The Drag Reduction Phenomenon. Observed Characteristics, Improved Agents, and Proposed Mechanisms Ralph C. Little, Robert J. Hansen;
Donald L. Hunston, Oh-Kil Kim, Robert L. Patterson, and Robert Y. Ting
Naval Research Laboratory, Washington, D.C. 20375
An overview is presented of experimental and theoretical work conducted at the Naval Research Laboratory on the reduction of hydrodynamic drag. The work establishes the effectiveness of certain chemical additives in reducing drag and defines the character of laminar, transitional, and turbulent flows when such additives are present. A variety of new additives for reducing drag have been developed during the program, and their unique properties are also summarized. Finally, the theoretical work is used as a basis for assessing the relative merits of proposed explanations for the drag reduction phenomenon.
I. Introduction Few discoveries of recent years in the area of fluid mechanics have created such intense interest as the drag reducing effect of certain chemical agents in turbulent flows. The reductions in pressure drop in turbulent pipe flow due to a high molecular weight polymer additive (88) and a dissolved micellar structure (67) were first reported in the mid 1940's, but the phenomenon was ignored until again observed by Westco Research in 1964. Since that time a wide variety of experimental and theoretical studies on the effect have been conducted at universities, government laboratories, and private corporations. The scope and quantity of work conducted to date on the phenomenon are evident from overview of the topic such as those offered by Hoyt ( 3 5 ) ,Lumley (62),and Granville ( 1 8 ) . The present paper is a review of research conducted a t the Naval Research Laboratory (NRL) on drag reduction. Many (though not all) of the results presented herein have been previously published in a variety of journal articles and reports. The present article provides a coherent picture of all NRL research on drag reduction to date and draws conclusions as to what additional work is required to fully understand and successfully utilize the phenomenon. I t must be emphasized that no attempt has been made to provide a comprehensive review of work conducted elsewhere. Research on the drag reduction phenomenon was initiated a t NRL in 1964, originally for the purpose of establishing the possible role of an interfacial or adsorbed polymer film in lowering skin friction drag. While it was soon shown that adsorbed polymer molecules played a minor role (if any) in the drag reduction mechanism, the assignment of the problem to this laboratory has allowed the development of a significant amount of work in new areas of the drag reduction problem; it is this work over the period of the past nine years which is the subject of this report. The major areas discussed in this paper are as follows: (a) factors important in drag reduction, (b) specially designed additives, and (c) proposed drag reduction mechanisms. Experimental studies to ascertain the factors which govern the magnitude and character of the drag reduction phenomenon have constituted a significant portion of the NRL effort. Such work is necessary to explain, apply, and optimize the phenomenon. The details of the experimental equipment and procedures employed have a pronounced effect on the results of such experiments, as evidenced by the many contradictory results reported in the literature for ostensibly similar conditions. From the outset an attempt has been made at NRL to employ well-characterized
polymer additives, using solution techniques which minimize molecular degradation, followed by tests in flow facilities which were carefully designed and instrumented. These precautions, for example, have made possible the acquisition of information on the effects of polymer adsorption and surface compliance on hydrodynamic drag, the character of the onset of drag reduction, the laminar-toturbulent transition in pipe flows of polymer solutions, and the influence of polymer homology, structure, and solvent on the observed drag reduction.
11. Possible Factors Involved in Drag Reduction A. Surface Effects. Some workers (12, 15,57) 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. The suggestion that adsorbed polymer layers play an important role in the drag reduction mechanism is a t first glance a thoroughly intriguing one and interest in it still persists (3, 1 0 ) . Early data obtained at this laboratory seemed to corroborate the polymer solution displacement experiments ( 5 3 ) .However, a serious objection to the adsorbed layer hypothesis as an explanation for the observed persistence of the effect arose when the experiments were repeated with polymer solutions to which a trace of dye had been added. Three flow regions were observed corresponding to (a) bulk displacement of polymer solution, (b) removal of viscous solution adhering to the wall, and (c) percolation of polymer solution out of the pressure tap connections. Thus, the observations suggest that the polymer is not tied to the surface alone. Ellipsometric and rotating disk drag reduction experiments (71, 72) were also run on aqueous solutions of several drag reducing polymers in contact with a wide variety of substances, and it was found that while the adsorption of drag reducing polymers is significant, the thickness of these adsorbed films is very small, suggesting that the polymers lie in a flat configuration on the surface ( 3 4 ) .On the other hand, while the addition of MgS04 greatly increases the film thickness, it is well known that added salt decreases the observed drag reduction ( 5 0 ) . In addition, surface energy effects seem unimportant since wall composition had virtually no effect. It seems improbable, then, that an adsorbed layer can perform a major role in the drag reduction mechanism. Drag reducing polymer molecules apparently lie flat or nearly so with respect to the surface and therefore cannot be expected to interact with the flow. Further indirect eviInd. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
'
283
e
30 0 SOLID DISK
MOWLUS=52Xldynes/mz SHEAR MCQULUS=22300dynes/cm2 THEORETICAL PREMCTION FOR SOLID DISK
0 SHEAR
-
.0
zot
-E
$e1 5 1 In
0
2
IO
10-
E i
60
I00
S H E A R STRESS
300
600 IMX) ZOO0
DYNES SIC^^)
Figure 3. The onset of early turbulence in a solution of polyox WSR-205 in a water-glycerine solvent (solvent viscosity = 0.0786 P, pipe diameter = 0.553 cm) (ref 1 7 ) .
5t
u
O-O --l
200 303 400 ROTATIONAL SPEED (rprnl
IO0
500
600
Figure 1. The torque on a 20.92-cm disk with rigid and with compliant surfaces.
501
Li
l
1 5 r12
a
I5
20
40 50 WALL SHEAR STRESS (dyneslcm' )
30
60
I
70
Figure 2. The variation in the onset wall shear stress for drag reduction with polymer concentration in a 0.660-cm diameter pipe (ref 2 5 ) .
