# Endoreversible Modeling of a Hydraulic Recuperation System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Endoreversible Formalism

## 3. Model Description

#### 3.1. Hydraulic Fluid

#### 3.2. Pipes and Hydraulic Fluid Tank

#### 3.3. Hydraulic Unit

#### 3.4. Bladder Accumulator

#### 3.5. Pressure Control Valve

#### 3.6. Composite Model

#### 3.7. Driving Dynamics of the Truck

## 4. Energy Savings

#### 4.1. Dynamical Behavior of the System

#### 4.2. Variation of Selected Parameters

#### 4.2.1. Bladder Accumulator Volume and Displacement of Hydraulic Unit

#### 4.2.2. Heat Transfer within the Bladder Accumulator

#### 4.2.3. Heat Transfer to the Environment

#### 4.2.4. Pipe Diameter

## 5. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\alpha $ | inclination of the street | ° |

${\alpha}_{V}$ | coefficient of volumetric thermal expansion | k^{−1} |

$\beta $ | compressibility | Pa^{−1} |

$\gamma $ | displacement factor | - |

$\theta $ | relief flux coefficient | m^{3} s^{−1} |

$\eta $ | efficiency | - |

$\lambda $ | thermal conductivity | W K^{−1} |

$\mu $ | chemical potential | J mol^{−1} |

$\nu $ | kinematic viscosity | m^{2} s^{−1} |

$\rho $ | density | kg m^{−3} |

a | cohesion pressure | Pa m^{6} mol^{−2} |

A | area | m^{2} |

b | co-volume | m^{3} mol^{−1} |

${c}_{\mathrm{d}}$ | drag coefficient | - |

${c}_{p}$ | specific heat capacity at constant pressure | J kg^{−1} K^{−1} |

${c}_{\mathrm{rr}}$ | coefficient of rolling resistance | - |

${c}_{V}$ | specific heat capacity at constant volume | J kg^{−1} K^{−1} |

${\widehat{c}}_{V}$ | dimensionless heat capacity at constant volume | - |

${C}_{p}$ | heat capacity at constant pressure | J K^{−1} |

${C}_{V}$ | heat capacity at constant volume | J K^{−1} |

${f}_{\mathrm{D}}$ | Darcy friction factor | - |

${f}_{\mathrm{m}}$ | mass factor | - |

g | gravitational acceleration | m s^{−2} |

h | specific heat transfer coefficient | W K^{−1} m^{−1} |

I | energy flux | W |

J | extensity flux | * |

k | heat transfer coefficient | W K^{−1} m^{−2} |

K | overall heat transfer coefficient | W K^{−1} |

l | length | m |

m | mass | kg |

M | molar mass | kg mol^{−1} |

n | mole number | mol |

${n}_{\mathrm{cyc}}$ | rotational speed | s^{−1} |

p | pressure | Pa |

P | power | W |

Q | volumetric flow rate | m^{3} s^{−1} |

R | universal gas constant | J mol^{−1} K^{−1} |

Re | Reynolds number | - |

s | energy savings | - |

t | time | s |

T | temperature | K |

u | mean velocity of the fluid | m s^{−1} |

U | internal energy | J |

v | velocity | m s^{−1} |

V | volume | m^{3} |

${V}_{m}$ | molar volume | m^{3} mol^{−1} |

${V}_{\mathrm{d}}$ | displacement | m^{3} |

X | extensity | * |

Y | intensity | * |

* varying unit |

## References

- Sciarretta, A.; Guzzella, L. Control of hybrid electric vehicles. IEEE Contr. Syst. Mag.
**2007**, 27, 60–70. [Google Scholar] - Peng, D.; Zhang, Y.; Yin, C.L.; Zhang, J.W. Combined control of a regenerative braking and anti-lock braking system for hybrid electric vehicles. Int. J. Automot. Technol.
**2008**, 9, 749–757. [Google Scholar] [CrossRef] - Nejabatkhah, F.; Danyali, S.; Hosseini, S.H.; Sabahi, M.; Niapour, S.M. Modeling and Control of a New Three-Input DC-DC Boost Converter for Hybrid PV/FC/Battery Power System. IEEE T. Power Electr.
**2012**, 27, 2309–2324. [Google Scholar] [CrossRef] - Moreno, J.; Ortúzar, M.E.; Dixon, J.W. Energy-Management System for a Hybrid Electric Vehicle, Using Ultracapacitors and Neural Networks. IEEE Trans. Ind. Electron.
**2006**, 53, 614–623. [Google Scholar] [CrossRef] - Ahn, J.K.; Jung, K.H.; Kim, D.H.; Jin, H.B.; Kim, H.S.; Hwang, S.H. Analysis of a regenerative breaking system for hybrid electric vehicles using an electro-mechanical brake. Int. J. Automot. Technol.
**2009**, 10, 229–234. [Google Scholar] [CrossRef] - Pourmovahed, A.; Beachley, N.H.; Fronczak, F.J. Modeling of a Hydraulic Energy Regeneration System: Part I— Analytical Treatment. J. Dyn. Syst. Meas. Control
**1992**, 114, 155–159. [Google Scholar] [CrossRef] - Shan, M. Modeling and Control Strategy for Series Hydraulic Hybrid Vehicles. Ph.D. Thesis, University of Toledo, Toledo, Spain, 2009. [Google Scholar]
- Hui, S.; Ji-hai, J.; Xin, W. Torque control strategy for a parallel hydraulic hybrid vehicle. J. Terramech.
