An Optimum Design for a Fast-Response Solenoid Valve: Application to a Limaçon Gas Expander

Abstract

Organic Rankine Cycle (ORC)–based small-scale power plants are becoming a promising instrument in the recent drive to utilize renewable sources and reduce carbon emissions. But the effectiveness of such systems is limited by the low efficiency of gas expanders, which are the main part of an ORC system. Limaçon-based expansion machines with a fast inlet control valve have great prospects as they could potentially offer efficiencies over 50%. However, the lack of a highly reliable and significantly fast control valve is hindering its possible application. In this paper, a push–pull solenoid valve is optimized using a stochastic optimization technique to provide a fast response. The optimization yields about 56–58% improvement in overall valve response. A performance comparison of the initial and optimized valves applied to a limaçon expander thermodynamic model is also presented. Additionally, the sensitivity of the valve towards a changing inlet pressure and expander rotor velocity is analyzed to better understand the effectiveness of the valve and provide clues to overall performance improvement.

1. Introduction

The world is currently at the center of an unprecedented and complex energy crisis affecting worldwide economies due to the overreliance on fossil fuels [1]. This has prompted a renewed global campaign to decarbonize and adopt more sustainable options [2]. In addition to the introduction of more photovoltaic and wind power plants, a reliable utilization of low- and medium-temperature heat sources such as biomass combustion, geothermal reservoirs, and industrial waste heat should be ensured to address this accruing crisis and limit global warming [3,4]. Another approach is to increase the overall efficiency of traditional energy conversion systems as eliminating them completely is not feasible. The typical efficiencies of conventional power generation systems range from 30% to 40% and can be increased up to 55% by using combined cycle arrangements [5]. Therefore, at least 45% of useful energy is wasted in the form of heat. This waste heat from diverse industries including power generation is mostly available at temperatures less than 200 °C [6]. The Organic Rankine Cycle (ORC) can potentially be an effective instrument to tap into this wasted energy and convert it into electricity. The ORC uses a low boiling point organic fluid giving it the capacity to operate at temperatures as low as 73.3 °C [7,8,9]. Thus, ORC systems can also be employed in renewable-based energy conversion as most renewable heat sources are available at temperatures of 90–500 °C [10]. In that context, the ORC is preferable to both gas and steam cycles when heat sources and plant sizes are low to medium in range as pointed out by Macchi [11].

ORC systems are often subject to high pressure ratios and limited mass flow as pointed out by Lemort et al. [12]. Additionally, ORC-based micro–power plants typically less than 100 kW utilize low-grade heat sources, which most often results in multi-phase fluid flow. This condition results in the formation of fluid droplets making them unsuitable for turbine technology. In addition, turbines operate at high rotating speeds, and their manufacturing costs are also high for these arrangements [13]. Positive displacement expanders are more suitable in that regard as they can operate at low speeds and high-pressure ratios and can handle low flow rates under multi-phase conditions [12,14].

Positive displacement expanders have thus gained popularity recently for small-scale power generation and waste heat recovery applications based on ORC. Experimental investigations of ORC-based power plants using positive displacement expanders of various types such as scroll [15,16,17,18,19], rotary vane [20,21], swash-plate [22], and screw [23,24,25] have been realized of late. As the primary component of the thermal-to-power conversion process of the ORC, the expander plays a vital role in the overall performance of the ORC [26,27]. Recently, a notable effort has been directed towards improving expander performance. For instance, Wang et al., in their study, showed that the specific power output of a twin-screw air expander could be increased by 10.8% by adopting two-phase expansion [28]. Apart from the operational optimization of the expander, an improvement in the design aspects of the expander has also been studied. Li et al. investigated the effect of profile modification of a scroll expander on the isentropic efficiency and power output [29]. In a similar work by Fabio et al., the authors showed that the power output of a scroll expander could be increased by 25% using dual intake port technology [30]. These studies are primarily concentrated on popular expander technologies like scroll, screw, and vane. However, expanders with various embodiments of limaçon machines have largely been overlooked. Although limaçon machines were proposed long ago, the problem of rotor-housing interference has hindered their industrial development until now. However, Sultan’s works provided a solution to this constraint and paved the way for their possible application in efficient thermal-to-power conversion [31,32]. Sultan et al. also provided optimized designs for limaçon machines in [14,33] and illustrated that a cam-operated inlet valve could theoretically improve its overall efficiency by a minimum of 24% [34]. However, cam-operated valves, which operate at a fixed angular position of the rotor, lack flexibility and controllability making them impractical for variable operating conditions or loads. It is evident that a control valve capable of regulating the fluid flow irrespective of the rotor position would be more beneficial in that regard. In that context, Phung et al. proposed a push–pull solenoid valve for a limaçon gas expander and demonstrated the valve response during machine operation [35]. However, the valve is not optimized in conjunction with the application of the limaçon gas expander as the speed of response of the valve was less compared to that of the machine. Additionally, the effect of the valve response on the performance of the expander is worth investigating. Therefore, there are ample incentives to optimize a valve for a gas expander, which would provide faster response and improve the overall performance. Moreover, a fast-acting valve would provide better controllability and performance to the operation of the expander system.

