Appendix E

CLAY-ONLY LAB TEST DATA

E-1

E-2

E-3

E-4

E-5

E-6

E-7

E-8

E-9

E-10

E-11

E-12

E-13

E-14

E-15

Appendix F

MODELLING PIPELINE FLOW OF NON-NEWTONIAN SLURRIES WITH SETTLING PARTICLES

F-1

MODELLING PIPELINE FLOW OF NON-NEWTONIAN SLURRIES WITH SETTLING PARTICLES

by

Kenneth C. Wilson

Professor Emeritus, Department of Civil Engineering, Queen’s University, Kingston, ON Canada, wilson@civil.queensu.ca

and

Robert J. Visintainer

Chief Engineer, GIW Industries Inc., Grovetown, GA USA rvisintainer@giwindustries.com

Many commercial slurries appear to flow as non-Newtonian materials. However, some of the solid particles in such slurries will settle, forming a sliding contact-load layer near the bottom of the pipe. In these cases, the tendency has been to use a two-layer force-balance model that requires considerable computer processing and thus is not ‘transparent’ to the user. For engineering applications it is preferable to have a simpler model that maintains the two basic components of ‘fluid effect’ and ‘solids effect’; and a model of this sort is developed here. The fluid-effect component is based on typical behaviour of non-Newtonian fluids in either laminar or turbulent flow. The solids-effect component increases with the overall volumetric concentration of solids, but not necessarily in the direct proportion assumed for settling slurries in Newtonian flow. The model includes a best-fit method for estimating the numerical parameters, and produces graphical representations of pressure gradient and specific energy consumption.

INTRODUCTION

Many concentrated slurries appear to flow as laminar non-Newtonian materials. However, some solid particles settle under these conditions, forming a sliding contact-load layer near the bottom of the pipe (Thomas et al. 2004). In analyzing this flow configuration [Clarke & Charles (1993), Maciejewski et al. (1993) and Wilson et al. (1993], the tendency has been to use a two-layer force-balance model that requires considerable computer processing and thus is not ‘transparent’ to the user.

For industrial applications it is preferable to have a simpler model that maintains the two basic components of ‘fluid effect’ and ‘solids effect’; and a model of this sort is developed here.

THE MODEL

The hydraulic gradient im of the mixture (expressed in height of water per unit length of pipe) has a fluid-effect component written if and a solids-effect component, written is. For given properties of fluid and solids, it is expected that both of these components may depend on volumetric solids concentration Cv, mixture velocity (discharge divided by area of pipe cross-section) Vm and internal pipe diameter D. Within the range of variables of commercial interest, it can be assumed that the effects of each of these variables can be approximated by a power law, as shown below.

The fluid-effect component is based on typical behavior of non-Newtonian fluids as described by, e.g., Wilson et al. (2006). For laminar flow, a basic relationship expresses the shear stress at the pipe wall as a power of (8Vm/D). This leads to the following expression for if.

if = A1(Vm)n/(ρwgD(1+n)) (1)

F-2

where A1 is coefficient, n is a power (usually considerably less than unity), ρw is the density of water and g is gravitational acceleration. The coefficient A1 is strongly dependent on the concentration of solids, and can be written as A2(Cv)m, where a representative value of m might be 3 or 4. As ρw and g are constants, they can be incorporated into the coefficient, which is now written simply A, giving

ifL = A(Cv)m(Vm)n/D(1+n) (2)

This equation is applicable for laminar flow, which will often occur for non-Newtonian materials.

In other circumstances the flow will be turbulent, and in this case the fluid-effect component if will be proportional to [Vm

2/(2gD)] and to the mixture density effect [1+X(Ss-1)Cv]. Here X is the fraction of the solids effective for this type of flow (X lies between 0 and 1.0, and is equivalent to the coefficient A’ in Wilson et al. (2006). Introducing a coefficient F to represent the friction factor, one obtains the expression for if in turbulent flow as

ifT = F[Vm2/(2gD)] [1+X(Ss-1)Cv] (3)

The next step is to obtain an expression for the solids-effect component, is. This will increase with the overall volumetric concentration of solids, Cv, but not necessarily in the direct proportion assumed for settling slurries in turbulent flow. As noted in references cited in the introductory paragraph, particles settle slowly in non-Newtonian media, and thus settling will still be taking place for a considerable time after a given portion of the slurry passes through a pump. For a particular section of the pipeline this time will be proportional to the mean velocity Vm and to the average distance that a particle falls, which in turn is proportional to the pipe diameter D. The proposed power-law relationship expressing this behavior can be written

is = B(Cv)p/(VmD)q (4) where B is a coefficient, the power p in the order of unity and the power q is expected to be considerably smaller. Combining Eq. (2) or (3) with (4) gives the expression for im as

im = MAX[ifL . ifT] + B(Cv)p/(VmD)q (5)

Like the relationship for particulate slurries in turbulent flow, Eq. (5) produces a minimum when im is plotted against Vm (with other quantities held constant). This minimum can be obtained by setting ∂im/∂Vm equal to zero, but the details will not be given here. As with the particulate case, it is not expected that im will show a minimum when Cv is varied. However this is not necessarily the case when specific energy consumption (SEC) is considered (for particulate slurries see Wilson 2004). For laminar flow, the variable portion of SEC is the ratio im/Cv, which for Eq. (5) becomes

im/Cv = A(Cv)(m-1)(Vm)n/D(1+n) + B(Cv)(p-1)/(VmD)q (6)

On differentiating this expression with respect to Cv, it is found that a minimum only exists if p < 1.

