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Research developments in pipeline transport of settling slurries
Powder Technology 156 (2005) 43 – 51 www.elsevier.com/locate/powtec
Research developments in pipeline transport of settling slurries Va´clav Matouxek Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Received 9 November 2004; received in revised form 22 March 2005; accepted 18 May 2005
Abstract The designing of slurry pipelines requires prediction of pipeline hydraulic performance. The preferable predictive tool is a model based on a physical description of slurry flow behavior in a pipeline. Various mechanisms that affect pipeline flow behavior of settling slurries with a Newtonian conveying liquid are described. Recent developments in measuring, understanding and modeling of the mechanisms are discussed. An implementation of recent research results to current versions of a predictive model for stratified flows of settling slurries (a two-layer model) is surveyed. D 2005 Elsevier B.V. All rights reserved. Keywords: Slurry flow; Solids friction; Two-layer model
1. Introduction In the hydraulic transport of solids using pipelines the liquid is used as a vehicle that carries solid particles to their destination at the end of a slurry pipeline. In practice, slurries of different liquids and solids are conveyed through pipelines. This paper discusses the transport of ‘‘settling’’ slurries with a Newtonian carrier. Settling slurries are mixtures in which the solid particles tend to separate from the carrying liquid owing to the action of gravity force. If solid particles cannot be maintained in suspension by the turbulent diffusive action of the conveying liquid during the transportation through horizontal or inclined slurry pipelines, they settle down to the bottom of the pipe. A sand – water mixture is typical settling slurry. Such slurry is discrete, i.e. even the finest sand particles are too coarse to combine with the conveying water to produce a stable homogeneous medium. A settling-slurry flow tends to stratify. In horizontal and inclined pipes, flows of settling slurries at the velocities usually used during practical operations are characterized by a non-uniform distribution of particles across a pipe cross-section. More particles gather near the bottom than near the top of a pipe cross-
E-mail address: [email protected]. 0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2005.05.054
section. The flow is considered stratified if a portion of solids forms a granular bed (stationary or sliding) at the bottom of a pipe. The flow is fully stratified if all particles occupy the granular bed (particles maintain a virtually permanent contact with other particles within the bed) or partially stratified if only a portion of particles occupies the bed and the rest of particles maintains either sporadic contact or no contact with other particles. The design of slurry pipelines carrying settling slurries requires prediction of hydraulic performance of the pipeline. This is given by a relationship between the slurry flow rate and the total pressure differential developed over a pipeline at the flow rate in question. For settling slurries, the pressure drop due to friction over the length of a pipeline is closely associated with the internal structure of the slurry flow. Since the early 1950s systematic research on the flow of settling slurries in pipelines has been conducted. In the pioneer era, the investigation was focused primarily on a collection of basic experimental data (pressure drops at different velocities) for flows of different pipe and particle sizes. The collected databases served as a basis for a construction of empirical models obtained by formulating suitable dimensionless numbers and finding the best fit by using the data regression. The 1970s and 1980s brought works giving an insight into fundamental principles of pipeline flows for settling slurries—basics of a macroscopic
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two-layer model by Wilson and of a microscopic slurry-flow model by Shook and Roco. Contemporary research makes use of the rapid development of computational and measuring techniques permitting the examination of slurry flow processes in still closer details and leading to better understanding of the complex nature of the behavior of settling-slurry flows in pipelines. Currently, practical interests focus attention on research of settling slurries that are highly concentrated (volumetric concentrations of solids higher than 35% approximately). Furthermore, the contemporary research tends to focus on analyzing settling-slurry flow behavior under specific conditions. A typical approach to the investigation of a pipeline flow of settling slurries is composed of the following steps: – observation of slurry flows under certain specific, welldefined conditions in the closest possible detail and using modern measuring techniques (including concentration and/or velocity distributions) – on the basis of the observations, determination of the prevailing mechanisms governing the observed slurry flow – modeling of the processes and sub-processes in order to simulate the slurry-flow behavior and flow parameters important in practice. Despite a fair degree of progress made over the years, current understanding of the complex behavior of a pipe flow of settling slurries is still far from complete.
2. Measuring techniques 2.1. Distribution of solids A radiometric method has been most often used to determine a distribution of solids across vertical crosssections of laboratory test pipes. At present, radiometric density meters adapted as concentration profilers for slurry pipes are in use in the laboratories of Saskatchewan Research Council (SRC) and Delft University of Technology (DUT, [1]). A radiometric density meter requires a two-point calibration at each measuring position in a pipe cross-section. Delft Hydraulics uses a set of conductivity probes mounted as pairs of electrodes along the perimeter of the pipe wall. The temperature and salinity of the carrying liquid strongly influences conductivity and thus the accuracy of the instrument. A similar electro-resistance sensor was used by Simkhis et al. [2] to detect dunes in a slurry pipe. Recently, the magnetic resonance imaging (MRI) method has been starting to gain a footing in laboratory slurry-pipe experiments [3]. An advantage of the method is that it can collect an entire concentration profile in one moment, a disadvantage is a relatively complex reconstruction process
required to obtain a concentration-profile image from the measured signal. As yet, the high price of the instrument reduces its application to small pipes. The measuring principle sets a maximum limit velocity at which a profile can be measured in a pipe. Pullum and Graham [3] reported 1 m/s as the approximate maximum mean velocity for a 100-mm pipe. 2.2. Distribution of solids velocity In general, it is difficult to measure the distribution of the local velocity of solid particles. Usually, settling slurries are transported at concentrations too high for the use of optical methods like LDA and PIV. In addition to a concentration profile, the MRI method also provides a liquid-velocity profile of a flow. The crosscorrelation of the electrical resistance (or conductivity) signals from two sensors located near each other is the measuring principle most often used to determine local solids velocity in a slurry pipe. The SRC uses an intrusive conductivity probe as a solids velocity profiler (e.g. [4]).
3. Flow-friction mechanisms 3.1. Friction due to permanent contact of solid particles with pipe wall Particles traveling within a granular body in contact with a pipe wall exert a solids stress against the pipe wall that contributes significantly to the total friction of the slurry flow. Wilson’s method [5,6] for the determination of the normal solids stress at the pipe wall is based on the simplified assumption that there is a hydrostatic-type of distribution of the normal stress in the vertical direction through a granular bed submerged in liquid. Furthermore, it assumes that the normal stress at the wall, r swC, produces the solids shear stress, s swC, at the pipe wall according to Coulomb’s law (the normal and shear stresses are mutually related through the friction coefficient, l s), s swC = l s(q s q l)gC vb(D / 2b) (sin b b cos b). It is not easy to verify this method, since it is difficult to eliminate other effects and measure only the effects caused by the Coulombic solids stresses during pipe tests. Tests with extremely high concentrated flows (plug flows) by Wilson et al. [6] justified the proposed solids-stress theory. The tests with extremely highly concentrated sand slurries conducted by Korving [7] confirmed that the Wilson’s plug model based on the solids-stress theory can successfully interpret Korving’s data (Fig. 1) if the coefficient of mechanical friction between the plug and the pipe wall is considered dependent on the mean concentration of solids in the pipe (i.e. in the plug). Another suitable condition for the verification of the theory is a fully stratified flow through a descending pipe inclined at a small angle (say 25 –35-), in which the velocity
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1 0.9 0.8 Plug 1800 kg/m3 Plug 1780 kg/m3 Plug 1730 kg/m3 1800 kg/m3 1780 kg/m3 1730 kg/m3 water
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0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
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Fig. 1. Comparison of experiments with the plug model for the 0.103-mm sand in the 158-mm pipe (from Korving [7]).
of the sliding bed is almost the same as the velocity of the water current above the bed. Under these conditions the solids effect on the total hydraulic gradient is exclusively due to the acting solids stress at the pipe wall in contact with the sliding bed. The test presented by Matousek [8] showed that mechanical friction at the boundary between the sliding granular bed and the pipe wall was successfully predicted by Wilson’s friction law with the friction-coefficient value of about 0.55 valid for all the sands and gravels that were tested. A problem arises in the interpretation of the verification tests if a drop in local concentration occurs at the bottom of the sliding bed adjacent to the pipe wall (see Fig. 2). This phenomenon seems to occur if the velocity of the sliding bed is high. The simulation of a fully stratified flow of coarse particles using Discrete Element Method suggests that this phenomenon is the result of interactions among particles rolling over each other at the bottom of the sliding bed [9] (Fig. 3). Coulomb’s law assumes a constant value of the friction coefficient relating the normal and shear stresses. However, variations in the coefficient value for a pipe wall with a local
concentration of solids or the velocity of the bed can be expected. Experimental evidence is rather limited and further investigation is required to provide better understanding of the exact relationship between the solids normal and shear stresses at a pipe wall. 3.2. Friction due to sporadic or zero contact of solid particles with pipe wall Basically, friction due to the presence of solid particles suspended in a flow is a result of processes in a relatively thin layer near the pipe wall. To identify interactions between particles and the conveying liquid and interactions Volumetric Concentration and Velocity Profiles V = 4 [m/s] Velocity [m/s]
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Fig. 2. Concentration profile and thickness of the sliding bed in the pipe inclined to 35- (the 1.4 – 2.0-mm-sand mixture) (from Matousek [8]).
