T
Technical Report
Prepared For
Office of Naval Research
Contract Nonr 2285
(03)Task NR 062-294
(1967)
DEPARTMENT OF CHEMICAL ENGINEERING
UNIVERSITY OF DELAWARE
NEWARK
DELAWARE
etii..
.1 U.S. Naval Rcarek
London
ç'
14
L:
V.
.TURBULENT FLOW PROPERTIES
OF VISCOELASTIC FLUIDS
F. A. Seyer and A. B. Metzner
Turbulent Flow Properties of
Viscoelastic Fluids
F. A. SEYER and A. B. METZIVER
Univeritv of Delaware, ]Vet ark. Del.. U.S.A.
Reprinted in Canada from
Il TE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 45: 121-126: June 1967
A Publication of The Chemical Institute of Canada 151 Slater Street, Ottawa 4, Canada.
Turbulent Flow Properties of Viscoelastic Fluids
The abnormally high resistance of viscoelastic fluids to sudden deformations and lo stretching may be expected to
cause the structure of turbulent velocity fields in these
systems to differ appreciably from those of Newtonian
fluids. An analysis based on these considerations is
pre-sented and supported using friction factor data for three
concentrations of a water soluble polymer. Pressure losses predicted by use of this correlation are within 15% or less of the experimental values, but as all dimensionless groups which influence the results were not varied independently, the correlation developed may not he universally applicable to all fluids. Suggestions for further and more more specific analyses are made.
E\perimental
studies published in recent years (see, for ex-ample,u7)) have documented but not interpreted mech-anistically the following apparent paradox: dilute solutions of polymeric materials which possess viscosity coefficients greater than those of the solvent nevertheless neid drag coefficients which may be as much as one or more orders-of-magnitude lower than those of the solvent, under conditions of turbulent flow. The purpose of the present paper is to consider an analysis by means of which these superficially-paradoxical characteristics of the viscoelastic polymeric solutions in question may be inter-preted. In common with most analyses of the momentum trans-port characteristics of turbulent fields it does not yield quail-titative a prioiz predictions. lt is, however, sufficiently explicit to be put to quantitative test and this is done in the experimental portion of the paper. Provided also are descriptions of those areas in which further, more specific analyses may be fruitful, and suggestions are made concerning the continuum properties of the fluid which might be optimized in order to minimize turbulent drag.
Comparison with Behavior of Purely-Viscous Fluids
For purely viscous fluids (i.e. time-independent and non-elastic materials(8)) the generalized Reynolds number:
D'' ('''p
NR. =
'y
serves, together with the flow behavior index of the fluid, n', as an adequate similarity criterion for both the laminar and tur-bulent flow regimes'9). The drag coefficients or friction factors of purely viscous fluids are a unique function of these dimensionless variables, i.e.:
fp = «N'e,, n')
(2)\nalytic expressions for Equation (2), based on a direct general-zation of the usual expressions for turbulent Newtonian fluids
F. A. SEVER and A. B. METZNER
University of Delaware, Newark, Del., U.S.A.
(1)
On peut s'attendre ce que la résistance extraordinaire-ment élevée des fluides visco-élastiques aux déformations soudaines
et à la distension puisse produire dans ces
systèmes une structure des champs de vélocité turbulente bien différente de celle des champs des fluides Newtoniens.On présente une analyse basée sur ces considérations,
laquelle est corroborée par les résultats obtenus pour les
facteurs de friction dans le cas de trois concentrations d'un polymère soluble dans l'eau. Les pertes de pression prédites
par l'emploi de cette corrélation n'excèdent pas 15% des
valeurs expérimentales, mais, étant donné qu'on n'a pas
fait varier indépendamment tous les groupes sans
dimen-ions qui influencent les résultats, la corrélation qu'on a
mise au point ne peut s'appliquer à tous les fluides. On
fait des suggestions pour faire des analyses plus poussées et plus spécifiques.
were developed previously(9 and subsequently verified inde-pendently by several investigators(46.'°. Thus the usual von Karman expression becomes:
(')O.Thlog (n')L2 (3)
Additionally, for these fluids the critical value of the Rey-nolds number defining the onset of turbulence is only a weak function of the fluid properties and appears to be independent of rube diameter9°"2. Thus in general the curves represented by Equation (3), and shown as solid lines for n' values of 04 and 0.6 in Figure 1, deviate only moderately from the Newtonian curve (nj' = 1.0) and are seen to retain the characteristics shape of the latter.
