Modeling and Pareto optimization of gas cyclone separator performance using RBF type artificial neural networks and genetic algorithms

Powder Technology 217 (2012) 84–99

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Modeling and Pareto optimization of gas cyclone separator performance using RBFtype artificial neural networks and genetic algorithms

Khairy Elsayed ⁎, Chris LacorVrije Universiteit Brussel, Department of Mechanical Engineering, Research Group Fluid Mechanics and Thermodynamics, Pleinlaan 2, 1050 Brussels, Belgium

⁎ Corresponding author. Tel.: +32 26 29 2368; fax: +E-mail address: [email protected] (K. Elsayed).

0032-5910/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.powtec.2011.10.015

a b s t r a c t
a r t i c l e i n f o
Article history:Received 26 July 2011Received in revised form 3 October 2011Accepted 9 October 2011Available online 18 October 2011

Keywords:Cyclone separatorArtificial neural network (ANN)Genetic algorithm (GA)Multi-objective optimizationPareto front

Both the pressure drop and the cut-off diameter are important performance parameters in the design of the cy-clone separator. In this paper, a multi-objective optimization study of the gas cyclone separator is performed. Inorder to predict accurately the complex non linear relationships between the performance parameters (pressuredrop and cut-off diameter) and the geometrical dimensions, two radial basis function neural networks (RBFNNs)are developed and employed to model the pressure drop and the cut-off diameter for cyclone separators. The ar-tificial neural networks have been trained and tested by the experimental data available in literatures for the pres-sure drop and the Iozia and Leith model for the cut-off diameter. The results demonstrate that artificial neuralnetworks can offer an alternative and powerful approach tomodel the cyclone performance parameters. The anal-ysis indicates the significant effect of the vortex finder diameterDx, the vortexfinder length S, the inletwidth b andthe total height Ht. The response surface methodology has been used to fit a second-order polynomial to theRBFNN. The second-order polynomial has been used to study the interaction between the geometrical parameters.The two trained artificial neural networks have been used as two objective functions to get new optimal ratios forminimum pressure drop andminimum cut-off diameter using themulti-objective genetic algorithm optimizationtechnique. Sometimes, themain concern is minimizing the pressure drop, so a single objective optimization studyhas been performed to obtain the cyclone geometrical ratio for minimum pressure drop. The comparison of nu-merical simulation of the new optimal design and the Stairmand design confirms the superior performance ofthe new design.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Cyclones are one of the most widely used separators, which relyon centrifugal forces to separate particles from a gas stream. The pri-mary advantages are the economy, simplicity in construction andadaptability to a wide range of operating conditions.Reversed flowcyclones with a tangential inlet are the most common cyclone design[1](Fig. 1). It consists of seven main geometrical parameters: inletsection height a and width b, cylinder height h, cyclone total heightHt, dust exit diameter (cone tip diameter) Bc, gas outlet diameter(also, called the vortex finder diameter) Dx and vortex finder heightS. All these parameters always given as a ratio of the cyclone body di-ameter D. It is generally known that these seven dimensions charac-terize the collection efficiency (cut-off diameter) and pressure dropof the cyclone separator [2-5]. Both the pressure drop and the cut-off diameter in a cyclone separator can be decreased or increased byvarying the cyclone dimensions. For an accurate optimal design of acyclone, it is quite necessary to use a reliable model for its perfor-mance parameters. Optimization of gas cyclone is, indeed, a multi-

32 26 29 2880.

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objective optimization problem rather than a single objective optimi-zation problem that has been considered so far in the literature [5,6].Both the pressure drop and the collection efficiency in gas cyclonesare important objective functions to be optimized simultaneously insuch a real-world complex multi-objective optimization problem[7]. These objective functions are either obtained from experiments,empirical models or computed using very timely and high-cost com-putational fluid dynamic (CFD) approaches. Modeling and optimiza-tion of the parameters are investigated in the present study, byusing radial basis function artificial neural networks and multi-objective genetic algorithm optimization technique in order to maxi-mize the collection efficiency (minimize the cut-off diameter) andminimize the pressure drop.

1.1. Previous optimization studies

In 1951, Stairmand [8] presented the geometrical ratios for high ef-ficiency cyclones. Until now, these ratios are still in use (cf. Fig. 1 andTable 1). Elsayed and Lacor [3] reported the following shortages in theStairmand model for pressure drop calculation [9] which has beenused to obtain these geometrical ratios: (1) the velocity distributionhas been obtained from a moment-of-momentum balance, estimatingthe pressure drop as entrance and exit losses combined with the loss

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(a) 3-D view (b) 2-D view

Fig. 1. Schematic diagram for cyclone separator in 3-D and 2-D.

85K. Elsayed, C. Lacor / Powder Technology 217 (2012) 84–99

of static pressure in the swirl, i.e., neglecting the entrance loss by assum-ing no change of the inlet velocity occurs at the inlet area; (2) assuminga constant friction factor; (3) the effect of particle mass loading on thepressure drop is not included [10].

Due to the wide range of industrial applications of cyclone separa-tors, they are already the subject of detailed studies for several de-cades. However, optimization studies are quite limited in theliterature. Moreover, many of these studies are not coherent studies.Ravi et al. [11] carried out a multi-objective optimization of a set ofN identical reverse-flow cyclone separators in parallel by using thenon-dominated sorting genetic algorithm (NSGA). Two objectivefunctions were used: the maximization of the overall collection effi-ciency and the minimization of the pressure drop. Non-dominatedPareto optimal solutions were obtained for an industrial problem inwhich 165 m3/s of air was treated. In addition, optimal values of sev-eral decision variables, such as the number of cyclones and eight geo-metrical parameters of the cyclone, are obtained. Their study showsthat the barrel diameter, the vortex finder diameter, and the numberof cyclones used in parallel, are the important decision variablesinfluencing the optimal solutions. Moreover, their study illustratesthe applicability of NSGA in solving multi-objective optimizationproblems involving gas-solid separations. The main drawbacks oftheir study are: (1) They used the model of Shepherd and Lapple[12] for predicting the dimensionless pressure drop (Euler number).In Shepherd and Lapple model, the Euler number depends on onlythree factors (Eu=16ab/Dx

2) and they used it to optimize the sevengeometrical parameters. (2) The barrel diameter, number of parallelcyclones and the gas velocity have been included into the optimiza-tion design space. Consequently, it is not devoted to the geometricalratio. (3) They used many side constraints on the geometrical values(0.4≤a/D≤S/D, 0.15≤b/D≤(1−Dx/D)/2 if 0.5≤Dx/D≤0.6) theseconstraints prevent searching for the global optimization geometricalratios for the seven geometrical parameters. (4) No table for the non-dominated Pareto front points is presented from which the designercan select a certain geometrical ratio set (optimal solution).

Table 1The geometrical parameters values for Stairmand design.

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/D

Stairmand design 0.5 0.2 0.5 4 1.5 0.5 0.375

Swamee et al. [13] investigated the optimum values of the numberof cyclones to be used in parallel, the diameter of cyclone barrel D andexit pipe Dx, when a specified flow rate of gas is to be separated fromsolid particles, and the cut-off diameter is already specified. Theyused Stairmand model for calculation of pressure drop and Gerrardand Liddle formula for the cut-off diameter [13] which is not a widelyused model. Instead of handling two objective functions, they blendedthe two objectives into a single objective problemwhich is not a suitablemethod when considering two conflicting objectives (the pressure dropand cut-off diameter).

Safikhani et al. [14] performed a multi-objective optimization ofcyclone separators. First, they simulated many cyclones to obtainthe pressure drop and the cut-off diameter and then used the artificialneural network approach to obtain the objective function values. Fi-nally, multi-objective genetic algorithms are used for Pareto basedoptimization of cyclone separators considering two conflicting objec-tives. However, the design variables are only four (instead of seven):the barrel height, the cone height, the vortex finder diameter andlength. So they ignored the effect of inlet dimensions, which hasbeen acknowledged by other researchers as significant geometricalparameters for the cyclone flow field and performance (cf. Elsayedand Lacor [3-5,7]). Moreover, they did not explain why they selectedthese particular parameters. Furthermore, they applied four side con-straints on the four tested variables, which prevents searching for theglobal optimization.

