Study on Performance of a Modified Two-Stage Piston Expander Based on Spray Heat Transfer

Abstract

To fully use high-pressure air, the two-stage piston expander (TSPE) has been widely studied. The following factors obstruct the use of the TSPE: A high expansion ratio will inevitably result in a lower air temperature in the cylinder, causing adverse effects such as ice blockage and lubricating oil freezing; the residual air from the I-stage cylinder will all flow into the II-stage cylinder, causing a large reverse force to the I-stage piston during the working process. To address the above problems, a modified two-stage piston expander (M-TSPE) based on spray heat transfer is proposed. Firstly, the working principle of the M-TSPE is introduced, followed by the construction of a mathematical model of the M-TSPE. Secondly, the valve-timing of the M-TSPE is determined and compared with the output power and efficiency of the TSPE. The output power and efficiency of the M-TSPE are increased by 57.58% and 13.28%, respectively. Then, the performance parameters of the M-TSPE with and without spray are compared and analyzed. Finally, parameter analysis is carried out on the air/water mass ratio and water mist particle size. Results show that when the intake pressure and load torque are set to 3 MPa and 150 N·m, respectively, the output power of the M-TSPE without spray is 14.22 kW and the output power of the M-TSPE with spray is 16.08 kW, which is a 13.08% increase in output power. The average air temperatures in the I-stage cylinder of the M-TSPE with and without spray are 321 K and 263 K, respectively, and the average air temperature in the I-stage cylinder is enhanced by 58 K. The output performance of the M-TSPE can be improved by increasing the mass ratio of the water mist in the cylinder and decreasing the particle size of the water mist.

1. Introduction

According to the International Energy Agency (IEA) [1], China has become the world’s largest energy consumer since 2010. To address climate change and air pollution caused by energy consumption, China is striving to promote the growth of renewable energy, which has emerged as the country’s fastest-growing energy source. According to the National Energy Administration (NEA) [2], by the end of October 2021, China’s cumulative installed power generation capacity from renewable energy had reached 1.002 billion kW. However, renewable energy generation has inherent intermittency and instability, resulting in considerable challenges in ensuring the flexibility and stability of the power system supply and demand balance [3]. Energy storage is the critical step to solving the problem of renewable energy generation quality and grid stability. CAES is a relatively novel type of energy storage technology with the advantages of no pollution, a long energy storage time, and high safety [4]. Currently, CAES technology has been studied and developed by many research institutions. Some energy storage systems with CAES technology have entered the commercial stage, including CAES power stations in Germany and the United States [5,6]. Thus, CAES has excellent prospects for commercial application.

As the terminal power output link of the CAES, the efficiency and operation characteristics of the expander have a decisive influence on the overall operating performance of the CAES system. With the vigorous development of fluid transmission and control technology [7,8,9], the expansion machine of different structures has been extensively studied. Because the piston expander inherits the mature technology of the reciprocating compressor structure, it is more conducive to transformation and manufacturing. The research shows [10] that a piston-type expander is more suitable for a small-scale CAES system. However, the research mainly focuses on the single-stage piston expander (SSPE).

Under high intake pressure, the SSPE generally has high exhaust pressure and significant energy loss to avoid the adverse effects of ice blockage and lubricating oil freezing caused by the high expansion rate. Especially in the case of high output power, the SSPE needs a higher intake pressure, which will produce a higher residual pressure gas after expansion. The residual pressure gas cannot be effectively used, leading to energy waste. Therefore, to achieve the broad application of piston expanders, it is essential to improve the energy usage efficiency of air.

To solve the above problems, some companies and university research institutions have mainly carried out research from two directions: (1) isothermal (constant temperature) gas cycling, which is accomplished by mixing the liquid (spray or foam) with the air in the cylinder for heat exchange. (2) multi-stage gas expansion scheme, the expanded gas in the I-stage cylinder that enters the II-stage cylinder continues to expand and release energy to achieve full gas expansion. In terms of isothermal (constant temperature) gas cycling, Zhang et al. [11] designed a reciprocating expander that can be applied to isothermal CAES, established a near-isothermal expansion model for the expander, and conducted a comparative analysis with the adiabatic expansion model. The results show that the specific work output of the near-isothermal process increases by 15.7%; compared with the inlet and outlet temperature difference of adiabatic expansion, the inlet and outlet temperature difference of the quasi-isothermal expansion process is significantly reduced. Based on the theoretical and experimental study of the piston expander prototype with a single cylinder and double inlet and exhaust valves [12,13], our research group established the quasi-isothermal expansion model of the piston expander prototype by introducing the water mist–air direct heat transfer equation. It was concluded [14] that heat transfer between water mist and air can increase the air temperature in the cylinder while decreasing the air temperature difference. The pressure increase in the quasi-isothermal expansion model’s intake and expansion stages was less than that in the exhaust stage, resulting in a 7.7% reduction in output power compared to the adiabatic expansion model. Jia et al. [15] and Srivatsa et al. [16] analyzed the effect of moisture content on the isothermal compression process. Jia et al. designed an algorithm to regulate the water spray flow to match different compression heats. The results showed that the efficiency of the compressor can be improved effectively. Srivatsa et al. established 0-D and 1-D compression process models, they showed that the primary mechanism of low energy consumption was the increase in the specific heat capacity of the air surface due to the increased propensity of water to evaporate. Vikram et al. [17] conducted water spraying experiments on liquid piston compressors. The results showed that water spraying could effectively reduce air temperature rise during compression under different injection pressures, injection angles, and stroke times. Braasch et al. [18] conducted an experimental study and numerical simulation of the isothermal expansion process. The results showed that the heat transfer effect between water mist and air can be significantly improved under the high-pressure ratio, and the work transferred in the isothermal process can reach up to 94.2%. In terms of increasing the temperature of the working medium in the air intake tract, Poongavanam et al. [19,20] studied an automotive radiator with multi-walled carbon-based nanofluids. The results showed a maximal enhancement of the Nusselt number “Nu” by 18.39% with an inlet temperature of 70 °C, 0.6% of MWCNTs nanomaterials, and a mass flow rate of 90 g/s. SustainX [21] developed an innovative concept that uses foam injection to achieve constant temperature. However, the generation of foam and its decomposition into air and water after the heat transfer process will be a key challenge for this concept. Recently, in the field of diesel engine development, Nachippan et al. [22] and Elumalai et al. [23] have studied the premixed charge compression ignition (PCCI) engine. The results showed that the PCCI engines operated with ethanol as the induction fuel and the direct injection of 100 ppm multi-walled carbon nanotube (MWCNT)-blended Tamanu methyl ester (TME) showed an effectively improved thermal efficiency and reduced NOx and smoke emissions compared to conventional diesel CI engines. In terms of multi-stage expansion design, LIU Hao et al. [24] proposed a two-stage expansion pneumatic engine. The principle is that compressed gas first expands in the I-stage cylinder to carry out work, and the residual gas, after expansions, flows through the heat exchanger for heat exchange. Then the gas enters the II-stage cylinder for expansion to carry out work. The performance of the two-stage expansion pneumatic engine is improved compared to that of the SSPE, but the heat obtained through the heat exchanger only accounts for 0.019% of the engine’s total mechanical work output. Liu CM et al. [25] conducted experimental and simulation studies of the expansion air engine with a two-cylinder series structure. The results showed that the two-cylinder expansion engine has good output performance at low speed. Our research group has also proposed an innovative two-stage piston expander [26]. This expander perforates near the BDC of the I-stage cylinder. When the I-stage piston runs to the BDC, the gas in the I-stage cylinder flows into the II-stage cylinder through the small hole to expand and carry out work to realize the reuse of the remaining energy of compressed air. Compared with the SSPE, this expander mainly improves the utilization of air expansion energy, and the efficiency of the expander is about 15% higher than that of the SSPE.

