Adiabatic modeling

Models based on oscillatory flow characteristics without pressure gradients are widely used for Stirling cycle characterization. The Stirling engine is divided into five chambers (as shown in Fig. 5), namely, expansion space, heater, regenerator, cooler, and compression space. The second-order analysis method is based on the ideal cycle, where the mass and energy conservation equations and the gas equation of the state of each chamber are solved to obtain an analytical solution. This approach allows further calculation of the various power losses and heat losses, yielding the output power and the required heat that are closer to the actual situation. During the development of the ideal adiabatic model, the following assumptions were made.

Fig. 5Full-screen View

(Color online) The diagrammatic representation of the model in the adiabatic analysis methodology

(1) The working fluid was assumed to be an ideal gas.

(2) There was no leakage of the working fluid in the cycle, and its mass remained constant.

(3) The temperatures of the working gas in the cooler and heater were considered to be equal to the wall temperature and were held constant.

(4) Instantaneous pressures throughout the system are equal.

(5) The heat leakage between the compression and expansion spaces and the heat transfer to the environment are negligible.

(6) The kinetic and potential energies of the gas flow are ignored.

The adiabatic model differential equation set is presented in Table 1, where the subscripts c, k, r, h, and e represent the compression cavity, cooler, return heaters, heaters, and expansion cavity, respectively. The subscripts ck, kr, rh, and he represent the four interfaces of the following five components: compression, cooler, heat return, heater, and expansion spaces, respectively.

Modification of the Adiabatic Model

The simple analysis method considers only the effects of non-ideal heat transfer and pressure loss in the heat exchanger. In this study, based on a simple analysis method, the mass leakage loss term and distribution piston shuttle loss are coupled in the energy conservation equation.

(1)Shuttle heat loss. The shuttle loss is primarily caused by the reciprocating movement of the displacer piston between the hot and cold cylinders. There was a significant temperature gradient at both ends of the displacer piston, leading to heat loss, because some of the heat was directly conducted to the cold end through the piston. The differential expression for this phenomenon is expressed as follows: $\text{d}{Q}_{\text{shu}}=\frac{\pi S^2{k}_{\text{g}}{D}_{\text{d}}}{8J{L}_{\text{d}}}\left(T_{\text{e}}-T_{\text{c}}\right),$ (1) where S is the displacer stroke, k_g is the gas thermal conductivity, D_d is the displacement diameter, J is the gap between the displacer and cylinder.

Calculation process

(2)Seal leakage loss. In the actual operation of the engine, a certain pressure difference emerges between the compression space and the buffer space, which leads to a portion of the gas passing through the gap between the piston and cylinder wall in the compression space and the buffer space back and forth, resulting in leakage losses. The leakage into the buffer space is calculated as follows: $m_{\text{leak}}=\pi D\frac{P+P_{\text{buffer}}}{4R{T}_{\text{g}}}\left(u_{\text{p}}J-\frac{J^3}{6\mu }\frac{P-P_{\text{buffer}}}{L_{\text{p}}}\right)$ (2) where P is the pressure in the compression space, D is the diameter of the cylinder, P_buffer is the pressure in the buffer space, J is the gap between the piston and cylinder, and L_p is the length of the piston.

