Research Article Multiobjective Optimization for the ...

Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2014, Article ID 614581, 13 pages http://dx.doi.org/10.1155/2014/614581

Research Article Multiobjective Optimization for the Impeller of Centrifugal Fan Based on Response Surface Methodology with Grey Relational Analysis Method Fannian Meng, Quanlin Dong, Pengfei Wang, and Yan Wang School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China Correspondence should be addressed to Quanlin Dong; [email protected] Received 8 January 2014; Accepted 14 April 2014; Published 23 June 2014 Academic Editor: Michal Kuciej Copyright © 2014 Fannian Meng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The theories of weighted grey relational analysis combined with RSM are used to optimize the centrifugal fan impeller parameters. In order to evaluate centrifugal fan efficiency (Eff) and total pressure (Tp), two objective parameters Eff and Tp are selected as multiple responses to optimize. The parameters evaluated are inlet blade angle, outlet blade angle, and blade number. Simulations format are based on Box-Behnken design (BBD) method and the values of multiple responses are acquired using computational fluid dynamics (CFD). The values of grey relational grade (GRG) for the multiple responses are acquired based on weighted grey relational analysis. The particularity of weighted grey relational analysis is that the weight factors can be calculated quantificationally from the weightiness of the multiple responses. RSM is used to construct the regression model between GRG and input variables. Based on weighted grey relational analysis and RSM, the optimal centrifugal fan impeller parameters can be obtained, and the acquired results indicate that both Eff and Tp increase after optimization. The results also indicate that the proposed optimization method is a very useful tool for multiobjective optimization of centrifugal fan impeller parameters.

1. Introduction As one of universal machines, large centrifugal fans have been broadly used in lots of departments of the national economy, such as cement stove ventilating system and mile ventilating system. Centrifugal fan is important auxiliary equipment in many companies, and 30 percent of plant electrical consumption is consumed by it. So the research and optimization of centrifugal fan are important for the energysaving of plant. With the development of CFD technology, more and more researchers use CFD software to simulate turbomachinery engineering such as centrifugal fan and pumps. Lin and Huang [1] researched one forward-curved blade centrifugal fan using CFD method; the simulation method combined with experiment was used for optimum studying of the fan. Visser et al. [2] presented potential flow approximation and CFD simulation to optimize one centrifugal pump impeller. Goto and Zangeneh [3] introduced an inverse design method combined with CFD to redistribute the load on the vane of

a pump diffuser. Both numerical simulation and experiment proved that the optimization could improve the performance of the pump. Besides, the CFD method using standard k𝜀 model has been successfully applied in the research for multiblade centrifugal fans [4], centrifugal pumps [5–7], and centrifugal compressors [8]. Most published articles focus on parameters optimization for centrifugal fan impeller applying optimization technology. Yu et al. [9] analyzed the effects of impeller gap and blade inlet angle. They found that the impeller gap and blade inlet angle played an important role for the centrifugal fan performance. Dai [10] introduced an optimizing design method for the centrifugal fan impeller. Wang et al. [11] developed the least squares method to optimize centrifugal fan with different outlet blade angle and blade numbers. Han and Maeng [12] presented a new method to optimize the cutoff in a multiblade fan by using CFD and a neural network. Chunxi et al. [13] investigated the influence of enlarged impeller on the type of G4-73 centrifugal fan performance. They found that total pressure rise, the flow rate, sound pressure level, and

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Advances in Mechanical Engineering Table 1: Impeller dimensions.

Number 1 2 3 4 5 6 7 8 9

Geometric structures Impeller blade outlet diameter, 𝐷2 (mm) Impeller blade inlet diameter, 𝐷1 (mm) Impeller inlet diameter, 𝐷0 (mm) Impeller outlet width, 𝑏2 (mm) Impeller inlet width, 𝑏1 (mm) Blade thickness, 𝑡 (mm) Inlet blade angle 𝛼 Outlet blade angle 𝛽 Blade number 𝑛

Dimensions 2880 1152 1334 187 682 24 (37∘ –41∘ ) (61∘ –65∘ ) (10–14)

shaft power have increased, while the efficiency has decreased when the centrifugal fan G4-73 operated with larger impeller. Yang et al. [14] suggested using three-dimensional inverse design method to optimize centrifugal fan impeller. Zhang et al. [15] used genetic algorithms and artificial neural networks method to optimize a centrifugal fan. Besides, many previous studies [16–19] have focused on multiobjective turbomachinery designs using evolutionary algorithms which can deal with conflicting relations among multiple objectives, and other articles optimized the centrifugal fan using Kriging method [20, 21], RSM method [22–24], and neural networks [12, 25]. The theory of grey systems is a new technique for performing prediction, relational analysis, and decision making in many areas. In the engineering field, numerous optimization problems can be solved by grey relational analysis; for instance, Tosun [26] optimized the drilling process parameters grey relational analysis. Lin and Ho [27] optimized chemical-mechanical polishing process parameters using grey relation and ANOVA, and other optimizations [28, 29] have to be done by grey relational analysis. So the grey relational analysis is very useful in engineering field. However, there is no published article evaluating the effect of centrifugal fan impeller parameter by using grey system theory in centrifugal fan design. Therefore, one purpose of the paper is to introduce the grey relational analysis in selecting the optimal centrifugal fan impeller parameters. In order to improve centrifugal fan efficiency and ensure high total pressure, it is necessary to choose the most appropriate centrifugal fan impeller parameters. To avoid consuming time in trial and error tryout procedure and facilitate modification, computational fluid dynamics (CFD) simulations are used to evaluate the centrifugal fan performance. CFD method is very useful in the turbomachinery engineering research just as the above literatures [1–6]. However, it is still not a very easy problem to obtain the optimum result by running the simulation code in the case that many parameters need to be considered, because each running of simulation will involve much computation time for a complicated centrifugal fan simulation. In order to minimize these simulation times, experiment design method is used to arrange simulation process. BBD is an experiment method which can make scientific arrangement of the optimization parameters.

