# Effect of a Vibrating Blade in a Channel on the Heat Transfer Performance

^{*}

## Abstract

**:**

^{−1}, the heat transfer can be improved by 16%. When the maximum amplitude of the blade is 8 mm and the inlet velocity is 0.8 m s

^{−1}, the heat transfer can be improved by 15%.

## 1. Introduction

## 2. Methodology

#### 2.1. Computational Domain

#### 2.2. Governing Equations and Boundary Conditions

_{p}, k, and T denote the specific heat, thermal conductivity, and temperature of the air, respectively.

_{k}and G

_{ω}denote the turbulent kinetic energy and the generic term of ω, respectively; Y

_{k}and Yω denote the turbulent kinetic energy dissipation term and the specific dissipation rate, respectively; Γ

_{k}and Γ

_{ω}denote the effective diffusivity of k

_{t}and ω, respectively; and D

_{ω}is the cross-diffusion term.

^{−1}, 0.8 m s

^{−1}, and 1.0 m s

^{−1}, respectively. The pressure outlet boundary condition is adopted for the channel outlet. The constant heat flux boundary condition is adopted for the heated surface, which is 1500 W m

^{−2}. For transient simulation of fan blade reciprocating vibrations, a larger time step results in worse mesh quality and reduces the accuracy of the simulation. In contrast, a smaller time step results in longer computational cycles [45]. After several tests, 1.0 × 10

^{−3}s was selected as a time step size in the present work. The SIMPLE algorithm is used to solve the coupling between pressure and velocity. The second-order upwind discretization scheme is adopted for all terms except diffusion terms.

#### 2.3. Data Reduction

_{inlet}is the inlet velocity of air, m s

^{−1}; D

_{h}is the equivalent diameter of the inlet, m; ρ is the density of air, kg m

^{−3}; and μ is the dynamic viscosity of air, kg m

^{−1}s

^{−1}.

^{−1}K

^{−1}; h is the convection heat transfer coefficient, W m

^{−2}K

^{−1}; q

_{h}is the heat flux of the heated surface, W m

^{−2}; and ΔT is the temperature difference between the heated surface and the air, K.

_{a}is defined as the ratio of the maximum amplitude of the blade A and the equivalent diameter of the inlet D

_{h}, i.e.,

_{h}is defined as:

_{1}… x

_{n−1}, x

_{n}. The phase space is reconstructed as follows:

_{j}is the point of phase space; Y

_{j}

_{0}is the nearest point of Y

_{j}; d

_{j}(0) is the distance between Y

_{j}and Y

_{j}

_{0}; q is the number of non-zero d

_{j}(i); and y(i) is the average value of ln[d

_{j}(i)]. The slope of the regression line made by the least square method is the maximum Lyapunov exponent [46,47].

#### 2.4. Grid Independence Validation and Model Validation

## 3. Analysis of Heat Transfer Characteristics

## 4. Analysis of Chaotic Characteristics

#### 4.1. Poincaré Map

_{inlet}= 0.5 m s

^{−1}to create a Poincaré map, as shown in Figure 11. It can be found that the particles are concentrated in the middle region without diffusion for the channel without a blade. The chaos is found when f = 5 Hz and A = 5 mm. When f = 10 Hz and A = 5 mm, strong chaos is found. This reveals that the vibrating blade significantly generates chaos.

#### 4.2. Power Spectral Density

#### 4.3. Phase Space Reconfiguration

#### 4.4. Maximum Lyapunov Exponent

_{inlet}= 0.8 m s

^{−1}, f = 7 Hz, and A = 5 mm. The maximum Lyapunov exponent is 1.1269 for v

_{inlet}= 1.0 m s

^{−1}, f = 7 Hz, and A = 5 mm. Obviously, the degree of chaos is greater when the inlet velocity is 0.8 m s

^{−1}.

