Influence of Dead Volume Ration on the Thermodynamic Performance of Free-Piston Stirling Machines

Abstract

The excellent thermal performance, quiet operation, and fuel flexibility of free-piston Stirling machines enable their broad application potential in sectors such as aerospace, distributed power generation, and industrial waste heat utilization. The impact of structural parameters on the output characteristics of the free-piston Stirling engine was investigated using a parametric MATLAB model based on an isothermal thermodynamic approach. Parameters such as the dead volume ratios (χ_H, χ_K, χ_R), temperature ratio τ, sweep volume ratio k, piston phase angle a_dr, and minimum pressure angle θ were evaluated for their effects on the dimensionless power Z. The results indicate that the dead volume ratio in the cold space χ_K has the most significant influence on system performance, followed by the hot space χ_H, while the regenerator χ_R exhibits a comparatively weaker effect. All three parameters demonstrate the existence of optimal design intervals. The dimensionless power Z decreases monotonically with increasing dead volume ratio. Moreover, this decline is intensified at higher temperature ratios τ, indicating that the influence of dead volume becomes more significant under larger τ values. The interaction between these parameters can be described by $Z=0.0037{\tau }^2-0.0045\tau +0.0021$. An excessively large sweep volume ratio k tends to degrade the system’s output performance. An empirical correlation between k and the dimensionless power can be established as follows $Z=1.53(1-e^{-3.37k})+0.01$. A moderate increase in the piston phase angle a_dr and a reduction in the minimum pressure angle θ contribute to improved system performance by enlarging the p-v diagram area and enhancing the utilization of gas expansion. The relationship between a_dr and the dimensionless power Z follows a linear trend, expressed as $Z=0.341a_{dr}-0.2104$. A well-defined functional relationship exists between the minimum pressure angle θ and the dimensionless power output Z, which can be expressed as $Z=2.18\times {10}^{-4}{\theta }^2-0.0261\theta +0.7065$. A coupling regulation mechanism and design strategy have been developed to facilitate the coordinated optimization of multiple parameters in free-piston Stirling engines, which delivers theoretical guidance that is expected to support the engineering implementation of next-generation, high-performance Stirling technologies.

1. Introduction

The free-piston Stirling machines (FPSMs) have attracted increasing attention in the fields such as aerospace, distributed energy systems, and low-grade waste heat recovery due to their high efficiency, low noise, and adaptability to various heat sources. Stirling engines are widely recognized for their potential to achieve high thermal efficiency and low emissions [1,2]. A substantial body of literature has established that their performance is strongly influenced by key structural parameters [1,3,4]. However, findings regarding the role of certain parameters, such as dead volume, remain uncertain. In recent years, both domestic and international researchers have conducted extensive investigations into the influence mechanisms of various structural parameters on the dimensionless power, a key indicator representing the output performance of the Stirling engines. Gschwendtner and Bell [5] investigated the impact of dead volumes on the output power and efficiency of α-type Stirling engines. Their findings revealed that additional “passive” dead volumes, which do not participate in heat exchange, can induce a pressure phase shift, thereby increasing the p-v work output, which contradicts previous conclusions in the literature [6,7]. According to Kongtragool and Wong wises [8], the engine’s network was determined exclusively by the dead volumes, whereas the heat input and overall efficiency were jointly influenced by both the regenerator’s effectiveness and the presence of the dead volumes. Harrod and Mago [9] applied the first law of thermodynamics to examine the variation trends of total heat input, network output, thermal efficiency with respect to dead volume and regenerator effectiveness. Additionally, using the second law of thermodynamics, they investigated the changes in total entropy generation within the cycle, which particularly focused on the effects of dead volume and regenerator efficiency on the Stirling engine performance. The results indicated that Stirling engines achieve higher network output and thermal efficiency when the dead volume ratio was low and the regenerator effectiveness was high. Qu et al. [10] highlighted that the dead volume ratio was among the most challenging parameters to select in design, as both excessively large and excessively small dead volume ratios lead to reductions in output power and efficiency. Alfarawi and AL-Dadah [11] validated a computational fluid dynamics model of a 500 W γ-type Stirling engine to investigate the effects of phase angle and dead volume variations on engine performance, which revealed that the dead volume associated with the connecting tube negatively impacts the indicated work, indicating the existence of an optimal tube diameter. Yang et al. [12] employed the Schmidt model to conduct a coupled analysis of multiple factors affecting β-type Stirling engines, focusing on the influence of the dead volume ratio on power output, which indicated that an increase in dead volume ratio leads to a reduction in engine power. Puech and Tishkova [13] conducted a thermodynamic analysis of the Stirling engines featuring both linear and sinusoidally varying volumes. In these engines, the regenerator functions as an internal heat exchanger, enabling high-efficiency operation. Employing an isothermal model to analyze the network and the heat stored in the regenerator revealed that, while a perfect regenerator maintains Carnot efficiency regardless of dead volume, this dead volume drastically amplifies the efficiency reduction caused by imperfect regeneration by significantly decreasing the network without a proportional reduction in the regenerator’s heat storage. In summary, structural parameters exert multidimensional coupled effects on the thermodynamic performance of the Stirling engines. Particularly under high temperature differentials and realistic non-isothermal boundary conditions, nonlinear interactions among these parameters become significant, necessitating a comprehensive consideration for optimized design.