In summary, optimization of fan performance researched mainly focuses on blade profile [18] or single impeller structural parameter [9, 13]. However, there is little research on the comprehensive effect of the key parameters such as inlet blade angle, outlet blade angle, and blade numbers on the fan performance. In the paper, BBD method combined with weighted grey relational analysis is used to optimize the centrifugal fan impeller parameters (inlet blade angle, outlet blade angle, and blade numbers). Most scholars use equal weight or choose a weight subjectively to emphasize the response for multiresponse optimization process in the research, which lack reasonable quantitative foundation. Therefore, it is important to adopt a reasonable standard for the calculation of weight factors objectively in order to distribute appropriate weight values to different responses in the optimization process. In order to improve the centrifugal fan Eff and Tp scientifically, this paper put forward a multiobjective optimization method based on grey relational analysis and the weight factors for multiple responses (Eff and Tp) are determined objectively and quantificationally. Besides, RSM is used to construct relationships between GRG and input variables and the optimization parameter can be obtained from the regression model. The first research process is to design parameters based on BBD method and the multiple responses are calculated by CFD, the second main step is grey relational analysis including calculation of grey relational grade and weight factor for multiple responses, the third main step is multiple regression modeling and variance analysis, and the last step is choosing the optimization parameter. The detailed optimization process is shown in Figure 1.

2. Simulation Method and Theory 2.1. Basic Parameters of Impeller. In this research, one type of the centrifugal fan (shown in Figures 2 and 3) coded with GD288 is used in the research. Table 1 shows the main dimensions of the centrifugal fan impeller. Because the centrifugal fan is mainly affected by inlet blade angle 𝛼, outlet blade angle 𝛽, and blade number 𝑛, so 𝛼, 𝛽, and 𝑛 are selected as optimization parameters. 2.2. Computational Domain and Computational Grid. In this part, the three-dimensional centrifugal fan models were first built in PROE (computer-aided design software) and translated into ∗.x-t type files. The ∗.x-t files were then imported into the mesh generator software (ICEM). In ICEM, model construction and split were processed. It consists of the inlet, impeller, and the volute. In order to simulate the true inlet condition, an extended cylinder is added to the inlet region for the purpose of imposing the real inlet boundary conditions. Grid independency tests were investigated for each model researched in this part. The static pressure which is equal to the atmospheric pressure was then defined as the inlet boundary condition. The mass flow rate was given as outlet boundary condition. Unstructured tetrahedral cells are used for the meshing of impeller and volute, while hexahedron cells are used for the meshing of inlet region.


Advances in Mechanical Engineering

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Identification and selection of control factors and response variables of the study

Selection of array and factor levels for experiment

Box-Behnken design

Computation of response variables

Response variables are calculated by CFD

Grey generation of raw data Computation of grey relational coefficient of response variables

Grey relational analysis

Calculate of weight factors

Computation of grey relational grade

Multiple regression modeling (RSM used) Analysis of variance Optimization and validation of optimal centrifugal fan parameters

Figure 1: The optimization process used in the paper. Control valve of the entrance

Impeller Gas tank Inlet ring

Shaft

Volute

Figure 2: The internal structure of centrifugal fan.


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Advances in Mechanical Engineering b2 D2

𝛼

t

𝛽

D0 b1

D1

(a)

(b)

Figure 3: Geometry of the backward-curved centrifugal fan.

Table 2: The grid number. Item Number

Inlet region 127832

Impeller 756849

Volute 308887

Convergence of the solution is determined by monitoring the normalized residuals for each dependent variable. The solution is considered converged when these normalized residuals reduce to 1 × 10−3 and the mass residual decreases to 1 × 10−6 . 2.4. Mathematical Models. For steady incompressible turbulent flows, the Reynolds-averaged Navier-Stokes equations are shown as follows:

Volute

𝜕 (𝜌𝑢𝑖 ) = 0, 𝜕𝑥𝑖 Impeller

𝜕𝑝 𝜕 𝜕 (𝜌𝑢𝑖 𝑢𝑗 ) = − + 𝜕𝑥𝑗 𝜕𝑥𝑖 𝜕𝑥𝑗

Inlet ring

× [𝜇 (

Extended cylinder

+ 𝑠𝑖𝑢 ,

Figure 4: Meshes of the centrifugal fan.

The grid number of the original centrifugal fan is listed in Table 2. A general view of the geometry and grids is shown in Figure 4. 2.3. Boundary Conditions Setting. Centrifugal fan geometry including three different domains: the inlet region, the impeller, and the volute. The volute and inlet region are defined as two stationary domains, whereas the impeller zone is researched as a rotating domain. Surfaces between the impeller and the entry zone (surfaces corresponding to the blades inlet) and surfaces between the volute and the impeller (surfaces corresponding to the blades outlet) are given as domain interfaces.

(1)

𝜕𝑢𝑖 𝜕𝑢𝑗 2 𝜕𝑢𝑘 + − 𝛿 ) − 𝜌𝑢𝑖󸀠 𝑢𝑗󸀠 ] 𝜕𝑥𝑗 𝜕𝑥𝑖 3 𝜕𝑥𝑘 𝑖𝑗 (2)

where 𝑢𝑖 and 𝑢𝑖󸀠 are mean and fluctuating velocities, respectively, and 𝑠𝑖𝑢 is source term. Governing equations with standard 𝑘-𝜀 turbulence model are transformed to nonorthogonal curvilinear coordinates and are discretized with finite volume approximations. As a numerical scheme for the convection terms, a linear upwind differencing scheme is used, and for the diffusion terms, a central differencing scheme is used. The strongly implicit procedure (SIP) was used to solve linear algebraic equations. Also the SIMPLEC algorithm is used to match pressure and velocities. 2.5. Validity for the Simulation Method. In the article [30], I simulated the internal flow field of the 9-19No.4A type centrifugal fan under different operating conditions; the research results indicated that the fan performance curves



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3700

3500

Efficiency (%)

Total pressure (Pa)

3600

3400 3300 3200 3100 800

1000

1200

1400

1600

1800

78 77 76 75 74 73 72 71 70 69 800

1000

1200

1400

1600

1800

Flow rate (m3 /h)

Flow rate (m3 /h)

Test Simulation

Test Simulation (a) Performance curves of total pressure and flow

(b) Performance curves of efficiency and flow

Figure 5: Comparison of test and simulation in literature [30]. Table 3: Impeller parameters and their levels. Factors A B C

Parameters Inlet blade angle 𝛼 Outlet blade angle 𝛽 Blade number 𝑛

Range 37∼41 61∼65 10∼14

of the experiment agree with those of numerical simulations. Figure 5 shows performance curves obtained through numerical simulation and experiment. It can be seen that the error of the total pressure between test and simulation is less than 4.5%, while the efficiency error is less than 3%, so conclusion can be drawn that the calculation results derived by numerical simulation are accurate enough to predict the inner flow of the fan. The CFD simulation technology in this paper is the same as the method used in the literature [30], so the CFD simulation can be applied to calculate the response under different centrifugal fan parameters. 2.6. Multiobjective Optimization of Centrifugal Fan Parameters. Centrifugal fans use a rotating impeller to increase the velocity of an airstream. When the air moves from the impeller hub to the blade tips, it gains kinetic energy. This kinetic energy is then translated to a static pressure increase. Its main purpose is to provide enough flow and pressure. So we select Tp as one optimization object. As we all know, 30 percent of plant electrical consumption is consumed by the fan. So the research and optimization of centrifugal fan are important for the energy-saving of plant. So Eff is one of the other important objective parameter to optimize. The Tp is defined as the average total pressure difference between the inlet section and the outlet section of the fan. The calculation equation is shown as follows: ∗ − 𝑃in∗ , Tp = 𝑃out

(3)

∗ is the outlet total pressure, Pa; 𝑃in∗ is the inlet total where 𝑃out pressure, Pa.