Fig. 3. Distribution of solids concentration and velocity in the sliding bed at V m = 4 m/s simulated using DEM (from Stienen et al. [9]).
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among particles in the region near a pipe wall and to describe their effect on flow friction is one of the greatest challenges to those researching the pipeline flow of settling slurries. As yet, available measuring techniques are not capable of recognizing the interactive processes within a narrow region near a pipe wall with sufficient distinction. Crucial questions must be answered before the pipe-wall friction of suspension flow can be determined. Friction models use the wall shear stress as a measure of suspension friction in a pipe flow. Basically, two wall shear stresses are recognized: the Fliquid-like_ (or viscous) shear stress and the shear stress caused by the contacts (impingements) of traveling suspended particles with a pipe wall. In contrast to the Coulombic solids stress, the solids stress due to impingements is velocity-dependent (kinetic). For different flow situations, it is difficult to determine whether or not solid particles traveling near the pipe wall contribute to the wall shear stresses, and if so to which shear stress they contribute and with which portion of their total concentration. Essentially, friction behavior of suspended coarse particles differs from that of fine particles. The threshold for particle diameter between the coarse-slurry behavior and the fine-slurry behavior is rather vague but it is assumed that the threshold particle size is related to the thickness of the viscous sub-layer adjacent to the pipe wall. Coarse particles in non-stratified flows (slurry flows without Coulombic stresses) have contacts with each other and with a pipe wall and so contribute to the total friction of slurry flow. The contacts between particles and the pipe wall are sporadic rather than permanent and they are a result of turbulent dispersive action (turbulent eddies tend to disperse solid particles to all directions) and collisional dispersive action (particles are impelled in the direction of the pipe wall by the collision with other particle of different velocity in the flow). The solids stress due to particle collisions with a pipe wall can be diminished by the hydrodynamic off-wall lift. Settling slurries composed of fine particles tend to exhibit a rather different friction pattern. Near the pipe wall, fine particles travel submerged within the viscous sublayer. Within this layer the effects of carrier turbulence and hydrodynamic lift on solid particles are negligible. The increase in the wall shear stress due to the presence of solid particles in the near-wall region is due to increased viscous friction rather than mechanical contact friction. Particles locked inside the viscous sub-layer increase the density of the carrying liquid and so increase the viscous shear stress at the pipe wall. Nevertheless, the solids dispersive stress seems to occur in the fine slurries if the concentration of solids near the wall becomes high. 3.2.1. Collisional stress In the near-wall region the velocity gradient is steep and particles with different local velocities collide with each other and with the pipe wall. The collisions generate the solids dispersive stress (first observed and described by
Bagnold [10]) acting against the pipe wall. This stress is a source of solids friction, additional to the liquid-like friction, in a pipe. The most appropriate geometry for an observation of the collisional stresses is a vertical pipe in which the flow is axisymmetric and there is no Coulombic stress. On the other hand, the need to subtract the hydrostatic part from the measured pressure differential is a potential source of inaccuracy. Shook and Bartosik [11,12] observed three coarse particle fractions (1.37-mm, 1.5-mm, 3.4-mm) in two narrow vertical pipes (D = 26-mm and 40-mm) and interpreted the measured solids effects on the total frictional loss as being due entirely to the Bagnold dispersive force. They adapted the original equation for Bagnold’s stress in the inertial regime of the sheared annular flow, s sB = K Bq sd 250k 2(dv / dy)2 (K B is equal to 0.013 according to Bagnold’s measurements, dv / dy is velocity differential) to the axisymmetric vertical-pipe flow and calibrated the equation by using pressure-drop data, s sw = (8.3018 107 / Re 2.317)D 2q sd 250k 1.5(s lw / l l)2 in which Re = V mq lD / l l and s lw = ( f l / 8)q lV m2. The solids effect due to Bagnold stresses predicted by Bartosik and Shook in a vertical pipe is virtually negligible for particles finer than coarse sand of the diameter 1 mm. The database of very coarse vertical flows was extended by the Ferre’s tests with glass spheres of 1.8 and 4.6 mm in a 40.9-mm pipe and conveying liquids of two different viscosities [13]. The calibration gave s sw = 0.0214 / Re 0.36 (d 50 / D)0.99k 1.31q sV m2 in s which Re s = V mq sd 50 / l l. Schaan et al. [14] found pressure drops higher than those due to viscous liquid-like friction in highly concentrated fine slurries (particle size 0.085-mm, 0.090-mm and 0.100mm) in horizontal pipes of two diameters at high velocities (the mean velocity V m = 4 m/s in Fig. 4). At this velocity any effects of flow stratification and Coulombic friction were
Fig. 4. Pressure gradient ratio I m/I l as a function of density ratio S m = q m / q l at velocity 4 m/s. Parity line I m = S mI l. Index m = slurry and l = liquid. (from Schaan et al. [14]).
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very small. The estimation of the thickness of the viscous sub-layer led the authors to an assumption that the fine-sand particles were too large to be submerged within the layer and to affect the carrier properties there. Instead, the solids friction was considered to be a result of the Bagnold’s particle-wall interactions. The total wall shear stress in axially symmetric flow and negligible Coulombic friction was considered as composed of two components: the mixture shear stress (determined as viscous shear stress for the ‘‘equivalent liquid’’ having density of slurry and viscosity of conveying liquid) and the solids shear stress of Bagnold type, s sw = Kq sk 2V m2 . Gillies and Shook [15] obtained K = 0.00002 from pressure-drop data for the 175Am sand slurries at high velocities in a horizontal 495-mm pipe. Korving and Matousek [16] observed the same effect as Schaan et al. (i.e. pressure drops higher than the Fequivalentliquid_ prediction at high velocities) in highly concentrated slurries of the fine sand (the mass-median diameter d 50 = 0.102 mm) in a horizontal 158-mm pipe of the MTI laboratory in Kinderdijk, the Netherlands. They attributed the high friction to the slurry viscous effects. A comparison of the MTI test results with the tests in a pipe of the similar size (the inner pipe diameter D = 150 mm, Laboratory of Dredging Engineering of DUT) and only slightly coarser sand (d 50 = 0.12 mm), showed a rather different behavior [17]. At high velocities the 0.12-mm sand slurry exhibited pressure drops lower than those of the equivalent-liquid friction model, I m = S mI l, in both horizontal and vertical pipe sections (see Fig. 5 for the results for vertical flows, in Fig. 5 the I m for slurry flow and I l for water flow are both in meters water column over 1-m length of the pipe). Furthermore, the relative hydraulic gradient I m / I l appeared to be relatively weakly dependent on the mean delivered solids concentration C vd (particularly in the vertical flow). This indicated that the lift force might be involved (see next paragraph). The entire solids effect I m I l was considered to be the result of kinetic solids
friction and this assumed a combined effect of the stress due to collisions between particles and the pipe wall and liquid lift force acting on solid particles in the near-wall zone. Considering the axisymmetric flow and no Coulombic friction s sw = (I m I l)q lgD / 4 = 0.991k 0.81V m0.99 in the vertical flow and s sw = (I m I l)q lgD / 4 = 0.499k 1.36V m1.02 in the horizontal flow at velocities equal to and higher than 3.5 m/ s. The same tests with the coarser sand (d 50 = 0.37 mm) exhibit even lower solids friction and weaker effects of solids concentration on the friction at high velocities (Fig. 6) indicating that the repelling effect of the hydrodynamic lift is more intensive on the 0.37-mm particles than on the 0.12mm particles. As mentioned above, the problem with a verification (at least for particles of medium size) of a collision-friction model is that pressure drops due to solids presence in nonstratified flows are a product of a combined effect of the Bagnold collisional force and liquid lift force acting on solid particles in the near-wall zone of the slurry flow rather than a product of a pure effect of collisions between particles and the pipe wall.
Fig. 5. Pressure gradient ratio and density ratio for the 0.12-mm sand flow in the vertical 150-mm pipe (from Matousek [17]).
3.2.2. Hydrodynamic lift The vertical-pipe experiments discussed above suggest that flows of particles of moderate size in slurries of low and moderate concentrations exhibit lower friction than finer slurries. This is a confirmation of the trend reported first by Newitt et al. [18] who observed that vertical flows of fine slurries behave essentially as homogeneous Fequivalent_ liquids and slurries of coarser particles (but smaller than about 1 mm in diameter) essentially as conveying liquids (there was virtually no effect of solids on flow friction). The test results in Figs. 5 and 6 show low slurry friction (much lower than predicted by the Fequivalent-liquid_ friction model) and only weak variation in solids concentration. Horizontal flows at high velocities showed the same trend [17]. If it is assumed that the collisional stress is exclusively responsible for the pressure drop due to solids presence in flow and this stress tends to increase with the
Fig. 6. Pressure gradient ratio and density ratio for the 0.37-mm sand flow in the vertical 150-mm pipe (from Matousek [17]).
