MODELING AND MULTI-OBJECTIVE OPTIMIZATION OF FORWARD-CURVED BLADE CENTRIFUGAL FANS USING CFD AND NEURAL NETWORKS

8/17/2019 MODELING AND MULTI-OBJECTIVE OPTIMIZATION OF FORWARD-CURVED BLADE CENTRIFUGAL FANS USING CFD AN…

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MODELING AND MULTI-OBJECTIVE OPTIMIZATION OF FORWARD-CURVED

BLADE CENTRIFUGAL FANS USING CFD AND NEURAL NETWORKS

Abolfazl Khalkhali, Mehdi Farajpoor, Hamed SafikhaniDepartment of Mechanical Engineering, East Tehran Branch, Islamic Azad University, Tehran, Iran

E-mail: [email protected]

Received May 2010, Accepted January 2011

No. 10-CSME-27, E.I.C. Accession 3190

ABSTRACTIn the present study, multi-objective optimization of Forward-Curved (FC) blade centrifugal

fans is performed in three steps. In the first step, Head rise (H R) and the Head loss (H L) in a setof FC centrifugal fan is numerically investigated using commercial software NUMECA. Two

meta-models based on the evolved group method of data handling (GMDH) type neural

networks are obtained, in the second step, for modeling of H R and H L with respect togeometrical design variables. Finally, using the obtained polynomial neural networks, multi-

objective genetic algorithms are used for Pareto based optimization of FC centrifugal fans

considering two conflicting objectives, H R and H L.

Keywords: forward-curved blade centrifugal fan; multi-objective optimization; CFD; GMDH;

genetic algorithms.

MODÉLISATION ET OPTIMISATION MULTI-OBJECTIF D’UN VENTILATEUR

CENTRIFUGE À AUBES INCLINEÉS VERS L’AVANT, UTILISANT LA

MÉCANIQUE DES FLUIDES NUMÉRIQUES (MFN) ET DES RÉSEAUX DENEURONES

L’objectif de cette étude, est l’exécution en trois étapes de l’optimisation multi-objectif d’un

ventilateur centrifuge à aubes inclinées vers l’avant. Dans un premier temps, l’augmentation de

charge et la perte de charge dans un ensemble de ventilateurs centrifuges à aubes inclinées, sont

examinées numériquement utilisant le logiciel commercial NUMECA. Dans un deuxième

temps, deux méta-modèles basés sur la méthode de traitement de données par groupe (MTDG)

de type de réseaux de neurones, sont obtenus pour la modélisation de l’augmentation de charge

et de la perte de charge, par rapport aux variables géométriques. Finalement, en utilisant les

réseaux de neurones polynômes obtenus, des algorithmes génétiques multi-objectifs sont utilisés

pour l’optimisation de Pareto d’un ventilateur centrifuge en prenant en considération ces deux

objectifs conflictuels l’augmentation de charge et la perte de charge.

Mots-clés : ventilateur centrifuge à aubes inclinées vers l’avant; optimisation multi-objectif;

MFN; MTDG; algorithmes génétiques.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, No. 1, 2011 63


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1. INTRODUCTION

Forward-Curved (FC) blade centrifugal fans or squirrel cage fans are the group of turbo

machines which occur in industry in large scales. Nowadays, increasing demands and competition

require the use of good models to describe the operation of FC centrifugal fans. Kim and Seo [1]

presented a response surface method using three dimensional Navier-Stokes analyses to optimize

the shape of a forward-curved blade centrifugal fan and finally improved the efficiency of the fan.Lu et al. [2] numerically investigated the internal flow field of centrifugal fans. They could

decrease the head loss and increase the total pressure using splitter blades in centrifugal fans.

