A new analysis of the turbulent flow of non-Newtonian fluids

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A New Analysis of the Turbulent Flow of

Non-Newtonian Fluids

KENNETH C. WILSON

Department of Civil Engineering, Queen's University, Kingston, Ontario, Canada K7L 3N6

and

ALLAN D. THOMAS

Consulting Engineer, Newcastle N.S.W., Australia

effects at the small time and length scales of the dissipative micro-eddies, the analysis predicts a thickening of the viscous A new analysis has been developed for the turbulent flow of non-Newtonian fluids. Based on enhanced viscosity sub-layer tending to increase throughput velocity and thus promoting drag reduction. The required flow parameters can be determined directly from rheograms, without employing correlations based on pipe-flow data. The results give good descriptions of flows of fluids with both power-law and Bingham behaviour and explain a number of peculiarities of l'augmentation de la viscosité dans les microtourbillons (échelle de Kolmogorov) et permet de prédire un épaississement On propose une nouvelle méthode d'analyse des écoulements de fluides non newtoniens. Cette analyse est fondée sur résistance due à la friction. Les paramètres nécessaires à cette analyse peuvent être directement obtenus à partir des de la sous-couche visqueuse. Il s'ensuit une augmentation de la vitesse d'écoulement et donc une réduction de la de type plastique de Bingham et loi de puissance; elle permet de décrire avec réalisme le comportement des fluides ainsi rhéogrammes, l'utilisation de corrélations empiriques étant rendue inutile. Cette méthode a été testée avec des fluides que d'expliquer certaines de leurs particularités.

example, in the laminar range it is possible (at least for understood only at low Reynolds numbers — for The shear-stress intercept o, is called the yield stress, and the slope ("tangent" viscosity n.) has a constant value. Bingham plastics and power-law fluids) to define an equiv- Hence, for any shear stress, o, which is greater than oo, the velocity gradient can be obtained as alent viscosity which produces the same relationship be- Newtonian flow. In the early stages of the study of turbulent tween friction factor and Reynolds number that is found for du/dy = (0 - o.)/n.... (1) turbulent viscosity could likewise be defined in such a way flow of non-Newtonian fluids, it was hoped that an effective follows that Since the left-hand side of this equation also equals o/n, it friction factor for smooth-walled turbulent flow (D. G. as to give a unique relation between Reynolds number and 7 = n/(1 - 0/0) (2) Thomas, 1963b). Analysis of recent experimental evidence 4. D. Thomas, 1983) shows that there is little likelihood straight-line fit can be applied to the logarithms of o and As an alternate to the Bingham-plastic model, the that such a unique correlation can exist. du/dy, giving the power-law formulation non-Newtonian flow it is appropriate to outline some of the Before considering the velocity distribution for turbulent 0 = K (du/dy)" (3) in terms of rate of strain, du/dy. Figure 1 shows the shape features of non-Newtonian relationships for shear stress, o, or of a typical curve of this type and nomenclature has been du/dy = (o/K)'/" (4) made compatible with Dealey (1984). As distinct from the On differentiating to obtain the tangent viscosity n, it is linear plot of a Newtonian fluid, for which o = n du/dy, the found that, for all locations on the curve non-Newtonian fluid does not have a constant viscosity. The viscosity, n, is still defined as the ratio of o to du/dy; this n. = nn. represents the slope of a secant line joining the origin to any point of interest on the curve. Another quantity related to linearly from zero at the centre-line to a maximum, 0w, at In pipe flow (laminar or turbulent) the shear stress varies du/dy, denoted here as n, and referred to as "tangent viscosity is the slope of a tangent to the curve of o versus the pipe wall. Thus the velocity gradient, as given by Equaviscosity" or "incremental viscosity" laminar flow of a Newtonian fluid, the variation of gradient tions (1) or (4), is greatest at the wall where o = ow. For non-Newtonian behaviour can be found in the literature, Both two-parameter had three-parameter expressions for across the section gives the well-known parabolic velocity with the two-parameter models having a considerable ad- 0 < n < 1) the laminar profile will be somewhat blunter, profile, but for typical non-Newtonian behaviour (o. > 0 or the Bingham-plastic model, equivalent to a linear approxivantage in simplicity. An early and well-known instance is the wall. As a result the "effective" viscosity ne, used to giving a smaller average velocity for the same conditions at mation to the actual shape of the curve linking o and du/dy. predict the discharge by means of the formula for laminar

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•POSSIBLE

EDDY

1.0 DISSIPATIVE SIZES

EDDIES

SHIFTED

MICROSCALE

1.0 DISTANCE FROM WALL, y (KOLMOGOROV) MACRO EDDIES

du/dy

Figure 1 - Typical non-Newtonian rheogram. (THICKNESS OF

VISCOUS SUBLAYER)

