Modeling and multi-objective optimization of square cyclones using CFD and neural networks

chemical engineering research and design 8 9 ( 2 0 1 1 ) 301–309

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Modeling and multi-objective optimization of square cyclones using CFD and neural networks H. Safikhani a,∗ , M.A. Akhavan-Behabadi a , N. Nariman-Zadeh b , M.J. Mahmood Abadi b a b

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran

a b s t r a c t Increasing of collection efficiency () and decreasing of the pressure drop (p), simultaneously, are important purpose in the design of cyclone separators. In the present study, multi-objective optimization of square cyclones is performed at three steps. At the first step, collection efficiency () and the pressure drop (p) in a set of square cyclones are numerically investigated using CFD techniques. Two meta-models based on the evolved group method of data handling (GMDH) type neural networks are obtained, at the second step, for modeling of  and p with respect to geometrical design variables. Finally, using obtained polynomial neural networks, multi-objective genetic algorithms are used for Pareto based optimization of square cyclones considering two conflicting objectives,  and p. It is shown that some interesting and important relationships as useful optimal design principles involved in the performance of square cyclones can be discovered by Pareto based multi-objective optimization of the obtained polynomial meta-models. Such important optimal principles would not have been obtained without the use of both GMDH-type neural network modeling and the Pareto optimization approach. Crown Copyright © 2010 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers. All rights reserved. Keywords: Square cyclone; Gas–solid; Multi-objective optimization; GMDH; CFD

1.

Introduction

Cyclones are widely used in filtration and separation industries as a result of their low setup and maintenance cost. Conventional cyclone which has circular cross section was the commonly used cyclone for the circulating fluidized bed (CFB) boiler. With the development of large CFB boilers, the huge body of the conventional cyclone became a major shortcoming because of the thick refractory wall that needs a long period to start the boiler. An alternative way to overcome these problems is the use of square cyclone. A square cyclone has more advantages over the conventional cyclone including convenient construction, easier membrane wall arrangement, and shorter start–stop time and at the same time easy integration with the boiler (Wang et al., 1999; Raoufi et al., 2009). Several attempts have been made in the last decade to improve the square cyclone performance. Wang et al. (1999)

studied the separation mechanism of a square cyclone at high inlet particle concentration. They proposed an instantaneous separation model based on experimental observation and measurement. Junfu et al. (1999) investigated the square cyclone at 75 t/h CFB boiler and presented a model to study the cyclones. Ravi et al. (2000) investigated a multi-objective optimization process on cyclone separators using NSGA algorithm. They tried to minimize pressure drop and maximize the collection efficiency in cyclone separators. They did not use CFD in their optimization procedure and used analytical function for pressure drop and collection efficiency. Effects of different parameters including length, diameter of vortex finder, and inlet velocity on performance of square cyclones were studied by Qiang et al. (2003). Su and Mao (2006) used a three-dimensional particle dynamic analyzer (3D-PDA) to understand the nature and characteristics of the suspension flow in the square cyclone separator and found out the fac-

∗ Corresponding author at: School of Mechanical Engineering, College of Engineering, University of Tehran, P.O. Box: 11155-4563, Tehran, Iran. Tel.: +98 918 957 3706; fax: +98 218 801 3029. E-mail addresses: h.safi[email protected], safikhani [email protected] (H. Safikhani). Received 7 April 2010; Received in revised form 3 June 2010; Accepted 6 July 2010

0263-8762/$ – see front matter Crown Copyright © 2010 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers. All rights reserved.

doi:10.1016/j.cherd.2010.07.004

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Nomenclature u x P Ne Vin VP

Y X R D50

velocity position pressure number of effective turns inlet velocity velocity pressure set of decision variables set of objective functions inlet width inlet height Reynolds stress tensor cut-point

