fi ( X ) ,$ i ( X ) = dimensionless velocity and its ith-order approximation = constitutive functional
5
Literature Cited
Ballman, R. L., Rheol. Acta 4, 137 (1965). Cogswell, R. N., Plastics Polymers 36, 109 (1968). Coleman, B. D., Noll, W., Arch. Rat. Mech. Anal. 3, 289 (1959). Coleman, B. D., Noll, W., Arch. Rat. Mech. Anal. 6, 366 (1960). Coleman, B. D., Noll,. W.,. Phys.,Fluids 6, 840 (1962). Lodge, A. S.,“Elastic Liquids, Academic Press, New York, 1964. Lodge, R. &I., “Fibers by Melt Extrusion,” in “Fibers from Synthetic Polymers,” R. Hill, ed., Elsevier, New York, 1953. Matovich, M. A., Ph.D. thesis, University of Cambridge, 1966. Oldroyd, J. G., Proc. Roy. Soc. A200, 523 (1950). Pearson, J. R , A,, Lubrication .4pproximation Applied to Non-
Newtonian Flow Problems,” in “Non-Linear Partial Differential Equations,” W. F. Ames, ed., Academic Press, Xew York, 1967. Pearson, J. R. A., “Mechanical Principles of Polymer Melt Processing,” Pergamon, London, 1966. Slattery, J. C., Phys. Fluids 7, 1913 (1964). Trouton, F. T., Proc. Roy. SOC.A77, 426 (1906). Van Dyke, M., “Perturbation Methods in Fluid Mechanics,” Academic Press, New York, 1964. Ziabicki, A., Bull. Acad. Pol. Sci., Ser. Sci. Tech. 12, 717, 725, 821, 925 (1964). Ziabicki, A., KolEoid Z. 176, 14 (1961); 179, 116 (1961). Ziabicki, A., Cybulski, A., Gromadowski, J., Bull. h a d . Po/ Sci., Ser. Sci. Tech. 13, 565, 681 (1965). Ziabicki, A., Kedzierska, K., Kolloid Z. 171, 51, 111 (1961). Ziabicki, A., Takserman-Krozer, R., Kolloid 2. 198, 60 (1964); 199, 9 (1964). RECEIVED for review December 14, 1967 ACCEPTED February 7, 1969
DISPLACEMENT OF AQUEOUS DRAG-REDUCING POLYMER SOLUTIONS RALPH C.
LITTLE
Naval Research Laboratory, Washington, D. C. 20590 Displacement experiments do not demonstrate significant persistence of a drag-reducing adsorbed film when a carboxymethylcellulose solution is displaced from a glass pipe by pure water in turbulent flow. The times needed to reach constant pressure gradient, upon which the existence of drag-reducing layers has been implied, can be greatly influenced by such factors as viscosity and the migration of drag-reducing fluid into the pressure tap connections.
E C E N T reports (Davies and Ponter, 1966; El’perin and Smolskii, 1965) have suggested that adsorbed polymer molecules might play a significant role in drag-reduction phenomena. T o explain the persistent drag-reduction effect observed when a polymer solution was displaced by the base solvent, these authors have postulated that an adsorbed layer of polymer molecules at the interface of flowing liquid and wall continues to function in a drag-reduction sense when the bulk of the polymer solution has been displaced by the base solvent. Davies and Politer (1966) report drag-reduction effects lasting up to 15 minutes in the presence of flowing base solvent. The effect ivas dependent upon the Reynolds number for the flow and the initial concentration of the drag-reducing solution. (No details of the apparatus used were given.) El’perin and Smolskii (1965) reported a persistent decrease in the friction factor u p to the “fourth change of water” in their system in which a glass flow tube 1.05 cm. in diameter and 128.5 cm. in length was used (no other information on the apparatus was given). The concept of a drag-reducing layer of adsorbed polymer molecules has been independently proposed as a possible rationale of the initial curvature of fluid parameter-polymer concentration plots on the basis of a quasi-BET analysis (Little, 1967). However, this latter model implies a system of flowing polymer solution in equilibrium with an adsorbed polymer layer in contrast to the model of a persisting layer of adsorbed polymer molecules which maintain their dragreducing role in the presence of a turbulently flowing base solvent. It was decided to try to duplicate the displacement 520
I&EC
FUNDAMENTALS
experiments to verify whether this technique might be directly applied to study adsorbed polymer molecules of the types effective in drag reduction. Experimental
Pipe flow measurements were made in a precision bore glass tube having a n internal diameter of 0.9525 f 0.0005 cm. The first pressure t a p was placed 70 diameters downstream from the tube entrance and the second tap was placed 100 diameters from the first one. Total length of the tube was 183 cm. The fluid was gravity-fed into the pipe test section from a constant-head device. Rate of flow was determined from the recorded output of a load cell arrangement on the collecting bottle. Pressure drop between taps was sensed by a Statham differential pressure transducer connected to the other channel of the Varian recorder. The pressure head could be varied from 0 to 8 feet of water. The water-soluble polymer selected for the displacement experiments was CMC (sodium salt of carboxymethylcellulose, type 7H, Hercules Powder Co.). The CMC sample chosen has been shown to be a good drag-reducing agent (Ripken and Pilch, 1963) and is considered similar to the Tylose sample (also the sodium salt of a carboxymethylcellulose) used by El’perin and Smolskii (1965). Results
Data derived from recorder plots of pressure transducer response vs. time are summarized in Table I. Table I shows that the time needed to reach constant pressure gradient appears to be a function of polymer concentration, residence
TIME (SECONDS)
Figure 1.