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solids concentration (according to both Bagnold and Bartosik and Shook), the data suggest that there is a lift force repelling particles from the wall (and thus diminishing the collisional stress at the pipe wall) and that this lift force increases with the solids concentration (up to the concentration value of about 0.43). The increase in the lift force with the concentration seems to be of the same order as the increase in the concentration of the normal force due to collisional stress acting at the pipe wall. Clear evidence of the lift effect is the presence of locus on a concentration-profile curve near the bottom of the horizontal pipe occupied by a heterogeneous flow without the bed. Concentration profiles with a local drop in concentration just above the bottom of the pipe were measured in different test loops in fast flows of slurries of medium to coarse solids. The lift force is a product of the interaction between a solid particle and liquid flow of the steep velocity gradient near the pipe wall. The velocity gradient across the particle causes its rotation and develops a pressure differential over the particle. This is responsible for the hydrodynamic force acting on the particle in the direction perpendicular to the pipe wall (the Magnus effect). Wilson and Sellgren [19] analyzed the criteria governing the interaction of a solid particle with the liquid velocity profile and leading to the repulsion of the particle from the wall of a horizontal pipe. According to the analysis the lift is governed by the product of the dimensionless velocity function (actually a combination of velocity differentials), n, and the third power of the dimensionless distance from wall, ( y +)3. The n varies with the vertical position above the pipe wall and reaches the maximum value somewhere within the buffer layer (Fig. 7). Fig. 7 shows that the off-the-wall force (force of inertia lift associated with carrier turbulence) is effective only within a certain range of vertical positions above wall, namely in a certain portion of the region of turbulent log velocity profile and in the adjacent buffer layer. Thus the force can act only on particles of sizes that fit the range. For sand –water slurries, the lift force is not
Fig. 7. Dimensionless velocity function n versus dimensionless distance from the wall y+ (from Wilson and Sellgren [19]).
Fig. 8. Relative bed roughness, k s/d 50, for the Nikuradze’s friction equation as function of particle mobility number h b, acryl particles of w* = 0.72 (from Sumer et al. [24]).
effective for particles smaller than approximately 0.15 mm and larger than of about 0.4 mm [19]. For a solid particle interacting with the logarithmic velocity profile ( y + > 30) in a horizontal water flow, the value of the off-the-wall force, F L, is estimated as ( F L / F W)max = 0.27f lV m2 / 8(S s 1)gd 50. Throughout the buffer layer (5 < y + < 30) the lift drops to a negligible value at the top of the viscous sub-layer. Whitlock et al. [20] discuss the method for the lift-reduction determination in the buffer layer. It is rather difficult to create conditions in a slurry pipe under which the quantification of the lift force would not be ambiguous. As mentioned above in connection with the determination of the collisional forces, it is impossible to separate the two effects on the observed slurry friction. Since the lift effect seems to be associated primarily with fast flows of slurry, i.e. with velocities above those used usually during practical operations, its impact in practice is limited. 3.3. Friction at the top of granular bed In stratified flows of settling slurries the top of the granular bed is an additional boundary at which friction must be determined. If the top of the bed is not sheared-off it can be considered a rough boundary with a roughness related to the particle size. For the sheared-off beds, however, the particle size is not an appropriate boundaryroughness parameter. Instead the friction law is related to another characteristic size of the boundary—the thickness of the sheared portion of the bed. This thickness depends on the bed shear stress (e.g. [4]). The most appropriate condition for the evaluation of the bed friction is the stratified flow with the stationary bed. In this flow, the position of the top of the bed and the mean velocity differential between the current and the bed can be accurately determined. Wilson and co-workers formulated the friction law on basis of observations in closed conduits with various fractions of sand and bakelite. For I m / (S s 1) > 0.0167,
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Fig. 9. Relative bed roughness, k s/d 50, for the Nikuradze’s friction equation as function of particle mobility number h b, sand particles of w* = 0.62 (+ data from Matousek [25]; line-prediction).
the proposed bed friction-law [21] reads f b = 0.87(I m / 2 ( S s 1))0.78. The bed shear stress, s b = ( f b / 8)q 1V ma . Ribberink [22] used a rough-wall concept to determine the bed shear stress in his generalized bed-load formula for steady flows and unsteady oscillatory flows. Based on Wilson’s close-conduit data in the range 1 < h b < 7, Ribberink adapted the Wilson’s earlier relationship between the Nikuradze particle-roughness height, k s, and the particle mobility number for the top of the bed, h b, ((k s/d 50) = 5h b , [23]) to the form (k s/d 50) = 1 + 6(h b 1). Sumer et al. [24] also found that their bed-friction data (e.g. for the acryl particles w* = 0.72, Fig. 8) processed by the Nikuradze’s presistance relation for a rough ffiffiffiffiffiffiffiffiffi boundary Vma =u4b ¼ 8=fb ¼ 2:46lnð14:8Rhb =ks Þ satisfy the relation, k s / d 50 = fn(h b). Data in Sumer et al. cover a relatively narrow range of particle mobility numbers (0.8 < h b < 5 approximately). Our tests with the 0.2 – 0.5-mm sand [25] revealed a trend that was very similar to that observed by Sumer et al. and showed that there was a clear correlation between k s / d 50 and h b also in flow regimes with much larger h b (up to almost 25). The data suggest the following relation in the range 4 < h b < 25, k s / d 50 = 0.13(h b + 1.38)2.34 (Fig. 9).
4. Predictive models The complex behavior of settling-slurry flows is not well understood yet. Therefore existing tools for its simulation and prediction are still far from accurate and versatile. For difficult conditions (e.g. a complex structure of transported solids), tests are recommended to determine the pressure drop due to friction and the deposition-limit velocity in a field pipe. If the field-pipe test is not possible, a test in a laboratory pipe (usually smaller than the field pipe) is an option. The laboratory results are scaled-up to a pipe of full
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size using an appropriate scale-up model. If no test is possible for the slurry in question one must rely on results from a predictive model. In practice, empirical models deriving from the 1950s and 1960s are still widely used. Dutch dredging companies, for instance, use modified versions of the empirical correlations of Durand (or of Fu¨hrbo¨ter or Jufin-Lopatin) to predict settling-slurry flows in large horizontal/inclined pipes. They calibrate the correlations to different conditions using their own data collected during dredging works. The correlations are simple to use and modify. However, a still better understanding of an internal structure of settling-slurry flow and a wider access to user-friendly computational techniques make it more attractive for practical engineers to adopt more complex models with a physical background. A two-layer model (the basis of which was formulated by Wilson in 1970s [5,26]) is now also becoming a standard way of analyzing fully or partially stratified flows in practice. Besides its physical background, the value of the two-layer model as compared to empirical models is in the fact that that it predicts more than just one slurry-flow parameter. The model outputs are the pressure drop, the deposition-limit velocity, slip between phases, the sliding-bed velocity and the thickness of the bed. This makes it a suitable tool for the prediction not only of pipeline hydraulic performance but also of the flow pattern for an estimation of pipe-wall wear. Various versions of a layered model are in use. Generally, they differ in the way of modeling of friction on boundaries and the division of solids into layers. The development of a two-layer model has been associated with the application of measuring techniques that make it possible to observe the internal structure of the flow (concentration and velocity profiles) and link it with the idealized two-layer structure handled by the model. The two-layer model developed in the Saskatchewan Research Council (the SRC model) assumes the presence of suspended particles in the contact layer (the lower layer in the model) and the buoyancy effect of suspension on the contact bed. This idealization is justified for flows of particles of medium size in which the granular bed is virtual rather than real or in flows of broad particle size distribution with a portion of fines. The recent modification of the SRC model to highly concentrated slurries [15] has introduced the dispersive stress in the suspension layer (the upper layer in the model) and modified the function for distribution of solids into two layers (the stratification-ratio function) to simulate increased solids friction observed in high concentrated settling slurries. The model developed at the Delft University of Technology [27] distinguishes between two suspension mechanisms in the layer above the contact bed. The proportion of particles contributing to suspension is found to be primarily dependent on the ratio of the mean velocity of slurry and the settling velocity of particle if solid particles above the bed are suspended by the diffusive effect of conveying-liquid turbulence. If the particles above the bed
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are dispersed due primarily to the dispersive effect of interparticle collisions, the stratification ratio varies with the bed shear stress. However, the laboratory test results showed that the overall exponential formula relating the stratification ratio and the ratio of slurry velocity and particle settling velocity (as proposed by Gillies et al. [28]) provides also reasonable results, if calibrated with own data for horizontal flows of different fractions of sand. As shown in Matousek [29], a heterogeneous flow with turbulent suspension can be successfully modeled as a two-layer system by linking a theoretical concentration profile (the modified Rouse – Schmidt turbulent-diffusion model) with the top of the virtual contact bed. A two-layer model is also a suitable tool for simulation of a stratified flow with a non-Newtonian carrier [30].