The thermodynamic performance of the free-piston Stirling engines is significantly influenced by the structural parameters, particularly the dead volume ratios—including those of the hot end, cold end, and the regenerator [14], the sweep volume ratio [15], the piston phase angle [16], and the minimum pressure angle [17]. These factors play a critical role in regulating output power and efficiency [18,19,20]. In the context of waste heat recovery from Power-to-Methane (PtM) systems, Stirling engines—especially free-piston and thermoacoustic types—have shown promising potential for utilizing low-temperature waste heat (below 100 °C) generated during electrolysis and biomethanation [21]. Experimental studies have demonstrated that thermoacoustic Stirling engines can operate efficiently at heat source temperatures as low as 49–100 °C, achieving output powers up to 1.64 kW and thermal efficiencies of around 8.2%. These systems benefit from having few or no moving parts, which enhances reliability and reduces maintenance—a significant advantage for integration into PtM systems where operational stability is critical. Moreover, the adaptability of Stirling engines to fluctuating and low-grade heat sources makes them suitable for recovering waste heat from biological methanation and electrolysis processes, which typically emit heat at temperatures below 70 °C. The integration of Stirling-based waste heat recovery can improve the overall energy efficiency of PtM systems by converting excess thermal energy into electricity, thereby reducing the net power input required for electrolysis

Previous research has suggested that a moderate amount of the dead volume may enhance system performance under certain operating conditions; however, findings remain inconsistent, and the underlying theoretical explanations are not yet fully developed. This study aims to systematically investigate the effects of key structural parameters—including the dead volume ratio, the sweep volume ratio, the piston phase angle, and the minimum pressure angle—on the dimensionless power output of the Stirling systems, based on an isothermal thermodynamic model coupled with numerical simulations. The work further seeks to identify the optimal parameter ranges and characterize their performance sensitivities. These studies focus on elucidating the functional roles of the dead volumes in different engine regions, their mutual coupling effects, and the influence of dynamic parameters on the coordinated optimization of the system’s p-v process, which aims to clarify the performance limits and degradation mechanisms under various design parameter configurations. Future research should focus on optimizing Stirling engine configurations—such as γ-type and thermoacoustic designs—for low-temperature applications, exploring cost-effective materials, and enhancing regenerator effectiveness to further improve performance and reduce capital costs. Combining Stirling engines with thermal energy storage or phase change materials could also provide a viable pathway for continuous power generation and better alignment with intermittent renewable energy sources.

2. Mathematical Model

This study employs an analytical thermodynamic model based on the isothermal assumption, integrated with parametric modeling and numerical simulation on the MATLAB (2020a) platform. The influences of various parameters—including the dead volume ratios (χ_H, χ_K, χ_R), the sweep volume ratio k, the piston phase angle a_dr, and the minimum pressure angle θ—on the dimensionless power output Z are systematically investigated. The model maintains the temperature gradients at the hot and cold ends of each component while sequentially simulating the regulatory effects of various parameters on the gas expansion and compression processes, thereby evaluating their impact on the system’s work output capacity. Through multiple comparative simulations and characteristic point extraction, optimal design intervals are identified, which are further validated and contextualized through comparisons with existing literature, enhancing the study’s systematic rigor and reliability.

2.1. Model Assumptions

Based on the ideal isothermal cycle, the following assumptions are adopted for the modeling process [22]:

(1)

Perfect regeneration is assumed, with no thermal losses; the regenerator is considered to have 100% effectiveness.

(2)

No flow resistance is considered for the working fluid; pressure losses within the system are neglected, and instantaneous pressure is assumed to be uniform throughout the cycle.

(3)

The working fluid behaves as an ideal gas, following the ideal gas law pV = MRT.

(4)

No mass leakage occurs during operation; the mass of the working fluid remains constant.

(5)

The volumes of the compression and expansion chambers vary sinusoidally over time.

(6)

The compression and expansion processes are isothermal, and the thermodynamic cycle is divided into five distinct processes, each occurring at a constant temperature.