In recent times, efforts have been made to optimize the valve performance for various applications. For instance, Zhong et al. used the NSGA-II genetic algorithm to optimize a high-speed on/off valve (HSV) and found that the optimization reduced the solenoid volume by 47.1% and improved the closing and opening response performance by 43% and 14.8%, respectively [36]. In a similar approach adopted by Qingtong et al., a reduction of 9% and 17% in the closing and opening response times, respectively, of an HSV was reported [37]. Wu et al. used a different algorithm called the particle swarm optimization method to optimize the design parameters of a hollow plunger-type solenoid HSV [38]. In a recent study by Yu et al., the authors used a back propagation neural network (BPNN) to optimize an HSV [39]. They reported a 15.67% and 22.49% reduction in drive and joule energy, respectively; however, the opening time response was increased by 6.24%.

In this study, we optimized a push–pull solenoid valve to provide a faster response. For this purpose, a cost function composed of valve opening and closing delays is minimized using the Simultaneous Perturbation Stochastic Approximation (SPSA) approach. A total of 12 valve parameters are optimized in this process. A performance comparison between the optimized and the initial design is also presented. In addition, a theoretical simulation of the limaçon expander is provided to analyze the effect of the inlet valve response on its performance indices. In the next section, some background and mathematical models of the valve and the limaçon expander are presented for a better understanding of the overall system and its components.

2. Limaçon Expander with Inlet Valve

Limaçon gas expanders, a variant of a rotary positive displacement machine, are constructed on the basis of Limaçon curves. Figure 1 shows a basic structure of a limaçon expander composed of a rotor that can slide and rotate inside the chamber housing due to the expansion of the compressed working fluid. The expander is equipped with a push–pull solenoid valve to regulate the fluid flow.

2.1. Push–Pull Solenoid Valve

This study aims to optimize a solenoid valve utilizing the lever-plunger push–pull arrangement as shown in Figure 2. The valve is composed of two solenoids to aid in opening and closing, respectively. Solenoid 2 and solenoid 1 are used to open and close the valve, respectively. The valve is designed to be normally open with the aid of a spring of stiffness $k_s$ that keeps the valve in the open position as depicted in Figure 2. This design aims to produce a fast response and ensures that the expander connected to the valve remains in operation even if any unforeseen solenoid malfunction occurs. A lever mechanism with an arm ratio of $\frac{a_1}{a_2}=b$ is used to amplify the force generated by the solenoids. The pivot point of the lever is placed closer to the valve spool to have an arm ratio $>1$. A mathematical model of the valve is developed as follows.

The system can be divided into mechanical and electrical subsystems for ease of modeling. The mechanical subsystem composed of a plunger, lever, and valve can be considered a forced vibration with damping; thus, the motion of the system can be represented by the Lagrange equation of motion as below:

$$\frac{d}{dt}(\frac{\partial KE}{\partial {\dot{x}}_v})+\frac{\partial PE}{\partial x_v}+\frac{\partial DE}{\partial {\dot{x}}_v}=F_{tot}$$

(1)

where $x_v$ is the vertical displacement of the valve spool; $KE$, $PE$, and $DE$ are the kinetic, potential, and damping energy, respectively. These energies can be derived as follows:

$$\begin{array}{}\mathrm{(2a)}& KE=\frac{1}{2}{m}_s{x}_s^2+\frac{1}{2}{m}_l{x}_l^2+\frac{1}{2}{m}_v{x}_v^2+\frac{1}{2}{I}_l{\dot{{\theta }_l}}^2\mathrm{(2b)}& PE=\frac{1}{2}{k}_s{({\delta }_s+x_v)}^2+m_{s}g{x}_s+m_{l}g{x}_l-m_{v}g{x}_v\mathrm{(2c)}& DE=\frac{1}{2}C{\dot{x_v}}^2\end{array}$$

where $m_s$, $m_l$, and $m_v$ are the masses of the plunger, lever, and valve spool, respectively,

• $x_s$ is the vertical displacement of the plunger,
• $x_l$, $I_l$, and ${\theta }_l$ are the displacement, inertia, and rotation of the lever at its center of gravity,
• $k_s$ is the spring stiffness,
• ${\delta }_s$ is the initial spring deflection,

and C is the damping coefficient. To represent $x_s$, $x_l$, and ${\theta }_l$ in terms of $x_v$, the movement of the plunger, lever, and spool is utilized as shown in Figure 3.

