CFD modeling and multi-objective optimization of cyclone geometry using desirability function, artificial neural networks and genetic algorithms

Applied Mathematical Modelling 37 (2013) 5680–5704

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

CFD modeling and multi-objective optimization of cyclone geometry using desirability function, artificial neural networks and genetic algorithms Khairy Elsayed ⇑, Chris Lacor Vrije Universiteit Brussel, Department of Mechanical Engineering, Research Group Fluid Mechanics and Thermodynamics, Pleinlaan 2, 1050 Brussels, Belgium

a r t i c l e

i n f o

Article history: Received 2 August 2011 Received in revised form 1 October 2012 Accepted 15 November 2012 Available online 14 December 2012 Keywords: Cyclone separator Response surface methodology (RSM) Computational fluid dynamics (CFD) Multiobjective optimization Desirability function Genetic algorithms

a b s t r a c t The low-mass loading gas cyclone separator has two performance parameters, the pressure drop and the collection efficiency (cut-off diameter). In this paper, a multi-objective optimization study of a gas cyclone separator has been performed using the response surface methodology (RSM) and CFD data. The effects of the inlet height, the inlet width, the vortex finder diameter and the cyclone total height on the cyclone performance have been investigated. The analysis of design of experiment shows a strong interaction between the inlet dimensions and the vortex finder diameter. No interaction between the cyclone height and the other three factors was observed. The desirability function approach has been used for the multi-objective optimization. A new set of geometrical ratios (design) has been obtained to achieve the best performance. A numerical comparison between the new design and the Stairmand design confirms the superior performance of the new design. As an alternative approach for applying RSM as a meta-model, two radial basis function neural networks (RBFNNs) have been used. Furthermore, the genetic algorithms technique has been used instead of the desirability function approach. A multi-objective optimization study using NSGA-II technique has been performed to obtain the Pareto front for the best performance cyclone separator. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Gas cyclones are widely used gas–solid separators using centrifugal forces. Cyclones are popular because of economy, simplicity in construction and adaptability to a wide range of operating conditions [1]. Reversed flow cyclones with a tangential inlet are the most common cyclone design as shown in Fig. 1. It consists of seven main geometrical parameters: inlet section height a and width b, cylinder height h, cyclone total height Ht , dust exit diameter (cone tip diameter) Bc , gas outlet diameter (also, called vortex finder diameter) Dx and vortex finder length S. All these parameters always given as a ratio of the cyclone body diameter D. It is generally known that these seven dimensions characterize the collection efficiency (cut-off diameter) and pressure drop of the cyclone separator [1–3]. 1.1. Performance parameters The cyclone performance is characterized by two parameters: the pressure drop and the collection efficiency. For low mass loading cyclone separators, the collection efficiency can be represented by the cut-off diameter (the particle diameter ⇑ Corresponding author. Tel.: +32 26 29 2368; fax: +32 26 29 2880. E-mail addresses: [email protected], [email protected] (K. Elsayed). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.11.010

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Fig. 1. Schematic diagram for Stairmand cyclone separator.

which produces 50% collection efficiency). Elsayed and Lacor [1] classified the different approaches to model the cyclone performance into four classes. 1. The theoretical and semi-empirical models, e.g., Shepherd and Lapple [4], Alexander [5], First [6], Stairmand [7], Barth [8], Avci and Karagoz [9] and Zhao [10]. 2. Statistical models through multiple regression analysis, e.g., Casal and Martinez-Benet [11], Dirgo [12] and Ramachandran et al. [13]. 3. Computational fluid dynamics (CFD) approach, e.g., Gimbun et al. [14], Zhao [15], Elsayed and Lacor [1,3,16]. 4. Artificial neural networks come into picture as an efficient approach to model the cyclone performance, e.g., Zhao and Sub [17] and Elsayed and Lacor [18,19].

1.2. Previous optimization studies In 1951, Stairmand [7] presented the geometrical ratios for high-efficiency cyclones. Until now, these ratios are still in use (cf. Fig. 1 and Table 1). Elsayed and Lacor [1] reported the following shortages in the Stairmand model for pressure drop calculation [20] which has been used to obtain these geometrical ratios: (1) the velocity distribution has been obtained from a moment-of-momentum balance, estimating the pressure drop as entrance and exit losses combined with the loss of static pressure in the swirl, i.e., neglecting the entrance loss by assuming no change of the inlet velocity occurs at the inlet area; (2) assuming a constant friction factor [21]. Due to the wide range of industrial applications of the cyclone separator, it was a matter of study for decades. However, the optimization studies on it is quite limited in literature. Moreover, many of these studies are not coherent studies or focus on a specific problem.

Table 1 The 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

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Ravi et al. [22] carried out a multi-objective optimization study on a set of N identical reverse-flow cyclone separators in parallel by using the non-dominated sorting genetic algorithm (NSGA). Two objective functions were used: the maximization of the overall collection efficiency and the minimization of the pressure drop. Non-dominated Pareto optimal solutions were obtained for an industrial problem in which 165 m3/s of air was treated. In addition, optimal values of several decision variables, such as the number of cyclones and eight geometrical parameters of the cyclone, are obtained. Their study shows that the diameters of the cyclone body and the vortex finder, and the number of cyclones used in parallel, are the important decision variables influencing the optimal solutions. Moreover, their study illustrates the applicability of NSGA in solving multi-objective optimization problems involving gas–solid separations. The main drawbacks are: (1) They used the model of Shepherd and Lapple [4] for predicting the dimensionless pressure drop (Euler number). In Shepherd and Lapple model, the Euler number depends on only three factors (Eu ¼ 16ab=D2x ) and they used it to optimize the seven geometrical parameters. (2) The barrel diameter, number of parallel cyclones and the gas velocity have been included into the optimization design space. Consequently, it is not devoted to the geometrical ratio. (3) They used many side constraints on the geometrical values (0:4 6 a=D 6 S=D; 0:15 6 b=D 6 ð1  Dx =DÞ=2 if 0:5 6 Dx =D 6 0:6) these constrains prevent searching for the global optimization geometrical ratios for the seven geometrical parameters. (4) No table for the non-dominated Pareto front points is presented from which the designer can select a certain geometrical ratios set. Swamee et al. [23] investigated the optimum values of the number of cyclones to be used in parallel, and the diameter of cyclone barrel D and exit pipe Dx , when a specified flow rate of gas is to be separated from solid particles, when the cut-off diameter is already specified. They used Stairmand model for calculation of pressure drop and Gerrard and Liddle formula for the cut-off diameter [23] which is not a widely used model. Instead of handling two objective functions, they blended the two objective into a single-objective problem which is not the suitable method to considering two conflicting objectives (the pressure drop and cut-off diameter). Safikhani et al. [24] performed a multi-objective optimization study on cyclone separators. First, they simulated many cyclones to obtain the pressure drop and the cut-off diameter and used artificial neural network approach to obtain the objective function values. Finally, a multi- objective genetic algorithms are used for Pareto based optimization of cyclone separators considering two conflicting objectives. However, the design variables were only four (instead of seven), the barrel height, the cone height, the vortex finder diameter and length. So they ignored the effect of the inlet dimensions, which has been acknowledged by other researchers as significant geometrical parameters for the cyclone flow field and performance (cf. Elsayed and Lacor [1,3,19,25]). Moreover, they did not explain why they selected these particular parameters. Furthermore, they applied four side constraints on the four tested variables, which prevent searching for the global optimization. Pishbin and Moghiman [26] applied the genetic algorithm technique to obtain an optimum cyclone design. They studied the seven geometrical parameters. The data used for optimization was obtained from 2-D axisymmetric simulations. However, the flow in the cyclone separator is 3-D unsteady. Instead of using multi-objective genetic algorithm (e.g., nondominated sorting genetic algorithm II (NSGA-II) [27]) they used the weighted-sum genetic algorithm. In this technique, a weighting factor is assigned for each objective function based on the user preference. The main shortage of the Pishbin and Moghiman [26] study are: (1) How to select the weighting factor, in scientific and engineering problems, it is a non-trivial task to find the one solution of interest to the decision maker [28]. The decision maker’s weight (no matter how defined) could be greater than necessary as more acceptable solutions are missed. Optimizing mostly profit could lead to poor quality or reliability, not a good compromise [28]. The weighted-sum genetic algorithm usually does not find all Pareto front points of interest. Nevertheless, this approach is a simple approach for handling multi-objective optimization problem, another simple but better result can be obtained using the desirability function approach [25,29]. (2) No table for the non-dominated Pareto front points are presented from which the designer can select a certain geometrical ratio. Safikhani et al. [30] carried out a multi-objective optimization study using the genetic algorithms to obtain the best vortex finder dimension (diameter and length) and shape (convergent and divergent). Four design variables have been investigated; vortex finder diameter, angle, the upper-part and lower-part lengths of the vortex finder. They applied neural networks to obtain a meta-model for the pressure drop and collection efficiency from CFD dataset. The main shortage of the Safikhani et al. [30] study are: (1) They used dimensional values instead of dimensionless. Moreover, they applied side constraints which prevent the optimization procedure from obtaining global optimization. (2) The selection of only the vortex finder dimensions as the design variables and neglecting the interaction with the other dimensions, specifically the inlet dimensions [3,19,31]. 1.3. Design variables and approaches Elsayed and Lacor [1,32] investigated the effect of seven geometrical parameters on the cyclone performance (pressure drop). They reported that the most significant factors affecting the performance are, the vortex finder diameter Dx , the inlet width b, the inlet height a and the total cyclone height Ht . The effects of the barrel height h, the vortex finder length S and the cone-tip diameter Bc are insignificant. Therefore, only four significant geometrical factors have been used in this study. The three other factors are fixed based on the Stairmand high efficiency design, i.e. Bc =D ¼ 0:375; h=D ¼ 1:5 and S=D ¼ 0:5. The selection of the values for both h and S are also based on the conclusion of Zhu and Lee [33] (when both the pressure drop and the particle collection efficiency are considered, a cyclone which has (h  S)/D of 1.0 would be an optimum design). Optimization of a gas cyclone is, indeed, a multi-objective optimization problem rather than a single objective optimization problem that has been considered so far in the literature [34]. Both the pressure drop and the cut-off diameter in gas

