Simulation and experimental improvement on a small-scale Stirling thermo-acoustic engine

Abstract

Compared with the traditional engines, the thermo-acoustic engines are relatively new and can act as the linear compressors for refrigerators. Many institutes have shown great interest in this kind of machine for its absence of moving mechanical part. In this paper, the influence of the dimensions of the main parts of the small-scale Stirling thermo-acoustic engine was numerically simulated using a computer code called DeltaEC. The resonator and the resonator cavity were found to be the most convenient and effective in improving the performance of the engine. Based on the numerical simulation, a small-scale Stirling thermo-acoustic engine were constructed and experimentally investigated. Currently, with a resonator length of only 1 m, the working frequency of the engine was decreased to 90 Hz and the onset temperature difference was decreased to 198.2 K.

Keywords

thermo-acoustic Stirling engine / small-scale / simulation / experiment

Introduction

Mechanical moving parts such as pistons are indispensable for the conventional heat engines to operate with a high efficiency. However, they often result in high production and maintenance costs and need advanced technical skills on sliding seals. Thermo-acoustic technologies that deal with the thermodynamic conversion between thermal energy and acoustic power by relying on the so-called thermo-acoustic effect completely eliminate the existence of mechanical moving parts. Besides, these thermo-acoustic technologies have obvious advantages in using environmentally-friendly working media and great potential in utilizing low quality heat sources. Thus, extensive efforts have been exerted to develop various configurations of thermo-acoustic engines (TEs). The appropriately phased pressure and velocity oscillations enable the compressible gas parcel to undergo a useful thermodynamic cycle in the vicinity of a solid material and thus convert thermal energy to acoustic power. According to the characteristics of the sound wave generated in thermo-acoustic engines, the engines can be categorized into traveling-wave and standing-wave TEs. For standing wave devices, the phase difference between the pressure and the velocity oscillations is close to 90° within the thermo-acoustic core while for traveling-wave engines, the phase difference is close to zero [ 1]. That is the reason why traveling wave thermo-acoustic engines, also called thermo-acoustic Stirling heat engines (TASHEs), have theoretically a higher efficiency than standing wave ones and have been paid more attention to in recent years.

Small-scale traveling wave thermo-acoustic engines have promising applications in space exploration and domestic occasions although their general performances are in great necessity of improvement, compared with large-scale ones. Helium is the common working media in the field of refrigeration and cryogenics, the performance of TASHEs with helium as the working media is, therefore, paid more attention to. For a small-scale TASHE with a whole longitudinal length of less than 1.5 m, the working frequency and the onset temperature difference are relatively high when using helium as the working media, which cause obstacles in combining it with a pulse tube cooler (PTC) or practically applying low quality heat sources. The literature on the small TASHEs is limited. Ueda et al. [ 2] have designed a small TASHE whose resonator length is 1.04 m with a huge square tank of 20 L as the resonator cavity. Using atmospheric air as the working media, their working frequency is 41 Hz and the onset temperature is over 210°C. Yu et al. [ 3] have investigated a small TASHE with a resonator length of 1.42 m. The working frequency of the TASHE is 80 Hz and 200 Hz using nitrogen and helium gas as the working media, respectively. The onset temperature difference is often more than 400°C with nitrogen gas. Zhou et al. [ 4] have designed the smallest TASHE with a resonator length of 0.65 m using helium as the working gas with high frequency of 282 Hz and onset temperature difference of 418°C.

In this paper, efforts are exerted to achieve better performances on the small-scale TASHEs via simulations and experimental investigations. The numerical and experimental results indicate that the resonator tube and the resonator cavity in the end play a very important role in improving the performance of a small TASHE because the change of these two acoustical parts bring obvious change of the engine performance and they are the most convenient parts for altering and re-assembling.

Numerical simulation

The linear thermo-acoustics theory has initially been formed by Rott [ 1] and later developed by many researchers. Nowadays, it has been widely adopted for low amplitude thermo-acoustic oscillations. Equations (1)–(3) quantitatively describes the interaction between the temperature field and acoustic field, known as Rott’s wave equation [ 1]. They are the thermo-acoustic versions of momentum, continuity and energy equations, respectively. These equations form the basis of the numerical calculation.