Newtonian flow, must be somewhat larger than the vis- EDDY SIZE cosity n which is applicable to conditions in the immediate vicinity of the wall (D. G. Thomas, 1963a). Figure 2 — Eddy scales in turbulent flow. For turbulent pipe flow, considerable idealization is employed, even for a Newtonian fluid (Reynolds, 1974). Thus the flow immediately adjacent to the pipe wall (assumed produce a smaller friction factor. Their choice generally fell here to be hydraulically smooth) is analysed specifically as on the "tangent" viscosity n, which, as seen above, is a viscous sub-layer. This thin sub-layer accounts for a large smaller than n. Nevertheless, no cogent reason has ever change in velocity because of its high strain rate, which is been adduced for the use of such a viscosity in the viscous based on wall shear stress and viscosity and thus resembles sub-layer (or elsewhere in the flow). Recent work (A. D. the near-wall part of a laminar flow. Once the viscous sub- Thomas, 1983) has shown that use of the tangent viscosity layer is passed, however, the turbulent velocity profile is does not lead to correct prediction of the friction factor, but much blunter than that of the laminar equivalent. The restill gives overpredictions. It was also found that other atduced velocity gradient is caused by turbulent momentum tempts to deal with this discrepancy (to be discussed later in interchange, which is an inertial process, not a viscous one. this paper) have not proved to be successful. Beyond the viscous sub-layer this produces an inertial layer typified by a logarithmic velocity profile. Drag reduction mechanisms If, for the same wall shear stress, the Newtonian fluid is replaced by a non-Newtonian one of the same density, major Better results may be sought by using what is now known changes in velocity gradient would not be expected in the of drag reduction mechanisms. In effect, two different types logarithmic zone. Also, the velocity gradient in the viscous of drag reduction may occur, one affecting the logarithmic sub-layer can be determined readily from the wall shear portion of the velocity profile and the other the thickness of stress and the viscosity associated with this stress. If the the viscous sub-layer. The first type, often described as a thickness of the viscous sub-layer is assumed to be unchanged von Karman coefficient, has been associated with changed, then it would be expected that the friction factor the action of gravity on a cross-flow density gradient. In the for the non-Newtonian flow would be nearly the same as atmosphere this typically arises from a temperature gradient that for a Newtonian flow at the same Reynolds number, (Webb, 1970), and in flumes it can be caused by varying pDV/n. This appoach was followed by Eissenberg and concentration of suspended sand particles (Vanoni, 1946). Bogue (1964) and by Edwards and Smith (1980), and their The non-Newtonian fluids which are of interest here do not papers will be discussed later. exhibit detectable density gradients, and thus will not be In fact, experimentally-determined friction factors often susceptible to this mechanism of drag reduction. lie significantly below the Newtonian equivalents. This The remaining mechanism - thickening of the viscous would now be expressed by the statement that typical nonsub-layer — is associated with the dramatic drag reduction Newtonian fluids have a significant drag-reduction effect. obtained in aqueous flows by the addition of small quantities However, in the 1960's, when much of the early work on of certain long-chain molecules. It appears that these subturbulent non-Newtonian flow was carried out, drag reducstances act by increasing the size of the smallest, dissition studies were only beginning and there was no general pative, turbulent eddies. The phenomenon, as described by awareness of this phenomenon. As a result, the rheologists Lumley (1973, 1978), is illustrated on Figure 2, which of the day generally did not look to drag reduction, but graphs the distance from the wall, y, in the vertical direction simply sought some reduced effective viscosity which and representative eddy sizes in the horizontal direction. As would give a larger Reynolds number and hence should shown, the size of the largest eddies (the macro scale of

