Extending the Wilson-Judge deposit-velocity equation to finer particles and larger pipes

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17th International Conference on Transport & Sedimentation of Solid Particles, 22-25 September 2015, Delft, The Netherlands

A MODIFICATION OF THE WILSON & JUDGE DEPOSIT VELOCITY EQUATION, EXTENDING ITS APPLICABILITY TO

FINER PARTICLES AND LARGER PIPE SIZES

Allan Thomas Slurry Systems Pty Limited, Sydney, Australia PO Box 424, Northbridge, NSW, 2321, Australia [email protected] Abstract: This paper is concerned with predicting the deposit velocity of turbulently flowing, fine particle slurries in a horizontal pipe. The Wilson and Judge (1976) deposit velocity correlation is widely used to predict the deposit velocity in turbulent pipe flow of sand in water slurries. However the W&J correlation is limited to medium size particles as the pipe size increases. For example, for silica sand in water in a 1000 mm pipe, the W&J correlation only predicts Vd adequately for d50 particle sizes 150 µm and coarser. Thomas (2014) showed how for most wide size distribution, viscous slurries pumped in the mining industry, the equivalent sand in water particle size is less than 100 µm. This means that the W&J correlation is generally of limited applicability for these slurries, especially in large pipe sizes. A Modified Wilson & Judge (MW&J) correlation is presented. Predicted deposit velocities using MW&J follow smooth curves up to 1000 mm pipe diameter for silica sand particles in water down to 30 µm. Some experimental results from the literature for sands in viscous fluids are compared with predicted deposit velocities using the modified equation. The relevance of the modified equation to deposit velocity predictions for the typical wide size distribution, viscous concentrates and tailings slurries pumped in the mining industry, is discussed.

1. INTRODUCTION The deposit velocity (Vd) is defined as the velocity at which a stationary bed of solids first appears as the velocity is progressively reduced. The Wilson and Judge (W&J) (1976) deposit velocity correlation involves a parameter ∆ which is a function of particle settling velocity, pipe diameter and solids SG. The W&J correlation applies to ∆ > 10-5. The 10-5 minimum on ∆ limits the applicability of the W&J correlation to medium size particles as the pipe size increases. For sand in water slurries this particle size limit may not seem too restrictive but it severely affects the applicability of the W&J correlation for most slurries pumped in the mining industry. Thomas (2014) showed how the d50 particle size of these wide size distribution slurries is generally less than 100 µm. Because these slurries possess a viscosity higher than the viscosity of water, the particle settling

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velocity of a 100 µm d50 size particle is equivalent to a much finer sand particle in water. For example the d50 = 100 µm particle could be equivalent to a 50 µm or 60 µm sand particle in water. The W&J correlation is therefore of limited use for the majority of wide size distribution slurries pumped in the mining industry.

2. CURRENT DEPOSIT VELOCITY PREDICTION METHODS FOR FINE PARTICLES

2.1 WILSON AND JUDGE (1976) The W&J (1976) deposit velocity correlation was the basis for the deposit velocity predictions for medium size sands in the nomograph presented by Wilson and Judge (1978). The deposit velocity, Vd, is the maximum deposit velocity for the particular particle size, solids SG, and pipe diameter. This maximum deposit velocity will relate to a particular concentration and Vd for other concentrations is always less than that given by the nomograph, although the dependence on concentration for fine particle slurries is small, (see Section 3). Vd is given in terms of the familiar Durand (1953) equation, where g is the gravitational constant, D is pipe diameter (m), and S is the ratio of density of particles to density of fluid (ρp/ρf ). The W&J FL parameter is as given below where W is the settling velocity of particle in quiescent fluid (m/s) and d is particle diameter (m).

