An Analysis Model Combining Gamma-Type Stirling Engine and Power Converter

Abstract

Waste heat is a potential source for powering our living environment. It can be harvested and transformed into electricity. Ohmic heat is a common type of waste heat. However, waste heat has the following limitations: wide distribution, insufficient temperature difference (ΔT < 70 K) for triggering turbines, and producing voltage below the open voltage of the battery. This paper proposes an energy harvester model that combines a gamma-type Stirling engine and variable capacitance. The energy harvester model is different from Tavakolpour-Saleh’s free-piston-type engine [7.1 W at ΔT = 407 K (273–680 K)]. The gamma-type Stirling engine is a low-temperature-difference engine. It can be triggered by a minimum ΔT value of 12 K (293–305 K). The triggering force in the variable capacitance is almost zero. Furthermore, the gamma-type Stirling engine is suitable for harvesting waste heat at room temperature. This study indicates that 21 mW of energy can be produced at ΔT = 30 K (293–323 K) for a bias voltage of 70 V and volume of 103.25 cc. Because of the given bias voltage, the energy harvester can break through the open voltage of the battery to achieve energy storage at a low temperature difference.

1. Introduction

Many forms of waste heat are produced in our living environment, such as the heat of combustion from factories, ohmic heating of computers, and heat from compressors. These heat sources increase the ambient temperature and cause global warming if we not do suitably handle them. Common methods of waste heat recovery include turbocharging, turbocompounding, thermoelectrics, and cabin heating and cooling [1]. Therefore, waste heat should be harvested and stored as electric power. The most suitable method for achieving this objective involves thermoelectric material use. However, the costs of semiconductor thermoelectric material are too high for typical thermoelectric power generation applications at ΔT < 135 °C [2]. This study aims to convert heat into electric power. To achieve this target, there exist three procedures. The first procedure involves converting heat into mechanical energy, the second involves converting mechanical energy into electric energy, and the third involves a combination of first two procedures. The three procedures are discussed in the following text.

The Stirling engine, which was created by Robert Stirling in 1816, has high efficiency for transforming heat into mechanical energy [3]. The Stirling cycle involves the cyclic expansion and compression of the working fluid by maintaining a temperature difference between the hot and cold plates. The heat energy produced can be converted to mechanical work. Stirling engines can be classified into three types: alpha, beta, and gamma machines [4]. The alpha-type Stirling engine has two pistons for compression and expansion. Both pistons are connected to the same crankshaft. The beta-type Stirling engine has one piston and one displacer, which share the same air room. The piston and displacer are linked by a wheel with two crankshafts. In the gamma-type engine, the piston and displacer are housed within two air rooms. The gamma-type engine (low temperature difference (LTD) engine) is the most suitable option for triggering the Stirling engine with a temperature difference from the environment (ΔT = 10–30 °C) [4]. The gamma-type engine has a high rotating speed, is triggered at LTDs, and has a small torque. The rotating speed is inversely proportional to the torque [5].

To calculate the performance of the Stirling engine, researchers developed computer fluid dynamic modes in commercial software [6]. Robson et al. created a dynamic model which forms with each motion of component [7]. Egas and Clucas compared the net work values of five types of Stirling engines by using the thermodynamics model [4]. Abovementioned studies have assumed that the instantaneous pressure in a Stirling engine is constant. However, a variety of instantaneous pressures should be considered in reality. Kim et al. used thermal flow to analyze high and low pressures in an LTD Stirling engine [8].

For transforming mechanical energy to electrical energy, a conversion method that affects the mechanism of harvester should be selected. Vibration energy harvesters (VEHs) can be classified into three types according to the principle of electron transfer: (i) electrostatic, (ii) electromagnetic, and (iii) piezoelectric VEHs. The electromagnetic method is unsuitable for microelectromechanical systems because the magnetic coil requires complicated manufacturing processes [9]. The piezoelectric material is highly brittle and hence is limited to a small deflection range [10]. In the electrostatic method, a bias voltage is provided between two metal plates so that they become a capacitance component. The capacitance component then releases current from the elevated potential energy through vibration [11]. The electrostatic method is more convenient than the electromagnetic and piezoelectric methods for harvesting energy from the environment. The capacitance component can be manufactured in any geometric shape and size to maximize the capacitive value. The component is widely used to increase the overlap area between two electrodes in the design concept of the interdigitated electrode [12,13]. Moreover, the designer adjusts the resonance frequency of the VEH according to the vibration source [14]. In addition to changing the gap of the variable capacitance, Moon et al. build a reverse electrowetting device with changing overlap area [15]. Hsu et al. proposed the bubbler structure, which causes air bubbles to move into the gap between two electrodes. The entry of air bubbles changes the dielectric coefficient of the variable capacitance [16].