Greek symbols  density  viscosity Subscripts p particle i,j,k 1,2,3 t turbulent Superscripts mean variables

tors affecting particle motion. The turbulent flow field inside a square cyclone was experimentally investigated by Su (2006) to study the mechanism of particle separation and provide guidance for the optimization of its structure. Junfu et al. (2007) evaluated the performance of advanced water-cooled square cyclone with curved inlet. The results were compared with other cyclones through fly ash analysis and showed that the overall performance of the square cyclone in such capacity is compatible with the conventional cyclones. Raoufi et al. (2009) obtained detailed flow information by CFD simulation within those cyclones tested by Wang et al. (1999) and Su and Mao (2006). Optimization of square cyclones is, indeed, a multiobjective optimization problem rather than a single objective optimization problem that has been considered so far in the literature. Both the pressure drop and the collection efficiency in square cyclones are important objective functions to be optimized simultaneously in such a real world complex multi-objective optimization problem. These objective functions are either obtained from experiments or computed using very timely and high-cost computer fluid dynamic (CFD) approaches, which cannot be used in an iterative optimization task unless a simple but effective meta-model is constructed over the response surface from the numerical or experimental data. Therefore, modeling and optimization of the parameters is investigated in the present study, by using GMDH-type neural networks and multi-objective genetic algorithms in order to maximize the collection efficiency and minimize the pressure drop. System identification and modeling of complex processes using input–output data have always attracted many research efforts. In fact, system identification techniques are applied in many fields in order to model and predict the behaviors of unknown and/or very complex systems based on given input–output data (Astrom and Eykhoff, 1971). In this way,

soft-computing methods (Sanchez et al., 1997), which concern computation in an imprecise environment, have gained significant attention. The main components of soft computing, namely, fuzzy logic, neural network, and evolutionary algorithms have shown great ability in solving complex non-linear system identification and control problems. Many research efforts have been expended to use of evolutionary methods as effective tools for system identification (Kristinson and Dumont, 1992). Among these methodologies, Group Method of Data Handling (GMDH) algorithm is a self-organizing approach by which gradually complicated models are generated based on the evaluation of their performances on a set of multiinput–single–output data pairs (Xi ,Yi ) (i = 1, 2, . . ., M). The GMDH was first developed by Ivakhnenko (1971) as a multivariate analysis method for complex systems modeling and identification, which can be used to model complex systems without having specific knowledge of the systems. The main idea of GMDH is to build an analytical function in a feed forward network based on a quadratic node transfer function (Farlow, 1984) whose coefficients are obtained using regression technique. In recent years, however, the use of such self-organizing networks leads to successful application of the GMDH-type algorithm in a broad range of areas in engineering, science, and economics (Farlow, 1984). Moreover, there have been many efforts in recent years to deploy genetic algorithms (GAs) to design artificial neural networks since such evolutionary algorithms are particularly useful for dealing with complex problems having large search spaces with many local optima (Porto, 1997). In this way, GAs has been used in a feed forward GMDH-type neural network for each neuron searching its optimal set of connection with the preceding layer (Nariman-Zadeh et al., 2003). In the former reference, authors have proposed a hybrid genetic algorithm for a simplified GMDH-type neural network in which the connection of neurons are restricted to adjacent layers. Moreover a multi-objective genetic algorithm has also been recently used by some of authors to design GMDH-type neural networks considering some conflicting objectives (Jamali et al., 2009). In this paper pressure drop (p) and the collection efficiency () in a set of square cyclones are numerically investigated using CFD techniques and compared with those of experimental results of Wang et al. (1999) and the validated CFD simulation of Raoufi et al. (2009). Next, genetically optimized GMDH-type neural networks are used to obtained polynomial models for the effects of geometrical parameters of the square cyclones on both p and . Such an approach of meta-modeling of those CFD results allows for iterative optimization techniques to design optimally the square cyclones computationally affordably. The obtained simple polynomial models are then used in a Pareto based multi-objective optimization approach to find the best possible combinations of p and , known as the Pareto front.

2. CFD simulation and validation of the results 2.1.

CFD simulation

For an incompressible fluid flow, the equation of continuity and balance of momentum are given as: ∂u¯ i =0 ∂xi

(1)

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∂u¯ i v 1∂P¯ ∂2 u¯ i ∂ = +v − R + u¯ j ∂t ∂xj ∂xi ∂xj ∂xj ∂xj ij

(2)

where Rij = ul uj is the Reynolds stress tensor. Here, ui = ui − u¯ i is the ith fluctuating velocity component. The RSTM provides differential transport equations for evaluation of the turbulence stress components where the turbulence production terms are defined as: ∂ ∂ ∂ R + u¯ k R ∂t ij ∂xk ij ∂xk − C1







vt Rij  k ∂xk

 