Level 1 37 61 10

Level 2 39 63 12

The Eff is defined as follows: Tp × 𝑄 , Eff = 𝑊 × 3600

Level 3 41 65 14

(4)

where 𝑄 is the volume flow, m3 /h; 𝑊 is the shaft power, W. According to the optimization problem, the multiobjective optimization can be written as follows: ∘ ∘ {37 ≤ 𝛼 ≤ 41 max Eff (𝛼, 𝛽, 𝑛) { ∘ 61 ≤ 𝛽 ≤ 65∘ max Tp (𝛼, 𝛽, 𝑛) { { {10 ≤ 𝑛 ≤ 14.

(5)

2.7. Sample Point Design. First, inlet blade angle 𝛼, outlet blade angle 𝛽, and blade number 𝑛 are selected as optimization parameters; Eff and Tp are selected as objective parameter. That is to say, there are 3 factors to consider in the optimization process, each factor having 3 levels. The factors and their levels are listed in Table 3. In this research, the BBD method is used to design parameters. Based on the BBD theory, an array with 13 rows is used for the simulation experiments. The numerical simulation results are listed in Table 4.

3. Grey Relational Analyses Grey relational analysis is one of the most important contents in grey system theory which is put forward by Julong Deng in 1989, and it can be used to solve the intricate correlations among the multiple responses effectively. Grey relational analysis can be used to measure the approximate relationships among sequences. In grey relational analysis, the simulation results of Eff and Tp are first


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Advances in Mechanical Engineering Table 4: The Box-Behnken design and corresponding results (target function value). Inlet blade angle 𝛼 37 41 37 41 37 41 37 41 39 39 39 39 39

Exp. number 1 2 3 4 5 6 7 8 9 10 11 12 13

Outlet blade angle 𝛽 61 61 65 65 63 63 63 63 61 65 61 65 63

normalized in the scope between zero and one, which is called grey relational generation operation. Second, the grey relational coefficients are calculated in order to express the relationship between the desired and actual results. Then, the grey relational grades are calculated by dealing with the GRC. Here, multiple objectives (Eff and Tp) optimization can be converted into single objective optimization (only GRG consideration), and this is the advantage of grey relational analysis theory. The optimal level of the process parameters is the level with the highest GRG. With the grey system theory, the optimal combination of the centrifugal fan impeller parameters can be calculated. 3.1. Data Preprocessing. The range and unit in one data sequence is different from the others, so data preprocessing is required. Data preprocessing is also necessary when the directions of the target are different or the sequence scatter range is too large. There are many methods of data preprocessing available for grey relational analysis. When the original sequence owns the characteristic “higher is better,” then the original sequence can be normalized as follows: 𝑥𝑖 (𝑘) =

𝑥𝑖0 (𝑘) − min𝑖 𝑥𝑖0 (𝑥) . max𝑖 𝑥𝑖0 (𝑘) − min𝑖 𝑥𝑖0 (𝑘)

(6)

When the original sequence owns the characteristic “lower is better,” then the original sequence can be normalized as follows: 𝑥𝑖 (𝑘) =

max𝑖 𝑥𝑖0 (𝑥) − 𝑥𝑖0 (𝑘) . max𝑖 𝑥𝑖0 (𝑘) − min𝑖 𝑥𝑖0 (𝑘)

(7)

However, if there is a definite target value (desired value) to be achieved, the original sequence can be normalized in form as follows: 󵄨󵄨 0 󵄨 󵄨󵄨𝑥𝑖 (𝑘) − 𝑥0 󵄨󵄨󵄨 󵄨 󵄨 , (8) 𝑥𝑖 (𝑘) = 1 − max𝑖 𝑥𝑖0 (𝑘) − 𝑥0 where 𝑖 = 1, . . . , 𝑚; 𝑘 = 1, . . . , 𝑛. 𝑚 is the number of experimental data items and 𝑛 is the number of parameters.

Blade number 𝑛 12 12 12 12 10 10 14 14 10 10 14 14 12

Eff (%) 92.395 91.755 92.4214 92.4398 93.2409 92.1942 93.4758 92.7515 91.1253 92.6324 93.4941 92.7085 92.6665

Tp (Pa) 15125.6 15021.4 15761.5 15763.4 15184.6 14788.9 16367.2 16048.3 14088.6 15238.4 15997.2 16503.3 15535.2

Table 5: The sequence after data preprocessing. Exp. number 1 2 3 4 5 6 7 8 9 10 11 12 13

Eff 0.53601 0.265831 0.547155 0.554922 0.89311 0.451241 0.992275 0.686508 0 0.636229 1 0.668355 0.650625

Tp 0.429453 0.386301 0.692798 0.693585 0.453887 0.290015 0.943637 0.811571 0 0.476167 0.790409 1 0.599081

𝑥𝑖0 (𝑘) is the original sequence, 𝑥𝑖 (𝑘) is the sequence after data preprocessing, max𝑖 𝑥𝑖0 (𝑘), min𝑖 𝑥𝑖0 (𝑘) denote the largest value and the smallest value of 𝑥𝑖0 (𝑘) respectively, and 𝑥0 is the desired value of 𝑥𝑖0 (𝑘). In the present research, higher fan Eff and Tp are indications of better result. For data preprocessing in the grey relational analysis procedure, centrifugal fan Eff and Tp are considered as the “higher is better.” Let the result of 13 simulations be the comparability sequence 𝑥𝑖0 (𝑘), 𝑖 = 1 ∼ 13, 𝑘 = 1 ∼ 2. All the sequences after data preprocessing according to (6) are shown in Table 5.

