**1. Introduction**

The internal combustion engine is one of the most common engines. With this motor, chemically stored energy is converted into mechanical energy. The energy conversion unfortunately produces exhaust gases and losses, which is the focus of research of many scientific elaborations [1]. To improve efficiency [2,3], various components were optimized [4], control strategies were adapted, valve timing was varied [5], alternative fuels [6] were investigated and much more.

Due to the shortage of fossil resources and the environmental impact of the combustion engine, alternative concepts for optimal energy conversion are being investigated. One possible future-oriented concept here is the free-piston engine [7–9]. The free-piston engine consists of an internal combustion engine, where the piston is connected to a linear electrical machine [10–12]. This means that no crankshaft is required, which makes it possible to vary the stroke trajectory, and thus, also the compression ratio. Various prototypes have already been developed [13,14]. With these prototypes, it could be shown that the operation of a two- or four-stroke free-piston engine is possible [15–17].

Tempelhagen,A.; Benecke, S.; Klepatz, K.; Leidhold,R.; Rottengruber, H. Investigations fora Trajectory Variation to Improve the Energy Conversion for a Four-Stroke Free-Piston Engine. *Appl. Sci.* **2021**, *11*, 5981. https://doi.org/10.3390/ app11135981

 R.; Gerlach,

**Citation:**

Academic Editors: Cinzia Tornatore and Luca Marchitto

Received: 31 May 2021 Accepted: 24 June 2021 Published: 27 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The free piston engine offers different research focuses. One major focus is the optimization of the control strategy [16,18–21]. Another focus is the analyzation of the combustion process [22–24]. As a third major research focus, the modeling investigation of the modeling of a free-piston engine can be defined [25,26]. The fourth research topic is the optimal design of electric machines [27–29].

In addition to varying the compression ratio, it is also possible to adjust the stroke trajectory if the linear machine is adequately controlled [8]. The influence of the piston trajectory on the combustion is also a very interesting research topic [30,31]. In the present work, different stroke trajectories are proposed and analyzed with respect to the losses in the combustion process and conversion of mechanical energy to electric energy with an electric machine. The novelty of this work is the analyzation of the system efficiency of a free-piston engine with respect to the piston trajectory. There was particular interest in how the losses can be varied with a variation in the stroke amplitude, the acceleration of the piston and a time shift based on the TDCF. It could be proved that the stroke course has a major influence on the efficiency of the internal combustion engine. In Section 2, it is explained how the chemically stored energy is converted into electrical energy. Section 3 explains how the experimental prototype is constructed. For the prototype, a thermodynamic model and an electromechanical model were designed, which are described in Sections 4 and 5, respectively. With these models, it is possible to determine the energy conversion losses. The same piston stroke curves were specified for the models. The generation of trajectories is explained in Section 6. The results are presented in Section 7 and summarized in Section 8.

#### **2. Consideration of the Energy Losses for Converting Chemically Stored Energy into Electrical Energy**

To justify the pros and cons for different piston trajectories regarding the thermodynamic and electric efficiency of the free-piston engine, a loss analysis is used. Depending on the piston trajectory, different losses may result in terms of combustion characteristics, heat losses, gas exchange, compression and expansion characteristics.

The chemical fuel energy *Ef uel*, which enters the engine via the injector, represents the maximal supplied energy for the engine and is the basis of the loss analysis. This energy only depends on the injected mass *mfuel* and the lower heat value *LHV* according to:

*Ef uel* = *mf uel*·*LHV* . (1)

This chemical energy is transformed to thermal energy during the combustion process. Due to the fact that an ideal combustion is not realizable, there is always an amount of unburned fuel *munb*, which leads to an energy loss during the transformation from chemical to thermal energy *Eth*. Besides the imperfect combustion, incomplete combustion could occur when injected fuel is not completely thermally converted and oxidized. When fuel reaches the cylinder wall or the upper face of the piston during combustion, the flame extinguishes and is, thus, not available for further thermal conversion. Out of the difference of the incomplete converted fuel and the injected fuel, incomplete combustion efficiency can be derived. The parameter *ηcomb* describes the efficiency of the combustion based on the converted fuel mass, whereby the thermal energy yields to:

$$E\_{\rm th} = \left(m\_{\rm fuel} - m\_{\rm unb}\right) \cdot LHV = \eta\_{\rm comb} \cdot E\_{\rm fuel} \,. \tag{2}$$