In previous studies of the TSPE [24,25], the I-stage exhaust channel was connected to the II-stage intake channel, so the gas of the I-stage cylinder in the exhaust process, the gas of the II-stage cylinder in pushing the second stage piston output effective work. To solve the problem that the TSPE’s stage I piston will be subjected to a large reverse force during the positive work of the stage II cylinder gas pushing the piston, we modified the structure of the TSPE. Moreover, two-stage gas expansion inevitably leads to a lower air temperature in the cylinder, which affects the regular operation of the expander. To boost the TSPE’s output power and efficiency even further, and to solve the adverse effects of freezing blockage and lubricating oil freezing caused by a low exhaust temperature, we realize the best application in a small-scale CAES system, and an M-TSPE based on spray heat transfer technology is proposed. This study aims to design a modified two-stage piston expander, optimize the valve timing, and analyze the mechanism of the output performance improvement after the water spray. Up to now, there has been no related research on the M-TSPE based on spray heat transfer.

The rest of the paper is organized as follows: the system structure and working principle are described in detail in Section 2. In Section 3, the mathematical model of the system is developed. Then, the simulation calculation and result analysis are presented in Section 4. Section 5 is devoted to the summary and conclusions.

2. Working Principle and Structural Characteristics

Figure 1 is the schematic diagram of the M-TSPE. The cylinders of the M-TSPE are connected in series on the same crankshaft, with the same connecting rod ratio and piston stroke. The two stages of the cylinder are completed with the intake expansion and exhaust two strokes, but the phase difference between the I-stage and II-stage is 180°. The TDC position of the I-stage cylinder is provided with a small hole leading to the intake valve channel of the II-stage cylinder, and the top position of the I-stage cylinder is provided with a double inlet and exhaust valve channel leading to the atmosphere. To ensure fast response, the M-TSPE adopts a cam mechanism with double inlet and exhaust valves to control the gas flow on–off.

The M-TSPE mainly consists of four inlet valves, four exhaust valves, two power pistons, two cylinders, a spray system, a water mist filter system, and a crankshaft. The schematic diagram of the M-TSPE principle is illustrated in Figure 2. The working process of the M-TSPE can be divided into four parts (see the four subgraphs in Figure 2) by the stroke of the I-stage expander: I-stage intake stroke, I-stage expansion stroke, I-stage exhaust utilization stroke, and I-stage exhaust stroke.

During the I-stage intake stroke, the I-stage intake valve is opened. The air source gas flows into the I-stage cylinder through the I-stage intake valve to push the I-stage piston to move from TDC to BDC. At this point, the exhaust valve of the II-stage cylinder is opened. The II-stage piston moves from BDC to TDC to discharge the residual air of the II-stage cylinder into the atmosphere.

During the I-stage expansion stroke, the I-stage intake valve is closed. The air in the I-stage cylinder expands to carry out work, pushing the piston to move toward BDC until it reaches BDC. At this time, the II-stage piston is still moving toward TDC until it reaches TDC, and all the residual air in the II-stage cylinder is discharged into the atmosphere. During the I-stage intake and expansion strokes, the nozzle constantly injects micron-level water mist into the cylinder, exchanging heat with the I-stage gas in the cylinder.

During the I-stage exhaust utilization stroke, the II-stage intake valve is opened. The water mist and residual air in the I-stage cylinder are first separated by the water mist filter system, after which the water mist flows into the liquid storage tank, and the residual air flows into the II-stage cylinder to push the II-stage piston to operate. The function of the water mist filtration system is to prevent the water mist from entering the II-stage cylinder, to avoid the heat exchange between the water mist and the air in the II-stage cylinder, and prevent the temperature of the water mist from dropping to the freezing point.

During the I-stage exhaust stroke, when the piston of the I-stage cylinder rises to a certain position, the I-stage exhaust valve is opened, and the II-stage intake valve is closed. The residual air and water mist are discharged into the atmosphere through the I-stage piston movement and reach the TDC directly. At this point, the air in the II-stage cylinder expands to carry out work and pushes the II-stage piston to reach the BDC. The M-TSPE completes a working cycle.