Optimization of FC fans is indeed a multi-objective optimization problem rather than a single

objective optimization problem that has been considered so far in the literature. Sugimura et al. [3]

investigated a multi-objective optimization process on centrifugal fans using multi-objective

robust design exploration method (MORDE). They tried to determine the design variables which

have the minimum turbulent noise level and the maximum efficiency. Besides applications to fan

optimization, in recent years there have been many efforts to increase the performance of different

types of turbo machines. Safikhani and Nourbakhsh [4] investigated a multi-objective

optimization approach on centrifugal pumps. They finally presented the Pareto front for

centrifugal pumps and defined five optimum points which had the best efficiency and cavitation

behavior. Derakhshan et al. [5, 6] optimized a pump as a turbine machine (PAT) for increasing the

efficiency using the genetic algorithms (GAs) and incomplete sensitivities method.

In centrifugal fans there are some objective functions which are not independent of each other,

like efficiency, head rise and input shaft power, so these parameters are not suitable for multi-

objective optimization process. Head rise and the head loss are important and independent

objective functions which can be used in a multi-objective optimization process. These objective

functions are either obtained from experiments or computed using very timely and high-cost

computational 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 head rise and minimize the head loss.

System identification and modeling of complex processes using input-output data have

always attracted many research efforts. System identification techniques are applied in many

fields in order to model and predict the behavior of unknown and/or very complex systems

based on given input-output data [7]. In this way, soft-computing methods [8], 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 developed that make use of evolutionary

methods as effective tools for system identification [9]. 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 performance on aset of multi-input-single-output data pairs X i , yi ð Þ (i 51, 2, …, M ). The GMDH was firstdeveloped by Ivakhnenko [10] 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 [11] whose coefficients are

obtained using regression techniques.



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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 [12]. Moreover, there have been many efforts in recent years to deploy 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 [13]. In this way, Gas

have been used in a feed forward GMDH-type neural network for each neuron searching its

optimal set of connections with the preceding layer [14]. In the former reference, authors haveproposed 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 [15, 16].

In this paper, the head rise and the head loss in a set of forward-curved blade centrifugal fans

are numerically investigated using NUMECA. Genetically optimized GMDH type neural

networks are then used to obtain polynomial models for the effects of geometrical parameters of

the FC fans on both H R and H L. This approach of meta-modeling of those CFD results allows

the use of iterative optimization techniques. The obtained simple polynomial models are then

used in a Pareto based multi-objective optimization approach to find the best possible

combinations of H R and H L, known as the Pareto front. The corresponding variations of designvariables, namely, geometrical parameters, known as the Pareto set, constitute some important

and informative design principles.

2. CFD SIMULATION OF FC BLADE CENTRIFUGAL FANS

The governing equations of incompressible flow are as follows:

Continuity equation

LV i

Lxi ~0 ð1Þ

Reynolds averaged momentum equation

DV i

Dt ~{

1

r

L p

Lxi zn

L2V i

Lx j Lx j {

L

Lx j ui u j ð2Þ

Standard k– e model

Dk

Dt~

L

Lx j C k

k 2

e zn

Lk

Lxi

{ui u j

LV i

Lx j

DeDt~

L

Lx j C k k

2

e zn

Le

Lx j

{C e1 ek u

i u j L

V i Lx j {C e2 e

2

k

ð3Þ

The dimensions of the case study in the present paper and some operating conditions for the

simulations are shown in Table 1. The simulations are performed using Numeca software.

Firstly one blade is modeled in Auto Blade 3.6 and then the Design 3D environment of Numeca

automatically generates the database with different design variables.



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To parameterize the camber line curve, the simple Bezier method is used [17]. A schematic

definition of simple Bezier method is shown in Fig. 1. The design variables in this method are

leading edge angle (b1), trailing edge angle (b 2) and the stagger angle (c). In the present paper

two sections are defined in the blades, one on hub and one on shroud as shown in Fig. 2. It is

supposed that b1 , b 2 and c are equal at hub and shroud section due to the 2D nature of FC

blades centrifugal fan, which can mathematically be given by

b1Hub~ b1Shroud ~Design Variable ð4Þ

b 2Hub~ b 2Shroud ~ Design Variable ð5Þ

cHub~ cShroud ~ Design Variable ð6Þ

The design variables and their range of variations are shown in Table 2. By changing the

geometrical independent parameters according to the Table 2, various designs will be generated

Table 1. Dimensions and operating conditions of FC centrifugal fan case study.