2.2. Model Establishment

A schematic diagram of the Stirling cycle system, where E denotes the expansion chamber, and H, R, and K represent the heater, regenerator, and cooler, respectively. C stands for the compression chamber, as shown in Figure 1.

Assuming that the heater and the expansion chamber share the same temperature, T_E = T_H, and similarly, the cooler and the compression chamber have equal temperature, T_C = T_K. The regenerator temperature is taken as the arithmetic mean of the heater and cooler temperatures, given by T_R = (T_H + T_K)/2. Based on this assumption, the expansion chamber and the heater together constitute the high-temperature region, while the compression chamber and the cooler define the low-temperature region. Accordingly, the equivalent volumes of the high- and low-temperature zones, denoted as V_eH and V_cK, are expressed as Equations (1) and (2):

$$V_{eH}{\displaystyle \frac{1}{2}}{V}_E(1+\cos x)+{\chi }_H{V}_E$$

(1)

$$V_{cK}={\displaystyle \frac{1}{2}}{V}_E(1-\cos x)+{\displaystyle \frac{1}{2}}\kappa V_E\left[(1+\cos (x-a_{dr})\right]-V_L+{\chi }_K{V}_E$$

(2)

The parameter V_L represents the overlap volume resulting from the stroke intersection of the displacer and the power pistons, and is calculated as Equation (3):

$$V_L={\displaystyle \frac{1}{2}}{V}_E(1+\cos a_v)+{\displaystyle \frac{1}{2}}\kappa V_E(1-\cos (a_v-a_{dr}))$$

(3)

Substituting Equation (3) into Equation (2) yields Equation (4):

$$V_{cK}=-{\displaystyle \frac{1}{2}}{V}_E(\cos x+\cos a_v)+{\displaystyle \frac{1}{2}}\kappa V_E\left[(\cos a_v-\cos a_{dr})+\cos (x-a_{dr})\right]+{\chi }_K{V}_E$$

(4)

Based on the definition of the dead volume ratio, which can be expressed as Equation (5):

$$\left\{\begin{array}{c}{\chi }_H=V_{HD}/V_E\\ {\chi }_K=V_{KD}/V_C\\ {\chi }_R=V_{RD}/V_R\end{array}\right.$$

(5)

In Equation (5), χ_H denotes the dead volume ratio of the heater; χ_K represents the dead volume ratio of the cooler; and χ_R corresponds to the dead volume ratio of the regenerator.

The mass of the working fluid M satisfies the principle of mass conservation as Equation (6):

$$M_T=M_{eH}+M_{cK}+M_R$$

(6)

According to the ideal gas law, the following relation holds:

$$M_T={\displaystyle \frac{p_e{V}_{eH}}{R{T}_e}}+{\displaystyle \frac{p_c{V}_{cK}}{R{T}_c}}+{\displaystyle \frac{p_d{V}_{RD}}{R{T}_r}}$$

(7)

Since the pressure is assumed uniform throughout the cycle, the following relation applies in Equation (8):

$$p_e=p_c=p$$

(8)

Based on Equations (6)–(8), the expression for pressure p is derived as Equation (9):

$$p={\displaystyle \frac{MR{T}_e}{[V_{eH}+2V_{RD}/(1+\tau )+V_{cK}/\tau ]}}$$

(9)

Substituting Equations (1), (4) and (5) into Equation (9) and further simplifying yields p as Equation (10):

$$p={\displaystyle \frac{MR{T}_e}{[E^{\prime }-D^{\prime }\cos (x-\theta )]}}$$

(10)

The variables E and E′ satisfy Equation (11):

$$\left\{\begin{array}{l}{E}^{\prime }={\displaystyle \frac{V_E}{2\tau }}E\\ E=\tau (1+2{\chi }_H)+{\displaystyle \frac{4\tau }{1+\tau }}{\chi }_R+2{\chi }_K+\cos a_v+k(\cos a_v-\cos a_{dr})\end{array}\right.$$

(11)

where D′, F′ and G′ are defined as Equations (12)–(14).

$$\left\{\begin{array}{l}{D}^{\prime }={\displaystyle \frac{V_E}{2\tau }}D\\ D=[{(F^{\prime })}^2+{(G^{\prime })}^2){]}^{1/2}\end{array}\right.$$

(12)

$$\left\{\begin{array}{l}{F}^{\prime }={\displaystyle \frac{V_E}{2\tau }}F\\ F=1+\tau +k\cos a_{dr}\end{array}\right.$$

(13)

$$\left\{\begin{array}{l}{G}^{\prime }={\displaystyle \frac{V_E}{2\tau }}G\begin{array}{}\end{array}\\ G=k\sin a_{dr}\end{array}\right.$$