By coupling the shuttle and leakage losses with the differential equations in the adiabatic analysis method, the pressure and pressure differential terms are changed to $\begin{array}{ll}M=\hfill & \frac{p\left(\frac{V_{\text{c}}}{T_{\text{c}}}+\frac{V_{\text{k}}}{T_{\text{k}}}+\frac{V_{\text{r}}}{T_{\text{r}}}+\frac{V_{\text{h}}}{T_{\text{h}}}+\frac{V_{\text{e}}}{T_{\text{e}}}\right)}{R}\hfill \\ \hfill & -\pi D\frac{P+P_{\text{buffer}}}{4R{T}_{\text{g}}}\left(u_{\text{p}}J-\frac{J^3}{6\mu }\frac{P-P_{\text{buffer}}}{L_{\text{p}}}\right)\hfill \end{array}$ (3) $\text{d}p=\frac{-p\left(\frac{\text{d}{V}_{\text{c}}}{T_{\text{ck}}}+\frac{\text{d}{V}_{\text{e}}}{T_{\text{he}}}\right)+\frac{\text{d}{Q}_{\text{shuttle}}R}{4R{T}_{\text{g}}}\left(\frac{T_{\text{he}}-T_{\text{ck}}}{T_{\text{he}}{T}_{\text{ck}}}\right)+R\text{d}{m}_{\text{leak}}}{\frac{V_{\text{c}}}{\gamma T_{\text{ck}}}+\left(\frac{V_{\text{k}}}{T_{\text{k}}}+\frac{V_{\text{r}}}{T_{\text{r}}}+\frac{V_{\text{h}}}{T_{\text{h}}}\right)+\frac{V_{\text{e}}}{\gamma T_{\text{he}}}}$ (4) The mass differential equation is as follows: $\text{d}{m}_{\text{c}}=\frac{p\text{d}{V}_{\text{c}}+V_{\text{c}}\text{d}p/\gamma }{R{T}_{\text{ck}}}-\frac{\text{d}{Q}_{\text{shuttle}}}{c_{\text{p}}{T}_{\text{ck}}}$ (5) $\text{d}{m}_{\text{e}}=\frac{p\text{d}{V}_{\text{e}}+V_{\text{e}}\text{d}p/\gamma }{R{T}_{\text{he}}}-\frac{\text{d}{Q}_{\text{shuttle}}}{c_{\text{p}}{T}_{\text{he}}}$ (6)

Non-idea heat transfer

The heat recovery performance of the Stirling engine was evaluated based on the efficiency of the regenerators. The efficiency of the regenerator is defined as the ratio of the actual enthalpy change to the maximum enthalpy change of the working fluid in the regenerator. When the gas flowed from the cooler to the heater, the temperature at which it exited the return heater was slightly lower than that of the heater. The gas absorbs more heat from the heater (external heat source), resulting in a less efficient heat cycle. Thus, the actual heat absorption and heat release can be written as follows: $\begin{array}{l}{Q}_{\text{h}}=Q_{\text{hi}}+Q_{\text{rloss}}=Q_{\text{hi}}+Q_{\text{ri}}\left(1-\epsilon \right)\\ Q_{\text{k}}=Q_{\text{ki}}-Q_{\text{rloss}}=Q_{\text{ki}}+Q_{\text{ri}}\left(1-\epsilon \right)\end{array}$ (7) where Q_h is the heat added to the working fluid in the heater, Q_k is the heat rejected to the cooler, Q_rloss is the heat loss in the imperfect regenerator, Q_ri is the amount of regenerator heat transferred in the ideal process, and ϵ is the effectiveness of the regenerator, calculated as follows: $\begin{array}{l}\epsilon =\frac{NTU}{1+NTU}\\ NTU=\frac{St\cdot A_{\text{wg}}}{2A}\end{array}$ (8) where NTU is the number of heat transfer units, A_wg is the internal wetted area of the regenerator, A is the cross-sectional area of the regenerator, and St is the Stanton number, which is calculated as follows: $St=0.46R{e}^{-0.4}P{r}^{-0.1}$ (9) where Re is the Reynolds number and Pr is the Prandtl number. In this study, Pr was considered as a constant of 0.71. The hydraulic diameter in the regenerator is calculated as $D_{\text{h}}=\frac{d_{\text{m}}\phi }{1-\phi }$ (10) where dm is the diameter of the regenerator wire and φ is the porosity coefficient. The heater and cooler are also non-ideal, and the actual temperature of the gases in the two heat exchangers depends on the heat losses in the return heaters and convective heat transfer between the gases and walls in the heater and cooler. The actual gas temperature in the heat exchanger is expressed as follows: $T_{\text{gh}}=T_{\text{wh}}-\frac{Q_{\text{ach}}}{h_{\text{h}}{A}_{\text{wh}}}$ (11) $T_{\text{gk}}=T_{\text{wk}}-\frac{Q_{\text{ack}}}{h_{\text{k}}{A}_{\text{wk}}}$ (12) where Q_ach is the actual heat absorbed by the heater, Q_ack is the actual heat rejection of the cooler, W_ac is the actual indicated power output, and h is the heat transfer coefficient in the heater and cooler. The heat transfer coefficient is expressed as follows: $h=\frac{f_{\text{r}}\mu C_{\text{p}}}{2D_{\text{h}}Pr}$ (13) In the analysis of the Stirling engine cycle, T_wh and T_wk denote the wall temperatures of the heater and cooler, respectively, and are generally set to constant values.

