TY - JOUR
AU1 - Farikhah,, Irna
AB - Abstract Waste heat is an environmental issue in the world. There are some technologies that can be used to recovery the waste heat, one of which is thermoacoustic cooler technology. Thermoacoustic technology can be divided into two parts: one is thermoacoustic engine and cooler. To design the cooler system having high efficiency and lower onset heating temperature, the effect of mean pressure is investigated. By increasing mean pressure from 0.5 to 3 MPa, the heating temperature generating acoustic power can be decreased from 831 to 580 K. Moreover, 15% of Thermodynamic upper limit value of the whole cooler system is achieved. 1. INTRODUCTION Thermoacoustic technology, which is a combination of acoustic and thermal energy, has attracted scientists for conducting some investigations related to energy conversion employing pistonless power [1–12]. This power can be generated by a lot of energy resources available in the world such as solar, geothermal and waste heat energy [13]. There is a current issue on heat recovery due to manufacture processes in industries resulting waste heat [14]. Therefore, it is important to be properly tackled. Moreover, the waste heat has the range level below than 673 K [7]. Using this technology, heat can be converted into acoustic power. In addition, the generated power can be used for some application such as driving a heat pump [7, 9, 11]. In the thermoacoustic field, there are two types of a heat-driven cooler regarding the number of engine regenerator; one is heat pump driven by one engine i.e. single-stage engine [4–8, 10], and another one is heat pump driven by some engines namely multi stage engine [9, 11]. Some researchers have been attracted by the multistage heat-driven cooler instead of single heat-driven due to the ability to reach lower onset heating temperature in the engine. However, this configuration is more complicated than the single-stage type, employing only one engine, in which the construction is simpler, has a high onset heating temperature [4–6, 8, 10, 12]. Furthermore, both types of the heat-driven cooler have low Carnot COP, particularly Carnot’s COP of a single heat-driven cooler has 2% of the Carnot’s COP [5, 10]. Some parameters have an impact on having low onset temperature and high performance. Zang et al. [11] conducted an investigation related to multi stage heat-driven cooler and found that by increasing mean pressure in this system, the onset heating temperature for generating acoustic power in the engine is decreasing [11]. However, they need more engine regenerators. To make it simpler, a single-stage heat-driven thermoacoustic cooler is investigated. In 2002, Yazaki et al. constructed the single-stage heat-driven thermoacoustic cooler. However, they did not measure the efficiency of the cooler system [4]. Moreover, they also did not optimize the parameter to get high efficiency of the whole system. Then, in 2017 based on the experimental investigation done by Yazaki et al., Farikhah et al. [12] calculated numerically efficiency of the cooler system. In addition, they also optimized the parameters especially the ratio of narrow flow-channel radii and the penetration depth |$r/\delta$| and