Energy harvesting and storage can be achieved at room temperature. Some researchers select the vibrations of pavements, which are caused by human walking and car movement [17,18]. The frequency of human walking is 1.41–2.13 Hz [19], and the frequency of bridge vibration is 0–30 Hz [20]. The vibration source should have a high frequency, provide continuous waves, and have a stable amplitude for VEHs. The vibration source above has a low frequency (resonance frequency of the capacitive VEH is 6–13 GHz [21]), is discontinuous, and is unstable. Consequently, the vibration source is unsuitable for the electrostatic method. Therefore, the gamma-type Stirling engine can provide a stable and high-frequency source for capacitive VEHs. The bias voltage from the harvester can break through the open circuit voltage of the battery (open circuit voltage of lithium-ion batteries is 2.7–3.6 V during the charging process [22]). The Stirling generator has many applications. To accelerate the free-piston Stirling engine for reaching the resonance state (9.2 Hz), Tavakolpour-Saleh et al. [23] installed an additional DC motor on the Stirling engine. The obtained output power was 7.1 W at ΔT = 407 K (273–680 K). The combined heat and power system is an efficient energy recovery system. It can generate electricity and useful thermal energy and provide warm bath water, warm air, and electricity (10–36 kW) in the house [24,25]. The dish Stirling generator can obtain solar energy through a concentrator and reflector. The range of output obtained with this generator is 2–50 kW. The dish Stirling generator is built in the United States, Japan, and Russia [26].

A mathematical model is proposed for evaluating the performance of the gamma-type Stirling engine with a capacitive harvester (GSCH). The governing equation of the energy harvester is constituted according to the conservation of energy. The governing equation includes the heat transfer, dynamic model, and electric model. If a temperature difference is applied between the hot plate and cold plate in the GSCH (input), the heat energy from the hot plate enters the work system of the Stirling engine. The energy is assigned into three parts, namely the mechanical, air, and friction parts. The mechanical part provides mechanical energy to generate power (output). According to the simulation results, the wheel of the energy harvester has a rotation frequency of 3.35 Hz at ΔT = 30 K (298–328 K) and can produce 21 mW electric power under unit area at a bias voltage of 70 V. Moreover, the optimized size of the engine can be determined by changing the geometric shape and size of the gamma-type Stirling engine, which has maximum conversion efficiency in a specific temperature range.

2. Modeling

2.1. Heat Transfer

We assume that the temperature function of the GSCH only depends on the distance of the z-axis. The heat transfer model can be simplified by considering an ideal cylinder. Consider that the thermal conductivity (ignoring convection and radiation) and temperature have a linear relationship. Therefore, the heat transfer rate is q (Figure 1). The heat transmitted through the hot plate, air room, and cold plate can be written as follows:

$$q=\frac{T_1-T_4}{\frac{\Delta z_{Hot}}{k_{Hot}{A}_{Dc}}+\frac{\Delta z_{Air}}{k_{Air}{A}_{Dc}}+\frac{\Delta z_{Cold}}{k_{Cold}{A}_{Dc}}},$$

(1)

where $\Delta z_{Hot}$, $\Delta z_{Air}$, and $\Delta z_{Hot}$ are the thicknesses of the hot plate, air room, and cold plate, respectively, and A_Dc the cross-sectional area of the air room.

The thermal conductivity of the hot (k_Hot) and cold (k_Cold) plates can be obtained easily. However, the rigorous theory for the determination of the thermal conductivity of the air room does not exist. A forced convection enclosed flow field occurs in the air room of the Stirling engine. In this study, the heat transfer contribution of the displacer was not considered. The empirical relations for free convection are provided in the enclosures [27].

2.2. Kinematics

Figure 2 displays the major components and strokes of the gamma-type Stirling engine. The displacement functions of the piston and displacer are represented using the angular displacement of the engine shaft as follows:

$$\begin{array}{l}{z}_p=\frac{L_p}{2}\left[1+\cos \left(\theta -\frac{\pi }{2}\right)\right],\\ z=\frac{L}{2}\left[1+\cos \theta \right],\end{array}$$

(2)

where z_p and z are the vertical positions of the piston and displacer, respectively, and θ the rotation angle of the shaft. The displacer is in its highest position (z = L) for at θ = 0. The phase difference between the movements of the displacer and piston is π/2.

The displacer and piston perform periodical vertical movement when the Stirling engine functions with an angular velocity ω. The relationships between the vertical movement of the piston and displacer and the angular velocity are given as follows:

$$\begin{array}{l}{U}_z=-\frac{L}{2}·\sin \theta ·\omega ,\\ U_{zp}=-\frac{L_p}{2}·\sin \left(\theta -\frac{\pi }{2}\right)·\omega .\end{array}$$

(3)

Therefore, the Stirling engine performs three functions: (i) gas compression and expansion, (ii) mechanical motion, and (iii) friction loss when the engine is running. According to the first law of thermodynamics, the work of engine is given as follows:

$$W_{eng}=W_{gas}+W_{mech}+W_{loss},$$

(4)

where W_gas and W_mech are the work of the gas and mechanical component, respectively, and W_loss the work of friction loss.