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turbulence) is directly proportional to the distance from the rithmic) zone of the flow are known (Tennekes and Lumley, wall, while the size of the smallest eddies (the dissipative, 1972, Chapter 5), but it is sufficient here to consider condior Kolmogorov, scale) is known to have a non-linear retions at the interface between the intertial layer and the lation with distance from the wall, as indicated on the viscous sub-layer. At this point the length scales of the figure. At large values of y the inertial macro-eddies are largest and smallest eddies effectively coincide, as shown on much larger than the dissipative micro-eddies, and between Figure 2, and the same applies to the typical velocities of them is a whole range of turbulent eddy sizes, shown shaded these eddies, which must be proportional to the shear velocon Figure 2. As the wall is approached, the range of possible ity u*, equal to (ow/p).'. As this typical velocity can be eddy sizes shrinks until the size of the largest and smallest considered as constant, and as it was seen that the product eddies are equal, which indicates the boundary of the turof velocity and eddy size varies with a, then it follows that bulence. At this point y equals &v, the thickness (in a statisthe minimum eddy size for a non-Newtonian fluid is larger tical sense) of the viscous sub-layer. than that for a Newtonian fluid (of the same n) by the factor The dashed line on Figure 2 shows that increasing the size a. Based on the geometric relationship shown on Figure 2, of the dissipative micro-eddies leads to an increase in the it can be seen directly that the same factor applies to the viscous sub-layer thickness. If other quantities are unthickness of the viscous sub-layer, &.. affected this, in turn, will produce a higher mean velocity for the same wall shear stress, giving a lower friction factor. Effect on velocity The explanation for the increase in the scale of dissipative eddies in non-Newtonian fluids lies in the nature of turbulent Next, it is necessary to determine the effect of the dissipation. It is generally accepted that new eddies tend to thickened viscous sub-layer on the velocity profile. For this form at the macro scale, and at this size they experience very purpose the transition, or buffer layer, between the viscous little viscous dissipation. Smaller eddies are produced from sub-layer and the logarithmic zone need not be considered; larger ones, mostly by stretching along the eddy axis instead the usual engineering approximation will be made, ennekes and Lumley, 1972, Chapter 3). This "vortex assuming that the linear profile of the viscous sub-layer stretching" process not only reduces eddy size but also inmeets the logarithmic profile where y = 8, (Reynolds, 1974, creases angular velocities, and the combined effect results in Chapter 4). Within the viscous sub-layer conditions are rapid increases in strain rates and viscous dissipation for the taken as essentially steady-state, so that the viscosity n, smallest eddy sizes. Related to this is the concept of the evaluated at or = ov is appropriate for a non-Newtonian "energy cascade" by which energy introduced at the largest fluid, and the velocity distribution, both Newtonian and eddy sizes moves from these through the inertial size ranges non-Newtonian, is given by to the smallest eddies, where it is dissipated From this description it can be seen that energy dissiй/и, = ри„y/n pation in turbulent flow is not a true steady-state process. The intercept with the logarithmic profile (at yv = 8 and Unlike the steady conditions found in laminar flow, and й =й.) is given, for the Newtonian case, by /u, = 11.6 approximated in the viscous sub-layer, turbulent flow inand 8, = 11.6 n/pu,. For the non-Newtonian case, with its volves a series of rapid dissipative events, as eddies drop thickened viscous sub-layer, 11.6 must be replaced by from the inertial to the dissipative size range. There is an 11.6a. analogy between these two processes in fluid mechanics and In the logarithmic zone the general form of the velocity steady-state versus dynamic loading in strength of materials. profile is given by Just as the strain energy on applying a load is determined from the integral of the stress-strain curve (Timoshenko and й/и* = 2.5 In (y/б,) + й./U* (7) MacCullough, 1949, Chapter 11), so the typical rate of where 2.5 is the inverse of the von Karman coefficient. i-*egral of the curve of shear stress versus strain rate. Of energy dissipation of a turbulent eddy is represented by the Substitution of /u*= 11.6 and the value given above for urse, for a Newtonian fluid this integral is merely the area 8, yields the classic expression for the Newtonian case of a triangle, given by 0.5 o (du/dy) or 0.5 m (du/dy), but й/и* = 2.5 In (pu„y/n) + 5.5 (8) for a non-Newtonian fluid the area involved will generally be larger, as shown on Figure 1. The ratio of non-Newtonian For the non-Newtonian case this becomes to Newtonian areas (for the same values of o and du/dy at й/и* = 2.5 In (puzy/n) + 5.5 + 11.6(a - 1) of rheologic parameters will be dealt with later. the upper limit) is denoted by a, and its evaluation in terms - 2.5 In (a) (9) The increased area under the non-Newtonian curve does Since the area ratio a (like the viscosity n) is evaluated for not imply a greater rate of energy dissipation, since the G = Ow, the last two terms of Equation (9) will be constant energy cascade dictates the energy dissipation rate which is throughout the pipe, and thus their contribution to the velocindicates that, as far as dissipative eddies are concerned, the imposed on the smallest eddies. Instead, the increased area ity ratio/u, will apply equally to the ratio of the throughput velocity V to u. (Any constant velocity increase in the non-Newtonian fluid acts as if it were a Newtonian fluid logarithmic zone will produce an equal increase in the core. with viscosity an (which will give the same area under the or velocity-defect zone, and the viscous sub-layer comprises curve, and hence the same dissipation rate as that for the such a minute portion of the total area that variations here non-Newtonian). Since the dissipation process imposes a have no influence on the throughput velocity.) constant-Reynolds-number relation on the smallest eddies, The velocity Vy represents the mean velocity for equivit follows that the product of eddy size and typical velocity alent Newtonian flow, i.e. flow with the same ow for a for these eddies will be proportional to an/p. Newtonian fluid with n corresponding to the non- The scaling laws by which the length scale (and the veloc- Newtonian value at o = o„. A final term, denoted here by ity scale) of the micro-eddies vary within the inertial (loga- S, should be included to represent the effect on mean veloc-

