper are reported here. Drag reduction was obtained with polyisobutylene (PIB) L-80 ( M , = 720,000) in cyclohexane a t 25’ C. and in benzene a t 24’ C., PIB L M M H ( M , c 46,000) in cyclohexane a t 25’ C., and poly(methy1 methacrylate) (PMMA) G ( M , = 1,500,000) in toluene a t 30’ C. Commercial grade solvents were used, and complete specifications are available (77). Pumping experiments were carried out in two recirculation systems, the large system and the small system. The large system consisted of a 100-gallon reservoir, two Viking pumps (0 to 35 and 0 to 200 gallons per minute) in parallel, each equipped with a variable speed drive, two turbine meters (1.5 to 15 and 10 to 140 gallons per minute) in parallel, and I/*-, 1-, and 2-inch i.d. smooth-bore steel tubing test sections in parallel. Appropriate manometer systems permitted pressure drops from 0.01 to 50 p.s.i. to be measured. Test section
Table 1.
Tube Diameter, Inches 0.03254 0,04625 0.06290 0.1033 0.509
calculations of non-Newtonian Reynolds numbers for the results of other workers are usually not available, and since none of these Reynolds numbers is of general applicability, all friction factor data in this study have been plotted against the solvent Reynolds number:
'Test Section Dimensions Entrance Test Section Length, Length, Diameters Diameters
...
744 524 385 465
...
... ... 100
300
0.999
75
200
1.998
50
100
(5)
L I D and entrance region L I D ' S are listed in Table I. Friction factors could be obt,ained in the Reynolds number range of 15,000 to 307,000 using toluene in the large unit. The small system consisted of a reservoir, one of tivo Zenith gear pumps (0 to 50 o:r 0 to 550 ml. per minute) equipped with a variable speed drive, and stainless steel capillary tubing test sections. The capillary stream was discharged horizontally into a specially designed funnel and returned to the reservoir (77). Pressure drops across the capillaries were measured by manometers or Bourdon gages and covered the range of 0.005 to 300 p.s.i. T h e capillary tubing dimensions are also listed in Table I. The miniimum L I D was 385, and appropriate entrance corrections were applied to all data. Friction factor data could be obtained in the laminar region and up to Reynolds number of about 15,000 for toluene. Temperature control to 1 0 . 1 ' C. \vas maintained in the test sections in both systems. Details of the apparatus and test procedures are given in Hershey's thesis (77). Proof of the validity of the measurements in the large system is shown in Figure 1, where friction factor data for cyclohexane are plotted. The average deviation a'f these data from Equation 2 is 1.9%. Friction factors in the large apparatus were rechecked with a Nejvtonian solvent before each new polymer-solvent system was introduced and studied. All solutions reported .here were determined to be apparently Kewtonian over the shear rate range of interest by means of the capillary viscometer.
where qs is solvent viscosity. Plots of this type remove all of the objections raised above and furthermore enable the reader to ascertain immediately the presence of drag reduction in the turbulent region when the friction factors lie below the curve of Equation 2. Fully Developed T u r b u l e n t Flow. Figures 2 through 5 show typical curves for drag reduction in fully developed turbulent flow. Equation 2 is plotted on these figures and the subsequent ones from iVRe = 3000 to iVRe = 590,000 and the equation: 16 f=-
-YRe
from -VR, = 420 to S R e = 2100. Figure 2 shows the data for the O.lyoPIB L-80 in cyclohexane and illustrates the effect of diameter and flow rate on drag reduction. No significant drag reduction was obtained in the 1- or 2-inch tubes at the flow rates attainable in the apparatus. In the '/*-inch tube a t low Reynolds numbers, the friction factors follon.ed Equation 2 for ordinary fluids. However, a t some point, called the critical Reynolds number, turbulence suppression and then drag reduction began. At the highest flow rates, a drag ratio of 0.73 was obtained in the 'jz-inch tube. The data for the 0.062-inch tube go through a normal aminar-turbulent transition. The critical Reynolds number I
Results and Discussion
n
After drag reduction data have been obtained, the problem of how to present the results in a meaningful manner must be solved. Rothfus and Prengle studied the laminar-turbulent transitions in \\ater and found that for a highly disturbed entrance turbulent eddies were formed between Reynolds numbers of 1500 and 3500 ( 2 9 ) . Since the von Karman equation. Equation 2, applies only to Reynolds numbers above 4000, the range between 1500 and 4000 i q a region where the friction factor-Reynold, number curve is poorly defined. T h u s Equation 1 is indet,zrminant in this region. Since many of the non-Ne\+tonian Reynolds numbers proposed in the literature tend to crolvd data points together (20, 25) and mask important transitions, since data needed for
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