Numerical Prediction of Particle Distribution of Solid-Liquid Slurries in Straight Pipes and Bends

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1 Engineering Alications of Comutational Fluid Mechanics ISSN: (Print) X (Online) Journal homeage: htt:// Numerical Prediction of Particle Distribution of Solid-Liquid Slurries in Straight Pies and Bends Gianandrea Vittorio Messa & Stefano Malavasi To cite this article: Gianandrea Vittorio Messa & Stefano Malavasi (2014) Numerical Prediction of Particle Distribution of Solid-Liquid Slurries in Straight Pies and Bends, Engineering Alications of Comutational Fluid Mechanics, 8:3, , DOI: / To lin to this article: htts://doi.org/ / Coyright 2014 Taylor and Francis Grou LLC Published online: 19 Nov Submit your article to this journal Article views: 347 View related articles View Crossmar data Citing articles: 3 View citing articles Full Terms & Conditions of access and use can be found at htt:// Download by: [ ] Date: 24 November 2017, At: 00:52

2 Engineering Alications of Comutational Fluid Mechanics Vol. 8, No. 3, (2014) NUMERICAL PREDICTION OF PARTICLE DISTRIBUTION OF SOLID- LIQUID SLURRIES IN STRAIGHT PIPES AND BENDS Gianandrea Vittorio Messa* and Stefano Malavasi Dit. I.C.A., Piazza Leonardo da Vinci, 32, Milano, Italy * (Corresonding Author) ABSTRACT: Turbulent solid-liquid slurry flows in ies are encountered in many engineering fields, such as mining. In articular, the distribution of the solids is a serious concern to engineers, but its determination involves considerable technical and economic difficulties. A two-fluid model for the numerical rediction of this arameter is resented. The model is robust and numerically stable, and requires relatively short comuter time to rovide a converged steady-state solution. The novelty of the roosed model and its better erformance comared to similar ones reside in the method of accounting for some ey hysical mechanisms governing these flows, namely turbulent disersion, interhase friction, and viscous and mechanical contributions to friction. The model is first validated by comarison with many exerimental data available in literature regarding the horizontal ie case over a wide range of oerating conditions: delivered solid volume fraction between 9 and 40%; slurry velocity between 1 m/s and 5.5 m/s; and ie diameter between 50 and 160 mm. A further comarison was erformed with resect to recent exeriments concerning a horizontal 90 bend. Keywords: 1. INTRODUCTION two-fluid model, numerical analysis, slurries, two-hase flow, horizontal ies, ie bends Pie flows of solid-liquid mixtures in the form of slurry are commonly encountered in many alications. A significant examle is given by the slurry ielines, used to transort mineral concentrate from a mineral rocessing lant near a mine. Pressure gradient and concentration distribution have been the most serious concern of researchers, as they dictate the selection of um caacity and may be used to determine arameters of direct imortance (mixture and solid flow rates) as well as secondary effects lie wall abrasion and article degradation. The flow of solid-liquid mixtures is very comlex. Doron and Barnea (1996) identified the flow atterns that characterize the flow of slurries through horizontal ies. If the flow rate is sufficiently high, turbulence is effective in eeing all the solids susended (fully susended flow); otherwise the articles accumulate at the ie bottom and form a aced bed, either sliding (flow with a moving bed) or not (flow with a stationary bed). The transitions between flow atterns are not always so clear and they are usually identified by ost-rocessing analysis of the measured data in terms of solid volume fraction rofile and ressure gradient (Albunaga, 2002). Numerous exerimental investigations have been carried out to determine the main features of slurry flows in ies. Almost all of them concern the case of horizontal ies; the disersed hase is usually sand (Roco and Shoo, 1983; Colwell and Shoo, 1988; Shaan et al., 2000; Matouse, 2000 and 2002; Gillies et al., 2004; Sudarnov et al., 2004; Kim et al., 2008), but sherical glass beads (Kaushal and Tomita, 2003 and 2007; Kaushal et al., 2005 and 2012), ash (Kumar et al., 2003), and solid nitrogen articles (Jiang and Zhang, 2012) have also been considered. Only few studies were focused on vertical ies (Shoo and Bartosi, 1994) and bends (Hsu, 1981; Turian et al., 1998; Kaushal et al., 2013). The distribution of the solids is very hard to determine exerimentally. Local values of solid volume fraction can be measured by isoinetic robe samling, but these techniques may roduce significant errors near both the ie wall (Nasr-el-Din et al., 1984) and the ie axis (Colwell and Shoo, 1988). More accurate results but with uncertainties of a few ercent are obtained using exensive gammaray density gauges, which are used to determine chord-average values of solids concentration. The mean volumetric concentration of the slurry is characterized in different ways by researchers. Kaushal and Tomita (2003 and 2007) and Kaushal et al. (2005) considered an overall-area average concentration, evaluated by integrating the local concentration rofile measured by an isoinetic samling robe. Matouse (2000 and 2002) measured the delivered solid volume Received: 31 Jan. 2013; Revised: 25 Feb. 2014; Acceted: 5 Mar