2.3. Gas Work

The source of power of the Stirling engine is the pressure gradient in air room, which is formed because of the difference in the temperature of the cold and hot plates. In this section, we refer to the method of Kim et al. to calculate the gas work of the Stirling engine [8]. Because of the low speed and small size of the gamma-type Stirling engine, the airflow in the air room can be reasonably assumed to be laminar. Moreover, at 10 RPS, the maximum Reynolds number is 985 as per the mean velocity and hydraulic diameter for the flow through the annulus between the displacer and air room wall. The air room comprises a compression space (V_hD + V_hd) and an expansion space (V_cD + V_cd), as displayed in Figure 3.

In Figure 3, V_hD and V_cD change with the displacer movement and V_p changes with the piston movement. These relationships are expressed as follows:

$$\begin{array}{l}{V}_{hD}=A_{Dc}·\frac{L}{2}\left(1+\cos \theta \right),\\ V_{cD}=A_{Dc}·\left[L-\frac{L}{2}\left(1+\cos \theta \right)\right],\\ V_p=A_p\frac{L_p}{2}\left[1+\cos \left(\theta -\frac{\pi }{2}\right)\right],\end{array}$$

(5)

where A_Dc and A_p are the cross-sectional area of the air room and piston, respectively. The rate of airflow from the compression space to the expansion space depends on the pressure gradient across the displacer and the viscous force on the lateral surface of the displacer. According to approximation, the flow rate of the flow passing the annulus of the displacer is given as follows:

$$Q_v=\pi D\left(\frac{U_{z}a}{2}-\frac{1}{12\mu }\frac{dp}{dz}{a}^3\right)=-A_{Dc}{U}_z,$$

(6)

where μ is the dynamic viscosity of air. Through Equation (6), the pressure gradient in the displacer annulus written as follows:

$$\frac{dp}{dz}=\frac{6\mu L}{\pi D{a}^3}(A_{Dc}+\frac{\pi Da}{2})\sin \left(\theta \right)\omega .$$

(7)

We assume that the pressure gradient in the displacer annulus can be approximated using the following equation:

$$\frac{dp}{dz}\simeq \frac{p_c-p_h}{l_D},$$

(8)

where p_c and p_h are the pressure of the cold and hot regions in the air room, respectively. The relationship between the cold and hot pressure is given as follows:

$$p_h\simeq p_c+\frac{6\mu L}{\pi D{a}^3}(A_{Dc}+\frac{\pi Da}{2})l_D\sin \left(\theta \right)\omega .$$

(9)

The gas in the air room is assumed to be an ideal gas, and no air is assumed to leak. The value of p_c depends on the geometry of the air room, temperature, rotation angle of the shaft, angular velocity, dynamic viscosity of air, and gas constant.

$$p_c=\frac{M-\left[\left(a_6+a_9\right)\sin \left(\theta \right)\omega +a_7\cos \left(\theta \right)\sin \left(\theta \right)\omega \right]}{a_1+a_4+a_8+\left(a_2+a_5\right)\cos \left(\theta \right)+a_3\cos \left(\theta -\frac{\pi }{2}\right)},$$

(10)

where M is the total mass of air in the air room and a₁–a₉ the coefficients of the Stirling engine (

Table A2. The coefficient of Equation (10).

a1	$\frac{V_{pd}+V_{cd}+0.5\left(A_{Dc}L+A_p{L}_p\right)}{R{T}_{ca}}$
a2	$\frac{-0.5A_{Dc}L}{R{T}_{ca}}$
a3	$\frac{0.5A_p{L}_p}{R{T}_{ca}}$
a4	$\frac{V_{hd}+0.5A_{Dc}L}{R{T}_{ha}}$
a5	$\frac{0.5A_{Dc}L}{R{T}_{ha}}$
a6	$c_0{l}_D\frac{V_{hd}+0.5A_{Dc}L}{R{T}_{ha}}$
a7	$c_0{l}_D\frac{0.5A_{Dc}L}{R{T}_{ha}}$
a8	$2l_D\frac{A_{Dc}-A}{R\left(T_{ca}+T_{ha}\right)}$
a9	$c_0{l}_D\frac{\left(A_{Dc}-A\right)l_D}{R\left(T_{ca}+T_{ha}\right)}$

). Therefore, the gas work of the Stirling engine per cycle is given as follows:

$$\begin{array}{l}{W}_{gas}=W_{pp}+W_{pd}\\ ={\displaystyle \int p_c{A}_{p}d{z}_p}+{\displaystyle \int A\left(p_c-p_h\right)}dz\\ =-\frac{L_p}{2}{\displaystyle {\int }_0^{2\pi }{A}_p{p}_c\sin \left(\theta -\frac{\pi }{2}\right)}d\theta -\frac{L}{2}{\displaystyle {\int }_0^{2\pi }A\left(p_c-p_h\right)\sin \left(\theta \right)}d\theta ,\end{array}$$

(11)

where W_pp and W_pd are the moving-boundary work produced by the piston and displacer per cycle, respectively.