dence that adsorbed polymer films do not function in a drag reduction sense was developed in experiments with a proprietary polymer product manufactured by 3M Company. The surface of this material consists of fibers which become erected when immersed in water for a sufficient period of time. They extend a distance of 300-400 into the aqueous phase. The surface of the wetted material is extremely slippery to the touch and conveys the sensation of lubricity. However, further rotating disk experiments with coatings of these materials yielded results identical with experiments with the Teflon, nylon, and stainless steel disks. Thus, the case for drag reducing adsorbed polymer layers does not seem t o be convincing on the basis of the experimental work done a t this laboratory. Another surface effect proposed for reducing hydrodynamic drag has been the use of a compliant coating. Some theoretical treatments (9, 20, 40, 4 7 ) have indicated that favorable effects occur in some circumstances, but conflicting experimental findings have been reported (47, 7 5 ) . An experimental program was initiated a t NRL to answer definitively the question of how surface compliance with isotropic mechanical properties affects the hydrodynamic drag exerted by liquids. A rotating disk apparatus was instrumented to measure torque vs. rotational speed for disks coated with compliant materials (polyvinyl chloride plastisols) over a range of shear moduli. Results are shown in Figure 1 for a 20.92 cm diameter disk with coatings corre284
30
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
sponding to two different shear moduli and for a rigid disk of the same external dimensions. Surface compliance has no effect on hydrodynamic drag below a critical rotational speed, which depends on the shear modulus of the compliant material. Above this critical condition, however, drag is significantly higher on a compliant surface than on a rigid one. A similar effect has been observed in all experiments conducted to date (23, 24). Photographic and visual studies of the compliant material-liquid interface show that undulations appear a t the disk perimeter when the critical speed is exceeded. The effect of these undulations on the flow appears to be similar to that of a roughened rigid surface in that the rates of turbulence production and dissipation are so altered as to increase drag. B. The Onset of D r a g Reduction a n d E a r l y T u r b u lence. The NRL studies of the onset of drag reduction were conducted with aqueous solutions of polyethylene oxide (Polyox). Four molecular weights were employed. Particular attention was given to polymer characterization, solution preparation, and other factors which may have affected earlier work. Tests were conducted in two singlepass pipe flow systems, with test section diameters of 0.660 and 2.53 cm. Other details of the experimental methods are given in ref (25). An example of the effect of polymer concentration on T*, the onset of non-Newtonian behavior, is given in Figure 2. The additive in this case (Polyox WSR 205) had a molecular weight of 9.0 X lo5. T * was observed to decrease with increasing concentration in all cases studied in contradiction to Virk's hypothesis which requires a constant onset point (91). I t increased with increasing molecular weight and decreased slightly with increasing pipe diameter. The pipe diameter effect was explained qualitatively by consideration of the Reynolds numher dependence of the turbulent boundary layer structure. The concentration dependence of T * was orders of magnitude too large to be explained in terms of the Rouse and Zimm theories (76, 93). It was concluded from the data that molecular aggregation or elongation might be occurring in drag reducing flows. The observation was first made a t NRL (60) in 1969 that the onset on non-Newtonian behavior can occur a t subcritical Reynolds numbers in flows of aqueous drag reducing polymer solutions through capillary tubes. This onset point was also shown to be influenced by the presence of salts ( 5 0 ) . The suggestion was made (60) that this unusual behavior would be observed in all pipe flows of these solutions where sufficiently large shear stresses could be realized in the laminar flow regime. This hypothesis was confirmed by subsequent experiments in larger diameter pipes with drag reducing additives dissolved in water-glycerine mixtures (17, 26). Figure 3 shows a plot of flow rate VS. shear stress for one such a flow. (The pipe diameter in this
case is 0.553 cm, the solvent viscosity is 0.786 P, and the concentration and molecular weight of the additive are 100 ppm and 9.0 x IO5, respectively.) Solution and solvent behavior are identical up to the onset wall shear stress T * , which is in the laminar flow regime. Between this onset condition and the laminar-to-turbulent transition for the solvent alone the wall shear stress is increased by the polymer additive. Further increase in the flow rate causes a continuous transition from this increased drag regime, which has been termed “early turbulence”, to turbulent flow with reduced drag (17).I t is apparent from this figure that early turbulence may be important in both applying and understanding the drag reduction phenomenon. A possible explanation for early turbulence is that the polymer additive exerts a destabilizing influence on the flow in appropriate circumstances. That is t o say, the Reynolds number a t which small-amplitude disturbances to the flow are amplified, rather than damped by viscous forces, may be decreased by the additive. To determine the merit of this proposal a theoretical study of high-phase-velocity disturbances in flows of dilute polymer solutions has been conducted (21),and some of the results are presented in part IV below. The additive is shown to exert a destabilizing influence on disturbances when the solvent viscosity and polymer molecular weight are sufficiently large or when pipe diameter is sufficiently small. Since the same trends in solvent viscosity and pipe diameter are conducive to the observation of early turbulence in experiments, the theoretical results appear to support a hydrodynamic stability explanation of the phenomenon. Additional theoretical and experimental work is now in progress to further explore the relationship of early turbulence and the effects of polymer additives on hydrodynamic stability. C. Effect of Homology a n d Solvent. The development of even an empirical function to relate drag reduction to relevant solution properties would greatly reduce the amount of time spent in evaluation procedures. Based on literature data (36), it was suggested ( 5 4 ) that the polymer critical concentration--defined as the concentration where random coils begin to touch (80)-might be used to normalize the drag reduction data since the same fraction of critical concentration appeared to produce the same degree of drag reduction irrespective of polymer molecular weight. However, the use of critical concentration as a normalizing factor seems to be limited to capillary tube flows and restricted to the higher molecular weight homologs of a drag reducing family (52). Virk ( 9 1 ) ,in developing an empirical correlation between concentration and drag reduction, defined a characteristic intrinsic concentration as
[ r ] = DR,/lim ( D R / c ) C-
n
( 1)
where DR is the percent drag reduction, c is polymer concentration, lim, -0 (DR/c) is intrinsic drag reduction, and DR, is maximum drag reduction for a given flow rate. The parameter [c] was found useful in superposing all the experimental data onto a single universal drag reduction curve well represented by
c-
0
For experimental use ( 7 0 ) , it was found more suitable t o simplify eq 2 into C DR DR, c + [e]
Rearrangement of eq 3 leads to
0.20
50
0.15
2
0.10
5
POLYOX C O A G U L A N T R e = 9000
0
I
I
2
4
1
1
I
I
6 8 IO 12 CONCENTRATION ( P P M )
I
14
0 16
Figure 4. Drag reduction of polyox coagulant, concentration de-
pendence and conformity to the universal drag reduction relation (ref 86).
C - -[ C I -
DR
DR,
C + __ DR,
Equation 4 indicates that a linear relationship exists between c/DR and c when the concentration dependence of drag reduction obeys eq 3. This drag reduction equation has been shown to be applicable to most drag reducing polymers (52, 86). As a typical example, Figure 4 presents the concentration dependence of Polyox Coagulant a t Reynolds number 9000. A plot of c/DR vs. c is linear and shows the validity of eq 3. The intercept value at c/DR = 0 yields the intrinsic concentration [c] and this quantity divided by the intercept a t c = 0 permits evaluation of DR,. It is readily seen from eq 3 that DR/c = DR,/([c] c). So drag reduction becomes more efficient on a unit concentration basis as the concentration decreases. Henry’s law conditions are reached when c/[c] values approach 0.01. This corresponds to a drag reduction of l%,well inside the ”error limit of all current drag reduction equipment. The distance between the random coils of the polymers a t this level of drag reduction may be estimated by using a relation developed by Paterson (70). For a Polyox compound having a molecular weight of 7 X IO6, the random coils, a t the 1%drag reduction level, are 20 diameters apart. Only a t these intermolecular distances is percent drag reduction a linear function of polymer concentration. When conditions close to an optimum drag reduction condition are approached, the polymer molecules are only a few diameters apart, or virtually touch. Equation 3 successfully describes the drag reduction results up to concentrations somewhat below that needed to produce an optimum drag reduction and only accounts for dilute solution behavior. Further increases in concentration bring decreases in drag reduction, hence the equation will fail. The maximum drag reduction, DR,, to be obtained as c is therefore not really a value attainable in the experiment. As cited earlier ( 5 4 ) , the polymer critical concentration had limited use in normalizing the drag reduction data. Its application breaks down a t lower polymer molecular weights. I t also fails as the critical concentration approaches a value above which no drag reduction takes place. On the other hand, [c] was found to be extremely useful in successfully normalizing the drag reduction data of different molecular weight compounds in one homolog series. Figure 5 shows the example of the Polyox family. The modified universal drag reduction equation, eq 3, is in nature an empirical correlation including two adjustable constants [c] and DR,. However, these parameters, being constants cKaracteristic of a given polymer compound, serve as a measure of the drag reduction effectiveness. The
+
-
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
285
80
0 WSR-35
0 WSR - 205 A COAGULANT
,
I
2 3 4 CONCENTRATION/ INTRINSIC CONCENTRPrTlON
I
5
Figure 5. The intrinsic concentration as a normalizing factor in capillary tube flows of Polyox solutions ( R = 12,000) (ref 5 2 ) .
t
80 -
RECIPROCAL OF CRITICAL CONCENTRATION x I
70 -
Figure 7. The reciprocal of Virk’s “intrinsic concentration” vs. the reciprocal of the polymer critical concentration ( R = 12,000) (ref
60 -
6ot
52).
50 -
0
POLYOX-WSR SERIES R e = 9000
ize the drag reduction data even for a homologous series of polyethylene oxide polymers it provides an apparent route for the development of intrinsic concentration-intrinsic viscosity relations (30). For example, the critical concentration, C,, for the touching of polymer random coils has been defined by Shin (80) as
M,X10’6
Figure 6. Drag reduction index as a function of molecular weight for Polyox materials (ref 86).
physical significance of these parameters becomes very clear if the limit of eq 3 at zero concentration is examined.