3. Mathematic Model

The modeling of the M-TSPE is mainly composed of three parts: air expansion, heat exchange of water mist, and piston motion. Figure 3 is the schematic diagram of air expansion and water mist heat exchange modeling of the M-TSPE. Make the following assumptions:

(1)

The compressed air is ideal, and the gas flows into or out of the cylinder in quasi-steady flow;

(2)

The M-TSPE has no gas leakage during the working process;

(3)

Ignoring the mass transfer between water mist and air;

(4)

All mechanical components are rigid bodies;

(5)

The pipeline volume connecting the I-stage cylinder and the II-stage cylinder is not considered;

(6)

The throttling effect of water mist filter on air is not considered;

(7)

The volume of water only accounts for 0.09–2.3% of the cylinder volume, so the influence of water mist volume on cylinder volume can be ignored.

Figure 3. Schematic diagram of air expansion and water mist heat exchange modeling.

Figure 3. Schematic diagram of air expansion and water mist heat exchange modeling.

3.1. Air Expansion Model

For the M-TSPE, the intake and exhaust processes are an open system, and the expansion process is a closed system. The I-stage and the II-stage expander working state of gas in the process of change should satisfy the equation of conservation of energy because, for the ideal of compressed air, the gas thermodynamics can change only related to the gas pressure and temperature. Reference [27] solves the energy conservation equation, and the in-cylinder gas temperature with the crankshaft angle can be obtained

$$\frac{d{T}_{\mathrm{I}}{}_{-a}}{d\theta }=(\frac{d{Q}_{\mathrm{I}}{}_{-a}}{d\theta }+\frac{d{Q}_{\mathrm{I}}{}_{-w}}{d\theta }-\frac{d{W}_{\mathrm{I}}{}_{-a}}{d\theta }+h_{\mathrm{I}}{}_{-a,in}{G}_{\mathrm{I}}{}_{-a,in}-h_{\mathrm{I}}{}_{-a,out}{G}_{\mathrm{I}}{}_{-a,out}-u_{\mathrm{I}}{}_{-a}{G}_{\mathrm{I}}{}_{-a})/(m_{\mathrm{I}}{}_{-a}{C}_{a,}{}_v)$$

(1)

$$\frac{d{T}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}=(\frac{d{Q}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}-\frac{d{W}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}+h_{\mathrm{I}}{}_{-a,out1}{G}_{\mathrm{I}}{}_{-a,out1}-h_{\mathrm{II}}{}_{-a,out}{G}_{\mathrm{II}}{}_{-a,out}-u_{\mathrm{II}}{}_{-a}{G}_{\mathrm{II}}{}_{-a})/(m_{\mathrm{II}}{}_{-a}{C}_{a,}{}_v)$$

(2)

where m_I-a and m _II-a are the gas mass in the cylinders of I-stage and II-stage, respectively, Q_I-a and Q_II-a are the heat transfer between the air and the I-stage cylinder wall and the II-stage cylinder wall, respectively, Q_I-w is the heat exchange between air and water mist, G_I-a,out is the mass flow rate of gas discharged from I-stage cylinder; it consists of two parts, one part is the gas mass flow discharged into the II-stage cylinder through the water mist filter, and the other part is the gas mass flow discharged from the I-stage cylinder into the atmosphere.

The mechanical power calculation for each cylinder on the piston can be described

$$\frac{d{W}_{\mathrm{I}}{}_{-a}}{d\theta }=p_{\mathrm{I}}{}_{-a}\frac{d{V}_{\mathrm{I}}{}_{-a}}{d\theta }$$

(3)

$$\frac{d{W}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}=p_{\mathrm{II}}{}_{-a}\frac{d{V}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}$$

(4)

where $\frac{d{V}_{\mathrm{I}}{}_{-a}}{d\theta }$ and $\frac{d{V}_{\mathrm{II}}{}_{-a}}{d(\theta +\pi )}$ are the changes in the working volume of the I-stage and II-stage cylinders with the crankshaft angle, respectively.

The change in gas mass in each cylinder satisfies the mass conservation differential equation

$$\frac{d{m}_{\mathrm{I}}{}_{-a}}{d\theta }=G_{\mathrm{I}}{}_{-a,in}-G_{\mathrm{I}}{}_{-a,out1}-G_{\mathrm{I}}{}_{-a,out2}$$

(5)

$$p_{\mathrm{II}}{}_{-a}{V}_{\mathrm{II}}{}_{-a}=m_{\mathrm{II}}{}_{-a}R{T}_{\mathrm{II}}{}_{-a}$$

(6)

When the pressure difference between the upstream and downstream of the system and the opening area of the valve is determined, the mass flow rate into and out of the M-TSPE can be calculated by referring to the mass flow rate of the shrinking spray [12]. The change rate of the mass of the inlet and outlet I-stage and II-stage cylinders with the crankshaft angle are

$$G_{\mathrm{I}-a,in、out}=\{\begin{array}{l}\begin{array}{l}\frac{1}{w}{S}_{\mathrm{I}}{{}_{-a}}_{,}{}_H(\theta )p_{\mathrm{I}}{{}_{-a}}_{,}{}_H\sqrt{\frac{k}{R{T}_{\mathrm{I}}{{}_{-a}}_{,}{}_H}{(\frac{2}{k+2})}^{\frac{k+1}{k-1}}},\\ \begin{array}{cc}& \end{array}\frac{p_{\mathrm{I}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{I}}{{}_{-a}}_{,}{}_H}\le 0.528\end{array}\hfill \\ \begin{array}{l}\frac{1}{w}{S}_{\mathrm{I}}{{}_{-a}}_{,}{}_H(\theta )p_{\mathrm{I}}{{}_{-a}}_{,}{}_H\sqrt{\frac{2k}{k-1}\frac{1}{R{T}_{\mathrm{I}}{{}_{-a}}_{,}{}_H}[{(\frac{p_{\mathrm{I}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{I}}{{}_{-a}}_{,}{}_H})}^{\frac{2}{k}}-{(\frac{p_{\mathrm{I}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{I}}{{}_{-a}}_{,}{}_H})}^{\frac{k+1}{k}}]},\\ \begin{array}{cc}& \end{array}\frac{p_{\mathrm{I}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{I}}{{}_{-a}}_{,}{}_H}>0.528\end{array}\hfill \end{array}$$

