Swirling Flow Hydrodynamics in Hydrocyclone - American Chemical ...

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Swirling Flow Hydrodynamics in Hydrocyclone Chandranath Banerjee,† Kaustav Chaudhury,‡ Arun Kumar Majumder,† and Suman Chakraborty*,‡ †

Department of Mining Engineering, Indian Institute of Technology Kharagpur, Kharagpur−721302, India Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur−721302, India

‡

ABSTRACT: Performance of a hydrocyclone as a size separation unit is popularly judged using empirical or phenomenological models, with a predefined design under consideration. In contrast, here we develop a fundamental basis for analyzing the classification behavior of any hydrocyclone using experiment, simulation, and concurrent theory. Considering the G force (defined as the ratio of the centrifugal force to the weight of the suspended particle under consideration) distribution that has implications in creating separation of the suspended particles, here we bring out its consistent dependence on the essential governing parameters. The present estimation seems to agree well with the existing notions and available literature data, irrespective of the hydrocyclone size. Thus, our analysis is expected to provide a consistent basis for design of any tailormade hydrocyclone.

1. INTRODUCTION Ever since its introduction to industry in the year 1891, the hydrocyclone serves as a widely popular size separation unit in various industrial practices. In-depth understanding of the physics of water motion inside a hydrocyclone is a prerequisite toward unveiling the hydrocyclone behavior. Considering the fundamental hydrodynamics in action in a hydrocyclone, one important consideration is the radial distribution of the tangential velocity. This eventually decides the distribution of the G force acting on a suspended particle, resulting in separations. Practicing professionals, and researchers as well, are, therefore, making constant efforts to maintain pertinent G force distribution inside the hydrocyclone to meet the specific classification requirements.1−5 From an operational point of view, this adjustment is so far achieved by tuning the flow rate at the inlet, cone angle, cone ratio, and the diameter of the cylindrical section, the vortex finder, and the spigot.6−13 It needs to be emphasized that rigorous parametric studies have also been made in correlating the performance of a hydrocyclone with the mentioned tuning parameters.5−7,9,10,14−16 Thus, empirical correlations or parametric studies are the present level of acquired benefits, nevertheless limited for particular design constraints. In view of the above perspective, thus, it is now becoming imperative that the characteristic dependence of the G force distribution on the mentioned tuning parameters must be established, however, following more fundamental approach. Toward this end, in the present study we begin with the fundamental theory of swirl flow dynamics and apply it pertinently for a hydrocyclone. Subsequently, we conduct experiments and corroborate simulations on a 50.8 mm hydrocyclone and also rely on the data available in literature whenever we require corresponding information on larger size (specifically classifying) hydrocyclones. Our analysis brings out explicit scaling relationships between the G force and the aforementioned parameters. The present scaling relationships sufficiently underpin the classifying nature of the hydrocyclones studied so far. Furthermore, following the explicit nature of the scaling relationships, we extend the potential of the present © 2014 American Chemical Society

paradigm to envisage the classifying nature of the hydrocyclones.

2. METHODS Figure 1 describes a schematic of the present problem under consideration. Here we consider single flow analysis with water as the working medium. Water is fed tangentially into the hydrocyclone, as shown in Figure 1. The inlet condition is specified by setting a specific pressure. Subsequently, a swirling

Figure 1. Schematic depiction of the problem domain. Both Cartesian (x, y, z) and cylindrical (r, θ, z) reference frames are considered for the present study, for convenience. Accordingly, the components of the velocity vector (u) are given as (ux, uy, uz) and (ur, uθ, uz), respectively. Received: Revised: Accepted: Published: 522

August 19, 2014 December 2, 2014 December 22, 2014 December 22, 2014 DOI: 10.1021/ie503307x Ind. Eng. Chem. Res. 2015, 54, 522−528

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Industrial & Engineering Chemistry Research

flow features, required for subsequent analysis. We primarily consider results from large eddy simulation (LES); details are delineated in the Appendix. It is worth mentioning that flow field data regarding a 50.8 mm hydrocyclone is rather obscure. Thus, here we rely on our in house simulation. For corresponding information on other hydrocyclone, we consider data from available literature.