Using fundamental geometric formulas, the following relationships are drawn:

$$\begin{array}{c}\hfill \begin{array}{cc}& \hfill \frac{x_s}{a_1}=\frac{x_v}{a_2}\approx {\theta }_l\hfill \\ & \hfill \Rightarrow x_s=b{x}_v\hfill \\ & \hfill \mathit{And}\frac{x_l}{\frac{a_1-a_2}{2}}=\frac{x_v}{a_2}\hfill \\ & \hfill \Rightarrow x_l=\frac{a_1-a_2}{2a_2}{x}_v=(b-1)\frac{x_v}{2}\hfill \end{array}\end{array}$$

(3)

Equation (2) can be rewritten by substituting $x_s$, $x_l$, and ${\theta }_l$ in terms of $x_v$ as follows:

$$\begin{array}{}\mathrm{(4a)}& KE=\frac{b^2}{2}{m}_s{x}_v^2+\frac{{(b-1)}^2}{8}{m}_l{x}_v^2+\frac{1}{2}{m}_v{x}_v^2+\frac{1}{2a_2^2}{I}_l{\dot{x_v}}^2\mathrm{(4b)}& PE=\frac{1}{2}{k}_s{({\delta }_s+x_v)}^2-m_{v}g{x}_v+m_{s}gb{x}_v+m_{l}g(b-1)\frac{x_v}{2}\mathrm{(4c)}& DE=\frac{1}{2}C{\dot{x_v}}^2\end{array}$$

Differentiating and substituting into Equation (1) gives the following:

$$\begin{array}{c}\left[b^2{m}_s+{(\frac{b-1}{2})}^2{m}_l+m_v+\frac{I_l}{a_2^2}\right]\ddot{x_v}+\left[k_s{\delta }_s+k_s{x}_v+b{m}_{s}g+(\frac{b-1}{2})m_{l}g-m_{v}g\right]+C\dot{x_v}\hfill \\ \hfill =b{F}_{sol}+F_p+F_{sur}\\ \Rightarrow \left[b^2{m}_s+{(\frac{b-1}{2})}^2{m}_l+m_v+\frac{I_l}{a_2^2}\right]\ddot{x_v}+C\dot{x_v}+k_s{x}_v\hfill \\ \hfill =b{F}_{sol}+F_p+F_{sur}-k_s{\delta }_s-\left[b{m}_{s}g+(\frac{b-1}{2})m_{l}g-m_{v}g\right]\\ \Rightarrow \left[M\right]\ddot{x_v}+C\dot{x_v}+k_s{x}_v=b{F}_{sol}+F_p+F_{sur}-k_s{\delta }_s-\left[W\right]\hfill \\ \Rightarrow \ddot{x_v}=\frac{1}{M}\left[b{F}_{sol}+F_p+F_{sur}-C\dot{x_v}-k_s{x}_v-k_s{\delta }_s-W\right]\hfill \end{array}$$

(5)

where M and W are the equivalent mass and weight of the valve system;

• $F_{sol}$ is the force generated by solenoids;
• $F_p$ is the pressure force acting on the valve disk; and
• $F_{sur}$ is the force exerted on the valve disk or spool when in contact with the seat or stopper.

The solenoid force for this particular two solenoid system can be found from the equation below courtesy of the work by Yuan et al. [40]:

$$F_{sol}=\sum _j\frac{{\beta }_{s_j}}{2}{\left(\frac{i_{s_j}}{x_{s_j}+d_0}\right)}^2$$

(6)

where $i_{s_j}$ is the current in solenoid $j=1,2$;

• $x_{s_j}$ is the plunger displacement for solenoid $j=1,2$;
• ${\beta }_{s_j}$ and $d_0$ are constants given by Yuan et al. [40]. The displacements and velocity due to solenoids 1 and 2 are given by:
• Solenoid 1: $x_{s_1}=b{x}_v$ and $\dot{x_{s_1}}=b\dot{x_v}$;
• Solenoid 2: $x_{s_2}=b(S-x_v)$ and $\dot{x_{s_2}}=-b\dot{x_v}$; where S is the stroke of the valve spool.