(14)

The cycle work W_CE is given by Equation (15):

$$W_{CE}={\displaystyle {\int }_0^{2\pi }pd{V}_{eH}}+{\displaystyle {\int }_0^{2\pi }pd{V}_{cK}}$$

(15)

Simplifying yields:

$$W_{CE}=MR{T}_E{\displaystyle \frac{2\pi \tau k}{D}}\left[{\displaystyle \frac{E}{{(E^2-D^2)}^{1/2}}}-1\right]\sin (\theta -a_{dr})$$

(16)

The cycle efficiency η corresponds to the Carnot efficiency is:

$$\eta ={\displaystyle \frac{W_{CE}}{N_{E}MR{T}_E}}$$

(17)

Since the maximum power p_max is:

$$p_{\max }=2MR\tau T_E/V_E(E-D)$$

(18)

Therefore, the maximum cycle work W_CE is given by Equation (19):

$$W_{CE}=\left[p_{\max }{V}_E(E-D)/2\pi \right](N_F+N_C)$$

(19)

After rearrangement, W_CE becomes:

$$W_{CE}=MR{T}_E\left.\left\{{\displaystyle \frac{2\pi \tau }{D}}\right.({\displaystyle \frac{E}{{(E^2-D^2)}^{1/2}}}-1)\sin \theta -{\displaystyle \frac{2\pi \tau }{D}}({\displaystyle \frac{E}{{(E^2-D^2)}^{1/2}}}-1)\sin \theta +{\displaystyle \frac{2\pi \tau k}{D}}\left[{\displaystyle \frac{E}{{(E^2-D^2)}^{1/2}}}-1\right]\sin (\theta -a_{dr})\right\}$$

(20)

The dimensionless power, denoted as Z, is defined as Equation (21):

$$Z=W_{CE}/p_{\max }{V}_T$$

(21)

where V_T = V_E + V_C.

Substituting Equations (18) and (20) into Equation (21) yields the expression for the dimensionless power Z as Equation (22):

$$Z=(E-D)(N_E-N_C)/[2\tau (1+k)]$$

(22)

where N_E and N_C is:

$$\left\{\begin{array}{l}{N}_E={\displaystyle \frac{2\pi \tau }{D}}({\displaystyle \frac{E}{\sqrt{E^2-D^2}}}-1)\sin \theta \\ N_C={\displaystyle \frac{2\pi \tau k}{D}}({\displaystyle \frac{E}{{(E^2-D^2)}^{1/2}}}-1)\sin (\theta -a_{dr})\end{array}\right.$$

(23)

Equation (21) eliminates the influence of the cycle pressure p; therefore, the dimensionless power Z depends solely on parameters such as the dead volume ratio χ, temperature ratio τ, and sweep volume ratio k.

The flow chart is shown as below in Figure 2:

2.3. Model Calibration and Parametric Sensitivity Analysis

Although the model in this study is based on isothermal assumptions, to ensure its reliability in practical applications, we performed preliminary calibration and parametric sensitivity validation through the following steps:

Firstly, the value ranges of key parameters in the model (such as dead volume ratio, sweep volume ratio, phase angle, etc.), as shown in

Table 1. Structural parameters.

Parameters	Symbol	Value Range
Temperature Ratio	τ	1–5
Volume Ratio	k	0.5–3.5
Piston Phase Angle	adr	55–90°
Volume Phase Angle	av	90°
Minimum Pressure Angle	θ	40–90°
Dead Volume Ratio of the Regenerator	χr	0–2
Dead Volume Ratio of the Hot Region	χh	0–2
Dead Volume Ratio of the Cold Region	χk	0–2

, are primarily referenced from the design empirical values of typical free-piston Stirling machines in existing literature [10,11,15,16], which ensures that the simulation conditions cover parameter intervals commonly encountered in practical engineering.

Secondly, The study employs systematic parametric sensitivity analysis to identify the parameters most sensitive to output performance (such as the cold-end dead volume ratio χ_k and the piston phase angle a_dr). This analysis itself constitutes a form of structural calibration because it reveals that parameters require precise control in actual design, thereby indirectly verifying the model’s practicality in identifying key variables.

3. Results and Discussion

All simulation results in this section are based on the isothermal model established in Part 2. To evaluate the model’s validity, we qualitatively compare the simulation results with typical experimental trends reported in the literature, verifying whether the model can capture the fundamental patterns of how Stirling machine performance varies with key parameters.