− Rik



∂u¯ j ∂xk

+ Rjk



∂u¯ i ∂xk



ε 2 2 2 Rij − ıij K − C2 Pij − ıij P − ıij ε K 3 3 3

∂u¯ j

∂u¯ + Rjk i Pij = − Rik ∂xk ∂xk

 , P=

1 P 2 ij

(3)

(4)

with P being the fluctuating kinetic energy production.  k = 1, C1 = 1.8, C2 = 0.6 are empirical constants (Launder et al., 1975). The transport equation for the turbulence dissipation rate, ε, is given as ∂ε ∂ ∂ε = + u¯ j ∂t ∂xj ∂xj



v+

vt ε

∂ε  ∂xj

− Cε1

ε ∂u¯ ε2 Rij i − Cε2 K ∂xj K

(5)

In Eq. (5), K = 12 ul ul is the fluctuating kinetic energy, and ε is the turbulence dissipation rate. The values of constants are  ε = 1.3, Cε1 = 1.44 and Cε2 = 1.92. The dispersion of small particles is strongly affected by the instantaneous fluctuation of fluid velocity. The turbulence fluctuations are random functions of space and time. In this study, a discrete random walk (DRW) model is used for evaluating the instantaneous velocity fluctuations. The values of u , v and w that prevail during the lifetime of the turbulent eddy, Te are sampled by assuming that they obey a Gaussian probability distribution. In this model the instantaneous velocity in the ith direction is given as ui =



ui u i

(6)

In Eq. (6), is a zero-mean, unit-variance, normally dis

tributed random number,

ui u i is the local root mean-square

(RMS) fluctuation velocity in the ith direction, and the summation convention on i is suspended. The characteristic lifetime of the eddy is defined as a constant given by Te = 2Tl

(7)

where, Tl is the eddy turn over time given as, Te = 0.3 k/ε in the RSTM. The other option allows for a long –normal random variation of eddy lifetime that is given by Te = −Tl log r

(8)

where, r is a uniform random number between 0 and 1. The particle is assumed to interact with the fluid fluctuation field, which stays fixed over the eddy lifetime. When the eddy lifetime is reached, a new value of the instantaneous velocity is obtained by introducing a new value of in Eq. (6).

Table 1 – Comparison of collection efficiency and pressure drop for 3 different grid numbers. Total no. of cells

 (%)

182,282 212,398 280,716 Max. diff.

90.07 91.42 88.12 3.601

p/0.5 2 1.98 2.02 1.93 4.452

There are two main approaches for modeling multiphase flows that account for the interactions between the phases. These are the Eulerian–Eulerian and the Eulerian–Lagrangian approaches. The former is based on the concept of interpenetrating continua, for which all the phases are treated as continuous media with properties analogous to those of a fluid. The Eulerian–Lagrangian approach adopts a continuum description for the liquid phase and tracks the discrete phase using Lagrangian particle trajectory analysis. In the present study, one way coupling method is used to solve the two phase flow and the Eulerian–Lagrangian approach is implemented for simulation of second discrete phase (particles). In this model, air is the continuous phase and the particles are treated as the dispersed discrete phase. The volume-averaged and steady state Navier–stokes equation is solved for the gas phase. The particle motions are simulated by the Lagrangian trajectory analysis procedure. Forces acting on the dispersed phase include drag and gravity. The discrete phase equations are solved using Runge–Kutta method for particles. The particle loading in a cyclone separator is typically small, and therefore, it can be safely assumed that the presence of the particles does not affect the flow field (oneway coupling). In this paper collisions between particles and the walls of the cyclone were assumed to be perfectly elastic (Gimbun et al., 2005; Safikhani et al., 2010a). Also, particle–particle collision is negligible. To test for grid independency, three grid types with increasing grid density are studied. The computational results of 3 grid types are compared in Table 1. As seen the maximum difference between the results is less than 5% so the grid template 182,282 is used for all computations in present study. Fig. 1 shows the details of the computational grid for the cyclones. The simulations are performed using commercial software FLUENT. Inlet mass flow boundary condition is used at the cyclone inlet and a fully developed boundary condition is used at the outlet (Safikhani et al., 2010a,b). The computation is continued until the solution converged with a total residual of less than 0.00001.