5. Conclusions Pipe-wall friction generated by solid particles in permanent contact with the pipe wall is better understood than friction deriving from sporadic contact (collisions) or zero contact of particles with the wall. Recent tests with plug flows and inclined flows with sliding beds confirmed that Wilson’s particle-wall friction concept is appropriate to describe the solids friction caused by permanent contact between particles and a pipe wall. No suitable general concept is available to describe pipewall friction resulting from sporadic or zero contacts of particles with a pipe wall when traveling in the near-wall region of a slurry flow. Little is known about mechanisms of particle friction at the pipe wall and particle dispersion in the near-wall region of a non-stratified flow (flow with no permanent contact between particles and a wall). One of the important aspects of the mechanisms governing particle movement in the near-wall region of non-stratified slurry flows is the effect of the particle size on the solids friction. Recent tests with non-stratified flows showed that fine slurries exerted higher solids friction than coarse slurries. The fine slurries (sand particles smaller than approximately 100 Am) exhibited hydraulic gradients higher than those predicted by the model handling the slurries as Fequivalentliquids_, while the coarse slurries (fine to medium- and medium sand particles) exhibited lower hydraulic gradients than the modeled Fequivalent-liquid_ slurries. Furthermore, an interesting effect of the particle size and the solids concentration on the solids friction was detected in flows of different coarse slurries. Surprisingly low friction was detected in highly concentrated medium-sand flows; the friction was lower than that in flows of fine to medium sand for the same flow conditions. The recently developed concept of hydrodynamic off-the-wall lift suggests a possible explanation of the observed low solids friction in the medium-sand slurry. Friction at the top of a granular bed has been studied extensively in recent years. New data from several closed
conduits has opened the possibility to formulate and verify friction laws for plane beds that together cover a broad range of Shields numbers (0.8 < h b < 25). The verification of friction and lift theories, the development of which is currently in progress, will require measuring techniques that can sense important parameters of slurry flow in smaller control volumes (adjacent to a pipe wall) than is currently possible. Notation cv local volumetric solids concentration in pipe crosssection Cv mean volumetric solids concentration in pipe cross-section C vb mean volumetric solids concentration in bed (spatial) C vd delivered C v C vi spatial C v C vmax maximum volumetric solids concentration in bed (spatial) d 50 median particle diameter [m] D pipe diameter [m] fb Darcy –Weisbach friction coefficient for the top of bed fl Darcy – Weisbach friction coefficient for liquid flow over pipe wall FL hydrodynamic off-the-wall force (lift force) [N] FW submerged weight of particle in liquid, (S s 1) q lgd 350p / 8 [N] g gravitational acceleration [m/s2] Il hydraulic gradient of liquid flow Im hydraulic gradient of slurry flow ks characteristic grain roughness height (by Nikuradze) [m] K proportional coefficient in Gillies – Shook equation for solids shear stress KB proportional coefficient in Bagnold’s equation for solids shear stress R hb hydraulic radius of area associated with the top of bed [m] Re Reynolds number, V mDq l / l l Sm relative density of slurry, q m / q l Ss relative density of solids, q s / q l u *b shear velocity at the top of bed [m/s] u *l shear velocity at the wall of pipe, V m( f l / 8)0.5 [m/s] v local velocity of liquid vt terminal settling velocity of solid particle [m/s] v+ dimensionless local velocity, v / u *l Vm mean velocity of mixture in pipe cross-section [m/s] V ma mean velocity of mixture above contact bed [m/s] w* dimensionless terminal settling velocity, v t / ((S s 1)gd 50)0.5 y vertical distance from pipe wall [m] y+ dimensionless distance from pipe wall, yu *lq l / l l b angle defining position of top of bed [rad]
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hb k ll ls n ql qs r swC sb s lw s sB s sw s swC
particle mobility number (Shields number), u *b2 / ((S s 1)gd 50) linear concentration of solids (by Bagnold), 1 / ((C vmax / C vi)0.33 1) dynamic viscosity of liquid [Pa s] mechanical friction coefficient of solids against pipe wall lift parameter (by Wilson and Sellgren), dv + / dy +( d2v + / dy + 2) density of liquid (water) [kg/m3] density of solid particle [kg/m3] solids normal stress exerted by sliding particles at pipe wall [Pa] shear stress at the top of bed [Pa] liquid shear stress at pipe wall [Pa] solids shear stress in sheared slurry in inertial regime (by Bagnold) [Pa] solids shear stress exerted by colliding particles at pipe wall [Pa] solids shear stress exerted by sliding particles at pipe wall [Pa]
Abbreviations DEM Discrete Element Method DUT Delft University of Technology LDA Laser Doppler Anemometry MRI Magnetic Resonance Imaging PIV Particle Image Velocimetry SRC Saskatchewan Research Council References [1] Matousek, Pressure drops and flow patterns in sand-mixture pipes, Experimental Thermal and Fluid Science (26) (2002) 693 – 702. [2] M. Simkhis, D. Barnea, Y. Taitel, Dunes in solid – liquid flow in pipes, in: J.F. Richardson (Ed.), Proc. 14th Int. Conf. on Hydrotransport, 1999, pp. 51 – 61. [3] L. Pullum, L.J.W. Graham, The use of magnetic resonance imaging (MRI) to probe complex hybrid suspension flows, in: J. Sobota (Ed.), Proc. 10th Int. Conf. on Transport and Sedimentation of Solid Particles, Wroclaw, Poland, 2000 (4 – 7 September), pp. 421 – 433. [4] F.J. Pugh, K.C. Wilson, Velocity and concentration distributions in sheet flow above plane beds, Journal of Hydraulic Engineering 125 (2) (1999) 117 – 125. [5] K.C. Wilson, Slip point of beds in solid – liquid pipeline flow, Journal of the Hydraulics Division 96 (HY1) (1970) 1 – 12. [6] K.C. Wilson, M. Streat, R.A. Bantin, Slip-model correlation of dense two-phase flow, Proc. 2nd Int. Conf. on Hydrotransport, BHRA, Warwick, UK, 1972 (20 – 22 September), pp. B1-1-10. [7] A.C. Korving, High-concentrated fine-sand slurry flow in pipelines: experimental study, in: N. Heywood (Ed.), Proc. 15th Int. Conf. on Hydrotransport, BHRG, Banff, Canada, 2002 (3 – 5 June), pp. 769 – 776. [8] V. Matousek, Distribution and friction of particles in pipeline flow of sand – water mixtures, in: A. Levy, H. Kalman (Eds.), Handbook of Conveying and Handling of Particulate Solids, Elsevier, 2001, pp. 465 – 471. [9] F. Stienen, V. Matousek, C. van Rhee, Simulation of solids behaviour at wall of circular slurry pipe using DEM, Proc. 12th Int. Conf. on
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Transport and Sedimentation of Solid Particles. Prague, 20 – 23 September, 2004, pp. 621 – 630. R.A. Bagnold, Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear, Proceedings of the Royal Society of London, Series A (225) (1954) 49 – 63. C.A. Shook, A.S. Bartosik, Particle-wall stresses in vertical slurry flows, Powder Technology 81 (1994) 117 – 124. A.S. Bartosik, Modelling the Bagnold stress effects in vertical slurry flow, Journal of Hydrology and Hydromechanics 44 (1) (1996) 49 – 58. A.L. Ferre, C.A. Shook, Coarse particle wall friction in vertical slurry flows, Particulate Science and Technology 16 (1998) 125 – 133. J. Schaan, R.J. Sumner, R.G. Gillies, C.A. Shook, The effect of particle shape on pipeline friction for Newtonian slurries of fine particles, Canadian Journal of Chemical Engineering 78 (2000) 717 – 725. R.G. Gillies, C.A. Shook, Modelling high concentration settling slurry flows, Canadian Journal of Chemical Engineering 78 (2000) 709 – 716. A.C. Korving, V. Matousek, Modeling of fine-slurry flow in pipeline: the effect of turbulent diffusion and slurry viscosity, Proc. CEDA Dredging Day, Amsterdam, Netherlands, 15 November, 2001, pp. 13-1 – 13-10. V. Matousek, Non-stratified flow of sand – water slurries, in: N. Heywood (Ed.), Proc. 15th Int. Conf. on Hydrotransport, BHRG, Banff, Canada, 2002 (3 – 5 June), pp. 563 – 574. D.M. Newitt, J.F. Richardson, B.J. Gliddon, Hydraulic conveying of solids in vertical pipes, Transactions of the Institution of Chemical Engineers 39 (1961) 93 – 100. K.C. Wilson, A. Sellgren, Interaction of particles and near-wall lift in slurry pipelines, Journal of Hydraulic Engineering 129 (1) (2003) 73 – 76. L. Whitlock, K.C. Wilson, A. Sellgren, Effect of near-wall lift on frictional characteristics of sand slurries, in: N. Heywood (Ed.), Proc. 16th Int. Conf. on Hydrotransport, BHRG, Santiago, Chile, 2004 (26 – 28 April), pp. 443 – 454. K.C. Wilson, F.J. Pugh, Real and virtual interfaces in slurry flows, Proceedings of the 8th Int. Conf. on Transport and Sedimentation of Solid Particles, Prague, 1995, pp. A4-1 – A4-10. J.S. Ribberink, Bed-load transport for steady flows and unsteady oscillatory flows, Coastal Engineering 38 (1998) 59 – 82. K.C. Wilson, Mobile-bed friction at high shear stress, Journal of Hydraulic Engineering 115 (6) (1989) 825 – 830. B.M. Sumer, A. Kozakiewicz, J. Fredsøe, R. Deigaard, Velocity and concentration profiles in sheet-flow layer of movable bed, Journal of Hydraulic Engineering 122 (10) (1996) 549 – 558. V. Matousek, Erosion of plane bed by sand slurry current in pipe, Journal of Hydrology and Hydromechanics 52 (3) (2004) 156 – 161. K.C. Wilson, A unified physically based analysis of solid – liquid pipeline flow, Proc. 4nd Int. Conf. on Hydrotransport, BHRA, Banff, Canada, 1976 (18 – 21 May), pp. A1-1 – A1-16. V. Matousek, Flow Mechanism of Sand – Water Mixtures in Pipelines, Delft University Press, 1997. R.G. Gillies, C.A. Shook, K.C. Wilson, An improved two layer model for horizontal slurry pipeline flow, Canadian Journal of Chemical Engineering 69 (1991) 173 – 178. V. Matousek, Partially stratified flow with turbulent suspension in a pipe, in: J. Sobota, R. Verhoeven (Eds.), Proc. 11th Int. Conf. on Transport and Sedimentation of Solid Particles, 2002 (9 – 12 September), pp. 391 – 398 (Gent, Belgium). L. Pullum, L.J.W. Graham, P. Slatter, A non-Newtonian two-layer model and its application to high density hydrotransport, in: N. Heywood (Ed.), Proc. 16th Int. Conf. on Hydrotransport, BHRG, Santiago, Chile, 2004 (26 – 28 April), pp. 579 – 593.
