Endoreversible Thermodynamics of a Hydraulic Recuperation System

von der Fakultät für Naturwissenschaften der Technischen Universität Chemnitz genehmigte Dissertation zur Erlangung des akademischen Grades

doctor rerum naturalium (Dr. rer. nat.)

vorgelegt von Robin Masser, M. Sc. geboren am 28.06.1991 in Gera eingereicht am 25.03.2019

Gutachter: Prof. Dr. Karl Heinz Hoﬀmann Prof. Dr. Peter Salamon

Tag der Verteidigung: 15.05.2019

Bibliographische Beschreibung

Masser, Robin Endoreversible Thermodynamics of a Hydraulic Recuperation System Technische Universität Chemnitz, Fakultät für Naturwissenschaften Dissertation (in englischer Sprache), 2019 113 Seiten, 40 Abbildungen, 9 Tabellen, 104 Literaturzitate

Referat

In dieser Arbeit verwende ich den Formalismus der endoreversiblen Thermodynamik um ein hydraulisches Rekuperationssystem für Nutzfahrzeuge zu modellieren und zu untersuchen. Dafür führe ich verlustbehaftete Übergänge extensiver Größen zwischen Teilsystemen eines Systems ein. Diese können einerseits der Modellierung von Lecka- gen und Reibungsverlusten, welche als Partikel- oder Drehmomentverluste dargestellt würden, dienen. Andererseits ermöglichen sie die Modellierung einer endoreversiblen Maschine, welche – durch Definition eines solchen verlustbehafteten, internen Über- ganges – ein gegebenes Wirkungsgradkennfeld und daraus resultierende Entropiepro- duktion inne hat. Diese wird infolge zur Modellierung der Hydraulikeinheit des Reku- perationssystems verwendet. Desweiteren basiert die Beschreibung des Rekupera- tionssystems auf der Modellierung der Hydraulikflüssigkeit als Van-der-Waals-Fluid, sodass Druckverluste im endoreversiblen Sinne konsistent berücksichtigt werden kön- nen. Von gegebenen Materialparamtern werden die dafür notwendigen Van-der-Waals- Parameter hergeleitet. Weitere Aspekte sind Wärmeverluste an die Umgebung sowie Wärmeübergänge zwischen Teilsystemen. Auf Grundlage realer Fahrdaten der Nutz- fahrzeuge werden verschiedene dynamische und thermodynamische Effekte im Reku- perationssystem analysiert. Ihr Einfluss auf die resultierenden energetischen Einspa- rungen beim Abbremsen und Beschleunigen wird durch Variation zugehöriger Pa- rameter aufgezeigt. Zuletzt wird mit einem vereinfachten Modell ohne Druck- und Wärmeverluste, aber unter Einbeziehung des Verbrennungsmotors des Fahrzeuges, eine Optimierung der Steuerung des hydraulischen Rekuperationssystems mit Hin- blick auf minimalen Kraftstoffverbrauch durchgeführt. Hier zeigt sich eine erhebliche Verbesserung durch die Leistungsaufteilung zwischen Verbrennungsmotor und Reku- perationssystem nach deren Betriebsbereichen mit maximalem Wirkungsgrad.

Schlagworte

Nichtgleichgewichtsthermodynamik, Endoreversible Thermodynamik, Energierückge- winnung, Hydraulische Speichersysteme, Kompressibles Fluid, Druckverluste, Van- der-Waals-Gas, Optimierung, Wirkungsgradkennfeld

Abstract

In this work I use the formalism of endoreversible thermodynamics to model and investigate a hydraulic recuperation system for commercial vehicles. For that, I introduce lossy transfers of extensive quantities between subsystems of an endoreversible system. On the one hand, these allow modeling of leakages and friction losses, which can be represented as particle or torque losses. On the other hand, they can be used as internal extensity transfers in endoreversible engines which, as a result, have a given efficiency or efficiency map and among other things give an expression for their entropy production. Such an engine is used to model the hydraulic unit of the recuperation system. Furthermore, the description of the recuperation system is based on the modeling of the hydraulic fluid as a van der Waals fluid, so that pressure losses can be taken into account in a consistent endoreversible fashion. From given material parameters the necessary van der Waals parameters are derived. Other aspects of the modeling include heat losses to the environment and heat transfers between subsystems. On the basis of real driving data, various dynamic and thermodynamic effects within the recuperation system are observed and their influence as well as the influence of selected parameters on the resulting energy savings for both acceleration and deceleration are shown. Finally, using a simplified model neglecting pressure and heat losses, but including the internal combustion engine of the vehicle, an optimization of the control strategy for the hydraulic recuperation system with regard to minimum fuel consumption is performed. Here, a significant improvement due to a power distribution between combustion engine and recuperation system according to their high efficiency operating ranges can be achieved.

Keywords

Non-Equilibrium Thermodynamics, Endoreversible Thermodynamics, Energy Recov- ery, Hydraulic Storage Systems, Compressible Fluid, Pressure Losses, Van der Waals Gas, Optimization, Eﬃciency Map

Contents

1 Introduction9

2 Endoreversible Systems 13 2.1 General Formalism...... 13 2.1.1 Subsystems...... 14 2.1.2 Interactions...... 16 2.1.3 Multi-Extensity Fluxes...... 19 2.1.4 Enthalpy Equivalence...... 23 2.2 Endoreversible Engine Setups...... 25 2.2.1 Reversible Engines...... 25 2.2.2 Irreversible Engines...... 28 2.2.3 Engine with Lossy Extensity Transfer...... 31 2.2.4 Dissipative Engine with Given Eﬃciency...... 34 2.3 Van der Waals Gas...... 39 2.4 Summary...... 44

3 Recuperation System 47 3.1 Model Description...... 47 3.1.1 Hydraulic Fluid...... 49 3.1.2 Pipes and Hydraulic Fluid Tank...... 52 3.1.3 Hydraulic Unit...... 56 3.1.4 Bladder Accumulator...... 60 3.1.5 Pressure Control Valve...... 62 3.1.6 Composite Model...... 64 3.1.7 Driving Dynamics of the Truck...... 66 3.2 Energy Savings...... 67 3.2.1 Dynamical Behavior of the System...... 68 3.2.2 Variation of Selected Parameters...... 71 3.3 Summary...... 76

4 Optimizing Fuel Savings 79 4.1 Model Description...... 79 4.1.1 Combustion Engine and Transmission...... 83 4.1.2 Optimization...... 87

7 4.2 Optimized Control...... 88 4.3 Summary...... 93

5 Conclusion 95

Nomenclature 99

Bibliography 103

8 1 Introduction

In the early 19th century, Sadi Carnot was the one who for the first time described the relationships between operating temperatures, heat transfer and mechanical work in the steam engines emerging at that time [1]. By abstracting the then-known essential features of those engines to an ideal and more general model, he was able to make clear predictions about their maximum attainable efficiency. He showed that this maximum efficiency for his idealized cycle – both later named after him – depends only on the operating temperatures T1 > T2 and results to ηC = 1 − T2/T1. This is often referred to as the beginning of classical thermodynamics, which deals with reversible processes and, more precisely, with thermal systems in equilibrium. Due to the reversibility within the systems considered with this classical equilibrium thermodynamics approach, statements about theoretical upper bounds for efficiencies and power outputs could be made. However, this also meant that the processes described would have to take place in infinite time and at zero rates in order to achieve the reversibility and thus the predicted efficiencies. Of course, real processes take place in finite time and it was necessary to incorporate irreversible effects in order to be able to describe them more precisely. Since then, a variety of approaches have been developed to take irreversibilities into account in thermodynamic descriptions. There have been developments named Onsager theory [2], Eckart’s theory [3–5], extended thermodynamics or finite-time thermodynamics, just to name a few. A comprehensive overview of these and other approaches has been given by Müller et al. [6]. A particularly interesting ansatz, which was already used by Reitlinger [7], Novikov [8] and Chambadal [9], as well as Curzon and Ahlborn [10] was given the name “endoreversible” thermodynamics by Rubin in 1979 [11,12]. The basic idea is to consider subsystems as reversible acting engines and as equilibrated reservoirs, and to link them together using irreversible interactions [13]. Thus, while all essential irreversibilities can be incorporated into the system in question, the knowledge of equilibrium thermodynamics remains applicable. As a result of that, entropy production is restricted to occur only in interactions between the subsystems. For these interactions typically the phenomenological relationships for heat transfer, friction, pressure loss and the like are used. Furthermore, the degree of detail can be increased almost arbitrarily by adding additional subsystems. Endoreversible thermodynamics thus represents a

9 1 Introduction suitable tool for the modeling of complex and dissipative systems, which, however, remain mathematically manageable while providing reliable predictions. Numerous scientists used this approach to model and investigate technical and natural systems of all types. The Carnot cycle has been studied in many variations, e. g. with different [14, 15] and generalized heat transfer laws [16–20]. Further, Chen et al. and Rubin et al. modeled sets of two and more staged Carnot engines [21–23]. Additionally heat leaks causing a loop-type characteristic in the power efficiency plot have repeat- edly been subject of research regarding the Carnot engine [24–27]. The minimum entropy production subject to thermal resistance losses was studied for heat engines in general in [28]. Andresen and Salamon modeled binary distillation processes [29] and a broad investigation on the effects of stochastic fluctuations in the heat baths of a Novikov engine was recently done by Schwalbe and Hoffmann [30–33]. More but equally large fields of endoreversible modeling are spanned by investigations on thermoelectric generators [34–39] and solar thermal heat engines [40–43]. In 2004 Fischer and Hoffmann showed that by the right choice of parameters the complex “Otto cycle can be accurately rendered by a Novikov engine with heat leak” emphasizing the usefulness and the capabilities of endoreversible thermodynamics [44]. The optimization of engines or systems with regard to performance, efficiency, entropy production or economic aspects is often done via adjusting the parameters describing transfers, reservoirs or engine operation. In the latter case, a very interesting and promising option to optimize heat engines with pistons is the optimization of the piston trajectories [20,45,46]. In the past, such optimization has been done for engines with Otto [47–49], Diesel [50–53], Miller [54] and Brayton [55] cycles as well as light- driven engines [56–58] showing substantial improvements of the engine efficiencies or power outputs. To do this, investigators typically rely on control theory to obtain the optimized piston trajectories. However, the application of endoreversible thermodynamics is by no means limited to heat engines. Gordon and Zarmi investigated the upper bound on global wind energy production [59]. The model they introduced was extended by other scientists [60, 61] and was even applied to planets in the solar system [62]. Chemical reactions have also been studied using endoreversible thermodynamics, initially without modeling the reaction processes themselves [63], later by introducing a chemical reactor as an endoreversible engine managing particle fluxes of reactants and products during the reactions [64,65]. Using economic parameters, De Vos even modeled the flows of goods at a market in analogy to the description of heat engines [66]. Irreversibilities in thermodynamic systems are generally not limited to finite heat transfer, but can be caused by friction or regenerator losses in Stirling engines, for instance. In order to add entropy generation to the system independent of external heat losses, engine models with internal losses have been investigated in the past. For example, using only one parameter, internal entropy production of a heat engine may be

10 proportional to an incoming entropy flux, which causes qualitatively different effects than external heat losses. Among many others, internally irreversible engines have been modeled and investigated in [67–75]. Such internal irreversibilities are mathematically easy to implement by replacing the engines balance equation. However, the entropy production within an engine contradicts the initial idea of endoreversible thermodynamics. The goal to improve the performance of the systems in question usually leads to a reduction in costs and consumption of resources, and thus has a positive effect on the environmental sustainability of these systems which is becoming increasingly important today. A global trend towards energy conservation and recovery of otherwise wasted energy has reached all areas of research, engineering and everyday life. Cer- tainly the most popular movement – because it is the most perceptible one in everyday life – is driven by the vehicle manufacturers. The reputation of combustion engines is slowly decreasing and hybrid techniques or the complete replacement by e. g. electric motors are gaining in interest. This is not least due to the fact that hybrid or fully electric drive systems are best suited to recover braking energy and reuse it for acceleration – tremendously enhancing the overall efficiency of the vehicle. The starting point of this work is a research project that deals with a hybrid drive system for commercial vehicles with tipper bodies. More specifically, the hybrid propulsion system is a hydraulic energy recovery system that extends the internal combustion engine drive train. On the one hand, the use of a hydraulic system makes sense since the vehicle already has a hydraulic system propelling the tipping body. On the other hand, it has a particularly high energy and power density compared to electrical systems which makes it suitable to absorb high energy fluxes when braking. The brake energy, which is absorbed using a hydraulic unit mounted on the cardan shaft, is stored in a hydraulic accumulator. This not only reduces the wear of the conventional disc brakes, but the energy stored can be reused to accelerate the vehicle as well as to operate the auxiliary units. Electric and mechanical hybrid systems have been extensively studied in recent decades [76–80]. Hydraulic hybrid systems have also been part of a number of stud- ies [81–85], as well as their individual components such as the hydraulic accumulator [86–88]. The focus of those investigations, however, was less on an overall thermodynamic description of the system, but rather on appropriate control strategies for different hybrid types. An endoreversible model of a hydraulic storage system to investigate the influence of dissipative heat transfers within the hydraulic accumulator was designed by Schwalbe et al. in 2014 [89]. However, this model is a first, not very detailed model, for which some simplifying assumptions have been made that may neglect some very important effects regarding the performance of the overall system. In this work, I will develop a more detailed endoreversible description of a hydraulic recuperation system in terms of heat losses, pressure losses, and loss sources of the

11 1 Introduction individual hydraulic components such as the hydraulic unit and a pressure relief valve. This allows an analysis of the effects of the individual loss terms on the overall system and its performance as well as their mutual influences. Additionally, in order to simplify the modeling of dissipative engines in endoreversible systems, I will introduce a dissipative engine model that allows for incorporation of internal dissipations in a consistent manner. Here, the approach is to define the dissipations occurring in a real engine, such as friction, leakages or heat leaks, as lossy interactions. Finally, a fuel consumption minimization will be carried out. For this, the endoreversible model is coarsened again and certain loss terms are neglected, while on the other hand the efficiency map of the internal combustion engine of the vehicle is taken into account. Thus, the interplay between the hydraulic unit and the internal combustion engine will be decisive for maximizing the efficiency of the recuperation system. To work through the points mentioned above, I structured this document as follows. In Chapter2, I will give a brief description of the endoreversible formalism as described in the review article of Hoffmann et al. [13], and of Wagner’s extension for multi- extensity fluxes given in [65]. Furthermore, I will introduce the dissipative engine model and explain its suitability to model dissipative engines with given efficiencies. Finally, the principal equations of state for the van der Waals gas will be derived, which are necessary for the modeling in the following Chapter3. Here, I will describe the individual parts of the hydraulic recuperation system as well as the overall system and its functionality. Endoreversible models for this will be presented and the system will be investigated, in particular with regard to estimated energy savings using real driving data. In Chapter4, on the one hand, I will simplify the previous presented model of the recuperation system and, on the other hand, add the combustion engine of the vehicle to the system. Operation strategies that lead to a fuel consumption minimization for a given driving profile will be investigated. In the last chapter, Chapter5, a summary of the findings of my work as well as an outlook on further applications of the presented endoreversible modeling strategies for hydraulic systems will be given.

12 2 Endoreversible Systems

Endoreversible thermodynamics is a formalism of reversible subsystems which are connected by reversible and irreversible interactions. Hence, entropy production occurs in interactions only. In this chapter a description of the endoreversible thermodynamics is given, following the review article by Hoﬀmann [13] and the work of Wag- ner [64, 65]. The equivalence of enthalpy transfer to combined particle and entropy transfer is shown, and a model to incorporate dissipative engines with given eﬃciency is introduced. Lastly, the principal equation of state for the van der Waals gas is derived.

2.1 General Formalism

The formulation of endoreversible thermodynamics is based on reversible acting subsystems and fluxes of energy or extensive quantities between them. Extensive quantities, short extensities, are quantities that scale with the system size, such as entropy, volume or particle number. In this work, the extensities of a subsystem i are denoted α by Xi , where α defines the specific type of extensity. Further, for each extensity there α is a conjugate intensive quantity Yi which does not scale with the system size, such as temperature, pressure or chemical potential.

If a subsystem i has an energy Ei, according to the Gibbs equation the change in its energy can be expressed as the sum of changes in extensities times the corresponding intensities: X α α dEi = Yi dXi . (2.1) α α A change in extensities of the subsystem can e. g. be caused by an extensity flux Ji out of or into the subsystem. Analogous to Eq. (2.1), the accompanying flux of energy that is carried by an extensity flux is then given by

α α Ii = Yi Ji . (2.2)

It should be mentioned that energy ﬂuxes never occur on their own without a carrying extensity ﬂux. For example, heat can only be transferred when an entropy transfer

13 2 Endoreversible Systems is involved. However, in endoreversible descriptions, it may happen that the carrying extensity ﬂux is not considered in detail, i. e. it does not matter which type of extensity is carrying the transferred energy. On the other hand, an extensity ﬂux can be transferred without energy being transmitted if the accompanying intensity is zero.

2.1.1 Subsystems

Each subsystem of an endoreversible model has a set of contact points through which it exchanges energy with other subsystems. These contact points are numbered by k. Since the transferred energy is carried by an extensity – or more than one extensities, α as will be discussed in Sec. 2.1.3 – the corresponding intensity Yi,k is assigned to that contact point. Furthermore, the subsystems are divided into three functionally distinct categories, namely reservoirs, engines and chemical reactors. In the case of reservoirs, the associated intensity at a contact point is the intensive quantity of the reservoir. Reservoirs act as storages for energy and extensities, and represent equilibrated systems. We can further distinguish between finite and infinite reservoirs. When a reservoir is finite, its state is given by its energy as a function of its extensive quantities α Ei = Ei(Xi ), (2.3) and its intensive quantities can be calculated as the partial derivatives of the energy

α α ∂Ei(Xi ) Yi = α . (2.4) ∂Xi Thus, the state of the subsystem varies with the variation of its extensities which in general can be expressed as ˙ α X α α Xi = Ji,k + Σi , (2.5) k α α where Ji,k are fluxes into or out of the subsystem at contact points k and Σi is an extensity generation or destruction term. The latter might be necessary for non-conserved extensive quantities e. g. when chemical reactions occur within the subsystem. If we consider an infinite reservoir, the intensities remain constant independent of the fluxes into or out of the reservoir. It represents an infinite large system with fixed intensive quantities and its state is thus described by them. Typical examples for such infinite systems are heat baths with constant temperature or reservoirs with constant pressure.

14 2.1 General Formalism

Engines, on the other hand, can neither store energy nor extensities. They are meant to convert energy between diﬀerent incoming and outgoing extensity ﬂuxes. Since engines work reversibly, no energy or extensity is generated or destroyed within them. Thus, the balance equations for extensities and energy for the engine i can be written as

X α 0 = Ji,k for all α and (2.6) k X X α α 0 = Ii,k = Yi,kJi,k, (2.7) k α,k respectively, where k denotes the k-th contact of the engine. Note that the summation over α in Eq. (2.7) is necessary if more than one extensities are transferred at a contact point (see Section 2.1.3). It is also possible to consider cyclically operating engines, for which energy and extensity transfers balance over the cycle time tcyc. In this case, the following set of equations holds for the cyclic engine i:

Z tcyc X α 0 = Ji,k dt for all α and (2.8) 0 k Z tcyc Z tcyc X X α α 0 = Ii,k dt = Yi,kJi,k dt. (2.9) 0 k 0 α,k

Unlike in reservoirs, the intensities associated with the contact points of an engine are freely selectable for each contact point. This is an important feature that allows a reversible connection between reservoirs with different states, as needed e. g. to model a perfect regenerator in a Stirling engine. The chemical reactor which is the third type of endoreversible subsystems and which was introduced by Wagner [64] is slightly different from the endoreversible engine. However, this subsystem type as well as chemical reactions in principal are not considered and hence not explained in detail in this work. Another point that should be mentioned here is that the endoreversible formalism is by no means limited to classical thermodynamic quantities entropy, volume and particle number. Among others, mechanical quantities such as momentum and angular momentum, or electrical quantities such as charge can easily be incorporated, too. α Setting the properties of a subsystem i still occurs via the definition of Ei(Xi ) which can include e. g. translational energy, rotational energy and electric potential energy. And since Eq. (2.4) applies, the corresponding intensities velocity, angular velocity or electric potential, respectively, can be derived.