This thermal energy is the basis for the volume change work. Due to the fact that an energy loss analysis is used for the comparison, the volume change work is called here the volume change energy *Evc*. The combustion process creates high temperature gradients, which lead to an energy loss through the walls *Ewal*. This is described by the heat flux through the walls of the combustion chamber, which is defined by:

$$E\_{wall} = \mathfrak{a} \cdot A \cdot \Delta T \cdot \Delta t \text{ .} \tag{3}$$

where *α* is the heat transfer coefficient (HTC), *A* is the area of the combustion chamber, Δ *T* is the temperature gradient between the surrounding solid material and the gas inside the combustion chamber and Δ*t* is the time in which the process is performed. All parameters are space- and time-dependent. The parameters in Equation (3) are mean values over one work cycle to simplify the formulas for a better understanding [32]. The time-dependent temperature at the wall, which is causing Δ *T*, is influenced by all three ways of heat transfer. Due to the very low thermal conductivity coefficient of air, the influence of the conduction within the combustion chamber is low as well. The influence of the radiation to the wall temperature depends on the used ignition strategy and the corresponding injection system. Diesel engines with direct injection are more influenced as gasoline engines in general and especially with port fuel injection. The last influence is the convection. Due to the fast motion of the gas mixture inside the combustion chamber, it has strongest influence.

Not all of the thermal energy can be transformed into volume change energy within the given time for the expansion stroke. Therefore, thermal energy *Eexh* is lost within the exhaust gas when leaving the combustion chamber over the exhaust valve, as defined as:

$$E\_{\rm exh} = E\_{\rm int\ out} - E\_{\rm int\ in} = (T\_{\rm out} \cdot c\_{p\_{\rm out}} - T\_{\rm in} \cdot c\_{p\_{\rm in}}) \cdot m\_{\rm gas} \, . \tag{4}$$

where *Eint out* is the internal energy within the exhaust gas, *Eint in* is the internal energy within the supplied air-fuel-mixture, T is the mean temperature, *cp* is the mean specific heat capacity and *mgas* is the gas mass which is supplied and exhausted. The values for *T* and *cp* of the supplied and exhaust gas are averaged over the time during which the specific valve is open.

The volume change energy *Evc* is the balance of the thermal energy, the wall heat losses and the thermal energy losses through the exhaust. The losses can be described by the thermal efficiency *ηth*. The volume change energy, thus, results in:

$$E\_{\rm rc} = E\_{\rm th} - E\_{\rm vall} - E\_{\rm exh} = \eta\_{\rm th} \cdot E\_{\rm th} \,\,. \tag{5}$$

The volume change energy leads to a motion of the piston, which in turns is always subject to friction. The friction losses arise mainly in our prototype in the piston rings, bearings and the valve train. The mechanical energy *EM* is the subtraction of the friction energy from the volume change energy and can be described by the friction efficiency loss *ηFr* as follows:

$$E\_M = E\_{\rm rc} - E\_{Fr} = \eta\_{Fr} \cdot E\_{\rm rc} \tag{6}$$

As described in Equation (6), friction occurs, which can also be described as:

$$P\_{Fr} = \upsilon^2 \cdot u = F\_{Fr} \cdot v \,, \tag{7}$$

where *μ* the coefficient of friction and *v* is the speed of the translator.

The mechanical power *PM* for the electrical machine can be expressed as:

$$P\_M = F\_{EM} \cdot \upsilon\_\prime \tag{8}$$

where *FEM* is the force of the electric machine. Due to the oscillation of the translator, the system requires acceleration power *Pacc* in many operating points. The acceleration power can be expressed with:

$$P\_{\rm acc} = m \cdot \frac{dv}{dt} \cdot v \,, \tag{9}$$

where *m* is the accelerated mass of the system and *dvdt* the time derivative of the speed. The power output from the internal combustion engine *PICE* can be described with:

$$P\_{ICE} = F\_p \cdot v = A\_{cyl} \cdot p \cdot v,\tag{10}$$

where *p* is the cylinder pressure, *Acyl* is the area of the piston and *Fp* is the gas force. In addition, in the electrical machine the reluctance power *PRel* arises, which can be described as a function of position:

$$P\_{\rm Rel} = f(\mathbf{x}) \cdot \mathbf{v}. \tag{11}$$

The power equilibrium on the mechanical transmission element results as follows:

$$P\_{ICE} + P\_M + P\_{Fr} + P\_{Rel} = P\_{acc} \, . \tag{12}$$

In the case of a directly driven free-piston engine [17], the trajectory can be varied highly dynamically with the electric machine using the field oriented control. This is possible with the electrical machine force *FEM*, which is proportional to the current *iq* via the force constant *kF*. The force of the electric machine can be expressed with:

$$F\_{EM} = \frac{3}{2} k\_F i\_{\emptyset}.\tag{13}$$

The mechanical energy can be converted into electrical energy with the help of the electrical machine.

The special feature of the free-piston engine is that it does not have a crankshaft.The required oscillation movement of the piston can be realized with the help of a linear electric machine. The electrical machines can be constructed in different ways [27,33,34].Depending on the machine type, different power losses can arise. The power loss occurs when the iron is re-magnetized, which is referred to as hysteresis power loss *PHys*. Eddy current power dissipation *PFt* can also arise. The sum of the hysteresis power loss and the eddy current power loss is referred to as the iron power loss *PFe*:

$$P\_{\rm Fc} = P\_{\rm FI} + P\_{\rm Plys} \; . \tag{14}$$

In addition to the iron power loss, there is also a power loss due to the current *iq* in a coil winding with ohmic resistance *R*. This power loss is referred to as the copper power loss. The copper power losses are calculated in the d/q-reference frame as follows:

$$P\_{Cu} = \frac{3}{2} \left( i\_q^2 + i\_d^2 \right) \cdot R \,. \tag{15}$$

Due to the iron and copper power losses, the entire mechanical power on the electrical machine cannot completely be converted into electrical power. This can be expressed stationary as follows:

$$P\_M + P\_{Cu} + P\_{Fe} = P\_{el}.\tag{16}$$

The electrical power *Pel* can be determined with the voltages *ud*,*<sup>q</sup>* and currents *id*,*q*.

$$P\_{cl} = \frac{3}{2} \left( \mu\_d \cdot i\_d + \mu\_q \cdot i\_q \right) \tag{17}$$

By integrating the calculated powers over a cycle, it is possible to calculate the energies.

## **3. Experimental Prototype**

The experimental measurements for the validation of the simulation model were obtained with the help of the prototype in Figure 1 and is shown schematically in Figure 2.

**Figure 1.** Experimental Prototype of the free-piston engine.

**Figure 2.** Schematic representation of the free-piston engine.

The internal combustion engine was originally used in a lawn mower. The crankshaft was removed and replaced with a piston and rod. This rod was passed out of the crankcase and connected to a linear electrical machine. The electrical machine is designed by the manufacturer Tecnotion (2xUXX6) as a permanent magne<sup>t</sup> excited synchronous machine (PMSM).

The mover of the electrical machine is ironless, where the primary part is moved. In this way, no cogging forces are present and the mass to be moved is low. The position of the electric machine was controlled using an inverter, a linear sensor and a microcontroller from Texas Instruments F28M35H52C. This microcontroller is able to calculate different trajectories, evaluate sensors and calculate manipulated variables (such as ignition sparks, injection times and transistor switch-on times). The data were then sent to a PC, saved and evaluated. The camshaft is connected to a highly dynamic PMSM. With the help of a valve drive, it is possible to operate the valves independently of the piston position. The camshaft and piston are synchronized digitally. To lubricate the cylinder surfaces, a nozzle was inserted which sprays a thin layer of oil under the piston. The most important parameters of the free-piston engine are shown in Table 1.



The problem with the electrical machine was that high copper losses occur when the four-stroke internal combustion engine is operated. It will be replaced with a selfdeveloped electric machine [29]. The primary part of the electrical machine is not moved. It has a flat geometry, a short stator and is double-sided. The essential characteristic data is illustrated in Figure 3 and shown in Table 2.