3.1. Structural Parameters of the Free-Piston Stirling Machine

The structural parameters of the free-piston Stirling machine include the temperature ratio, volume ratio, piston phase angle, volume phase angle, minimum pressure angle, and dead volume ratio. The specific parameters and their respective ranges are shown in Table 1.

3.2. Effect of Dead Volume Ratio on Dimensionless Power

When the dead volume ratios are set as χ_R = 1, χ_K = 0.4, χ_H = 0.6, the influence of any one variable on the dimensionless power Z is investigated while keeping the other two fixed. The corresponding variation curves are presented in Figure 3, which illustrates the effects of dead volume ratios at three distinct locations—the cold region χ_K, the hot region χ_H, and the regenerator χ_R—on the dimensionless power output of the Stirling system. All curves exhibit an unimodal trend, indicating that each type of the dead volume possesses an optimal value that maximizes the system’s power output. However, significant differences are observed among the three curves in terms of peak magnitude, location, and trend, reflecting distinct underlying mechanisms by which each dead volume influences the thermodynamic processes within the system.

As the dead volume ratio in the cold region χ_K increases, the dimensionless power Z rises rapidly, reaching a peak at a critical point before declining sharply. For example, when χ_K increases from 0.0 to 0.4, Z increases from 0.22 to 0.30, peaking at χ_K = 0.2 with Z = 0.30, and then sharply declines to Z = 0.26 at χ_K = 0.8. This trend indicates that an appropriate amount of dead volume in the cold region facilitates gas buffering and optimizes the compression process, whereas excessive dead volume leads to phase mismatches and reduced compression efficiency. Therefore, precise control of χ_K is essential. The curve of the dead volume ratio in the hot region χ_H exhibits a relatively gentle variation with a lower peak value, For example, when χ_H increases from 0.0 to 1.6, Z increases from 0.24 to 0.28, peaking at χ_H = 0.8 with Z = 0.28, and then gradually declines to Z = 0.26 at χ_H = 1.6., which indicates that the system is less sensitive to the dead volume changes in the hot region; nevertheless, an optimal point still exists. Near this point, the coordination between heat input and piston motion is optimized, which contributes to enhanced system efficiency. The dead volume ratio curve of the regenerator χ_R changes most gradually, exhibiting a broad and flat peak. For example, when χ_R increases from 0.0 to 1.6, Z gradually decreases from 0.30 to 0.22, with a broad peak between χ_R = 0.4 and χ_R = 0.8 where Z remains around 0.28, which indicates a higher tolerance to variations in dead volume. Since the regenerator primarily facilitates heat transfer rather than directly participating in the compression and expansion processes, its influence on the system’s output power is comparatively indirect.

In summary, the system’s output power is most sensitive to the dead volume ratio in the cold region χ_K, followed by the hot region χ_H, and least affected by the regenerator χ_R. Therefore, practical design and control efforts should primarily focus on optimizing χ_K, while allowing some flexibility in the design of χ_R.

The conclusion of this study—that “the cold-end dead volume ratio χ_K is the most sensitive to system performance”—aligns with the findings of Alfarawi et al. [11] from their CFD simulations and experimental studies on a γ-type Stirling engine. They concluded that the dead volume in the connecting tube (analogous to the cold-end volume in this paper) has a significant negative impact on the indicated work, and there exists an optimal tube diameter (corresponding to an optimal dead volume), which indicates that the model presented in this paper possesses a certain predictive capability in identifying key sensitive parameters.

3.3. Effects of the Other Factors on the Dimensionless Power and the Dead Volume Ratio

Once the dead volume ratios are determined, it is necessary to examine the influence of the temperature ratio τ, the swept volume ratio k, the piston phase angle α_dr, and the minimum pressure angle θ on the dimensionless power output.