15 2 Endoreversible Systems

2.1.2 Interactions

Interactions in endoreversible thermodynamics are used to describe how energy and extensities are exchanged between the subsystems. An interaction is always assigned to exactly one contact point of each subsystem that is linked via this interaction. It is thus characterized by the set of contact points that belong to the interaction and by the speciﬁc type (or types) of extensity carrying the exchanged energy. Since interactions are not meant to produce or destroy energy, too, energy is balanced over all contact points linked by an interaction. The same holds for all extensities except entropy. Whether an interaction is reversible or irreversible, i. e. generates entropy or not, depends on its deﬁnition. Here, we have the following two main options. First, the intensities at the contact points connected via the interaction obey

α α Yi,k = Yj,l for all α, (2.10) where k and l denote connected contact points of the subsystems i and j, respectively. In this case the interaction is reversible and the transfer rates between the subsystems can become infinitely large in order to ensure an instantaneous equilibration of the intensities. Second, a transport law is used to define either the flux of the interaction’s extensity or the flux of energy. This transport law may be a function of intensities, extensities (in the case of reservoirs) and additional parameters. A typical example is Newton’s heat transfer law which is a function of the intensities, the temperatures of the connected subsystems, with an additional parameter, the heat transfer coefficient, defining the energy that is exchanged between the subsystems. While in the case of a reversible interaction, as shown in Fig. 2.1a, the amounts of both the extensity and the energy fluxes are equal at all contact points

α α α J1 = J = −J2 , (2.11) α α I1 = Y J = −I2, (2.12) in the case of an irreversible heat transfer, displayed in Fig. 2.1b, only energy conversion holds

I1 = K(T2 − T1) = −I2, (2.13) where K is the heat transfer coeﬃcient and T1 and T2 are the temperatures of reser- S voir 1 and 2, respectively. With the entropy ﬂux J1 = I1/T1 entering or leaving

16 2.1 General Formalism

(a) (b) J α J S J S α α 1 2 Y Y T1 T2 1 2 1 2

Figure 2.1: Reversible interaction with arbitrary extensity (a) and irreversible interaction with heat transfer (b) between two subsystems. subsystem 1, the entropy ﬂux into or out of reservoir 2 is given by

S I2 S T1 S S T2 − T1 J2 = = −J1 = −J1 + J1 , (2.14) T2 T2 T2 | {z } σ where σ is the entropy generation due to the difference in intensities of the subsystems. To clarify the conventions in this work, the fluxes associated to the contact points are defined so that a positive flux corresponds to an increase in the extensive quantity of the reservoir or to an incoming flux into an engine. Hence, there is no specific direction assigned to the interaction since the actual direction of transfer can be reversed, e. g. if the intensities change. In illustrations, therefore, a double-headed arrow is used unless the transfer is physically feasible in one direction only. Further, as already done above, the index k is omitted, if the subsystem has only one contact point. For the Newtonian heat transfer we saw that the entropy generated is automatically transferred to the contact point with lower temperature. However, if contact points with extensity type α other than entropy are involved in an irreversible interaction, as shown in Fig. 2.2a, an additional entropy reservoir is needed to receive the entropy generated. In this case, the extensity flux is conserved

α α α J1 = J = −J2 (2.15) but the energy balance is not fulﬁlled

α α α 0 6= I1 + I2 = J (Y1 − Y2 ) = ∆I. (2.16)

Hence, the excess energy ∆I escapes into a reservoir with temperature T3 carried by the entropy ﬂux S ∆I J3 = − = σ, (2.17) T3 where σ, again, is the entropy generated.

17 2 Endoreversible Systems

(a) (b) α α α α J α α J1 J2 α Y1 Y2 Y Y

1 2 1 S 2 J S α 4 J3 J3 T4 α T3 Y3 3 3 4

Figure 2.2: Irreversible interaction between reservoirs with diﬀerent intensities (a) and lossy interaction between reservoirs with equal intensities where a loss extensity transfer occurs towards a third reservoir (b). In both cases generated entropy is transferred to another additional reservoir.

Yet, there is a third option to define an interaction between two subsystems. We set this interaction so that reservoir 1 and 2 have the same intensity, but the transfer of extensities or the transfer of energy is lossy to some amount. Hence, the fluxes entering or leaving those reservoirs do not balance and a reservoir receiving the excess extensity flux α α α − J3 = J1 + J2 (2.18) is needed, as shown in Fig. 2.2b. Additionally, if this reservoir has a different intensity, energy balance is not yet fulfilled

α α α α α α α α 0 6= Y (J1 + J2 ) + Y3 J3 = J3 (Y3 − Y ) = ∆I (2.19) and in analogy to the previous case, generated entropy σ is transferred to – in general

– another subsystem 4 with temperature T4 by the entropy ﬂux

S ∆I J4 = − = σ. (2.20) T4

Such an interaction may find application especially in mechanical systems. Consider two components with angular momentum, e. g. two rotating flywheels, which are connected by a solid shaft and hence have equal angular velocities. We assume that there is no torsion within the shaft, so an reversible interaction transferring angular momentum between two reservoirs with the same angular velocities would describe this setup. However, if the shaft is supported by a bearing causing friction loss, the above described lossy interaction can be used. The lost angular momentum is transferred to the bearing which can be described by an infinite reservoir with zero angular velocity, since the bearing is typically mounted to a housing or to the floor. Then, the dissipated energy is represented by an entropy flux into a heat bath with ambient

18 2.1 General Formalism

α α α J1 J2 α Y1 Y2 1 2 α S J3 J4 α Y3 T4 3 4

Figure 2.3: Lossy interaction between reservoirs with different intensities. The loss extensity and the generated entropy are transferred towards a third and fourth reservoir, respectively. temperature. If a combination of both different intensities and different fluxes at the contacts of an interaction is used, as shown in Fig. 2.3, the balance equations for this interaction are given by

α α α J1 + J2 = −J3 , (2.21) α α α α α α 0 6= Y1 J1 + Y2 J2 + Y3 J3 = ∆I, (2.22) α α α α α α S ∆I J2 (Y1 − Y2 ) J3 (Y1 − Y3 ) J4 = − = + = σ, (2.23) T4 T4 T4 where the generated entropy σ has to be positive for physically feasible processes. I want to emphasize here, that for the described lossy interaction an additional law α defining the loss flux of the specific extensity – J3 in the figures above – is needed. In the case of the bearing with friction considered above, this could be a function of the intensities of the reservoirs involved. Of course, dependencies on other extensities or additional parameters are generally possible, too. Here, an important feature is that the resulting energy loss occurs regardless of the direction of the main energy exchange between two subsystems.

2.1.3 Multi-Extensity Fluxes

In particular, when considering heat engines, one typically has to deal solely with heat transports and thus only with single-extensity ﬂuxes. In many real systems, however, multi-extensity ﬂuxes must be taken into account. A stream of gas or liquid, for instance, may combine a particle transfer with an entropy transfer, a volume transfer

19 2 Endoreversible Systems

(a) β β α (b) S S β J = f J β J1 J2 Y Y T1 T2 α α Y α Y µ1 n n µ2 J J1 J2 1 2 1 2

Figure 2.4: Reversible interaction with multi-extensity flux of arbitrary coupled extensities (a) and irreversible interaction with combined particle and entropy transfer (b) between two reservoirs. or others. Those transferred extensities are coupled and have a fixed ratio which can be expressed as β β α Ji,k = fi,kJi,k, (2.24) α where Ji,k is the reference flux of extensity α entering or leaving subsystem i at contact β point k and Ji,k is a flux of extensity β that is coupled to α. The coupling coefficient, β which is determined by the choice of α and β is fi,k. An often suitable choice for the reference flux is the particle flux expressed in mol s−1. For this choice, to give a few examples, the following corresponding fluxes would result from multiplication of the reference flux by the coefficients:

S n Ji,k = Sm,iJi,k, (2.25) m n Ji,k = MiJi,k, (2.26) Q n Ji,k = ziFJi,k, (2.27) V n Ji,k = Vm,iJi,k, (2.28)

S m Q V where Ji,k, Ji,k, Ji,k and Ji,k are fluxes of entropy, mass, charge and volume, respectively, entering or leaving a reservoir i at contact point k. They are coupled to the n reference particle flux Ji,k at named contact point, and their coefficients are the mo- S m lar entropy fi,k = Sm,i, the molar mass fi,k = Mi, the charge number times Faraday Q V constant fi,k = ziF and the molar volume fi,k = Vm,i of the reservoir i, respectively. In the case of a reservoir, the coupling coefficients are equal for all contact points and represent the molar extensive quantities describing the properties of the matter of that reservoir. However, in the case of an engine, these coefficients can be chosen freely for each contact point. Only in the case of a reversible interaction, where we have the same intensities at both ends of the interaction, as shown in Fig. 2.4a, the coupling coefficients at the connected contact points are equal. Often, however, at least one of the intensive quantities of the connected subsystems differ and the interaction is irreversible and hence generates entropy. In analogy to the single-extensity transfers discussed earlier, this generated

20 2.1 General Formalism entropy is then transferred to a third subsystem, or to one of the connected subsystems if an entropy transfer is involved. In Fig. 2.4b, a multi-extensity interaction transferring particles and entropy between reservoirs with different chemical potentials and temperatures is shown. Here, for instance, the difference in the chemical potentials ∆µ = µ2 − µ1 could drive the n n n particle flux J1 = J1 (∆µ) = −J2 . If this leads to a flow direction from reservoir 1 to reservoir 2, we would logically define the coupled entropy flux leaving reservoir 1 as

S n J1 = Sm,1J1 , (2.29) where Sm,1 = S1/n1 is the molar entropy of reservoir 1. For the entropy ﬂux entering reservoir 2, however, in general one has

S n J2 6= Sm,2J2 , (2.30) with Sm,2 = S2/n2 being the molar entropy of reservoir 2. Here, the actual entropy ﬂux results from the energy balance of the interaction to

S S n Sm,1(T2 − T1) + (µ2 − µ1) J2 = −J1 + J1 , (2.31) T2 | {z } σ where σ is the entropy generated. Thus, the definition of this interaction actually S n S n depends on the flow direction. The latter decides whether J1 = Sm,1J1 or J2 = Sm,2J2 is set, and the generated entropy is transferred to reservoir 2 or 1, respectively. S n S n Note that in the case of setting both J1 = Sm,1J1 and J2 = Sm,2J2 for the example discussed above, neither entropy nor energy balance are fulfilled. Therefore, an additional entropy reservoir for necessary or excess entropy and energy has to be added. However, modeling such a process using an engine might be a more appropriate approach. Another point to be mentioned here regards the peculiarity of the endoreversible description: The described multi-extensity transfer can be defined e. g. for a pressure dependent gas flow between two gas containers which are connected by a tube. The gas flow consists of particles and the entropy transferred with them. What one typically has in mind is an influx of the gas from one container into the other, and a subsequent mixing of the incoming gas and the previously present gas. However, since equilibrated subsystems are considered in endoreversible modeling, the mixing process is considered to be completed at the moment of entry. Hence, the gas leaves and enters the containers in their respective states and the entropy generated by this state change – representing the mixing process – is transferred to the subsystem the gas flows into.

21 2 Endoreversible Systems

Comparing the entropy generation given in Eq. (2.31) with that of the irreversible heat transfer given in Eq. (2.14), we notice an additional term due to the diﬀerence in the chemical potential. In general, for every additional coupled ﬂux of extensity being transferred in an irreversible multi-extensity interaction, a corresponding term for entropy generation is added leading to an overall entropy production

β β X Yj,l − Yi,k σ = J α f β , (2.32) i,k i,k T β m,n

α where Ji,k is the reference ﬂux of extensity α leaving from contact point (i, k) to contact β β point (j, l), and Yi,k and Yj,l are the intensities at those contact points, respectively. α Of course, for β = α a coupling coeﬃcient fi,k = 1 has to be used. In general Tm,n is the temperature at a contact point n of a third subsystem m, to which the generated entropy is transferred. However, if the interaction transfers entropy and the generated entropy is transferred to subsystem j – as in the example above – this temperature is the temperature Tj,l. Furthermore, I want to take a look at lossy multi-extensity interaction, for which a simple case with particle and entropy transfer is shown in Fig. 2.5. Here, “lossy” means n n that the particle transfer that leaves reservoir 1 with J1 and enters reservoir 2 with J2 n is subject to some sort of leakage J3 resulting in

n n n J1 = −J2 − J3 . (2.33)

The total amount of the reference extensity ﬂux leaving one subsystem is not entering a second subsystem, since some leakage occurs towards a third subsystem. And thus, an amount of the coupled extensities – here entropy – corresponding to the share of leakage is transferred along. S S Now, the simplest approach to calculate J1 and J2 is to split the energy balance of the interaction given by

n n n S n S 0 = J1 µ1 + J1 Sm,1T1 + J2 µ2 + J2 T2 + J3 µ3 + J3 T3, (2.34)

S n where J1 = Sm,1J1 was used, into

n n n S 0 = −J2 µ1 − J2 Sm,1T1 + J2 µ2 + J2 T2, (2.35) n n n S 0 = −J3 µ1 − J3 Sm,1T1 + J3 µ3 + J3 T3. (2.36)

Here, we used Eq. (2.33) and separated the energy components. Solving for the entropy

22 2.1 General Formalism

S S J1 J2 T1 T2 µ1 n n µ2 J1 J2 1 2 n S J3 J3

µ3 T3 3

Figure 2.5: Lossy interaction with multi-extensity flux of particles and entropy between two reservoirs. The loss flux which is a combined particle and entropy flux occurs towards a third reservoir.

flux into reservoir 2 and 3 one obtains S n T1 (µ1 − µ2) n n Sm,1 (T2 − T1) + (µ2 − µ1) J2 = J2 Sm,1 + = Sm,1J2 + J2 , (2.37) T2 T2 T2 | {z } σ1 S n T1 (µ3 − µ1) n n Sm,1 (T3 − T1) + (µ3 − µ1) J3 = J3 Sm,1 + = Sm,1J3 + J3 , (2.38) T3 T3 T3 | {z } σ2 respectively. The entropy production terms σ1 and σ2 fit well with the expression given in Eq. (2.32) if we consider two independent fluxes from reservoir 1 to reservoirs 2 and from reservoir 1 to reservoir 3 – which is what we mathematically did. Thus, a two- part modeling of this interaction is just as expedient. However, modeling them as one lossy interaction can provide a pedagogical advantage and make actual processes more understandable.

2.1.4 Enthalpy Equivalence

We will now take a brief look at Gibbs free energy and the equivalence of enthalpy transfers and combined particle and entropy ﬂuxes. The Gibbs free energy G is a thermodynamic potential with the natural variables temperature T , pressure p and mole numbers ni composing the i-th component of the system. It describes a system in contact with reservoirs that maintain the temperature and pressure of the system at constant values. Resulting from a Legendre transformation of the internal energy U we can write the Gibbs free energy as

G = U + pV − TS, (2.39)

23 2 Endoreversible Systems where V and S are the volume and entropy of the system. According to this, the total diﬀerential can be expressed as

dG = dU + pdV + V dp − T dS − SdT. (2.40)

Using the phenomenological fundamental equation of thermodynamics X dU = T dS − pdV + µidni, (2.41) i where µi is the chemical potential of the i-th component of the system, Eq. (2.40) can be written as X dG = V dp − SdT + µidni. (2.42) i

The extensive quantities S, V and ni allow a simple integration of dU given by Eq. (2.41). Thus, we obtain the expression for the internal energy of a homogeneous system X U = TS − pV + µini. (2.43) i Inserting this into the deﬁnition of the Gibbs free energy leads to the equation X G = µini. (2.44) i This can also be derived using Euler’s homogeneous function theorem and the expression ∂G = µ (2.45) ∂n i i T,p,nj6=i which can be concluded from Eq. (2.42). For a one-component system and with n = 1, Eq. (2.44) can by simpliﬁed to

Gm = µ, (2.46) where Gm is the molar Gibbs free energy of the system. Another expression for Gm can be derived by taking Eq. (2.43) in molar terms, namely

Gm = Um + pVm − TSm, (2.47) where Um, Vm and Sm are the molar internal energy, molar volume and molar entropy of the system in question. Now, if we consider a subsystem i with mole number and entropy as extensive quan- n S tities, their time derivatives owing to a coupled particle ﬂux Ji and entropy ﬂux Ji

24 2.2 Endoreversible Engine Setups leaving this subsystem are given by

n n˙ i = Ji , (2.48) ˙ S n Si = Ji = Sm,iJi (2.49) where the molar entropy of this subsystem Sm,i is the coupling coeﬃcient. Note that the index i now refers to the subsystem, again. Utilizing Equations (2.46) and (2.47) the time derivative of the subsystem’s energy can thus be formulated as

˙ n S Ei = Ji µi + Ji Ti (2.50) n = Ji (µi + TiSm,i) (2.51) n = Ji (Um,i + piVm,i) (2.52) n = Ji Hm,i. (2.53)

Here, µi and Ti are the chemical potential and the temperature of the subsystem i. Its molar internal energy, pressure, molar volume and molar enthalpy are denoted by Um,i, pi, Vm,i and Hm,i respectively. The last step was done since H = U + pV as well as its molar expression hold. This equivalence is of particular importance because in engineering applications enthalpy flows are usually used to describe movements of gases or fluids. As shown above, these movements can be described in endoreversible modeling as coupled entropy and particle fluxes without loss of generality. Or vice versa, coupled entropy and particle fluxes in an endoreversible thermodynamic description might be replaced by enthalpy fluxes, if beneficial. I will refer to this option in Chapter4.

2.2 Endoreversible Engine Setups

Below I will briefly explain reversible and irreversible engine configurations using the Carnot engine and the Novikov engine as simple examples, respectively. One way of extending such engine setups to further dissipative effects will be explained using the example of the Novikov engine with added heat leak. Furthermore, we will deal with non-heat engine setups, especially an engine setup with a lossy extensity transfer, and based on that a model which allows the use of any given efficiency function.

