The effect of the dust outlet geometry on the performance and hydrodynamics of gas cyclones

The effect of the dust outlet geometry on the performance and hydrodynamics of gas cyclones

Computers & Fluids 68 (2012) 134–147 Contents lists available at SciVerse ScienceDirect Computers & Fluids j o u r n a l h o m e p a g e : w w w . e...

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Computers & Fluids 68 (2012) 134–147

Contents lists available at SciVerse ScienceDirect

Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

The effect of the dust outlet geometry on the performance and hydrodynamics of gas cyclones Khairy Elsayed ⇑, Chris Lacor Vrije Universiteit Brussel, Department of Mechanical Engineering, Research Group Fluid Mechanics and Thermodynamics, Pleinlaan 2, 1050 Brussels, Belgium

a r t i c l e

i n f o

Article history: Received 12 October 2011 Received in revised form 19 June 2012 Accepted 30 July 2012 Available online 14 August 2012 Keywords: Cyclone separator Dust outlet geometry Dustbin Dipleg Reynolds stress turbulence model (RSM)

a b s t r a c t Four cyclone separators with different dust outlet geometries (without dustbin, with dustbin, with dipleg and with dustbin plus dipleg) have been computationally investigated. There are two purposes of this study. First, to study the effect of changing the dust outlet geometry on the flow field pattern and performance of air cyclones. Second, to demonstrate whether simulation of the flow without accounting for the dust outlet geometry (i.e., simulate only the cylinder on cone region) is sufficient for performance estimation. Results show that the maximum tangential velocity is almost the same for the four tested cyclones. No radial acceleration occurs in the cyclone space (the maximum tangential velocity is nearly constant throughout the cyclone). Three cyclones (without dustbin, with dipleg and with dustbin plus dipleg) exhibit the inverted W axial velocity profile. An inverted V axial velocity profile is observed for the cyclone with dustbin. Although the absence of the dust outlet geometry from the simulation domain will result in large savings in the computational effort, the present study indicated it can affect the estimated pressure drop and cut-off diameter significantly. If the cyclone performance parameters are estimated without the dust outlet geometry, an error margin of around 10% in the calculations of the Euler number and 35% in the calculation of the cut-off diameter should be taken into account. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Cyclones are one of the most widely used separators, which rely on centrifugal forces to separate particles from a gas stream. The primary advantages are economy, simplicity in construction and adaptability to a wide range of operating conditions. Reversed flow cyclones with a tangential inlet are the most common cyclone design as shown in Fig. 1. It consists of seven main geometrical parameters: inlet section height a and width b, cylinder height h, cyclone total height Ht, dust outlet diameter (cone tip diameter) Bc, gas outlet diameter (also, called the vortex finder diameter) Dx and vortex finder height S. All these parameters always given as a ratio of the cyclone body diameter D. The dimensions and the process conditions determine the collection efficiency (cut-off diameter) and the pressure drop of the cyclone separator [1–9]. The highly swirling turbulent flow in cyclone separators is the main mechanism for the separation of different density phases. The Stairmand high-efficiency cyclone is the typical geometrical layout of a gas cyclone [10] (Fig. 1a). The flow in the cyclone separator is extremely complicated three dimensional swirling flow. This type of flow inspired the researcher over decades to use experimental techniques (e.g., [11–15]) and theoretical and computational

⇑ Corresponding author. Tel.: +32 26 29 2368; fax: +32 26 29 2880. E-mail addresses: [email protected], [email protected] (K. Elsayed). 0045-7930/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2012.07.029

studies (e.g., [16–20]) to investigate the effect of different geometrical and operating parameters on the performance and flow pattern. Conventional cyclones always have a dustbin attached to the cone to collect the separated solid particles. When a gas flow stream enters the dustbin (closed at bottom), some of the flow will return the cone and carry some of the separated particles. This phenomena is called ‘‘re-entrainment’’ and it will affect the separation efficiency of the cyclone [21]. However, many works have been carried out to investigate the influence of different geometrical parameters such as cyclone length, inlet and outlet pipe geometries. On the performance of cyclones [17,22–25], there has been little work concerning the dust outlet geometries [26–29]. Regarding this influence, discrepancies and uncertainties exist in the literature. Xiang and Lee [30] reported that the dustbin connected to the cyclone should be incorporated in the flow domain as it affects the results. On the other hand, numerous studies were performed without dustbin (e.g., [31,32]) with good matching with experimental results. Obermair et al. [26] performed cyclone tests with five different dust outlet geometries to find the influence of the dust outlet geometry on the separation process. They showed that separation efficiency can be improved significantly by changing the dust outlet geometry, and they reported that further research is needed to

