Influence of the dipleg and dustbin dimensions on performance of gas cyclones: An optimization study

Separation and Purification Technology 239 (2020) 116553

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Influence of the dipleg and dustbin dimensions on performance of gas cyclones: An optimization study

T

⁎

Khairy Elsayeda,b, Farzad Parvazc, Seyyed Hossein Hosseinid, , Goodarz Ahmadie a

Mechanical Engineering Department, College of Engineering and Technology-Smart Village Campus, Arab Academy for Science, Technology and Maritime Transport (AASTMT), P.O. Box 12676, Giza, Egypt b Mechanical Power Engineering Department, Faculty of Engineering at El-Mattaria, Helwan University, Masaken El-Helmia P.O., Cairo 11718, Egypt c Department of Mechanical Engineering, Semnan University, P.O. Box 35131-191, Semnan, Iran d Department of Chemical Engineering, Ilam University, Ilam 69315–516, Iran e Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699-5725, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Optimization CFD Dipleg Dustbin Gas cyclone

Gas cyclones have numerous industrial applications. Typically, each cyclone has a dustbin to collect the trapped particles and the dimensions of the dustbin affect the cyclone performance. This paper aims to optimize the dustbin geometry via numerical simulations. The surrogate-based optimization approach has been applied in this study. The Latin-hyper cube sampling plan is used to generate thirty test cases. An artificial neural network with radial basis function has been used as a surrogate model trained by the CFD simulations. Here three design parameters (the dipleg length, the dustbin height, and the dustbin diameter) and two performance parameters, namely, the Euler number and the Stokes number are used. The fitted surrogate model shows that the variations of the dustbin geometry have a larger effect on the Stokes number than that on the Euler number. Both singleobjective and bi-objective optimization studies are carried out using the artificial neural network. It is shown that the resulting optimum design of the dustbin and the dipleg leads to better performance than the conventional cyclones.

1. Introduction Cyclone separators are widely used in gas-solid separation due to their simplicity and lack of moving parts. In cyclone separators, the tangential inlet generates the swirling motion of the gas stream, which forces particles toward the outer wall where they spiral in the downward direction. Eventually, the particles are collected at the cyclone wall and in the dustbin; the gas flow reverses its downward motion and moves upward and escapes from the cyclone top via the vortex finder. Swirl and turbulence are the two phenomena that significantly affect the separation process. The swirl induces a centrifugal force on the solid phase, which moves the particles toward the wall and is the driving force behind the separation. Turbulence disperses the solid particles and enhances the probability that particles get caught in the exit stream as well as particle deposition on the wall. Both swirl and turbulence effects are functions of particle size and the flow conditions in the cyclone [1]. While the cyclone geometry is simple, the airflow is a rather complicated three-dimensional swirling flow [2]. The complexity of the gas-solid flow pattern in cyclones has long been a matter of many

⁎

experimental and theoretical studies [3–6]. More recently, Laser Doppler anemometry (LDA) (e.g., [7,8]) and particle image velocimetry (PIV) (e.g., [9–11]) are used to experimentally study the flow structure in cyclones. As for theoretical work, computational fluid dynamic (CFD) codes have proven to be a useful tool for simulation of flows in cyclones (e.g., [12,13]). The geometry of the cyclone significantly affects the flow pattern and separation performance [14,15]. The cyclone geometry is described by nine geometrical parameters, which are the inlet length a and width b, the vortex finder diameter Dx and length S, the cylinder height h, the cyclone total height Ht and the cone-tip diameter Bc, the cyclone dipleg length ldp, and the cyclone Dustbin diameter d. These parameters are given as a ratio of the cyclone diameter, D, as shown in Fig. 1a. A dustbin is typically attached to the dust outlet of conventional cyclones. Earlier experiments indicated that much gas flow enters into the dustbin through the dust outlet opening. However, because the bottom of the dustbin is closed, the gas flow would turn and re-enter into the separation space. The returning gas may carry some separated particles from the dustbin and bring them back into the inner cone

Corresponding author. E-mail address: [email protected] (S.H. Hosseini).

https://doi.org/10.1016/j.seppur.2020.116553 Received 29 September 2019; Received in revised form 24 December 2019; Accepted 10 January 2020 Available online 12 January 2020 1383-5866/ © 2020 Elsevier B.V. All rights reserved.

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Fig. 1. Cyclone geometry and surface mesh.

