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1 Oen Archive Toulouse Archive Ouverte (OATAO) OATAO is an oen access reository that collects the work of Toulouse researchers and makes it freely available over the web where ossible. This is an author-deosited version ublished in: htt://oatao.univ-toulouse.fr/ Erints I: 5319 To link to this article: htt://dx.doi.org/ /j.seur To cite this version : availles, Aurélien and Climent, Eric and Bourgeois, Florent Fondamental understanding of swirling flow attern in hydrocyclones. (2012) Searation and Purification Technology, vol ISSN Any corresondence concerning this service should be sent to the reository administrator: staff-oatao@in-toulouse.fr

2 Fundamental understanding of swirling flow attern in hydrocyclones Aurélien availles a,b,, Eric Climent a,b, Florent Bourgeois c a Université de Toulouse, INPT, UPS, Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula, F Toulouse, France b CNRS, Fédération de recherche FERMAT, Toulouse, France c Université de Toulouse, INPT, UPS, Laboratoire de Génie Chimique, 4 allée Emile Monso BP 44362, Toulouse Cedex 4, France abstract Keywords: Hydrocyclone Swirling flow Air core Turbulence model This work is concerned with establishing and validating a hysics-based model that describes the swirling flow inside hydrocyclones. The hysics is embedded in a Comutational Fluid ynamics (CF) simulation model whose key features are resented and justified in the aer. Some features are selected in such a way that the model can eventually be used to simulate dense flow inside hydrocyclones. Nevertheless, its underlying hysics is here within validated against dilute flow conditions. The model alies a Eulerian multi-fluid modelling aroach for fluid article turbulent flows, and is comuted using the semi-industrial code NEPTUNE_CF. Simulation results are successfully comared to water slit, velocity rofiles inside the hydrocyclone and artition function measurements, either roduced using our own exerimental setu or from the literature. The work finds velocity rofiles to be the most discriminating arameter for validation of the hysics that describes the swirling flow inside the hydrocyclone. Water slit on the other hand shows no relation to the choice of turbulence model and hence cannot be used to validate a mechanistic model of the hydrocyclone. The hysics-based model resented here is the first building block towards describing and understanding hydrocyclone flow under dense regime. 1. Introduction 1.1. escrition of article searation A hydrocyclone is a size classifier used to rocess slurries. The searation mechanism is based on enhanced gravity and takes advantage of article size and density. Although the hydrocyclone may be used to different ends, this study is concerned with solid liquid searation. The slurry is tangentially injected into the cylindrical zone, which rovides a very high rotation rate to the slurry. Solid articles, with a higher density than the liquid, are efficiently dragged towards the outer wall by centrifugal forces induced by the curvature of streamlines (see Fig. 1). The centrifugal acceleration can be locally several thousand times the acceleration of Earth s gravity, tyically between 500 and 5000 g deending on dimensions and oerating conditions of the hydrocyclone (see Section 4.5 for G-force rofiles). The articulate bed that forms on the wall of the cylindrical body flows down the conical section, which ends through a narrow section tube (sigot). Corresonding author at: Université de Toulouse, INPT, UPS, Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula, F Toulouse, France. addresses: aurelien.davailles@imft.fr (A. availles), eric.climent@imft.fr (E. Climent), florent.bourgeois@ensiacet.fr (F. Bourgeois). The overflow (which carries fine and light articles) exits through a tube (vortex finder) that dis into the cylindrical body. The rogressive downward reduction in section of the cone and the continuous increase of the solid fraction yield a significant ressure dro increase, which dictates the flow through this tube. Most hydrocyclones oerate with a central air core resulting from air asiration through the sigot; this forces the fluid and lighter articles to exit through the vortex finder. It follows that the recovery of water to overflow or water slit is generally high (around 90%). It follows that the coarser articles exit through the underflow as a dense slurry Comutational Fluid ynamics: a literature overview Steing from hysical descritions of the inner workings of the hydrocyclone, as er Bradley s 1965 classic reference work [1], Comutation Fluid ynamics (CF) simulations of hydrocyclones did start in the 1990s. One of the early works in this field is that of Hsieh and Rajamani [2,3]. Initially, the axisymmetric assumtion was imosed to reduce the simulation domain to two dimensions. The simulated water flow matched closely the fluid motion measured by Laser oler velocimetry inside the hydrocyclone. Qualitatively, the redictions were correct: fluid motion in rotation downward along the wall and the resence of a strong uward central vortex. Close insection of the measurements revealed

3 1.3. Objectives of the study Fig. 1. Schematic of the hydrocyclone flow structure. the resence of asymmetrical fluid flow atterns inside the device (resulting from single injection or multile tangential injections). Consequently, the simlified symmetrical geometry was eventually surassed by truly three-dimensional simulations. Ever since this early work, three-dimensional simulations of hydrocyclones have been imroving steadily due to the continuous imrovement of CF simulation, due to imroved comuter technology, hysical models and numerical algorithms. Nevertheless, CF models of hydrocyclones are yet to make significant contributions in ractice. There remains a great deal of scoe for imroving their redictive caability, which deends mostly on the quality of the hysics used in the models. All asects of the flow cannot realistically be catured by numerical simulations due to strong turbulence, anisotroy and the three-hase nature of the flow (articles, liquid and air inside the air core). Nevertheless, the question of selecting a suitable turbulence model is an imortant issue. Over time, a number of significant turbulence models have been roosed: renormalization Grou (RNG) [4], Reynolds Stress Model (RSM) [5] and more recently the Large Eddy Simulation (LES). Because of the strong anisotroy of the flow, the RSM model must be referred over models that assume local isotroy of the Reynolds stress tensor. However, the recession of the air core is roblematic because it contributes to the global unsteadiness of the flow. Thus, fluctuating velocity (or turbulence level) rofiles are generally not accurately redicted in the central art of the flow. As shown by Slack et al. [6], LES is robably the most accurate turbulence modelling aroach. It simulates the motion of high-energy vortices at large scales and alies a turbulence model at small scales. Over the ast decade, numerous ublications on hydrocyclones have highlighted the imortance of the resence of the air core [7 12]. A suitable method for deformable interface simulations (VOF Volume Of Fluid) have been couled to a RANS turbulence model or to large eddy simulation [13 16]. A number of mature emirical models have been develoed over the years for assisting engineers with design or otimisation of hydrocyclones for industrial alications. It is the authors belief that these models have evolved to a oint where they can no longer be exected to hel achieve significant develoment in hydrocyclone technology. A better understanding of the hysics of flow inside hydrocyclones, which can eventually be embedded into existing emirical models is exected however to rovide ractitioners with new information that will hel them better the erformance of hydrocyclones. Enacted through CF simulations, sound hysics-based models of flow can give satial information about hydrodynamics, like velocity rofiles or solids concentration anywhere inside the hydrocyclone. Information about the inner workings of the hydrocyclone, which is difficult to obtain exerimentally, will eventually hel imrove our understanding of hydrocyclone erformance to the oint where ractical value can be gained from it. The aim of our work is to roose a systematic rocedure for hydrocyclones CF simulation. According to revious work described in Section 1.2, turbulence and air core modelling have to be taken into account through accurate models. Moreover, in order to simulate industrial cases with high feed solids content, articles should be considered. Unfortunately, the most