## 5. Conclusions

- A larger frequency or amplitude is beneficial to improve heat transfer at the same inlet velocity. When the frequency is 10 Hz, the heat transfer can be increased by 16%. When the maximum amplitude of the blade is 8 mm, the heat transfer can be increased by 15%.
- The vibrating blade forms the longitudinal vortices. Hence, the heat transfer is enhanced.
- More than four incommensurable frequencies are in the power spectrum, indicating that the system has reached a chaotic state. The system reaches a chaotic state when the vibrating frequency is over 5 Hz.
- As the amplitude increases, the system gradually changes from a steady state to a weakly chaotic one. The amplitude increases further to a periodic state and finally to a chaotic state. The degree of chaos becomes more intense when the amplitude is 8 mm.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Nomenclature | |

A | amplitude, mm |

c_{p} | specific heat of air, J kg^{−1} K^{−1} |

Da | dimensionless amplitude |

D_{h} | the equivalent diameter, mm |

f | frequency, Hz |

k | thermal conductivity, W m^{−2} K^{−1} |

l | length of the blade, mm |

m | mass, kg |

h | convective heat transfer coefficient, W m^{−2} K^{−1} |

h_{av} | average convective heat transfer coefficient, W m^{−2} K^{−1} |

p | pressure, Pa |

q | the heat flow density |

T | temperature, K |

v_{in} | inlet velocity, m s ^{−1} |

u | velocity in x-direction, m s ^{−1} |

V | volume |

v | velocity in the y-direction, m s ^{−1} |

w | velocity in the z-direction, m s ^{−1} |

Greek symbols | |

Γ | diffusion coefficient |

λ | thermal conductivity, Wm^{−1} K^{−1} |

ρ | fluid density, kg m^{−3} |

τ | delay time, s |

μ | the dynamic viscosity, kg m^{−1} s^{−1} |

Dimensionless groups | |

Nu | Nusselt number |

Pr | Prandtl number |

Re | Reynolds number |

MLE | Maximum Lyapunov exponent |

Subscript | |

a | air |

av | average value |

blade | vibrating blade |

max | maximum value |

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**Figure 4.**Model validation. Numerical results compared with the experimental results of research [48].

**Figure 5.**The effects of the natural frequency and the amplitude of the blade on the heat transfer performance: (

**a**) frequency at A = 5 mm; (

**b**) amplitude at f = 5 Hz.

**Figure 7.**Temperature contour of the heated surface: (

**a**) without blade; (

**b**) Phase 0π; (

**c**) Phase π/2.

**Figure 8.**Velocity vector at different phases of z = 0 mm: (

**a**) Phase 0π; (

**b**) Phase π/2; (

**c**) Phase π; (

**d**) Phase 3π/2; (

**e**) Phase 2π.

**Figure 12.**The partial velocity w against time: (

**a**) f = 1 Hz; (

**b**) f = 3 Hz; (

**c**) f = 5 Hz; (

**d**) f = 10 Hz.

**Figure 13.**Variation of power spectral density: (

**a**) f = 1 Hz; (

**b**) f = 3 Hz; (

**c**) f = 5 Hz; (

**d**) f = 10 Hz.

**Figure 15.**Phase space trajectories for A = 4 mm: (

**a**) point 1; (

**b**) point 2; (

**c**) point 3; (

**d**) point 4.

**Figure 16.**Phase space trajectories for A = 8 mm: (

**a**) point 1; (

**b**) point 2; (

**c**) point 3; (

**d**) point 4.

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## Share and Cite

**MDPI and ACS Style**

Yuan, X.; Lan, C.; Hu, J.; Fan, Y.; Min, C. Effect of a Vibrating Blade in a Channel on the Heat Transfer Performance. *Energies* **2023**, *16*, 3076.
https://doi.org/10.3390/en16073076

**AMA Style**

Yuan X, Lan C, Hu J, Fan Y, Min C. Effect of a Vibrating Blade in a Channel on the Heat Transfer Performance. *Energies*. 2023; 16(7):3076.
https://doi.org/10.3390/en16073076

**Chicago/Turabian Style**

Yuan, Xinrui, Chenyang Lan, Jinqi Hu, Yuanhong Fan, and Chunhua Min. 2023. "Effect of a Vibrating Blade in a Channel on the Heat Transfer Performance" *Energies* 16, no. 7: 3076.
https://doi.org/10.3390/en16073076