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-n = 0.2 -n =1.0 -n= 0.01 -n = 0.8 0.08 -n = 0.6 0.06- -n =04 -n =0.2 0.04- - =0.1 -n =0.0 -14

0.01- 0.02 n= 0,8

f= 64/ Re' 0.01- 1:0.4-

LEGEND:

10° 10° pVD/n 100 10° - - - DODGE - METZNER (1959) - EQUATION (12)

Figure 3 — Friction-factor prediction for power-law fluids.

ity, if any, of possible blunting of the velocity profile in the logarithmic or core regions of the flow. This gives the 00O 103 10] 10 expression Ré: D"V 12-n/к 8 (n-1), V/u= = VN/u* + 11.6(a - 1) - 2.5 In a - S.. (10) for power-law fluids. Figure 4 - Comparison with Dodge -Metzner (1959) correlation As the flow in the logarithmic and core regions is governed by inertia, not viscosity, non-Newtonian viscosity per se Figure 3, which displays f versus Re (Re = pDV/n, should have virtually no effect here, implying a zero value where n is evaluated at the wall shear stress), gives a more for S2. However, non-Newtonians with a yield stress will direct indication of the differences between Newtonian (n = have a velocity profile with a flattened core, and in this case 1) and non-Newtonian (n ‡ 1) behaviour which are implied convenience this ratio will be denoted by §). One reasonable S2 will be greater than zero and should depend on o./o„ (for in Equation (12). In constructing this figure, Vv/ u, has been

evaluated using the smooth-wall expression which can be

way of approximating this effect is to assume that the logawritten rithmic profile Equation (9) extends inwards only until the radius at which o = o, is reached. From there to the cenx/4* =2.5 In (pDu/n).... (13) treline, the velocity is assumed to remain constant. For such a profile it can be shown that and the turbulent-flow lines have been terminated where

they intercept the appropriate laminar-flow line. Figure 3

2 = -2.5 In (1 - 5) - 2.5 § (1 + 0.5 %)..... (11) shows clearly that the predicted friction factor for turbulent

flow of power-law fluids with n< 1 will be less than that

An alternate evaluation based on a parabolic velocity-defect for a Newtonian fluid at the same Reynolds number Re. law near the pipe centreline (Reynolds, 1974, p. 138) gave As data from experiments on non-Newtonian flow are a different form for the expression for S, but did not differ often correlated using the Bingham-plastic equation, it is significantly in numerical value throughout the expected also important to express Equation (10) in terms of the range of 5. It can be seen that for any fluid with o, = 0, Bingham parameters. It can easily be shown that (a - Equation (11) gives S = 0, as expected. (For a Newtonian equals o./ow, or E. When this, in combination with Equafluid it is also the case that a = 1.00, so that all terms but tion (11), is substituted into Equation (10) a solution can be the first drop from the right-hand side of Equation (10).) obtained without difficulty in cases where o, is known in Substitution of Equation (11) into Equation (10) gives a advance. For other cases it is desirable to recast the equation relationship of a general nature, which can readily be apin terms of two dimensionless parameters, the "plastic"

plied to those problems for which the wall shear stress O„ is shear Reynolds number Re* = pDu,/ n, and the Hedstrom

known in advance. The area ratio a can, if necessary, be number He = pD'o./n}. Using these parameters, the ratio determined directly from the curve of shear stress versus 0./Ow is expressed as He/Re!?. The Newtonian velocity shear rate; using the upper limit associated with the wall ratio V/u, of Equation (12) is then rewritten as shear stress ow, which is appropriate since the typical eddy velocity is u, or (ow/p)°s • As the ratio V/u, equals (8/f)s Vx/4* = 2.5 In (Re, n./n) (14) where f is the Stanton- Moody friction factor, the process or, using Equation (2), and with §= 0/0, just described is equivalent to evaluating the friction factor. For the particular case of the power law fluid discussed Vx/u* = 2.5 In Re, + 2.5 In (1 - 8) (15) spective of the location of the upper limit for integration, previously, it can be shown that a equals 2/(1 + n), irre- Combining Equations (10), (1I) and (15) gives, for the and likewise S = 0. It follows that, for the power law fluid, Bingham-plastic formulation [for which (a - 1) = {l

Equation (10) can be rewritten as V/u, = 2.5 In Re, + 2.5 In [(1 - 8)/(1 + 4))

V/u*= VN/4* + 11.6(1 - n)/(l + n) + 5(14.1 + 1.255).. (16) + 2.5 In [(1 + n)/2] (12) In this form of the equation, the term in Re, can be thought