3.3.1. The Temperature Ration τ

The overall level of the dimensionless power is higher at elevated temperature ratios, as shown in Figure 4a, but it also exhibits increased sensitivity. The effects of the dead volume ratios at different locations vary, with the dead volume ratio at the hot end χ_H having the most significant impact on the dimensionless power. As χ_H increases, the curve declines rapidly, particularly at higher temperature ratios, indicating that an increase in the dead volume at the hot end significantly impairs the Stirling machine’s heat absorption and work output efficiency. For instance, at a high temperature ratio of τ = 3, the dimensionless power Z drops from approximately 0.23 to below 0.14 as χ_H increases from 0.4 to 2.0. For the dead volume ratio at the cold end χ_K, the trend is similar but less pronounced compared to the hot end. Although it also exhibits a decreasing tendency, the curve is relatively flat, indicating a comparatively smaller impact of the cold-end dead volume. Specifically, at τ = 3, Z decreases from about 0.195 to 0.14 over the same range of χ, a decline of 0.055, which is less severe than that of χ_H (a decline of over 0.09). Regarding the dead volume ratio of the regenerator χ_R, its impact is relatively minor, manifested by a gradual decrease or even a plateau in the dimensionless power. Nevertheless, it should not be overlooked, as it primarily affects the efficiency of heat recovery. Furthermore, in terms of extreme and performance inflection points, certain curves exhibit peak characteristics when the dead volume ratio is relatively low, corresponding to a maximum dimensionless power output. As the dead volume ratio further increases, the performance drops rapidly, indicating the existence of an “optimal dead volume ratio range” beyond which the thermodynamic performance degrades significantly. With the increase in the temperature ratio, the overall level of the dimensionless power can effectively enhance, but it also intensifies the sensitivity of performance to the structural parameters. At high temperature ratios, the curves exhibit significantly steeper slopes compared to those at low temperature ratios, particularly in response to variations in the dead volume ratio at the cold end χ_K and the hot end χ_H, which indicates that under high-temperature conditions, a more stringent control of the dead volume design is required.

It is worth noting that the intersection of the three curves at a single point indicates that, at this specific condition, the output power Z of the Stirling machine exhibits consistent behavior with respect to variations in the dead volume ratios of different components, which is characterized by strong thermodynamic coupling robustness, low parameter sensitivity, and high system synergy. Therefore, it represents a critical “characteristic point” or “equilibrium point” with significant value for both theoretical analysis and engineering design. The influence of different temperature ratios on the dimensionless power is illustrated in Figure 4b. As the temperature ratio τ increases from 1 to 5, the dimensionless power Z demonstrates a clear and sequential decline, decreasing from approximately 0.230 to 0.205. The relationship between the two follows the functional expression given in Equation (24).

$$Z=0.0037{\tau }^2-0.0045\tau +0.0021$$

(24)

This monotonically decreasing trend is nearly linear within the observed range, with only a slight curvature, indicating a strong inverse correlation between the temperature ratio and the output power. The rate of decline is most pronounced in the initial phase; for instance, as τ increases from 1 to 2, Z drops by about 0.01. However, as τ continues to rise, the rate of decrease moderates, suggesting a diminishing negative impact of further increasing the temperature ratio on the system’s power output. This overall downward trend underscores a fundamental performance constraint: while a higher temperature ratio is a key driver for thermodynamic efficiency in theory, it also introduces increased losses or irreversibilities in practice, which ultimately dominate and lead to a net reduction in the usable power output Z. To achieve stable high-performance output, it is recommended that the operating temperature difference of the Stirling machine should be set in the range of τ > 2.0.

3.3.2. The Dead Volume Ratio of the Regenerator χ_R

As the regenerator dead volume ratio χ_R increases from 0.5 to 1, the dimensionless power exhibits a clear decreasing trend under different values of hot-end χ_H and cold-end dead volume ratio χ_K, as shown in Figure 5, which indicates that a larger regenerator dead volume has a significant negative impact on the system’s performance. This is evidenced by the entire family of curves (for both χ_H and χ_K) shifting downward. For example, at a dead volume ratio of χ = 1.0, the dimensionless power for all components drops from a range of approximately 0.18–0.195 (when χ_R = 0.5) to about 0.165–0.18 (when χ_R = 1). Specifically, an increase in both hot-end and cold-end dead volume ratios leads to a reduction in dimensionless power. This decreasing trend becomes more pronounced when χ_R = 1, as indicated by the steeper decline of the solid lines compared to the dashed lines. For instance, the cold-end dead volume curve (χ_K) under χ_R = 1 plummets from around 0.215 at χ = 0.4 to below 0.15 at χ = 2.2, a much more dramatic drop than its counterpart when χ_R = 0.5. This phenomenon is primarily attributed to the extended residence time of the working fluid within the regenerator due to the increased dead volume, which effectively reduces the proportion of active heat exchange area. Consequently, the degree of irreversibility in the system is intensified, and the heat transfer efficiency between the high-enthalpy gas at the hot end and the low-temperature gas at the cold end deteriorates. In addition, the expansion of the hot-end dead volume reduces the energy utilization efficiency of the high-temperature gas, while an increase in the cold-end dead volume raises the resistance to heat rejection. Under the condition of a relatively large regenerator dead volume, these negative effects are further compounded, resulting in a significant reduction in the system’s output power. The chart clearly shows that for any given χ_H or χ_K value, the system’s power output is consistently lower when χ_R = 1 than when χ_R = 0.5. Therefore, properly controlling the regenerator dead volume ratio is of great importance for enhancing the thermodynamic performance of the Stirling system.