The pressure force $F_p$ can be written as $F_p=\Delta P{C}_d{A}_e$; where $\Delta P=P_{in}-P_a$ is the pressure difference between the inlet and valve antechamber; $C_d$ is the drag coefficient; and $A_e$ is the dynamic effective area of the inlet port subject to the valve operation given by Tymer et al. [41] as shown in Equation (7).

$$A_e=\sqrt{\frac{1}{\frac{1}{{\left(0.85\pi g_o(S-x_v)\right)}^2}+\frac{1}{A_o^2}}}$$

(7)

where $g_o$ and $A_o=\frac{\pi }{4}{g}_o^2$ are the diameter and area of the expander inlet orifice, respectively. The contact force $F_{sur}$ due to contact with the seat and stopper can be found as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{sur}& =-k_{seat}(x_v-S)-C_{seat}\dot{x_v}\mathit{if}{x}_v>S\hfill \\ & =-k_{stop}\left(x_v\right)-C_{stop}\dot{x_v}\mathit{if}{x}_v<0\hfill \\ & =0\mathit{if}0\le x_v\le S\hfill \end{array}\end{array}$$

where $k_{seat}$ and $k_{stop}$ are the stiffness of the seat and stopper, respectively, and $C_{seat}$ and $C_{stop}$ are the damping coefficients of the seat and stopper, respectively. The equation for the electrical subsystem can be written as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v_{s_j}& =i_{s_j}{R}_{s_j}+\frac{d}{dt}\left(L_{s_j}{i}_{s_j}\right)\hfill \\ & =i_{s_j}{R}_{s_j}+L_{s_j}\frac{d{i}_{s_j}}{dt}+i_{s_j}\frac{d{L}_{s_j}}{dt}\hfill \\ \hfill \Rightarrow \frac{d{i}_{s_j}}{dt}& =\frac{1}{L_{s_j}}\left[v_{s_j}-i_{s_j}{R}_{s_j}-i_{s_j}\frac{d{L}_{s_j}}{dt}\right]{|}_{j=1,2}\hfill \end{array}\end{array}$$

(8)

where $v_{s_j}$ is the voltage, $i_{s_j}$ is the current, and $L_{s_j}$ is the inductance of the solenoids $j=1,2$, respectively.

Inductance $L_{s_j}$ is given by Yuan et al. as $L_{s_j}=\frac{{\beta }_{s_j}}{x_{s_j}+d_0}{|}_{j=1,2}$; thus, $\frac{d{L}_{s_j}}{dt}$ can be found as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{L}_{s_j}}{dt}& =\frac{d{L}_{s_j}}{d{x}_{s_j}}\frac{d{x}_{s_j}}{dt}\hfill \\ & =\frac{-{\beta }_{s_j}}{{(x_{s_j}+d_0)}^2}\dot{x_{s_j}}{|}_{j=1,2}\hfill \end{array}\end{array}$$

Substituting the value of $L_{s_j}$ and $\frac{d{L}_{s_j}}{dt}$ into Equation (8), we obtain the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d{i}_{s_j}}{dt}& \hfill =\frac{x_{s_j}+d_0}{{\beta }_{s_j}}\left[v_{s_j}-i_{s_j}{R}_{s_j}+i_{s_j}\frac{{\beta }_{s_j}\dot{x_{s_j}}}{{(x_{s_j}+d_0)}^2}\right]{|}_{j=1,2}\hfill \end{array}\end{array}$$

(9)

The nonlinear differential equations of (5) and (9) are sufficient to describe the valve actuation system. However, it is essential to analyze the valve about the motion of the expander crankshaft. Therefore, these equations can be rewritten in terms of the expander crankshaft rotational angle $\theta$ and valve spool displacement $x_v$ to model the operation of the valve with the connected expander as follows:

$$\begin{array}{}\mathrm{(10)}& \frac{d\dot{x_v}}{d\theta }=\frac{1}{\omega M}\left[\frac{b{\beta }_{s_1}}{2}{\left(\frac{i_{s_1}}{b{x}_v+d_0}\right)}^2-\frac{b{\beta }_{s_2}}{2}{\left(\frac{i_{s_2}}{b(S-x_v)+d_0}\right)}^2+\Delta P{C}_d{A}_e+F_{sur}-C\dot{x_v}-k_s{x}_v-k_s{\delta }_s-W\right]\hfill \mathrm{(11)}& \frac{d{i}_{s_1}}{d\theta }=\frac{b{x}_v+d_0}{\omega {\beta }_{s_1}}\left[v_{s_1}-i_{s_1}{R}_{s_1}+i_{s_1}\frac{b{\beta }_{s_1}\dot{x_v}}{{(b{x}_v+d_0)}^2}\right]\mathrm{(12)}& \frac{d{i}_{s_2}}{d\theta }=\frac{b(S-x_v)+d_0}{\omega {\beta }_{s_2}}\left[v_{s_2}-i_{s_2}{R}_{s_2}-i_{s_2}\frac{b{\beta }_{s_2}\dot{x_v}}{{(b(S-x_v)+d_0)}^2}\right]\end{array}$$

Note: $\ddot{x_v}=\frac{d\dot{x_v}}{dt}=\frac{d\dot{x_v}}{d\theta }\frac{d\theta }{dt}=\frac{d\dot{x_v}}{d\theta }\omega$ and $\frac{d{i}_{s_j}}{dt}=\frac{d{i}_{s_j}}{d\theta }\frac{d\theta }{dt}=\frac{d{i}_{s_j}}{d\theta }\omega$; where $\omega$ is the angular velocity of the crank shaft.