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of as representing the "Newtonian" value of V/u... When the Bogue (1964) and Edwards and Smith (1980) also used the sum of the remaining terms was plotted against 5, a maxsecant viscosity. For mildly non-Newtonian fluids (n ≥ imum value of approximately 3.2 was found at § of about 0.89) they found that this approach generally resulted in 0.65. It follows that, for a particular value of Re*, the velocity profiles and friction factors identical to the Newlargest Vu, is given by tonian values, whereas for more highly non-Newtonian V/u* = 2.5 In (Rex) + 3.2... (17) These authors postulated the existence of some wall-slip or fluids (n< 0.89) this agreement was generally not found. This is so provided that § = 0.65 is included in the portion drag-reduction effect, but did not attempt to analyze the of the constant-Hedstrom curve to the right of the laminarmechanism involved. It will now be shown how most of turbulent intercept, which is the case for all He > 4.5 × 105 these apparent anomalies can be explained by the new (corresponding to Re > 830 and pDV/n, > 1.7 × 10'). The right limb of the envelope on the diagram of friction On considering Figure 3 it is seen that for n > 0.89 the factor versus plastic Reynolds number has been plotted on predicted friction factors fall below the Newtonian value, this basis on Figure 5. For smaller values of He the envelope but the divergence is no more than 6 or 7 percent. Experi-

can be based on the intercept of the turbulent expression mental uncertainty would often be of this order, and it is

given above and the well known laminar expression understandable that the Newtonian value could be assumed

to apply. The results obtained by Eissenberg and Bogue (1964) for their high concentration thoria suspensions can be

Points obtained at this intercept were used to plot the left dealt with next. For these slurries the measured turbulent limb of the envelope shown on Figure 5. velocity profiles indicate V/u, consistently exceeding the

Newtonian value by about 2.4 (see their Figure 6). The

Comparison with experimental data and earlier theories laminar flow curves for these slurries can be approximated

by a power law with n = 0.66, for which Equations (10) and

The theory just presented is applicable to all purely vis- (11) predict a constant increase in turbulent V/u* by 1.91. cous fluids (Category I of Metzner, 1961), and it is not In view of the considerable scatter of the data this can be necessary to fit a rheological model since the area under the considered as essential agreement with the measured inflow curve can, if necessary, be evaluated numerically. crease. Similarly, the observed friction factors which were However the fitting of a model will often be convenient and some 15% less than the Newtonian line are in good agreethe most common models - power-law and Binghamment with the 18% reduction predicted by the new analysis plastic — will be used below in comparing the existing (see Figure 3). corpus of experimental results with the predictions of the The apparently anomalous data of other authors cited by new theory which has been developed above. Although Edwards and Smith (1980) are likewise explained. Thus the three-parameter models such as the yield-power-law and the data from Shaver (1957), with n = 0.78 fell some 15 percent Powell-Eyring, together with more unusual two-parameter below the Newtonian friction factor line, which is in excelmodels such as the Casson, appear suitable for analysis by lent agreement with present predictions. Similarly Dodge's the new theory, discussion of these models will not be (1961) data for 0.2% and 0.3% carbopol (n = 0.726) and undertaken here. In addition, it should be noted that the n = 0.617) which fell respectively some 5% and 18% below theory cannot be applied immediately to suspensions conthe Newtonian line are in reasonable agreement with present taining coarse particles, for which additional effects must be predictions (16% and 23%). taken into account (A. D. Thomas, 1983). In summary, it can be concluded that the new analysis

provides an adequate prediction of the behaviour of power-

POWER-LAW MODEL

For power law fluids the classic work of Dodge and BINGHAM-PLASTIC MODEL Metzner (1959) provides a valuable datum for comparison. Figure 4 shows a plot of friction factor versus the Metzner- For the Bingham model the predictions have been given Reed (1955) Reynolds number Re', given by in Figure 5 as friction factor versus Reynolds number based Re' = pD"V2-") /k'8("- 1) (19) on the plastic viscosity, and with the Hedstrom number, He,

