POSSIBLE INTERPRETATION OF THE MECHANISM OF DRAG REDUCTION IN VISCOELASTIC LIQUIDS G l A N N l A S T A R I T A Istituto di Chimica Industrial? Lnizersiiy of I hples. .Yofiles. Italy
An interpretation of the mechanism of drag reduction in the turbulent flow of viscoslastic liquids i s proposed, based on the consideration of the frequencies of the noninviscid eddies, which are shown to be higher than the inverse of the relaxation times of those liquids which are usually considered viscoelastic.
indications have recently appeared in the literature, concerning the reduction of the friction loss in turbulent flow of a liquid which can be obtained by the addition of some high molecular weight additives, even in very low concentration. T h e mechanism by which the reduction in friction is obtained is not understood. T h e aim of this paper is to review available knowledge on the phenomenon considered, and to propose a possible interpretation of its mechanism. OME
Review of Available Knowledge
T h e first indication of the anomalous behavior of some polymeric solutions in t u r b d e n t flow has been given by Dodge (9),who observed that friction factors measured in the turbulent flow of aqueous carboxymethylcellulose (CMC) solutions through pipes were abnormally low as compared to the curves correlating the results for other non-Newtonian purely viscous fluids. I t was hinted (9, 70) that viscoelasticity could be the cause of the phenomenon. At the same time, turbulent friction factors for non-Newtonian fluids were correlated by Shaver and hlerrill (43). predicting friction factors systematically lower than the ones proposed by Dodge and Metzner (70). Shaver and Merrill’s results refer to different liquids, the only solution used in both investigations being the C M C solution. This suggests that all of Shaver and h4errill’s liquids are in the category of the anomalous liquids, whose existence was acknoivledged by Dodge and Metzner. Indications of drag reduction obtained through the addition of additives appeared in the early sixties (73, 79, 28, 32, 36, 39). I t is very probable that drag reduction is the same type of phenomenon as the anomalous behavior observed by Dodge. More precise indications, and some tentative interpretation, have been recently published ( 7 , 72. 27, 33-35. 40). Three possible explanations have been considered : a particulate effect of the kind discussed with relation to the flow of dilute solid suspensions (22, 44, 4 6 ) ; a wall-slip effect, by analogy with analogous effects observed in laminar flow ( 7 , 37, 42, 45) ; and a direct influence of viscoelasticity, as a continuum effect influencing the conditions of turbulent flo\v. Particulate effects are of minor importance (44, 4 6 ) , and sometimes of opposite direction ( 2 2 ) ; they cannot be considered in the case of solutions of high polymers (3-2,38). T h e wall-slip effect is more difficult to eliminate, particularly because the Toms effect (a shift of the friction factor-Reynolds number curve with tube diameter, which is predicted in the case of wall slip) has been observed together with drag reduc354
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FUNDAMENTALS
tion. Xonetheless, there are t\vo arguments by which the \vall-slip interpretation can be dismissed. The velocity distribution for the turbulent flow of purely viscous fluids through round tubes is well known (4, 8. 27: 24, 2.9, 30. 37, 47) and has been adequately correlated ( 4 ) in dimensionless form. It has been shown that the same correlation can be extended to non-Newtonian purely viscous liquids by simply evaluating the viscosity a t the wall shear rate (4, 77). I n contrast with this, the same correlation does not fit the velocity distributions measured by Shaver and Merrill (43); this is not surprising,. because once the velocity distribution is known. the friction ,factor-Reynolds number curve can be calculated, and vice versa ( 7 8 ) . T h e velocity profiles measured by Shaver and Merrill are markedly steeper than the purely viscous ones; should the wall slip be the cause of the lower friction loss, flatter velocity profiles would be observed. An equivalent wall-slip effect has been measured for CMC solutions ( 7 ) . The wall velocities measured by Astarita, Marrucci, and Palumbo are approximately correlated by the equation : u p = 0.02ro
(1)
Ivhere up is in centimeters per second, and r, in dynes per sq. cm. Dodge’s data (9) correspond to values of the wall shear stress much higher than the ones encountered in the experiments leading to Equation 1, but a n order of magnitude analysis of the possible effect of wall slip on Dodge’s data can be made by using Equation 1. T h e correction on Dodge’s data calculated through Equation l amounts to no more than 27” of the Reynolds number, and no more than 4% of the friction factor. The discrepancy between the uncorrected data and the curve correlating the data for purely viscous fluids is as much as 707,, so that the wall slip cannot be claimed as a quantitative explanation. The same conclusion can be drawn from the analysis of the pressure-drop data of Ferrari-Bravo (74). O n this basis, it is believed that the phenomenon considered should be interpreted on the assumption that it is caused by viscoelasticity. Proposed Interpretation
T h e phenomenon considered in this paper has sometimes been referred to as “turbulence suppression,” although large scale turbulence has been observed (43) and the laminar flow of viscoelastic liquids is, if different, less stable than for purely viscous liquids (20).