accurate turbulence modelling (LES) is very costly in comuting time and requires algorithms with low numerical diffusion that ose numerical stability roblems. Also, from a hysics oint of view, the validity of LES models for twohase dense flow is still a much debated issue. The air core modelling should be done with a VOF aroach, but there is no validation to date that roves the soundness of the simultaneous use of all these simulation aroaches for multihase flow. Many questions remain oen regarding the reliability of couled simulations LES- VOF with solid articles. The work that is resented here is concerned with embedding a sound hysics-based model in a CF simulation environment that is suitable for simulation of industrial-size hydrocyclones. This imlies a degree of comromise between the accuracy of the hysics that is built into the model and the comutational seed for solving the model; this led us to choose Eulerian multi-fluid model with RANS turbulence modelling. Simulation results have been comared to exerimental data obtained on a dedicated test rig that is resented in Section 3.1. Velocity rofiles ublished by Hsieh [3] have also been used to validate the hydrodynamics inside the hydrocyclones. 2. CF model resentation 2.1. NEPTUNE_CF resentation The three dimensional numerical simulations were erformed with NEPTUNE_CF V1.07@Tlse (see [17 20]). NEPTUNE_CF is a multihase flow software develoed in the framework of the NEP- TUNE roject, financially suorted by CEA (Commissariat à l Énergie Atomique), EF (Électricité de France), IRSN (Institut de Radiorotection et de Sûreté Nucléaire) and AREVANP. The behaviour of multihase flows can be modelled using the general Eulerian multi-fluid balance equations. It may corresond to distinct hysical materials (e.g. gas, liquid and solid articles) which can be slit into different grous (e.g. water and several grous of different articles diameter); different thermodynamic hases of the same comonent (e.g. liquid and its vaour) or hysical comonents, where some of which may be slit into different grous. Assuming constant temerature, the following multi-fluid balance equations are obtained from the fundamental conservation laws of Physics, restricted to Newtonian mechanics:

4 mass conservation momentum balance These two conservation laws are written under differential form which is valid for arbitrary time and location within the continuum, excet across the interfaces between two hysical hases. At the interfaces, jum conditions derived from the continuous equations are written and integrated through source and sink terms in the equations. Equations for m fluids that can be a hysical hase or a model field of a hysical hase, are written in a symbolic coordinate-free notation. The articular Cartesian coordinate system is used only when it makes things clearer. The algorithm, based on the ellitic fractional ste method, enforces mass conservation with original ressure ste actualization Transort equations The multi-fluid mass balance equation for field k ða kq k ða k k U k;i Þ¼0 ð1þ i with a k, q k, U k, the volumetric fraction, the density and the mean velocity of hase k. The multi-fluid momentum balance equation for hase kis defined ða kq k U k;i ða k k U k;j U k;i Þ ¼ a k þ I k;i þ a k k g i T k;ij ð2þ j with P the mean ressure, I k;i the average interfacial momentum transfer, g i acceleration due to gravity and T k;ij the effective stress tensor. I k;i, which accounts for momentum transfer rate from liquid to solid hase, can be modelled using an estimate of the drag force between hases (when k = l, we refer to the liquid and k ¼ to the class of articles). I l;i ¼ I ;i ¼ a q V s F r;i with 1 ¼ 3 l s F 4 l s F l q l hc i hjv q r ji d is the viscous article relaxation time scale and it reresents the characteristic time of the interaction between fluid and articles through the drag force [21], <> the ensemble average oerator over the articulate hase. V r;i is the average of the local relative velocity and can be exressed in terms of the averaged velocity between hases and drift velocity (which is modelled). The mean drag coefficient of a single article, hc i can be written as a function of articulate Reynolds number. With several articles and in the multi-fluid aroach, the article volume fraction should be taken into account. Under these conditions, Gobin et al. [22] justifies using a combination of relations roosed by Wen and Yu [23] for dilute cases and by Ergun [24] for concentrate area regime. 