# Pre-Charge Pressure Estimation of a Hydraulic Accumulator Using Surface Temperature Measurements

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## Abstract

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## 1. Introduction

## 2. Accumulator Model

#### 2.1. Mechanical Model

#### 2.2. Gas Model

#### 2.3. Thermal Model

#### 2.3.1. End Cap

#### 2.3.2. Accumulator Piston

#### 2.3.3. Accumulator Wall

## 3. Model Validation

## 4. Estimator Design

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EKF | Extended Kalman Filter |

SAEKF | State Augmented Extended Kalman Filter |

BWR | Benedict–Webb–Rubin |

MW | Mega-Watt |

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**Figure 1.**Mean surface temperatures for different pre-charge pressures. The fluid inlet is located on the left-hand side of the plot (also seen in Figure 2), which also denotes the initial piston position. The vertical black bar shows the final piston position. The surface temperature values are interpolated from eight evenly distributed sensors along the length of the accumulator.

**Figure 3.**Accumulator wall schematic divided into eight elements including the piston position notation.

**Figure 7.**Simulated and experimental piston position for a test sequence emulating the operation of a supply accumulator. The position residual is defined as the difference between the simulated and experimental values.

**Figure 8.**Simulated and experimental wall temperature of two wall elements. The upper sub-figure shows the piston position during the test. The dashed lines indicate the wall elements.

**Figure 9.**Estimated and experimentally measured piston position and molar value of gas for the low-load test sequence. The accumulator is pre-charged to 125 bar/134 mol.

**Figure 10.**Estimated and experimentally measured piston position and estimated molar value of gas for the high-load test sequence. The accumulator is pre-charged to 125 bar/134 mol.

**Figure 11.**Estimated pre-charge pressures at selected levels for the low-load and high-load test sequences.

**Figure 12.**Estimated and experimental molar value for the low-load test sequence at different initial conditions. The accumulator is pre-charged to 125 bar/134 mol.

**Figure 13.**Estimated and experimental molar value for the low-load test sequence under natural and forced convection conditions. The accumulator is pre-charged to 125 bar/134 mol.

Notation | Description | Value | Unit |
---|---|---|---|

${m}_{p}$ | Piston mass | 3.97 | (kg) |

B | Viscous friction coefficient | 5 × ${10}^{3}$ | $\left(\frac{\mathrm{N}\xb7\mathrm{s}}{\mathrm{m}}\right)$ |

C | Coulomb friction coefficient | $2\times {10}^{3}$ | (N) |

R | Gas constant | 8.31 | (N) |

${A}_{0}$ | BWR constant | 0.11 | () |

${B}_{0}$ | BWR constant | $4.07\times {10}^{-5}$ | () |

${C}_{0}$ | BWR constant | 816.58 | () |

a | BWR constant | $2.54\times {10}^{-6}$ | () |

b | BWR constant | $2.33\times {10}^{-9}$ | () |

c | BWR constant | $7.38\times {10}^{-2}$ | () |

$\alpha $ | BWR constant | $1.27\times {10}^{-13}$ | () |

$\gamma $ | BWR constant | $5.3\times {10}^{-9}$ | () |

${c}_{N2}$ | Specific heat capacity of nitrogen | 1040 | $\left(\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right)$ |

${M}_{N2}$ | Molar mass of nitrogen | $2.8\times {10}^{-2}$ | $\left(\frac{\mathrm{kg}}{\mathrm{mol}}\right)$ |

${V}_{a,0}$ | Non-displaceable accumulator volume | $1.4\times {10}^{-3}$ | (m${}^{3}$) |

${m}_{e}$ | End-cap mass | 28.45 | (kg) |

${c}_{steel}$ | Specific heat capacity of steel | 490 | $\left(\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}\right)$ |

${\alpha}_{a}$ | Heat transfer coefficient (air to steel) | - | $\left(\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right)$ |

${\alpha}_{o}$ | Heat transfer coefficient (oil to steel) | - | $\left(\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right)$ |

${\alpha}_{g}$ | Heat transfer coefficient (gas to steel) | - | $\left(\frac{\mathrm{W}}{{\mathrm{m}}^{2}\xb7\mathrm{K}}\right)$ |

${A}_{eg}$ | Area between end-cap and gas | $2.54\times {10}^{-2}$ | (m${}^{2}$) |

${A}_{ea}$ | Area between end-cap and air | $9.18\times {10}^{-2}$ | (m${}^{2}$) |

${A}_{pg}$ | Area between piston and gas | $2.71\times {10}^{-2}$ | (m${}^{2}$) |

${A}_{p}$ | Piston area | $2.54\times {10}^{-2}$ | (m${}^{2}$) |

${r}_{is}$ | Inner radius of accumulator | $9\times {10}^{-2}$ | (m) |

${r}_{os}$ | Outer radius of accumulator | $11\times {10}^{-2}$ | (m) |

n | Number of wall elements | 8 | (-) |

${d}_{x}$ | Length of wall element | 0.123 | (m) |

${l}_{p}$ | Piston length | 0.11 | (m) |

${l}_{a}$ | Accumulator length | 0.983 | (m) |

$\rho $ | Density of steel | 7800 | $\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)$ |

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**MDPI and ACS Style**

Asmussen, M.F.; Liniger, J.; Sepehri, N.; Pedersen, H.C.
Pre-Charge Pressure Estimation of a Hydraulic Accumulator Using Surface Temperature Measurements. *Wind* **2022**, *2*, 784-800.
https://doi.org/10.3390/wind2040041

**AMA Style**

Asmussen MF, Liniger J, Sepehri N, Pedersen HC.
Pre-Charge Pressure Estimation of a Hydraulic Accumulator Using Surface Temperature Measurements. *Wind*. 2022; 2(4):784-800.
https://doi.org/10.3390/wind2040041

**Chicago/Turabian Style**

Asmussen, Magnus F., Jesper Liniger, Nariman Sepehri, and Henrik C. Pedersen.
2022. "Pre-Charge Pressure Estimation of a Hydraulic Accumulator Using Surface Temperature Measurements" *Wind* 2, no. 4: 784-800.
https://doi.org/10.3390/wind2040041