Multi-objective optimization of a gas cyclone separator using genetic algorithm and computational fluid dynamics

Powder Technology 325 (2018) 347–360

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Multi-objective optimization of a gas cyclone separator using genetic algorithm and computational fluid dynamics Xun Sun a, Joon Yong Yoon b,⁎ a b

Department of Mechanical Design Engineering, Hanyang University, 55, Hanyangdaehak-ro, Sangnok-gu, Ansan, Gyeonggi-do 15588, Republic of Korea Department of Mechanical Engineering, Hanyang University, 55, Hanyangdaehak-ro, Sangnok-gu, Ansan, Gyeonggi-do 15588, Republic of Korea

a r t i c l e

i n f o

Article history: Received 3 August 2017 Received in revised form 12 September 2017 Accepted 3 November 2017 Available online 8 November 2017 Keywords: Cyclone separator Multi-objective optimization Response surface methodology Genetic algorithm Computational fluid dynamics

a b s t r a c t In the present study, multi-objective optimization of a gas cyclone is performed using a genetic algorithm (GA) and computational fluid dynamics (CFD) techniques to minimize pressure drop and maximize its collection efficiency. The reference model is a well-optimized cyclone from a previous study. First, a screening experiment for seven factors is performed to determine the statistically significant factors. Then, to define the fitness function used in the GA, four of the factors are studied using the central composite design in the response surface methodology. The second-generation non-dominated sorting genetic algorithm is utilized to optimize the four significant factors of the cyclone according to the well-defined fitness functions, and 53 non-dominated optimum cyclone design points are suggested. The reasonable accuracy of the results from the GA is confirmed via CFD validation of five representative optimum points. The obtained Pareto front comprises important design information for the new cyclones. Finally, the performance and flow field of a representative optimal design are compared with those of the classical Stairmand model and the reference model. The optimal design reduces the pressure drop and cut-off size by 7.38% and 9.04%, respectively, compared to the reference model. In addition, compared to the Stairmand model, decreases of 19.23% and 42.09% are achieved for the pressure drop and cut-off size, respectively. © 2017 Elsevier B.V. All rights reserved.

1. Introduction A gas cyclone separator is a kind of mechanical equipment that removes solid particles from polluted gas via the centrifugal force resulting from a tangential flow, rather than gravity. Compared with filtration and wet scrubbers, a cyclone has the advantages of a simple structure, lack of moving parts, ease of maintenance, low-operating cost, high-reliability, and high-collection efficiency. It is one of the most robust, efficient, and economical methods of fine dust collection, and has been widely utilized in a large number of heavy and light industrial applications, even under harsh operating conditions. In general, two critical parameters of the cyclone are used to evaluate its performance and are of great importance to engineers and designers: the pressure drop through the cyclone and collection efficiency. These two parameters represent the performance of the collection operating process, with respect to economic efficiency and effectiveness, respectively. They are greatly dependent on the geometrical and operational factors, and are generally in direct contrast with each other.

⁎ Corresponding author. E-mail addresses: [email protected] (X. Sun), [email protected] (J.Y. Yoon).

https://doi.org/10.1016/j.powtec.2017.11.012 0032-5910/© 2017 Elsevier B.V. All rights reserved.

In order to extensively study the effects of the geometrical and operational factors on performance, namely, pressure drop and collection efficiency, researchers have conducted a large number of experimental investigations regarding cyclones [1–18]. To predict cyclone performance more conveniently, several important empirical models have been established according to massive experimental data, including the Lapple model (1950) [19], Barth model (1956) [20], Dietz model (1981) [21], Iozia and Leith model (1989) [22], and Muschelknautz model (2008) [23]. Because they are only able to predict acceptable results compared with experimental data for limited factors in the model, their utilization is not convenient for the mutable cyclone geometrical design. Recently, the computational fluid dynamics (CFD) technique became a powerful tool for predicting cyclone performance with the advantages of low cost, high robustness, and accuracy [24–31]. Many numerical studies on the geometrical or operational effects on cyclone performance have been proposed over the last twenty years [17,25,32–49]. However, in general, the experimental and numerical works have been concentrated only on the individual effect of a single factor on cyclone performance, and analyses on the interaction effects between two factors have been absent. The performance improvement resulting from a single factor or part optimization is very limited. To ascertain the interaction effects of several factors and further enhance