relative position between engine and cooler regenerators |$L/{L}_{\mathrm{loop}}$| where L is the length between engine and cooler regenerators and |${L}_{\mathrm{loop}}$| is the length of the looped tube. However, another parameter such as mean pressure of the working gas was not evaluated. Therefore, in this study, we have numerically investigated the effect of mean pressure on the onset temperature of the engine by using the thermoacoustic theory proposed by Rott [15, 16] and advanced by Swift [17] and Tominaga [18, 19]. 2. METHOD 2.1. Calculation model In Figure 1, the model of the thermoacoustic cooler consists of two main parts of thermoacoustic components, namely engine regenerator and heat pump (cooler) regenerator. The engine regenerator is sandwiched with hot and ambient heat exchanger. The initial point of x is set at the ambient side of the engine regenerator [see Figure 1] The temperature of ambient and cold heat exchanger denotes as |${T}_a$| and |${T}_c$|⁠, respectively. |${T}_a$| is 28°C and |${T}_c$| is −22°C. This investigation is following the numerical conditions outlined by Farikhah [12]. The working gas is Helium varied |${P}_m$| from 0.5 to 3 MPa. Figure 1 Open in new tabDownload slide Schematics of the heat-driven thermoacoustic cooler. Figure 1 Open in new tabDownload slide Schematics of the heat-driven thermoacoustic cooler. Moreover, in the looped tube there are two thermal buffer tubes: the first is connected to a hot heat exchanger where the temperature change gradually from hot temperature |${T}_h$| to |${T}_a$| and the other is near a cold heat exchanger where the temperature change gradually from hot temperature |${T}_c$| to |${T}_a$|⁠. The length of the regenerator is 40 mm, and their porosity is set to 0.77. In addition, according the research conducted by Farikhah et al. [12], they found the optimum dimensionless radii |${(r/\delta)}_{\mathrm{opt}}$| and |$L/{L}_0$|are about 1 and 0.53, respectively [12]. Therefore, we used the optimum values. All of these components are installed into a looped tube having a diameter of 40 mm and length of 2.8 m. These values are similar to those Yazaki’s experimental setup [4]. The value of the mean pressure of the working gas |${P}_m$| filled inside the looped tube is varied because it will be used as a parameter to reveal the dependence of low-onset temperature for generating the engine and the performance of the whole thermoacoustic cooler system. We varied the mean pressure from 0.50 to 3.0 MPa. 2.2. Calculation method In the computation, we use the following two equations derived from Rott [15]. $$\begin{equation} \frac{dP}{dx}=-\frac{i\omega{\rho}_m}{1-{\chi}_{\nu }}U \end{equation}$$(1) $$\begin{equation} \frac{dU}{dx}=-\frac{i\omega \left[1+\left(\gamma -1\right){\chi}_{\alpha}\right]}{\gamma{P}_m}P+\frac{\chi_{\alpha }-{\chi}_{\nu }}{\left(1-{\chi}_{\nu}\right)\left(1-\sigma \right)} \end{equation}$$(2) where |$x$| is the axial coordinate along the looped tube, |${\chi}_{\alpha }$| and |${\chi}_{\nu }$| are spatially averaged thermal and viscous function, respectively. |$\sigma$|⁠, |$\gamma$|and |${\rho}_m$| are the prandtl number, ratio of isobaric to isochoric specific heats and mean density of the working gas, respectively. All of these are mentioned in Table 1. |$P$| and |$U$| are oscillatory pressure and cross-sectional velocity. The gas properties. Table 1 The gas properties. |${P}_m$| (MPa) . |${\chi}_{\alpha }$| . |${\chi}_{\nu }$| . |$\sigma$| . |$\gamma$| . |${\rho}_m$||$(\mathrm{kg}/{\mathrm{m}}^3)$| . 