**2009**, 46, 259–265. [Google Scholar] [CrossRef] - Hui, S.; Junqing, J. Research on the system configuration and energy control strategy for parallel hydraulic hybrid loader. Automat. Constr.
**2010**, 19, 213–220. [Google Scholar] [CrossRef] - Chen, Y.L.; Liu, S.A.; Jiang, J.H.; Shang, T.; Zhang, Y.K.; Wei, W. Dynamic analysis of energy storage unit of the hydraulic hybrid vehicle. Int. J. Automot. Technol.
**2013**, 14, 101–112. [Google Scholar] [CrossRef] - Rupprecht, K.R. Hydrospeicher, Experimentelle und Analytische Untersuchungen zur Energiespeicherung. Ph.D. Thesis, Technische Hochschule Aachen, Aachen, Germany, 1988. [Google Scholar]
- Pourmovahed, A.; Baum, S.A.; Fronczak, F.J.; Beachley, N.H. Experimental Evaluation of Hydraulic Accumulator Efficiency With and Without Elastomeric Foam. J. Propulsion
**1988**, 4, 185–192. [Google Scholar] [CrossRef] - Pourmovahed, A.; Otis, D.R. An Experimental Thermal Time-Constant Correlation for Hydraulic Accumulators. J. Dyn. Syst. Meas. Control
**1988**, 112, 116–121. [Google Scholar] [CrossRef][Green Version] - Jou, D.; Casas-Vázquez, J.; Lebon, G. Extended Irreversible Thermodynamics, 4th ed.; Springer: New York, NY, USA, 2010. [Google Scholar]
- Lebon, G. Heat conduction at micro and nanoscales: A review through the prism of Extended Irreversible Thermodynamics. J. Non-Equilib. Thermodyn.
**2014**, 39, 35–39. [Google Scholar] [CrossRef] - Lebon, G.; Jou, D.; Grmela, M. Extended Reversible and Irreversible Thermodynamics: A Hamiltonian Approach with Application to Heat Waves. J. Non-Equilib. Thermodyn.
**2017**, 42, 153–168. [Google Scholar] [CrossRef] - Truesdell, C.; Bharatha, S. The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines; Springer Science & Business Media: Berlin, Germany, 1977. [Google Scholar]
- Matolcsi, T. Ordinary Thermodynamics; Society for the Unity of Science and Technology: Budapest, Hungary, 2017. [Google Scholar]
- Janečka, A.; Pavelka, M. Gradient Dynamics and Entropy Production Maximization. J. Non-Equilib. Thermodyn.
**2018**, 43, 1–19. [Google Scholar] [CrossRef][Green Version] - Hoffmann, K.H.; Burzler, J.M.; Schubert, S. Endoreversible Thermodynamics. J. Non-Equilib. Thermodyn.
**1997**, 22, 311–355. [Google Scholar] - Salamon, P.; Nitzan, A.; Andresen, B.; Berry, R.S. Minimum Entropy Production and the Optimization of Heat Engines. Phys. Rev. A
**1980**, 21, 2115–2129. [Google Scholar] [CrossRef] - Rubin, M.H.; Andresen, B. Optimal Staging of Endoreversible Heat Engines. J. Appl. Phys.
**1982**, 53, 1–7. [Google Scholar] [CrossRef] - De Vos, A. Reflections on the power delivered by endoreversible engines. J. Phys. D Appl. Phys.
**1987**, 20, 232–236. [Google Scholar] [CrossRef] - Gordon, J.M. Observations on Efficiency of Heat Engines Operating at Maximum Power. Am. J. Phys.
**1990**, 58, 370–375. [Google Scholar] [CrossRef] - Hoffmann, K.H.; Burzler, J.M.; Fischer, A.; Schaller, M.; Schubert, S. Optimal Process Paths for Endoreversible Systems. J. Non-Equilib. Thermodyn.
**2003**, 28, 233–268. [Google Scholar] [CrossRef] - Fischer, A.; Hoffmann, K.H. Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak? J. Non-Equilib. Thermodyn.
**2004**, 29, 9–28. [Google Scholar] [CrossRef] - Huleihil, M.; Andresen, B. Optimal piston trajectories for adiabatic processes in the presence of friction. J. Appl. Phys.
**2006**, 100, 114914. [Google Scholar] [CrossRef][Green Version] - Aragón-González, G.; Canales-Palma, A.; León-Galicia, A.; Morales-Gómez, J.R. Maximum Power, Ecological Function and Efficiency of an Irreversible Carnot Cycle. A Cost and Effectiveness Optimization. Braz. J. Phys.
**2008**, 38, 1–8. [Google Scholar] [CrossRef][Green Version] - Kojima, S. Maximum Work of Free-Piston Stirling Engine Generators. J. Non-Equilib. Thermodyn.
**2017**, 42, 169–186. [Google Scholar] [CrossRef] - Paéz-Hernández, R.T.; Chimal-Eguía, J.C.; Sánchez-Salas, N.; Ladino-Luna, D. General Properties for an Agrowal Thermal Engine. J. Non-Equilib. Thermodyn.
**2018**, 43, 131–139. [Google Scholar] [CrossRef] - Andresen, B.; Salamon, P. Distillation by Thermodynamic Geometry. In Thermodynamics of Energy Conversion an Transport; Sieniutycz, S., De Vos, A., Eds.; Springer: New York, NY, USA, 2000; chapter 12; pp. 319–331. [Google Scholar]
- Gordon, J.M. Generalized Power Versus Efficiency Characteristics of Heat Engines: The Thermoelectric Generator as an Instructive Illustration. Am. J. Phys.
