Particle size effects in turbulent pipe flow of solid-liquid suspensions

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808 VITAL SAN

Reprint

of a Paper Presented at a Technical Conference of

THE INSTITUTION OF ENGINEERS, AUSTRALIA

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6th Australasian Hydraulics and Fluid Mechanics Conference Adelaide, Australia, 5-9 December 1977

Particle Size Effects in Turbulent Pipe Flow of Solid-

Liquid Suspensions

A. D. THOMAS

Senior Research Engineer, M.D. Research Co., North Ryde

SUMMARY The classic work in the field of hydraulic transport of solids is that due to Durand (1953) but since then numerous workers have pointed out shortcomings of his pressure gradient prediction method. Using carefully selected experimental data the limitations of the Durand approach are revealed. It is found that a mostly neglected parameter, the ratio of particle size d to viscous sub-layer thickness 8 importance. Lack of consideration of this parameter explains many of the reported discrepancies of the Durand equation. The result is a relatively simple empirical procedure which is shown to be applicable over the whole range of particles sizes pipe diameters and which can be used to confidently predict the pressure gradient of narrow size distribution suspensions. 1 INTRODUCTION 2 "homogeneous" portion pertaining to energy loss

due to particle-particle and particle-fluid in-

Figure 1 shows results obtained by the author for teractions. mm and 1.20 mm in two different pipe diameters, two different sands of median particle size of 0.13 THE DURAND EQUATION & Ø VERSUS · PLOTS 18.9 mm and 105mm. Both sands are of narrow size range, the first having 98% of solids between.070 Durand (1953) proposed the equation and.210 mm and the second 98% between 0.60 mm and 2.00 mm and the delivered concentration in both K·-1.5 (1) cases is nominally 12% by volume. The exhibited where = J-J (2) behaviour is typical of the behaviour obtained in horizontal pipes with suspensions of discrete part-

icles in liquids. develop a method of predicting such behaviour given The purpose of this paper is to and · = Y/Ca

the particle size and solids specific gravity. 8D (S-1) (3) 2 NOTATION Durand did not give a value for the constant K and

it has been variously reported as 81 and 150.

C Volume concentration of solids Ca Particle drag coefficient d Particle size

4 09 D gravitational constant pressure gradient of suspension Pipe diameter metre) 10.0 7.5

velocity as the suspension. Pressure gradient of water flow at same mean per 5.0L

v S specific gravity of solids critical velocity below which a sliding bed of water

mean velocity in pipe

d ary bed of particles appears critical deposit velocity below which a station- 1° V* friction velocity for water W particles appears variable defined by equation 8 viscous friction velocity at critical conditions settling velocity of a single particle viscosity of suspension defined by equation 5 = sub-layer thickness 5tw/ (ewV*) (See Hinze (1959)) for water Gradient (m of Pressure 1. 2.5L, 75 20000 viscosity of water ew density of water non-dimensional excess pressure gradient defined by equation 2 modified Froude number defined by equation 3 0.5 2 3 4

Subscripts Velocity (ms"',

1 "heterogeneous" portion pertaining to energy Figure 1 Typical behaviour, Sand C= 12%, • 0.13 mm required to maintain particles in suspension 1.20 mm

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The data for the two different sands in Figure 1 rather than equation 1. This lower slope has also replotted in Figures 2 and 3 using been noted by Babcock (1968), Hayden and Stelsun Durand variables D and ·. (1968) and Toda et al (1969). author using a 53.8 mm pipe have also been included. for a nominal concentration of 12% by Equation 4 appears to adequately correlate the data the actual concentrations ranged for the 1.2 mm sand but data on a 0.82 mm sand, not from 7.4% to 13.5% and these were used in presented here, revealed that the inclusion of the to calculate D. In calculating the particle drag particle drag co-efficient in equation 4 overcompen-

co-efficient, Cy, the median particle size was used sated for particle size and that in fact for Ca≤1

along with the calculated settling velocity of a the pressure gradient is independent of particle sphere of that size in water at that temperature. size as has been suggested by Newitt et al (1955) The resulting values were 0.71 and 13.3 for the 1.2 and Babcock (1968). This fact immediately explains and 0.13mm sands respectively. some of the scatter which has been evident on the