(1) {\displaystyle \frac{\text{d}{p}_{\mathrm{1}}}{\text{d}x}}=-{\displaystyle \frac{\text{i}\varpi {{\rho }_m}_{}}{A(\mathrm{1}-{f_v}_{})}}{U}_{\mathrm{1}},

(2) {\displaystyle \frac{{dU}_{\mathrm{1}}}{dx}}=-{\displaystyle \frac{\text{i}\varpi A}{\gamma {p_m}_{}}}[\mathrm{1}+(\gamma -\mathrm{1}){f_k}_{}]p_{\mathrm{1}}+{\displaystyle \frac{({f_k}_{}-{f_v}_{})}{(\mathrm{1}-\sigma )(\mathrm{1}-{f_v}_{})}}{\displaystyle \frac{\mathrm{1}}{{T_m}_{}}}{\displaystyle \frac{{d{T}_m}_{}}{\text{d}x}}{U}_{\mathrm{1}},

(3) {\displaystyle \frac{{\mathrm{d}T}_{\mathrm{m}}}{\mathrm{d}x}}={\displaystyle \frac{\dot{H}-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathrm{Re}\left\{p_{\mathrm{1}}\widetilde{U_{\mathrm{1}}}\left(\mathrm{1}-\left(f_{\mathrm{k}}-{\tilde{f}}_{\mathrm{v}}\right)/\left(\mathrm{1}+\sigma \right)\left(\mathrm{1}-{\tilde{f}}_{\mathrm{v}}\right)\right)\right\}}{{\displaystyle \frac{{\rho }_{\mathrm{m}}{c}_{\mathrm{p}}{\left|U_{\mathrm{1}}\right|}^{\mathrm{2}}}{\mathrm{2}A\varpi \left(\mathrm{1}-{\sigma }^{\mathrm{2}}\right){\left|\mathrm{1}-f_{\mathrm{v}}\right|}^{\mathrm{2}}}}\mathrm{Im}\left(f_{\mathrm{k}}+\sigma {\tilde{f}}_{\mathrm{v}}\right)-\left(Ak+A_{\mathrm{s}}{k}_{\mathrm{s}}\right)}},

where A, A_s, k_s, H, p, T, U, ρ, γ,

\varpi

, a, σ, and i are the cross-sectional area of the channel, the cross sectional area of the solid, the thermal conductivity of the solid, the total power, pressure, temperature, volumetric velocity, density, the ratio of specific heat capacities of the gas, the angular frequency, the sound speed, the Prandtl number and the imaginary unit, respectively. f_k, f_v are complex functions. The subscript “1” indicates the first order of a variable with complex amplitude. Re, Im and the superscript “~” are the real part, the imaginary part and the conjugation of a complex quantity, respectively.

DeltaEC [ 5], a specialized computer code referred to as a design tool for low-amplitude thermo-acoustic energy conversion, has been developed by Los Alamos National Laboratory based on the linear thermo-acoustic theory and has been used for modeling many thermo-acoustic devices including prime movers and refrigerators by many researchers [ 6– 9]. The oscillating amplitude of the pressure of a small TASHE is mostly less than 10% of the mean pressure, which is fit for the low oscillating amplitude acoustic approximation. So DeltaEC has been employed in this paper for basic calculations. DeltaEC applies the shooting method and integrates Rott’s equation numerically segment by segment throughout the whole device using the sinusoidal time dependence of the variables. Figure 1 shows the schematic diagram of the small thermo-acoustic Stirling engine that will be studied in this paper.

