Parametric Simulations on Leakage and Performance of a Miniature Free-Piston Generator (MFPG)

Featured Application

This work proposes a newly designed single-piston miniature free-piston generator, which features simple configuration and better sealing. This device has promising applications as the power source for small-scale electrical equipment and portable devices. This work carried out the preliminary parametric study of its performance and the findings provide a theoretical standpoint for the research on piston trajectory control.

Abstract

The miniaturization of electrical equipment and popularization of portable devices is an appealing motivation for the development of small-scale heat engines. However, the in-cylinder charge leaks severely as the engine dimension shrinks. The free-piston engine on a small scale provides better sealing than other miniature heat engines. Therefore, a miniature free-piston generator (MFPG) with a single-piston internal combustion engine (ICE) and a voice coil motor (VCM) was proposed in this work. A dynamic model with special attention on the heat transfer and leakage was established accordingly, upon which parametric studies of leakage and its effects on the performance of the MFPG system were performed. Four key parameters, including scavenging pressure, ignition position, combustion duration and piston mass, were considered in the model. The results showed that the mass leakage during the compression decreases with the rise of the motoring current. The indicated thermal efficiency can be improved by boosting scavenging pressure and increase motoring current. The critical ignition position is 2 mm before the top dead center. When ignition occurs later than that, the MFPG system is incapable of outputting power. The chemical to electric energy conversion efficiency is about 5.13%, with an output power of 10~13 W and power density around 4.7~5.7 W/cc.

1. Introduction

The design and manufacture of micro/small scale heat engines have drawn wide attention in all the works due to the relatively low energy density of chemical batteries [1,2,3]. The energy density of hydrocarbon fuels is 75 times higher than that of an ordinary lithium battery [4,5]. A heat engine with an energy conversion efficiency of no less than 2.5% can achieve a higher energy density than all the existing chemical batteries [6]. The experimental investigation of a 20 mm ultra-micro scale gas turbine obtained 15 W at 60,000 rpm. Thermal cyclic efficiency of 18.6% was attained [7]. A micro linear power generation system was proven to reach an energy density of 180 Whr/kg [8], which is on the upper limit of rechargeable lithium batteries [9]. A variety of small-scale heat engine prototypes have been developed and investigated in the form of micro gas turbines [7,10,11,12,13], micro rotor engines [14,15,16], micro internal combustion swing engines [17,18,19], micro free-piston generators [20,21,22,23], micro steam turbine [24,25,26] and micro P³ (Personal Power Pack) external combustion engines [27,28].

Free-piston heat engines do not involve the conventional rod–crankshaft mechanism, so its piston motion is determined by the resultant force of external forces acting on the piston. Compared to other micro heat engines, the free-piston configuration has advantages of a simple structure, better sealing and controllable compression ratio [29,30]. Although the operation frequency is relatively low, it also degrades the requirements on materials and bearings. In addition, the wider fuel adaptability and flexible piston trajectory provide a new way to improve the thermal efficiency of the system [31,32]. Aichlmayr et al. [33,34] carried out a fundamental study of the combustion process of a free-piston combustor as small as 3 mm in diameter and 0.3 mm in length. They stated that to generate 10–20 W power in 10³ mm³, the time scale of compression and expansion stroke must be much smaller than that of thermal diffusion, indicating a completely adiabatic combustion process. Therefore, a small-scale free-piston engine with ultra-high frequency could achieve higher theoretical thermal efficiency. However, with the reduction in the scale, the heat loss caused by the resulting enormous area-to-volume ratio and the leakage caused by the sealing problem between the piston and cylinder both increase severely, which limits the improvement of engine efficiency.

Lungu et al. [20] proposed a small-scale high-aspect-ratio free-piston engine. They applied two different sealing methods. Their experiment demonstrated that the better sealing resulted in a longer stroke, 6.35 mm, compared to a 3.175 mm stroke obtained by the worse sealing method. A significant improvement in power generation is obtained with the better sealing method. Suzuki et al. [35] manufactured a 0.0095 cc miniature free-piston generator on a silicon wafer. Single ignition combustion was successfully realized with an output power of 29.1 mW. However, the in-cylinder peak pressure was only 142.6 kPa, whereas the theoretical value was 349 kPa, indicating a serious leakage and heat loss. Lee et al. [36] also proposed an ultra-miniature free-piston generator using the sandwich structure to reduce heat loss. The single ignition stroke was 2.7 mm compared to a designed piston stroke of 8 mm, due to the serious leakage and heat loss. In addition, Tian et al. [37] contributed the low conversion efficiency of around 0.6% of their free-piston compressor to the combined effect of severe leakage, heat transfer and friction. Huang et al. [21] tested the performance of a reciprocating free-piston linear engine with a pair of 22.4 mm diameter combustion chambers. The engine output was 31.6 W power and it attained 42.8% thermal efficiency with an external resistance of 6.3 Ω. However, the researchers did not mention the sealing method in their paper.

Some theoretical analyses on the leakage and heat transfer of miniature heat engines have been carried out [38,39,40,41,42,43]. Sher et al. [38] pointed that the leakage gap is essentially the same regardless of the engine size. It is mainly determined by the fabrication tolerance, which is around 10~20 μm. The gas leakage per cycle is two orders of magnitude higher than that of conventional internal combustion engines (ICEs) because there are no piston rings between the piston and the cylinder wall in micro ICEs. Tian et al. [37] fitted Sher’s analytic simulation with a leakage discharge coefficient of 0.12, then argued that their work exaggerated the leakage by not considering the sealing effect of lubricant oil. Instead, they proposed a smaller value of 0.06 and reasonably reproduced their own experimental cylinder pressure. Annen et al. [40] stated that leakage loss becomes very important when the size of the micro-heat engine is smaller than 20 W. Various solutions are proposed for different prototypes, including the usage of sealing parts and lubricant [41], adopting more advanced bonding technology for the sandwich structure [36], applying liquid piston structure [42] and improving the fabrication technology to reduce the gap width [43].