APPLICATION TO REPRESENTATIVE LIMITING CASES

The proposed model has eight numerical parameters – the coefficients A, B, F and X, and the four powers m, n, p and q. These parameters will differ for various types of slurry, and it is expected that the coefficients A (or F if the flow is turbulent) and B will be subject to more

F-3

variability than the powers. This model has been incorporated into spread sheets developed at GIW Industries, and these have been used to produce graphical representations of pressure gradient and specific energy consumption.

Before considering the application of the model to complex slurries that require the full suite of parameters, it is appropriate to examine two representative limiting cases. The first is a sand slurry, such as would be pumped in dredging operations. Characteristic curves for a slurry of this type are shown on Figure F-1. As with the figures for other examples that follow, Figure F-1 has two panels. The first panel plots the mixture pressure gradient im (expressed in m of water per m of pipe) versus throughput velocity Vm (discharge/cross sectional area of pipe). This panel illustrates the typical concave-upward shape for contours of constant solids concentration.

Figure F-1. Characteristic Curves for Sand Slurry.

The second panel of the figure is derived from the same information but presented in a different form, as a plot of specific energy consumption (SEC) in kW-hr/tonne-km versus production of solids in tonnes/hr. The two sets of contour lines represent solids concentration by weight (or mass) Cw and throughput velocity Vm. The specific energy consumption is a quantity of great economic interest because low SEC indicates low energy cost for the system. In the second panel of Figure F-1, both sets of contour lines are concave upward (typical of a sand slurry), and the conditions near minimum SEC can be read easily off the graph.

Figure F-2 maintains the same axis system for each panel, but presents representative behaviour for a different type of slurry–a non-Newtonian viscoplastic material such as clay slurry.

F-4

Figure. F-2. Characteristic Curves for Non-Newtonian Slurry.

As shown on the first panel of Figure F-2, this type of slurry displays both turbulent and laminar flow, and these behave quite differently. The turbulent flow, seen primarily to the right of the panel, has a rapid rise of im with Vm, with lines of equal concentration spaced closely. For laminar flow, seen to the left, the behaviour is quite distinct, showing large differences in im with increasing Cw, but only small increases with increasing Vm.

The second panel of Figure F-2 shows an interesting feature of this type of non-Newtonian flow; an apparent ‘fold’ in the contour pattern that indicates the transition from turbulent to laminar flow. This ‘fold’ has a significant influence on the location and value of the point of minimum SEC.

MODEL CALIBRATION FOR PHOSPHATE-MATRIX SLURRIES

It is now time to turn from the simpler cases dealt with above to instances of complex slurries that require evaluation of the full set of parameters in the model. The spread sheets developed at GIW Industries include a best-fit method for estimating these parameters. This technique was applied to phosphate-matrix slurries tested experimentally in the GIW Hydraulic Laboratory. Two examples will be presented here, denoted simply as ‘low-friction” and ‘high-friction’.

Figure F-3 shows characteristic curves for the ‘low-friction’ case, using the axis systems employed on the previous figures. It is appropriate to begin by considering the second panel of Figure F-3, which shows the laboratory data for specific energy consumption, compared with the

F-5

corresponding predictions of the model as calibrated. The scatter is gratifyingly small, and the set of contour lines for this type of slurry are not greatly dissimilar to those of the sand slurry discussed above.

Figure F-3. Characteristic Curves for ‘Low Friction’ Slurry.

The im versus Vm characteristics for the calibrated ‘low-friction’ slurry model are displayed on the first panel of Figure F-3. This shows that the lines of constant concentration are rather evenly spaced and rise monotonically with increasing Vm. As with the sand plot on Figure F-1, a typical value of im is roughly 0.04.

Figure F-4. Characteristic Curves for ‘High Friction’ Slurry.

F-6

Corresponding plots for the ‘high-friction’ phosphate-matrix slurry are shown on Figure F-4. Here again, the first matter to be considered is the calibration shown on the second panel of the figure. The scatter between the experimental points and those of the calibrated model are acceptably small, but the sets of contour lines are shaped quite differently from those of the ‘low-friction’ slurry. Specifically, the ‘fold’ in the lies near the bottom is reminiscent of that noted for the non-Newtonian slurry of Figure F-2. In general the values of SEC for this slurry exceed those for the ‘low-friction’ slurry discussed above.