The TASHEs with a whole length of less than 1.5 m has been focused on. Table 1 lists the basic dimension data of the main parts of the engine used in the numerical model, i.e., main ambient heat exchanger (MAHE), regenerator, hot heat exchanger (heater), buffer tube, second ambient heat exchanger (SAHE), feedback inertance, compliance, and resonator. The loop length has been kept around 1.2 m. As helium is the common working media in the field of refrigeration and cryogenics, it is chosen as the working media for the small TASHE studied in this paper. Besides, the influence of the dimensions of the key parts on the main performances such as the working frequency, the heating temperature and the amplitude of the pressure has been studied. Figure 2 depicts the DeltaEC model of the engine.

Simulation results

In the numerical simulation, the effect of almost all the dimensions of the physical parts on the performances of the small TASHE has been modeled. The numerical results demonstrate that the length and diameter of the resonator, the diameter of the cavity of the resonator, the diameter of the compliance, the screen mesh and the length of the regenerator have the most significant influences on the performances. The calculation conditions are the mean pressure of 2 MPa and heating power of 400 W.

Effects of resonator and its cavity

Figures 3 and 4 show the calculated influence of the length of the resonator (within 1.5 m) on working frequency, heating temperature, and the amplitude of the pressure (including amplitudes of the pressure above MAHE and in the resonator cavity), respectively. The inner diameter of the resonator and the cavity of the resonator is 30 mm and 102 mm, respectively. It is seen that as the length of the resonator increases, the working frequencies monotonously decreases and the heating temperature has its lowest value around 1 m. The amplitudes of the pressure above MAHE and in the cavity of the resonator both increase.

Figures 5 and 6 display the variation of working frequency, heating temperature and the amplitude of the pressure as the inner diameter of the resonator increases. With the inner diameter of the resonator increasing, the working frequency monotonously increases while the heating temperature has its lowest value around 34 mm. The amplitude of the pressure experiences a maximum around 26 mm and then begins to decrease with the inner diameter of the resonator increasing. This indicates that the diameter of the resonator should be neither too small nor too large with given a structure of the loop.

The calculated influence of the inner diameter of the cavity of the resonator on the performances of the small TASHE is exhibited in Figs. 7 and 8 while the length of the cavity of the resonator is kept around 305 mm including the conical length. It can be seen that the working frequency and the heating temperature both decrease as the inner diameter of the cavity of the resonator increases. The working frequency decreases from 120 Hz to 75.6 Hz when the inner diameter of the cavity of the resonator increases from 83 mm to 300 mm. The amplitude of the pressure in the cavity of the resonator decreases while the amplitude of the pressure above MAHE increases with the inner diameter of the cavity of the resonator increasing. When the resonator cavity diameter was smaller than 117 mm, the amplitude of the pressure was higher than that above MAHE. When the inner diameter of the cavity of the resonator was over 117 mm, the amplitude of the pressure above MAHE surpassed that in the cavity of the resonator. This means that for a small-scale TASHE, the volume of the cavity of the resonator relative to the volume of the loop determined the part that plays the role of the cavity of the real resonance.

Effects of regenerator and compliance in the loop

The dimensions of compliance and the regenerator have more influence than the acoustic parts in the loop on the performances of the TASHE. The compliance and the feedback tube play an important role in transporting part of the acoustic work amplified on the hot side of the thermo-acoustic core back to the cold side. From Figs. 9 and 10, it can be observed that the working frequency decreases from 115.5 Hz to 103.3 Hz with the diameter of the compliance increasing from 50 mm to 80 mm. The heating temperature is gradually increased with the increasing of the inner diameter of the compliance tube. The amplitude of the pressure in the cavity of the resonator changes slightly while the amplitude of the pressure above MAHE gradually decreases as the diameter of the compliance increases.

The regenerator is the most important part in the thermo-acoustic core. It is consisted of a pile of stainless steel screen, so the specifications of the screen influence the porosity and hydraulic radius of the regenerator, thus influence the performances of the engine. Figures 11 to 14 show the effect of different meshes of the screen and the length of the regenerator on the performances of the TASHE. It is seen that the screen mesh and the length of the regenerator have relatively little influence on the working frequency. The lowest heating temperature is for mesh #100 and the heating temperature increases significantly with the mesh and its length of the regenerator. The amplitude of the pressure gradually increases with both the increasing of the mesh of the screen and the length of the regenerator.

