# Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills

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## Abstract

**:**

## 1. Introduction

## 2. Research Model

#### Configuration of Model Parameters

## 3. Research Methodology

#### 3.1. Numerical Analysis

#### 3.1.1. Governing Equations

#### 3.1.2. Theory of Turbulence Model

#### 3.1.3. Standard k–ε Turbulence Model

_{k}indicates the turbulence kinetic energy generated by the velocity gradient of laminar flow. G

_{b}is the turbulence kinetic energy generated by the buoyancy. In compressible turbulent flows, Y

_{M}is the fluctuation generated by the excessive diffusion. σ

_{k}and σ

_{ε}are the turbulent Prandtl numbers of turbulence kinetic energy and turbulent dissipation; and C

_{1ε}, C

_{2ε}, and C

_{3ε}are the empirical constants. The recommended values of these coefficients are shown in Table 3 [24].

#### 3.1.4. RNG k−ε Turbulence Model

#### 3.2. Performance Testing Equipment for Fans

#### 3.2.1. Calculation of Flow Rates

_{5}and PL

_{6}can be obtained. The flow rates on the cross-sections of the nozzles can be determined with varying nozzle coefficients as shown in Figure 5. If there is a need to calculate the outlet flow rate of the fan under test, then the effect of density variations must be considered, the measurement of which is as follows [28].

^{3}; is the expansion factor; ${C}_{n}$ is the discharge coefficient of the nth nozzle; and ${A}_{6n}$ is the cross-sectional area of the nth nozzle’s throat, m

^{2}.

#### 3.2.2. Calculation of Air Pressures

_{s}) and total pressure (ΔP

_{t}). The static pressure, defined as the difference between the fan’s static pressure at typical pressure readings, can be directly measured by instruments, but understanding (P

_{s2}) and the static pressure at inlet (P

_{s1}) is required. The total pressure is the difference between the fan’s total pressure at outlet (P

_{t2}) and the total pressure at inlet (P

_{t1}). The equations for measurement and calculation are explained respectively as follows.

_{2}and PL

_{1}, respectively, they can be defined as follows [29]:

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

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

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

_{t2}is the total pressure at the fan’s outlet (or plane $P{L}_{2}$); and P

_{t1}is the total pressure at the fan’s inlet (or plane $P{L}_{1}$).

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

_{t7}. The Type A method (a test method with no duct at either the outlet or the inlet) that was carried out at the outlet test box is a special case of testing. When carrying out different types of tests or different equipment, the equation of the static pressure of the fan under test is thus more complicated. The calculation of dynamic pressure is as follows [30]:

_{ν2}is the outlet dynamic pressure of the fan under test, mm-Aq; V

_{2}is the outlet air velocity of the fan under test, m/s; ρ

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

^{3}; and ${V}_{2}=\frac{{Q}_{2}}{60{A}_{2}}=\frac{Q}{60{A}_{2}}\cdot \frac{\rho}{{\rho}_{2}}$$=\frac{Q}{50{\rho}_{2}{A}_{2}}$ where Q

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

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

^{2}; ρ is the density of air at STP (1.2 kg/m

^{3}); and ${P}_{t}={P}_{s}+{P}_{v}={P}_{s}+{P}_{{v}_{2}}$.

#### 3.2.3. Fan Performance Power and Efficiency

## 4. Numerical Simulation

^{−3}for convergence.

## 5. Verification of the Case Study and Numerical Analysis

#### 5.1. Verification between Numerical Simulation and Experiment Testing

#### 5.2. Design Cases and Comparison between Simulation Results

_{2}O. The result static pressure of Idea-A was 1.74 mm-H

_{2}O. From the aspect of overall air flow rate, the simulation results indicated that Idea-D had the largest air flow rate.

## 6. Conclusions

- The distribution at the inlet can be smoother.
- After passing through the inlet, the air pressure will increase at some portions about 1/2 of the impeller height.
- The maximum velocity at each cross-sectional plane occurs closer to the outlet.
- The change in the outlet location makes the air velocity increase and move toward the outlet direction.
- At the outlet plane of Idea-B, C, and E, many regions were found to have a low air velocity and recirculation. This phenomenon indicates inferior outlet conditions.
- Idea-D, with the honeycomb shape, had the most uniform air velocity among the six opening patterns. From the aspect of the leaving flow rate, it was also the most optimal opening pattern.
- If a fan is not improved by the airfoil design, the assessment can be carried out on the design factors of the outlet pattern.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Configuration of the fan performance testing system and specifications of the major instruments.