To summarize, the present study aims at the optimization of the dustbin geometry via numerical simulations. The surrogate-based optimization approach applied the Latin-hyper cube sampling plan to generate thirty test cases. An artificial neural network with radial basis function is used as a surrogate model trained by the CFD simulations. Accordingly, three design parameters, namely, the dipleg length, the dustbin height, and the dustbin diameter are varied to optimize two performance parameters (the Euler number and the Stokes number).

vortex, which leads to the so-called re-entrainment. This process will reduce the separation efficiency of the cyclone [16]. One possible solution to minimize the particle re-entrainment is using a cone insert underneath the separation space. 1.1. Literature review Many studies have been carried out to investigate the influence of different geometric parameters such as cyclone length, inlet and outlet pipe dimensions, and etc. on the performance of cyclones (e.g., [15–39]). However, little work concerning the dust outlet geometries has been reported in the literature [23–26]. For example, the optimum dimension and location of the apex-cone in the dustbin to enhance the collection efficiency has not been explored. Moreover, the gain in the collection efficiency and the corresponding pressure drop increase due to the installation of apex-cone in the dustbin are not known. In addition to the lack of relevant literature, there are discrepancies and uncertainties in the literature. For example, Xiang and Lee [27] reported that the dustbin connected to the cyclone should be included in the flow simulation domain as it affects the results. On the other hand, numerous studies on cyclone performance were performed without dustbin (e.g., [28,29]) that showed good matching with the experimental data. On the other hand, Obermair et al. [23] performed cyclone tests with five different dust outlet geometries to investigate the influence of the dustbin geometry on the separation process. They reported that the separation efficiency could be significantly improved by changing the dust configuration. They also noted that future research is needed to clarify the effects of dustbin geometry on cyclone performance. The relevant publications concerning the dust outlet are listed in Table 1 with a short description of the work. In the literature cited in Table 1, different apex designs were investigated briefly but at different testing conditions, which makes the comparison difficult if not impossible [23]. Furthermore, the main focus of the previous studies was on the cyclone collection efficiency with only a few studies on the effect of apex-cone on the flow pattern and the pressure drop in the cyclones.

2. Numerical model description 2.1. Governing equation of gas phase The gas flow is assumed to be incompressible and at a constant temperature. It is also known that gas flow in the cyclones is in the turbulent flow regime. Accordingly, the governing equations are the continuity and the Reynolds averaged Navier-Stokes (RANS) equations that are given by,

∂u¯i =0 ∂x i

(1)

∂u¯i ∂u¯ 1 ∂P¯ ∂2u¯i ∂ + u¯i i = − +ν − Rij ∂t ∂x i ρ ∂x i ∂x i ∂x j ∂x j

(2)

where x i stands for the position vector, u¯i is the mean velocity vector, P¯ is the time-averaged static pressure, ν is the gas kinematic viscosity, Rij represents the Reynolds shear stresses, and ρ stands for the constant gas density. The turbulence Reynolds Stress transport model (RSMT) is used to study the mean flow pattern inside the cyclone. The corresponding transport equation of this model is given as,

2

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Table 1 Earlier publication on dust outlet in a chronologic order. Authors

Comments

Barth [30,31]

Barth studied the re-entrainment of the separated dust caused by the central vortex. · He proposed installing an apex cone underneath the dust outlet to reduce the re-entrainment from the dustbin into the cyclone. Kecke experimentally confirmed that the flow pattern and the secondary flows in the lower part of the cyclone are of prime importance for the re-entrainment phenomena. He suggested to use a long connection tube to the dust bin (downcomer tube, aka dipleg) Muschelknautz stated that the apex diameter should be 1.2 to 1.4 times the vortex finder diameter. Mothes stated that by installing an apex cone the transport of the dust from the bin into the cyclone chamber is reduced. The position of the apex cone must be selected in such a way that the separated dust from the cyclone wall is easily collected into the bin, and prevent the gas flow from the bin back in the cyclone chamber. Kirch studied the effect of the apex cone on the flow pattern in the lower part of the cyclone. He stated that the apex cone affects the flow pattern in the whole cyclone. Cyclones with apex cone exhibit better collection efficiency especially when the dustbin is full. Rentschler compared three test cases: (i) cyclone without an apex cone, (ii) apex cone below the cyclone cone tip and (iii) apex cone in the cyclone cone tip (apex). He reported that the apex cone makes a significant enhancement in the separation efficiency when compared to a cyclone without an apex cone. Muschelknautz proposed putting the apex cone a little above the apex-end [24] Obermair and Staudinger stated that when a cone is inserted underneath the separation space the separation efficiency increases when compared to a cyclone with a dustbin only. They reported that the separation efficiency is further improved by using a downcomer tube or an additional step-cone, located between the cyclone and the dust bin (no cone insert). As a consequence of the high tangential velocities in the downcomer tube there is an additional separation. Therefore, particles which have once reached the downcomer tube will hardly be reentrained back into the cyclone chamber. Obermair et al. experimentally measured the velocity using LDA for the clean gas flow of a gas cyclone with three different dust outlet geometries. They stated that both the flow in the dust outlet geometry and the flow within the lower part of the cyclone change with outlet geometry. They reported that with a cyclone with a dust bin only, the potential vortex continues from the cyclone cone into the dustbin which will cause re-entrainment of the collected particles whereas for a cyclone with an apex cone, separation improvement is achieved compared to that found in a cyclone without an apex cone. They stated that the separation efficiency can be improved significantly by changing the dustbin geometry, and they reported that further research is needed to clarify precise effects of the dustbin geometry on the cyclone performance. Qian et al. investigated the prolonged cyclone (dustbin with dipleg). They applied CFD simulations using the Reynolds stress transport model (RSM) to predict the gas flow fields of the conventional and prolonged cyclones and to test the separation performances of three cyclones with different dipleg lengths. They indicated that the tangential velocity, axial velocity and turbulent kinetic energy in the dustbin reduced greatly when the prolonged vertical tube attaching into the dust outlet. I.e., the dustbin with dipleg can avoid the re-entrainment of already separated dust effectively. Furthermore, the prolonged vertical tube increases the separation space of dusts and consequently the separation performance at the cost of a slightly increased pressure drop. The downward flow rate into the dustbin of the prolonged cyclone decreases compared with the conventional cyclone. Kaya and Karagoz numerically investigated the particle separation process inside two cyclones prolonged with a dipleg. They stated that the length of the dipleg considerably influences the cyclone separation efficiency by providing more separation space and minimizing the reentrainment of separated particles) rather than the cyclone pressure drop, especially for lower inlet velocities in relatively short cyclones. They employed CFD tool to study the influence of dipleg shape on the performance of gas-solid cyclone. Among the models with various types of diplegs, the diamond dipleg and the inverted cone dipleg predicted the lowest pressure drop and highest collection efficiency, respectively.

