Turbulent pipe flow of Bingham plastics

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Third National Conference on Rheology - Melbourne, May 16-17, 1983.

Turbulent Pipe Flow of Bingham Plastics

A.D. THOMAS

Consulting Engineer, Hilldale, Newcastle, N.S. W. 2420

1. INTRODUCTION

When the particles in a slurry are either wholly or partly colloidal in size, say less than 10 microns, the slurry will generally behave as a Bingham plastic. For such slurries it should be possible to perform bench scale viscometer tests. The problem then becomes one of predicting the pipeline behaviour from the laminar rheological properties. The laminar flow behaviour in any pipe can be obtained with relative ease but when it comes to turbulent flow prediction the situation is not so There have been numerous theories advanced but none is entirely satisfactory. Indeed as recently as last year Techchandani & Shook (1) stated that "prediction of turbulent pipe flow behaviour is not yet an exact science". One theory which has achieved some prominence is that due to Hanks & Dadia (2) and more recently The present paper critically examines this theory by comparing it with a range of data. An unusual approach is taken whereby a distinction is made between those slurries having all colloidal sized particles, e.g. clay slurries; and those having both colloidal sized and coarser particles,

2. FULL COLLOIDAL SLURRIES

Early workers (4, 5, 6,) found that the turbulent flow data were correlated by the Reynolds number yield stress had little or no influence. More recently Cheng (7) and Vocadlo (8) reached the same conclusions. Russian workers apparently agree also ( (9), page 161) and (10), All of the above found that when the plastic viscosity was used in the Reynolds number the friction factors plotted parallel to but generally below the Newtonian line. A very comprehensive series of tests were performed by Thomas (11, 12) who found that all of his data were correlated by the Reynolds number but that the friction factor - Reynolds number relationship had a different slope than the Newtonian line. This slope depended on the degree of non-Newton ian behaviour as measured by either the yield stress or the plastic viscosity and was increasingly less negative than the Newtonian line as the yield stress increased. If data from any one pipe size are diameter effect over and above that due to the Newtonian line this means in effect that there is an additional pipe Thomas' correlation also indicates any given Reynolds number the friction factor falls increasingly below the Newtonian line as the yield stress increases. in contrast to workers before him Thomas was in effect saying that both the yield stress and the pipe diameter were important in addition to the influence of

A convenient method of studying the problem is to plot f/fw versus Ty. Here f is the observed friction is the calculated value if the slurry is assumed to behave as a Newtonian fluid of viscosity equal to the plastic viscosity. Ty is the yield stress. Fig. l is such a plot for a range of data (4, 11, 12, 13, 14, 15, 16, 17). The numbers beside some data points indicate pipe diameters in mm. Two things are obvious. Firstly there is a gradual reduction in f/En as Ty increases in agreement with Thomas' (11, 12) conclusions. Secondly a slight pipe diameter effect is apparent with E/EN increasing as the pipe size increases once again in agreement with Thomas. A study of this pipe diameter dependence shows that it is more pronounced at lower yield stress values becoming zero and indeed slightly negative at very high yield stresses. Industrial slurries very rarely have a yiela D. greater than 30 For data below this the increase in f/Ew is roughly proportional to f/EN with yield stress is approximated by the drawn line. but with a wide variation in the exponent between.02 and.12. For Ty<30 the variation of In 1971 Hanks & Dadia (2) published a theory of Bingham plastic flow in which it was assumed that the turbulent behaviour depended on both the Reynolds number and the Hedstrom number, He, given by

не = D7,0/7?1 (1)

where D is the pipe diameter, is the density and pi is the plastic viscosity.

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More recently Hanks (3) has modified the theory slightly in the light of additional data. theory indicates that the friction factor almost parallels the Newtonian line but that the position below that line depends on the Hedstrom number rising closer to it as He increases. From (3) it is possible to plot his predicted number range from 3000 to 1x0 Variation in f/Er with He. which covers the range of all of the data of Fig. 1. Fig. 2 is such a plot for the Reynolds been replotted on Fig. 2 as f/EN versus He. The numbers beside some data points indicate yield stress in Pascals.