Ø -* plots of, for example, Zandi & Govatos (1967). However, it is not a major source of scatter, since the inclusion of Ca in equation 4 only causes a relatively small error in Ø in the range 0.44<C, <

Q 1.0.

by considering Figure 3. This shows that for fine particles not only do the data points

0-70-001 collected data from numerous sources, The departure from the Durand equation is greatest at high values of different pipe diameters are displaced markedly. deviate from a straight line but the paths for The main source of scatter in the Ø - plots is Zandi and Govatos ( 1967)

points in all, and also found that the scatter was greatest at high V values. The change in slope at VAlO for fine particles was noted by Zandi and Govatos but the displacement of the curves for different pipe diameters appears to have gone unnoticed except by Thomas (1976). Low velocity region near the critical velocity the Durand equation underpredicts the pressure gradient ·-1.25 for small pipe sizes but increasingly overpredicts further illustrated by the data of Schriek et al for 0.175 mm sand in 52 and 315 mm diameter pipes where, at low velocities there is a difference of some 400% in the value of Ø for the two pipe sizes for identical · values. 20 30

Figure 2 Typical behaviour of coarse (1.2 mm) 4 FLOW IN HIGH VELOCITY REGION sand, C = 12% • D = 18.9 mm., x 53.8 mm, 0 105 mm 4.1 Basic Differences between Fine and Coarse 15 Particles

An inspection of Figures 1 and 3 will reveal that the change in slope on the Ø - plot att≥10 is related to the paralleling of the pressure gradient to the water line. Inherent in the Durand equation is the assumption that the excess pressure gradient w› approaches zero as the velocity increases or in other words the slurry pressure gradient approaches the water only pressure gradient.

xx ure 1 the results for the 18.9 mm pipe suggests

that this is the case for the coarse sand but not for the fine sand, the data of which appear to parallel the water line. Qualitatively this different behaviour can be explained by the fact that particles smaller than the viscous sub-layer can enter the sub-layer and so raise the effective viscosity in the wall region where (see for example

10 50 100 Hinze (1959)) the major part of turbulence generat- Figure 3 Typical behaviour öf fine (0.13 mm) sand, ion and direct viscous dissipation occurs. trast, coarse particles have little effect on the = 12% • D = 18,9 mm, x 53,8 mm, 0 105 mm sub-layer so that the effective viscosity in the

wall region will be essentially

Figure 2 shows that a definite correlation exists of the fluid medium. between Ø and y for the 1.2 mm sand although the data appear to best fit an equation of lower slope The region of high velocity flow where pseudo homosuch as geneous flow prevails is similar to the flow of

suspensions in vertical pipes and to the flow of

ø = 80 · -1.25 (4) neutrally buoyant particles. In both these cases

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the importance of the ratio of particle size to 4.3 Relation toØ -4 plots viscous sub-layer thickness (d/б) has been recognized (Durand (1953)), Maude & Whitmore (1958), The change in slope and displacement of data Newitt et al (1961) and Daily & Roberts (1966) and different pipe sizes evident in Figure 3 at high it is generally agreed that fine particles produce values of y can now be explained. For this fine higher pressure gradients than do coarse particles. 0.13 mm sand the d/ô ratio is around 3 to 7 which 4.2 Assumed Model for High Velocity Flow Regime are of importance. means that the viscosity effects in the wall region For the same value of · these

effects will be more significant in the smaller

The model for the pseudo-homogeneous region can now pipe diameters giving higher Ø values. be outlined. For d/8 < 1 as a first approximation ues to be increased eventually in all pipe sizes the pressure gradient can be calculated using the d/5 > 10 so that the data for different pipe sizes usual friction factor - Reynolds number relationshould converge at very high values of, a trend ship with the viscosity calculated using any of the which is suggested by suitable equations proposed in the literature, e.g. Thomas (1965). For the coarse 1.2 mm sand of Fig. 2 d/ô # / = 1 + 2.5c + 10.05c +.062exp 1.875C 50 and for a concentration of 12% equation 6 gives the Result is the same for all pipe d1ameters so = 1.07 at high velocities, i.e. Ø = 0.58 and

