# Pressure Losses in Hydraulic Manifolds

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discussion of Past Literature

_{m}and with different orientation of the second elbow compared to the first one, thereby changing the relative angle between the axis of the first and last part of the pipe (twisting angle). The geometry varies from a U shape (twisting angle 0°) to an S shape (twisting angle 180°). The authors found that, if the distance between the elbows is lower than five times the diameter, the upstream elbow can strongly influence the downstream one and the loss coefficient is no longer constant with the twisting angle, reaching a maximum for values spanning in the range of 120°–140°. They found that the distance at which the presence of the elbows stops influencing the fluid flow is lower for the U and S shapes compared to the others. The U shape is the one that introduces the lowest pressure drop. In [15], the authors confirm these results applying CFD analysis to an hydraulic manifold with the bend geometries described in [14]. These results were also confirmed in [16,17], where, moreover, CFD analysis was used to analyze the pressure drop through a complicated internal passage with multiple elbows, for which the empirical formulation cannot be applied. Comparing the numerical results with the experimental ones, CFD was assessed as a good method to evaluate pressure drop according to the authors, even if a certain gap between the numerical and experimental results is evident.

## 3. Computational Fluid Dynamic (CFD) Preliminary Analysis

^{3}; it is the same fluid available in the test rig described in the following and used for the experimental analysis. The imposed boundary conditions were atmospheric pressure at the outlet of the system and a flow rate at the inlet. An additional portion of the channel (entry region) was also considered in the analysis in a way that, after this region, the velocity profile became fully developed and continued unchanged. An estimate of the entry length for different types of flow depends on the Reynolds number Re and the channel diameter d and is given by ~0.06∙Re∙d for laminar flow while it is approximated with different formulas for turbulent flow, in which the influence of Re number is weaker; in many pipes’ flow of practical engineering interest, the entrance effect are insignificant after a pipe length equal to 10∙d [19].

## 4. Experimental Setup

- 15 kW electric motor, 4-pole, operating at 380 V, 50 Hz ((1) in Figure 5);
- A variable displacement swashplate axial piston pump with flow and pressure compensator (maximum flow 65 L/min, maximum pressure 28 MPa) ((2) in Figure 5);
- An external gear pump feeding the off-line circuit dedicated to the cooling and filtering of the fluid (not represented in figure); the heat exchanger allows maintaining a controlled temperature (38 °C ± 2–3 °C).

## 5. Single-Elbow Geometries

#### 5.1. Simple 90° Elbow

#### 5.2. Expanding and Contracting Elbow

#### 5.3. Offset Elbow

#### 5.4. General Considerations

- Elbow with expansion: in the first experimental tests, cavitation and aeration strongly affect the pressure drop-flow rate curves. After repeating the tests restricting with the downstream orifice, the trends maintain a parabolic shape only for the lower expansion ratio, 1.25; whereas for higher expansion ratio values, the pressure drop trends are very low (and the accuracy of the measurements are very small in these cases). Instead, CFD results for the elbow with expansion show that the pressure drop-flow rate trend is always parabolic.
- With an expansion ratio of 1.25, the pressure drop is drastically reduced with respect to the normal elbow and both experimental and CFD results show this trend;
- In the case of a contraction elbow, there is no significant variation of the pressure drop with the contraction ratio (this is true both for CFD and experimental results) and CFD results still follow the experimental trends in always overestimating them;
- Looking at the offset elbow, attention must be paid to the fact that, with relatively high distances s and t, the intersection area between the channels may be drastically reduced, thereby causing very high pressure drop values.

## 6. Two Elbows

^{−2}(MPa) (same order of the measurement accuracy), i.e., the pressure drop is not sensitive to the backpressure if cavitation is not occurring.

## 7. Discrepancy between Numerical and Experimental Results

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Computational Fluid Dynamic (CFD) analysis of a 90° bend, results presented as in [14]: velocity map and profiles on the curved and normal plane at a distance 0.8·d from the curvature (

**a**) and 3·d from the curvature (

**b**).

**Figure 2.**Qualitative velocity map (yellow is highest speed value) at different distance L/d from the curvature.

**Figure 6.**(

**a**) The manifold block CAD Model and the position of the pressure transducers in the single-elbow case (pressure field is shown); (

**b**) image of the test rig and the connections.

**Figure 8.**Pressure drop trends coming from: experimental test—with and without the restriction of the downstream orifice; CFD simulations; calculated according to the semi-empirical formulation of [13].