0.5 0.9579 0.9158 0.6627 1.6660 0.801 1.0 0.9462 0.8946 0.6616 1.6652 1.597 1.5 0.9172 0.8457 0.6605 1.6649 2.549 2.0 0.8999 0.8188 0.6597 1.6646 3.180 2.5 0.8766 0.7846 0.6586 1.6640 4.123 3.0 0.8594 0.7608 0.6579 1.6632 4.749 |${P}_m$| (MPa) . |${\chi}_{\alpha }$| . |${\chi}_{\nu }$| . |$\sigma$| . |$\gamma$| . |${\rho}_m$||$(\mathrm{kg}/{\mathrm{m}}^3)$| . 0.5 0.9579 0.9158 0.6627 1.6660 0.801 1.0 0.9462 0.8946 0.6616 1.6652 1.597 1.5 0.9172 0.8457 0.6605 1.6649 2.549 2.0 0.8999 0.8188 0.6597 1.6646 3.180 2.5 0.8766 0.7846 0.6586 1.6640 4.123 3.0 0.8594 0.7608 0.6579 1.6632 4.749 Open in new tab Table 1 The gas properties. |${P}_m$| (MPa) . |${\chi}_{\alpha }$| . |${\chi}_{\nu }$| . |$\sigma$| . |$\gamma$| . |${\rho}_m$||$(\mathrm{kg}/{\mathrm{m}}^3)$| . 0.5 0.9579 0.9158 0.6627 1.6660 0.801 1.0 0.9462 0.8946 0.6616 1.6652 1.597 1.5 0.9172 0.8457 0.6605 1.6649 2.549 2.0 0.8999 0.8188 0.6597 1.6646 3.180 2.5 0.8766 0.7846 0.6586 1.6640 4.123 3.0 0.8594 0.7608 0.6579 1.6632 4.749 |${P}_m$| (MPa) . |${\chi}_{\alpha }$| . |${\chi}_{\nu }$| . |$\sigma$| . |$\gamma$| . |${\rho}_m$||$(\mathrm{kg}/{\mathrm{m}}^3)$| . 0.5 0.9579 0.9158 0.6627 1.6660 0.801 1.0 0.9462 0.8946 0.6616 1.6652 1.597 1.5 0.9172 0.8457 0.6605 1.6649 2.549 2.0 0.8999 0.8188 0.6597 1.6646 3.180 2.5 0.8766 0.7846 0.6586 1.6640 4.123 3.0 0.8594 0.7608 0.6579 1.6632 4.749 Open in new tab Eq. (1) and Eq. (2) can be solved analytically if |${dT}_m/ dx=0$|⁠. However, if |${dT}_m/ dx\ne 0$|⁠, the equations have to be computationally integrated. The temperature of the heat exchanger |${T}_h$| is determined as a result of stability limit calculation developed by Ueda [20]. We modified Equations (1) and (2) become $$\begin{equation} \frac{d}{dx}\left(\genfrac{}{}{0pt}{}{P\left(x,t\right)}{U\left(x,t\right)}\right)=B(x)\left(\genfrac{}{}{0pt}{}{P\left(x,t\right)}{U\left(x,t\right)}\right) \end{equation}$$(3) $$ B(x)\equiv \left(\begin{array}{cc}0& \frac{- i\omega{\rho}_m}{1-{\chi}_{\nu }}\\{}\frac{- i\omega \left[1+\left(\gamma -1\right){\chi}_{\alpha}\right]}{\gamma{P}_m}& \frac{\chi_{\alpha }-{\chi}_{\nu }}{\left(1-{\chi}_{\nu}\right)\left(1-\gamma \right)}\frac{1}{T_m}\frac{dT_m}{dx}\end{array}\right) $$ Using the fourth-order Runge–Kutta method to Equation (3), a forward different scheme is applied as follows: $$\begin{equation} \left(\genfrac{}{}{0pt}{}{P\left(x+\Delta x,t\right)}{U\left(x+\Delta x,t\right)}\right)=\left(E+\Delta x{C}^{\prime }(x)\right)\left(\genfrac{}{}{0pt}{}{P(x)}{U(x)}\right) \end{equation}$$(4) $$ {C}^{\prime }(x)=\frac{1}{6}\left({RK}_1+2{RK}_2+2{RK}_3+{RK}_4\right) $$ $$ {RK}_1=B(x) $$ $$ {RK}_2=B\left(x+\frac{\Delta x}{2}\right)\left(E+\frac{\Delta x}{2}{RK}_1\right) $$ $$ {RK}_3=B\left(x+\frac{\Delta x}{2}\right)\left(E+\frac{\Delta x}{2}{RK}_2\right) $$ $$ {RK}_4=B\left(x+\Delta x\right)\left(E+\Delta x{RK}_3\right) $$ Here E is a unit