An interpretation based on the assumption that turbulence is not suppressed is here offered. T h e interpretation tries to show that the turbulence in viscoelastic liquids is less dissipative; the discussion of the mechanism of energy dissipation in turbulent flow follows Landau and Lifshitz (23) and Levich
(25). T h e largest eddies which are present in a circular pipe have a scale of motion equal to the pipe diameter, d, and a velocity equal to the average velocity of the fluid, U . T h e Reynolds number for these eddies is equal to the Reynolds number of the flow :
Rex,,,
= Re
>> 1
For smaller eddies, Re;, = __A
< Re
P
(3)
As long as Rex is larger than 1, the eddies may be considered to be substantially coniiervative-i,e,, their motion is inviscid. These eddies receive energy from the larger ones, and give it u p to the smaller ones, but their own motion is nondissipating. These concepts can be extended to the non-Newtonian fluids, because in both the inertial and the energy-containing range the concept of a n effective viscosity is valid also for non-Newtonian fluids, as pointed out by Lumley (27). Let us now consider those eddies whose scale of motion is smaller than A, where A, is defined by:
(4) In purely viscous liquids, these eddies are, in contrast with larger ones, essentially dissipative ; in fact, viscous forces are important when the Reynolds number is less than 1. Levich (25) shows that, for the inviscid eddies, UA
0
U(A/d)’I3
(5)
where the symbol 0 means “proportional to, the proportionality factor being of the order of 1.’’ Substituting Equation 5 into 4, the scale of motion of the largest noninviscid eddies is obtained: ub z
U Re-’14; A,
z
d Re-314
(6)
Thus, the lowest frequency of the noninviscid eddies is: yxo
=
uxo/h, =
d
Re1iz
applied long enough to make viscous dissipation possible ; but in the turbulent flow the shear in noninviscid eddies is reversed at such a high frequency that the elastic stresses cannot relax, and the noninviscid eddies are nondissipative, or a t least less dissipative than in purely viscous liquids. Even if the maxwellian picture of a liquid as a stress-relaxing solid is accepted for ordinary liquids, those liquids which are usually considered as purely viscous-e.g., water-have relaxation times of the order of second (77, 2 6 ) , so that viscous dissipation is possible in noninviscid eddies. If this interpretation is accepted, the drag reduction phemenon should not be considered as turbulence suppression. Two facts have led us to think in terms of turbulence suppression: The friction factors lie close to the extrapolated laminar curve, and the velocity profile is steeper, and hence “more laminar,” than in purely viscous fluids. Both facts can easily be accounted for on the basis of the proposed interpretation. No matter how conservative the structure of turbulence, there is a lower limit to the energy dissipation, and this is the laminar dissipation : Observed friction factors in viscoelastic flow are close, but slightly greater than the corresponding laminar friction factors, in agreement with the principle of maximum dissipation. T h e total energy dissipation is related to the wall-shear stress by the equation:
dE/’dz = 4 r,/d
(8)
which, being a statement of the energy conservation principle as applied to the isothermal flow of an incompressible fluid through a constant section conduit, is clearly valid whatever the rheology of the fluid. Hence, if dE/dz is lowered by any mechanism, the wall-shear stress is also lowered: and so the wall-shear rate; this means that, a t equal flow rate, the bulk velocity profile needs to be steeper. T h e fact that the velocity profile depends on the energy dissipation can be proved in general (5, 6). Conclusions
Analysis of the mechanism of energy dissipation in turbulent flow offers the possibility of interpreting the drag reduction phenomenon observed in the turbulent flow of viscoelastic liquids. This interpretation is offered on a tentative basis; although the author is aware that a much deeper investigation is needed than given here, it is hoped that this communication may stimulate fruitful discussion.