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cyclones are important objective functions to be optimized simultaneously in a multi-objective optimization problem. These objective functions values are either obtained from experiments, empirical models or computed using computational fluid dynamic (CFD) approaches. CFD modeling and optimization of the parameters are investigated in the present study and multi-objective Nelder–Mead optimization algorithms are used in order to maximize the collection efficiency (minimize the cutoff diameter) and minimize the Euler number (dimensionless pressure drop). The desirability function approach has been used to handle the two objective function. The application of the Nelder–Mead technique suffers from one big disadvantage. The optimal point may be local minimum because the technique depends on the starting point. This was the motivation to use also the genetic algorithm technique. Furthermore, the radial basis function neural networks can attain high accuracy as curve fitting approach than the response surface [19]. Consequently, the application of the radial basis function neural networks to model the relationship between the performance parameters and the four geometrical parameters will result in more accurate results. Elsayed and Lacor have presented many optimization studies (e.g., [1,19,35]). These studies was applied for the seven geometrical parameters but the current study is focusing only on the most significant four geometrical parameters according to Elsayed and Lacor [1]. The above mentioned studies used meta-models using mathematical models and experimental measurements but this study is based only on CFD simulations performed on sampling cyclone with a diameter of 31  103 m. 1.4. Study objectives There are four objectives of this study. (1) Investigation of the effect of the four geometrical parameters on the cyclone separator performance based on CFD simulations results. (2) Study the possible interaction between the four geometrical parameters using the response surface methodology. (3) Application of the multi-objective optimization technique to obtain new geometrical ratios for minimum pressure drop and minimum cut-off diameter, followed by a comparison of the numerical simulations for the new design and the Stairmand design using the Reynolds stress turbulence model. (4) Application of the genetic algorithm technique instead of the desirability function approach using the two trained radial basis function neural networks as fitness functions. 1.5. Study outline This study is performed in six stages. The first stage is the application of the response surface methodology (Box–Behnken design) to design an experiment to study the effect of four geometrical parameters (the inlet height and width, the vortex finder diameter, and the cyclone total height based on the study of Elsayed and Lacor [1]) on the cyclone performance using Statgraphics statistical software. Secondly, the obtained 27 test cases (designs) have been computationally simulated using the Reynolds stress turbulence model and discrete phase modeling with the Fluent solver. In the third stage, the Euler numbers and the cut-off diameters obtained are used to fit a second order polynomial (response surface) for each response (the Euler number and the cut-off diameter). The obtained polynomials have been used to study the variation of the two responses with the four geometrical parameters. Furthermore, the obtained polynomials have been used to obtain new cyclone geometrical ratios using the multi-objective optimization between the two conflicting objectives (the Euler number and the cut-off diameter) using the desirability function approach. The fourth stage is a computational investigation: a numerical comparison between the new design and the Stairmand design has been performed. Fifthly, replace the response surface methodology with the artificial neural networks approach and study the effect of each geometrical parameter on the cyclone separator performance. The last stage is an optimization study using the genetic algorithm instead of desirability function. 2. Design of experiment 2.1. The response surface technique In order to model a complex multivariate process where the responses are influenced by several variables, the response surface statistical technique seems the best approach [1]. The steps are as follow: (1) construct the design of experiment by identifying the four tested geometrical parameters (minimum and maximum values) and also to decide upon the dependent variables (the Euler number and the cut-off diameter). Statgraphics commercial statistical software gives 27 runs to be performed (cf. Table 2) using the Box–Behnken design of experiment method [36]. (2) Once the runs have been conducted (using CFD simulations) and recorded data inserted in the table; Statgraphics software fits a second order polynomial to this data (one response surface per dependent variable) [37]. The second-order polynomial (response surface) has the form [1]:

Y k ¼ b0 þ

4 4 X X XX bi X i þ bii X 2i þ bij X i X j ; i¼1

i¼1

ð1Þ

i
where b0 ; bi ; bii , and bij are the regression coefficients for intercept, linear, quadratic and interaction terms, respectively. X i and X j are the independent variables, and Y k is k the response variable (k ¼ 1 for the Euler number and k ¼ 2 for the cut-off diameter). (3) The third step, is the analysis of the response surface plot, main effect plots, Pareto chart and interaction plots.

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Table 2 The Box–Behnken’s design matrix and the responses of the Euler number and cut-off diameter.a Run No.

X1 a=D

X2 b=D

X3 Dx =D

X4 Ht =D

Euler number (–)

Cut-off diameterb (lm)

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

0.250 0.500 0.375 0.375 0.375 0.250 0.375 0.375 0.375 0.500 0.250 0.375 0.375 0.375 0.500 0.375 0.250 0.375 0.500 0.250 0.375 0.375 0.500 0.250 0.500 0.375 0.375

0.2625 0.2625 0.3750 0.2625 0.2625 0.2625 0.1500 0.1500 0.2625 0.3750 0.2625 0.1500 0.1500 0.3750 0.2625 0.3750 0.2625 0.2625 0.2625 0.3750 0.2625 0.3750 0.1500 0.1500 0.2625 0.2625 0.2625

0.50 0.75 0.25 0.50 0.75 0.50 0.50 0.50 0.25 0.50 0.75 0.50 0.75 0.50 0.50 0.50 0.25 0.25 0.25 0.50 0.50 0.75 0.50 0.50 0.50 0.50 0.75

3 4 4 4 3 5 4 5 5 4 4 3 4 3 5 5 4 3 4 4 4 4 4 4 3 4 5

3.500 2.827 27.257 3.475 2.333 2.952 2.726 2.530 17.712 9.086 1.413 3.000 1.211 7.500 5.904 6.326 12.720 21.000 25.440 4.543 4.770 3.029 3.634 1.817 7.000 4.770 1.968

1.546 2.541 1.158 1.683 2.444 1.364 1.353 1.284 0.956 2.163 1.939 1.455 1.826 2.081 1.787 1.836 0.860 1.084 1.127 1.651 1.683 2.610 1.513 1.155 2.025 1.683 2.156

a The values of h=D; S=D and Bc =D are identical to that of Stairmand high efficiency design given in Table 1, so the variation in the total cyclone height is due to the variations of the cone height. b The value of cut-off diameter depends on the gas velocity and density, particle density. In this study, D ¼ 31  103 m, gas volume flow rate Q in ¼ 50 l/ min, gas density q ¼ 1:0, gas viscosity l ¼ 2:11  105 Pa s and particle density qp ¼ 860 kg/m3.