flow is developed inside the core body. Water eventually comes out through the bottom opening (known as the spigot) and through the top opening (known as the vortex finder) as well depending on the inlet pressure. Here rvf, rc, and rsp denote the radius of the vortex finder, cylindrical section and spigot respectively; their diameters are denoted by dvf, dc, and dsp, respectively. Here our primary objective is to unveil the implication of the swirling flow hydrodynamics on the behavior of a hydrocyclone, irrespective of the design constraints on the hydrocyclone size. We rely on the data available in literature whenever we require corresponding information on larger size (specifically classifying) hydrocyclones. For understanding the behavior of a smaller size hydrocyclone, for which pertinent literature data are rather obscure, we conduct experiments using water in a 50.8 mm hydrocyclone attached with a closed circuit test rig consisting of pump and sump assembly, as shown in Figure 2.

3. RESULTS From our experimentations, we obtain data for percentage of water split reported at overflow. The data set is shown in Figure 3. In the figure, we also present the water split data obtained

Figure 3. Comparison of the experimental (markers) and simulated (lines) water split data. The error bars shows the ±2.5% (i.e., total 5%) deviation from the experimental observation. The water split is measured with respect to the overflow.

from our simulations. From the comparisons, presented in Figure 3, it appears that our simulations are satisfactory in predicting the water split data. However, we must emphasize at this juncture that water split is primarily dependent on the pressure drop across the length of the hydrocyclone.3 Therefore, so long as the hydrodynamic features inside a hydrocyclone are grossly captured by a simulation setup, one can obtain the water split data in fairly close agreement with the experimental observations, notwithstanding the underlying details of the local flow structures.3 Thus, the agreement between the present single phase simulations with the experimental observations, as shown in Figure 3, establishes, at least, the validity of the present simulation setup in capturing the gross hydrodynamic features of a 50.8 mm hydrocyclone. The basic essence of the swirling flow hydrodynamics for a 50.8 mm hydrocyclone is shown in Figure 4. From the velocity vectors on different horizontal planes (z as the unit normal), it is evident that swirling flow prevails over the entire cross section. However, the significance of the prevailing flow field on particle classification can be well understood from the distribution of G force (G = uθ2/rg, with r measured from the central axis), as shown in Figure 4. From the G force contour, it is plain that near the central axis, G assumes high enough value. Needless to say that it is this Gmax (maximum value of G) that plays the dominant role in throwing the suspended particles toward the wall. In Figure 4, we also show the radial distribution of G force at different cross section in the cylindrical section. The feature of the G force profile is reminiscent of a composite vortex flow, realized by the combination of a free and a force vortex flows. Needless to say that the swirling flow characteristics, as discussed here, is generic for hydrocyclones of other sizes. The composite vortex

Figure 2. (a) Photograph of the experimental hydrocyclone closed circuit test rig with the corresponding (b) illustrative sketch: (OF) overflow stream; (UF) underflow stream; (BP) bypass stream; (V1) bypass valve ; (V2) feed inlet valve; (P1) pressure gauge.

We calculate the volumetric flow rate at each experimental condition by collecting timed samples, simultaneously, reporting through overflow and underflow. A series of tests are conducted to quantify the effects of spigot diameter, vortex finder diameter, and feed inlet pressure on water split behavior. One variable at a time is adopted during experimentation. Repeat experiments are also carried out at arbitrarily selected operating condition. The range of variables used in experiments is illustrated in Table I. Accordingly, we prepare a simulation setup. The objective of the present simulation is to get certain details of the prevailing Table I. Range of Parameters variable

variable range

vortex finder diameter (mm) spigot diameter (mm) inlet pressure (kPa)

14, 11, 8 3.2, 4.5, 6.4 68.95, 137.90, 206.84, 275.79, 344.74 523

DOI: 10.1021/ie503307x Ind. Eng. Chem. Res. 2015, 54, 522−528

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swirling flow dynamics inside the conical section as well. Nevertheless, restricting our attention to bring out the essential physics of interest, in this article, we primarily consider the theoretical analysis for the cylindrical section only. However, corresponding simulations are conducted for the entire hydrocyclone. Regarding the remaining factors in eq 1, Uavg can be estimated as Uavg =