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(a) Cyclone I

(b) Cyclone II

(c) Cyclone III

(d) Cyclone IV

(e) The surface mesh Fig. 1. Schematic diagrams and surface meshes for the four tested cyclone separators.

clarify precise effects of dust outlet geometry. The effect of a dipleg (a vertical tube between the cyclone and the dustbin) was posed and investigated by several researchers [27,33]. The previous studies on the effect of dust outlet geometry on the cyclonic flow can be classified into the following categories: 1. Comparison between cyclones with dustbin and that with dustbin plus dipleg [21,26,27]. Obermair et al. [26] investigated experimentally the effect of different dust outlet geometries on the flow pattern. Nevertheless, the given flow pattern was limited to the cyclone bottom. The effects of the dust outlet geometry on the velocity profiles are not given. 2. Comparison between the cyclone with and without dipleg (prolonged cyclones). Kaya and Karagoz [33] numerically investigated the flow characteristics and particle collection efficiencies of conventional (without dipleg) and prolonged cyclones. 3. Comparison between cyclone with and without dustbin [28]. Elsayed and Lacor [28] numerically investigated two cyclones with and without dustbin. They reported

a negligible effect of the dustbin on the performance. They reported further research is needed with the cyclone with dipleg and dust hopper (dustbin plus dipleg). All studies above do not contain enough information about the effect of the dust outlet geometry on the flow pattern, velocity profiles and separation performance. Furthermore, there is no clear comparison between the four cases: separation space only (cylinder on cone), cyclone with cylindrical shaped dustbin, cyclone with vertical tube (dipleg) and cyclone with dustbin plus dipleg. The present numerical investigation aims at examining the influence of dust outlet geometry on the flow pattern and the cyclone performance. This serves two purposes: (1) to demonstrate whether simulation of the separation space only without including the dustbin or the dipleg is sufficient to estimate the cyclone performance. (2) to give information for the designer about the effect of the configuration under the separation space (the dustbin or the dipleg) on the flow pattern and performance.

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2. Mathematical modelling description

Table 2 The details of the four tested cyclones.

2.1. Selection of the turbulence model

Cyclone a

The flow in the cyclone separators is three dimensional and highly swirling (non-isotropic). Consequently, neither the standard k  e, RNG k  e nor Realizable k  e turbulence models are suitable for such type of flow [20,34–37]. On the other hand, many researchers reported the suitability of the Reynolds stress model (RSM) in modelling the cyclonic flow (e.g., [2,10,36–48]). The Reynolds stress turbulence model (RSM) [49] has been used in this study to reveal the effect of the cyclone dust outlet geometry on the turbulent flow in the cyclone separator. In RSM, Fluent solves (in addition to the continuity and three momentum equations) three transport equations for the turbulent shear stresses and three transport equations for the turbulent normal stresses. Moreover, Fluent solves another transport equation for the turbulent dissipation rate e. For the values of the turbulent kinetic energy k at the boundaries another transport equation for k is also solved. For all interior points, the turbulent kinetic energy equals the trace of the Reynolds stress tensor k ¼ 12 u0 u0 [50]. In other words, k and e values are computed to model some of the terms which appear in the Reynolds stress model [51]. For the detailed governing equations for both the Reynolds averaged Navier–Stokes equations (RANS) and the Reynolds stress model transport equations, the interested reader can refer to Elsayed and Lacor [10] and the Fluent manual [50]. 2.2. Configurations of the four tested cyclones The numerical simulations were performed on four cyclones. Cyclone I has only the cylinder on cone shape (separation space only), cyclone II has a dustbin, cyclone III has a dipleg while cyclone IV has a dust hopper (dustbin plus dipleg). Fig. 1 and Table 1 gives the cyclones dimensions. Table 2 gives more details for the used cyclones, including the number of cells, cyclone volume and the flow residence time for each cyclone. Nine sections are used to plot the velocity profiles as shown in Table 3. 2.3. Solver settings Based on the study of Kaya and Karagoz [45] for the best selection of numerical schemes to be used with RSM model, the following discretization schemes have been used. The PRESTO scheme has been used for the pressure interpolation, the SIMPLEC algorithm for pressure velocity coupling, the QUICK scheme for momentum equations, the second-order upwind for the turbulent kinetic energy and the first-order upwind discretization scheme for the Reynolds stresses [10]. The standard wall functions are used to model the near-wall region of the flow [50,52].