Pishbin and Moghiman [15] applied the genetic algorithm for opti-mum cyclone design. They studied the seven geometrical parameters.The data used for optimization was obtained from 2-D axisymmetricsimulations. However, theflow in the cyclone separator is 3-D unsteady.Instead of using multi-objective genetic algorithm (e.g., non-dominatedsorting genetic algorithm II (NSGA-II) [16]) they used theweighted-sumgenetic algorithm. In this technique, a weighting factor is assigned foreach objective function based on the user preference. The main short-ages of the Pishbin and Moghiman [15] study are: (1) How to selectthe weighting factor, in scientific and engineering problems, it is anon-trivial task to find the one solution of interest to the decisionmaker [17]. The decision maker's weight (no matter how defined)could be greater thannecessary asmore acceptable solutions aremissed.Optimizing mostly profit could lead to poor quality or reliability, not agood compromise [17]. The weighted-sum genetic algorithm usuallydoes not find all Pareto front points of interest. But this approach is asimple approach for handling multi-objective optimization problem,

86 K. Elsayed, C. Lacor / Powder Technology 217 (2012) 84–99

another simple but better result can be obtained using the desirabilityfunction approach [7,18]. (2) No table for the non-dominated Paretofront points is presented from which the designer can select a certaingeometrical ratio.

Safikhani et al. [19] carried out a multi-objective optimizationusing the genetic algorithm technique to obtain the best vortex finderdimension (diameter and length) and shape (convergent and diver-gent). Four design variables have been investigated; the vortex finderdiameter, angle, upper-part length and lower-part length of the vor-tex finder. They applied neural networks to obtain a meta-model forthe pressure drop and collection efficiency from CFD dataset. Themain shortages of the Safikhani et al. [19] study are: (1) They used di-mensional values instead of dimensionless, and applied side con-straints, which prevent the optimization procedure from obtainingglobal optimization. (2) The selection of only the vortex finder di-mension as the design variables and neglecting the interaction withthe vortex finder diameter with the other dimensions, especially theinlet dimensions [4,5].

Elsayed and Lacor [3,5,7,20-22] have performed several optimizationstudies (single and multi-objective) on cyclone separator geometry.They presented new designs for optimum performance, but in all oftheir studies, they used the Nelder–Mead technique [23] which suffersfrom two drawbacks, (1) the final solution may be affected by the startpoint, (2) the obtained optimum may be a local minimum. To avoidthis, the application of evolutionary methods like the genetic algorithmis a must.

1.2. Study objectives

There are four objectives of this study. (1) Investigation of the ef-fect of the seven geometrical parameters on the cyclone separatorperformance (the pressure drop and cut-off diameter) based on theexperimental data for the pressure drop and the most robust mathe-matical models for the cut-off diameter. (2) Study the possible inter-action between the seven geometrical parameters affecting thecyclone performance using response surface methodology.(3) Multi-objective optimization to obtain new geometrical ratiosfor optimum performance (minimum pressure drop and minimumcut-off diameter). (4) Obtaining the optimum design (geometrical ra-tios) of the cyclone separator for minimum pressure drop using thegenetic algorithm optimization technique, followed by a comparisonof the numerical simulations of the optimal design and the Stairmanddesign using the Reynolds stress turbulence model.

2. Artificial Neural Network (ANN) approach

With the development of modern computational technologies, ar-tificial neural networks (ANNs) have become an attractive approachfor modeling highly complicated and nonlinear system [5,24,25]. Themost widely used types of ANNs for solving the regression problemare back propagation neural network (BPNN), radial basis functionneural network (RBFNN) and generalized regression neural network(GRNN). Zhao and Su [25] conducted a detailed comparison betweenthese three artificial neural networks. They developed and employedthese ANNs to model the Euler number for cyclone separators. Themain conclusions of Zhao and Su are: (1) The BPNN results are closeto that of the RBFNN in the testing process but BPNN takes a largercomputational cost. (2) The GRNN has fast conversion but the pre-dicted result for the tested sample was not satisfactory. (3) Comparedwith the BPNN and GRNN, the RBFNN provides superior predictionand a high robustness. In this study, the radial basis function neuralnetwork (RBFNN) has been used tomodel the effect of cyclone dimen-sions on the cyclone performance parameters. For more details aboutthe radial basis function neural network, the interested reader canrefer to Elsayed and Lacor [5].

2.1. The Euler number

The pressure drop across the cyclone essentially depends on thecyclone dimensions and operating conditions. Generally, it is propor-tional to the average dynamic pressure at the inlet and is often de-fined as [25]

ΔP ¼ Eu12ρgV

2in

� �ð1Þ

Where Eu is the Euler number (the dimensionless pressure dropalso called pressure drop coefficient [25]).

In order to determine the Euler number more accurately, all eightdimensions of the cyclone are selected to establish the ANN models.They all affect the Euler number but to different extents [25,26]. Usu-ally, these dimensions can be characterized by the barrel diameter Dand expressed as seven dimensionless geometric ratios [25]:

Eu ¼ fDx

D;aD;bD;SD;Ht

D;hD;Bc

D

� �ð2Þ

According to Eq. (2), seven independent dimensionless geometri-cal variables and one dependent variable (the Euler number of the cy-clone) are selected as respectively the input and output parameters intheANNmodel. For simplicity, the division of each factor by the barreldiameter D will be dropped.

A dataset of 98 samples obtained from the measurements of pres-sure drop for different cyclone designs [25-27] is used in the presentinvestigation to evaluate the prediction performance of the ANNmodels. Table 2 presents more details about the used dataset includ-ing the minimum, mean, maximum, range and the standard deviationof the seven dimensionless geometrical ratios. Due to the large differ-ence in the order of magnitude of the value (cf. Table 2), the availabledataset is transformed into−1 to 1 interval using the Matlab intrinsicfunction; mapminmax in order to avoid solution divergence [25], cf.Elsayed and Lacor [5] for more statistical details for the used dataset.The ANN calculations have been performed using the neural networktoolbox available from Matlab commercial software 2010a.

2.2. The cut-off diameter (Stokes number)

The source of the training data for the cut-off diameter has beenobtained from the application of Iozia and Leith model [28]. Thismodel has been approved as an acceptable approach for calculatingthe cyclone cut-off diameter [28,29]. The cut-off diameter x50 is theparticle diameter which produces 50% collection efficiency.

A question may appear here, why the authors employed the cut-off diameter instead of the collection efficiency, like in the study ofRavi et al. [11]. Firstly, For low mass loading cyclone separator, thecut-off diameter can replace the collection efficiency, since one canfit the grade efficiency curve using the cut-off diameter via some cor-relations, cf. Hoffmann and Stein ([10], page 91) for more details.Moreover, many models can predict well the cut-off diameter but ex-hibit different grade efficiency curves ( [10], page 97). Secondly, thecut-off diameter (instead of the collection efficiency) has been usedas an objective function in many recent publications e.g. [3,14]. More-over, the selection of the cut-off diameter or the collection efficiencyfor low mass loading cyclones can be considered as a researcherchoice.

Based on Iozia and Leith model [28], the cut-off diameter is a func-tion of the inlet gas velocity (i.e., a function of both gas volume flowrate, Barrel diameter, inlet section height and width), gas viscosityand particle density.

The cut-off diameter x50 for a cyclone separator is always given inunits of μm. Another way to represent x50 is using a dimensionlessnumber, the Stokes number. The Stokes number based on the cut-

Table 2Descriptive statistical parameters for the seven geometrical factors used to train the RBFNN.