The ratio of the heating temperatures at the ends of the regenerator and the amplitude of the pressure above MAHE can be expressed as

(4) \tau \approx \mathrm{1}+{\displaystyle \frac{R}{{\varpi }^{\mathrm{2}}LC{R}_{}}},

{\left|p_{1,c}\right|}^2\approx \frac{2R}{{\tau \varpi }^{2}LC}{Q}_h

where τ, R, R_res,

\varpi

, L, C, and Q_h denote the ratio of heating temperatures at the ends of the regenerator, the resistive impedance of the regenerator, the resistive impedance of the resonator, the frequency, the feedback inertance, the compliance, and the heating power, respectively.

Most of the simulation graphs regarding the heating temperatures and the pressure amplitude can be connected with Eqs. (4) and (5). Taking the case of the influence of the length of the resonator on the performance of the small engine for instance, when the length of the resonator increases, the working frequency decreases and the resistive impedance of the resonator increases, too. Similarly, as the diameter of the resonator increases, the working frequency increases while the resistive impedance of the resonator decreases. These make the curves of the heating temperature over the length of the resonator and the diameter of the resonator have inflection points as shown in Figs. 3 and 5. Meanwhile, the amplitudes of the pressure in the small engine system shown in Fig. 4 increase with the increasing length of the resonator because of the obvious decreasing working frequency. However, the effect of the cavity of the resonator of a small thermo-acoustic engine has not been involved in Eqs. (4) and (5). In this paper, the effect of the cavity of the resonator on the performance of a small thermo-acoustic engine has been investigated. The regenerator and the compliance are located in the loop. It is not convenient to unload and assemble regenerator and the compliance frequently. It is much easier to change the resonator and the cavity of the resonator and the performance of the small-scale TASHE are more sensitive to these two parts, which means that the resonator and the cavity in the end have more advantages to have better performance on the TASHE. The following experiments will focus on the resonance, including the resonator and the cavity of the resonator.

Experiments

Based on the above calculated results, a prototype of a small-scale thermo-acoustic engine has been constructed with an overall size of approximately 1.5 m in length and 0.5 m in height. The central length of the loop is approximately 1.2 m, almost equivalent to that of the resonator. Three resonators with different diameters (30 mm, 34 mm and 39 mm) have been fabricated with the same length of 1 m. Two cavities of the resonator with an inner diameter of 102 mm and 153 mm have also been fabricated with a volume of 1.96 L and 4.34 L, respectively. The thickness of the stainless vessel ducts forming the engine is no less than 3 mm. The calculated volume porosity and the hydraulic radius of the regenerator made of stainless steel screens of mesh #100 are 0.67 µm, 50 µm, respectively. The engine system has been designed to handle a mean pressure of up to 3.0 MPa and a maximum temperature of 700°C at the hottest zone in the loop. In the thermo-acoustic engine, three thermocouples have been placed axially across the regenerator through a tiny hole to obtain the characteristics of the temperature. Seven dynamic pressure sensors have been used to measure the pressure signals, as shown in Fig. 1. Besides, a PT100 has been used to measure the cooling temperature in the case when the engine is connected with a cooler.

Figure 15 shows the experimental working frequencies of the engine with different diameters of the resonator. It is seen that the experimental working frequencies are in good agreement with the calculated ones for three different diameters of the resonator with a resonator cavity of 1.96 L. When the diameter of the resonator is decreased from 39 mm to 30 mm, the working frequency is decreased from 130 Hz to 107 Hz. So, when the resonator cavities with larger diameters have been investigated, the resonator with a diameter of 30mm is chosen. For the resonator with a cavity of 4.34 L, the experimental working frequency is also in accordance with the calculated result. For the resonator with a cavity of 17.8 L, only the calculated result was given and the large designed resonator cavity will be used in the future experimental device.