Vd = FL [2gD(S-1)]0.5 (1)

FL = 2.0 + 0.3 log10∆ (2)

∆ = 0.75 W2/[g D (S-1)] = d/(D Cd ) (3) Cd = particle drag coefficient = 4gd(S-1)/(3W2) (4) The particle settling velocity of a granular particle is assumed as for a sphere. Schriek et al (1973) conducted settling tests on a range of individual sand particles and found that the particle drag coefficients were essentially the same as for a sphere. The mean of each mesh size range represented the characteristic particle size. In a quiescent fluid a particle will align itself so as to result in the highest drag and therefore lowest settling velocity. However in a turbulent flow field the particle will present all alignments to the rapidly changing velocity field, thereby effectively giving an average equivalent settling velocity even closer to that of a sphere.

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2.2 VISCOUS SUB-LAYER DEPOSITION – THOMAS (1979) Thomas (1979) developed a deposit velocity prediction method for particles smaller than the viscous sub-layer, based on the Wilson sliding bed theory. Thomas argued that Equation 5 provides a lower limit to the deposit velocity. Vdδ

* = 1.1 [g µf(ρp - ρf)/ρf 2 ]1/3 (5) The dδ subscript identifies it as the viscous sub-layer deposit velocity. Vdδ

* is friction velocity at deposition = Vdδ√(f/2), (m/s), f = Fanning friction factor at the deposit velocity, and µf = viscosity of fluid (Pa.s). Thomas (1979) defined the viscous sub-layer thickness, δ, as: δ = 5 µf /(ρf V*) (6) where V* is friction velocity = V√(f/2), and f is friction factor at velocity V. Therefore, according to Eqn 5, the thickness of the viscous sub-layer at deposition conditions (δd) is given by:

δd = 5 µf /(ρf Vdδ

*) (7)

2.3 SANDERS ET AL (2004) Sanders et al (2004) modified Thomas’ (1979) viscous sub-layer theory to include the effect of particle size and concentration. Their deposit velocity, given in terms of the friction velocity at deposition, (termed here VdSa

*) is as follows:

1.1 VdSa */ Vdδ * = [ 0.76 + 0.15 dρf VdSa */µf ] / [(Cmax – Cv)0.88]1/3 (8) Cmax is the maximum packing concentration by volume and Cv is the volume concentration of interest. Vdδ

* is the viscous sub-layer friction deposit velocity of Thomas (1979) given by Eqn 5. Equation 8 applies to d+ (= d ρf VdSa

*/ µf ) < 5.

3. NEW MODIFIED WILSON & JUDGE CORRELATION Using an informed trial and error approach, and after trying many different variations, a modification to Equation 2 was arrived at. The new, Modified Wilson & Judge (MW&J) correlation for FL is:

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FL = 2 + 0.305 log10∆ + 1.1x 10-4 ∆-0.489 – 0.044 (1x107 ∆)-1.06 (9) In considering Equation 9, it should be noted that Wilson & Judge’s equation for FL (Eqn 2) was not a direct theoretically derived relationship but represented a best fit correlation to the results from their individual computer analyses for various combinations of particle size and pipe size. With this consideration taken into account the form of Equation 9 would seem equally valid. FL from Eqn 9 is inserted into Eqn 1 to give the predicted MW&J deposit velocity, Vd∆. The d∆ subscript identifies it as the deposit velocity predicted using MW&J based on ∆. Figure 1 compares MW&J Vd∆ predictions for 200 µm, 150 µm, 100 µm, 75 µm and 55 µm sand in water, with the W&J predictions (Eqn 2). The thick grey lines are the W&J predictions and the thin black lines are the MW&J predictions. The W&J predictions for 200 µm and 150 µm particles are shown as being applicable up to the 1000 mm pipe diameter although it is likely that W&J (and hence MW&J) under-predict for particles greater than 150 µm in a 1000 mm pipe. The ∆<10-5 criterion limits the applicable W&J pipe size for particles less than 150 µm as indicated, i.e. maximum 325 mm pipe size for 150 µm particle size, maximum 115 mm pipe size for 75 µm particle size, and maximum 35 mm pipe size for 55 µm particle size. The thick black curve near the bottom of the graph is the viscous sub-layer prediction, Vdδ of Thomas (1979), Eqn 5.