(7)

$$G_{\mathrm{II}-a,in、out}=\{\begin{array}{l}\begin{array}{l}\frac{1}{w}{S}_{\mathrm{II}}{{}_{-a}}_{,}{}_H(\theta +\pi )p_{\mathrm{II}}{{}_{-a}}_{,}{}_H\sqrt{\frac{k}{R{T}_{\mathrm{II}}{{}_{-a}}_{,}{}_H}{(\frac{2}{k+2})}^{\frac{k+1}{k-1}}},\\ \begin{array}{cc}& \end{array}\frac{p_{\mathrm{II}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{II}}{{}_{-a}}_{,}{}_H}\le 0.528\end{array}\hfill \\ \begin{array}{l}\frac{1}{w}{S}_{\mathrm{II}}{{}_{-a}}_{,}{}_H(\theta +\pi )p_{\mathrm{II}}{{}_{-a}}_{,}{}_H\sqrt{\frac{2k}{k-1}\frac{1}{R{T}_{\mathrm{II}}{{}_{-a}}_{,}{}_H}[{(\frac{p_{\mathrm{II}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{II}}{{}_{-a}}_{,}{}_H})}^{\frac{2}{k}}-{(\frac{p_{\mathrm{II}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{II}}{{}_{-a}}_{,}{}_H})}^{\frac{k+1}{k}}]},\\ \begin{array}{cc}& \end{array}\frac{p_{\mathrm{II}}{{}_{-a}}_{,}{}_L}{p_{\mathrm{II}}{{}_{-a}}_{,}{}_H}>0.528\end{array}\hfill \end{array}$$

(8)

where w is the crankshaft angle velocity, and the subscripts H and L are the compressed air in the upstream and downstream, S(θ) is the instantaneous effective cross-sectional area of the inlet and outlet. S(θ) is calculated according to the shape of the channel and the diameter of the valve rod. Reference [28] shows the effective cross-sectional area is calculated as follows

$$S(\theta )=\{\begin{array}{cc}\pi l_v(\theta )\cos \gamma \left[d_f+\frac{l_v(\theta )}{2}\right]\hfill & \hfill ,\hspace{1em}{l}_v(\theta )\le 0.31d_f\\ \frac{\pi }{4}(d_f^2-d_v^2)\hfill & \hfill ,\hspace{1em}{l}_v(\theta )>0.31d_f\end{array}$$

(9)

where l_v(θ) is valve lift, γ is valve cone angle, d_f is flow passage diameter, d_v is stem diameter. The valve lift and distribution phase angle of class I-stage and II-stage are introduced in the following section.

The gas state change of each cylinder satisfies the equation of state

$$p_{\mathrm{I}}{}_{-a}{V}_{\mathrm{I}}{}_{-a}=m_{\mathrm{I}}{}_{-a}R{T}_{\mathrm{I}}{}_{-a}$$

(10)

$$p_{\mathrm{II}}{}_{-a}{V}_{\mathrm{II}}{}_{-a}=m_{\mathrm{II}}{}_{-a}R{T}_{\mathrm{II}}{}_{-a}$$

(11)

The air expansion model of the M-TSPE can be constructed by combining the above equations.

3.2. Heat Exchange Model of Water Mist

In this paper, it is considered that the water mist–air only conducts heat exchange, and it is considered that the conventional spherical water mist with a uniform diameter enters the cylinder and quickly fills the entire cylinder. The pressure of the water mist does not work on the moving piston. The state change of the water mist in the cylinder satisfies the energy conservation equation. Solving the energy conservation equation of the water mist and the temperature of the water mist in the cylinder with the crankshaft angle can be obtained [11]

$$\frac{d{T}_{\mathrm{I}}{}_{-w}}{d\theta }=\frac{1}{m_{\mathrm{I}}{}_{-w}{C}_{w,p}}(h_{\mathrm{I}}{{}_{-w}}_{,in}{G}_{\mathrm{I}}{{}_{-w}}_{,in}-h_{\mathrm{I}}{{}_{-w}}_{,out1}{G}_{\mathrm{I}}{{}_{-w}}_{,out1}-h_{\mathrm{I}}{{}_{-w}}_{,out2}{G}_{\mathrm{I}}{{}_{-w}}_{,out2}-\frac{d{Q}_{\mathrm{I}}{}_{-w}}{d\theta }-u_{\mathrm{I}}{}_{-w}{G}_{\mathrm{I}}{}_{-w})$$

(12)

where m_I-w is the mass of water mist in the I-stage cylinder, and C_w,p is the specific heat capacity of water mist.

The mass flow balance equation of water mist is calculated as follows

$$G_{\mathrm{I}}{}_{-w}=G_{\mathrm{I}}{}_{-w,in}-G_{\mathrm{I}}{}_{-w,out1}-G_{\mathrm{I}}{}_{-w,out2}$$

(13)

where, G_I-w,in is the mass flow rate of water mist entering the cylinder, G_I-w,out1 is the water mist mass flow rate discharged into the filter, G_I-w,out2 is the mass flow of water mist discharged into the atmosphere.