matrix. Hence, $$\begin{equation} \left(\genfrac{}{}{0pt}{}{P\left(x,t\right)}{U\left(x,t\right)}\right)={M}_{II}\left(x,{x}_0\right)\left(\genfrac{}{}{0pt}{}{P_0\left({x}_0,t\right)}{U_0\left({x}_0,t\right)}\right) \end{equation}$$(5)|${M}_{II}\left(x,{x}_0\right)\equiv (E+\Delta x{C}_{n-1}^{\prime})(E+\Delta x{C}_{n-2}^{\prime})\ldots (E+\Delta x{C}_1^{\prime})(E+\Delta x{C}_0^{\prime})$| where n is the number of partitions between |${x}_0$| and |$x$| and |$\Delta x$| denote as |$(x-{x}_0)/n$|⁠, and |${C}_j^{\prime }$| is. |${C}^{\prime }$| at |$x={x}_0+j\Delta x$|⁠. As shown in Figure 1, the thermoacoustic system was divided into 10 parts: (1) ambient heat exchanger, (2) engine regenerator, (3) hot heat exchanger, (4) thermal buffer tube, (5) wave guide 1, (6) ambient heat exchanger, (7) cooler regenerator, (8) cold heat exchanger, (9) thermal buffer tube 2 and (10) wave guide 2. Then, we did a numerical integration along the parts and we obtained the transfer matrices. Since the areas of the whole flow-path of each part are different, so we found the connecting matrix |${S}_{q,r}$| and the total of the transfer matrices as follows: $$ {M}_{\mathrm{all}}={M}_{10}{S}_{10,9}{M}_9{S}_{9,8}{M}_8{S}_{8,7}{M}_7{S}_{7,6} $$ $$\begin{equation} {M}_6{S}_{6,5}{M}_5{S}_{5,4}{M}_4{S}_{4,3}{M}_3{S}_{3,2}{M}_2{S}_{2,1}{M}_1 \end{equation}$$(6) where |${S}_{q,r}$| is $$\begin{equation} {S}_{q,r}=\left(\begin{array}{cc}1& 0\\{}0& {A}_q/{A}_r\end{array}\right) \end{equation}$$(7) The total path area is denoted as |$A$|⁠. The number of the component is denoted as the subscript |$q$| and |$r$|⁠. Here |${M}_{\mathrm{all}}$| is used in the calculation the pressure |${P}_{a,e}$| and |${U}_{a,e}$| at the first ambient heat exchanger as follows: $$\begin{equation} {M}_{\mathrm{all}}\left(\genfrac{}{}{0pt}{}{P_{a,e}}{U_{a,e}}\right)=\left(\genfrac{}{}{0pt}{}{P_{a,e}}{U_{a,e}}\right) \end{equation}$$(8) If the matrix determinant |$\Big({M}_{\mathrm{all}}-E\Big)$| is zero, the solution |$({P}_{a,e},{U}_{a,e})$| of Equation (6) is non zero. $$\begin{equation} \left({m}_{11}-1\right)\left({m}_{22}-1\right)-{m}_{12}{m}_{21}=0 \end{equation}$$(9) where the elements of |${M}_{\mathrm{all}}$| is denoted |${m}_{ij}$|⁠; hence, Equation (9) is solved numerically. Then, the stability limit condition of the spontaneous gas oscillation induced in the thermoacoustic system. Then, we can find the onset heating temperature. In this integration of |${dT}_m/ dx$|⁠, two temperature conditions are used. The first is that |${dT}_m/ dx$| takes a constant value. Using this, the value of |${T}_h$| can be obtained. The second is |${dT}_m/ dx$| which is calculated with the assumption that the tube is insulated from its surroundings. Therefore, the enthalpy flow |$\dot{H}$| along the regenerators must be set to be constant [17]. Rott [16] suggests |$\dot{H}$| can be expressed as $$\begin{equation} \dot{H}=\dot{W}-\dot{Q} \end{equation}$$(10) where |$\dot{H}$|⁠, |$\dot{W}$| and |$\dot{Q}$| are total energy flow, acoustic power and heating power, respectively. |$\dot{W}$| and |$\dot{Q}$| are as follows [17]: $$\begin{equation} \dot{W}=\frac{A}{2}\mathit{\operatorname{Re}}[P\overset{\sim }{U}] \end{equation}$$(11) $$\begin{align} \dot{Q}&=\frac{A}{2}\mathit{\operatorname{Re}}\left[P\overset{\sim }{U}\left(\frac{{\overset{\sim }{\chi}}_{\nu }-{\chi}_{\alpha }}{\left(1+\sigma \right)\left(1-{\overset{\sim }{\chi}}_{\nu}\right)}\right)\right]\nonumber\\[3pt] &\qquad-\frac{A{\rho}_m{c}_p{\left|U\right|}^2}{2\omega \left(1-{\sigma}^2\right){\left|1-{\chi}_{\nu}\right|}^2}\mathit{\operatorname{Im}}\left[{\chi}_{\alpha }+\sigma{\overset{\sim }{\chi}}_{\nu}\right]\frac{dT_m}{dx} \end{align}$$(12) Then, Equations 11 and 12 can be substituted into Equation 10 and can be expressed as $$\begin{equation} \frac{dT_m}{dx}=\frac{\dot{H}-\frac{A}{2}\mathit{\operatorname{Re}}\left[P\overset{\sim }{U}\left(1-\frac{{\overset{\sim }{\chi}}_{\nu }-{\chi}_{\alpha }}{\left(1+\sigma \right)\left(1-{\overset{\sim }{\chi}}_{\nu}\right)}\right)\right]}{\frac{A{\rho}_m{c}_p{\left|U\right|}^2}{2\omega \left(1-{\sigma}^2\right){\left|1-{\chi}_{\nu}\right|}^2}\mathit{\operatorname{Im}}\left[{\chi}_{\alpha }+\sigma{\overset{\sim }{\chi}}_{\nu}\right]} \end{equation}$$(13) where |${T}_m$| is the mean temperature, |$x$| is the axial coordinate along the regenerators, |$\overset{\sim }{U}$| is the conjugate value of velocity, |$A$| is the cross-sectional area and |$\omega$| is the angular frequency. Temperature gradient along the regenerators |${dT}_m/ dx\ne 0$| can be calculated when the boundary conditions |$P$| and |$U$| are given, by coupling Equations 1–2 and 10–13. In our calculation, the acoustic streaming [3, 17] and the thermal conduction along the |$x$| axis are neglected. In this calculation, the performance of the whole devices is verified. It includes efficiency of the engine |${\eta}_{2,e}$|⁠, looped tube |${\eta}_{\mathrm{tube}}$| and heat pump |${\eta}_{2,c}$|⁠. First, to find the formula for the engine efficiency, acoustic power generated in the engine must be defined. The power can be expressed as $$\begin{equation} \Delta{\dot{W}}_e={\dot{W}}_{e,H}-{\dot{W}}_{e,A} \end{equation}$$(14) where |$\Delta{\dot{W}}_e$|⁠, |${\dot{W}}_{e,H}$| and |${\dot{W}}_{e,A}$| are the gain of acoustic power in the engine and acoustic power at the ambient end of the engine and at the hot end of the engine, respectively. Then, the efficiency can be defined as $$\begin{equation} {\eta}_{2,e}=\frac{\Delta{\dot{W}}_e/{\dot{Q}}_h}{\eta_{Carnot}} \end{equation}$$(15) where |${\eta}_{2,e},\Delta{\dot{W}}_e\ \mathrm{and}{\dot{Q}}_h$| are second law efficiency of the engine, gain of acoustic power and heating power, respectively. Here, |${\dot{Q}}_H$| is the heating power of engine imposed by the hot temperature at hot side, and |${\eta}_{Carnot}$| is the thermodynamic upper limit of |${\eta}_e$| as follows [21]. $$\begin{equation} {\eta}_{car}=1-\frac{T_a}{T_h} \end{equation}$$(16) where |${\eta}_{car}$| is the Carnot efficiency of the engine, |${T}_a$| is the ambient temperature and |${T}_h$| is the heating temperature. The second is the efficiency of the looped tube |${\eta}_{\mathrm{tube}}$|⁠. It can be expressed as $$\begin{equation} {\eta}_{\mathrm{tube}}=\frac{{\dot{W}}_{hp,a}-{\dot{W}}_{hp,c}}{\Delta{\dot{W}}_e} \end{equation}$$(17) where |${\eta}_{\mathrm{tube}},{\dot{W}}_{hp,a},{\dot{W}}_{hp,c}$| are the efficiency of the looped tube and acoustic power at the ambient end of the cooler and at cooler end, respectively. Finally, the