2.2.

Definition of the objective functions

Both the collection efficiency and pressure drop in cyclones are important objective functions to be optimized simultaneously. The collection efficiency statistics are obtained by releasing a specified number of particles at the inlet of the cyclones and by monitoring the number escaping through the underflow. The range of particles size is 0.205 mm of a material whose density is equal to 2250 kg/m3 , moreover some other operating conditions are shown in Table 2. The first theory for collection efficiency in conventional cyclones was developed by Shepherd and Lapple (1939). It is based on the assumption of a plug flow. In order to calculate the efficiency, first the particle size with 50% collection efficiency (DP50% ) needs to be determined, according to the

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Table 3 – The components of Wang’s pressure drop theory. Component Pe = C2 VPin Pk = VPin − VPout Pf = C VPin Pr = V2 in (R/r0−1 ) P0 = C3 VPout

Definition Entry loss, C2 ≈ 1 Kinetic energy loss Frictional loss, C1D3D = 0.14, C2D2D = 0.28, C1D2D = 0.15 Rotational loss, r0 = radius of the vortex interface, R = cyclone body radius Pressure loss in the inner vortex and exit tube, C3 ≈ 1.8

Fig. 1 – Grid generation for CFD analysis of square cyclones. following equation:

 DP 50% =

Fig. 2 – Definition of design variables.

9 Y 2Ne Vin p

(9)

The collection efficiency for any other particle size (DPj ) can then be determined by =

1 1 + (DP50% /DPj )

Many empirical models have been proposed for the pressure drop in the conventional cyclones (Shepherd and Lapple, 1939; Casal and Martinez, 1983; Coker, 1993; Wang, 2004). In Wang’s model the total pressure loss in the cyclone is obtained by summing up the five pressure drop components as follows:

(10)

2

Ptotal = Pe + Pk + Pf + Pr + Po

Table 2 – Dimensions and some operating conditions. Parameter

Value

Inlet velocity (m/s) Temperature (k) Particle feed rate (g/min) Inlet width (Y/a) Inlet height (X/a) Gas outlet length (S/a)

26 300 3 0.25 0.5 0.53

(11)

where, the components of Eq. (11) are explained in Table 3.

2.3.

Definition of the design variables

The design variables in present paper are: the dimensionless vortex finder diameter (De /a), the dimensionless upside and downside body of square cyclone (Lup /a) and (LDown /a). The design variables and their range of variations are shown in Fig. 2 and Table 4. By changing the geometrical independent

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Table 4 – Design variables and their range of variations. Design variable

From

De /a Lup /a LDown /a

To

0.2 0.9 0.9

0.6 1.6 1.6

parameters according to the Table 4, various designs will be generated and evaluated by CFD. Consequently, some metamodels can be optimally constructed using the GMDH-type neural networks, which will be further used for multi-objective Pareto based design of such cyclones. In this way, 80 various CFD analyses have been performed due to those different design geometrics.

2.4.

Validation of the CFD results

Samples of numerical results, using CFD are shown in Table 5. Moreover a typical velocity vector and static pressure contour in one of CFD simulations are shown in Fig. 3. To attain the confidence about the simulation, it is necessary to compare the simulation result with the available data. The comparison of CFD prediction for dynamic and static pressure drop is shown in Fig. 4. As seen, our numerical data predicts the dynamic and static pressure drop with the acceptable deviation from Raoufi et al. (2009) numerical data. The comparison of experimental and CFD prediction for collection efficiency is also shown in Fig. 5. It is obvious that numerical simulation can properly adapt with pattern of experimental efficiency curves. The results obtained in such CFD analysis can now be used to build the response surface of both the collection efficiency and the pressure drop for those different 80 geometries using GMDH-type polynomial neural networks. Such meta-models will, in turn, be used for the Pareto based multi-objective optimization of the square cyclones. A post analysis using CFD is also performed to verify the optimum results using the meta-modeling approach. Finally, the solutions obtained by the approach of this paper exhibit some important trade-offs among those objective functions which can be simply used by a designer to optimally compromise among the obtained solutions.

Fig. 3 – Velocity vector and static pressure contour, Vin = 26 m/s, De /a = 0.5, Lup /a = 1.093, LDown /a = 1.093.

