# Multi-Factor Design for a Vacuum Ejector Improvement by In-Depth Analysis of Construction Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Numerical Method

#### 2.1.1. The Solver

#### 2.1.2. The Mesh

#### 2.1.3. Boundary Conditions

#### 2.1.4. Simulation Performance

#### 2.2. Factor Evaluation Design of the Numerical Simulations

#### 2.2.1. Criterion for Improvement

#### 2.2.2. Single-Factor Evaluation Method

#### 2.2.3. Multi-Factor Design

## 3. Results

#### 3.1. Influence of Parameters on Overall Improvement

#### 3.1.1. Analysis of the Eight Relevant Initial Parameters

#### 3.1.2. Single-Factor Analysis for the Selected Parameters

#### 3.2. Multi-Factor Design Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A | Area |

α1 | Convergent angle of the first nozzle (°) |

α2 | Divergent angle of the first nozzle (°) |

α3 | Convergent angle of the second nozzle (°) |

α4 | Divergent angle of the second nozzle (°) |

${c}_{v}$ | Specific heat at constant volume ($\mathrm{J}/\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) |

${C}_{p}$ | Specific heat capacity at constant pressure ($\mathrm{J}/\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) |

$F{R}_{E}$ | Secondary flow rate enhancement (%) |

l1 | Horizontal distance for the convergent section of the first nozzle ($\mathrm{mm}$) |

l2 | Horizontal distance for the throat section of the first nozzle ($\mathrm{mm}$) |

l3 | Horizontal distance for the divergent section of the first nozzle ($\mathrm{mm}$) |

l4 | Horizontal distance for the mixing chamber section of the second nozzle ($\mathrm{mm}$) |

l5 | Horizontal distance for the constant area section of the second nozzle ($\mathrm{mm}$) |

l6 | Horizontal distance for the diffuser section of the second nozzle ($\mathrm{mm}$) |

${\dot{m}}_{p}$ | Primary flow rate ($\mathrm{kg}/\mathrm{s}$) |

${\dot{m}}_{s}$ | Secondary flow rate ($\mathrm{kg}/\mathrm{s}$) |

M | Molecular mass ($\mathrm{g}/\mathrm{mol}$) |

μ | Normalized entrained flow ratio (-) |

${\mu}_{0}$ | Dynamic viscosity ($\mathrm{kg}/\mathrm{s}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$) |

$NXP$ | Nozzle exit position ($\mathrm{mm}$) |

${O}_{E}$ | Overall enhancement (%) |

${p}_{a}$ | Atmospheric pressure ($\mathrm{Pa}$) |

${p}_{s}^{*}$ | Normalized secondary pressure (-) |

${p}_{s}$ | Pressure in secondary inlet ($\mathrm{Pa}$) |

r1 | Radius for the throat of the first nozzle ($\mathrm{mm}$) |

r2 | Radius for the throat of the second nozzle ($\mathrm{mm}$) |

$\rho $ | Density (${\mathrm{kg}/\mathrm{m}}^{3}$) |

${R}^{\prime}$ | Constant for gases ($\mathrm{J}/\mathrm{mol}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) |

RV | Reference value |

$S{P}_{E}$ | Secondary pressure enhancement (%) |

T | Temperature ($\mathrm{K}$) |

$\left|\mathbf{u}\right|$ | Magnitude of velocity |

${u}_{i,j}$ | Velocity components in directions i and j ($\mathrm{m}/\mathrm{s}$) |

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**Figure 1.**Diagram of ejector geometry with internal geometry showing the two most important characteristics. The flow direction is from the left (main inlet) to the right (outlet).

**Figure 4.**Single influence per each relevant parameter on the flow rate, pressure and overall enhancement of the ejector.

**Figure 11.**Geometry with the reference values. (

**a**) Results for the maximum entrained flow simulation. (

**b**) Results for the minimum secondary pressure simulation.

**Figure 12.**Geometry with the parameter combinations that enhance the performance. (

**a**) Results for the maximum entrained flow simulation. (

**b**) Results for the minimum secondary pressure simulation.

Property | Value |
---|---|

Dynamic viscosity, ${\mu}_{0}$ | 1.8 ($\mathrm{kg}/\mathrm{s}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$) |

Specific heat capacity, ${C}_{p}$ | 1004.5 ($\mathrm{J}/\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$) |