2.2.1 Reversible Engines

Fig. 2.6a shows a full model of a reversible engine converting energy from extensity α to β or vice versa. The engine itself (subsystem 5 which is drawn as a circle) is

25 2 Endoreversible Systems

(a) (b) α α Y1 1 Y1 1 J α J α Y α Y α β 5,1 β 5,1 β J β β J β P Y3 Y5,3 5 Y5,4 Y4 5 Y α Y α 3 5,2 4 5,2 J α J α α α Y2 Y2 2 2

Figure 2.6: Reversible engine setups with specified extensity flux J β as power output carrier (a) and unspecified power output P (b) between four and two reservoirs with different intensities, respectively. between the reservoirs 1 to 4 which are reversibly connected to it. The intensive α α β β quantities of these reservoirs are Y1 , Y2 , Y3 and Y4 , respectively. The intensities α α β at the contact points of the engine towards reservoir 1, 2, 3 and 4 are Y2,1, Y2,2, Y2,3 β and Y2,4, respectively. Since the subsystems are reversibly connected, the intensities on both ends of each interaction are equal

α α Y5,1 = Y1 , (2.54) α α Y5,2 = Y2 , (2.55) β β Y5,3 = Y3 , (2.56) β β Y5,4 = Y4 . (2.57)

Due to the fact that extensities are conserved in interactions as well as in endoreversible engines, the following equations hold

α α α J1 = −J = −J5,1, (2.58) α α α J2 = J = −J5,2, (2.59) β β β J3 = −J = −J5,3, (2.60) β β β J4 = J = −J5,4. (2.61)

26 2.2 Endoreversible Engine Setups

Then, the energy balance within the engine results to

α α α β β β J (Y1 − Y2 ) = J Y3 − Y4 . (2.62)

We now want to calculate the eﬃciency of this engine which is considered to operate in a steady state, and we assume that energy is converted from extensity α to β. Hence, α α for Y1 > Y2 the input energy transfer, or input power, can be deﬁned as the energy transfer entering the engine from reservoir 1

α α α α Pin = J5,1Y5,1 = J Y1 , (2.63)

β β while with Y3 < Y4 the power output can be defined as the energy per time added to the extensity flux of extensity β flowing trough the engine

β β β β β β β Pout = J5,3Y5,3 + J5,4Y5,4 = J Y3 − Y4 . (2.64)

The eﬃciency of this engine setup which is the ratio of output power to input power can thus be expressed as

β β β J Y − Y α α α α Pout 3 4 J (Y1 − Y2 ) Y2 η = = α α = α α = 1 − α , (2.65) Pin J Y1 J Y1 Y1 where we used Eq. (2.62) in the first step. Of course, the definitions of power input and output as well as the definition of the efficiency depend on the goal of the process and can thus be chosen differently. The efficiency may also be based on the ratios of the extensities rather than energy transfers that flow into and out of the engine, which is typically the case e. g. considering volumetric efficiencies. The energy-based definition chosen here represents a case in which the extensity transfer towards reservoir 2 is of no use, and the extensity transfer from reservoir 3 to engine 5 has to be taken into account. Since the power output of this fully reversible engine can also be expressed as α α α Y2 Pout = Pinη = J Y1 1 − α , (2.66) Y1 neither power output nor efficiency are actually dependent on the second extensity flux β β or the involved intensities Y3 and Y4 . Therefore, it is reasonable to take these out of the equation and even out of the engine setup when considering reversible transfer of the second extensity flux. Fig. 2.6b shows a simplified reversible engine setup with unspecified power output P , where the two reservoirs 4 and 5 were removed. So, if this

27 2 Endoreversible Systems engine receives or delivers some amount of energy, this energy is transferred via the power output P without speciﬁcation of the extensity that is carrying the energy.

In case of a heat engine with entropy fluxes as well as a high temperature TH and a low temperature TL in the reservoirs 1 and 2, respectively, Eq. (2.65) gives the well known Carnot efficiency TH ηC = 1 − . (2.67) TL The Carnot efficiency is the upper bound on efficiency for any heat driven engine. Similarly, Eq. (2.65) represents an upper bound for energy converting engines using an arbitrary extensity α.

2.2.2 Irreversible Engines

We remain, for the sake of simplicity, in the consideration of heat machines and take a look at examples that extend the reversible Carnot engine by simple dissipation terms. A Novikov engine which is a Carnot engine with irreversible heat transfer into the engine from the high temperature heat bath is shown in Fig. 2.7a. The temperature of heat bath 1 is TH and the temperature of heat bath 2 is TL. The irreversible interaction is deﬁned by Newton’s heat transfer law

q1 = K(TH − TM), (2.68) where q1 is the heat transfer, K is the heat transfer coeﬃcient and TM is a medium temperature at the upper contact point of engine 3. The interaction between engine 3 and the low temperature heat bath 2 remains reversible. For named temperatures the following assumption holds

TH ≥ TM ≥ TL (2.69) and physically feasible ﬂow directions can be considered in the description and balance equations according to Fig. 2.7a. From the energy balance equation within the engine we obtain the power output

P = q1 − q2 (2.70) and the entropy balance equation can be expressed as q q 1 = 2 . (2.71) TM TL

28 2.2 Endoreversible Engine Setups

(a) (b)

TH 1 TH 1

q1 q1

TM TM P P 3 3 q3 TL TL

q2 q2

TL TL 2 2

Figure 2.7: Novikov engine (a) and Novikov engine with heat leak (b) between heat

baths 1 and 2 with temperatures TH and TL, respectively. The upper working temperature of the Novikov engine is denoted by TM.

The resultant power output of the Novikov engine can thus be written as TL TLTH P = q1 1 − = K TH − TM − + TL . (2.72) TM TM

Considering q1 as input energy transfer and P as output energy transfer, we obtain the efficiency P T η = = 1 − L . (2.73) q1 TM Here, it is interesting to see, that the efficiency depends only on the high and medium temperatures TH and TM, respectively, but not on the heat transfer coefficient K.

In Fig. 2.8 the power output over efficiency of the Novikov engine for TH = 400 K, −1 TL = 300 K and K = 1 W K is shown. For this, the medium temperature TM was varied from TL to TH. From Eqs. (2.72) and (2.73) we see, that the power output vanishes for the extreme values of TM and has a maximum in between, while the efficiency monotonically increases with increasing TM from zero to the Carnot efficiency ηC = 25 % for given temperatures. The result of both effects is the plot shown. The maximum of the curve is the point of maximum power output Pˆ of the Novikov engine with 2 ˆ p p P = K TH − TL (2.74) at an efficiency s T ηˆ = 1 − L . (2.75) TH

29 2 Endoreversible Systems

6 [W]

P Novikov r = 0.01 4 r = 0.05 r = 0.1 2 power output

0 0 5 10 15 20 25 efficieny η [%] Figure 2.8: Power output over efficiency for Novikov engine and Novikov engine with heat leak using different values of r. Here, r describes the proportions of the heat transfer coefficients defining the heat transfer to and past the engine, respectively. An increasing r leads to decreasing heat transfer to the engine and increasing heat leak past the engine.

This point provides a much better prediction for real heat engines than the Carnot efficiency [90] and is therefore a simple yet essential and effective improvement towards the description of real engines. Another interesting possibility of adding dissipation to the engine setup is an additional heat leak past the engine as proposed in [91] resulting in the Novikov engine with heat leak shown in Fig. 2.7b. Here, two Newtonian heat transfers are defined

q1 = K1(TH − TM), (2.76)

q3 = K3(TH − TL). (2.77)

While the resultant power output differs only in the heat transfer coefficient TL TLTH P = q1 1 − = K1 TH − TL − + TL , (2.78) TM TM the efficiency changes to

P K (T − T )(T − T ) η = = 1 H M L M . (2.79) q1 + q3 TM (K1(TM − TH) + K3(TL − TH))

30 2.2 Endoreversible Engine Setups

The power output over eﬃciency for this engine setup is also shown in Fig. 2.8, where the two heat transfer coeﬃcients were set to

K1 = (1 − r)K, (2.80)

K3 = rK, (2.81) so that r describes the split of the heat transfer coefficient K into the two coefficients K1 and K3 which define the amount of heat transferred to and past the engine, −1 respectively. Here, again, K = 1 W K was chosen. Now, for TM → TL the same behavior as for the standard Novikov engine can be observed in Fig. 2.8 because the contribution of the heat leak is vanishingly small. However, for TM → TH not only the power output but also the efficiency decreases to zero, generating the loop in the plot.

Further, when decreasing the ratio of K1 to K3 the loop shrinks, while increasing it leads to an enlargement of the loop and approach to the Novikov engine curve. A lot of other extensions to the Carnot engine with additional dissipative heat transfers have been investigated and have shown their usefulness and importance in modeling heat engines. However, the above described models as well as other heat engine models may seem inappropriate for the use of engines with extensity fluxes other than entropy. One reason for that is that the influence of a low intensity reservoir is less significant, since its intensity can be close to zero or even equal to zero if we consider e. g. angular momentum reservoirs. This is not the case in reservoirs representing heat baths since their temperatures are rarely close to zero. Furthermore, loss terms of other extensive quantities, such as losses of momentum or angular momentum, may be assigned to the engine setup using quite different relations.

2.2.3 Engine with Lossy Extensity Transfer

In the following, we will consider an engine setup with a lossy interaction representing losses of extensity, which could be more in line with the intuitive modeling of endoreversible non-heat engines. In Fig. 2.9 this setup is shown. A lossy interaction as it is described in Section 2.1.2 connects the reservoir 1 with the engine. Loss of extensity α of this interaction is transferred to reservoir 4 and generated entropy to reservoir 5. Engine 3 is further reversibly connected to reservoir 2. The intensive quantities of the α α α reservoirs 1, 2, 4 and 5 are Y1 , Y2 , Y4 and T5, respectively, while the intensities at α α the engines contact points are Y3,1 and Y3,2, respectively. In general extensities are conserved within the lossy interaction as well as within the

31 2 Endoreversible Systems

4 α Y1 1 α α Y J1 4 α J4 T5 J α J S 3,1 5 α Y3,1 5 P 3 α Y3,2 J α α Y2 2

Figure 2.9: Engine setup with lossy extensity transfer. The upper reservoir is connected with engine 3 via a lossy interaction with loss extensity and entropy ﬂux towards reservoirs 4 and 5, respectively, while the lower reservoir is reversibly connected to the engine which further has an unspeciﬁed power output. engine so that the equations

α α α J3,1 = −J1 − J4 , (2.82) α α J3,2 = −J3,1 (2.83) hold. From the reversibility of the interaction between engine 3 and reservoir 2 we can further conclude

α α α J3,2 = −J = −J2 , (2.84) α α Y3,2 = Y2 . (2.85)

The power output P of the engine can thus be written as

α α α α P = −J3,1Y3,1 − J3,2Y3,2 (2.86) α α α = −J3,1 Y3,1 − Y3,2 (2.87) α α α α = (J1 + J4 ) Y3,1 − Y3,2 . (2.88)

α α α Note that if we deﬁne P as a power output we would typically assume Y1 ≥ Y3,1 ≥ Y2 , α α α α and J1 and J4 to have opposite signs. By deﬁning I1 = J1 Y1 as the input energy

32 2.2 Endoreversible Engine Setups

α 6 J4 = 0 [W] c = 5×10−6 P c = 1×10−5 4 c = 2×10−5 g = 0.1 g = 0.2 2 g = 0.3 power output

0 0 5 10 15 20 25 efficieny η [%] Figure 2.10: Power output over efficiency for dissipative engine with lossy extensity transfer. The loss extensity flux was chosen to be zero, proportional to α α the intensity of reservoir 1 as J4 = cY1 or proportional to the engine α α α setup’s influx J1 so that J4 = −gJ1 holds. The values for c and g were varied. transfer to the engine setup, its efficiency can be expressed as

α α α α (J1 + J4 ) Y3,1 − Y3,2 η = α α . (2.89) J1 Y1

α α Since Y1 and Y3,1 do not have to be equal we can deﬁne a transfer law with the transfer coeﬃcient Kα for extensity α as

α α α I1 = K Y3,1 − Y1 . (2.90)

α We further have to define the loss flux J4 which can be done, for instance, by setting it proportional to the intensity of the first reservoir and hence constant as

α α J4 = cY1 (2.91)

α or by setting it proportional to J1 which can be expressed as

α α I1 J4 = −gJ1 = −g α . (2.92) Y1 Here, c and g are the proportionality factors.

33 2 Endoreversible Systems

Fig. 2.10 shows the power over efficiency for these two examples with different values α α α for c and g as well as for zero loss flux. We chose K1 = 1 W, Y1 = 400 and Y3 = 300 in order to achieve comparable results – assuming a dimensionless extensity. Then, α α α α α α Y3,1 was varied from Y2 to Y1 so that Y1 ≥ Y3,1 ≥ Y2 holds. α If we compare the plot with zero loss flux J4 with the plot of the Novikov engine in Fig. 2.8, the lower power output immediately stands out. The reason for this is that the entropy generated by the irreversible transfer does not enter engine 3 as in the case of the Novikov engine, but is discharged into the reservoir 5. The thus reduced energy transfer into engine 3 decreases the power output. As far as the efficiency is α concerned, again, it decreases from the Carnot efficiency to zero with increasing Y3,1. Now, if we take a look at the curves for the loss flux proportional to the first reservoir’s intensity shown in red, we can see the same behavior as for the Novikov engine with α heat leak. This is because the loss flux here is constant since Y1 is constant, and in case of the Novikov engine the heat leak past the engine is also constant for given TH and TL. Similarly, increasing the value of c and hence the loss flux leads to a decrease α of the size of the loop. However, if we consider a loss flux proportional to the J1 a qualitatively different behavior can be observed. Here, the green curves in Fig. 2.10 retain their shape but maximum power output and maximum efficiency decrease with increasing proportionality factor g. These two effects illustrated by the red and green plots are comparable to those achieved with the rather complex model with internal irreversibility used in [92]. How- ever, the model introduced here does not rely on internal irreversibilities and hence fully complies with the idea of endoreversible modeling.

2.2.4 Dissipative Engine with Given Eﬃciency

As in this work, endoreversible modeling can be used to study large composite systems. In such cases, individual components may either have already been studied or may not necessarily be part of the study. For example, engines could be used for which output powers and efficiency are already present as functions of various parameters. These engines must then be incorporated into the endoreversible model considering the given power output and efficiency functions. To implement this, the previously described engine with lossy transfer is suitable (see α α α Fig. 2.9). If we set Y1 = Y3,1, we can easily define the loss flux J4 so that the overall efficiency of the engine setup equals a given efficiency ηG. Let us assume, energy is transferred from extensity α to an unspecified power output by transferring it from a

34 2.2 Endoreversible Engine Setups

α α high intensity reservoir to a low intensity reservoir and Y1 ≥ Y2 holds. In analogy to Eq. (2.89), the given eﬃciency of the engine setup can be expressed as

α α α α (J1 + J4 ) Y3,1 − Y3,2 ηG = α α (2.93) J1 Y1 α α α α (J1 + J4 )(Y1 − Y2 ) = α α (2.94) J1 Y1 α α α J4 Y1 − Y2 = 1 + α α , (2.95) J1 Y1

α α α α α where we used Y3,1 = Y1 and Y3,2 = Y2 . Solving Eq. (2.95) for J4 we obtain α α α Y1 J4 = J1 ηG α α − 1 (2.96) Y1 − Y2 as the desired loss ﬂux. The given eﬃciency can of course also apply to the opposite direction of action, where the power input causes a transfer of extensity α from a lower intensity reservoir to a α α higher intensity reservoir. In this case, where we can also assume Y1 ≥ Y2 , the power output of the engine setup would be

α α α α Pout = J1 Y1 − J3,2Y3,2 (2.97) α α α α = J1 Y1 + J3,1Y2 (2.98) α α α α α = J3,1Y2 − J4 + J3,1 Y1 (2.99) α α α α α = J3,1 (Y2 − Y1 ) − J4 Y1 , (2.100) where Eq. (2.82) was used. The energy transferred into the engine setup is given by

α α α Pin = P = J3,1 (Y2 − Y1 ) . (2.101)

Combining the equations above, the given eﬃciency can be written as

α α α α α Pout J3,1 (Y2 − Y1 ) − J4 Y1 ηG = = α α α (2.102) Pin J3,1 (Y2 − Y1 ) α α J4 Y1 = 1 + α α α (2.103) J3,1 (Y1 − Y2 )

α which can be solved for J4 leading to α α α Y2 J4 = (1 − ηG) J3,2 1 − α . (2.104) Y1

35 2 Endoreversible Systems

Note that the power input might not be dependent on the intensity of reservoir 1, but the intensity of the contact point of engine 3 towards this reservoir may dependent on the power input and can be calculated from energy balance within engine 3 as

α α α Pin + J3,2Y3,2 α Pin Y3,1 = − α = Y3,2 + α . (2.105) J3,1 J3,2 Here, we assumed reservoir 1 to have that intensity. Furthermore, for both directions of energy conversion, the loss extensity transfer has to occur between the reservoir with the higher intensity and the engine while the reservoir with the lower intensity is reversibly connected. α For Y2 = 0, which e. g. might occur in engines with angular momentum transfer, Equations (2.96) and (2.104) simplify to a form that agrees with an intuitively described loss ﬂux for a simple lossy engine setup. Of course, the given eﬃciency is not α α restricted to be constant but can be a function intensities {Yi }, extensities {Xj } or additional parameters {zk}

α α ηG = ηG({Yi }, {Xj }, {zk}). (2.106)

The generated entropy of the overall engine setup for both conversion directions yields α α S α Y1 − Y4 σ = J5 = J4 (2.107) T5 and is transferred to reservoir 5 for which in most cases an infinite entropy reservoir with ambient pressure is suitable. In principle, this also allows a kind of black box view of this engine setup as a dissipative engine with given efficiency as illustrated in Fig. 2.11. Using the above equations for loss fluxes and entropy generation, non-heat engines with arbitrary efficiency functions can be incorporated into an endoreversible description of large composite systems with ease. In particular for engineering applications where many different components are assembled and synergy effects or thermodynamic effects have to be investigated, this black box model can be very beneficial. It is easy to handle while it precisely maps the relevant performance parameters of the engines. Of course, the above description can also be applied to engines with specified power output in form of an extensity transfer with extensity β. The endoreversible model for this engine setup is shown in Fig. 2.12a. Here, the flux of extensity α from reservoir 1 to reservoir 2 drives a flux of extensity β from reservoir 3 to reservoir 4, or vice versa. The reservoirs 2 and 3 are reversibly connected to engine 5 while the reservoirs 1 and 4 are connected via a lossy extensity transfer. The latter are those reservoirs with higher α α β β intensities, hence Y1 ≥ Y2 and Y4 ≥ Y3 . Loss extensity fluxes enter the reservoirs 6 and 7 and generated entropy is transferred to the heat bath 8 with temperature T8. As

36 2.2 Endoreversible Engine Setups

(a) (b) α Y1 1 4 α J1 α Y1 1 α J α Y4 1 4 α α α J1 J4 J4 T5 α α S S J3,1 α α Y1 J5 J5 Y4 Y4 α α P Y J4 ηG 5 1 P P T5 T5 α 3 S Y2 J5 Y α α 2 J2 α 5 J3,2 α Y2 α J2 2 α Y2 2

Figure 2.11: Simpliﬁcation of the engine with lossy extensity transfer (a) to a black box dissipative engine with given eﬃciency (b). The resulting black box model (rectangle with rounded corners) operates between reservoirs 1

and 2 with eﬃciency ηG while the loss extensity is reversibly transferred to reservoir 4 and the generated entropy to reservoir 5. illustrated, the intensities at the contact points of the engine are equal to those of the linked reservoirs. From the extensity balance equation of the reversible interactions and engine 5 we obtain

α α α α J2 = J = −J5,2 = J5,1, (2.108) β β β β J3 = J = −J5,3 = J5,4, (2.109)

α α β β where J5,1, J5,2, J5,3, and J5,4 are the fluxes entering or leaving engine 5 at the contact α β points towards the reservoirs 1 to 4, respectively, and J2 and J3 are the fluxes into or out of the reservoirs 2 and 3, respectively. The relations of the extensity fluxes of the lossy interactions are given by

α α α 0 = J1 + J5,1 + J6 , (2.110) β β β 0 = J4 + J5,4 + J7 . (2.111)

37 2 Endoreversible Systems

(a) (b) Y α 1 1 α β α 1 Y1 Y4 4 Y α J1 6 6 α J α J β J6 1 4 T8 S 8 α α β J J J8,1 α 6 5,1 S Y1 Y4 Y α α S J8 6 6 Y J8,2 α β β β 1 β T8 η = ηGη J7 J J G β β β β 4 β Y Y 5 Y Y 8 α Y β Y7 3 3 4 β 4 Y2 3 α J5,4 Y2 α β 7 3 4 J2 J α β 3 J J7 α β Y2 Y3 Y α β 2 Y7 2 3 2 7

Figure 2.12: Engine setup with lossy extensity transfer for both extensities α and β (a) and corresponding black box model with given eﬃciencies (b). The engine operates between reservoirs 1 and 2, and reservoirs 3 and 4 with eﬃciency η while loss extensity transfers of extensities α and β occur towards reservoirs 6 and 7, respectively, and the generated entropy is transferred to reservoir 8.

