ADVANCEMENT OF SMALL-SCALE THERMOACOUSTIC ENGINE

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1 ADVANCEMENT OF SMALL-SCALE THERMOACOUSTIC ENGINE By SUNGMIN JUNG Masters in Mechanical Engineering WASHINGTON STATE UNIVERSITY School of Mechanical and Material Engineering MAY 2009

2 To the Faculty of Washington State Uniersity: The members of the Committee appointed to examine the thesis of SUNGMIN JUNG find it satisfactory and recommend that it be accepted. Konstantin I. Matee, Ph.D., Chair Jeongmin Ahn, Ph.D. Michael J. Anderson, Ph.D. ii

3 Acknowledgements I would like to express profound gratitude to my adisor, Dr. Konstantin Matee, for his inaluable guidance and support during my graduate research. Great deelopment and adancement in my work were done with his help and suggestions. I am also grateful to my committee members, Dr. Jeongmin Ahn and Dr. Michael J. Anderson, since they helped me learn fundamental knowledge and come up with useful ideas. While I was designing components of the thermoacoustic engine, Mr. Dae M. Saage and Mr. John Rutherford, manager and machinist, respectiely, at The Physical Science Shop, aided me to build more precise fittings and suggested materials for better performance of the engine. With their skills and effort, I could get a suitable system for my research and sae much more time. Finally I would like to thank my friends, Najmeddin Shafiei-Tehrany and Andrew Brian Eans Wekin. Their preious research of thermoacoustics encouraged me and suggested aluable solutions. iii

4 ADVANCEMENT OF SMALL-SCALE THERMOACOUSTIC ENGINE Abstract by Sungmin Jung, M.S. Washington State Uniersity May 2009 Chair: Konstantin I. Matee Thermoacoustic engines are energy-conersion deices that produce acoustic power using heat flowing from a high-temperature source to a low-temperature sink. Thermoacoustic engines can be made without moing parts and using arious gases as working fluids. The simplicity of manufacturing such engines results in low cost and low maintenance and, therefore, is desirable in industry. A recently proposed candidate for small-scale electricity generation inoles a thermoacoustic engine coupled with a piezoelectric transformer. A simple thermoacoustic engine is composed of a resonator with one end closed and the other end open with a piece of porous material, referred to as a stack, placed inside the resonator at a specific location. In this research, reticulated itreous carbon was used as the stack material and atmospheric air as the working fluid. The engine was tested with resonators of ariable lengths in the range mm. The temperature difference across the stack and the acoustic pressure amplitudes inside and outside of the engine were measured and compared with theoretical alues. The engine starts generating sound at the temperature differences of C between the hot and cold parts of the system. The acoustic pressure amplitudes up to 2 kpa i

5 are measured inside the resonator in the excited regimes. The acoustic pressure amplitude in the quasi-steady tests shows a monotonic change and no significant hysteresis. In the test with fast change of the heat supply rate, the pressure amplitude shows strong hysteresis. A simplified energy-balance theory adequately predicts a trend in the temperature onset, while underestimating actual alues. Model estimations show that the stack-generated acoustic power reaches 00 mw with the stack-based efficiencies of seeral percent. Optimized geometries and conditions of the engine yield the lowest critical temperature difference of 53 and optimized efficiency is about twice the result from experimental data in 67-mm engine.

6 TABLE OF CONTENTS Acknowledgements...iii Abstract...i List of Symbols...ii List of Tables...ix List of Figures...x. Introduction..... Background Objecties Experimental system Experimental results and discussion Modeling Theory Comparison with test data Optimized design Conclusions Bibliography...48 i

7 A b B c c c p wire List of Symbols resonator cross-sectional area porosity-dependent parameter in the heat transfer coefficient systematic uncertainty speed of sound porosity-dependent parameter in the friction factor gas specific heat at constant pressure D diameter of one wire of the screen E & mesh acoustic power absorbed at the resonator walls E & rad E & res f k acoustic power radiated from the open end thermoiscous power loss at mesh layer thermal acoustic function f iscous acoustic function (spatial aerage of h ) h complex function depending on pore geometry H spacing of a parallel plate H & thermoacoustic enthalpy flow through the stack κ fluid thermal diffusiity k fluid thermal conductiity k S effectie heat conductiity of RVC matrix L length of the resonator L ' internal length of the resonator n number of data points or mesh number of the screen p acoustic pressure amplitude p m mean pressure ( = Pm ) p acoustic pressure waeform P pressure amplitude at closed end of the resonator p acoustic pressure amplitude (measured) PR random uncertainty change of acoustic pressure r k r r h R R p e specific acoustic resistance due to thermal effects specific acoustic resistance due to iscous effects hydraulic radius of the stacked screens radius of the resonator readability uncertainty ii