150 µm 200 µm

100 µm

75 µm 55 µm 50 µm 45 µm 40 µm

Deposit Velocity (m/s)

55 µm

0.175 mm, Schriek et al, 1973 & Shook et al, 1982 0.18 mm, Thomas, 1979

0.1

0.13 mm, Thomas, 1979

10 100 1000

Pipe Diameter (mm)

Figure 1 Deposit velocity vs pipe diameter – Comparisons MW&J and W&J predictions

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The MW&J predictions are similar (within 1%) to the original W&J predictions up to the original limit of ∆ = 10-5. The dashed curves are the extensions to the original W&J predictions for ∆ < 10-5 and show increasing deviation from the obvious behaviour trends as the particle size decreases and pipe size increases. In a 1000 mm pipe this deviation is significant for particle sizes less than 100 µm. Also shown in Figure 1 are MW&J Vd∆ predictions for 50 µm, 45 µm and 40 µm particles. For industrially important pipe sizes from 100 mm to 1000 mm, the predicted Vd∆ for 45 µm particle size is similar to the viscous sub-layer (Eqn 5) prediction. For 40 µm particle size Vd∆ is approximately 0.1 m/s below Vdδ. Since, for 45 µm particle size, Vd = Vdδ, therefore for particles smaller than 45 µm, Vd = Vdδ since Vdδ represents a lower limit to the deposit velocity according to Thomas (1979). Also shown in Figure 1 are observed deposit velocities (all for Cv = 0.12) for 0.18 mm sand in 53.8 mm and 105 mm, 0.175 mm sand in pipe sizes from 52.2 mm to 495 mm, and 0.13 mm sand in pipe sizes from 9.41 mm to 105 mm. These data points are consistent with the MW&J predictions for these particle sizes and follow the predicted slope. Of course these data also follow the original W&J prediction. Note that the predicted slope and slope of the data, is significantly less than the slope of the Maximum Maximorum line of Wilson applying to coarse particles (see Wilson et al, 2010). So far no mention has been made of the effect of concentration on the deposit velocity. In the author’s experience (e.g. Thomas, 1979), the maximum Vd for medium to fine sand in water slurries almost always occurs for volume concentrations around 0.12. The wide ranging data of Schriek et al (1973) show a similar result. The two sets of results indicate that for volume concentrations in the range 0.12 to 0.38, Vd is generally within 10% of the value at Cv = 0.12. Hence the MW&J predictions for sand-water slurries in this paper are assumed to apply for concentrations in this range. The exact effect of concentration on MW&J predictions for more viscous slurries has not yet been investigated. In Figure 1 the predictions of MW&J and W&J were plotted against pipe diameter. The predictions of MW&J with W&J can also be plotted against particle size for a particular pipe diameter. Figure 2 shows such a plot for silica sand in water for pipe diameters 1000 mm, 325 mm and 115 mm. (Solids density 2650 kg/m3, fluid density 1000 kg/m3, Cv = 0.12, viscosity 1 mPas, pipe roughness 0.01 mm). The thick grey curves indicate the predictions of W&J down to the particle size for which ∆ = 10-5, with the thin line extensions for ∆ < 10-5 representing the MW&J predictions. Also shown as horizontal lines in Figure 2, are the viscous sub-layer (Vdδ) predictions from Equation 5, for the three pipe sizes, ranging from 0.70 m/s in the 1000 mm pipe to 0.55 m/s in the 115 mm

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pipe, as well as the Vdδ prediction for a 18.9 mm pipe (0.43 m/s). For convenience these are shown extending across to 160 µm particle size although once Vd∆ exceeds Vdδ, the latter no longer applies. Observed deposit velocities in an 18.9 mm pipe for 17 µm and 26 µm sand, (Thomas, 1979), agree well with the Vdδ prediction as shown by the two open circle data points.