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the performance, multi-objective optimization with trade-offs for a number of factors must be implemented. Lastly, many optimization studies have been presented using the response surface methodology (RSM) or COMPLEX method [50–54]. Alternatively, the genetic algorithm (GA) is also a powerful method for cyclone optimization. The GA has been successfully utilized in many industrial applications. It is based on the process of natural selection with mutation, crossover, and selection, and was first presented by Holland in 1975 [55]. Unlike traditional optimizations, the GA normally provides a set of several equally desirable, optimal points. Recently, some cyclone optimization studies using the GA have been conducted. Ravi et al. [56] presented a multi-objective optimization of an industrial cyclone with eight geometrical parameters using the nondominated sorting genetic algorithm (NSGA). Prediction of the performance for the various factors was obtained from the empirical model. It was found that the optimal geometry under the reference conditions is significantly different from that of the Stairmand high-efficiency cyclone under the suggested conditions. Elsayed and Lacor [57] performed a multi-objective cyclone optimization study for seven factors using the radial basis function neural network (RBFNN) and secondgeneration NSGA (NSGA-II). Two RBFNNs for each objective were trained and tested using the experimental data and empirical model. Both studies used the empirical model to evaluate the performance. Even though the use of the empirical model for optimization is much more convenient and time efficient than using CFD, owing to the lowaccuracy limitation of this type of model for complex geometries, the use of the empirical model is not suitable for cyclone optimization. Safikhani et al. [58] utilized the group method of data handling (GMDH) type of artificial neural networks (ANNs) and NSGA-II to optimize the geometry of a classical cyclone based on CFD simulation results. Safikhani [59] proposed a successful multi-objective optimization of a new Karagoz cyclone using a combination of CFD, GMDHtype ANNs, and NSGA-II techniques. Both works obtained very important information and trade-offs in the optimum designs according to the Pareto fronts of objective functions. Nevertheless, these two works have only studied four factors; some other significant factors were not included, and therefore improvement of the cyclones was relatively limited. Elsayed and Lacor [60] used the RSM and desirability function to optimize four factors of a Stairmand design cyclone using CFD. The new design achieved superior performance. In addition, they also utilized two RBFNNs as an alternative approach for applying the RSM as a meta-model to implement a multi-objective GA optimization. Because the error between the performance of the new design resulting from the RSM model and that resulting from CFD for two responses in the RSM portion reached 17.20% and 20.23%, respectively, the quality of the obtained RSM models is very limited and the accuracies of the models and results are dissatisfactory. Moreover, both the RSM and GA optimization processes only included four geometrical factors: inlet height, inlet width, vortex finder diameter, and cyclone height. Brar and Elsayed [61] performed single-objective and multiobjective optimizations of a multi-inlet gas cyclone using the RBFNNs and GA for three variables, viz., the flow rate through the secondary inlet, cross-sectional area of the secondary inlet, and location of the top of the main inlet from the cyclone roof. The responses of the Euler number, cut-off size, and global collection efficiency were carried out using the large eddy simulation (LES). Although the accuracy of the numerical simulation was validated before the optimization process, the correctness of the GA results was not confirmed by CFD simulations of several representative optimal points after the optimization process. To ensure the results' accuracy, the validation should be implemented. In the history of the development of gas cyclone separators, the amount of published optimization research is still very small compared to the experimental and numerical cyclone studies. Limited information on the interaction effects has been obtained. Furthermore, for the cases of cyclone optimization research using the most popular optimization

methods, RSM and GA, the accuracy of the response model or fitness function is extremely critical for the reliability of the optimization process and final optimization result, but previous studies have paid less attention to this accuracy. In addition, all the previous optimization works derived the optimal model from an original cyclone design, such as the Stairmand cyclone. To further improve the performance, optimization starting with an optimal model is required because a huge improvement in performance can be easily achieved using any modern optimization techniques rather than a classical, ancient model. The present paper performed a multi-objective cyclone optimization using the RSM, GA, and CFD techniques. The chosen objectives of the optimization process were to minimize the pressure drop and maximize the total efficiency. The cyclone design that was well-optimized by the RSM in the author's previous work [53] was selected as the reference model instead of a classical cyclone design. Fig. 1 and Table 1 present the geometrical configuration and the values of the corresponding

Fig. 1. Schematic diagram of the optimal cyclone separator in the previous study [53] as the reference model.

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2.2. Discrete phase model

Table 1 Geometrical factors of the reference cyclone [53] (cyclone diameter D = 290 mm). Factor

a/D

b/D

Dx/D

Ht/D

h/D

S/D

Bc/D

Hc/D

Dc/D

Lc/D

Value

0.566

0.2

0.451

6

1

0.2

0.5

1

0.5

1.5

factors in the reference model, respectively. Seven factors, which include the inlet height a, inlet width b, vortex finder diameter Dx, cylinder height h, vortex finder length S, cone tip and collector diameter Bc(Dc), and surface roughness e [44], were considered in the optimization. Because increasing Ht gives rise to the natural vortex turning phenomenon which leads to a sudden drop in efficiency for overlong cyclones [14,17,23], Ht was not considered in the optimization and was set as 6D. In addition, the particles carried in the downward flow in the cyclone can be significantly disturbed by inconsistent values of the cone tip diameter Bc and collector diameter Dc [53], deteriorating the collection efficiency; thus, Bc and Dc were combined as a single factor. Before implementing the RSM, four statistically significant factors were filtered from the seven factors. The central composite design (CCD) was applied to establish two second-order response surface models for the responses as the fitness functions used in the GA. A set of optimal design points was obtained from the Pareto front of the GA optimization with a well validation using CFD. Finally, the performance and flow field of the Stairmand model, reference model, and a representative optimal model were compared and analyzed.

To achieve the collection performance of the cyclone, particle trajectories in the cyclone are modeled using the discrete phase model (DPM) in ANSYS Fluent 17.1, which is based on the Eulerian–Lagrangian approach. The fluid phase is treated as a continuum by solving the Navier-Stokes equations, while the dispersed phase is solved by tracking certain numbers of particles through the calculated gas flow field [64]. Even though previous studies [49,65] have found the existence of interaction between particle motion and gas flow field, because the particle loading of this work is low and to simplify the simulation process, it was assumed that the particles do not affect the flow field. According to the Eulerian-Lagrangian approach (one-way coupling), the equation of particle motion can be written as [53,64]
 ! ! ! ! ! dup u − u p g ρp −ρ ¼ þ þ F ρp dt τr

ð2Þ

! where F is an additional acceleration (force per unit particle mass) ! ! u−up is the drag force per unit particle mass, and the droplet term, τr or particle relaxation time is defined as. 2

τr ¼

2. Numerical method 2.1. Turbulence model In CFD simulations, to accurately predict the flow behavior for various categories of fluid flows, it is very important to select an appropriate turbulence model. The Reynolds-averaged Navie-Stokes (RANS) turbulence models, which are based on the Boussinesq eddy-viscosity assumption, have been widely used in a great number of industrial applications and have shown clear advantages, especially for wallbounded flows. In cyclone simulations, however, the performances of these models are very limited. They generally result in an unrealistic tangential velocity profile, similar to a solid-body rotation, instead of the measured near-Rankine vortex profile [23]. This is because the flow field of cyclones is dominated by a high-swirling flow that has significant turbulent transport or non-equilibrium effects, and the RANS models are under the isotropic eddy-viscosity hypothesis [62, 63]. As a result, all the RANS models fail to satisfactorily capture the flow behavior and predict the performance of cyclones [23,28,34,37, 38,42,44,47]. The Reynolds stress model, which abandons the isotropic hypothesis, can predict a much more reasonable flow field and cyclone performance compared to the RANS models, with an acceptable amount of additional computing time. Although the LES gives better quantitative agreement with the laser Doppler anemometry (LDA) experimental data [26,29], considering that the LES requires several orders of magnitude greater computing resources than the Reynolds stress model, the latter is more suitable for cyclone optimization. A description of the Reynolds stress model can be found in many cyclone research papers and is not repeated here. To avoid deterioration of the standard wall functions under a grid refinement with y∗ b 11, scalable wall functions were used for near-wall treatment. The y∗ for the standard wall functions is replaced by ~
¼ MAX ðy
; y
limit Þ y where y
limit ¼ 11:225 [64].