Once again, the im versus Vm characteristics for the ‘high-friction’ model are displayed on the first panel of the figure. At low solids concentration the contours are rather close together and show little change with Vm. As the concentration is increased further the effect of Vm becomes variable, and im rises rapidly to values considerably above those for the ‘low-friction’ slurry.

CONCLUSION

A simple multi-parameter model has been developed to simulate flow characteristics of complex slurries. For such slurries, particles settle slowly in non-Newtonian media, and the settling will still be taking place for a considerable time after a given portion of the slurry passes through a pump. The model was initially demonstrated for limiting cases of simple slurries – one with sand and the other a non-Newtonian fluid. Finally, the model was calibrated successfully for complex slurries, using two phosphate-matrix slurries from Florida that differ greatly in pumpability.

REFERENCES

Clarke, P.F. and Charles, M.E. 1993. A flow sedimentation model for the laminar pipeline transport of slowly-settling concentrated suspensions. Hydrotransport 12, BHR, Brugge, Belgium, 615-628.

Maciejewski, W., Oxenford, J. and Shook, C.A. 1993. Transport of coarse rock with sand and clay slurries. Hydrotransport 12, BHR, Brugge (Belgium), 705-724.

Thomas, A.D., Pullum, L. and Wilson, K.C. 2004. Stabilised laminar slurry flow: review, trends and prognosis. Hydrotransport 16, BHR, Santiago (Chile), 701-716.

Wilson, K.C. 2004. Energy consumption for highly-concentrated particulate slurries. 12th Intn’l Conf, on Transport and Sedimentation of Solid Particles, Prague (Czech Republic), 681-688.

Wilson, K.C., Pugh, F.J., Addie, G.R, Visintainer, R.J. and Clift, R. 1993. Slurries with non-Newtonian carrier fluids: up, down and sideways. Hydrotransport 12, BHR, Brugge (Belgium), 657-670.

Wilson, K.C., Addie, G.R., Sellgren, A. and Clift, R. 2006. Slurry Transport Using Centrifugal Pumps, 3rd Ed., Springer, Norwell, MA, USA.

Appendix G

VALIDATION OF A FOUR-COMPONENT PIPELINE FRICTION-LOSS MODEL

G-1

VALIDATION OF A FOUR-COMPONENT PIPELINE FRICTION-LOSS MODEL

A. Sellgren

Luleå University of Technology, Sweden

K.C. Wilson

Queen’s University, Kingston, Ontario, Canada

Abstract

Recent experiments and analyses have provided greater understanding of friction losses for the size distribution of the particles forming the slurry. Wilson & Sellgren (2001), Wilson et al. (2006) and Sellgren & Wilson (2006) have proposed a multi-component model consisting of carrier fluid and pseudo-homogeneous, heterogeneous and fully-stratified solids. The technique presented here was found to estimate friction losses well when compared to reported experimental results with pipelines with diameters mainly from 0.2 to 0.5m for various mixtures of fine sand to coarse gravel and products from mineral processing and phosphate mining.

Introduction and Background

For broad-graded slurries with a modest fraction of fine particles (to be defined below) the combination of the fines and the liquid form a carrier fluid which typically approximates Newtonian behaviour, and the flow will generally be turbulent. The technique that follows is an extended version of that presented by Wilson & Sellgren (2001), Wilson et al. (2006) and Sellgren & Wilson (2006). In the new four-component model, the mass fraction of the fine particles is designated Xf, and the remaining fraction is divided into three segments – pseudo-homogeneous, Xp, heterogeneous, Xh, and fully-stratified, Xs. The liquid plus the fraction Xf acts as the carrier fluid, and the Xp- fraction (called the ‘middlings’) is considered to combine with the carrier fluid to produce a pseudo-homogeneous mixture. The next-larger fraction, Xh (the ‘grits’), follows the heterogeneous behaviour described below, while the largest fraction (the ‘clunkers’) represents the fully-stratified load. Heterogeneous and fully-stratified solids Wilson & Sellgren (2001) proposed a three-component model for broadly-graded slurries, consisting of carrier fluid (typically water), heterogeneous solids and fully-stratified solids. The subsequent work has shown that an additional component is required for pseudo-homogeneous solids in the size range between fine particles included in the carrier fluid and heterogeneous solids. It is appropriate to begin by outlining the behaviour of the heterogeneous load – the “grits” or particles in the sand size range, followed by that of the coarser fully-stratified solids. The carrier fluid and the pseudo-homogeneous component will be introduced thereafter.