1-1.595C (5) there is no data separation for different pipe siz-

For coarse particles much larger than the viscous es, even at high values of. sub-layer the majority of evidence from both vert- 5 FLOW IN LOW VELOCITY REGION ical fipe flow (Durand (1953), Newitt et al (1961) ally buoyant particles (Dailey & Roberts (1965)), and Toda et al (1969)) and from the flow of neutr- Although the displacement of the data points for indicates that the pressure gradient is near the different pipe sizes evident in Figure 3 has been water value. There is very little explained in the region of high Y values the disthe flow of coarse, settling suspensions at very placement 1w values is as yet unexplained. high velocities in horizontal pipes. One exception To study this region it is helpful to return to the is the work of Murphy et al (1955) who suggested Durand parameter · which has been seen to successthe following equation. fully corelate the data for coarse sands.

observed by Durand 1953) that the critical velo-

52/3 = 1.07 city for these suspensions was given by

which seems to agree with the data obtained by the equations 5 and 6 J = 28(-1) (7)

determined for both very high and bery low d/d Figure 4 shows the resulting values for where F close to 1.3. is a weak function of concentration but is Examination of equation 3 will reve- C =.12,.20 and.30 for particles of S.G. 2.65. al that the variable · does in fact contain V and Also shown are experimental obtained by so in equation 4 · could be replaced by the more author and from the data of Shriek et al (1973) general 5 given by settling suspensions at high velocities in horizontal pipes where settling effects are negli- (8) Equation 6 • seems to apply for d/8> 10. dashed lines show possible transition paths for in-

termediate values of d/s. The use of V/V, has been suggested by Yufin et al (1975).

For coarse particles V is given by equation 7 and {=·. For fine partičles smaller than the viscous sub-layer (i.e. d/8 < 1) D.G. Thomas (1962) has proposed the following equation

= 0.010 dv* e (9)

proportional to D' This equation indicates that V compared with the D is approginately

experimentally observed

(Shriek et al 1973, A.D. Thomas 1975). les midway between the fine and coarse extremes such as the 0.13 mm sand of Fig 3 the critical velocity could be expected to vary approximetely as D" where m might lie somewhere bewteen o.1 and 0.5, actual critical deposit velocities for this sand in the 18.9, 53.8 and 105mm pipes at C = 12% were

cates that V 0.80, this, and d:33 m/s respectively which i because V= Va

10 Fig 3 Shows that the displacement of the data points depending on D in this manner examination of

d/8 at low · values could be eliminated if $ were used with d/8 S.G. 2.65 ferent pipe sizes in this Figure are still displace-

*Ex 4 variation of. 13 * 45,.48, 1.20 mme = 12% d at high values but this can be explained and alinstead of ·. Of course the data points for dif- •C = 30% lowed for as discussed in Section 4.

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6 COMBINED MODEL FOR HIGH AND LOW VELOCITY REGIONS ion method which can explain most of the scatter To obtain a smooth continuous equation describing evident on previous Ø - * plots. The pressure gradient for any size particle in any pipe size can the pressure gradient for all velocity values above be obtained by applying equation 11. it is convenient to assume an additive form as is calculated using equations 12 and 8

(11) along with an estimate for V using equation 5 or 6 along with figure 4.

Low and high and Pelority regions respectively. Followare increasingly important at the 9 ACKNOWLEDGEMENT

ing the discussion in Section 5 a suitable equation The author wishes to thank M.D. Research Co. Pty.

Ltd. for permission to publish this paper.