ð1Þ

ρp dp 24 18μ C d Re

ð3Þ

! ! Here, u is the fluid phase velocity, u p is the particle velocity, μ is the molecular viscosity of the fluid, ρ is the fluid density, ρp is the density of the particle, dp is the particle diameter, and Re is the relative Reynolds number defined by. Re ≡

! ! ρdp  u p − u  μ

ð4Þ

For smooth particles, the drag coefficient can be defined by the spherical drag law, taking the constants from Morsi and Alexander [66]. 2.3. Modeling of the surface roughness The effect of the surface roughness can be modeled using the modified “law of the wall.” In ANSYS Fluent [44,64,67], the law of the wall for the mean velocity modified for the roughness is defined as   ρu
yp up u
1 ¼ ln E ‐ΔB τw =ρ κ μ

ð5Þ

where up is the mean velocity of the fluid at the wall-adjacent cell centroid, yp is the distance from the centroid of the wall-adjacent cell to the wall, τw is the wall shear stress, ρ is the fluid density, κ is the von Kármán constant (= 0.4187), E is an empirical constant (9.793), and μ is the dynamic viscosity of the fluid. The friction velocity can be written as u
¼ C 1=4 μ k

1=2

ð6Þ

where Cμ is a constant and k is the turbulence kinetic energy. Furthermore, ΔB is correlated to the non-dimensional roughness ∗ height, K+ s = ρKsu /μ, where Ks is the physical roughness height. For the hydrodynamically smooth regime (K+ s ≤ 2.25): ΔB ¼ 0

ð7Þ

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For the transitional regime (2.25b K+ s ≤ 90): ΔB ¼

 þ  

1 K −2:25 þ ln S þ CSK þ S  sin 0:4258 ln K S −0:811 κ 87:75

ð8Þ

where CS is a roughness constant (=0.5 in this study). In the fully rough regime (K+ s N 90): ΔB ¼

 1 ln 1 þ C S K þ S κ

ð9Þ

2.4. Solver For obtaining the responses, the performances of cyclones in the optimization process were evaluated by the commercial CFD code ANSYS Fluent 17.1. The three-dimensional, incompressible NavierStokes equations were discretized using the finite volume method. To couple the pressure and velocity, the SIMPLE algorithm was used. The second-order upwind scheme was used for the momentum, turbulence kinetic energy, and turbulence dissipation rate equations. The firstorder upwind scheme was used for the Reynolds stress equations together with the PRESTO interpolation scheme for pressure [53,68]. Because the pressure drop through the cyclone increases with the progressing iteration, to obtain a reasonable, well-converged result, it is necessary to implement sufficient time steps in the transient simulation to reach a stable pressure drop between the inlet and outlet (i.e., Δp = constant) after a certain steps of steady-state iteration [68]. Therefore, every numerical simulation of all the experimental runs in the optimization was obtained by 1000 steady-state iterations followed by 100 time step transient iterations. The time step size and iteration number per time step were specified as 0.01 s and 100, respectively. The maximum number of steps and tolerance of particle tracking for the DPM were set as 106 and 1e− 6, respectively.

Fig. 2. Grid independence test for the reference model for vin = 14.37 m/s (x = 0, y = 6.25D).

10−5 N × m2 for general room temperatures, respectively. The velocity inlet condition was implemented at the cyclone inlet and velocity was set as a fixed air volume flow rate Q in = 0.1374 m3/s (for the reference model, vin = 14.37 m/s). The turbulence quantities at the inlet were uniformly specified by the turbulence intensity I = 0.16(ReDH)−1/8 and hydraulic diameter DH ¼ AP, where A is the cross-sectional area and P is

2.5. Grid generation and grid independence test Because of better accuracy and shorter computing time compared to a tetrahedral grid, the structure grid (or hexahedral grid) has been widely used in many applications, especially for cyclone simulations. In cyclone optimization, a large number of simulations for each run with different geometries are required. Then, these structure grids for each different case should be individually generated, which costs a large amount of time. The polyhedral grid is an alternative option that could be generated automatically. It has been also shown that the polyhedral grid is more accurate and costs less CPU time than the hexahedral grid in cyclone simulations, with the additional advantage of the automatic grid generation process [53,69]. All meshes used in the simulations were generated as polyhedral grids using ANSYS Meshing 17.1 and ANSYS Fluent 17.1. Because the cyclone geometries for each run are not identical, the maximum edge size and face size of each cell were chosen as the index of the grid's resolution, instead of the total cell number. Fig. 2 presents a grid independence test that was carried out for five various grid resolutions for the reference model, corresponding to 7.5 mm, 8.0 mm, 8.5 mm, 9.0 mm, and 9.5 mm for the maximum edge size and face size of each cell. For the current study, a grid resolution of 8.0 mm for the maximum edge size and face size was utilized to predict the grid-independent simulation results with a consideration of economy and grid sensitivity. Fig. 3 shows the generated polyhedral grid for the reference model. 2.6. Boundary conditions The boundary conditions for all the runs can be found in Table 2. The air density and viscosity were specified as 1.204 kg/m3 and 1.81 ×

Fig. 3. Polyhedral grid generation for the cyclone simulation.