Other losses effects

(1)Thermal conductivity loss: The regenerator was connected to the cooler and heater, and the heat from the hot end flowed through the cylinder wall of the return heaters to the cold end, which was the main source of heat conduction loss. This can be evaluated using Fourier’s law[37]: $Q_{\text{w}}=\frac{k_{\text{m}}{A}_{\text{w}}\text{Δ}T}{l}$ (14) where k_m is the heat conduction coefficient of the material, A_w is the thermal conductivity cross section, and l is the conductivity length.

(2) Flow-resistance loss. Pressure loss is caused by friction owing to the viscosity of the fluid, and the loss caused by the pressure drop is known as flow resistance loss. The multilayered annular mesh structure within the regenerator is the primary cause of the flow resistance loss. The pressure drop and flow resistance loss can be calculated using the following semi-empirical formulas: $\text{Δ}p=\frac{-2C_{\text{ref}}\mu uV}{D_{\text{h}}{}^{2}A}$ (15) $W_{\text{fr}}={\displaystyle {\int }_0^{2\pi }}\left(\text{Δ}p\frac{\partial V}{\partial \theta }\right)\text{d}\theta$ (16) where, C_ref is the Reynolds friction factor; μ is the dynamic viscosity; V is the volume of the working area; d is the hydraulic diameter; A is the flow cross-section area. The Reynolds friction factor C_ref is calculated as: $C_{\text{ref}}=\{\begin{array}{cc}\mathrm{16,}& Re<2000\\ {7.343\times 10}^{-4}R{e}^{1.3142}\mathrm{,}& 2000<Re<4000\\ 0.7091R{e}^{0.75}\mathrm{,}& Re>4000\end{array}$ (17) (3) Finite piston speed and mechanical friction loss. Relevant studies indicated that the instantaneous pressure on a piston surface differs from the instantaneous average effective pressure within an engine cylinder [38]. The actual work of compression and expansion is different from the theoretical calculations based on classical thermodynamics. This pressure loss is caused by the finite-speed motion of the pressure waves generated by the piston within the working space. The formula for calculating the work loss due to the finite speed of the piston and mechanical friction can be calculated as follows: $\begin{array}{l}{W}_{\text{FST}-\text{MF}}={\displaystyle \int }{P}_{\text{m}}\left(\pm \frac{a{u}_{\text{p}}}{c}\pm \frac{f\text{Δ}{P}_{\text{mf}}}{P_{\text{m}}}\right)\text{d}V\\ a=\sqrt{3\gamma }\\ c=\sqrt{3R_{g}T}\end{array}$ (18) where P_m is the mean effective working pressure, u_p is the linear speed of the piston, f is the frequency, and $\text{Δ}{P}_{\text{mf}}$ is the pressure loss caused by the mechanical friction of the components. The ± symbol represents the compression process (+) and the expansion process (-), respectively. P_m is calculated as follows: $\begin{array}{l}\text{Δ}{P}_{\text{mf}}=\frac{\left(0.04+0.0045u_{\text{p}}\right){\times 10}^5}{3\mu }\left(1-\frac{1}{r_v}\right)\\ \mu =1-\frac{1}{3r_v}\end{array}$ (19) where rv is the compression ratio and $r_v=\frac{V_{\text{max}}}{V_{\text{min}}}$.

The algorithm of the model used in this study is shown in Fig. 6. In the first step, the Stirling engine geometry and hydrodynamic and thermodynamic parameters were initialized. The geometric equations for V_c, V_e, dV_c and dV_e as functions of the rotation angle were obtained from the engine configuration. Hydrodynamic parameters, such as the cross-sectional area, wet area, and hydraulic diameter. based on the geometric parameters. Initialize the thermodynamic parameters, such as the specific heat capacity, gas constant, and thermal conductivity.