Energy conversion within engine 5 leads to

α α α α β β β β J5,1Y1 + J5,2Y2 = − J5,3Y3 + J5,4Y4 . (2.112)

Considering an energy conversion direction from extensity α to β the above model can be seen as a series of two smaller setups. The first one converts energy from extensity α α to power output Pout and the second one converts energy from its energy input which β α is the output of the first one, Pin = Pout, to extensity β. Now, if these smaller setups α β have the efficiencies ηG and ηG, respectively, one would expect them to multiply for the overall setup. To show that this applies to the described engine setup, we define the loss fluxes according to the Equations (2.96) and (2.104) as

α α α α Y1 J6 = J1 ηG α α − 1 , (2.113) Y1 − Y2 β ! β β β Y3 J7 = ηG − 1 J3 1 − β . (2.114) Y4

38 2.3 Van der Waals Gas

The power input into the engine setup is the (negative) energy ﬂux leaving reservoir 1

β β Pin = −J1 Y1 (2.115) and the power output is expressed and can be rearranged as follows

β β β β β β β β β β Pout = J3 Y3 + J4 Y4 = J3 Y3 − J5,4Y4 − J7 Y4 (2.116) β β β β β β β β = J3 Y3 − J5,4Y4 − ηG − 1 J3 Y4 − Y3 (2.117)

β β β β β β β β β = J3 Y4 − J5,4Y4 − ηG J3 Y4 − J3 Y4 (2.118) β β β β β β α α α α = −ηG J5,4Y4 + J5,3Y3 = ηG J5,1Y1 + J5,2Y2 (2.119) β α α α β α α α α = ηGJ5,1 (Y1 − Y2 ) = −ηG (J1 + J6 )(Y1 − Y2 ) (2.120) α β α α α Y1 α α = −ηG J1 + J1 ηG α α − 1 (Y1 − Y2 ) (2.121) Y1 − Y2 α α β α = −J1 Y1 ηGηG, (2.122) where we used Equations (2.108) to (2.114). Therefore, the resultant engine efficiency is given by Pout α β η = = ηGηG. (2.123) Pin Such a multiplicative decomposition of the overall efficiency of an engine is by no means uncommon. Examples of these are thermal or volumetric efficiencies covering heat losses and leakages, respectively, which occur in combination with a mechanical efficiency that generally covers friction occurring in the system. For the sake of simplicity, a black box model can also be used for the above described rather complex engine setup with two extensity fluxes. Fig. 2.12b illustrates this model where the reservoirs and the engine with given efficiency are left over. The entropy generated and transferred to reservoir 8 can be written as α α α α S S S α Y1 − Y6 β Y4 − Y7 σ = J8 = J8,1 + J8,2 = J6 α + J7 α . (2.124) Y1 Y4

2.3 Van der Waals Gas

In this section, we will derive the equations of state we need to model ﬂuids in the following chapters. For this, the van der Waals gas was chosen as a simple and well investigated model describing the qualitative behavior of real gases. The derived principle equation of state, as introduced by Essex and Andresen in [93], will be the basis for endoreversible modeling in this work.

39 2 Endoreversible Systems

Van der Waals introduced his equation as an extension to the ideal gas law by incorporating attractive and repulsive forces within a fluid, in the late 19th century. Using two parameters describing these effects, he modeled a fluid with interacting particles having a volume, which provides a qualitatively better description of the behavior of real gases. It leads to a simple understanding of the liquefaction and many other properties in which real gases differ from the ideal gas. The van der Waals gas equation can be expressed as n2a p + (V − nb) = nRT, (2.125) V 2 where p, T and n are the pressure, temperature and particle number of the fluid and R is the universal gas constant. The van der Waals parameters a and b can be interpreted as the cohesion pressure and the co-volume of the gas, respectively. For zero a and b this equation of state reduces to that of the ideal gas equation given by

pV = nRT. (2.126)

It can also be written in molar terms a p + 2 (Vm − b) = RT. (2.127) Vm Here, the molar volume of the gas V V = (2.128) m n was used. The internal energy of the van der Waals gas can be calculated as

an2 U =c ˆ nRT − . (2.129) V V

The dimensionless speciﬁc heat capacity at constant volume cˆV is typically chosen to be half of the degrees of freedom of the molecules in the gas. Here, cˆV = 3/2 gives a good approximation for monatomic and cˆV = 5/2 for diatomic gases. At high temperatures two additional degrees of freedom for diatomic gases have to be considered and hence cˆV = 7/2 is used. In contrast to the ideal gas which lacks the second term in Eq. (2.129), the internal energy of the van der Waals gas not only depends on the mole number and temperature but also on the volume of the gas. This is due to the fact that the particles have intermolecular attraction forces causing the negative second term. The internal energy of the van der Waals gas is thus lower than that of the ideal gas while temperature dependence of U is the same. Accordingly, the expression for the heat capacity of the

40 2.3 Van der Waals Gas van der Waals gas is the same as for the ideal gas and results to

∂U CV = =c ˆV nR. (2.130) ∂T V,n

In order to derive the internal energy of the van der Waals gas as a function of its extensive quantities entropy, volume and mole number, which is the preferred description for endoreversible descriptions, we need the entropy of the gas given by T V − nb n S0 S(T, V, n) = nR cˆV ln + ln − ln + n , (2.131) T0 V0 − n0b n0 n0 where S0(T0,V0, n0) is a given reference state of the entropy at temperature T0, volume V0 and mole number n0 [64]. Solving this equation for T results to

1 ! cˆ (V − n b)T cˆV n S S V T (S, V, n) = 0 0 0 exp − 0 (2.132) n0 V − nb nR n0R which can be inserted into the equation for the internal energy, Eq. (2.129). We then obtain the internal energy in dependence on the extensive quantities S, V and n

1 cˆV ! cˆV 2 (V0 − n0b)T0 n S S0 an U(S, V, n) =c ˆV nR exp − − . (2.133) n0 V − nb nR n0R V

This equation has been called the principle equation of state by Essex and Andresen due to the fact that it captures all physical content in the relationship between the thermodynamic variables of the system [93]. Further, all other equations – which are typically referred to as equations of state – can be deduced from this expression. From the derivations of the principle equation of state with respect to the extensities S, V and n we obtain the intensities

1 ! cˆ ∂U (V − n b)T cˆV n S S V T (S, V, n) = = 0 0 0 exp − 0 , (2.134) ∂S V,n n0 V − nb nR n0R ∂U nR an2 p(S, V, n) = − = T (S, V, n) − 2 , (2.135) ∂V S,n V − nb V ∂U RV S 2an µ(S, V, n) = = cˆV R + − T (S, V, n) − , (2.136) ∂n S,V V − nb n V respectively, which are also functions of S, V and n.

41 2 Endoreversible Systems

A reduced form of the van der Waals equation can be written using the critical pressure pc, critical temperature Tc and critical volume Vc which are calculated from the point of zero slope and zero curvature in a p-V diagram [94]. In terms of a, b and n, these are given by a p = , (2.137) c 27b2 8a T = , (2.138) c 27bR Vc = 3bn. (2.139)

Using the deﬁnitions of reduced pressure, reduced temperature and reduced volume p pred = , (2.140) pc T Tred = , (2.141) Tc V Vred = , (2.142) Vc respectively, the resulting reduced van der Waals equation can be expressed as

3 pred + 2 (3Vred − 1) = 8Tred. (2.143) Vred As we see from Eq. (2.143), this description is independent on the parameters a and b and is therefore universal for all van der Waals gases. In Fig. 2.13 the reduced pressure over reduced volume is shown for different values of the reduced temperature. The point of zero slope and zero curvature is, according to the definition of the critical and reduced variables, at (1, 1) with Tred = 1. The graphs with T < 1 show the typical shapes with an increasing pressure for increasing volume between a local minimum and maximum, describing a rather unphysical behavior of the gas. However, this behavior can be explained considering phase changes and an equilibrium of gas and liquid phase during isothermal expansion below the critical temperature. During this process the liquid starts to evaporate, and the pressures of the liquid phase and the gas phase are equal and only dependent on temperature. In other words, the states on the relevant part of the van der Waals isotherm are unstable and a transition to states that have the lowest possible Gibbs free energy at that temperature and pressure occurs. The result is a straight line between two volumes in the pred-Vred diagram, as shown in Fig. 2.14. The Maxwell construction is a simple method to find these two volumes, so that the two areas (red and blue) between the straight line and the isotherm are equal [95].

42 2.3 Van der Waals Gas

4 Tred = 1.05 T = 1.00 3 red

red Tred = 0.95 p Tred = 0.90 2 Tred = 0.85 Tred = 0.80 T = 0.75 1 red

reduced pressure 0

−1 0 1 2 3 4 5

reduced volume Vred Figure 2.13: Reduced pressure over reduced volume for the van der Waals gas using diﬀerent values of reduced temperatures.

1 Tred = 0.85 Maxwell constr. 0.8 red p

0.6

0.4

reduced pressure 0.2

0 0 1 2 3 4 5

reduced volume Vred Figure 2.14: Reduced pressure over reduced volume with Maxwell construction. The latter equalizes the sizes of the two areas (blue and red) between the isotherm and the line representing the phase transition (green).

Calculating these volumes for Tred < 1, the coexistence curve can be plotted, as it is done as a dashed line in Fig. 2.15. Below this curve, the liquid and gas phase are present at the same time. Above this curve, the vapor and liquid phase for Vred < 1 and Vred > 1, respectively, are stable. Further, above the curve of Tred = 1 which is

43 2 Endoreversible Systems

2 Tred = 1.05 Tred = 1.00

red 1.5 Tred = 0.95 p Tred = 0.90 Tred = 0.85 1 Tred = 0.80 Tred = 0.75 Coex. curve 0.5 reduced pressure

0 0 1 2 3 4 5

reduced volume Vred Figure 2.15: Reduced pressure over reduced volume for diﬀerent values of reduced temperature. Below the coexistence curve the isotherms are replaced by the Maxwell constructions for these temperatures. tangent to the coexistence curve in (1, 1), a supercritical phase exists, where gas and liquid phase are indistinguishable. In Chapter3, when modeling the hydraulic ﬂuid, we will refer back to this point.

2.4 Summary

In this chapter, I gave a brief introduction to endoreversible thermodynamics or rather endoreversible modeling as it is not restricted to thermodynamic processes. The key elements to build endoreversible models, reservoirs and engines, were explained as well as how they can be connected using reversible and irreversible interactions. Here, I introduced a lossy interaction which is not solely irreversible due to different intensities at both ends of the interaction but due to loss of extensity transferred. Defining a loss flux within this interaction that is transferred to a third reservoir generating entropy is a suitable extension to endoreversible modeling especially when it comes to non- heat engines. We further took a look at multi-extensity fluxes and the equivalence of combined particle and entropy fluxes to enthalpy fluxes which will be important in Chapter4. Then, with the examples Carnot engine and Novikov engine as well as the Novikov engine with heat leak, we looked at reversible and irreversible heat engines. Using a

44 2.4 Summary lossy interaction, I modeled an engine setup that has a comparable characteristic in the power over efficiency plot as the Novikov engine with heat leak and as can be seen e. g. in [92]. This engine set up is particularly useful for modeling and incorporating engines with given efficiencies in large composite systems. The loss fluxes to achieve a desired efficiency for either operation directions were derived and entropy production of the setup was given. This was done for both the simplified model with unspecified power output or input and for the comprehensive model regarding two extensity fluxes. Since this model can also be viewed as a kind of black box model for engines with given efficiencies it is especially suitable for engineering applications, and will also be used in the following chapter. Lastly we looked at the van der Waals gas equation and derived its principle equation of state. This will be used in the following chapter to model the gas inside the hydraulic accumulator as well as the hydraulic fluid. In particular, in the latter case, I will briefly return to the characteristics of the van der Waals gas, which were discussed here.

3 Recuperation System

In this work, a hydraulic recuperation system is investigated and optimized under the assumption of potential application for medium heavy trucks with tipper bodies or similar skip lifting mechanisms. The reason for this is the particular suitability of a hydraulic recuperation system for such commercial vehicles. They are typically operated in urban areas which offer a great potential for recovering brake energy due to the frequent velocity changes in city traffic. Furthermore, their design already provides them with a hydraulic system, which reduces the installation costs and the additional space and weight requirements of the recuperation system. Although I will only focus on drive support and brake wear reduction in this work, the energy stored by the recuperation system can also be used to power auxiliaries, to support the thermal management of the vehicle, and to operate the hydraulic mechanism to lift the skip. In all these cases, the use of the internal combustion engine, which is necessary in particular in the latter case, can be reduced or even omitted. The use of a recuperation system thus promises a high potential to reduce fuel consumption as well as air and noise pollution. In this chapter I will describe the recuperation system and its functioning in more detail. The endoreversible modeling of the individual components as well as the entire composite system will be shown and the simulation results, especially with regard to energy savings and the influence of selected parameters, are presented.

3.1 Model Description

The considered hydraulic system consists of the following components: a hydraulic fluid tank, a hydraulic pump/motor which is mounted to and driven by the vehicle’s cardan shaft, a hydraulic bladder accumulator, a pressure control valve, a heat exchanger between the hydraulic fluid cycle and the vehicle’s cooling cycle, and the pipes connecting named components. A scheme of the system is shown in Fig. 3.1. The hydraulic pump/motor, which I will refer to as hydraulic unit, has the ability to act both as a pump and as a motor. This means that input power from the cardan shaft can be used to propel flow of hydraulic fluid, or a pressure driven fluid flow is converted to a power output put into the cardan shaft, respectively. Here, the cardan shaft is the

47 3 Recuperation System

bladder accumulator pressure cooling control circuit valve hydraulic pump/motor cardan shaft heat exchanger hydraulic ﬂuid tank

Figure 3.1: Scheme of the hydraulic recuperation system shaft connecting the transmission of the vehicle with its differential. The hydraulic unit is also able to switch between these two operating modes during operation, as further explained Section 3.1.3. When energy is to be stored, the hydraulic unit works as a pump. Then, the hydraulic fluid is pumped from the low pressure hydraulic fluid tank towards the high-pressure bladder accumulator. When that happens, the fluid compresses the gas inside the bladder of the accumulator leading to a pressure increase. The power that is needed to compress the gas is taken from the cardan shaft. As a consequence, the cardan shaft and hence the vehicle is decelerated. This supports the conventional disc brakes of the vehicle which therefore have to absorb less energy resulting in reduced wear. Once the gas is compressed and a certain amount of hydraulic fluid is in the bladder accumulator, the hydraulic unit can be operated as a motor. Now, a fluid flow in opposite direction generated by the gas pressure propels the hydraulic motor. Hence, the power carried by this fluid flow from a high to a low pressure is transferred to the cardan shaft accelerating the vehicle. A pressure control valve is used to avoid that the pressure exceeds the maximum value that is suitable for the system. The application of this pressure control valve offers another interesting possibility: If the system has reached the maximum pressure, it can switch from the energy saving mode into a retarder mode. In this mode, the pressure control valve generates just enough fluid flow to maintain the pressure. Hence, the hydraulic unit can continue to operate and decelerate the cardan shaft. In this case, the pressure drop at the valve generates heat that warms the hydraulic fluid and the valve. The heat exchanger is used to cool the hydraulic fluid by transferring heat to the vehicle’s cooling circuit. This heat exchange might then be used to speed up the warming of the cooling circuit after a cold start so that optimal operation conditions are reached faster and a reduction of wear and fuel consumption can be achieved.

48 3.1 Model Description

3.1.1 Hydraulic Fluid

In the description of the individual components, we start with the modeling of the hydraulic fluid itself. In order to consistently incorporate pressure losses in the endoreversible description, we need a pressure-dependent equation of state able to model a fluid. The van der Waals equation is suitable for this, since it can map the pressure behavior of liquids quite well while being relatively easy to handle. We determine the parameters of the van der Waals equation for the hydraulic fluid using the known values of the coefficient of volumetric thermal expansion and the compressibility of the hydraulic fluid. The coefficient of volumetric thermal expansion and the compressibility of a fluid are defined as

1 ∂V αV = , (3.1) V ∂T p 1 ∂V β = − , (3.2) V ∂p T respectively, where the volume, temperature and pressure of the ﬂuid are denoted by V , T and p, respectively. Solving the van der Waals equation, Eq. (2.125), for p and calculating its total diﬀerential we obtain

nR nRT 2n2a dp = dT − − dV (3.3) V − nb (V − nb)2 V 3 with the mole number n, the universal gas constant R and the van der Waals parameters a and b. By setting dp = 0 we can derive the expression of the volumetric thermal expansion of the van der Waals gas V − nb αV = , (3.4) 2an V − nb2 VT − R V which can also be written in molar terms as V − b α = m , (3.5) V 2 2a Vm − b VmT − R Vm where Vm is the molar volume. The same can be done for the compressibility by setting

49 3 Recuperation System

Table 3.1: Parameters of the van der Waals hydraulic ﬂuid −3 p0 = 101 325 Pa T0 = 293.15 K ρ = 860 kg m −4 −1 −10 −1 −1 −1 αV = 7 × 10 K β = 7 × 10 Pa cp = 1900 J kg K 6 −2 −2 −1 a = 0.950 Pa m mol b = 4.817 × 10 l mol cˆV = 9.707 −2 −1 Vm = 5.648 × 10 l mol pc = 151.711 bar Tc = 703.1 K dT = 0 in Eq. (3.3) and we obtain

(V − nb)2 β = . (3.6) V − nb2 nRV T − 2an2 V

Expressed in molar terms the above equation reads

(V − b)2 β = m . (3.7) V − b2 RVT − 2a m Vm

By inserting the values of the volumetric thermal expansion and the compressibility as well as the corresponding reference pressure and temperature into Equations (3.5) and (3.7), and the van der Waals equation in molar terms (Eq. (2.127)), we can determine the parameters of the van der Waals equation a and b, and the volumetric volume Vm for the reference state. Using these parameters we are able to model the physical properties of the hydraulic fluid as a van der Waals fluid. What is missing for a complete description is the heat capacity of the fluid. Using Eq. (2.130) the dimensionless specific heat capacity at constant volume of the van der Waals fluid can be expressed as C cˆ = V . (3.8) V nR

Hence, for a given specific heat capacity at constant pressure cV = CV /m and m = nM = nρVm, where M is the molar mass, the above equation can be written as ρV cˆ = c m . (3.9) V V R Here, ρ is the density of the fluid. Since not the specific heat capacity at constant volume but constant pressure cp is usually given for liquids, we further need the relation

50 3.1 Model Description

2 0.012

0.010

red 1.5 T = 1.00

p red 0.008 Tred = 0.90 0.006 Tred = 0.80 1 0.385 0.390 0.395 Tred = 0.60 Tred = T0/Tc Coex. curve 0.5 Ambient cond. reduced pressure

0 0 1 2 3 4 5

reduced volume Vred Figure 3.2: Reduced pressure over reduced volume for diﬀerent values of reduced temperature as well as the coexistence curve. The inset diagram shows a zoom to the point of ambient condition in reduced variables for the van der Waals hydraulic ﬂuid. This point is slightly below the coexistence curve. between these two quantities given by

α2 T c = c − V . (3.10) V p ρβ

Using the above derived parameters we can represent the hydraulic ﬂuid as a van der Waals ﬂuid. The values used and the resulting van der Waals parameters are given in Tab. 3.1. Here, if we take a look at the critical temperature Tc, calculated using Eq. (2.138), we can see that it will not be exceeded during normal operation. However, when looking at the coexistence curve, which is shown in Fig. 3.2 again without the

Maxwell constructions, it is noticeable that a state with ambient pressure p0 and ambient temperature T0 is slightly below the coexistence curve. This state is shown as the black point at ambient condition for n = 1 and the isotherm with ambient temperature was added. However, since this state is still in the descending arm of the isotherm and not in the vicinity of unphysical behavior, we ignore the fact that a real gas with the same material properties would already start to evaporate as discussed in the previous chapter. In addition, it should be noted that for given pressure and temperature around this state, three non-complex solutions for the volume are obtained from the van der Waals equation, and in our case the smallest one is to be chosen. Of course, numerical

51 3 Recuperation System methods like Newton’s method will lead to the proper solution using suitable starting values.

