New analysis of non-Newtonian turbulent flow - Yield power law fluids

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NOTES

New Analysis of Non-Newtonian Turbulent Flow - Yield-Power-Law Fluids

A. D. THOMAS

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

and

K. C. WILSON

Department of Civil Engineering, Queen's University, Kingston, Ontario, Canada K7L 3N6 of the exponent of the yield-power law decreases from unity, the calculated friction factor begins by converging toward The writers' new analysis of non-Newtonian turbulent flow is extended to the yield-power-law model. As the value the Newtonian line at high velocity, then parallels it, and finally diverges downward. This prediction agrees with previously unexplained experimental results. de puissance avec seuil. À mesure que la valeur de l'exposant de la loi de puissance avec seuil décroit à partir de la L'analyse originale, que font les auteurs, de l'écoulement turbulent non-newtonien est étendue au modèle de la loi valeur un, le facteur de friction calculé commence par converger vers la droite newtonienne à haute vitesse, puis suit parallèlement cette ligne pour finalement diverger vers le bas. Cette prédiction est en accord avec des résultats expérimentaux qui n'ont pu être expliqués antérieurement. Keywords: turbulence, non-Newtonian flow, yield-power-law slurries, slurry pipelines. on a recent paper (Wilson and Thomas, 1985) a new analvsis of the turbulent flow of non-Newtonian fluids was 2/(n + 1), which is the relation for a power-law fluid. fluid. On the other hand, for o. = 0 it reduces to a = The effect of the predicted thickening of the viscous subpresented. Based on enhanced micro-scale viscosity effects, the analysis predicts a thickened viscous sub-layer, with conlayer with a is to increase the throughput velocity V (for a factor. The analysis indicates that the thickness of the vissequent increased throughput velocity and reduced friction given shear velocity *). This effect is expressed by Equation (10) of Wilson and Thomas (1985) cous sub-layer is proportional to the area ratio a, defined as the ratio of the area under the non-Newtonian rheogram Vlu*= VN/u* + 11.6 (a - 1) - 2.5 In a -&.. (3) to that for a Newtonian fluid with the same strain rate and wall shear stress. As these areas can be obtained from the In this equation Vv represents the throughput velocity for an rheogram there is no necessity for a rheological model to be fitted. However, a model is often convenient, especially equivalent Newtonian flow, i.e. flow with the same wall shear stress, Ow, for a Newtonian fluid with viscosity n vious paper the theory was applied to two-parameter formuwhen extrapolation of the rheogram is necessary. In the prewhich corresponds to the non-Newtonian value at shear stress o = Ow. It should be noted that n is the "secant" viscosity, lations - the power-law and the Bingham-plastic model. It i.e. n = o/(du/dy). The area ratio a, now calculated by law model (Also known as the Herschel-Bulkley model). is now extended to a three-parameter case — the yield-power- Equation (2), is also evaluated at o = Ow- The final term, S, of Equation (3) gives the effect on Application of the analysis to the yield-power-law model throughput velocity of any blunting of the velocity profile in the core of the flow caused by the presence of a yield stress

in non-Newtonian fluids. It was proposed (Wilson and

The equation which defines yield-power-law rheological can reasonably be approximated by the expression Thomas, 1985) that S2 depends only on the stress ratio §, and behaviour is

du (1) 0 = 0, +k Note that the plus sign in this equation wrongly appeared dy where o is shear stress, o, is yield shear stress and du/dy as a minus sign in Wilson and Thomas (1985). is velocity gradient. For this model it is found that the area ratio a is given by Comparison with experimental results a = 2(1 + [n)/(1 + n) (2) slurries (Thomas, 1981), based on data obtained in a 7.2 mm Figure 1 shows laminar-flow rheograms for kaolin clay where = 010w, with ow representing the shear stress at Thomas (1978), other tests in this tube viscometer using clays diameter tube viscometer by Tuft (1977). As noted by reduces to a = 1 + §, which is the relation for a Bingham the pipe wall. It can be seen that when n = 1, Equation (2) from the same source showed excellent agreement with

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 65, APRIL 1987 335

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LEGEND LEGEND

25 C(%) 0, (Pa) MEASURED, D= 18.9 mm MEASURED, D= 105 mm

3.8 1.60 0.0046l 0.93 FRICTION FACTOR f - - - BINGHAM, n= 1.00

• 7. 5 7.44 0.0130 0.875 YIELD POWER LAW, n=0.875

9. 8 8.6l 0.0268 0.80 - - NEWTONIAN

20

0.01

(Pa) 0.006L

2x10 10° 105 2x105

kaolin.