Heat loss is another major factor leading to inefficiency or flame extinction of micro-engines [44,45,46]. As the engine dimension shrinks, the area-to-volume ratio increases, resulting in greater heat transfer loss per unit volume. Drost et al. [46] carried out many tests on the extinguishing distance of propane/oxygen mixture in order to determine the suitable combustion chamber size. Annen et al. [40] simulated the heat loss of a miniature reciprocal ICE, and determined the cylinder wall temperature required to limit the heat loss under 20% and 30% of the total heat released by fuel combustion at different scales.

Attracted by the advantages of the free-piston scheme, a single-piston miniature free-piston generator (MFPG) prototype, which directly converts hydrocarbon fuel into electricity, is proposed in this work. The prototype is constructed by combining a miniature internal combustion engine (ICE) with a compatible voice coil motor (VCM). To better understand the influence of leakage and heat transfer on the efficiency of this MFPG, a dynamic model, including heat transfer and leakage sub-models, was established and the parametric study of leakage and its effects on the performance of the MFPG were performed. The energy conversion in the MFPG prototype is also investigated by analyzing the transient power flow among each component force. This study provides a theoretical analysis for the subsequent system optimization.

2. MFPG Prototype

The schematic configuration and test bench of the proposed MFPG system are shown in Figure 1a,b.

As is seen from the picture, the proposed MFPG consists of a reciprocating internal combustion engine (ICE), a voice coil motor (VCM), a control unit and a super capacitor-battery energy storage device pack (SC-Batt ES). The conventional rod–crankshaft unit is diminished, and the ICE and the VCM are coupled directly through an integrated piston-valve rod. Thus, the piston-valve rod moves without the constraint of the crankshaft and its movement is governed by the external force acting on the rod. It has been proven that the quenching effect increases with the decrease in cylinder diameter [47] and the piston frequency soars with the reduction in piston mass, which is adverse to trajectory control [48]. Therefore, in order to obtain better combustion and easier trajectory control for future work, a 2.5 cc small scale ICE cylinder is selected. It is a smaller version of a conventional two-stroke ICE. Kim et al. [49] built a prototype on the same cylinder structure with a larger diameter and explained the structure in detail.

For a conventional engine, thanks to the rod–crankshaft unit and the flywheel, the energy and torque generated during the compression stage are stored and released in the other stages. Thus, the energy is transferred continuously and smoothly during a complete working cycle. In contrast, the energy flow in a free-piston engine is balanced during a single stroke, so continuous motion is a big challenge for free-piston structures. Usually, an extra piston rebounding unit is a necessary part of a free-piston heat engine. However, the proposed MFPG prototype is designed as simply as possible to accommodate size reduction and reduce the risk of failure [22]. The major simplifications include: (1) the common used spark plug is replaced by a platinum glow plug; (2) A miniature atomizer is used for the fuel atomization instead of the electronic fuel injection system; (3) A self-designed linear valve-port scavenging case without extra electrical and moving parts is applied; (4) The VCM serves as both the energy storage and piston rebounding component, compared to other configurations usually contain separate parts for the piston rebounding.

The MFPG is essentially an internal combustion engine and needs to follow the basic thermodynamic cycle to output work. Thus, a complete working cycle of the MFPG also consists of four main stages, which are scavenging, compression, expansion and exhaust. Figure 2 shows a complete working cycle of the MFPG system. The scavenging stroke starts when the scavenging port opens (SPO), the piston is driven by the in-cylinder pressure to the BDC and motored back by the VCM to the scavenging port closing (SPC). When the piston moves negatively to the position of exhaust port closing (EPC), the combustion chamber III is sealed and the system enters the compression stroke. When the piston stops at the top dead center, with the assistant of the glow plug, the combustible mixture ignites and the in-cylinder pressure rises rapidly to push the piston back. The MFPG enters the expansion stroke, in which the VCM is working on the generator mode. The exhaust stroke starts when the piston reaches EPO and ends after the piston stops at BDC. These entire four strokes constitute a complete working cycle, then the piston is ready to start the next cycle. The piston continuously moves back and forth to transform the fuel energy into electricity.

The MFPG specifications are listed in

Table 1. Specifications of the miniature free-piston generator (MFPG) prototype.

Parameters	Symbol	Value
Bore/mm	D	15.5
Top limit position/mm	xtlim	0.5
Bottom limit postion/mm	xblim	16
Ignition position/mm	xign	3
Exhaust port opening/mm	xEPO	10
Exhaust port height/mm	xEh	4
Scavenging port opening/mm	xSPO	11.5
Scavenging port height/mm	xSh	4
Scavenging temperature/K	Tsc	310
Scavenging pressure/bar	psc	1.15
Combustion duration/ms	tcd	3.5
lower heating value/J/mg	Hu	19.66
Combustion quality factor	m	1.5
Air–Fuel Ratio of methanol	AFmeth	6.45
Adiabatic exponent	γ	1.35
Piston mass/g	mp	156
Motor Constant/N/A	kmotor	7.24
Motoring Current/A	imotor	5.5
Motor Inductance/mH	Ls	2.38
Back EMF constant/V/m/s−1	kemf	7.24
Output current/A	iout	9
Motor resistance/Ω	Rs	5.26

and the configuration of the piston coordinates is shown in Figure 3. The MFPG has a shorter residence time at the TDC than conventional ICEs, so the fuel adopted in the MFPG prototype has to be faster in flame speed [50,51]. Methanol (CH₃OH) has a flame propagation speed of 0.523 m/s, which is much higher than 0.337 m/s and 0.39 m/s of gasoline and diesel [52], respectively. As an oxygenated fuel, its ignition limit is 6.7~36%, which is wider than that of traditional fuels, in addition to the lower minimum ignition energy, which is conducive to the rapid formation of a flame core. Moreover, methanol can burn stably when the exhaust gas recirculation (EGR) is more than 23% [53]. This feature can tolerate the low scavenging efficiency of a two-stroke engine and is therefore beneficial in complete combustion and reducing cycle variation. Methanol is then selected as the fuel of this MFPG prototype.