Table 1 The geometrical dimensions of the four tested cyclones.a Dimension

Length (m)

Dimension ratio (Dimension/D)

Body diameter, D Inlet height, a Inlet width, b Gas outlet diameter, Dx Gas outlet duct length, S Cone tip-diameter, Bc Cylinder height, h Cyclone height, Ht

0.205 0.105 0.041 0.105 0.105 0.076875 0.3075 0.82

1 0.5 0.2 0.5 0.5 0.375 1.5 4

a The outlet section is above the cyclone surface by Le = 0.5 D. The inlet section located at a distance Li = 0.75 D from the cyclone center, the height of the dustbin and the dipleg, LD = 2 D.

Number of cells Cyclone volume  102 (m3) tres (s)b

I

II

III

IV

260230 1.99 0.237

593125 3.35 0.399

322286 2.18 0.259

441062 2.76 0.328

a

The total number of hexahedral cells after the mesh sensitivity study. The average residence time, tres = V/Qin where V is the cyclone volume and Qin is the gas flow rate. b

Table 3 The position of different plotting sections.a

a

Section

S1

S2

S3

S4

S5

S6

S7

S8

S9

z‘/D

2.75

2.5

2.25

2

1.75

1.5

1.25

1.0

0.75

z‘ is measured from the top of the inlet section (cf. Fig. 1(a)).

Chuah et al. [36] stated that the time step should be selected as a tiny fraction of the residence time tres. From Table 2, the value of tres varies between 0.237 and 0.399 s. Therefore a time step of 1E-4 is an acceptable value for the current simulations [10]. The simulations have been performed using Fluent 6.3.26. 2.4. CFD grid and boundary conditions The mesh sensitivity study has been performed for the four tested cyclones with three levels for each cyclone, to be sure that the obtained results are grid independent. For example, three different meshes with respectively 130596, 260230 and 478980 cells have been used for cyclone I. The computational results on the three grids are presented in Table 4. As the maximum difference between the results is less than 5%, so the grid template 230596 produces the grid independent results [10,53]. It has been observed that even 130596 grid provides a sufficient grid independency. However, for excluding any uncertainty, computations have been performed using the 260230 cells grid, where the total number of grid points was not that critical with respect to the computation overhead [54]. Fig. 1e shows the surface mesh of the four cyclones. The hexahedral computational grids were generated using the GAMBIT grid generator. The boundary condition at inlet section is the velocity inlet (i.e., the velocity vector and the scalar properties of flow at inlet (the turbulent intensity and the length scale) are given to the solver [55,56, p. 19]). The velocity distribution at the inlet is assumed uniform. The turbulence intensity and turbulent length scale are given to the solver. According to the Fluent manual [50], the turbulent intensity can be estimated as I ¼ 0:16ðReDh Þ1=8 , where ReDh is the Reynolds number based on the inlet section hydraulic diameter and the average inlet velocity Uin. In the current study, the turbulent intensity equals 5% and the characteristic length ‘ equals 0.07 times the inlet width [38]. The turbulent kinetic energy k and the turbulent dissipation rate e at the inlet section can be calculated 3=2 as k ¼ 32 ðU in IÞ2 and e ¼ C l3=4 k =‘, where Cl is an empirical constant equals 0.09. The computed value of k is used to estimate the Reynolds stresses at the inlet from the assumption of isotropic turbulence, i.e. the normal stresses are set to 2k/3, and all shear stresses are zero [50]. The solutions were hardly sensitive to the exact inlet values for k and e [38]. An outflow boundary condition is used at the outlet (i.e., zero normal gradients of all the solution variables except for pressure with a flow rate weighing of one was applied [57]. The solver extrapolates the required information from interior. Furthermore, an overall mass balance correction is applied between the inlet and the gas exit sections [55]). The no-slip (wall) boundary