(7)
T h e question now ,arises: What is a viscoelastic liquid? There is probably no clear-cut division between viscous and viscoelastic liquids (3, 76, 25) ; one usually considers viscoelastic those liquids which have stress-relaxation times of a measurable order of magnitude, say of the order of a t least 10-3 second (2, 75, 77). If an order-of-magnitude analysis of Equation 7 is made, the lowest noninviscid frequency is seen to be, in any practical situation, a t least l o 3 sec.-l or even more; thus, the frequency of noninviscid eddies in the liquids which are usually considered viscoelastic is, in turbulent flow through pipes, larger than the inverse of the stress-relaxation time. With respect to phenomena of such a high frequency, the elastic properties of the liquid may well make the noninviscid eddies conservative. In the limit, the shape oscillation of a perfectly elastic solid is a nondissipative phenomenon, while the shape oscillation of a viscous liquid is dissipative. When a viscoelastic liquid is in laminar motion, the shear is
Acknowledgment
Discussion with G. Marrucci made this work possible. Nomenclature
d
= diameter of tube, cm.
E = energy per unit volume, dynes per sq. cm. U = mean velocity, cm. per sec. up = wall slip velocity, cm. per sec. z = distance along tube axis, cm. Re = Reynolds number X = scale of motion of eddy, cm. A, = smallest scale of inviscid eddies, cm. @ = viscosity, gram per cm., sec. Y = frequency, 1 sec. p = density, grams per cc. T, = shear stress at wall, dynes/sq. cm. I
SUBSCRIPTS X
= eddy of scale X
A,
=
eddy of scale X, max = maximum VOL. 4
NO. 3
AUGUST
1965
355
literature Cited
(1) Astarita, G., Marrucci, G., Palumbo, G., IND. ENG. CHEM. FUNDAMENTALS 3, 333 (1964). (2 Benbow, J. J., Howells, E. R., Polymer 2, 429 (1961). (31 Bergen, J. T., “Viscoelastivity,” Academic Press, New York, 1960. (4) Bogue, D. C., Metzner, .A. B., IND.ENG.CHEM.FUNDAMENTALS 2, 143 (1963). (5) Corcoran, \\’. H., Opfell, J. B., Sage, B. H., “Momentum Transfer in Fluids,” Academic Press, New York, 1956. (6) Corrsin, S., J . Aeron. Sci.20, 357 (1953). (7) Crawford, H. R., Rept. to Bur. of Naval Weapons, U. S. Navy, 1962. (8) Deissler, R. G., Natl. Advisory Comm. Aeron., Rept. 2138 (1950). (9) Dodge, D. LV., Ph.D. thesis, University of Delaware, Newark, 1957. (10) Dodge, D. W., Metzner, A . B., A . I. Ch. E . J . 5 , 189 (1959). (11) Eissenberg, D. M., Bogue, D. C., .4m. Inst. Chem. Engrs. Meeting, Houston, Tex.. 1963. (12) Fabula, A. G., International Congress on Rheology, 1963. (13) Fabula, .4.G., Naval Ordinance Rept., 1961 (unclassified). (14) Ferrari-Bravo, A , . Chem. Eng. thesis, University of Naples, Naples, 1964. (15) Ferry, J. D., “Viscoelastic Properties of Polymers,” \Viley, London. 1961. (16) Fredrickson, A. G., “Principles and Applications of Rheology,” Prentice-Hall, Englewood Cliffs, N. J., 1964. (17) Frenkel, J., “Kinetic Theory of Liquids,” Dover, New York, 1955
(24) Laufer, J., Natl. Advisory Comm. Aeron., Rept. 1174 (1954) : 2954 11953). (25) L&ch. V. ‘G., “Physico-chemical Hydrodynamics,” Prentice-Hall, Englewood Cliffs, N. J., 1962. (26) Lodge, A. S., “Elastic Liquids.” Academic Press, New York, 1O A A
(2jj”LUmley, T. J., Phys. Fluids 7, 335 (1964).