8 R g R p S t T T m T T c T id u U U U r W & x x S x s x m y z air gas constant characteristic size of pore standard deiation of data t-distribution coefficient gas temperature mean temperature temperature difference across the stack critical temperature difference across the stack ideal critical temperature difference acoustic elocity amplitude olumetric elocity fluctuation or total uncertainty olumetric elocity waeform change of acoustic elocity ector elocity acoustic power flow coordinate along the resonator stack position stack length mesh layer thickness coordinate perpendicular to sound-propagation direction coordinate perpendicular to sound-propagation direction ϕ porosity of stacked screens γ specific heat ratio η s thermoacoustic energy conersion efficiency σ Prandtl number σ nine-component iscous stress tensor ω angular frequency λ acoustic waelength λ dimensionless thermal disturbance number λ k dimensionless shear wae number ρ mean air density ( = ρ m ) ν gas kinetic iscosity µ gas dynamic iscosity δ k δ thermal penetration depth of the fluid iscous penetration depth of the fluid iii

9 List of Tables. Experimental results for mm engines Example of measured data and the time step in 67-mm engine Example of the uncertainty calculation of pressure amplitude in 67-mm engine Optimization results for the lowest critical temperature difference in 67-mm engine Optimization results for acoustic power flow and efficiency in 67-mm engine...44 ix

10 List of Figures. (a) The Sondhauss tube...2 (b) Schematic of the prime moer (a) Schematic of a standing-wae thermoacoustic engine...4 (b) Thermal interaction between a gas parcel and a stack plate during one acoustic cycle in the presence of high temperature gradient Pressure-olume graph of gas parcel in the standing-wae engine Schematic of a closed-end refrigerator Sketch of 67-mm engine (a) Photo of the 67-mm engine... (b) Fully assembled 00-mm engine Experimental setup of a thermoacoustic engine Example of pressure amplitude dependence on temperature difference for fast change in heat supply in 57-mm engine with RVC stack (a) Example of pressure amplitude dependence on temperature difference for quasi-steady in 67-mm engine with RVC stack...6 (b) Example of pressure amplitude dependence on temperature difference for fast change in heat supply in 67-mm engine with RVC stack (a) Example of pressure amplitude dependence on temperature difference for quasi-steady in 00-mm engine with RVC stack...7 (b) Example of pressure amplitude dependence on temperature difference for quasi-steady in 00-mm engine with steel wool stack...7. Example of pressure amplitude dependence on temperature difference for quasi-steady in heat supply in 24-mm engine with RVC stack (a) Aerage pressure amplitude recorded in 67-mm engine...2 (b) Aerage pressure amplitude recorded in 00-mm engine...2 (c) Aerage pressure amplitude recorded in 24-mm engine...2 (d) Aerage pressure amplitude recorded in 57-mm engine...2 x

11 3. (a) Example of temperature differences for electric power in 67-mm engine...23 (b) Example of pressure amplitude for electric power in 67-mm engine Relationship between f k, and λ ' k, Proportions of E & rad, E & res, E & mesh and change of W & for different lengths Critical temperature difference ersus engine length (a) Estimations for acoustic power produced in the stack...39 (b) Estimations for stack thermoacoustic efficiency (a) Critical temperature difference for radius of resonator...4 (b) Critical temperature difference for stack position...4 (c) Critical temperature difference for mean pressure...4 (d) Critical temperature difference for pore radius Critical temperature difference for arious gases in 67-mm engine Acoustic power flow and efficiency for ariable p / P m in 67-mm engine...44 xi

12 Chapter Introduction

13 . Introduction.. Background Thermoacoustics is a research field that studies heat and sound interactions. The history of thermoacoustics is old but sparsely known. The beginning of thermoacoustic approach started from simple acoustic oscillations noticed in early studies. A reiew of Putnam and Dennis describes experiments of acoustic oscillations in an organ-pipe []. It refers to experiments of Byron Higgins in 777 in which acoustic oscillations in a large pipe were excited by suitable placement of a hydrogen flame inside [2]. The Sondhauss tube, shown in Figure (a), was built in 850. It is considered as the earliest thermoacoustic engine that is a direct antecedent of the prime moers [3]. Sondhauss found out that when hot glass came in contact with a cool open ended glass tube, the sound waes were generated. He also inestigated the relation between the pitch of the sound and the dimensions of the apparatus. The frequency of the obsered tonal sound was equal to the natural frequency of the tube [4]. Figure : (a) The Sondhauss tube. (b) Schematic of the Prime moer. 2

14 In 859, Rijke extended Higgins work and noted that an open-ended tube which has hot gauze in the lower half of the tube produced sound [5]. He postulated that sound generation could be explained by the expansion of air caused by hot gauze and contraction of cooling air toward the open end [6]. Subsequently, Lord Rayleigh explained the Sondhauss tube qualitatiely in 896. He suggested that thermoacoustic instabilities caused this phenomenon: At the phase of greatest condensation heat is receied by the air, and at the phase of greatest rarefaction heat is gien up from it, and thus there is a tendency to maintain the ibrations [7]. Thermoacoustic engines are energy-conersion deices that produce acoustic power using heat flowing from a high-temperature source to a low-temperature sink. Thermoacoustic engines are also called prime moers and a schematic of the prime moer is shown in Figure (b). A simple thermoacoustic engine is composed of a resonator with one end closed and the other end open with a piece of porous material, referred to as a stack, placed inside the resonator at a specific location (Figure 2 (a)). The most important part of this system is a stack where a temperature gradient is maintained externally. Thermoacoustic engines comprise two major categories: standing-wae and traeling-wae engines. In a standing-wae thermoacoustic engine, heat is supplied to oscillating gas parcels at the moment of their compression and remoed at the moment of their rarefaction (Figure 2 (b)). Then, these self-sustained oscillations satisfy Rayleigh s criterion (Rayleigh 945) [8] and acoustic power is generated. Acoustic modes of the engine resonator define the motion of gas parcels. From the pressure-olume graph (Figure 3), we can find out the net work done per cycle by the parcel of gas on its surroundings. The shaded area of the ellipse in the figure, which is the cycle integral of pressure with respect to olume, is the net work [9]. 3