3 fraction in the ieline by a counter flow meter. Other authors (Shaan et al., 2000; Gillies et al., 2004) have made reference to a mean in-situ volume fraction, obtained by adding weighted quantities of solids to the loo, whose volume was nown. In all cases, the uncertainty about this arameter must be considered when maing reference to literature data. In the ast simlified models have been develoed based on a global formulation to redict the ressure gradient of slurries in horizontal ies for all the flow configurations: equivalent liquid models for fully-susended flow (Matouse, 2002; Pecer and Helvaci, 2008), two-layer models for flows with a moving bed (Gillies and Shoo, 2000; Gillies et al., 2004; Doron et al., 1987), three layer models for flows with a stationary deosit (Doron and Barnea, 1993 and 1995; Matouse, 2009). Imroved versions have been roosed to account for the resence of multi-sized articles (Kumar et al., 2003), the influence of article shae (Shaan et al., 2000), the additional stresses due to article-wall interactions (Pecer and Helvaci, 2008), and the reulsion of articles from the wall observed under certain conditions (Wilson et al., 2010). Using those models, the major losses in horizontal ies can be estimated easily and the redictions agree with the exerimental evidence over a wide range of oerating conditions. Therefore, these models reresent a very owerful tool for most engineering alications. However, their global formulation maes them unsuitable for redicting the solids concentration distribution as well as for alication to more comlex flows; meeting these needs requires the develoment and validation of distributed models. CFD has been used to investigate slurry flows in ies, mostly with regard to the horizontal ie case. Anyway, the develoment of a model that is both reliable over a wide range of flow conditions and comutationally economical therefore attractive to engineers is a goal which has not been comletely reached yet. The majority of existing CFD models emloys an Eulerian-Eulerian aroach, since Eulerian- Lagrangian models are not alicable to dense mixtures due to their excessive comutational cost. Some worers studied the roblem by means of the Algebraic Sli Model (ASM), which solves the momentum equation for the mixture rather than for both hases, thereby saving comutational time. However, the ASM assumes that local equilibrium is achieved between the hases over short satial length scales. Therefore, it can be used only for very low values of the Stoes number. Also when alicable, the ASM roved inadequate to estimate the ressure dro even for fully-susended flows in straight ies (Kaushal et al., 2012), and it does not seem very accurate in redicting the solid volume fraction distribution (Ling et al., 2003; Kaushal et al., 2012; Patha, 2011). Other authors made use of an Eulerian two-fluid model with closures derived either from emirical or semi-emirical relations (Chen, 1994), or from inetic theory of granular flow (KTGF) (Chen et al., 2009; Eambara et al., 2009; Lahiri and Ghanta, 2010; Kaushal et al., 2012 and 2013). Anyway, even for straight ie flows, the existing two-fluid models show some roblems which may comlicate their alication to more comlex flows of engineering interest, such as those through bends and ieline fittings. The first imression is that these models are easily suscetible to numerical instabilities, which often result in solutions characterized by non-hysical asymmetry (Kaushal et al., 2012) or oscillations (Lahiri and Ghanta, 2010). In some cases, the simulations are very time-consuming; for examle, Eambara and co-worers (2009) attained a stable steady-state solution erforming a U-RANS simulation and then averaged the solution over a considerable time interval. A similar rocedure may not be easily alicable when dealing with comlex geometries, since the calculation time would robably become rohibitively exensive. In other cases, the validation of these models with resect to the exerimental evidence is often rather oor, in the sense that the comarison is either limited to a few flow conditions (Chen et al., 2009; Kaushal et al., 2012) or highlights a occasionally excellent caacity of the model to describe adequately the main features of the flow (Lahiri and Ghantha, 2010). Within the flow conditions commonly encountered in slurry ielines, the recent wor of Kaushal et al. (2013) seems the only alication of a two-fluid model to a more comlex flow configuration (i.e. through a 90 bend in a horizontal ie), with the redictions comared with exerimental data. In the resent wor a mathematical model is resented for the numerical rediction of the article distribution of solid-liquid slurry flows in ies, which is based on an Euler-Euler aroach that uses the Inter-Phase Sli Algorithm (IPSA) of Salding (1980). The roosed model shows comarable or better agreement with the exerimental evidence than similar models (Eambara et al., 2009; Lahiri and Ghanta, 2010; 357