3.1.2 Pipes and Hydraulic Fluid Tank

The pipes of the recuperation system and the hydraulic fluid within the tank are modeled as reservoirs with constant volume and constant pressure, respectively. Fig. 3.3 shows a pipe segment, reservoir 2, and the hydraulic fluid tank, reservoir 1. They are connected by a combined particle and entropy flux, and are coupled with other pipe segments as well, and have entropy fluxes to the environment which is drawn as a line rather than a rectangle. Since the volume of the fluid inside that tank can vary but its pressure is constant at ambient pressure, reservoir 2 has an additional reversible connection transferring volume from or to the infinite environment reservoir. We assume that the pipe segments do not run empty, but are always completely filled with hydraulic fluid and therefore the corresponding equations of state for the van der Waals fluid representing the hydraulic fluid apply with constant volume. In order to model pressure losses within the pipes, the volumetric flow rate Q of hydraulic fluid between the reservoirs shall be a function of the pressure difference ∆p so that

n Q = Q(∆p) = VmJ (∆p) (3.11)

n with the molar volume Vm and the particle ﬂux J holds. To determine a suitable relation for that, we consider the Darcy-Weisbach equation expressing the relationship between pressure loss and mean ﬂow velocity u as

ρu2 l ∆p = fD , (3.12) 2 di where fD, ρ, l and di are the Darcy friction factor, the density of the fluid, and the length and the inner diameter of the pipe, respectively. The Darcy friction factor is given by 64 f = (3.13) D,lam Re for laminar flow and can be approximated using the Blasius correlation by 0.316 fD,tur = √ (3.14) 4 Re for a turbulent flow within the pipes. Here, Re is the Reynolds number, and laminar flow occurs roughly up to Re ≈ 2300 while turbulent flow occurs above this value [96].

52 3.1 Model Description

S S S S J2,2 T2 J2,1 J1,2 T1 J1,1 p2 p1 = p0 n n n n J2,2 µ2 J2,1 J1,2 µ1 J1,1 2 1 S S V J2,3 J1,3 J1,4

T0 T0 p0

Figure 3.3: 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.

During the transition from laminar to turbulent flow, the flow actually is unstable and varies between laminar and turbulent behavior. For fluid flow in circular pipes, which is assumed here, the Reynolds number can be calculated as ud Re = i (3.15) ν and the mean flow velocity is given by Q u = , (3.16) A where ν and A are the kinematic viscosity of the fluid and the circular section of the pipe, respectively. If we consider volumetric flow rates up to about Q = 250 l min−1, that can be achieved within the recuperation system, we obtain values of the mean flow velocity and the Reynolds number of about u > 6 m s−1 and Re > 3900, respectively, using the above equations and the parameters given in Tab. 3.2. Accordingly, we can not consider pressure losses with laminar flow only. Using Equations (3.12) to (3.15) the pressure loss in dependence on the mean flow velocity yields to 32 ρνl ∆plam = 2 u, (3.17) di s 7 4 νu ∆ptur = 0.158 ρνl 5 , (3.18) di for laminar and turbulent flow, respectively, which is shown in Fig. 3.4 for l = 1 m. This clearly shows that the turbulent flow causes a strong increase in pressure loss

53 3 Recuperation System

Table 3.2: Parameters used for friction and heat transfer 2 −1 ν = 46 mm s di = 30 mm do = 38 mm −2 −1 −2 −1 −1 −1 ki = 1000 W m K ko = 70 W m K λ = 50 W m K −1 K1 = 650 W K compared to the laminar flow. In order to take this into account as well as to ensure numeric stability, we will use the Reynolds number where the two Darcy friction factors given in Equations (3.13) and (3.14) are equal, to define the transition between laminar and turbulent flow. This intersection lies at Reint = 1189.39 and we can thus write 32 ρνl  u if u < uint  d2 ∆p = i s (3.19) νu7  4 0.158 ρνl 5 otherwise,  di where uint = Reint ν/di is the corresponding mean flow velocity. From this we can derive an expression for the mean flow velocity in dependence on pressure loss as

 2 di  ∆p if ∆p < ∆pint 32 ρνl u = (3.20) s 4 5  7 ∆p di  otherwise,  0.158 ρl ν with the pressure loss at the intersection point given by

2 32 Reint ρν l ∆pint = 3 . (3.21) di Using Equations (3.11) and (3.16) we can now write the resultant particle flux J n between two reservoirs as  Ad2  i  ∆p if |∆p| < ∆pint 32 Vmρνl J n = s (3.22) sgn(∆p)A ∆p 4 d5  7 i  otherwise, Vm 0.158 ρl ν where the molar volume Vm depends on the temperature and pressure of the fluid. Thus, the particle flux between pipe segment 2 and hydraulic fluid tank 1 can be

54 3.1 Model Description

0.4 ∆plam ∆ptur transition regime 0.3 [bar] laminar ﬂow turbulent ﬂow p ∆ 0.2

0.1 pressure loss

0 1 int 2 3 4 5 6 7 8 u mean flow velocity u [m/s] Figure 3.4: Pressure loss for laminar and turbulent flow over mean flow velocity. The

intersection of the two pressure loss curves lies at uint. Between the areas where laminar and turbulent flow typically occur is a transition regime as shown in gray. written as n n n J2,1 = J = −J1,2 (3.23) n with J from Eq. (3.22) and ∆p = p1 − p2. As length l we use the distance li,j between two hydraulic components i and j which are connected by the pipe segment. S n The entropy flux that is coupled to the particle flux is defined by J1,2 = Sm,1J1,2 for S n p1 > p2 or J2,1 = Sm,2J2,1 otherwise. Note that the viscosity of the fluid is actually strongly temperature-dependent and thus so is the pressure loss. However, for the sake of simplicity and because of the mostly rather small temperature fluctuations in the hydraulic fluid we assume it to be constant. Furthermore, we incorporate heat dissipation to the environment for which we assume a Newtonian heat transfer given by

I2,3 = hl2(T2 − T0), (3.24) where h is the speciﬁc heat transfer coeﬃcient per length and l2 is the length of that pipe segment. In order to take the heat conduction through the pipe wall into account, we set π h = . (3.25) 1 + 1 ln do + 1 kidi 2λ di kodo

55 3 Recuperation System

Here, λ, ki, ko, di and do are the thermal conductivity of the pipe wall, the inner and outer heat transfer coeﬃcients, and the inner and outer diameter of the pipe, respectively. The heat transfer from the hydraulic ﬂuid within the tank to the environment is also assumed to be Newtonian and can be written as

I1,3 = K1(T1 − T0), (3.26) where K1 is the estimated overall heat transfer coeﬃcient given in Tab. 3.2.

3.1.3 Hydraulic Unit

The hydraulic unit used in the recuperation system has the ability to act both as a hydraulic pump and as a hydraulic motor. The model for that hydraulic unit is a radial piston pump which has a movable floating eccentric ring around the pump shaft that determines the stroke of the pump’s pistons. If this ring is in its neutral position, there is no eccentricity and thus no piston movement even if the pump shaft rotates. In this state, the pump is nonfunctional. When the floating ring is rotated in one direction, which can be done up to a maximum value, the eccentricity increases and so do the pistons’ stroke, up to reaching the maximum displacement of the hydraulic unit Vd. The pump can also be operated reversibly, as a motor, by moving the floating ring in the opposite direction. This way, the amount as well as the direction of the generated flow of the hydraulic fluid can be controlled. In this work, for simplicity, a displacement factor γ defining the ratio of current displacement to maximum displacement, so that −1 ≤ γ ≤ 1 holds, is used to describe the operation mode of the hydraulic unit. A negative displacement factor means that hydraulic fluid is pumped from the ambient pressure fluid tank to the high pressure bladder accumulator. A positive displacement factor means that a fluid flow from the high pressure bladder accumulator to the ambient pressure fluid tank generates a power output. The hydraulic unit is mounted directly on the cardan shaft, which means that the cardan shaft actually is the shaft of the hydraulic unit. Since the unit does not move any pistons when the eccentric ring is in its neutral position and no clutch is needed, an additional moment of inertia is negligible, unlike in other hybrid systems such as e. g. those described in [83,84]. Fig. 3.5 shows the endoreversible model of the hydraulic unit with adjacent pipe segments. Here, the flow directions are drawn according to the hydraulic unit acting as a pump, hence from reservoir 2 to reservoir 4. The hydraulic unit itself, engine 3, is modeled as a dissipative engine with given volumetric efficiency ηvol. In comparison to

56 3.1 Model Description

S P S J2,3 J4,3

S S S S S S J2,1 T2 J2,2 J3,1 J3,2 J4,1 T4 J4,2 T2 T4 p2 ηvol p4 n n n µ2 µ4 n n n J2,1 µ2 J2,2 J3,1 J3,2 J4,1 µ4 J4,2 2 µ4T4 3 4 n S J3,3 J3,3 n S J1,5 J1,5

S S J1,2 T1 J1,1 p1 = p0 n n J1,2 µ1 J1,1 1 S V J1,3 J1,4

Figure 3.5: Endoreversible model of hydraulic unit as 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 hydraulic unit operating in pump mode. the dissipative engine setup introduced in Section 2.2.4 (see Fig. 2.11), the extensity transfer of extensity α is now a multi-extensity flux with particle and entropy transfer, and the loss extensity fluxes are transferred to the hydraulic fluid tank, reservoir 1. As mentioned previously, the hydraulic unit can generate a variable volumetric flow rate Q. Depending on the displacement factor γ and the rotational speed of the shaft ncyc, this is given by Q = γncycVdηvol (3.27) where, Vd and ηvol are the maximum displacement and the volumetric efficiency of the unit. The input flow rate can thus be assumed as the efficiency-free term of the above equation, while density differences between the low and high pressure side of the system can be neglected due to the low compressibility of the fluid. In the case of the pump mode, the incoming particle flux which is assumed to be reversible yields

n n γncycVd J2,2 = −J3,1 = , (3.28) Vm,2 where Vm,2 is the molar volume of reservoir 2. The coupled entropy ﬂux can thus be

57 3 Recuperation System written as S S n J2,2 = −J3,1 = Sm,2J2,2 (3.29) with Sm,2 being the molar entropy of reservoir 2.

Since the hydraulic unit pumps hydraulic fluid from pressure p2 to pressure p4 in pump mode, the chemical potentials and temperatures at the contact points towards these reservoirs are chosen to equal their intensive quantities. As a consequence, engine 3 transfers the multi-extensity flux from the state of the reservoir 2 to the state of reservoir 4. The necessary energy input for this conversion is given by the unspecified power transfer P . The volumetric efficiency of the hydraulic unit defines the loss of particle and entropy transfer, or in other words the leakage of the operating hydraulic unit. This loss transfer in case of the hydraulic unit in pump mode is given by

n n Vm,2 J3,3 = (ηvol − 1) J3,1 , (3.30) Vm,4 S n J3,3 = Sm,2J3,3, (3.31) where Vm,2 and Vm,4 are the molar volumes of reservoir 2 and 4, respectively. Due to the fact that the loss transfer towards reservoir 1 has the same intensities as the one towards reservoir 4, there is no entropy production within the dissipative engine with given efficiency, which is a special case of the general description in Section 2.2.4. However, the entropy production is thus shifted to the interaction between engine 3 and reservoir 1, which is why it is irreversible. This means, that the entropy generated due to the loss flux results in a transfer of heated hydraulic fluid to reservoir 1 instead of a heat transfer e. g. to the environment. When the hydraulic unit is operating in the opposite direction, as a hydraulic motor, the above equations can be easily adapted. The interaction of engine 3 and reservoir 4 is then defining the influx of the hydraulic unit so that

n n γncycVd J4,1 = −J3,2 = − , (3.32) Vm,4 S S n J4,1 = −J3,2 = Sm,4J4,1, (3.33) holds, where Sm,4 is the molar entropy of reservoir 4. Note that, in this case the loss transfer towards the tank still has the intensities µ4 and T4 of the higher intensity reservoir 4 as it has been deﬁned for the dissipative engine with given eﬃciency in

58 3.1 Model Description

1,200 70 . η η 0 mech vol 0 .80 1

− 1,000 0 .90 0 min

. 75 90 . 80 85 0 . . 800 0 0 cyc 0 n .95 600

90 . 0

0 400 80 0 . . 85 . 0 90 0.95 0.90 200 0.80 0.80

rotational speed 0.70 0.70 0 0.50 0.50 0 100 200 300 0 100 200 300 pressure difference ∆p [bar] pressure difference ∆p [bar] Figure 3.6: Mechanical and volumetric efficiency of the hydraulic unit over rotational speed and pressure difference for γ = 0.5

Section 2.2.4. The loss transfer of the hydraulic unit in motor mode is deﬁned as

n n J3,3 = J3,2 (ηvol − 1) , (3.34) S n J3,3 = Sm,4J3,3. (3.35)

In order to incorporate the mechanical eﬃciency of the hydraulic unit ηmech, in pump mode, we simply consider the input power transferred to the recuperation system Prec from the cardan shaft to be given by

P = Precηmech. (3.36)

In case of the hydraulic unit acting as a motor, the power transferred to the cardan shaft is calculated as

Prec = P ηmech. (3.37)

Furthermore, I will use a realistic efficiency map which depends on the displacement factor γ, the rotational speed of the cardan shaft ncyc and the difference of the applied pressures ∆p, in this work. Hence, assuming the efficiencies to be identical for both directions of operation, for the mechanical and the volumetric efficiency applies

ηmech = ηmech(|γ|, ncyc, ∆p), (3.38)

ηvol = ηvol(|γ|, ncyc, ∆p), (3.39)

59 3 Recuperation System

Table 3.3: Parameters used for the hydraulic unit −3 −9 −1 c1 = −7.248 836 × 10 s c2 = −1.987 551 × 10 Pa c3 = −1.637 043 −6 3 −1 c4 = −0.127 307 0 s c5 = 5.844 646 1 × 10 s c6 = −20.384 61 s −3 c7 = 1.298 063 × 10 s c8 = 0.825 991 9 c9 = −1.238 341 s −7 3 −1 c10 = 0.433 235 8 c11 = 2.125 879 × 10 s c12 = 0.818 416 6 s 3 −1 −9 −1 c13 = 1.359 079 × 10 s c14 = −4.644 669 × 10 Pa c15 = −1.069 270 3 c16 = −533.1668 Vd = 307 cm respectively. Fig. 3.6 shows both eﬃciency maps over the relevant range of rotational speed and pressure diﬀerence for displacement factor γ = 0.5. The equations used for this are

3 ηmech = c1ncyc + c2∆p(c3 + c4ncyc) + c5(ncyc + c6) + c7|γ|ncyc + c8, (3.40) 3 ηvol = c9ncyc + c10 ln(ncyc + 1) + c11(ncyc + c12|γ| + c13) + c14∆p + c15|γ| + c16, (3.41) with the parameters given in Tab. 3.3. These are not based on any hydraulic pump/ motor model such as e. g. in [97], but are least squares fits on measured data of a real hydraulic unit with the same functionality and similar properties. Here, on the one hand, the use of interpolated tabulated values shall be avoided, since smooth functions improve the numerical handling of the system. On the other hand, the use of the dissipative engine with given efficiency allows us to use any conceivable efficiency characteristics without having to model the processes within the engine.

3.1.4 Bladder Accumulator

The bladder accumulator which is used in the recuperation system to store brake energy can be described as a cylinder with two chambers that are separated by a bladder totally enveloping one of the chambers. Within this bladder is an inert gas, usually nitrogen, that has a precharge pressure ppre whose value depends on the application. The second chamber, outside of the bladder, is filled with hydraulic fluid and is connected to the hydraulic line. In order to prevent the bladder from being damaged when touching the outlet valve of the hydraulic fluid, a minimum volume of hydraulic

ﬂuid Vmin has to remain within the accumulator. The bladder accumulator is typically characterized by its pressure range and its total volume Vacc. The endoreversible model of the bladder accumulator and the adjacent pipe segment is shown in Fig. 3.7. Reservoir 5 is the pipe segment that is connected to the accumulator

60 3.1 Model Description

n S S J5,1 J5,1 J9,3 J V J V n n 9,1 10,1 T5 J5,4 J9,4 T9 T10 S S p5 p9 J9,2 J10,2 p10 = p9 J S S S 5,3 µ5 J5,4 J9,4 µ9 µ10 5 9 10 n S J5,2 J5,2

Figure 3.7: 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. and the reservoirs 9 and 10 represent the hydraulic fluid, and the inert gas within the bladder, respectively. Here, and also in the previous sections, the numbering was chosen in accordance with the composite model of the recuperation system that will be assembled in Section 3.1.6. Since, within the accumulator, hydraulic fluid and inert gas have the same pressure p9 = p10 and both share the total volume of the accumulator

Vacc = V9 + V10, (3.42) the two reservoirs representing them are reversibly connected by a volume transfer, V V where J9,1 = −J10,1 holds. The inert gas itself which is assumed to be nitrogen is modeled as a van der Waals gas with the parameters given in Tab. 3.4. Since the heat losses of a hydraulic accumulator have a great influence on its efficiency, these should not be neglected. This is due to the fact that the gas heats up when energy is stored and it is compressed, and transfers some amount of that energy in form of heat to the hydraulic fluid. This amount of energy reduces the compression work of the accumulator. On the other hand, the hydraulic fluid can warm the gas after expansion which, however, might have a considerably smaller influence. We assume that the bladder is always entirely surrounded by hydraulic fluid within the accumulator and can hence describe the thermal losses as heat transfers from the gas to the hydraulic fluid and from the hydraulic fluid to the environment. Modeled

61 3 Recuperation System

Table 3.4: Parameters used for the gas and the bladder accumulator 6 −2 −2 −1 a = 0.141 Pa m mol b = 3.910 × 10 l mol cˆV = 2.39 ppre = 200 bar Vacc = 200 l Vmin = 5 l −1 −1 Kacc = 5 W K K9 = 35 W K as Newtonian heat transfers these can be expressed as

I9,2 = Kacc (T9 − T10) = −I10,2, (3.43)

I9,3 = K9 (T9 − T0) , (3.44) where Kacc and K9 are the corresponding estimated heat transfer coeﬃcients.

