# Application of Gray Relational Analysis and Computational Fluid Dynamics to the Statistical Techniques of Product Designs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model for Investigation

#### 2.1. Development of the Model

#### 2.2. Fan Model for Investigation

#### 2.3. Fan Parameters Affecting the Performance Curve

- Blade pitch angle: The larger the pitch angle, the larger the pressure difference between the blade’s upper and lower surfaces. Under the same rotation speed, the air pressure is also larger with a larger pitch angle. However, when the pressure of the lower surface is too large, the phenomenon of recirculation may occur, and this instead reduces the fan’s performance. Therefore, the blade pitch angle should also be increased to a certain extent.
- Blade spacing: When the distance between the blades is too small, this leads to air-flow disturbance, which increases the friction on the blade surfaces and reduces fan efficiency. When the distance between blades is too large, this leads to an increase of pressure loss and insufficient air pressure [18].
- The number of blades: This affects other specifications of fan blades, such as the sectional curve and pitch angle. The width of each blade usually depends on its height. To guarantee that blade spacing will not affect the air pressure, the approach of increasing the number of blades is usually adopted as a remedy in comparatively thinner fans.

## 3. Research Methods

#### 3.1. Gray Relational Theory

_{0}(k) is taken as the reference sequence, and X

_{i}(k), i ≠ 0 is the comparison sequence [8]:

_{i}(k) on X

_{0}(k) is defined as

_{i}on X

_{0}is

_{i}(k) on X

_{0}(k) is defined as

#### 3.2. Governing Equations

#### 3.3. Standard k−ε Turbulence Model

#### 3.4. Performance Testing Equipment for Wind Turbines

#### 3.5. Calculation of Flow Rates

${Q}_{5}$ | the total flow rate measured by a bank of nozzles, CMM |

$\Delta P$ | the pressure difference across the nozzles, mm-Aq |

${\rho}_{5}$ | the air density upstream of the nozzles, $kg/{m}^{3}$ |

$Y$ | expansion factor |

${C}_{n}$ | the discharge coefficient of the nth nozzle (Nozzle Discharge Coefficient |

${A}_{6n}$ | the cross-sectional area of the nth nozzle’s throat, ${m}^{2}$ |

#### 3.6. Method of Measurements

_{s}is the static pressure of the fan under test;

_{t}is the total pressure of the fan under test;

_{v}is the dynamic pressure of the fan under test;

_{7}. Therefore, ${P}_{{s}_{2}}={P}_{{s}_{7}}$.

_{2}is the outlet air density of the fan under test, kg/m

^{3};

_{2}is the outlet flow rate of the fan under test, CMM;

_{2}is the outlet cross-sectional area of the fan under test, m

^{2};

^{3}).

#### 3.7. Method of Measuring the Performance Curves of Fans

#### 3.8. Fan Performance Test Equipment

## 4. Case Study

#### 4.1. Analysis of the Correlation Degree of Gray Information

#### 4.2. Configuration of the Numerical Model

#### 4.3. Settings of Model Parameters

- The flow field is at a steady state and the fluid is non-compressible air.
- The turbulence model that is used in this case study is k–ε with an eddy correction.
- The influence of gravitation is neglected.
- Relevant fluid properties, including the viscosity coefficient, density, and specific heat, are constants.
- A rotation speed of 2000 RPM is set for the fluid in the rotating zone.
- The fluid velocity at the surface of a solid is zero, and this is the no-slip condition.
- The heat radiation term and the buoyancy term are neglected, while physical properties are independent of temperature. This is because when the temperature of fluid is different at different locations, the buoyancy force is generated due to the variation in its density. However, air is driven by fans under forced convection while the natural convective effect is much less effective; therefore, the buoyance term can be neglected. On the other hand, the heat convective term due to the fluid’s sensible heat and latent heat is much larger than the heat radiation term, and therefore the radiation term can be neglected.

#### 4.4. Simulation Results of Fans

_{1}~x

_{n}are determined by the following equation. By applying the weighted averages to the flow rate and the static pressure of each fan design, the resulting values of maximum flow rate and maximum static pressure are shown in Table 11.

#### 4.5. Comparison Between the Results of Simulation and Experiment

- Preparatory work for measurements
- Turn on the thermometer, hygrometer, barometer, fiber-optic tachometer, and inverter one hour before measurement. Make sure the equipment operates at a stable state. A testing workbench with a wind tunnel is shown in Figure 10a. The fan to be tested is mounted on the front plate of the main chamber. Care should be taken to ensure that the fan is sealed adequately to prevent leakage.
- Turn on the test fan and the auxiliary blower for several minutes until both of them run stably. Adjust the blast gate from fully open to fully closed and check the air flow through the chamber. Check the readings of each of the equipment.
- Measure the pressure difference between the free-flow condition (free deliver) and the no-flow condition (shut off). Divide the pressure difference into nine segments for determining the pressure increment and the data acquisition points.