The parameter DRm/[c], the Henry’s law constant, defines the “efficiency” of the polymer additives on a unit concentration basis at infinite dilution. Figure 6 shows a correlation between this drag reduction “index” and the viscosityaverage molecular weight for the Polyox family (6). The plot is surprisingly linear and a least-square fit yields a The intercept value of M , = 2.46 X slope of 18.35 X IO5 suggests a cutoff point in molecular weight below which no drag reduction takes place at this Reynolds number (9000). This cutoff molecular weight is higher than the values reported by Little ( 5 2 ) and Hoyt and Soli (36), probably due to the difference in tube diameters used in these experiments (84). The slope in such a plot indicates the rate of increasing drag reduction effectiveness with increasing polymer molecular weight. Another method of rating family efficiencies has been developed through consideration of the critical concentration and intrinsic concentration concepts. To a good approximation it has been shown graphically that l/[c] is a linear function of l/C,. From such a plot (Figure 7 ) it was determined that C = B[1 - CC“,/C,l)] where B = 2.22 X lo2 (in NRL capillary rheometer a t Re = 12,000) and C,‘ = 6900 ppm, the critical concentration above which no drag reduction will occur. While the critical concentration concept fails to normal286
Ind. Eng. Chem., Fundam., Vol. 14,
No. 4, 1975
where p = packing constant (0.74), = Flory’s universal constant, N A = Avogadro’s number, [17] = intrinsic viscosity, and y = D,ff/v’/P2, the ratio of polymer effective diameter to the r.m.s. end to end distance where D,ff = r when P(r)/P” = 0.9. P(r)/P” relates to the probability of locating all segments of a given polymer molecule from that of a second molecule so that its center of mass is at a distance r from the center of mass of the other molecule. This relation for C, may be simplified to C, = p/[7] since p, a, and N A are all constants and y has only a small dependence on intrinsic viscosity a t least for Polyox polymers in water (80). Substitution of this relation into eq 6 yields
or where /3 = @/NA(3/6ay3)
where D = /3/B and [7’]is the intrinsic viscosity associated with the parameter C,’. Plots of l/[c] vs. [7]should be essentially linear if the approximations leading to eq 8 are valid. Figure 8 shows a least-squares plot for four polymer families-polyethylene oxide, polyacrylamide, polyacrylic acid, and an unknown polyelectrolyte family of commercial origin, polymer “PX”. All data fit the simple two-constant equation surprisingly well with correlation coefficients ranging from 0.988 to 0.999 and the usefulness of such plots to catalog family drag reduction efficiencies is emphasized particularly well in the case of unknown polymers such as polymer “PX”. Table I lists the constants appropriate to each polymer family and Table I1 contains their molecular weights. In the case of the polyelectrolytes it was of course necessary to determine intrinsic viscosities in ionic media. All determinations of intrinsic concentration, however, were per-
Table I. Equation Constants for Polymer Families 4
~
~~~
PX -
i*o}
Polymer
PAM PAA I
0 10
20
30
Polymer
40
INTRINSIC V l S C O S I T Y l d l / ~ l
Figure 8. Reciprocal of the intrinsic concentration vs. the intrinsic viscosity for polyethylene oxide (PEO), polyacrylic acid (PAA), polyacrylamide (PAM),and polymer series “X” (PX) (ref 4 4 ) .
formed in distilled water. Specifically, one constant, [$I, indexes the minimum intrinsic viscosity needed to observe a drag reduction effect under the given turbulent flow conditions for a given family. The second constant, D, indexes the drag reduction index growth of that particular family. The general characteristics of successful drag reducing polymeric additives-high molecular weight, extended molecular structure, and good dispersibility in the solventstrongly imply that solvent effects will have a n important influence on the observed drag reduction. For nonpolar polymers in nonpolar solvents, for example, the mismatch between solute and solvent represented by the difference in their respective solubility parameters (30) might be expected to be important. On the other hand, for neutral polymers in aqueous salt solutions the Debye-McAulay relation (27) might help in assessing salt effects. In the case of aqueous polyelectrolyte solutions, however, the insolubilizing power of a given cation added to the system will be determined not only by its degree of binding but also by the extent to which the binding process diminishes the solvent affinity of the ionized polymer segment (2). While Hershey and Zakin (29) found 40% less DR for polyisobutylene in benzene, a poor solvent, than in cyclohexane, a good solvent, no additional information is available for other nonpolar systems with respect to solvent power, temperature, and viscosity on the extent of drag reduction. In a series of rotating disk experiments (71) on solutions of polystyrene (M.W. = 1.8 X lo6) in various nonpolar solvents the following results were obtained. (1) Drag reduction is significantly greater in good solvents. (2) In any one solvent, at a given temperature, % drag reduction increases with increasing Reynolds number. (3) In any one solvent, DR increases as the temperature decreases. (4) Percent drag reduction is greatest in the most viscous solvent when comparisons are made at similar Reynolds number and solvent power. Some limited success in correlating solvent, viscosity, and temperature effects was obtained by casting the results in a time-based correlation where the time of the disk system (l/rpm) is combined with the polymer relaxation time (cu[go(gt, - l)]/T&), that is
%DR = K‘(rpm)
& (77, TK
- 1) (Re
- 3.9 x lo5) Re
(10)
where K = a constant, 70 = solvent viscosity, vr = reduced viscosity, Re = Reynolds number, TK = temperature, and C = concentration. Such a correlation, however, while certainly not novel provides the barest suggestion that solvent effects might be accounted for by cataloging polymer reduced viscosities or intrinsic viscosities together with some sort of drag reduction index such as intrinsic concentration. The NRL work with the capillary rheometer has shown
[q’], dl/g
x
H2O
2.99
21.9
HZO
8.47
aq. NaOH (2 N ) aq. NaCl (2 N)
0.545
40.9
22.2
5.80
11.4
66.4
XD
D X
~
PEO
I
Solvent (for [ q ] )
Type
~
= %
La’]
n
‘‘PX”
Nonionic Nonionic Polyelectrolyte Polyelectrolyte
6.79
65.4 57.5
Table 11. Molecular Weight of Polymers
Sample no.
[a]
Type of polymer
-
M , x 10-60
02 PAM 5.54 1.5 17-2 PAM 8.05 2.4 18 PAM 13.20 4.4 05 PAM 13.75 4.7 32-a PAM 16.20 5.8 6.7 32-b PAM 18.30 22-1 HPAM ... -4.7 ... -4.7 23-2 HPAM 23-3 H PAM ... -4.7 43-1 Gly/PAM(1/2) 11.48* >4.7 Gly/ PAM(1/3) 13.17b 4.7 43-2 44 PAA 3.59 3.7 63 PAA 5.72 a a Molecular weight (viscosity average) was computed according to the following relationship: for PAM: [ a ] = 6.31 X 10-5 x ?dvo.80 (dl/g) in water, 25°C (25); for PAA: [ q ] = 3.38 x 10-3 x (l./g); in 2 N NaOH, 30°C (26). The applicability of the reso that the molecular lationship is limited up to d.p. = 5 X weights of PAA samples are probably incorrect. Reduced viscosity ( c = 0.296g/dl, in water, 25°C). through a combination of literature ( 5 ) and experimental (55, 58) data that the effect of salt concentration on both the intrinsic concentration and the intrinsic viscosity can be described by equations of the following type-at least for Polyox materials
-1 - -
[‘I
-
[VI =
1 [‘lHZO
[7?1H*O
(1 -
g)
M (1 - ;i?.)
where M’ is the molarity a t [c] = m and M” is the molarity at [ q ] = 0. The M’ and M” constants are, of course, analytical conveniences. These equations may be combined to express [c] as a function of [v] as follows
4 [ c I = A2 (13) where A1 = (M’ - M ” ) [ ~ ] H ~ o / and M ” A2 = M‘/M”[v]H20[C]H20. It is clear that when M’ = M” the relation ([VI
+
simply reduces to
=
[7?1H2O[‘lH~O
(14)
which was approximately true for the low molecular weight polymers. This relation suggests an interesting new avenue in the characterization of drag reducing polymers in various solvent systems; that is if
[7?1e[cl =
[VI[Cl
Ind. Eng. Chem., Fundam.. Vol. 14. No. 4, 1975
287
80,
70 POLYACRYLIC ACID (#63) AP-5-e 60
..
a
E b
0 +
A b A
50
R e = 9000
20
d
0 W O
40
2 5 PPMW
0 Lz
n
2 12 12
0 W LT
20 MIN
20 MIN
5 1 '
a 20
I
I
I
I
I
I
I
I
20
25
30
35
40
45
50
55
FLOW RATE
30
w
60 MIN
Q (CC/SEC)
Figure 9. Percent drag reduction of glyoxal-modified polyacrylamides for different compositions and reaction times (ref 4 3 ) .
IO
0
3
Figure 11. Drag reduction vs. pH for several concentrations of polyacrylic acid (ref 4 3 ) .
e $
40-
0
5
A
3 0 ~
0
.. *.* .@
M W X IO&
0
8 1 f
201
I.1
0 HPAK (HIGH!
' I
20
-47 -47 -47 4 7 37
T HPAM (MEGIVM! A HPAM (LOW1 0 PAM
2 5 PPVW
I '15
. .. .