Water displacement of
Table I. Water Displacement of CMC Solutions Equilibrium Solvent Flow Rate, Cc./Sec.
Approximate Time to Displace Bulk of Fluid, Sec.
... ...
30 65 95 135
5 3 2 1.5
6 8 10 11
0.05
5 15
30 30
5 5
27 42
0.5
5 15
65 65
6 6
29 47
1.o
15 15
95 95
8
75
30
95 ..
8 8-
5 60
135 135
6 6
Concn.,
9i 0
Residence Time of Fluid, Min
Total Time t o Reach Constant Press., Grad., Sec.
75
>90 59 >70
time of the drag-reducing fluid in the pipe before displacement, and flow rate (corresponding to a Reynolds number range of 4000 to 20,000). These data appeared to confirm the results of Davies and Ponter, (1966) and El’perin and Smolskii (1965). The times necessary to reach constant pressure gradient ranged from four to more than ten times the bulk displacement time. Given even more concentrated solutions, longer fluid residence times, and favorable flow conditions, time delays of even greater length might be observed. A serious objection to the adsorbed layer hypothesis as an explanation for the “time delays” observed in the present work arose when the experiments were repeated with C M C solutions which a trace of dye had been added. When the glass pipe was filled with the colored C M C solution, t h a t solution convected slowly into the fluid (water) contained in the pressure t a p connections. The curves observed in the recorder plots of these displacement experiments take on a new significance when correlated with the behavior of the flowing colored solutions. Figure 1 represents a typical displacement experiment. It shows three regions, the relative magnitudes of which depend 011 the viscosity of the fluid being displaced, the amount of fluid entrapped in the pressure t a p connections,
170CMC solution
and the flow rate. Region I corresponds to the bulk displacement of the fluid by the turbulently flowing base solvent. Region I1 appears to terminate when colored viscous material adhering to the wall is no longer visible. Region 111, which can vary greatly in its contribution to the delay, corresponds to the slow bleeding out of entrapped colored CMC solution from the pressure tap connections to maintain a very low polymer concentration in the flowing solvent. These results illustrate the complications which must be considered when the displacement technique is used to investigate the existence of a drag-reducing interfacial layer of adsorbed polymer. Region I1 can probably be used to estimate the approximate lifetime of the polymer solution film adhering to the pipe wall in the presence of the turbulent displacing solvent. Flow rate data simultaneously recorded (not shown) with the pressure data indicated that an equilibrium flow rate corresponding to that of the displacing solvent alone had nearly been reached at the end of region 11. When proper precautions were taken to minimize penetration of dragreducing fluid into the pressure tap connections, region I11 was virtually eliminated. I n general, depending on flow rate and initial polymer concentration, the residual polymer solution adhering to the pipe walls appears to be swept out in a time period only three to four times greater than that required to displace the bulk of the polymer solution. It may be concluded for the present CMC sample at least, that any drag-reducing adsorbed layer has a n extremely brief existence in the presence of the flowing base solvent. It is believed that the presence of a tenacious drag-reducing interfacial layer of adsorbed polymer molecules which maintains its drag-reducing function in the presence of a turbulent displacing base solvent remains to be rigorously demonstrated. literature Cited
Davies, G. A., Ponter, A. B., Nature 212, 66 (1966). El’perin, I. T., Smolskii, B. M., Vestsi Akad. Navuk Belarusk. S S R Ser. Fiz-Tekhn., Navuk 2, 39 (1965). Little, R. C., Naval Research Laboratory, Rept. 6642 (May 31, 1967).
Ripken, J. F., Pilch, M., St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Tech. Paper 42, Ser. B, April 1963. RECEIVED for review August 2, 1968 ACCEPTED March 5, 1969 VOL.
8
NO.
3 AUGUST
1969
521