Parameter Value

Outer diameter ( mm) 333

Inner diameter (mm) 210

Width of blades (mm) 150

Mass flow rate (kg/s) 0.34

Rotational velocity (rpm) 634

Inlet k (m2/s2) 5

Inlet e (m2/s3) 30000

Outlet static pressure (atm) 1

Fig. 1. Blade camber line parameterization using simple Bezier method.



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and evaluated using CFD. Consequently, some meta-models can be optimally constructed using

GMDH-type neural networks, which will be further used for multi-objective Pareto based

design of such centrifugal fans. In this way, 132 various CFD analyses have been performed dueto those different design geometrics.

For CFD grid generation, the Auto Grid environment of Numeca is coupled with the Auto

Blade environment. To test for grid independency, three grid types (named A, B , and C ) with

increasing grid density are studied and their details are listed in Table 3. The computational

results of three grid types for different mass flow rates are compared in Table 4. As can be seen,

the maximum difference between the results is less than 6 % so the grid type ( A) is used for all

computations in the present study. Figure 3 shows the details of the computational grid for the

centrifugal fans. The physical model used in the solver is the Reynolds-Averaged Navier–Stokes

equations and the k-e turbulence model. Mass flow, k and e are imposed at the fan inlet. A

static pressure outlet boundary condition is used at the outlet and finally periodic boundary

condition is applied between two blades. The computation is continued until the solutionconverged with a total residual of less than 25. Samples of numerical results, using CFD are

shown in Table 5. A typical pressure contour in one of the simulations is shown in Fig. 4.

The results obtained in such CFD analysis can now be used to build the response surface of

both the head rise and the head loss for those different 132 geometries using GMDH-type

polynomial neural networks. Such meta-models will, in turn, be used for the Pareto-based

multi-objective optimization of the FC fans. A post analysis using the CFD software

NUMECA 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

Fig. 2. Defining two sections on centrifugal fan blade.

Table 2. Design variables and their range of variations.

Design Variable From To

c (deg) 7 25

b1 (deg) 12 47

b 2(deg) 25 60

N (no. of blades) 25 40



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General connection between inputs and output variables can be expressed by a complicateddiscrete form of the Volterra functional series in the form of

y~a0zXni ~1

ai xi zXni ~1

Xn j ~1

aij xi x j zXni ~1

Xn j ~1

Xnk ~1

aijk xi x j xk z . . . ð10Þ

Which is known as the Kolmogorov-Gabor polynomial [18]. 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

Fig. 3. CFD structured grid generation for centrifugal fans.

Table 5. Samples of numerical result using CFD.

Input Data Output Data

Num c(deg) b1(deg) b 2(deg) N H R(m) H L(m)

1 7.45 12.14 42.77 40 14.236 1.559

2 7.45 46.14 59.77 40 18.777 2.056

3 9.54 46.14 59.77 40 15.587 1.707

4 24.45 46.14 59.77 35 18.89 2.008

5 7.45 12.14 42.77 35 13.951 1.481

6 24.45 29.14 25.77 35 12.530 1.331

7 7.45 29.14 25.77 30 10.841 1.1688 24.4 46.14 25.77 30 11.360 1.220

9 7.45 12.14 25.77 30 10.373 1.113

…

130 9.54 12.14 59.77 25 9.488 0.745

131 7.45 46.14 42.77 25 15.427 1.729

132 9.54 12.14 42.77 25 5.768 0.645



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^ y y~G xi ,x j

~a0za1xi za2x j za3xi x j za4xi 2za5x j

2 ð11Þ

There are two main concepts involved within GMDH-type neural networks design, namely,

the parametric and the structural identification problems. In this way, some authors presented 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 H R and H L. 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 the CFD simulation discussed in Section 2. Both of the tables consist of four variables as

inputs, namely, the geometrical parameters of the FC fans c, b1, b 2 (Fig. 1) and N (number of

blades) and outputs, which are H R and H L. The tables consist of a total of 132 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 112 out of the 132 input–

output data pairs for H R and H L, is used for training the neural network models. The testing

set, which consists of 20 unforeseen input–output data samples for H R and H L 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 head rise and head loss 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 H R and H L. The corresponding

polynomial representation for Head Rise (H R) is as follows:

Fig. 4. A typical contour of pressure in one of simulations.