I -
n
25
b ,PAA 1
1
1
30 35 40 45 FLOW RATE 0 I C C / S E C )
1
50
1
55
Figure 10. Effect of hydrolysis on the drag reduction of polyacrylamide (ref 85)
then
where a is the polymer expansion factor and 0 refers to a 0 solvent. This would then imply that one need only characterize the drag reduction efficiency of a given polymer under 0-solvent conditions. Drag reduction research in other solvent systems might, in principle, merely be accomplished through use of a viscometer. In practice, eq 12 rather than eq 13 is obeyed and the dissimilarity between M' and M" increased with molecular weight. However, if one subscribes to the idea that high molecular weight polymers are more easily degraded in good solvents than in poor solvents at high dilutions since they are in a more extended state, then the discrepancy between M' and M" can be rationalized. Then combining eq 8 and 12 leads to
which approximates the effect of homology and solvent in terms of the intrinsic viscosity where the subscripts refer to specific solvents. 111. Specially Designed Additives A. Structural Effects. 1. Poly(acry1amide) PAM, and Its Modifications. In order to observe the effect of polyacrylamide (PAM) molecular weight on the degree of drag reduction a number of straight chain PAM samples were synthesized and drag reduction taken. Plots of the reciprocal 288
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
of the intrinsic concentration vs. molecular weight were linear as with the Polyox samples but a cut-off point of the order of 2 X lo6 in molecular weight was observed below which no drag reduction took place. This cut-off point is an order of magnitude greater than that observed for the Polyox family and appears to be greater than what might be ascribed to differences in chain length and flexibility. The chemical nature of a particular polymer of this series was then changed by modifying the amide side chains of a PAM sample (fi, = 4.7 X lo6) to nonionic, strongly hydrophilic groups by reacting the amide groups with glyoxal (Gly). Figure 9 shows a plot of percent drag reduction of this glyoxal-modified PAM against flow rate. Samples prepared using long reaction times (60 min) exhibited a definite decrease in the drag reduction compared to the original PAM; on the other hand, samples prepared using a reaction of 20 min duration demonstrated a distinct increase in the drag reduction-especially a t lower flow rates. In the case of the short reaction time preparation, the drag reduction of the glyoxal-modified PAM molecule actually reduced the maximum drag reduction asymptote (91).I t is believed that the hydrated bulky hydrophilic side groups contribute considerably to the enhanced drag reduction not only by promoting expansion of the polymer coil as a result of interference between side groups but also by increasing the solubility of the polymer by increasing the amount of hydrophilic groups. An interesting comparison in drag reduction behavior was made by introducing ionic groups into PAM ( M , = 4.7 X lo6) by partial hydrolysis thus converting the amide groups of PAM into carboxyl groups. In this manner, (acrylamide/acrylic acid) copolymers having three different molar fractions of acrylic acid (AA) in the PAM molecules, 63 (high), 55 (medium), and 45 (low) % were prepared. Since the expansion of macromolecules in solution can be promoted by the incorporation of ionic groups in the chain molecules the drag reduction of these modified PAM samples was expected to surpass the original PAM sample as the degree of hydrolysis increased. As illustrated in Figure 10, the drag reduction of PAM was indeed enhanced by hydrolysis to a significant extent (85).It reached a maximum around 55% hydrolysis of amide to acid groups and then showed a tendency to decline with further hydrolysis. The enhanced drag reduction by the hydrolyzed PAM (PAA fractions were 100% neutralized with sodium hydroxide)
350
300
c t
150
I
1
1
1
3
4
5
I 6
1
I
7
8
I
9
DH
F i g u r e 13. Reduced viscosity of polyacrylic acid solutions vs. p H for two selected concentrations ( 2 5 O C ) (ref 43). 1 0 ISPPMW (3
~
r,c L -.
~
0
PAA#I
IOPPMW PAA # 2
L . .
50
00
c
,50
(PPM)
F i g u r e 12. Reduced viscosity of polyacrylic acid solutions vs. concentration for different degrees of neutralization (25'Cj (ref 4 3 ) . e
can therefore he interpreted as a result of molecular expansion due to the electrostatic repulsion between like charges and osmotic effects in the chain molecules. The existence of an optimum ionic content for drag reduction seems analogous to reported changes in the reduced viscosity of PAA (78) and PMAA (19) which increased sharply with an increase of neutralization up to 40% and then decreased gradually. 2. Polyacrylic Acid (PAA)-pH Effect. PAA homopolymer is a highly flexible chain molecule, the expansion a n 3 hydration of which is sensitive to pH changes in solution. A PAA sample ( [ q ]= 5.72) was carefully purified to eliminate any ionic contamination and then partially neutralized prior to drag reduction evaluation. Figure 11 shows a plot of percent drag reduction vs. the pH of the sample. I t may be noted that drag reduction increases sharply with increasing ionization up to p H -6 (corresponding to 45% neutralization), and then the effect levels off with further increases of pH. Similar drag reduction results for PAA polymers were reported recently by Parker and Hedley (69),who suggested that the large increase in effectiveness a t higher pH is due to an extended molecular structure. Since the increase in viscosity is a major consequence of molcular expansion (X3), one might attempt to correlate the observed drag reduction results for PAA with its viscosity data. A plot of the reduced viscosity against PAA concentration (partially neutralized) is illustrated in Figure 12. The dependence of viscosity on concentration is in general agreement with early reports ( 7 8 ) , and has been qualitatively explained in terms of molecular expansion (64). However, a clearer comparison between the drag reduction data and the viscosity data can be made by Figure 13 where the reduced viscosity is plotted vs. ph' for two different P A A concentrations. It can he seen that the reduced viscosity passes through a maximum at a different p H for different concentrations. The decrease in reduced viscosity a t higher pH may result from an increasing shielding effect caused by a larger number of counterions in the domain of polymer molecule. This phenomenon seems to suggest that molecular expansion of PAA diminishes a t higher p H after
e
lot
NUMBER OF P45SES
F i g u r e 14. Degradation behaviors of PEO and PAA:0, PAA sample 1,l.i ppm (wtj; 0 , PAA sample 2 , l O ppm (wt); 0 , PEO,5 ppm (wt.); Re, 9000; d , 0.62 cm (ref 87).
the maximum value attained near p H 5-6. Similar effects were also observed earlier with PMAA (19,41). A comparison between the drag reduction and the viscosity data (Figures I1 and 13) shows that there is a favorable correspondence between them up to pH 5.5. The reason for the differing behavior a t higher p H is not yet clear. I t should be noted, however, that the viscosity data of PAA were obtained a t fairly low shear rates, while the turbulent drag reduction took place a t much higher shear rates. One possible explanation is that the strong shear field caused by the turbulent flow conditions might have disturbed the shielding effect of counterions allowing repulsive forces to maintain the molecules in a more extended state than would have otherwise occurred. Another factor which may be involved in the extended drag reduction plateau of Figure 11 is perhaps a more favorable hydration effect as the PAA is increasingly neutralized. B. Shear Stability of Polymers. The NRL drag reduction program has devoted part of its effort to develop polymers with improved shear stabilities in turbulent flows using two basic approaches. The first approach concerns the effect of charged groups in the polymer backbone on turbulent shear stability. Two samples of PAA-sodium salt were synthesized and tested. Sample 1 had a molecular weight of 3.7 X lo6 and sample 2,6.0 X lo6. The sample solutions were repeatedly passed through the test pipe and the degradation history recorded. Figure 14 shows the degradation history of the PAA samples, as compared with the results of Polyox-WSR-301 (PEO) on an equal percentage drag reduction basis. The fresh PEO solution is more effective than PAA because a lower concentration is required. Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
289
70
c
6or=-== 70 b
R e :9000
Re
5+-**03
I
NUMBER OF PASSES
2
3
4
5
6
7
8
9
80
:
/I
5730
12
1314
NUMBER OF PASSES
F i g u r e 15. Degradation behaviors of the 100 ppm solutions of linear PAM (closed points) and branched PAM (open points) (ref 45 )
F i g u r e 16. Degradation behaviors of the 100 ppm solutions of Polyox (closed points) and branched PAM (open points) (ref 4.5).
