# Thermo-Hydraulic Modelling and Experimental Validation of an Electro-Hydraulic Compact Drive

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## Abstract

**:**

## 1. Introduction

## 2. Control Volume Dynamics

#### 2.1. Lumped Pressure Dynamics

#### 2.1.1. Mechanical Elasticity

#### 2.1.2. Diaphragm and Bladder Accumulators

#### 2.2. Lumped Temperature Dynamics

## 3. Fluid Properties

#### 3.1. Temperature Independent Density

#### 3.2. Comparing Modelled and Measured Fluid Properties

## 4. Electro-Hydraulic Compact Drive Prototype

## 5. Thermo-Hydraulic Model Formulation

#### 5.1. Benchmark Model

#### 5.2. Reduced Model

## 6. Thermal Resistance Networks

#### 6.1. Heat Conduction

#### 6.2. Convection

#### 6.3. Radiation

#### 6.4. Thermal Resistances in the Reduced Model

## 7. Component Models

#### 7.1. Cylinder and Orifice

#### 7.2. Electric Motor

#### 7.3. Pump

## 8. Results

#### 8.1. Loss Behaviour

#### 8.2. Static Temperatures

#### 8.3. Transient Temperature

## 9. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

CV | Control Volume |

ISV | Inverse Shuttle Valve |

RPM | Revolutions per Minute |

CS | Control Surface |

ECD | Electro-Hydraulic Compact Drives |

MBW | Moving Boundary Work |

## Appendix A. Nomenclature

Symbol | Description | Unit | Symbol | Description | Unit |

$\alpha $ | Isobaric expansion coefficient | K${}^{-1}$ | $\dot{H}$ | Enthalpy Flow | W |

$\beta $,${\beta}_{{}_{\mathrm{eff}}},{\beta}_{{}_{\mathrm{mech}}}$ | Isothermal, Effective and Mechanical | Pa | k | Thermal Conductivity | $\mathrm{W}/\mathrm{mK}$ |

Bulk Modulus | ${L}_{{}_{\mathrm{c}}}$ | Characteristic Length | m | ||

$\u03f5$ | Emissivity | - | $\dot{m}$ | Mass Flow | |

$\mu $ | Dynamic Viscosity | $\mathrm{Ns}/{\mathrm{m}}^{2}$ | $\widehat{\mathbf{n}}$ | Normal Vector | |

$\nu $ | Specific Volume | ${\mathrm{m}}^{3}/\mathrm{kg}$ | p | Pressure | Pa |

$\rho ,{\rho}_{{}_{\mathrm{A}}},{\rho}_{{}_{\mathrm{F}}}$ | Mixture, Air, Oil density | $\mathrm{kg}/{\mathrm{m}}^{3}$ | $\dot{Q}$ | Heat Flow | W |

$\omega $ | Shaft Speed | $\mathrm{rad}/\mathrm{s}$ | ${R}_{{}_{\mathrm{th}}}$ | Thermal Resistance | $\mathrm{K}/\mathrm{W}$ |

$\tau $,${\tau}_{{}_{\mathrm{L}}}$,${\tau}_{{}_{\mathrm{T}}}$ | Shaft Torque, Torque Losses, | Nm | S | Entropy | $\mathrm{J}/\mathrm{K}$ |

Theoretical Torque | ${T}_{{}_{\mathrm{s}}}$, T | (Surface) Temperature | K | ||

$\overline{\u2022}$ | Property evaluated at average conditions | u | Specific Internal Energy | $\mathrm{J}/\mathrm{kg}$ | |

$\tilde{\u2022}$ | Temperature Indepedent Property | U | Internal Energy | J | |

${\u2022}_{{}_{\mathrm{A}}}$,${\u2022}_{{}_{\mathrm{B}}}$,${\u2022}_{{}_{\mathrm{C}}}$ | Quantify evaluated in CV A, B or C | $\widehat{V}$,$\widehat{\mathbf{V}}$ | Fluid Speed & Velocity Vector | $\mathrm{m}/\mathrm{s}$ | |

A, ${A}_{{}_{\mathrm{s}}}$ | Area & Surface Area | ${\mathrm{m}}^{2}$ | V, ${V}_{{}_{\mathrm{x}}}$ | Volume & Volume at ${p}_{{}_{0}}$ | ${\mathrm{m}}^{3}$ |