2.4. Mechanical Component Work

The engine work comprises three components, namely the work of the (i) flywheel, (ii) shaft assembly, and (iii) piston and displacer systems. The work of the engine is represented as follows:

$$W_{mech}=W_{wheel}+W_{shaft}+\left(W_{dis}+W_{pis}\right),$$

(12)

where W_wheel is the work of the flywheel, W_shaft the work of the shaft, W_dis the work of the displacer system, and W_pis the work of the piston system. As shown in Figure 4a, there exist two holes in the first quadrant of the flywheel. The two holes cause the center of mass to be located in the third quadrant. Therefore, the work of the flywheel is written as follows:

$$\begin{array}{l}{W}_{wheel}=M_{wheel}·g·Z_{wheel}+0.5\left(0.5M_{wheel}{R}_{wheel}^2+M_{wheel}{R}_{new}^2\right){\omega }^2,\\ Z_{wheel}=R_{new}[1+\cos \left(\theta +1.5\pi \right)],\end{array}$$

(13)

where M_wheel is the mass of the flywheel, g the gravity, R_new the distance between the center of the flywheel and mass of the center point, and Z_wheel the vertical distance (when θ = 1.5π, Z_wheel is maximum). Moreover, a bearing links the rotation of the shaft and flywheel (Figure 4a). The work of the shaft is written as follows:

$$W_{shaft}=0.5\left(M_c{R}_c{}^2+2M_s{R}_s{}^2\right){\omega }^2,$$

(14)

where M_c and M_s are the mass of the bearing and shaft, respectively, and R_c and R_s the radii of the bearing and shaft, respectively.

For convenient calculation, all the components (displacer, piston, screws, and links) are set up in two mass systems: displacer and piston systems (Figure 4b). The work of the displacer and piston systems is then given as follows:

$$\begin{array}{l}{W}_{dis}=M_1·g·z+0.5\left(M_1{U}_z^2\right),\\ W_{pis}=M_2·g·z+0.5\left(M_2{U}_{zp}^2\right),\end{array}$$

(15)

where M₁ and M₂ are the mass of the displacer and piston systems, respectively. As indicated by Equations (12)–(15), the work of the engine depends on the angular velocity of the engine.

2.5. Friction Loss

The work of friction falls into two broad categories: solid and gas friction loss. Among them, solid friction loss occurs at contact surfaces between the bearing and shafts. Furthermore, the volume of solid friction depends on the torque acting on the bearing. The work of friction in one cycle can be represented as follows:

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{shaft}& ={\mu }_k·{\tau }_{shaft}·V_{\tan }·{\mathrm{T}}_{\mathrm{per}}\hfill \\ & =2\pi f{\mu }_k\left(\left|F_{piston}\right|+\left|F_{displacer}\right|\right)R_c·{\mathrm{T}}_{\mathrm{per}}\hfill \\ & =-\left(\pi {\mu }_k{R}_{c}L\right)\{{\omega }^2\left[\left|M_1{\displaystyle {\int }_0^{2\pi }\cos \theta d\theta }\right|+\left|M_2{\displaystyle {\int }_0^{2\pi }\cos (\theta -\frac{\pi }{2})d\theta }\right|\right]\hfill \\ & +\alpha \left[\left|M_1{\displaystyle {\int }_0^{2\pi }\sin \theta d\theta }\right|+\left|M_2{\displaystyle {\int }_0^{2\pi }\sin (\theta -\frac{\pi }{2})d\theta }\right|\right]\},\hfill \end{array}$$

(16)

where μ_k is the coefficient of kinetic friction, τ_shaft the load, V_tan the tangential velocity of the shaft, T_per the period, f the frequency, and α the angular acceleration.