However, the effectiveness of the PEO solution drops more than 10%in terms of the percent drag reduction after only one pass through the pipe. After three passes it becomes even less effective than the lower molecular weight PAA. By way of contrast, PAA solutions show remarkable resistance to shear degradation. After 13 passes, the drop in the drag reduction effectiveness of PAA sample 1 is only a few percent. The same is true of PAA sample 2 after seven passes through the system, With pH values between 7.1 and 7.6, the PAA molecules in the dilute solutions are presumably highly ionized. It was suggested (87)that the electrostatic interactions between adjacent charged groups may result in the formation of a polymer ladder-type structure with associative bonds, which, if broken, could again be reformed in regions of lower shear. Such an effect has already been postulated for the shear stable drag reducing phenylstearate soap solutions ( 7 ) . The second approach undertaken is to improve the shear stability by polymer structural variation. A branched PAM sample was synthesized by grafting PAM chains onto a small nucleus molecule. The nucleus was prepared by reacting tetraethylene pentamine with epichlorohydrin to provide a typical backbone structure as follows
dation behaviors of 100 ppm solutions of these polymers, compared on an equal concentration basis with approximately equal initial percent drag reduction ( 4 5 ) . At three different Reynolds numbers, the solutions of the linear PAM and PEO show a very rapid decline in percent drag reduction with increasing number of passes through the capillary. The rate of the decrease in percent drag reduction is much slower for the branched PAM. Such a contrast clearly demonstrates the superior shear stability gained by molecular branching. Physically, this may be understood based on the molecular scission concept. According to Bueche's midpoint break theory (11) polymer chain breaking does not take place at random along the chain but occurs predominantly in the central portion of the chain. In the case of a linear polymer, this kind of chain breakage will immediately reduce the polymer molecular weight to approximately one half, and hence the drag reducing ability is greatly reduced ( 4 2 ) .In the case of the branched PAM sample, however, the reduction in molecular weight is apparently less dramatic probably because the shear forces acting on the polymer are distributed among the individual branches. Any reduction in the molecular weight, when it occurs, will probably correspond to that of an individual branch rather than one half of the molecule. Because of the poor resistance of PEO and PAM to mechanical shearing, the concept of drag reduction so far has not been considered for applications in closed loop pumping systems or the conventional recirculating type water tunnel. But the improved turbulent shear stability of these specially designed polymers may now offer great potentials in these applications as well as others in chemical engineering and marine technology where closed circuit flow systems are commonly used. C. Novel Agents. 1. Association Colloids. Association colloids consisting of linear extended structures of high molecular weight (7) possess the unique virtue of being able to reheal their drag-reducing linear structure after passing through regions of intense shear and would thus be admirably suited for use in closed systems. In preliminary hydraulic fluid experiments it was established that association colloids could provide the dual function of developing both the necessary viscosity and drag-reduction characteristics of the fluid (7). Several salts of selected fatty acids were studied, and of these, lithium phenylstearate appeared to be of greatest interest. Repetitive passes of lithium phenylstearate dispersions in 5606 hydraulic fluid base
ROH
HOR
\NCHzCH:NCH?CH,~CH,CH2P;CH,CH:N
I
HOR'
I
ROH
ROH
/
ROH
'ROH
where ROH = CH,CHCHZCI
I
OH There was a total of seven potential chain-grafting sites in this molecule. Through the redox reaction of the molecule with ceric ion, the graft polymerization of acrylamide was initiated on the ROH groups (65).The intrinsic viscosity of this PAM compound was 6.13 dllg, leading to an estimated minimum molecular weight of 1.7 X lo6 if a known relationship for linear PAM was employed (81). The degradation study was performed in a capillary flow system (65). At Reynolds number 9000 the branched PAM had a value of DR,/[c] = 4.14. Commercial linear PEO (Polyox-WSRN-3000, DRm/[c] = 4.15) and PAM (Magnifloc-905N, DRm/[c] = 3.78) with similar effectiveness were selected for comparison. Figures 15 and 16 show the degra290
Ind. Eng. Chern., Fundarn., Vol. 14, No. 4, 1975
BO 1=200 F
1
I
I
I
P
80
40
ALUMINUM MPALMITATE
1000
. A 2000
3000
4000
5000
C(ppm1
Figure 18. Test of Virk’s Iiniversal Drag Reduction Relation for two aluminum soaps (ref 5 5 ) . 0
100
200
300 TIME ( H O U R S )
400
500
Figure 17. The extended drag reduction life uf a protected lithium
soap dispersion (ref28). stock (a kerosene-type fluid) through a 0.561-cm pipe demonstrated both the resistance of the fluid to shear breakdown and its substantial drag-reducing qualities. On the negative side, the deleterious effects of water, air, and carbon dioxide on such soap-thickened fluids were explicitly pointed out. In addition, association colloids like polymers are also affected by solvent environment (51, 59) and the colloid used must be compatible with the fluid to fulfill its role as a drag reducing agent. While the utility of such dispersions will be dictated by the degree of interaction of these fluids with reactive gases, they apparently remain the most promising shear breakdown resistant fluids to date. A demonstration of the resistance of association colloids to shear breakdown in a hydraulic fluid test set-up is shown in Figure 17 for a flow rate of 3.0 GPM using a 0.2% dispersion of lithium phenylstearate in 5606 base stock to which 1%antioxidant has been added (28). While the change in drag-reduction properties is a t first rapid, a practical level of 40% drag reduction is soon reached within the first 10 hr of operation. The loss of drag-reduction effectiveness follows virtually a linear relation with respect to time. Extrapolation of this curve to intersect the 20% level of drag reduction leads to an estimated fluid lifetime of approximately 1000 hr. The apparent viscosity of the soap-thickened fluid-estimated a t a shear rate of 1150 sec-’--decreased from 2.61 to 2.11 CP(measured a t ZOOOF) after 500 hr of operation. The soap dispersion thus remained about three times more viscous than the 5606 base stock alone, even after this extended period of continuous operation. By comparison, the lifetime of a polyisobutylene-thickened fluid of the same concentration was only 2 hr. It was evident, however, that the loss of drag reduction activity is primarily due to oxidative breakdown of the colloid rather than shear forces in spite of attempts made to exclude reactive gases from the test system through dry nitrogen purges and the use of an aircraft accumulator to adsorb volume changes in the fluid during heating-cooling cycles. Nonaqueous association colloids formed from metal soaps in nonpolar solvents can be expected to have very low critical micelle concentrations in the absence of competing polar substances. Therefore, association colloids should behave very much like chemical polymers with respect to the growth of drag reduction with increase in concentration. Specifically, drag reduction tests made for two aluminum soaps showed that these soaps conformed to Virk’s Universal Drag Reduction Relation (56) as indicated by the linear plots in Figure 18. This linearity indicates no change in col-
loid size or species over the concentration range studied, in agreement with Sheffer’s molecular weight measurements (49, 79). Thus, the drag reduction data confirm what the recorded physicochemical data explicitly state: that these colloidal soaps do not change their micellar molecular weight with concentration. The drag reduction efficiencies of the dilaurate and dipalmitate soaps corresponded respectively to Polyox polymers with molecular weights of 130,000 and 275,000 (52). Thus, it would seem that any soap “critical concentration” required for drag reduction (not to be confused with critical micelle concentration) in nonaqueous systems must be the result of events taking place on a molecular level, i.e., the competition between the disoap anions and the polar impurities for the soap cations as the soap concentration is varied in agreement with Singleterry’s work ( I , 32, 32, 33). The practical application of drag reducing fluids to closed systems containing a multitude of fittings, valves, pipe diameter changes and the like may require considerable additional research. For example, the flow of viscoelastic fluids into sudden enlargements is characterized by larger head losses than for Newtonian fluids (4). (It may be of interest to note that this is not necessarily the case in certain capillary tube experiments ( 6 1 ) . Surface roughness effects also may take their toll on the observed drag reduction (92).In other words, the remarkable drag reduction effects observed in smooth pipes may be somewhat reduced in practical working environments such as a complex hydraulic system. Specifically, if drag-reducing additives are to be used to improve pumpability, head losses in fittings as well as expansions and contractions should be thoroughly studied to properly assess the gross benefits conferred by the candidate fluids. Because of the uncertainty of these corrections in the case of viscoelastic fluids, preliminary measurements of candidate fluids should be made in systems closely simulating those actually used in military or industrial applications. 2. Polyphosphates. Polyphosphates offer an interesting alternative to the organic polymers normally used ( 3 8 ) . The synthesis of linear or branched polymers with extremely high molecular weights is usually difficult and expensive. The synthesis of potassium polyphosphate, however, is simple and can easily be modified to produce a wide variety of molecular structures (71, 90). I t proceeds through dehydration of potassium monobasic with heat.