=

= =

∫A uz dA ∫A dA ∫A uz dA + ∫A uz dA u

o

∫A

u + Ao

dA

Qu − Qo Ac (1 − x wsp)Q in − x wspQ in

⇒ Uavg =

Ac (1 − 2x wsp)Q in Ac

(2)

A basic scheme for this analysis is shown in Figure 5. Here A represents the area having z axis as the unit normal; the z

Figure 4. G force and velocity distributions. The G force contour is shown on a plane passing midway through the hydrocyclone with x as the unit outward normal. The results are shown for vortex finder and spigot diameter of 14 and 4.5 mm respectively at inlet pressure of 206.84 kPa. The unit of G force measurement is Newton.

flow inside a hydrocyclone has far ranging consequence in the performance of a hydrocyclone, as discussed subsequently.

4. CHARACTERISTICS OF SWIRL FLOW HYDRODYNAMICS AND G FORCE DISTRIBUTION It is worth mentioning that it is essentially the G force distribution that lead toward the subsequent events within the swirling flow field inside a hydrocyclone. Now, the G force distribution can be given as G=

2

Figure 5. Schematic illustration of the flow splitting at any cross section of the cylindrical section of the hydrocyclone.

2⎞

Uavg ⎛ Uθ uθ ⎟ ⎜ = rg rcg ⎝ R ⎠ 2

(1)

direction is chosen as the axial direction here. We consider Au and Ao as the area at any cross section of the cylindrical part through which the direction of the flows are toward underflow and toward overflow respectively, as shown in Figure 5. Thus, the cross section area of the cylindrical section is obtained as Ac = Au + Ao. In estimating the volume flow rates Qu = ∫ Au uz dA and Qo = ∫ Ao uo dA, we assume that Qo = xwspQin, with xwsp as the fraction of the volume flow rate reported to overflow and Qin being the volume flow rate at the inlet of the hydrocyclone. It is important to note that in reality, xwsp is measured at the outlet of the vortex finder section. At any axial location, the relation Qo = xwspQin needs to be replaced by Qo = x′wspQin where xwsp ′ denotes the fraction of Qin that goes toward the overflow at the chosen axial location under consideration. Nevertheless, understanding of x′wsp requires a detailed analysis of the actual flow pattern. For the time being, we proceed with the relationship Qo = xwspQin; a detailed rigorous analysis of xwsp ′ is preserved as scope of future work.

Here Uavg is the average axial velocity with R = r/rc and Uθ = uθ/Uavg being the normalized radial coordinate and the tangential velocity, respectively. The normalization scheme is generic for swirling flow dynamics in confinements.17,18 Among all the factors in eq 1, estimation of Uθ2/R requires the knowledge of the radial distribution of the tangential velocity. Following an analysis of the pertinent eigen value problem,17,18 an analytical estimation is possible for Uθ2/R. It is important to note, however, that the analytical estimation is based on an axisymmetric swirl flow analysis, which may deviate significantly from the physical reality in hydrocyclones. Nevertheless, we adopt the basic notions from the theoretical analysis to form the basis for unveiling the essential physics of interest. It also needs to be appreciated in this context that the development of the prevailing swirling flow essentially takes place inside the cylindrical section. Therefore, it is prudent to analyze the swirling flow features inside the cylindrical section. However, for complete analysis it is essential to consider the 524


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sizes. With this accounting, we observe the variation of Gmax with rc, as shown in Figure 7. The essential information

Proceeding with the estimation presented in eqs 1 and 2, the estimation of G force from eq 1 can be obtained as [(1 − 2x wsp)Q in]2 ⎛ Uθ 2 ⎞ ⎜ ⎟ G= π 2rc 5g ⎝ R ⎠

(3)

The estimation of G force following eq 3 has far ranging consequences in the operation of a hydrocyclone, as discussed in the subsequent section.

