# Analyzing the Energy Efficiency of Fan Systems by Using the Dimensional Analysis (DA) Method

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## Abstract

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## 1. Introduction

## 2. Literature Review

^{3}/s]; $\mathrm{p}$ is the total pressure of the fan, measured in [Pa]; η is the total coefficient of efficiency of the fan system.

## 3. Research Methodology

- fan pressure—$\mathrm{p},\text{\hspace{0.17em}\hspace{0.17em}}[\mathrm{P}\mathrm{a}]$;
- fan flow rate—$\mathrm{Q},\text{\hspace{0.17em}\hspace{0.17em}}[{\mathrm{m}}^{3}/\mathrm{s}]$;
- speed—$\mathrm{n},\text{\hspace{0.17em}\hspace{0.17em}}[{\mathrm{s}}^{-1}]$;
- fan system coefficient of efficiency—$\mathsf{\eta}$.

- fluid density—$\mathsf{\rho},\text{\hspace{0.17em}}[\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}]$;
- gas viscosity—$\mathsf{\mu},\text{\hspace{0.17em}}[\mathrm{P}\mathrm{a}.\mathrm{s}]$.

- $\mathrm{D},\text{\hspace{0.17em}\hspace{0.17em}}[\mathrm{m}]$—pipe conventional diameter (in case of a pipe with not-round cross-section area, a hydraulic diameter is used). At first admission for the calculations, it can be used the size of the fan suction inlet or discharge outlet.
- ${\mathrm{L}}_{\mathrm{T}\mathrm{P}},\text{\hspace{0.17em}\hspace{0.17em}}[\mathrm{m}]$—referred length of the pipe system (network), including its geometric sizes, as well as the hydraulic resistance in the pipes.

- density: ρ = [M
^{1}L^{3}T^{0}]; - speed: n = [M
^{0}L^{0}T^{−1}]; - equivalent pipe diameter: D = [M
^{0}L^{1}T^{0}].

_{j}, b

_{j}, and c

_{j}(j = 1…5), the following system of equations is used:

M | L | T | |

$\mathrm{p}$ | $1={\mathrm{c}}_{1}$ | $-1={\mathrm{a}}_{1}-3\text{\hspace{0.17em}}{\mathrm{c}}_{1}$ | $-2=-{\mathrm{b}}_{1}$ |

$\mathrm{Q}$ | $0={\mathrm{c}}_{2}$ | $3={\mathrm{a}}_{2}-3\text{\hspace{0.17em}}{\mathrm{c}}_{2}$ | $-1=-{\mathrm{b}}_{2}$ |

$\mathsf{\mu}$ | $1={\mathrm{c}}_{3}$ | $-1={\mathrm{a}}_{3}-3\text{\hspace{0.17em}}{\mathrm{c}}_{3}$ | $-1=-{\mathrm{b}}_{3}$ |

${\mathrm{L}}_{\mathrm{T}\mathrm{P}}$ | $0={\mathrm{c}}_{4}$ | $1={\mathrm{a}}_{4}-3\text{\hspace{0.17em}}{\mathrm{c}}_{4}$ | $0=-{\mathrm{b}}_{4}$ |

$\mathsf{\eta}$ | $0={\mathrm{c}}_{5}$ | $0={\mathrm{a}}_{5}-3\text{\hspace{0.17em}}{\mathrm{c}}_{5}$ | $0=-{\mathrm{b}}_{5}$ |

$\mathrm{p}$ | ${\mathrm{a}}_{1}=2$ | ${\mathrm{b}}_{1}=2$ | ${\mathrm{c}}_{1}=1$ |

$\mathrm{Q}$ | ${\mathrm{a}}_{2}=3$ | ${\mathrm{b}}_{2}=1$ | ${\mathrm{c}}_{2}=0$ |

$\mathsf{\mu}$ | ${\mathrm{a}}_{3}=2$ | ${\mathrm{b}}_{3}=1$ | ${\mathrm{c}}_{3}=1$ |

${\mathrm{L}}_{\mathrm{T}\mathrm{P}}$ | ${\mathrm{a}}_{4}=1$ | ${\mathrm{b}}_{4}=0$ | ${\mathrm{c}}_{4}=0$ |

$\mathsf{\eta}$ | ${\mathrm{a}}_{5}=0$ | ${\mathrm{b}}_{5}=0$ | ${\mathrm{c}}_{5}=0$ |

$\left[\mathrm{p}\right]={\left[\mathrm{D}\right]}^{2}{\left[\mathrm{n}\right]}^{2}{\left[\mathsf{\rho}\right]}^{1}$ | or given as dimensionless parameters | ${\mathsf{\pi}}_{1}^{\prime}=\frac{\mathrm{p}}{\mathsf{\rho}\text{\hspace{0.17em}}{\mathrm{n}}^{2}\text{\hspace{0.17em}}{\mathrm{D}}^{2}}$ |

$\left[\mathrm{Q}\right]={\left[\mathrm{D}\right]}^{3}{\left[\mathrm{n}\right]}^{1}{\left[\mathsf{\rho}\right]}^{0}$ | ${\mathsf{\pi}}_{2}^{\prime}=\frac{\mathrm{Q}}{\mathrm{n}\text{\hspace{0.17em}}{\mathrm{D}}^{3}}$ | |