The heat transfer equations of air and water mist in the cylinder are widely used in quasi-isothermal compression/expansion [11,29,30,31,32,33]. This paper is also deduced in detail in the previously published literature [14]. Relevant empirical formulas are given as follows

$$\frac{d{Q}_{\mathrm{I}}{}_{-w}}{d\theta }=c_{w,ht}{S}_{\mathrm{I}}{}_{-w}(\theta )\frac{(T_{\mathrm{I}}{}_{-w}-T_{\mathrm{I}}{}_{-a})}{\theta }$$

(14)

$$S_{\mathrm{I}}{}_{-w}=\frac{6m_{\mathrm{I}}{}_{-w}}{{\rho }_w{d}_{\mathrm{I}}{}_{-w}}$$

(15)

$$c_{w,ht}=\frac{Nu\cdot {\lambda }_w}{d_{\mathrm{I}}{}_{-w}}$$

(16)

$$Nu=2+0.6{\mathrm{Re}}^{1/2}{\mathrm{Pr}}^{1/3}$$

(17)

$$\mathrm{Re}=\frac{{\rho }_a{d}_{\mathrm{I}}{}_{-w}\left|v_{\mathrm{I}}{}_{-a}-v_{\mathrm{I}}{}_{-w}\right|}{h_{a,dv}}$$

(18)

$$v_{\mathrm{I}}{}_{-w}=\sqrt{\frac{4d_{\mathrm{I}}{}_{-w}{\rho }_{w}g}{3{\rho }_a{C}_D}}$$

(19)

$$C_D=\{\begin{array}{ccc}24/\mathrm{Re}& & \mathrm{Re}\le 0.2\\ 18.5/{\mathrm{Re}}^{0.6}& & 0.2<\mathrm{Re}\le 500\\ 0.44& & 500<\mathrm{Re}\le {10}^{-5}\end{array}$$

(20)

where λ_w is the thermal conductivity, d_I-w and v_I-w are the diameter and velocity of the water mist, respectively, and C_D is drag coefficient.

3.3. Piston Motion Model

The commonly used dynamic analysis methods include the Virtual Displacement method, Lagrange method, Newton–Euler method, etc. [34,35]. The theoretical basis of the Lagrange method is the principle of energy conservation, which usually calculates the kinetic energy and potential energy of the system, respectively, and then establishes the Lagrange function. Referring to the dynamics model of the single-crankshaft connecting rod mechanism established in the literature [36], the mechanical analysis of the system’s double-crankshaft rod mechanism is carried out, and the dynamics model is established by using Lagrange’s equation. Figure 4 shows the force diagram of the crankshaft and connecting rod mechanism under the OXY coordinate system.

On the established OXY plane, the crankshaft angle θ and connecting rod angle β of the I-stage expander have the following constraints

$$r\sin \theta =L\sin \beta$$

(21)

Through the above constraints, the relationship between β and θ is

$$\beta ={\sin }^{-1}(\frac{r}{L}\sin \theta )$$

(22)

To obtain the kinetic energy and potential energy, the position of the center of mass of the crankshaft, connecting rod, and piston must be calculated first, and the position of the center of mass of the crankshaft, connecting rod, and piston are considered to be at the geometric center position. The position of the center of mass of the crankshaft is at the origin of the established coordinate system, and the centroid position of the connecting rod and piston head is calculated as follows

$$\{\begin{array}{c}{x}_{\mathrm{I}}{}_{-cr}=r\cos \theta +\frac{1}{2}L\cos {\displaystyle [}{\sin }^{-1}(\frac{r}{L}\sin \theta )],\\ y_{\mathrm{I}}{}_{-cr}=\frac{1}{2}L\sin \left[{\sin }^{-1}(\frac{r}{L}\sin \theta )\right]\\ x_{\mathrm{I}}{}_{-ph}=r\cos \theta +L\cos \left[{\sin }^{-1}(\frac{r}{L}\sin \theta )\right],\\ y_{\mathrm{I}}{}_{-ph}=0\end{array}$$

(23)

$$\{\begin{array}{c}{x}_{\mathrm{II}}{}_{-cr}=r\cos (\theta +\pi )+\frac{1}{2}L\cos \left[{\sin }^{-1}(\frac{r}{L}\sin (\theta +\pi ))\right],\\ y_{\mathrm{II}}{}_{-cr}=\frac{1}{2}L\sin \left[{\sin }^{-1}(\frac{r}{L}\sin (\theta +\pi ))\right]\\ x_{\mathrm{II}-ph}=r\cos (\theta +\pi )+L\cos \left[{\sin }^{-1}(\frac{r}{L}\sin (\theta +\pi ))\right],\\ y_{\mathrm{II}}{}_{-ph}=0\end{array}$$

(24)

The kinetic energy of the M-TSPE system includes the kinetic energy of the flywheel, crankshaft, I-stage, and II-stage connecting rod and the piston head. The system’s kinetic energy is computed as follows

$$\{\begin{array}{c}{E}_{k,fc}=\frac{1}{2}{J}_{fc}{\dot{\theta }}^2\\ \begin{array}{ll}{E}_{k,cr}& =\frac{1}{2}{J}_{cr}({\sin }^{-1}(\frac{r}{L}\sin \theta ){)}^{\prime }2+\frac{1}{2}{m}_{cr}({\dot{x}}_{\mathrm{I}-cr}^2+{\dot{y}}_{\mathrm{I}-cr}^2)+\\ & \frac{1}{2}{J}_{cr}{\left[{\sin }^{-1}(\frac{r}{L}\sin (\theta +\pi ))\right]}^{\prime }{}^2+\frac{1}{2}{m}_{cr}({\dot{x}}_{\mathrm{II}-cr}^2+{\dot{y}}_{\mathrm{II}-cr}^2)\end{array}\\ E_{k,ph}=\frac{1}{2}{m}_{\mathrm{I}-ph}{\dot{x}}_{\mathrm{I}-ph}^2+\frac{1}{2}{m}_{\mathrm{II}-ph}{\dot{x}}_{\mathrm{II}-ph}^2\end{array}$$

(25)

where J_fc is the moment of inertia of the flywheel and the crankshaft, J_cr is the moment of inertia of the connecting rod.