3. Modeling of p and  using GMDH-type neural networks By means of GMDH algorithm a model can be represented as set of neurons in which different pairs of them in each layer are connected through a quadratic polynomial and thus produce new neurons in the next layer. Such representation can be used in modeling to map inputs to outputs. The formal definition of the identification problem is to find a function fˆ so that can be approximately used instead of actual one, f in order to predict output yˆ for a given input vector X = (x1 , x2 , x3 , . . . ,xn ) as close as possible to its actual output y. Therefore, given M observation of multi-input–single–output data pairs so that

Table 5 – Samples of numerical result using CFD. Input data Num. 1 2 3 4 5 6 7 8 9 10 ... 79 80

Output data p/0.5 2

 (%)

De /a

Lup /a

LDown /a

0.6 0.4 0.6 0.2 0.5 0.5 0.3 0.5 0.6 0.2

0.9 1.6 1.3 1.3 1.1 1.6 0.9 1.3 1.6 1.3

0.9 0.9 1.3 1.6 1.3 1.6 1.3 1.1 1.3 1.1

2.362 2.751 0.868 1.270 1.431 1.175 2.521 1.810 2.331 1.184

92.058 99.363 45. 740 52.417 76.511 76.389 96.222 82.440 85.555 56.409

0.5 0.6

1.1 0.9

1.1 1.3

0.8901 2.2170

50.023 49.125

yi = f (xi1 , xi2 , xi3 , ..., xin )

(i = 1, 2, . . . , M),

(12)

It is now possible to train a GMDH-type neural network to predict the output values yˆ i for any given input vector X = (xi1 , xi2 , xi3 , ..., xin ), that is yˆ i = fˆ (xi1 , xi2 , xi3 , ..., xin )

(i = 1, 2, . . . , M),

(13)

The problem is now to determine a GMDH-type neural network so that the square of difference between the actual output and the predicted one is minimized, that is M

i=1

fˆ (xi1 , xi2 , xi3 , ..., xin ) − yi

2

→ min

(14)

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Fig. 4 – Comparison of numerical result for dynamic and static pressure drop. General connection between inputs and output variables can be expressed by a complicated discrete form of the Volterra functional series in the form of

y = a0 +

n i=1

ai xi +

n n i=1 j=1

aij xi xj +

n n n

ajki xi xj xk + · · ·,

i=1 j=1 k=1

(15) where is known as the Kolmogorov–Gabor polynomial (Farlow, 1984). This full form of mathematical description can be represented by a system of partial quadratic polynomials consisting of only two variables (neurons) in the form of yˆ = G(xi , xj ) = a0 + a1 xi + a2 xj + a3 xi xj + a4 xi2 + a5 xj2 .

(16)

There are two main concepts involved within GMDH-type neural networks design, namely, the parametric and the structural identification problems. In this way, some works by Nariman-Zadeh et al. (2003), present a hybrid GA and singular value decomposition (SVD) method to optimally design such polynomial neural networks. The methodology in these references has been successfully used in this paper to obtain the polynomial models of the p and . The obtained GMDH-type polynomial models have shown very good prediction ability of

unforeseen data pairs during the training process which will be presented in the following sections. The input–output data pairs used in such modeling involve two different data tables obtained from CFD simulation discussed in Section 2. Both of the tables consists of three variables as inputs, namely (De /a), (Lup /a) and (LDown /a) (Fig. 2) and outputs, which are p and . The tables consist of a total of 80 patterns, which have been obtained from the numerical solutions to train and test such GMDH-type neural networks. However, in order to demonstrate the prediction ability of the evolved GMDH-type neural networks, the data in both input–output data tables have been divided into two different sets, namely, training and testing sets. The training set, which consists of 60 out of the 80 input–output data pairs for p and , is used for training the neural network models using the method presented in Section 2. The testing set, which consists of 20 unforeseen input–output data samples for p and  during the training process, is merely used for testing to show the prediction ability of such evolved GMDH-type neural network models. The GMDH-type neural networks are now used for such input–output data to find the polynomial models of p and  with respect to their effective input parameters. In order to design genetically such GMDH-type neural networks described in the previous section, a population of 10 individuals with a crossover probability (Pc ) of 0.7 and mutation probability (Pm ) 0.07 has been used in 500 generations for p and . The corresponding polynomial representation for dimensionless pressure drop is as follows:

Y1 = 6940 − 3708 + 874

L

L

Down

up

a

2

a

Y2 = 7276 − 12, 096 + 874

L

Down

Fig. 5 – Comparison of numerical and experimental (Wang et al., 1999) results for collection efficiency.