- Measurement procedure
- Start the measurement from the free-flow condition with a static pressure of 0. Pay attention to the pressure difference between the nozzle array. The pressure difference needs to be in the range of 0.5–2.5 mm-Aq or the measured air-flow rate could be incorrect. In this case, it is required to select another nozzle from the nozzle array for a different range of air-flow rates.
- After the data under the free-flow condition are determined, use the blast gate and the inverter of the auxiliary blower to adjust the pressure to a desired range.
- Increase the pressure to the next range of these nine segments by swapping the nozzle and adjusting the blast gate and the inverter. Use the data-acquisition system to take data of system readings after it has been stabilized. Repeat this step for all of these nine segments.
- Pull out the data that are recorded in files and calculate the air-flow rate, air pressure, and efficiency of the computer program.
- Summarize the calculation results in the performance curve of the fan.

## 5. Results and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bredell, J.R.; KrÃÃger, D.G.; Thiart, G.D. Numerical investigation of fan performance in a forced draft air-cooled steam condenser. Appl. Therm. Eng.
**2006**, 26, 846–852. [Google Scholar] [CrossRef] [Green Version] - Hurault, J.; Kouidri, S.; Bakir, F.; Rey, R. Experimental and numerical study of the sweep effect on three-dimensional flow downstream of axial flow fans. Flow Meas. Instrum.
**2010**, 21, 155–165. [Google Scholar] [CrossRef] - Rhee, D.H.; Cho, H.H. Effect of vane/blade relative position on heat transfer characteristics in a stationary turbine blade: Part 2. Blade surface. Int. J. Therm. Sci.
**2008**, 47, 1544–1554. [Google Scholar] [CrossRef] - Lai, H.H.; Chen, C.H.; Chen, Y.C.; Yeh, J.W.; Lai, C.F. Product design evaluation model of child car seat using gray relational analysis. Adv. Eng. Inform.
**2009**, 23, 165–173. [Google Scholar] [CrossRef] - Wei, G.W. Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information. Expert Syst. Appl.
**2011**, 38, 4824–4828. [Google Scholar] [CrossRef] - Qiu, B.; Wang, F.; Li, Y.; Zuo, W. Research on Method of Simulation Model Validation Based on Improved Grey Relational Analysis. Phys. Procedia
**2012**, 25, 1118–1125. [Google Scholar] [CrossRef] [Green Version] - Li, Q.X. Grey dynamic input–output analysis. J. Math. Anal. Appl.
**2009**, 359, 514–526. [Google Scholar] [CrossRef] [Green Version] - Trivedi, H.V.; Singh, J.K. Application of Grey System Theory in the Development of a Runoff Prediction Model. Biosyst. Eng.
**2005**, 92, 521–526. [Google Scholar] [CrossRef] - Li, L.; Huang, G.; Chen, J. Aerodynamic characteristics of a tip-jet fan with a large blade pitch angle. Aerosp. Sci. Technol.
**2019**, 91, 49–58. [Google Scholar] [CrossRef] - Wang, Z.; Wu, Y.; Lu, S.; Meng, X.; Zhang, J. A study on model experiment and aerodynamic match of Wind Energy Fan. Sustain. Cities Soc.
**2019**, 49, 101618. [Google Scholar] [CrossRef] - Wu, Y.; Pan, D.; Peng, Z.; Hua, O. Blade force model for calculating the axial noise of fans with unevenly spaced blades. Appl. Acoust.
**2019**, 146, 429–436. [Google Scholar] [CrossRef] - Villiers, D.J.; Mathews, M.J.; Maré, P.; Kleingeld, M.; Arndt, D. Evaluating the impact of auxiliary fan practices on localised subsurface ventilation. Int. J. Min. Sci. Technol.
**2019**, 29, 933–941. [Google Scholar] [CrossRef] - Tournier, J.M.; El-Genk, M.S. Axial flow, multi-stage turbine and compressor models. Energy Convers. Manag.
**2009**, 51, 16–29. [Google Scholar] [CrossRef] - Qian, X.; Deba, D. Design of heterogeneous turbine blade. Comput. Aided Des.
**2003**, 35, 319–329. [Google Scholar] [CrossRef] - Lin, H.H. Application of Fuzzy Decision Model to the Design of a Pillbox for Medical Treatment of Chronic Diseases. Appl. Sci.
**2019**, 9, 4909. [Google Scholar] [CrossRef] [Green Version] - Lin, S.C.; Huang, C.L. An integrated experimental and numerical study of forward–curved centrifugal fan. Exp. Therm. Fluid Sci.
**2002**, 26, 421–434. [Google Scholar] [CrossRef] - Li, Y.L.; Liu, J.; Ou, H.; Du, Z.H. Internal flow mechanism and experimental research of low pressure axial fan with forward-skewed blades. J. Hydrodyn. Ser. B
**2008**, 20, 299–305. [Google Scholar] [CrossRef] - Niu, M.; Zang, S. Experimental and numerical investigations of tip injection on tip clearance flow in an axial turbine cascade. Exp. Therm. Fluid Sci.
**2011**, 35, 1214–1222. [Google Scholar] [CrossRef] - Yao, C. Application of Gray Relational Analysis Method in Comprehensive Evaluation on the Customer Satisfaction of Automobile 4S Enterprises. Phys. Procedia
**2012**, 33, 1184–1189. [Google Scholar] - FLUENT User’s Guide. Available online: https://www.ansys.com/products/fluids/ansys-fluent (accessed on 1 January 2019).
- Lin, H.H.; Huang, Y.Y. Application of ergonomics to the design of suction fans. In Proceedings of the 1st IEEE International Conference on Knowledge Innovation and Invention, Jeju Island, South Korea, 23–27 July 2018; pp. 203–206. [Google Scholar]
- Hsiao, S.W.; Lin, H.H.; Lo, C.H.; Ko, Y.C. Automobile shape formation and simulation by a computer-aided systematic method. Concurr. Eng. Res. Appl.
**2016**, 24, 290–301. [Google Scholar] [CrossRef] - Lin, H.H. Improvement of Human Thermal Comfort by Optimizing the Airflow Induced by a Ceiling Fan. Sustainability
**2019**, 11, 3370. [Google Scholar] [CrossRef] [Green Version] - Lin, H.H.; Hsiao, S.W. A Study of the Evaluation of Products by Industrial Design Students. Eurasia J. Math. Sci. Technol. Educ.
**2018**, 14, 239–254. [Google Scholar] [CrossRef] - Hsiao, S.W.; Lin, H.H.; Lo, C.H. A study of thermal comfort enhancement by the optimization of airflow induced by a ceiling fan. J. Interdiscip. Math.
**2016**, 19, 859–891. [Google Scholar] [CrossRef] - Lin, H.H.; Cheng, J.H. Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills. Symmetry
**2019**, 11, 959. [Google Scholar] [CrossRef] [Green Version]