Variable Dx a b S Ht h Bc

Minimum 0.25 0.113 0.067 0.39 1.158 0.501 0.14Mean 0.428653 0.629653 0.21148 0.89101 3.28309 1.18908 0.341847Maximum 0.667 1.0 0.4 3.052 10.97 3.5 1.0Range 0.417 0.887 0.333 2.662 9.812 2.999 0.86Standard deviation 0.110117 0.261828 0.0935936 0.428851 2.09556 0.672901 0.149841


off diameter; Stk50 ¼ ρpx250V in= 18μDð Þ[30]. It is the ratio between the

particle relaxation time; ρpx502 /(18μ) and the gas flow integral timescale; D/Vin.

2.3. Fitting the ANNs

Tables 3 and 4 presentmore details about the validation of the usedRBFNNs. Both the average, standard deviation, minimum, maximumand range of the input and the predicted values are given. It is clearfrom Tables 3 and 4 that the ANNs preserved the descriptive statisticalparameters of the input data. The correlation coefficient between theinput and the output and the mean squared error are given for eachRBFNN. The intercept and the slope of the adjusted line between theinput and the predicted value of the ANN are also given.

The configured RBFNN predictions versus experimental data forthe Euler number and the Iozia and Leith mathematical model forthe Euler number and cut-off diameter are shown in Fig. 2. Accordingto Fig. 2, it can be seen that the ANNmodels are able to attain the hightraining accuracy. The training mean square errors for the experimen-tal values and the Iozia and Leith model have the values 1.311E-4 and3.258E-4 respectively (Fig. 2). This indicates that, compared with tra-ditional models of curve fitting, the models based on an artificial in-telligence algorithm have a superior capability of nonlinear fitting.Especially, the RBFNN has its unique and optimal approximationcharacteristics in the learning process [25].

Fig. 2 illustrates the agreement between the ANNs input and out-put. The obtained relation is a typical linear relation with a coefficientof correlation close to 1 (R>0.999). The agreement between theinput and output of the ANN is also clear from the value of themean squared error E2. That means, the trained neural networks pre-dict very well both the Euler number and cut-off diameter values andcan be used in cyclone design and performance estimation. Tables 3, 4and Fig. 2 present different performance indicators as a validation ofthe proposed model for experimental values.

2.4. The effect of geometrical parameters on the cut-off diameter basedon RBFNN

The effect of the geometrical parameters on the Euler numberbased on the trained RBFNN has been investigated by Elsayed andLacor [5]. They acknowledge the significant effect of the vortex finder

Table 3Validation of the used RBFNN to model the Euler number⁎.

x y

Average 23.2684 23.2684Standard deviation 32.8858 32.874Minimum 2.3 1.74544Maximum 155.3 155.985Range 153.0 154.24Correlation Coefficient, R 0.99964Mean squared error, E2 1.311E-4Intercept 0.0167Slope 0.999

Both x and y represent the Euler number.The values of R, E2, intercept and slope are that for the testing stage.⁎ x is the input to the RBFNN and y is the predicted value.

diameter Dx and the vortex finder length S, the inlet width b and thetotal height Ht. Less effect is due to the cylinder height h (for h>2.5)and the inlet height a (for a>0.55) (cf., Elsayed and Lacor [5]Fig. 6(a)).

For this particular study presented in Fig. 3, the following valueshave been used: Barrel diameter D=0.1 m, air flow rate =0.8333 l/s, air viscosity 1.0E-5 Pa s and particle density 860 kg/m3.This meansthat the obtained results will be valid for this particular case. But,the authors believe the variation of cut-off diameter due to variationsof cyclone geometrical dimensions is superior to the effect of theseoperating parameters, which is quite difficult to cover their range ofoperating conditions.

The effects of the geometrical parameters on the cut-off diameterare depicted in Fig. 3. To study the effect of each parameter, the testedRBFNNmodel has been used by varying one parameter at a time fromits minimum tomaximum values of the available 98 dataset, while theother parameters are kept constant at their mean values (cf. Table 2).Fig. 3 indicates the significant effect of the vortex finder diameter Dx

and the vortex finder length S, the inlet width b, the inlet height aand the total height Ht. Less effect is due to the cylinder height h andthe cone tip diameter Bc. More analysis is given in Table 5.

2.5. The significant geometrical parameters on the cut-off diameter (Stokesnumber) using the response surface methodology (RSM) approach

In this study only the effect of the geometry was taken into ac-count. The effect of flow rate on the performance was not considered.Overcamp and Scarlett [33] studied the effect of changing Reynoldsnumber on the cut-off diameter (Stokes number) and found that forReynolds number values beyond 1E4, the effect of increasing Reynoldsnumber is very limited. Furthermore, Karagoz and Avci [34] studiedthe effect of increasing the Reynolds number on the pressure dropand found that beyond Reynolds number of 2E4 any increase in theReynolds number has nearly no effect on the pressure drop. As theReynolds number for all cases considered is above 2E4. (All the testedcyclones have the same flow rate.), the effect of flow rate can safely beneglected. As this optimization study does not include changing thecyclone diameter or the number of cyclones to get the optimum cy-clone diameter. So no need to add the capital cost of the cyclone sep-arator as a design parameter in this study [13]. Consequently, onlythe effect of changing the geometrical parameters on the performancewill be considered.

Table 4Validation of the used RBFNN to model the cut-off diameter⁎.

x y

Average 8.21939 8.21939Standard deviation 2.55998 2.5578Minimum 3.64 3.7157Maximum 15.3 15.4048Range 11.66 11.6891Correlation Coefficient, R 0.99915Mean squared error, E2 3.258E-4Intercept 0.014Slope 0.999

Both x and y represent the cut-off diameter.The values of R, E2, intercept and slope are that for the testing stage.⁎ x is the input to the RBFNN and y is the predicted value.

Input value (x)

Pre

dic

ted

val

ue

(y)

Pre

dic

ted

val

ue

(y)

40 80 120

40

80

120

Data pointLinear fit

Data pointLinear fit

y = 0.999 x + 0.0167R = 0.99964

E2 = 1.311E-4

(a) The Euler number

Input value (x)4 6 8 10 12 14

4

6

8

10

12

14y = 0.998 x + 0.014R=0.99915

E2 = 3.258E-4

(b) The cut-off diameter

Fig. 2. Linear regression of the RBFNNs for the Euler number and the cut-off diameter.

Table 5The variation of the cut-off diameter with cyclone dimensions using the RBFNN model(cf., Fig. 3).

Factor Analysis

Dx The vortex finder diameter has the most significant effect on the cut-offdiameter x50 (the highest slope in Fig. 3). The slope is very high untilDx=0.5 and any further increase in Dx produces a small change in x50. Ingeneral, increasing Dx increases x50 (decreasing the collection efficiency),this is one of the main reasons of the trade-off between the Euler numberand the cut-off diameter objectives. This makes the optimization of cy-clone geometry a multi-objective procedure.

b The variation of x50 with the inlet width is similar in trend andsignificance to that for Dx but here the slope changes at b=0.25.

S and a The effect of the vortex finder length and the inlet section height on thecut-off diameter is almost paralleled up to S=1.5 and a=0.6 afterwardsthey lose their significance and become nearly constant.

h Increasing the barrel height slightly decreases the cut-off diameter withnearly linear relation. This trend has been reported by other researchersusing CFD simulations, e.g., Elsayed and Lacor [31].

Ht The effect of the cyclone total height is basically due to two effects thecone height and barrel height. The curve can be subdivided into four mainregions. Sharp decrease in x50 up toHt=3.25, no valuable differencebetween 3.25 and 5.25, sharp increase between 5.25 and 8, andinsignificant effect beyond 8⁎.

Bc The effect of the cone-tip diameter on the cut-off diameter is quite small.First, increasing the cone-tip diameter slightly decreases the cut-off di-ameter up to Bc=0.55 and any further increment increases the cut-offdiameter. This trend has been reported by other researchers, e.g., Elsayedand Lacor [22,32].