3.2. Calculation of GRC. In the grey system theory, the measure of relevancy between two sequences is defined as GRG. Suppose the reference sequence after data preprocessing is 𝑋0 = {𝑥0 (𝑘), 𝑘 = 1, 2, . . . , 𝑛}; target vector sequence 𝑋𝑖 = {𝑥𝑖 (𝑘), 𝑘 = 1, 2, . . . , 𝑛}, 𝑖 = 1, . . . , 𝑚, 𝑚, is the number of target vectors. So the grey relation coefficient 𝜉𝑖 (𝑘) for the



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Δ 0𝑖 (1) 0.46399 0.734169 0.452845 0.445078 0.10689 0.548759 0.007725 0.313492 1 0.363771 0 0.331645 0.349375

Exp. number 1 2 3 4 5 6 7 8 9 10 11 12 13

Δ 0𝑖 (2) 0.570547 0.613699 0.307202 0.306415 0.546113 0.709985 0.056363 0.188429 1 0.523833 0.209591 0 0.400919

𝑘th performance characteristics in the 𝑖th experiment can be expressed as: 𝜉𝑖 (𝑘) =

Δ min + 𝜌Δ max , Δ 0𝑖 (𝑘) + 𝜌Δ max

󵄨 󵄨 Δ min = min min 󵄨󵄨󵄨𝑥0 (𝑘) − 𝑥𝑖 (𝑘)󵄨󵄨󵄨 , ∀𝑘

(10)

󵄨 󵄨 Δ max = max max 󵄨󵄨󵄨𝑥0 (𝑘) − 𝑥𝑖 (𝑘)󵄨󵄨󵄨 , ∀𝑗∈𝑖

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Average percentage deviation = 7.27%

0

1

2

3

4

5 6 7 8 9 10 11 12 13 14 Experiment number

Simulated Predicted

Figure 6: Comparison of predicted and simulated values for GRG.

value for distinguishing coefficient 𝜌 is reckoned as 0.5. The grey relational coefficients are calculated by applying (9). The calculation results of GRC are listed in Table 7.

(9)

where Δ 0𝑖 (𝑘) is called the deviation sequence between the comparability sequence and the reference sequence. Consider 󵄨 󵄨 Δ 0𝑖 (𝑘) = 󵄨󵄨󵄨𝑥0 (𝑘) − 𝑥𝑖 (𝑘)󵄨󵄨󵄨 , ∀𝑗∈𝑖

GRG

Table 6: The deviation sequence.

∀𝑘

where 𝑋0 = {𝑥0 (𝑘), 𝑘 = 1, 2, . . . , 𝑛} is called the reference sequence and 𝑋𝑖 = {𝑥𝑖 (𝑘), 𝑘 = 1, 2, . . . , 𝑛} is called the comparability sequence. Δ 0𝑖 (𝑘) is called the deviation sequence between the reference sequence 𝑥0 (𝑘) and the comparability sequence 𝑥𝑖 (𝑘). 𝜌 ∈ [0, 1] is called the distinguishing coefficient or identification coefficient. 𝜌 = 0.5 is usually used. Now we denote 𝑥𝑖 (𝑘) and 𝑥0 (𝑘) as comparability sequence and reference sequence, respectively. The deviation sequences Δ 0𝑖 can be computed as follows: 󵄨 󵄨 Δ 01 (1) = 󵄨󵄨󵄨𝑥0 (1) − 𝑥1 (1)󵄨󵄨󵄨 = |1 − 0.53601| = 0.46399, 󵄨 󵄨 Δ 01 (2) = 󵄨󵄨󵄨𝑥0 (2) − 𝑥1 (2)󵄨󵄨󵄨 = |1 − 0.429453| = 0.570547. (11) So Δ 01 = (0.46399, 0.570547). The same calculation method is executed for 𝑖 = 1 ∼ 13 and the calculation results of all Δ 0𝑖 for 𝑖 = 1 ∼ 13 are shown in Table 6. According to Table 6, Δ max and Δ min can be obtained as follows: Δ max = Δ 09 (1) = Δ 09 (2) = 1, Δ min = Δ 011 (1) = Δ 012 (2) = 0.

(12)

The identification coefficient 𝜌 can be substituted into (9) in order to calculate the grey relational coefficient 𝜉𝑖 (𝑘). The

3.3. Calculation of GRG. After the GRC is obtained, the GRG is calculated as follows: 𝑅𝑖0 =

1 𝑛 ∑ 𝜆 𝜉 (𝑘) 𝑛 𝑘=1 𝑘 𝑖

𝑛

∑ 𝜆 𝑘 = 1,

(13)

𝑘=1

where 𝜆 𝑘 shows the weight factor of various responses and the GRG expresses the correlation between the comparability sequence and the reference sequence. The evaluated GRG takes values in the range from 0 to 1 and equals 1 when these two sequences accord profoundly. That is to say, the level owning the maximal GRG is the optimal level of the centrifugal fan. Therefore, if one certain comparability sequence is more vital than the other comparability sequences corresponding to the reference sequence, then the GRG for that comparability sequence and reference sequence would be higher than the other grey relational grades. The GRC and GRG values for each simulation of the array are computed by applying (9) and (13) (shown in Table 7). Table 7 lists the GRG for each simulation using factorial design method. Bigger GRG represents that the corresponding simulation result is closer to the ideally normalized value. Simulation 7 has the best multiperformance characteristics among all the 13 simulations because it owns the biggest GRG as listed in Table 7 and Figure 6. It can be found that, in the present research, optimization of the complex multiperformance characteristics of centrifugal fan impeller parameters has been transformed into optimization of a GRG. 3.4. Calculation of Weight Factors. The weightiness of various responses is different from each other for a certain engineering problem. The GRG is different when unequal weight is carried for the various responses, which shows that weight factors are vital to the calculation results. In general, researchers calculate the GRG of numerous responses using


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Advances in Mechanical Engineering Table 7: The calculated GRC and GRG for 13 comparability sequences.

Exp. number 1 2 3 4 5 6 7 8 9 10 11 12 13

GRC (Eff) 0.518677 0.405131 0.524744 0.529057 0.823873 0.476754 0.984784 0.614634 0.333333 0.578857 1 0.601218 0.588668

GRC (Tp) 0.467051 0.448954 0.619424 0.620028 0.47796 0.413228 0.898694 0.726291 0.333333 0.488361 0.704631 1 0.554989

GRG 0.492286 0.427533 0.573144 0.575561 0.647042 0.44428 0.940775 0.671713 0.333333 0.532595 0.849007 0.805075 0.571451

Order 10 12 7 6 5 11 1 4 13 9 2 3 8

Table 8: Mean values for each parameter at each level.