3. Modelling of the MFPG System

The modeling of the MFPG system is a multi-physics problem since it couples a linear motor with an internal combustion engine [54]. Therefore, a dynamic model of the MFPG prototype is proposed here, consisting of a heat release model, gas exchange model and electromagnetic force model, etc. Special attention is drawn to the modeling of the mass leakage through the cylinder–piston gap and the heat transfer through the cylinder wall.

3.1. Dynamic Model

The piston assembly of the MFPG prototype is governed by the resultant of all the external forces [55], thus, the dynamic model of the piston assembly is described as:

$$m_p\ddot{x}=F_{gas}-F_{ele}-F_f$$

(1)

where, m_p is the piston mass, kg; $\ddot{x}$ is the piston acceleration, m·s⁻²; F_f is the frictional force, N; F_ele is the electromagnetic force of the voice coil motor, N; F_gas is the gas force inside the combustion chamber, N.

3.2. Heat Transfer Model

The heat convection across the cylinder wall is the dominant mode of heat dissipation in the MFPG system [17,44,45]. Therefore, in order to simplify the model and accelerate the calculation, the composition inhomogeneity and temperature gradient inside the combustion cylinder can be reasonably neglected [33,34]. Wu et al. [55] and Li et al. [56] adopted these assumptions of uniform charge in modelling the heat transfer. Their simulation showed good agreement against test results. Thus, the heat loss of the MFPG system is simply approximated by Newton’s law of cooling [57]:

$$\frac{d{Q}_w}{dt}={\alpha }_w\cdot A_w\cdot (T-T_w)$$

(2)

where Q_w is the heat loss across the cylinder wall, J; A_w is the effective area of heat transfer, m²; T is the in-cylinder gas temperature, K; T_w is the wall temperature, K; a_w is the convective heat transfer coefficient, W/(m²·K), which is usually estimated by experiments or simulations.

Two commonly used estimation formulas of a_w are the Woschni model [58] and the Hohenberg model [59], as shown in Equations (3) and (4).

$${\alpha }_w=820p^{0.8}{T}^{-0.53}{D}^{-0.2}{\left[C_1{v}_m+C_2\frac{T_{EPC}\cdot V_h}{p_{EPC}\cdot V_{EPC}}(p-p_0)\right]}^{0.8}$$

(3)

$${\alpha }_w=130V_{}^{-0.06}{\left(\frac{P_{EPC}}{{10}^5}\right)}^{0.8}{T}_{EPC}{}^{-0.4}{\left[v_m+1.4\right]}^{0.8}$$

(4)

where p, T, V are the pressure, temperature and volume of the in-cylinder gas, with the unit of Pa, K, m³, respectively; p_EPC, T_EPC and V_EPC are the pressure, temperature and volume of the in-cylinder gas at the exhaust port closing, with the unit of Pa, K and m³, respectively; V_h is the swept volume, m³; v_max is the max piston velocity, m/s.

Figure 4a compares the effects of the two heat transfer models on the in-cylinder pressure. Compared to the Woschni model, the Hohenberg model predicts higher in-cylinder gas pressure and shows better agreement against the test data. Since the real in-cylinder gas temperature of the MFPG is unavailable due to the technical obstacles in the field of internal combustion design [60,61], Figure 4b only displays the simulated in-cylinder gas temperatures of the two models, in conjunction with the adiabatic condition. It shows that the Hohenberg model also predicts higher in-cylinder gas temperature than that of the Woschni model. This explains the cause of the higher in-cylinder gas pressure calculated by the Hohenberg model. The better agreement for the Hohenberg model on the in-cylinder pressure indicates that this model is more suitable for the MFPG prototype studied in this paper.

3.3. Mass Leakage Model

The leakage process is modeled by a simplified scenario, in which the combustion chamber is regarded as a control volume and the mass flow rate of the in-cylinder gas through the cylinder–piston gap is considered as an orifice flow [62]:

$$\frac{d{m}_{leak}}{dt}={\mu }_{cdl}{A}_{leak}{\Psi }_{lsub,lsup}\sqrt{p_{\mathrm{I}}{\rho }_{\mathrm{I}}}$$

(5)

$${\Psi }_{lsub,lsup}=\{\begin{array}{ll}\sqrt{\frac{2\gamma }{\gamma -1}\left[{\left(\frac{p_{\mathrm{II}}}{p_{\mathrm{I}}}\right)}^{\frac{2}{\gamma }}-{\left(\frac{p_{\mathrm{II}}}{p_{\mathrm{I}}}\right)}^{\frac{\gamma +1}{\gamma }}\right]}& \frac{p_{\mathrm{II}}}{p_{\mathrm{I}}}>{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}\\ {\left(\frac{2}{\gamma +1}\right)}^{\frac{1}{\gamma -1}}\sqrt{\frac{2\gamma }{\gamma +1}}& \frac{p_{\mathrm{II}}}{p_{\mathrm{I}}}\le {\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}\end{array}$$

(6)

where m_leak is mass of the leaked gas, kg; μ_cdl is discharge coefficient of the leakage passage; A_leak is leakage area of the piston–cylinder unit, m²; Ψ_lsub, Ψ_lsup are subsonic and supersonic flow functions of the in-cylinder gas leakage; p_I, and p_II are upstream and downstream pressures of the throttling orifice, Pa; ρ_I and ρ_II are upstream and downstream densities of the throttling orifice, kg/m³; γ is the adiabatic exponent.