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Table 2 Boundary conditions used in the numerical simulation. Boundary

Boundary condition

Inlet Outlet Wall (except for the bottom of the dust collector) Bottom of the dust collector

Category

Discrete phase model

Velocity inlet Pressure outlet Wall Wall

Escape Escape Reflect Trap

the wetted perimeter [64,70]. The outlet condition was set as the atmospheric pressure [53]. Because the surface roughness was taken as a factor in this work, the roughness heights of the cyclone walls were set to various values and were modeled as mentioned in Section 2.3. In addition, the walls were set as a no-slip condition, and collisions between particles and the walls were assumed to be perfectly elastic. For obtaining the total efficiency, a fixed number of 2500 particles with a density of 2700 kg/m3, which was consistent with the validated experiment, were released at the cyclone inlet for each case. The particle size distribution was defined as the Rosin-Rammler diameter distribution with 10 diameter classes. The mass fraction of the particles with diameters greater than d is given by. n

Y d ¼ e−ðd=dÞ

(a) Tangential velocity

ð10Þ

where the mean diameter d and spread parameter n were set as 5 μm and 3.5, respectively, and the minimum and maximum diameters of the distribution were 1 μm and 5 μm, respectively [53]. 2.7. Validation of the numerical simulation To ensure the accuracy of the numerical simulation, it is important to achieve consistency between the results from available experimental data and the corresponding numerical simulation under the same conditions. In this work, the flow pattern in the cyclone was validated by the LDA experimental data from Hoekstra (2000) [54]. Fig. 4 shows a comparison between the LDA data and the corresponding simulated tangential and axial velocity at the location of x = 0, y = 5.25D for vin = 16.34 m/s for the Stairmand cyclone separator. Furthermore, the pressure drop and grade-efficiency resulting from the simulation were also validated by the experimental data results in Hoekstra (2000) [54] and Zhao (2005) [71], as presented in Fig. 5. It was found that the flow pattern, pressure drop, and collection efficiency derived from the numerical simulations are in reasonable agreement with the corresponding experimental data. This demonstrates that the numerical simulations can predict the cyclone performance with accuracy and reliability.

(b) Axial velocity Fig. 4. Comparison of the simulated flow pattern and LDA experimental data forvin =16.34 m/s (x = 0, y = 5.25D) [54].

3. Multi-objective optimization In this study, the multi-objective cyclone optimization was performed combining the design of experiments in the Minitab 17 statistical software and the genetic algorithm of NSGA-II [72] in MATLAB 2016a. The flow chart of the optimization is shown in Fig. 6. The responses of the pressure drop (Δp) and total efficiency (ηt) of the cyclone were defined as shown below [53]:

where Ntotal is the number of the particles released form the inlet and Ntrapped is the number of the particles trapped at the bottom of the dust collector.

Δp ¼ pin −pout

Design of experiments (DOE) is a mathematical technique for studying complicated problems that involve one or more responses affected by a main effect and an interaction effect from several factors (or independent variables). DOE has the capability to arrange experiments in the most efficient way and maximally assimilate information about a system from its results. DOE can provide convincing results for a complex system with scientific and reasonable expenditure of time and resources [73–75].

ð11Þ

where pin and pout are the static pressures of the inlet and outlet, respectively, which are numerically measured in the same way as the referenced experiment [54], and ηt ¼

Ntrapped  100% N total

ð12Þ

3.1. Design of experiments

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(a) Pressure for various inlet velocities

Fig. 7(a) shows the Pareto charts of the effects for the responses of pressure drop and total efficiency at the α = 0.05 significance level. Other high-order terms, not included in the charts, have very small effects, and thus, they were ignored. For the pressure drop, the inlet width b and vortex finder diameter Dx have significant effects on pressure. Meanwhile, the significant effects of b, Dx, inlet height a, and surface roughness e on total efficiency were identified by ANOVA. Plots of the main effects of the factors on the responses are presented in Fig. 7(b). The magnitudes of the main effects in the plots, which are proportional to the absolute values of the slopes of the lines, are revealed to be similar to the effects in the Pareto charts. In addition, the plots also clearly indicate either positive or negative effects for each factor. For the pressure drop, apart from the insignificant factors of the cylinder height h, vortex finder length S, and cone tip and collector diameter Bc (Dc), the significant factors have negative effects. Meanwhile, these significant factors also show the tendencies of negative effects for the total efficiency. Therefore, the values of the significant factors (b, Dx, a, and e) should be considered in the RSM and be taken as trade-offs in the following GA optimization process. Furthermore, h, S, and Bc (Dc) were selected as 362.5 mm, 217.5 mm, and 188.5 mm, respectively, by comprehensively balancing the positive and negative effects. 3.2. Response surface methodology The RSM is a collection of statistical and mathematical techniques based on identifying and fitting the response surface models from the experimental design data. It is useful for developing, improving, and optimizing processes in industrial applications. The RSM can be used for three main objectives: mapping of a response surface over a particular region of interest, optimization of the response, and selection of the operating conditions to achieve specifications or customer requirements [76,77]. Table 4 presents the selected significant factors in the RSM and the corresponding levels. 3.2.1. Central composite design CCD was presented by Box and Wilson in 1951 [78] and is the most popular method used in RSM design. The present study utilized a full uniformly rotatable CCD that consists of several cube points, star (or axial) points, and center points: 2

N ¼ k þ 2k þ n

(b) Grade-efficiency Fig. 5. Comparison of the simulated results and experimental data [54,71].

3.1.1. Screening experiments (fractional factorial experiments) Because the number of factors in a factorial experiment increases, the number of runs required for a full design rapidly increases and easily exceeds most experimenters' resources. For a complex problem, the response(s) is likely to be dominated by certain main effects and the low-order interaction effects resulting from a portion of the factors. The remaining factors can be neglected. Therefore, before implementing an RSM, it is advisable to screen the factors that have a statistically significant effect on the response(s) from all the factors using fractional factorial experiments (FFE) with only a fraction of the full design of runs [76]. Table 3 presents the selected factors for the FFE and the corresponding levels. To separate the main effects of each factor from all the effects, FFE design with IV resolution level and 16 runs was conducted a 2k−p IV by CFD. To analyze the statistical significance of each factor, the results obtained from the FFE were processed with analysis of variance (ANOVA).