3.3.3. The Sweeping Volume Ratio k

When other parameters are held constant, an increase in the sweeping volume ratio k leads to a decrease in the dimensionless power Z. Figure 6a illustrates the influence of the dead volume ratio χ on the dimensionless power, where the dashed lines correspond to k = 1 and the solid lines correspond to k = 2. The chart unequivocally shows that for any given component (χ_H, χ_R, or χ_K) and at any given dead volume ratio, the Z value for k = 2 is consistently and significantly lower than that for k = 1. For example, at χ = 1.0, the dimensionless power for all components under k = 2 is suppressed to below 0.20, whereas under k = 1, the values are generally above 0.22. Overall, as the dead volume ratio increases, the dimensionless power Z initially rises and then either declines or decreases gradually. Under the same conditions, the Z values for k = 2 are generally lower than those for k = 1, indicating that a larger sweeping volume ratio does not favor the improvement of system performance. Among the dead volume ratios at different locations, the hot-end dead volume ratio χ_H has the most significant impact on Z. This is evident from the steep slopes of both the χ_H1 and χ_H2 curves. Notably, the χ_H1 curve exhibits a sharp peak, rising from about Z = 0.24 at χ = 0.4 to a maximum exceeding 0.28 near χ = 0.8, before falling rapidly. A moderate increase in χ_H is beneficial for raising the average gas temperature, whereas excessive increases lead to a decline in heat utilization efficiency. The regenerator dead volume ratio χ_R can optimize the regenerative heat exchange process within a certain range, as seen in the χ_R1 curve, which maintains a relatively high and stable Z value above 0.22 across a broad range from χ = 0.4 to 1.2; however, excessive dead volume leads to increased energy losses. The cold-end dead volume ratio χ_K has a relatively minor effect on Z, but its increase also results in a decline in compression efficiency. Overall, proper control of the dead volume ratios of various components—particularly the hot end and the regenerator—combined with optimization of the sweeping volume ratio, constitutes an effective approach to enhancing the dimensionless power of the Stirling machine. The data suggests that operating with a lower k value (k = 1) and carefully tuning the hot-end dead volume to its optimal point (around χ = 0.8 for k = 1) can yield the highest system performance, with Z reaching nearly 0.30.

The effect of varying the sweeping volume ratio k on the Stirling machine’s output performance is shown in Figure 6b. Simulation results indicate that as k increases, the dimensionless power Z rapidly decreases and approaches saturation, demonstrating a typical nonlinear thermodynamic coupling effect. Specifically, as k rises from 0.5 to 3.5, a final value of about 0.17. This decline is particularly pronounced in the low-k region; for instance, when k increases from 0.5 to 1.5, Z plummets by nearly 0.07, accounting for about 70% of the total decrease. In contrast, as k continues to increase beyond 2.0, the curve flattens significantly, with Z decreasing by less than 0.02 over the range from k = 2.0 to k = 3.5, clearly indicating a saturation trend. By performing exponential curve fitting on the extracted data, the following empirical formula is obtained, as shown in Equation (25):

$$Z=1.53(1-e^{-3.37k})+0.01$$

(25)

This study found that an excessively large sweep volume ratio k leads to performance degradation. This is consistent with the phenomenon observed by Qu et al. [10] in their experiments with a γ-type Stirling heat engine, namely that there exists an optimal range for the sweep volume ratio, and deviation from this range results in decreased output power and efficiency.

3.3.4. The Minimum Pressure Angle θ

Figure 7a illustrates the influence of the minimum pressure angle (θ = 60°, 70°, 80°) on the dimensionless power Z of the Stirling engine, when θ increases from 60° to 80°, the dimensionless power Z decreases by approximately 0.12–0.15. The results show that Z decreases significantly with increasing θ, indicating that a smaller minimum pressure angle is beneficial for enhancing system output performance, for example, when θ = 60° and χ_h1 = 60°, Z drops from 0.22 at χ = 0.4 to 0.17 at χ = 2.0. The underlying mechanism is that a smaller minimum pressure angle θ can effectively separate the compression and expansion processes, optimize the phase relationship of gas pressure variation, and reduce energy cancellation. This enhances the p-v diagram area, thereby increasing the effective work output per unit scavenging volume. In contrast, a larger θ leads to earlier compression and delayed expansion, resulting in lower thermal energy utilization efficiency and a significant reduction in Z. The response trends of different dead volume ratios (χ_H, χ_R, χ_K) to variations in θ are consistent. The improvement in Z resulting from optimizing θ is particularly pronounced when the dead volume ratios are small.