Fig. 6Full-screen View

Calculation process

Second, the computational parameters are initialized. The temperatures of the expansion and compression spaces were initialized to wall temperatures T_wh and T_wk. The initial mass is computed using the following first-order model:

Third, initialize angle θ as 0 and the calculation step is set to 1. The angle is initialized at the end of each cycle, and for the initial value problem, the system of differential equations is solved using the fourth-order Runge–Kutta method with θ=360 to complete the calculation of one cycle. It is determined whether the cavity temperature at the end of the cycle is equal to the temperature at the beginning of the cycle until the gas temperature approaches the limit of convergence to turn on the next step of the calculation.

The regeneration efficiency of the regenerator and convection heat transfer coefficients of the heater and cooler are described in detail in [24]. The actual heat exchange quantity between the heat exchanger wall and the working fluid is derived based on the ideal heat exchange amount obtained from the previous step. The new temperature of the heat exchanger is computed and used as the initial value to return to the third step of the ideal adiabatic model for calculation. This process is repeated until the convergence criteria are met and then proceeds to the fifth step of the calculation.

Finally, other losses were calculated to obtain the actual heat and output work. The method for solving The classical fourth-order Runge–Kutta method is solved as follows: $\{\begin{array}{l}{k}_1=f\left(x_i\mathrm{,}{y}_i\right)\hfill \\ k_2=f\left(x_i+\frac{1}{2}h\mathrm{,}{y}_i+\frac{1}{2}{k}_1\right)\hfill \\ k_3=f\left(x_i+\frac{1}{2}h\mathrm{,}{y}_i+\frac{1}{2}h{k}_2\right)\hfill \\ k_4=f\left(x_i+h\mathrm{,}{y}_i+h{k}_3\right)\hfill \\ y_{i+1}=y_i+\left[\frac{1}{6}\left(k_1+2k_2+2k_3+k_4\right)\right]h\hfill \end{array}$ (20)

Average pressure convergence

In this study, an iterative method was applied for pressure convergence. While solving the Stirling cycle, the average pressure must be calculated to satisfy the condition of average pressure convergence. The average pressure in the compression space over one cycle should be equal to the set average pressure. $\left|{\displaystyle {\int }_0^{\delta }}\left[{\displaystyle \sum _{n=1}^N}\frac{p_n}{N}\right]\text{d}\delta -p_{\text{mean}}\right|<p_{\epsilon }{p}_{\text{mean}}$ (21) pε represents the residual value for determining the pressure convergence. After the first cycle of the solution was completed, the initial pressure of the model was adjusted until the calculated average pressure satisfied the periodically stable solution, as described by the following formula: $p_{\text{start}\_\text{new}}=p_{\text{start}\_\text{last}}-p_{\epsilon }{p}_{\text{start}\_\text{last}}$ (22) Employing this iterative method allows the rapid identification of a solution that meets the target average pressure across the entire system.

Verification and validation

3.5.1

Validation of the GPU-3 Engine

The GPU-3[23, 39] Stirling engine, a portable generator set developed by General Motors for the military sector, has well-established equations of motion for the rhombic drive mechanism to calculate the expansion and compression space volume changes, as shown in Fig. 7. The detailed design dimensions and parameters of GPU-3 are presented in Table 2.

Fig. 7Full-screen View

The geometric relationship of the drive equation

Table 2Full-screen View

GPU-3 engine dimensions and parameters

Parameters		Values	Parameters	Values
Clearance volumes			Swept volumes
	Compression space	28.68 cm3	Compression space	113.14 cm3
	Expansion space	30.52 cm3	Expansion space	120.82 cm3
Heater			Cooler
	Tube number	40	Tube number	312
	Tube inside diameter	3.02 mm	Tube inside diameter	1.08 mm
	Tube length	245.3 mm	Tube length	46.1 mm
	Void volume	70.38 cm3	Void volume	13.8 cm3
Regenerator			Drive
	Void volume	50.55 cm3	Connecting rod length	46 mm
	Length	22.6 mm	Crank radius	13.8 mm
	Internal diameter	22.6 mm	Eccentricity	20.8 mm
	No. per cylinder	8	Displacer stroke	31.2 mm
	Diameter of wire	0.04 mm	Internal diameter of cylinder	69.9 mm
	Porosity	0.697	Working fluid	Helium
	Material	Stainless steel	Frequency	41.72 Hz

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To validate the model, simulations (Helium, T_wh = 922 K, T_wk = 286 K) were performed against the GPU-3 Stirling engine experiment with two different pressure conditions (P_mean = 2.76 MPa and P_mean = 4.14 MPa). The computational results were compared with the simulation data of the available second-order analysis models (Simple-II, CAFS, ISAM, and simple models) and the third-order analysis software Sage. Experimental data were obtained from NASA experiments on a GPU-3 Stirling engine.