The simulation of the leakage characteristics of the MFPG prototype is shown in Figure 5 along with that of two other micro heat engines, micro conventional crank shaft engines and micro rotary engines. The three types of micro engines are of the same displacement of 2.5 cc. The leakage characteristics of the micro conventional crank shaft engines and micro rotary engines are collected from the literature [38,39], while that of the MFPG is calculated by the proposed model. The gap widths are set to be 20, 15 and 10 μm, respectively. The discharge coefficient μ_cdl is assumed to be 0.06 to account for the sealing effect of the lubricant oil in the piston–cylinder gap, as suggested by Tian et al. [37].

The revolution of the other two micro engines is around 6000~14,000 rpm, while the equivalent engine revolution of the MFPG prototype varies from 600 to 2000 rpm. In comparison with these two micro engines, the lower operation frequency of the MFPG would increase the mass leakage to some extent. However, the much lower compression ratio, compared with the micro conventional crank shaft engines, would help reduce the mass leak of the MFPG significantly. Thus, it is reasonable for the calculated leakage of the MFPG, with the suggested μ_cdl 0.06, to be smaller than that of the micro conventional crank shaft engines. Likewise, with the same displacement and with an identical compression ratio, micro rotary engines leak much more severely than the reciprocating engines [39]. Thus, with the suggested μ_cdl 0.06, the calculated leakage of the MFPG is smaller than that of the micro rotary engines, which also makes sense. Therefore, the proposed leakage model along with the selected leakage discharge coefficient μ_cdl is considered to be a reasonable estimation of mass leakage characteristics of the proposed MFPG prototype.

3.4. Model Validation

The proposed heat transfer model and the leakage model of in-cylinder gas were validated against the experiment data and literature data, as shown in Figure 4 and Figure 5, respectively. The proposed simulation model was validated against the preliminary test under three different scavenging pressures p_sc, as is illustrated in Figure 6. It is seen that the pressure prediction is generally lower than the tested data. As the p_sc increases, the piston top dead center (TDC) shifts farther from the cylinder head and the in-cylinder pressure trace becomes sharper indicating a higher degree of isochoric combustion. The predicted in-cylinder pressure seems not able to capture these variations of TDC and combustion process. It is because these parameters are set to be constant in the model, while these variables are strongly coupled with the piston dynamics, which hardly excludes their effects to conduct the test of a single parameter. However, the simulation well predicts the pressure increase trend and agrees with the pressure trace reasonably. The maximum relative errors were 2.9%, 5.6% and 10.5%, respectively, at p_sc = 1.05, 1.10 and 1.15 bar. The results confirmed that the proposed model is able to reasonably captures the characteristics of the MFPG system, especially if it takes more parameters into consideration.

4. Parametric Study on Leakage and Performance

In this section, the effects of scavenging pressure p_sc, ignition position x_ign, combustion duration t_cd and piston mass m_p are investigated based on the established dynamic model. In addition, the indicated thermal efficiency η_i and effective thermal efficiency η_e are calculated as well to evaluate the performance of the MFPG system.

The mass of the in-cylinder gas at exhaust port closing is denoted as m_gepc, which implied the mass sealed in cylinder. The gas mass at combustion start, denoted as m_gsoc, is the mass of charge that actually participated in combustion. Thus, the difference between these two is the mass leaked during compression, denoted as:

m_lcp = m_gepc − m_gsoc

(7)

where, m_lcp is the mass leakage of the in-cylinder gas during compression, kg; m_gepc is the mass of the in-cylinder gas at EPC, kg; m_gsoc is the mass of the in-cylinder gas at SOC, kg.

Likewise, the mass leaked during combustion and expansion, which affects the work performed by the in-cylinder gas, is denoted as m_lexp. It can be calculated by deducting the gas mass at the exhaust port opening (EPO) from that of the start of combustion (SOC).

m_lexp = m_gsoc − m_gepo

(8)

where, m_lexp the mass leakage of the in-cylinder gas during combustion and expansion, kg; m_gepo is the mass of the in-cylinder gas at EPO, kg; m_gsoc is the gas mass at SOC, kg.

To investigate the effects of leakage loss and heat loss on system performance, the efficiencies are calculated under three hypotheses:

Hypothesis 1 (H1).

The cylinder was completely sealed and combustion was adiabatic, which is the ideal case and baseline, denoted as η_i,a and η_e,a.

Hypothesis 2 (H2).

Combustion is still adiabatic, but the gap between the piston and the cylinder is considered, denoted η_i,b and η_e,b. The indicated thermal efficiency difference between Hypothesis 1 and Hypothesis 2, Δη_i,a-b = η_i,a − η_i,b characterizes the effect of leakage loss, Q_lcp.

Hypothesis 3 (H3).

Account for both the leakage loss and heat loss, with the denotation of η_i,c and η_e,c. Similarly, the indicated thermal efficiency difference between Hypothesis 2 and Hypothesis 3, Δη_i,b-c = η_i,b − η_i,c characterizes the effect of heat loss, Q_w.

In the three hypotheses, Hypothesis 1 is the baseline hypothesis. It does not have energy loss, which is an ideal circumstance. By taking the two energy loss into consideration one by one, the two thermal efficiencies in Hypotheses 2 and 3 decrease successively. Therefore, the efficiency drops of the two adjacent hypotheses (Δη_i,a-b and Δη_i,b-c) are regarded as the indicators of leakage loss, Q_lcp, and heat loss, Q_w, respectively.