5. DISCUSSION We have presented a theoretical estimation of G force through eq 3. According to the equation, we can expect the maximum value of the G force (Gmax) to vary as [(1−2xwsp)Qin]2. This perception is compared against the data obtained from our simulations, as shown in Figure 6. This comparison establishes

Figure 7. Comparison of the scaling relationship G ∼ rc−5 with the present simulation data and the data available from literature.

regarding the discrete data points in the figure are provided in Table II. The solid line in Figure 7 represents a trend-line with Table II. Dataset Used for Comparing the Scaling Relationship G ∼ rc−5a set

dc

dvf

z

Chiné and Concha21 Hsieh and Rajamani22 Delgadillo and Rajamani14 present simulation

102 75 250 50.8

32 25 100 8

105 60 203 75

Figure 6. Comparison of the scaling relationship G ∼ [(1−2xwsp)Qin]2 with the present simulation data. Here [Gmax] = N and [Qin] = m3/s.

a

the validity of the present theoretical depiction in describing the hydrodynamic features inside a hydrocyclone. Using experimental data and taking advantage of the regression analysis, Bradley19 proposes d50 ∼ Qin−m with m varying from 0.53 to 0.64. Here d50 represents the cut-size denoting the diameter of the particles suspended inside a hydrocyclone those are having 50% probability to report to underflow. Our analysis brings out a relationship Gmax ∼ Qin2. This signifies that increasing inlet flow rate results in increase in G force so that the tendency of the particles to move toward the wall increases. Thus, at higher flow rate, we are left with finer particles around the central axis hence the reduction in d50. Additionally our analysis shows that Gmax ∼ (1−2xwsp)2. This is an interesting notion. It shows that there exists a strong correlation between the water split (denoted by xwsp) and the G force. Equivalently, we may also consider that by looking at the water split data one can also obtain an idea about the separation efficiency. Needless to say that for this purpose one has to undertake the analysis of the dynamics of the suspended particles taking the present form of the G force estimation into consideration. In the present article we, however, only discuss this issue. The detailed analysis is preserved as a scope of future work. Our theoretical estimation also shows an inverse power five relationship between the G force and the radius of the cylindrical section (see eq 3). We perform experiments, and supporting simulations, considering a 50.8 mm hydrocyclone. We take data from literature, for hydrocyclones of different

Gmax ∼ rc−5. From the comparison it appears that the present inverse power five relationship gives at least a rationalization against the decreasing trend of G force with the radius of the cylindrical section. It is a well-known fact that increase in the size of hydrocyclone (essentially increase in rc) results in an increase in the separation efficiency and elevation in the d50 as well.20 Our scaling relationship Gmax ∼ rc−5 pertinently shows a tremendous increase in the Gmax with reduction in cyclone radius. Thus, for larger size hydrocyclones, we can obtain more crowding of the coarser particles around the central axis, resulting in an increase in d50. In smaller size hydrocyclones, on the other hand, the prevailing tremendous magnitude of Gmax allows only very fine particles around the central axis. Thus, it is expected that very small size hydrocyclone can be good alternative for filtering out the suspended particles from the water. In Figure 8a, we show the radial distribution of the G force whereas Figure 8b represents Gmax as obtained from different settings of the radius of vortex finder (rvf); here all the results are presented for an inlet pressure of 206.84 kPa. The theoretical trend-line presented in Figure 8b is reminiscent of the theoretical estimation of the variation of (Uθ2/R)max as a function of cutoff radius demarcating the free and forced vortex zone at the reference axial location in a Rankine vortex flow (see refs 17,18 for details). Here we make an assumption that the tip of the vortex finder protruded inside acts as the reference axial location to control the cutoff radius for the 525

All dimensions in millimeters.


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Figure 8. (a) Radial distribution of the G force at different dvf. (b) Variation of Gmax (the maximum value of G force) at different dvf. The solid line in part b is the theoretical trend line obtained from axisymmetric swirl flow analysis. Here the results are shown for dc = 50.8 mm and at inlet pressure 206.84 kPa.

imposed Rankine vortex flow profile; this consideration seems to be rational by comparing the close matching between the trend line with our simulation data, as shown in Figure 8b. Following the arguments, thus, we can infer that the role of vortex finder is to decide the basic flow pattern by controlling the cutoff radius at the reference axial location. Now, taking our simulation data and the concurrent fundamental analysis into consideration, we can infer that reduction of rvf (or dvf) causes tremendous increase in Gmax, resulting in propelling of the suspended particles toward the wall of the hydrocyclone leading toward poor classification. This is in tune with the commonly observed fact that increase in the vortex finder diameter increases the d50.23