3.1.5 Pressure Control Valve

In the later application, the pressure control valve would be, in the best case, an electrically controlled valve that generates a discharge flow which keeps the system pressure constant, if it exceeds a predetermined control pressure. For optimal support of the thermal management of the vehicle, this control pressure could be set to the minimum system pressure after a cold start. However, in this work, I restrict my investigations to the recuperation system’s function of drive support. The pressure control valve can be modeled as a simple pressure relief valve with a fixed control pressure and a high relief flux coefficient avoiding additional differential equations for valve control. In addition, for the sake of simplicity, I also refrain from modeling a heat exchanger. Instead, I consider the valve housing as an additional reservoir that absorbs heat from the oil and releases it to the environment. In practice, it can be observed that in fact a large portion of the heat is released directly to the valve, which is why the valve housing should, if possible, be included in the heat exchange process in a later application. Since the pressure control valve simply transfers hydraulic fluid from a high pressure state to a low pressure state, and due to the fact that, unlike the hydraulic unit, there are no other energy conversions or transfers involved, an irreversible interaction is perfectly suitable to describe this process. The pressure relief through the valve is an isenthalpic process which, as we can derive from Equations (2.50) to (2.53), can be expressed by an energy conserving combined particle and entropy flux. The pressure drop that leads to an increase in internal energy during the isenthalpic process is expressed by the change in chemical potential and the generated entropy heating the fluid, here. Fig. 3.8 shows the endoreversible model consisting of the irreversible

62 3.1 Model Description

S J8,2 T0 T8 8 S J8,1 S J7,4

S S S S J6,1 T6 J6,2 J7,1 T7 J7,2 p6 p7 n n n n J6,1 µ6 J6,2 J7,1 µ7 J7,2 6 7 S S J6,3 J7,3

Figure 3.8: 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. interaction which is between reservoirs 6 and 7, as well as the valve housing represented by reservoir 8 which is connected to reservoir 7 via irreversible heat transfer. The relief of hydraulic fluid through the valve is determined by  θ p6 n  − 1 if p6 > pvalve J6,2 = Vm,6 pvalve (3.45) 0 otherwise, where θ and pvalve are the relief flux coefficient and the relief pressure of the valve, respectively. The molar volume of the hydraulic fluid of reservoir 6 is denoted by Vm,6 and its pressure by p6. As previously mentioned, this flow rate corresponds to the model of a simple pressure relief valve. However, by choosing a high value for θ, the behavior of the control valve is mapped sufficiently well. The valve housing is modeled as a solid body whose entropy is given by

T8 S8(T8) = Cp ln + S8,0, (3.46) T8,0 where Cp is the heat capacity of the valve housing at constant pressure, and T8,0 and S8,0 are reference values for its temperature and corresponding entropy. Thus, the

63 3 Recuperation System

Table 3.5: Parameters used for the pressure control valve 3 −1 θ = 10 m s pvalve = 300 bar mvalve = 0.79 kg −1 −1 −1 −1 cp = 502 J kg K Kvalve = 5 W K K8 = 0.7 W K temperature of reservoir 8 in dependence on the entropy can be expressed as S8 − S8,0 T8(S8) = T8,0 exp , (3.47) Cp where we use Cp = cpmvalve with the speciﬁc heat capacity of the valve housing material cp and its mass mvalve. The heat transfer between the valve housing and the hydraulic ﬂuid as well as that between the valve housing and the environment are assumed to obey Newton’s heat transfer law so that

I8,1 = Kvalve (T8 − T7) = −I7,4, (3.48)

I8,2 = K8 (T8 − T0) , (3.49) hold, with the estimated overall heat transfer coeﬃcients Kvalve and K8, respectively. The values for these coeﬃcients and other parameters introduced in this section are given in Tab. 3.5.

3.1.6 Composite Model

Having described the endoreversible models of all the individual components of the recuperation system, we can now take a look at the whole composite model shown in Fig. 3.9. As mentioned earlier, the numbering is the same as that used in the previous sections. Five reservoirs, namely the subsystems 2, 4, 5, 6 and 7, are used to model the hydraulic pipes. The spatial resolution associated with that could be increased at will by increasing the number of reservoirs representing the pipes. The inﬁnite environmental reservoir is again represented by lines instead of a rectangle, and the intensive quantities of all reservoirs are shown. Labeling the ﬂuxes has been omitted for readability, but they are consistent with the descriptions of the previous sections. Here, the reservoirs above the hydraulic unit, engine 3, represent the high pressure side and those below engine 3 represent the low pressure side of the recuperation system. The parameter values of the composite endoreversible model which were not given in the previous sections are shown in Tab. 3.6. Those include the lengths of the pipe segments li and the lengths needed to calculate the pressure losses li,j between two subsystems i and j.

64 3.1 Model Description

entropy flux T9 T10 particle flux T0 p9 p10 = p9 volume flux µ9 9 µ10 10

T4 T5 T6 T0 p4 p5 p6 T0 µ4 4 µ5 5 µ6 6

T P 0 3

T2 T1 T7 p p = p p T 2 1 0 7 8 8 µ2 2 µ1 1 µ7 7

T0 T0 p0 T0 T0

Figure 3.9: Composite endoreversible model of the recuperation system with the hydraulic components as described in the previous sections and the pipe segments represented by the reservoirs 2 and 4 to 7. Lines (instead of rectangles) represent the environment, and intensities as well as all incorporated ﬂuxes are shown.

Table 3.6: Additional parameters used for friction and heat loss in pipes

l2 = 1 m l4 = 2 m l5 = 1 m l6 = 1 m l7 = 2 m l1,2 = 1 m l4,5 = 2.5 m l5,6 = 1.5 m l7,1 = 2 m l5,9 = 0.5 m

At this point I would like to mention that all heat transfer parameters were estimated using the data and equations given in [96] and [98]. Other properties of the hydraulic components were estimated using product brochures [99–101] or were based on actual components used in the project I mentioned in the introduction.

65 3 Recuperation System

3.1.7 Driving Dynamics of the Truck

If we are interested in the dynamical behavior of the system and possible energy savings, a given driving profile of a vehicle which can be equipped with the recuperation system provides a suitable basis for investigation. From such a driving profile we primarily need the velocity v of the vehicle and the height profile h or the angle of inclination α of the street as a function of time. Using these variables, the power that is needed to accelerate or decelerate the truck can be expressed as dv ρ P = f m v + mgc cos(α)v + air c Av3 + mg sin(α)v. (3.50) truck m dt rr 2 d

Here, the first term is the acceleration component, where fm and m are the mass factor and the mass of the vehicle, respectively. The mass factor is used to take moments of inertia of transmission and combustion engine into account and is actually dependent on the selected gear. However, here it is approximated as constant, since we ignore the influence of gear switching operations and thus the calculation of suitable gears with and without recuperation for the sake of simplicity. The second and third term represent the rolling resistance and aerodynamic drag of the vehicle, where g and crr are the gravitational acceleration and the coefficient of rolling resistance, respectively. The parameters of the third term ρair, cd and A are the density of air, the drag coefficient and the cross sectional area of the truck. The last term of Eq. (3.50) represents the change in potential energy at inclines.

With these deﬁnitions we can conclude that if Ptruck is positive, the vehicle needs power to accelerate or to maintain its velocity. This power is then provided by the recuperation system and the combustion engine so that

Ptruck = Prec + Pcomb (3.51) holds. Accordingly, if Ptruck is negative, that power can be stored in the recuperation system or transferred to the discs brakes where it is dissipated, and hence

Ptruck = Prec + Pbrake. (3.52)

A simple control strategy of the recuperation system can therefore be based on the following principle: For a given driving proﬁle, Ptruck is an upper bound of energy transfer into or out of the recuperation system. If this power can not be provided or absorbed completely by the recuperation system, the recuperation system is supported by the combustion engine and the conventional brakes, respectively.

66 3.2 Energy Savings

Table 3.7: Parameters used for the driving dynamics of the truck

fm = 1.05 mtruck = 21 000 kg mrec = 550 kg −3 crr = 0.01 ρair = 1.293 kg m cd = 0.51 2 A = 8.25 m Rfd = 2.962 Cw = 2.879 m

Furthermore, from the velocity of the vehicle the rotational speed of the cardan shaft can be calculated as v ncyc = Rfd , (3.53) Cw where Cw is the eﬀective circumference of the wheel and Rfd is the ﬁnal drive ratio, which is the ratio of angular velocity of the cardan shaft to the angular velocity of the wheels. The angular velocity in rad s−1 can be calculated by

ω = 2πncyc (3.54) and the torque at the cardan shaft can be expressed as P τ = truck . (3.55) ω This torque – at least to some extent – enters or leaves the hydraulic unit as an extensity ﬂux transferring angular momentum at intensity ω. For the comparison of energy consumption of the vehicle with and without the recuperation system we consider the mass of the recuperation system mrec as additional weight in Eq. (3.50) with m = mtruck + mrec or m = mtruck, respectively. The values for all parameters of this section are given in Tab. 3.7.

3.2 Energy Savings

This section shows the numerical results obtained with the composite endoreversible model of the hydraulic recuperation system. The dynamical behavior of the system and its individual components will be discussed and a selection of parameters will be varied to investigate their influence on the energy saving potential of the system. For the initial state of the recuperation system the temperatures are at ambient temperature and the bladder accumulator is at its precharge pressure with the minimum amount of hydraulic fluid Vmin within. The hydraulic system supports the conventional disc brakes while storing energy or being in retarder mode and acts as a drive support as soon as energy has been stored, and until Vmin of the fluid is reached again.

67 3 Recuperation System

80 h] / 60 [km

v 40 20

velocity 0 340

[m] 320 h 300 280 height 260 96 97 98 99 100 101 102 103 104 time t [min] Figure 3.10: Section of investigated velocity and height proﬁle over time.

A section of the velocity and height profile over time that will serve as basis for the calculations in this section is shown in Fig. 3.10. The height profile was generated from mapped topographic data using the recorded GPS signal while the speed of the vehicle was read from its CAN bus. The data used to create the 120 minute profile was recorded on a truck collecting glass from bottle banks and therefore represents the typical daily operation and provides realistic results.

3.2.1 Dynamical Behavior of the System

Fig. 3.11 shows the key variables describing the dynamic behavior of the system. At the top we see the displacement factor which determines the generated hydraulic fluid flow which is pumped by or which propels the hydraulic unit at given pressure difference and rotational speed of the cardan shaft. A positive value stands for the fact that a hydraulic fluid flow from the bladder accumulator drives the hydraulic unit, which acts as a motor. This results in the falling pressures of the gas within the accumulator, which can be seen in the diagram below. In contrast, negative values indicate that the hydraulic unit is operating as a pump, transporting the hydraulic fluid from the tank to the high pressure side of the system. An increase of the pressure up to the relief value of the pressure control valve is the result. In the middle of Fig. 3.11 we can see the mechanical and volumetric efficiency of the hydraulic unit, which depends on the displacement factor, rotational speed and applied

68 3.2 Energy Savings

1 γ 0.5 0 −0.5 displ. factor −1

300 [bar] 10 p 250

200 pressure 1

η 0.8 0.6 0.4 eﬃciency η 0.2 mech ηvol

s] 100 n n n n / J1,2 J3,2 J3,3 J6,2

[mol 50 n J 0 −50 ﬂuxes

400 Pcomb Pbrake Prec Pvalve 200 [kW]

P 0 −200 power

96 97 98 99 100 101 102 103 104 time t [min] Figure 3.11: 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).

69 3 Recuperation System pressure difference. While the mechanical efficiency remains relatively constant and appears to be largely determined by the pressure curve of the high pressure side of the system, the volumetric efficiency is subject to strong fluctuations, which are mainly related to the vehicle’s velocity and thus to the rotational speed of the cardan shaft. n The latter determines the loss particle flux J3,3 (see Eq. (3.30)) to the hydraulic fluid tank, which is illustrated in red in the diagram below. n n n The fluxes J1,2, J3,2 and J6,2 are the ones from the tank towards the hydraulic unit, from the hydraulic unit towards the bladder accumulator, and the particle flux through n n the pressure control valve, respectively. The signs of J1,2 and J3,2 thus correspond to the sign of the displacement factor and depend on the operation mode of the hydraulic unit. It can be seen that the amount of particle flux out of the engine (positive green, negative blue) is always decreased by the amount of the loss flux compared to the particle flux into the engine. Furthermore, around t = 98 min when the gas pressure has reached the relief pressure of the valve, the operation of the pressure control valve n n can be recognized. Here, J3,2 and J6,2 seem to be identical, which they are actually not due to a remaining minimal flow into the accumulator which we will come to later.

The bottom diagram shows the power shares which together give the total power Ptruck needed to accelerate or decelerate the vehicle. In addition, the portion that could be achieved due to the use of the pressure control valve was marked. Here, it can be seen that for 97 min < t < 100.5 min large parts of the energy requirement of the truck can be covered by the recuperation system, whereas e. g. from t = 100.5 min to around t = 102 min the drive can hardly be supported. The reason for this difference is the velocity profile and, particularly in this case, the height profile. Until t = 100.5 min, the track is slightly downhill, whereas it is steeply uphill after that. This results in the increased power demand of the truck that can only be covered by the combustion engine.

Fig. 3.12 shows the temperatures T1 and T7 to T10 which belong to the hydraulic fluid tank, the pipe segment after the valve, the valve housing, and to the hydraulic fluid and gas within the bladder accumulator, respectively. It is particularly noticeable here how the temperature of the gas rises and falls with its pressure changes. At the same time, the temperatures of the hydraulic fluid within the bladder accumulator approaches the temperature of the gas. Particularly good to see is this exponential approach when – next to small interruptions – the amount of hydraulic fluid within the bladder accumulator remains constant after t = 101 min. In contrast, the hydraulic fluid does not heat up but cools down around t = 98 min when the pressure control valve is active. This is due to the fact that the gas slowly gives off heat to the hydraulic fluid. The resulting decreasing volume of the gas provides space for more hydraulic fluid to enter the accumulator. This minimal remaining flux – which I mentioned earlier – causes a slight cooling of the hydraulic fluid.

70 3.2 Energy Savings

T 60 1 T7 C] ◦ [ T8 T T9 40 T10

temperature 20

96 97 98 99 100 101 102 103 104 time t [min] Figure 3.12: Temperatures of selected components over time (section).

Meanwhile, the fluid flow through the pressure control valve heats up itself and thus also the valve housing, which slowly cools down to ambient temperature, thereafter. Furthermore, the amount of heated hydraulic fluid that enters the tank slightly increases its temperature. In order to obtain an estimate for the reduction in fuel consumption and brake wear, we look at the relative reduction in total energy that has to be provided by the combustion engine or absorbed by the disc brakes, respectively, during the whole driving cycle.

In this example, savings of energy to accelerate the vehicle of sac = 10.16 % could be achieved due to the use of the recuperation system. The braking energy which has to be absorbed by the disc brakes is reduced by sbr = 58.12 %. Just as these values depend on the considered driving proﬁle, they also depend on the chosen parameter values of the hydraulic recuperation system. In the next section we will deal with the variation of certain parameters and the consequences.

3.2.2 Variation of Selected Parameters

Bladder Accumulator Volume

Energy savings accelerating and decelerating the truck, sac and sbr, respectively, are shown over the bladder accumulator volume Vacc in Fig. 3.13. Here, we can see, as one would expect, an increase in energy savings for acceleration with increasing accumulator volume. However, this increase is slightly reduced after Vacc = 300 l. The reason for this is that with this volume already a large part of the braking processes are covered in the best possible way and the use of the pressure valve is reduced.

71 3 Recuperation System

20 70

[%] sac s 15 sbr 65 10 60 5 55

energy savings 0 50 200 400 600

accumulator volume Vacc [l] Figure 3.13: Energy savings both accelerating and decelerating the vehicle over bladder accumulator volume.

A further increase in accumulator volume only helps in a few cases of long braking processes, such as the one at t = 98 min discussed previously. The reusable amount of energy, however, does not increase for shorter braking processes. If we look at the reduction in braking energy that has to be absorbed by the disc brakes, we even see a negative eﬀect. Here, as the storage volume increases, the energy savings decrease. This is mainly due to the use of the pressure control valve, which allows to continue to support the braking process at relief pressure in retarder mode. Further, since a larger volume also causes a slower increase in pressure, the braking power which can be approximated by

P = Q∆p (3.56) with the volumetric ﬂow rate Q and the pressure diﬀerence ∆p is reduced.

Displacement of Hydraulic Unit

From Eq. (3.56) we can conclude two possibilities to increase the energy saving potential of the recuperation system. The first one is an increase of the displacement of the hydraulic unit Vd in order to generate higher volume flow rates at given rotational speeds. Fig. 3.14a shows the energy savings over Vd. In this case, for both acceleration and deceleration of the vehicle an increase in the hydraulic unit’s displacement tremendously increases the energy savings. While when increasing the accumulator volume, weight and space requirements have to be carefully weighed against benefits, here, a general solution seems to be found. However, the space under the vehicle at the cardan shaft is limited, which is why 3 the initial value of Vd = 307 cm was used. This value is the actual displacement of

72 3.2 Energy Savings

20 100 20 100 [%] sac [%] sac s s s 15 br 75 15 sbr 75 10 50 10 50 5 25 5 25

energy savings 0 0 energy savings 0 0 200 400 600 100 200 300 3 (a) displacement Vd cm (b) precharge pressure ppre [bar] Figure 3.14: Energy savings both accelerating and decelerating the vehicle over hydraulic unit displacement (a) and precharge pressure of the gas (b). the radial piston pump used in the project mentioned earlier, and an enlargement to higher displacements would require a revised construction of the hydraulic unit.

Precharge Pressure

The second possibility that can be concluded from Eq. (3.56) is to change the precharge pressure ppre of the gas within the bladder accumulator and hence changing the applied pressure difference at the hydraulic unit. In Fig. 3.14b the effect of varying precharge pressure on the energy savings is shown. Here, the relief pressure of the valve pvalve was varied accordingly so that pvalve = ppre + 100 bar holds. We can see that increasing the precharge pressure also leads to an increase in both energy savings for vehicle acceleration and deceleration. Compared to the behavior for increasing Vd, little to no difference can be observed and hence a beneficial effect can be achieved here, too. The reason why this diagram is comparatively limited in variation of the parameter of interest, is the lack of efficiency map data for higher pressure differences. In fact, the increase in precharge pressure is an option that comes with no extra space requirements. However, with higher operating pressures, additional safety relevant properties and the robustness of the individual components must be taken into account.

Heat Transfer within the Bladder Accumulator

The improvement of energy savings aside, the variation of parameters can also indicate the effects of loss factors such as heat and pressure losses. In Fig. 3.15a the energy savings over heat transfer coefficient Kacc is shown. This coefficient determines the heat transfer between the hydraulic fluid and the gas within the bladder accumulator.

73 3 Recuperation System

10.6 58.4 10.6 58.4 [%] sac [%] sac s s s 10.4 br 58.2 10.4 sbr 58.2 10.2 58 10.2 58 10 57.8 10 57.8

energy savings 9.8 57.6 energy savings 9.8 57.6 0 50 100 150 200 0 10 20 30 40

(a) heat transfer coefficient Kacc [W/K] (b) heat transfer factor φ Figure 3.15: 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.

One might assume, as previously indicated, that energy savings accelerating the vehicle decrease due to the decrease in gas pressure that results from the heat exchange with hydraulic fluid within the accumulator, and that this effect is enhanced at higher heat transfer coefficients. However, a contrary behavior can be observed and the lowest energy savings are obtained with zero heat transfer. One reason for this is that with decreasing pressure and hence decreasing volume of the gas within the bladder accumulator, there is – albeit very little – more space for hydraulic fluid to be pumped to the high pressure side of the recuperation system. In particular when the pressure control valve is active, as in the case discussed earlier, the amount of hydraulic fluid that can drive the hydraulic unit thereafter is increased while the high pressure remains. Another reason for this effect is the heat transfer in the opposite direction, from the hydraulic fluid at ambient or even higher temperatures to the cold gas after expansion. This increases the gas pressure and thus the pressure on the high pressure side of the system, adding a small amount to the usable energy stored within the accumulator. Note, that this does not contradict the observations and explanations in [89], since the driving profile investigated here leads to substantially different effects than the driving profile with long waiting times investigated by Schwalbe et al. Although the effects just mentioned seem to positively influence the amount of reusable energy, the effect of pressure loss outweighs them if we consider pure braking performance. Here we can observe a decrease in energy savings sbr with increasing heat transfer coefficient, which is due to the lower power inputs with decreasing pressures.