15 (a) Hot heat exchanger Stack Cold heat exchanger Resonator (b) Heat First half cycle Closed end Acoustic motion Open end Hot Second half cycle Cold 0 XS L x Heat Figure 2: (a) Schematic of a standing-wae thermoacoustic engine. (b) Thermal interaction between a gas parcel and a stack plate during one acoustic cycle in the presence of high temperature gradient. Figure 3: Pressure-olume graph of gas parcel in the standing-wae engine. A quasi-one-dimensional theory for low-amplitude thermoacoustic systems has been preiously deeloped and alidated. A detailed presentation of this theory and examples of practical systems are gien in the textbook by Swift [9]. In a traeling-wae thermoacoustic engine, the gas oscillates with traeling-wae time phasing in a channel with a steep axial 4

16 temperature gradient, the lateral thermal contact between the gas and the stack surface being as perfect as possible [0]. Since Ceperley s 979 realization [] that Stirling engines are of the traeling-wae class, both categories of engines hae been under deelopment. Hence, acousticians can play a significant role in the deelopment of robust, efficient heat engines. In today s engines, the heat exchangers are being imbedded in a ariety of acoustic caities and systems, creating the time phasings and acoustic modes needed for realization of heat engines with the simplicity and suitability of sound waes. One of promising applications of thermoacoustic engines is for electricity generation. Medium-scale thermoacoustic prototypes of this kind hae been tested [2]. Recently, a lot of efforts hae been undertaken to deelop arious types of miniature energy sources as alternaties to low-energy-density batteries [3]. These sources are important for applications of MEMS, sensor networks, unmanned ehicles, and other systems. Howeer, all traditional and recently deeloped small-scale power systems are not generally satisfactory. For example, scaled down rotating machinery is challenging to fabricate. A recently proposed promising candidate for small-scale electricity generation inoles a thermoacoustic engine coupled with a piezoelectric transformer [4]. Adantages of thermoacoustic systems include enironmental friendliness, potentially high reliability due to simple structure and minimal number of moing parts, and reasonable efficiency. There also exists a reersed system of thermoacoustic engine called thermoacoustic heat pump or refrigerator. This system uses sound to pump heat against a temperature gradient [5]. A simple thermoacoustic refrigerator consists of a resonator with a speaker attached to one end. 5

17 The other end is open or closed, and stack is placed at a specific location inside the resonator [7]. A schematic of a closed-end refrigerator is shown in Figure 4. Figure 4: Schematic of a closed-end refrigerator [5]. Scaling down thermoacoustic systems is challenging due to increased role of thermoiscous losses, thermal management and fabrication issues, and difficulty in integrating with heat sources and electroacoustic transformers. On the other hand, potential adantages of small scales include unfaorable conditions for turbulence, minor losses, and heat leaks by acoustic mass streaming. There were seeral attempts aimed at deeloping miniature thermoacoustic engines. For example, the construction and performance of a relatiely small 4- cm Hofler tube was documented [6]. Much smaller systems, down to a few centimeters in 6

18 length, were also built [7], but their design was not reported in details sufficient for reproduction..2. Objecties The main goal of the present study is to experimentally demonstrate a feasibility of a robust small-scale thermoacoustic engine of seeral centimeters in size and to characterize its performance. The open-closed tube system is adopted as a simple standing-wae configuration. The deelopment of this deice and experimental approach are described in the following section. Measured parameters, including the critical temperature difference across the stack and the acoustic pressure amplitudes inside and outside the engine are reported. These measurements were obtained by arying a controlled parameter which was heat applied to the engine with uniform time step. A simplified thermoacoustic theory is applied for estimating the critical temperature difference and generated acoustic power in the excited regime. Modeling results include prediction of critical temperature difference across the stack and estimations for acoustic power and efficiency. The optimizations for low critical temperature difference and high efficiency are also carried out. 7