as parameter. The first thing to note is that the theory pre-

where k' = K[(3n + 1)/4n]". dicts a maximum deviation below the Newtonian line of the new theory and the dashed lined are plots of Dodge and The full lines on Figure 4 indicate the predictions using between 15 and 25%, occurring at or shortly after laminarturbulent transition. Thereafter the predicted curve apment is evident, with both sets of curves roughly paralleling Metzner's empirical correlation formulas. Significant agree- Reynolds number some five times the transition value. This proaches the Newtonian line, effectively reaching it at a the Newtonian line (n = 1.0) but showing increasing devirather rapid convergence to the Newtonian line is in very ation below that line with decreasing n. close agreement with the prediction of Tomita (1959, 1961), The predictions of the new analysis fall 5 to 15 percent and the data given by Bain and Bonnington (1970) for chalk below the Dodge and Metzner (1959) correlations, and are slurries show similar behaviour (see Figure 5). rather similar to the formulations of Clapp (1961) and The same trend has also been found in a sizeable body of Torrance (1963), both of which fall some 15% below Dodge suspension data which has been analysed, not as a Bingham and Metzners' for n = 0.6 and some 20% below for n = 0.4, plastic, but as a power-law fluid. For example, Metzner and based on a range of Reynolds number from 104 to 105. Reed (1955) show data for a variety of suspensions for The alternate presentation of new analysis shown on Figwhich the friction factor virtually joins the Newtonian line ure 3 uses the secant viscosity based on the prevailing wall at a generalized Reynolds number about 3.5 to 5.5 times the shear stress. It was noted previously that Eissenberg and laminar-turbulent transition value. Similarly, Kemblowski

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LEGEND

He=10 • - - LAMINAR LINE

• He=lOY • TURBULENT LINE

.06 TT ENVELOP

04- CHALK HeML3, 25, 8.7% BAIN & BONNINGTON. Figure 5 - Comparison with Bingham- 03- He = 10* plastic data and theory of Hanks (1978)..02 - He= 10°→

He = 10 He= 10'

.O1 10°

PVD/T.

.I0 bears a slight similarity to the "extended transition" region 09 LEGEND predicted by Hanks and Dadia (1971) for Hedstrom numbers 08- DATA PRESENT THEORY in excess of about 5 × 10°. However the behaviour which.07.06 • - - HANKS & DADIA (1971) NEWTONIAN THEORY they predict is significantly different, showing the beginning of deviation from the laminar line at friction factors well He = 2.76 × 106 above the Newtonian turbulent line, as shown on Figure 6..05 It should be noted that the data plotted on Figure 6 do not

show this type of transition region. Mishra and Tripathi

04 (1971) had already remarked on the discrepancy between f Hanks' theory and published data. It is also significant.03 ROUGH WALL that Bowen (1961), who considered both power law and

Bingham plastics, concluded that no significant transitio. zone existed for Bingham plastics.

02 SMOOTH WALL' The predicted change in behaviour at He = 4.5 × 10° is

partly substantiated by the data of D. G. Thomas (1960, 1962) who correlated data from a number of different slur-

ROUGH WALL ries tested in pipes from 3 to 26 mm in diameter. He found SMOOTH WALL differing behaviour for yield stress values above and below

25 Pa. For slurries having yield stress below 25 Pa the

.OI 10° 3 4 105 friction factor-Reynolds number plot converged towards the Newtonian line whilst for high-yield-stress slurries it

PVD/n+ diverged. Using typical values of density and plastic vis-

cosity for his 25 Pa slurry, the relevant Hedstrom numbers

Figure 6 — Comparison with data for limestone in 206 mm pipe. for the pipe sizes of these experiments are found to range

from 1 x 10ª to 1 × 10°. These limits encompass the value of 4.5 × 10° at which the new analysis predicts a behaviour

and Kolodziejski (1973) presented data for a 30% kaolin change, and thus it appears that the present analysis can suspension which indicate a sevenfold increase between explain the change in behaviour observed by D. G. Thomas transition velocity and that required to reach the Newtonian (1960, 1962). All these sets of data are in good accord with the Another feature of Hanks and Dadia's (1971) theory predictions of the new analysis. which is not in agreement with the new theory is the pre- The theory of Hanks and Dadia (1971) predicts a different dicted effect of Hedstrom number on the shape of the curv pattern with the Bingham line nearly paralleling the Newof friction factor versus plastic Reynolds number. As notec tonian line, and other experimental data, e.g. Caldwell and previously, their theory predicts a turbulent friction factor Babbitt (1941), Wilhelm et al. (1939), D. G. Thomas relation which becomes parallel to the Newtonian line at (1960, 1962) and A. D. Thomas (1981) show the same Reynolds numbers larger than the transition range. For a trend. (The data of the latter for a kaolin clay slurry in 7.2, Hedstrom number of 10' their prediction (for large Reynolds 19 and 105 mm pipes is shown in Figure 5.) The existence number) lies some 40% below the Newtonian line, but the of two populations differing in behaviour implies that more predicted difference decreases with increasing Hedstrom detailed analysis is needed, possibly requiring a threenumber, and for He = 10° their prediction coincides with the parameter rheologic model. This problem will be addressed Newtonian line. The original prediction indicated a further in the near future. rise of f above the Newtonian line at higher Hedstrom Another feature of the present theory is, as previously numbers, but more recently Hanks (1978) reappraised the discussed, the different behaviour predicted for Hedstrom data used and modified the theory to give friction factors numbers above and below 4.5 × 10°. Below this value of He below the Newtonian line for all values of He. The new the predicted friction factor for turbulent flow shows a moanalysis that has been presented above does not predict any notonous approach towards the Newtonian line with inequivalent rise with increasing He, and although A. D. creasing Reynolds number. For He > 4.5 × 105 the pre- Thomas (1983) has shown that available data indicate a dicted initial trend following the transition to turbulence is slight effect when the change in He is due to a change in pipe to drop away from the Newtonian line until a maximum diameter, as seen for the kaolin data on Figure 5, the data deviation is reached (at § = 0.65) whence the trend alters, do not correspond to Hanks' (1978) prediction. It was noted move back up toward the Newtonian line. This behaviour by Lumley (1973, p. 270) that a similar diameter effect is