Figure A1. Geometry and mesh for the present problem domain.

it clear that the grids remain aligned with the swirling flow field. This can be appreciated from the velocity vector distribution, as shown in the main text. This grid and flow field alignment helps us avoiding any numerical diffusion. We conduct simulations with different grid resolutions characterized by number of control volumes. In Table AI, we show three sets specifically considered for resolution test. Table AI. Grid Data

6. CONCLUSIONS In this study, we have demonstrated how the fundamental hydrodynamics of vortex flows inside a cylindrical tube can be translated into the hydrodynamic analysis of a hydrocyclone. Our study bring out consistent relationship between the G force (G) with the fraction of the flow rate reported to overflow (xwsp), volume flow rate at the inlet (Qin), radius of the cylindrical portion of the hydrocyclone (rc), and the vortex finder (rvf). Here we have also argued that the vortex finder acts as to decide the basic pattern of the radial distribution of the G force. This in turn decides the distribution of the G force and the subsequent classification of the suspended particles. In contrast to the existing notions those are limited for a predefined set of design condition, the estimation from our analysis can be aptly adopted for hydrocyclone of any size.

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number of control volumes set 1 set 2 set 3

250000 400000 550000

LES Model and Implementation Strategy

Governing equations ∂ρ + ∇· (ρ u̅ ) = 0 ∂t ∂ Momentum: (ρ u̅ ) + ∇·(ρ u̅ u ̅ ) ∂t Continuity:

= −∇p ̅ + ∇·T div + ∇·Tsgs

APPENDIX

(A1)

Geometry and Mesh Generation

Figure A1 shows the grid distribution for the present problem. The domain is divided into a number of control volumes using body fitted grids. Grid refinements are also employed to capture the near wall flow features. This can be appreciated from the grid distribution on an arbitrary horizontal plane (with z as the unit outward normal) at the cylindrical section, as shown in Figure A1. The grid distribution on such plane makes

where F̅ = Ωcv−1∫ Ωcv F dΩ denotes the filtrated version of any quantity F (= u, p) and where Ω being the volume with Ωcv as the volume of the control volume under consideration. Description of the stress terms 526


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Industrial & Engineering Chemistry Research μ Deviatoric viscous stress tensor: T div = (∇u̅ + ∇u̅ T) 2 2μ − ∇·u̅ 3

simulation is run for 3 s of real time which takes around 72 h of computation time on a 64 bit computer with 16 GB ram and 3.40 GHz Intel Core i7 processor. After every simulation we compute the wall y+ values. In order to maintain the wall y+ value within 30−300, we also need to make further grid refinements around the wall. This way we obtain realizable flow features. In Figure A2 we present the radial distribution of the tangential velocity and the G force at different set of grids. The

Subgrid scale stress tensor: Tsgs = − (ρ uu − ρ u̅ u ̅ ) (A2)

where μ is the molecular viscosity of the medium. At this juncture, it important to mention that some sort of modeling is required for capturing the subgrid scale turbulence. With finer grid resolution, it is possible to capture the inherent turbulence directly in LES formalism, without invoking any empiricism. However, a consistent subgrid scale modeling is nevertheless essential so as to obtain the complete spectrum of turbulence. Towards this end, the subgrid scale stress tensor is modeled as Tsgs = −2μt D

(A3)

where D = 1/2(∇u̅ + ∇u̅ ) and μt is the subgrid scale turbulent viscosity. The genesis of μt is modeling purpose, similar to that of Reynolds averaged Navier−Stokes (RANS) equation. With this understanding in background it now remains to model μt so as to capture the subgrid scale turbulence, to the extent necessary for the present problem. It has been reported in the literature that the Smagorinsky− Lilly24 model, along with pertinent grid resolution, is sufficient to capture the essential physics of interest for the present problem under consideration.14,15,25,26 According to this model, μt at the computational cell is estimated as T

μt = ρls 2 2D: D

Figure A2. Grid resolution test.

measurement is taken on a horizontal plane (z as the unit outward normal) at a distance 0.075 m from the roof of the hydrocyclone. From the figure it is evident that with the mentioned grid resolutions, we get almost grid independent results, to the extent necessary. From this observation, we prefer to choose set 2 (400 000 number of control volumes) grid resolution for subsequent analysis.