Research developments in pipeline transport of settling slurries Va´clav Matouxek Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Received 9 November 2004; received in revised form 22 March 2005; accepted 18 May 2005
Abstract The designing of slurry pipelines requires prediction of pipeline hydraulic performance. The preferable predictive tool is a model based on a physical description of slurry flow behavior in a pipeline. Various mechanisms that affect pipeline flow behavior of settling slurries with a Newtonian conveying liquid are described. Recent developments in measuring, understanding and modeling of the mechanisms are discussed. An implementation of recent research results to current versions of a predictive model for stratified flows of settling slurries (a two-layer model) is surveyed. D 2005 Elsevier B.V. All rights reserved. Keywords: Slurry flow; Solids friction; Two-layer model
1. Introduction In the hydraulic transport of solids using pipelines the liquid is used as a vehicle that carries solid particles to their destination at the end of a slurry pipeline. In practice, slurries of different liquids and solids are conveyed through pipelines. This paper discusses the transport of ‘‘settling’’ slurries with a Newtonian carrier. Settling slurries are mixtures in which the solid particles tend to separate from the carrying liquid owing to the action of gravity force. If solid particles cannot be maintained in suspension by the turbulent diffusive action of the conveying liquid during the transportation through horizontal or inclined slurry pipelines, they settle down to the bottom of the pipe. A sand – water mixture is typical settling slurry. Such slurry is discrete, i.e. even the finest sand particles are too coarse to combine with the conveying water to produce a stable homogeneous medium. A settling-slurry flow tends to stratify. In horizontal and inclined pipes, flows of settling slurries at the velocities usually used during practical operations are characterized by a non-uniform distribution of particles across a pipe cross-section. More particles gather near the bottom than near the top of a pipe cross-
E-mail address: [email protected]. 0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2005.05.054
section. The flow is considered stratified if a portion of solids forms a granular bed (stationary or sliding) at the bottom of a pipe. The flow is fully stratified if all particles occupy the granular bed (particles maintain a virtually permanent contact with other particles within the bed) or partially stratified if only a portion of particles occupies the bed and the rest of particles maintains either sporadic contact or no contact with other particles. The design of slurry pipelines carrying settling slurries requires prediction of hydraulic performance of the pipeline. This is given by a relationship between the slurry flow rate and the total pressure differential developed over a pipeline at the flow rate in question. For settling slurries, the pressure drop due to friction over the length of a pipeline is closely associated with the internal structure of the slurry flow. Since the early 1950s systematic research on the flow of settling slurries in pipelines has been conducted. In the pioneer era, the investigation was focused primarily on a collection of basic experimental data (pressure drops at different velocities) for flows of different pipe and particle sizes. The collected databases served as a basis for a construction of empirical models obtained by formulating suitable dimensionless numbers and finding the best fit by using the data regression. The 1970s and 1980s brought works giving an insight into fundamental principles of pipeline flows for settling slurries—basics of a macroscopic
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two-layer model by Wilson and of a microscopic slurry-flow model by Shook and Roco. Contemporary research makes use of the rapid development of computational and measuring techniques permitting the examination of slurry flow processes in still closer details and leading to better understanding of the complex nature of the behavior of settling-slurry flows in pipelines. Currently, practical interests focus attention on research of settling slurries that are highly concentrated (volumetric concentrations of solids higher than 35% approximately). Furthermore, the contemporary research tends to focus on analyzing settling-slurry flow behavior under specific conditions. A typical approach to the investigation of a pipeline flow of settling slurries is composed of the following steps: – observation of slurry flows under certain specific, welldefined conditions in the closest possible detail and using modern measuring techniques (including concentration and/or velocity distributions) – on the basis of the observations, determination of the prevailing mechanisms governing the observed slurry flow – modeling of the processes and sub-processes in order to simulate the slurry-flow behavior and flow parameters important in practice. Despite a fair degree of progress made over the years, current understanding of the complex behavior of a pipe flow of settling slurries is still far from complete.
2. Measuring techniques 2.1. Distribution of solids A radiometric method has been most often used to determine a distribution of solids across vertical crosssections of laboratory test pipes. At present, radiometric density meters adapted as concentration profilers for slurry pipes are in use in the laboratories of Saskatchewan Research Council (SRC) and Delft University of Technology (DUT, [1]). A radiometric density meter requires a two-point calibration at each measuring position in a pipe cross-section. Delft Hydraulics uses a set of conductivity probes mounted as pairs of electrodes along the perimeter of the pipe wall. The temperature and salinity of the carrying liquid strongly influences conductivity and thus the accuracy of the instrument. A similar electro-resistance sensor was used by Simkhis et al. [2] to detect dunes in a slurry pipe. Recently, the magnetic resonance imaging (MRI) method has been starting to gain a footing in laboratory slurry-pipe experiments [3]. An advantage of the method is that it can collect an entire concentration profile in one moment, a disadvantage is a relatively complex reconstruction process
required to obtain a concentration-profile image from the measured signal. As yet, the high price of the instrument reduces its application to small pipes. The measuring principle sets a maximum limit velocity at which a profile can be measured in a pipe. Pullum and Graham [3] reported 1 m/s as the approximate maximum mean velocity for a 100-mm pipe. 2.2. Distribution of solids velocity In general, it is difficult to measure the distribution of the local velocity of solid particles. Usually, settling slurries are transported at concentrations too high for the use of optical methods like LDA and PIV. In addition to a concentration profile, the MRI method also provides a liquid-velocity profile of a flow. The crosscorrelation of the electrical resistance (or conductivity) signals from two sensors located near each other is the measuring principle most often used to determine local solids velocity in a slurry pipe. The SRC uses an intrusive conductivity probe as a solids velocity profiler (e.g. [4]).
3. Flow-friction mechanisms 3.1. Friction due to permanent contact of solid particles with pipe wall Particles traveling within a granular body in contact with a pipe wall exert a solids stress against the pipe wall that contributes significantly to the total friction of the slurry flow. Wilson’s method [5,6] for the determination of the normal solids stress at the pipe wall is based on the simplified assumption that there is a hydrostatic-type of distribution of the normal stress in the vertical direction through a granular bed submerged in liquid. Furthermore, it assumes that the normal stress at the wall, r swC, produces the solids shear stress, s swC, at the pipe wall according to Coulomb’s law (the normal and shear stresses are mutually related through the friction coefficient, l s), s swC = l s(q s q l)gC vb(D / 2b) (sin b b cos b). It is not easy to verify this method, since it is difficult to eliminate other effects and measure only the effects caused by the Coulombic solids stresses during pipe tests. Tests with extremely high concentrated flows (plug flows) by Wilson et al. [6] justified the proposed solids-stress theory. The tests with extremely highly concentrated sand slurries conducted by Korving [7] confirmed that the Wilson’s plug model based on the solids-stress theory can successfully interpret Korving’s data (Fig. 1) if the coefficient of mechanical friction between the plug and the pipe wall is considered dependent on the mean concentration of solids in the pipe (i.e. in the plug). Another suitable condition for the verification of the theory is a fully stratified flow through a descending pipe inclined at a small angle (say 25 –35-), in which the velocity
V. Matous˘ek / Powder Technology 156 (2005) 43 – 51
45
1 0.9 0.8 Plug 1800 kg/m3 Plug 1780 kg/m3 Plug 1730 kg/m3 1800 kg/m3 1780 kg/m3 1730 kg/m3 water
Hydraulic gradient [-]
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
1
2
3 4 Mean slurry velocity Vm [m/s]
5
6
7
Fig. 1. Comparison of experiments with the plug model for the 0.103-mm sand in the 158-mm pipe (from Korving [7]).