1-0

033

0,250 •8 - 035 XX77

860

•7

•x0+40 Ref Hanks •x0+ | Rey. 13:15 17

10° 10* 10 10°

a/ 10 100 He Ty (Pa)

Figure 1: Variation of f/En with yield Figure 2: Variation of f/Ev with Hedstrom stress for colloidal slurries. number for colloidal slurries. It is obvious that Hanks' theory describes this data very poorly. The main reason for this is due to the role of the yield stress. Consider tests on different slurries in a single pipe size. constant pipe diameter He depends on ply /na. For any particular colloidal slurry this parameter remains fairly constant for all concentrations and hence vield stress values (it is in fact a the degree of coagulation) although it decreases at very low values of ty. is progressively decreased He will at first remain constant and then at best Hanks theory predicts a constant value of f/En as Ty decreases when that f/En increases. More fundamentally as ly is decreased to very low values He decreases and his theory predicts a lowering of f/f away from the Newtonian value of 1 when it would likely that it should approach 1 as is shown in Fig. 1. Next consider data for a particular slurry (pTy /72 constant) in different pipe sizes. case He increases as D increases. Consideration of Fig. 2 indicates the data sizes do in fact show an increase in f/fy as He increases. Hanks' theory predicts such an increase at least for He below 10° increases roughly as He Thus his theory does predict the correct trend as far as pipe diamgter, i.e. f/f, increases as D' • greater than the do 8gge He=10 iS concerned, although it appears to overestimate the effeg: 20

Thus it has been seen how for colloidal slurries Hanks' theory does not describe the effect of yield stress but does describe the pipe diameter effect although possibly to an exaggerated degree. This latter point will be returned to in the following discussion on coarser slurries.

3. COARSER SLURRIES

These slurries are those having only some of the particles colloidal in size with the rest ranging sometimes up to mm size. Fig. 3 shows an analagous plot to Fig. 1 for this type of slurry with the data taken from Refs. 13, 14, 18, 19, 20, 21, 22 & 23. Numbers beside data points indicate pipe diameter in mm. Once again allows comparison with colloidal slurries, (Fig. 1). a general reduction in f/En as ty increases is evident. Consideration of the data for different pipe The full line sizes shows that there is less in fluence of pipe diameter except for the loam data which show a very pronounced effect. Plotting of the data as f/En versus He (Fig. 4) indicates even more scatter than stress in Pascals. occurred with colloidal slurries (Fig. 2). Note however that the pipe diameter effect observed with the loam, £/EN this Figure numbers bestde data points indicate y1g. is identical to the Hanks prediction. An interesting observation is that unlike colloidal slurries Closely more data is needed but, available data fox coarse slurries does not appear to influence this effect but particle size does. To study this effect more the yield stress has been measured is very limited. Fortunately there is an alternative means of studying this without requiring a knowledge of the yield stress. This is as follows. If a particular slurry has been tested in a number of different pipe sizes a plot of J/Jw the existence or otherwise gradient for water at the same velocity. If there is no pipe diameter effect"s /J the slurry pressure gradient and J is the pressure will be essentially independent of D but if there is an Actually, because of the change in slope of to show a slight negative dgpendence even for a Newtonian fluid e.g. for a viscosity of the friction factor - Reynolds number plot with increasing Reynolds number, J/J. would be expected • 030 Pas J/9 would vary roughly as D • On the other hand according to the observation of Fig. 3

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19 7 86105

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Ret. Iron are Cemeat sand тав то-6

3 Cool Chalk Loam Cool F/SN Hanks Re-34s $ 10 8'23

Seuage shrdge Iren ore Legend - See Fig 3 13 Sand in clay

523

0./ Ty (Pa) I0 100 10° 10° 105 10°

He

Figure 3: stress for coarser slurries. Variation of f/f with yield Figure 4: Variation of f/E with Hedstrom

number for coarser slurries.

coarse particle slurries could be expected to show a positiye, dependegion D, e.g. the two sets of loam data of Fig. 3 indicate that J/J is proportional to There is a large body of data which can be used in this manner to study this effect. Fig. 5 shows some available data from a range of sources (24, 25, 26, 27) plotted as J/J of lines drawn through each set of data have been measured and are shown in "Fig. 6 plotted against the versus D. The slopes to around +0.20 median particle size. for coarser slurries. The slope is seen to be tending below zero for fine particles but rising up It is apparent from fig. 6 that the influence of pipe diameter increases sharply as the median particle size increases from 30 to 100 microns. Thus, whereas with colloidal slurries the yield stress influenced the pipe diameter effect, for these coarser slurries particle size appears to be the controlling parameter. None of the published theories of Bingham plastic flow include particle size as a parameter so none would be expected to fully describe the behaviour of these slurries.