Level 1 Level 2 Level 3 Range ∑ range Weight

𝛼 0.71302 0.620415 0.506394 0.206626

Eff 𝛽 0.564285 0.697743 0.558469 0.139274 0.632803 48.88%

𝑛 0.553204 0.513255 0.800159 0.286904

the same weight or choosing a certain weight to emphasize the target subjectively, which seems unscientific because the method lacks reasonable quantitative analysis. However, it is necessary to research a reasonable method to calculate weight factors objectively to distribute appropriate factor to various responses. Therefore, one feasible method is developed to assign appropriate weight factors based on the influence degree between Eff and Tp. Mean values of the GRC for each parameter at each level for Eff and Tp can be calculated according to Table 8 and the range of GRC (calculated from GRCmax − GRCmin ) of each response for each parameter can be acquired. For instance, the mean value of GRC for inlet blade angle (𝛼) at level 1 can be obtained by taking the average of the GRC for the simulations 1, 3, 5, and 7. Using the same method, the mean values of the GRC for other variables in each level can be calculated. Table 8 shows the mean values of the GRC for every parameter at each level. The range of GRC is higher if the influence degree of centrifugal fan capability on each response becomes larger. On the contrary, if the response sustains in a specified value when the centrifugal fan’s capability vary, this indicates that there is no relationship between the response and the centrifugal fan’s capability; in this case, the weight factor will be equal to zero. Therefore, the weight factor should be higher if the response is sensitive to the alteration of the centrifugal fan’s capability. The quantitative value of the influence degree for each response is computed by the sum-average of the GRC range

Tp 𝛽 0.488492 0.614232 0.681953 0.193461 0.661782 51.12%

𝛼 0.615782 0.616263 0.552125 0.064138

𝑛 0.428221 0.542089 0.832404 0.404184

(max.–min.). The ratio of quantitative value for each response is the criterion for the calculation of weight factors we used. The range of GRC and weight factors are calculated according to (14) and (15), respectively. The last row means the weight factor for each response in Table 8. Consider 𝑅𝑖𝑗 = max {𝐾𝑖,𝑗,1 , 𝐾𝑖,𝑗,2 , . . . , 𝐾𝑖,𝑗,𝑘 } − min {𝐾𝑖,𝑗,1 , 𝐾𝑖,𝑗,2 , . . . , 𝐾𝑖,𝑗,𝑘 } ,

(14)

𝑝

𝜆𝑖 =

∑𝑗=1 𝑅𝑖,𝑗 𝑝

∑𝑚 𝑖=1 ∑𝑗=1 𝑅𝑖,𝑗

,

(15)

where 𝑖 = 1, 2 ⋅ ⋅ ⋅ 𝑚, 𝑗 = 1, 2 ⋅ ⋅ ⋅ 𝑝, 𝑘 = 1, 2 ⋅ ⋅ ⋅ 𝑙. 𝑚 means the number of responses; there, it is equal to 2, 𝑝 is the number of optimal parameters (inlet blade angle, outlet blade angle, and blade number), l is the number of experimental levels, 𝑅 is the range of GRC, 𝐾 is the average GRC for each parameter at each level of each response, and 𝜆 is the weight of each response. The weight factors of responses are determined by the proposed method and the expression of GRG is as follows: GRG = 0.4888 GRC(Eff) + 0.5112 GRC(Tp) ,

(16)

where GRC(Eff) is GRC of Eff and GRC(Tp) is GRC of Tp. The GRG values of two responses are listed in Table 7.



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Table 9: ANOVA for the regression model and respective model terms. Source

Sum of squares

Degree of freedom

Mean square

𝐹-Value

𝑃 value prob. > 𝐹

Model

0.34

9

0.038

8.75

0.0046

A

0.036

1

0.036

8.17

0.0244

B

0.018

1

0.018

4.23

0.0788

C

0.21

1

0.21

49.11

0.0002

AB

1.128𝐸 − 003

1

1.128𝐸 − 003

0.26

0.6268

AC

1.099𝐸 − 003

1

1.099𝐸 − 003

0.25

0.6312

BC

0.015

1

0.015

3.39

0.1082

A2

7.374𝐸 − 005

1

7.374𝐸 − 005

0.017

0.9002

B2

0.011

1

0.011

2.43

0.1634

2

0.050

1

0.050

11.40

0.0118

Residual

0.031

7

4.364𝐸 − 003

Lack of Fit

0.031

3

0.010

Pure Error

0.000

4

0.000

Cor Total

0.37

16

C

𝑅-Squared = 0.9183

4. Process Modeling and ANOVA Analysis

calculated using Design-Expert.8.05b software based on the simulation data is as follows:

In order to optimize the centrifugal fan, it is necessary to construct relationships between GRG and input variables. RSM is used to construct the regression model. RSM is a set of mathematical and statistical techniques that are useful for modeling and predicting the response of interest affected by a number of input variables with the aim of optimizing this response. RSM also specifies the relationships among one or more measured responses and the essential controllable input factors. In the practical application of RSM, it is necessary to develop an approximating model for the true response surface. The approximating model is based on observed data from the process or system and is an empirical model. Multiple regression analysis is a collection of statistical techniques useful for building the types of empirical models required in RSM. Usually, a second-order polynomial equation is used in RSM: 𝐾

𝐾

𝑖=1

𝑖=1

𝑦𝑃 = 𝑐0 + ∑ 𝑐𝑖 𝑥𝑖 + ∑ 𝑐𝑖𝑖 𝑥𝑖2 𝐾−1

𝐾

+ ∑ ∑ 𝑐𝑖𝑗 𝑥𝑗 𝑥𝑖 ,

(17) 𝑃 = 1, . . . , 𝑛,

𝑗=1 𝑖=𝑗+1

where 𝑦𝑃 is response (dependent variable), 𝑐0 is a constant coefficient, 𝑐𝑖 , 𝑐𝑖𝑖 , and 𝑐𝑗𝑖 represent coefficients for the linear, quadratic, and interaction effect independently, and 𝑥𝑖 , 𝑥𝑗 are factors (independent variables). The regression model