Figure 1 also illustrates typical drag coefficient results for the fluids studied in this work. Data for other polymer con-centrations are similar, as indicated in Figure 2. In contrast to the purely viscous behavior described above the drag coefficients for the fluids studied in this work are nearly an order of magni-tude lower than the Newtonian values, and show a pronounced effect of pipe diameter. For all concentrations the data show a strong tendency to follow an extension of the laminar line to Reynolds' numbers in excess of 2100, though in fact the effects of tube diameter, when shown on a larger scale, clearly reveal the flow is not laminar. In Figure 1 data for the l-in, and 2-in. pipes depart gradually from the laminar line before going through small hut fairly well defined "transitions" at Reynolds' numbers roughly equal to 15000 and 6000 respectively. These effects have been noted for a variety of materials and in most cases are strongly dependent on molecular weight and concentration of the polymeric constituent'3'459°'4'-5'.
's 's4D zL 00 0.2% ET-597 o "PIPE I. o 2 NRe
Figure iDrag CoefficientReynolds number curves,
var-ious pipe diameters. Analysis
One or more of the following phenomena may account for the major differences between viscoelastic and purely-viscous turbulent fields:
I. Particulate effects due to the large size of the polymeric molecules (and as present in dusty gases) may promote the stability of the laminar field or dampen turbulent fluctuations, or both.
The dissipative frequencies of the turbulence may be substantially suppressed by the solid-like resistance of the fluid to sudden deformations or by the relatively large re-sistance of the fluid to stretching modes of deformation. Similar considerations may also be applied to the generation of turbulence by small eddies in the wall region.
A significant effect may be the stabilization of the laminar regime, so that transition is delayed to much higher Reynolds numbers than for purely viscous fluids.
Each of these will be discussed in turn; other possible mech-anisms which have been rejected previously(4) are not con-sidered here.
Particulate Effects
The predicted stabilization of laminar fields by the presence of particulate marter(l7.18> suggests changes in the turbulent momentum transport rates in. the direction noted in polymeric solutions. While, as also noted carlier4'9'20, the effects ob-served in many systems of suspended solids are much smaller that those of interest herein this may not be generally true as major (comparable) effects have been reported21. Thus the particulate behavior of the added polymeric molecules may well play a role, and it seems that particulate and continuum analyses may be equally appropriate and perhaps equivalent. The present paper will pursue a continuum approach but it is to be expected that similar results may be derived on the basis of particulate behavioral models.
Solid-Like Characteristics of the Fluid
It has recently been shown2224 quantitatively that the cor-rect asymptotic forms of constitutive equations which describe the continuum properties of viscoelastic materials are, in the limit of rapidly-changing deformation rates, those which retain only the contributions of the elastic material properties. That is to say, in materials capable of both elastic and viscous re-sponses only the former are operative if the deformation rates are changing sufficiently rapidly.
At the other asymptotic extreme, that of a steady laminar shearing flow field, neither the velocity profile nor the viscous drag are at all influenced by the presence or absence of elastic properties, both being dependent only on the apparent viscosity
o-3 C0NCENTRATIO O 0.2%ET-97 0.3% o 04% . O 06% N6
Figure 2 - Drag Coefficients - Reynolds number curves,
denicting concentration effects.