4.1. Effect of the Scavenging Pressure

The scavenging pressure p_sc influences the scavenging efficiency φ_s and the mass of gas sealed in the cylinder at the exhaust port closing, thus affecting the mass of fuel supplied per cycle m_fuel. Figure 7 shows the simulation results as p_sc increases from 1.05 to 1.20 bar. It is seen from Figure 7a that the mass of the charge decreases nonlinearly as the piston moves towards the cylinder head in all the cases. For instance, when the scavenging pressure is 1.2 bar, the gas mass at the scavenging port closing (SPC) is around 3 mg. As the piston moves to the exhaust port closing (EPC), it decreases to 2.73 mg and further to 2.08 mg at the start of combustion (SOC), respectively. As suggested in Equation (7), the difference between 2.73 mg at EPC and 2.08 mg at SOC is the mass leakage during the compression stroke, m_lcp. It is about 0.65 mg.

After the onset of the combustion, the piston moves backward from SOC to the exhaust port opening position (EPO), due to the high-pressure produce by the burning fuel. The in-cylinder gas still leaks nonlinearly during this period and the mass leakage is the m_lexp. It can be calculated according to Equation (8). The value of mass leakages during the compression and expansion strokes is displayed in Figure 7b. It shows that the m_lexp is greater than the m_lcp, indicating a more serious mass leakage at the expansion stroke. This is because that the in-cylinder pressure rises rapidly due to the fuel combustion, as inferred in in Figure 7c, compared to the almost unchanged pressure on the other side of the piston. Therefore, the pressure difference between the two sides of the piston rod increases enormously, which deteriorates the mass leakage as can be inferred from Equation (7). Thus, the mass leaks asymmetrically with respect to the SOC.

The boosting of p_sc increases the mass flow at both scavenging and exhaust port simultaneously, but the increment decreases gradually. This observation means that there is a limit to the promoting effect of the scavenging pressure on the mass trapped in the cylinder, which is referred as a “saturated trend” here. The in-cylinder gas mass at SOC m_gsoc shows a similar saturation trend, which explains the phenomenon of increment saturation of the in-cylinder peak pressure p_max shown in Figure 7c.

In terms of the mass leakage during the compression m_lcp, as described in Equation (7), is the mass difference between the EPC and SOC. It increases from 0.61 mg to 0.65 mg as p_sc increases from 1.05 to 1.20 bar, as shown in Figure 7b. The reason is that the time consumed during the compression increases, which prolongs the time of leakage, and the elevated in-cylinder gas pressure, which increases the pressure difference between the two piston ends. However, the mass leak ratio m_lcp/m_gepc decreases, indicating a better sealing performance under higher the scavenging pressure condition.

As to the top dead centre (TDC), it increases slightly with the increase in p_sc, as shown in Figure 7d. This results in a gentle increase in the effective compression ratio ε_ec, and therefore, a relatively higher combustion pressure to be compensated by the piston movement. Thus, the piston stroke increases and the bottom dead centre (BDC) expands. The geometric compression ratio ε_c increases accordingly. Since the enhanced in-cylinder pressure imposes more hindrance effect on the piston movement, the piston velocity v decreases more rapidly in the later stage of the compression stroke with the increase in scavenging pressure p_sc, as inferred from the shorter time elapsed from the maximum compression speed to zero velocity. The peak velocity v_max also rises with the increase in p_sc for the same reason. In addition, the piston frequency f_p decreases with the increase in p_sc, which is a combined result of the enhanced in-cylinder pressure and prolonged piston stroke, which exert opposite effects on system stiffness.

As indicated from Figure 7e, the indicated thermal efficiency under Hypothesis 1 decreases slightly as p_sc increases, due to the reduced in-cylinder trapping efficiency φ_trap. Under Hypothesis 2, since the reduced gas leakage ratio offsets the effect of φ_trap decreasing, the corresponding η_i,_b increases slightly. Therefore, the effect of leakage loss on the indicated thermal efficiency characterized Δη_i,a-b = η_i,a − η_i,b decreases, shown in Figure 7f. Hypothesis 3 takes heat loss into consideration. The heat transfer across the cylinder wall is mainly determined by the in-cylinder gas temperature T and heat transfer time. Since the heat transfer mainly occurs in the expansion stroke, the expansion duration t_exp is thus considered as the heat transfer time. The gas temperature enhances with the increase in p_sc, while t_exp shrinks. Thus, in general, the effect of heat loss, characterized as Δη_i,b-c = η_i,b − η_i,c increases at low p_sc and changes little at high p_sc.

The effective thermal efficiency is mainly affected by the indicated work W_i and three consumption works: the motoring work W_consm, the pumping work W_pump and the frictional work W_f. With the boosting of p_sc, W_pump increases. Meanwhile, the extension of the piston stroke and the acceleration of the piston velocity lead to the rise of the W_consm and the W_f, respectively. As a result, η_e,a decreases monotonously. With the contribution of positive increment of W_i in Hyp. B, η_e,b increases at first and starts to decay, while the increment of energy consumption is too large to offset, leading to a local extremum η_e,b at 1.15 bar. Since the variation of heat loss is slight, the effective thermal efficiency under Hyp. C η_e,c retains the same trend with η_e,b.

4.2. Effect of the Start of Combustion

In reality, the ignition time is controlled by adjusting the temperature of the glow plug and the SOC is characterized by the ignition position x_ign. In the simulation, the value of x_ign is varied from 1~4 mm, while all the other parameters remain unchanged. The increase in x_ign indicates the advancing of SOC and the decrease in x_ign means the delay of SOC.

According to Figure 8a, when the x_ign varies, the in-cylinder gas mass at EPC m_gepc shows little difference, while the in-cylinder gas mass at SOC m_gsoc varies largely. When the combustion takes place between 2~4 mm, m_gsoc increases with the advance of SOC, whereas m_gsoc decreases with the advancing of SOC after the piston is compressed less than 2 mm. The reason is that the piston frequency f_p increases sharply with the delay of SOC, reduces the time of leakage and offsets the effect of the increase in pressure difference at this range. The corresponding leakage characteristics are presented in Figure 8b. In most cases, the mass leaked during the expansion stroke is more than that during the compression stroke. As seen from the mass leakage ratio curve, the leakage peaks at x_ign = 2 mm indicate that the worst sealing performance occurs there. Better sealing could be achieved either by advancing ignition time or postponing the ignition until the piston passes 2 mm.