ð13Þ

where k is the number of factors and n is the number of replicate center points. In this study including four factors (k = 4), a total of 31 runs with 16 cube, 8 star, and 7 center points were carried out by CFD. The distance of each star point from the center in the design domain in the full uniformly rotatable CCD can be determined by. α ¼ 2k=4 ¼ 2

ð14Þ

Therefore, all the factors were studied in five levels: − 2, − 1, 0, 1, +2. 3.2.2. Analysis of variance and model fit The performance and accuracy of the GA optimization are highly dependent on whether the fitness function(s) is appropriate enough to describe the problem. If the quality of the fitness function is poor, the GA will either converge on an inaccurate solution or have difficulty achieving convergence. To define the functions, a secondorder response surface model was used, and it can be written as

y ¼ β0 þ

k X i¼1

βi xi þ

k X i¼1

βii x2i þ

k X X i b j¼2

βij xi x j þ ε

ð15Þ

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353

Fig. 6. Illustration of the flow chart for the cyclone multi-objective optimization.

where y is the response; β0, βi, βii, and βij are the regression coefficients for the intercept, linear, quadratic, and interaction terms, respectively; xi and xj are the factors; and ε is the residual according to the experiments

Table 3 Values and levels of the factors in the screening experiments. Factor

Minimum

Basic design

Maximum

a/D b/D Dx/D h/D S/D Bc (Dc)/D e (mm)

0.416 0.1 0.301 0.75 0.25 0.35 0.075

0.566 0.2 0.451 1 0.5 0.5 0.15

0.716 0.3 0.601 1.25 0.75 0.65 0.225

[73,76,77]. The parameters in Eq. (15) were obtained by the CCD experiments for each response. The performance of the response surface model generally can be assessed by considering R2, Adj. R2, and Pred. R2 at the same time. The coefficient of determination R2 can be used to estimate the goodness of fit of the model. The adjusted coefficient of determination Adj. R2 is a useful parameter for comparing the explanatory power of models with different numbers of predictors and is used for penalizing R2. The predicted coefficient of determination Pred. R2represents how well the model predicts responses for new observations [53,73,76,79]. Table 5 presents the ANOVA results for the response surface regression of the pressure drop. Except for the square term x4 ∗ x4 and interaction term x1 ∗ x4, all the terms have a significant effect on the response because their p-values are less than 0.05. Even though the non-linear terms, including the surface roughness, have relatively high p-values,

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(a) Pareto chart

(b) Main effect Fig. 7. Analysis of the screening experiments for the responses.

the obtained model for the pressure drop also achieved reasonable performance: R2 = 99.20%, Adj. R2 = 98.49%, and Pred. R2 = 95.37%. R2 indicates that only 0.80% of the total variations are not explained and is only 0.71% greater than Adj. R2. Nevertheless, it is not circumstantial to the excellent performance of the model in the previous work [53], which indicates that it is not easy to fit the effect of the surface roughness on the response. Table 6 shows the ANOVA results for the response surface regression of the total efficiency. Unfortunately, the model achieved a unsatisfactory quality, given that the difference between R2 and Adj. R2 is 2.48% and Pred. R2 is 83.68%. In addition, apart from the terms of x3 ∗ x3 and x3 ∗ x4, the p-values of all the square and interaction terms are far greater than 0.05. The result demonstrates that the model must be modified by deleting insignificant terms; otherwise, the model will lead to inaccurate GA results. At the same time, this operation should be done very carefully, as any change in the terms gives rise to the performance deterioration of the pressure drop model. In the end, x1 ∗ x1, x4 ∗ x4, x1 ∗ x3, and x1 ∗ x4 were deleted from the original models by making a compromise between the two models with comprehensive

Table 4 Values and levels of the significant factors used in the RSM. Factor

Minimum

Center

Maximum

a (a/D), x1 b (b/D), x2 Dx (Dx/D), x3 e, x4

116 (0.4) 43.5 (0.15) 116 (0.4) 0.1

145 (0.5) 58 (0.2) 145 (0.5) 0.15

174 (0.6) 72.5 (0.25) 174 (0.6) 0.2

consideration. Hence, the final R2, Adj. R2, and Pred. R2 for the pressure drop were 98.24%, 97.37%, and 94.37%, whereas for the total efficiency, the corresponding values were 96.97%, 95.45%, and 89.17%, respectively. The performances of the models are highly acceptable.

Table 5 ANOVA for the response surface regression of the pressure drop. Source

Degree of freedom

Adjusted sums of squares

Adjusted means squares

F-Value

p-Value

Model Linear x1 x2 x3 x4 Square x1 ∗ x1 x2 ∗ x2 x3 ∗ x3 x4 ∗ x4 2-way interaction x1 ∗ x2 x1 ∗ x3 x1 ∗ x4 x2 ∗ x3 x2 ∗ x4 x3*x4

14 4 1 1 1 1 4 1 1 1 1 6 1 1 1 1 1 1

2,316,202 2,136,111 393,767 883,696 816,070 42,579 105,960 11,903 33,868 75,356 27 74,131 26,459 7278 2983 23,863 6737 6811

165,443 534,028 393,767 883,696 816,070 42,579 26,490 11,903 33,868 75,356 27 12,355 26,459 7278 2983 23,863 6737 6811

140.92 454.87 335.40 752.71 695.11 36.27 22.56 10.14 28.85 64.19 0.02 10.52 22.54 6.20 2.54 20.33 5.74 5.80

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.000 0.000 0.881 0.000 0.000 0.024 0.131 0.000 0.029 0.028

S = 34.2639; R2 = 99.20%; Adj. R2 = 98.49%; Pred. R2 = 95.37%.