• Ref. 24 Iron ore Legend - See Fig 5 24 Copper ore 24 Pyrrotine ore •20H + 14 Loam * 26 Iron ore

J/T 2 27 Coal 51% 27 Coal 47% 25 Potash — O Slope -10 +

10 100 10 100 1000 D (om) d5o (um)

Figure 5: Variation of J/J with pipe Figure 6: Change in pipe diameter diameter for coarser slurries. dependence with median particle size

for coarser slurries.

For the coarsest slurries having median particle sizes above 2g° microns, ghe dependence of f/f, on D is much greater than was the case with colloidal slurries (D c.f. D ). Russian workers are apparently aware of this strong pipe diameter effect since for coal slurries they claim that the effective viscosity increases directly with pipe size ( (9), page 161, (10), page 170). mean that f/fw would vary roughly as D the exponent depending slightly on the Reynolds number. This would This is therefore in agreement with Fig. 6. Fig. 6 is an improvement however since it describes the transition as the particle size becomes finer. From the above discussion it is clear that the Hapks theory more correctly describes diameter effect for the coarsest slurries (f/E,*D ) than for ejtber the full colloidal slurries the pipe ) or for the finest part colloidal slurries (E/E*D ). Obviously Thomas' (11, 12)

celad¿on will not deseribe these coarser slurries since it correlated only data for which

Regarding the effect of yield stress. For colloidal slurries it was noted how the term p ty /n? for which this term tends to increase as remains either constant or decreases as T decrease. Thus f/fu will increase as He increases, Such is not the case for coarser slurries

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more in keeping with Hanks' theory. This can be seen by considering the two sets of loam data in the same pipe size on Fig. 4. Because of this difference in the rheological behaviour of colloidal and coarse slurries which hitherto appears to have been overlooked in the literature, Hanks' theory

tend to describe the effect of yield stress changes for coarse slurries. this behaviour of ply /el changes tends more towards

and Hanks' theory fails. This can be seen by considering the iron ore data of Ref. 18

5. THE INFLUENCE OF COARSE PARTICLES

Both the change in the behaviour of the term ply /øl and the increased pipe diameter effect are obviously somehow caused by the presence of coarse particles. behaviour of fly /pl is almost certainly a reflection of the purely mechanical role of coarse particles in increasing the yield stress and plastic viscosity of a colloidal slurry. This has previously been discussed by the author (Ref. 15) the most obvious effect being the dramatic difference in the variation of yield stress with concentration. For full colloidal slurries Ty varies roughly as volume concentration cubed whereas for part colloidal slurries it can be to the eightth or ninth power. Since both types of slurries obvious that the Hedstrom umber is not the correct correlating tool. Ty is preferable, e.g. Figs.1 & 3. The cause of the pipe diameter effect is not so obvious but it is possible that it is due to a

fluids, the behaviour being different depending on whether the particle smaller or larger than the viscous sub-layer - (28, 29, 30). Thomas (13) showed that this effect also occurred when the carrier fluid was non-Newtonian. showing the variation of J/Jw (for silica sand slurries) with d/d where & is the thickness of the

(2)

v* is the friction velocity.

From this graph it can be inferred that for the highest concentration tested, Now for a Newtonian fluid & varies 308 by volume125/ varies approximately invergelyswith a/d.

for colloidal slurries (where no d/s effect can occur) This is consistent with ith the for ene contest stures. this exponent from.08

concentration increases Ref (30) indicates a greater dependence on d/S so that the exponent of D would be expected to increase with concentration. This does in fact occur as can be seen coal data (27) of Fig. 5. That this d/& effect is feasible is reinforced by the following calculations. the middle of the transition Lies, at about d/8 = 3. equation 2 gives N =.0012 Pas i.e. of the same order as the viscosity of water. particle size. = 23 microns. For typical values of = 1700 Kg m-3 The relevant viscosity would be the viscosity, not of the total slurry, but of the colloidal portion only. This would often be of similar order as that of water. The concept is therefore feasible. One further point of interest regards the transition to laminar flow. For high Hedstrom numbers Hanks' theory predicts an "extended transition region" with the friction factor rising slowly and crossing the Newtonian line before going into the laminar The present author has not this with colloidal slurries. For example tests with a kaolin clay (15) showed that even this effect did not occur. However the present author has sometimes observed it with coarser slurries. intriguing possibility is that this effect is also due to d/§ effects. It has been shown (28) that with Newtonian slurries the friction gradual rise above the Newtonian line as the velocity is decreased when the particle size is the same order as 8. If conditions were such that this effect occurred at velocities just above flow it could be mistaken for an extended transition region.

