# Scaling Criteria for Axial Piston Machines Based on Thermo-Elastohydrodynamic Effects in the Tribological Interfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Scaling of Swashplate-Type Axial Piston Machines

#### Scaling of Lubricating Interfaces in Swashplate-Type Axial Piston Machines

- The pressure difference across the gap pushes the fluid through the interface, generating fluid shear and energy dissipation. In this way, the energy dissipation has a positive correlation with gap height, as well as with the gap flow rate.
- The relative motion of the solid boundaries of the lubricating gap causes the fluid to shear, and dissipates energy into heat. In this way, the energy dissipation has a negative correlation with gap height.

## 3. Fluid–Structure–Thermal Interaction Model

## 4. Size-Dependence of Physical Phenomena in Tribological Interfaces

#### 4.1. Linear Scaling Method (Conventional Approach)

#### 4.2. Analysis of the Governing Equations Describing the Fundamental Physical Phenomena in Scaled Tribological Interfaces

#### 4.2.1. Fluid Pressure Distribution

#### 4.2.2. Solid Body Elastic Deformation

**C**:

**C**contains the isothermal elastic modulus $E$ and the Poisson’s ratio $\nu $:

**k**scales as:

#### 4.2.3. Multi-Domain Heat Transfer

## 5. Findings and Scaling Guides

- Pressure distribution and heat transfer in both the fluid domain and the solid domain are the only contributions to the size-dependence of the axial piston machine’s lubricating interface performance.
- As long as the pressure distribution and temperature distribution are proportionally scaled, the deformation due to the pressure and thermal load stays constant, and therefore does not contribute to the size-dependence of the lubricating interface performance.

- A size-independent fluid pressure distribution can be achieved by scaling the viscosity with the linear scaling factor. The simulation results in Figure 10, Figure 11 and Figure 12 demonstrated this by showing an identical normalized performance for all three lubricating interfaces when using a scaled viscosity and no heat transfer.
- The size-independent temperature distribution in both the fluid and solid domains can be achieved by scaling the fluid and solid conductivity with the linear scaling factor. The simulation results in Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 demonstrated this by showing an identical normalized performance for all three lubricating interfaces with a scaled viscosity and conductivity.

- When scaling a swashplate-type axial piston machine to a different size, the size-dependent pressure distribution-induced performance bias can be eliminated by using hydraulic fluid at a different viscosity grade.
- ○
- Use higher viscosity for up-scaling.
- ○
- Use lower viscosity for down-scaling.
- ○
- Choose the viscosity based on the linear scaling factor as in Equation (1).

- The viscosity of the fluid can also be controlled by:
- ○
- Increasing operating temperature for down-scaling
- ○
- Decreasing operating temperature for up-scaling.

- When using a fluid of different viscosity is not feasible, design modifications should compensate for the size-dependent sealing function of the scaled lubricating interfaces:
- ○
- For the piston/cylinder interface, use a lower normalized clearance for up-scaling, and use a higher normalized clearance for down-scaling [28].
- ○
- For the cylinder block/valve plate interface, increase the sealing land area for up-scaling, and decrease the sealing land area for down-scaling [27].
- ○
- For the slipper/swashplate interface, increase the sealing land area for up-scaling, and decrease the sealing land area for down-scaling.

- To compensate for the size-dependent heat transfer, the design of the lubricating interfaces needs to be modified:
- ○
- When up-scaling, the lubricating interface design should be modified in order to increase the cooling performance of the lubricating gap, e.g., adding a flow channel beneath the valve plate to smooth the temperature distribution.
- ○
- When down-scaling, the lubricating interface design should be modified to create more thermal deformation, e.g., by using a bi-material solid body [30].

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbols | Denotation | Unit |

$B$ | Strain-displacement matrix | $[1/\mathrm{m}]$ |

$C$ | Constitutive matrix | $[\mathrm{Pa}]$ |

$d$ | Diameter | $[\mathrm{m}]$ |

$f$ | Vector of nodal force | $[\mathrm{N}]$ |

$h$ | Gap height | $[\mathrm{m}]$ |

$H$ | Maximum stroke | $[\mathrm{m}]$ |

$k$ | Element stiffness matrix | $[\mathrm{N}/\mathrm{m}]$ |

$l$ | Length | $[\mathrm{m}]$ |

$P$ | Power | $[\mathrm{W}]$ |

$p$ | Pressure | $[\mathrm{Pa}]$ |

$q$ | Rate of heat flux | ${[\mathrm{W}/\mathrm{m}}^{2}]$ |

$s$ | Stroke | $[\mathrm{m}]$ |

$T$ | Temperature | $[\mathrm{K}]$ |

U | Strain energy | $[\mathrm{J}]$ |

$u$ | Vector of nodal displacement | $[\mathrm{m}]$ |

$V$ | Applied load potential energy | $[\mathrm{J}]$ |

$v$ | Speed | $[\mathrm{m}/\mathrm{s}]$ |

$w$ | Normal squeezing velocity | $[\mathrm{m}/\mathrm{s}]$ |

$\beta $ | Swashplate angle | $[\mathrm{rad}]$ |

$E$ | Elastic modulus | $[\mathrm{Pa}]$ |

$\upsilon $ | Poisson’s ratio | $[-]$ |

$\mathsf{\Gamma}$ | Diffusion coefficient | $[\mathrm{kg}/\mathrm{m}\xb7\mathrm{s}]$ |