${c}_{{}_{\mathrm{p}}}$, ${c}_{{}_{\mathrm{v}}}$ | Isobaric & Isochoric Specific Heat | $\mathrm{J}/\mathrm{kgK}$ | $\dot{W}$ | Rate of Work | W |

d, ${d}_{\mathrm{i}}$ | Diameter, Internal Diameter | m | x | Piston Position | m |

${F}_{{}_{\mathrm{L}}}$, ${F}_{{}_{\mathrm{F}}}$ | Load & Cylinder Friction Force | N | z | Height in Gravitational Field | m |

h | Convection Heat Transfer Coefficient | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ | |||

h | Specific Enthalpy | $\mathrm{J}/\mathrm{kg}$ |

## Appendix B. Modelling Parameters

Symbol | Description | Value | Symbol | Description | Value |

${\alpha}_{{}_{0}}$ | Expansion Coefficient at ${p}_{{}_{0}},{T}_{{}_{0}}$ | 0.00067 K${}^{-1}$ | ${B}_{{}_{\mathrm{L}}}$ | Cylinder Viscous Friction | 285 Ns/m |

${\beta}_{{}_{0}}$ | Bulk Modulus at ${p}_{{}_{0}},{T}_{{}_{0}}$ | 1.65 GPa | ${C}_{{}_{\mathrm{d}}}$ | Orifice Discharge Coefficient | 0.7 |

${\beta}_{{}_{\mathrm{mech}}}$ | Mechanical Bulk Modulus | 0.3 GPa | ${c}_{{}_{\mathrm{p}0}}$ | Oil Specific Heat Parameter | 657 J/kgK |

$\u03f5$ | Volumetric Air Ratio | 0.01 | ${D}_{{}_{\mathrm{p}}}$ | Pump Displacement | 6.3 cm${}^{3}$/rev |

$\kappa $ | Polytropic process coefficient | 1.4 | ${F}_{{}_{\mathrm{s}}}$ | Static Friction | 90 N |

${\rho}_{{}_{\mathrm{F}0}}$ | Oil Density at ${p}_{{}_{0}},{T}_{{}_{0}}$ | 873 kg/m${}^{3}$ | ${F}_{{}_{\mathrm{c}}}$ | Coulomb Friction | 15 N |

$\sigma $ | Stefan-Boltzmann Constant | g | Gravitational Acceleration | 9.82 m/s${}^{2}$ | |

$\tau $ | Thermal Time Constant | 3.5 s | |||

${A}_{{}_{\mathrm{A}}}$ | Cylinder Piston Area | 12 cm${}^{2}$ | ${K}_{{}_{\mathrm{cp}}}$ | Oil Specific Heat Parameter | 4.21 J/kgK${}^{2}$ |

${A}_{{}_{\mathrm{B}}}$ | Rod Chamber Area | 8.8 cm${}^{2}$ | ${K}_{{}_{\mathrm{P}}}$ | Friction Parameter | 0.9 N/bar |

${A}_{{}_{\mathrm{o}}}$ | Orifice Area | 14 mm${}^{2}$ | ${M}_{{}_{\mathrm{L}}}$ | Load Mass | 112.5 kg |

${a}_{{}_{\mathrm{k}1}}$ | Conductivity Parameter | 0.17 W/mK | ${p}_{{}_{0}}$ | Atmospheric Pressure | 1.01 bar |

${a}_{{}_{\mathrm{k}2}}$ | Conductivity Parameter | 97 W/mK${}^{2}$ | ${p}_{{}_{\mathrm{acc}0}}$ | Precharge Pressure | 2.4 bar |

${a}_{{}_{\mu 1}}$ | Viscosity Model Parameter | 63 Ns/mm${}^{2}$ bar | R | Air Gas Constant | 287 J/kgK |

${a}_{{}_{\mu 2}}$ | Viscosity Model Parameter | 880 K | ${T}_{{}_{0}}$ | Reference Temperature | 288.15 K |

${a}_{{}_{\mu 3}}$ | Viscosity Model Parameter | 178 K | ${T}_{{}_{\mathrm{acc}0}}$ | Precharge Temperature | 288.15 K |