4 Kaushal et al., 2012 and 2013), and it also overcomes the main limitations inferred from insection of these earlier aers, namely suscetibility to numerical instability and high comutational cost. In fact, the new model requires relatively short comuter time to attain a converged steady-state solution and is caable of roviding a numerical solution without nonhysical asymmetries or oscillations. The novelty of the roosed model, which is the basis for its good erformance, resides in the combined use of modelling strategies reviously develoed but never emloyed simultaneously to the flows considered in this aer: hase diffusion fluxes are introduced in all conservation equations to reroduce the effect of the turbulent disersion of articles; the resence of other articles on the interfacial momentum transfer is taen into account by considering their effect on a mixture viscosity; a wall function is emloyed to model the viscous (due to the fluid) and mechanical (due to the articles) contributions to the wall shear stress. The model is considerably simler and solves one transort equation fewer than those based on the KTGF. In the interests of guaranteeing the widest ossible alicability, the model redictions of the concentration distribution are validated with resect to various sets of exerimental data from the literature. The measurements from Roco and Shoo (1983), Shaan et al. (2000), Matouse (2000 and 2002), and Gillies et al. (2004) allow establishing the redictive caacity of the model in the horizontal ie case over different flow configurations (fully-susended flow, and flow with a moving bed) and a large range of oerating conditions: delivered solid volume fraction between 9 and 40%; article size between 90 and 520 μm; slurry velocity between 1 m/s and 5.5 m/s; ie diameter between 50 and 160 mm. Finally, the model was alied to a more comlex flow, which is that through a 90 bend in a horizontal ie, and the redictions were successfully comared with the exeriments from Kaushal et al. (2013). The uncertainties of both comutations and measurements are discussed when comaring the numerical results with exerimental data. 2. MATHEMATICAL MODEL 2.1 Conservation equations The two-hase flow is reresented by using an Eulerian aroach in which both hases are treated as interenetrating continua. The flow is assumed to be statistically steady in the sense that Reynolds-averaging has been alied and so the continuity equation for hase C, taes the following form: U D (1) where the subscrit is a hase indicator arameter which is equal to C for the carrier fluid and for the articles. Moreover is the volume fraction; is the density; U is the velocity vector; and D is a hase diffusion coefficient, which aears in the hase diffusion term that reresents the turbulent flux associated with correlations between fluctuating velocity and volume fraction. The hase diffusion fluxes are modeled in terms of a gradient diffusion aroximation with the hase-diffusion coefficient D given by: tc, D (2) where tc, is the turbulent inematic viscosity of the carrier fluid hase, determined by turbulence modeling; and is the turbulent Schmidt number for volume fractions. The turbulent Schmidt number for volume fractions is not well established, in the sense that no single constant value of σ α can be used in the numerical simulations to match the various sets of exerimental data (Shirolar et al., 1996), but rather revious worers (Chen, 1994; Chen et al., 2011) have found that different constant values are needed for different cases. These values tyically fall in the range of 0.2 to 0.9. In the resent wor, a constant value was used in the simulations, and as will be discussed later, the choice of a unique value rocured good overall agreement with the various sets of exerimental data. The resence of hase diffusion fluxes in all conservation equations, which has the advantage of romoting numerical stability, distinguishes the resent model from similar ones alied to slurry flows. The mean global continuity is given by the equation that states that the two volume fractions must sum to unity. The momentum equation for hase C, is: UU P T, T t, g (3) M DU where P is the ressure, shared by the hases; g is the gravitational acceleration; T, and T t, are the viscous and turbulent stress tensors resectively; and M is the generalized drag force er unit volume, which will be discussed later. 358

5 The stress tensors are given by: T 2 D, T 2 D, l, t, t, (4) where l, and t, are the laminar and turbulent inematic viscosity of hase resectively, and D is the deformation tensor, equal to: D 0.5 U U (5) where the suerscrit + indicates that the transose of the dyadic U is taen. Two viscosities aear in Eq. (4) for the article hase, namely, the turbulent eddy viscosity t,, and the laminar viscosity l,. The former is determined from a turbulence model, as will be discussed later. The latter is commonly associated with the inter-granular stresses which characterize bed flows (Chen et al., 2011); the model of Ahilan and Sleath (1987) for l, was briefly exlored but finally l, was set to 0 after discovering the negligible influence of the term T, on the redictions of the solids volume fraction for all the flow conditions simulated. This can be exlained by the fact that the laminar viscosity of the articles is small comared to the eddy viscosity, excet very close to the ie bottom, where however, the interfacial momentum transfer term dominates. The interfacial momentum transfer term accounts for the momentum transfer between hases, and is given by stationary drag, lift, added mass, history and other forces (Ishii and Mishima, 1984). The two-fluid model reresents the turbulent disersion of articles by means of a turbulent diffusion term in the hasic continuity equation (Eq. (1)), and so an exlicit turbulent-disersion force term does not aear in the momentum equation. A literature review revealed that the history force is negligible for the flows considered here (Chung and Troutt, 1988), and therefore was not included in this model. Under the assumtion of mono-disersed sherical articles, the interfacial momentum transfer term is given by: 6 C 3 d l vm d M M F F F (6) where Fd, Fl, and Fvm are the drag, lift, and virtual mass forces, calculated resectively as follows: 2 1 d Fd = CdC U UC 2 4 U U C (7) 3 Fl = ClCd U UC U C (8) 3 4 d Fvm = CvmC 3 8 U U UC U C (9) where d is the article diameter; C d is the drag coefficient, which will be discussed later; C l is the lift coefficient; and C vm is the virtual mass coefficient. As er the indications of Kaushal et al. (2012), both C l and C vm were set equal to 0.5. The drag coefficient is given by the well nown Schiller and Naumann (1935) formula: Cd max 10.15Re, 0.44 (10) Re in which Re is the article Reynolds number. The use of alternative correlations (Clift et al., 1978; Ishii and Mishima, 1984) does not seem to have significant imact on the results, as they do not differ very much from Eq. (10) in the range of Re considered. Following a well-established aroach (Barnea and Mizrahi, 1973; Ishii and Mishima, 1984) to account for the resence of other articles, the article Reynolds number is defined as Re = ρ Cd Ur /μ m, where μ m is the viscosity of the mixture. Several correlations for the mixture viscosity are available in literature (Clift el al., 1978). They are basically emirical, or semi-emirical, and they deend on arameters that account for the shae and size distribution of the articles. In the resent wor use is made of the Mooney (1951) formula: m Cl, C ex 1 / (11) m in which the two fitting arameters are the maximum acing concentration α m and the intrinsic viscosity [η]. The former accounts for the shae and size distribution of the articles, as well as the shear rate (Pecer and Helvaci, 2008); whereas the latter accounts for article shae. For the first time, the mixture viscosity aroach is emloyed in a two-fluid model for the simulation of slurry flows in ies. In articular, the asymtotic behaviour of the viscosity of the mixture, which tends to infinity as the solids volume fraction aroaches the maximum acing one, sets an uer limit to the concentration of articles, reventing the solids from over-acing. This avoids the need to introduce a collisional ressure term in the disersed hase momentum equations. The absence of this term contributes to the numerical stability of the resent model. Some authors have argued for the existence of a wall lubrication force in the generalized drag term to account for the reulsion of articles from the ie wall observed in some exeriments (Matouse, 2002; Kaushal and Tomita, 2007), but this effect is not considered in the resent wor. A semi-theoretical model for this force was derived 359