Keck [32] Muschelknautz [33] Mothes [34]

Kirch [35] Rentschler [36]

Muschelknautz [37] Obermair and Staudinger [24]

Obermair et al. [23b]

Qian et al. [16]

Kaya and karagoz [38]

Parvaz et al. [23a]

Table 2 Cyclones dimensions.

Table 3 Used numerical schemes.

Dimension

Length (m)

Dimension rate (dimension/ D)

Barrel diameter (D) Inlet height (a) Inlet width (b) Gas outlet diameter (Dx) Vortex finder length (S) Cone-tip diameter (Bc) Cyclone height (Ht) Duct length (Li) Outlet tube length (Le) Dustbin height (Ldu)

0.205 0.105 0.041 0.105 0.105 0.076875 0.82 0.15375 0.1025

1 0.5 0.2 0.5 0.5 0.375 4 0.75 0.5 ⎧ 0.3075 0.41 ⎨ 0.5125 ⎩

Dustbin diameter (Dd)

Dipleg height (Ldp)

⎧ Ldu = 1.5D Dd = D Ldu = 2D ⎨ ⎩ Ldu = 2.5D

⎧ Dd = 1.5D Ldu = D Dd = 2D ⎨ ⎩ Dd = 2.5D ⎧ Ldp = 1.5D Ldu = Dd Ldp = 2D ⎨ L = 2.5D ⎩ dp

Scheme

Numerical setting

Body force weighted Simple Quick Second-order upwind Second-order upwind First-order upwind

Pressure discretization Pressure velocity coupling Momentum discretization Turbulent kinetic energy Turbulent dissipation rate Reynolds stresses

Table 4 Grid independency study results.

⎧ 0.3075 0.41 ⎨ 0.5125 ⎩ ⎧ 0.3075 0.41 ⎨ 0.5125 ⎩

Total number of cells

Static pressure drop

Total pressure drop

672,832 796,352 919,872 Difference (%)a

1253.15 1300.25 1313.65 4.8

1146.97 1174.29 1184.36 3.2

a

3

The percentage difference between the coarsest and the finest grid.

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Fig. 2. Current simulation validation by comparison with the experimental data of Hoekstra [47] and Zhao [51].

∂ ∂ Rij + u¯k Rij ∂t ∂xk =

2.2. Discrete phase modeling settings In the Eulerian-Lagrange approach followed in the present study, 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 [39]. This approach is broadly used in the numerical study of cyclone separator due to the low computational cost [25,39]. For low particle loading, it can be safely assumed that the presence of the particles does not affect the flow field and the one-way coupling assumption can be used [39] In this study, we also assumed that the collisions among particles are negligible. The high value of the particle to gas density

∂u¯j ε 2 ∂u¯ ∂ ⎛ υt ∂ Rij ⎞ − ⎡Rik + Rjk i ⎤ − C1 ⎡Rij − δij K⎤ − C2 ⎥ ⎢ k⎣ 3 ∂x i ∂xk ⎦ ∂xk ⎝ σ k ∂xk ⎠ ⎣ ⎦ 2 2 ⎡Pij − δij P⎤ − δij ε (3) 3 3 ⎣ ⎦ ⎜