${a}_{{}_{\mu 4}}$ | Viscosity Model Parameter | 334 bar | ${T}_{{}_{\mathrm{rig}}}$,${T}_{{}_{inf}}$ | Test-rig & Ambient Temp. | 296.15 K |

${a}_{{}_{\mu 5}}$ | Viscosity Model Parameter | 3.26 bar/K | ${V}_{{}_{\mathrm{acc}}}$ | Volume of Accumulator | 1.4 L |

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**Figure 1.**(

**a**) Valve-compensated ECD. (

**b**) Pump-compensated ECD. (

**c**) ECD with two electric motors. (

**d**) ECDs may rely solely on passive cooling.

**Figure 2.**Illustration of a lumped control volume. Pressure, temperature and specific volume are assumed to be homogeneous. Mass transfer may occur across the control surface (CS).

**Figure 3.**Fluid properties evaluated with a volumetric air content of 1%. (

**a**) Dashed lines are modelled oil densities and solid lines are oil–air mixture densities. Measurements are obtained from [55]. (

**b**) Modelled oil–air mixture bulk moduli. Dashed lines include the effect of mechanical elasticity with ${\beta}_{\mathrm{mech}}=2$ GPa. $\tilde{\beta}$ is the temperature independent bulk modulus. (

**c**) Modelled thermal expansion coefficient of oil–air mixture.

**Figure 4.**ECD prototype manufactured at TU-Dresden. The table lists the main components of the system.

**Figure 5.**(

**a**) Diagram showing the main components of the ECD prototype. (

**b**) Simplified schematic, valid for the conducted experiments, and used during formulation of the thermo-hydraulic model. The simplified schematic also shows the position of the oil temperature sensors on the prototype.

**Figure 6.**(

**a**) In the reduced model, natural convection and radiation to the surroundings are modelled to occur from five shapes. (

**b**) In the benchmark model, natural convection and radiation are modelled to occur from 18 shapes (three not shown).

**Figure 8.**Thermal resistance network for the reduced model. The natural convection and radiation resistance are found as a parallel connection of the convection and radiation resistances.

**Figure 9.**(

**a**) The heat conduction from the cylinder barrel to the test-rig is modelled as a serial connection of two conduction and two contact resistances. (

**b**) Bladder type gas-loaded accumulator. (

**c**) The bladder accumulator is approximated as a sphere when calculating the natural convection heat transfer from the outer surface. (

**d**) Forced convection on the inner surface is approximated as internal pipe flow, with the length of the pipe being the oil height in the virtual sphere and the pipe radius being evaluated as the radius of the spherical cap at $l/2$.

**Figure 10.**(

**a**) Simplified schematic valid for the conducted experiments. (

**b**) Simple schematic showing the quantities used for modelling mass and enthalpy flow of the inverse shuttle valve/orifice. (

**c**) The inputs to the static pump model are the fluid properties in the adjacent chambers, and the shaft speed $\omega $. The shaft torque as well as mass and enthalpy flows are model outputs.

**Figure 11.**(

**a**) Measured efficiency map of the electric motor, as a function of torque and speed. Dotted lines are power levels. (

**b**) Measured efficiency map of the pump as a function of pressure difference, speed and temperature.

**Figure 12.**(

**a**) Measured and simulated piston position. (

**b**) Measured and simulated pressure in the piston chamber ${p}_{{}_{\mathrm{A}}}$. (

**c**) Measured and simulated mechanical output power $(\dot{W}=\dot{x}{F}_{{}_{\mathrm{cyl}}})$. (

**d**) Measured and simulated pressure in the rod chamber ${p}_{{}_{\mathrm{B}}}$. (

**e**) Measured and simulated shaft torque and shaft power ($\dot{W}=\omega \tau $). (

**f**) Measured and simulated accumulator pressure.

**Figure 13.**(

**a**) Simulated steady state temperatures of the benchmark model compared to the measured static temperatures of the ECD prototype. (

**b**) Simulated steady state temperatures of the reduced model compared to the measured static temperatures of the ECD prototype.