6 by Antal et al. (1991) for air-water bubbly flow in the laminar regime, but it roved unsuitable for slurry flows, confirming the observations of Eambara et al. (2009). Wilson and Sellgren (2003) and Wilson et al. (2010) roosed a model for the wall-lubrication force in slurry flows, but the global nature of its formulation recludes its imlementation in a CFD code. 2.2 Turbulence modeling The following modified form of the -ε model is used for turbulence modeling of the fluid hase: U C C C tc, CC l, C CC P C U C C C tc, C tc, CC l, C (12) (13) tc, CC C1P C2 C C 2 tc, C (14) in which P 2 t, C D C : UC is the volumetric roduction rate of due to the woring of the Reynolds stresses against the mean flow. The usual values of the model constants are emloyed, namely 1.0, 1.314, C 0.09, C1 1.44, and C There aears to be no simle model of general validity for evaluation of the article eddy viscosity t, in dense article flows. Nevertheless, even the simle model of t, t, C indicated by Issa and Oliveira (1997) was found to yield accurate redictions of the solid volume fraction distribution, which is the focus of this aer. 2.3 Comutational domain and boundary conditions Two validation cases are considered in this wor, namely a straight horizontal ie and a horizontal 90 ie bend. The two comutational domains are reorted in Figs. 1 and 2 resectively. In the straight ie case (Fig. 1), the flow and geometrical symmetry of the henomenon about the vertical axis (Roco and Shoo, 1983; Kaushal et al., 2005) has been exloited by solving only over one half of the ie section. In both cases, a fully-develoed turbulent flow rofile is secified at the ie inlet, with the distribution of the axial velocity, turbulence inetic energy and dissiation rate determined from Niuradse's boundary-layer theory (Schlichting, 1960) for single-hase flow in straight ies. No sli is assumed between the hases at the inlet section, therefore the same velocity distribution is alied to the fluid and the articles. 1/ N in in N 1 2N 1 2r 1 uc, z u, z Vs 1 2 N 2N D f 2 VD s f 1.82 log Re 1.64 Re lc, (15) Fig. 1 Comutational domain and boundary conditions for horizontal ie. Fig. 2 Comutational domain, boundary conditions, and reference sections for 90 bend (S1=bend inlet; S2=bend centre; S3=bend exit; S4=5D from bend exit; S5=25D from bend exit; S6=50D from bend exit). 360

7 l in in m 3 2 f 2 2r 10 2r V s D 3 D l 3/2 m 4 D 2r 2r D D (16) (17) In Eq. (15), V s is the suerficial velocity of the slurry, which is the ratio between the volumetric slurry flow rate and the ie area. The inlet volume fractions of both hases are taen as uniformly distributed. At the outlet, the normal gradients of all variables and the value of the ressure are set to zero. The length of the comutational domain in the straight ie case is 100D to ensure that fully-develoed flow conditions are attained, which tyically occurs some 50D downstream of the inlet, confirming the results of revious worers (Ling et al., 2003). For this reason 50D of straight ie were simulated ustream the 90 bend (Fig. 2) whilst the outlet boundary is located about 50D downstream the bend exit in order to analyze the develoment of the flow downstream the bend and comare the numerical results with the exerimental evidence. At the ie wall, no sli conditions are imosed to both hases, and the equilibrium wall function of Launder and Salding (1972) is emloyed to evaluate the velocity comonent arallel to the wall for the two hases u w C,, the turbulence w w inetic energy, and its dissiation rate in the near wall cells: u u w * w w * 1 u ln y E l, C (18) * uc (19) C C 3/4 y 3/2 w (20) * where u is the friction velocity of hase C, ; 0.41 is von Karman constant; E is a roughness arameter; and y is the normal distance of the first grid oint from the wall. The arameter E is considered as a function of the * roughness Reynolds number Re r ur C / l, C, where r is the equivalent sand-grain roughness height. The formula for E, develoed by Jayatillia (1969), is as follows. Em Rer Rer 1 a E a 3.7 Re r b Em b Rer 100 Re r (21) where Em 8.6 is the value of E aroriate for smooth walls, and a and b are given by: Rer a 1 2x 3x, x b , Rer (22) 2.4 Comutational methodology and consistency of numerical solution The general-urose, commercial CFD code PHOENICS was emloyed for the numerical solution of the finite-volume analogue of the mathematical described above. This was done by using the built-in Eulerian, two-fluid, Inter-Phase Sli Algorithm (IPSA) of Salding (1980) together with user-defined functions for imlementation of secific constitutive equations and boundary conditions. The calculations are erformed following the ellitic-staggered formulation in which the scalar variables are evaluated at the cell centers and the velocity comonents at the cell faces. Central differencing is emloyed for the diffusion terms, while the convection terms are discretized using the hybrid differencing scheme of Salding (1972) for the straight ie case and the MINMOD scheme (Versteeg and Malalaseera, 2007) for the 90 bend. The finite-volume equations are solved iteratively by means of the SIMPLEST and IPSA algorithms of Salding (1980). The calculation rocedure is organized in a slab-by-slab manner, in which all the deendent variables are solved at the current slab before the solver routine moves to the next slab. The PHOENICS solver was run until the sum of the absolute residual sources over the whole solution domain is less than 1 er cent of reference quantities based on the total inflow of the variable in question. An additional requirement is that the values of the monitored deendent variables at a selected location do not change by more than 0.1% between subsequent iteration cycles. The consistency of the numerical solution with resect to the domain discretization is now discussed for the two scenarios simulated. A cylindrical-olar structured mesh was emloyed 361