8 if a > 0:3 E E >< Min C Wen&Yu ; C Ergun hc i ¼ ð3þ if a 6 0:3 >: hc Wen&Yu i with E C Ergun E C Wen&Yu ¼ ¼ 200 a þ 7 Re 3 8 if Re < 1000 >< a 1:7 24 l Re ð1 þ 0:15Re 0:687 Þ >: Re ¼ a lq l d hjv r ji l l if Re P :44a 1:7 l 2.3. Turbulence modelling Ensemble averaging is usually alied to instantaneous Navier Stokes equations in order to study industrial flows. It is convenient to analyse the flow into two arts: the mean (or average) flow field and fluctuations ð~u i ¼ U i þ u i Þ. These new equations are called Reynolds Averaged Navier Stokes (RANS) equations. The effective stress tensor, T k;ij, in Eq. (2) writes: T k;ij ¼ a k q k hu k;i u k;j iþh k;ij It contains two contributions: a collisional or molecular viscosity term H k;ij, resectively, for solid or liquid hases, and the turbulent kinetic stress hu k;i u k;j i due to turbulence or fluctuations in hase k q 2 k - k model The q 2 k - k model is an extension of the classical k- model, used for single hase flow. It is a simle and widely used turbulence model that can be couled to diserse hases (bubbles, dros or articles), but it assumes isotroy of turbulence. Using a Boussinesq hyothesis for water, the Reynolds stress tensor is exressed as follows: hq l u l;i u l;j i l ¼ l l;i l i þ 2 3 d i;j q l q 2 l þ l t m introducing l t l, the turbulent viscosity and the turbulent kinetic energy of the water q 2 l ¼ 1 hu 2 l;iu l;i i l. As roosed by Balzer and Simonin in [25]: l t l ¼ C l q 2 l C l ¼ C l " # 1 1 þ C 12 a q a l q l q 2 l s F l 1 q l 2q 2 l with C l ¼ 0:09 and C 12 = For the solid hase, the kinetic stress tensor, hq u ;i u ;j i, reresents the transort of momentum by article velocity fluctuations. The collisional stress tensor, H k;ij, accounts for transort and sink of the momentum. The constitutive relations for viscosity and diffusivity are derived in the framework of the kinetic theory of granular flows R ij - k model The Reynold s stress model (RSM) is a second order turbulence model. With RSM, the turbulent viscosity aroach has been comlemented and the Reynolds stresses are comuted directly. The exact Reynolds stress transort equation accounts for the directional effects of the Reynolds stress fields. The Reynolds stress model involves calculation of the individual Reynolds stresses, hq k u k;i u k;j i k, using differential transort equations and leads to higher comutational costs. The individual Reynolds stresses are then used to obtain closure of the Reynolds-averaged momentum equation. ð4þ ð5þ ð6þ ð7þ

5 2.4. Comutational domain The origin of the air core comes from air being sucked through the sigot that is oen to the atmoshere. The air accumulates in the area of low ressure, which forms the air core. One consequence of the resence of the air core is that it reduces the effective area available for the fluid to exit through the sigot. This reduction in section contributes to increasing the ressure dro across the hydrocyclone, which has a major effect on water slit. In this work, two ossible aroaches have been used to model the air core. A realistic one with real gas and an economical one with a gas liquid boundary condition relacing air in the middle of the hydrocyclone Real air-core The first air core modelling aroach we used consisted in letting the CF model calculate the size and osition of the air core inside the hydrocyclone. This was achieved by allowing air to flow uward through the sigot, as required by the Physics of the mass and momentum balance equations embedded into the CF model. Actually, this aroach mimics reality as far as the formation of the air core is concerned with a real oerating hydrocyclone. This required that the entire volume of the hydrocyclone be meshed, as shown in Fig. 2. The Simail software, which is associated with NEPTUNE_CF V1.07@Tlse, was used to mesh the hydrocyclone. The 3 mesh was first generated by a full rotating extrusion of a vertical slice of the cyclone. Then, in order to eliminate the singularity that such a mesh generation creates in the centre of the cyclone, the central volume of the hydrocyclone was meshed as a cylinder with a concentric mesh Air-core modelling by a vertical tube In several studies [7,11,12], the