The method of [8] is used to analyze the work of gas friction loss. The analytical method is applicable for (i) steady flow, (ii) incompressible laminar flow, and (iii) completely developed flow in the air room. The shear stresses on the lateral surfaces of the displacer, piston, and displacer rod are given as follows:

$$\begin{array}{l}\tau =\mu \frac{U_z}{a}+\frac{a}{2}\frac{dp}{dz},\\ {\tau }_p=\frac{\mu U_{zp}}{r_p\ln \left(\frac{r_p}{r_p+a_p}\right)},\\ {\tau }_{DR}=\frac{\mu U_z}{r_{DR}\ln \left(\frac{r_{DR}}{r_{DR}+a_{DR}}\right)}.\end{array}$$

(17)

Equation (17) is applied to obtain the work of friction loss from the viscous drags in the displacer, piston, and displacer rod per cycle.

$$\begin{array}{l}{\mathsf{\Omega }}_s={\displaystyle \int F_{s}dz=-\frac{L}{2}{\displaystyle {\int }_0^{2\pi }\pi ·D·l_D·\tau ·\sin \left(\theta \right)d\theta }},\\ {\mathsf{\Omega }}_{sp}={\displaystyle \int F_{sp}d{z}_p=-\frac{L_p}{2}{\displaystyle {\int }_0^{2\pi }\pi ·D_p·l_p·{\tau }_p·\sin \left(\theta -\frac{\pi }{2}\right)d\theta }},\\ {\mathsf{\Omega }}_{sDR}={\displaystyle \int F_{sDR}dz=-\frac{L}{2}{\displaystyle {\int }_0^{2\pi }2\pi ·r_{DR}·l_{Dr}·{\tau }_{DR}·\sin \left(\theta \right)d\theta }}.\end{array}$$

(18)

where F_s, F_sp, and F_sDR are the shear forces on the lateral surfaces of the displacer, piston, and displacer rod, respectively.

By integrating Equations (18), the relationship between the temperature difference (ΔT = T₁ − T₄) and the frequency of the gamma-type Stirling engine movement can be obtained:

$$\begin{array}{l}q·{\mathrm{T}}_{\mathrm{per}}=W_{eng}\\ \Rightarrow \Delta T=c_1{f}^3+c_2{f}^2+c_{3}f,\end{array}$$

(19)

where c₁, c₂, and c₃ are functions such as the mass, geometry, angle, force, kinetic friction coefficient, and dynamic viscosity (

Table A1. The coefficient of Equation (19).

$\mathsf{\Delta }\mathit{T}={\mathit{c}}_1{\mathit{f}}^3+{\mathit{c}}_2{\mathit{f}}^2+{\mathit{c}}_3\mathit{f}$
Coefficient	Function	Remark
c1	$\left|\frac{{\pi }^2}{2}\left[M_1{L}^2{\displaystyle {\int }_0^{2\pi }{\sin }^2\theta d\theta +M_2}{L}_p^2{\displaystyle {\int }_0^{2\pi }\sin \left(\theta -\frac{\pi }{2}\right)}d\theta \right]\right|\left(R_{con}\right)$	Kinetic energy of M1 and M2 systems.
	$\left|+4{\pi }^3{\mu }_k{R}_{c}L\left[M_1{\displaystyle {\int }_0^{2\pi }\cos \theta d\theta +M_2}{\displaystyle {\int }_0^{2\pi }\cos (\theta -\frac{\pi }{2})}d\theta \right]\right|\left(R_{con}\right)$	Kinetic friction loss M1 and M2 systems.
	$\left|+2{\pi }^2\left(0.5M_{wheel}{R}_{wheel}^2+M_{wheel}+R_{new}^2\right)\right|\left(R_{con}\right)$	Kinetic energy of the flywheel component.
	$\left|+2{\pi }^2\left(m_s{R}_s^2\right)\right|\left(R_{con}\right)$	Kinetic energy of the shaft component.
	$\left|+{\pi }^2{M}_c{L}_c^2\right|\left(R_{con}\right)$	Kinetic energy of the bearing component.
c2	$\left|L_p{A}_p\pi {\displaystyle {\int }_0^{2\pi }\frac{\left(a_6+a_9\right)\sin \theta +a_7\cos \theta \sin \theta }{a_1+a_4+a_8+\left(a_2+a_5\right)\cos \theta +a_3\cos \left(\theta +\frac{\pi }{2}\right)}}\sin \left(\theta +\frac{\pi }{2}\right)d\theta \right|\left(R_{con}\right)$	Moving-boundary work of piston.
	$\begin{array}{l}|-L\{{\pi }^2·D·l_D·{\displaystyle {\int }_0^{2\pi }\sin \theta [\frac{\mu }{a}\left(-\frac{L}{2}\sin \theta \right)}\dots \\ +\frac{2}{a}\left(-\frac{6\mu L}{\pi ·D·a^3}\left(A_{Dc}+\frac{\pi ·D·a}{2}\right)\right)]d\theta \}|\left(R_{con}\right)\end{array}$	Friction loss from the viscous drags in the displacer.
	$\left|+L_p\left[{\pi }^2·D_p·l_p\frac{\mu }{2r_p\ln \left(r_p/r_p+a_p\right)}{\displaystyle {\int }_0^{2\pi }{\sin }^2\left(\theta +\frac{\pi }{2}\right)d\theta }\right]\right|\left(R_{con}\right)$	Friction loss from the viscous drags in the piston.
	$\left|+L\left[{\pi }^2·r_{DR}·l_{DR}\frac{\mu }{r_{DR}\ln \left(r_{DR}/r_{DR}+a_{DR}\right)}{\displaystyle {\int }_0^{2\pi }{\sin }^2\theta d\theta }\right]\right|\left(R_{con}\right)$	Friction loss from the viscous drags in the rod.
	$\left|\frac{6\mu L}{\pi D{a}^3}(A_{Dc}+\frac{\pi Da}{2})\left(L·A·l_D·\pi \right){\displaystyle {\int }_0^{2\pi }{\left(\sin \theta \right)}^2}d\theta \right|\left(R_{con}\right)$	Moving-boundary work of displacer.
c3	$\left|-\frac{L_p}{2}{A}_p{\displaystyle {\int }_0^{2\pi }\frac{M}{a_1+a_4+a_8+\left(a_2+a_5\right)\cos \theta +a_3\cos \left(\theta +\frac{\pi }{2}\right)}}\sin \left(\theta +\frac{\pi }{2}\right)d\theta \right|\left(R_{con}\right)$	Moving-boundary work of piston.
	$\left|+\alpha \pi {\mu }_k{R}_{c}L\left[M_1{\displaystyle {\int }_0^{2\pi }\sin \theta d\theta +M_2}{\displaystyle {\int }_0^{2\pi }\sin \left(\theta -\frac{\pi }{2}\right)}d\theta \right]\right|\left(R_{con}\right)$	Kinetic friction loss M1 and M2 systems.
	$\left|+g\left[\left(M_1·z\right)+\left(M_2·z_p\right)+\left(M_{wheel}·Z_{wheel}\right)\right]\right|\left(R_{con}\right)$	Potential energy of the M1, M2 systems, and flywheel component.
Rcon	$\frac{\Delta z_{Hot}}{k_{Hot}A}+\frac{\Delta z_{Air}}{k_{Air}A}+\frac{\Delta z_{Cold}}{k_{Cold}A}$	Thermal resistance in series of the Stirling engine components.