2.2. Limaçon Gas Expander

The dual lobe rotor chord $p_1{p}_2$ of length 2L (m) rotates and slides about the limaçon pole z at an angle $\theta$ as depicted in Figure 1. The center of the rotor travels along the circumference of the limaçon base circle. The thermodynamic model that is used to describe the behavior of such a gas expander can be expressed by the following set of highly nonlinear stiff differential equations: The working fluid flowing into the chamber through the inlet valve expands inside the chamber. The rate of change of fluid volume, $V\left({\mathrm{m}}^3\right)$, and density, $\rho (\mathrm{gm}/{\mathrm{m}}^3)$, due to the expansion in each cycle can be described with the following set of differential equations:

$$V\nabla \rho =\nabla (m_i-m_o)-\rho \nabla V$$

(13)

$$\rho VT\nabla S=(h_i-h_c)\nabla m_i-(h_c-h_o)\nabla m_o$$

(14)

where m (gm) and h (J/gm) are the mass and enthalpy of fluid and the subscripts i, c, and o denote inlet, chamber, and outlet, respectively; T(K) and S are the chamber temperature and entropy, respectively. The mass flow rate $\nabla m_i$ (gm/sec) depends on the effective port area, $A_e$ (${\mathrm{m}}^2$), as defined in (7); the pressure difference, $\Delta P$ (kPa); and the fluid density, $\rho$ ($\mathrm{gm}/{\mathrm{m}}^3$). The following equation can represent this:

$$\nabla m_i=f(A_e,\Delta P,\rho )=f(x_v,\Delta P,\rho )$$

(15)

Equations (13) and (14) are solved iteratively in conjunction with the valve model until the below condition is met:

$$\sqrt{({({\rho }_c-{\rho }_0)}^2+{(S_c-S_0)}^2)}\le \epsilon$$

(16)

where 0 and c denote the start and end of the cycle and $\epsilon$ is the precision error for the simulation. The expansion of the fluid inside the chamber converts the potential energy of the fluid into usable mechanical energy, which can be described as follows:

$$E_c={\int }_{t_0}^{t_c}{P}_c\nabla vdt$$

(17)

The isentropic efficiency is the ratio of the energy converted to the total potential energy in the fluid:

$${\eta }_i=\frac{E_c}{(h_i-h_o){\rho }_i{V}_i}$$

(18)

where ${\rho }_i$ ($\mathrm{gm}/{\mathrm{m}}^3$) is the fluid density at the inlet orifice and $V_i$$\left({\mathrm{m}}^3\right)$ is the fluid volume inside the chamber as the valve is closed. The volumetric efficiency or filling factor given below is the ratio of the actual mass admitted into the chamber to the possible mass that could populate the chamber volume in a given operating cycle.

$${\eta }_v=\frac{m_i}{{\rho }_i{V}_i}$$

(19)

3. Optimization Process

The primary performance indicator of the valve is its speed of response. Figure 4 shows the ideal operation of the inlet valve of the gas expander. At the start of every half cycle of the rotor rotation, the valve should remain open to allow fluid to enter the chamber till ${\theta }_c$, where it is closed to cut off the flow and opened again at ${\theta }_p$ for the next half cycle. However, the valve will be subject to some amount of delay due to the time required to build up currents in the solenoid coils and actuate the system. The optimization process aims to reduce these closing and opening delays denoted as $D_o$ and $D_c$, where $D_c={\theta }_{c_{actual}}-{\theta }_{c_{ideal}}$ and $D_o={\theta }_{p_{actual}}-{\theta }_{p_{ideal}}$. Based on these two performance indices of the valve, the following cost function, $f(\Psi )$, is minimized:

$$f_c(\Psi )=\sqrt{D_o^2+D_c^2}$$

(20)

where $\Psi$ denotes the design vector comprising different design parameters of the valve as given below:

$$\Psi ={\left[m_v{m}_s{m}_l{I}_{l}b{a}_2{v}_{s_1}{v}_{s_2}{R}_{s_1}{R}_{s_2}{k}_{s}C\right]}^T$$

(21)