Continuity | 1 |
---|---|

X-momentum | u |

Y-momentum | v |

Z-momentum | w |

${C}_{1\epsilon}$ | ${C}_{2\epsilon}$ | ${C}_{u}$ | ${C}_{k}$ | ${C}_{\epsilon}$ |

1.44 | 1.92 | 0.09 | 1.0 | 1.3 |

Fan Design No. | No. 1 | No. 2 | No. 3 |
---|---|---|---|

Rotation speed | 2000 | 2000 | 2000 |

Leading-edge radius | 61 | 63 | 65 |

Blade count | 7 | 9 | 11 |

Outside diameter of the fan | 70 | 72 | 74 |

Hub incidence angle | 1 | 3 | 6 |

Tip incidence angle | 0 | 3 | 6 |

Hub stagger angle | 1 | 2 | 3 |

Tip stagger angle | 0.75 | 0.85 | 0.95 |

No. 1 | No. 2 | No. 3 |
---|---|---|

7 blades | 9 blades | 11 blades |

Fan Design No. | 1 | 2 | 3 | |
---|---|---|---|---|

Rotation speed | X_{0} | 2000 | 2000 | 2000 |

Leading-edge radius | X_{1} | 61 | 63 | 65 |

Blade count | X_{2} | 7 | 9 | 11 |

Outside diameter of the fan | X_{3} | 70 | 72 | 74 |

Hub incidence angle | X_{4} | 1 | 3 | 6 |

Tip incidence angle | X_{5} | 1 | 3 | 6 |

Hub stagger angle | X_{6} | 1 | 2 | 3 |

Tip stagger angle | X_{7} | 0.75 | 0.85 | 0.95 |

**Table 6.**Initialization of design parameters for gray relational analysis (GRA) ${X}_{0}=\left[{x}_{0}\left(1\right),{x}_{0}\left(2\right),\cdots ,{x}_{i}\left(k\right)\right]$.