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Y 1~{27:356z0:338 b2z1:677 N -0:00345 b22{0:0234 N 2z0:00291 N b2 ð12aÞ

Y 2~{20:196886{0:00586b1z1:7555 N z0:00054b12{0:0234N 2z0:00198b1N ð12bÞ

Y 3~{19:7199{0:10393cz1:73866N z0:00507c2{0:02340N 2z0:0039480c N ð12cÞ

Y 4~{1:552z0:18001 b1z0:4275 b2{0:001652b12{0:003306 b2

2{0:00013b1 b2 ð12dÞ

Y 5~8:78175z0:502008Y 2{1:1355Y1{0:04922Y 22zz0:000709Y1

2z 0:1217Y 2Y1 ð12eÞ

Y 6~{6:2660z1:05304Y 4{0:044003Y3{0:016115Y 42z0:026433Y3

2z0:02430Y 4Y3 ð12f Þ

H R~1:898441{0:45095Y 5z1:18816Y6{0:14790 Y 52{0:17717Y6

2z0:3359Y 5Y6 ð12gÞ

Similarly, the corresponding polynomial representation of the model for Head Loss (H L) is inthe form of

Y ’1~1:2905{0:0799c z0:01905b1z0:00382c2z3:383e{005b1

2{0:000769c b1 ð13aÞ

Y ’2~{2:8676z0:02454b2 z0:1849 N {0:00025b22{0:002538N 2z0:000339N b2 ð13bÞ

H L~{2:93563z4:11599Y ’1z0:4002Y’2{1:4661Y ’12{0:10065Y’2

2z0:602920Y ’1Y’2 ð13cÞ

The very good behavior of such GMDH type neural network model for head rise is also

depicted in Fig. 5, both for the training and testing data. Such behavior has also been shown for

the training and testing data of head loss in Fig. 6. It is evident that the evolved GMDH typeneural 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. The models

obtained in this section can now be utilized in a Pareto multi-objective optimization of the FC

centrifugal fans considering both HR and HL as conflicting objectives. Such a 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.



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4. MULTI-OBJECTIVE OPTIMIZATION OF FC CENTRIFUGAL FANS USING

POLYNOMIAL NEURAL NETWORK MODELS

Multi-objective optimization, which is also called multi criteria optimization or vector

optimization, has been defined as finding a vector of decision variables satisfying constraints to

give acceptable values to all objective functions. In these problems, there are several objective or

cost functions (a vector of objectives) to be optimized (minimized or maximized)

simultaneously. These objectives often conflict with each other so that improving one of them

will deteriorate another. Therefore, there is no single optimal solution as the best with respect to

all the objective functions. Instead, there is a set of optimal solutions, known as Pareto optimal

solutions or Pareto front [19] for multi-objective optimization problems. The concept of Pareto

front or set of optimal solutions in the space of objective functions in multi-objective

optimization problems (MOPs) stands for a set of solutions that are non-dominated to each

other but are superior to the rest of solutions in the search space. This means that it is not

possible to find a single solution to be superior to all other solutions with respect to all

objectives so that changing the vector of design variables in such a Pareto front consisting of these non-dominated solutions could not lead to the improvement of all objectives

simultaneously. Consequently, such a change will lead to deteriorating of at least one objective.

Thus, each solution of the Pareto set includes at least one objective inferior to that of another

solution in that Pareto set, although both are superior to others in the rest of search space. Such

problems can be mathematically defined as:

Fig. 5. CFD vs. Network for H R..