⎟

where, the turbulent production is given by,

∂u¯j ∂u¯ 1 + Rjk i ⎤, P = Pij Pij = −⎡Rik ⎢ ∂xk ⎥ 2 ⎦ ⎣ ∂xk

(4)

ratio

hereP and νt are, respectively, the turbulence kinetic energy production and turbulent kinematic viscosity and σ k = 1, C1 = 1.8 and C2 = 0.6 are empirical constants. The transport equation for the dissipation rate is given as,

∂ε ∂ε ∂ ⎡ υ ∂ε ⎤ ε ∂u¯ ε2 ⎛υ + t ⎞ + u¯j = − C ε1 Rij i − C ε 2 ε ⎢ ⎥ ∂t ∂x j ∂x j ⎣ ⎝ σ ⎠ ∂x j ⎦ K ∂x j K 1

where k = 2 u′i¯u′i is the turbulence kinetic energy. σ ε = 1.3, C ε1 = 1.44 , and C ε2 = 1.92 are model constants, and.

(

ρp ρ

=

2770 1.225

)

≅ 2261 implies that the Basset force and the added

mass term are small and are therefore neglected [25,40,41]. For particles with small diameter, the rotational lift force and the Saffman lift force (due to the effect of shear on the particles) can be neglected in comparison to the drag force [25,40]. Moreover, the Brownian motion of the particles is not considered since the particles under study are larger than a few micron [25]. Particle tracking was done as a postprocessing step after obtaining the RANS solution where particle trajectories were individually computed [42]. In terms of the EulerianLagrangian approach (One Way coupling), the equation of particle

(5) Here

4

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Fig. 3. Tangential velocity at sections S1 and S9 for V = 19.5 m/s.

dx pi

motion is given by [13,23,25,39]:

gi (ρp − ρ) 18μ CDRep (ui − upi ) + = 2 dt ρp dp 24 ρp

dt

dupi

(6)

= upi

where the term 5

(7) 18μ CD Rep (ui 24

ρp dp2

− upi ) is the drag force per unit particle

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Fig. 4. Tangential velocity contours at Y = 0 for V = 19.5 m/s.

2.3. Computational geometry

mass[23,13]. Here CD is the drag coefficient, ui and upi are the gas and particle velocity in ith direction, respectively, gi is the gravitation acceleration in ith direction, Rep is the relative Reynolds number defined as [23,39]

Rep =

The cyclone geometry consists of two main parts. (1) The separation zone (the cyclone barrel and the cyclone cone) and (2) The dipleg and dustbin. The details of gas cyclone geometry dimensions are listed in Table 2.

ρp dp |u − up | μ

(8)

The drag coefficient for spherical particles is calculated by the correlation of Morsi and Alexander [43]. 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) [25]. The discrete phase modeling (DPM) study is performed by injecting 7000 particles from the inlet surface with a particle density of 2770 kg/m3 and with a particle diameter ranging from 0.5 to 6 µm with a velocity equal to the local gas velocity. The turbulent dispersion of small particles is taken into account by using the instantaneous fluid velocity ui = u¯i + ui′ In the drag term of particle equation motion. In this study, the stochastic discrete random walk (DRW) model is used to determine the fluctuating gas velocity [21,22]. The values of fluctuating velocities that prevail during the lifetime of a turbulent eddyτe , are assumed to have a Gaussian probability distribution and is evaluated as [45]:

u'i = ς u'i¯u'i

2.4. Boundary condition At the inlet section, the velocity inlet boundary condition is used. That is, the velocity vector and the turbulence intensity and the length scale are specific [25,45]. The velocity distribution at the inlet is assumed uniform. The turbulence intensity is estimated using −1

I = 0.16(ReDH ) 8 [25,46], where ReDh is the Reynolds number based on the inlet section hydraulic diameter and the average inlet velocity Uin In the present study, a turbulence intensity 5% and a characteristic length ℓ equal to 0.07 times the inlet width, are used [25,47]. The computed value of turbulent kinetic energy, k , is used to estimate the component of Reynolds stresses at the inlet under the assumption of isotropic turbulence; that is, the normal stresses are set to 2k 3 , and all shear stresses are set to zero [25,46]. An outflow boundary condition is imposed at the outlet (i.e., zero normal gradients of all solution variables except for pressure) [25,48,49]. The no-slip (wall) boundary condition is used at the cyclone solid surfaces [50]. Assuming an air inlet velocity of 19.5 m/s, air density of 1.225 kg/m3, and dynamic viscosity of 1.7894 × 10−5 kg/(m·s), leads to a Reynolds number of 3.18 × 104 based on the cyclone diameter and the area-averaged inlet velocity.