**Figure 3.** (**a**) CAD illustration of the developed electrical machine. (**b**) Photo of the developed electric machine.

The electrical machine was designed based on a measured force curve through the first prototype. A special feature of the electrical machine is that the primary part has the same length as the secondary part. This creates reluctance forces when the secondary part moves out of the stator. This force acts like a mechanical spring and can minimize copper losses. The reluctance force for this electrical machine is shown in Figure 4a. However, iron losses occur due to the magnetic structure. The iron power loss *PFe* was measured as a

function of the electrical frequency *fel* at various operating points and a function for this was approximated [35]. The function here corresponds to a Steinmetz model:

$$P\_{Fe} = \mathbb{C}\_{hy} \cdot \hat{B}^{1,6} \cdot f\_{el} + \mathbb{C}\_{wb} \cdot \hat{B}^2 \cdot f\_{cl}^2 \tag{18}$$

where *B* ˆ is the magnetic flux density and *Chy* and *Cwb* are two coefficients. The iron power loss is shown in Figure 4b.

**Table 2.** Characteristic data of the developed electrical machine.

**Figure 4.** (**a**) Reluctance force *FRel* depending on the position *x*. (**b**) Iron power losses *PFe* depending on frequency *fel*.

These special properties were inserted into a simulation model in order to determine the losses in the respective trajectories. For the simulations that were carried out, the parameters of the newly developed machine were used.

Since the secondary part moves out of the stator and the proportion of covered magnets to the windings is reduced, the magnetic flux is also reduced. This leads to a reduction in the induced voltage as well as to a reduction in the force per ampere ratio. Therefore, the force of the electrical machine depends on the position and current and is shown in Figure 5. It can be seen in Figure 5 that the force decreases at the stroke limits.

**Figure 5.** Force *FEM* depending on position *x* and current *iq*.

#### **4. Thermodynamic Model of the Free-Piston Engine**

Based on the data of Table 1, a one-dimensional thermodynamic model for the freepiston engine is built up. The combustion engine is working with an external mixture formation via a gasoline injector with a constant injected fuel mass in the intake manifold. The fuel mass is chosen this way so that the equivalence ratio *φ* = 1. This model of the free-piston engine is quite similar to an engine with a conventional crank train. To model the combustion process and the corresponding heat release, the predictive combustion model, the so-called Spark-Ignition Turbulent Flame Model (SITurb) from GT-Suite, was used. This model can be used for several operating points after it has been parameterized once. Additionally, it is normally used for gasoline engines, which is the test bench engine. The heat release is calculated by the burned fuel mass *mb*:

$$\frac{dm\_b}{dt} = \frac{m\_a - m\_b}{\tau} \tag{19}$$

where *me* is the entrained mass and *τ* is the time constant.

The entrained mass is calculated by:

$$\frac{dm\_{\text{ef}}}{dt} = \rho\_{u^{\prime}} \cdot A\_{\text{t}^{\prime}} \cdot (S\_T + S\_L) \tag{20}$$

where *ρu* is the unburned density, *Ae* is the surface area at flame front, *ST* is the turbulent flame speed and *SL* is the laminar flame speed.

The turbulent flame speed is calculated by:

$$S\_T = C\_{TFS} \cdot u' \cdot \left(1 - \frac{1}{1 + C\_{FKG} \cdot \left(\frac{R\_f}{L\_i}\right)^2}\right) \tag{21}$$

where *CTFS* is the turbulent flame speed multiplier, *u* is the turbulent intensity, *CFKG* is the flame kernel growth multiplier, *Rf* is the flame radius and *Li* is the integral length scale. 

The laminar flame speed is calculated by:

$$S\_L = \left(B\_{\rm un} + B\_{\theta} \cdot \left[\phi - \phi\_{\rm tr}\right]^2\right) \cdot \left(\frac{T\_u}{T\_{ref}}\right)^{a\_\ell} \cdot \left(\frac{p}{p\_{ref}}\right)^\beta \cdot f(Dilution) \tag{22}$$

where *Bm* is the maximum laminar speed, *Bφ* is the laminar speed roll-off value, *φ* is the equivalence ratio, *φm* is the equivalence ratio at maximum speed, *Tu* is the unburned gas temperature, *Tref* is defined as 298 K, *αe* is the temperature exponent, *p* is the pressure, *pref* is defined as 101,325 Pa and *β* is the pressure exponent.