The onset temperature difference determines whether a thermo-acoustic engine has the actual potential in utilizing low quality heat sources. The heating temperature in the numerical simulation has often been used to help design the experimental device, which is different from the onset temperature of the engine because the onset process of a thermo-acoustic engine is abruptly unstable while the numerical simulation was conducted according to the linear thermo-acoustic theory based on the stable assumption and the low oscillating amplitude acoustic approximation. This experimental onset temperature difference of the small-scale TASHE as a function of mean pressure with three different resonator diameters is plotted in Fig. 16. It can be seen that as the diameter of the resonator decreases from 39 mm to 30 mm, the lowest onset temperature difference significantly decreases by 150–200°C. For the resonator with a diameter of 30 mm and with helium as the working gas, the engine achieves its lowest onset temperature of 313.6°C at a mean pressure of 1.4 MPa. For convenience, the mean pressure for the lowest onset temperature difference with a certain resonator is denoted as p_opti. It can also be seen that the value of p_opti gets higher as the resonator diameter decreases, which is beneficial for the engine to manage to work in a much higher mean pressure range.

The influence of the resonator diameter on the pressure amplitude was also experimentally investigated with three different resonators, as shown in Fig. 17. It can be seen that among the three different resonators, the resonator with the inner diameter of 30 mm achieved the largest pressure amplitude. Note that the pressure amplitude in the previous calculated figures was half of the peak to peak amplitude. Here, the pressure amplitude was the peak to peak amplitude in Fig. 17. The experimental pressure amplitude was much lower than the simulated result for all kinds of energy losses in the practical device, but the experimental variation trend of the pressure amplitude as a function of the resonator diameter was in reasonable agreement with the simulated ones.

The onset temperature difference is demonstrated in Fig. 18 with a diameter resonator of 30 mm but with two different resonator cavities. It can be seen that the engine experimentally achieves its best performance with a bigger resonator cavity of 4.34 L. The onset temperature difference drops below 200°C at 198.2°C, which enhances the confidence in practical utilization of low quality heat sources like solar power. Besides, with a resonator cavity of 4.34 L, the working frequency is also significantly decreased to 90 Hz and the efficiency of the heating power to acoustic work (at resonator entrance, i.e., T-junction) is estimated to be near 22% with a total heating power of 800 W. The amplitudes of the pressure at different points within the small TASHE system are given in Fig. 19 with the two resonator cavities. P1 is located just above the main ambient heat exchanger. P2 is placed near the entrance of the feedback tube at the lower-left corner of the loop. P3 is installed near the entrance of the compliance at the upper-left corner of the loop. P4 is put at the tee. P5 is installed near the left end of the main resonator. P6 and P7 are placed in the middle of the main resonator and the resonator cavity, respectively. It can be seen that, for the resonator cavity with a relatively smaller diameter of 102 mm, the highest amplitude of the pressure is in the resonator cavity while for a larger resonator cavity with a diameter of 153 mm, the acoustical field in the engine is changed and the highest amplitude of the pressure is above MAHE. This indicates that the resonator cavity with a diameter of 153 mm has a better performance in simulating as an opening than the other one with a diameter of 102 mm. The experimental results show that the predictions in Fig. 8 are convincible.

Conclusions

To make small-scale thermo-acoustic Stirling engines practically utilize low quality heat or new energy like solar power to drive refrigerators or generators with a low working frequency in a limited space, some effort have been exerted to improve the performance of small TASHEs. A small TASHE has been numerically and experimentally investigated, focusing on the influence of the dimensions of the main parts on the performance of the engine. The numerical and experimental results demonstrate that the resonator tube and its cavity play a very important role in improving the performance of a small TASHE. The small TASHE experimentally achieves a working frequency of 90 Hz and an onset temperature difference of 198.2 K. A co-axial pulse tube cooler has been connected to the small TASHE for a trial, and the cooling temperature reaches 194.5 K using helium at a mean pressure of 2.5 MPa. Further work should be devoted to putting this kind of promising machine into practice in the future.

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