0.10

LEGEND

SHEAR STRESS 5 Figure 2 - Turbulent-flow friction factor for slurry of 7.5 percent • - NEWTONIAN YIELD POWER LAW, n= 0.94 BINGHAM, n= 1.00 MEASURED, C = 3.56 %

5 FACTOR f 0.0 0.10 LEGEND

FRICTION - BINGHAM, n= 1.00 NEWTONIAN YIELD POWER LAW, n=0.84 MEASURED, C= 8.95 %

1000 2000 TRUE SHEAR RATE du/dy (s-') Figure 1 - Rheograms for kaolin slurries at various 0.0IL1 concentrations. 6x103 104 105 3x105

REYNOLDS NUMBER Re = pVD/n

Figure 3 - Turbulent-flow friction factors for slurries of 3.56 and

laminar flow data from recirculating pipe systems with 8.95 percent kaolin in 105mm pipe. internal diameter 18.9 mm and 105 mm. Specifically there was no indication of the "slip" effects sometimes reported for rheological tests of clay slurries. In the previous paper with n = 0.0875 is most appropriate. The same material (Wilson and Thomas, 1985) the slurry with 7.5% clay was was tested at other concentrations by Tuft (1977), and analysed as a Bingham plastic (n = 1.0), but it can be seen Figure 1 also shows plots based on his tube viscometer data that a yield-power-law model with n = 0.875 fits the data for concentrations 9.8%, and 3.8%. The yield-power-law better. (Correlation coefficient 0.9983 cf. 0.9971 for a curves for these slurries which are shown on Figure 1

represent the best fit as determined by linear regression (with

Figure 2 shows turbulent-flow behaviour predicted by the value of n selected at intervals of 0.01). It can be seen Equation (3) for the 7.5% kaolin slurry in two pipe sizes, that the concentration affects the parameters of the yield- 18.9 mm and 105 mm. The experimental data obtained by power-law, with the best-fit value of n decreasing with Thomas (1981) for these pipe sizes have also been plotted concentration; and it is suggested that this based on the viscosity n, evaluated at =w on the figure. The Reynolds number used for this figure is behaviour is typical of many slurries. Slurries of the same kaolin clay were tested by Thomas turbulent-flow data are in better agreement with the yield- (1977) at a number of different concentrations in the 105 mm power-law prediction, n = 0.875, than with the prediction pipe with turbulent flow conditions. Figure 3 shows the preof the Bingham model. Note that for such a value of n the dictions of the present theory for two concentrations yield-power-law prediction shows less rapid convergence 3.56% and 8.95% by volume. The appropriate yield-powertowards the Newtonian line than does the Bingham (n = 1) law parameters were found by interpolation from the three

values already discussed in connection with Figure 1. Also

It was noted above that for the kaolin clay slurry at shown, as dashed lines, are the predictions of the present 7.5% concentration by volume, a yield-power-law model theory using the Bingham model. For the lower concentration

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0.04 for n = 0.4, corresponding to the power law for the present

example diverge downward with increasing plastic Reynolds number (as before, the plastic Reynolds number is based on the "observed" tangent viscosity of 0.0040 Pa • s).

•n = 1.0 It may be noted in particular that for n = 0.8 the curva- 0.02 ture of the rheogram is modest, and the yield stress o, of

-NEWTONIAN LINE 5.0 Pa is not greatly different from the "Bingham" value

f of 6.0 Pa. In such a case the approximation using the

Bingham model would normally appear quite appropriate,

n= 0.8 but the present analysis shows that the behaviour of the

turbulent-flow friction factor could be strongly affected. As

0.0I n= 0.6 noted previously, the exponent n of the yield-power-law rheo-

n= 0.4 gram has a tendency to increase with increasing

concentration. The analysis indicates that this would result in a change of behaviour with increasing concentration -

105 6 × 105 it, or even diverging. This prediction is in full accord with from converging towards the Newtonian line to paralleling PVD/7+ the behaviour noted above in connection with Figure 3.

Figure 4 - Predicted turbulent-flow friction factors for numerical Agreement is also found in the work of D. G. Thomas example. (1963), who noted a similar change in behaviour with

increasing concentration for slurries of kaolin and thorium oxide, including downward divergence from the Newtonian