function*. Hovvtr. even in this case other effects - in par-ticula, the well-known normal stress phenomena - arise and may influence some measurements, such as those of an impact pressure profile125'
From the analyses in the above papers it is clear that two new dimensionless groups may be helpful in assessing the be-havior of viscoelastic fluids. In the first place the ratio of the elastic to the viscous forces may be important as it determines the innate capacity of the material to respond in several distinct manners. Thus, in the case of stead' laminar flow through tubes, the deviations of impact readings from their usual values, 'at a given Reynolds number, would be determined by this ratio. Secondly, having defined the inherent ability of the fluid to respond in a given manner the character of the velocity field must also be considered in relation to the fluid properties, as it may influence the extent to which either the elastic or viscous effects predominate. The first of these dimensionless groups has been used in a number of previous analyses1422'20251 and has been termed the \Veissenberg number in honor of the man who first suggested its possible significance in fluid mechanics. The second necessary dimensionless grouping has been noted to represent the ratio of a characteristic time parameter of the fluid to that of the flow process, in which the latter must be a parameter which represents either the duration of a given deformational state or. equivalently, the reciprocal of the rate at which the deformation process This grouping, termed the Deborah number, in turn defines whether the flow process is one in which the elastic or the viscous properties of the fluid are likely to predominate.
In order to carry these arguments further it is helpful to introduce an explicit formulation for the physical properties of the fluid being considered. The convected Maxwell model, in which the stresses r" are related to the deformation rates d" by a viscosity function i and a relaxation time function û, appears to portray the most important properties of the materia] in as simple a fashion as possible and will be chosen for the present study. It may be written as32:
T" =
2I.sd"û--'
(4)ör
The parameters describing the material, z and O, may be evalu-ated using laminar shearing flow measurements of the shearing
stress r12 and of one normal stress difference r
-°This may be verified directly by means of suitable momentum balances. Savins 120) and White and Metzner () amongst many others, discuss this point in the engineering literature.
fIt has been pointed Out elsewhere 20. ) that earlier attempts to define such a dimensionless grouping using the fluid shear rate as a time con-stant characteristic of the flow process, while dimensionally identical to that used here, are incorrect conceptually as they are unrelated to the kind of response offered by the fluid.
F
i»
-fr, - fi
(8)is a function of the same variables since the laminar drag co-efficient is only a function of the generalized Reynolds number81:
= 16/N'i,
(9)while the purely-viscous coefficient f, is given by Equation (3). The frequency characterizing the dissipative portion of the turbulent spectrum may be related to the gross characteristics of the turbulent velocity field employing arguments concisely summarized by Hinze for the case of Newtonian fluids. Since these are primarily ordering arguments they will be used
as a first approximation for viscoelastic fluids as well. Thus we assume:
poe u'k4 (10)
By dimensional arguments Hinze shows that k, the wave-number characterizing the dissipative portion of the spectrum may be estimated by:
(,
3/4A /0)1
1,
in which i, is the length scale of the largest eddies and (N/0')i, is the Reynolds number based on 1,. Substituting Equation (Il) into Equation (10) yields:
u' .1/4
p oe (NR,) (12)
Into Equation (12) one introduces the further order of magni-tude approximations that:
u' U
oe
(13)i, D and
(J\TR,)i, oe (14)
Thus one obtains for a frequency characterizing the dis-sipative portion of the spectrum:
ii = (N'10)a (15)
and the Deborah number becomes:
ND,b = O () (N'A,)31' (16)
This result is generally of sinn lar form to that proposed by
Astarita(4:fl.
It makes no difference in the above development whether or not the fluid property parameters are taken as constants or variables. However, in the latter event additional dimensionless groups may arise. For example, the generalized Reynolds number of Equations (I) and (2) may be replaced by one in-volving the apparent viscosity evaluated at some arbitrary stress state (e.g. at the tube wall) but n' remains as a para-meter001. Similarity, if in the present analysis O is not taken as a constant but as a variable at least one additional dimensionless term must be included. Assuming as in an earlier analysis(21 that a power-law function will suffice over a sufficient range of the variables involved, a second power law exponent enters and Equations (7), (8), and (16) may be combined with these considerations to give:
F=(N'R,,Nv,o,n',$)
(17)= ('2-' p
(N'R)", n' ,
(1 7a)For the very dilute polymeric solutions of primary interest in turbulent drag reduction studies the fluid viscosity is usually a constant058m31. If one assumes that O, the relaxation time, is also constant, the fluid properties reduce to those of a "second order" fluid under steady flow conditions, with n' = I, s = 2.00. Therefore, Equation (l7a) reduces, in this special case, to:
(DUp OU(DUp)3°
/2
'D
jaSome interpretation of Equations (17) would appear to be OU
in order. If the turbulent Deborah number, (N',)314 is a D
small number, the usual Newtonian arguments (Equations (lO)(I 5)) would appear to apply generally to viscoelastic materials as the response will be primarily of a viscous nature.