D



a

Y3 = −550 + 584 2

+ 1466

e

a

e

a

− 005(Y23 ) − 3.88

Down



a

 L 2 up

+ 418

a

L L up Down

(17.a)

a

a

− 4597

+ 3210

D

L

− 4766

L

Down

a



 D 2

+ 6663

D L e Down a

(17.b)

a

 D 2

+ 2.94(Y23 ) + 82, 212

D e

a

(Y23 )

e

a

e

a

− 8.6180e (17.c)

chemical engineering research and design 8 9 ( 2 0 1 1 ) 301–309

Y4 = 151.66 + 1.9123 Y2 + 71.70

 L 2 up

− 26.021

a

L up

+ 0.00016 Y22

a

− 1.2012 Y2

307

L up

(17.d)

a

p = −0.03992 + 1.55e − 3 Y3 + 9.87e − 4 Y4 + 7.4e − 7Y3 2 0.5 v2 + 2.07e − 6Y4 2 − 2.96e − 6Y3 Y4

(17.e)

Similarly, the corresponding polynomial representation of the model for collection efficiency is in the form of Y1 = −2.44 + 0.26

D e

 L 2 up

− 3.661

L

L up

a

Down

2

a

L

Down

+ 0.22

2

a

a

L

 D 2 e

− 0.860

a

D L up e a

(18.a)

a

L

Down



a

− 3.66

a

L

Down



a

L

L up

a

up

a (18.b)

a

+ 0.131 Y1

Down

2

(18.c)

a

 L 2

+ 2.582Y2 + 2.131 2

up

up

a (18.d)

a

Fig. 6 – Variation of dimensionless pressure drop with input data.

4. Multi-objective optimization of square cyclone using polynomial neural network models

 L 2

L L up Down

− 0.241 Y 

Y4 = 8.302 − 8.292Y2 −8.431 + 4.721Y2

up

+ 0.29

Y3 = −0.330 + 1.071 Y1 + 0.731 − 0.331

L

+ 0.262123

a

Y2 = −2.43 + 6.86 − 0.33

a

+ 7.01

In order to investigate the optimal performance of the square cyclones in different geometrical parameters the polynomial neural network models obtained in section 3 are now deployed in a multi-objective optimization procedure. The two conflicting objectives in this study are p and  that are to be simultaneously optimized with respect to the design variables (De /a), (Lup /a) and (LDown /a). The 2-objective optimization problem can be formulated in the following form:

⎧ Maximize collection efficiency () = f1 (De /a, Lup /a, LDown /a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Minimize pressure drop (p) = f2 (De /a, Lup /a, LDown /a) Subject to : 0.2 < x1 = De /a < 0.6(19)

⎪ ⎪ ⎪ 0.9 < x2 = Lup /a < 1.6 ⎪ ⎪ ⎩

0.9 < x3 = LDown /a < 1.6 (19)

 = 4.823 + 75.911 Y3 + 11.310 Y4 − 601.31 Y3 − 582.119 Y4 2

+ 1194.2109 Y3 Y4

2

(18.e)

The good behavior of such GMDH-type neural network model for dimensionless pressure drop is also depicted in Fig. 6, both for the training and testing data. Such behavior has also been shown for the training and testing data of collection efficiency in Fig. 7. It is evident that the evolved GMDH-type neural network in terms of simple polynomial equations successfully model and predict the outputs of the testing data that have not been used during the training process. It should be noted that these polynomials valid just for design variables in the range of present case study (Table 4). The models obtained in this section can now be utilized in a Pareto multi-objective optimization of the square cyclone considering both p and  as conflicting objectives. Such study may unveil some interesting and important optimal design principles that would not have been obtained without the use of a multi-objective optimization approach.

Fig. 7 – Variation of collection efficiency with input data.