The primary factor in determining the molecular structure Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
291
NEWTONIAN
MAXWELL
OLDROYD
n E C Figure 20. The spring-dashpotsystems correspondingto the Newtonian, Maxwell, and Oldroyd models are shown.
/
I
TEMPERATURE
/
Figure 19. A typical drag reduction efficiency profile for potassium polyphosphate is shown. The efficiency, as measured by the reciprocal of intrinsic concentration, is plotted as a function of K/P ratio and synthesis temperature. The synthesis time was 2 hr and the polymer was dissolved in a 1%sodium metaphosphate solution.
of the polymer is the number of reaction sites (-OH groups) per molecule in the starting material. This in turn is reflected by the ratio of the number of potassium atoms to the number of phosphorus atoms; molecules of KzHP04 (K/P = 2) will terminate chain propagation whereas molecules of H3P04 (K/P = 0) will serve as potential branching points. Consequently, if the starting material has an average K/P ratio close to 1.00, the reaction will tend to produce linear molecules of high molecular weight. If the ratio is increased by adding potassium hydroxide, linear polymers of lower molecular weight will result. When the ratio is decreased by adding phosphoric acid, a branched polymer will be produced. This wide variety of possible structures and the ease of synthesis give polyphosphates a unique potential. The friction reducing ability of this material depends on two factors: the solution preparation technique and the molecular structure of the polymer. In order to realize the maximum drag reducing efficiency, a study was initiated to optimize these two factors. While a complete discussion of this study will be presented elsewhere, two results are of interest here. First, a special low-shear, solution preparation technique was developed using sodium metaphosphate as a solubilizing agent. With this technique the intrinsic concentration of a particular sample was reduced from 125 ppmw to 6.5 ppmw; this represents a 20-fold increase in efficiency. Secondly, a systematic synthesis program revealed that an optimum molecular structure was obtained by heating a K/P = 0.95 sample for 2 hr a t 750°C (Figure 19). This material has an intrinsic concentration of less than 1 ppmw and thus is equivalent to the most efficient material presently known. Additional study with these unique materials should lead to valuable knowledge concerning the relationship between molecular structure and drag reduction.
IV. The Mechanism of Drag Reduction A. Turbulent Flow and Drag Reduction. Sophisticated flow visualization experiments have been made on the turbulent boundary layer structure in Newtonian fluids (46, 66) and the results of these studies have established a relationship between the production of turbulence and the behavior of long narrow segments of liquid (streaks) which travel a t low velocities in the region near the wall. Periodi292
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
cally, individual streaks are ejected away from the wall in a process which is termed the bursting phenomenon. This process can be described as a continuous chain of events leading from a relatively quiescent wall flow to the formation of relatively large and chaotic fluctuations. Measurements (42) of turbulence production during bursting and nonbursting periods show that essentially all of the production occurs during the bursting period. As a result it is now generally believed that bursting is the mechanism for the generation of turbulence. While the preceding sections provide valuable information regarding the relationship between the molecular structure of a polymer additive and its drag reducing properties, this does not explain how the polymer molecules produce such a dramatic change in flow. Recently, flow visualization experiments (14) have been carried out in dilute solutions of drag reducing polymers. These experiments indicate that the spatially averaged bursting rate is greatly decreased by the addition of the polymer. This evidence strongly suggests that the turbulence production is decreased by the presence of the polymeric additive through inhibition of the formation of low speed streaks or the attenuation of the bursting. T o understanding the mechanism of drag reduction, therefore, it is necessary to determine how small amounts of polymer can alter the bursting process. Since the flow patterns associated with bursting are very complex, it cannot be treated quantitatively. One alternative is to examine a number of simple well defined flows that closely model the types of motion which appear to be associated with the bursting phenomenon. Since drag reduction is characterized by large changes in flow produced by the addition of very small amounts of polymer, the objective in these studies is to seek a simple flow in which the behavior of a very dilute solution is significantly different from that of the solvent. R. Proposed Mechanisms. Several studies of this kind have been performed and the results suggest that three different flow geometries satisfy the above criteria: (1) simple shear disturbances, either transient (22, 7 7 ) or steady-state (89), (2) elongational flow ( 1 3 ) , and (3) oscillatory shear disturbances to steady shear flows (8). The mathematical models utilized in these studies, however, were very simple and consequently NRL initiated a program (39,53) to reexamine these three types of flow using a more realistic model. 1. Models. The solvent is usually assumed to be a Newtonian fluid; that is, the stress tensor is directly related to the strain rate tensor. Schematically, this model can be represented as a dashpot, Figure 20A. The addition of polymer molecules to the solvent not only increases the viscosit y but also adds an elastic contribution to the total response. As a result some previous studies have described solution behavior with a Maxwell model since it combines both viscous and elastic behavior, Figure 20B. While this model has the advantage of mathematical simplicity, it does not provide a very good description for polymer solutions because the responses of the solvent and the polymer
-
IO'
W v)
\ v)
IO'
a w
-z I' 0
z ,?
+
2
5
IO2
+ +
1.0 lo2
IO'
FREOUENCY
10.
IO'
IO'
10'
'01
a
I
I
I
I
I
1
I
I
lRADIANS/SEC)
01 FREQUENCY
Figure 21. The frequency (radians/secj dependence of the amplitude attenuation coefficient (neperdcm) is shown. Water is represented by the dashed line while the solid curve corresponds to calculated values for a 100 ppmw aqueous solution of Polyox WSR301.
do not combine as a simple sum. Consequently, a two-element model such as that developed by Oldroyd (68) is needed. The Oldroyd model combines a Newtonian element for the solvent with a Maxwell element for the polymer, Figure 20C. The stress tensor, u, and the strain rate tensor, d, are related by
where vs is the solvent viscosity (Newtonian element), 7' and G are the viscous and elastic contributions of the polymer molecules (Maxwell element), and a / D t represents a convected time derivative. The response time of such a system to an applied stress can be characterized by a time constant called the relaxation time, 7 = v'/G. For an actual material the parameters in these equations can be determined by measuring solvent viscosity, solution viscosity, polymer molecular weight, and concentration. This model provides a good qualitative description of solution behavior for most flows but it must be modified to include solvent relaxation times when considering deformations a t very short response times or a t very high frequencies where the solvent behavior is no longer Newtonian. By keeping this limitation in mind, however, it is now possible to consider the three types of deformations thought to be important in drag reduction. 2. Simple Shear Disturbances. One theory that has been proposed to explain drag reduction involves transient or steady-state shear disturbances. It has been suggested that the propagation velocity for such disturbances in very dilute polymer solutions is substantially less than that in the solvent. Since a certain stage of the bursting phenomenon may involve a deformation of this type, a large decrease in the propagation velocity may result in less bursting and thus less turbulent drag. The propagation of a transient or steady-state shear disturbance is dependent on the behavior of simple sinusoidal shear waves. Consequently, the propagation of a shear disturbance in solution will be different from that in the solvent only if there is a corresponding difference in the propagation of simple shear waves. It is of interest therefore to evaluate the phase velocity and amplitude attenuation coefficient for shear wave propagation as a function of frequency. For illustrative purposes a 100 parts per million by weight (ppmw) solution of a commercial sample (Polyox WSR-301, Union Carbide Corp.) will be used as an example. The weight average molecular weight of this polymer is about 2.4 X lo6 while the solution viscosity is 1.21 cP. The
(RADIANS/SEC)
Figure 22. The frequency (radiandsecj dependence of phase velocity (cm/secj is shown. Water is represented by the dashed line while the solid curve corresponds to calculated values for a 100 ppmw solution of Polyox WSR-301.
results of calculations for this solution and water are shown in Figures 21 and 22. These figures clearly indicate that no significant differences between solvent and solution behavior are predicted, even a t the relatively high concentration of 100 ppmw. Furthermore, the behavior a t high frequencies is completely dominated by the solvent and thus the inclusion of solvent relaxation times will not alter this conclusion. To test these predictions, shear wave propagation was measured in water and a series of drag reducing solutions using the Layered Waveguide Technique (37). In all cases the results are in agreement with the theoretical predictions. Clearly, these results indicate that in its present form a mechanism based on the propagation of a shear disturbance is unsatisfactory. The theory asserts that the propagation velocity in solution is much smaller than that in solvent. The theoretical and experimental evidence indicates, however, that the differences are very small and are in the hrong direction; that is, the velocity in solution is slightly higher than that in solvent at low frequencies. Of course, it might be possible to develop an alternative theory based on a high velocity in solution, but the very small magnitude of the differences involved would still be a severe limitation. 3. Elongational Flow. A second possible explanation for drag reduction involves elongational flow. The bursting process contains a stage which is characterized by a stretching motion similar to elongational flow. It has been suggested that the addition of small amounts of polymer to a solvent substantially increases its resistance to elongational flow. I t is proposed, therefore, that the increased resistance to stretching results in less bursting and thus less turbulent drag. Elongation flow is defined by a strain rate tensor of the following form.