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of e dominates the total efficiency. Therefore, the pressure drop can be retained by increasing roughness at a high vortex finder diameter.

Table 6 ANOVA for the response surface regression of the total efficiency. Source

Model Linear x1 x2 x3 x4 Square x1*x1 x2*x2 x3*x3 x4*x4 2-way interaction x1 ∗ x2 x1 ∗ x3 x1 ∗ x4 x2 ∗ x3 x2 ∗ x4 x3 ∗ x4

355

Degree of freedom

Adjusted sums of squares

Adjusted means squares

F-Value

14 4 1 1 1 1 4 1 1 1 1 6 1 1 1 1 1 1

1166.69 1116.34 106.23 369.84 555.92 84.35 27.07 0.17 0.71 25.28 0.13 23.27 4.22 0.56 1.52 1.26 2.38 13.33

83.335 279.086 106.233 369.841 555.917 84.353 67.768 0.17 0.711 25.276 0.13 3.879 4.217 0.564 1.523 1.257 2.383 13.328

39.19 131.23 49.95 173.91 261.41 39.67 3.18 0.08 0.33 11.89 0.06 1.82 1.98 0.27 0.72 0.59 1.12 6.27

p-Value

3.3. Multi-objective genetic algorithm optimization 0.000 0.000 0.000 0.000 0.000 0.000 0.042 0.781 0.571 0.003 0.808 0.158 0.178 0.614 0.410 0.453 0.306 0.024

S = 45830; R2 = 97.17%; Adj. R2 = 94.69%; Pred. R2 = 83.68%.

During the CCD in the RSM, the final second-order response surface models for the responses were defined as below: Pressure drop ðPaÞ ¼ 6948−10:03x1 −62:67x2 −30:81x3 −4547x4 þ 0:1539x22 þ 0:05860x23 þ 0:0967x1 x2 þ 0:0918x2 x3 þ 28:3x2 x4 þ 14:23x3 x4

The GA is a search and optimization procedure that is motived by the principles of natural genetics and natural selection, and it is used to solve both constrained and unconstrained problems. The GA repeatedly revises a population of individual solutions. In each step, the GA randomly selects individuals from the current population to be parents that produce the children for the next generation. In the end, an optimal solution can be produced during the successive generations [80–82]. The multi-objective GA function in MATLAB 2016a uses a controlled elitist genetic algorithm that is a variant of NSGA-II [72,81]. The fitness functions resulting from the RSM and the constraints used in the GA optimization are shown below: Minimize pressure drop : f 1 ða; b; Dx; eÞ Minimize negative total efficiency : − f 2 ða; b; Dx; eÞ ð87≤a ≤203Þ Subject to ð29 ≤b≤87Þ ð87 ≤Dx ≤203Þ ð0:05≤e≤0:25Þ

ð18Þ

For convenience, maximizing the total efficiency was replaced by minimizing the negative total efficiency. The options and parameters used in the GA optimization are presented in Table 7. 4. Results and discussion

ð16Þ Total efficiency ð%Þ ¼ 114:8−0:0017x1 −0:176x2 þ 0:030x3 −159:6x4 −0:00082x22 −0:001134
x23 −0:001221x1 x2 þ 0:000667x2 x3 þ 0:532x2 x4 þ 0:629x3 x4 ð17Þ

3.2.3. Analysis of the response surfaces To visualize the effects of the factors on the responses, the obtained regression models can be used to generate a number of twodimensional contour plots and three-dimensional response surface plots for all possible pairs of factors. The plots are constructed by varying all possible pairs of factors within the experimental range while holding the other factors at their center value [51,53,79]. Because the interaction effects among the inlet height a, inlet width b, and vortex finder diameter Dx were well-studied in the previous work [53], in this work, only the interaction effects of the surface roughness e were investigated. Fig. 8(a) illustrates the contour plots and response surface plots of the surface roughness for the pressure drop. It was found that a and e have no interaction effects on the pressure as the isolines in the contour plot are completely straight. However, for b versus e and Dx versus e, at first, the effects of e make almost no difference on the pressure drop for low b and Dx. As the magnitudes decrease, the effect of e on reducing the pressure drop continually increases. In addition, for the three cases, the pressure drop increase rapidly increases with decreased a, b, and Dx. Fig. 8(b) shows the interaction effects among the four factors. The total efficiency increases with decreased a, b, and Dx. From the steeper isolines in the three-contour plot, the effect of e on the efficiency is considerably larger than that on the pressure drop. In the case of the pressure drop, there is no interaction effect between a and e. In other words, the effect of e is identical when a is varying. In the case of b versus e, the isolines become bent as b decreases. This trend becomes more significant in Dx versus e. At the low level of Dx, the isolines are nearly parallel to the e axis. Meanwhile, at the high level of Dx, the lines are perpendicular to the e axis, which demonstrates that the effect

4.1. Optimization result A Pareto front of 53 non-dominated optimum design points for the two objective functions was carried out by the GA, as shown in Fig. 9. It clearly demonstrates the trade-off between pressure drop and total efficiency, and the appropriate cyclone design can be obtained by considering a compromise. A higher value for one of these objectives of optimum design always corresponds to a lower value for the other objective. Table 8 shows the 53 optimum points with their geometrical factors resulting from the GA. It was found that the values of a, b, and Dx of the chosen individuals were widely distributed in the whole corresponding regions. However, the individuals with higher e were eliminated by the natural selection process and do not appear in the table despite the fact that this can considerably reduce the drop in pressure, indicating that a better cyclone performance can be achieved by minimizing the surface roughness. To confirm the accuracy and reliability of the GA result, the objectives of five representative points, namely, A, B, C, D, and E, were evaluated by CFD. Table 9 shows a comparison between the objectives obtained from the numerical simulation and the GA. The CFD results agree well with the GA results, and this reveals a reasonable performance of the fitness functions and high reliability of the GA results. Although points A and E in the Pareto front achieved the best pressure drop and total efficiency, because the other objectives are too extreme, they can hardly be used in real applications. Some of the compromise points such as B and C, by contrast, have relatively high realistic values for practical application. In comparison to A, point B obtained an 89.17% higher total efficiency with an additional pressure drop of only 301 Pa. Point C is a trade-off optimal design resulting from the mapping method. To exhibit the amount of performance improvement, a comparison of the Stairmand model, reference model, designs in the RSM, and optimal model are presented in Fig. 10. In spite of the fact that a huge improvement has been already obtained from the Stairmand design to the reference design in the previous work [53], this work offered a series

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(a) Pressure drop

(b) Total efficiency

Fig. 8. 2D contour and 3D response surface plots for the responses.