of the sliding bed is almost the same as the velocity of the water current above the bed. Under these conditions the solids effect on the total hydraulic gradient is exclusively due to the acting solids stress at the pipe wall in contact with the sliding bed. The test presented by Matousek [8] showed that mechanical friction at the boundary between the sliding granular bed and the pipe wall was successfully predicted by Wilson’s friction law with the friction-coefficient value of about 0.55 valid for all the sands and gravels that were tested. A problem arises in the interpretation of the verification tests if a drop in local concentration occurs at the bottom of the sliding bed adjacent to the pipe wall (see Fig. 2). This phenomenon seems to occur if the velocity of the sliding bed is high. The simulation of a fully stratified flow of coarse particles using Discrete Element Method suggests that this phenomenon is the result of interactions among particles rolling over each other at the bottom of the sliding bed [9] (Fig. 3). Coulomb’s law assumes a constant value of the friction coefficient relating the normal and shear stresses. However, variations in the coefficient value for a pipe wall with a local
concentration of solids or the velocity of the bed can be expected. Experimental evidence is rather limited and further investigation is required to provide better understanding of the exact relationship between the solids normal and shear stresses at a pipe wall. 3.2. Friction due to sporadic or zero contact of solid particles with pipe wall Basically, friction due to the presence of solid particles suspended in a flow is a result of processes in a relatively thin layer near the pipe wall. To identify interactions between particles and the conveying liquid and interactions Volumetric Concentration and Velocity Profiles V = 4 [m/s] Velocity [m/s]
0
1
2
3
4
5
0.6 volumetric concentration
0.5
solids velocity
y/D [-]
0.4
0.3
0.2
0.1
0.0 0.0
0.2
0.4
0.6
0.8
10.
Volumetric Concentration [-]
Fig. 2. Concentration profile and thickness of the sliding bed in the pipe inclined to 35- (the 1.4 – 2.0-mm-sand mixture) (from Matousek [8]).
Fig. 3. Distribution of solids concentration and velocity in the sliding bed at V m = 4 m/s simulated using DEM (from Stienen et al. [9]).
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among particles in the region near a pipe wall and to describe their effect on flow friction is one of the greatest challenges to those researching the pipeline flow of settling slurries. As yet, available measuring techniques are not capable of recognizing the interactive processes within a narrow region near a pipe wall with sufficient distinction. Crucial questions must be answered before the pipe-wall friction of suspension flow can be determined. Friction models use the wall shear stress as a measure of suspension friction in a pipe flow. Basically, two wall shear stresses are recognized: the Fliquid-like_ (or viscous) shear stress and the shear stress caused by the contacts (impingements) of traveling suspended particles with a pipe wall. In contrast to the Coulombic solids stress, the solids stress due to impingements is velocity-dependent (kinetic). For different flow situations, it is difficult to determine whether or not solid particles traveling near the pipe wall contribute to the wall shear stresses, and if so to which shear stress they contribute and with which portion of their total concentration. Essentially, friction behavior of suspended coarse particles differs from that of fine particles. The threshold for particle diameter between the coarse-slurry behavior and the fine-slurry behavior is rather vague but it is assumed that the threshold particle size is related to the thickness of the viscous sub-layer adjacent to the pipe wall. Coarse particles in non-stratified flows (slurry flows without Coulombic stresses) have contacts with each other and with a pipe wall and so contribute to the total friction of slurry flow. The contacts between particles and the pipe wall are sporadic rather than permanent and they are a result of turbulent dispersive action (turbulent eddies tend to disperse solid particles to all directions) and collisional dispersive action (particles are impelled in the direction of the pipe wall by the collision with other particle of different velocity in the flow). The solids stress due to particle collisions with a pipe wall can be diminished by the hydrodynamic off-wall lift. Settling slurries composed of fine particles tend to exhibit a rather different friction pattern. Near the pipe wall, fine particles travel submerged within the viscous sublayer. Within this layer the effects of carrier turbulence and hydrodynamic lift on solid particles are negligible. The increase in the wall shear stress due to the presence of solid particles in the near-wall region is due to increased viscous friction rather than mechanical contact friction. Particles locked inside the viscous sub-layer increase the density of the carrying liquid and so increase the viscous shear stress at the pipe wall. Nevertheless, the solids dispersive stress seems to occur in the fine slurries if the concentration of solids near the wall becomes high. 3.2.1. Collisional stress In the near-wall region the velocity gradient is steep and particles with different local velocities collide with each other and with the pipe wall. The collisions generate the solids dispersive stress (first observed and described by
Bagnold [10]) acting against the pipe wall. This stress is a source of solids friction, additional to the liquid-like friction, in a pipe. The most appropriate geometry for an observation of the collisional stresses is a vertical pipe in which the flow is axisymmetric and there is no Coulombic stress. On the other hand, the need to subtract the hydrostatic part from the measured pressure differential is a potential source of inaccuracy. Shook and Bartosik [11,12] observed three coarse particle fractions (1.37-mm, 1.5-mm, 3.4-mm) in two narrow vertical pipes (D = 26-mm and 40-mm) and interpreted the measured solids effects on the total frictional loss as being due entirely to the Bagnold dispersive force. They adapted the original equation for Bagnold’s stress in the inertial regime of the sheared annular flow, s sB = K Bq sd 250k 2(dv / dy)2 (K B is equal to 0.013 according to Bagnold’s measurements, dv / dy is velocity differential) to the axisymmetric vertical-pipe flow and calibrated the equation by using pressure-drop data, s sw = (8.3018 107 / Re 2.317)D 2q sd 250k 1.5(s lw / l l)2 in which Re = V mq lD / l l and s lw = ( f l / 8)q lV m2. The solids effect due to Bagnold stresses predicted by Bartosik and Shook in a vertical pipe is virtually negligible for particles finer than coarse sand of the diameter 1 mm. The database of very coarse vertical flows was extended by the Ferre’s tests with glass spheres of 1.8 and 4.6 mm in a 40.9-mm pipe and conveying liquids of two different viscosities [13]. The calibration gave s sw = 0.0214 / Re 0.36 (d 50 / D)0.99k 1.31q sV m2 in s which Re s = V mq sd 50 / l l. Schaan et al. [14] found pressure drops higher than those due to viscous liquid-like friction in highly concentrated fine slurries (particle size 0.085-mm, 0.090-mm and 0.100mm) in horizontal pipes of two diameters at high velocities (the mean velocity V m = 4 m/s in Fig. 4). At this velocity any effects of flow stratification and Coulombic friction were
Fig. 4. Pressure gradient ratio I m/I l as a function of density ratio S m = q m / q l at velocity 4 m/s. Parity line I m = S mI l. Index m = slurry and l = liquid. (from Schaan et al. [14]).
V. Matous˘ek / Powder Technology 156 (2005) 43 – 51
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very small. The estimation of the thickness of the viscous sub-layer led the authors to an assumption that the fine-sand particles were too large to be submerged within the layer and to affect the carrier properties there. Instead, the solids friction was considered to be a result of the Bagnold’s particle-wall interactions. The total wall shear stress in axially symmetric flow and negligible Coulombic friction was considered as composed of two components: the mixture shear stress (determined as viscous shear stress for the ‘‘equivalent liquid’’ having density of slurry and viscosity of conveying liquid) and the solids shear stress of Bagnold type, s sw = Kq sk 2V m2 . Gillies and Shook [15] obtained K = 0.00002 from pressure-drop data for the 175Am sand slurries at high velocities in a horizontal 495-mm pipe. Korving and Matousek [16] observed the same effect as Schaan et al. (i.e. pressure drops higher than the Fequivalentliquid_ prediction at high velocities) in highly concentrated slurries of the fine sand (the mass-median diameter d 50 = 0.102 mm) in a horizontal 158-mm pipe of the MTI laboratory in Kinderdijk, the Netherlands. They attributed the high friction to the slurry viscous effects. A comparison of the MTI test results with the tests in a pipe of the similar size (the inner pipe diameter D = 150 mm, Laboratory of Dredging Engineering of DUT) and only slightly coarser sand (d 50 = 0.12 mm), showed a rather different behavior [17]. At high velocities the 0.12-mm sand slurry exhibited pressure drops lower than those of the equivalent-liquid friction model, I m = S mI l, in both horizontal and vertical pipe sections (see Fig. 5 for the results for vertical flows, in Fig. 5 the I m for slurry flow and I l for water flow are both in meters water column over 1-m length of the pipe). Furthermore, the relative hydraulic gradient I m / I l appeared to be relatively weakly dependent on the mean delivered solids concentration C vd (particularly in the vertical flow). This indicated that the lift force might be involved (see next paragraph). The entire solids effect I m I l was considered to be the result of kinetic solids
friction and this assumed a combined effect of the stress due to collisions between particles and the pipe wall and liquid lift force acting on solid particles in the near-wall zone. Considering the axisymmetric flow and no Coulombic friction s sw = (I m I l)q lgD / 4 = 0.991k 0.81V m0.99 in the vertical flow and s sw = (I m I l)q lgD / 4 = 0.499k 1.36V m1.02 in the horizontal flow at velocities equal to and higher than 3.5 m/ s. The same tests with the coarser sand (d 50 = 0.37 mm) exhibit even lower solids friction and weaker effects of solids concentration on the friction at high velocities (Fig. 6) indicating that the repelling effect of the hydrodynamic lift is more intensive on the 0.37-mm particles than on the 0.12mm particles. As mentioned above, the problem with a verification (at least for particles of medium size) of a collision-friction model is that pressure drops due to solids presence in nonstratified flows are a product of a combined effect of the Bagnold collisional force and liquid lift force acting on solid particles in the near-wall zone of the slurry flow rather than a product of a pure effect of collisions between particles and the pipe wall.