**Figure 14.**(

**a**) Measured and simulated oil temperature in the piston chamber ${T}_{{}_{\mathrm{A}}}$. (

**b**) Measured and simulated oil temperature in the rod chamber ${T}_{{}_{\mathrm{B}}}$. (

**c**) Measured and simulated oil temperature and pressure in the accumulator ${T}_{{}_{\mathrm{C}}}$ and ${p}_{{}_{\mathrm{C}}}$. (

**d**) Measured and simulated surface temperatures of the motor and cylinder barrel. (

**e**) Measured and simulated surface temperatures of the pump and accumulator shell. (

**f**) Measured and simulated surface temperatures of mounting plate II. (

**g**) Measured and simulated surface temperatures of mounting plate III. (

**h**) Measured and simulated surface temperatures of the cylinder head, flange and foot. This is only available for the benchmark model, as these components are not included in the reduced model.

Model Complexity | ||
---|---|---|

Benchmark | Reduced | |

Fluid Properties | Temperature Dependent | Temperature Independent |

(Equations (31)–(33)) | $\tilde{\alpha}=0$ and Equations (38) and (39) | |

Number of Thermal Capacities | 3 Oil Capacities | 1 Combined Oil/Solid Capacity |

18 Solid Capacities | 1 Oil Capacity & 1 Solid Capacity | |

Number of Thermal Resistances | 18 Natural Convection & Radiation | 5 Natural Convection & Radiation |

21 Conduction Resistances | 1 Conduction Resistance | |

23 Forced Convection Resistances | 1 Forced Convection Resistance |

**Table 2.**Nusselt numbers for convective heat transfer problems in the reduced model. ${A}_{{}_{\mathrm{proj}}}$ is the projected area of the cube on a horizontal flat surface below the cube. ${d}_{\mathrm{i}}$ is internal diameter. The Nusselt number for internal pipe flow is valid for laminar flows during hydrodynamic and thermal flow development.

Geometry | Nusselt Number | Prantdl Function | L${}_{{}_{\mathbf{c}}}$ | |
---|---|---|---|---|

Natural | Horizontal | Nu = ${\left(0.752+0.387{\left(\mathrm{Ra}\phantom{\rule{0.166667em}{0ex}}f\left(\mathrm{Pr}\right)\right)}^{1/6}\right)}^{2}$ | f(Pr) = ${\left(1+{\left(\frac{0.559}{\mathrm{Pr}}\right)}^{9/16}\right)}^{-16/9}$ | L${}_{{}_{\mathrm{c}}}$ = $\frac{1}{2}\pi d$ |

Cylinder | ||||

Sphere | Nu = $0.56{\left(\left(\frac{\mathrm{Pr}}{0.846+\mathrm{Pr}}\right)\mathrm{Ra}\right)}^{1/4}+2$ | - | L${}_{{}_{\mathrm{c}}}$ = d | |

Cube | Nu = $5.748+0.752{\left(\frac{\mathrm{Ra}}{f\left(\mathrm{Pr}\right)}\right)}^{0.252}$ | f(Pr) = ${\left(1+{\left(\frac{0.492}{\mathrm{Pr}}\right)}^{9/16}\right)}^{16/9}$ | L${}_{{}_{\mathrm{c}}}$ = $\frac{A}{d}$ | |

$d=\sqrt{4\frac{{A}_{{}_{\mathrm{proj}}}}{\pi}}$ | ||||

Forced | Internal Pipe Flow | Nu = ${\left(49.37+{\left({f}_{{}_{1}}-0.7\right)}^{3}+{f}_{{}_{2}}^{3}\right)}^{1/3}$ | ${f}_{{}_{1}}=1.615{\left(\mathrm{RePr}\frac{{d}_{\mathrm{i}}}{l}\right)}^{1/3}$ | L${}_{{}_{\mathrm{c}}}$= ${d}_{\mathrm{i}}$ |

${f}_{{}_{2}}={\left(\frac{2}{1+22\mathrm{Pr}}\right)}^{1/6}\left(\mathrm{RePr}\frac{{d}_{\mathrm{i}}}{l}\right)$ |

**Table 3.**Thermal resistances, according to Figure 8, evaluated at oil and solid temperatures of 60 °C, an ambient temperature of 20 °C and for fluid velocities present for a piston speed of 150 mm/s.