**Figure 9.**Resulting plots of CFD simulation for the numerical fan model. (

**a**) Pressure distribution on top and bottom covers; (

**b**) pressure distribution on the fan; (

**c**) distribution of stream lines; (

**d**) distribution for flow field; and (

**e**) program convergence plot.

**Figure 10.**Non-dimensional velocity V/Uref at nine planes along the vertical centerline of the model for the numerical simulation.

**Figure 11.**Non-dimensional velocity V/Uref at nine planes along the vertical centerline of the model for the numerical simulation for these six opening patterns.

Airfoil Type | Airfoil Name | Blade No. | Radius of Hub | Tip of Blade | Radius of Shroud |
---|---|---|---|---|---|

NACA65 | NACA65-Parabolic | 7 | 8 | 36 | 40 |

Thickness of hub | Section No. | Tip clearance | Incidence angle of the blade at the hub | Incidence angle of the blade at the tip | Blade width at the hub |

23.5 | 31 | 0.75 | 50 | 35 | 11 |

Equation | ψ |
---|---|

Continuity | 1 |

X-momentum | u |

Y-momentum | v |

Z-momentum | w |

Energy | I or T |

C_{1ε} | C_{2ε} | C_{μ} | C_{k} | C_{3ε} |
---|---|---|---|---|

1.44 | 1.92 | 0.09 | 1.0 | 1.3 |

Inlet boundary condition [9,32] | The inlet condition is for the initial calculation, this research simulates the fan in an infinite domain condition, therefore, at the inlet, it selects and adopts normal atmospheric pressure P0. |

Outlet boundary condition [33] | The flow generated by the rotation of the fan is the simulated flow toward the ambient atmosphere. Therefore, the outlet boundary condition of the normal atmospheric pressure P0 is also adopted. |

Wall boundary condition | Except for the non-permeable condition to be satisfied when a fluid flows through a wall, it also needs to satisfy the no-slip condition), i.e., u = v = w = 0. k and ε are determined by the near-wall model. |

Assumption that is made to reduce the complexity of flow field calculation [34] | The flow field is at the steady state and the fluid is incompressible air. |

The turbulence model is the standard k–ε model with eddy rectification. | |

The influence of gravity is neglected. | |

Related fluid properties such as viscosity, density, and specific heat are all constants. | |

A rotation speed of 2000 RPM is set for the MRF fluid rotating region. | |

The relative velocity between the solid surface and the fluid is zero, which is the no-slip condition. | |

The influence of radiation and buoyancy is neglected. Moreover, physical properties do not vary with temperature. | |

Rotating speed of the fan | Configured to be 2000 RPM. |

Numerical Simulation | Experimental Result | Deviation | |
---|---|---|---|

Air flow rate | 16.8CFM | 16.3CFM | 3% |

Static pressure | 1.75 mm-H_{2}O | 1.71 mm-H_{2}O | 2% |

Idea-A (a) | Idea-B (b) | ||

1(a) Pressure distribution on the fan | 1(b) Pressure distribution on the outer cover | 2(a) Pressure distribution on the fan | 2(b) Pressure distribution on the outer cover |

1(c) Velocity distribution | 1(d) Streamline distribution | 2(c) Velocity distribution | 2(d) Streamline distribution |

Idea-C (c) | Idea-D (d) | ||

3(a) Pressure distribution on the fan | 3(b) Pressure distribution on the outer cover | 4(a) Pressure distribution on the fan | 4(a) Pressure distribution on the fan |

3(c) Velocity distribution | 3(d) Streamline distribution | 4(c) Velocity distribution | 4(c) Flow field distribution |

Idea-E (e) | Idea-F (f) | ||

5(a) Pressure distribution on the fan | 5(b) Pressure distribution on the outer cover | 6(a) Pressure distribution on the fan | 6(b) Pressure distribution on the outer cover |

5(c) Velocity distribution | 5(d) Streamline distribution | 6(c) Velocity distribution | 6(d) Streamline distribution |

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**MDPI and ACS Style**

Lin, H.-H.; Cheng, J.-H.
Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills. *Symmetry* **2019**, *11*, 959.
https://doi.org/10.3390/sym11080959

**AMA Style**

Lin H-H, Cheng J-H.
Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills. *Symmetry*. 2019; 11(8):959.
https://doi.org/10.3390/sym11080959

**Chicago/Turabian Style**

Lin, Hsin-Hung, and Jui-Hung Cheng.
2019. "Application of the Symmetric Model to the Design Optimization of Fan Outlet Grills" *Symmetry* 11, no. 8: 959.
https://doi.org/10.3390/sym11080959