According to the established OXY coordinate system, the plane of the OY axis is the zero reference plane of potential energy. The gravitational potential energy of the connecting rod and piston of the system is expressed as

$$\{\begin{array}{c}{E}_{p,cr}=m_{cr}g{x}_{\mathrm{I}-cr}^{}+m_{cr}g{x}_{\mathrm{II}}{}_{-cr}\\ E_{p,ph}=m_{\mathrm{I}}{}_{-ph}g{x}_{\mathrm{I}}{}_{-ph}+m_{\mathrm{II}-ph}g{x}_{\mathrm{II}-ph}\end{array}$$

(26)

The Lagrange function M_L is calculated as follows

$$\begin{array}{ll}{M}_L& \equiv E_k-E_p\\ & =E_{k,fc}+E_{k,cr}+E_{k,ph}-E_{p,cr}-E_{p,ph}\end{array}$$

(27)

In the M-TSPE system, the forces acting on the two-stage crankshaft connecting rod include F_I-a and F_II-a, the force of gas on the piston in two cylinders; F_I-f and F_II-f, the friction force; T_D, the damping torque; T_L, the load torque, which is provided by the brake. Through the above force on the two crankshaft connecting rods, the sum of virtual work can be obtained as

$$d{W}_v=(F_{\mathrm{I}}{}_{-a}-F_{\mathrm{I}}{}_{-f})(-d{x}_{\mathrm{I}}{}_{-ph})+(F_{\mathrm{II}}{}_{-a}-F_{\mathrm{II}}{}_{-f})(-d{x}_{\mathrm{II}}{}_{-ph})-(T_D+T_L)d\theta$$

(28)

The air force of I-stage and II-stage pistons is calculated as follows

$$F_{\mathrm{I}}{}_{-a}=(p_{\mathrm{I}}{}_{-a}-p_e)A_{\mathrm{I}}{}_{-ph}$$

(29)

$$F_{\mathrm{II}}{}_{-a}=(p_{\mathrm{II}}{}_{-a}-p_e)A_{\mathrm{II}}{}_{-a}$$

(30)

where A_I-ph and A_II-ph are the stressed areas of I-stage and II-stage piston heads, respectively.

In this paper, the two cylinders friction generated by the piston ring group are considered, which are expressed as

$$F_{\mathrm{I}}{}_{-f}=c_f{p}_{\mathrm{I}}{}_{-a}\pi d_{\mathrm{I}}{}_{-ph}{d}_r\mathrm{sgn}({\dot{x}}_{\mathrm{I}}{}_{-ph})$$

(31)

$$F_{\mathrm{II}}{}_{-f}=c_f{p}_{\mathrm{II}}{}_{-a}\pi d_{\mathrm{II}}{}_{-ph}{d}_r\mathrm{sgn}({\dot{x}}_{\mathrm{II}}{}_{-ph})$$

(32)

where d_r is piston ring thickness.

The damping torque T_D generated by coupling the output shaft with the brake is expressed as [12]

$$T_D=c_d\dot{\theta }$$

(33)

where c_d is the damping coefficient.

The formulas about the piston motion model can be represented by an independent variable θ, so the generalized coordinates of the M-TSPE system take θ, then according to the Lagrange equation

$$\frac{d}{dt}(\frac{\partial M_L}{\partial \dot{\theta }})-\frac{\partial M_L}{\partial \theta }=d{W}_v$$

(34)

The M-TSPE system’s output power is calculated as follows

$$P_{out}=\frac{T_{out}\times n}{9550}$$

(35)

where T_out is the average output torque of the M-TSPE, n is crankshaft.

The high-speed movement of the water flow hitting the nozzle produces the water mist. Reference [37] gives the expression of water mist energy consumption as

$$W_{\mathrm{I}}{}_{-w}=p_w\frac{G_{\mathrm{I}}{}_{-w,in}}{{\rho }_w}{t}_w$$

(36)

where t_w is the spray time of water mist in I-stage cylinder.

The energy efficiency of the M-TSPE with spray is calculated as follows

$$\eta =\frac{{\displaystyle {\int }_0^{2\pi }{p}_{\mathrm{I}}{}_{-a}{V}_{\mathrm{I}}{}_{-a}d\theta +{\displaystyle {\int }_0^{2\pi }{p}_{\mathrm{II}}{}_{-a}{V}_{\mathrm{II}}{}_{-a}d\theta }}}{m_{\mathrm{I}}{}_{-a,in}R{T}_{\mathrm{I}}{}_{-a,in}\ln (p_{\mathrm{I}}{}_{-a,in}/p_{a,out})+W_{\mathrm{I}}{}_{-w}}$$

(37)

4. Simulation Calculation and Result Analysis

4.1. Simulation Parameter Setting

The above mathematical model can be written as a θ function, in which the crankshaft angle can be calculated by the product of the crankshaft angular velocity and time. Therefore, the overall mathematical model can be constructed by t. Since the analytical solution cannot be obtained by direct integration, the above mathematical model can only be solved using numerical integration using a computer. The mathematical model is constructed in the MATLAB/Simulink simulation environment. The Runge–Kutta fourth-order method with fixed steps is applied to calculate and solve the above system of equations by integration. Figure 5 shows the dynamic modeling framework of the M-TSPE structure. The heat transfer model of water mist mainly provides the heat transfer of water mist for the air expansion model of the I-stage cylinder. The air expansion models of the I-stage and II-stage cylinders interact and provide in-cylinder pressure for the piston motion model. The crankshaft angle is obtained by quadratic integration of the piston motion model, and the overall connection is established through the crankshaft angle.

The structural dimensions of the M-TSPE should be determined before performing the simulation calculations and analyzing the results. The structural dimensions of the I-stage expander can be selected from reference [14]. The structural dimensions of the I-stage expander and II-stage expander are different, including cylinder diameter, valve structure, and inlet and exhaust channel diameter. According to the conclusion obtained in the literature [38], the working efficiency of the two-stage expansion pneumatic engine reaches its peak when the cylinder diameter ratio reaches about 1.4. Therefore, the II-stage cylinder diameter can be selected regarding this conclusion. The valve structure of the expander and the diameter of the intake and exhaust passages are constrained by the second-class cylinder diameter, which can be set by comparing the cylinder diameter. The simulation parameters are shown in

Table 1. Main parameters of the M-TSPE.