**1991**, 59, 551–555. [Google Scholar] [CrossRef] - Wu, C. Maximum Obtainable Specific Cooling Load of a Refrigerator. Energy Convers. Manag.
**1995**, 36, 7–10. [Google Scholar] [CrossRef] - Chen, J.; Andresen, B. The Maximum Coefficient of Performance of Thermoelectric Heat Pumps. Int. J. Power Energy Syst.
**1996**, 17, 22–28. [Google Scholar] [CrossRef] - Wagner, K.; Hoffmann, K.H. Endoreversible modeling of a PEM fuel cell. J. Non-Equilib. Thermodyn.
**2015**, 40, 283–294. [Google Scholar] [CrossRef] - Marsik, F.; Weigand, B.; Thomas, M.; Tucek, O.; Novotny, P. On the Efficiency of Electrochemical Devices from the Perspective of Endoreversible Thermodynamics. J. Non-Equilib. Thermodyn.
**2019**, 44, 425–437. [Google Scholar] [CrossRef] - De Vos, A. Is a solar cell an edoreversible engine? Sol. Cells
**1991**, 31, 181–196. [Google Scholar] [CrossRef] - Schwalbe, K.; Hoffmann, K.H. Optimal Control of an Endoreversible Solar Power Plant. J. Non-Equilib. Thermodyn.
**2018**, 43, 255–271. [Google Scholar] [CrossRef] - Schwalbe, K.; Hoffmann, K.H. Stochastic Novikov engine with time dependent temperature fluctuations. Appl. Thermal Eng.
**2018**, 142, 483–488. [Google Scholar] [CrossRef] - Schwalbe, K.; Hoffmann, K.H. Performance Features of a Stationary Stochastic Novikov Engine. Entropy
**2018**, 20, 52. [Google Scholar] [CrossRef][Green Version] - Schwalbe, K.; Hoffmann, K.H. Novikov engine with fluctuating heat bath temperature. J. Non-Equilib. Thermodyn.
**2018**, 43, 141–150. [Google Scholar] [CrossRef] - Schwalbe, K.; Fischer, A.; Hoffmann, K.H.; Mehnert, J. Applied endoreversible thermodynamics: Optimization of powertrains. In Proceedings of the ECOS 2014—27th International Conference on Efficiency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems, Turku, Finland, 15–19 June 2014; pp. 45–55. [Google Scholar]
- Wagner, K.; Hoffmann, K.H. Chemical reactions in endoreversible thermodynamics. Eur. J. Phys.
**2016**, 37, 015101. [Google Scholar] [CrossRef] - Masser, R.; Hoffmann, K.H. Dissipative Endoreversible Engine with Given Efficiency. Entropy
**2019**, 21, 1117. [Google Scholar] [CrossRef][Green Version] - Muschik, W.; Hoffmann, K.H. Endoreversible Thermodynamics: A Tool for Simulating and Comparing Processes of Discrete Systems. J. Non-Equilib. Thermodyn.
**2006**, 31, 293–317. [Google Scholar] [CrossRef][Green Version] - Wagner, K. An Extension to Endoreversible Thermodynamics for Multi-Extensity Fluxes and Chemical Reaction Processes. Ph.D. Thesis, Technische Universität Chemnitz, Chemnitz, Germany, 2014. [Google Scholar]
- Essex, C.; Andresen, B. The principal equation of state for classical particles, photons, and neutrinos. J. Non-Equilib. Thermodyn.
**2013**, 38, 293–312. [Google Scholar] [CrossRef][Green Version] - Ho, T.H.; Ahn, K.K. Modeling and simulation of hydrostatic transmission system with energy regeneration using hydraulic accumulator. J. Mech. Sci. Technol.
**2010**, 24, 1163–1175. [Google Scholar] [CrossRef] - Stephan, P.; Kabelac, S.; Kind, M.; Mewes, D.; Schaber, K.; Wetzel, T. VDI-Wärmeatlas; Springer: Berlin/Heidelberger, Germany, 2013. [Google Scholar]
- Schweizer, A. Formelsammlung und Berechnungsprogramme Anlagenbau. Available online: https://www.schweizer-fn.de/ (accessed on 18 February 2019).
- Addinol. Sicherheitsdatenblatt ADDINOL Hydrauliköl HLP 46. Available online: http://addinol.oilfinder.net/show_msds.php?id=64764&download=1 (accessed on 18 February 2019).
- Bosch Rexroth AG. Drehzahlvariables Druck- und Förderstrom-Regelsystem Sytronix DFEn 5000. Available online: http://static.mercateo.com/c0/986f0946fb5c4a09a9ec8ea992d97080/pdf/boschrexroth-rd62241.pdf?v=2 (accessed on 18 February 2019).
- Hydac. Proportional-Druckbegrenzungsventil Schieberausführung, vorgesteuert Einschraubventil UNF - 350 bar. Available online: https://www.hydac.com/fileadmin/pdb/pdf/PRO0000000000000000000059913010011.pdf (accessed on 18 February 2019).