F =
(lib)
123
11w relative importance of the several terms of Equation (4) depends not only upon the relative values of the material pro-to which the material is subjected. In turbulent Newtonian perties a and O but also upon the kind of deformation process fluids one of the significant features of the How is the "stretch-mg" of line or arca elements by the continuous relative move-ments of the fluid . Batchelor°3 shows the time rate of
change of the size of line and arca elements is directly related to thc energy dissipation rate per unit mass of fluid and that the stretch rate of these elements is determined by the dissipative frequency of the turbulence. lt is thus of interest to inquire whether materials described by means of Equation (4) respond significantly differently, in such modes of deformation, from Newtonian fluids (Equation (4) with a = const. and O = O). The particular case of the steady stretching of line and area elements of viscoelastic fluids has been treated both analyti-cally'237.383940/ and experimentally (0), and it has been shown
that the resistance to this kind of deformation depends upon the magnitude of the Deborah number. For example, in the case of the two-dimensional deformation of an area element it may be shown that the ratio of stresses required for a given stretch rare of a fluid dc-ined by Equation (-I.) as compared to those developed in a Newtonian fluid is given by:
(r11 )Visco2lastic
(5) (r'1)Newtonian
- (
in which ¿k"/x' denotes the stretch rate. Thus, if the fluid is to remain continuous (i.e. if the stresses remain bounded) the viscoelastie fluid possesses a finite maximum stretch rate, a conclusion which may be shown to be independent of the form of constitutive equation used to portray the fluid properties(24).
l'ue prediction that the stress ratio given by Equation (5) is ouch greater than unity at significant stretch rates has been carefully documented in the experimental study of Ballman40.
In the case of a given turbulent field in which the energy levels available to deform a fluid element are fixed at some finite value by the fluid inertial effects it follows directly that the stretch rates will be far lower in the viscoelastic material, hence the velocity fluctuations may be expected to bc repressed significantly.
The magnitude of this turbulence repression is seen to depend övi
upon the magnitude of the Deborah number O ; following a .1
Batchelor's relating of the stretch rate avt/8x1 to the dissipative frequency y the l)eborah number describing this velocity field becomes:
ND6 = O (6)
and the drag coefficient or friction factor becomes a function of this group, in addition to those employed for purely viscous fluids, i.e.:
f =
(N', n', ND,b)
(7) Alternately, the fractional turbulent drag reduction at a given Reynolds number:on the other hand, ifvery 'arge values (N»6 » 1) are computcd. the fluid response would be expected to be predominantly elastic rather than viscous and Equations (1O)(l 5) may no longer apply. The same dimensionless groups as shovn in Equations (17) may still serve to correlate friction factors, but presumably a strong dampening of the turbulent fluctuations would he expected, duc to the larger resistance of viscoelastic materials to stretching. This effect would bC expected ro be felt first ofall at the highest wave numbers, introducing thereby : progressive "cutoff' in the turbulent spectrum as the tevnolds number is increased.