GRG (𝛼, 𝛽, 𝑛) = −51.16186 − 0.16653𝐴 + 1.62193𝐵 + 0.5489𝐶 + 0.00419812𝐴 × 𝐵

(18)

− 0.00414375𝐴 × 𝐶 − 0.0152𝐵 × 𝐶 − 0.00104625𝐴2 − 0.012534𝐵2 + 0.027172𝐶2 , where 𝐴 is the value of inlet blade angle, 𝐵 is the value of outlet blade angle, and 𝐶 is the value of blade number. Table 9 shows the summary of ANOVA results for regression model. According to the statistical analysis results, the coefficient of 𝑅-Squared for this model is equal to 91.83%, which demonstrates that the RSM model has good compatibility to the calculation data. Therefore, this regression model based on BBD method and grey relational analysis for the optimization of the multiresponse problems can be used as a prediction model. In order to measure how well the suggested model fits the simulation data, the parameters 𝐹-value, 𝑅Squared, 𝑃 value, and Lack of Fit can be used. According to Table 9, 𝐹-value is equal to 8.75 and expresses that the quadratic model is significant. Furthermore, each term in the model is also tested. It is clear that the linear terms for blade number (𝐶) and the inlet blade angle (𝐴) have large effects on GRG because of owning high 𝐹-values. Besides, the linear term for outlet blade angle (𝐵) is also significant but has a smaller effect on GRG because of its smaller 𝐹value than the other linear terms. However, the quadratic


10

Advances in Mechanical Engineering Table 10: Results with different centrifugal fan parameters.

term for blade number (𝐶2 ) has a large 𝐹-value. Thus, the effect of the blade number on the GRG is most strongly modeled with the quadratic term. The quadratic term for outlet blade angle (𝐵2 ) is also significant but has smaller 𝐹values than their corresponding linear terms. The quadratic term for the inlet blade angle (𝐴2 ) is less significant than the other quadratic terms. The only significant coupling term on the GRG is between outlet blade angle and blade number (𝐵𝐶), indicating an interaction on the GRG between those two variables. The optimization result can be calculated by the sequential quadratic programming (SQP) algorithm using MATLAB software. The optimal solution computed by the software for GRG is 0.927317 which corresponds to the following centrifugal fan parameter: inlet blade angle 𝛼 is 37, outlet blade angle 𝛽 is 63, and blade number 𝑛 is 14. The optimal setting of centrifugal fan parameters that optimize the multiple objectives is also the calculation number 7, which reaches the highest GRG shown in Figure 6. Figure 6 shows the values of GRG calculated from simulation and prediction, in which the predicted values are calculated from (18). Traditionally the design of a centrifugal fan is determined based on conventional method, which is according to oneor two-dimensional ideal flow and designer’s experience, so the designed centrifugal fan parameters may not be the optimal values. The initial designated levels of centrifugal fan parameters are A2 , B2 , and C2 which is corresponding to the simulation number 13 shown in Table 4. For the comparison between initial centrifugal fan parameters and the optimization result, Eff increased from 92.6665% to 93.4758% and Tp increased from 15535.2 Pa to 16367.2 Pa., which is shown in Table 10.

5. Results and Discussion 5.1. Factor Effects Analysis. The main effects analysis is employed to investigate the contribution and effects of centrifugal fan’s capability on the two responses Eff and Tp as shown in Figures 7 and 8. The plots show the variation of individual response with the parameters, that is, inlet blade angle, outlet blade angle, and blade angle, separately. In the plots, the 𝑥-axis shows the values of each parameter at three levels and 𝑦-axis the values of GRC of each response. The horizontal line in the plot shows the mean value of each response. From Figures 7 and 8, we can get the message that the main effect factor is blade number while the inlet blade angle and outlet blade angle seem to have a slightly lower effect than blade number. The trend of changing for GRC between Eff and Tp is nearly alike under the parameter inlet blade angle. The trends are different for the other two parameters. The optimal performance for GRC of Eff can be

𝛼 39 37

Tp 15535.2 Pa 16367.2 Pa

GRC of Eff

Eff 92.6665% 93.4758%

0.81 0.78 0.75 0.72 0.69 0.66 0.63 0.6 0.57 0.54 0.51 0.48

37

39 𝛼

41

𝛽 63 63

61

63 𝛽

65

𝑛 12 14

10

12 n

14

Figure 7: Effect of impeller parameters on Eff (evaluating by GRC).

GRC of Tp

Items Initial centrifugal fan parameters The optimal centrifugal fan

0.85 0.82 0.79 0.76 0.73 0.7 0.67 0.64 0.61 0.58 0.55 0.52 0.49 0.46 0.43 0.4 37

39 𝛼

41

61

63 𝛽

65

10

12 n

14

Figure 8: Effect of impeller parameters on Tp (evaluating by GRC).

obtained for inlet blade angle 𝛼 (level 1), outlet blade angle 𝛽 (level 2), and blade number 𝑛 (level 3) combination. The optimal performance for GRC of Tp can be obtained for inlet blade angle 𝛼 (level 2), outlet blade angle 𝛽 (level 3), and blade number 𝑛 (level 3) combination. 5.2. Effect of Impeller Parameters on Performance. The response surface figures are built by using the RSM software Design-Expert.8.05b. Essentially, the bigger the GRG value is, the better the multiperformance characteristics are. Figure 9 shows the response surface of GRG between inlet blade angle and outlet blade angle. It is clear from Figure 9 that the inlet blade angle and outlet blade angle are the most significant factors that affect the GRG. With an increase in inlet blade angle, the GRG decreases; while with an increase of the outlet blade angle, the GRG first increases and then decreases. Figure 10 shows the response surface of GRG between inlet blade angle and blade number. From Figure 10, it can be seen that with an increase in blade number, GRG values increase. Figure 11 shows the response surface of GRG between outlet blade angle and blade number. From Figure 11, it can be also


11

0.65

1

0.6

0.9

0.55

0.8

0.5

0.7 GRG

GRG


0.45

0.6

0.4

0.5

0.35

0.4

65.00

41.00 40.00

64.00

B: o utle t

39.00

63.00

bla d

e an gle

62.00 61.00 37.00

38.00

n A: i

let

gle e an blad

41.00

14.00

13.00

C: bl

40.00

12.00 39.00 le ang de en a 11.00 l 38.00 b um t ber inle 10.00 37.00 A:

ad

Figure 9: Response surface of GRG for combined effect of inlet blade angle and outlet blade angle.