Figures 8 and 9 illustrates the relationship between the working fluid temperature variation with angle in the expansion and compression cylinders, as well as the P-V (pressure-volume) diagram of the cycle. The temperature of the cavity returns to the initial point after one complete rotation of the Stirling rhombic drive mechanism, and the cycle is repeated periodically in accordance with the basic characteristics of the Stirling cycle.

Fig. 8Full-screen View

Pressure-volume diagram

Fig. 9Full-screen View

Variation of the working fluid temperature

Figure 10 presents a comparison of the output work results between the model in this paper and the existing second-order model at various frequencies. Figure 11 displays a comparison of the cycle efficiency results between the model in this paper and the existing second-order model across different frequencies. Compared with the simple model (P_mean=4.14 MPa), the maximum power error of the proposed model was reduced by 152.5%, and the maximum efficiency error (as a difference) decreased by 10.67%. The average reduction in the power error across all frequency conditions was 110.0%, and the average reduction in the efficiency error was 8.43%. Figure 12 illustrates the comparison of the simulation values between the model in this study and the existing second-order model under the standard operating conditions of 4.14 MPa pressure and 41.67 Hz frequency. Compared with the simple model, the simulation error for the output work was reduced by 46.2%, and the simulation error (as a difference) for cycle efficiency was reduced by 8.7%. Compared with the third-order analytical Sage model, the output work exhibited maximum and minimum errors(as differences) of 28.3% and 4.5%, respectively, with an average error of 11.5%. The cycle efficiency demonstrated maximum and minimum errors (as differences) of 5.43% and 0.30%, respectively, with an error of 1.55%.

Fig. 10Full-screen View

Comparison of output power between different models and experimental values (Helium). (a) T_wh=922 K, T_wh=288 K, P_mean=2.76 MPa (b) T_wh=922 K, T_wh=288 K, P_mean=4.14 MPa

Fig. 11Full-screen View

Comparison of thermal efficiency between different models and experimental values (Helium). (a)T_wh=922 K, T_wh=288 K, P_mean=2.76 MPa (b) T_wh=922 K, T_wh=288 K, P_mean=4.14 MPa

Fig. 12Full-screen View

Bar Chart of Simulation Results Comparison

A comparison of the simulated and experimental values allowed for the analysis of certain working characteristics of the Stirling engine. When helium was used as the working fluid, the output work of the Stirling engine decreased as the frequency increased under high-frequency conditions (with a working frequency greater than 41.72 Hz), and the cycle efficiency exhibited an overall downward trend within the frequency analysis range defined in this study, which was particularly pronounced in the high-frequency region. In contrast, in the subsequent parameter sensitivity analysis, when hydrogen was used as the working fluid, both the output work and cycle efficiency of the Stirling engine exhibited significant upward trends. This difference is caused by the different material properties of the two substances. In addition, as the working frequency increased, the error between the experimental and simulated values in the engine gradually increased because of other undetermined losses caused by high-frequency oscillating flows.

Effect of frequency and different working fluids

The selection of working fluids for Stirling engines predominantly focuses on gases, with commonly used working fluids being N₂, H₂, CO₂, He, and air. Owing to the superior heat exchange characteristics of small-molecular-weight gases, most technologically advanced high-power Stirling engines employ He or H₂ as the working fluid.

To investigate the effects of frequency and different work materials on the output work and cycle efficiency, hydrogen is selected as the work material (T_wh=977 K, T_wk=288 K), with two different pressure variables set at 1.38 MPa and 2.76 MPa. Figures 13a and 13b plot the changes in the output work with respect to the frequency. Figures 14a and 14b illustrate the changes of the cycle efficiency with the frequency.