⁎ Due to the interaction between the geometrical parameters, especially between Ht

with S and Dx (cf., Fig. 5 and Table 7), the obtained conclusions may not be applicablegenerally and the application of response surface methodology to analysis the effect ofeach particular parameter must.


2.6. Why RSM?

The usualmethod of optimizing any experimental set-up is to adjustone parameter at a time, keeping all others constant, until the optimumworking conditions are found. Adjusting one parameter at a time is nec-essarily time consuming, and may not reveal all interactions between

Dx, a, b, Bc

h, S

Ht

X50

[m

icro

n]

0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3 3.5

2 4 6 8 10

0

2

4

6

8

10

12

14

DxabSHthBc

Fig. 3. The effect of geometrical parameters on the cut-off diameter based on the Ioziaand Leith model [28]. Note: The plotted curves are obtained for a test case with the fol-lowing settings, Barrel diameter=0.1 m, air flow rate=0.8333 l/s, air viscosity=1.0E-5 Pa s, particle density=860 kg/m3.

the parameters. In order to fully describe the response and interactionsof any complex system a multivariate parametric study must be con-ducted [35]. As there are seven geometrical parameters to be investigat-ed, the best technique is to perform this study via the response surfacemethodology (RSM). Response surface methodology has been used toprepare a new design of experiment (due to the correlation betweenthe 98 input data set used for training of the ANNs, cf. Elsayed andLacor [5]).

In order to conduct a RSM analysis, one must first design the ex-periment, identify the experimental parameters to adjust, and definethe process response to be optimized. Once the experiment has beenconducted and the recorded data tabulated, RSM analysis softwaremodels the data and attempts to fit second-order polynomial to thisdata [35]. The generalized second-order polynomial model used inthe response surface analysis was as follows [3,5]:

Y ¼ β0 þX7i¼1

βiXi þX7i¼1

βiiX2i þ∑∑

ibjβijXiXj ð3Þ

where β0, βi, βii, and βij are the regression coefficients for intercept,linear, quadratic and interaction terms, respectively. While Xi and Xj

are the independent variables, and Y is the response variable (Stokesnumber Stk50).

2.6.1. Design of experiment (DOE)The statistical analysis is performed through threemain steps. Firstly,

construct a table of runs with a combination of values of the indepen-dent variables via the commercial statistical software STATGRAPHICScenturion XVI by giving the minimum and maximum values of theseven geometrical factors under investigation as input. Secondly, per-form the runs by estimating the cut-off diameter (Stokes number)using the trained artificial neural network (based on the Iozia andLeith model [28]). Thirdly, fill in the values of the Stokes number inthe STATGRAPHICSworksheet and obtain the response surface equation

Table 7Analysis of variance and the regression coefficients of the fitted quadratic equation forthe Stokes numbera.

Source Regression coefficient F-ratio P-value

β0 −0.0470554

Linearβ1 0.942933 6.65 0.0154β2 −1.3178 24.12 0.0000β3 2.10188 0 1β4 8.3493 1145.05 0.0000β5 −0.843633 0 1β6 −0.527695 243.98 0.0000β7 1.46453 137.75 0.0000

Quadricβ11 −1.17696 0.21 0.6489β22 1.44904 0.18 0.6737β33 −3.50314 0.37 0.5476β44 2.33086 9.38 0.0048β55 0.281211 1.49 0.232β66 0 39.06 0.0000β77 0.243505 7.33 0.0114

Interactionβ12 0.726218 0.03 0.8558β13 0 0 1β14 −2.09219 1.25 0.2732β15 0 0 1β16 0.121646 0.22 0.6401β17 −0.187708 0.09 0.7726β23 0 0 1β24 −4.84457 5.03 0.0330β25 0 0 1β26 0.262495 0.78 0.3842β27 −0.466053 0.39 0.5352β34 0 0 1β35 0 0 1β36 0 0 1β37 0 0 1β45 0 0 1β46 −0.869946 38.41 0.0000β47 1.44156 16.87 0.0003β56 0 0 1β57 0 39.06 0.0000β67 −0.370128 58.83 0.0000R2 0.984099

a Bold numbers indicate significant factors as identified by the analysis of variance(ANOVA) at the 95% confidence level.


with main effect plot, interaction plots, Pareto chart and response sur-face plots (we refer to Antony [36] chapter 4 for more details aboutthe definition of these plots and how they have been calculated).

Table 6 represents the parameters ranges selected for the sevengeometrical parameters. The study was planned using Box–Behnkendesign, with 64 combinations. A significant level of Pb0.05 (95% con-fidence) was used in all tests. Analysis of variance (ANOVA) was fol-lowed by an F-test of the individual factors and interactions [5].

2.7. Analysis of variance (ANOVA)

Analysis of variance (ANOVA) showed that the resultant quadraticpolynomial models adequately represented the experimental datawith the coefficient of multiple determination R2 being 0.984099(cf., Table 7). This indicates that the quadratic polynomial modelobtained was adequate to describe the influence of the independentvariables studied [37]. Analysis of variance (ANOVA) was used toevaluate the significance of the coefficients of the quadratic polyno-mial models (see Table 7). For any of the terms in the models, alarge F-value (small P-value) would indicate a more significant effecton the respective response variables [3,5].

Based on the ANOVA results presented in Table 7, the variablewith the largest effect on the Stokes number (cut-off diameter) wasthe linear term of vortex finder diameter, followed by the linearterm of the cyclone total height, the vortex finder length and theinlet width (Pb0.05); the other three linear terms (inlet height, barrelheight, and cone tip diameter) did not show a significant effect(P>0.05). The quadratic term of cyclone total height, vortex finderdiameter and vortex finder length also had a significant effect onthe pressure drop; however, the effect of the other four quadraticterms was insignificant. Furthermore, the interaction between Dx

with (Ht, S, b) and between S with (h, Ht) also had a significant effecton the Stokes number, while the effect of the remaining terms wasinsignificant.

2.8. Analysis of response surfaces

For visualization of the calculated factor, main effects plot, Paretochart and response surface plots were drawn. The slope of the maineffect curve is proportional to the size of the effect and the directionof the curve specifies a positive or negative influence of the effect[3,38] (Fig. 4(a)). Based on the main effect plot, the most significantfactors on the Stokes number are: (1) the vortex finder diameter Dx,with a second–order curve of direct relation. (2) the cyclone totalheight Ht inversely related to the Stokes number. (3) the vortex finderlength Swith direct relationship. (4) the inlet dimensions width b andheight a inversely related to the Stokes number. Whereas the otherfactors have an insignificant effect. The main effect plot supports theanalysis given in Table 5, except for Ht where the strong interactionbetween the cyclone total height and the vortex finder length affectedthe trend given in Fig. 3.

Pareto charts were used to summarize graphically and display therelative importance of each parameter with respect to the Stokesnumber [3]. The Pareto chart shows all the linear and second-ordereffects of the parameters within the model and estimates the

Table 6The values of the independent variables used in the design of experiment.

Variables Minimum Center Maximum

Inlet height, a=X1 0.4 0.55 0.7Inlet width, b=X2 0.14 0.27 0.4Cone tip diameter, Bc=X3 0.2 0.3 0.4Vortex finder diameter, Dx=X4 0.2 0.475 0.75Barrel height, h=X5 1.0 1.5 2.0Total cyclone height, Ht=X6 3.0 5.0 7.0Vortex finder length, S=X7 0.4 1.2 2.0

significance of each with respect to maximizing the Stokes numberresponse. A Pareto chart displays a frequency histogram with thelength of each bar proportional to each estimated standardized effect[35]. The vertical line on the Pareto charts judges whether each effectis statistically significant within the generated response surfacemodel; bars that extend beyond this line represent effects that arestatistically significant at a 95% confidence level. Based on the Paretochart (Fig. 4(b)) and ANOVA table (Table 7) there are five significantparameters at a 95% confidence level: the vortex finder diameter Dx,the total cyclone height Ht, the vortex finder length S and the inlet di-mensions a and b. Therefore, the Pareto chart is a perfect supplementto the main effect plot.