D = 1000 mm

2.5

D = 325 mm

D = 115 mm

1.5

Deposit Velocity (m/s)

D=1000, Sanders et al

D=325, Sanders et al D=115, Sanders et al

D = 1000 mm

Vdδ

0.5

D = 18.9 mm

0 20 40 60 80 100 120 140 160

d50 Particle Size (µm)

Figure 2 MW&J Predictions for Sand in Water c.f. Sanders et al (2004) and Thomas (1979) Also shown in Figure 2 are predictions using the method of Sanders et al (2004), Eqn 8, for 1000 mm, 325 mm and 115 mm pipes, shown extending to the d+ = 5 limit, as recommended by those authors. The Sanders et al predictions provide a smooth increase above Vdδ as the particle size increases but once they cross the MW&J curves the predicted trends are at variance with MW&J predicted trends. This discrepancy between the two models requires further investigation and comparison with appropriate data when it becomes available. It could be that the Sanders et al predictions may only apply until they cross the the MW&J curves, in which case, for the predictions shown (Cv = 0.12)

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they would apply for particle sizes from about 30 µm to 50 µm. For concentrations greater than 0.12, the Sanders et al predicted curves rise more steeply and will cut the MW&H curves at a particle size greater than the 50 µm indicated in Figure 2. The MW&J predictions cut the Vdδ predictions at 43 µm, 44 µm, and 45 µm respectively for the 1000 mm, 325 mm and 115 mm pipe diameters. These critical particle sizes are in agreement with the observation in regard to Figure 1, that Vd∆ is similar to Vdδ at approximately 45 µm particle size. From Eqn 7, for the particular case of silica sand in 200C water, at Vdδ the viscous sublayer thickness δd = 180 µm. Figure 2 shows that Vd∆ = Vdδ for a particle size around 45 µm for industrially relevant pipe sizes from 100 mm to 1000 mm. Hence at viscous sublayer deposition, the ratio d/δ = 45/180 = 0.25. Interestingly, it appears that according to the MW&J correlation, viscous sub-layer deposition becomes the controlling criterion when d/δ becomes less than 0.25, for a wide ranging combination of particle size, solids density, fluid viscosity and fluid density. For example Table 1 summarises MW&J predictions for four combinations of solids density, fluid density and fluid viscosity. For each case the particle size has been adjusted until the predicted deposit velocity using the MW&J correlation (Vd∆) equals the predicted viscous sub-layer deposit velocity (Vdδ ) for industrially relevant pipe sizes from 100 mm to 1000 mm. The particle size at which this occurs is shown in Table 1 together with the viscous sub-layer thickness calculated as per Equation 7. In each case the ratio of particle size to viscous sub-layer thickness (d/δd) equals 0.25. Thus, based on the predictions of MW&J, viscous sub-layer deposition becomes the controlling criterion when d/δ becomes less than 0.25. This is in reasonable agreement with the data of Thomas (1979) which indicated deposition was controlled by viscous sub-layer deposition for d/δ < 0.3.

Viscous Sub-layer

d/δd

Fluid Viscosity

Particle Size for Vd∆ = Vdδ

Thickness, δd

Table 1 Solids Density (kg/m3)

Fluid Density (kg/m3)

(mPas)

(µm)

2650 1000 1 45 180 0.25 2650 1200 3 93 367 0.25 5000 1000 1 33 134 0.25 1400 1050 2 120 471 0.25

