**7. Results**

This chapter shows the results of the efficiency loss analysis for the trajectory variation. The losses of the internal combustion engine and the electrical machine were shown together in a diagram in order to evaluate the influence of the trajectory. For the evaluation of the energy conversion from chemical to mechanical energy the courses for the piston motion, the corresponding pressure and temperature as well as a PV diagram of the compared trajectories will be shown and discussed. For the energy conversion from mechanical to electrical energy, the position and speed of the piston, the electrical machine force and the resulting power are shown and discussed.

The loss analysis for the reference is shown in Figure 14. The summation of all energy shares is the chemical energy from the fuel. This energy is divided into the losses for unburned fuel, the wall heat losses, the losses inside the exhaust gas, the friction losses, the iron losses and the copper losses. The last share is the electrical energy, which is issued by the free-piston engine.

**Figure 14.** Energy losses for trajectory REF.

Corresponding to the trajectory variation described in Section 6, the courses for the piston motion, the pressure and temperature are shown in Figure 15. The PV diagram for variant A1 and A2 in comparison to the reference is shown in Figure 16. The moved position in time of top dead center firing (TDCF) affects the pressure and temperature courses identically. The moved TDCF of variant A1 and A2 leads to earlier increasing pressure and temperature in comparison to the reference due to the fact that the combustion is initialized early. The decrease of the pressure and temperature is shifted in parallel. At the end of the expansion stroke all courses approach until the exhaust valve opens (EVO). After that, the courses are similar.

**Figure 15.** Comparison of trajectories REF, A1 and A2 related to the energy conversion of the internal combustion engine: illustration of the stroke, cylinder pressure and temperature depending on time.

**Figure 16.** Comparison of trajectories REF, A1 and A2 related to the energy conversion of the internal combustion engine: cylinder pressure depending on volume.

For the evaluation of the electrical energy conversion of the trajectory REF, A1, A2, the stroke, speed and force of the electrical machine are shown in Figure 17 and the volume change, copper, friction and electrical power in Figure 18.

**Figure 17.** Comparison of trajectories REF, A1 and A2 related to the energy conversion of the electrical machine: illustration of the stroke, velocity and electric machine force depending on time.

**Figure 18.** Comparison of trajectories REF, A1 and A2 related to the energy conversion of the electrical machine: volume change power, copper power losses, friction power losses and electric power depending on time.

The shift of the trajectory from REF to A2 is realized by further accelerating the drive train. The required trajectory can be implemented using the force *FEM* of the electrical machine. Due to the increased force *FEM*, the required current and, thus, the copper losses *PCu* increase. Since the speed is increased, the friction losses also increase. In addition to the volume change work, it can be seen that the electrical machine requires energy in the intake and exhaust cycle so that the four-cycle process can be implemented.

The longer expansion time with a higher temperature and similar pressure leads to the higher wall heat losses and reduced exhaust losses shown in Figure 19. This phenomenon can be seen when moving the TDCF to an earlier point in time. Due to the fact that the increase of the wall heat losses combined with the copper losses is higher than the decrease of the exhaust losses, both variants show a lower effective efficiency than the reference.

**Figure 19.** Energy losses for trajectory A1 and A2 with respect to REF.

The courses for the piston motion, the pressure and temperature are shown in Figure 20. The PV diagram for variant B1 and B2 in comparison to the reference is shown in Figure 21.

**Figure 20.** Comparison of trajectories REF, B1 and B2 related to the energy conversion of the internal combustion engine: illustration of the stroke, cylinder pressure and temperature depending on time.

**Figure 21.** Comparison of trajectories REF, B1 and B2 related to the energy conversion of the internal combustion engine: cylinder pressure depending on volume.

The variant B1 has an increased acceleration of the piston during the expansion close to the TDCF and a reduced acceleration near the BDC. Variant B2 is exactly flipped. This leads to a decrease of the peak pressure near TDCF for B1 and an increased peak pressure for B2. Near BDC, the opposite behavior can be observed. The maximum temperature is not influenced by the different variation of the piston acceleration. The decrease of the temperature from TCDF to BDC is slower for variant B1 and faster for variant B2 in comparison to the reference.

For the evaluation of the electrical energy conversion of the trajectory REF, B1, B2, the stroke, speed and force of the electrical machine are shown in Figure 22 and the volume change, copper, friction and electrical power are shown in Figure 23.

**Figure 22.** Comparison of trajectories REF, B1 and B2 related to the energy conversion of the electrical machine: illustration of the stroke, velocity and electric machine force depending on time.

**Figure 23.** Comparison of trajectories REF, B1 and B2 related to the energy conversion of the electrical machine: volume change power, copper power losses, friction power losses and electric power depending on time.

Due to the slow change in the piston stroke in the trajectory B2, the electric machine must compensate for the force of the internal combustion engine so that a slow movement is possible. In contrast, in the case of trajectory B1, the piston is accelerated almost freely. As a result, the copper losses are high in case B2 and low in case B1. The friction losses are also approximately the same because the speed curve of B1 and B2 can be mirrored. Because of the reduced copper losses in case B1, the electrical energy conversion is good.

The loss analysis for B1 and B2 with respect to the reference is shown in Figure 24. Due to the fact that the maximum pressure has a strong influence on the fluid properties, and therefore, on the HTC, the wall heat losses for B1 are lower and for B2 higher in comparison to the reference. For the exhaust losses, this behavior is the other way around. Due to the higher temperature course for B1, especially when the exhaust valve opens, the exhaust losses increase in comparison to the reference. For B2, the temperature is lower when the exhaust valve opens. This leads to lower exhaust losses.