Fig. 5. Pressure gradient ratio and density ratio for the 0.12-mm sand flow in the vertical 150-mm pipe (from Matousek [17]).
3.2.2. Hydrodynamic lift The vertical-pipe experiments discussed above suggest that flows of particles of moderate size in slurries of low and moderate concentrations exhibit lower friction than finer slurries. This is a confirmation of the trend reported first by Newitt et al. [18] who observed that vertical flows of fine slurries behave essentially as homogeneous Fequivalent_ liquids and slurries of coarser particles (but smaller than about 1 mm in diameter) essentially as conveying liquids (there was virtually no effect of solids on flow friction). The test results in Figs. 5 and 6 show low slurry friction (much lower than predicted by the Fequivalent-liquid_ friction model) and only weak variation in solids concentration. Horizontal flows at high velocities showed the same trend [17]. If it is assumed that the collisional stress is exclusively responsible for the pressure drop due to solids presence in flow and this stress tends to increase with the
Fig. 6. Pressure gradient ratio and density ratio for the 0.37-mm sand flow in the vertical 150-mm pipe (from Matousek [17]).
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solids concentration (according to both Bagnold and Bartosik and Shook), the data suggest that there is a lift force repelling particles from the wall (and thus diminishing the collisional stress at the pipe wall) and that this lift force increases with the solids concentration (up to the concentration value of about 0.43). The increase in the lift force with the concentration seems to be of the same order as the increase in the concentration of the normal force due to collisional stress acting at the pipe wall. Clear evidence of the lift effect is the presence of locus on a concentration-profile curve near the bottom of the horizontal pipe occupied by a heterogeneous flow without the bed. Concentration profiles with a local drop in concentration just above the bottom of the pipe were measured in different test loops in fast flows of slurries of medium to coarse solids. The lift force is a product of the interaction between a solid particle and liquid flow of the steep velocity gradient near the pipe wall. The velocity gradient across the particle causes its rotation and develops a pressure differential over the particle. This is responsible for the hydrodynamic force acting on the particle in the direction perpendicular to the pipe wall (the Magnus effect). Wilson and Sellgren [19] analyzed the criteria governing the interaction of a solid particle with the liquid velocity profile and leading to the repulsion of the particle from the wall of a horizontal pipe. According to the analysis the lift is governed by the product of the dimensionless velocity function (actually a combination of velocity differentials), n, and the third power of the dimensionless distance from wall, ( y +)3. The n varies with the vertical position above the pipe wall and reaches the maximum value somewhere within the buffer layer (Fig. 7). Fig. 7 shows that the off-the-wall force (force of inertia lift associated with carrier turbulence) is effective only within a certain range of vertical positions above wall, namely in a certain portion of the region of turbulent log velocity profile and in the adjacent buffer layer. Thus the force can act only on particles of sizes that fit the range. For sand –water slurries, the lift force is not
Fig. 7. Dimensionless velocity function n versus dimensionless distance from the wall y+ (from Wilson and Sellgren [19]).
Fig. 8. Relative bed roughness, k s/d 50, for the Nikuradze’s friction equation as function of particle mobility number h b, acryl particles of w* = 0.72 (from Sumer et al. [24]).
effective for particles smaller than approximately 0.15 mm and larger than of about 0.4 mm [19]. For a solid particle interacting with the logarithmic velocity profile ( y + > 30) in a horizontal water flow, the value of the off-the-wall force, F L, is estimated as ( F L / F W)max = 0.27f lV m2 / 8(S s 1)gd 50. Throughout the buffer layer (5 < y + < 30) the lift drops to a negligible value at the top of the viscous sub-layer. Whitlock et al. [20] discuss the method for the lift-reduction determination in the buffer layer. It is rather difficult to create conditions in a slurry pipe under which the quantification of the lift force would not be ambiguous. As mentioned above in connection with the determination of the collisional forces, it is impossible to separate the two effects on the observed slurry friction. Since the lift effect seems to be associated primarily with fast flows of slurry, i.e. with velocities above those used usually during practical operations, its impact in practice is limited. 3.3. Friction at the top of granular bed In stratified flows of settling slurries the top of the granular bed is an additional boundary at which friction must be determined. If the top of the bed is not sheared-off it can be considered a rough boundary with a roughness related to the particle size. For the sheared-off beds, however, the particle size is not an appropriate boundaryroughness parameter. Instead the friction law is related to another characteristic size of the boundary—the thickness of the sheared portion of the bed. This thickness depends on the bed shear stress (e.g. [4]). The most appropriate condition for the evaluation of the bed friction is the stratified flow with the stationary bed. In this flow, the position of the top of the bed and the mean velocity differential between the current and the bed can be accurately determined. Wilson and co-workers formulated the friction law on basis of observations in closed conduits with various fractions of sand and bakelite. For I m / (S s 1) > 0.0167,
V. Matous˘ek / Powder Technology 156 (2005) 43 – 51
Fig. 9. Relative bed roughness, k s/d 50, for the Nikuradze’s friction equation as function of particle mobility number h b, sand particles of w* = 0.62 (+ data from Matousek [25]; line-prediction).
the proposed bed friction-law [21] reads f b = 0.87(I m / 2 ( S s 1))0.78. The bed shear stress, s b = ( f b / 8)q 1V ma . Ribberink [22] used a rough-wall concept to determine the bed shear stress in his generalized bed-load formula for steady flows and unsteady oscillatory flows. Based on Wilson’s close-conduit data in the range 1 < h b < 7, Ribberink adapted the Wilson’s earlier relationship between the Nikuradze particle-roughness height, k s, and the particle mobility number for the top of the bed, h b, ((k s/d 50) = 5h b , [23]) to the form (k s/d 50) = 1 + 6(h b 1). Sumer et al. [24] also found that their bed-friction data (e.g. for the acryl particles w* = 0.72, Fig. 8) processed by the Nikuradze’s presistance relation for a rough ffiffiffiffiffiffiffiffiffi boundary Vma =u4b ¼ 8=fb ¼ 2:46lnð14:8Rhb =ks Þ satisfy the relation, k s / d 50 = fn(h b). Data in Sumer et al. cover a relatively narrow range of particle mobility numbers (0.8 < h b < 5 approximately). Our tests with the 0.2 – 0.5-mm sand [25] revealed a trend that was very similar to that observed by Sumer et al. and showed that there was a clear correlation between k s / d 50 and h b also in flow regimes with much larger h b (up to almost 25). The data suggest the following relation in the range 4 < h b < 25, k s / d 50 = 0.13(h b + 1.38)2.34 (Fig. 9).