To visualize the effect of the independent variables on the depen-dent ones, surface response of the quadratic polynomial models weregenerated by varying two of the independent variables within the ex-perimental range while holding the other factors at their central values(cf., Table 6) [37] as shown in Fig. 5. Thus, Fig. 5(a) was generated byvarying the total height Ht and the vortex finder length Swhile keepingthe other five factors constant. The response surface plots presented inFig. 5 illustrate the strong interactions between Ht with (S and Dx) andDx with (S and b).

(a) Main effects plot

(b) Paretochart. A=a,B=b, C=Bc, D=Dx, E=h, F=Ht, G=S,FG=Ht S, etc.

Fig. 4. Analysis of design of experiment for the Stokes number.

(a) Ht versus S

(b) Dx versus Ht

(c) Dx versus S

(d) b versus Dx

Fig. 5. The response surface plots for the Stokes number. Note: the stokes numbervalues are multiplied by 1000.


3. Optimization using genetic algorithms

The genetic algorithm is an optimization technique for solvingboth constrained and unconstrained optimization problems that isbased on natural selection, the process that drives biological evolution[39]. The genetic algorithm repeatedly modifies a population of indi-vidual solutions. At each step, the genetic algorithm selects individualsat random from the current population to be parents and uses them toproduce the children for the next generation. Over successive genera-tions, the population evolves toward an optimal solution. The geneticalgorithm can be used to solve a variety of optimization problemsthat are not well suited for standard optimization algorithms, includ-ing problems in which the objective function is discontinuous, no dif-ferentiable, stochastic, or highly nonlinear [39].

In case of cyclone separator geometry optimization for minimumEuler number and minimum cut-off diameter, the objectives are con-flicting with each other. There is no solution for which all objectivesare optimal simultaneously [40]. The increase of one objective willlead to the decrease of other objectives. Then, there should be a setof solutions, the so-called Pareto optimal set or Pareto front, inwhich one solution cannot be dominated by any other member ofthis set [40].

Recently, a number ofmulti-objective genetic algorithms (MOGAs)based on the Pareto optimal concept have been proposed. The wellknown nondominated sorting genetic algorithm II (NSGA-II) pro-posed by Deb et al. [16] is one of the most widely used MOGAs since

it provides excellent results as compared with other multi-objectivegenetic algorithms proposed [41].

4. Results and discussion

4.1. Optimal cyclone design for minimum pressure drop

The genetic algorithm optimization technique has been applied toobtain the geometrical ratios for minimum pressure drop (Eulernumber). The objective function is the Euler number (using thetrained radial basis function neural network). The design variablesare the seven geometrical dimensions of the cyclone separator (cf.,Fig. 1(a)).

4.1.1. GA settingsTable 8 presents the settings used to obtain the optimum design

for minimum pressure using global optimization Matlab toolbox in

Table 8Genetic operators and parameters for single objective optimization.

Population type: Double vectorInitial range: [0.2 0.1 0.1 0.3 2.0 0.65 0.05 ;

0.8 0.8 0.6 0.8 8.0 2.5 0.75]Fitness scaling: RankSelection operation: Tournament (tournament size equals 4)Elite count: 2Crossover fraction: 0.8Crossover operation: Intermediate crossover with the default

value of 1.0Mutation operation: The constraint dependent defaultMaximum number of generations: 1400Population size: 200

Table 10The values of geometrical parameters for the two designs (D=0.205 m)⁎.

Cyclone a/D b/D Dx/D Ht/D h/D S/D Bc/D

Stairmand design 0.5 0.2 0.5 4 1.5 0.5 0.375New design 0.595 0.201 0.549 4.549 1.411 0.595 0.275

⁎ The outlet section is above the cyclone surface by Le=0.618D. The inlet section lo-cated at a distance Li=D from the cyclone center.


Matlab 2010a commercial package. Table 9 gives the optimum valuesfor cyclone geometrical parameters for minimum pressure drop esti-mated by the artificial neural network using the genetic algorithm op-timization technique. It is clear from Table 9 that the new optimizeddesign is very close to the Stairmand design in many geometrical pa-rameters, whereas the new ratios will result in minimum pressuredrop. To understand the effect of this small change in the geometricalratios on the flow field pattern and performance, a CFD study for thetwo designs is needed [5].

4.2. Comparison between the two cyclone designs using CFD

4.2.1. Solver settingsThe air volume flow rate Qin=0.08 m3/s for the two cyclones

(inlet velocity for Stairmand design is 19 m/s and 16 m/s for thenew design), air density 1.0 kg/m3 and dynamic viscosity of 2.11E-5 Pa s. the turbulent intensity equals 5% and characteristic lengthequals 0.07 times the inlet width [42]. Velocity inlet boundary condi-tion is applied at inlet, outflow at gas outlet and wall boundary condi-tion at all other boundaries [5].

The finite volume method has been used to discretize the partialdifferential equations of the model using the SIMPLEC (Semi-ImplicitMethod for Pressure-Linked Equations-Consistent) method for pres-sure velocity coupling and QUICK scheme to interpolate the variableson the surface of the control volume. The implicit coupled solution al-gorithm was selected. The unsteady Reynolds stress turbulencemodel (RSM) was used in this study with a time step of 0.0001 s.

The grid refinement study using different levels of grid shows thata total number of 134759 hexahedral cells for the Stairmand cycloneand 378963 hexahedral cells for the new design are sufficient to ob-tain a grid-independent solution, and further mesh refinement yieldsonly small, insignificant changes in the numerical solution. The hexa-hedral meshes have been obtained using the GAMBIT commercialsoftware. These simulations were performed on an 8 nodes CPUOpteron 64 Linux cluster using FLUENT 6.3.26 commercial software.The geometrical values for the two cyclones (cf. Fig. 1(b)) are givenin Table 10.

In order to validate the obtained results, the predictions are com-pared with the measurements of Hoekstra [43] on the Stairmand cy-clone using Laser Doppler Anemometry (LDA). For the validation of

Table 9The optimized cyclone separator design for minimum pressure drop.

Factor Low High Stairmand design Optimum design

Dx 0.2 0.75 0.5 0.549a 0.5 0.75 0.5 0.595b 0.14 0.4 0.2 0.201S 0.4 2.0 0.5 0.595Ht 3.0 7.0 4.0 4.549h 1.0 2.0 1.5 1.411Bc 0.2 0.4 0.375 0.275

the axial and tangential velocity in addition to the pressure dropand cut-off diameter, we refer the reader to Elsayed and Lacor [4, 5].

4.2.2. The pressure fieldFig. 6 shows the contour plot at Y=0. In the two cyclones, the

time-averaged static pressure decreases radially from the wall to cen-ter. A negative pressure zone appears in the forced vortex region(central region) due to high swirling velocity. The pressure gradientis largest along the radial direction, while the gradient in the axial di-rection is very limited. The cyclonic flow is not symmetrical as is clearfrom the shape of the low-pressure zone at the cyclone center (twistedcylinder). However, the two cyclones have almost the same flow pat-tern, but the highest pressure of the Stairmand design is nearly 1.5times that of the new design, implying that the new design has alower pressure drop.

The pressure distribution presented in Figs. 7 and 8 of the two cy-clones at sections S1–S6 depict the two parts pressure profile (for

Fig. 6. The contour plots for the time averaged flow variables at sections Y=0 (cf. Fig. 1(b)). From top to bottom: Stairmand design and the new design respectively. From leftto right: the static pressure (N/m2), the tangential and axial velocity (m/s). Note: bothcyclones have the same barrel diameter and air volume flow rate.