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found in dilute polymer flow, where it may be associated = velocity, m/s

= time-averaged velocity at a point, m/s

Although all examples presented above have assumed = velocity at edge of viscous sub-layer (time averaged), m/s smooth-wall turbulent flow, this present analysis since the effect of wall roughness (in the = shroughput velocity discharge / pipe area, m/s = (ow/p)) transition zone between the smooth and fully rough cases) Vv = throughput velocity for equivalent Newtonian flow, m/s = distance from boundary, m can be incorporated in the first term of Equation (10). It is

results for a minus 300 micron limestone slurry.' These of interest to note the effect of wall roughness on some Greek letters

data, for a 206 mm pipe, are shown in Figure 6. This = ratio of rheogram areas: non-Newtonian/Newtonian particular slurry fitted the Bingham model with a yield stress = thickness of viscous sub-layer, m of 7.92 Pa and a plastic viscosity of 14.1 mPas. The predic- = viscosity (n = o/(du/dy)). Pa • s tion of the present theory with a smooth-pipe assumption is = "effective" viscosity based on equivalent laminar Newseen to lie well below the data. However tests with water tonian flow, Pa•s revealed a relative roughness for this pipe of 0.00079 which, n. = "tangent" or "incremental" viscosity, Pa•s = stress ratio (§ = 0./o.) in this Reynolds number range, puts the water friction factor = fluid density, kg/m* some 12% above the smooth pipe value. Increasing the = shear stress, Pa smooth pipe value by the appropriate ratio results in the = yield stress, Pa predicted "rough wall" curve shown. Agreement is now = shear stress at pipe wall, Pa within about 8%, and the data are seen to follow a trend very = term in dimensionless throughput velocity equation (see similar to that of the predicted curve. For comparison, the Equations (10), (11)) Hanks' (1978) prediction for this Hedstrom number (2.76 x 10° is shown with the pronounced "extended transition" References egion evident. As with the data shown on Figure 5, these data do not follow Hanks' prediction. Bain, A. G. and S. T. Bonnington, "The Hydraulic Transport of Although there is lack of complete agreement both be- Solids by Pipeline", Pergamon Press, Oxford (1970). tween different theories and between different sets of data Bowen, R. L., Scale-up for Non-Newtonian Fluid Flow: Part 4, which have been fitted to the Bingham model, the new 24, 143-150 (1961). Designing Turbulent-Flow Systems", Chem. Eng. 68(15), July analysıs appears to explain a number of experimentally observed phenomena which occur with Bingham fluids. The Caldwell, D. H. and H. E. Babbitt, "Flow of Muds, Sludges and remaining problem - why some data are seen to converge Suspensions in Circular Pipe", Ind. Eng. Chem. 33, 249 (1941). rather rapidly towards the Newtonian line as predicted by Clapp, R. M., "Turbulent Heat Transfer in Pseudo-Plastic Non- Newtonian Fluids", Int. Developments in Heat Transfer, the theory whilst other data tend to lie parallel - may A.S.M.E., Part III, Sec. A, 652 (1961). require consideration of time-dependent effects or the use of Dealy, J., "Official Nomenclature for Material Functions Dethree-parameter rheologic models. This matter will be adscribing the Response of a Viscoelastic Fluid to Various dressed in the near future. Shearing and Extensional Deformations", J. Rheol. 28, 181