2.2. Analysis of response surface plots Analysis of variance (ANOVA) showed that the resultant quadratic polynomial models adequately represent the used data with the coefficient of determination R2 , being 0.94284 and 0.973468 for the Euler number and cut-off diameter respectively. This indicates that the quadratic polynomial models obtained were adequate to describe the influence of the independent variables studied [38]. Analysis of variance (ANOVA) was used to evaluate the significance of the coefficients of the quadratic polynomial models (cf. Tables 3 and 4). For any of the terms in the models, a large F-value (small P-value) would indicate a more significant effect on the respective response variables. Based on the ANOVA results presented in Table 3, the variable with the largest effect on the pressure drop (Euler number) was the linear term of vortex finder diameter (b3 ), the linear terms of inlet height and width (P < 0.05) (b1 and b2 respectively), whereas the linear term of cyclone total height did not show a significant effect (P > 0.05). The quadratic term of vortex finder diameter also had a significant effect on the pressure drop; however, the effect of the other three quadratic terms was insignificant. Furthermore, the interaction between the inlet dimensions and vortex finder diameter also had a significant effect on the pressure drop (b13 and b23 ), whereas the effect of the remaining terms was insignificant. Table 4 confirms the significant effect of all linear terms of the inlet dimensions, vortex finder diameter and total cyclone height on the cut-off diameter. Moreover, the interaction between the inlet width and the vortex finder diameter also had a significant effect on the cut-off diameter, whereas the effect of the remaining terms was insignificant. For the visualization of the results of the analysis, main effects plot, Pareto chart and response surface plots were drawn. The slope of the main effect curve is proportional to the size of the effect, and the direction of the curve specifies a positive or negative influence of the effect [1,39] (Fig. 2(a)). Based on the main effect plot and the Pareto chart shown in Fig. 2, the most significant factors on the Euler number are: (1) the vortex finder diameter Dx , with a non-linear relation with a wide range where it is inversely proportional to Euler number and a narrow range with a direct proportional relation. (2) the inlet width b and the inlet height a almost linearly related to the Euler number, (3) the cyclone total height Ht has an insignificant effect. Pareto charts were used to summarize graphically and display the relative importance of each parameter with respect to the Euler number [1]. The Pareto chart shows all the linear and second-order effects of the parameters within the model and estimates the significance of each with respect to maximizing the Euler number response. A Pareto chart displays a frequency histogram with the length of each bar proportional to each estimated standardized effect [37]. The vertical line on the Pareto charts judges, whether each effect is statistically significant within the generated response surface model; bars

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Regression coefficient

b0

1.88508

F-ratio

P-value

Linear b1 b2 b3 b4

41.3522 134.36 49.1776 4.14082

14.21 33.78 209.03 0.94

0.0012 0.0000 0.0000 0.3430

Quadric b11 b22 b33 b44

16.9991 44.8511 87.9024 0.368311

0.1 0.5 44.97 0.2

0.7509 0.4878 0.0000 0.6602

Interaction b12 b13 b14 b23 b24 b34 R2

48.4622 90.448 1.096 185.69 1.56444 2.923 0.94284

0.41 7.01 0.02 28.5 0.03 0.47

0.5306 0.0155 0.8992 0.0000 0.8708 0.5017

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

Table 4 Analysis of variance and regression coefficients for the cut-off diameter.a Variable

Regression coefficient

b0

0.365539

F-ratio

P-value

Linear b1 b2 b3 b4

1.84206 1.23737 0.494019 0.00682157

67.21 89.76 524.79 15.14

0.0000 0.0000 0.0000 0.0009

Quadric b11 b22 b33 b44

2.25164 0.708116 0.307782 0.00841817

0.82 0.06 0.25 0.05

0.3763 0.8152 0.6237 0.8308

Interaction b12 b13 b14 b23 b24 b34 R2

2.73778 2.68 0.112 7.07071 0.164444 0.16 0.973468

0.59 2.77 0.08 18.61 0.14 0.63

0.4531 0.1116 0.7837 0.0003 0.7170 0.4360

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

that extend beyond this line represent effects that are statistically significant at a 95% confidence level. Based on the Pareto chart (Fig. 2(b)) there are six significant parameters at a 95% confidence level: the vortex finder diameter Dx ; the inlet width b; the inlet height a and the combinations aDx ; bDx and D2x . Therefore, the Pareto chart is a perfect supplement to the main effects plot. To visualize the effect of the independent variables on the dependent ones, response surfaces of the quadratic polynomial models were generated by varying two of the independent variables within the experimental range whereas holding the other factors at their central values [38] as shown in Fig. 2. Thus, Fig. 2(e) was generated by varying the inlet height a and the inlet width b whereas keeping the other factors fixed at their central values. The trend of the curve is linear, with a more significant effect for the inlet width b, and a weak interaction between the inlet height a and width b. The response surface plots of Figs. 2(c), (d) and (f) show that there are strong interactions between the vortex finder diameter Dx and the inlet dimensions a and b. From the analysis of the design of experiment for the cut-off diameter, the effect of variation of the vortex finder diameter Dx on the cut-off diameter is opposite to that on the Euler number (cf. Fig. 3(a)). The Pareto chart given in Fig. 3(b) indicates

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Fig. 2. Analysis of design of experiment for the Euler number. Note: In (c)–(f), for each plot of two independent variable, all other variables are hold at their central values.

the significance effect of the cyclone total height Ht on the cut-off diameter, but its effect is minor in comparison with the three other factors. The significant interaction exists only between the vortex finder diameter and the inlet dimensions.

3. Multi-objectives optimization using the desirability function The downhill simplex method is a widely used nonlinear optimization technique proposed by Nelder and Mead [40] and is a technique for minimizing an objective function in a many-dimensional space. It requires only function evaluations

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Fig. 3. Analysis of design of experiment for the cut-off diameter. Note: In (c)–(f), for each plot of two independent variable, all other variables are hold at their central values.

without any derivatives calculation [41] that motivate many researchers to use it [42,43], cf. Elsayed and Lacor [1] for more details. The target in this study is to obtain the global optimum values. Consequently, no linear constraints are applied. 3.1. The desirability function approach From the previous analysis, it is observed that the optimal values for the geometrical parameters that minimize the pressure drop are different from the values that minimize the cut-off diameter (cf. Figs. 2 and 3). As a result, a multiobjective optimization procedure is needed. The utilization of desirability function proposed by Harrington [44] is the most popular and strongly suggested method for multiple response optimization problems [45] to convert the problem into

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single-objective. The Statgraphics statistical package uses this desirability function approach for optimization of multiple response problems. The desirability function is first defined for each response. The desirability function dðyÞ expresses the desirability of a response value equal to y on a scale of 0 (if the response value is in an unacceptable range) to 1 (for the optimum value), for minimization of response variable. The procedure will then find the settings of the experimental factors that maximize a combined desirability function, which is a function that expresses the desirability of a solution involving m, where m here equals 2 (one for the Euler number and the other for the cut-off diameter), responses through the function of the form [46],