).

2.6. Electrostatic Capacitance

The capacitive energy harvester consists of two electrode plates: piston and piston cylinder (Figure 5). According to the capacitive formula, the capacitance value is proportional to the overlap area of the two electrode plates and the frequency of engine rotation. Therefore, changes in the capacitance value depend on the Stirling engine movement.

Krupenkin and Taylor proposed a method to simulate the energy harvester with periodic movement [28]. The equivalent circuit is displayed in Figure 5. The circuit has a bias voltage of V_bias, resistive load of R, and variable capacitance of C. According to Kirchhoff’s circuit laws, the RC (resistance and capacitance) circuit is represented as follows:

$$\begin{array}{l}\frac{d{Q}_c}{dt}+\frac{Q_c}{RC}-\frac{V_{bias}}{R}=0,\\ Q_c(t=0)=V_{bias}{C}_0,\end{array}$$

(20)

where Q_c is the charge on the variable capacitor, C₀ the maximum capacitance of the variable capacitor, V_bias the bias voltage, and t the time. The variation in the capacitance with time is expressed as follows:

$$\begin{array}{l}C(t)=0.5C_0(1+\cos (\omega t)),\\ C_0=\frac{{\epsilon }_0{k}_{die}{A}_{over}}{h},\end{array}$$

(21)

where A_over is the maximum overlap area between the outer part of the piston surface and the piston cylinder during the Stirling engine cycle, h the dielectric thickness of the piston cylinder, k_die the dielectric constant of the thin film, and ε₀ the vacuum permittivity.

Equation (20) is transformed into a dimensional equation as follows:

$$\begin{array}{l}{a}_0\frac{d{\widehat{Q}}_{ch}}{d\widehat{t}}+\frac{2{\widehat{Q}}_{ch}}{1+\cos \widehat{t}}-1=0,\\ {\widehat{Q}}_{ch}(\widehat{t}=0)=1,\end{array}$$

(22)

where ${\widehat{Q}}_{ch}=Q_h{V}_{bias}^{-1}{C}_0^{-1}$, $a_0=\omega R{C}_0$, and $\widehat{t}=\omega t$. The solution of Equation (22) is given as follows:

$${\widehat{Q}}_{ch}=\frac{1}{a_0}{e}^{-\frac{zi}{a_0}-\frac{z\tanh [\frac{t}{2}]}{a_0}}\left(a_0{e}^{\frac{zi}{a_0}}+i{e}^{\frac{4i}{a_0}}\mathrm{Ei}\left[-\frac{2i}{a_0}\right]-iEi\left[\frac{2i}{a_0}\right]-i{e}^{\frac{4i}{a_0}}\mathrm{Ei}\left[\frac{2(-i+\tanh [\frac{t}{2}])}{a_0}\right]+i\mathrm{Ei}\left[\frac{2(i+\tanh [\frac{t}{2}])}{a_0}\right]\right),$$

