# Numerical Investigation of Pressure Loss in a Rectangular Channel with a Sharp 180-Degree Turn: Influence of Design Variables and Geometric Shapes

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## Abstract

**:**

## 1. Introduction

## 2. Geometrical Details and Mathematical Method

#### 2.1. Model Description

#### 2.2. Numerical Method

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#### 2.3. Grid Sensitivity

#### 2.4. Validation of Simulation

## 3. Parameter and Optimization Study

## 4. Results and Discussion

#### 4.1. Basic Study Results

#### 4.2. Results of Parameter and Optimization Studies

#### 4.2.1. Response Surface

#### 4.2.2. Regression Analysis

#### 4.2.3. Adjoint Method

## 5. Conclusions

- The contribution of the channel geometry design variables to the TPLC at four Reynolds numbers was investigated by the response surface and regression analysis. The influence of the divider tip-to-wall clearance on TPLC prediction was the highest, and this influence increased with an increasing Reynolds number. Two candidate points satisfying the objective function of minimizing the total pressure drop were obtained from the response surface for each Reynolds number, with a maximum improvement in the TPLC of 20.87% at the best candidate point compared to the original model.
- Regression analysis was used to derive predictive models consisting of design variables and Reynolds numbers based on the data set. The prediction based on the proposed model was improved using the interaction terms. The maximum difference between the predictive model with interaction terms and the CFD was 2.29%.
- Using the adjoint solver, the TPLC was improved by approximately 26% and 11% when compared to the original model at the representative Reynolds numbers of 24,130 and 77,901 respectively; it was more effective at lower Reynolds numbers than higher Reynolds numbers. The adjoint solver showed more improvement in the TPLC with less computational cost and less geometric change when compared to the effect of the candidate points from the response surface.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Comparison of the TPLC for different Reynolds numbers [2].

**Figure 9.**Flow characteristics of the upstream and downstream channels (Re = 24,230): (

**a**) velocity distribution; (

**b**) pressure distribution.

**Figure 12.**Comparison of the TPLC between the design point and two candidate points for four Reynolds numbers.

**Figure 16.**Velocity contour at the center of the x-coordinate using the adjoint solver (Re = 24,230).

**Figure 17.**Flow characteristics in the upstream and downstream channels by the adjoint solver (Re = 24,230): (

**a**) velocity distribution; (

**b**) pressure distribution.

Reynolds Number | Experiment | CFD | % Error |
---|---|---|---|

14,074 | 3.128 | 3.219 | 5.220 |

24,230 | 3.141 | 3.210 | 2.194 |

46,031 | 2.704 | 3.020 | 11.710 |

77,901 | 2.837 | 2.854 | 0.574 |

Design Variables | Values | ||
---|---|---|---|

Lower | Initial | Upper | |

Channel width (m) | 0.02381 | 0.03175 | 0.03969 |

Divider width (m) | 0.00455 | 0.00607 | 0.00759 |

Divider tip-to-wall clearance (m) | 0.02385 | 0.03180 | 0.03975 |

Design Variables | Design Point | Candidate Point 1 | Candidate Point 2 |
---|---|---|---|

Channel width (m) | 0.03175 | 0.03969 | 0.03969 |

Divider width (m) | 0.00607 | 0.00759 | 0.00455 |

Divider tip-to-wall clearance (m) | 0.03180 | 0.03975 | 0.03975 |

Input Parameters | Reynolds Number | |||
---|---|---|---|---|

14,074 | 24,230 | 46,031 | 77,901 | |

Channel width | −0.45 | −0.39 | −0.37 | −0.35 |

Divider width | −0.23 | −0.23 | −0.22 | −0.23 |

Divider tip-to-wall clearance | −0.78 | −0.81 | −0.82 | −0.83 |

Coefficients for | Reynolds Number | ||||
---|---|---|---|---|---|

14,074 | 24,230 | 46,031 | 77,901 | ||

Intercept | ${\beta}_{0}$ | 8.64223 | 8.29657 | 7.93911 | 7.65969 |

Channel width | ${\beta}_{1}$ | −0.04760 | −0.04116 | −0.03797 | −0.03508 |

Divider width | ${\beta}_{2}$ | −0.12993 | −0.12575 | −0.11837 | −0.11839 |

Divider tip-to-wall clearance | ${\beta}_{3}$ | −0.08305 | −0.08449 | −0.08387 | −0.08239 |

R-squared | 0.8626 | 0.858 | 0.8581 | 0.8579 | |

p-value | 4.82 × 10^{−5} | 5.75 × 10^{−5} | 5.74 × 10^{−5} | 5.79 × 10^{−5} |

Reynolds Number | R-Squared | % Difference | |
---|---|---|---|

No Interaction Terms | With Interaction Terms | ||

14,074 | 0.8626 | 0.9951 | 15.36 |

24,230 | 0.858 | 0.9905 | 15.44 |

46,031 | 0.8581 | 0.9881 | 15.15 |

77,901 | 0.8579 | 0.9863 | 14.97 |

**Table 7.**Comparison of TPLC prediction between the numerical method and the regression model with design variables.

Design Points | Numerical Method | Regression Model | |
---|---|---|---|

No Interaction Terms | With Interaction Terms | ||

Design point | 3.210 | 3.539 | 3.229 |

Candidate point 1 | 2.562 | 2.350 | 2.562 |

Candidate point 2 | 2.743 | 2.732 | 2.677 |

**Table 8.**Comparison of TPLC prediction between the numerical method and the regression model with design variables at the design point and Reynolds numbers.

Reynolds Number | CFD | Regression Model | |
---|---|---|---|

No Interaction Terms | With Interaction Terms | ||

14,074 | 3.291 | 3.646 | 3.367 |

24,230 | 3.210 | 3.569 | 3.231 |

46,031 | 3.020 | 3.406 | 3.015 |

77,901 | 2.854 | 3.166 | 2.886 |

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**MDPI and ACS Style**

Kim, B.; Kim, S.
Numerical Investigation of Pressure Loss in a Rectangular Channel with a Sharp 180-Degree Turn: Influence of Design Variables and Geometric Shapes. *Energies* **2023**, *16*, 3050.
https://doi.org/10.3390/en16073050

**AMA Style**

Kim B, Kim S.
Numerical Investigation of Pressure Loss in a Rectangular Channel with a Sharp 180-Degree Turn: Influence of Design Variables and Geometric Shapes. *Energies*. 2023; 16(7):3050.
https://doi.org/10.3390/en16073050

**Chicago/Turabian Style**

Kim, Byunghui, and Seokho Kim.
2023. "Numerical Investigation of Pressure Loss in a Rectangular Channel with a Sharp 180-Degree Turn: Influence of Design Variables and Geometric Shapes" *Energies* 16, no. 7: 3050.
https://doi.org/10.3390/en16073050