Molecular mass, M | 28.96 ($\mathrm{g}/\mathrm{mol}$) |

Prandtl number, ${P}_{r}$ | 0.7 (-) |

Simulation Type | Entrained Flow | Secondary Pressure | ||
---|---|---|---|---|

Patch Name | p Patch Type | Pressure (MPa) | p Patch Type | Pressure (MPa) |

Main Inlet | uniformTotalPressure | 0.7 | uniformTotalPressure | 0.7 |

Secondary Inlet | totalPressure | 0.1 | zeroGradient | – |

Outlet | characteristicFarfieldPressure | 0.1 | characteristicFarfieldPressure | 0.1 |

Mesh | Number of Cells | ${\dot{\mathit{m}}}_{\mathit{p}}$ (kg/s) | ${\dot{\mathit{m}}}_{\mathit{s}}$ (kg/s) | ${\mathit{p}}_{\mathit{s},\mathbf{min}}$ (kPa) |
---|---|---|---|---|

Coarse mesh | 13,000 | 0.00945 | 0.00932 | 21.57 |

Study mesh | 20,300 | 0.00946 | 0.00938 | 21.73 |

Fine mesh | 29,250 | 0.00948 | 0.00936 | 21.90 |

Parameter | Original Shape (%) | ${\mathit{FR}}_{\mathit{E}}$ (%) | ${\mathit{SP}}_{\mathit{E}}$ (%) | ${\mathit{O}}_{\mathit{E}}$ (%) |
---|---|---|---|---|

$\alpha 3$ | −33 | 2.7 | 0.8 | 3.5 |

$NXP$ | 100 | 1 | 3.9 | 4.9 |

$l4$ | −100 | 0.6 | 9.4 | 10 |

$l5$ | −33 | 1.9 | 0.5 | 2.4 |

**Table 5.**Level A shows the reference value (RV), the start point and level B shows the selected values for each parameter.

Parameter | Level A ^{1} | Level B |
---|---|---|

$\alpha 3$ | RV | 2RV/3 |

$NXP$ | RV | 2RV |

$l4$ | RV | 0 |

$l5$ | RV | 1RV/3 |

^{1}Exact values deleted for confidentiality purposes.

**Table 6.**Full factorial combination design. Both levels, A and B, are shown in Table 5.

Run # | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$NXP$ | A | A | A | A | A | A | A | A | B | B | B | B | B | B | B | B |

$l4$ | A | A | A | A | B | B | B | B | A | A | A | A | B | B | B | B |

$l5$ | A | A | B | B | A | A | B | B | A | A | B | B | A | A | B | B |

$\alpha 3$ | A | B | A | B | A | B | A | B | A | B | A | B | A | B | A | B |

**Table 7.**Fractional factorial combination design. Both levels, A and B, are shown in Table 5.

Run # | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

$NXP$ | A | A | A | A | B | B | B | B | B |

$l4$ | A | A | B | B | A | A | B | B | B |

$l5$ | A | A | B | B | B | B | A | A | B |

$\alpha 3$ | A | B | A | B | A | B | A | B | B |

Enhancement | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

$F{R}_{E}$ (%) | 2.7 | −0.8 | 0.4 | 1.2 | 1.2 | −1.8 | −1.8 | −1.8 |

$S{P}_{E}$ (%) | 0.8 | 10.4 | 10 | 3.9 | 3.9 | 10.7 | 10.7 | 9.2 |

${O}_{E}$ (%) | 3.5 | 9.6 | 10.4 | 5.2 | 5.2 | 8.9 | 8.9 | 7.4 |

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**MDPI and ACS Style**

Macia, L.; Castilla, R.; Gamez-Montero, P.J.; Raush, G.
Multi-Factor Design for a Vacuum Ejector Improvement by In-Depth Analysis of Construction Parameters. *Sustainability* **2022**, *14*, 10195.
https://doi.org/10.3390/su141610195

**AMA Style**

Macia L, Castilla R, Gamez-Montero PJ, Raush G.
Multi-Factor Design for a Vacuum Ejector Improvement by In-Depth Analysis of Construction Parameters. *Sustainability*. 2022; 14(16):10195.
https://doi.org/10.3390/su141610195

**Chicago/Turabian Style**

Macia, Llorenç, Robert Castilla, Pedro Javier Gamez-Montero, and Gustavo Raush.
2022. "Multi-Factor Design for a Vacuum Ejector Improvement by In-Depth Analysis of Construction Parameters" *Sustainability* 14, no. 16: 10195.
https://doi.org/10.3390/su141610195