(23)

where $\mathrm{Ei}(x)=-{\displaystyle {\int }_{-x}^{\infty }\frac{e^{-j}}{j}dj}$ is an exponential integral. The energy harvested per Stirling engine cycle is expressed as

$$\widehat{E}={\displaystyle \underset{\mathrm{period}}{\int }{a}_0\left(\frac{d{\widehat{Q}}_{ch}}{d\widehat{t}}\right)}d\widehat{t}\simeq \frac{5}{4}\left[1-\tanh (0.5(1-\log (a_0))\right],$$

(24)

and Equation (24) can be written with the original variables as follows:

$$E=\frac{5}{4}{V}_{bias}^2{C}_0\left[1-\tanh (0.5(1-\log (\omega R{C}_0))\right].$$

(25)

Therefore, the average generated power is given as follows:

$$P=\frac{5}{4}{V}_{bias}^2{C}_0{T}^{-1}\left[1-\tanh (0.5(1-\log (\omega R{C}_0))\right].$$

(26)

3. Result and Discussion

The temperature difference is used to change the volume of the air room, which in turn induces a change in the piston position. As indicated by Equation (5), the sizes of the compression space (V_hD + V_hd) and expansion space (V_cD + V_cd) depend on the phase angle of the wheel. In a gamma-type Stirling cycle, the isothermal compression is θ = 13~3(π/8), constant volume heating is θ = 4 (π/8), isothermal expansion is θ = 5–11 (π/8), and constant volume cooling is θ = 12 (π/8). When θ = 4 (π/8), the air room has maximum volume. The pressure difference between the hot and cold regions varies with the sinusoid motion of the displayer at fixed temperature differences (Figure 6). The variations in p_h and p_c for θ = 1–4 (π/8), θ = 9–12 (π/8), and ΔT = 0–30 K are calculated using Equations (9) and (10), where ω = 2πf for each ΔT is the result obtained from Equation (19). The relative pressure (compared with atmospheric pressure) of the cold region is 2921.8–2985.7 Pa for θ = π/2.

The pressure difference in the air room of the gamma-type Stirling engine is very small because the thickness of the air room is only 10 mm. Consequently, hot and cold air mix quickly in the air room. Thus, the Stirling engine works in a steady state, which only requires a pressure difference of less than 0.1 Pa in the air room.

The relationship between the temperature differences of the Stirling engine and the frequency of the wheel can be obtained using Equation (19). Assume that the cold and hot plates have an initial temperature of 293 K (ΔT = 0 K) and the hot plate is then heated to 323 K (ΔT = 30 K). In this scenario, Equation (19) has one real root and two nonreal complex conjugate roots for each temperature difference. The solid line (real root) in Figure 7 represents the simulated results obtained using Equations (1)–(19), whereas the dotted line represents the fitting results obtained from an experiment. For the experimental results, the wheel rotation is determined in revolutions per minute, whereas for the theoretical results, the wheel rotation is determined in revolutions per second (Hz). There exist 63 experiment points, and the fitting line is a cubic equation, which is represented as follows: $\Delta T=-0.0003f^3+0.0105f^2+0.0975f$.

As displayed in Figure 7, the triggered temperature difference of the Stirling engine is at least 12 K because the engine must overcome the friction and mass of the mechanical system (M₁, M₂, M_s, M_c, and M_wheel). The activated resistance force is [M₁+μ_k(M₂+M_s+M_c+M_wheel)]g = 0.1731 N. Furthermore, through Equation (10), the difference in the pressures at 293 (P₀) and 305 (P₁₂) K is calculated to be 27.4 Pa for θ = 0. The pressure on the displacer is equal to 0.1743 N.

The work system of the Stirling engine is divided into the mechanical, gas, and friction parts (Figure 8a). The work of the mechanical part comprises W_dis, W_pis, W_shaft, and W_wheel. The motion of the displacer and piston is vertical simple harmonic motion through wheel rotation. Therefore, the lines of W_dis and W_pis depend on the position functions of the displacer and piston. Because the Stirling engine works in a steady state, the angular velocity is independent of time. Therefore, W_shaft is a constant volume. Moreover, there exist two holes in the first quadrant of the flywheel. Consequently, the moment of inertia depends on the phase angle (Figure 8b). The work of the gas contains W_pp and W_pd. Both the lines displayed in Figure 8c are sine functions. When W_pp is a positive volume, the fluid in the engine outputs work to the piston; otherwise, the piston compresses the fluid in the engine. The displacer only moves the fluid in the air room and cannot compress or expand the fluid. Therefore, the volume of W_pd is positive. Two peak volumes exist for W_pd. These peak volumes represent the displacer touching the clearance in the hot and cold areas (Figure 8c). At the friction work part (Figure 8d), when F_displacer and F_piston drive the shaft rotation, the work of friction (Ω_shaft) reduces the mechanical work of gas expansion. Ω_sp and Ω_sDR are produced when the piston and displacer rod friction fluid (negative value). Ω_s is produced when the fluid is reflected from the cold plate inside friction the lateral surface of displacer (positive value, reverse direction).