$\left[\mathsf{\mu}\right]={\left[\mathrm{D}\right]}^{2}{\left[\mathrm{n}\right]}^{1}{\left[\mathsf{\rho}\right]}^{1}$ | ${\mathsf{\pi}}_{3}^{\prime}=\frac{\mathsf{\rho}\text{\hspace{0.17em}}\mathrm{n}\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{D}}^{2}}{\mathsf{\mu}}$ | |

$\left[{\mathrm{L}}_{\mathrm{T}\mathrm{P}}\right]={\left[\mathrm{D}\right]}^{1}{\left[\mathrm{n}\right]}^{0}{\left[\mathsf{\rho}\right]}^{0}$ | ${\mathsf{\pi}}_{4}^{\prime}=\frac{{\mathrm{L}}_{\mathrm{T}\mathrm{P}}}{\mathrm{D}}$ | |

$\left[\mathsf{\eta}\right]={\left[\mathrm{D}\right]}^{0}{\left[\mathrm{n}\right]}^{0}{\left[\mathsf{\rho}\right]}^{0}$ | ${\mathsf{\pi}}_{5}^{\prime}=\mathsf{\eta}$ |

^{3}air, and the main characteristics of a given system, such as: the system main geometric parameter—the equivalent diameter of the pipe system and the properties of the transported gas (air).

## 4. Results

_{a}= 101,325 Pa and relative humidity hu = 0.4) respectively are: density ${\mathsf{\rho}}_{\mathrm{o}}=1.205\text{\hspace{0.17em}}\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\text{\hspace{0.17em}}$ and dynamic viscosity—${\mathsf{\mu}}_{\mathrm{o}}=15.11\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{10}^{-6}\text{\hspace{0.17em}}\mathrm{P}\mathrm{a}\text{\hspace{0.17em}}\mathrm{s}$.

#### 4.1. Throttle Regulation of the Flow Rate

#### 4.2. Frequency Regulation of the Flow Rate

_{TP}is constant. The fan system’s total coefficient of efficiency is also a constant, i.e., η is a constant. These two parameters are determined by using Equations (19) and (28), involving the obtained before regulation system output flow rate.

#### 4.3. Determining the Impact of the Air Temperature on the Energy Consumption of Fan Systems

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Change of the specific energy consumption criterion depending on the selected method of flow rate regulation (Source: Author own calculation, based on Excel).

**Figure 2.**Change of the specific energy consumption criteria depending on the temperature variation (Source: author’s own calculation, based on Excel).

Parameter | Equation |
---|---|

$\mathrm{p}$ | $\left[{\mathrm{M}}^{1}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\right]$ |

$\mathrm{Q}$ | $\left[{\mathrm{M}}^{0}{\mathrm{L}}^{3}{\mathrm{T}}^{-1}\right]$ |

$\mathrm{n}$ | $\left[{\mathrm{M}}^{0}{\mathrm{L}}^{0}{\mathrm{T}}^{-1}\right]$ |

$\mathsf{\eta}$ | $\text{\hspace{0.17em}}\left[{\mathrm{M}}^{0}{\mathrm{L}}^{0}{\mathrm{T}}^{0}\right]$ |

$\mathsf{\rho}$ | $\left[{\mathrm{M}}^{1}{\mathrm{L}}^{-3}{\mathrm{T}}^{0}\right]$ |

$\mathsf{\mu}$ | $\left[{\mathrm{M}}^{1}{\mathrm{L}}^{-1}{\mathrm{T}}^{-1}\right]$ |

${\mathrm{L}}_{\mathrm{TP}}$ | $\left[{\mathrm{M}}^{0}{\mathrm{L}}^{1}{\mathrm{T}}^{0}\right]$ |

$\mathrm{D}$ | $\left[{\mathrm{M}}^{0}{\mathrm{L}}^{1}{\mathrm{T}}^{0}\right]$ |

$\mathbf{p}=\mathbf{f}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}$ | $\mathbf{\eta}=\mathbf{f}\mathbf{\left(}\mathbf{Q}\mathbf{\right)}$ | ||||

${\mathrm{a}}_{\mathrm{p}}$ | ${\mathrm{b}}_{\mathrm{p}}$ | ${\mathrm{c}}_{\mathrm{p}}$ | ${\mathrm{a}}_{\mathrm{e}}$ | ${\mathrm{b}}_{\mathrm{e}}$ | ${\mathrm{c}}_{\mathrm{e}}$ |

2765 | −10,492 | −58,578 | 0.1311 | 0.1496 | −0.00878 |

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**MDPI and ACS Style**

Popov, G.; Klimentov, K.; Kostov, B.; Dimitrova, R.
Analyzing the Energy Efficiency of Fan Systems by Using the Dimensional Analysis (DA) Method. *Symmetry* **2020**, *12*, 537.
https://doi.org/10.3390/sym12040537

**AMA Style**

Popov G, Klimentov K, Kostov B, Dimitrova R.
Analyzing the Energy Efficiency of Fan Systems by Using the Dimensional Analysis (DA) Method. *Symmetry*. 2020; 12(4):537.
https://doi.org/10.3390/sym12040537

**Chicago/Turabian Style**

Popov, Gencho, Kliment Klimentov, Boris Kostov, and Reneta Dimitrova.
2020. "Analyzing the Energy Efficiency of Fan Systems by Using the Dimensional Analysis (DA) Method" *Symmetry* 12, no. 4: 537.
https://doi.org/10.3390/sym12040537