D¼

n

I I d11 d22

I . . . dmm

.  o 1 mm I j j¼1

;

ð2Þ

where dj is the calculated desirability of the jth response and Ij is an impact coefficient that ranges between 1 and 5 [46]. It represents the importance relative of each response over the other responses [47]. Where the default value is 3. In this study, more importance is given to the Euler number (I1 ¼ 5 and I2 ¼ 3). When a response is to be minimized, the desirability of a ^j is defined as, predicted response equal to y

dj ¼

8 1 > > < > > :

^j highj y lowj highj

0



^j < lowj y ^j 6 highj ; lowj 6 y

ð3Þ

^j > highj y

where lowj and highj are the minimum and maximum values of jth response. In this study, the minimum and maximum values obtained in the data sheet have been used (cf. Table 2). The desirability plots are given in Fig. 4. For more information about statistical model used for multiple response optimization refer to Ref. [46]. Table 5 presents the optimum values of geometry parameters that minimize the values of the Euler number and the cut-off diameter, which gives optimum desirability D ¼ 0:83 to minimize the Euler number and cut-off diameter. It is clear from Table 5 that the new optimized design differs from the Stairmand design in many geometrical parameters, whereas the new ratios will result in minimum pressure drop and minimum cut-off diameter. The ratio of inlet width to height b=a ¼ 0:589 lays in the optimal cyclone lies in the recommended range of b=a from 0.5 to 0.7 proposed by Elsayed and Lacor [3]. To understand the effect of these changes in the geometrical ratios on the flow field pattern and performance, a CFD study for the two designs is needed. 3.2. CFD comparison between the Stairmand and optimal design The Fluent solver has many turbulence models available for simulating turbulent flow. It is generally recognized that only the Reynolds stress model (RSM) and large eddy simulation (LES) can capture the main features of the highly complicated swirling flow in cyclone separators [1,3,14,48–56]. The Reynolds stress turbulence model has been used in this study to reveal the turbulent flow in the two cyclone separators. For the detailed governing equation for both the Reynolds averaged Navier–Stokes equation (RANS) and the discrete phase modeling (DPM) the reader can refer to Elsayed and Lacor [3,57]. The geometrical values are given in Table 6 for the two cyclones (cf. Fig. 5). Numerical settings The air volume flow rate Q in = 50 l/min for the two cyclones, air density is 1.0 kg=m3 and dynamic viscosity 2:11  105 Pa s. The turbulent intensity equals 5% and characteristic length equals 0.07 times the inlet width [58]. A velocity inlet boundary condition is applied at inlet, outflow at gas outlet and wall boundary conditions at all other boundaries. The finite volume method has been used to discretize the partial differential equations of the model using the SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations-Consistent) method for pressure velocity coupling and QUICK scheme to interpolate the variables on the surface of the control volume. The implicit coupled solution algorithm was selected. The unsteady Reynolds stress turbulence model (RSM) was used in this study with a time step of 0.0001 s. 3.3. Grid independency study The grid independence study has been performed for the tested cyclones. Three levels of grid for each cyclone have been tested, to be sure that the obtained results are grid independent. The hexahedral computational grids were generated using Gambit grid generator and the simulations were performed using Fluent 6.3.26 commercial finite volume solver on a 8 nodes CPU Opteron 64 Linux cluster. To evaluate accuratelyto estimate accurately the numerical uncertainties in the computational results, the concept of grid convergence index (GCI) was adopted using three grid levels per cyclone. Grid convergence index (GCI) The grid convergence index (GCI) proposed by Roache [59–61] was employed to test the grid independence of the simulations. The GCI are computed using three levels of grids in order to estimate accurately the order of convergence and check that the solution is within the asymptotic range of convergence [62]. The GCI is based upon a grid refinement error estimator derived from the theory of the generalized Richardson extrapolation [31,62]. The GCI is a measure of how far the computed value is

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Fig. 4. The desirability plots. Note: For each plot of two independent variable, all other variables are hold at their optimal values.

Table 5 The optimized cyclone separator design for best performance using the desirability function. Factor

Minimum

Center

Maximum

Stairmand design

Optimum

a b Dx Ht

0.25 0.15 0.25 3

0.375 0.2625 0.5 4

0.5 0.375 0.75 5

0.5 0.2 0.5 4.0

0.256 0.151 0.415 4.56

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Table 6 The values of geometrical parameters for the two designs (D = 31  103 m). Cyclone

a=D

b=D

Dx =D

Ht =D

h=D

S=D

Bc =D

Li =D

Le =D

Stairmand design New design

0.5 0.256

0.2 0.151

0.5 0.415

4 4.56

1.5 1.5

0.5 0.5

0.375 0.375

1.0 1.0

0.5 0.5

Fig. 5. The cyclone geometry and the surface mesh for the new design.

away from the value of the asymptotic numerical value. Consequently, it indicates how much the solution would change with a further refinement of the grid. A small value of GCI indicates that the computation is within the asymptotic range. The GCI on the fine grid is defined as [31]:

GCIfine ¼

Fs j e j ; ð r p  1Þ

ð4Þ

where F s is a factor of safety. F s ¼ 3 for comparison of two grids and 1.25 for comparison over three grids or more. For the coarse grid:

GCIcoarse ¼

F s j e j rp ; ð r p  1Þ

ð5Þ

e is a relative error measure of the key variable f between the coarse and fine solutions,

e¼

f2  f1 f1

ð6Þ

where f2 is the coarse-grid numerical solution obtained with grid spacing h2 . f1 is the fine-grid numerical solution obtained with grid spacing h1 . r is the grid refinement ratio (r ¼ h2=h1 > 1). For complicated geometries r is replaced by the ratio of the number of control volumes in the fine and coarse mesh [63] which is the case in this study,

r12 ¼

 D1 N1 ; N2

ð7Þ

where D ¼ 2 and 3 for two-dimensional and three-dimensional geometries respectively [59, p. 410]. N 1 and N 2 are the number of control volumes in the fine and coarse mesh respectively.  13  13 p is the order of the discretization method. r 12 ¼ NN12 ; r23 ¼ NN23 ; e12 ¼ f2  f1 ; e23 ¼ f3  f2 , where ei;iþ1 ¼ fiþ1  fi is the difference in the key variable f resulting from the use of different grids. For the details of the iterative procedure used to estimate the value of p, the reader can refer to Elsayed and Lacor [31]. Now one can calculate,

fine fine fine 1:25je12 j 1:25je23 j e12 ¼ f2ff1 1 ; e23 ¼ f3ff2 2 ; GCIfine 12 ¼ ðr p 1Þ and GCI23 ¼ ðr p 1Þ. GCI12 should be smaller than GCI23 . 12

23

To check if the solution is in the asymptotic range, a  1 (cf. Eq. (8)):

a¼

r p12 GCIfine 12 GCIfine 23

:

ð8Þ

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The Richardson extrapolation can be used to obtain the value of f when the grid spacing h vanishes (h ! 0) [59,64]:

  fexact ¼ f1 þ ðf1  f2 Þ= rp12  1 :