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Fig. 8 – Multi-objective Pareto result for square cyclones. Table 6 – The values of objective functions and their associated design variables of the optimum points. Point

De /a

Lup /a

LDown /a

p/0.5

A B C D E

0.412 0.489 0.544 0.533 0.520

0.970 0.990 1.009 1.166 1.294

0.941 1.058 1.176 1.490 1.529

3.048 2.014 1.503 1.080 0.856

2

 (%) 100.0 96.04 91.50 67.45 43.57

The evolutionary process of Pareto multi-objective optimization is accomplished by using the NSGA-II approach as Ravi et al. (2000), Coello Coello et al. (2002), Rangaiah (2009) and Jamali et al. (2009) suggested. Population size of 1024 and a generation number of 300 have been chosen in different runs with crossover probability Pc and mutation probabilities Pm are 0.9 and 0.01, respectively. Fig. 8 depicts the obtained non-dominated optimum design points as a Pareto front of those two objective functions. There are five optimum design points, namely, A, B, C, D and E whose corresponding design variables and objective functions are shown in Table 6. These points clearly demonstrate trade-offs in objective functions  and p from which an appropriate design can be compromisingly chosen. It is clear from Fig. 8 that all the optimum design points in the Pareto front are non-dominated and could be chosen by a designer as optimum square cyclone. Evidently, choosing a better value for any objective function in the Pareto front would cause a worse value for another objective. The corresponding decision variables of the Pareto front shown in Fig. 8 are the best possible design points so that if any other set of decision variables is chosen, the corresponding values of the pair of objectives will locate a point inferior to this Pareto front. Such inferior area in the space of the two objectives is in fact bottom/right side of Fig. 8. In Fig. 8, the design points A and E stand for the best collection efficiency and the best pressure drop. Moreover, the other optimum design points, B and D can be simply recognized from Fig. 8. The design point, B exhibits important optimal design concepts. In fact, optimum design point B obtained in this paper exhibits a decrease in  (about 7.1%) in comparison with that of point A whilst its p improves about 46.6% in comparison with that of A, similarly optimum design point D

Fig. 9 – Optimal variations of pressure drop with respect to design variables. exhibits an increase in p (about 6.45%) in comparison with that of point E whilst its  improves about 44.6%. It is now desired to find a trade-off optimum design points compromising both objective functions. This can be achieved by the method employed in this paper, namely, the mapping method. In this method, the values of objective functions of all non-dominated points are mapped into interval 0 and 1.Using the sum of these values for each non-dominated point, the trade-off point simply is one having the minimum sum of those values. Consequently, optimum design point C is the trade-off points which have been obtained from the mapping method. In a post numerical study, the design points of the obtained Pareto front have been re-evaluated by CFD. The results of such CFD analysis re-evaluations have been compared with those of numerical results using the GMDH model in Fig. 8 as A , B , C , D and E . As seen the GMDH data agree well with the CFD re-evaluation results. There are some interesting design facts which can be used in the design of square cyclones. The optimal variations of

Fig. 10 – Optimal variations of collection efficiency with respect to design variables.

chemical engineering research and design 8 9 ( 2 0 1 1 ) 301–309

collection efficiency and pressure drop with respect to design variables are shown in Figs. 9 and 10, respectively. It is clear from these figures, from point A to B, (Lup /a) is almost constant whereas (De /a) and (LDown /a) vary linearly. Similarly from B to C, all of the design variables vary linearly. From C to D, (De /a) is constant and the other design variables vary with a quadratic relation. Finally from D to E, (De /a) and (LDown /a) are constant whereas (Lup /a) varies linearly. These useful relationships that indefeasible between the optimum design variables of square cyclones cannot be discovered without the use of multi-objective Pareto optimization process presented in this paper.

5.

Conclusion

Genetic algorithms have been successfully used both for optimal design of generalized GMDH-type neural networks models of p and  in square cyclones and for multi-objective Pareto based optimization of such processes. Two different polynomial relations for p and  have been found by evolved GS-GMDH-type neural networks using some numerically validated CFD simulations for input–output data of the square cyclones. The derived polynomial models have been then used in an evolutionary multi-objective Pareto based optimization process. Some important facts in the optimum design of square cyclones have been obtained and proposed based on the Pareto front of two conflicting objective functions.

Acknowledgement One of the authors (Hamed Safikhani) would like to thanks Mr. M. Javanshir, the engineer of KIASA CO. for valuable discussions.

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