**Figure 9.**Experimental pressure trends coming from the experimental test with and without the restriction of the downstream orifice.

**Figure 11.**Pressure drop experimentally measured restricting the downstream orifice, expansion (

**a**) contraction (

**b**).

**Figure 12.**Pressure drop experimentally measured without restricting the downstream orifice, expansion (

**a**) contraction (

**b**).

**Figure 14.**Pressure drop calculated with CFD and experimentally measured in the offset connection test, in both the directions of flow ((

**a**) direction 1; (

**b**) direction 2).

**Figure 15.**Pressure drop calculated with CFD in the offset connection geometries with variable distance s.

**Figure 16.**Velocity map in the two-elbow geometries: (

**a**) twisting angle equal to zero; (

**b**) twisting angle equal to zero, more distance between the elbows with respect to case (

**a**); (

**c**) twisting angle equal to 180°.

**Figure 17.**Pressure drop trends for different twisting angle values: 0° (

**a**); 90° (

**b**); 180° (

**c**); the elbows are at a constant distance L = 1.5·d.

**Figure 18.**Pressure drop trends as function of the distance L/d between the elbows, for different twisting angle values (0°, 90° and 180°), calculate with CFD analysis (

**a**); measured (

**b**).

**Figure 19.**Pressure drop trends as function of the flow for the two CFD analyses performed and the experimental test.

**Figure 20.**Relation between the experimental and numerical pressure drops in the contraction case (

**a**) and expansion case (

**b**).

Poiseuille Law | μ (Pa·s) | d (m) | Q (m^{3}/s) | L (m) | Δp (MPa) |

0.03956 | 0.01 | 0.0005 | 0.1 | 0.0081 |

Mesh Ratio | 0.2 | 0.1 | 0.08 | 0.06 | 0.05 |
---|---|---|---|---|---|

N° of elements | 41 K | 256 K | 508 K | 1460 K | 1746 K |

N° of nodes | 16 K | 84 K | 145 K | 384 K | 484 K |

Δp from CFD analysis with k-ε (MPa) | 0.895 | 0.89 | 0.87 | 0.81 | 0.8 |

Δp from CFD analysis with k-omega (Mpa) | 0.8 | 0.805 | 0.775 | 0.8 | 0.775 |

Δp from CFD analysis with low-Re k-ε (MPa) | 0.845 | 0.775 | 0.905 | 1.2 | 1.5 |

Sensor/Data Acquisition | Range | Accuracy |
---|---|---|

Piezoresistive pressure sensor | 0–6 MPa, relative | 0.5% FS (0.03 MPa) |

Piezoresistive pressure sensor | −0.1–0.6 MPa, relative | 0.5% FS (0.0035 MPa) |

Turbine flow meter | 7.5–75 L/min | 2.5% FS (1.69 L/min) |

MULTI SYSTEM 5060 PLUS | 8 channels | - |

Kinematic viscosity ν | 46 | cSt |

Dynamic viscosity μ | 39.56 | cP |

Temperature T | 40 | °C |

Density ρ | 860 | kg/m^{3} |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zardin, B.; Cillo, G.; Rinaldini, C.A.; Mattarelli, E.; Borghi, M.
Pressure Losses in Hydraulic Manifolds. *Energies* **2017**, *10*, 310.
https://doi.org/10.3390/en10030310

**AMA Style**

Zardin B, Cillo G, Rinaldini CA, Mattarelli E, Borghi M.
Pressure Losses in Hydraulic Manifolds. *Energies*. 2017; 10(3):310.
https://doi.org/10.3390/en10030310

**Chicago/Turabian Style**

Zardin, Barbara, Giovanni Cillo, Carlo Alberto Rinaldini, Enrico Mattarelli, and Massimo Borghi.
2017. "Pressure Losses in Hydraulic Manifolds" *Energies* 10, no. 3: 310.
https://doi.org/10.3390/en10030310