Parameter	Value	Parameter	Value
I-stage cylinder diameter	85 mm	II-stage cylinder diameter	127 mm
I-stage inlet and exhaust channel diameter	12 mm	II-stage inlet and exhaust channel diameter	18 mm
I-stage inlet and exhaust valve stem diameter	7 mm	II-stage inlet and exhaust valve stem diameter	7 mm
I-stage intake valve lift	3.5 mm	II-stage intake valve lift	6 mm
I-stage exhaust valve lift	5.72 mm	The valve cone angle	45°
I-stage inlet opening and closing angle	0–90°	II-stage exhaust opening and closing angle	0–180°
Radius of the crankshaft	44 mm	Connecting rod mass	0.526 kg
Connecting rod length	139 mm	Piston mass	1.011 kg
Damping coefficient	0.145	Piston ring friction coefficient	0.05
Moment of inertia of flywheel and crankshaft	0.43 kg·m2	Inlet air temperature	343 K
Atmospheric temperature	293 K	Atmospheric pressure	0.1013 MPa

. Secondly, we need to determine the valve-timing of the M-TSPE. According to the simulation parameters given in Table 1, we also need to determine the opening and closing phase angle of the II-stage intake valve and the opening and closing phase angle of the I-stage exhaust valve. The opening angle of the II-stage intake valve is set as 180°, and the closing angle can be selected through simulation. Because the simulated M-TSPE adopts a pressure compensation valve mechanism that can reduce the inertia force of the valve, it can work under a high pressure of 2–3 MPa [28].

In the actual process, the opening and closing of the valve must have a buffer gradient process, similar to a parabolic shape. If the I-stage exhaust valve is opened after the II-stage intake, the valve is completely closed. In that case, the residual high-pressure air will be compressed during the exhaust process of the I-stage cylinder, resulting in large compression resistance. In order to avoid large compression resistance caused by gas compression, the phase angle of the II-stage intake process and the I-stage exhaust process should form a certain overlap. To determine the II-stage intake valve-timing, the output performance of the M-TSPE without spray is simulated under different II-stage intake valve timings, as shown in Figure 6. When the closing angle of the II-stage inlet valve is about 280°, the output power/efficiency of the M-TSPE without spray reaches the peak value, indicating that the gas has a strong power capacity and high gas utilization rate at this point. Under the condition that the intake pressure is 2.5 MPa and the load torque is 110 N·m, when the closing angle of the II-stage intake valve is 360°, the output power of the M-TSPE without spray is 7.45 kW, the output efficiency is 49.53% (under this condition, the M-TSPE without spray works in the same way as the TSPE [24,25]). When the closing angle of the stage II intake valve is 280°, the output power of the M-TSPE without spray is 11.74 kW, the output efficiency is 56.11%, the output power is increased by 57.58%, and compared to this, the output efficiency is increased by up 13.28%. The above analysis shows that the reasonable utilization of the residual high-pressure air of the I-stage cylinder can effectively improve the output performance of the M-TSPE. When conducting the performance study of the M-TSPE in the next step, the setting of the cam lift curve is shown in Figure 7.

4.2. Performance Analysis

To better prove the effectiveness of the proposed M-TSPE based on spray heat transfer, we built and compared two expander models. The two expander models are as follows: (1) the M-TSPE without spray model; (2) the M-TSPE with spray model. The heat exchange model has been applied by many researchers in the isothermal compression/expansion model. Combined with the calculation of the water mist state in my previous paper [14], it is concluded that the water mist is in a laminar flow state in the I-stage cylinder of the M-TSPE. Therefore, it can be assumed that the change in water mist temperature is uniform during the heat exchange between the water mist and air. The boundary conditions of the two models are set as shown in

Table 2. Boundary condition setting of two models.

Parameter	The NT-TSPE without Spray	The NT-TSPE with Spray
GI-a,in	3 MPa
TI-a,in	343 K
TL	150 N·m
GI-w,in	-	0.06 kg/s
dI-w	-	5 × 10−5 m
TI-w,in	-	373 K

.

Figure 8 shows that the crankshaft velocity of the two models all have a start-up stage and a stable fluctuation stage. In the stable fluctuation stage, the crankshaft velocity of the M-TSPE fluctuates with a certain amplitude, and the crankshaft velocity fluctuation of the M-TSPE with spray is smaller than that of the M-TSPE without spray, which means that the operation of the M-TSPE with spray is more stable. The average crankshaft velocity of the M-TSPE without spray is 905 r/min, and the average crankshaft velocity of the M-TSPE with spray is 1022 r/min, an increase of 13.93% compared with the average crankshaft velocity. The results show that the heat transfer between the water mist and air makes the M-TSPE more capable of carrying out work on the piston, so the output speed of the M-TSPE with spray is faster than that of the M-TSPE without spray.

Figure 9 shows the p-V diagram of the two models in the stable fluctuation stage. During the I-stage intake stroke, the I-stage cylinder gas pressure of the M-TSPE without spray is greater than that of the M-TSPE with spray. This is because, in one cycle, the charging time of the M-TSPE with spray is shorter than that of the M-TSPE without spray, resulting in a smaller charging volume of the M-TSPE with spray, so the cylinder pressure of the M-TSPE without spray is greater than that of the M-TSPE with spray. During the I-stage expansion stroke, the I-stage cylinder gas pressure of the M-TSPE with spray is greater than that of the M-TSPE without spray. This is because the water mist provides heat for the air in the expansion stage, which reduces the energy loss caused by the heat release, converting more of the air’s energy into the mechanical energy acting on the piston. During the I-stage exhaust utilization stroke, the I-stage cylinder gas pressure of the M-TSPE with spray is greater than that of the M-TSPE without spray. This is because the air temperature in the cylinder decreases more slowly after the water mist mixes with the air in the intake and expansion stage, which is much larger than the air temperature in the cylinder of the M-TSPE without spray. Hence, more of the air in the cylinder of the M-TSPE with spray is converted into mechanical energy, which hinders the movement of the piston to the TDC. Figure 9 shows that the area enclosed by p_II-a and V_II-a of the M-TSPE with spray is larger than that enclosed by p_II-a and V_II-a of the M-TSPE without spray. This shows that the heat exchange between the water mist and air in the I-stage cylinder can effectively improve the working ability of the gas in the I-stage cylinder to the piston.