X. Sun, J.Y. Yoon / Powder Technology 325 (2018) 347–360 Table 7 Options and parameters used in the multi-objective GA optimization. Population type Population size Creation function Selection function Crossover fraction Mutation function Crossover function Stopping criteria

Double vector 150 Feasible population Tournament (Tournament size: 10) 0.95 Adaptive feasible Heuristic (Ratio: 1.6) Function tolerance: 1e-5

of cyclone designs with further improvement in performance, especially point F (No. 12), compared with previous designs and the CCD used in the RSM. To observe the improved performance in depth, the Stairmand model, reference model, and point F were compared. Their parameters are shown in Table 10. 4.2. Comparison between the reference model and optimal model F 4.2.1. Performance A performance comparison of the Stairmand model, reference model, and optimal model F are presented in Table 11. The optimal model can reduce the pressure drop by 7.38%, increase the total efficiency by 3.11%, and decrease the cut-off size by 9.04% compared to the reference model. In addition, compared to the Stairmand model, the optimal model decreases the pressure drop by 19.23% and cut-off size by 42.09%, and increases the total efficiency by 11.67%. Fig. 11 shows a comparison of the grade-efficiencies between the reference model and optimal model. For fine particles with diameters less than 3 μm, the optimal model achieved an increase of roughly 3–10% in the grade-efficiency compared to the reference model. Although the improvement during the optimization is seemingly quite limited, because the reference model has been obtained from the successful RSM optimization in the previous study [53], such improvement also demonstrates a reasonable progress in the performance and effectiveness of the RSM and GA optimization processes in the present work. 4.2.2. Gas flow field Because there is not a large geometrical difference between the reference model and the optimal model, the gas flow fields would be very similar. To compare the two models in detail, instead of using a

357

Table 8 Optimal points resulting from the GA. Point

a

b

Dx

e

Pressure drop (Pa)

Overall efficiency (%)

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

87.00 202.97 198.58 200.08 144.95 200.83 194.11 197.56 131.75 90.30 197.96 198.57 201.23 197.35 102.50 202.28 200.57 121.24 195.82 199.52 106.37 201.32 195.48 132.50 91.38 123.12 155.28 202.42 198.73 198.07 143.26 178.41 161.45 201.05 188.34 199.43 175.71 201.42 202.97 196.84 112.87 197.41 195.73 141.79 201.10 196.25 161.48 87.00 200.70 199.89 199.01 101.50 202.29

29.00 75.76 54.90 40.39 29.10 51.48 38.96 31.60 29.79 29.77 47.19 49.95 43.55 34.28 29.60 42.74 41.44 29.00 53.28 49.27 30.10 38.85 32.23 29.25 29.00 29.24 29.22 36.10 48.53 30.52 29.76 29.48 29.15 34.99 29.04 35.91 29.34 41.06 75.76 37.69 29.01 53.99 46.33 29.00 57.01 29.35 29.28 29.00 58.83 48.10 45.25 29.60 61.64

87.00 196.86 151.29 119.68 91.86 144.06 116.81 101.29 92.22 88.91 134.60 140.52 127.23 107.15 89.51 125.08 122.99 89.75 147.44 138.98 90.91 116.68 102.70 91.17 87.35 90.23 92.79 111.20 137.56 98.66 92.89 95.35 93.30 109.14 95.22 110.58 94.61 121.47 196.86 113.94 89.10 149.50 132.83 91.40 156.26 96.26 93.35 87.00 160.05 136.79 130.95 89.51 165.46

0.0500 0.0546 0.0526 0.0512 0.0500 0.0524 0.0511 0.0503 0.0501 0.0501 0.0519 0.0521 0.0515 0.0507 0.0501 0.0514 0.0514 0.0500 0.0525 0.0521 0.0501 0.0510 0.0504 0.0501 0.0500 0.0501 0.0501 0.0509 0.0521 0.0503 0.0501 0.0501 0.0501 0.0507 0.0501 0.0509 0.0501 0.0512 0.0546 0.0510 0.0500 0.0525 0.0518 0.0500 0.0528 0.0501 0.0501 0.0500 0.0529 0.0520 0.0517 0.0501 0.0532

2502 131 432 951 1997 519 1052 1406 2065 2418 682 582 799 1260 2326 831 889 2207 494 601 2258 1010 1385 2094 2464 2178 1904 1131 629 1469 1972 1688 1853 1192 1630 1163 1723 909 131 1097 2278 461 725 2031 364 1549 1848 2502 325 636 744 2333 265

97.02 49.79 71.29 83.50 94.23 74.16 84.86 90.02 94.47 96.47 78.07 75.69 80.79 88.10 95.98 81.46 82.45 95.41 73.01 76.24 95.52 84.57 89.65 94.74 96.81 95.21 93.69 86.51 76.87 90.79 93.95 92.39 93.42 87.35 92.20 86.83 92.64 82.85 49.79 85.71 95.80 72.15 78.90 94.44 69.07 91.65 93.37 97.02 67.42 77.14 79.51 96.01 64.78

flow field contour plot, it is desirable to utilize the xy plot at the location of x = 0, y = 5.5D. 4.2.2.1. Pressure field. In a cyclone, static pressure normally decreases radially and rapidly from wall to center, and a negative pressure region

Table 9 Validation of the five representative optimal points using CFD. Point (No.)