19 Chapter 2 Experimental System 8

20 2. Experimental System The main purpose of this section is to describe the thermoacoustic engine setup in detail. A modular arrangement of a small-scale thermoacoustic engine is chosen for conenience of modifying the system. A schematic of assembled engine is shown in Figure 5. There are four principal parts that form the resonator of the engine. Two flanged copper tubes form a 4-mmdiameter quarter-waelength resonator with one end closed and the other end open to the atmosphere. The wall thickness of the copper tubes is mm. Ceramic and copper fittings are placed between the copper tubes to hold the stack and a miniature pressure transducer, respectiely. Copper material with high heat conductiity is selected to proide efficient heat addition and rejection. In contrast, a ceramic stack holder with low heat conductiity is chosen to minimize the heat conduction leak between the hot and cold parts of the engine. Electric band heaters (supplied by Omega Engineering, Inc.) and a flame created by a butane torch sere as heat sources for the side of the stack located at the closed-end tube section. A cooling jacket with flowing water is placed around the open-end tube to proide effectie cooling of the opposite side of the stack. A heat exchange between the gas inside the resonator and the hot and cold parts is distributed oer the tube sections. For better uniformity of temperature across the engine crosssection, two layers of 30x30-size copper mesh with wire diameter 0.30 mm are placed on each side of the stack. The flanged tube sections, ceramic stack holder, and copper fitting are held together by three screws going through the bores in these parts. As discoered in preious testing, the elimination of tiny gaps between the parts is critically important for the system performance. Inserting graphite gaskets in each junction is found to proide adequate sealing. The only ariable dimension in the present tests is the length of the open-ended part of the 9

21 resonator (right from the stack in Figure 5). Four engines are experimentally studied with total lengths 57, 67, 00, and 24 mm. Figure 5: Sketch of 67-mm engine. The most important element in a thermoacoustic engine is the stack. It is known that for efficient engine performance a characteristic pore size in the stack should be a few thermal penetration lengths, which is defined as follows, 2κ δ k =, () ω where κ is the thermal diffusiity of the fluid and ω is the angular frequency of acoustic oscillations. In quarter-wae resonators, such as shown in Figures 2 and 5, the angular frequency can be estimated as follows c π c ω = 2π, (2) λ 2L 0

22 where c is the aeraged speed of sound, λ is the waelength of the fundamental acoustic mode in a quarter-waelength resonator, and L is the resonator length. In high-frequency miniature thermoacoustic engines the thermal penetration depth becomes so small that it is difficult to find or to construct a regular-geometry stack, such as a parallel-plate type. Howeer, random porous materials with small pores are aailable, including fine metal wools and open-cell foam known as reticulated itreous carbon (RVC). Although it is likely that randomness increases iscous losses in the stack and reduces efficiency of thermoacoustic energy conersion, these materials are found to perform well in some thermoacoustic systems [8]. RVC samples of 80 pore-perinch grade (supplied by Energy Research & Corporation, Inc.) are employed as the stack material in most of tests presented here. The stack length is 7 mm and located 5 mm away from the closed end. An additional test sequence was carried out with a stack made of a super-fine steel wool with fiber diameter 50 µm. Figure 6 (a) and (b) show pictures of the resonator of the 67mm engine and fully assembled 00 mm engine, respectiely. As shown in Figure 6 (b), thermal insulation coers the electric heater at the closed end of the resonator in order to minimize heat losses to surroundings. Figure 6: (a) Photo of the 67-mm engine. (b) Fully assembled 00-mm engine.

23 The main measured ariables in our tests include temperatures at the hot and cold sides of the stack and the acoustic pressure inside and outside the engine. For temperature measurements, two 0.02-inch type-k thermocouples are inserted between the ceramic stack holder and the copper tube flanges into the copper mesh layers on the stack sides. A direct measurement of the acoustic pressure inside a small resonator is accomplished by a flush-mounted miniature piezoresistie Endeco pressure transducer 850C-5 with sensitiity of 2.04 mv / kpa. The transducer is screwed into a tapped hole in a specially built copper fitting located near the cold side of the stack (Fig. 3). A signal from the pressure transducer is amplified by Endeco Model 36 amplifier. To monitor a sound pressure leel outside the engine, a LinearX M-52 microphone is also employed. It has 70dB SPL capability, wide frequency response, low oltage power supply requirements, and a sensitiity of.2 mv / Pa. It is installed at 30 cm from the open end of the engine on the resonator axis. The microphone data are collected solely for qualitatie estimation of the external acoustic field, since it is sensitie to the surroundings. The uncertainties in measured temperatures and pressure amplitude inside the resonator are estimated using the standard procedure that combines bias and random uncertainties [9]. Repeated tests are carried out to determine random errors using Student s t-distribution. In some thermoacoustic systems a transition to the excited regime (onset of sound) and behaior in excited states depend on history of ariation of controlled parameters, such as a heat supply rate [20]. To proide a consistent method for acquiring test data, which is also conenient for comparison with aailable theories, a quasi-steady approach is utilized in this study. Using a regulated power supply, the heat release rate on the electric heater is increased at a sufficiently slow rate to ensure a quasi-steady temperature field in the system and to aoid possible nonlinear 2