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K. Elsayed, C. Lacor / Computers & Fluids 68 (2012) 134–147 Table 4 The details of the grid independence study for cyclone I.

a

Total number of cells

Static pressure drop (N/m2)

Total pressure drop (N/m2)

Cut-off diameter (N/m2)

130596 260230 478980 % Differencea

955.51 960.25 961.12 0.584

967.76 972.5 973.37 0.576

1.48 1.5 1.51 1.987

The percentage difference between the coarsest and finest grid.

condition is used at the other boundaries [10]. The air volume flow rate Qin = 0.08405 m3/s for all cyclones, air density 1.225 kg/m3 and dynamic viscosity of 178.940E-6 Pa s. The Reynolds number based on the barrel diameter and the inlet velocity equals 2.8  104. 2.5. DPM settings The Lagrangian discrete phase model in Fluent follows the Euler–Lagrange approach. The fluid phase is treated as a continuum by solving the time-averaged Navier–Stokes equations, while the dispersed phase is solved by tracking a large number of particles through the calculated flow field [9]. A fundamental assumption made in this approach is that the dispersed second phase occupies a low volume fraction (usually less than 10–12%, where the volume fraction is the ratio between the total volume of particles and the volume of fluid domain), even though high mass loading is acceptable. The particle trajectories are computed individually at specified intervals during the fluid phase calculation. This makes the model appropriate for the modeling of particle-laden flows [9]. For modeling of heavily loading cyclones, particle–particle and particle turbulence interaction should be considered. The particle loading in this study is small, and therefore, it can be safely assumed that the presence of the particles does not affect the flow field (one-way coupling) [9]. 2.5.1. Settings, forces and assumptions Particle tracking was done as a post-processing step after obtaining the RANS solution [58]. Each particle’s motion was individually computed [58]. It is assumed that collisions among particles are negligible. The high value of the particle to gas density ratio qp/q implies that the Basset force and the added mass term are small and are therefore neglected [59,60]. The effect of shear on the particles generates the so-called Saffman lift force. For particles with small diameter and low inertia, the Saffman lift force can be neglected in comparison to the drag force [59]. Brownian motion of the particles was not considered due to the diameter of the particles under study. In terms of the Eulerian–Lagrangian approach (one way coupling), the equation of particle motion is given by [9,19]

g i ðqp  qÞ dupi 18l C D Rep ¼ ðui  upi Þ þ dt qp qp d2p 24

ð1Þ

dxpi ¼ upi dt

ð2Þ

where the term

18l C D Rep ðui qp d2p 24

 upi Þ is the drag force per unit particle

mass [19]. q and l are the gas density and dynamic viscosity respectively, qp and dp are the particle density and dimeter respectively, CD is the drag coefficient, ui and upi are the gas and particle velocity in i direction respectively, gi is the gravitational acceleration in i direction, Rep is the relative Reynolds number [9].

Rep ¼

qp dp ju  up j l

ð3Þ

In Fluent, the drag coefficient for spherical particles is calculated by using the correlations developed by Morsi and Alexander [61]. The equation of motion for particles was integrated along the

trajectory of an individual particle. Collection efficiency statistics were obtained by releasing a specified number of mono-dispersed particles at the inlet of the cyclone and by monitoring the number escaping through the outlet. Collisions between particles and the walls of the cyclone were assumed to be perfectly elastic (coefficient of restitution is equal to 1). A discrete phase modeling (DPM) study has been performed by injecting 104 particles from the inlet surface with a particle density of 860 kg/m3 and with a particle diameter ranging from 0.025 until 5 lm at a velocity equals to the gas velocity. Since the particles injected in this study are very small, the effect of instantaneous fluctuation of gas velocity on turbulent dispersion of small particles cannot be neglected. The turbulent dispersion of small particles was taken into account by integrating the trajectory equations for individual particles using the instanta  neous fluid velocity ui ¼ U i þ u0i . In this study, a discrete random walk (DRW) model based on stochastic method was used to determine the fluctuating gas velocity [62]. The values of fluctuating velocities that prevail during the lifetime of a turbulent eddy se, are sampled by assuming that they obey a Gaussian probability distribution using correlation as [63]:

qffiffiffiffiffiffiffiffi u0i ¼ f u0i u0j

ð4Þ

where f is a normally distributed random number and the term qffiffiffiffiffiffiffiffi u0i u0j is the local root mean square (RMS) of the velocity fluctuations. In the present study, the characteristic lifetime se of an eddy was calculated as random variation about the Lagrangian integral time scale TL, using se = TL ln (r) where r is a random number between 0 and 1 and TL = 0.3 k/e [63]. 2.5.2. How the particles are affected by turbulence? The particles was released at the inlet section where they were uniformly distributed and the particle velocity at inlet is assumed equals the gas velocity. There are 17 particle sets (0.025, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 3, 4, 5) lm. each set contains 10,000 identical particles. Feeding particles starts at t = 0 in a fully developed flow [64]. Derksen [64] investigated different approaches to handle the solid particle motion in cyclone separators, here we will focus on only two approaches which can be used with commercial software. The first approach is the concurrent simulation method and the cheaper one is the frozen field method. In the first approach, the particles are continuously fed in the simulation domain from t = 0 to t = 18 Tint (where Tint = D/Uin is the integral time scale, D is the cyclone diameter and Uin is the average inlet velocity). At every time step, the particle positions are updated by solving the particles equation of motion using a particle time step that equals the flow time step. The simulation continues until no more particles can be observed that escape via the vortex finder. Then the grade efficiency curve can be plotted. The long time span of the concurrent simulation of gas flow and particle motion prompted Derksen [64] to look for another alternative way for particle transport modeling, which can be referred to as a frozen field realization method. In this method, the particles are released in statistically independent flow field realizations (frozen fields) [65]. Derksen [64] suggested to use 16 realizations, each separated 2 Tint

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in time. In each realization (stored flow field data), the particles sets are released and tracked throughout the flow field. The procedure for updating the particle position is identical to that in the concurrent simulation method except that the flow is frozen. Derksen [64] stated that the frozen field realization method saves 99% of the running time needed for the concurrent simulation method. The shape of the grade efficiency curves obtained from the two methods are similar, but the cut-off diameter obtained by the frozen field realization method is smaller than the concurrent simulation method. The frozen field realization method has been applied in this study using 16 realizations separated at 0.005 s. The presented values of Stokes numbers in the subsequent sections are averaged over the 16 realizations.

 1  xnþ1 ¼ xnp þ Dt u2p þ unþ1 p p 2

ð8Þ

The integration of the above-mentioned equations starts from n = 1 at the inlet section to n = Nmax, where Nmax is the maximum number of time steps for particle simulation. If Nmax is sufficiently large, all particles either trapped or exit the domain before n reaches Nmax. If n reaches Nmax, the particle is designated as incomplete and we exclude the incomplete particles from the efficiency calculation. In these simulations Nmax = 9E5 and no incomplete particle has been recorded for all cases. Fig. 2 represents a flow chart demonstrating the steps involved in tracking one injected particle. 2.6. Validation of the numerical model

2.5.3. Particle histories Each particle is tracked until one of the abort criteria is satisfied or the particle escapes from the simulation domain. When a particle strikes a boundary face, one of several fates may arise in case of cyclone separator (escape, reflected or trapped): the particle may be reflected via an elastic or inelastic collision (at the cyclone wall). The particle may escape through the boundary. The particle is lost from the calculation at the point where it impacts the boundary (at the gas outlet). The particle will be considered trapped at the dust exit, when the center of the particle reaches to a distance equal to its radius [66] from the dust exit (cone bottom).

Fig. 3 compares the predicted and the experimental tangential 0 and axial velocity profiles at a station located at Z /D = 0.75 (cf.

2.5.4. Integration of particle’s equation of motion The equation of motion for particles was integrated along the trajectory of an individual particle. Collection efficiency statistics were obtained by releasing the mono-dispersed particles at the inlet of the cyclone and by monitoring the number escaping through the outlet. Collisions between particles and the walls of the cyclone were assumed to be perfectly elastic, i.e., coefficient of restitution equals unity [9]. The prediction of grade efficiency curve with trapezoidal scheme was selected [63]. The equation of motion of dispersed particles were solved by stepwise integration over discrete time steps. The integration of Eq. (1) yields the velocity of the particle at each point along the trajectory and the position of the trajectory can be determined using Eq. (2). The equation of motion of a particle (Eq. (1)) can be written as [63]:

dupi 1 ¼ ðui  upi Þ þ a dt sp

ð5Þ

q d2

where sp ¼ 18p lp is the particle response time and a is acceleration due to all other forces except the drag force. The trapezoidal discretization schemes is recommend by Shukla et al. [63] to solve the set of Eqs. (2) and (5). When the semi-implicit trapezoidal discretization is applied to solve Eq. (5), one gets [63]

 unþ1  unp 1  p ¼ þ u  up þ an Dt sp

up

ð6Þ

Here u ¼ 12 ðun þ unþ1 Þ is the average fluid velocity,   is the average fluid velocity. a is the acceleration ¼ 12 unp þ unþ1 p

due to other forces held as constant. unþ1 ¼ un þ Dtunp . Dun is the fluid velocity at the new location. Using these expressions the particle velocity at the new location n + 1 is obtained as [63]:

unþ1 p

    unp 1  12 Dspt þ Dspt un þ Dtunp  Dun þ Da ¼ 1 þ 12 Dspt

ð7Þ

The new particle location is computed by a trapezoidal discretization of Eq. (2) as:

Fig. 2. Flow chart demonstrating the steps involved in tracking one injected particle [67,68].

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LDA RSM

0.5

1.5

Axial velocity / inlet velocity

Tangintial velocity / inlet velocity

1.75

1.25

1

0.75

0.5

0.25

0

LDA RSM

0.25

-0.25 -1

-0.75

-0.5

-0.25

0

0.25

0.5

Radial position /cyclone radius

0.75

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

Radial position /cyclone radius 0

Fig. 3. Comparison of the time averaged tangential and axial velocity between the experimental measurements by Hoekstra [38] at Z /D = 0.75. Note: The barrel diameter D = 29 cm for the validation.

Fig. 4. The radial profile for the time averaged tangential and axial velocity at different sections.

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Fig. 4. (continued).

Hoekstra [38] and Shukla et al. [57] for more details about the used cyclone in the validation). Fig. 3 indicates a good matching between the experimental and numerical results for both the axial and tangential velocity. The comparison indicates the underestimation of the maximum tangential velocity, and overestimation of the axial velocity at the central region. The agreement between the simulation and measurements are considered to be quite acceptable due to the complexity of the turbulent swirling flow in the cyclones [2,10].

3. Results The dominant velocity component of the gas flow in cyclones is the tangential velocity, which results in the centrifugal force for particle separation [30]. The axial velocity is responsible more than the gravity for the transport of particles to the collection devices [69,70]. These velocity components will be discussed in details in order to investigate the effect of the dust outlet geometry on the flow properties. Moreover, the pressure distribution in the swirling flow in these four cyclone separators will be discussed in details.

3.1. The variation of flow properties in the axial direction Figs. 4 and 5 present the radial profiles of the time-averaged tangential and axial velocity and static pressure at nine axial stations. As expected, the tangential velocity profiles exhibit the socalled Rankine vortex, which consists of two parts, an outer free vortex and an inner solid rotation in the center (Fig. 4). The tangential velocity distribution in the inner region is rather similar at different sections for the same cyclone. In the outer region, due to the sharp drop in velocity magnitude in the near wall region, the distribution is different but the maximum tangential velocity is similar for all sections. Generally, the tangential velocity distribution varies only slightly with axial positions for the same cyclone, which is also reported in other articles (e.g., [11,30,71,72]). This means that, if the tangential velocity increases at one section of the cyclone, it will increase at all other sections. The axial velocity profiles at nine different stations are shown in Fig. 4. The axial velocity in the inner vortex is either reported as an inverted V or inverted W-shaped profile, i.e., with a maximum (Vshaped) or a dip (W-shaped) at the symmetry axis [70,73]. The two types of axial velocity profiles are observed. Cyclone I, III and

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141

Fig. 5. The radial profiles for the time-averaged static pressure at different sections for the four cyclones.