Measurements of the fluctuating velocity components or ise of dye tracer techniques would directly enable the verifi-cation of these predictions. Unfortunately the uc of either hot wire techniques or of impact tubes having low time constants would not necessarily be expected to lead to reliable results in these fluids, at least at high wavcnunsbcrs, in part because of the tendency toward "solidlike" properties of the fluid when sufficiently high Deborah numbers are reached. In this case, as an element of fluid approaches an object in the fluid stream, such as a probe, the Deborah number ofthe fluid element would be expected to become very large, as discussed elsewhere°". The necessary consequence ofthis large Deborah number is the development of a region of markedly reduced fluid velocities, a fact which, in turn, implies increases in the response time of the probe. In the case ofthermal probes this would be expected to lead to low rates of heat transfer from the probe even under liminar flow conditions, an effect which has indeed been re-ported: Lindgrenr 14), when towing a hot 61m probe through a solution containing 100 ppm. of polymer, found a 10% re-duction in heat transfer rares at all velocity levels studied (1.0-10.0 cm/sec.). Correspondingly greater effects were noted in a more concentrated polymeric solution. Thus one may con-clude that the direct infcrral of hih frequency fluctuating velocity components from impact or hot film probes would he suspect ori theoretical grounds, and the few available data serve to support the suspicion. A further analysis of this problem has recently been prepared by Mctzner and Astarita44.
The use of dye filament techniques is not subcct to the above restriction on use of probes but unfortunately tends to be less quantitative. Such visual studies°45464° have proven to be of qualitative value, however. The spreading of a dye filament in Newtonian fluids is, of course, known to be largely due to the large scale fluctuations. Nevertheless it frequently appears as a continuous spred of the material owing to the presence of many small, high-frequency eddies. In contrast, turbulent diffusion measurements in drag reducing systems show that much of the small scale turbulence must be inhibited as the spreading filament breaks into large lumps which retain their idntity for rather large diffusion times. The large scale fluctuations were also found to be far less frequent than in Newtonian systems, at a given level of Reynolds number. Thus these visual observations suggest that the turbulent field may be of far lower intensity than in Newtonian systems and, in addition, the fine structure appears to be largely absent.
Stabilization of Laminar Flow
Recent analvses1849 of the stability of laminar velocity fields suggests that fluid elasticity may exert a destabilizing effect on the laminar field, rather than increasing the value of the transitional Reynolds number; Savins6'50 has noted this prediction to be in conformity with the present observations of the effect of tube diameter (Figure 1) as well as with earlier data, though a contrary view has also been given(3). In the somewhat more carefully studied case of instabilities in rota-tional flows (Taylor instabilities) it is clear that either stabiliz-ation51 or gross dcstabilizstabiliz-ation5152 may occur, depending on the sign of the second normal stress difference and therefore in some detail on the precise nature of the fluid properties. Thus it is not clear at present to what extent this factor con-tributes to the low values of the turbulent drag coefficients noted experimentally.
Summary: Theoretical Concepts
-I he presence of viscoelastic properties is predicted to result in a reduction of the intensity of turbulence, due to both a greater resistance to fluid "stretching ' and to the predominance of the elastic properties at high Deborah numbers. Detailed analytic investigations of both phenomena, and of fluid stability, would appear to be in order. Until these become available, Equations (17) provide a framework for interpretation of experimental results as all three of these naajor phenomena would be expected to depend on the same four dimensionless groups.
Experimental
Both turbulent pipe flow data and rheological measurements were made for 0.2%, 0.3%, 0.4% and 0.6% solutions (by weight) of ET-597 in water. This material is a high molecular weight partially hydrolyzed experimental polyacrylamide sup-plied by the Dow Chemical Company. It vas chosen as re-presenting the best known compromise between high elasticity and reasonable resistance to degradation at the high shear rate levels ofinterest. The solutions were far more concentrated than desired for optinauna turbulent drag reduction; however, as no means appeared to have been developed by rheologists for measurement of the physical properties of more dilute solutions at the high shear rate levels of interest use of such concentrated systems was considered necessary when the study was under-taken. Since that time, however, a significant advance in physical property determinations has been reported by Oliver(o3), and subsequent studies may perhaps be performed on systems cf greater direct practica I interest.
Care was taken to prepare each of the fluids used in ;n identical manner. lt was found that the sniutions were stable .vjth respect to further degradation after mixing and pumping the fluid through the test loop for a 24 hour period. This un-doubtedly reduced the molecular weight of the original polymer; no estimate of the actual molecular weight of the polymer "as used" is thus available.
Turbulent pressure drop-flowrate measurements were made after careful calibration procedures with standard fluids had checked the reliability of both the equipment and the experi-mental procedures employed. These have been described in detail elsewhere0 5I) After obtaining a set of data on each solution (approximately 100 runs) several initial points were repeated to cheek for the presence of any further fluid degrada-non. In each case none was found.