air core has been relaced by a metal rod in order to imrove searation erformances. This aroach has been reroduced but with gas liquid interface boundary conditions (shear free condition allowing sli). In this case, the air core is modelled by a fixed tube located at the centre of the hydrocyclone. The justification for using this air-core modelling aroach is that it is clearly less comuter time consuming than the first solution, which requires solving two-hase (gas liquid) flow equations over an even greater number of cells. Mesh generation roceeded as before with Simail, meshing a vertical slice of the hydrocyclone and then a 360 degree rotation. The central hollow tube can be seen in Fig Additional details about simulation conditions The inlet flow assumes uniform solid concentration. In the inlet section, the Reynolds number reaches 70,000 and fully turbulent Fig. 3. Mesh with air-core modelling by a vertical tube. conditions are used for the boundary conditions. With the tube aroach for modelling the air core, shear free boundary conditions are imosed as er a gas liquid interface (zero flux for all variables d:: ¼ 0 with cancellation of the normal velocity comonent U n ¼ 0). Outlet boundary conditions are similar for all cases dn and are adated for water and articles or for water and gas simulations. The ressure rofile is imosed dynamically by coying the internal rofile and resetting around a fixed ressure (atmosheric ressure) at both outlets, which allows secies to enter through the underflow and overflow sections if the ressure inside the device becomes lower than atmosheric ressure. With liquid gas simulation, only gas is allowed to enter the domain (which may occur at the sigot end of the hydrocyclone), whereas with liquid solid simulation, the vertical tube relaces the air core and no secies can flow in when the tube diameter is given its correct value. Standard friction functions are used for the liquid hase (law of the wall) while a zero fluxcondition on the velocity vector is du alied for articles ¼ 0. dn The flow in the narrow section of the conical art is a critical oint with meshing of the domain. Indeed, meshing the region between the air core and the sigot wall requires care. The size of the cells is almost uniform within the grid while the smallest dimension is about m. The mesh is, resectively, comosed of roughly 250,000 and 450,000 hexahedral and entahedral structured cells for the Hsieh s hydrocyclone and for our own device. At the beginning of a simulation, the hydrocyclone is full of water. uring the transient art of the simulation, water and articles are fed into the domain. Eventually, a steady-state is reached, and time averages are formed for all system variables to analyse their statistics. 3. Exerimental data Validation of the model requires access to a number of data on hydrocyclone erformance. Model validation, which is covered in Section 4, relies uon measurements of water slit, velocity rofiles inside the hydrocyclone and artition functions. We used water slit and artition functions from both our own test rig and from Hsieh [3]. Our own test rig consists of a 100 mm diameter olyurethane hydrocyclone, whereas Hsieh s used a 75 mm glass hydrocyclone. Velocity rofiles used to test the hysics of our CF model are those measured by Hsieh [3] mm diameter hydrocyclone Fig. 2. Mesh with real air-core modelling. The Neyrtec s HC100 hydrocyclone is made of interchangeable olyurethane arts, allowing various geometrical configurations to be tested (see technical secifications in Fig. 4 and in Table 1).

6 Table 1 General features of Neyrtec s HC100 hydrocyclone. Model HC 100 iameter (mm) 100 Feed rate (m 3 /h) Feed ressure (bar) Cut size ðlmþ 7 18 Material Polyurethane Overflow (mm) 33 Underflow (mm) 18 The length of the cylindrical body can be adjusted by three 100 mm extensions, and several diameter sigots (6 18 mm) are available. Only one 100 mm cylindrical extension has been installed for the data reorted here, and the 18 mm diameter sigot was used. The hydrocyclone is fed using a centrifugal um that is connected to a 1 m 3 agitated sum. Only oerating arameters and global balance (water slit and artition function) are accessible with this set-u, which is not fitted with any otical means for evaluation of air core diameter mm diameter hydrocyclone [3] The work by Hsieh and Rajamani rovides the scientific community with several sets of high quality exerimental data on the flow field inside the hydrocyclone along with the first CF model of the hydrocyclone. Not surrisingly, their exerimental set-u has served as a reference configuration in many CF studies [13,14,26 29]. The characteristics of the hydrocyclone he used are listed in Table 2. The exerimental test rig being couled to a Laser oler Velocimetry measurement system, Hsieh was able to measure many velocity rofiles along the height of the hydrocyclone, which we used to validate our CF model arameter settings (ure water, 1.1 L s 1 ). Finally, test series Nos. 7 and 8 corresond to exeriments run with water and articles. Hsieh also measured and reorted all the standard corresonding hydrocyclone erformance data, such as water recovery to underflow or artition functions. The hydrocyclone geometry we used in our CF simulation work corresonded recisely to that used by Hsieh (Cf. Table 2). The cylindrical inlet tube used by Hsieh was simulated by a rectangular tube with an equivalent surface area (see [13,28,29]), which is about 8% of the cross-sectional area of the feed chamber. A sensitivity study on the inlet geometry has shown no influence of the length of the inlet ie on hydrodynamics. The air core diameter was estimated from the exerimental velocity rofiles available in Hsieh s Ph.. aer. The air core diameter for series No. 1 (water only) is about 11 mm (88% of the sigot diameter) and 9 mm (72% of the sigot diameter) for series No. 5 (water and glycerol). were tested with both k-e and R ij -e models. The first art of our model derivation work consisted in simulating Hsieh s No. 1 series for single-hase flow, and comaring the water slit and velocity rofiles he measured with our CF model redictions. Then, sensitivity tests of the numerical methodology have been realised on the two exerimental geometries and the numerical and exerimental searation erformances have been comared with dilute slurries Water-slit For a hydrocyclone oerating under continuous rocessing conditions with fixed inlet arameters, we first comared exerimentally measured water slit values with simulation results (see Table 3). The ressure dro across the hydrocyclone is caused by the section restriction in the conical section and sigot, and by the resence of the air core. Most of the feed water flows through the vortex finder and is recovered to the overflow. For the real air-core with k-e turbulence model, the redicted behaviour of the hydrocyclone is not relevant because there is no air sucked by the underflow. This leads to a single hase flow simulation with an equivalent CPU time (9 h calculation on 24 rocessors) to that with the tube aroach. With R ij -e turbulence model, the real air core redicted is about 10.5 mm (84% of the sigot diameter, see Fig. 5). It is very similar to air core measurements done by Hsieh and described in Section 3.2. The CPU time of this two-hase flow simulation is increased by the high gas velocity at the underflow. With the tube aroach, the simulation of 2.5 hysical seconds takes about 10 h on 24 rocessors while with the air core, the simulation of one real second takes about 17 h on the same number of rocessors. Excet with the combination of the real air-core model and the k-e turbulence model, which is inaccurate, all combinations of air core and turbulence models yielded values of water slit that are close to that measured by Hsieh (Table 3). These results indicate that water slit carries little value for testing the validity of a CF model Table 3 Comarison of exerimental and numerical water-slit. Exerimental results from [3] 95.1 Real air-core and k-e model 73.7 Real air-core and R ij -e model 94.1 Vertical tube and k-e model 99.0 Vertical tube and R ij -e model 92.1 Series No. 1 (water) (%) 4. Model validation and analysis The selection of both turbulence and air-core models was carried out simultaneously, as both air-core modelling techniques Table 2 imensions of Hsieh s hydrocyclone. Cyclone diameter 75 mm Feed diameter 25 mm Vortex finder diameter 25 mm Sigot diameter 12.5 mm Length of the vortex finder 50 mm Length of the cylindrical section 75 mm Cone angle 20 ( ) Length of the conical section 186 mm Fig. 4. Geometric features of Neyrtec s HC100 hydrocyclone.