ø = 80 € - 1.25 can be obtained by the method of Section 4

d/8, the first indirectly through the dependence of thất both d are dependent on the ratio 10 REFERENCES

and the second directly as given by the BABCOCK, HA. (1968). Heterogeneous flow of Heteroof calculating for all d/8 valves is not yet geneous solids. Int. Symposium on Solid Liquid Flow in pipes, Univ. of Penn. Phil. U.S.A. and lower limits while the available, however Equations 7 and 9 provide upper results for the 0.13 mm icle DAILY, W.D. and ROBERTS, P.R. (1969). Rigid partsuspensions in turbulent shear flow. Tappi, sand provide data intermediate between these ex- Vol. 49, No. 3, PP 115 - 125.

DURAND, R. (1953). Basic relationships of the trans

Using the observed critical velocity to calculate portation of solids in pipes. Proc. Minn. Int. Hydr } values Ø has been calculated for the 0.13 mm Conv., Int. Assoc, for Hydr. Res. PP 89-103. sand in the 18.9, 53.8 and 105 mm pipe sizes, and HAYDEN, J.W. & STELSUN, T.E. (1968). Hydraulic conthe results are shown in Figure 5. Although exact veyance of solids in pipes. Int. Symposium on Solid agreement is not obtained the general trends of Liquid flow in Pipes, Univ. of Penn, Phil. haviour - namely data from all three pipe sizes the predicted curves agree with the observed be- HINZE, J.O. (1959) Turbulence, McGraw-Hil1, New converging at the low and high 5 regions but being York. displaced at intermediate values of 5. MAUDE, A.D. & WHITMORE, R.L. (1958). The turbulent mum equivalent error in pressure gradient predictflow of suspensions in tubes. Trans Inst Chem.

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MURPHY G. YOUNG, D.F. & BURIAN, R.J. (1955). Pro-

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NEWITT, D.M. RICHARDSON, J.F. & GLIDDON, G.J. (1961)

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* я* SCHRIEK, W. SMITH, L.G. HAAS, D.B. & HUSBAND, W.H.W.

1. 0u (1973). Saskatchewan Research Council of Canada,

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SHOOK, C.A. SCHRIEK, W. SMITH, L.G. HAAS, D.B. &

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15 50 100 200 300 cil of Canada. Report E73-20. Fig. 5 Variation of with 5, C = 12%, THOMAS, A.D. (1976). Scale-up methods for pipeline 0.13 mm sand • D = 18.9 mm, X 53.8 mm, O 105 mm transport of slurries. Int. J. Miner. Process. Vol.

7 DEPOSIT VELOCITY 3, pp 51-69.

The pressure gradient prediction method THOMAS, A.D. (1975). Factors affecting the hydraulhere involves an estimate of V great practical interest is V c Obviously also of For fine particles ic performance of slurries. Thermofluids conf.Bris. Dec 3-5, Inst. of Eng. Aust. No. 75/9, PP 96-100. can be considerate less ican v but for coarse particles in small pipes W • When a single suspensions. Part VIII. J. Colloid Sci., Vol. 20, THOMAS, D.G. (1965). Transport characteristics of layer of particles deposits in the pipe the fiow PP 267. area is reduced. The deposit will only remain if a pipe of diameter D-d. Further consideration will the mean velocity above the bed is less than V suspensions: Part VI. A. I.Ch.E. Jnl, Vol 8, No. 3 THOMAS, D.G. (1962). Transport characteristics of

PP 373-378.

= { (D-d)/D (13) Int. Chem EngnI. TODA, M., KONNO, H., SAITO, S. & MAEDA, Vol. 9, No. 3, pp 553-560. S. (1969)

where m varies from 0.5 for coarse particles to 0.1 for fine particles. YUFIN, A.P., FILIMONOVA, I.V. TARASOV, V..K., DANIL' 8 CONCLUSIONS CHENKO, N.V. & BELOVA, N.T. (1975). Fluid Mechanics

- Soviet Research, Vol 4, No. 2 pp 5-8.

its main A careful study of the Durand equation has enabled inadequacies to be isolated and studied. ZANDI, I & GOVATOS, G. (1967). Heterogeneous flow The result is a simple but rationally based predict of solids in pipelines. A,S.C.E. H73, PP 145-157.

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