8 in the straight ie case. A grid sensitivity study was erformed to determine the otimum discretization of the domain by reference to the redicted solids volume fraction rofiles. For this urose the following case was taen from Roco and Shoo (1983): ie diameter = 50.7 mm; article diameter = 520 μm; delivered solid volume fraction = ; slurry suerficial velocity = 2.00 m/s. The ie is assumed hydraulically smooth ( r 0 ). This case exhibits a high degree of slurry stratification which is exected to enhance the effect of mesh resolution. Three different meshes were emloyed, as follows: 9 angular by 12 radial by 85 axial (Grid 1), 12 by 19 by 126 (Grid 2), 15 by 30 by 200 (Grid 3), and 20 by 40 by 300 (Grid 4) cells. For consistency with measurements made using γ-ray density gauges, the redicted solids volume fraction at a distance y from the ie bottom is evaluated by using the chord-average value, i.e: x chord 1 ( y) ( x, y) dx x (23) 0 where x accounts for the horizontal variation of concentration in the ie cross-section (Fig. 1). The redicted volume fraction rofiles for each grid are shown in Fig. 3. The solutions obtained on the three finest meshes (Grids 2, 3 and 4) are very close to each other, whilst that for Grid 1 deviates slightly from the finer grid solutions. The Grid 3 solution is considered adequate for comarison with the exeriments. For each vertical location y, the difference chord chord,4 ( y),3 ( y) - in which chord,3 and chord,4 are the values obtained on Grids 3 and 4 resectively - is assumed an indicator of the effect of further mesh refinement. The quantity chord chord,4 ( y),3 ( y) was found to be lower than for all y/d, less than the error usually associated with the measurements. The alication of the method roosed by Eça and Hoestra (2006) for estimating the grid discretization error confirmed the reliability of Grid 3. A curvilinear structured mesh in body-fitted coordinates was instead emloyed in the 90 bend case. The grid indeendence study was erformed with resect to the following flow condition: article diameter = 2645 g/m 3 ; delivered solid volume fraction = ; slurry suerficial velocity = 2.67 m/s; ie roughness = mm. The four meshes used consisted of: 300 slabwise by 300 axial (Grid 1), 600 by 350 (Grid 2), 1040 by 450 (Grid 3, details of which are shown in Fig. 4), and 1170 by 500 (Grid 4) cells. For consistency Fig.3 Fig.5 Solid volume fraction rofiles calculated on three different grid levels for straight ie case. Fig.4 Details of mesh for 90 bend. Vertical solid volume fraction rofiles at distance of 5D downstream bend exit (Section S4 in Fig. 2) calculated on different grid levels. The Grid 3 solution has error bars corresonding to the grid discretization error estimated by the method of Eça and Hoestra (2006). 362

9 with the samling robe measurements of Kaushal et al. (2013), the redicted solids volume fraction at a distance y from the ie bottom is now evaluated by using the value along the vertical diameter, i.e.: axis ( y) 0, y (24) The redicted volume fraction rofiles for each grid at a distance of 5D downstream the bend exit (Section S4 in Fig. 2) are shown in Fig. 5, and indicate that Grid 3 is caable to rocure a consistent numerical solution, the difference axis axis,4 ( y),3 ( y) being less than for all y. In Fig. 5, the Grid 3 solution has error bars corresonding to the grid discretization error estimated following the method of Eça and Hoestra (2006). Grid 3 was used in the simulations. 2.5 Governing terms and calibration of numerical constants In the two-fluid model different emirical submodels are emloyed for the inter-hase rocesses and closure of the various turbulence correlations. These submodels contain adjustable coefficients which are in ractice treated as calibration arameters, their values being determined by matching comutations with exeriments. The submodels which mainly affect the solid volume fraction distribution refer to two main terms. The former is the hase diffusion of each conserved roerty ( 1 for mass conservation), equal to D, in which the hase-diffusion coefficient D is given by Eq. (2). The hase diffusion, besides maing the model numerically more stable and thereby contributing to its seed in attaining convergence, has a smoothing effect on the solids concentration rofile, reducing its variation along the vertical direction. The imact of hase diffusion is controlled by an emirical arameter, which is the turbulent Schmidt number for volume fractions which aear in Eq. (2). A sensitivity analysis to quantify the influence of on the solid volume fraction rofile was erformed. The results are deicted in Fig. 6 for horizontal ie flows with very different flow conditions, both taen from Roco and Shoo (1983): A) ie diameter = 51.5 mm; article diameter = 165 μm; article density = 2650 g/m 3 ; delivered solid volume fraction = 0.09; slurry suerficial velocity = 3.78 m/s; B) ie diameter = 50.7 mm; article diameter = 520 μm; article density = 2650 g/m 3 ; delivered solid volume fraction = ; slurry suerficial velocity = 2.00 m/s. In both cases, the ie is regarded is hydraulically smooth. It is observed that a reduction of results in a flatter rofile as a consequence of the increase in diffusivity. The latter term which most influences the solid volume fraction rofile is the interfacial momentum transfer (Eq. (6)) and, articularly, the drag force (Eq. (7)). As already noticed, the definition of the article Reynolds number with resect to the viscosity of the mixture m, instead of that of the fluid hase C, uts an uer limit to the solids concentration due to the asymtotic trend of m (Eq. (11)). The mixture viscosity, as derived from the correlation of Mooney (Eq. (11)), deends on two emirical arameters, the and the maximum acing intrinsic viscosity concentration m. The effect of and m on Fig. 6 Effect of turbulent Schmidt number for volume fractions on solid volume fraction rofiles for two flow conditions (: exeriments from Roco and Shoo (1983); : 0.3; : 0.5 ; : 0.7 ; : 0.9 ). 363