7 From these results, we can conclude that velocity rofiles are the right arameters for selection of a suitable turbulence model, here R ij -e. In addition, since redictions with the real air-core or with the vertical tube are very close, we can also conclude that with the aroriate turbulence model modelling the air core as a constant diameter tube is suited for modelling the hysics of the hydrocyclone Air-core: sensitivity to the tube diameter In this section, we tested the sensitivity of the CF model to the tube aroximation of the air core for both our own hydrocyclone and that of Hsieh. Fig. 5. Gas volume fraction with real air-core modelling and R ij -e model. as long as the air core diameter is correctly redicted or set, as water slit is a simle function of the ressure dro, which bears no relationshi with the local turbulence level and mean flow structure Velocity rofiles Axial and tangential velocity rofiles of the series No. 1 are resented in Fig. 6. Although we saw earlier that water slit redictions bore no relation to turbulence models, velocity rofiles on the other hand are very much deendent on the turbulence model used in the simulations. The R ij -e model reroduces accurately axial and tangential velocity evolutions, but the k-e model underestimates axial velocity in the centre of the cyclone and yields a oor rediction of tangential velocity resulting in a oor rediction of the ressure distribution in the central art of the device. That is the hysical reason of the absence of the air core with the real air core modelling mm diameter hydrocyclone The question of choosing the right value of the diameter used to model the air core for any given simulation is an imortant issue. For the model validation above, the tube diameter was given a constant value taken from Hsieh s velocity rofiles, although variations of diameter could be read along the vertical axis of the hydrocyclone (Fig. 5). The sensitivity of the searation to the value of this diameter is an imortant issue for the relevance of simulation results to rediction of searation erformance. The diameter of the tube was varied from 70% to 95% of the sigot diameter with the R ij -e turbulence model. The air core diameter for series No. 1, estimated through exerimental rofiles from Hsieh s Ph.. thesis was about 11 mm (88% of the sigot diameter). Influence of air core diameter modelled by a rigid tube on water slit can be seen in Fig. 7 under dilute conditions (1 wt% = 0.4% of silica). With too small a diameter, say less than 70% of the sigot diameter, comutations lead to very low ressure at the exit (air suction). This leads to water being sucked through the sigot and reorting to the overflow, which exlains the redicted increase of water recovery to overflow with small tube diameters. Increasing the diameter of the tube reduces this water suction u to a oint where it becomes nil. This oint, which corresonds to the right of the minimum water slit value that is found in Fig. 7, is the correct tube diameter. As the diameter increases beyond this value, the surface area available for the underflow becomes too small, associated with a strong increase of the ressure dro, and the water recovery to overflow increases raidly. Partition curves associated with various tube diameter values within the range shown in Fig. 7 were calculated to assess the influence of the air core diameter on searation erformance. We found that searation curves were only marginally sensitive to this arameter in the range 80 95%. Hence, the choice of tube diameter does not aear to be critical for rediction of hydrocyclone searation erformance from CF simulation. Fig. 6. Series No. 1: axial and tangential velocity rofiles. Fig. 7. Influence of tube diameter on water-slit.

8 mm diameter hydrocyclone We have alied our CF model without modifications to data of our own using a ilot test rig resented in Section 3.1. The air core has been modelled by a vertical tube whose diameter is estimated through several simulations with different tube diameters, as exlained in Section It was eventually set equal to 90% of the sigot diameter (16.2 mm) (see Fig. 8) Two-hase flow simulations The addition of articles to our numerical simulations allowed the comarison between exerimental and numerical searation erformances through artition functions mm diameter hydrocyclone Two different exeriments with articles are reorted in [3] for dilute conditions, namely 4.88 and wt%. CF simulations were erformed for both cases with the R ij -e turbulence model and with a vertical tube to model the air-core. The artition functions measured by Hsieh, those redicted with our CF model are lotted in Fig. 9. The searation curves reorted here have been corrected for the water slit. For instance, the total CPU time for the calculation with 4.8 wt% of silica was about 29 days. Comuted on 24 rocessors, it reresents about a calculation day. Overall, we find that our CF redictions are in good agreement with the behaviour of the hydrocyclone measured by Hsieh, from the velocity rofiles measured locally inside the