Fan Design No. | 1 | 2 | 3 | |
---|---|---|---|---|

Rotation speed | X_{0} | 0.3333 | 0.3333 | 0.3333 |

Leading-edge radius | X_{1} | 0.3228 | 0.3333 | 0.3439 |

Blade count | X_{2} | 0.2593 | 0.333 | 0.4074 |

Outside diameter of the fan | X_{3} | 0.3241 | 0.333 | 0.3426 |

Hub incidence angle | X_{4} | 0.1 | 0.3 | 0.6 |

Tip incidence angle | X_{5} | 0.1 | 0.3 | 0.6 |

Hub stagger angle | X_{6} | 0.1667 | 0.3333 | 0.5 |

Tip stagger angle | X_{7} | 0.2941 | 0.3333 | 0.3725 |

**Table 7.**Difference sequence ${\Delta}_{0,i}=\left|{X}_{0}\left(k\right)-{X}_{i}\left(k\right)\right|$.

Fan Design No. | 1 | 2 | 3 | |
---|---|---|---|---|

Leading-edge radius | X_{1} | 0.0106 | 0.0000 | 0.0106 |

Blade count | X_{2} | 0.0741 | 0.0000 | 0.0741 |

Outside diameter of the fan | X_{3} | 0.0093 | 0.0000 | 0.0093 |

Hub incidence angle | X_{4} | 0.2333 | 0.0333 | 0.2667 |

Tip incidence angle | X_{5} | 0.2333 | 0.0333 | 0.2667 |

Hub stagger angle | X_{6} | 0.1667 | 0.0000 | 0.1667 |

Tip stagger angle | X_{7} | 0.0392 | 0.0000 | 0.0392 |

Fan Design No. | 1 | 2 | 3 | |
---|---|---|---|---|

Leading-edge radius | X_{1} | 0.9265 | 1.0000 | 0.9265 |

Blade count | X_{2} | 0.6429 | 1.0000 | 0.6429 |

Outside diameter of the fan | X_{3} | 0.9351 | 1.0000 | 0.9351 |

Hub incidence angle | X_{4} | 0.3637 | 0.8000 | 0.3334 |

Tip incidence angle | X_{5} | 0.3637 | 0.8000 | 0.3334 |

Hub stagger angle | X_{6} | 0.4445 | 1.0000 | 0.4445 |

Tip stagger angle | X_{7} | 0.7727 | 1.0000 | 0.7727 |

Factor | Correlation Degree | |
---|---|---|

Leading-edge radius | X_{1} | 0.9510 |

Blade count | X_{2} | 0.7619 |

Outside diameter of the fan | X_{3} | 0.9567 |

Hub incidence angle | X_{4} | 0.4990 |

Tip incidence angle | X_{5} | 0.4990 |

Hub stagger angle | X_{6} | 0.6297 |

Tip stagger angle | X_{7} | 0.8485 |

Fan Design No. | 1 | 2 | 3 |
---|---|---|---|

Rotation speed when the maximum flow rate occurs (RPM) | 2000 | 2000 | 2000 |

Maximum static pressure Ps (mm-Aq) | 1.75 | 1.92 | 1.83 |

Maximum flow rate Q (CFM, Cubic feet per minute) | 37.3 | 40.4 | 38.2 |

Fan Design No. | 1 | 2 | 3 |
---|---|---|---|

Maximum static pressure Ps (mm-Aq) | 0.3182 | 0.3491 | 0.3327 |

Maximum flow rate Q (CFM) | 0.3218 | 0.3486 | 0.3296 |

Results from the Simulation | Results from the Experiment | |
---|---|---|

Rotation speed at the maximum flow rate (RPM) | 2000 | |

Maximum static pressure Ps (mm-Aq) | 1.92 | 1.75 |

Maximum flow rate Q (CFM) | 40.4 | 38.3 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lin, H.-H.; Cheng, J.-H.; Chen, C.-H.
Application of Gray Relational Analysis and Computational Fluid Dynamics to the Statistical Techniques of Product Designs. *Symmetry* **2020**, *12*, 227.
https://doi.org/10.3390/sym12020227

**AMA Style**

Lin H-H, Cheng J-H, Chen C-H.
Application of Gray Relational Analysis and Computational Fluid Dynamics to the Statistical Techniques of Product Designs. *Symmetry*. 2020; 12(2):227.
https://doi.org/10.3390/sym12020227

**Chicago/Turabian Style**

Lin, Hsin-Hung, Jui-Hung Cheng, and Chi-Hsiung Chen.
2020. "Application of Gray Relational Analysis and Computational Fluid Dynamics to the Statistical Techniques of Product Designs" *Symmetry* 12, no. 2: 227.
https://doi.org/10.3390/sym12020227