G-2

The heterogeneous load is carried partly by sliding contacts and partly by turbulent suspension, and the statistical nature of turbulence indicates that the fraction of particles carried by fluid suspension will vary in an essentially probabilistic fashion, based on, for example, the integrated log-normal distribution. The mathematical form of this distribution is rather awkward, and it is convenient to employ a simple power law function as a working approximation as shown by Wilson et al.(2006). The equation for this approximating power law is expressed in terms of a reference velocity V50, and a coefficient, M. It gives the excess gradient ∆ih as;

−=∆

VV S0.22 i

50

-M

mh )1( (1)

Here Sm is the relative density of the mixture, and the coefficient 0.22 represents the value of the relative solids effect where the mean flow velocity V equals V50. The coarsest particles or “clunkers” are not susceptible to turbulent support, but travel as fully-stratified load, which produces an additional solids effect ∆is. In the absence of other solids, this is given by:

∆is = (Sm – 1)B’[V/0.55Vsm]-0.25 (2)

where Vsm is the velocity at the limit of stationary deposition and B’ is a coefficient that may be as high as unity. As noted in Wilson et al. (2006), flow tends to shift from heterogeneous to fully-stratified at values of d/D of 0.015, assigning all particles with d > 0.015D to the fully-stratified load, where d is a representative particle size and D is the pipeline diameter. Carrier-fluid and pseudo-homogeneous components The carrier fluid is comprised of the true fluid and the fine particles that interact with it. In earlier models, Wilson et al. (1997), all particles finer than 75 µm were assigned to the carrier fluid, with larger particles considered to be heterogeneous load. Research has advanced in the interim, see Wilson et al.(2006). The carrier fluid is now considered to include only particles finer than 40 µm, and particles between 40 µm and about 200 µm are now called the pseudo-homogenous load. The fraction Xp comprises particles that are small enough to be engulfed in the viscous sub-layer. The upper size limit for Xp can be taken as 200 µm for cases where the properties of the carrier fluid do not differ significantly from those of water. If the carrier fluid has a Newtonian viscosity greater than that of water, the value of 200 µm is to be multiplied by the viscosity ratio νr (the ratio of the kinematic viscosity of the carrier fluid to that of water at 20oC). The heterogeneous fraction spans particle diameters between 200 µm and 0.015D is used to define the upper particle diameter for heterogeneous flow, with larger particles comprising the fully-stratified solids fraction Xs. Pseudo-homogeneous flow assumes that the solids have little effect on friction factor, and that the mixture acts as a liquid as far as the relative-density effect is concerned. The resulting hydraulic gradient for pseudo- homogeneous mixture flow, iph, depends on Sm, and iw, which is the hydraulic gradient for water flow only. An appropriate general equation for the hydraulic gradient is:

i1)]-S(A+[1=i wmph ′ (3)

where A´ is a coefficient that equals unity for the equivalent-fluid case, which means that iw is identical with the hydraulic gradient expressed in m of slurry per m of pipe, jm.

Four-Component Model The hydraulic gradient for the mixture as a whole, im (m water/m pipe), is considered to have four components, the carrier fluid, the pseudo-homogeneous mixture, the heterogeneous load and the fully-stratified load. As indicated by Eq. 3, the hydraulic gradient for pseudo-homogeneous flow of an aqueous slurry is given by iw[1 + A´(Sm - 1)].However, (Sm - 1), i.e. Cv(Ss – 1), refers to the excess relative density of the slurry as a whole, and as only the fraction Xp is part of the homogeneous flow, (Sm - 1) must now be replaced by CvXp(Ss – Sf), where Sf is the relative density of the carrier fluid. Likewise, the hydraulic

G-3

gradient for water, iw, must be replaced by that for the carrier fluid, if. Here if may be somewhat larger than iw because of increases in density and viscosity produced by the fine particles. Thus the pseudo-homogeneous component of the hydraulic gradient, iph (m water/m pipe), is given by:

iph = if[1 +A’CvXp(Ss – Sf)] (4) In dealing with the component due to heterogeneous flow, we begin with Eq. 1, which gives this type of solids effect as 0.22(Sm – Sf)[V50/V]M. This term must now be multiplied by the fraction of homogeneous particles Xh, with (Sm – 1) replaced by Cv(Ss – Sfp) where Sfp represents the relative density of the combination of carrier fluid and the pseudo-homogeneous fraction, typically the relative density of the mixture of fluid and all particles less than 200 µm in size. This expression will be denoted ∆ih , i.e.

∆ih = 0.22 CvXh(Ss – Sfp)[V50/V]M (5) The reference velocity V50 depends on the size and other quantities and where M is here taken approximately as unity, see Wilson & Sellgren (2001) who presented a curve for the effect of relative viscosity on V50. In most cases it should be sufficient to evaluate V50 by using the simple relationship:

[ ] 25.0rfps

0.450.35h50 )/1.65S - S( d 3.93 V −ν≈ (6)

where dh ( mm) represents the mean of the percentages passing the upper and lower limits of the heterogeneous fraction, see Figure G-1. Here Ss is the relative density of the solid particles and νr is the relative viscosity of the carrier fluid (compared with water at 200 C).V50 is given in m/s in Eq.(6). Figure G-1 shows an example of how the defined particle fractions are applied to a cumulative representation of the particle size distribution.

Figure G-1. Grading curve showing size fractions used in the modelling in a pipeline with diameter, D=0.333m, Xf=20%, Xp=20%, Xh=40%, Xs=20% and dh=1000 µm.