Figure 10 shows the variation in the output torque with the crankshaft angle in the stable fluctuation stage. The main difference between the M-TSPE with and without spray is marked with boxes. When the crankshaft angle is within the range of 0°~80°, the output torque of the M-TSPE without spray is greater than that of the M-TSPE with spray. This is because the intake volume of the M-TSPE without spray is greater than the M-TSPE with spray during a cycle. When the crankshaft angle ranges from 180° to 290°, the output torque of the M-TSPE with spray is greater than that of the M-TSPE without spray. The above analysis shows that the output torque amplitude of the M-TSPE with spray is smaller than that of the M-TSPE with spray under one cycle, which indicates that the M-TSPE with spray operates more stably.

Figure 11 shows the air temperature variation in the I-stage cylinder with the crankshaft angle. When the crankshaft angle is 180°~360°, the air temperature in the without spray M-TSPE’s I-stage cylinder is less than 273 K. When the air enters the II-stage cylinder, and it needs to push the piston to expand to carry out work, this results in cooler air temperatures in the II-stage cylinders. In long-term operation, the M-TSPE without spray causes icing, lubricating oil freezing, and other adverse effects. The M-TSPE with spray has a small gas temperature difference in the I-stage cylinder, and the average temperature of the gas in the I-stage cylinder is calculated to be 321 K. This shows that the heat transfer of the water mist and air can significantly increase the air temperature in the cylinder. When the crankshaft angle is 180°~280°, the average temperature of the M-TSPE with spray I-stage cylinder is 319 K, and the air with the average temperature of 319 K enters the secondary cylinder to expand and conduct work, which will not cause the air temperature of the II-stage cylinder to be too low. It can avoid freezing and lubricating oil freezing in the M-TSPE with spray.

Figure 12 shows the output power/efficiency comparison of the M-TSPE without spray and the M-TSPE with spray at different load torques. When the load torque increases continuously, the output power of the two models first increases to the peak value and then decreases continuously. The trend of the output power can be explained by Equation (35). The output power increases with the load torque because the rate of decrease n is less than the rate of increase T_out. In the decreasing stage of output power with load torque, the decreasing rate of n is greater than the increasing rate of T_out. When the load torque is 150 N·m, the output power of the M-TSPE without spray is 14.22 kW, and the output power of the M-TSPE with spray is 16.08 kW, an increase of 13.08%. When the load torque is in the range of 100~200 N·m, the output efficiency of the M-TSPE with spray is lower than that of the M-TSPE without spray because the power consumption of spray is considered.

4.3. Influence of Air/Water Mass Ratio and Water Mist Particle Size on Performance of the M-TSPE

To better analyze the influence of the mass of the water mist injected into the cylinder on the performance of the M-TSPE, we can define the ratio of the air mass entering the I-stage cylinder to the injected water mist mass in one cycle, i.e.,

$$M_{a/w}=\frac{m_{\mathrm{I}}{}_{-a,in}}{m_{\mathrm{I}}{}_{-w,in}}$$

(38)

Without changing the conditions of air expansion, by changing the water mist flow rate in the I-stage cylinder, the ratio of the air quality into the I-stage cylinder and the water mist quality into the cylinder in one cycle is reached. Then the influence of spray quality on the performance of the two-stage piston expander is analyzed. Figure 13 shows the influence of water mist quality on the performance of the M-TSPE. As shown in Figure 13, when M_a/w increases, the average air temperature in the I-stage cylinder decreases gradually, and the output power and efficiency also decrease. This is because the decrease in water mist quality in one cycle leads to the decrease in heat transfer between the water mist and air, which affects the output performance of the M-TSPE.

The boundary conditions of the M-TSPE with spray are set as shown in Table 2. The influence of the particle size of the water mist on the performance of the M-TSPE is analyzed by changing the particle size of the water mist, as shown in Figure 14. As shown in Figure 14, with the increase in particle size, the in-cylinder temperature of the I-stage cylinder increases, and the output power decreases. This is because the same quality of water mist flows into the cylinder, and the smaller the water mist particle size, the greater the surface area of the water mist, the water spray and air heat transfer into the cylinder more fully, the output of the two levels of the piston expander goes at a faster speed, resulting in a reduction in the quality of the water mist entering the I-stage cylinder in one cycle. The above analysis shows that the smaller the particle size of the water mist, the greater the heat exchange between the water mist and the air, and the greater the output power of the M-TSPE.

5. Conclusions

This paper proposes the M-TSPE based on spray heat transfer technology. Firstly, the air expansion model, water mist heat exchange model, and piston motion model of the expander are established, and the intake and exhaust valve timing of the M-TSPE are determined. Secondly, under the given boundary conditions, the M-TSPE with spray and the M-TSPE without spray are compared and discussed. Finally, parameter analysis is carried out on the air/water mass ratio and water mist particle size. Some conclusions are made as follows:

• Compared with the TSPE without spray, the output performance of the M-TSPE without spray is better under reasonable setting of the intake/exhaust valve timing. The output power and efficiency of the M-TSPE without spray increased by 57.58% and 13.28%, respectively.
• When the crankshaft velocity is in the stable stage, the crankshaft velocity amplitude of the M-TSPE with spray is smaller than that of the M-TSPE without spray, indicating that the output velocity of the M-TSPE with spray is more stable. The crankshaft velocity of the M-TSPE with spray increased by 13.93%.
• Through the comparative analysis of the p-V diagram, output torque, and in-cylinder air temperature under the single cycle of the M-TSPE, the reasons for the improved output performance of the M-TSPE with spray are disclosed.
• When the load torque is 150 N·m, the output power of the M-TSPE with spray is 16.08 kW, which is 13.08% higher than that of the M-TSPE without spray, the output efficiency of the M-TSPE with spray is lower than that of the M-TSPE without spray.
• Improving the quality of the water mist and reducing the water mist particle size during a cycle can improve the output performance of the expander.

In this study, the output performance of the M-TSPE is analyzed in detail, which provides an important reference for further realizing the efficient application in small-scale CAES systems. A further task will be to optimize the structural parameters of the M-TSPE and design an algorithm to control the water spray flow to match the expansion heat in the expansion process which will further realize its good application in small-scale CAES systems.