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4. RELEVANCE TO SLURRIES IN THE MINING INDUSTRY The MW&J prediction method is aimed at predicting the deposit velocity for fine particle sizes equivalent to sand in water particles 100 µm to 45 µm so as to be relevant to wide size distribution, viscous slurries of interest to the mining industry. Unfortunately the author has been unable to find any sand-water data for which ∆ < 1.E-5 and Vd∆ > Vdδ. Such data is more likely to apply to large pipe diameters so is understandably scarce. However there is some limited data for sand in fine particle carrier fluids. Table 2 summarises relevant data from Sanders et al (2004), with their Figure 5 being relevant to Slurries 1 and 2, and their Figure 10 being relevant to Slurry 3. Additional information regarding Slurry 3 was provided by Sanders (2015). For Slurries 1 and 2 the observed Vd are not much higher than Vdδ predicted by Thomas (1979), Eqn 5, namely 0.43 m/s and 0.23 m/s higher respectively, and the predicted Vd by MW&J and Sanders et al could be considered equally relevant. For slurry 3 the observed Vd range is 0.78 m/s to 0.98 m/s above Vdδ. The MW&J predicted 1.82 m/s is 0.22 m/s higher than the mean observed Vd whilst the 1.35 m/s predicted by Sanders et al is 0.25 m/s less than the mean observed Vd. So once again MW&J and Sanders et al could be considered equally relevant. Further resolution of these comparisons will require more data in the relevant range for which ∆ < 1.E-5 and Vd∆ > Vdδ.

Observed

Predicted

Table 2 Data for which ∆ < 10-5 and Vd∆ > Vdδ No.

D mm

d50 µm

ρf kg/m3

µf mPas

∆

Vd Sanders et al

Vd m/s

Vd MW&J

m/s

m/s 1 264 169 1145 3.5 9.7E-6 1.18 1.31 1.02 2 264 169 1170 6 3.4E-6 1.07 0.98 0.98 3 495 158 1123 2.12 9.7E-6 1.5-1.7 1.82 1.35 Thomas (2014) considered the properties of 39 concentrates and 105 tailings slurries at 60% concentration and analysed the data using the inherent viscosity approach of Thomas (2010). For the concentrates he found that the equivalent sand in water particle size was 23 µm for the average size concentrate and 113 µm for the coarsest concentrate. Figure 2, applicable to sand in water, indicates that for particles less than about 45 µm, Vd is given by Vdδ. It could be concluded that Vd for the average size concentrate would similarly be given by Vdδ and Vd for the coarsest concentrate would be given by MW&J, Vd∆. Actual predictions of Vd for concentrates in a typical 300 mm diameter pipe confirmed these conclusions. The cross over occurs at a weighted mean particle size (dm) ≅ 50 µm. For the average concentrate, the predicted Vd = Vdδ = 1.1 m/s and for the

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coarsest concentrate, Vd = Vd∆ = 1.75 m/s. These predictions do not take into account laminar/turbulent transition, which depends on slurry yield stress, and will often control deposition. For the tailings, Thomas (2014) found that the equivalent sand in water particle size was 22 µm for the average size tails and 128 µm for the coarsest tails. Prediction of Vd for tailings in a typical 600 mm diameter pipe, confirmed that Vd for the average tails is determined by Vdδ, and for the coarsest tails by Vd∆, i.e. by MW&J. The cross over occurs for dm ≅ 125 µm. For the average tails, the predicted Vd = 1.35 m/s and for the coarsest, Vd = 2.2 m/s. As noted above, all these calculations assume 60% concentration in all cases. For concentrations less than 60% Vd∆ will become more relevant than Vdδ.