(9)

here ς is a normally distributed random number and the term u′i¯u′i is the local root mean square (RMS) of the velocity fluctuation in ith direction. In the present study, the characteristic life-time τe of an eddy is calculated as randomly varying about the Lagrangian integral time scaleTL , using τe = −TL Ln (r ) , where r is a random number between 0 and 1 and TL = 0.3k / ε [44].

2.5. Solver settings The CFD code ANSYS-FLUENT is used to solve the governing equations of fluid motion. The SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) method is used for velocity/pressure 6

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Fig. 5. The axial velocity profiles at sections S1 and S9.

model are listed in Table 3. The implicit coupled solution algorithm was also selected. The unsteady Reynolds stress transport turbulence model (RSM) was used in this study with a time step of 10−4 s.

coupling. In addition, to discretize the momentum term, the convective terms of the transport equations including the turbulence dissipation rate, and the Reynolds stresses the QUICK interpolation method, second-order upwind scheme, and the first-order upwind scheme, respectively. The details of applied discretization schemes in the present 7

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Fig. 6. The axial velocity contours of cyclones at Y = 0.

computer with sixteen nodes Intel Core i7- 2630QM, 2HZ. Fig. 1 shows the block structured grid of the computational domain. Table 4 presents the variations of the pressure drop and the cut-off diameter of cyclone I using the grid sizes of. 672,832, 796,352 and 919,872 tetrahedral cells. The resulting percentage differences between the tested grids reported in Table 4 indicate that the CFD results obtained with the 796,352 cells are essentially grid-independent. To accurately evaluate the numerical uncertainties in the computational results, the concept of grid convergence index (GCI) was adopted using three grid-density levels per cyclone. 2.7. Validation of the numerical model

Fig. 7. Variations of computed Euler number for different geometrical dimensions.

The model validation is done to ensure the reliability of the computational model. Accordingly, a comparison between the model predictions and the corresponding measured data of Hoekstra [47] and Zhao [51], respectively, for the continuous and discrete phases, and the results are shown in Fig. 2. This figure shows a closed agreement between the CFD model and the experimental data, which indicates that the computation model is capable of predicting the hydrodynamic of the gas cyclone with reasonable accuracy. Additional verification of the computation model was reported by the authors in their earlier works [21,22].

2.6. Grid independency study

3. Results and discussion

A grid independence study was performed for the computational mesh of the examined cyclones. Three levels of grid-density were tested for each cyclone, to make sure that the obtained results are grid independent. The tetrahedral computational grids were generated using the grid generator Workbench. The simulations were performed using the commercial CFD code ANSYS-Fluent 16 finite volume solver on a

3.1. Tangential velocity The swirling flow inside the cyclone has both tangential and axial velocity components that significantly affect the particle separation. The tangential velocity is the most important component in gas cyclones. In order to provide a physical description of the tangential velocity in different cyclone models, the tangential velocity at two 8

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Fig. 8. Grade efficiency curves for different geometries.

change in cyclone performance for particle separation.

sections S1 = z/D = 0.75 and S9 = z/D = 2.75 for different geometries of the dipleg and dustbin are computed. Fig. 1 shows that the nearest and furthest levels from the cyclone top are, respectively, sections S1 and S9. The axial velocity profiles on these levels for different cases are also evaluated and discussed. The effect of changing the geometrical/design parameters (the dustbin diameter/height and the dipleg height) on the tangential velocity at sections S1 and S9 is presented in Fig. 3. It is seen that the variations of dustbin diameter, dustbin height, and the dipleg height noticeably affect the tangential velocity profiles at sections S1 and S9, which are in the cyclone particle separation zone. Fig. 4 shows the contour plots of tangential velocity at a section across different cyclone geometries. It is seen that the behavior of the tangential velocity for different dustbin and dipleg dimensions is roughly the same. However, it is also seen that the tangential velocity for different cases is quite different in the dipleg and dustbin regions. Therefore, the variation in height of dipleg and height and diameter of dustbin may lead to a

3.2. Axial velocity The axial velocity in the cyclone is influenced by the operating conditions and geometrical factors such as the inlet velocity and the cyclone configuration. The axial velocity could be downward or upward. The maximum and minimum axial velocities appear typically in the regions near the core vortex finder and near the wall, respectively. Fig. 5 shows the mean axial velocity at sections S1 and S9 for different dustbin and dipleg dimensions. It is observed that the dimension of dustbin and dipleg significantly affect the axial velocity profiles, especially at level S9, that is at the middle of the conical section of the cyclone. The contour plots of axial velocities at a section across the cyclone (Y = 0) for different cases are shown in Fig. 6. This figure reveals that 9

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Table 5 Input data to the surrogate model. No.

x1 (Ldp)

x2 (Ldu)

x3 (Dd)