The delay of SOC is realized by controlling the combustion that takes place closer to the cylinder head, so the effective compression ratio ε_ec monotonically increases with the delay of SOC. The in-cylinder peak pressure p_max, on the other hand, varies more complexly, as shown in Figure 8c. When x_ign is in the range of 3~4 mm, p_max increases with the delay of SOC, as being straightforwardly influenced by the increased ε_ec. When x_ign decreases to the range of 2~3 mm, the mass leakage exacerbates and overtakes the promoting effect of ε_ec increase. Thus, the p_max decreases with a delay of SOC in this range. As x_ign further decreases to the range of 1~2 mm, p_max increases with the delay of SOC, again due to the combined effects of the increased ε_ec and the aforementioned slight increase in m_gsoc in this range.

As shown in Figure 8d, the more advanced the SOC, the farther the piston trajectory form the cylinder head. This means that the TDC and BDC deviate from the cylinder head further at the same time with the advancing of SOC. However, compared to the given range of SOC, the position difference of the BDC is little. Therefore, the geometric compression ratio ε_c decreases with the advancing of SOC. The variation of piston stroke S and peak piston velocity v_max show the same trend as p_max. Since the effect of p_max enhancement on system dynamic stiffness is partly counteracted by the effect of piston stroke S elongation, the piston frequency f_p decreases with the advance of SOC, in general.

Figure 8e displays the predicted indicated and effective thermal efficiencies of the MFPG prototype. The η_i,a decreases with the advancing of SOC because of the relatively lower ε_ec. The η_i,b obtains a local extremum at x_ign = 3 mm due to the opposite effect of the decrease in mass leak and reduction of ε_ec. The η_i,c keeps the same trend with η_i,b and the heat loss Δη_i,b-c also has a local extremum at x_ign = 3 mm as shown in Figure 8f. In addition, as the SOC timing advances, the in-cylinder gas force decreases exponentially. The consumption work of piston motoring drops accordingly. Thus, the effective thermal efficiencies, under the three hypotheses η_e,a, η_e,b, η_e,c, all increase with the advancing of the SOC. As indicated by the negative value of η_e,c the ignition position is less than 2 mm, the system is incapable of outputting energy at this range.

4.3. Effect of the Combustion Duration

Combustion duration t_cd is an important parameter reflecting the combustion process. The t_cd is varied from 2.5~4 ms in the simulation while all the other parameters are kept the same. As implied by Figure 9a,b, the curve of the in-cylinder gas during compression overlaps each other and the mass leak m_lcp is nearly flat, meaning that the variation of t_cd has no significant effect on the fuel loss. On the other hand, the mass of in-cylinder gas at exhaust port opening m_gepo decreases with the extending of t_cd, which means the mass leakage during expansion m_lexp increases. Although the prolongation of t_cd lowers the p_max, as shown in Figure 9c, which helps to reduce the m_lexp, the increased expansion time t_exp is mainly responsible for the increase in m_lexp.

With a shorter t_cd, a system releases the heat in a shorter time, which is closer to the ideal Otto cycle. The p_max is therefore enhanced, as presented in Figure 9c. The TDC increases, as shown in Figure 9d, as a result of elevated in-cylinder gas pressure and the effective compression ratio ε_ec hence decreases slightly. The BDC also increases due to the enhanced p_max. By taking the difference between the positions of TDC and BDC, the piston stroke S shows an increasing trend and so does the compression ratio ε_c. The increased piston peak velocity v_max and piston frequency f_p are also the results of enhanced p_max.

Figure 9e shows that the indicated thermal efficiencies, under the three hypotheses, all increase monotonously with the reducing of combustion duration t_cd, mainly due to the aforementioned higher degree of isochoric combustion. There are few effects of t_cd on the three major consumption work, W_consm, W_pump, and W_f, so the effective thermal efficiencies η_e,a, η_e,b and η_e,c show nearly the same trends with their corresponding indicated thermal efficiencies. The Δη_i,a-b, which reflects the effect of leakage loss, decreases insignificantly with respect to the increase in t_cd as implied by Figure 9f. The result is consistent with the aforementioned conclusion that t_cd has no significant effect on m_gsoc. The variation of t_cd brings in two effects on heat loss. As t_cd increases, the maximum combustion temperature T_max decreases, which is beneficial for heat loss reduction. On the other hand, time consumed by expansion t_exp increases, which extends the heat transfer time and intensifies the heat loss. The Δη_i,b-c, which reflects the effect of the heat loss, decreases slightly with respect to the prolongation of t_cd, indicating that the influence of T on the heat loss is dominant in these cases.

4.4. Effect of the Piston Mass

Piston mass m_p is another important factor affecting the natural frequency of the piston f_p. The piston mass of the physical MFPG prototype is 156 g, so the adjusting range of m_p is set to be 100~184 g with the other parameters unchanged. According to Figure 10a, the mass of the in-cylinder gas at EPC m_gepc is almost the same in the four cases and so is the gas mass at SOC m_gsoc, which means that the variation of m_p has no significant effect on m_lcp as directly presented in Figure 10b. This is mainly attributed to the insignificant differences of the in-cylinder gas pressure during combustion, as shown in Figure 10c, and nearly the same compression time t_cp as implied by Figure 10d. During the expansion, the peak velocity v_max of the heavier piston is obviously lower than that of the light ones, which results in the extended time of expansion t_exp and leads to the enlarged mass leakage m_lexp in this range.