ð9Þ

Table 7 presents the grid convergency calculations using GCI method for three grid levels for each cyclone. The following conclusions have been obtained from the GCI analysis [31]:  The results are in the asymptotic range, because the obtained values for a are close to unity.  The ratio R is less than unity this means monotonic convergence [64]. fine  There is a reduction in the GCI value for the successive grid refinements (GCIfine 12 < GCI23 ) for the two variables (Eu and X 50 ). This indicates that the dependency of the numerical results on the cell size has been reduced. Moreover, a grid independent solution has been achieved. Further refinement of the grid will not give much change in the simulation results. For the two variables (Eu and X 50 ), the extrapolated value is only slightly lower than the finest grid solution. Therefore, the solution has converged with the refinement from the coarser grid to the finer grid [64]. Fig. 6 presents a qualitative proof that the obtained results are in the asymptotic range.  The value of ei;iþ1 represent the relative change in each value from coarse to medium and from medium to fine mesh. For example, e1;2 ¼ 0:0256 for the Euler number in the new design means the percentage change in the Euler number when the mesh becomes 986,748 cells instead of 717,353 cells equals 2.56%. This means no need to use the fine mesh and the usage of the medium mesh of 717,353 cells is sufficient. Another example, e1;2 ¼ 0:0052 for the cut-off diameter in the Stairmand design means the percentage change in the cut-off diameter when the mesh becomes 848,783 cells instead of 622,253 cells equals 0.52%. This means no need to use the fine mesh and the usage of the medium mesh of 622,253 cells is sufficient. In summary, the grid refinement study shows that a total number of about 622,253 hexahedral cells for Stairmand cyclone and 717,353 hexahedral cells for the new design are sufficient to obtain a grid-independent solution, and further mesh refinement yields only small, insignificant changes in the numerical solution.

Table 7 Grid convergency calculations using GCI method using three grid levels for each cyclone. i

Ni

fi

New design Euler number 0c 1

986748

2.5195 2.5380

2

717353

2.6030

3

362679

4.6170

Cut-off diameter 0 1

986748

0.6621 0.6710

2

717353

0.6900

3

362679

0.9820

Stairmand design Euler number 0 1

848783

5.4310 5.4860

2

622253

5.6060

3

356181

6.8330

Cut-off diameter 0 1

848783

1.7187 1.7210

2

622253

1.7300

3 a b c

356181

r i;iþ1

ei;iþ1

ei;iþ1

GCIi;iþ1 %

1.1121

0.0650

0.0256

0.9112

1.2553

2.0140

0.7737

4.0099

1.1121

0.0190

0.0283

1.6647

1.2553

0.2920

0.4232

5.0609

1.1090

0.1200

0.0219

1.2521

1.2044

1.2270

0.2189

3.9010

1.1090

0.0090

0.0052

0.1678

1.2044

0.1850

0.1069

0.8173

Ra

ab

0.0331

1.0256

0.0669

1.0283

0.0999

1.0219

0.0489

1.0052

1.9150

R = e12 =e23 .   a = rp12 GCI12 =GCI23 . The value at zero grid space (h ! 0). i = 1, 2 and 3 denote the calculations at the fine, medium and coarse mesh respectively.

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7

Euler number

1.5 5 New design (Eu) New design (X50) Stairmand design (Eu)

4

Cut-off diameter

6

Stairmand design (X50)

1 3

2

0

1E-06

(h --> 0 )

2E-06

N

-1

Fig. 6. Qualitative representation of the grid independency study. The Euler number and the cut-off diameter for the two cyclones using the three grid levels. N 1 is the reciprocal of the number of cells, h ! 0 means the value at zero grid size (cf. Table 7). To obtain a smooth curve; the spline curve fitting has been applied in Tecplot post-processing software.

In order to validate the obtained results, it is necessary to compare the prediction with experimental data. The comparison is performed with the measurements of Hoekstra [49] using the Laser doppler anemometry (LDA) technique. The present simulations are compared with the measured axial and tangential velocity profiles at an axial station located at 0.75D below the inlet section top [[65], Fig. 2] where D is the cyclone barrel diameter. The interested reader can refer to Shukla et al. [65] and Hoekstra [49] for more informations about the used cyclone in the validation. Fig. 7 shows an acceptable matching between the RSM simulation and the experimental velocity profiles. Considering the complexity of the turbulent swirling flow in the cyclones, the agreement between the simulations and measurements is considered to be quite acceptable [66]. 3.3.1. Flow field pattern All subsequent figures present the flow variables in a dimensionless form. This approach results in a better study how the velocity varies inside the cyclone in terms of the average inlet velocity. Moreover, This allows to compare the velocity profiles for different cyclones working at different inlet velocities (at a fixed volume flow rate). The dimensional value of the

2 1.8

Experiment CFD

0.4

1.6

Axial velocity/ Inlet velocity

Tangential velocity/ Inlet velocity

0.6

Experiment CFD

1.4 1.2 1 0.8 0.6

0.2

0

-0.2

0.4 0.2 -0.5

0

0.5

Radial position / Cyclone radius

-0.4

-0.5

0

0.5

Radial position / Cyclone radius

Fig. 7. Comparison of the time averaged tangential and axial velocity profiles between the LDA measurements [49,65] and the RSM simulation. Note: The cyclone barrel diameter D ¼ 0:29 m for the validation.

K. Elsayed, C. Lacor / Applied Mathematical Modelling 37 (2013) 5680–5704

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velocity can be calculated using the given volume flow rate and the inlet section dimensions. In case of static pressure, the pressure value can be estimated using the dynamic pressure at the inlet (12 qV 2in ), where q is the gas density, V in is the average inlet velocity. The approach of representing the velocity and pressure profiles in dimensionless forms is widely used in the cyclone separator studies, e.g., Derksen et al. [67, p. 65], [68, p. 876] and Gronald and Derksen [69, p. 166]. Fig. 8 shows the contour plot at Y = 0 and throughout the inlet section. In the two cyclones, the time-averaged dimensionless static pressure (Euler number) decreases radially from the wall to center. A negative pressure zone appears in the forced vortex region (central region) due to high swirling velocity. The pressure gradient is largest along the radial direction, whereas the gradient in the axial direction is very limited. The cyclonic flow is not symmetrical as is clear from the shape of the low-pressure zone at the cyclone center (twisted cylinder). The flow asymmetry is more pronounced in the new cyclone. However, the two cyclones have almost the same flow pattern, but the Euler number of the Stairmand design is nearly twice that of the new design.

Fig. 8. The contour plots for the time averaged flow variables at sections Y = 0 and throughout the inlet section. From top to bottom: The dimensionless static pressure (divided by the dynamic pressure at inlet), the dimensionless tangential velocity, axial velocity. Note: Both cyclones have the same air volume flow rate.

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New design Stairmand design

4

New design Stairmand design

1.2

Tangential velocity / Inlet velocity

Static pressure / Dynamic pressure at inlet

4.5

3.5 3 2.5 2 1.5 1 0.5

1

0.8

0.6

0.4

0.2 0 -0.5 -1

-0.5

0

0.5

Radial position / Cyclone radius

1

0 -1

-0.5

0

0.5

1

Radial position / Cyclone radius

Fig. 9. The radial profiles of the time averaged static pressure and tangential velocity at z0 /D = 1.5.