It has been shown that for Bingham plastics the turbulent flow behaviour, as measured by f/En. depends on both the yield stress and the pipe diameter. E/EN decreases as Ty increases. whilst for the coarsest slurries (median size above 200 microns) it varies as intermediate particle sizes the exponent ranges between +.2 and zero. Hanks' theory was shown to roughly predict the correct trends for the coarsest slurries as the effect of Ty and D although with often large absolute errors. However for progressively finer slurries it was shown to increasingly overpredict the effect of D and to predict the reverse trend as regards Ty. latter reversal reflects the different behaviour of the term pTy/ 7,

as the concentration is changed which hitherto does not appear to have been noted in

For coarser slurries it has been postulated that the ratio of particle size to viscous sub-layer thickness is of importance. In any event it appears that to be successful a prediction method must include particle size as well as the rheological parameters. Much further work is obviously required to clarify the In the interim

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REPERENCES

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HANKS, R.W., DADIA, B.H., A.I.Ch.E.Jnl., 17, 3, 554 (1971)

3. HANKS, R.W., Hydrotransport 5 Conf., BHRA, paper C2, (1978) 4. CALDWELL, D.H., BABBITT, H.F., Ind. Eng. Chem., 33, 249 (1941) 5. ALVES, G.E., et al, Chem Eng. Progress, 48, 385 (1952) 6. HEDSTROM, B.O.A., Ind. Eng. Chem., 44, 651 (1952) 7. CHENG, D.C.-H, Hydrotransport 1 Conf., BHRA, paper J5 (1970) 8. VOCADLO, J.T., Hydrotransport 4 Conf., BHRA, paper D4 (1976) 9. TRAINIS, V. Parameters and Flow Regimes for Hydraulic Transport of Coal by Pipelines, Terraspace, Rockville, Md., U.S.A. (1977) 10. SMOLDYREV, A.Y., SAFANOV, Y.K., Pipeline Transport of Concentrated Slurries, Terraspace,

Rockville, Md., U.S.A. (1979)

11. THOMAS, D.G., A.I.Ch.E. Jnl., 6, 4, 631 (1960)

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13. THOMAS, A.D., Hydrotransport 5 Conf., BHRA, paper D5 (1978) 14. THOMAS, A.D., Proc. 4th Int. Tech. Conf. on Slurry Transport, 196, Slurry Transport Assoc., U.S.A. (1979) 15. THOMAS, A.D., 2nd National Conf. on Rheology, Sydney (1981) 16. HISAMITSU, N., et al, Hydrotransport 5 Conf., paper D3, BHRA (1978) 17. KENCHINGTON, J.M., Hydrotransport 5 Conf., paper D7, BHRA, (1978) 18. STEVENS, G.S., CHARLES, M.E., Hydrotransport 2 Conf., paper E3, BHRA (1972) 19. WILHELM, R.H., et al, Ind. Eng. Chem., 31, 5, 622(1939) 20. DUCKWORTH, R.A. et al, 4th Int. Symp. on Freight Pipelines, Atlantic City, N.J., U.S.A. (1982) 21. RIGBY, G.R., et al, Hydrotransport 8 ConE., BHRA (1982) 22. WASP., E.J., et al, Hydrotransport 1 Conf., paper H4, BHRA (1970) 23. CHENG, D.C.-H., WHITAKER, W., Hydrotransport 2 Conf., paper C3, BHRA (19 72) 24. KARASIK, V.M. et al, Slurry Hydrotransport of Minerals and Tailings, Terraspace, Rockville, Md., U.S.A. (1979) 25. SMITH, L.G. et al, Experimental Studies on the Hydraulic Transport of Potash, Saskatchewan Research Council, Canada (1973) 26. SCHRIEK, W. et al, Experimental Studies on the Hydraulic Transport of Iron Ore, Saskatchewan

Research Council, Canada, (1973)

27. SCHRIEK, W., et al, Experimental Studies on the Hydraulic Transport of Coal, Saskatchewan

Research Council Canada (1973)

28. MAUDE, A.D., WHITMORE, R.L., Trans. Instn. Chem. Engrs, 36, 296 (1958)

29. DAILY, J.W., ROBERTS, P.R., TAPPI, 49, 3, 115 (1966)

30. THOMAS, A.D., 6th Australasian Hydraulics and Fluid Mechanics Conf., 113, I.E. Aust. (1977)