(1984). Dodge, D. W., Ph.D. Thesis, Univ. of Delaware, Newark, Dela-

A new analysis has been developed for the turbulent pipe Dodge, D. W. and A. B. Metzner, "Turbulent Flow of Non- Newtonian Systems", AIChE J. 5, 189 (1959). flow of any viscous non-Newtonian fluid. It is based on Edwards, M. F. and R. Smith, "The Turbulent Flow of Nonvelocity profile alteration (drag reduction) associated with Newtonian Fluids in the Absence of Anomalous Wall Effects", thickening of the viscous sub layer, and the flow parameters J. Non-Newtonian Fluid Mech. 7, 77 (1980). in be determined directly from rheograms, without pipe- Eissenberg, D. M. and D. C. Bogue, "Velocity Profiles of Thoria flow data. It has been shown to give a good description of Suspensions in Turbulent Pipe Flow", AIChE J. 10, 723 (1964). flows of fluids with both power-law and Bingham behaviour Hanks, R. W., "Low Reynolds Number Turbulent Pipeline Flow of Pseudo Homogeneous Slurries", Proc. Hydrotransport 5 and to explain a number of peculiarities of such flows. The Conf., Hanover, GFR C2-23 (1978). authors recommend its use for non-Newtonian fluids and Hanks, R. W. and B. H. Dadia, "Theoretical Analysis of the colloidal suspensions. Turbulent Flow of Non-Newtonian Slurries in Pipes"

17, 554 (1971). Kemblowski, A. and J. Kolodziejski, "Flow Resistances of Non-

Nomenclature Newtonian Fluids in Transitional and Turbulent Flow", Int.

Chem. Eng. 13, 265 (1973).

D = internal diameter of pipe, m Lumley, J. L., "Drag Reduction in Turbulent Flow by Polymer f He = Hedstrom number (He = pD'o./n.) = Stanton - Moody friction factor Additives", J. Poly Sci., Macromol. Rev. 7, A. Peterlin (Ed.), Interscience, New York, pp. 263-290 (1973). = coefficient in power-law model (see Equation (3)) Lumley, J. L., "Two-Phase and Non-Newtonian Flow", in "Turk' = coefficient in Equation (19) (k' = K|(3n + I)/4nl") bulence", P. Bradshaw (Ed.), Topics in Applied Physics, Vol. = exponent in power-law model 12, Springer-Verlag, Berlin (1978), Chap. 7. Re = Reynolds number (Re = pDV/n) Metzner, A. B., "Flow of Non-Newtonian Fluids", Section 7, Re' = Metzner-Reed Reynolds number (see Equation (13)) "Handbook of Fluid Mechanics", Ed. V. Streeter, McGraw-

Re* = "plastic" shear Reynolds number (Rc, = pDu /n.) Metzner, A. B. and J. C. Reed, "Flow of Non-Newtonian Fluids Hill, New York (1961).

- Correlation of the Laminar, Transition and Turbulent Flow

'Proprietary data of Slurry Systems Pty. Ltd., North Sydney, Australia. Regimes", AlChE J. 1, 434 (1955).

Page 8

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Chem. Eng. Sci. 26, 915 (1971). Timoshenko, S. and G. H. MacCullough, "Elements of Strength Reynolds, A. J., "Turbulent Flows in Engineering", John Wiley & of Materials", 3rd Ed. Van Nostrand, Princeton, N.J. (1949). Sons, New York (1974). Tomita, Y., "A Study of Non-Newtonian Flow in Pipelines" Shaver, R. G., D.Sc. Thesis, M.I.T., Cambridge, Mass. (1957). Bulletin J.S.M.E. 2, 469 (1959); 4, 77 (1961). Tennekes, H. and J. L. Lumley, "A First Course in Turbulence". Torrance, B. Mck., "Friction Factors for Turbulent Non- MIT Press, Cambridge, Mass. (1972). Newtonian Fluid Flow in Circular Pipes", S. Afr. Mech. Eng- Thomas, A. D., "Slurry Pipeline Rheology", 2nd National Conf. 13, 89 (1963). on Rheology, Sydney, Aust. (1981). Vanoni, V. A., "Transportation of Suspended Sediment by Thomas, A. D., "Turbulent Pipe Flow of Bingham Plastics", 3rd Water", Trans. Am. Soc. Civ. Eng. 5, 111 (1946). National Conf. on Rheology, Melbourne, Aust. (1983). Webb, E. K., "Profile Relationships, the Log-Linear Range and Thomas, D. G., "Heat and Momentum Transport Characteristics Extension to Strong Stability", Q.J.R. Meteorol. Soc. 96, of Non-Newtonian Aqueous Thorium Oxide Suspensions", 67-90 (1970). AlChE J. 6, 631(1960). Wilhelm, R. H., D. M. Wroughton and W. F. Loeffel, "Flow of Thomas, D. G., "Transport Characteristics of Suspensions: Part IV Suspensions through Pipes", Ind. Eng. Chem. 31, 622 (1939). Friction Loss of Concentrated Flocculated Suspensions in Turbulent Flow", AIChE J. 8, 266 (1962) Thomas, D. G., "Non-Newtonian Suspensions: Part I, Physical Manuscript received June 28, 1984; revised manuscript received Properties and Laminar Transport Characteristics", Ind. and January 28, 1985; accepted for publication March 1, 1985. Eng. Chem. 55, 18 (1963a).