Finally, there is the fully-stratified load (particles larger than 0.015D), which will produce an additional solids effect ∆is. It will be recalled that, in the absence of other solids, this is given by Eq. 4 as ∆is = (Sm –

G-4

1)B’[V/0.55Vsm]-0.25. For the case dealt with here, (Sm – 1) must be replaced by CvXs(Ss – Sfph), where Sfph is the relative density of the mixture of fluid, pseudo-homogeneous and heterogeneous particles. Thus:

∆is = CvXs(Ss – Sfph)B’[V/0.55Vsm]-0.25 (7) The sum of the terms given by Eq. 4, 5 and 7 gives the gradient for the mixture as a whole, im (m water/m pipe), thus:

im = iph + ∆ih + ∆is (8) The technique presented here in Eqs. 4 to 8 with the relative solids effect expressed by the coefficients A’ and B’ taken as unity in Eqs. 4 and 7, and Eq.5 with 0.22 have been found to be conservative, especially for slurries in the range 0.1 to 0.25 mm, with few particles over 1 mm, in pipelines with diameters exceeding about 0.15m. The objective is to compare the model with a variety of experimental results covering pipeline diameters and particle sizes up to about 0.5 m and 65 mm, respectively.

Remarks Regarding Near-Wall Hydrodynamic Lift

The aim here is to discuss various mechanisms and introduce subsequent modifications of the model. Setting the coefficient A′ equal to unity gives the relative-density effect of an equivalent-fluid model, whereas A′ = 0 gives the behaviour observed by Carstens & Addie (1981) for pseudo-homogeneous sand-slurry flows (Cv=3.6%) where im did not exceed iw at all for velocities in excess of 3.5 m/s in a 0.2 m diameter pipeline. Intermediate types can be represented by values of A′ between zero and unity, using concepts introduced in papers by Wilson et al. (2000), Wilson & Sellgren (2002) and Whitlock et al. (2004).

Matoušek (2006) found from experiments in a vertical 0.15 m-diameter pipeline with narrowly graded sands (Average particle sizes of 0.12, 0.37, 1.84 mm) and concentrations by volume up to about 35% that the medium-sand particles gave less pipe wall friction than both the coarse-sand and fine-sand particles. He discussed various mechanisms and confirmed that the medium-sand particles showed stronger hydrodynamic repelling force off the wall than the others at velocities of practical interest, in accord with theoretical estimations by Wilson & Sellgren (2002).

Within the turbulent portion of horizontal flow, particle support is dominated by turbulent diffusion, which produces characteristic concentration profiles. The diffusion concept implies that any horizontal plane in the flow acts as a ‘mirror’ which ejects upward a flux of particles equal to those that move down through the plane. This condition is met in the fully-turbulent part of the flow, but causes difficulties near the lower part of the boundary. Here turbulence is ineffective in the viscous sub-layer and begins to be felt in the adjacent buffer layer. Thus the ‘mirror’ effect will be inhibited in these near-wall regions, and here the particles that fall out of the turbulent flow must be replaced by a flux of particles driven away from the wall by a force (the off-wall force) that arises from hydrodynamic lift.

Wilson & Sellgren (2002) proposed that solid particles in a turbulent fluid flow influence the energy spectrum of turbulence. For flows behind stationary cylinders in a wind tunnel, it is known that spectral peaks occur at vortex-shedding frequencies. In that configuration, turbulent energy is produced, but it is expected that moveable particles will have the opposite effect, damping eddies of approximately their own size. The result should be a 'blip' of reduced energy density at the corresponding frequency. If the particle size is near-uniform, and the concentration is sufficiently high, this 'blip' will tend toward a gap or 'hole' in the spectrum. On the other hand, for broadly-graded solids there will be a series of 'blips' in the turbulent spectrum that combine to form a broad 'trough' rather than a narrow 'hole'. If the solids include both particles directly susceptible to the lift force and somewhat larger particles, a benefit may be obtained, at least at large overall solid concentrations. Instead of a particle subject to lift being completely removed from the near-wall area, it will tend to rise only until it hits a larger particle, thus contributing to the support of the larger particle and simultaneously reflecting the smaller particle back into the high-lift zone near the wall, where the process can be repeated. It is believed that this mechanism is associated with the reduction of A’ and B’ for broadly-graded slurries.