Figure 10c indicates that m_p has little effect on the in-cylinder gas pressure during the compression. However, p_max increases when a heavier piston is applied, which is mainly attributed to the increased effective compression ratio ε_ec, since the mass of the in-cylinder gas at SOC m_gsoc is almost the same and hardly affects the p_max. The heavier the piston, the closer the TDC with respect to the cylinder head, so the ε_ec increases accordingly and promotes the p_max. In addition, the inertia effect becomes more pronounced with the increase in m_p, the piston, therefore, responds slowly when expansion. Thus, the degree of isovolumetric combustion is hence increased and further enhances the p_max. It is notable that the heavier the piston, the faster the in-cylinder gas pressure decreases in the expansion stroke, which is due to the more severe leakage of m_lexp with increase in the piston mass.

As the piston mass increases, the piston stops farther with a larger value of BDC, as is indicated in Figure 10d. This is because the heavier piston leads to higher p_max, which produces more work and needs a longer distance to offset. For the heavier pistons, the longer BDC implies increased acceleration length in the next cycle and therefore the higher speed at SOC. The piston kinetic energy is, therefore, greater and the piston is closer to the cylinder head, resulting in a decrease in TDC. Hence, the piston stroke S and compression ratio ε_c both increase accordingly. In contrast, when the pistons expand back towards the BDC, all the pistons start at almost the same position, so the advantage of acceleration distance for the heavier pistons disappears and results in lower v_max. It is known that the natural frequency is inversely proportional to the square root of the mass, thus the piston frequency f_p decreases as the piston mass increases.

The effective compression ratio ε_ec and the degree of isochoric combustion both increase with an increase in the piston mass, so the η_i,a increases, as presented in Figure 10e. The η_i,b keeps the same trend with η_i,a because the fuel leakage indicator m_lcp is little affected by the variation of m_p. The change of Δη_i,a-b, shown in Figure 10f, is therefore unobvious. The increase Δη_i,b-c indicates an aggravated heat loss due to the prolonged expansion stroke and enhances combustion temperature for heavier pistons. In terms of effective thermal efficiency, the elongated piston stroke S for the heavier piston consumes more electric energy W_consm during the compression. The increment of W_i is not enough to compensate the W_consm, so the net output power W_net reduces and leads to a decrease in η_e for all the cases. The influence of leakage and heat transfer on η_e is about the same as that on η_i, so the three effective thermal efficiencies η_e,a, η_e,b and η_e,c all decrease monotonously as the m_p increases.

4.5. Parameter Optimization

The piston dynamics of the MFPG system are more complexed than that of traditional micro ICEs and different combinations of system parameters define different performances of an MFPG system.

Table 2. Influences of parameters on the MFPG performance.

Parameters	Symbol	${\mathit{p}}_{\mathit{s}\mathit{c}}$1	${\mathit{x}}_{\mathit{i}\mathit{g}\mathit{n}}$1	${\mathit{t}}_{\mathit{c}\mathit{d}}$1	${\mathit{m}}_{\mathit{p}}$1
Top dead center/m	TDC	↑	↑	↓	↓
Bottom dead center/m	BDC	↑	↑	↓	↑
Piston stroke/m	S	↑	↓↑↓	↓	↑
Compression ratio	εc	↑	↓	↓	↑
Effective compression ratio	εec	↓	↓	↑	↑
Max piston velocity/m/s	vmax	↑	↓↑↓	↓	↓
Piston frequency/Hz	fp	↓	↓	↓	↓
Mass of in cylinder gas at SOC/kg	mgsoc	↑	↓↑	→	→
Peak pressure of in-cylinder gas/Pa	pmax	↑	↓↑↓	↓	↑
Leakage loss through piston-cylinder gap/J	Qlcp	↓	↓	↑	↑
Heat Loss across cylinder wall/J	Qw	↑↓	↑↓	↓	↑
Indicated thermal efficiency 2	ηi,c	↑	↓↑↓	↓	↓
Effective thermal efficiency 2	ηe,c	↑↓	↑	↓	↓
Optimum value		1.15	3	3.5	156

¹ The parameters in the table are set to increase, the effects in the table are listed accordingly. ² Only the result under Hypothesis 3 is considered in this table, which accounts for both the leakage loss and heat loss.

summarizes the influences of the controllable external parameters on the performance of the MFPG prototype.

Based on that, a set of optimal parameter combinations was preliminarily determined, which is listed in Table 2, as well. The 1.15 bar of scavenging pressure and 3 mm of SOC are selected because they correspond to the local extremum of the thermal efficiencies. The combustion duration is controlled by chemical reaction kinetics and is hard to obtain. When 3.5 ms is chosen, the simulation best approximates the experimental results. A lightweight piston is more favorable in improving thermal efficiency. Its high operation frequency, however, poses more challenges to the design of the control system. For this reason, the first generation of MFPG prototype chooses a relatively larger piston mass, which is about 156 g, to compromise the control system.

5. Power Flow and Energy Distribution

In order to predict the optimal energy conversion efficiency of the proposed MFPG, the optimum values listed in Table 2 are applied in the power and energy calculations of the MFPG. Figure 11 shows the transient powers of the major forces within an entire cycle. The positive power represents that the piston absorbs power from the external force, whereas the negative power indicates that the piston outputs power to the external force. The shadowed areas are the works performed by the corresponding forces.

The simulation starts from the BDC. In the second half of the scavenging stroke, the voice coil motor drives the piston from BDC to exhaust port closing (EPC). The electromagnetic force transfers its power into the piston, so the power of electromagnetic force P_ele is positive during this period. At the same time, the in-cylinder gas power P_p and frictional power P_f are relatively minor due to the lower scavenging pressure and piston velocity. Thus, only a small portion of P_ele is used to balance the P_p and P_f and the major portion of the P_ele is converted to the kinetic energy of the piston, which is the inertial power of piston P_a.