The rheologieal properties of the fluids (O, ¡a (or '). n', s) were measured by means of capillary tube reehniques(534. As with other rheogoniometrie techniques data must be provided for a number of geometries (capillary tube diameter in this case) to validate the correctness of assumptions made in in-terpreting the experimental measurements, but ir is important to emphasize that no a priori assumptions concerning the form of the constitutive relationships exhibited by the fluid are necessary
Results, Discussion
Figures 3 atad 4 depict the measured physical properties of the fluids used in this work. The shear stress - shear rate curves* are shown on Figure 3. on which the laminar pipe flow measure-ments have also been included to show the excellent agreement obtainable over a wide range of tube diameters. The solid lines through the data represent second order polynomials as obtained from a suitable numerical regression analysis. At high shear rates the "end corrections", to account for the fact that only the overall pressure drops and not the pressure gradi-ents were measured, arc not negligible. The usual Newtonian corrections were made but as these are inadequate for visco-elastic fluids27 and no vel1-defined alternare exists at present, rFor laminar well-developed Row through round tubes the wall shear rate is directly related to 8V/D <e>. As a result either of the two may be used and as the latter represents the experimental measurements more directly
8, SEC. lo rw 0 (PS F) DI TUBE DIAMETER 05422CM 002668 00.1146 0.0835 i-PIPE A i' 2' WATE H 0.6%
,
04% 2% 02Figure 3Flow curves for ET-597 solutions.
SV -I
,SEC.
Figure 4Relaxation time for ET-597 soIution.
«2
o D o D D 8 CONCENTRATION o 0.2% ET-597 ¿ OE4% D 0.6%some of these points deviate apprccialtiv. However as the consideration of these deviations would cause the value of n' to change by less than 5%(54) they are in fact not very important
in the overall problem.
Using relationships given elsewhere32 the relaxation time O is obtainable, as a function of shear rate, through the above shear stress results and the measured normal stress difference r11 - T23. The normal stresses were obtained from
measure-ments of the axial thrust of a laminar fluid jet emerging from a smooth round tube55 as a difference between this thrust and the calculated momentum flux. Accordingly the desired results incorporate all the difficulties involved in taking modest dif-ferences between two large quantities and the data occasionally scatter excessively1541. In spite of such scatter, however, the curves are adequately well-defined and, most importantly, the absence of any systematic deviations of the data with tube diameter validates the assumptions used in analysing the noriiial
stress measurementst.
Relaxation times for the fluids used are shown in Figurc 4. One sees that the change in the relaxation time of the fluid with concentration is hardly outside the limits of experimental error, except at low shear rates. Therefore the lack of any effect of concentration on the turbulent behavior of the fluids (Figure 2) is hardly surprising: over the concentration ranges studied the elasticity and viscosity of the fluid vary so similarly tThe results are also validated by independent measurements using a rotational rheogoniometric device on solutions generally similar to those used here F .0 0.8 0.6 0.4 0.2 0.0 C
i
-D H G 5,000 7,500 10,000 15,000 20,000 30,000 40,000 60,000that the ratio of these effects as measured by. the relaxation time Is only a very weak function of concentration. This sanie observation also tends to explain a fact noted elsewhere"'5,6'1151: viz. that the optimum concentration of polymer required to niaximizc turbulent drag reduction is very low. Evidently the first small amounts of polymer increase the relaxation time of the fluid very appreeiably51. and further increases in concentration serve primarily to increase the viscosity level of the fluid while affecting the relaxation time to only a slight degree. In fact the low shear rare points of Figure 4- support an earlier
observa-tion1111 using polyisoburylene solutions that above some fairly
low concentration level the increases in viscosity with concentra-tion level may be so great as to actually cause the relaxaconcentra-tion time to decrease with such further increases in concentration
level of the polymeric solution.