efficiency of the heat pump can be defined as $$\begin{equation} {\eta}_{2,c}=\frac{{\dot{Q}}_c/\left({\dot{W}}_{hp,a}-{\dot{W}}_{hp,c}\right)}{COP_{Carnot}} \end{equation}$$(18) where |${\eta}_{2,c},{\dot{W}}_{hp,a},{\dot{W}}_{hp,c}$|are the second law efficiency of the heat pump and acoustic power at the ambient end and the cold end, respectively. |${COP}_{Carnot}$| is the thermodynamic upper limit of the heat pump [21]. It can be expressed as follows: $$\begin{equation} {COP}_{Carnot}=\frac{T_c}{T_a-{T}_c} \end{equation}$$(19) Thus, the second law thermodynamic of the total efficiency of this device is as follows $$\begin{equation} {\eta}_{2,\mathrm{total}}=\frac{{\dot{Q}}_c/{\dot{Q}}_h}{\eta_{car.}{COP}_{car not}} \end{equation}$$(20) where |${\eta}_{2,\mathrm{total}}$| is the total efficiency of the cooler system. Moreover, it can be written as $$\begin{equation} {\eta}_{2,\mathrm{total}}={\eta}_{2,e}.{\eta}_{2, hp}.{\eta}_{\mathrm{tube}} \end{equation}$$(21) 3. RESULTS AND DISCUSSION Zang et al. [11] reported that there is a dependence of onset heating temperature on a multi-heat-driven cooler. Therefore, we investigated the effect of mean pressure |${P}_m$| on a single-stage heat-driven thermoacoustic cooler. Moreover, it is known that the design parameters of the regenerators for instance |$r/\delta$| have an impact on the performance of a cooler [12, 22]. In this investigation, therefore, we selected the optimum |$r/\delta$| at each variation of |${P}_m$|⁠. Thus, the result was not only low- and medium-onset heating temperature |${T}_h$| of the single-stage engine, but also the high efficiency of the whole cooler system. As shown in Figure 2, the total efficiency |${\eta}_{2,\mathrm{total}}$| increases as |${P}_m$| increases from 0.50 to 3.0 MPa. It can also be seen clearly in Figure 2 as |${P}_m$| increases |${T}_h$| decreases. This result is consistent with the results found by Zhang et al. [11]. They found that by increasing the mean pressure from 0.5 to 3 MPa the onset temperature different is decreasing. However, they did not evaluate the efficiencies. Moreover, we also compare them with the results found by Yu et al. [23]. In our results, we selected the optimum |$r/\delta$| [12] before we investigate the effect of |${P}_m$| on |${T}_h$|⁠. Likely, the results by Yu et al. [23], they also found the optimum |$r/\delta$| for finding the effect of |${P}_m$| on |$\Delta{T}_h$|⁠. As we can see in Figure 3 [23], the optimum dimensionless radius|${(r/\delta =2{r}_h/\delta )}_{\mathrm{opt}}$|where |${r}_h$| is the hydraulic radius.|${(r/\delta )}_{\mathrm{opt}}$| at r = 142.2, |$r=44.6$| and |$r=36.4\ \mu \mathrm{m}$| are 0.67, 0.43 and 0.49, respectively [23]. Moreover, it can be seen that in Figure 2 [23] the optimum |${P}_m$| was found. As we can see from this figure, we know that with keeping optimum |${(r/\delta )}_{\mathrm{opt}}$|the increasing mean pressure |${P}_m$| leads the decreasing temperature difference of the heating temperature |$\Delta{T}_h$|⁠. Figure 2 Open in new tabDownload slide Total efficiency and heating temperature as a function of mean pressure. Figure 2 Open in new tabDownload slide Total efficiency and heating temperature as a function of mean