5. CONCLUSIONS The MW&J predicted curves of Vd∆ versus D in Figure 1 for sand in water, have the following properties:

• The predictions are within 1% of the Wilson & Judge (1976) predictions for ∆ > 10-5 for the range of D relevant to each particle size. • The MW&J predictions appear to be reasonable extrapolations of the relevant W&J predictions. • As the particle size decreases, the predicted Vd∆ approaches the same slope as the viscous sub-layer prediction, Vdδ, of Thomas (1979), at least for the industrially meaningful pipe size range of 100 mm to 1000 mm. • The MW&J predicted deposit velocity, Vd∆, equals, and then becomes less than Vdδ, at a particles size around 45 µm. Thus the MW&J correlation extends the W&J correlation to provide a good connection between the approaches of Wilson and Judge (1976) and Thomas (1979). On a plot of Vd versus particle size (Figure 2), the MW&J predicted curves cross the viscous sub-layer predictions at about 45 µm equating to d/δ = 0.25. Table 1 shows that the same value of d/δ at cross-over applies for a range of solid and fluid densities and fluid viscosities. This indicates that viscous sub-layer deposition becomes the controlling criterion when d/δ becomes less than 0.25. Figure 2 also compares predictions of Sanders et al (2004) with MW&J. For particle sizes from about 30 µm to 50 µm the Sanders et al correlation could be considered to be providing a transition between Vdδ and Vd∆. The Sanders et al predictions in Figure 2

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apply to Cv = 0.12. The Sanders et al correlation includes concentration as a parameter and for concentrations greater than 0.12, the Sanders et al predicted curves rise more steeply and will cut the MW&H curves at a particle size greater than the 50 µm indicated in Figure 2. Finally it must be emphasised that the MW&H correlation has not been tested against data from large pipes and therefore caution must be exercised when making predictions in very large pipes. Also the MW&H correlation does not include coarse solids concentration as a parameter. For medium to fine sands in water, experimental data suggest that the MW&H predicted Vd will be within 10% of the maximum Vd in the practical concentration range. For more viscous slurries this may not be the case.

6. REFERENCES 1. Durand, R. (1953), Basic relationships of the transportation of solids in pipes – experimental research. Proc. 5th Minneapolis Int. Hydr. Conv., Int. Assoc. Hydr. Res. pp 89-103. 2. Sanders, R.S., Gillies, R.G., McKibbon, M.J., Litzenberger, C. and Shook, C.A. (2004) Deposition velocities for particles of intermediate size in turbulent flow. Hydrotransport 16 Conf., Santiago, Chile, 26-28 April. 3. Sanders, R.S., (2015), Private communication 4. Schriek, W., Smith, L.G., Haas, D.B. and Husband, W.H.W. (1973), Experimental studies on the transport of two different sands in water in 2, 4, 6, 8, 10 and 12 inch pipelines. Experimental Studies on Solids Pipelining of Canadian Commodities for The Canadian Transport Commission and The Transportation Development Agency, E73-21, Saskatchewan Research Council. 5. Shook, C.A., Gillies, R., Haas, D.B., Husband, W.H.W, and Small, M. (1982), Flow of coarse and fine sand slurries in pipelines, Jnl of Pipelines, 3, pp 13-21. 6. Thomas, A.D. (1979), Predicting the deposit velocity for horizontal turbulent pipe flow of slurries, Int. Jnl, Multiphase Flow, Vol. 5, pp 113-129 7. Thomas, A.D. (2010) Method of determining the inherent viscosity of a slurry and other rheological trends as illustrated by a data bank of over 200 different slurries. Hydrotransport 18 Conf., Rio de Janeiro, Brazil, Sept 22-24 8. Thomas, A.D. (2014), Slurries of most interest to the mining industry flow homogeneously and the deposit velocity is the key parameter. Hydrotransport 19 Conference, Golden, Colorado, 24-26 Sept 2014. 9. Wilson, K.C. and Judge, D.G. (1976), New techniques for the scale-up of pilot plant results to coal slurry pipelines. The Int. Symp. On Freight Pipelines, Washington D.C., USA, Dec., Uni. of Penn. Phil, USA. 10. Wilson, K.C. and Judge, D.G. (1978), Analytically based nomographic charts for sand water flow. Proc. 5th Int. Conf. on Hydraulic Transport of Solids in Pipes, Hannover, Germany. BHRA Cranfield, U.K. A1/1-12 11. Wilson, K.C., Addie, G.R., Sellgren, A. and Clift, R. (2010), Slurry transport using centrifugal pumps, 3rd Edition, Springer Science+Business Media, Inc.