The cost function of Equation (20) is minimized with the aid of the SPSA optimization algorithm. In this arrangement, the design parameters are updated in an iterative process as per Equation (22) until a certain number of iterations or a specified goal threshold of the cost function is met.

$$\begin{array}{cc}\hfill {\Psi }_i^{j+1}& ={\Psi }_i^j-a_k^j\frac{f_c^{k^{+}}-f_c^{k^{-}}}{2C_k^j{\Delta }^j}\hfill \\ \hfill & \hfill ={\Psi }_i^j-a_k^j\frac{f_c({\Psi }^j+C_k^j{\Delta }^j)-f_c({\Psi }^j-C_k^j{\Delta }^j)}{2C_k^j{\Delta }^j}\hfill \end{array}$$

(22)

here i denotes the individual position of the parameters in the design vector $\Psi$;

• j is the iteration step number where $1<j<N$, N is the number of iterations;
• $f_c^{k^{+}}$ and $f_c^{k^{-}}$ are the cost function evaluation at two adjacent perturbed points;
• ${\Delta }^j$ is the random perturbation vector of the same size as $\Psi$ assigned either +1 or −1 values in a binary Bernoulli distribution;
• $C_k^j$ is the perturbation size at each iteration;
• and $a_k^j$ is the adaptive step size at each iteration;

A limiting condition is imposed as follows to keep the parameters within the design limits:

$${\Psi }_i^{j+1}=\left\{\begin{array}{cc}\hfill \hfill & \hfill {\Psi }_{i,max}if{\Psi }_i^{j+1}>{\Psi }_{i,max}\hfill \\ \hfill \hfill & \hfill {\Psi }_{i,min}if{\Psi }_i^{j+1}<{\Psi }_{i,min}\hfill \end{array}\right.$$

(23)

The values of $C_k^j$ and $a_k^j$ are updated at each iteration as follows:

$$\begin{array}{cc}\hfill C_k^j& =\frac{C_o}{j^{\gamma }}\hfill \\ \hfill a_k^j& =\frac{A}{{(j+A+1)}^{\alpha }}\hfill \end{array}$$

(24)

where $C_o$, A, $\gamma$, and $\alpha$ are constants of the values 0.2, 100, 0.2, and 0.94, respectively. These values are determined through a grid search approach, updated within a range as follows: $C_o\approx \{0.05:0.01:0.2\}$, $A\approx \{10:6:100\}$, $\gamma \approx \{0.01:0.01:0.2\}$, and $\alpha \approx \{0.1:0.06:1\}$. A grid is constructed with these values as per Equation (25a). The best values of these constants are found by evaluating the SPSA for each point in the grid, and the constants giving the minimum $f_c$ are selected according to Equation (25b).

$$\begin{array}{}\mathrm{(25a)}& C_{grid}={\left[C_{o_1}^{n1}{A}_1^{n2}{\gamma }_1^{n3}{\alpha }_1^{n4}\right]}^T\hfill \mathrm{(25b)}& C_{best}=arg{\displaystyle \underset{C_{grid}}{min}}{f}_c\end{array}$$

A process flow diagram is depicted in Figure 5 representing the computational process of the optimization procedure adopted in this paper.

4. Results

The computational results are presented as follows: first, the valve parameters are optimized to achieve a faster response, and the optimized valve is applied to a thermodynamic model of a limaçon gas expander to evaluate its performance.

4.1. Optimized Valve Dynamics

The initial and optimized design vector parameters are stipulated in

Table 1. Design vector parameters.

Parameters	Initial Value	Minimum Value	Maximum Value	Optimized Value
Mass of valve, $m_v$ ($\mathrm{kg}$)	34 × 10−3	20 × 10−3	50 × 10−3	48.22 × 10−3
Mass of plunger, $m_s$ ($\mathrm{kg}$)	45 × 10−3	30 × 10−3	70 × 10−3	30 × 10−3
Mass of lever, $m_l$ ($\mathrm{kg}$)	30 × 10−3	10 × 10−3	50 × 10−3	15.6 × 10−3
Inertia of lever, $I_l$ ($\mathrm{kg}.{\mathrm{m}}^2$)	2 × 10−4	1 × 10−4	3 × 10−4	1.75 × 10−4
Arm ratio, b	3	2	5	4.6
Length of the shorter lever arm, $a_2$ (m)	0.015	0.005	0.003	0.029
Solenoid 1 input voltage, $v_{s_1}$ ($\mathrm{volt}$)	100	100	500	388.3
Solenoid 2 input voltage, $v_{s_2}$ ($\mathrm{volt}$)	75	20	300	150.38
Solenoid 1 resistance, $R_{s_1}$ ($\mathrm{ohm}$)	40	50	100	94.98
Solenoid 2 resistance, $R_{s_2}$ ($\mathrm{ohm}$)	78	50	100	91.92
Spring stiffness, $k_s$ ($\mathrm{N}/\mathrm{m}$)	10	5	15	15
Damping coefficient, C	50.45	20	75	26.82

.