where r is the stretching rate which is taken to be a constant here. The reduced elongational viscosity, i j , is a measure of the resistance to this type of flow compared t o the resistance of the solvent to simple shear flow and will be defined as i j = ( u l ~- u22)/qSr. To examine elongational flow in solvent and solution, eq 19 is combined with the appropriate constitutive equation and the correct values for the relevant parameters. Such calculations were made, again using a 100 ppmw solution of WSR-301 as an example. Figure 23 shows a plot of reduced Ind. Eng. Chem., Fundam., Vol. 14. No. 4, 1975
293
2--
NORMALIZED I
'
""'"I
'
".'"'
-T
Figure 23. The reduced elongational viscosity is plotted against flow time for the Oldroyd model. The lower curve is the calculated behavior for values of r less than l , while ~ the upper curve corresponds to a value of r greater than ST.The dashed line is the solvent behavior. I I I I ................. . .. .. NEWTONIAN FLUID
I
I I I I I
- - - -. .K = 2
-. . . .
___
l -
K=5 K=IO K=CO
Figure 25. The normalized amplitude of a disturbance is shown as a function of normalized time in a 100 ppmw aqueous solution of Polyox WSR-301 (solid line) and a Newtonian liquid with the same steady-shear viscosity (dashed line). The value of E for this plot is 0.498, where the stabilizing influence of the additive is a maximum for K = 1.21.
cillatory disturbances to steady shear flows. One stage of the bursting phenomenon appears to contain an oscillatory disturbance superimposed on the basic motion which is a steady shear flow. The rate of growth or decay of such a disturbance, or in conventional hydrodynamic terms its stability, may well affect the bursting rate. It is proposed, therefore, that the addition of a small amount of polymer to a fluid will increase the rate of decay for oscillatory disturbances and thereby decrease bursting and the associated generation of turbulence. By combining eq 18 with the momentum equations and specializing the result for disturbances to a shear flow, it is found that the effects of the polymer additive become large when the quantity [~'/(v' ~ , ) ] T is~ of S the ~ order of or larger, where S denotes shear rate. This criterion can be realized for sufficiently large values of S even if, as in dilute solutions, the difference between solution and solvent viscosity is small. To illustrate the results obtained in such an analysis, Figure 24 shows the rate of decay for a highphase-velocity, axisymmetric, disturbance in a pipe flow. E is a normalized parameter involving the polymer relaxation time divided by the Reynolds number of the flow; K is solution viscosity divided by solvent viscosity. The rate of decay of a disturbance is increased by the polymer additive when E is in the range of approximately 0.5 or less and decreased for larger values of E . Figure 25 shows how a disturbance with a normalized amplitude of 1.0 decays with time in a 100 ppmw aqueous Polyox WSR-301 solution ( K = 1.21) and in a Newtonian liquid of the same steady-shear viscosity; if the corresponding comparison were to be made between the solution and water, the differences would be even larger. Of course, at the concentrations actually used in drag reduction experiments (10 ppmw or less) these differences will be smaller; nevertheless, they may be large enough to produce a reduction in turbulent drag. In addition, other studies (48) using different models and formulations have predicted that even larger differences may be present. Thus, these stability effects must be included as a possible mechanism for drag reduction. C. Evaluation. I t is now possible to draw some conclusions about the relative merits of the proposed mechanisms. Based on the results presented here, it would seem most likely that the mechanism involves elongational flow or an oscillatory disturbance to a shear flow. The advantage of the latter theory is that it may also provide an explanation for early turbulence. As indicated earlier in this paper, there is some evidence that drag reduction (a stabilizing ef-
+
Figure 24. The rate of decay for an oscillatory disturbance is plotted against a normalized parameter, E , which involves the relaxation time of the polymer and the Reynolds number of the flow. The parameter K is solution viscosity divided by solvent viscosity.
elongational viscosity vs. normalized flow time ( t / ~for ) solution and solvent a t several different stretching rates. As seen in this figure, the Oldroyd equation predicts that the solution behavior will be significantly different than the solvent behavior when the flow time and stretching rate exceed certain critical values (16, 39). Moreover, several recent treatments (82) of elongational flow using more sophisticated models have reached the same conclusions. The experimental evidence for this type of behavior is quite limited but studies such as those by Metzner and Metzner (63) seem to indicate that large values of 3 can be obtained a t low polymer concentrations. Although a great deal more experimental work is needed before quantitative comparisons with theory are possible, it is clear that elongational flow must be given serious consideration as an explanation for drag reduction. 4. Oscillatory Disturbances to Steady Shear Flows. A third proposed mechanism for drag reduction involves os294
Ind. Eng. Chem., Fundam., Vol. 14, No. 4. 1975
TIME
fect) and early turbulence (a destabilizing effect) may be related. I t is possible therefore that the same mechanism is responsible for both phenomena. As shown in Figure 24, the rate of decay for an oscillatory disturbance in solution may be smaller than or larger than that in a Newtonian fluid. It is conceivable, therefore, that a mechanism based on such a disturbance could explain both drag reduction and early turbulence. Before such a mechanism can be proposed, however, two things must be established through additional study. First, it is necessary to demonstrate that the stability analysis effects are large enough to produce the observed alterations in flow at the very low concentrations used in drag reducing experiments. Secondly, it must be shown that the conditions necessary to achieve the stability analysis effects-i.e., the proper values of E, K , and c-are actually present when drag reduction (or early turbulence) is observed. By contrast, a mechanism based on elongational flow has the advantage that the effects are predicted to be very large; i j for a 1 ppmw Polyox WSR-301 solution may exceed that for water by a factor of lo3 or more. With effects this large it is not difficult to understand how small quantities of polymer can produce large changes in flow. In this regard elongational viscosity must be considered the most promising of the three proposed theories. Before a mechanism can be established, however, two things must be studied further. First, the large values of i j that are predicted for dilute solutions must be demonstrated experimentally. Secondly, the stretching rate and flow time during the bursting process must be shown to exceed the critical values needed for the onset of a large elongational viscosity.
V. Overview The major thrust of the NRL drag reduction program has been in three basic areas: (1) the development of empirical equations to relate drag reduction to various macroscopic and molecular parameters, (2) the synthesis of new materials with unique advantages for drag reduction studies, and (3) the evaluation of proposed mechanisms for this unusual phenomenon. As this paper illustrates, significant progress has been made in all three areas. The combination of these results with those of other groups has now provided a picture of many aspects of drag reduction. Nevertheless, some additional work is needed before the final details of the phenomenon can be completely understood; in particular, four general areas for research can be designated. First, results such as those described in this paper indicate that unique molecular structures, such as a star shaped polymer, have significant promise as superior drag reducing agents. More tests with these and other structures are needed to fully realize the potential benefits of variations in molecular structure. The second area of importance is the characterization of the polymeric additives. There is a need for direct experimental evaluation of both macroscopic properties such as viscoelastic relaxation times and microscopic properties such as molecular weight distribution. Without such information a definitive picture of drag reduction is not possible. Third, there is a need for better models both to describe turbulent flow and to predict the viscoelastic properties of polymer solutions. The excellent work of several groups has now provided a good qualitative description of turbulence. In order to detail a mechanism for drag reduction, however, it will be necessary to have quantitative models for turbulence. In addition more realistic constitutive equations are needed to predict the behavior of polymer solutions. The use of more complete models and equations will eliminate some of the uncertainties presently surrounding the mechanism of drag
reduction. Finally, more information is needed concerning the behavior of polymer molecules in flow situations. For example, it is important to know how exposure to turbulent flow will affect molecular factors such as polymer aggregation, r.m.s. end-to-end distance, molecular weight distribution (i.e., degradation effects), etc. Such knowledge is essential to the formulation of a molecular level picture for drag reduction. The acquisition of additional information in these four areas should make it possible to develop a detailed explanation for the drag reduction phenomenon. Acknowledgment The authorship of this review serves to identify those workers a t this laboratory who are currently active in the area of drag reduction research. However, the authors also gratefully acknowledge the past contributions of their coworkers as follows: H. R. Baker, R. N. Bolster, R. Darby, C. M. Henderson, P. B. Leach, P. Peyser, and C. R. Singleterry. The authors further acknowledge the support of the Office of Naval Research, the Naval Air Systems Command, and the Naval Ship Systems Command. L i t e r a t u r e Cited ( 1) Arkin, L. S., Singleterry. C. R.. J. Colloidlnterface Sci.. 4, 537 (1949).