74 3.2 Energy Savings

Heat Transfer to the Environment

The heat transfer of the hydraulic fluid in pipes, tank and bladder accumulator towards the environment should have an influence on the effect just observed. An increased heat transfer might result in lower temperatures of the hydraulic fluid entering the accumulator and enhance named effect. In order to investigate this, we varied the coefficients of the heat transfers from the pipe segments, the hydraulic fluid tank and the hydraulic fluid within the bladder accumulator towards the environment using a scaling factor φ. The resulting heat transfer coefficients are thus φ times the values used before.

The energy savings sac and sbr over this heat transfer factor are shown in Fig. 3.15b. In fact, we can see that for increasing φ the energy savings during acceleration improve and the energy savings during deceleration decrease, and it can be assumed that the same reasoning applies here. However, in comparison to the variation of Kacc the magnitude of the eﬀect is lowered and even approaches a limit, here, and it can be assumed that this limit is determined by the heat transfer between the hydraulic ﬂuid and gas within the accumulator.

Pipe Diameter

Finally, we investigate the influence of the pipe diameter and thus the influence of the resulting pressure losses within the recuperation system. The energy savings over the inner diameter di of the pipes can be seen in Fig. 3.16. Without elaborating on Barlow’s formula, the outer pipe diameter is varied so that do = 1.1 di holds. It should be noted that as the diameter of the pipe increases, so does the heat transfer of hydraulic fluid within the pipes to the environment which might slightly enhance the effect of the hydraulic fluid cooling the gas. Although the relative influence is much smaller here, the effect is similar to the that of the bladder accumulator volume. In case of the hydraulic unit acting as a motor accelerating the vehicle, an increase in the inner pipe diameter leads to an increase in energy savings sac, which asymptotically approach a maximum value. This maximum value corresponds to the pressure loss free ideal case. In contrast, in the case of the hydraulic unit in pump mode, the energy savings sbr decrease with increasing diameter. This is due to the fact that, as a result of the pressure losses, the pressure that the hydraulic unit has to overcome is higher than the pressure within the bladder accumulator or at the pressure control valve in retarder mode. According to Eq. (3.56) this higher pressure generates higher power rates that can be received by the hydraulic recuperation system when braking and thus enhances the energy savings.

75 3 Recuperation System

10.6 58.6

[%] sac s 10.4 sbr 58.4 10.2 58.2 10 58

energy savings 9.8 57.8 20 40 60

pipe diameter di [mm] Figure 3.16: Energy savings both accelerating and decelerating the vehicle over inner pipe diameter.

However, the pressure losses at the lower end of the plotted diameter range already cause negative pressures in the first pipe segment between the hydraulic fluid tank and the hydraulic unit when the latter is operating in pump mode. This is an indica- tion that cavitation would occur at the hydraulic unit, permanently wearing out this component. Therefore, if smaller pipe diameters are used, a low pressure accumulator should be placed on the low pressure side of the hydraulic recuperation system instead of the hydraulic fluid tank, as can be seen in other publications [82,83].

3.3 Summary

In this chapter, the endoreversible modeling of the hydraulic recuperation system was shown and explained. Here, both the working gas within the bladder accumulator and the hydraulic fluid are modeled as van der Waals fluids. In the latter case, the necessary van der Waals and caloric parameters were determined using the coefficient of volumetric thermal expansion, the compressibility and the specific heat capacity at constant pressure of the hydraulic fluid. Furthermore, pressure losses within the pipe segments where incorporated in the modeling for both laminar and turbulent flow regimes. Heat dissipation to the environment is taken into account for the hydraulic fluid tank, the pipes segments, the valve housing and the hydraulic fluid within the bladder accumulator using Newton’s heat law. Additionally, heat transfer between the gas and hydraulic fluid within the bladder accumulator was incorporated. The modeling of the hydraulic unit was done using the dissipative engine with given efficiency introduced in Section 2.2.4. The corresponding mechanical and volumetric efficiency maps were incorporated as nonlinear model fits on the basis of mapped data. After the endoreversible models of the individual components were introduced, the composite model of the recuperation system was shown and the power needed or provided by the truck when accelerating or decelerating, respectively, was calculated.

76 3.3 Summary

Using the velocity and height profile of recorded driving data, simulations of the dynamical behavior of the system have been carried out to investigate the influence of certain parameters on the ability to save energy during acceleration and to reduce energy transferred to the disc brakes while braking. On a section of that driving profile, interesting effects in the temperature curves and fluid flows, especially when braking with active pressure control valve, were observed and explained. Subsequently, the parameters accumulator size, displacement of the hydraulic unit and precharge pressure of the gas were varied to investigate and discuss their influence and potential for improving energy savings. Finally, the coefficients of the heat transfers within the bladder accumulator and towards the environment as well as the inner pipe diameter were varied to investigate the influence of their associated loss terms. In particular, in the case of the heat transfer coefficient, the rather unexpected effect that its increase leads to increasing energy savings accelerating the vehicle could be observed.

4 Optimizing Fuel Savings

In the last chapter, we assumed that the hydraulic recuperation system supports the conventional power train, in which it is switched on in parallel, or even replaces the propulsion by the combustion engine as soon as possible. This strategy is called baseline strategy and is typically the reference point to which control optimization approaches are compared to. Here, publications often aim at deterministic control strategies that support the combustion engine in its low efficiency areas while taking the efficiency map of the hydraulic unit into account [83,84,102,103]. However, thermodynamic effects such as heat or pressure losses depending on the dynamics of the system are typically neglected. In this chapter, I will introduce a simplified model of the recuperation system based on the endoreversible formalism. Here, the modeling will be extended to the internal combustion engine and transmission to consider not only power sharing and resulting energy savings, but also the efficiency of the diesel engine and thus the resultant reduction of fuel consumption. Using this model and a realistic driving profile section, an optimization of the control strategy will be carried out, and its potential to reduce fuel consumption will be shown. The introduced model shall serve as the basis for further investigations in which dissipation and other energy loss factors can be incorporated with arbitrary degree of detail.

4.1 Model Description

As already mentioned, we include the combustion engine in the modeling, for which we additionally use the cardan shaft as the connecting link. Since we focus on efficient propulsion support by the recuperation system, not on the energy storing mode or the retarder mode, as far as the optimization is concerned, the overpressure line with the pressure control valve can be removed from the model. The resultant model is shown in Fig. 4.1. On the left side of the figure we can see the subsystems 1 to 4, which represent the recuperation system. Here, reservoirs 1 and 2 are the gas and the hydraulic fluid within the bladder accumulator, the dissipative engine 3 represents the hydraulic unit, and reservoir 4 represents the hydraulic fluid tank. It can be seen that the coupled particle

79 4 Optimizing Fuel Savings

T T 1 4 J V p1 p4 = p0 p0 µ1 1 µ4 4 7 L V n S Jtruck J J J Pdiesel L T2 S ω L J Jhyd Jcomb p2 = p1 ηhyd ω 5 ω ηcomb J n µ2 2 3 ω 6 L S T L L S Jhyd Jhyd 0 Jbrake Jcomb Jcomb

ω0 T0 ω0 ω0 T0

Figure 4.1: Endoreversible model used for fuel consumption minimization. The recuperation system is represented by the gas 1 and the hydraulic fluid 2 within the bladder accumulator, the hydraulic unit 3 and the hydraulic fluid tank 4. Engine 5 represents the cardan shaft and engine 6 is the combustion engine connected to a work reservoir 7 representing the diesel fuel tank. Energy transfers via the cardan shaft are carried by angular momentum fluxes. The angular momentum flux towards the chassis of the

vehicle with angular velocity ω0 = 0 represents energy dissipation at the conventional disc brakes. and entropy fluxes are now reversible and heat losses as well as the heat transfer within the bladder accumulator have been eliminated. On the one hand, this can partly be justified by the fact that the effects of pressure loss and heat transfer within the accumulator may have opposite influence on the resultant energy savings – as seen in the previous chapter – and thus may cancel each other out. On the other hand, it simply serves as simplification of the model for optimization. Further, by eliminating the pressure and heat losses, the modeling of the pipes connecting the hydraulic components is no longer necessary. In case of the hydraulic unit – the dissipative engine 3 – we now consider both extensity transfers namely combined particle and entropy transfer as well as angular momentum L transfer. Here, the angular momentum flux Jhyd connects the hydraulic unit with engine 5 which represents the cardan shaft of the vehicle with angular velocity ω. Since the latter is assumed to not store but only transfer energy, the modeling as an endoreversible engine is appropriate. Further, it is connected to the dissipative engine 6 which represents the internal combustion engine. The irreversible angular momentum L flux Jbrake towards an infinite reservoir with intensity ω0 = 0 represents the energy absorption and dissipation at the conventional disc brakes, and a reversible angular L momentum flux Jtruck is carrying the transferred energy to accelerate or decelerate the

80 4.1 Model Description truck. The internal combustion engine is modeled as a dissipative engine with torque L output represented by the flux Jcomb and has an unspecified power input Pdiesel from the work reservoir 7. Note that there is actually no direct angular momentum transfer between the cardan shaft and the disc brakes. The cardan shaft transfers the angular momentum of the hydraulic unit and the combustion engine towards the wheels which then transfer angular momentum to the disc brakes when braking. However, the simplified representation here suffices for the modeling. Having an overview of the model, we begin again from the left side of Fig. 4.1 describing the interactions as well as the gas and the hydraulic fluid within the system. The modeling of the bladder accumulator is similar to that described in Section 3.1.4, where the gas is modeled as a van der Waals gas and there is equilibrated pressure p1 = p2 and a constant overall volume Vacc = V1 + V2 within the accumulator. Hence we can write V V V J1,1 = J = −J2,1 (4.1) for the volume flux between the gas and the hydraulic fluid. The hydraulic fluid, however, is modeled as an incompressible liquid instead of a van der Waals fluid, which is a suitable assumption for liquids in general. From Section 2.1.4 we know, that the fluid flow described by the combined particle and entropy flux can also be described as an enthalpy flux since

n n J (µ + TSm) = J (Um + pVm) (4.2) holds, where Um and Vm are the molar internal energy and the molar volume of the ﬂuid. Doing so, the volume change of reservoir 2 can simply be expressed as

˙ n ˙ V2 = J2,2Vm = −V1 (4.3)

n with the particle ﬂux J2,2 towards engine 3. As far as the hydraulic unit is concerned, we use the overall eﬃciency of the hydraulic unit

ηhyd = ηmechηvol, (4.4) where ηmech and ηvol are functions of the displacement factor γ, the rotational speed of the cardan shaft ncyc and the pressure difference ∆p = p2 −p4, as given in Section 3.1.3. Furthermore, we can deduce from previous assumptions regarding heat transfers that the loss flux of the combined particle and entropy flux can be neglected. Instead we use the overall efficiency of the hydraulic unit to define a loss flux of angular momentum and the corresponding generated entropy which is transferred to the environment.

81 4 Optimizing Fuel Savings

In case of the hydraulic unit acting as a pump, according to the description of the dissipative engine with given eﬃciency in Section 2.2.4 the loss angular momentum and entropy ﬂux can be written as

L L Jhyd,loss = J3,1(ηmech − 1), (4.5) S L ω Jhyd = Jloss,hyd , (4.6) T0

L where we inserted the angular velocity ω0 = 0 of the vehicle’s chassis, to which Jloss,hyd L is transferred. In Fig. 4.1, this ﬂux is a portion of the ﬂux Jhyd towards the reservoir with ω0, while the other portion is the excess angular momentum resulting from the angular momentum balance of the hydraulic unit. Assuming T2 = T4 so that Um,2 = Um,4, we can thus write

L S n 0 = Jhydω + JhydT0 + J (p2 − p4)Vm (4.7) L n = Jhydωηhyd + J (p2 − p4)Vm, (4.8) where we used Eq. (4.2) and

n n n n n J = J4,1 = −J3,4 = J3,2 = −J2,2 (4.9) which can be derived from balance equations. The same procedure can be done for the hydraulic unit in motor mode and we obtain

L n 0 = Jhydω + ηhydJ (p2 − p4)Vm (4.10) as the governing equation. In accordance with Equations (3.36) and (3.37) we can now write L Prec = Ihyd = Jhydω (4.11) for both pump and motor mode, respectively. Also, according to the energy balance in engine 5, Equations 3.51 and 3.52 apply here if we deﬁne the energy transfers

L Pbrake = Ibrake = Jbrakeω, (4.12) L Pcomb = Icomb = Jcombω, (4.13) towards the conventional disc brakes and from the combustion engine, respectively. As mentioned before, the internal combustion engine is also modeled as a dissipative engine with given eﬃciency but with unspeciﬁed power input Pdiesel from the work reservoir 7 which represents the diesel tank. For named engine, as described in Sec-

82 4.1 Model Description tion 2.2.4 we can simply write Pcomb Pdiesel = , (4.14) ηcomb where Pdiesel is the power equivalent of the diesel fuel flux. Thus, the actual mass flow of diesel fuel into the combustion engine can be calculated by P m˙ = diesel , (4.15) diesel LHV where LHV is the lower heating value of diesel fuel. Lastly, we have to take look at the efficiency of the combustion engine.

4.1.1 Combustion Engine and Transmission

Fig. 4.2 shows a typical brake specific fuel consumption map of a large diesel engine used in a medium heavy truck. Here, the brake specific fuel consumption BSFC which represents the value of current fuel consumption is shown over the mean effective pressure pme and the rotational speed ncyc,comb of the engine. The mean effective pressure is a variable to assess the efficiency of reciprocating engines regardless of the

205 215 250 25 200 210 BSFC[g/kWh] 215 205 220 [bar] 230 200

me 20 p 200 195

210 215

220 15 205

200 200 20510 210 215 205 230 210 205 210 210 215 220 230 250 2155 215 220 220 230 270

mean effective pressure 250 230 230 300 250 250 270 270 270 300 800 1,000 3001,200 1,400 1,600 1,800 300 −1 rotational speed ncyc,comb min Figure 4.2: Brake specific fuel consumption map over mean effective pressure and rotational speed of the combustion engine. For this, a typical map for large diesel engines was used and scaled to an actual minimum fuel consumption value of 193 g/kWh. This value belongs to the engine of the vehicle with which the data used in Chapter3 was recorded.

83 4 Optimizing Fuel Savings

Table 4.1: Gear ratios of the transmission

Rt,1 = 15.86 Rt,2 = 12.33 Rt,3 = 9.57 Rt,4 = 7.44 Rt,5 = 5.87 Rt,6 = 4.57 Rt,7 = 3.47 Rt,8 = 2.70 Rt,9 = 2.10 Rt,10 = 1.63 Rt,11 = 1.29 Rt,12 = 1.00 engine size. In a four-stroke engine with two revolutions per power stroke, it can be converted to the output torque by V τ = p d,comb , (4.16) comb me 4π where we will use Vd,comb = 10.5 l as the displacement of the combustion engine. However, the rotational speed of the engine is related to that of the cardan shaft only via the transmission and therefore varies with the gear engaged. Same applies to the torque transferred to the cardan shaft τcard, while their relations are given by

τcard = Rt,i τcomb, (4.17) ncyc,comb ncyc = , (4.18) Rt,i respectively, where Rt,i are the gear ratios depending on the selected gear i. Since we want to avoid the modeling and simulation of the gear shifting, we assume that the selected gear always yields the lowest possible BSFC for the desired torque output and rotational speed of the cardan shaft. Using the gear ratios of a twelve- speed automatic transmission given in Tab. 4.1, we obtain the scheme which is shown in Fig. 4.3. Here, we can see the optimal gear with regard to minimal fuel consumption over desired torque output and vehicle velocity (see Eq. (3.53)). As expected with a transmission, it can be seen that the low gears transmit high torques at low speeds, and the high gears at high speeds can transmit only little torque. The empty area at high torques and high velocity exceeds the maximum power of the internal combustion engine. Using this optimal gear scheme, we can now plot the BSFC over the vehicle’s velocity and desired power output as shown in Fig. 4.4. Here, one notices that the jumps caused by gear changes, along the typical diagonal lines, are relatively small. The main dependence is the power output, while for increasing velocity the BSFC can be kept nearly constant by changing gears. Greater deviations from this only occur for high power outputs, which are more likely to occur when driving on highways than in urban area. Note, that the low values in the lower right corner of the image correspond to areas which exceed the maximum power, and were therefore set to zero.

84 4.1 Model Description

8 gear 4 0 0 10 0 20 [kNm] 40 20 τ velocity 60 v [km/h] torque

Figure 4.3: Optimal gears to achieve lowest brake speciﬁc fuel consumption over velocity and desired torque output of the combustion engine. The empty area exceeds the engines maximum power output.

300 kWh] /

[g 250

BSFC 200

velocity 60

[km 20

/ h] 0 0 50 100 150 200 250 300 power output P [kW] Figure 4.4: Lowest achievable brake speciﬁc fuel consumption using optimal gears (Fig. 4.3) over velocity and desired power output of the combustion engine. Low values in the lower right correspond to areas exceeding the maximum power output of the combustion engine and were therefore set to zero.

85 4 Optimizing Fuel Savings

0.5

η 0.4

0.3 efficiency ηcomb ηcomb 0.2 0 50 100 150 200 250 300 power output P [kW] Figure 4.5: Averaged values and polynomial fit of combustion engine efficiency over desired power output using optimal gears for lowest brake specific fuel consumption (Fig. 4.3).

Table 4.2: Parameters used for the combustion engine −3 −4 −1 c0 = 0.2607 kW c1 = 8.004×10 c2 = −2.041×10 kW −6 −2 −8 −3 −10 −4 c3 = 3.126×10 kW c4 = −2.995×10 kW c5 = 1.811×10 kW −13 −5 −15 −6 −18 −7 c6 = −6.680×10 kW c7 = 1.362×10 kW c8 = −1.169×10 kW

Given that, averaging the BSFC values over speed should yield a sufficiently good approximation. Using the lower heating value of diesel fuel LHV = 11.953 kWh/kg and the averaged values BSFC we can calculate the efficiencies of the engine for given power output as 1 η = (4.19) comb BSFCLHV and use a polynomial fit to reproduce the resultant curve as shown in Fig. 4.5. This polynomial fit is given by 8 X i ηcomb = ciPcomb (4.20) i=0 with the coefficients ci given in Tab. 4.2. Note that the transmission itself has an efficiency close to 1, which is therefore not considered. Furthermore, engine starts or shutdowns are not considered, which is the case in some other publications. To be precise, the engine does not use diesel fuel at idle in this model.

86 4.1 Model Description

4.1.2 Optimization

The objective function of the optimization is the total work

Z tend W = Pdiesel dt (4.21) t0 which the combustion engine received from the work reservoir 7. This represents the total diesel fuel consumption for a driving proﬁle over the time interval t = [t0, tend] and shall be minimized. The control variable is the displacement factor γ which determines the power input and output of the recuperation system. By deﬁnition it is constrained by − 1 ≤ γ ≤ 1. (4.22)

Furthermore, we deﬁne the pressure of the gas as a state variable for which the van der Waals equation of state given by

2 n1a p1 + 2 (V1 − n1b) = n1RT1 (4.23) V1 with the van der Waals parameters given in Tab. 3.1 has to be fulﬁlled at any time. Additionally, the following conditions apply to ensure reasonable functioning of the recuperation system:

V1 ≤ Vacc − Vmin, (4.24)

p1 ≤ pvalve, (4.25)

Ptruck ≥ Prec ≥ 0 if Ptruck ≥ 0 and (4.26)

Ptruck ≤ Prec ≤ 0 otherwise. (4.27)

These set the limits for gas volume and gas pressure, and ensure that the recuperation system’s power input or output does not exceed the power necessary to decelerate or accelerate the truck, respectively. The constraints given by (4.26) and (4.27) can actually be replaced by

Ptruck ≥ Prec ≥ Ptruck − Pcomb,max if Ptruck ≥ 0 and (4.28)

Ptruck ≤ Prec ≤ 0 otherwise, (4.29) where Pcomb,max is the maximum power output of the internal combustion engine. This allows that trough increased power output to the cardan shaft, the combustion engine can simultaneously transfer energy to the recuperation system to be stored within the bladder accumulator, and accelerate the vehicle. We will refer to the solu-

87 4 Optimizing Fuel Savings tions with these two diﬀerent constraints as optimal and optimal enhanced strategy, respectively.