IV show an inverted W profile. Only cyclone II (with dustbin) has an inverted V profile, cf. Section 3.2 for more details. The radial profiles of the time averaged static pressure are given in Fig. 5 for the four cyclones. Like for the tangential velocity, the axial variations of static pressure are very small for the same cyclone. An exception is the cyclone I which shows some variations in the central part. Fig. 6 compares the static pressure profiles for the four cyclones at sections S7–S9 (located at the cylindrical part of the cyclone, which is the most effective part of the cyclone in the separation process, the location of the highest area average tangential velocity). The plots at sections S7–S9 are also representative for the other sections, as the axial variations in the flow variables are small. From the comparison between the radial profiles of the four cyclones, the minimum pressure at the cyclone center is almost the same for all cyclones. The static pressure radial profiles of cyclones I, III and IV almost coincide. Cyclone II (with dustbin) depicts fewer gradients in the radial direction. It has the lowest maximum pressure value. Referring to Fig. 6, there is no difference between the radial pressure profiles at the three sections for the same cyclone.

3.2. The velocity profiles Fig. 7 compares the tangential and axial velocity profiles at sections S7–S9. The variation of velocity profiles (both axial and tangential) from cyclone to cyclone is mainly located at the central region for both the tangential and axial velocity. Since the effect of the dust outlet geometry on the tangential velocity (centrifugal force) is minor, it is expected that the collection efficiency (cut-off diameter) of the four cyclones will be comparable. As mentioned before, the axial velocity exhibits either an inverted V or inverted W-shaped profile, i.e., with a maximum (Vshaped) or a dip (W-shaped) at the symmetry axis [70]. Hoekstra et al. [52] stated that the shape of the axial velocity profile is affected by the cyclone geometry. They referred the dip in the inverted W profile to the loss of swirl in the vortex finder (the friction force of the vortex finder wall attenuates the swirling flow), which results in an adverse pressure gradient at the centerline [27]. Hence, fluid with less swirl is drawn back from the exit pipe into the cyclone. This core flow prevails throughout the entire separation space of the cyclone in spite of the attenuation of swirl

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Fig. 6. The radial profile for the time-averaged static pressure at different sections for the four cyclones.

in the conical part of the cyclone [27]. This explains the reason behind the inverted W-shaped profile exhibited by cyclones I (cylinder on cone), cyclone III (dipleg) and IV (dustbin plus dipleg). But, why cyclone II (with dustbin) exhibit the inverted V-shaped profile? In cyclone II, the dustbin has an equal diameter to the cyclone barrel, which means sudden expansion to the downward flow and sudden contraction to the upward flow (the flow inside the dustbin also has two streams due to the absence of a vortex stabilizer [52] which can prevent the re-entrainment of the collected particles). The upward gas flow (directed from the end of dustbin) has a higher kinetic energy especially at the cone tip diameter (sudden contraction) which can overcome the adverse pressure gradient at the centerline (caused by the swirl attenuation in the vortex finder) and results in the inverted V axial velocity profile. One more question may appear now, why cyclones III (dipleg) and IV (dustbin plus dipleg) did not exhibit the inverted V-shaped profile? The reason can be referred to the diameter of the dustbin directly connected to the cyclone. For cyclones III and IV there is no change in the flow area at the connection, consequently no flow acceleration happens. The effect of dustbin dimensions (diameter, height) still need more investigations. The authors believe, if a cone is in-

serted at the entrance of the cone-tip inside the dustbin of cyclone II, the axial velocity will become inverted W-shaped. But the dimensions and location of this cone still need more investigations (cf. Obermair et al. [26] for more details). Another factor which was not taken into consideration in this study which may also affect the axial velocity profile is the effect of the cone tip diameter. However, the authors studied that before in another article [9]. It was found that the effect of Bc is minor but it may have an interaction with the dustbin or dipleg. So, the authors think that, if the cone tip diameter is much smaller than the vortex finder diameter, the inverted V profile might be observed in all cases, whereas, if Bc is very close to the vortex finder diameter, all cyclones may show the inverted W profile. This is however to be confirmed by a future study. 3.3. The flow pattern Regarding the effect of neglecting including the dust outlet geometry (dustbin or dipleg or dustbin plus dipleg) in the simulating domain on the flow field pattern, the following comments can be drawn (Fig. 8).

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Fig. 7. The radial profile for the time averaged tangential and axial velocity at different sections on the X–Z plane (Y = 0) at the inlet region (sections S7, S8 and S9). From top to bottom: section S7–S9.