Figure 25 shows the courses for the piston motion, the pressure and temperature. The PV diagram for variant C1 and C2 in comparison to the reference is shown in Figure 26. The extended maximum stroke leads to no significant variance in the pressure and temperature courses. However, based on the different acceleration of the piston from TDCF to BDC, because of the different maximum strokes, which has to be reached in the same time, the PV diagram shows an extension of the included area to the end of the expansion stroke as well as a small decrease of the peak pressure.

**Figure 24.** Energy losses for trajectory B1 and B2 with respect to REF.

**Figure 25.** Comparison of trajectories REF, C1 and C2 related to the energy conversion of the internal combustion engine: illustration of the stroke, cylinder pressure and temperature depending on time.

**Figure 26.** Comparison of trajectories REF, C1 and C2 related to the energy conversion of the internal combustion engine: cylinder pressure depending on volume.

For the evaluation of the electrical energy conversion of the trajectory REF, C1, C2, the stroke, speed and force of the electrical machine are shown in Figure 27 and the volume change, copper, friction and electrical power are shown in Figure 28.

**Figure 27.** Comparison of trajectories REF, C1 and C2 related to the energy conversion of the electrical machine: illustration of the stroke, velocity and electric machine force depending on time.

**Figure 28.** Comparison of trajectories REF, C1 and C2 related to the energy conversion of the electrical machine: volume change power, copper power losses, friction power losses and electric power depending on time.

Due to the stroke amplitude variation, the required speed increases in the expansion and exhaust cycle. The increase in speed leads to higher friction losses. However, the required acceleration in the expansion stroke is also increased. Since the combustion engine releases energy in this cycle, the electric machine does not have to brake the piston as hard. An increase in the stroke amplitude at this operating point has resulted in the copper losses being reduced. The iron losses have also increased, as the stroke variation leads to an increase in the electrical frequency.

The higher acceleration of the piston from TDCF to BDC leads to a small decrease of the peak pressure, which has a positive (reducing) influence on the wall heat losses. Against that, the higher maximum stroke leads to a higher surface area of the combustion chamber. This has a negative (increasing) influence on the wall heat losses. For the variant C1, the second phenomenon has a greater influence. This changes for the variant C2; thus, the first phenomenon has a greater influence. Therefore, the wall heat losses for C1 are higher and the wall heat losses for C2 are lower in comparison to the reference shown in Figure 29. The exhaust losses for C1 and C2 are lower than the exhaust losses from the reference. With higher maximum stroke, the benefit from the reduced exhaust losses decreases.

Figure 30 shows the courses for the piston motion, pressure and temperature. The PV diagram for variant D1 and D2 in comparison to the reference is shown in Figure 31. Because of the combination of moving the TDCF to an earlier point in time and the increased acceleration of the piston during the expansion near the TDCF, the courses of D1 and D2 are the superimposition of the variation of A and B1. This leads to an earlier increase of the pressure similar to the TDCF shift to an earlier time. Contemporaneous with the peak pressure, decreases similar to the increased piston acceleration in comparison to the reference. The temperature course is parallel shifted similar to the TDCF shift to earlier. However, the decrease of the temperature during the expansion is slower for the increased piston acceleration. This leads to higher temperatures when the exhaust valve opens.

**Figure 29.** Energy losses for trajectory C1 and C2 with respect to REF.

**Figure 30.** Comparison of trajectories REF, D1 and D2 related to the energy conversion of the internal combustion engine: illustration of the stroke, cylinder pressure and temperature depending on time.

**Figure 31.** Comparison of trajectories REF, D1 and D2 related to the energy conversion of the internal combustion engine: cylinder pressure depending on volume.

For the evaluation of the electrical energy conversion of the trajectory REF, D1, D2, the stroke, speed and force of the electrical machine are shown in Figure 32 and the volume change, copper, friction and electrical power are shown in Figure 33.

**Figure 32.** Comparison of trajectories REF, D1 and D2 related to the energy conversion of the electrical machine: illustration of the stroke, velocity and electric machine force depending on time.

**Figure 33.** Comparison of trajectories REF, D1 and D2 related to the energy conversion of the electrical machine: volume change power, copper power losses, friction power losses and electric power depending on time.

The combination of variants A and B1 shows that the electrical machine must provide an adapted force curve. In this case, the copper losses in D1 are smaller than in D2, but the friction losses have increased.

The associated loss analysis for D1 and D2 with respect to the reference is shown in Figure 34. Due to the higher temperature when the exhaust valve opens, D1 has higher exhaust losses than D2, which has higher exhaust losses than the reference. Because of the lower pressure, D1 and D2 have lower wall heat losses than the reference. D1 has the lowest peak pressure. Therefore, the decrease of the wall heat losses for D1 is higher than for D2.

**Figure 34.** Energy losses for trajectory D1 and D2 with respect to REF.

Figure 35 shows the electrical energy for all variants in comparison to the reference. The shifting of the TDCF to an earlier point in time shows a decrease of the effective efficiency. The lower exhaust losses are compensated by the increasing wall heat and copper losses. The increased piston acceleration shows the best results and the decreased acceleration the worst results. The decreased wall heat and copper losses are way higher than the increased exhaust losses for B1. This is the other way around for B2. The maximum piston stroke extension shows higher efficiency with increasing maximum piston stroke. This is based on the stronger decreasing wall heat losses in comparison to the increasing exhaust losses. The combination from A and B1 (D1 and D2) shows a decreasing efficiency with a higher shift of the TDCF identical to A. Due to the higher acceleration, the efficiency is on a higher level comparable to B1.

**Figure 35.** Comparison of the electrical energy conversion for the different trajectories.

Variant B1 shows the highest effective efficiency. The efficiency for the conversion from chemical energy within the fuel to electrical energy could be increased by 41%.