The optimization itself is carried out discretizing the variables γ and p1 in time, and i i use the obtained {γ } and {p1} of the temporal nodes i as independent variables. These variables are connected trough the equations given in the previous sections as well as the constraints and objective function given above. An interior-point method is used with a large number of diﬀerent randomly chosen starting points. This is to increase the probability of ﬁnding a global rather than a local minimum, which is usually indicated by the fact that the best solution has been found several times.

4.2 Optimized Control

The numerical results for the introduced simplified model will be presented in this section. The obtained optimal solutions will be compared to the baseline strategy with regard to the dynamical behavior of the system and the resultant fuel savings. For the initial state the pressure and temperature of the gas were chosen to be ◦ p1 = pvalve and T1 = 50 C which roughly corresponds to a bladder accumulator that has just been fully charged. The velocity of the driving profile and resultant power to accelerate or decelerate the truck is shown in Fig. 4.6. Here, Ptruck was calculated using Eq. (3.50) and the height profile is constantly zero.

80 h] / 60 [km

v 40 20

velocity 0 200 [kW] 0 truck P −200 power 0 20 40 60 80 100 120 140 160 180 time t [s] Figure 4.6: Velocity proﬁle and corresponding power needed to accelerate or decelerate the truck.

88 4.2 Optimized Control

γ 1 baseline 0.5 strategy 0 −0.5

displ. factor −1 300 1 p 250

200 pressure

200

[kW] 0 Ptruck

P Pcomb P −200 rec power

1 η 0.8 ηhyd 0.6 ηcomb η 0.4 cc eﬃciency 0.2 0 20 40 60 80 100 120 140 160 180 time t [s] Figure 4.7: From top to bottom: displacement factor, gas pressure, power and eﬃ- ciencies of hydraulic unit and combustion engine over time for baseline strategy.

In Fig. 4.7 the key variables of the system with the baseline strategy are shown. At the top of the picture the displacement factor γ can be seen, which varies almost only between its extreme values, so that propulsion can immediately be fully supported and the maximum amount of braking energy can be stored. The drive support always lasts until the gas pressure p1 has reached its minimum value, as seen below, which corresponds to the minimum amount of hydraulic ﬂuid within the accumulator, and hence an empty energy storage. Underneath, the resulting power distribution between the hydraulic unit and the internal combustion engine is shown, where, again, one can clearly see that the combustion engine is immediately supported as soon as there is energy stored within the bladder accumulator.

89 4 Optimizing Fuel Savings

At the bottom of Fig. 4.7 the efficiencies of the hydraulic unit ηhyd and the combustion engine ηcomb can be seen. Additionally, the efficiency ηcc which the combustion engine would have if there was no recuperation system involved, is shown. Particularly note- worthy here are the two facts that the efficiency of the hydraulic unit at low speeds reaches very low values, and that the efficiency of the internal combustion engine always decreases when there is propulsion support by the recuperation system. The latter is due to the fact, that the support by the hydraulic unit decreases the power output of the combustion engine and thus – in the cases observed here – its efficiency which has a maximum at around 170 kW (see Fig. 4.5). This is particularly evident at t = 70 s and t = 150 s. The resultant diesel fuel savings using the baseline strategy are 20.84 % and – to compare this with the results of the last chapter – the energy savings accelerating the truck are sac = 21.59 %. This means that the short driving profile and initial conditions chosen in this chapter offer a much higher energy saving potential than the real data profile used in Chapter3. However, the profile used here does not serve to estimate the recuperation potential but only to optimize the control for typical acceleration processes in urban traffic, for which it is well suited. Having the baseline strategy as a reference we can now look at the optimized control strategies. The optimal solution for the first control strategy introduced using the constraints given by (4.26) and (4.27) can be seen in Fig. 4.8. Immediately apparent here is that the displacement factor does not just switch between its extreme values. While energy is still being stored with maximum available power, the displacement factor remains roughly between 0.3 and 0.8 during drive support. As can be seen from the power plot this is no limitation in needed energy output to accelerate the vehicle. Rather, the efficiencies of both the hydraulic unit and the combustion engine are thereby kept at high levels, which is particularly clear to see around t = 75 s in the efficiency plot. At the beginning of the driving profile, the hydraulic unit no longer supports propulsion, since it can operate only with poor efficiency due to the low velocity of the truck and hence the low rotational speeds. On the other hand, while having high efficiency values at higher velocities around t = 50 s and t = 170 s the hydraulic unit supports or replaces the combustion engine which has low efficiency values there. The diesel fuel savings achieved with this optimal control strategy amount to 25.26 % which equates to an improvement of more than one fifth. Finally, we come to the optimal enhanced strategy, shown in Fig. 4.9, where the engine can both propel the vehicle and transfer energy into the recuperation system at the same time. We can see that this possibility is actually used twice, at t = 30 s and t = 130 s. In both cases, the internal combustion engine operates with improved efficiency due to the increased power output. In addition, before and after, more

90 4.2 Optimized Control

γ 1 optimal 0.5 strategy 0 −0.5

displ. factor −1 300 1 p 250

200 pressure

200

[kW] 0 Ptruck

P Pcomb P −200 rec power

1 η 0.8 ηhyd 0.6 ηcomb η 0.4 cc efficiency 0.2 0 20 40 60 80 100 120 140 160 180 time t [s] Figure 4.8: From top to bottom: displacement factor, gas pressure, power and efficiencies of hydraulic unit and combustion engine over time for optimal control strategy. energy from the recuperation system can be used in areas with high efficiency values of the hydraulic unit, which is in particularly visible around t = 20 s. While in the remaining sections the behavior is similar to that of the optimal strategy, diesel fuel consumption can be lowered a little further and we achieve total fuel savings of 25.72 %. This small improvement is explained by the increased fuel consumption when the accumulator is charged in addition to propulsion. All three strategies have in common that up to t = 87 s, the bladder accumulator is completely emptied which can be recognized by the minimum pressure value. The reason for this is that during the following braking process the bladder accumulator

91 4 Optimizing Fuel Savings

γ 1 optimal 0.5 enhanced 0 strategy −0.5

displ. factor −1 300 1 p 250

200 pressure

200

[kW] 0 Ptruck

P Pcomb P −200 rec power

1 η 0.8 ηhyd 0.6 ηcomb η 0.4 cc eﬃciency 0.2 0 20 40 60 80 100 120 140 160 180 time t [s] Figure 4.9: From top to bottom: displacement factor, gas pressure, power and eﬃcien- cies of hydraulic unit over time for optimal enhanced control strategy.

can be filled again and at t = 108 s the maximum pressure p1 = pvalve is reached. Therefore, with the accumulator size used here, a split of the driving profile at this point and a separate optimization is possible as it is done e. g. in [104]. However, for larger accumulator sizes it is easy to devise scenarios in which the stored energy should be saved for usage after a deceleration, comparable to the effect that occurs here with the shorter braking process at t = 130 s.

92 4.3 Summary

4.3 Summary

In this chapter, a simplified endoreversible model of the recuperation system for optimization purposes was introduced. In this model heat losses and transfers, as well as pressure losses were neglected and the hydraulic fluid was modeled as an incompressible fluid. Further, incorporation of a disc brake model as well as a model for the combustion engine and the transmission was done to investigate the power distribution between these components. Interactions with angular momentum transfers were used to allow modeling of the engine’s losses fully consistent with the endoreversible formalism. Here again, the dissipative engine with given efficiencies introduced in Sec- tion 2.2.4 was used to provide given efficiency maps to the modeled engines. In order to derive an efficiency map for the combustion engine independent of gear changes, an optimal gear assumption was made for the transmission. The resultant fuel consumption map over velocity and power output could be averaged over the velocity and was provided to the simulation. The optimization with regard to minimum fuel consumption showed significant improvements for the investigated driving profile. The main reason for this is that the recuperation system can support the propulsion of the vehicle in moments in which the combustion engine operates in a particularly inefficient area, or must not support the propulsion if same applies to the hydraulic unit. Especially at low velocities, the support of the hydraulic recuperation system is avoided, which is not the case with the baseline strategy. The additional optimal enhanced strategy which allows for simulta- neous propulsion of the vehicle and storing of energy within the bladder accumulator showed only little improvement compared to the previous optimization.

5 Conclusion

In this thesis the endoreversible formalism was applied on a hydraulic system for the first time. The initial motivation was the investigation of thermodynamic effects within a hydraulic recuperation system and their influence on the system’s dynamical behavior and energy saving potential. To implement this, while providing a simplifying tool and exemplifying the modeling as well as the possibility of optimizing a hydraulic system using endoreversible thermodynamics, the following three steps have been performed. First, lossy interactions were introduced which are defined by an additional extensity transfer towards a third reservoir, that diminishes the transfer between the two main reservoirs connected by the interaction. This not only extends the formalism of endoreversible thermodynamics to take account of further occurring loss effects of this kind, such as leakage or friction, but also offers the possibility to create an interesting engine setup for engines with predefined efficiencies or efficiency maps. This dissipative engine with given efficiency can be used to easily set up and investigate the thermodynamics of compound systems with a large number of various interacting components while remaining within the framework of endoreversible modeling. It thus represents a useful tool to model and simulate such systems in order to find synergy effects and main sources of energy loss and entropy generation. The second step was the detailed description of the individual components’ and the overall hydraulic system’s endoreversible modeling. Here, the rather unconventional approach of modeling the hydraulic fluid using the van der Waals equation was chosen in order to incorporate pressure losses within the fluid and along the pipes between the main hydraulic components of the system. The heart of the recuperation system, the hydraulic unit, was modeled using the introduced dissipative engine with given efficiency and fitted data of mechanical and volumetric efficiencies. Furthermore, heat losses to the environment and heat transfer between the gas and the hydraulic fluid within the bladder accumulator were incorporated, and the pressure control valve was modeled as a simple interaction and an additional reservoir representing the valve housing. The numerical results obtained with this complex endoreversible model were observed with regard to the dynamical behavior of the system and its energy saving potential both accelerating and decelerating the vehicle. Using an extensive driving profile

95 5 Conclusion of real driving data, interesting effects of the interacting components, e. g. when the pressure control valve is active, were examined. A subsequent analysis of the influence of selected parameters showed how they affect the dynamical behavior and thus the resultant energy savings. In particular, the counterintuitive influence of the heat transfer within the bladder accumulator can be mentioned as an outstanding finding. As for all investigations concerning hybrid technologies and fuel consumption, estimates for the energy saving potential are highly dependent on the examined driving profile. We have therefore used real driving data of a vehicle suitable for the installation of the recuperation system to obtain realistic results for energy savings during acceleration and deceleration which were found to be around 10 % and 58 %, respectively. Furthermore, the work done here can be viewed as an extensive application of the multi-extensity fluxes introduced by Wagner [64,65] which are necessary to model the fluid flows within the system. What has not been discussed in this work is that in particular the modeling of the hydraulic fluid as a van der Waals fluid in a number of reservoirs provides a much more detailed resolution of the pressure and temperature distribution along the pipes, but also causes an additional non-negligible numerical effort. Lastly, the optimization of the hydraulic system of interest was exemplified by incorporating simplifying assumption such as negligible pressure and temperature losses, as well as by adding models of the combustion engine, the conventional disc brakes, and the cardan shaft as the connecting unit. This way the additional main source of energy loss regarding propulsion of the vehicle was incorporated in the modeling in accordance with the endoreversible formalism, and the optimal interplay of the components hydraulic unit and internal combustion engine could be investigated. These two components were again modeled as dissipative engines with given efficiencies to incorporate their rather complex efficiency maps, where in the latter case, a simplified relationship between efficiency and power output was assumed and derived. The optimal solutions which were obtained for a fuel consumption minimizing recuperation strategy showed remarkable improvements in comparison to the simple baseline strategy. Here, the reduction in fuel consumption could be enhanced from roughly 21 % using the baseline strategy to almost 26 % for both of the optimized strategies. The two examined optimized strategies did not differ greatly with regard to fuel consumption. Of course, the modeling of the recuperation system was drastically coarsened, but the general feasibility was exemplified and the introduced model may serve as a basis for further investigations. These further investigations may then incorporate the loss terms, e. g. pressure and heat losses, whose modeling was presented in Chapter3, as well as other possible sources of energy dissipation. As a result, deterministic rules for a consumption- efficient control of the recuperation system might then be derived, or an online optimization solution could be established where driving profiles are recorded and used to improve energy storage and reuse during operation, similar to the work by Bender

96 at al. [104]. Regarding the detailed recuperation system model, heat exchange with the vehicle’s cooling circuit can be implemented, and the components bladder accumulator and pressure control valve may be described in more detail. Here, varying heat transfer in dependence on current bladder size and a more detailed entropy generation and heat dissipation model of the valve may be of interest. Also, while the presented model hopefully provides a suitable guideline for endoreversible modeling of hydraulic systems, following models of hydraulic systems can be extended by including components such as check valves, or eﬀects like ﬂuid hammer.

Nomenclature

Greek α inclination of the street ◦ −1 αV coefficient of volumetric thermal expansion K β compressibility Pa−1 γ displacement factor - θ relief flux coefficient m3 s−1 η efficiency -

ηC Carnot eﬃciency - λ thermal conductivity WK−1 µ chemical potential J mol−1 ν kinematic viscosity m2 s−1 ρ density kg m−3 σ entropy production WK−1 ω angular velocity rad s−1

Latin a cohesion pressure Pa m6 mol−2 A area m2 b co-volume m3 mol−1 BSFC brake speciﬁc fuel consumption g kW−1 h−1

cd drag coefficient - −1 −1 cp specific heat capacity at constant pressure J kg K crr coefficient of rolling resistance - −1 −1 cV specific heat capacity at constant volume J kg K cˆV dimensionless heat capacity at constant volume - −1 Cp heat capacity at constant pressure JK −1 CV heat capacity at constant volume JK Cw wheel circumference m d diameter m E energy J f coupling coefficient *

fD Darcy friction factor - fm mass factor -

99 Nomenclature

F Faraday constant C mol−1 g gravitational acceleration m s−2 G Gibbs free energy J h specific heat transfer coefficient WK−1 m−1 H enthalpie J I energy flux W J extensity flux * k heat transfer coefficient WK−1 m−2 K overall heat transfer coefficient WK−1 l length m L angular momentum kg m2 s−1 LHV lower heating value kW h kg−1 m mass kg τ torque N m M molar mass kg mol−1 n mole number mol −1 ncyc rotational speed s p pressure Pa

pme mean eﬀective pressure Pa P power W q heat transfer W Q volumetric ﬂow rate m3 s−1 R universal gas constant J mol−1 K−1 Re Reynolds number -

Rfd ﬁnal drive ratio - Rt,i gear ratio of the transmission in gear i - s energy savings - S entropy JK−1 t time s

tcyc cycle 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 m3 3 Vd displacement m X extensity * Y intensity * z charge number -

* varying unit

100 Subscripts 0 initial or reference value ac acceleration acc bladder accumulator air air br braking brake conventional disc brake c critical value comb combustion engine diesel diesel fuel G given value or function hyd hydraulic unit H high value i inner in input int intersection lam laminar loss loss L low value m molar quantity max maximum value mech mechanical min minimum value M medium value o outer out output pre precharge rec recuperation system red reduced truck truck tur turbulent valve valve vol volumetric

101

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110 Lebenslauf

Persönliche Daten Name: Robin Masser Geburtsdatum: 28.06.1991 Geburtsort: Gera

Bildungsweg 2002 – 2010 Zabel-Gymnasium Gera Allgemeine Hochschulreife mit Gesamtnote 1,3 2010 – 2013 TU Chemnitz Bachelorstudium Computational Science Bachelor of Science mit Gesamtnote 1,8 2013 – 2015 TU Chemnitz Masterstudium Computational Science Master of Science mit Gesamtnote 1,2 2015 – 2019 TU Chemnitz Promotionsstudium Physik

Berufserfahrung 2012 – 2014 Studentische Hilfskraft am Fraunhofer-Institut ENAS Bereich System Packaging 2014 – 2015 Wissenschaftliche Hilfskraft an der TU Chemnitz Professur für Technische Mechanik / Dynamik 2015 – 2019 Wissenschaftlicher Mitarbeiter an der TU Chemnitz Professur für Theoretische Physik, insbesondere Computerphysik

Tagungen

Mär 2016 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics Braunschweig, Deutschland (Vortrag) Jul 2017 30th International Conference on Eﬃciency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems San Diego, USA (Vortrag) Mär 2018 Frühjahrstagung der Deutschen Physikalischen Gesellschaft Berlin, Deutschland (Vortrag) Jun 2018 5th International Conference on Fluid Flow, Heat and Mass Transfer

111 Lebenslauf

Niagara Falls, Kanada (Poster) Jun 2018 31st International Conference on Eﬃciency, Cost, Optimization, Simulation and Environmental Impact of Energy Systems Guimarães, Portugal (Teilnahme)

Publikationen

Okt 2013 R. Masser Predicting the Reaction Behavior of Integrated Nanoscale and Reac- tive Multilayer Systems Bachelorarbeit, TU Chemnitz Jun 2014 R. Masser, J. Braeuer, and T. Gessner Modeling the reaction behavior in reactive multilayer systems on sub- strates used for wafer bonding Journal of Applied Physics, 115, 244311 Okt 2014 J. Braeuer, J. Besser, S. Hertel, R. Masser, W. Schneider, M. Wiemer, and T. Gessner Reactive Bonding with Integrated Reactive and Nano Scale Energetic Material Systems (iRMS): State-of-the-Art and Future Development Trends ECS Transactions, 64(5), 329–337 Sep 2015 R. Masser Entwicklung und Implementierung einer Zeitadaptivität für Galerkin- basierte und variationelle Integratoren Masterarbeit, TU Chemnitz Okt 2016 R. Masser, M. Bartelt, and M. Groß Structure analysis of higher-order variational integrators and imple- mentation of SUNDMAN adaptivity in time Proceedings in Applied Mathematics and Mechanics, 16, 221 Jul 2017 R. Masser, K. Schwalbe, and K. H. Hoffmann Energy Recuperation System for Skip Trucks Proceedings of ECOS 2017, July 2–6, 2017, San Diego, USA Jun 2018 R. Masser, K. Schwalbe, and K. H. Hoffmann Optimized Control of a Hydraulic Recuperation System for Skip Trucks Proceedings of FFHMT’18, June 7–9, 2018, Niagara Falls, Canada Sep 2018 J. Prehl, R. Masser, P. Salamon, and K. H. Hoffmann Modeling reaction kinetics of twin polymerization via diﬀerential scanning calorimetry Journal of Non-Equilibrium Thermodynamics, 43(4), 347–357

112 Mai 2019 R. Masser, A. Khodja, M. Scheunert, K. Schwalbe, A. Fi- scher, K. H. Hoffmann Optimized Piston Motion for a Stirling Engine Manuskript vorliegend

113