1. From the comparison between the static pressure contour plots of each cyclone and that of cyclone I, the highest value of static pressure is obtained in case of cyclone III (dipleg). The lowest value is given by cyclone II (dustbin). The highest

value of the static pressure in cyclone IV (dustbin plus dipleg) is in between that for cyclones III and II. This indicates a slight underestimation of the pressure drop by neglecting the effect of dipleg or dustbin plus dipleg (cyclone III, IV

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Fig. 8. The contour plots for the time averaged flow variables at sections Y = 0 and throughout the inlet section.

versus cyclone I). Furthermore, a slight overestimation of the pressure drop is observed by neglecting the effect of the dustbin (cyclone II versus cyclone I). 2. The tangential velocity pattern is very similar in the four cyclones (Rankine profile). The highest value is almost the same for the four cyclones. Consequently, the collection efficiency (cut-off diameter) of the four cyclones will be almost the same. Moreover, the contour plots for the tangential velocity of the four cyclones are quite similar in the main separation space (cylinder and cone).

3. The axial velocity patterns for cyclones I, III and IV have the shape of an inverted W profile while that of cyclone II has an inverted V profile, indicating different flow behavior. Close to the cone bottom there are different flow patterns as a consequence of different dust outlet geometry. Fig. 9 shows a qualitative view of the complex flow in the four tested cyclones with the streamtraces plots of the time averaged velocities colored by the time-average axial velocity. The swirling, downward flow at the outer region of the cyclone is clearly visible.

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Cyclone I Cyclone II Cyclone III Cyclone IV

Collection efficiency %

80

60

40

20

0

1

2

3

4 5

Particle diameter [micron] Fig. 10. The grade efficiency curves for the four cyclones.

The shapes of the grade collection efficiency curves of all models have a so-called S shape [74]. It is clear from Fig. 10 and Table 5 that the effect of the cyclone dustbin or dipleg, on the cut-off diameter (particle diameter of 50% collection efficiency) is not negligible. Neglecting the effect of dust outlet geometry in the cyclone simulation overestimates the cut-off diameter (when comparing it with cyclone I) as is clear from Table 5. A deeper look to the GECs indicates the variation of the collection efficiency for particles diameters larger than 1.5 lm, with a higher efficiency for cyclone II and the lower efficiency for cyclone I. This behavior is due to the increase in the separation space and the change in the maximum tangential velocity. For particles with diameters less than 0.8 lm, the collection efficiency tends to zero except in case of cyclone IV which depict a non zero collection efficiency. In order to estimate the effect of dust outlet geometry on the performance parameters, the pressure drop and cut-off diameter have been calculated. Table 5 shows the effect of the dustbin geometry on the pressure drop and the cut-off diameter. It can be concluded that if the dustbin is removed from the simulation domain, an error margin of around 10% in the calculations of the Euler number and 35% in the calculation of the cut-off diameter should be taken into account.

4. Conclusion Four cyclones have been simulated using the Reynolds stress model (RSM), to study the effect of the dust outlet geometry on the cyclone separator performance, flow pattern and velocity profiles. The following conclusions have been obtained. Fig. 9. The streamtraces plots for the time averaged flow variables, colored by the average axial velocity (m/s).

Near the bottom of the cyclone, it is diverted into an upward flow near the cyclone center. In cyclone II (with dustbin) and IV (with dustbin and dipleg), the flow behavior in the dustbin is quite different because of the different length of the dustbin.

3.4. The performance Fig. 10 presents the grade efficiency curves (GECs) for the four tested cyclones. As expected, the collection efficiencies of all the cyclones are seen to increase with the increase in particle size.

 The maximum tangential velocity in the four cyclones is very similar.  The axial velocity patterns obtained by the four cyclones are different.  Although the absence of the dust outlet geometry from the simulation domain will result in large savings in the computational effort, the present study indicated it can affect the estimated pressure drop and cut-off diameter significantly. If the cyclone performance parameters are estimated without the dust outlet geometry, an error margin of around 10% in the calculations of the Euler number and 35% in the calculation of the cut-off diameter should be taken into account.

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Table 5 The effect of dust outlet geometry on the cyclone performance.a

a

Cyclone

I

II (dustbin)

III (dipleg)

IV (dustbin plus dipleg)

Pressure drop (N/m2) Cut-off diameter (lm)

960 1.5

890 (7.29%) 1.0 (33%)

1017 (5.94%) 1.25 (16.7%)

1008 (5.0%) 1.2 (20%)

The values in parentheses is the percentage change with respect to cyclone I.

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