(A4)

Here ls is the subgrid scale mixing length and is computed as the minimum from κd and CsV1/3 at each computational cell. Further, d denotes the distance of the closest wall, where V is the volume of the working computational cell. κ and Cs are the von Karman and Smagorinsky constants. Here we use finite volume discretization invoking SIMPLE27 strategy for pressure−velocity coupling. For discretization of the flux functions, we consider the bounded central differing scheme.28 This method takes the advantage of central difference scheme along with the convection boundedness criterion, thus, making the scheme devoid of any unphysical oscillation in the solution. In the present simulation, semi implicit time discretization policy is employed for the temporal terms. Additionally, whenever the gradient operations are required, other than the convective flux terms, the Green− Gauss cell-based operation is employed. The discretized versions of the governing equations are then solved using a Gauss−Seidel policy along with the algebraic multigrid (AMG)29 method. We use the finite volume solver platform of FLUENT for solving the governing transport equations, following the strategies discussed above. It is important to mention that here we consider pressure specified at the inlet and the two outlets (spigot and vortex finder openings respectively). Under this situation, there is no need to specify the turbulent fluctuation. However, if we would specify the velocity at inlet, we could need to specify the realistic turbulent fluctuation at inlet.30

Comparison between RANS and LES Model

Since RANS based models contain some inherent empiricism, these require calibration for specific type of flow condition.14,15,25,26 Therefore, the lack of universality is the key limitation of RANS model.14,15,25,26 Intrinsically, the LES model is devoid of several such issues. However, some modelling is required for capturing the subgrid scale turbulence that is filtered out. It has been shown in literature that the LES model gives somewhat better prediction of the velocity profiles than that by the RANS model.14,15,25,26 Where the prediction from RANS model is based on the mean flow field, the LES model provides the flow features taking inherent turbulent features into account within the resolution larger than the grid sizes. Thus, by making the finer grid resolution, one can capture the turbulent flow field to finer extent. It is worth emphasizing that reported literature on the flow features of a smaller size hydrocyclone, as used in our simulation, is rather obscure. Thus, at this point it is legitimate to proceed with the LES model, resolving the flow features without invoking any empiricism to the extent possible. With this consideration in background, along with the grid resolution test and subsequent wall y+ consideration, we hope that LES model is likely to give better depiction of the inherent swirling flow features inside a hydrocyclone than that by RANS based models.

Model Assessment

We start the simulation, for a given grid resolution, at different time steps. We find that time step of 5 × 10−4 provides the converged solution within a realizable time frame. Every 527


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(23) Wang, B.; Yu, A. B. Numerical Study of the Gas−liquid−solid Flow in Hydrocyclones with Different Configuration of Vortex Finder. Chem. Eng. J. 2008, 135, 33−42. (24) Smagorinsky, J. General Circulation Experiments with the Primitive Equations. Mon. Weather Rev. 1963, 91, 99−164. (25) Narasimha, M.; Brennan, M.; Holtham, P. N. Large Eddy Simulation of Hydrocycloneprediction of Air-Core Diameter and Shape. Int. J. Miner. Process. 2006, 80, 1−14. (26) Delgadillo, J. A.; Rajamani, R. K. Computational Fluid Dynamics Prediction of the Air-Core in Hydrocyclones. Int. J. Comut. Fluid Dyn. 2009, 23, 189−197. (27) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Taylor & Francis, 1980. (28) Leonard, B. P. The ULTIMATE Conservative Difference Scheme Applied to Unsteady One-Dimensional Advection. Comput. Methods Appl. Mech. Eng. 1991, 88, 17−74. (29) Stüben, K. A Review of Algebraic Multigrid. J. Comput. Appl. Math. 2001, 128, 281−309. (30) Klein, M.; Sadiki, A.; Janicka, J. A Digital Filter Based Generation of Inflow Data for Spatially Developing Direct Numerical or Large Eddy Simulations. J. Comput. Phys. 2003, 186, 652−665.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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