As the minimum pressure angle θ varies within the range of 40° to 90°, the dimensionless power Z exhibits a clearly nonlinear behavior, with an overall decreasing trend, as shown in Figure 7b. The Fitting analysis indicates that this trend can be described by the following empirical Equation (26):

$$Z=2.18\times {10}^{-4}{\theta }^2-0.0261\theta +0.7065$$

(26)

3.4. Model Validation: Comparison with the Classical Schmidt Model

To validate the correctness of the isothermal model presented, we compared its prediction results with the widely accepted classical Schmidt isothermal model in the field of Stirling machines. Under identical input parameters (τ = 3, k = 1, a_dr = 90°, θ = 40°, and all dead volume ratios set to zero), the dimensionless power Z was calculated respectively.

Result from this model: Z = 0.125; Result from the Schmidt model: Z ≈ 0.121.

The relative error between the two is approximately 3.3%, which falls within an acceptable range for engineering purposes, which indicates that the model established in this paper is reliable in its core calculation of work output. The primary source of the minor deviation stems from our model’s assumption of using the arithmetic mean for the regenerator temperature, whereas the classical Schmidt model employs a more complex integral mean. This comparison validates the effectiveness of the model presented in this paper as a tool for rapid design and parametric sensitivity analysis.

4. Conclusions and Recommendations

Using an isothermal thermodynamic model, this study systematically investigates the influence of key structural parameters—including dead volume ratio, sweep volume ratio, piston phase angle, and minimum pressure angle—on the dimensionless power output Z of the Stirling system. The principal findings are summarized as follows:

Firstly, the dimensionless power output is significantly influenced by the dead volume ratios, with distinct effects observed across different engine sections. The cold space dead volume ratio χ_K exhibits the highest sensitivity to Z, often showing a sharp peak, as evidenced by its curve rising to Z ≈ 0.30 at an optimal χ_K near 0.4–0.8 before declining rapidly. This is followed by the hot space χ_H, which also demonstrates a significant impact, with its optimal point yielding Z > 0.28 at χ_H ≈ 0.8 under certain conditions. followed by the hot space χ_H, while the regenerator section χ_R shows a comparatively weaker impact, characterized by a broad, flat peak where Z remains relatively stable over a wider range of χ_R (e.g., Z ≈ 0.22–0.28 for χ_R between 0.4 and 1.2), indicating a higher design tolerance. Each parameter demonstrates an optimal design range. As the dead volume ratio increases, the system’s dimensionless power output Z exhibits a generally monotonic decline, increasing χ_K from 1.0 to 2.0 can lead to a performance drop of over 30%. Moreover, under higher temperature ratio conditions, the system demonstrates increased sensitivity to variations in dead volume, resulting in more pronounced performance degradation, as the curves for χ_H and χ_K exhibit steeper slopes at higher τ.

Furthermore, within the investigated range, an increase in the swept volume ratio k from 1 to 2 leads to a noticeable decline in the dimensionless power Z, with performance under k = 2 being consistently and significantly lower across all dead volume configurations, at a representative dead volume ratio of χ = 1.0, Z values are suppressed to below 0.20 for k = 2, whereas they remain above 0.22 for k = 1, which suggests that an oversized swept volume introduces greater flow-phase discrepancies, thereby impairing the system’s output efficiency.

Additionally, an increase in the piston phase angle α_dr enhances system performance by improving the synchronization of compression and expansion phases, leading to better gas expansion efficiency and higher dimensionless power Z, especially in the hot chamber and regenerator, and simulation results confirm that an increase in a_dr to a recommended value of 70° or higher significantly contributes to this improvement.

On the other hand, reducing the minimum pressure angle θ contributes to an improvement in the dimensionless performance index Z, mainly through the expansion of the p-v loop area and the alleviation of internal energy losses. The enhancement effect is especially significant when the dead volume ratio χ is small, as a lower system dead volume allows the benefits of a reduced pressure angle to be more fully realized in the thermodynamic cycle.

In summary, this study proposes a coupling regulation mechanism for structural parameters in free-piston Stirling systems and offers the following design recommendations:

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Priority should be given to maintaining the dead volume ratios at the cold and hot ends within their optimal ranges, while appropriately reducing the regenerator volume to mitigate irreversible losses, as its impact, while lesser, remains non-negligible, especially at higher swept volume ratios.

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The sweep volume ratio should remain within a moderate range (preferably closer to k = 1 than k = 2) to avoid performance degradation caused by excessive values.

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Additionally, adopting a larger piston phase angle (α_dr ≥ 70°) and a smaller minimum pressure angle (θ ≤ 60°) is beneficial for enhancing gas work capacity and improving the efficiency of the p-v cycle, this parameter coupling strategy ensures that the system operates close to its optimal thermodynamic state, maximizing the dimensionless power output Z.