10 Fig. 7 Effect of arameters [η] and m in mixture viscosity correlation of Mooney (Eq. 11) on solid volume fraction rofiles for two flow conditions (: exeriments from Roco and Shoo (1983); continuous lines: redictions). The deendence on [η] is studied for m 0.7 ; that on m for [η] = 2.5. Table 1 Flow conditions considered for comarison. Reference System D [mm] r / D [-] d [ m] C [%] V [m/s] s 3 [g/m ] Roco and Shoo (1983) Pie Gillies et al. (2004) Pie Matouse (2000 and 2002) Pie Shaan et al. (2000) Pie Kaushal et al. (2013) 90 Bend the solid volume fraction rofile is analyzed one at a time in Fig. 7 for the two above mentioned flow conditions from Roco and Shoo (1983). In articular, the results highlight that m affects the behaviour of the mixture only at high concentration, therefore as a direct effect on the revention of article over-acing. As already mentioned, this avoids the need to introduce a collisional ressure term in the momentum equation of the solid hase, further romoting the numerical stability of the model and the raid achievement of a converged solution. Our investigations showed that a unique combination of the three emirical arameters 0.7, 2.5 and m 0.7 (all within the range of variability found in literature) roduced estimations of the solids concentration rofile in good agreement with the exerimental evidence for all the flow conditions considered in the resent wor, referring to both the horizontal ie case and the 90 bend case. This feature, together with the numerical stability and the relatively short comuter time required, contributes to mae the model articularly useful for the alications. 3. RESULTS The numerical model is validated by comaring the results of the simulations against exerimental data reorted by the authors listed in Table 1. This 364

11 table rovides the flow conditions for each set of measurements. As already mentioned, two scenarios have been considered, namely a straight horizontal ie and a 90 bend, which will be discussed in searate sub-sections. 3.1 Straight horizontal ie The redictions of the model have been comared with exerimental data for 26 flow conditions, summarized in Table 2, which differ in terms of ie diameter, ie roughness, article size, slurry suerficial velocity, and delivered solid volume fraction. Each set of measurements allows an assessment of the model's caability for reroducing the influence of secific features affecting the distribution of the solid volume fraction. The data of Roco and Shoo (1983) and Gillies et al. (2004) are used to investigate the effect of article diameter and slurry velocity with different delivered solid volume fraction. The measurements of Matouse (2000, 2002) rovide the oortunity to extend the validity of the model u to a ie Reynolds number Re (Eq. (15)) of about Finally, the data of Shaan et al. (2000) are used to assess the erformance of the model in case of natural sands with irregularly-shaed grains. Almost all the data refer to configurations in which all the solids are et susended, but moving bed flows have also been briefly exlored to chec how far one can go in emloying the model for the urose of achieving useful information from an engineering oint of view. Measurements of solids concentration either bul-mean or local are subjected to errors deending mainly on the instrumentation used, as discussed in Section 1. All the data reorted in Table 2 are obtained using gamma-ray absortion methods, erhas the technique which rovides the highest accuracy attainable, with errors of the order of a few ercentage oints. Since this technique allows measuring chord-average values of solid volume fraction, the comuted volume fraction rofile was evaluated by means of Eq. (23). In the exeriments from Roco and Shoo (1983) cases A1 to A5 in Table 2 the disersed hase consists of three inds of sand, with narrow size distribution and mean article diameters of 165 and 520 μm. The solids density is 2650 g/m 3, the ie diameter is about 51 mm, Table 2 Details of exerimental conditions for straight ie case (A1 to A5: exerimental data from Roco and Shoo (1983); B1 to B12: exerimental data from Gillies et al. (2004); C1 to C6: exerimental data from Matouse (2000 and 2002); D1 to D4: exerimental data from Shaan et al. (2000); * values not declared by exerimenters). Test ID D [mm] r / D [-] d [ m] [%] 3 V s [g/m ] Flow attern A * FS A * FS A * FS A * MB A * MB B FS B FS B FS B FS B FS B FS B FS B FS B FS B FS B FS B FS C * * FS C * * - C * * - C * * FS C * * - C * * MB D FS D FS D FS D FS 365

12 the slurry bul-mean velocity varies from 1.90 to 4.33 m/s, and the ie Reynolds number from between and The delivered solid volume fraction ranges from 9% to 30%. The authors did not indicate the roughness of the ie, which was assumed hydraulically smooth. No evaluation of measurement uncertainty was reorted by the exerimenters, and they did not identify the flow atterns according to the more recent classification of Doron and Barnea (1996). However, for the smallest article diameter the data suggest susended flow, whereas for the largest diameter the indications are for movingbed flow. Fig. 8 comares the comuted and measured solid volume fraction rofiles. The good agreement between comutation and measurements reveals the ability of the model to reroduce the effect of article diameter on the article distribution for slurries with different delivered solid volume fraction. The simle twofluid model roosed in this wor aears caable to redict with reasonable accuracy the solid volume fraction rofile not only in case of fully-susended flow, but also in case of moving bed flows. This indicates that the asymtotic trend of the interfacial momentum transfer term Fig. 8 Solids concentration rofiles for flow conditions A1 to A5 in Table 2 ( : exeriments from Roco and Shoo (1983); : redictions). Fig. 9 Solids concentration rofile for flow conditions B1 to B12 in Table 2 ( : exeriments from Gillies et al. (2004); : redictions). 366