Figure 10: Response surface of GRG for combined effect of inlet blade angle and blade number.

seen that with an increase in blade number, GRG increase and that with an increase of the outlet blade angle, the GRG first increases and then decreases.

to say, a bigger GRG is desired for optimum condition and the optimal level of the centrifugal fan impeller parameters is the level with the highest GRG. Therefore, the optimal levels of centrifugal fan impeller parameters for improved Eff and Tp are acquired for inlet blade angle 𝛼 (level 1), outlet blade angle 𝛽 (level 2), and blade number 𝑛 (level 3) combination as shown in Table 11. Here, an asterisk (∗) denotes that the level value is bigger than the other two level values. Therefore, simulation 7, as listed in Figure 6 and Table 7, may be reckoned as very close to meet the optimal conditions. As shown in Table 11, the difference between the maximum value and the minimum value of the GRG of the centrifugal fan impeller parameters is as follows: 0.1351 for inlet blade angle 𝛼, 0.1296 for outlet blade angle 𝛽, and 0.3256 for blade number 𝑛 (Table 11 and Figure 13). The most impactful factor affecting performance characteristics on centrifugal fan impeller is determined by comparing these difference values. The most effective factor is the maximum value among these three values. In the paper, the maximum value among 0.1351, 0.1296, and 0.3256 is 0.3256. This expresses that the blade number 𝑛 has the strongest effect on the multiperformance characteristics than the other centrifugal fan impeller parameters.

5.3. The Analysis of Response for GRG. In addition to the determination of optimum centrifugal fan impeller parameters for Eff and Tp, the response table for the factorial design method is used to calculate the average GRG for each level of the centrifugal fan impeller parameters. The process is as follows. First Step. Group the grey relational grades by factor level for each column in the factorial design. Second Step. Take their average; for instance, the GRG for factor inlet blade angle 𝛼 at level 1 can be computed as follows: Level 1 (𝛼) =

1 ∗ (0.492286 + 0.573144 + 0.647042 + 0.940775) 4

= 0.663312. (19) The mean of the GRG values for each level of the centrifugal fan impeller parameters is computed using the above method. GRG represents the level of relationship between the comparability sequence and the reference sequence. The bigger value of the GRG hints that the comparability sequence has a better relationship to the reference sequence. The mean of the GRG for each level of the centrifugal fan impeller parameters is shown in Table 11. Figure 12 also shows the GRG acquired for different centrifugal fan impeller parameters. Essentially, the bigger the GRG value is, the closer the centrifugal fan quality to the ideal condition will be. That is

6. Conclusions The grey relational analysis combined with BBD and RSM is used to optimize the centrifugal fan parameters. The input variables are selected as inlet blade angle, outlet blade angle, and blade number, the centrifugal fan Eff and Tp are selected to be the optimization objective. The optimization results acquired in the paper prove that the optimization method is very useful in optimizing the centrifugal fan parameters. In addition, a feasible solution is proposed to compute weight


12

Advances in Mechanical Engineering Table 11: The response table for GRG.

Parameter

Average GRG by factor level Level 2 Level 3 0.615292 0.529772 0.621594 0.655052∗ 0.527995 0.816643∗

Level 1 0.663312∗ 0.52554 0.489313

Inlet blade angle 𝛼 Outlet blade angle 𝛽 Blade number 𝑛

Max.–Min. 0.13354 0.129512 0.32733∗

Total mean value of the GRG = 0.6049. ∗ Optimum levels.

0.35 0.3

0.8

0.25

(Max.–min.) grade

0.9

0.7 GRG

0.6 0.5

0.2 0.15 0.1

0.4

0.05

0.3

0 1

14.00

65.00 13.00

C: bl

64.00

ad

12.00

en um ber

63.00 11.00 10.00

62.00 61.00

e blad tlet u o B:

GRG

3

Figure 13: Effect of the impeller parameters on the multiperformance characteristics.

le ang

Figure 11: Response surface of GRG for combined effect of outlet blade angle and blade number. 0.84 0.81 0.78 0.75 0.72 0.69 0.66 0.63 0.6 0.57 0.54 0.51 0.48 0.45

2 Impeller parameters

of 37 degree, outlet blade angle of 63 degree, and blade number of 14 are the optimal combination of centrifugal fan parameters. (2) From the initial responses and optimization results, the Eff increases from 92.6665% to 93.4758%, and Tp increases from 15535.2 Pa to 16367.2 Pa. (3) Comparing to the initial values, the analysis results imply that multiresponses such as Eff and Tp can be improved simultaneously after optimization using the method proposed in the paper. (4) One rational method is proposed to calculate weight factors of each response quantificationally, which is more scientific than traditional method.

37

39 𝛼

41

61

63 𝛽

65

10

12

14

n

Conflict of Interests

Figure 12: Average GRG for the impeller parameters on three levels.

The authors declare that there is no conflict of interests regarding the publication of this paper.

factors of each response according to the influence degree of centrifugal fan parameters’ variations on multiple responses, which is a useful method to determine the influence degree for each response corresponding to traditional method. The optimization centrifugal fan parameters can be found from the calculation result analysis. Some conclusions are listed as follows.

Acknowledgment

(1) The multiobjective optimization solution we proposed in the paper shows that inlet blade angle

This work was financially supported by the program of Hebei Dunshi Engineering Technology Company (201017123).

References [1] S.-C. Lin and C.-L. Huang, “An integrated experimental and numerical study of forward-curved centrifugal fan,” Experimental Thermal and Fluid Science, vol. 26, no. 5, pp. 421–434, 2002.