Table 11The position of different sectionsa.

Section S1 S2 S3 S4 S5 S6

z`/Db 2.75 2.5 2.25 2.0 1.75 1.5

a Sections S1–S5 are located in the conical section, section S6 at the cylindrical part.b z′ is measured from the top of the inlet section (cf. Fig. 1(b)).


Rankine vortex). Again, the highest static pressure for Stairmand de-sign is more than 1.5 times that of the new design at all sections whilethe central value is almost the same for the two cyclones irrespectiveof the section location. This indicates that, the new design has a lowerpressure drop with respect to the Stairmand design.

4.2.3. The velocity fieldBased on the contour plots of the time-averaged tangential velocity

presented in Fig. 6, and the radial profiles at sections S1–S6 shown in

Distance from center (m)

Sta

tic

pre

ssu

re (

N/m

2 )

-0.1 -0.05 0 0.05 0.1Distance fr

-0.1 -0.050

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New designStairmand design


Fig. 7. The radial profile for the time averaged tangential and axial velocity at different sectioS1–S3. From left to right: time-averaged static pressure, tangential and axial velocity respe

Figs. 7 and 8, the following conclusions can be drawn. The tangential ve-locity distributions for the two cyclones are approximately nearly identi-cal in pattern and values (dimensionless), with the highest velocityoccurring at 1/4 of the cyclone radius for both cyclones. This impliesnearly equal collection efficiency for both cyclones, as the centrifugalforce is the main driving force for particle collection in the cyclone sepa-rator. The axial velocity profiles for the two cyclones are also very close,exhibiting the inverted W axial velocity profile.

4.3. Discrete phase modeling (DPM)

The discrete phase model in Fluent follows the Euler–Lagrange ap-proach. The fluid phase is treated as a continuum by solving the time-averaged Navier–Stokes equations, while the dispersed phase issolved by tracking a large number of particles through the calculatedflow field. A fundamental assumption made in this model is that thedispersed second phase occupies a low volume fraction (usually lessthan 10–12%, where the volume fraction is the ratio between the

om center (m)0 0.05 0.1

Distance from center (m)-0.1-0.1 -0.05 0 0.05 0.1


Axi

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om center (m)0 0.05 0.1


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om center (m)0 0.05 0.1


-0.5

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ns on the X–Z plane (Y=0) at sections S1–S3 (cf., Table 11). From top to bottom: sectionctively.

image of Fig.�7


Sta

tic

pre

ssu

re (

N/m

2 )

-0.1 -0.05 0 0.05 0.1Distance from center (m)


-0.1-0.1 -0.05 0 0.05 0.10

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Fig. 8. The radial profile for the time–averaged tangential and axial velocity at different sections on the X–Z plane (Y=0) at sections S4–S6 (cf., Table 11). From top to bottom: sec-tion S4–S6. From left to right: time-averaged static pressure, tangential and axial velocity respectively.


total volume of particles and the volume of fluid domain), eventhough high mass loading is acceptable. The particle trajectories arecomputed individually at specified intervals during the fluid phasecalculation. This makes the model appropriate for the modeling of

Table 12The performance parameters for the two cyclones.

Design Method Δp(N/m2)

Euler number x50(μm)

Stokesnumber×103

New design CFD 803 6.338 0.804 0.114ANN 584.4 4.613 2.938 1.815Ramachandranmodel [26]

877.98 5.523

Stairmanddesign

CFD 1190 6.567 1.0 0.209ANN 1015.8 5.606 3.314 1.931Ramachandranmodel [26]

699.66 4.846

particle-laden flows. The particle loading in a cyclone separator issmall (3–5%), and therefore, it can be safely assumed that the pres-ence of the particles does not affect the flow field (one-way coupling).Collection efficiency statistics were obtained by releasing a specified

Table 13Genetic operators and parameters for multi-objective optimization.

Population type: Double vectorPopulation size: 105 (i.e., 15* number of variables)Initial range: [0.2 0.1 0.1 0.3 2.0 0.65 0.05 ;

0.8 0.8 0.6 0.8 8.0 2.5 0.75]Selection operation: tournament (tournament size equals 2)Crossover fraction: 0.8Crowding distance fraction 0.35Crossover operation: Intermediate crossover with the default

value of 1.0Number of generations (iterations): 1400 (i.e., 200* number of variables)

Table 14The diameters, air flow rates and the particle densities for the sixteen test cases.

Case D [mm] Q [l/min] ρp [kg/m3]

1 205 50 8602 205 60 8603 205 70 8604 205 80 8605 205 50 10006 205 50 15007 205 50 17508 205 50 20009 31 50 86010 31 60 86011 31 70 86012 31 80 86013 31 50 100014 31 50 150015 31 50 175016 31 50 2000


number of mono-dispersed particles at the inlet of the cyclone and bymonitoring the number escaping through the outlet. Collisions be-tween particles and the walls of the cyclone were assumed to be

Euler number

Sto

kes

num

ber

x10

3

0 5 10 15 20 25

0.5

1

1.5

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3

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4

Case 1Case 2Case 3Case 4Case 5Case 6Case 7Case 8Case 9Case 10Case 11Case 12Case 13Case 14Case 15Case 16

(a) Pareto chart for 16 test cases, linear scale

Euler number

Sto

kes

num

ber

x10

3

5 10 15 20 25 30350.5

1

1.5

2

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3

3.54

4.55

5.56

6.57

Pareto pointsStk50=10 {̂0.3016[LOG10(Eu)]^2-0.9479 LOG10(Eu)-2.5154}

(b) Pareto chart for 16 test cases in log scale with the curve fittingformula, Stk50 = 10 0.3016(log

10 (Eu))2 −0.9479log

10 (Eu) −2.5154,

R2 = 0.98643

Fig. 9. Pareto charts for different test cases.(Cont

perfectly elastic (coefficient of restitution is equal to 1). For the equa-tion of particle motion and the DPM settings, the interested readercan refer to Elsayed and Lacor [5].

4.3.1. The DPM resultsIn order to calculate the cut-off diameters of the two cyclones, 104

particles were injected from the inlet surface with zero velocity and aparticles mass flow rate _mp of 0.001 kg/s (corresponding to inlet dustconcentration Cin _mp=Q

� � ¼ 11:891 gm=m3). The particle density ρpis 860 kg/m3 and the maximum number of time steps for each injec-tion was 200000 steps. The DPM analysis results and the pressuredrops for the two cyclones are depicted in Table 12. Although, the dif-ference between the two cyclone cut-off diameters is small, the sav-ing in pressure drop is considerable (nearly 32.5% the value ofStairmand cyclone).

Based on the flow pattern analysis and the DPM results, one canconclude that the cyclone collection efficiency for the two cyclonesis very close, with the advantage of low pressure drop in the new de-sign. The authors want to emphasis that only small changes in thegeometrical dimensions of the two designs lead to this improvementin the performance.

Euler number

Sto

kes

num

ber

x10

3

5 10 15 20 25

0.5

1

1.5

2

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3

3.5

4A

BC

(c) Pareto chart for test case 1

Euler number

Sto

kes

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ber

x10

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5 10 15 20 25

0.5

1

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2

2.5

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3.5

4A

BC

(d) Pareto chart for test case 9

inued) Pareto charts for different test cases.


4.4. Optimal cyclone design for best performance

4.4.1. NSGA-II settingsTable 13 presents the genetic operators and parameters for multi-

objective optimization. The Euler number values have been obtainedfrom the artificial neural network trained by experimental values. TheStokes number values are obtained from Iozia and Leith model [28]. Inorder to investigate the effect of different geometrical and operationalparameters on the Pareto front, sixteen test cases with different bar-rel diameter, gas flow rate and particle density have been tested, cf.Table 14. The sixteen test cases covers: 1) Two barrel diameters,31 mm and 205 mm. 2) Four levels of air flow rates, 50, 60, 70 and80 l/min. 3) Five values of particle density, 860, 1000, 1500, 1750and 2000 kg/m3.