13 roduces on the solid volume fraction rofile the same effect that is traditionally achieved by the collisional ressure and the tensor T,. Gillies et al. (2004) measured the concentration rofiles of sand-water slurries in a 103 mm diameter ie using gamma-ray absortion gauges, without reorting the uncertainty. The roughness of the ie is 2 nm. Two inds of sand were considered, with narrow size distribution and mean article diameter of 90 and 280 μm. In both cases the article density was 2650 g/m 3, and the slurry suerficial velocity varies from 1.33 to 5.4 m/s (corresonding to a ie Reynolds number between and ). The delivered solid volume fraction varies from about 10% to about 45%. Details of the flow conditions considered for comarison are reorted in Table 2, cases B1 to B12. From an examination of the measured data for the ressure gradient, it can be inferred that in all cases the flow is fullysusended. This is a good test for the model s ability to redict the combined effect of slurry velocity, solids loading, and article diameter in a larger ie. Fig. 9 shows comarison of the comutational and the exerimental results for the vertical distribution of the solid volume fraction. The model roduces good overall agreement with the data. The largest deviations are observed near the ie wall for the small articles at high concentration (cases B2 and B4 in Table 4), but as noted by the authors themselves these measurements are subject to the largest exerimental error. Matouse (2000 and 2002) measured the concentration rofiles of sand-water slurries in a 150 mm diameter ie using gamma-ray absortion method, quantifying a 4% uncertainty of the data. We reroduced numerically the flow conditions reorted in Table 2 (cases C1 to C6), in which the disersed hase consists of two inds of sand with narrow size distribution and mean article diameters of 120 and 370 μm. Unfortunately neither the density of the solid articles nor the ie roughness are reorted in the exeriments of Matouse (2000 and 2002); and so in the simulations the former was set to 2650 g/m 3, which is the value declared by Roco and Shoo (1983) and Gillies et al. (2004), and the ie was regarded as hydraulically smooth. The slurry suerficial velocity was 2 or 6 m/s, corresonding to a ie Reynolds number u to The delivered solid volume fraction ranged between 25 and 43%. According to the author, the flows are either fully susended or with a moving bed, but only for three cases the flow attern could be clearly identified from the lot of ressure gradient versus slurry suerficial velocity. Fig. 10 comares the comuted and measured solid volume fraction rofiles; each set of exerimental data has error bars indicating the uncertainty declared by the exerimenter. The rather good agreement between comutations and measurements confirm the reliability of the model for high ie Reynolds numbers. A closer insection reveals that for case C6 the comuted concentration is lower than the exerimental results in the lower art of the ie and higher in the uer region. However, the ressure gradient curve clearly reveals that in case C6 a moving bed of articles is formed. It is again confirmed that the roosed model is caable to rocure a rough indication of the distribution of the solid also for flows in which the inter-granular stresses lay an imortant role. Moreover, in the case of highly dense slurries lie case C6, the exerimental data are subject to the most uncertainty, and the numerical solutions are sensitive to the emirical arameters m and which aear in the mixture viscosity correlation (Eq. (11)). For the same case, the measured solids concentration rofile shows a dro near the ie bottom. This reversal is not redicted by the numerical model, which is robably due to the absence of the wall lift force discussed earlier in Section 2.1. The same limitation was reorted by Eambara et al. (2009). The aer of Shaan et al. (2000) focuses on the effect of article shae uon ressure gradient and deosition velocity, but these worers also reorted some measurements of the solid volume fraction rofiles. Shaan et al. (2000) quantified the article shae by reference to the following quantities, evaluated from hotos of the articles: the axes ratio, i.e. the ratio between the major and minor axes of an ellise circumscribing the grains; the circularity index, i.e. the ratio between the area of the article, multilied by 4π, and its erimeter. The sand articles have a very irregular shae, as can be inferred by comaring the measured values of axes ratio (1.6) and circularity (0.62) against tyical values reorted in the literature for sand grains. The mass-median value of the article size distribution curve is taen as the characteristic article diameter. The flow conditions reorted in Table 2 (cases D1 to D4) were reroduced to chec the behaviour of the model for the case of irregularly-shaed sand articles. The ie diameter (158.5 mm), delivered solid volume fraction (15% or 35%), slurry suerficial velocity (1.4 or 3 m/s), article density (2655 Kg/m 3 ), and article diameter (90 μm) are all aroximately within the range 367