In the compression stroke, from the exhaust port closing to top dead center, the VCM continues to accelerate the piston and outputs positive work. The positive P_ele is divided into three parts to compensate the piston inertia power P_a, to overcome the friction power P_f and to compress the in-cylinder gas P_p. In the later stage of the compression stroke, the in-cylinder gas pressure increases exponentially, so P_ele is not enough to maintain the acceleration of the piston. Thus, the piston decelerates and the inertia power P_a, stored as kinetic energy of the piston, is released. After all of the P_a is converted into the compression power of the in-cylinder gas P_p, the piston stops. After the onset of combustion, the gas pressure in the cylinder increases sharply and leads to a small spike between SOC and TDC on the P_a and P_p curves. Furthermore, the inertia power P_a balances within the range from BDC to TDC. That means that the areas on the two sides of the abscissa before TDC are equal. This indicates that P_a is only an intermediate state, used for the temporary storage of excessive power of P_ele, while the ultimate states of the power conversion are between P_ele and P_p.

In the expansion stroke, the in-cylinder gas starts to work on the piston and pushes the piston towards the BDC, so the value of P_p is positive. After overcoming the frictional power P_f and the electromagnetic power P_ele, the residual power is stored in the piston in the form of P_a. As the piston moves further, the in-cylinder pressure drops and the gas power P_p becomes insufficient to generate more P_ele. Then, the piston slows down to release the stored P_a power and becomes another source of P_ele. Thus, the value of P_a is negative. When the piston moves from EPO back to BDC, the P_p is further reduced. The P_a is responsible for overcoming the frictional power P_f and supplying the electric power generation at the same time. The power flow of the entire working cycle of the MFPG system indicates that the energy conversion of the system is a complex dynamic balance of different powers. The two ultimate power states in the process are P_ele and P_p. When they are not balanced, their residual power is stored temporarily in the form of P_a.

It is assumed that the major energy loss of the MFPG system includes the short-cut loss Q_scut caused by scavenging, heat transfer loss Q_w across the cylinder wall, fuel leakage loss Q_lcp through the piston–cylinder gap, the friction loss W_f between the moving/stationary units and mechanical loss W_pump for boosting the scavenging pressure. Based on this assumption, the energy distribution per cycle for this example is predicted and listed in

Table 3. Energy flow distribution per cycle in the MFPG prototype.

Parameters	Symbol	Value (J)	Ratio (%)	Descriptions
Heat input	Qfuel	11.38	100	Heat input through scavenging port at the very beginning
Short cut loss	Qscut	4.017	35.30	Heat lost by the fuel flowing away from the exhaust port
Leakage loss	Qlcp	1.955	17.18	Heat lost by the fuel leaked through the piston–cylinder gap
Heat transfer loss	Qw	1.858	16.33	Heat dissipated across the cylinder wall
Indicated work	Wi	1.737	15.26	Work performed by the in-cylinder gas
Friction loss	Wf	0.311	2.73	Energy lost by the frictional force
Gross electric power	Wgen	1.426	12.32	Electric energy generated by the linear motor
Scavenging loss	Wpump	0.282	2.49	Energy loss for boosting the scavenging pressure
Motoring energy	Wconsm	0.495	4.35	Electric energy consumed for motoring the piston
Net electric power	Wnet	0.649	5.70	Net electric energy produced in a single working cycle

.

The optimal parameters result in a compression ratio of 5.6. The friction loss takes up 17.9% out of the indicated work. About 82.1% of the indicated work is converted to the gross electric energy by the VCM, 34.71% of which is consumed to drive the piston from BDC to TDC in the next cycle, and 19.78% is used for the boosting of scavenging pressure. Only 45.51% of the gross electric energy is output as net electric energy. Therefore, the resulting effective energy efficiency is 5.7%. Assuming that the mechanical-electro conversion efficiency of the VCM is about 90%, the total chemical to electric energy conversion efficiency is about 5.13% for the proposed MFPG prototype. The operating frequency of the prototype is around 20~24 Hz, which is equivalent to 1200~1440 rpm, the MFPG prototype outputs a power of 10~13 W, with a power density around 4.7~5.7 W/cc.

6. Conclusions

This work centers on the influences of the system parameters on characteristics of the in-cylinder mass leakage and the corresponding system performance. The effects of scavenging pressure, ignition position, combustion duration and piston mass are investigated. A sample power flow and energy distribution are analyzed to understand the pattern of energy conversion in the MFPG prototype.

(1) The leakage loss is closely related to the mass leakage during the compression m_lcp, which increases with increased scavenging pressure p_sc. The combustion duration t_cd and piston mass m_p have little effect on the m_lcp. With the advancing of the onset of combustion, m_lcp maximizes at x_ign = 2 mm. The effect of leakage loss on the indicated thermal efficiency Δη_i,a-b can be straightforwardly inferred by the gas leakage ratio m_lcp/m_gecp.

(2) The indicated thermal efficiency η_i,c can be improved by boosting the scavenging pressure p_sc, whereas the increase in combustion duration t_cd and piston mass m_p deteriorates the η_i,c. The optimal position of SOC for the higher η_e,c is around 3 mm. The effective thermal efficiency η_e,c decreases monotonously with the delay of SOC and the MFPG system is incapable to output power when the x_ign is less than 2 mm.

(3) The piston mass m_p has the most significant effect on the operation frequency f_p and peak piston velocity v_max. The TDC and effective compression ratio ε_ec are most sensitive to the change of SOC. The promotion effect of p_sc boosting on the in-cylinder peak pressure p_max and indicated thermal efficiency η_i,c has a “saturated trend”.

(4) During the entire working cycle, the powers of the MFPG system are dynamically conserved and circulate among each other. The frictional power P_f absorbs energy from the piston all the time. When the transient states of the two ultimate power states P_ele and P_p are not balanced, the residual power is stored temporarily in the form of P_a.