The heat transfer is defined by the WoschniClassic model from GT-Suite. This model is used for engines without swirl flows, which correspond to the test bench engine. First, a convective heat transfer is calculated; then, it is calibrated to the measurement data by a multiplier to take the influences of conduction and radiation into account. The heat transfer coefficient is calculated by:

$$\alpha = \frac{K\_1 \cdot p^{0.8} \cdot w^{0.8}}{B^{0.2} \cdot T^{K\_2}} \tag{23}$$

where *K*1 and *K*2 are model constants, *p* is the cylinder pressure, *w* is the average cylinder gas velocity, *B* is the cylinder bore and *T* is the cylinder temperature. All multipliers are used to calibrate the model to the measurement data.

This heat transfer coefficient only takes the convection into account. To consider the influence of the radiation and conduction, an additional multiplier is used to calculate *α* for Equation (3).

The major difference of the model and the conventional engine models is the completely free definition of the piston motion. Specifically, the determination of the piston movement in the model is done via a table with free choice of the piston position as a function of time. In contrast to that, the piston motion normally is defined by the geometrical parameters of the crank train and the corresponding relation between crank angle and piston position. The thermodynamic model is illustrated in Figure 6. This model included the combustion and heat transfer process represented by the "cylinder" symbol and the mechanical components, which are represented by the "crank train" symbol. The boundary conditions of the incoming air and the outcoming exhaust gas are symbolized by the green symbols "inlet" and "outlet." The intake system consists of the inlet manifold, the inlet pipe, the throttle valve, the inlet channel, the injector and the inlet valve, as illustrated by the symbols on the left side of Figure 6. Thus, the engine works with an external mixture formation and a quantity control via the throttle valve. The symbols on the right side of Figure 6 represent the exhaust valve, the exhaust channel and the exhaust pipe.

**Figure 6.** Thermodynamic model of a free-piston engine.

To analyze the influence of different trajectories on the thermodynamic losses, a stationary operating point is chosen and validated with test bench data. In the highpressure area, which represents the combustion in this operation point, the measurement and simulation data fit quite well (see Figure 7).

**Figure 7.** Comparison of the measured and simulated cylinder pressure, the engine power and the volume change energy.

This point, which is shown in Table 3, is located at an engine speed of 764 min−<sup>1</sup> and an IMEP of 4.7 bar. The stroke is at an initial value of 0.03 m for the reverence trajectory, which leads to a compression ratio of 5.368. As fuel, the reference fuel indolene is used to describe automotive gasoline according to [36]. To model the combustion process, the already described predictive approach called "SI-Turb" is used, which describe the combustion by defining the turbulence intensity within the combustion chamber. The parameters, or rather the multipliers and their values which are used for this approach, are summarized in Table 3. The turbulent flame speed multiplier has influence on the turbulence level of the flame front during the combustion. The flame kernel growth multiplier describes how fast the flame kernel spreads during the combustion. With a higher multiplier of the Taylor length scale, the amount of time the air–fuel mixture is entrained into the flame front is decreased, which leads to short burning durations [37].


#### **5. Simulation Model of the Linear Permanent Magnet Synchronous Machine**

The mentioned properties of the electrical machine were implemented in a simulation model. The simulation model for this is shown schematically in Figure 8.

The acceleration force *FACC* is calculated with the acceleration *a* and the mass *m*. The gas force *FICE* is calculated from the product of cylinder pressure *p* and area *Acyl*. It is possible to determine the reluctance force *FRel* from the position *x* and the course from Figure 4a. The frictional force was neglected. With the help of these forces, the force of the electrical machine is calculated. The map from Figure 5 is then used to calculate the target current from the force *FEM* and the position *x*. The copper power loss *PCu* can be calculated from the current *iq* and ohmic resistance *R*. The mechanical power of the electrical machine *PM* can be determined from the force *FEM* and the speed *v*. The iron losses *PFe* can be calculated from the electrical frequency *fel* and the map from Figure 4. The sum of all powers results in the electrical power.

**Figure 8.** Simplified simulation model of the LMPSM.