14 Fig. 10 Solids concentration rofiles for flow conditions C1 to C6 in Table 2 ( : exeriments from Matouse (2000 and 2002); : redictions). Fig. 11 Solids concentration rofiles for flow conditions D1 to D4 in Table 2 ( : exeriments from Shaan et al. (2000); : redictions). covered by the earlier simulations. The numerical and exerimental results are comared in Fig. 11. The agreement between the numerical redictions and the exerimental evidence is fairly good; inadequacies of the numerical model esecially in the wall boundary conditions for the solid hase and measurements error can both contribute to the discreancies observed near the ie bottom for case D2. The results seem to indicate that the numerical model is a suitable tool for redicting the solid volume fraction distribution roduced by sands with a very irregular shae. 3.2 Horizontal 90 bend The redictions of the model have been comared with the recent exeriments of Kaushal et al. (2013) regarding a horizontal 90 bend. Among the different flow conditions considered by these authors, reference is made to those characterized by delivered solid volume fraction of 8.82%, within the range of interest in this wor. The exerimenters used a samling robe to measure the volume fraction rofile of the slurry in the 53 mm diameter ie along the vertical axis at distances of 5D, 25D, and 50D downstream the bend exit (referred to as sections S4, S5, and S6 in Fig. 2, resectively). They quantified a 2% uncertainty of the data. The solid hase consists of silica sand articles with narrow size distribution and mean article diameter of 450 μm. The slurry suerficial velocity is 1.78, 2.67, or 3.56 m/s. The flow within the bend is much more comlex comared to that through the straight ie since secondary motions develo in the crosswise direction. These motions affect significantly the distribution of the solids and, as a consequence, the volume fraction distribution is no longer constant over horizontal lanes. This is evident in Fig. 12, which shows the color lot of the solids volumetric concentration over the six cross sections highlighted in Fig. 2. As shown in Fig. 13, the redictions of our model generally agree with the exerimental data available. Greater discreancies are observed for the highest suerficial velocity (3.56 m/s), where the model seems to underestimate the quantity of articles dragged by the secondary motions. This is 368

15 of the secondary motions on the solids for the lowest velocities. Fig. 13 also reveals that our model tends to overestimate the solids volume fraction close to the ie bottom far from the bend exit. It is exected that the redictions could be imroved if the effect of the wall lubrication force, for which a suitable model does not exist at resent, were accounted for. Fig. 12 Solid volume fraction distributions on sections S1 to S6 in Fig. 2 for delivered solid volume fraction of 8.82%. Left hand side is the outer wall of the ie. oosite to the KTGF based model of Kaushal et al. (2013), which tends to over-enhance the effect 4. CONCLUSIONS A mathematical model has been described for the estimation of solid volume fraction distribution of solid-liquid slurry flows in horizontal ies and bends. The model is based on an Eulerian- Eulerian aroach, and uses the Inter-Phase Sli Algorithm (IPSA) of Salding (1980). The novelty of this model resides in the combined use of existing modelling strategies never reviously emloyed simultaneously for the redictions of slurry flows. In order to account for turbulent disersion of articles, hase diffusion terms have been included in the hase continuity equations, together with all the conservation equations. The mixture viscosity aroach is adoted to model the effect of multile articles on the inter-hase friction. A wall function is emloyed to model the flow in the near-wall region in case of both hydraulically smooth and rough walls. All these strategies result in a model which, desite being simler that those commonly used for addressing solid-liquid slurry flows, shows comarable or even better redictive caacity. Moreover, this model is robust and numerically stable, and requires relatively short CPU time to rocure converged steady-state solutions. The model has been first validated by comarison with exerimental data for the horizontal ie case over a wide range of oerating conditions: delivered solid volume fraction between 10% and 40% by volume; mean article diameter between 90 and 520 μm; slurry suerficial velocity between 1 m/s and 5.5 m/s; and ie diameter between 50 and 158 mm. The model redictions were found to be in good agreement with the exerimental evidence for all the flow conditions considered for comarison. Even if the model is originally intended to address fully-susended flows, the results indicated its caability to rovide rough indications of the solid volume fraction distribution also in case of bed flows. At last, the comarison with the recent measurements of Kaushal et al. (2013) revealed that this model is caable to rocure reliable redictions of the article distribution downstream a 90 bend in a horizontal ie. 369

16 Fig. 13 Solid volume fraction along vertical diameter on sections S4 to S6 in Fig. 2 for delivered solid volume fraction of 8.82% (: exeriments from Kaushal et al. (2013); : redictions). ACKNOWLEDGEMENTS The authors would acnowledge the CINECA and the Regione Lombardia award under the LISA initiative, for the availability of high erformance comuting resources and suort. The authors would lie to than CHAM Ltd for suorting this research and acnowledge Dr. Randy Gillies of Sasatchewan Research Council for roviding his exerimental data. REFERENCES 1. Ahilan RV, Sleath J (1987). Sediment transort in oscillatory flow over flat beds. Journal of Hydraulic Engineering 113: Albunaga BE (2002). Slurry Systems Handboo McGraw-Hill. 3. Antal S, Lahey R, Flaherty J (1991). Analysis of hase distribution in fully-develoed laminar bubbly two hase flow. International Journal of Multihase Flow 17(5): Barnea E, Mizrahi J (1973). A Generalized Aroach to the Fluid Dynamics of Particulate Systems. Chemical Engineering Science 5(2): Chen L, Duan Y, Pu W, Zhao C (2009). CFD simulation of coal-water slurry flowing in horizontal ielines. Korean Journal of Chemical Engineering 26(4): Chen RC (1994). Analysis of homogeneous slurry ie flow. Journal of Marine Science and Technology 2(1): Chen X, Li Y, Niu X, Li M, Chen D, Yu X (2011). A general two-hase turbulent flow model alied to the study of sediment transort in oen channels. International Journal of Multihase Flow 37(9): Chung JN, Troutt TR (1988). Simulation of article disersion in an axisymmetric jet. Journal of Fluid Mechanics 186: Clift R, Grace JR, Weber ME (1978). Bubbles, Dros and Particles. Academic Press. 10. Colwell JM, Shoo CA (1988). The Entry Length for Slurries in Horizontal Pieline Flow. Canadian Journal of Chemical Engineering 66(5):