**Figure 2.**Endoreversible model of pipe segment (Reservoir 2) and hydraulic fluid tank (Reservoir 1) with combined particle and entropy fluxes between them and towards further pipe segments. Additionally, both reservoirs have irreversible heat transfers to the environment (line below), and the fluid tank is connected to the environment via a reversible volume flux.

**Figure 3.**Endoreversible model of the hydraulic unit as a dissipative engine with given efficiency (Engine 3) and unspecified power input as well as hydraulic fluid tank and adjacent pipe segments (Reservoirs 1, 2 and 4, respectively). The flow directions are drawn according to the flow directions of the hydraulic unit operating in pump mode.

**Figure 4.**Mechanical and volumetric efficiency of the hydraulic unit over rotational speed and pressure difference for $\gamma =0.5$.

**Figure 5.**Endoreversible model of the bladder accumulator with adjacent pipe segment. Reservoirs 9 and 10 represent the hydraulic fluid and the gas within the bladder accumulator, respectively, which are connected via an irreversible heat transfer and a reversible volume flux. Further, there is an irreversible heat transfer from the hydraulic fluid to the environment.

**Figure 6.**Endoreversible model of the pressure control valve represented by an irreversible combined particle and entropy flux between two pipe segments (Reservoirs 6 and 7). The valve housing (Reservoir 8) is connected to Reservoir 7 and the environment via irreversible heat transfers.

**Figure 7.**Composite endoreversible model of the recuperation system with the hydraulic components as described in the previous sections and the pipe segments represented by Reservoirs 2 and 4–7. Lines (instead of rectangles) represent the environment, and intensities as well as all incorporated fluxes are shown.

**Figure 9.**From top to bottom: Displacement factor, gas pressure, mechanical and volumetric efficiency of the hydraulic unit, selected particle fluxes and power shares over time (section).

**Figure 11.**Energy savings both accelerating and decelerating the vehicle over bladder accumulator volume (

**a**) and hydraulic unit displacement (

**b**).

**Figure 12.**Energy savings both accelerating and decelerating the vehicle over heat transfer coefficient within the accumulator (

**a**) and heat transfer factor (

**b**). The latter scales the heat transfers of the pipes segments, the hydraulic fluid within the bladder accumulator and the hydraulic fluid tank to the environment.

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Masser, R.; Hoffmann, K.H.
Endoreversible Modeling of a Hydraulic Recuperation System. *Entropy* **2020**, *22*, 383.
https://doi.org/10.3390/e22040383

**AMA Style**

Masser R, Hoffmann KH.
Endoreversible Modeling of a Hydraulic Recuperation System. *Entropy*. 2020; 22(4):383.
https://doi.org/10.3390/e22040383

**Chicago/Turabian Style**

Masser, Robin, and Karl Heinz Hoffmann.
2020. "Endoreversible Modeling of a Hydraulic Recuperation System" *Entropy* 22, no. 4: 383.
https://doi.org/10.3390/e22040383