${\mathsf{\epsilon}}_{F}$ | Vector of elastic strain | $[-]$ |

$\mathbf{\kappa}$ | Conductivity | $[\mathrm{W}/\mathrm{m}\xb7\mathrm{K}]$ |

$\lambda $ | Linear scaling factor | $[-]$ |

$\mu $ | Fluid dynamic viscosity | $[\mathrm{Pa}\xb7\mathrm{s}]$ |

$\mathrm{\Pi}$ | Total potential energy | $[\mathrm{J}]$ |

$\rho $ | Density | ${[\mathrm{kg}/\mathrm{m}}^{3}]$ |

$\mathsf{\sigma}$ | Vector of elastic stress | $[\mathrm{Pa}]$ |

$\mathsf{\Phi}$ | Energy dissipation rate | $[\mathrm{W}]$ |

$\phi $ | Shaft angle | $[\mathrm{rad}]$ |

$\omega $ | Shaft speed | $[\mathrm{rad}/\mathrm{s}]$ |

Subscripts | Denotation | |

$B$ | Cylinder block | |

$b$ | Bottom surface | |

$cd$ | Conductive | |

$cv$ | Convective | |

$E$ | Pressure load | |

$fluid$ | Fluid domain | |

$k$ | Piston | |

$opt$ | Optimal | |

$solid$ | Solid domain | |

$SQ$ | Loss due to leakage | |

$ST$ | Loss due to friction | |

$t$ | Top surface | |

$T$ | Thermal load | |

$0$ | Pre-scaled |

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**Figure 11.**Normalized cylinder block/valve plate interface performance comparison for different sizes.

**Figure 12.**Normalized slipper/swashplate interface energy dissipation comparison for different sizes.

**Figure 14.**Normalized cylinder block/valve plate interface performance comparison for different sizes.

**Figure 15.**Normalized slipper/swashplate interface energy dissipation comparison for different sizes.

A | B | C | |
---|---|---|---|

Linear factor $\lambda $ | 0.5 | 1 | 2 |

Unit size [cc] | 9.375 | 75 | 600 |

Pressure [bar] | 400 | 400 | 400 |

Displacement [%] | 100 | 100 | 100 |

Speed [rpm] | 7200 | 3600 | 1800 |

A | B | C | |
---|---|---|---|

Linear factor $\lambda $ | 0.5 | 1 | 2 |

Unit size [cc] | 9.375 | 75 | 600 |

Pressure [bar] | 400 | 400 | 400 |

Displacement [%] | 100 | 100 | 100 |

Speed [rpm] | 7200 | 3600 | 1800 |

Fluid viscosity | $0.5\xb7{\mu}_{0}$ | ${\mu}_{0}$ | $2\xb7{\mu}_{0}$ |

A | B | C | |
---|---|---|---|

Linear factor $\lambda $ | 0.5 | 1 | 2 |

Unit size [cc] | 9.375 | 75 | 600 |

Pressure [bar] | 400 | 400 | 400 |

Displacement [%] | 100 | 100 | 100 |

Speed [rpm] | 7200 | 3600 | 1800 |

Fluid viscosity | $0.5\xb7{\mu}_{0}$ | ${\mu}_{0}$ | $2\xb7{\mu}_{0}$ |

Fluid conductivity | $0.5\xb7{\mathbf{\kappa}}_{fluid\_0}$ | ${\mathbf{\kappa}}_{fluid\_0}$ | $2\xb7{\mathbf{\kappa}}_{fluid\_0}$ |

Solid conductivity | $0.5\xb7{\mathbf{\kappa}}_{solid\_0}$ | ${\mathbf{\kappa}}_{solid\_0}$ | $2\xb7{\mathbf{\kappa}}_{solid\_0}$ |

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**MDPI and ACS Style**

Shang, L.; Ivantysynova, M.
Scaling Criteria for Axial Piston Machines Based on Thermo-Elastohydrodynamic Effects in the Tribological Interfaces. *Energies* **2018**, *11*, 3210.
https://doi.org/10.3390/en11113210

**AMA Style**

Shang L, Ivantysynova M.
Scaling Criteria for Axial Piston Machines Based on Thermo-Elastohydrodynamic Effects in the Tribological Interfaces. *Energies*. 2018; 11(11):3210.
https://doi.org/10.3390/en11113210

**Chicago/Turabian Style**

Shang, Lizhi, and Monika Ivantysynova.
2018. "Scaling Criteria for Axial Piston Machines Based on Thermo-Elastohydrodynamic Effects in the Tribological Interfaces" *Energies* 11, no. 11: 3210.
https://doi.org/10.3390/en11113210