The fact that the slopes of the curves in Figure 3 and 4 are very similar at all three concentration levels implies that the parameters n' and s of Equations (I 7-1 7a) were not varied appreciably. This is fortuitous in the sense that one may readily draw conclusions concerning the method of analysis when only a small tiumber of dimensionless groups controls the results; it is disappointing however in the sense that while the method of analysis and significance of the Deborah number term may he assessed no firm design method may he developed on the basis of these results alone.
Figure 5 depicts the drag reduction ratio as a function of Deborah number, with the Reynolds number as a parameter, as suggested by Equation (1 7a) when the parameters n' and s are all nearly constant. One sees the curves to be generally
well-125
0_' o0-0
D-L5 B F E CONCENTRATION o 0.2% ET-597 0.4% 0 0.6% CURVE N'Re A B C D E F O H 5,000 7,500 0,000 5,000 20,000 30,000 40,000 60,000 400 800 1200 600 2000 2400 NDebFigure 5Dependence of friction factor ratio on Deborah
saum ber.
2000 2400
400 800 200 1600 NDe b
Figure 6 - Drag Reduction Ratios - Dependence on Rey-nolds and Deborah numbers.
I.0 0,8 F 06 08 F 0.6 F 0.8 0.6 F 0.8 0.6
dehned; the fractional drag reductions achieved are as high as 96%, and all lie above 60%. In view of the fact that the Deborah numbers corresponding to these flows arc so high this is perhaps not surprising: with the relaxation times up to three orders of magnitude larger than the calculated time scales of the dis-sipative portion of the Newtonian turbulent spectrum thc velo-city fluctuations in the viscoelastic case, in this wavcnumber region, would be expected to be very appreciably rcstricted if not completely eliminated.
For high values of the Deborah number (ND,b > 1500) the
curves become very flat and must be viewed with caution since small vertical displacements correspond to large variations in a predicted friction factor or pressure drop. For lower values of the Deborah number discrepancies between measured pres-sure losses and those predicted from the curves on the figure
vere always less than I 5%(54).
In Figure 6 the results are summarized in order to portray more clearly the effects of Reynolds number. In order to apply this chart for design purposes, using other fluids, one requires estimates of the effects of the other two parameters, i, and s. As the effects of the flow behavior index are small in the case of purely viscous fluids° they are perhaps also negligible as a first approximation, in viscoelastic systems, but rather extensive data are required to define these influences more carefully.
concluding Remarks
The general method of analysis of turbulence in viseoelasric fluids (Equations (6)(1 7), inc.) appears to have merit in that a consistent design chart (Figure 6) serves to portray the results over wide ranges of the Reynolds and Deborah numbers and a modest but useful range of tube diameters. Further data are required to ascertain the quantitative influences of all four dimensionless groups.
The application of the Deborah number concepts as de-veloped herein would appear to be useful in a wide variety of other viscoclastic fluid processing problems. In particular, fluid atomization processes, some lubrication problems and some textile fiber spinning problems (depending on the geometry of the hardware employed) would be expected to represent high Deborah number flow fields in which the peculiar effects of fluid elasticity would tend to be maximized.
Acknowledgment
The authors have benefited directly from several discussions with J. G. Savins and G. Astarita.
This work has been supported by the Office of Naval Research, U.S. Navy, and reproduction of this work, in whole or in part, is permitted for any purpose of the United States Government. W. F. Seifert. Dow Chemical Company, arranged for a donation of the ET-597 polymeric moterial used.
.Vomencluture
rate of strain tensor, having components d = pipe or tube diameter
= drag coefficient (fraction factor)f r,,/3' pU2 = laminar drag coefficient, Equation (9)
= drag coefficient for purely-viscous fluids, Equation (3) = drag reduction ratio, Equation (8)
conversion factor
= wave number of turbulent fluctuations = consistency index of fluid'8
length scale of largest eddies flow behavior index18>
Deborah number, Equation (6)
= Reynolds number (generalized), Equation (1) = pressure gradient
= elasticity power law index122> = E utensity of turbu lance> 12) = hulk (average) veloiW
= bcal velocity component in the direction of the zt coordinate
= bulk velocity under laminar flaw conditions (i j) F g, kd K' n No,t N'R,
. P/L
u, VI V126
5( ill-y = consistency of non-Newtonian fluid. 'y =
o = fluid r,laxation time (variable), Equation (4)
a = fluid viscosity (variable), Equation (4)
V = characteristic fraquency of dissipative portion of turbulent spectrum
p = fluid density
T = stress tensor having components r'
r,, = shear s:ress r12 evolutad at tube wall conditions = unspecified function
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Meter, D. M. and Bird, R. B., A.I.Ch.E. Journal, 10, 881 11964). Shin, H., Sc.D. thesis, Mass. Inst. of Technology, Cambridge, Mass. (1965).