(28) Lummus, J. L., Fox, J. E., Anderson, D. B., Oil Gas J . 59, 87 (December 1961). (29) Millikan, C. B., “Proceedings of 5th International Congress in Applied Mechanics, p. 386, \\’iley, New York, 1939. (30) Nikuradse, J., Forsch. Gebiete Zngenieurru. 3B, 1 (1932). (31) Oldroyd, J. G., in “Rheology,” Vol. I, Academic Press, New York. 1956. (32) Ousterhout, R. S.: Hall, C. D., J . Petrol. Techno/. 13, 217 (1961). (33) Park, M. G.. 111, M.Ch.E. thesis, University of Delaware, iX;Cwark. . ..~ .. 1963. (34) Park. M. G., 111, Metzner, A. B., Fluid Mech. Div., Am. Phys. SOC.Meeting, Boston, 1963. (35) Pruitt, G. T., Crawford, H. R., A.1.Ch.E. Meeting, Houston, Tex.. 1963. (36) Randall, B. V., Pan American Petroleum Corp., Tulsa, Okla.. pri\ ate communication, 1960. (37) Reichardt, H., Z. Angezc. Math. Mech. 31, 208 (1951). (38) Saffmann, P. G.. J . FluzdMech. 13, 120 (1962). (39) Sailor, R. A , , Univ. of Delaware, Newark, Del., unpublished work 1960. (40) Savins, J. G., A.1.Ch.E. Meeting, Houston. Tex., 1963. (41) Schlichting, H., “Boundary Layer Theory,” Chap. 18-24: McGra\b-Hill York. 1960. ~ . - ~ Nrw . (42) Schultz-Grunow, F.,~Rheol.Acta 1, 289 (1958). (43) Shaver. R. G., Merrill, E. W., A . I . Ch. E . J . 5 , 181 (1959). (44) Sproul, \V, T., .Tuture 190, 976 (1961). (45j Tbms, B. A., J . ColloidSci.’4, 511 (1949). (46) Vanoni, V. A , Trans. Am. Soc. Czu. Engrs. 111, 67 (1946). ~
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(18) Friend, W. L., Metzner, A. B., A . I. Ch. E. J . 4, 392 (1958). (19) Granville, P. S., U. S. Navy Dept., David Taylor Model Basin Rept., 1961. (20) Herbert, D. M., J . Fluzd. Mech. 17, 353 (1963). (21) Hinze, J. O., “Turbulence,” McGraw-Hill, Kew York, 1958. (22) Kada, H., Hanratty, T. J., A . I. Ch. E. J . 6, 624 (1960). (23) Landau, L. D., Lifshitz, E. M., “Mekhanika Sploshnykh Sred,” Chap. 11, Mikrostruktura Turbulentnogo Potoka, Gostekhizdata, 1944.
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RECEIVED for review October 27, 1964 ACCEPTED December 29, 1964
LOCAL HEAT FLUX T O A WATER FILM FLOWING DOWN A VERTICAL SURFACE RAMESH C H A N D ‘ A N D HAROLD F. ROSSON Department of Chemical and Petroleum Engineering, Unicersity of Kansas, Lawrence, Kan.
A thin water film with ripples on its surface was caused to flow down a heated vertical plate. The local heat flux to the film was studied near the beginning of the heated section of the plate. The effect of the ripples on the local heat flux was dependent on the magnitude of the average heat flux. At higher intensity-e.g., 80,000 B.t.u./(hr.) (sq. ft.)-the local heat flux oscillated with the same wave frequency as the ripples on the film surface. At lower intensity-e.g., 30,000 B.t.u./(hr.) (sq. ft.)-but at the same ripple frequency, heat flux oscillations were less frequent and of smaller amplitude. At still lower heat flux, the oscillations disappeared entirely,
A qualitative explanation is offered to describe the mechanism
of the heat transport.
investigators ( 2 - 4 ) have observed that when a thin layer of liquid passes over a vertical plane, its surface is always disturbed by wave motion and ripples, even in the absence of air traction. Increases in heat transfer rates have been reported (6, 7 7) in systems in which rippling occurs. Portalski (70) has recently postulated the formation of circulating eddies under the troughs of a down-moving wave. T h e present study \\.as made to observe the effects of these ripples on the local heat flux to a laminar film. ANY
Present address, Analytic Systems Co., Pasadena, Calif 356
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FUNDAMENTALS
Experimental Equipment
A heat meter was used to measure point values of heat flux. This meter is based on an analogy between an electrical resistance-capacitance circuit and a body with thermal resistance and capacitance. T h e basis of the analogy lies in the similarity of the differential equations describing temperature and electrical potential distribution in thermal and electrical circuits. respectively. If the two are coupled, the current at any point in the electrical circuit is a measure of heat flux a t the corresponding point in the thermal body. A simplified schematic of the meter is sho\vn in Figure 1 . Myers ( 9 ) gives a detailed account. T h e thermal slab and the electrical circuit were coupled with nvo thermocouples, one on each side of the thermal