Fig. 9. Pareto front for the multi-objective GA.

A (02) B (03) C (16) D (37) E (01)

Pressure drop (Pa)

Total efficiency (%)

GA

CFD

Error (%)

GA

CFD

Error (%)

131 432 831 1723 2502

138 448 852 1738 2607

5.34 3.70 2.53 3.48 4.21

49.79 71.29 81.46 92.64 97.02

48.35 70.07 80.36 91.77 98.09

−2.89 −1.71 −1.35 −0.94 1.11

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Fig. 10. Comparison of the Stairmand model, RSM, reference model, and optimal model.

Fig. 11. Comparison of the grade-efficiency between the reference model and optimal model.

exists in the center region as a central, highly swirling flow. Fig. 12 shows a comparison of the static pressure of the Stairmand model, reference model, and optimal model. The static pressure distributions are very similar. In comparison to the reference model, the optimal model further decreased the static pressure of the outer region from the Stairmand model. Furthermore, it also relieved the phenomenon of “lip leakage” and transverse secondary flow owing to the decreased difference of the static pressure from the wall to the center. This resulted in an improvement in particle collection [23].

reference model, the optimal model has a slightly greater peak tangential velocity, and the location of the peak tangential velocity moved approximately 27 mm towards the wall. In addition, the tangential velocity in the near-wall region of the optimal model was also slightly increased with a lower pressure drop than the reference model, which is beneficial for the collection efficiency. Axial velocity is another important factor for particle collection in a cyclone. During the particle collection process, the gravity effect is usually assisted or can be ignored, as the particle's weight is extremely small, except for high solid-loaded operating conditions. Under the influence of the strong tangential velocity resulting from the circular cyclone structure, the particles are driven by the gas flow and rotate in the near-wall region. At the same time, the particles are also influenced by the axial velocity of the outer vortex and move downward by the effect. A comparison of the axial velocity of the Stairmand model, reference model, and optimal model is shown in Fig. 14. The axial velocity in the outer vortex region is negative (i.e., downward flow), and has a dominant effect for transporting the carried solid particles out of the bottom of the cyclone [23,28,38]. While in the inner vortex regions, the axial velocity is positive (i.e., upward flow), and the gas flow travels upward and passes through the vortex finder. The reference model and Stairmand model have similar distributions, but the optimal model has a different pattern in the inner vortex. The axial velocity reached a peak

4.2.2.2. Velocity field. Tangential velocity is one of the most important factors for particle collection in a cyclone. In general, the collection capability is proportional to the magnitude of the tangential velocity. This is because whether the particles will be led to the near-wall area or escape from the vortex finder is highly dependent on the magnitude of the radially directed centrifugal force, mv2θ /r. Therefore, better collection can be achieved in a cyclone with higher tangential velocity. However, higher tangential velocity generally results in higher pressure loss. Fig. 13 presents a comparison of the tangential velocity of the Stairmand model, reference model, and optimal model. The tangential velocity field can be divided into two regions: an outer region of quasi-free vortex flow and an inner region of quasi-forced vortex flow, which is a so-called “Rankine”-type vortex [23,28,31]. In contrast with the

Table 10 Comparison of the parameters of the Stairmand model, reference model, and optimal model F. Factor

a/D

b/D

Dx/D

e

Ht/D

h/D

S/D

Bc (DC)/D

Hc/D

Stairmand model [54] Reference model [53] Optimal model F

0.5 0.566 0.685

0.2 0.2 0.172

0.5 0.451 0.501

0.15 0.15 0.052

4 6 6

1.5 1 1.25

0.5 0.2 0.75

0.37 (1) 0.5 0.65

2 1 1

Table 11 Performance comparison between the Stairmand model, reference model, and optimal model. Model

Stairmand model [54] Reference model [53] Optimal model

Pressure drop (Pa)

Total efficiency (%)

Cut-off size (mm)

Value

Improvement (%)

Value

Improvement (%)

Value

Improvement (%)

729 643 589

– 11.85 19.23

67.97 73.79 75.90

– 8.56 11.67

2.78 1.77 1.61

– 36.33 42.09

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Fig. 12. Comparison of the static pressure of the Stairmand model, reference model, and optimal model (x = 0, y = 5.5D).

Fig. 14. Comparison of the axial velocity of the Stairmand model, reference model, and optimal model (x = 0, y = 5.5D).

value in the center. The optimal model has the largest axial velocity in the near-wall region, which gives rise to an increase in the collection efficiency.

The results exhibit an acceptable error (less than 5%) and demonstrate the reliability of the GA optimization process. The cyclone designs with important trade-offs provided a larger improvement in the performance than the original Stairmand design and a more reasonable improvement than the reference model. A representative optimal design decreases the pressure drop by 7.38% (from 643 Pa to 589 Pa) and the cut-off size by 9.04% (from 1.77 mm to 1.61 mm), and increases the total efficiency by 3.11% (from 73.79% to 75.90%), compared to the reference model. The optimum design points of the Pareto front obtained in this work assist in reducing the choices faced by the designer and decision maker for optimizing cyclone designs.

5. Conclusions A multi-objective cyclone optimization process was successfully performed using the RSM, GA, and CFD techniques. In this study, a well-optimized cyclone design resulting from previous work [53] was chosen as the reference model, and the objectives were defined to minimize the pressure drop and maximize the total efficiency. Two fitness functions for the objectives used in the GA were well defined during the CCD in the RSM after the screening experiments using numerical simulation. A total of 53 non-dominated optimum design points with respect to the control factors of the cyclone parameters were carried out by the GA. The accuracy of the GA results was validated using CFD.

Fig. 13. Comparison of the tangential velocity of the Stairmand model, reference model, and optimal model (x = 0, y = 5.5D).

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