Eu

Stk50 × 106

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.825 2.175 1.025 1.575 1.325 1.225 1.425 1.675 1.725 1.925 1.875 2.075 1.525 2.325 1.075 2.025 1.775 1.975 1.275 1.475 2.225 2.475 1.625 2.125 2.275 2.425 1.175 1.125 1.375 2.375

1.175 1.675 2.425 1.825 2.075 1.625 2.025 2.325 1.025 2.225 2.475 2.175 1.475 2.275 1.225 1.725 2.375 1.575 1.525 1.275 1.075 1.975 1.375 1.425 1.775 1.125 1.875 1.325 1.925 2.125

1.075 1.475 1.975 1.525 1.025 1.675 1.925 1.625 2.425 1.725 2.325 1.125 1.175 2.175 1.425 1.225 2.375 1.325 2.075 2.025 2.225 1.825 2.125 1.375 1.275 2.275 1.575 2.475 1.875 1.775

5.3662 5.2747 5.2320 5.2643 5.2907 5.2661 4.9954 5.0294 5.0352 5.0958 5.0190 5.0325 5.0087 5.0555 5.0613 5.0336 5.0465 5.0095 5.0656 5.0360 5.0487 5.0278 5.0116 5.0602 5.0108 5.0434 5.0648 5.0695 4.9882 5.0374

4.05 5.29 10.01 4.90 13.97 5.29 2.98 4.05 6.70 4.05 4.05 6.70 6.70 8.27 6.70 2.07 4.05 2.98 13.97 6.70 5.29 1.07 6.70 2.98 5.29 4.05 4.05 6.70 2.98 2.07

shows the performance of a cyclone is the separation efficiency. The cyclone collection efficiency is defined as,

Collection efficiency =

the axial velocity changes in the conical section of the cyclone with a change in the dustbin height and diameter as well as the dipleg height. It should be emphasized that the axial velocity in the dipleg markedly changes with the dipleg height. 3.3. Pressure drop

3.4. Dustbin geometry optimization

The highest static pressure for a cyclone occurs on the wall while the lowest pressure appears in the central axis of the cyclone, away from the wall. This is due to the high rotational velocity in the core region. The dimensionless group for the pressure drop is Euler number, which is defined as,

From the previous discussion, to obtain the optimum dimensions of the dustbin, an optimization study is required. Whereas ANSYSWorkbench has a tool for conducting the optimization directly, but this process will take a long time and the computational cost is quite high. An alternative approach that has been successfully used in many studies is the surrogate-based optimization (SBO) [52–54] The first step in SBO is the design of experiment (DoE) [55], which is a statistical approach to minimize the required number of experiments for studying the input-output relationship of a system. The fidelity of the surrogate model relies on the used sampling plan. The stratified sampling plans, such as full factorial and central composite design of experiments, are widely used for polynomial surrogates [56,57]. These plans are known as space-filling and require a uniform spread of points; consequently, it leaves a gap in the projections [57]. The Latin hypercube sampling (LHS) [57] is a better alternative to the stratified sampling plan. In LHS, the design space is split into equal-

ΔP 1 ρuin 2 2

(11)

Fig. 8 shows the collection efficiency curves for different dustbin heights and diameters and dipleg heights. Here the calcium carbonate particles are considered with the Rosin-Rammler distribution where the particle diameters are in the range of 0.05–8 µm. It is seen that the noted design parameters (dustbin heights and diameters and dipleg heights) affect the collection efficiency for different particle sizes. It can be found that in order to reach the best model in terms of grade efficiency the optimization of these parameters is required. In summary, Figs. 3–8 show that the dustbin height and diameter, as well as the dipleg height, affect the performance of a gas cyclone and their best values should be obtained by an optimization process.

Fig. 9. Design of experiment using LHS.

Eu =

The mass of the trapped particles The total mass of the injected particles

(10)

The Euler number (non-dimensional pressure drop) for different cyclones is computed and the results are shown in Fig. 7. This figure shows that the Eu number is significantly affected by the variation in dustbin diameter and height as well as the dipleg height. This figure shows that the pressure drop reaches a minimum when the dustbin diameter is twice the cyclone body diameter or when dustbin height is 2d. Therefore, an optimal design for the cyclone could be obtained. In addition to the pressure drop, the important parameter that 10

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Fig. 10. Agreement between the training data and the output of the surrogate models.