Natural Convection & Radiation | |||||||||
---|---|---|---|---|---|---|---|---|---|

ID | Geometry | $A{}_{{}_{\mathrm{s}}}$ | $A{}_{{}_{\mathrm{proj}}}$ | $L{}_{{}_{\mathrm{c}}}$ | $\u03f5$ | $h{}_{{}_{\mathrm{conv}}}$ | $h{}_{{}_{\mathrm{rad}}}$ | $h{}_{{}_{\mathrm{comb}}}$ | $R{}_{{}_{\mathrm{th}}}$ |

I | Horizontal | 980 cm${}^{2}$ | - | 79 mm | 0.92 | 7.1$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 6.4$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 13.5$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 0.76$\frac{\mathrm{K}}{\mathrm{W}}$ |

Cylinder | |||||||||

II | Cube | 1341 cm${}^{2}$ | 221 cm${}^{2}$ | 799 mm | 0.26 | 5.0$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 1.8$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 6.8$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 1.1$\frac{\mathrm{K}}{\mathrm{W}}$ |

(Manifold) | |||||||||

III | Cube | 2268 cm${}^{2}$ | 469 cm${}^{2}$ | 928 mm | 0.69 | 4.8$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 4.8$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 9.6$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 0.46$\frac{\mathrm{K}}{\mathrm{W}}$ |

(Electric Motor) | |||||||||

IV | Cube | 614 cm${}^{2}$ | 120 cm${}^{2}$ | 497 mm | 0.92 | 5.7$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 6.4$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 12.1$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 1.34$\frac{\mathrm{K}}{\mathrm{W}}$ |

(Pump) | |||||||||

V | Sphere | 707 cm${}^{2}$ | - | 150 mm | 0.92 | 5.9$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 6.4$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 12.4$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 1.14$\frac{\mathrm{K}}{\mathrm{W}}$ |

(Accumulator) | |||||||||

Heat Conduction to Test-Rig | |||||||||

ID | Geometry | $A{}_{{}_{1}}$ | $A{}_{{}_{2}}$ | $A{}_{{}_{3}}$ | $L{}_{{}_{1}}$ | $L{}_{{}_{2}}$ | k [46] | ${h}_{\mathrm{c}}$ [40] | $R{}_{{}_{\mathrm{th}}}$ |

VI | Plane Wall | 20 cm${}^{2}$ | 28 cm${}^{2}$ | 75 cm${}^{2}$ | 73 mm | 16 mm | 63.9 $\frac{\mathrm{W}}{\mathrm{m}\mathrm{K}}$ | 6.5 $\frac{\mathrm{k}\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 0.37 $\frac{\mathrm{K}}{\mathrm{W}}$ |

Forced Convection (Oil to Accumulator) | |||||||||

ID | Geometry | $A{}_{{}_{\mathrm{s}}}$ | l | ${d}_{\mathrm{i}}$ | $\widehat{V}$ | Re | $h{}_{{}_{\mathrm{conv}}}$ | $R{}_{{}_{\mathrm{conv}}}$ | |

VII | Internal | 243 cm${}^{2}$ | 51.6 mm | 113 mm | 7.6 $\frac{\mathrm{m}\mathrm{m}}{\mathrm{s}}$ | 28 | 51$\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ | 0.80$\frac{\mathrm{K}}{\mathrm{W}}$ | |

Pipe Flow |

**Table 4.**Pump torque, mass and enthalpy flows based on operation quadrant. $\Delta P=\left({p}_{{}_{\mathrm{A}}}-{p}_{{}_{\mathrm{B}}}\right)$. ${}^{{}^{\u2605}}$ Only valid for ${\dot{m}}_{{}_{\mathrm{A}}}>0$, ${}^{{}^{\u2020}}$ Only valid for ${\dot{m}}_{{}_{\mathrm{B}}}<0$.