24 effects, such as early triggering of excitation and hysteresis. A typical rate of ariation of the stack temperature difference in the quasi-steady tests performed in this study is about 2 C/min. To illustrate the effect of unsteadiness, an additional test sequence is conducted using a butane torch as a heat source, which proides fast change in the heat addition rate corresponding to the temperature difference ariation rate about 0 C/min. In Figure 7, the entire iew of our experimental setup is shown. This picture includes a fully assembled thermoacoustic engine, two thermometers connected with thermocouples, the microphone, an oscilloscope (BK PRECISION-220B) and the signal amplifier. Figure 7: Experimental setup of a thermoacoustic engine. 3

25 Chapter 3 Experimental Results and Discussion 4

26 3. Experimental Results and Discussion In this section all the experimental results for the thermoacoustic engines are presented and discussed. The temperature difference between the two sides of the stack was the main controlled parameter in the tests. In Figure 8, an example of the dependence of the pressure amplitude P at 30cm away from the open end of a 57-mm engine (with RVC stack) on the temperature difference of the stack T measured in one test with fast change in heat supply rate is shown. Since it is limited and difficult to reach a critical temperature difference Tc by electric band heater in a 57-mm engine test, we use a butane torch as a heat source. As we can see in Figure 8, points and crosses are scattered irregularly because of unsteadiness of increasing and decreasing temperature. For ease of comparison with the quasi-steady test, discussion of fast change in heat supply rate is dealt in following paragraphs. 5 4 P,out [Pa] T [deg C] Figure 8: Example of pressure amplitude dependence on temperature difference for fast change in heat supply in 57-mm engine with RVC stack. Points and crosses correspond to increasing and decreasing temperature differences, respectiely. 5

27 An example of the dependence of the pressure amplitude inside the engine on the stack temperature difference measured in one quasi-steady test of a 67-mm engine with RVC stack is shown in Figure 9 (a). The system becomes self-excited and the sound is generated upon achieing a critical temperature difference. In the excited regime the pressure amplitude monotonically increases with increasing the temperature difference. When T is decreased upon achieing a certain maximum, the pressure amplitude decreases as well, and the sound disappears at the temperature difference about the same as the onset Tc. Thus, no significant hysteresis in the pressure amplitude is noticed in the quasi-steady tests. For fast change in heat supply model, on the other hand, Figure 9 (b) shows that obious hysteresis in the pressure amplitude depending on the temperature difference appears and lower pressure amplitude is obtained in comparison with the amplitude in the quasi-steady test P [kpa].5 P [kpa] (a) T [deg C] 0 (b) T [deg C] Figure 9: Example of pressure amplitude dependence on temperature difference for (a) quasi-steady and (b) fast change in heat supply in 67-mm engine with RVC stack. Points and crosses correspond to increasing and decreasing temperature differences, respectiely. 6

28 To find out the influence of stack material, a steel wool stack is used in addition to RVC stack. An example of the pressure amplitude dependence on the stack temperature difference in quasi-steady tests for a 00-mm engine with two different stack materials is shown in Figure 0. The performance of the engine with a steel wool stack is inferior to that with RVC stack: the critical temperature difference is much higher and the pressure magnitude in the excited regime is lower. All subsequent data are reported for engines with RVC stacks P [kpa] P [kpa] (a) T [deg C] 0 (b) T [deg C] Figure 0: Example of pressure amplitude dependence on temperature difference for quasi-steady in 00-mm engine with (a) RVC stack and (b) steel wool stack. Points and crosses correspond to increasing and decreasing temperature differences, respectiely. 7

29 In Figure, an example of pressure amplitude dependence on temperature difference in quasi-steady tests for a 24-mm engine with RVC stack is shown. Een though its pattern is similar to that of shorter engine, each alue of pressure amplitude corresponding to specific temperature difference is not higher than the alue of shorter engine. From this result, we can find out that pressure amplitude is not proportional to the length of engine and guess there exist the most suitable length and geometry for the best performance. Detailed description is gien below..5 P [kpa] T [deg C] Figure : Example of pressure amplitude dependence on temperature difference for quasi-steady in heat supply in 24-mm engine with RVC stack. Points and crosses correspond to increasing and decreasing temperature differences, respectiely. 8

30 Obtained in repeated quasi-steady tests, the aerage dependence of the sound pressure amplitude on the stack temperature difference for the 67-mm engine is shown in Figure 2 (a). The aerage critical temperature difference is found to be about 256 C. Upon excitation, the pressure amplitude P monotonically increases with T and reaches 2 kpa at T 340 C. The sound frequency in the 67-mm engine is about.7 khz, which does not significantly ary in the studied range of T. Also shown in Figure 2 are the readings from the external microphone, which are more than two orders of magnitude smaller than the acoustic pressure amplitude inside the engine. Results for the 00-mm engine are shown in Figure 2 (b). The pressure-temperature dependence is qualitatiely similar to the engine with shorter length. Howeer, the onset temperature difference is significantly smaller (about 23 C). A scatter in the data and the total uncertainty are also smaller. The recorded sound frequency is about. khz, which makes thermoiscous losses smaller than in the shorter engine, in accordance with obsered decrease in T c. Howeer, the aerage slope of the pressure amplitude cure is smaller in this engine. When the engine length is increased up to 24 mm (Figure 2 (c)), the operating frequency decreases down to about 0.94 khz. Howeer, the critical temperature difference increases up to about 238 C and the pressure amplitude becomes smaller, especially at T in the icinity of Tc. Possible reasons for these phenomena include an increase of the energydissipating surface area of the resonator and a shift of the stack position relatie to the tube open end. The shortest engine configuration with length 57 mm is obtained by remoing the transducer copper fitting from the 67-mm engine. Only external acoustic pressure amplitude can 9