G-5

Comparison with Experimental Results The mechanisms discussed above are now to be quantified by comparison with a variety of reported experimental data, mainly for broad particle-size gradings. For the cases considered here, the fine-particle portion (below 40 µm) of the solids can be assumed to be rheologically inactive, i.e. not containing much clay-like minerals. Also, the maximum volumetric concentrations of the fine particle portion are less than 7% in these cases. Therefore, it is reasonable to assume in friction-loss calculations that the viscosity of this carrier fluid is approximately the same as for water. The hydrodynamic lift and turbulence effects for Xp (40-200 µm) are assumed to be effective for 25% of these particles, corresponding to a reduction from A’=1 to (1 – 0.25Xp). In the Xh-interval the coefficient 0.22 is related to that portion of the particles that contribute to mechanical friction at the pipe wall. Here, it is assumed that the effect of near-wall lift is simply related to a much smaller contribution to pipe wall friction for particles from 200 to 500 µm in the Xh-interval. As shown by Whitlock et al. (2004), this effect is significant for particle sizes up to about 500 µm. This has been quantified here by a linear increase of the constant from 0 to 0.22 for 200 to 500 µm. Simulated friction losses with the model are compared with experimental results at operating conditions of practical interest in Table G-1. Table G-1. Comparison of modeled friction losses (Eqs. 4-8) with various reported pipeline loop data, Simulated and measured friction loss gradients in m slurry per m pipe are denoted js and jm, respectively. Particle size-grading notations (%) are as on Figure G-1. Particle sizes are in mm. The parameters dmax and d50 express the maximum particle size and the particle diameter for which 50% of the particles are finer, respectively.

# D(m) V(m/s) Cv Ss d50 dh dmax Xf Xp Xh Xs jm js 1 0.305 4.5 15 2.65 0.7 0.90 12 2 23 60 15 0.060 0.063 2 0.305 4.5 27 3 0.85 0.85 65 20 15 30 35 0.075 0.075 3 0.1 2 13 2.65 0.085 0.23 0.25 - 98 2 - 0.034 0.034 4 0.438 4 38 2.65 0.2 0.40 0.9 18 32 50 - 0.029 0.028 5 0.263 3.1 26 2.65 0.17 0.27 1.5 28 30 42 - 0.026 0.027 6 0.206 2 30 2.71 0.085 0.33 0.6 25 65 10 - 0.016 0.015 7 0.206 3 32 2.65 0.2 0.32 0.5 4 46 50 - 0.030 0.033

Values of B’ of 0.25 and 0.20 were used for the rows marked 1 and 2, respectively, in Table G-1. The data for rows 1 to 4 were obtained from the GIW Hydraulic Laboratory, U.S.A, with 1 and 2 presented by Sundqvist et al. (1996a) and Sundqvist & Sellgren (2004), respectively. Whitlock et al. (2004) presented the fine sand results in 3, and 4 represents an oil-sand tailings product. No. 5 was obtained from Shook & Roco (1991), while 6 and 7 are gold-tailings results from Sauermann (1982).

Discussion Effect of high solids concentration The volumetric concentrations for the comparisons in Table G-1 varied from 13 to 38%.The lowest value for the #3-results corresponded to equivalent fluid behaviour. This behaviour remained up to 34%,which was the largest experimentally investigated Cv-value (Whitlock et al. 2004), with simulated results that matched this observation. However, results from Schaan et al. (2000) with a sand with a similar particle grading in a 0.15 m diameter pipeline showed a deviation of about 50% from equivalent fluid behaviour at Cv=32%. In a comparison with slurries with two other similar-sized products, the deviation from equivalent behaviour was related to the particle shape and loose-poured settled bed concentration. Such factors have not been considered in the preliminary model construction presented here. Modelled friction losses matched well the large diameter results for Cv= 38% for the oil-sand tailings in Table G-1. Maciejewski et al. (1993) compared pipe transport of very coarse particles (about 100 mm in size) in clay suspensions and in oil-sand tailings slurries. They found that the sand slurry was more

G-6

effective than the clay in reducing the solids effect of the coarse particles. The important role of particles with sizes 0.1 to 0.5 mm in diminishing friction was also demonstrated by Sundqvist et al. (1996), for products with d50 near 0.6 mm. An example of experimental results is shown on Figure G-2 (from Wilson et al. 2002). It gives plots of solids effect (im – iw) versus (Sm – 1) based on Sundqvist’s sand data with V ≈ 4.3 m/s. All three sands had d50 in the 0.6-0,7 mm range.; Sand 1 has a very narrow grading (d85/d50 = 1.2), for Sand 2 this ratio is 2.3 and for Sand 3 it is 7.1 (a very broad grading).It is Sand 3-results that is used here in Table G-1 (#1) for Cv=15%.The parameters d85 express the particle diameters for which 85% of the particles that are finer. Figure G-2 shows that the slurry of intermediate grading has a solids effect that is directly proportional to (Sm – 1), i.e. to Cv, up to the maximum Cv of about 0.37. As expected in view of the analysis put forward in this paper. Sand 1, with its very narrow grading, exhibits the commercially undesirable feature – a disproportionate increase in the solids effect at high concentrations. Sand 3, with its very broad grading, shows the opposite, a pronounced flattening of the solids-effect relationship at high Cv, a feature that has considerable commercial potential. In fact, experimental results for the operating conditions in Table G-1 for #1 (sand 3) for Cv = 39% showed that the jm-value remained nearly constant (0.060) up to this high solids concentration. Similar tendencies as discussed here were also found in experiments by Gillies & Shook (2000).

Figure G-2. Plots of solids effect for Sundqvist’s sands for average mixture velocity 4.3m/s. Sm-1 of 0.25 and 0.6 correspond to Cv 15 and 36%,respectively. From Wilson et al. (2002).