Fig. 13Full-screen View

The variation of output power with frequency (Hydrogen). (a)T_wh=922 K, T_wh=288 K, P_mean=1.38 MPa (b) T_wh=922 K, T_wh=288 K, P_mean=2.76 MPa

Fig. 14Full-screen View

The variation of efficiency with frequency (Hydrogen). (a) T_wh=922 K, T_wh=288 K, P_mean=1.38 MPa (b) T_wh=922 K, T_wh=288 K, P_mean=2.76 MPa

In scenarios with slower rotational speeds, the system efficiency increased with the rotation speed. However, as the rotational speed continued to increase, the system efficiency declined, resulting in a general trend of an initial increase followed by a decrease. When the work material is H₂, the power output of the Stirling engine is significantly enhanced with increasing rotational speed, displaying a steep upward trend. However, when the work material is He, a faster rotation speed has a detrimental impact on the output of the Stirling engine, with a less steep upward trend. Due to the screen-like structure of the regenerator, a substantial pressure drop occurs as the working mass flows through it, leading to a loss attributed to flow resistance. Figures 15 and 16 demonstrate the effect of the Reynolds number of the regenerator and different rotational speeds on the resulting pressure loss. Figure 17 demonstrates the variation of the maximum pressure drop within the heat exchanger with frequency. With an increase in the frequency, both the Reynolds number and pressure loss inside the heat exchanger increase. Compared to the pressure loss in the regenerator, the pressure losses in the cooler and heater were negligible, demonstrating that the wire mesh structure of the regenerator was the primary source of pressure loss. The high rotational speed of the Stirling engine causes an oscillating flow and heat exchange of the work material inside the body, which tends to be complicated and results in a significant increase in the flow resistance loss or local loss, which in turn affects the cycle efficiency. H₂, which has a lower dynamic viscosity coefficient, produces a smaller flow resistance loss when operating at high rotational speeds than He. However, H₂ exhibits poor sealing properties, rendering it susceptible to leakage and explosion, which can cause challenges such as hydrogen embrittlement in certain materials. Therefore, helium is commonly used as the working fluid.

Fig. 15Full-screen View

Effect of rotation speed on Reynolds number of regenerator

Fig. 16Full-screen View

Effect of rotation speed on pressure drop of regenerator

Fig. 17Full-screen View

The maximum pressure drop of the heat exchanger varies with frequency

Effect of regenerator parameters

The regenerator features an internal structure filled with a metal wire mesh. Porosity is represented by the ratio of the porous volume of the regenerator to its overall volume. Figure 18 depicts the variation of output work and cyclic efficiency with changes in porosity and wire mesh diameter. For metal wire meshes with different diameters, the cyclic efficiency tended to initially increase and then decrease. This trend is attributed to the fact that an appropriate increase in porosity increases the hydraulic diameter and reduces the wetted area of the mesh in contact with the working fluid. This reduction in the pressure drop across the regenerator leads to a decrease in the flow resistance losses, thereby enhancing the output work and cyclic efficiency of the engine. At lower porosities, a denser mesh configuration results in a smaller hydraulic diameter and an increased wetted area for the regenerator, which can significantly affect the flow resistance and adversely affect the cyclic efficiency and output work.

Fig. 18Full-screen View

The variation of (a) output power and (b) thermal efficiency with porosity and wire diameter

When the porosity was below 0.797, Stirling engines equipped with larger-diameter wire meshes exhibited higher cyclic efficiencies than those equipped with smaller-diameter meshes. This trend is attributed to the larger convective heat transfer area provided by the larger-diameter meshes under conditions of low porosity. In contrast, when the porosity surpassed 0.797, wire meshes with smaller diameters demonstrated superior cyclic efficiencies, attributable to their comparatively lower flow resistance losses. At a wire mesh diameter of 0.04 mm, a turning point in cyclic efficiency was observed at a porosity of 0.797. With a further increase in porosity to 0.847, there was a slight increase of 0.1% in the output power, whereas the cyclic efficiency decreased by 3%. In conclusion, the engine output and cyclic efficiency were optimal at a porosity of 0.797.