pressure. Figure 3 Open in new tabDownload slide Mean pressure as a function of temperature difference [23]. Figure 3 Open in new tabDownload slide Mean pressure as a function of temperature difference [23]. As mentioned above, in this calculation, we set |${T}_C=251\ K$|⁠. However, the value of |${T}_h$| is varied as a result of stability limit calculation by variation of |${P}_m$|⁠.It can be seen from Equation (13) that the cold constant temperature and the thermal relaxation dissipation power (namely, the second term in the fractional numerator on the right side) decrease as the mean pressure increases, resulting in a lower critical onset heating temperature difference. As mentioned above, |${\eta}_{2,\mathrm{total}}$| consists of efficiency of the engine |${\eta}_{2,e}$|⁠, efficiency of the heat pump |${\eta}_{2,c}$|⁠, and efficiency of the looped tube |${\eta}_{\mathrm{tube}}$|⁠. Therefore, it is essential to know those values. Figure 4 shows that |${\eta}_{2,e}$| remains constant at a high value at about 72%. This value is comparable to the efficiency of the most efficient thermoacoustic engine constructed [3, 24]. Like the value of |${\eta}_{2,c}$|, this value also remains constant at about 44%. The value of |${\eta}_{2,\mathrm{total}}$|is much higher than the value of that obtained in the previous research [4–6, 8, 10]. The highest efficiency found in the previous research is below 2% [5, 10]. On the other hand, employing multi-stage engine for driving cooler, the construction is more complicated and the efficiency reaches 32% [9]. Therefore, compared to the other heat-driven coolers [1–11], we can say that the present result has high performance while keeping in the lower heating temperature using a single stage type. $$\begin{equation} {\left[\frac{dT_m(x)}{dx}\right]}_{\mathrm{crit}}=\frac{\frac{1}{2}\omega{\rho}_m\frac{\mathit{\operatorname{Im}}\left[-{\chi}_{\nu}\right]}{A{\left|1-{\chi}_{\nu}\right|}^2}{\left|U\right|}^2+\frac{1}{2}\frac{\gamma -1}{\gamma{P}_m}\omega{A}_f\mathit{\operatorname{Im}}\left[-{f}_k\right]{\left|{p}_1\right|}^2}{\beta \frac{1}{2}\left|{p}_1\right|\left|{U}_1\right|\left[\mathit{\operatorname{Re}}\left[g\right]\cos \theta -\mathit{\operatorname{Im}}\left[g\right]\sin \theta \right]} \end{equation}$$(21) where g is the complex gain factor arising in the continuity equation. Figure 4 Open in new tabDownload slide Efficiency of the engine as a function of mean pressure. Figure 4 Open in new tabDownload slide Efficiency of the engine as a function of mean pressure. In this cooler, |${\eta}_{\mathrm{tube}}$| has a high impact on the value of |${\eta}_{2,\mathrm{total}}$| as shown in Figure 4 Since the value of |${\eta}_{2,e}$|and |${\eta}_{2,c}$|remains constant at high values, the dependence of |${\eta}_{2,\mathrm{total}}$| on |${P}_m$| is influenced by |${\eta}_{\mathrm{tube}}$| as shown in Figure 4. Therefore, it is essential to reveal the reason of the dependence. In Figure 4, as |${P}_m$| rises, |${\eta}_{\mathrm{tube}}$| goes up from 32 to 49%. This result can be attributed to the value of dissipation. The dissipation can be shown using a ratio of the value of intensity. In this device, there are two divisions of wave guide (looped tube). The first one is