The effect seen here in Figure G-2 shows that the grading strongly influences the friction losses for higher concentrations. Further experimental evidence of the diminishing effect when going from a narrow to a moderately broad grading are given in Table G-2, from Wilson and Sellgren (2002), based on Cv near 30%.

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Table G-2. Solids and pipeline data for moderately broad (B) and narrow (N) gradings resulting in about 35% reduction in friction losses for the broader size distribution. From Wilson and Sellgren (2002).

Ref. Shook et al. (1972) Sundqvist et al.(1996a) dmax(mm) 0.9 1.3 1.0 2.0 d50 (mm) 0.52 0.52 0.63 0.58 d15 (mm) 0.46 0.21 0.54 0.27 %<0.04mm 0 4 0 2 d85/d50 1.12 2.50 1.17 2.36 N/B N B N B D(m) 0.1 0.1 0.2 0.2

For both pipe diameters in Table G-2, the slurry friction gradients were about 35% lower for the broad-particle-size distribution at velocities of industrial interest. For the broadly-graded solids, a lot of particles are in the range where the effect of the off-the-wall force is large, as indicated in previous sections. At low total solids concentrations in the model, the total solids effect (im – iw) will be approximately linear in Cv, but as the concentration is increased Sfp and Sfph also increase, and the effect of increasing Cv is less than linear for broadly-graded solids. However, the observed strong non-linear effect for higher concentrations is not accounted for in the present model configuration. For example, with the measured jm-value of 0,06 for sand 3 at Cv=30%,the corresponding modelled value was 0.077, i.e. an overestimation of nearly 30%. The strong effect of Cv, particularly for the narrowest gradings in Figure G-2 and Table G-2 is mainly represented by a two-component formulation of the model with Xh approximately taken as 1. For this case, Wilson et al. (2002) extended an algebraic analysis beyond the linear increase in (im – iw) with Cv, introducing a parametric approach expressing the non-linearity. Effect of fine particles Large-scale experimental results from the 0.495 m-diameter pipeline loop system at the GIW Hydraulic Laboratory, U.S.A., have been compared to modelled friction losses for the addition of rock flour to a fine sand slurry, Whitlock & Sellgren (2004), and for phosphate matrix products containing clay (Addie et al. 2005). The friction losses for a narrow-graded sand (d50=0.35mm) transported at Cv=24% were reduced about 25% when 20% of the sand was replaced with rock flour particles with median size 0.04 mm, see Table G-3. Table G-3. Comparison of modeled friction losses (Eqs. 4 - 8) with reported pipeline loop data, for sand without (#1) and with (#2) addition of rock flour. Notations are given in Table 1 and Fig.1.D=0.495 m.

# V(m/s) Cv Ss d50 dh dmax Xf Xp Xh Xs jm js 1 4.5 24 2.65 0.35 0.3 0.9 - 5 95 - 0.038 0.037 2 4.5 24 2.65 0.3 0.35 0.9 10 15 75 - 0.028 0.027

The volumetric concentration of the rheologically-inactive rock flour particles less that 0.04 mm in the mixture was about 5%, which indicates that the reduction in friction loss gradients from 0.038 to 0.028 in Table G-3 can be related mainly to hydrodynamic mechanisms. The three phosphate-matrix products had various size distributions with portions of particles smaller than 0.04 mm of 10 to 17%. The portion of particles smaller than about 0.1 mm in the matrix product is called phosphate clay, and consists to about 70% of true clay mineral particles (less than 2 µm). Modelled and measured results are compared in Table G-4.

G-8

Table G-4. Comparison of modeled friction losses (Eqs. 4 - 8) with reported pipeline loop data, for three phosphate-matrix products with various size distributions. D=0.495 m.

# V (m/s) Cv Ss d50 dh dmax Xf Xp Xh Xs jm js 1 5 27 2.65 0.25 0.35 1.5 10 30 70 - 0.035 0.034 2 4.4 27 2.65 0.25 0.80 6 17 28 55 - 0.047 0.044 3 4.4 27 2.65 0.5 0.45 20 14 21 51 8 0.045 0.042

The portion of particles smaller than 0.04 mm here corresponded to volumetric concentrations of 2.7 and 4.6%, respectively, with # 1 having coarser particles (fine sand). The largest fine-particle content was for # 2, while # 3 had the broadest size distribution with maximum particles up to 20 mm. A phosphate clay at Cv about 4.6% is rheologically active and can be characterized as a non-settling non-Newtonian slurry with a yield stress of 5-10 Pa. The modelled values in Table G-4 for the phosphate products were also based on the viscosity for water for the carrier fluid. As seen, this approximation worked reasonably well for the considered concentration of 27%. There was a slight underestimation of the losses for the more clay-rich products (#2 and 3). Procedures to include non-Newtonian carrier-fluid behaviour in the model presented here are given by Wilson et al. (2006), and model comparisons for this type of slurries will be the subject of a future paper.