4.4.2. Pareto frontThe Pareto front (non dominated points) for the sixteen test cases

are presented in Fig. 9(a). Fig. 9(a) clearly demonstrate tradeoffs inobjective functions Euler number and Stokes number from which anappropriate design can be compromisingly chosen by the designer[14]. All the optimum design points in the Pareto front are non-dominated and could be chosen by a designer as optimum cycloneseparator [14]. The corresponding geometrical ratios of the Paretofront shown in Fig. 9(a) are given in Table 15 for test case 1 andTable 16 for test case 9. Three points A, B and C are indicated inFigs. 9(c) and (c) and Tables 15 and 16. Point A indicates the point

Table 15The seven geometrical parameters and the obtained Euler number and Stokes number for t

Point Dx a b S Ht

1 0.306 0.659 0.385 0.410 6.9572B 0.306 0.688 0.398 0.404 6.9933 0.618 0.229 0.213 0.411 6.7744 0.326 0.331 0.240 0.451 6.6685C 0.360 0.295 0.253 0.443 6.6646 0.670 0.230 0.213 0.410 6.7777 0.585 0.226 0.217 0.419 6.7638 0.306 0.685 0.299 0.401 6.9859 0.306 0.666 0.361 0.407 6.96510 0.305 0.516 0.271 0.439 6.84011 0.303 0.286 0.318 0.449 6.61112 0.476 0.240 0.217 0.434 6.74313 0.312 0.622 0.277 0.419 6.94314 0.308 0.686 0.379 0.427 6.97415 0.592 0.229 0.213 0.413 6.83116 0.320 0.570 0.273 0.423 6.89817 0.598 0.229 0.213 0.412 6.77018 0.309 0.441 0.266 0.450 6.76919 0.430 0.307 0.231 0.438 6.73520 0.306 0.667 0.389 0.417 6.95021 0.306 0.392 0.277 0.441 6.70822 0.326 0.331 0.209 0.451 6.66823 0.308 0.686 0.348 0.427 6.97424 0.680 0.229 0.213 0.409 6.78725 0.307 0.653 0.296 0.421 6.97226 0.307 0.368 0.338 0.439 6.70827 0.514 0.235 0.217 0.429 6.70928 0.310 0.596 0.275 0.424 6.91829 0.559 0.246 0.221 0.453 6.78530 0.399 0.308 0.229 0.438 6.70731 0.516 0.244 0.241 0.422 6.72832 0.413 0.256 0.280 0.444 6.66733 0.306 0.507 0.352 0.437 6.81134 0.307 0.644 0.333 0.423 6.96035 0.646 0.229 0.219 0.409 6.79736A 0.692 0.228 0.213 0.408 6.81937 0.558 0.216 0.230 0.439 6.802Minimum 0.303 0.216 0.209 0.401 6.611Maximum 0.692 0.688 0.398 0.453 6.993

A indicates the point of minimum Euler number andmaximum Stokes number. B indicates thpoint for the multi-objective optimization problem. (cf. Fig. 9(c)).

of minimum Euler number and maximum Stokes number. Point B in-dicates the point of maximum Euler number and minimum Stokesnumber. Point C indicates an optimal point for the multi-objective op-timization problem.

In order to obtain the Euler number–Stokes number relationship,Fig. 9(b) has been drawn. It indicates a general relationship (trend)between the two dimensionless numbers irrespective to the barrel di-ameter, gas flow rate, particle density. A second-order polynomial hasbeen fitted between the logarithms of Euler number and Stokes num-ber, Eq. (4). The obtained correlation can fit the data with a coefficientof correlation R2=0.98643 as shown in Fig. 9(b).

Stk50 ¼ 100:3016 log10 Euð Þð Þ2−0:9479log10 Euð Þ−2:5154 ð4Þ

4.4.3. Bubble plots for Pareto frontFor visual inspection of the effect of the seven geometrical parame-

ters on the two conflicting performance parameters, the bubble plots onPareto front points have been drawn for each geometrical parameter.However, only figures for test case 1 (Fig. 10) and 9 (Fig. 11) are pre-sented, but all other cases depict the same results (trend).

Fig. 10 indicates that: a) Decreasing the vortex finder diameter Dx

decreases the Stokes number and increases the Euler number, Fig. 10(b). b) Generally speaking, increasing the inlet height a increases theEuler number and decreases the Stokes number. c) A similar trend is

he nondominated points (Pareto-front) for test case 1 (cf. Table 14).

h Bc Euler number Stokes number×103

1.779 0.387 23.843 0.5521.779 0.317 27.322 0.5401.885 0.495 1.026 2.7851.885 0.444 6.485 0.7981.910 0.459 4.892 0.9371.859 0.496 0.879 3.4781.901 0.492 1.159 2.0891.789 0.425 18.474 0.5761.782 0.398 22.300 0.5581.868 0.428 12.977 0.6331.930 0.462 8.404 0.6991.901 0.471 1.915 1.5241.909 0.429 14.816 0.6191.781 0.459 23.145 0.5541.895 0.494 1.117 2.4911.909 0.434 12.767 0.6581.893 0.494 1.095 2.5651.881 0.435 10.629 0.6741.898 0.465 3.211 1.2291.779 0.367 24.941 0.5501.882 0.443 9.963 0.6761.900 0.471 5.515 0.8231.807 0.459 21.132 0.5641.857 0.495 0.849 3.6401.816 0.420 17.606 0.5881.895 0.430 11.399 0.6591.901 0.479 1.596 1.7181.899 0.429 14.387 0.6221.898 0.455 1.486 1.9151.891 0.460 3.732 1.1021.904 0.486 1.797 1.6741.927 0.472 3.510 1.1621.848 0.460 16.110 0.6021.807 0.401 19.929 0.5751.863 0.497 0.963 3.1131.855 0.498 0.815 3.8051.922 0.486 1.311 1.9451.779 0.317 0.815 0.5401.930 0.498 27.322 3.805

e point of maximum Euler number andminimum Stokes number. C indicates an optimal

Table 16The seven geometrical parameters and the obtained Euler number and Stokes number for the nondominated points (Pareto-front) for test case 9 (cf. Table 14).