Fig. 11. Elementary effect distribution.

sized bins and placing points in the bins (one in each), assuring that from each occupied bin one could exit the design space along any direction parallel with any of the axes without encountering any other occupied bins [57]. In this study, we utilized the Latin-hypercube sampling plan (LHS) for three design variables x1, x2, x3, namely, the dipleg length (Ldp), the dustbin length (Ldu) and the dustbin diameter (Dd), respectively. The range of non-dimensional values of these variables (divided to the cyclone diameter to become dimensionless) are from a minimum of 1 to a maximum of 2.5. Fig. 9 shows the output of the design of experiment (DoE) where the training point cover the domain, uniformly [56,57]. Thirty points have been generated in the DoE step, and these 30 cyclones geometries and meshes were generated following the same settings as described in the previous sections. After conducting the simulations, the next step is training the surrogate model. The widely used surrogate models in the literature are [57]: (1) Polynomial regression models that are suitable to mimic the input-output relationship of (maximum) quadratic relationships. (2) Artificial neural networks (ANNs) that have been used in numerous engineering studies [56,57]. The main advantages of ANN are: (a) It is available in many toolboxes such as Matlab. (b) It can handle highly nonlinear relationships. (c) It can easily handle a large number of inputs and outputs. In this study, the artificial neural network with radial basis function was used to

simulate the input-output relationship. Additional details of the approach are provided in [55–60]. The input data to the surrogate model is listed in Table 5. The accuracy of the fitted artificial neural network is checked in Fig. 10 using all the training dataset, where the mean squared error is less than 10−5. The third step is to use the fitted surrogate model in understanding the effect of changing the dustbin geometrical parameter on the cyclone performance. Here, we plotted the elementary effect distribution suggested by Forrester et al. [61] for identifying the most significant parameters, in Fig. 11. From Fig. 11, and since the sample mean for the three variables are not zero, all tested parameters affect the performance of the cyclone (both the Euler and the Stokes numbers). Moreover, from the value of the sample standard deviation, there are interactions between the variables that have higher-order (nonlinear) effect. The Euler number (pressure drop) is more sensitive to the dipleg length than to the other the dustbin length and diameter. The effect of the dustbin geometrical parameters on the Stokes number is stronger than that for the Euler number. This can be deduced from the y-axis values of the elementary effect distribution plot. Moreover, the effects of these parameters are the opposite of each other. The dustbin height and the dipleg length have a negative effect on the pressure drop and positive effect on the Stokes number. 11

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Fig 12. Three-dimensional surface plots.

Objective

x1 (Ldp)

x2 (Ldu)

x3 (Dd)

Value

mid-value of the range considered in this study. The last application of the surrogate model is the optimization study. Two optimization studies have been conducted, the single objective for each of the two performance parameters and multi-objective optimization using the genetic algorithms. Table 6 lists the used settings for single-objective optimization studies. Table 7 lists the optimized values of the dependent variables for the best values of Euler and Stokes numbers (using single-objective optimization). The multi-objective genetic algorithm (MOGA) optimization study has been conducted using the default settings for the non-dominated sorted genetic algorithm (NSGA-II). Additional details of the algorithm were reported in [55–59]. The Pareto front points are listed in Table 8, corresponding to Fig. 13, where a trade-off between the two objectives is obvious. The decision-maker can select from the Pareto front values based on his/her preferences and needs.

Euler number Stk50 × 106

1.38 1.56

1.925 1.53

1.88 1.76

4.7 0.02

4. Conclusions

Table 6 Genetic operators and parameters for single objective optimization. Double vector

Population type

Rank Tournament (tournament size equals 4) 2 0.8 The constraint dependent default 1000 200

Fitness scaling Selection operation Elite count Crossover fraction Mutation operation Maximum number of generations Population size

Table 7 Optimum values for single objective optimization.

The effect of the dustbin geometry on the performance of gas cyclones was studied. In the first stage of this study, seven cyclones with different values of the dipleg height, the dustbin length, and the dustbin diameter were simulated. The analysis of the results revealed that these parameters affect the cyclone performance. In the second stage of the work, a surrogate-based optimization study was performed. Thirty test cases were selected using the design of experiment approach, and the corresponding cyclone performances were simulated using the ANSYSFluent code. Then both single and bi-objective optimization studies were conducted. The obtained results can be summarized as follows: The three parameters (dipleg height, dustbin length and dustbin

Another application of the surrogate model is the 3D surface plots shown in Fig. 12 together with the contour plots. These plots support the choice of ANN instead of the polynomial regression, as the effects of the majority of the tested geometrical parameters are non-linear. The Euler number varies linearly (except at the mid-range values due to interaction between the geometrical parameters). The Stokes number (and consequently the collection efficiency) is also a nonlinear function in the dustbin diameter with an optimum value close to the mid-values of the dustbin diameter and the dustbin length. It is worth to mention that in the shown 3D plots, the third parameter is kept constant at their 12

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Table 8 Pareto solution points using MOGA. No.

x1 (Ldp)

x2 (Ldu)

x3 (Dd)