(2) Armstrong, R. W., Strauss. U. P., in "Polyelectrolytes," "Encyclopedia of Polymer Science and Technology," Vol. 10, Interscience. New York, N.Y., 1969. (3) Arunchalam, V. R.. Fulford. G. D., Chem. Eng. Sci., 26, 1065 (1971). ( 4 ) Astarita. G.. Nicodemo, L., lnd. Eng. Chem., Fundam., 5, 237 (1966). ( 5 ) Bailey, F. E., Jr., Callard, R. W., J. Appl. Polym. Sci., 1, 56 (1959). (6) Bailey, F. E., Kucera, J. L., Imhof, L. G., J. Appl. Polym. Sci., 32, 517 (1958). ( 7 ) Baker, H. R.. Bolster. R. N., Leach, P. B., Littie, R. C., lnd. fng. Chem., Prod. Res. Dev., 9, 541 (1970). ( 8 ) Betchov, R., Phys. Fluids, 8 , 1910 (1965). (9)Blick. E. R.. "Viscous Drag Reduction," p 409, Plenum Press, New York, N.Y., 1969. (70) Bryson, A. W., Arunchalam. V. R., Fulford, G. D., J. Fluid Mech., 47, 209 (1971). ( 7 7 ) Bueche, F.. J. Appl. Polym. Sci., 4, 101 (1960). ( 1 2 ) Davies. G. A,. Ponter. A. E., Nature(London), 212, 66 (1966). (73) Denn, M. M., Marruci, G., AlChEJ., 17, 101 (1971). (74) Donohue, G. L., Tiederman, W. G., Reischman, M. M., J. fluid Mech., 56, 559 (1972). (15) El'perin, I. T.. Smolskii. E. M., Vest; Akad. Navak Belarusk SSR Ser. Fiz.Tekhn. Nauk, 2, 39 (1965). (76)Everage, A. E., Gordon, R. J.. AlChEJ., 17, 1257 (1971). ( 7 7 ) Forame, P. C., Hansen. R. J., Little, R. C., AlChEJ., 18, 213 (1972). ( 78) Granville, P. S., "Progress in Frictional Drag Reduction-Summer 1972 to Summer 1973," NSRDC Rept. SPD 569-01 (1974). ( 79) Gregor, H. P., Gold, D. H., J. Polym. Sci., 23, 467 (1959). (20) Grosskreutz, R.. Proc. Max. Plank hst. of Flow Res. and the Aero. Exp. Station, No. 53, Gottingen (1971). (21)Hansen, R. J., AlChEJ., 19, 298(1973). (22) Hansen, R. J., J. FluidEng., 95, 23 (1972). (23) Hansen, R. J., Hunston. D. L.. Proc. Eighth lnt. Congr. Acoustics, 2, 579 (1974). (24)Hansen, R. J., Hunston, D. L., J. Sound Vibration, 34, 297 (1974). (25)Hansen, R. J., Little, R. C., Chem. Eng. Progr. Symp. Ser., 67, 93 (1971). (26) Hansen, R. J., Little, R. C.. Forame. P. C.. J. Chem. Eng. (Japan), 6, (1973). (27)Harned, H. S.,Owen, E. E., "The Physical Chemistry of Electrolytic Solutions," 2nd ed, pp 397-400, Reinhold. New York, N.Y., 1950. (28) Henderson, C. M., Little, R. C., Lubr. Eng., 30, 458 (1974). (29)Hershey, H. C.. Zakin, J. L., Chem. Eng. Sci., 22, 1847 (1967). (30)Hildebrand, J. H.. Scott, R. L., "The Solubility of Nonelectrolytes." Reinhold, New York, N.Y.. 1950. (37)Honig. J. G., Singleterry, C. R., J. Phys. Chem., 58, 201 (1954). (32)Honig, J. G.. Singleterry, C. R., J. Phys. Chem., 60, 1108 (1956). ( 3 3 ) Honig, J. G., Singleterry, C. R., J. Phys. Chem., 60, 1114 (1956). ( 3 4 ) Howard, G. J.. McConneli. P.. J. Phys. Chem., 71, 2974 (1967). (35) Hoyt, J. W., J. Basic Eng. 258 (1972). (36) Hoyt, J. W., Soli, G., Science, 149, 1509 (1965). (37) Hunston. D. L., Knauss, C., Palmer, M. E., Myers, R. R.. Trans. SOC. Rheol., 16, 45 (1972). (38) Hunston, D. L., Griffith, J. R., Little, R. C., Nature (London), Phys. Sci., 245, 140 (1973). (39)Hunston, D. L., Ting, R. Y., in press. (40)Kaplan. R. E.. "The Stability of Laminar Incompressible Boundary Layers in the Presence of Compliant Boundaries." MIT Report No. A SRLTR 116-1 (1964). ( 4 1 ) Katchalsky. A., Eisenberg, H., J. Polym. Sci.. 6, 145 (1951). ( 4 2 ) Kim, H.T., Kline, S . J., Reynolds, W. C., J. FluidMech., 50, 133 (1971). ( 4 3 ) Kim, 0. K., Little R. C.. Ting, R. Y., AlChE Symp. Ser., 69, 39 (1973). ( 4 4 ) Kim, 0. K., Little, R. C.. Ting. R. Y . , J. Colloid lnterface Sci., 47, 530 (1974).
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Received for reuiew December 16,1974 Accepted August 4,1975
Membrane Potentials and Transport Parameters for Alkali and Alkaline Earth Chlorides across Cellulose Acetate Membranes King Wai Choi and Douglas N. Bennion" Energy and Kinetics Department, School of Engineeringand Applied Science, University of California, Los Angeles, California 90024
The relative transport rates and associated membrane potentials across modified cellulose acetate membranes cured at 85OC have been measured for alkali and alkaline earth chloride solutions during reverse osmosis experiments. An effective cation mobility is defined as the product of the salt diffusion parameter, Le, times the transference number, t+m.The magnitudes of the mobilities can be ordered as follows: K+ > Rb+ > Cs+ > Na+ > Li' and Be2+ Sr2+ Ca2+ > Ba2+ > Mg2+. The mobility of the 3e2+ ion was exceptionally large, approximately the same as for the alkali ions. All other doubly charged ion mobilities were significantly smaller than for singly charged ions. The observed potential differences across the membranes are explained as a sum of contributions from a diffusion potential across the thin rejecting layer, a streaming potential across the porous sublayer, and an asymmetric double layer on both sides of the membrane. The theoretical interpretation indicates that for BeC12 solutions Be2+ ions are preferentially adsorbed on the membrane, but for all the other salts studied CI- ions were preferentiallyadsorbed.
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Introduction The mobilities of ions and water in modified cellulose acetate membranes are affected by the physical and chemical properties of the ions in solution, the properties of the membrane, and the mutual interactions between solvent, solutes, and membrane. In order to improve descriptions and understanding of the properties which affect mobilities, the relative transport rates of various ions are being measured and interpreted (Fayes, 1970; Re and Bennion, 1973). The present work is to extend the previous data on alkali metal ion relative transport rates to include the alkaline earth ions. In addition, the electrical potential differ296
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ences across the membranes during reverse osmosis are investigated. There have been a number of equations proposed, using various theoretical bases, to predict or a t least correlate transport rates of ions and water across cellulose acetate membranes. Comparisons of various approaches have been summarized elsewhere (Osborn and Bennion, 1971; Choi and Bennion, 1973). It has often been convenient to place the various models into two categories. The first are modifications of Fick's law or the Nernst-Planck equations, frequently with pore flow terms added to provide better correlation for the salt flux observations. Lonsdale et al. (1965), Sherwood e t al. (1967). and Govindan and Sourirajan