13

[2] F. C. Visser, R. J. H. Dijkers, and J. G. H. op de WoerdVisser, “Numerical flow-field analysis and design optimization of a high-energy first-stage centrifugal pump impeller,” Computing and Visualization in Science, vol. 3, no. 1-2, pp. 103–108, 2000. [3] A. Goto and M. Zangeneh, “Hydrodynamic design of pump diffuser using inverse design method and CFD,” Transactions of the ASME: Journal of Fluids Engineering, vol. 124, no. 2, pp. 319–328, 2002. [4] S.-J. Seo, K.-Y. Kim, and S.-H. Kang, “Calculations of threedimensional viscous flow in a multiblade centrifugal fan by modelling blade forces,” Proceedings of the Institution of Mechanical Engineers—Part A: Journal of Power and Energy, vol. 217, no. 3, pp. 287–298, 2003. [5] J. Gonz´alez, J. Fern´andez, and E. Blanco, “Numerical simulation of the dynamic effects due to impeller-scroll interaction in a centrifugal pump,” ASME Journal of Fluids Engineering, vol. 124, pp. 348–355, 2002. [6] L. Zhou, W. Shi, and S. Wu, “Performance optimization in a centrifugal pump impeller by orthogonal experiment and numerical simulation,” Advances in Mechanical Engineering, vol. 2013, Article ID 385809, 7 pages, 2013. [7] C. Wang, W. Shi, L. Zhou, and W. Lu, “Effect analysis of geometric parameters on stainless steel stamping multistage pump by experimental test and numerical calculation,” Advances in Mechanical Engineering, vol. 2013, Article ID 575731, 8 pages, 2013. [8] M. Zangeneh, M. Schleer, F. Pløger et al., “Investigation of an inversely designed centrifugal compressor stage. Part I: design and numerical verification,” Journal of Turbomachinery, vol. 126, no. 1, pp. 73–81, 2004. [9] Z. Yu, S. Li, W. He, W. Wang, D. Huang, and Z. Zhu, “Numerical simulation of flow field for a whole centrifugal fan and analysis of the effects of blade inlet angle and impeller gap,” HVAC and R Research, vol. 11, no. 2, pp. 263–283, 2005. [10] S. Dai, “Optimizing design method for centrifugal impeller its engineering application and numerical simulation,” Journal of Beijing University of Aeronautics and Astronautics, vol. 30, no. 3, pp. 267–271, 2004 (Chinese). [11] S. Wang, L. Zhang, Z. Wu, and H. Qian, “Optimization research of centrifugal fan with different blade number and outlet blade angle,” in Proceedings of the 2009 Asia-Pacific Power and Energy Engineering Conference (APPEEC ’09), pp. 2987–2990, March 2009. [12] S.-Y. Han and J.-S. Maeng, “Shape optimization of cut-off in a multi-blade fan/scroll system using neural network,” International Journal of Heat and Mass Transfer, vol. 46, no. 15, pp. 2833–2839, 2003. [13] L. Chunxi, W. S. Ling, and J. Yakui, “The performance of a centrifugal fan with enlarged impeller,” Energy Conversion and Management, vol. 52, no. 8-9, pp. 2902–2910, 2011. [14] W. Yang, H. Wang, and Y. Wu, “Aerodynamic optimization design of centrifugal fan impeller based on three-dimensional inverse design method,” Transactions of the Chinese Society of Agricultural Machinery, vol. 43, no. 9, pp. 43–48, 2012 (Chinese). [15] B. Zhang, T. Wang, C. Gu, and X. Shu, “Blade optimization design and performance investigations of an ultra-low specific speed centrifugal blower,” Science China Technological Sciences, vol. 54, no. 1, pp. 203–210, 2011. [16] S. Obayashi, T. Tsukahara, and T. Nakamura, “Multiobjective genetic algorithm applied to aerodynamic design of cascade airfoils,” IEEE Transactions on Industrial Electronics, vol. 47, no. 1, pp. 211–216, 2000.

[17] A. Oyama and M.-S. Liou, “Multiobjective optimization of rocket engine pumps using evolutionary algorithm,” Journal of Propulsion and Power, vol. 18, no. 3, pp. 528–535, 2002. [18] X. Liu and W. Zhang, “Multi-objective automatic optimization design of centrifugal impeller based on genetic algorithm,” Journal of Xi'an Jiaotong University, vol. 44, no. 1, pp. 31–35, 2010 (Chinese). [19] L. Zhang, S. Wang, and C. Hu, “Multi-objective optimization design and experimental investigation of centrifugal fan performance,” Chinese Journal of Mechanical Engineering, vol. 26, no. 6, pp. 1–9, 2013. [20] K. Sugimura, S. Jeong, and S. Obayashi, “Kriging-model-based multi-objective robust optimization and trade-off rule mining of a centrifugal fan with dimensional uncertainty,” Journal of Computational Science and Technology, vol. 3, no. 1, pp. 196–211, 2009. [21] X.-F. Wang, G. Xi, and Z.-H. Wang, “Aerodynamic optimization design of centrifugal compressor's impeller with Kriging model,” Proceedings of the Institution of Mechanical Engineers— Part A: Journal of Power and Energy, vol. 220, no. 6, pp. 589–597, 2006. [22] F. Meng, Q. Dong, Y. Wang, P. Wang, and C. Zhang, “Numerical optimization of impeller for backward-curved centrifugal fan by Response Surface Methodology (RSM),” Research Journal of Applied Sciences, Engineering and Technology, vol. 6, no. 13, pp. 2436–2442, 2013. [23] S.-J. Seo and K.-Y. Kim, “Design optimization of forwardcurved blades centrifugal fan with response surface method,” in Proceedings of the 2004 ASME Heat Transfer/Fluids Engineering Summer Conference (HT/FED ’04), pp. 551–556, July 2004. [24] K.-Y. Kim and S.-J. Seo, “Application of numerical optimization technique to design of forward-curved blades centrifugal fan,” JSME International Journal B: Fluids and Thermal Engineering, vol. 49, no. 1, pp. 152–158, 2006. [25] A. Khalkhali, M. Farajpoor, and H. Safikhani, “Modeling and multi-objective optimization of forward-curved blade centrifugal fans using CFD and neural networks,” Transactions of the Canadian Society for Mechanical Engineering, vol. 35, no. 1, pp. 63–79, 2011. [26] N. Tosun, “Determination of optimum parameters for multiperformance characteristics in drilling by using grey relational analysis,” International Journal of Advanced Manufacturing Technology, vol. 28, no. 5-6, pp. 450–455, 2006. [27] Z.-C. Lin and C.-Y. Ho, “Analysis and application of grey relation and ANOVA in chemical-mechanical polishing process parameters,” International Journal of Advanced Manufacturing Technology, vol. 21, no. 1, pp. 10–14, 2003. [28] S.-P. Lo, “The application of an ANFIS and grey system method in turning tool-failure detection,” International Journal of Advanced Manufacturing Technology, vol. 19, no. 8, pp. 564– 572, 2002. [29] J. Yan and L. Li, “Multi-objective optimization of milling parameters-the trade-offs between energy, production rate and cutting quality,” Journal of Cleaner Production, vol. 52, pp. 462– 471, 2013. [30] F. Meng, Q. Dong, N. Chen, and Y. Fan, “Numerical calculation of centrifugal fan 9-19No.4A,” Advanced Materials Research, vol. 614-615, pp. 536–540, 2013.