Conclusion For the comparisons listed in Tables G-1, G-3 and G-4, modelled values of slurry friction loss gradients are all within 10% of those determined from experimental results. Additional research presently underway, and to be incorporated in the model, includes effects of particle grading and other properties at high solids concentrations, plus possible influences of large pipe diameters. When maximum accuracy is required, the best approach is to perform pipe-loop tests with the actual slurry of interest, especially if the solids concentration is very high.

REFERENCES ADDIE G.R , SELLGREN A., WHITLOCK L., WILSON K.C. Centrifugal slurry pump concentration limit testing and evaluation-phase 1. Final report, Florida Institute of Phosphate Research, FIPR, 2005 Publication No. 04-069-215 CARSTENS, M.R. & ADDIE, G.R. A sand-water slurry experiment. Jour. Hydr. Div. ASCE Vol. 107, No. HY4, 1981. pp. 501-507. GILLIES R.G., SHOOK C.A. Modelling high concentration settling slurry flows, Can.. J. Chem.. Eng., 78, 2000. pp. 709-716 MACIEJEWSKI, W., OXENFORD, J. SHOOK, C.A. Transport of coarse rock with sand and clay slurries., Proceedings, Hydrotransport 12,Bruegge,Belgium, 1993. pp.705-724. MATOUŠEK V. Solids stress at wall of vertical slurry pipe, The 5th Int. Conference for conveying and handling of particulate solids, Sorrento, Italy, August 27-31,2006. SAUERMANN, H.B. The influence of particle diameter on the pressure gradients of gold slimes pumping, Proc. Hydrotransport 8, BHRA Fluid Engineering, Cranfield, UK, 1982. pp 241- 248. SCHAAN, J, SUMNER, R. J, GILLIES, R.G., SHOOK, C.A. Effect of particle shape on pipeline friction for Newtonian slurries of fine particles, Canada. J. Chem.Engrg.,78,4, 2000. pp.717-725. SELLGREN A., WILSON K.C. Pressure drops for pipeline transport of slurries with broad grading, Proceedings, The 5th Int. Conference for conveying and handling of particulate solids, Sorrento, Italy, August 27-31,2006

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SHOOK C.A., ROCO M.C. Slurry Flow, Principles and Practice, Butterworth- Heinemann, U.S.A, 1991. SUNDQVIST, Å., SELLGREN, A. & ADDIE, G.R. Pipeline friction losses of coarse sand slurries; comparison with a design model. Powder Technology, Elsevier, Vol. 89, 1996. pp.9-18. SUNDQVIST A., SELLGREN A. Large-scale testing of coarse waste rock pumping Proc. Hydrotransport 16, BHR Group, Cranfield, UK, 2004, pp. 127-137. WHITLOCK, L., WILSON, K.C., SELLGREN, A.Effect of near-wall lift on frictional characteristics of sand slurries, Proc. Hydrotransport 16, BHR Group, Cranfield, UK, 2004. pp. 443-454. WHITLOCK L., SELLGREN A. Energy requirement for pumping sand slurries. Comparison of large-scale loop and field results with a design model, Proceeding, The XIV WEDA Dredging conference, 2004 WILSON, K. C., ADDIE, G.R., SELLGREN, A., CLIFT, R. Slurry Transport Using Centrifugal Pumps,2nd

Ed., Blackie, London, UK.,1997 WILSON, K.C., SELLGREN, A., ADDIE, G.R Near-wall fluid lift of particles in slurry pipelines, Proc. 10th Conf. on Transport and Sedimentation of Solid Particles, Wrocław, Poland,2000 WILSON, K.C., SELLGREN, A.Hydraulic transport of solids. Pump Handbook, 3rd Editi McGraw-Hill, 2001, pp. 9.321-9.349. WILSON, K.C. , SELLGREN, A Effect of particle grading on pressure drops in slurry flows, Proc. 11th Intern’l Conf. on Transport and Sedimentation of Solid Particles, Ghent, Belgium,2002. pp. 277-287. WILSON, K.C., CLIFT, R., SELLGREN, A. Operating points for pipelines carrying concentrated heterogeneous slurries. Powder Technology, 23, 1, 2002, pp.19-24. WILSON, K. C., ADDIE, G.R., SELLGREN, A., CLIFT, R. Slurry Transport Using Centrifugal Pumps, 3rd Ed., Springer, New York, U.S.A.,2006

Appendix H

MICROSOFT EXCEL SPREADSHEET MODEL OUTPUT FOR ALL TYPES OF SLURRIES

H-1

H-2

H-3

H-4

H-5

H-6

H-7

H-8

H-9

H-10

H-11

H-12

H-13

H-14

H-15

H-16

H-17

H-18

H-19

H-20

H-21

H-22

H-23

H-24

H-25

H-26

H-27

H-28

H-29

H-30

H-31

H-32

H-33

H-34

H-35

H-36

H-37

H-38

H-39

H-40

H-41

H-42