To compare the effect of the area of heated plate with the Stirling engine, a dimensionless length ratio is defined. The length ratio is equal to the diameter of the displacer divided by the diameter of the wheel (D/2R_w). The length ratio is proportional to the heat flux of the air room if the gap between the displacer and displacer cylinder is fixed. Figure 9a displays the relationship between the length ratio, temperature difference, and frequency of the wheel. When the temperature of the hot plate is 30 K higher than that of the cold plate, the wheel rotation frequency equals 3.35 Hz. However, when the length ratio is increased by six times, the rotation speed increases by 2.69 times. Figure 9b illustrates power required (W_wheel + W_dis + W_shaft + W_pis + W_pd + W_pp + Ω_s + Ω_sp + Ω_sDR + Ω_shaft) by the Stirling engine in the steady state. When the length ratio increases, the mass of the wheel increases. Therefore, the engine requires additional energy to push the wheel.

The aforementioned frequency (Figure 7, Figure 8 and Figure 9) of the Stirling engine is simulated through Equations (25) and (26). The dielectric coefficient of acrylic (ε_a) is 3.7, and the dielectric thickness (h) is 1 μm. The energy and power produced can be calculated. As displayed in Figure 10a,b, different bias voltages (26, 44, 53, 61, and 70 V) are produced between the two plates of the capacitor. The variable capacitance produces power along the piston of the Stirling engine. As per the computational results, the energy harvester produces 6.2 mJ of energy per cycle per unit area at a bias voltage of 70 V and temperature difference of 30 °C. Thus, the engine produced 21 mW of energy per unit area.

The conversion efficiency of heat energy into electric power is calculated using the formula η_he = 100(P/q). The conversion efficiency of engine power into electric power is calculated using the formula η_ee = 100(P/η). In this study, η = 1 − (T₄/T₁), A_over = A_Dc, h = 1 μm, and V_bias = 70 V. A dimensional number (ss) is delineated to adjust the size of the Stirling engine. For instance, an ss value of 2 indicates that the length, mass, and gap between the displacer and displacer cylinder of the Stirling engine become two times the dielectric thickness. The efficiency of the heat engine power (η) is proportional to the temperature difference. However, the relationship between the frequency of the wheel and temperature is a concave curve. When ΔT > 10 K, the rate of increase decreases. Therefore, the conversion efficiency of engine power into electric power follows the trend of a convex curve (Figure 11a). Considering the dimensional efficiency, a large energy harvester is suitable for high temperature differences. For instance, η_he (ss = 150) is twice η_he (ss = 30) at ΔT = 60–90 K. In Figure 10, each curve of η_ee has a maximum value [η_ee,max(ss = 30) at ΔT = 7 K, η_ee,max(ss = 60) at ΔT = 18 K, and η_ee,max(ss = 90) at ΔT = 30 K], the corresponding temperature difference is the best temperature difference (T_bt) for the conversion efficiency of heat energy into electric power. Thus, η_ee exhibits positive growth for temperature differences below T_bt.

4. Conclusions

In this study, a mathematical model that considers the conversion of heat energy to mechanical energy and the mechanical energy to electric energy is proposed and the relationship between the frequency and temperature difference is verified. The simulated results indicate that the frequency of the wheel is 3.35 Hz at ΔT = 30 K (293–323 K) and the triggered temperature difference of the Stirling engine is 12 K (293–305 K). A small pressure difference exists between the hot and cold regions in the air room of the gamma Stirling engine ($\pm 0.1\mathrm{Pa}$ at ΔT = 30 K). When heat enters the work system of the Stirling engine, the energy can be divided into three parts: mechanical, air, and friction energy. The proportions of energy depend on the angular velocity of the engine [ω = 2π(3.35)]. The rotation of the wheel, which involves mechanical work, requires the highest heat energy. Moreover, the ratio between the power of work and resistance of work (D/2R_w) affects the frequency and moment of inertia. When the length ratio increases by six times, the rotation speed increases by 2.69 times. When a variable capacitor is applied to harvest energy at V_bias = 70 V and ΔT = 30 K, 21 mW electric power is produced, which can be harvested and stored at room temperature. The optimized conversion efficiency of heat energy into electric power is obtained through the ss. Every size of the Stirling engine energy harvester has a best temperature difference range. It can provide power for low-power sensors to be used in remote mountains, rainforests, and deserts.