point Dx a b S Ht h Bc Euler number Stokes number×103

1A 0.686 0.236 0.236 0.478 6.910 1.915 0.473 1.008 3.7432 0.308 0.655 0.390 0.423 6.902 1.995 0.471 21.688 0.5583 0.308 0.692 0.390 0.423 6.902 1.995 0.472 22.898 0.5514 0.585 0.235 0.235 0.418 6.925 1.727 0.471 1.355 2.4365 0.314 0.330 0.236 0.431 6.837 1.864 0.470 6.568 0.7506 0.309 0.486 0.335 0.426 6.869 1.931 0.471 13.947 0.6217 0.578 0.237 0.235 0.421 6.684 1.975 0.471 1.356 2.0088 0.309 0.272 0.347 0.424 6.897 1.967 0.472 8.025 0.6979 0.310 0.538 0.369 0.429 6.882 1.984 0.467 16.799 0.59710 0.473 0.238 0.235 0.478 6.923 1.971 0.471 2.105 1.48111 0.308 0.375 0.382 0.423 6.901 1.989 0.471 12.168 0.63412 0.432 0.238 0.236 0.445 6.785 1.947 0.472 2.494 1.30013 0.321 0.327 0.236 0.431 6.839 1.862 0.470 6.223 0.77714 0.308 0.606 0.383 0.424 6.896 1.990 0.471 19.736 0.57115 0.308 0.643 0.381 0.424 6.895 1.986 0.472 20.775 0.56516 0.407 0.255 0.240 0.470 6.899 1.907 0.472 3.119 1.16517 0.309 0.333 0.353 0.424 6.882 1.958 0.471 10.021 0.66518 0.452 0.236 0.236 0.448 6.793 1.945 0.472 2.255 1.39419 0.678 0.236 0.236 0.473 6.893 1.919 0.473 1.029 3.61520 0.625 0.237 0.236 0.449 6.800 1.944 0.472 1.187 2.89121 0.309 0.596 0.379 0.425 6.894 1.988 0.471 19.039 0.57822 0.313 0.325 0.262 0.430 6.847 1.884 0.471 7.197 0.73023 0.664 0.237 0.236 0.476 6.911 1.914 0.473 1.076 3.40424 0.309 0.365 0.345 0.425 6.876 1.946 0.471 10.740 0.65625 0.608 0.235 0.235 0.438 6.921 1.797 0.472 1.261 2.69126B 0.308 0.692 0.390 0.423 6.902 1.995 0.472 22.898 0.55127 0.309 0.436 0.381 0.424 6.899 1.989 0.464 14.120 0.61628 0.308 0.427 0.360 0.424 6.886 1.962 0.471 13.112 0.62629C 0.330 0.290 0.252 0.465 6.890 1.955 0.471 5.639 0.81830 0.658 0.245 0.239 0.477 6.910 1.924 0.473 1.146 3.29031 0.308 0.362 0.365 0.424 6.883 1.668 0.467 11.924 0.64832 0.520 0.251 0.244 0.470 6.906 1.684 0.472 1.996 1.67133 0.631 0.236 0.236 0.449 6.796 1.945 0.472 1.164 2.96434 0.549 0.242 0.240 0.430 6.909 1.783 0.471 1.613 1.82335 0.316 0.270 0.341 0.425 6.891 1.966 0.472 7.482 0.72636 0.686 0.236 0.236 0.478 6.910 1.915 0.473 1.008 3.74337 0.380 0.239 0.237 0.477 6.890 1.922 0.473 3.327 1.071Minimum 0.308 0.235 0.235 0.418 6.684 1.668 0.464 1.008 0.551Maximum 0.686 0.692 0.390 0.478 6.925 1.995 0.473 22.898 3.743

A indicates the point of minimum Euler number andmaximum Stokes number. B indicates the point of maximum Euler number andminimum Stokes number. C indicates an optimalpoint for the multi-objective optimization problem. (cf. Fig. 9(d)).


exhibited by the inlet width b but due to interaction with other geo-metrical and operational variables, one could see a range of bubblesizes in the region of best performance (lower values for both theEuler and Stokes numbers). d) The higher values of total cycloneheight Ht will produce less Stokes number, intermediate valuescould produce less Euler number, smaller-intermediate values couldproduce the optimum performance due to interaction with other var-iables. e) Short barrels will produce better collection efficiency (lowStokes number) and higher Euler numbers. Intermediate values re-sults in low Euler number values. Long barrels can produce the bestperformance. f) Short vortex finder may produce higher values ofEuler numbers or higher values of Stokes number due to strong inter-action with other variables. Long vortex finder can produce the opti-mum performance. g) Generally speaking, the variation of the cone-tip diameter Bc has no effect on the performance parameter. Theabove comments is restricted to the range of each geometrical vari-ables located on the Pareto front and not for the whole range of values(cf., Fig. 10 for the range of each geometrical parameters).

5. Conclusions

To predict the complex non-linear relationships between the perfor-mance parameters and the geometrical dimensions, two radial basisneural networks (RBFNNs) are developed and employed to model theEuler number and Stokes number for cyclone separators. The neural

networks have been trained and tested by the experimental data avail-able in literature for Euler number (pressure drop) and Iozia and Leithmodel [28] for the Stokes number (cut-off diameter). The effects ofthe seven geometrical parameters on the Stokes number have been in-vestigated using the trained ANN. To declare any interaction betweenthe geometrical parameters affecting the Stokes number, the responsesurface methodology has been applied. The trained ANN has beenused as an objective function to obtain the cyclone geometrical ratiosfor minimum Euler number using the genetic algorithms optimizationtechnique. A CFD comparison between the new optimal design andthe Stairmand design using the Reynolds stress turbulence model hasbeen performed. A multi-objective optimization technique usingNSGA-II technique has been applied to determine the Pareto front forthe best performance cyclone separator.

The following conclusions can be drawn from analysis of theobtained results:

• The result demonstrates that artificial neural networks can offer an al-ternative and powerful approach to model the cyclone performance.

• The analysis indicates the significant effect of the vortex finder di-ameter Dx and the vortex finder length S, the inlet width b, theinlet height a and the total height Ht on the cyclone performance.

• The response surface methodology has been used to fit a second-order polynomial to the RBFNN for the cut-off diameter. The analy-sis of variance of the cut-off diameter indicates a strong interactionbetween Dx with (Ht, S, b) and between S with (h, Ht).

(a) Dx, range: 0.303 - 0.692 (b) a, range: 0.216 - 0.688

(c) b, range: 0.209 - 0.398 (d) Ht, range: 6.611 - 6.993

(e) h, range: 1.779 - 1.930 (f) S, range: 0.401 - 0.453

(g) Bc, range: 0.317 - 0.498

Fig. 10. Bubble plots for different geometrical parameters for test case 1 (cf. Fig. 9(c) and Table 15).


(a) Dx, range: 0.308 - 0.686 (b) a, range: 0.235 - 0.692

(c) b, range: 0.235 - 0.390 (d) Ht, range: 6.684 - 6.925

(e) h, range: 1.668 - 1.995 (f) S, range: 0.418 - 0.478

(g) Bc, range: 0.464 - 0.473

Fig. 11. Bubble plots for different geometrical parameters for test case 9 (cf. Fig. 9(d) and Table 16).



• The trained RBFNN has been used to get a new optimized cyclonefor minimum pressure drop (Euler number) using the genetic algo-rithm optimization technique.

• A comparison between the new design and the standard Stairmanddesign has been performed using CFD simulation with the Reynoldsstress turbulence model to get a clear vision of the flow field patternand performance in the new design.

• CFD results shows that, the new cyclone design are very close to theStairmand high efficiency design in the geometrical parameter ratio,and superior in lowpressure drop at nearly the same cut-off diameter.

• The new cyclone design results in nearly 68% of the pressure dropobtained by the old Stairmand design at the same volume flowrate. This confirms that the obtained design using the genetic algo-rithm is better than that obtained using Nelder–Mead techniquewhich results in 75% of the Stairmand pressure drop [5].

• The two trained RBFNNs have been used in a multi-objective opti-mization process using NSGA-II technique. Sixteen test cases withdifferent barrel diameter, gas flow rate and particle density havebeen tested and plotted. The Pareto fronts for the 16 test cases arevery close. A second-order polynomial has been fitted betweenthe logarithms of Euler number and Stokes number to obtain a gen-eral formula, Stk50 ¼ 100:3016 log10 Euð Þð Þ2−0:9479log10 Euð Þ−2:5154 with a co-efficient of correlation R2=0.98643. This formula can be used toobtain the Stokes number if the Euler number is known.

As a future extension of this study, the following issues still needmore investigation. (1) Comparison between the suggested geometri-cal parameters ratios and the other cyclone designs available in litera-tures e.g. Elsayed and Lacor design [3] (they used MM model for theestimation of the pressure drop). (2) Create a neural network modelas a design tool for the cyclone separator and estimate its performanceparameters. (3) Generate performance curves for each geometricaland operating parameters that affect the cyclone performance tohelp the designer in predicting the change of the performance due tochange in the cyclone geometrical ratios and operating conditions.(4) Perform a Robust parameter design study to investigate the effectof uncertainty in the geometrical values and the optimization process

Acknowledgments

Part of this study has been published in Evolutionary and Deter-ministic Methods for Design, Optimization and Control (Eurogen2011) conference, Italy 2011.

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Documents

Modeling and Pareto optimization of gas cyclone separator performance using RBF type artificial neural networks and genetic algorithms