31 be measured in this setup. Results for this engine are shown in Figure 2 (d). The operating frequency is about.9 khz and the aerage critical temperature difference is about 288 C. Both T c and uncertainty in pressure magnitude monotonically increases with Tc are larger than those quantities in longer engines. The external T as in the preious cases. The important outputs measured in mm engines are arranged in Table. The lowest aerage critical temperature is obtained in 00-mm engine and the highest maximum pressure amplitudes in-and-outside of resonator are measured in 67-mm engine. Aerage sound frequency decreases as the length of the engine gets longer. 20

32 P [kpa].5 P [kpa] (a) T [deg C] (b) T [deg C] P [kpa].5 P [Pa] (c) T [deg C] (d) T [deg C] Figure 2: Aerage pressure amplitude recorded in engines with different lengths: (a) 67 mm, (b) 00mm, (c) 24 mm, (d) 57 mm. Circles show measurements by the pressure transducer inside the engine; squares correspond to external microphone. Microphone data in sub-figures (a), (b), (c) are multiplied by 00. Error bars indicate the total uncertainty. Dotted lines show the aerage critical temperature differences, which uncertainties are gien by horizontal error bars. 2

33 Engine length, L 57mm 67mm 00mm 24mm Estimated parameters: Aerage critical temperature difference, T c Aerage sound frequency, f.88 khz.66 khz.3 khz 0.94 khz Inside maximum pressure amplitude, 2.25 kpa.96 kpa.42 kpa P, in Outside maximum pressure amplitude, P, out 5.2 Pa 7.5 Pa 3.33 Pa 2.5 Pa Table : Experimental results for mm engines. As described in experimental section, the electric heater was used to supply heat at closed-end of the engine. An example of the changes of temperature differences and pressure amplitudes with increasing electric power in 67-mm engine are presented in Figure 3. Both graphs show the step-shape behaior of electric power since heat is supplied with specific time interal. Two measured parameters increases as time goes by, howeer, increasing pattern of pressure amplitude does not perfectly correspond to that of temperature difference and changes more dramatically. Een though thermal insulation is used to minimize heat losses to surroundings, most of the electric power is lost as heat leak. The example of measured data and the time step in 67-mm engine after sound is generated is shown in Table 2. 22

34 T [deg C] (a) Time[minute] Electric power [W] P [kpa] (b) Time[minute] Electric power [W] Figure 3: Example of (a) temperature differences and (b) pressure amplitudes with increasing electric power in 67-mm engine: solid, (a) temperature differences and (b) pressure amplitudes; dashed, electric power. 23

35 Electric power(w) Time(minute) Temperature difference [ ] Pressure amplitude [kpa] (Sound generated) min min min min min min Table 2: Example of measured data and the time step in 67-mm engine. 24

36 In any experiment it is important to the experimenter that the output of a measurement system presents the actual alue of the measurand. Since there is no perfect measurement system, some deiation between the actual alue of the measurand and measurement system output exist and the experimenter must estimate them to use the output for intended purpose. What the experimenter can estimate is the uncertainty interal (or simply uncertainty). In general, uncertainties in experiments consist of two main categories: systematic uncertainties and random uncertainties. Systematic uncertainties, denoted B (sometimes called bias uncertainty), are consistent, repeatable and estimated maximum fixed errors [9]. In this experiment, the pressure transducer accuracy and the range from manufacturer s manual are 0.% of range and kpa, respectiely. Random uncertainty, denoted P R (sometimes called precision uncertainty), is the imprecision in the measurements caused by a lack of repeatability in the output of the measuring system. Random uncertainties can be gien for mean alues as follows, S P R = ± t, (3) n where t comes from the t-distribution which depends on degrees of freedom and confidence leel [9], S is the standard deiation of the data, and n is the number of data points (or repeating times). In addition to these uncertainties, we also hae one more uncertainty which is called the readability uncertainty, denoted R e. The total uncertainty, denoted U, can be calculated as follows, U B + P R + R e =. (4) 25

37 One example of the uncertainty calculation for pressure amplitude is shown in Table 3. For uncertainty of the critical temperature difference, standard deiation is considered as the only uncertainty factor. Temp. [ ] Ag. press. Amplitude [kpa] Stand.de.(S) [kpa] n t P R [kpa] B[kPa] R e [kpa] U[kPa] Table 3: Example of the uncertainty calculation of pressure amplitude in 67-mm engine. 26