the wave guide between the hot side of the engine and the ambient side of the heat pump regenerator, denoted as a ratio of the first acoustic intensity ratio |${I}_{\mathrm{out}\ 1}/{I}_{\mathrm{in}1}$|⁠, which represents a ratio of output intensity from the engine to input intensity of the heat pump. The second one is the wave guide between the cold side of the heat pump and the ambient side of the engine regenerator, denoted as the second acoustic intensity ratio |${I}_{\mathrm{out}2}/{I}_{\mathrm{in}2}$|⁠, which represents the ratio of output intensity from heat pump to input intensity of engine. Figure 5 shows that |${I}_{\mathrm{out}\ 1}/{I}_{\mathrm{in}1}$| and |${I}_{\mathrm{out}2}/{I}_{\mathrm{in}2}$| decrease from 1.37 to 1.28 and 1.64 to 1.31, respectively. If the ratio of acoustic intensity decreases approximately 1, the dissipation is almost 0. It is indicated that the dissipation is decreased when |${P}_m$| increases. Figure 5 Open in new tabDownload slide Acoustic Intensity ratio as a function of mean pressure. Figure 5 Open in new tabDownload slide Acoustic Intensity ratio as a function of mean pressure. In addition, this result can also be explained using Equation (14) [17, 24]. $$\begin{equation} {E}_{\mathrm{loss}}=\frac{1}{4}\frac{{\left|p\right|}^2}{\gamma{P}_m}\left(\gamma -1\right){\delta}_k\omega \end{equation}$$(22) where |${E}_{\mathrm{loss}}$| is the energy dissipated by the thermal-relaxation dissipation and |${\delta}_k$| is the penetration depth. It is shown in Equation 22 that the value of |${E}_{\mathrm{loss}}$| is proportional to the value of |$1/{P}_m$|⁠. Therefore, the dissipation power decreases as the mean pressure increases. 4. CONCLUSION Increasing |${P}_m$| with keeping optimal |$r/\delta$| not only decreases onset heating temperature to generate the engine from 831 to 580 K but also increases the performance. The efficiency of the tube considerably increases from 32 to 49% while keeping the high efficiency of the engine and heat pump at 72 and 44%, respectively. As a result, 15% of the upper limit value is achieved. Therefore, using only one engine regenerator is possible to utilize lower waste heat for the engine while achieving high total efficiency of the heat-driven single-stage thermoacoustic cooler. Acknowledgements This work was supported by LPPM Universitas PGRI Semarang and Ministry of Research and Technology of the Republic of Indonesia. REFERENCES [1] Ceperley PH . A-piston-less Stirling engine . J Acoust Soc Am 1979 ; 66 : 1508 – 13 . Google Scholar Crossref Search ADS WorldCat [2] Swift GW . Thermoacoustic engine . J Acoust Soc Am 1988 ; 84 : 1145 – 80 . Google Scholar Crossref Search ADS WorldCat [3] Swift GW , Gardner DL, Backhauss S. A thermoacoustic Stirling engine . Nature. 1999 ; 399 : 335 – 8 . 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TI - The effect of mean pressure on the performance of a single-stage heat-driven thermoacoustic cooler
JF - International Journal of Low-Carbon Technologies
DO - 10.1093/ijlct/ctaa009
DA - 2020-08-19
UR - https://www.deepdyve.com/lp/oxford-university-press/the-effect-of-mean-pressure-on-the-performance-of-a-single-stage-heat-ynRPbPEIfm
SP - 471
EP - 476
VL - 15
IS - 3
DP - DeepDyve
ER -