The SPSA algorithm minimizes the cost function as shown in Figure 6a.

The cost function value at each iteration is found by iteratively solving the set of differential Equations (10)–(12) with respect to the expander angle $\theta$ where ${\theta }_c$ and ${\theta }_p$ are set as $90$° and $180$°, respectively. For this simulation, parameters other than that of Table 1 are kept constant as follows: $\omega$ = 800 rpm, $P_{in}$ = ${10}^3$ kPa, $P_a$ = 0.6 × $P_{in}$ kPa, $\beta$ = $2.64\times {10}^{-4}$, $d_0$ = 0.0017, S = $5\times {10}^{-3}$ m, $k_{seat}$ = $k_{stop}$ = $k_s$ × 10,000 N/m, $C_{seat}$ = $C_{stop}$ = $C$ × 1000, and $g_o$ = $25\times {10}^{-3}$ m. It is seen that the cost function value starting at $46.12$° reduces with each iteration and becomes minimum at a value of $19.5$° which is around half that of the initial estimation. Similarly, the components of the cost function, namely, the closing and opening delays, are reduced from an initial delay of $33.1$° and $32.1$°, respectively, to an optimized delay of $14.3$° and $13.3$°, respectively, as shown in Figure 6b. The reduction is about 56.80% and 58.57% for closing and opening delays, respectively. The reduction in the delay is due to the increased currents in the solenoid coils producing more force as seen in Figure 7.

In the optimized design, solenoids 1 and 2 have peak currents of 62.96 A and 16.35 A, whose values were at 30.28 A and 9.61 A at the start of the iteration. It is also seen that the current in solenoid 1 is greater compared to that in solenoid 2. This is because solenoid 1 is used for closing the valve against the pressure force in the inlet manifold, whereas solenoid 2 actuates the opening operation during which the pressure force and spring tension act in the direction of valve displacement. Similar to the currents, the force generated by the solenoids follows the changes in solenoid currents as shown in Figure 7b where the optimized version of solenoid 1 produces a force with a steep rising peak (4028 N) required for fast closing. The velocity of the valve spool is shown in Figure 8a where the optimized valve accelerates rapidly peaking at 0.06 m/s at the cutoff and 0.04 m/s at the pass angles.

Figure 8b provides a comparison of the valve area variations for initial and optimized valve designs. Although both the initial and optimized version of the valve has delays, it has been considerably reduced in the optimized version making it faster.

A faster response means less fluid leakage through the valve openings. Figure 9 shows a comparison of the amount of fluid leakage at different inlet pressures between the initial and optimized designs. It is evident that at a low pressure of 800 kPa, the leakage is small at 1.72 L and 2.23 L for optimized and initial designs, respectively. However, at a higher pressure of 1000 kPa and over, the leakage increases to 2.27 L and 3.47 L, respectively, and becomes almost saturated. The optimized valve performs better in terms of curtailing fluid leakage, which gives it better controllability.

4.2. Application to a Limaçon Gas Expander

The sensitivity of the initial and optimized valve designs to the expander performance is analyzed under varying rotor speeds and input pressures. For this purpose, the thermodynamic model of the gas expander described in Section 2.2 is simulated analytically along with the valve model. The parameters used to simulate the limaçon expander are stipulated in

Table 2. Data for the limaçon expander.

Parameters	Value
Rotor chord length ($2L$)	$92.8$ mm
Base circle radius (r)	$7.98$ mm
Limaçon aspect ratio ($R_l=\frac{r}{L}$)	0.171
Housing-rotor clearance ($C_h$)	$0.71$ mm
Clearance ratio ($C_l=\frac{C_h}{L}$)	0.0153
Design coefficient (c)	1.73
Depth of rotor housing (H)	$60.38$ mm
Fluid type	Air
Outlet pressure ($P_{out}$)	100 $\mathrm{k}$$\mathrm{Pa}$
Inlet port start angle	$-24.9$°
Outlet port start angle	140°
Inlet port end angle	$-5.9$°
Outlet port end angle	175°
Inlet port length	$13.35$ mm
Outlet port length	$21.47$ mm

.

Figure 10 shows the sensitivity of the valve designs to the isentropic efficiency of the expander. As seen in Figure 10a, both designs’ efficiency increases with increased rotor velocity and inlet pressure. However, the optimized version produces higher isentropic efficiencies at different velocities and pressures. In addition, it indicates that the isentropic efficiency increases with increased operating velocity and pressure. The contours of isentropic efficiencies for both designs are shown in Figure 10b. It is seen that to obtain the same isentropic efficiency, the optimized valve requires the expander to operate at less rotor speed and input pressure. Additionally, a 2.24% increase in isentropic efficiency is achieved with the optimized valve.