The dimensionless static pressure distribution presented in Fig. 9 for the two cyclones indicates that the highest dimensionless static pressure for the Stairmand design is more than twice that of the new design at all sections whereas the central value is almost the same for the two cyclones. This indicates that, the new design has a lower dimensionless pressure drop than the Stairmand design. However, these results are obtained at different inlet velocity for the two cyclones (to have the same air flow rate). The same Euler number values would be obtained if the two cyclones work at the same inlet velocity because the Euler number is not a function of flow velocity if the Reynolds number is higher than 2  104 [1]. The tangential velocity profile is composed of two regions. In the inner region, the flow rotates approximately like a solid body (forced vortex), where the tangential velocity increases with radius. After reaching its peak the velocity decreases with radius in the outer part of the profile (free vortex). The tangential velocity distributions for the two cyclones are nearly identical in pattern (Rankine profile). The inner part of the tangential velocity distribution of the two cyclones is very similar. The outer part for the new design is flatter in comparison with the Stairmand cyclone. This implies that there is more space in the optimal cyclone where the particles are subjected to high centrifugal force. Whereas, the maximum dimensionless tangential velocity for Stairmand cyclone is higher than that for the new design, the cyclone performance is not only affected by the maximum tangential velocity but also with the separation space (the new design is longer than the Stairmand design). The axial velocity profiles for the two cyclones are different in values and shape. Stairmand cyclone exhibit the inverted W axial velocity profile. The new design exhibit the inverted W axial velocity profile away from the inlet section and the inverted V elsewhere. This is due to the very high swirl exist at the inlet section in case of the new design. 3.3.2. Performance parameters To calculate the cut-off diameters of the two cyclones, 104 particles were injected from the inlet surface at a velocity equals the gas velocity. The particle density qp is 860 kg=m3 and the maximum number of time steps for each injection was 2109 steps [57]. To inject 104 particles from the inlet section, two options are available in Fluent 6.3.26, either using an injection file, or by creating a plane. In this study, a bounded plane is created at the inlet section. The number of points on each side are adjusted to have a uniformly distributed points (according to the ratio of inlet height to the inlet width for each cyclone). The interested reader can refer to the Fluent manual [70, Section 27.6]. In the current study, 104 mono-dispersed particles have been injected. In practical cyclones, the injected specimen has a wide range of particle diameters. If the particle size distribution is given from an experimental measurements as diameter ranges and corresponding mass fractions, the Rosin–Rammler diameter distribution method [71] can be used to represent the given particle distribution. Then the minimum, maximum and mean particle diameters in addition to the number of particle diameters and the spread parameter should be given to Fluent [70, Section 23.3.13], to mimic the given experimental particle distribution. Moreover, if the total mass of particles are known, the Swamee and Swamee approach [72, Eq. (29)] can be applied to obtain the number of particles to be injected. Since this study does not mimic any certain experimental data, the two above mentioned approaches are not applicable. For discrete phase modeling using one-way coupling, and for RANS simulations, a sufficient number of particles (in order to obtain a good statistical averaging) should be more than 5000 particles and for LES simulations 10,000 particles is the minimum [73, p. 88]. One should study the effect of the number of injected particles on the obtained cut-off diameter. This check has not been performed here but from many other similar studies, the above mentioned values, as a rule of thumb, turn out to be sufficient to have an independent solution [1,3,19,31,35,57,66]. The interested reader about the settings for the discrete phase modeling can refer to Elsayed and Lacor [57, pp. 137–138] and Shukla et al. [74].

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K. Elsayed, C. Lacor / Applied Mathematical Modelling 37 (2013) 5680–5704 Table 8 The cyclone performance parameters using CFD simulations and four mathematical models. Parameter

Cyclone

Statgraphics

CFD

Ramachandran

MM

Iozia

Ritema

Euler number

Stairmand design New design Stairmand design New design

5.673 2.221 1.706 0.865

5.606 2.603 1.73 0.69

4.85 2.71 – –

5.33 2.99 – –

– – 1.69 0.95

– – 1.55 0.71

Cut-off diameter,

lm

The DPM analysis results and the Euler number for the two cyclones are given in Table 8. Table 8 introduces also a comparison between the CFD results and four different mathematical models viz., the Ramachandran model [13], the Muschelknautz method of modeling (MM) [1,75,76], the Iozia model [77], the Ritema model [78] (cf. Hoffmann and Stein[21] for more details about these mathematical models). However, no-good matching between the two approaches (CFD and mathematical models) is obtained, they agree in the trend of superior performance of the new design. The Euler number and cut-off diameter for the new design is approximately half that of Stairmand design. 4. Artificial neural network (ANN) approach Artificial neural networks (ANNs) have become an attractive approach for modeling highly complicated and nonlinear system [17,19,79]. In this study, the radial basis function neural network (RBFNN) has been used to model the effect of cyclone dimensions on both the pressure drop and the cut-off diameter. For more details about the radial basis function neural networks, the reader can refer to Elsayed and Lacor [19,35]. Two RBFNNs have been trained using the 27 data set obtained from the CFD simulations for both the Euler number (the dimensionless pressure drop) and the cut-off diameter. The pressure drop is defined as DP = (the area- and time-averaged static pressure at the inlet section)  (the area- and time-averaged static pressure at the gas exit section). The pressure drop across the cyclone essentially depends on the cyclone dimensions and operating conditions. Generally, it is proportional to the average dynamic pressure at the inlet and is often defined as [17]:

DP ¼ Eu

  1 qg V 2in ; 2

ð10Þ

where Eu is the Euler number (the dimensionless pressure drop also called the pressure drop coefficient [17, p. 4132, 21, Chapter 2, p. 71, 80, p. 62]). The Euler number is a complex nonlinear function of the cyclone geometrical dimensions and is not affected by operating conditions in the high Reynolds number (Re > 5  104 ) [1,21]. The Euler number will be constant for any cyclone configuration regardless of size as long as the dimension ratios remain the same, although the pressure drop varies with different operating conditions (due to the effect of qg and V in ). Therefore, pressure drop can be established by determining experimentally or theoretically for a particular cyclone design and also be modified by the semi-empirical correlations to take the effect of solid loading [17]. In this study, the performance parameters are assumed as functions of only the four geometrical parameters, whereas all other parameters kept constant, Eq. (11):

 a b Dx Ht ; ; ; D D D D   a b Dx H t ¼ f2 ; ; ; : D D D D

Eu ¼ f1 x50



ð11Þ

Due to the large difference in the order of magnitude of the value (cf. Table 5), the available dataset is transformed into 1 to 1 interval using the Matlab intrinsic function; mapminmax in order to avoid solution divergence [17,19]. The RBFNN calculations have been performed using the neural network toolbox available from Matlab commercial software 2010a. The cut-off diameter x50 for cyclone separator is always given in units of lm. Another way to represent x50 is using a dimensionless number; Stokes number Stk50 ¼ qp x250 V in =ð18lDÞ [67]. It is the ratio between the particle relaxation time; qp x250 =ð18l) and the gas flow integral time scale; D=V in . 4.1. Fitting the RBFNNs The configured RBFNNs predictions versus the CFD data for the Euler number and cut-off diameter are shown in Fig. 10. It can be seen that the RBFNN models are able to attain the high training accuracy. The training mean square errors are zeros (i.e., identical matching between the input and output, the reason behind that may refer to the consistency between the used data in the simulation using the design of experiment, which is not the case for the study of Elsayed and Lacor [19] using experimental data set), Fig. 10. This indicates that, in comparison with traditional models of curve fitting, the models based on an artificial intelligence algorithm have a superior capability of nonlinear fitting. Especially, the RBFNN has its unique and optimal approximation characteristics in learning process [17,19].

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Data point Linear fit

26

Data point Linear fit

2.5

24 22

18

y=0.999 x -2.2E-15 R=0.99999 2 E = 0.0

Predicted value (y)

Predicted value (y)

20

16 14 12 10

y=0.999 x +4.03E-16 R=0.99999 2 E = 0.0

2

1.5

8 6 4

1

2 5

10

15

20

25

1

1.5

Input value (x)

2

2.5

Input value (x)

(a) The Euler number

(b) The cut-off diameter

Fig. 10. Linear regression of the RBFNNs for the Euler number and the cut-off diameter.

Fig. 10 illustrates the agreement between the RBFNNs input and output. The obtained relation is a typical linear relation with a coefficient of correlation close to 1 (R > 0:999). The agreement between the input and output of the RBFNN is also clear from the value of the mean squared error E2 . Consequently, the trained neural networks predict very well both the Euler number and cut-off diameter values and can be used in cyclone design and performance estimation. Fig. 10 present different performance indicators as a validation of the proposed models for the trained data.