Figure 19 illustrates the variation of output work and cycle efficiency with the length of the regenerator. At a mean pressure of 4.14 MPa, an inflection point appears in the cycle efficiency curve when the regenerator length is approximately 0.025 m. When the regenerator length is less than 0.025 meters, an increase in length leads to a larger heat exchange area, allowing for more sufficient heat transfer within the regenerator’s metal wire mesh, which in turn increases cycle efficiency with the increase in length. Moreover, because the pressure losses due to flow resistance inevitably increase with length, the output work will necessarily decrease as the regenerator lengthens. Consequently, when the regenerator length exceeds 0.025 m, the positive effect of the increased heat exchange area on cycle efficiency is insufficient to offset the negative effect caused by increased losses, resulting in a decline in cycle efficiency. At P_mean=4.14 MPa, the optimal regenerator length is approximately 0.025 m.

Fig. 19Full-screen View

The influence of the regenerator length

Effect of temperature of heat exchangers

During the actual operation of a space nuclear reactor, the hot-end temperature of the Stirling system fluctuates with changes in the core temperature of the nuclear reactor, and the performance of the radiative heat rejection system can also affect the cold-end temperature. Shifts in both the hot- and cold-end temperatures directly affect the quantity of heat absorbed and released by the working fluid gas, thereby influencing the output work and cyclic efficiency of the Stirling engine. Consequently, for space nuclear power reactors, it is necessary to assess the impact of cold- and hot-end temperatures on the performance of Stirling systems.

Figure 20 demonstrates the variation in output work and cyclic efficiency of the Stirling engine at different temperature ratio conditions as the cooler temperature increases from 280 K by increments of 20 K up to 440 K. At a temperature ratio of 0.25, the output work increased from 4874 to 5301 W, an increment of 2.3%. At a temperature ratio of 0.35, the output work increased from 3146 to 3480 W, which is an increase of 5.4%. When the temperature ratio was 0.45, the output work increased from 1773 to 2037 W, denoting an increase of 9.8%. When the temperature ratio was fixed, alterations in the cooler temperature had a negligible influence on the output work and cyclic efficiency. In actual scenarios, owing to material constraints, the heater of a Stirling engine cannot reach excessively high temperatures; therefore, an ideal case is considered here. At a constant cooler temperature, as the temperature ratio rose from 0.25 to 0.35, the output work decreased by an average of 34.8%, and the cyclic efficiency decreased by an average of 8.61%. As the temperature ratio increased from 0.35 to 0.45, the output work decreased by an average of 42.3%, and the cyclic efficiency decreased by an average of 10.0%. In particular, the lower the temperature ratio, the higher the output power and cyclic efficiency of the system. Conversely, the higher the temperature ratio, the lower the output power and cyclic efficiency of the system. At a constant cooler temperature, a higher heater temperature is more conducive to achieving satisfactory output work and cyclic efficiency. Similarly, under the condition of a constant heater temperature, the lower the cooler temperature, the better the engine performance. However, in practice, owing to the impact of environmental factors, coolers cannot achieve excessively low temperatures, and this limitation should be considered based on the actual conditions. The operating temperature of the working fluid in high-temperature heat pipes exceeds 730K. The most frequently used sodium heat pipes have a working temperature range of 700-1100K. The hot end of the Stirling engine transfers heat to a high-temperature heat pipe through solid-to-solid contact, whereas the cold end can only dissipate heat into the space environment through radiative cooling. Therefore, for space power devices, thermal efficiency is particularly important. A higher thermal efficiency under the same weight and cost conditions allows for higher power output and extended operation.

Fig. 20Full-screen View

The influence of the temperature ratio

In the parameter sensitivity analysis of the cold- and hot-end temperatures, we fixed the temperature boundaries to simulate steady-state operating conditions. However, temperature control and management are complex and critical issues for practical engineering applications. For example, during the reactor start-up process, there is a significant concern about controlling the thermal balance throughout the entire process, from the heat generated in the core to its transfer via high-temperature heat pipes to the Stirling engine, and finally to the heat rejection system. Heat pipes are often quite fragile, and initiating a cold start of a Stirling engine can lead to thermal imbalance, causing the heat pipes to fail and posing a risk to reactor safety. Therefore, to address the startup issue, it is necessary to devise a reasonable temperature control plan to balance the heat transfer between the various components. This challenge will be further explored in a subsequent study.