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Find the vector X ~ x1,x2, . . . ,x

n

T to optimize

F X ð Þ~ f 1 X ð Þ, f 2 X ð Þ, . . . , f k X ð Þ½ T

, ð14Þ

Subject to m inequality constraints

g i X ð Þƒ0, i~1 to m, ð15Þ

And p equality constraints

h j X ð Þ~0, j~1 to p, ð16Þ

Where X [


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Maximize Head Rise H Rð Þ~ f 1 g ,b1,b2,N ð Þ

Minimize Head Loss H Lð Þ~ f 2 g ,b1,b2,N ð Þ

Subject to 120 ƒb1ƒ470

250ƒb2ƒ600

25ƒN ƒ40

8>>>>>><

>>>>>>:ð17Þ

The evolutionary process of Pareto multi-objective optimization is accomplished by using the

recently developed algorithm, namely, the -elimination diversity algorithm by some of authors

[14] where a population size of 60 has been chosen in all runs with crossover probability P c and

mutation probability P m as 0.7 and 0.07 respectively.

Figure 7 depicts the obtained non-dominated optimum design points as a Pareto front of

those two objective functions. There are four optimum design points, namely, A, B, C and D

whose corresponding designs variables and objective functions are shown in Table 6. Moreover,

for more clarity, the design variables of optimum design points have been superimposed with

each other in Fig. 8. These points clearly demonstrate tradeoffs in objective functions head rise

and head loss from which an appropriate design can be compromisingly chosen. It is clear fromFig. 7 that all the optimum design points in the Pareto front are non-dominated and could be

chosen by a designer as optimum FC fan. Evidently, choosing a better value for any objective

Fig. 7. Pareto front of head rise and head loss.



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function in the Pareto front would cause a worse value for another objective. The

corresponding decision variables of the Pareto front shown in Fig. 7 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. 7.

In Fig. 7, the design points A and D stand for the best head loss and the best head rise.

Moreover, the other optimum design point, B can be simply recognized from Fig. 7. The design

point, B exhibit important optimal design concepts. In fact, optimum design point B obtained in

this paper exhibits an increase in head loss (about 23.58 %) in comparison with that of point A

whilst its head rise improves about 30.6 % in comparison with that of A.

It is now desired to find a trade-off optimum design point 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 point which has been obtained from the mapping method.

There are some interesting design facts which can be used in the design of such FC fans. It is

clear from Figs. 9 and 10 that from point A to B , design variables c, b1 and N are nearly

constant whereas b2 varies almost linearly. Similarly, from point B to point C , the geometrical

Fig. 8. Overlay graph of the design variables in optimum points.

Table 6. The values of objective functions and their associated design variables of the optimum

points.

Point c (deg) b1(deg) b 2(deg) N HR (m) HL (m)

A 13.48 12.36 25.24 25 6.5036 .52068B 12.54 12.14 57.82 25 10.695 .8093

C 13.33 12.01 57.57 33 14.376 1.156D 17.38 42.19 58.57 38 18.981 1.8272



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some CFD simulations for input–output data of the fans. The derived polynomial models have

been then used in an evolutionary multi-objective Pareto based optimization process so that

some interesting and informative optimum design aspects have been revealed for fans with

respect to the design variables such as geometrical parameters of c, b1, b2 (Fig. 1) and number of

blades (N ). Consequently, some very important tradeoffs in the optimum design of FC

centrifugal fans have been obtained and proposed based on the Pareto front of two conflicting

objective functions. Such combined application of GMDH type neural network modeling of

input–output data and subsequent non-dominated Pareto optimization process of the obtainedmodels is a very promising technique for discovering useful and interesting design relationships.

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Table 7. Re-evaluation of the obtained optimal Pareto front using CFD.

Point

HR (m) HL(m)

GMDH CFD Error (%) GMDH CFD Error (%)

A 6.5036 6.351 2.31 .52068 .501 3.79

B 10.695 10.09 5.99 .8093 .778 3.98C 14.376 14.11 2.87 1.156 1.10 5.09

D 18.981 18.25 4.01 1.8272 1.765 3.23



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MODELING AND MULTI-OBJECTIVE OPTIMIZATION OF FORWARD-CURVED BLADE CENTRIFUGAL FANS USING CFD AND NEURAL NETWORKS