Pump Model Outputs | |||||||
---|---|---|---|---|---|---|---|

Quadrant | $\tau $ | ${\dot{m}}_{{}_{\mathrm{A}}}$ | ${\dot{m}}_{{}_{\mathrm{B}}}$ | ${\dot{H}}_{{}_{\mathrm{A}}}$ | ${\dot{H}}_{{}_{\mathrm{B}}}$ | ${\dot{H}}_{{}_{\mathrm{L}}}$ | |

I | $(\omega \ge 0$) | ${\tau}_{{}_{\mathrm{T}}}+{\tau}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}-2{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}-{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{A}}}({h}_{{}_{\mathrm{B}}}+\Delta h-{h}_{{}_{\mathrm{A}}})$${}^{{}^{\u2605}}$ | ${\dot{m}}_{{}_{\mathrm{L}}}\Delta h$${}^{{}^{\u2605}}$ | ${\dot{m}}_{{}_{\mathrm{L}}}({h}_{{}_{\mathrm{B}}}+\Delta h-{h}_{{}_{\mathrm{C}}})$${}^{{}^{\u2605}}$ |

$(\Delta P\ge 0)$ | |||||||

II | $(\omega <0$) | $-{\tau}_{{}_{\mathrm{T}}}-{\tau}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}-2{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}-{\dot{m}}_{{}_{\mathrm{L}}}$ | 0 | ${\dot{m}}_{{}_{\mathrm{L}}}\left({h}_{{}_{\mathrm{A}}}-{h}_{{}_{\mathrm{B}}}\right)-$ | ${\dot{m}}_{{}_{\mathrm{L}}}({h}_{{}_{\mathrm{A}}}-{h}_{{}_{\mathrm{C}}})$ |

$(\Delta P>0)$ | ${\dot{m}}_{{}_{\mathrm{T}}}\left({h}_{{}_{\mathrm{A}}}+\Delta h-{h}_{{}_{\mathrm{B}}}\right)$ | ||||||

III | $(\omega <0$) | $-{\tau}_{{}_{\mathrm{T}}}-{\tau}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}+{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}+2{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{L}}}\Delta h$${}^{{}^{\u2020}}$ | $-{\dot{m}}_{{}_{\mathrm{B}}}\left({h}_{{}_{\mathrm{A}}}+\Delta h-{h}_{{}_{\mathrm{B}}}\right)$${}^{{}^{\u2020}}$ | ${\dot{m}}_{{}_{\mathrm{L}}}({h}_{{}_{\mathrm{A}}}+\Delta h-{h}_{{}_{\mathrm{C}}})$${}^{{}^{\u2020}}$ |

$(\Delta P<0)$ | |||||||

IV | $(\omega >0$) | ${\tau}_{{}_{\mathrm{T}}}+{\tau}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}+{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{T}}}+2{\dot{m}}_{{}_{\mathrm{L}}}$ | ${\dot{m}}_{{}_{\mathrm{L}}}\left({h}_{{}_{\mathrm{B}}}-{h}_{{}_{\mathrm{A}}}\right)-$ | 0 | ${\dot{m}}_{{}_{\mathrm{L}}}({h}_{{}_{\mathrm{B}}}-{h}_{{}_{\mathrm{C}}})$ |

$(\Delta P<0)$ | ${\dot{m}}_{{}_{\mathrm{T}}}\left({h}_{{}_{\mathrm{B}}}+\Delta h-{h}_{{}_{\mathrm{A}}}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Ketelsen, S.; Michel, S.; Andersen, T.O.; Ebbesen, M.K.; Weber, J.; Schmidt, L.
Thermo-Hydraulic Modelling and Experimental Validation of an Electro-Hydraulic Compact Drive. *Energies* **2021**, *14*, 2375.
https://doi.org/10.3390/en14092375

**AMA Style**

Ketelsen S, Michel S, Andersen TO, Ebbesen MK, Weber J, Schmidt L.
Thermo-Hydraulic Modelling and Experimental Validation of an Electro-Hydraulic Compact Drive. *Energies*. 2021; 14(9):2375.
https://doi.org/10.3390/en14092375

**Chicago/Turabian Style**

Ketelsen, Søren, Sebastian Michel, Torben O. Andersen, Morten Kjeld Ebbesen, Jürgen Weber, and Lasse Schmidt.
2021. "Thermo-Hydraulic Modelling and Experimental Validation of an Electro-Hydraulic Compact Drive" *Energies* 14, no. 9: 2375.
https://doi.org/10.3390/en14092375