(16) Uebler, E. A., M.Ch.E. thesis, Univ. Delaware, Newark, Del. (1964).
Michael, D. H., J. Fluid Mech., 18, 19 (1964). Saifman, P. G., i. Fluid. Mech., 13, 120 (1962).
Bobkowicz, A. J. and Gauvin, W. H., Pulp and Paper Research Inst. of Canada, Tech. Report No. 364, Montreal (1964).
) 20 ) Love, R. H., Technical Report 35:3-2 to Office of Naval Research,
Washington (1965).
Zandi, I., Journal Am. Water Works Assoc. (in press).
Metaner, A. B. and White, J. L., AJ.Ch.E. Journal, 11, 989 (1965). Metzner, A. B.. White, J. L. and Denn, M. M., A.1.Ch.E. Journal, 12, 863 (1966) and Cileni. Eng. Progr., 62, (No. 12), 81 (1966.) Astarita, G., md. Eng. Chem. Fundamentals, 6, 257 (1967). Savins, J. G, A.J.Ch.E. Journal, 11 673 (1965).
Astarita, G. and Nicodemo, L., A.I.Ch.E. Journal, 12, 478 (1966). White, J. L. and Metzner, A. B.. Prog. in tnt. Res. on Thermo-dvnamic.s and Transport Prop., p. 748, A.S.M.E. and Academic Press, New York (1962).
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White, J. L. and Metzner, A. B., A.I.Ch.E. Journal, il, 324 (1965). Astarita, G., Can. J. Diem. Eng., 44, 59 (1966).
Savins, J. G., Oral Discussion of Work Reported in (3), A.I.Ch.E. meeting (Dec. 1965).
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Reid. W. It., Proc. Camb. Phil. Soc., 51, 3.50 (1955). Taylor, G. I., Proc. Royal Soc., A164, 15 (1938).
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Slnttcry, J. C., A.I.CILE. Journal, 12, 456 (1966). Ballnsan, R. L., Rheologica Acta, 4, 137 (1985).
Hinze. J. O., "Turbulence", Mc-Graw-Hill,, New York (1958). Astarita, C., Jod. Eng. Chem. Fundamentals, 4. 354 (196.5). Lindgren, E. R., Report to David Taylor Model Basin. Washington
(1965). See also Metzner and Astarita, A.I.Ch.E. J., 13, 550 (1967). Gadd, G. E., Nature, 206, 463 (1965).
Shaver, R. G. and Merrill, E. W., A.I.Ch.E. Journal, 5, 181 (1959). Seyer, F. A., Ph.D Thesis, Univ Delaware, Newark, Del. (1987). Chan Man Fong, C. F. and Walters, K.. J. Mecanique, 4, 29
(1965).
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Oliver, D. R., Can. j. Chem. Eng., 44, 100 (1966).
Seyer, F. A., M.Ch.E. thesis, Univ. Delaware, Newark, Del. (1965). Shertzer, C. R. and Metzner, A. B., Proc. 4th Int. Cong. on Rheology, 603 (1965). See also Shertzer, C. R., Ph.D. thesis, Univ. Delaware. Newark. Del. (1965).
Frederickson. A. G., "Principles and Applications of Rheology". Prentice-Hall, Englewood Cliffs, N.J. (1964).
Manuscript received August 8, 1966: accepted February 7, 1967. Based on a paper presented to the 15th Canadian Chemical Engineering Conference, C.l.C., October 24-27, 1965.
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