searator to the macroscoic erformance of the hydrocyclone, as seen from the water slit and the artition function. These encouraging results confirm that the hyotheses and boundary conditions used for CF simulation of the hydrocyclone cature the key features of the Physics of the searation that takes lace inside a hydrocyclone under dilute conditions mm diameter hydrocyclone As our urose is the validation of our CF model under dilute conditions, we have restricted our resentation to the data we obtained with a 1 wt% (0.4 vol%) silica feed. Strictly identical artition curves (same cut-size and shaes) have been obtained with 5 wt% (1.9 vol%) and 10 wt% (4 vol%) feed solid content. Measured and redicted artition functions are lotted in Fig. 10 for one articular set of oerating conditions, ie for 2.25 kg s 1 inlet mass flow rate with 1 wt% of silica. All things considered, the agreement between the artition functions lotted in Fig. 10 is quite remarkable. Adding to the thorough validation of the local hydrodynamics we made against Hsieh s data, this confirms that the numerical aroach we have resented here for simulating hydrocyclones, which does not require any emirical arameter adjustment, yields accurate rediction of the behaviour of hydrocyclones Hydrodynamics of the 100 mm diameter hydrocyclone The centrifugal force induced by the high tangential inlet velocity controls article searation. It deends on both article density and size (see Eq. (8)). Fig. 8. Influence of tube diameter on water-slit. F c ¼ðq q l Þ d3 6 F g ¼ðq q l Þ d3 6 g U 2 h r A comarison with the sedimentation force alied on articles gives the G-force equals to U2 h rg. In the central art of the cyclone, where tangential velocity is highest (see tangential velocity rofiles in Fig. 11 and G-force rofiles at three heights for 35 lm diameter articles in Fig. 12), centrifugal force is maximum. Velocity rofiles are similar to those reviously discussed in Section 4.2 and exerimentally measured by Hsieh in [3]. ue to the high velocity and the low radius of curvature of the streamlines ð8þ Fig. 9. Partition curve for series Nos. 7 and 8. Fig. 10. Partition function for 1 wt% silica feed.

9 Fig. 11. Tangential velocity rofiles at three hydrocyclone heights. Fig. 12. G-force rofiles at three hydrocyclone heights. in the central art of the hydrocyclone, centrifugal force is very high and larger articles cannot flow towards the overflow. Moreover, with a small amount of articles, centrifugal force is high all along the height of the device which allows radial sedimentation in the entire hydrocyclone body. 5. Conclusions and ersectives This work aimed to derive a sound hysical basis on which to build a comlete CF model of the hydrocyclone. The hysics model was resented in some detail and validated against exerimental data from our own ilot test rig and data from the literature. This work is the initial steing stone towards derivation of a comlete hysics-based model of the hydrocyclone, and as such the roosed model is here within tested with either ure water or dilute slurries containing 1 wt% (0.4 vol%) solids. Velocity rofiles inside the hydrocyclone roved to be the best criterion for testing the validity of hysics assumtions. Whereas rediction of water slit roved to be insensitive to the choice of turbulence model, rediction of velocity rofiles on the other hand led to rejection of turbulence models which were deemed inadequate from a hysics analysis of the swirling flow inside the hydrocyclone. It was found that the R ij -e model is suited for simulating the behaviour of hydrocyclones under dilute conditions. esite the unsteady nature of the air core rofile, it was found that it is ossible as a first aroximation to model the air core using a solid cylinder with free sli boundary condition. This clearly is comutationally advantageous, and a method for choosing the correct tube diameter was roosed. It was found that the model redicted the measured artition curves, without requiring numerical fitting of the model arameters. As indicated above, this work will be followed by an ucoming ublication dealing with alication of the model resented here to hydrocycloning of dense slurries, whereby solids concentration has a direct influence on the erformance of hydrocyclones. Acknowledgements The authors acknowledge TOTAL S.A. for funding this study through a Ph.. CIFRE rogram and the scientific suort of C. Leroi and C. Yacono. The simulations were erformed at the High Performance Comuting centre CALMIP (Calcul en Midi-Pyrénées) under the roject P0406. References [1]. Bradley, The hydrocyclone, vol. 4, Pergamon, [2] K. Hsieh, R. Rajamani, Mathematical model of the hydrocyclone based on hysics of fluid flow, AIChE Journal 37 (1991). [3] K. Hsieh, Phenomenological model of the hydrocyclone, Ph.. Thesis, University of Utah, eartment of Metallurgy and Metallurgical Engineering, [4] V. Yakhot, eveloment of turbulence models for shear flows by a double exansion technique, Technical Reort, TIC ocument, 1991.

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