slurry the predictions using the yield-power-law and the Bingham models are similar - both showing a maximum error It would appear that an answer has now been provided to of about 6%. In the case of the higher-concentration slurry the question raised in the previous paper (Wilson and the difference between the two predictions is more Thomas, 1985), as to why data for some slurries with a yield pronounced, with the yield-power-law prediction following point tend to converge towards the Newtonian line whilst the data more closely, having a maximum error around 10%. other data more nearly parallel it. It is now seen that at low It can be seen that the lower-concentration data tend to conconcentrations the Bingham model is reasonably appropriate verge to the Newtonian line with increasing Reynolds (with consequent convergence towards the Newtonian line), number, whereas for the higher-concentration slurries the but as the concentration is increased the advantages of trend of the data remains significantly below the Newtonian yield-power-law model become more significant. line, approximately paralleling it. If the friction factor is plotted against the plastic Reynolds number (based on the viscosity n.), the difference in trends between the two concentrations becomes even more evident. The recent theory of Wilson and Thomas (1985) has been The nature of the trends can best be shown by numerical extended to the case of the yield-power-law model. The example. Suppose for this purpose that the rheogram for predictions using this model show a change in behaviour as some non-Newtonian material has been obtained for a limited the exponent n decreases. For n = 1.0 the plot of friction range of strain rate, say near du/dy = 1000 s-!, for which factor versus plastic Reynolds number converges towards the • = 10 Pa and the "tangent" viscosity n, = 0.0040 Pa • s. Newtonian line, but as n is decreased to around 0.8 the A series of yield-power-law rheologic relations can be fitted predicted curve lies below the Newtonian line and essentially to these data, corresponding to various values of n in Equaparallel to it. Further lowering of n causes divergence downtion (1). For this type of problem, once n has been selected ward from the Newtonian line. For the clay slurries tested the corresponding values of k and o, can be calculated. The it was shown that n decreases as the solids concentration is meaningful range of n in this case is from unity (Bingham increased. This implies that the predicted change in turbuplastic with o, = 6.0 Pa) to n = 0.4 (power-law fluid with lent behaviour - from convergence through paralleling to

eventual divergence - will take place as the concentration

With the limited data postulated above, a material of this of solids is increased. type would probably be treated as a Bingham plastic, using This behavioural characteristic also explains the phenothe plastic Reynolds number pVD/n, based on the value of menon mentioned in the earlier paper whereby the turbulentn, near du/dy = 1000 s-!. For p = 1100 kg/m?, flow friction factor of some ""Bingham" slurries converges D = 0.100 m, the results are shown on Figure 4, which is towards the Newtonian line whilst that of others more nearly a plot of turbulent-flow friction factor versus the plastic parallels the Newtonian line. The converging behaviour Reynolds number. As expected, the Bingham-plastic line would now appear to be exhibited by low-concentration or (calculated according to the present theory and marked weakly non-Newtonian slurries which are close to true n = 1.0 on the figure) lies below the line of Newtonian Bingham plastics. The diverging behaviour is shown by behaviour, but rises rapidly to converge with the Newtonian higher-concentration, strongly non-Newtonian slurries, for line. If, however, the assumed Bingham relation is not the which the yield-power-law analysis is superior. true one, and the actual value of n is significantly less than 1.0, the predicted behaviour is different, as shown on Figure 4. Note that the lines for the various values of n pass through a common point on the figure, and diverge to the right of C = volumetric concentration of solids in slurry (percent) this point. The prediction for n= 0.8 lies approximately D = internal diameter of pipe, m parallel to the Newtonian line, whilst that for n = 0.6 (and du/dy = velocity gradient (shear rate), s-!

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= Stanton-Moody friction factor References k l = coefficient in yield-power-law model (see Equation (1)) = exponent in yield-power-law model (see Equation (1)) Thomas, A. D., Unpublished data at M.D. Research Co., Sydney, Re = Reynolds Number (Re = pDV/n) Aust. (1977). = velocity, m/s Thomas, A. D., "Coarse Particles in a Heavy Medium — Turbu- = shear velocity (u* = V (ow/p)), m/s lent Pressure Drop Reduction and Deposition under Laminar = throughput velocity (discharge/pipe area), m/s Flow", Proc. Hydrotransport 5, BHRA Fluid Engineering, = throughput velocity for equivalent Newtonian flow, m/s Cranfield, U.K. (1978). = distance from boundary, m Thomas, A. D., "Slurry Pipeline Rheology", 2nd National

Conference on Rheology, Sydney, Aust. (1981).

Greek symbols Thoms, D. G., "Non-Newtonian Suspensions Part II. Turbulent

Transport Characteristics", Ind. Eng. Chem. 55(12), 27-35

= ratio of Rheogram areas: non-Newtonian/Newtonian (1963). = viscosity (n = o/(du/dy)), Pa • s Tuft, P. B. R., Unpublished data at M.D. Research Co., Sydney, = "tangent" viscosity, Pa • s Aust. (1977)- = stress ratio (§ Wilson, K. C. and A. D. Thomas, "A New Analysis of the Turbu- = density of slurry, kg/m lent Flow of Non-Newtonian Fluids", Can. J. Chem. Eng. = shear stress, Pa 63, 539-546 (1985). = yield shear stress, Pa = shear stress at pipe wall, Pa 2 = term in dimensionless velocity equation (see Equations Manuscript received May 26, 1986; revised manuscript received (3), (4)) September 22, 1986; accepted for publication September 24, 1986.

338 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 65, APRIL 1987