4.2. The effect of the four geometrical parameters on the cyclone performance based on RBFNNs The effects of the geometrical parameters on both the Euler number and the cut-off diameter are depicted in Figs. 11 and 12. To study the effect of each parameter, the tested RBFNNs models have been used by varying one parameter at a time from its minimum to maximum values of the available CFD dataset, whereas the other parameters are kept constant at their mean values (cf. Table 5). Figs. 11 and 12 indicate the significant effect of the vortex finder diameter Dx , the inlet width b, the inlet height a. Less effect is assigned to the total cyclone height Ht . More analysis is given in Tables 9 and 10.

Ht 3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

Euler number

15

10 Dx a b Ht

5

0.2

0.3

0.4

0.5

0.6

0.7

D x, a, b Fig. 11. The effect of geometrical parameters on the Euler number.

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Ht 2.4

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

2.2

X 50 [micron]

2 1.8 1.6 1.4 Dx a b Ht

1.2 1 0.8

0.2

0.4

0.6

D x, a, b Fig. 12. The effect of geometrical parameters on the cut-off diameter.

Table 9 The variation of the Euler number with cyclone dimensions using the RBFNN model (cf. Fig. 11). Factor

Analysis

Dx

The most significant effect is that of the vortex finder diameter Dx with inverse relationship when increasing Dx up to  0:65 after which the relation becomes direct, Fig. 11. This can be explained as follows: Although the pressure loss in the vortex finder decreases with increasing the vortex finder diameter like the case of viscous flow in a pipe, the Euler number in the cyclone body instead will increase due to the decrease of the flow area just after the flow entrance from the inlet region (the annular space between the barrel and the vortex finder). This analysis indicates the large contribution of the pressure loss in the vortex finder to the total Euler number (the pressure loss at the entrance, the pressure loss in the cyclone body, and the pressure loss in the vortex finder [1]) The inlet height a and width b is almost linearly related to the Euler number. But, why the relation is direct (at the same flow rate increasing the inlet dimensions decreases the inlet velocity. Consequently,the loss in the vortex finder will decrease)? The reason is the increase in the pressure drop at the inlet section, due to deviation of the inlet flow when it mixes with the swirling flow The effect of changing the total cyclone height Ht on the Euler number is very small

a and b

Ht

Table 10 The variation of the cut-off diameter with cyclone dimensions using the RBFNN model (cf. Fig. 12). Factor

Analysis

Dx

The vortex finder diameter has the most significant effect on the cut-off diameter x50 (the highest slope in Fig. 12). The slope is very high for Dx ¼ 0:4—0:55 and any further increase or decrease in Dx beyond the above range produces a small change in x50 . Increasing the vortex finder diameter decreases the swirling intensity in the cyclone (i.e. Reduction in the centrifugal force). Consequently, low collection efficiency (higher x50 ) is obtained. In the meantime, the increase of Dx decreases the pressure drop. This is one of the main reasons of the trade-off between the Euler number and the cut-off diameter objectives. This makes the optimization of cyclone geometry a multiobjective procedure The variation of x50 with the inlet width and height are similar in trend and significance to that for Dx but here the slope is high in the range of a ¼ 0:3—0:4 b ¼ 0:2—0:275 The effect of the cyclone total height on the cut-off diameter is due to the effect of the cone height as the barrel height is fixed in this study. The slope is very small, with a general trend of inverse relation (increasing the separation space, enhances the collection efficiency)

a and b Ht

5. Optimization using genetic algorithms Genetic algorithms (GAs) are one of the optimization methods finding wide application in optimization problems [81]. Genetic algorithm searches stochastically through the real space of the problem by generating a random initial population. GA technique is one kind of evolution-based systems which measures the fitness of each individual in population, and then selects the fittest individuals until reproducing the intermediate population. The genetic operators affect some individuals in this population and produce the next population for the new generation through selection, crossover, and mutation operations. The GA could optimize linear and nonlinear objective functions by exploring the space of the problem [81]. Fig. 13

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Fig. 13. Flow chart for the genetic algorithms process.

present the flow chart of GA. For more details about the genetic algorithm optimization technique, the reader can refer to Ref. [28,35,82–86].

5.1. Optimal cyclone design for minimum pressure drop The genetic algorithm optimization technique has been applied to obtain the geometrical ratios for minimum pressure drop (Euler number). The objective function is the Euler number (using the trained radial basis function neural network). The design variables are four geometrical dimensions, the inlet height a, the inlet width b, the vortex finder diameter Dx and the total cyclone height Ht . These four variables are the most significant factors which affect the cyclone performance [1]. Table 11 presents the settings used to obtain the optimum design for minimum pressure using global optimization Matlab toolbox in Matlab 2010a commercial package. Table 12 gives the optimum values for cyclone geometrical parameters for

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K. Elsayed, C. Lacor / Applied Mathematical Modelling 37 (2013) 5680–5704 Table 11 Genetic operators and parameters for single objective optimization. Population type

Double vector

Initial range

[0.25 0.15 0.25 3; 0.5 0.375 0.75 5] for a; b; Dx and Ht respectively Rank Tournament (tournament size equals 4) 2 0.8 Intermediate crossover with the default value of 1.0 The constraint dependent default 800

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

200

Table 12 The optimized cyclone separator design for minimum pressure drop using genetic algorithm. Factor

Low

High

Stairmand designa

Optimum designb

a b Dx Ht

0.25 0.15 0.25 3.0

0.5 0.375 0.75 5.0

0.5 0.2 0.5 4.0

0.4915 0.1578 0.6167 4.5384

5.606 1.706

2.369 1.7038

Euler number Cut-off diameter a b

The values for the Euler number and the cut-off diameter for the Stairmand design have been obtained from CFD simulations, cf. Table 8. The values for the Euler number and the cut-off diameter for the new optimal design have been obtained from the trained RBFNN.

Table 13 Genetic operators and parameters for multi-objective optimization. Population type Population size Initial range Selection operation Crossover fraction Crowding distance fraction Crossover operation Maximum number of generations

Double vector 60 (i.e., 15⁄ number of variables which is the default in Matlab) [0.25 0.15 0.25 3; 0.5 0.375 0.75 5] for a; b; Dx and Ht respectively tournament (tournament size equals 2) 0.8 0.35 Intermediate crossover with the default value of 1.0 800 (i.e., 200⁄ number of variables which is the default in Matlab)

minimum pressure drop estimated by the artificial neural network using the genetic algorithm optimization technique. It is clear from Table 12 that the new optimized design is very close to the Stairmand design in many geometrical parameters, whereas the new ratios will result in the minimum pressure drop.

5.2. Optimal cyclone design for best performance using NSGA-II In case of cyclone separator geometry optimization for minimum Euler number and minimum cut-off diameter, the objectives are conflicting with each other. There is no best solution for which all objectives are optimal simultaneously [87]. The increase of one objective will lead to the decrease of the other objective. Then, there should be a set of solutions, the so-called Pareto optimal set or Pareto front, in which one solution cannot be dominated by any other member of this set [35]. Recently, a number of multi-objective genetic algorithms (MOGAs) based on the Pareto optimal concept have been proposed. The well known nondominated sorting genetic algorithm II (NSGA-II) proposed by Deb et al. [27] is one of the most widely used MOGAs since it provides excellent results as compared with other multi-objective genetic algorithms proposed [88]. Table 13 presents the genetic operators and parameters for multi-objective optimization. The Euler number and the Stokes number values have been obtained from the artificial neural network trained by the CFD data set. The Pareto front (non dominated points) is presented in Fig. 13 and Table 14. Fig. 14(a) clearly demonstrate tradeoffs in objective functions (Euler number and Stokes number). All the optimum design points in the Pareto front are non-dominated and could be chosen by a designer as optimum cyclone separator [24]. This set of designs makes the Pareto front approach