4. Predictive models The complex behavior of settling-slurry flows is not well understood yet. Therefore existing tools for its simulation and prediction are still far from accurate and versatile. For difficult conditions (e.g. a complex structure of transported solids), tests are recommended to determine the pressure drop due to friction and the deposition-limit velocity in a field pipe. If the field-pipe test is not possible, a test in a laboratory pipe (usually smaller than the field pipe) is an option. The laboratory results are scaled-up to a pipe of full
49
size using an appropriate scale-up model. If no test is possible for the slurry in question one must rely on results from a predictive model. In practice, empirical models deriving from the 1950s and 1960s are still widely used. Dutch dredging companies, for instance, use modified versions of the empirical correlations of Durand (or of Fu¨hrbo¨ter or Jufin-Lopatin) to predict settling-slurry flows in large horizontal/inclined pipes. They calibrate the correlations to different conditions using their own data collected during dredging works. The correlations are simple to use and modify. However, a still better understanding of an internal structure of settling-slurry flow and a wider access to user-friendly computational techniques make it more attractive for practical engineers to adopt more complex models with a physical background. A two-layer model (the basis of which was formulated by Wilson in 1970s [5,26]) is now also becoming a standard way of analyzing fully or partially stratified flows in practice. Besides its physical background, the value of the two-layer model as compared to empirical models is in the fact that that it predicts more than just one slurry-flow parameter. The model outputs are the pressure drop, the deposition-limit velocity, slip between phases, the sliding-bed velocity and the thickness of the bed. This makes it a suitable tool for the prediction not only of pipeline hydraulic performance but also of the flow pattern for an estimation of pipe-wall wear. Various versions of a layered model are in use. Generally, they differ in the way of modeling of friction on boundaries and the division of solids into layers. The development of a two-layer model has been associated with the application of measuring techniques that make it possible to observe the internal structure of the flow (concentration and velocity profiles) and link it with the idealized two-layer structure handled by the model. The two-layer model developed in the Saskatchewan Research Council (the SRC model) assumes the presence of suspended particles in the contact layer (the lower layer in the model) and the buoyancy effect of suspension on the contact bed. This idealization is justified for flows of particles of medium size in which the granular bed is virtual rather than real or in flows of broad particle size distribution with a portion of fines. The recent modification of the SRC model to highly concentrated slurries [15] has introduced the dispersive stress in the suspension layer (the upper layer in the model) and modified the function for distribution of solids into two layers (the stratification-ratio function) to simulate increased solids friction observed in high concentrated settling slurries. The model developed at the Delft University of Technology [27] distinguishes between two suspension mechanisms in the layer above the contact bed. The proportion of particles contributing to suspension is found to be primarily dependent on the ratio of the mean velocity of slurry and the settling velocity of particle if solid particles above the bed are suspended by the diffusive effect of conveying-liquid turbulence. If the particles above the bed
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V. Matous˘ek / Powder Technology 156 (2005) 43 – 51
are dispersed due primarily to the dispersive effect of interparticle collisions, the stratification ratio varies with the bed shear stress. However, the laboratory test results showed that the overall exponential formula relating the stratification ratio and the ratio of slurry velocity and particle settling velocity (as proposed by Gillies et al. [28]) provides also reasonable results, if calibrated with own data for horizontal flows of different fractions of sand. As shown in Matousek [29], a heterogeneous flow with turbulent suspension can be successfully modeled as a two-layer system by linking a theoretical concentration profile (the modified Rouse – Schmidt turbulent-diffusion model) with the top of the virtual contact bed. A two-layer model is also a suitable tool for simulation of a stratified flow with a non-Newtonian carrier [30].
5. Conclusions Pipe-wall friction generated by solid particles in permanent contact with the pipe wall is better understood than friction deriving from sporadic contact (collisions) or zero contact of particles with the wall. Recent tests with plug flows and inclined flows with sliding beds confirmed that Wilson’s particle-wall friction concept is appropriate to describe the solids friction caused by permanent contact between particles and a pipe wall. No suitable general concept is available to describe pipewall friction resulting from sporadic or zero contacts of particles with a pipe wall when traveling in the near-wall region of a slurry flow. Little is known about mechanisms of particle friction at the pipe wall and particle dispersion in the near-wall region of a non-stratified flow (flow with no permanent contact between particles and a wall). One of the important aspects of the mechanisms governing particle movement in the near-wall region of non-stratified slurry flows is the effect of the particle size on the solids friction. Recent tests with non-stratified flows showed that fine slurries exerted higher solids friction than coarse slurries. The fine slurries (sand particles smaller than approximately 100 Am) exhibited hydraulic gradients higher than those predicted by the model handling the slurries as Fequivalentliquids_, while the coarse slurries (fine to medium- and medium sand particles) exhibited lower hydraulic gradients than the modeled Fequivalent-liquid_ slurries. Furthermore, an interesting effect of the particle size and the solids concentration on the solids friction was detected in flows of different coarse slurries. Surprisingly low friction was detected in highly concentrated medium-sand flows; the friction was lower than that in flows of fine to medium sand for the same flow conditions. The recently developed concept of hydrodynamic off-the-wall lift suggests a possible explanation of the observed low solids friction in the medium-sand slurry. Friction at the top of a granular bed has been studied extensively in recent years. New data from several closed
conduits has opened the possibility to formulate and verify friction laws for plane beds that together cover a broad range of Shields numbers (0.8 < h b < 25). The verification of friction and lift theories, the development of which is currently in progress, will require measuring techniques that can sense important parameters of slurry flow in smaller control volumes (adjacent to a pipe wall) than is currently possible. Notation cv local volumetric solids concentration in pipe crosssection Cv mean volumetric solids concentration in pipe cross-section C vb mean volumetric solids concentration in bed (spatial) C vd delivered C v C vi spatial C v C vmax maximum volumetric solids concentration in bed (spatial) d 50 median particle diameter [m] D pipe diameter [m] fb Darcy –Weisbach friction coefficient for the top of bed fl Darcy – Weisbach friction coefficient for liquid flow over pipe wall FL hydrodynamic off-the-wall force (lift force) [N] FW submerged weight of particle in liquid, (S s 1) q lgd 350p / 8 [N] g gravitational acceleration [m/s2] Il hydraulic gradient of liquid flow Im hydraulic gradient of slurry flow ks characteristic grain roughness height (by Nikuradze) [m] K proportional coefficient in Gillies – Shook equation for solids shear stress KB proportional coefficient in Bagnold’s equation for solids shear stress R hb hydraulic radius of area associated with the top of bed [m] Re Reynolds number, V mDq l / l l Sm relative density of slurry, q m / q l Ss relative density of solids, q s / q l u *b shear velocity at the top of bed [m/s] u *l shear velocity at the wall of pipe, V m( f l / 8)0.5 [m/s] v local velocity of liquid vt terminal settling velocity of solid particle [m/s] v+ dimensionless local velocity, v / u *l Vm mean velocity of mixture in pipe cross-section [m/s] V ma mean velocity of mixture above contact bed [m/s] w* dimensionless terminal settling velocity, v t / ((S s 1)gd 50)0.5 y vertical distance from pipe wall [m] y+ dimensionless distance from pipe wall, yu *lq l / l l b angle defining position of top of bed [rad]
V. Matous˘ek / Powder Technology 156 (2005) 43 – 51
hb k ll ls n ql qs r swC sb s lw s sB s sw s swC
particle mobility number (Shields number), u *b2 / ((S s 1)gd 50) linear concentration of solids (by Bagnold), 1 / ((C vmax / C vi)0.33 1) dynamic viscosity of liquid [Pa s] mechanical friction coefficient of solids against pipe wall lift parameter (by Wilson and Sellgren), dv + / dy +( d2v + / dy + 2) density of liquid (water) [kg/m3] density of solid particle [kg/m3] solids normal stress exerted by sliding particles at pipe wall [Pa] shear stress at the top of bed [Pa] liquid shear stress at pipe wall [Pa] solids shear stress in sheared slurry in inertial regime (by Bagnold) [Pa] solids shear stress exerted by colliding particles at pipe wall [Pa] solids shear stress exerted by sliding particles at pipe wall [Pa]
Abbreviations DEM Discrete Element Method DUT Delft University of Technology LDA Laser Doppler Anemometry MRI Magnetic Resonance Imaging PIV Particle Image Velocimetry SRC Saskatchewan Research Council References [1] Matousek, Pressure drops and flow patterns in sand-mixture pipes, Experimental Thermal and Fluid Science (26) (2002) 693 – 702. [2] M. Simkhis, D. Barnea, Y. Taitel, Dunes in solid – liquid flow in pipes, in: J.F. Richardson (Ed.), Proc. 14th Int. Conf. on Hydrotransport, 1999, pp. 51 – 61. [3] L. Pullum, L.J.W. Graham, The use of magnetic resonance imaging (MRI) to probe complex hybrid suspension flows, in: J. Sobota (Ed.), Proc. 10th Int. Conf. on Transport and Sedimentation of Solid Particles, Wroclaw, Poland, 2000 (4 – 7 September), pp. 421 – 433. [4] F.J. Pugh, K.C. Wilson, Velocity and concentration distributions in sheet flow above plane beds, Journal of Hydraulic Engineering 125 (2) (1999) 117 – 125. [5] K.C. Wilson, Slip point of beds in solid – liquid pipeline flow, Journal of the Hydraulics Division 96 (HY1) (1970) 1 – 12. [6] K.C. Wilson, M. Streat, R.A. Bantin, Slip-model correlation of dense two-phase flow, Proc. 2nd Int. Conf. on Hydrotransport, BHRA, Warwick, UK, 1972 (20 – 22 September), pp. B1-1-10. [7] A.C. Korving, High-concentrated fine-sand slurry flow in pipelines: experimental study, in: N. Heywood (Ed.), Proc. 15th Int. Conf. on Hydrotransport, BHRG, Banff, Canada, 2002 (3 – 5 June), pp. 769 – 776. [8] V. Matousek, Distribution and friction of particles in pipeline flow of sand – water mixtures, in: A. Levy, H. Kalman (Eds.), Handbook of Conveying and Handling of Particulate Solids, Elsevier, 2001, pp. 465 – 471. [9] F. Stienen, V. Matousek, C. van Rhee, Simulation of solids behaviour at wall of circular slurry pipe using DEM, Proc. 12th Int. Conf. on
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