38 Chapter 4 Modeling 27

39 4. Modeling 4. Theory The goals of simplified modeling in this study are to predict the critical temperature difference across the stack, corresponding to the sound appearance, to roughly estimate acoustic power generation in the stack and thermoacoustic efficiency, and to optimize them. We use the energy analysis similar to that of Swift [2]. In order to formulate theoretical equations to predict aboe parameters, first, we need to analyze the acoustic pressure and elocity oscillations in our system. The acoustic pressure and olumetric elocity fluctuations in the standing wae are written using the complex notation, iω t [ p ( x) e ] p( x, t) = Re, (5) iω t [ U ( x) e ] U ( x, t) = Re. (6) The pressure and elocity waeforms are approximated by the fundamental mode in a quarterwaelength resonator with uniform temperature distribution and the x-axis defined in Figure 2: p = π x P cos 2, (7) λ PA x U = i sin 2π, (8) ρ c λ where P is the pressure amplitude at the resonator solid-wall end at x = 0, A is the resonator cross-sectional area, ρ is the mean air density, and i accounts for the phase shift between pressure and elocity fluctuations. c is the speed of sound which can be written as follows: c= γ R T, (9) g 28

40 where γ is the specific heat ratio, waelength λ is a function of length of the resonator: R g is the air gas constant, and T is the gas temperature. The λ = 4( L+ 0.6R), (0) where L and R are the length and the radius of the resonator, respectiely. All fluid properties in this analysis are ealuated at the aerage temperature of the hot and cold ends of the stack. mechanisms: The thermoacoustic power generated in the stack W & is balanced by three damping W & = E& + E& + E&, () res rad mesh where E & res is the acoustic power absorbed at the resonator walls, radiated from the open end of the system, and E & rad is the acoustic power E & mesh is the thermoiscous power loss at the mesh layers. If we consider sound propagating in the x direction within a channel with constant crosssectional area A, the ariation of the time aeraged acoustic power flow W & along the channel with the area A can be written as follows, d W& d( pu ) = da, (2) dx dx A where p is the acoustic pressure amplitude, u is the acoustic elocity amplitude, and the oerbar stands for time aeraging. Rewriting Equation (2) in complex notation and expanding the x deriatie yields, W& dx ~ = Re U 2 dp + ~ p dx du dx d. (3) 29

41 Hence, the time-aerage acoustic power produced in a short stack can be estimated from general expression for acoustic power ariation in a channel [9]: ~ [ U p + ~ p U ] W & Re 2, (4) where p and U are the acoustic pressure and olumetric elocity amplitudes, the subscript signifies first order, a tilde stands for the complex conjugate, and designates the change across the stack. To derie p, we start with momentum equation which can be expressed as follows, r r r ρ + ( ) = p+ σ, (5) t where r is the ector elocity and σ is the nine-component iscous stress tensor. If we assume that gradients in the iscosities can be neglected and the fluid is incompressible with respect to momentum effects, then Equation (5) simplifies to, r r r 2 r ρ + ( ) = p+ t µ, (6) where µ is the gas dynamic iscosity. Following Rott s acoustic approximation [22], the pressure, elocity, temperature, and density are expressed as, iωt p= p + Re[ p ( x, y, z) e ], (7) m iωt u = Re[ u ( x, y, z) e ], (8) T iωt = T + Re[ T ( x, y, z) e ], (9) m iωt ρ = ρ m + Re[ ρ ( x, y, z) e ], (20) 30

42 where the subscript m denotes mean alue. With these approximations, the x component of Equation (6) becomes, 2 2 dp u u iω ρ = + + m u µ. (2) 2 2 dx y z Equation (2) is a differential equation for u ( y, ), with boundary condition u 0 at the solid surface, therefore the solution is, u z i dp = [ h ( y, z ], (22) ω ρ dx ) m where h ( y, z) is the complex function determined by the specific channel geometry [9]. Integrating Equation (22) with respect to y and z oer the cross-sectional area A of the resonator yields the olume flow rate U on the left side and changes h to its spatial aerage f (detailed description of f is shown later). Arranging for dp results in, = dp iωρ m dx = U. (23) A ( f ) Following a procedure similar to that for Equation (23) aboe, U can be deried as well. The continuity equation of the fluid is, ρ r + ( ρ ) = 0. (24) t Simplifying by Rott s approximations and spatially aeraging Equation (24) oer the crosssectional area A gies, d u iω ρ + ρ m = 0, (25) dx 3

43 where the symbol means the spatial aerage and, ρ m ρ ρ = +. (26) m T p Tm pm Equation (26) is obtained from ideal-gas equation of state with Rott s approximation. Deried by heat transfer equation, spatially aeraged oscillating temperature T can be written as, T dtm ( f k ) σ ( f ) = ( f k ) p U. (27) ρ c iω A dx ( f )( σ ) m p where c p is the gas specific heat at constant pressure and σ is the gas Prandtl number. Combining Equations (25-27) with spatially aeraged elocity u yields [9], du iω Adx f k f dtm = [ + ( γ ) f k] p+ U. (28) γ p T m ( f )( σ) m Therefore the changes of acoustic pressure and elocity across a high-porosity stack can be written with help of Equation (23) and Equation (28), p iωρ x A ( f ) s U, (29) U iω A x γ p m s f k f T k p + U, (30) T [ + ( γ ) f ] ( f )( σ) where xs is the stack length. Thermoacoustic functions f k and f depend on ariables λ k and λ, respectiely, which are defined as follows, R p λ k, = 2, (3) δ k, 32