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

A B C

point

a

b

Dx

Ht

h

S

Bc

Euler number

Stokes number  103

1 2 3 4A 5 6 7 8 9 10 11 12 13 14 15 16 17 18B 19 20 21C 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0.500 0.391 0.374 0.250 0.252 0.250 0.320 0.363 0.259 0.391 0.419 0.251 0.455 0.251 0.298 0.376 0.472 0.500 0.255 0.252 0.286 0.351 0.251 0.265 0.493 0.312 0.352 0.486 0.252 0.252 0.251 0.426 0.261 0.272 0.253 0.251 0.266

0.375 0.296 0.270 0.150 0.152 0.150 0.214 0.268 0.159 0.284 0.327 0.151 0.365 0.151 0.189 0.270 0.358 0.375 0.155 0.152 0.183 0.233 0.151 0.154 0.321 0.206 0.234 0.365 0.152 0.153 0.151 0.333 0.150 0.161 0.154 0.151 0.284

0.250 0.250 0.252 0.750 0.362 0.732 0.300 0.300 0.472 0.252 0.256 0.711 0.251 0.695 0.441 0.262 0.251 0.250 0.608 0.497 0.346 0.295 0.638 0.460 0.250 0.320 0.295 0.250 0.730 0.676 0.650 0.252 0.747 0.553 0.553 0.711 0.252

4.994 4.979 4.961 4.899 4.893 4.846 4.929 4.945 4.960 4.966 4.983 4.853 4.976 4.858 4.910 4.960 4.986 4.994 4.810 4.879 4.907 4.949 4.897 4.881 4.991 4.869 4.950 4.992 4.840 4.861 4.895 4.977 4.838 4.880 4.880 4.853 4.966

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375 0.375

33.750 20.806 17.883 0.756 3.326 0.799 8.617 12.160 2.086 19.737 23.514 0.854 29.693 0.895 3.277 16.657 30.324 33.750 1.214 1.759 4.954 10.568 1.050 2.193 28.441 7.136 10.728 31.833 0.817 0.957 1.013 25.126 0.798 1.630 1.451 0.854 13.428

0.511 0.569 0.595 6.958 1.262 6.446 0.840 0.778 2.080 0.581 0.568 5.903 0.528 5.504 1.554 0.630 0.525 0.511 3.820 2.380 1.103 0.789 4.354 1.780 0.531 0.941 0.784 0.518 6.387 5.071 4.565 0.551 6.822 2.945 3.022 5.903 0.633

Minimum Maximum

0.25 0.50

0.150 0.375

0.250 0.750

4.810 4.995

1.5 1.5

0.5 0.5

0.375 0.375

0.7555 33.7497

0.511 6.958

The point of minimum Euler number and maximum Stokes number. The point of maximum Euler number and minimum Stokes number. An optimal point for the multi-objective optimization problem (cf. Fig. 14(a)).

more preferred than the desirability function approach which gives only one design point. Three points A, B and C are indicated in Fig. 14(a). Point A indicates the point of minimum Euler number (maximum Stokes number). Point B indicates the point of maximum Euler number (minimum Stokes number). Point C indicates an optimal point for the multi-objective optimization problem. In order to obtain the Euler number-Stokes number relationship, Fig. 14(b) has been drawn. It indicates a general relationship (trend) between the two dimensionless numbers irrespective to the four geometrical parameters values. A second-order polynomial has been fitted between the logarithms of Euler number and Stokes number, Eq. (12). The obtained correlation can fit the data with a coefficient of determination R2 ¼ 0:99613 as shown in Fig. 14(b). Elsayed and Lacor [35] presented another correlation between the Euler number and the Stokes number, Eq. (13). Eq. (13) presents a good matching for only high values of Euler numbers. For smaller values of Euler number, there is underestimation of the Stokes number. The difference between the two correlations can be referred to two reasons. Firstly, the new correlation is based on the CFD simulations data for both Euler number and Stokes number, whereas Elsayed and Lacor correlation was obtained from experimental data for Euler number values and Iozia and Leith model for the Stokes number values. Secondly, the new correlation is limited to only four geometrical parameters. The other three factors are fixed, h ¼ 1:5; S ¼ 0:5 and Bc ¼ 0:375. Consequently, the new correlation (Eq. (12)) is valid only for these values: 2

1:1645log 10 ðEuÞ2:3198

2

0:9479log 10 ðEuÞ2:5154

Stk50 ¼ 100:3533ðlog 10 ðEuÞÞ Stk50 ¼ 100:3016ðlog 10 ðEuÞÞ

ð12Þ :

ð13Þ

K. Elsayed, C. Lacor / Applied Mathematical Modelling 37 (2013) 5680–5704

5701

Fig. 14. Pareto front plots.

6. Conclusions CFD simulations data have been used to understand the effect of four geometrical parameters on the cyclone performance and to optimize the cyclone geometry. Two meta-models have been used viz., the response surface and the radial basis function neural network approaches. Two optimization techniques have been applied, the desirability function with Nelder– Mead technique and the non-sorted dominated genetic algorithm NSGA-II.  The response surface methodology has been used to fit two second-order polynomials to the Euler number and cut-off diameter obtained from CFD simulations. The analysis of variance of the Euler number indicates a strong interaction between Dx with (a; b) and between Dx with b only for the cut-off diameter.  The bi-objective functions have been converted to single-objective function using the desirability function approach. A new optimal design has been obtained using the Nelder–Mead technique available in Statgraphics commercial software. The ratio of inlet width to height b=a ¼ 0:589 lays in the optimal cyclone lies in the recommended range of b=a from 0.5 to 0.7 proposed by Elsayed and Lacor [3]. The new design and the Stairmand design have been computationally compared to

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get a clear vision for the differences between the flow field pattern and performance in the two designs. The CFD simulations results and four mathematical models confirmed the better performance of the new design in comparison with the Stairmand design. The result demonstrates that artificial neural networks can offer an alternative and powerful approach to model the cyclone performance better than the response surface methodology. The used RBFNN presented zero mean squared error and almost unity coefficient of determination. The analysis indicates the significant effect of the vortex finder diameter Dx and the inlet dimensions a and b on the cyclone performance. Moreover, the range of high influence is given for each geometrical parameters using the trained RBFNNs. The trained RBFNN for the Euler number has been used to get a new optimized cyclone for minimum pressure drop (Euler number) using the genetic algorithm optimization technique. The new cyclone design is very close to the Stairmand high efficiency design in the geometrical parameter ratio, and superior in low pressure drop at nearly the same cut-off diameter. But, the optimal design obtained from the desirability function results in a better collection efficiency (smaller cutoff diameter) as is clear from Table 8 because there the cut-off diameter is included in the desirability function, whereas, the obtained result are for single objective (Euler number). The two trained RBFNNs have been used in a multi-objective optimization process using NSGA-II technique. The Pareto front is presented for the designer with a wide choice for selection. A second-order polynomial has been fitted between the logarithms of Euler number and Stokes number to obtain a gen2 eral formula, Stk50 ¼ 100:3533ðlog10 ðEuÞÞ 1:1645 log10 ðEuÞ2:3198 with a coefficient of determination R2 ¼ 0:99613. This formula can be used to obtain the Stokes number if the Euler number is known at h ¼ 1:5; S ¼ 0:5 and Bc ¼ 0:375.

As a recommendation for future work, the CFD simulations results given in Table 2 can be used with Kriging or gradientenhanced Kriging [89,90] to obtain two surrogate models (for both the Euler number and the cut-off diameter using the available data). The resulting meta-model can be used for performance prediction for new cyclone designs or to calculate the objective functions in optimization instead of the RBFNN models used in the current study. Moreover, Co-Kriging [91] can be used to fit a surrogate model using multi-fidelity data. The CFD data can be considered as high-fidelity data and the low fidelity data can be obtained from some robust mathematical models, e.g., Ramachandran et al. model [13] for pressure drop calculations and Rietema model [78] for cut-off diameter estimation. Alternatively, if some experimental data are available, it can serve as a high-fidelity data whereas, the CFD data given in Table 2 will be the source of the low-fidelity data.

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