As a limaçon expander is a volumetric machine, it is necessary to discuss the sensitivity of the isentropic efficiency on the pressure and volume ratio. Figure 11 shows the variation in the isentropic efficiency of the expander with the optimized valve at different pressure and volume ratios. It is seen that the efficiency is higher at higher pressure and volume ratios. The highest isentropic efficiency of 56.67% corresponds to a pressure ratio of 18 and volume ratio of 8. Therefore, the expander performs better in terms of isentropic efficiency at higher pressure and volume ratios.

Figure 12 depicts the volumetric efficiency or filling factor variation for the two designs. For a particular value of rotor velocity, the filling factor is always less for the optimized valve compared to that of the initial design as seen in Figure 12a. This is attributed to the faster response of the optimized design, which ensures that the volume of fluid entering the chamber is always less than that for the initial design. It can also be inferred that the filling factor generally has a decreasing trend for increased rotor velocity and input pressure. Figure 12b represents the contours of filling factors for different rotor speeds and pressures. It is seen that a higher value of the filling factor is achieved at a lower speed and pressure. In contrast, the optimized valve achieves the same filling factor at a lower speed. The optimized valve produces an average 5.04% reduction in the filling factor across all velocity and pressure ranges compared to the initial design. At this stage, it is necessary to discuss the nature of the change in the filling factor at different inlet pressures. As seen in Figure 12a, at a low pressure and low rotor velocity, the filling factor is over 1 for both designs. This is attributed to the fluid leakage through the valve due to the time delay during closing operation. This makes the numerator in Equation (19) larger than the denominator. However, as the pressure increases, the air density increases rapidly as shown in Figure 13 which increases the denominator, thereby reducing the filling factor.

Figure 14 shows the flow of mass for both designs at different input pressures and rotor speeds. The mass flow for the optimized valve is less regardless of the pressure and speed compared to the initial design. It is also seen in Figure 14a that the mass flow increases with pressure but decreases with speed. Moreover, the same mass flow is achieved at a lower speed with the optimized valve as seen in Figure 14b. The optimized design reduces the mass flow by about 5% compared to the initial design, thereby saving costly compressed fluid.

Figure 15 depicts a comparison between the energy converted with the initial and optimized valve. As the optimized valve is faster, less fluid is allowed into the chamber for expansion, and the energy is less for this case as seen in Figure 15a. Therefore, to produce the same amount of energy with the optimized valve as that of the initial design, the expander has to be operated at a higher pressure as seen in Figure 15b.

Similarly, the power output is also less with the optimized design as seen in Figure 16. The power and energy produced with the optimized design is about 2.91% less compared to the initial design. In light of the above discussions, it is evident that the optimized design provides a faster response, which improves both the isentropic and volumetric efficiency of the limaçon gas expander. However, a faster response means the flow of fluid into the chamber is blocked earlier, which contributes to a lower value of mass flow, energy, and power. As efficiency improvement of the expander is our primary goal, this reduction in energy and power is not concerning. Moreover, the energy and power can be increased by increasing the input pressure, which further improves the efficiency. It should, however, be noted that the maximum isentropic efficiency achieved with the optimized valve is 56.67%, which is lower than popular positive displacement machines such as scroll (68.3%) [15] and screw (64%) [23]. This is because this study aims to optimize the speed of response of the inlet valve only and provides a snapshot of how this optimization affects the expander performance. The optimization of the expander is planned for the next stage, which will include optimization of the expander design and operational parameters as stipulated in Table 2. This should be sufficient to increase the isentropic efficiency further.

5. Conclusions

Positive displacement gas expanders require reliable and fast-response inlet valves to operate these machines efficiently. In this study, the speed of response of a push–pull solenoid valve for a limaçon gas expander is optimized using the SPSA technique. It is found that the optimized valve is about 56.80% and 58.57% faster compared to the initial design during closing and opening operations, respectively. Later, the effect of the optimized valve on the performance of a limaçon gas expander is studied. It is found that the isentropic and volumetric efficiencies of the expander could be improved by 2.24% and 5.04%, respectively. In addition, the optimized valve can achieve similar performance at a lower input pressure than the initial design. This is crucial as it means a reduced load on the compressor of the ORC system, which will improve the overall performance of the system. The study provides good insight into the performance improvement of a gas expander using an inlet control valve. In the later stage of this research, the gas expander could be optimized to improve the overall efficiency of the ORC system.