Eu

Stk50 × 106

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

1.49995 1.49952 1.49984 1.49995 1.49994 1.49995 1.5 1.49931 1.49991 1.49977 1.49982 1.49973 1.4997 1.49969 1.49968 1.49982 1.49992 1.49981 1.49998 1.49989 1.49969 1.49983 1.49993 1.5 1.49984 1.49994 1.49973 1.49992 1.49982 1.49976 1.49983 1.49985 1.49998 1.49978 1.49991 1.49991 1.49976 1.49994 1.49984 1.49979 1.4998 1.49993 1.49956 1.49976 1.49974 1.49976 1.49932 1.49986 1.4999 1.49984 1.4999 1.49991 1.49976 1.49988 1.49981 1.49977 1.4997 1.49955 1.4998 1.49953 1.49992 1.49995 1.49996 1.49991 1.49977 1.49969 1.49998 1.49995 1.4997 1.49994

1.85789 1.85708 1.8597 1.85696 1.85773 1.85679 1.8565 1.85798 1.85785 1.85781 1.85733 1.85823 1.85886 1.86048 1.85967 1.8588 1.85804 1.86043 1.85681 1.86135 1.85924 1.86094 1.86021 1.8565 1.85682 1.85819 1.85741 1.85864 1.85992 1.86108 1.85996 1.85818 1.85664 1.86027 1.86145 1.86158 1.8576 1.85724 1.8574 1.85814 1.85776 1.85784 1.85967 1.85776 1.85822 1.85852 1.85793 1.86013 1.85936 1.85776 1.86052 1.85849 1.8583 1.85886 1.86028 1.85895 1.85828 1.86012 1.85974 1.85829 1.85845 1.85674 1.85962 1.86205 1.86136 1.85759 1.85923 1.857 1.85752 1.85962

1.99897 1.98225 1.97022 1.9915 1.99796 1.99996 1.85327 1.99512 1.99365 1.97872 1.99258 1.98138 1.96629 1.92915 1.94973 1.90618 1.98727 1.96088 1.87747 1.92804 1.9395 1.90804 1.962 1.85327 1.88159 1.88903 1.99181 1.95927 1.9575 1.9641 1.9332 1.9861 1.86421 1.94025 1.91441 1.97174 1.99469 1.87148 1.89116 1.96761 1.88764 1.97538 1.94686 1.91212 1.92059 1.90898 1.89799 1.94423 1.95226 1.94874 1.94568 1.96304 1.95846 1.9759 1.97378 1.95521 1.90996 1.94116 1.96902 1.95394 1.98125 1.88377 1.91615 1.90426 1.93451 1.91759 1.86031 1.98956 1.89618 1.91908

4.74589 4.77468 4.79458 4.75869 4.74762 4.7442 4.98153 4.75284 4.75502 4.78047 4.75691 4.77596 4.8013 4.86213 4.82869 4.89924 4.76587 4.81009 4.94469 4.86364 4.84554 4.89572 4.8082 4.98153 4.93838 4.92632 4.75827 4.81292 4.81575 4.80471 4.8556 4.76789 4.96501 4.84408 4.88544 4.79187 4.75332 4.95375 4.92329 4.79912 4.92869 4.78602 4.83348 4.8901 4.8764 4.89488 4.91266 4.83756 4.82444 4.83054 4.83511 4.80666 4.81439 4.7851 4.78859 4.81969 4.89342 4.84273 4.79661 4.82201 4.77608 4.93498 4.88308 4.90137 4.85325 4.8814 4.96992 4.762 4.91546 4.8784

1.15797 0.977493 0.838961 1.07201 1.14652 1.1707 0.150107 1.12795 1.09683 0.932489 1.08703 0.962169 0.802293 0.469097 0.642052 0.315005 1.02358 0.744478 0.189701 0.456168 0.552476 0.324565 0.752967 0.150107 0.204951 0.229965 1.08024 0.727492 0.711545 0.777151 0.4976 1.01194 0.160547 0.556193 0.361624 0.852071 1.11276 0.175003 0.240875 0.814195 0.227228 0.892797 0.618551 0.351646 0.407134 0.332424 0.281989 0.588655 0.660708 0.630898 0.60044 0.765078 0.723396 0.898561 0.876659 0.691536 0.339423 0.568953 0.827291 0.685196 0.956069 0.210463 0.372407 0.301687 0.508651 0.388311 0.155453 1.04989 0.265964 0.392217

Fig. 13. Pareto front for the two-objective optimization (cf. Table 8).

diameter) affected the cyclone performance, but their effect on the Stokes number was higher than that on the pressure drop. There were also couplings between these three variables which was more significant for the Stokes number.

• For minimum pressure drop, the dipleg height, the dustbin length • •

and the dustbin diameter should be 1.38, 1.925 and 1.88 times the barrel diameter, respectively. The optimization study resulted in a rather large improvement in the Stokes number (with respect to the minimum value in the training data) and marginal improvement in the Euler number. The multi-objective optimization study resulted in a Pareto front design points that facilitated the designer to make decisions according to the selection criterion.

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