44 where δ = 2ν / ω is the iscous penetration depth with ν being the gas kinematic iscosity, δ k is the thermal penetration depth defined by Equation (), and R p is the characteristic size of the pore. It was suggested by Wilen [23] that an RVC stack performs similar to a parallel-plate stack whose spacing H is 5% larger that the RVC pore size R p specified by the RVC manufacturer. The analytical expression for thermoacoustic functions in a parallel-plate stack is gien by Swift [9]: f k, where λ ' = ( iλ' / 2) tanh k, =, (32) iλ' / 2 k, k, 2H / δ k,. Hence, the acoustic power generation can be ealuated using Equations ( - 32) and selecting λ ' k, =. 5λk,. It should be noted that efforts toward more accurate (and more complicated) calculation of thermoacoustic processes in tortuous materials hae been recently undertaken by Roh et al. [24]. The relationship between f k, and ' k, λ can be illustratie as shown in Figure 4. 33

45 Re[f k, ] 0.5 Im[f k, ] λ' k, Figure 4: Relationship between f k, and ' k, λ : dashed, real part of f k, ; dotted, imaginary part of f,. k 34

46 The radiation from the open end and surface losses on the resonator walls are estimated using standard expressions [2], 2 4 π P R E& rad =, (33) 2 8 ρ cλ where E & 2 P R ωπ R L δ ( γ ) δ res = 4 ρ c 2 k + + L', (34) L ' is the internal length of the resonator. The losses at mesh layers are calculated using a lumped-element approximation [9], r 2 2 & mesh = U xm + p xm, (35) 2 2rk E where xm is the mesh layer thickness and r and r k are the specific acoustic resistances due to iscous and thermal-relaxation effects. These resistances for mesh layers are roughly estimated following Swift and Ward [25] and retaining terms significant at low amplitudes of acoustic motions (since we consider the onset of acoustic oscillations), µ c( ϕ) r, (36) 2 r ϕ A 8 h r iω ARe ρ T k c p ε h, (37) + ε h 2 ε = 8i rh h / 3 b( ϕ) σ δ, (38) 2 k where ϕ and r h are the porosity and the hydraulic radius of the stacked screens, which related to the wire diameter and mesh number Swift [9] and written as follows: 35

47 π n D ϕ = wire, (39) 4 ϕ r h = D wire, (40) 4( ϕ ) where n is mesh number of the screen in one inch, D wire is the diameter of one wire of the screen, and c and b are the porosity-dependent parameters in the friction factor and heat transfer coefficient that can be ealuated by empirical correlations of Swift and Ward [25] based on experimental data of Kays and London [26]. 4.2 Comparison with test data At low-amplitude acoustic oscillations, each term in Equation () is proportional to the square of the pressure amplitude which cancels out. Then, the critical temperature difference that appears in expression for W & can be found as a function of material properties, geometry, and aerage temperature. Calculations hae been carried out for all four test engine lengths. The experimental aerage temperature at the sound onset (in the range C) is applied for ealuating the air properties. It appears that the radiation and mesh losses are comparable but slightly smaller than the thermoiscous losses at resonator wall in the shortest engine, while and E & mesh become about ten times smaller than E & res in the longest engine. From Figure 5, the proportions of with each other. E & rad, E & res, E & rad E & mesh and change of W & for different lengths are readily compared 36

48 Proportion [W*0-2 /Pa 2 ] mm 67 mm 00 mm 24 mm Engine length Figure 5: Proportions of W & ; square, E & res ; circle, E & rad, E & res, E & mesh ; cross, E & rad. E & mesh and change of W & for different lengths: triangle, The calculated critical temperature difference Tc is presented by circles in Figure 6, where it is compared with experimental data shown by squares. The trend in ariation of this difference with the engine length is predicted satisfactorily, demonstrating a minimum Tc at the 00-mm engine. A reduction of frequency with increase of resonator length decreases losses in shorter engines, while the increase of the resonator surface area and shift of the relatie stack position increase results for Tc in the longest engine. The deiation between test data and modeling Tc increases from about 28 C to 60 C with increasing the engine length. The 37

49 discrepancy can be associated with a non-uniform temperature field, effect of stack on acoustic waeforms, and geometrical imperfections in the real system. These results can be also compared with the ideal critical temperature difference that is simply estimated for a single-plate standing-wae configuration and iniscid ideal gas [2] by the formula: Aω p Tid =, (4) ρ c U p The results for Tid are shown by triangles in Figure 6. As expected, the ideal critical temperature difference is much smaller than the test data and results of more complete theory. T c [deg C] L [cm] Figure 6: Critical temperature difference ersus engine length: squares, experimental data; circles, modeling results; triangles, ideal theory (Equation (4)). 38

50 The acoustic power generated in the stack in the regime aboe the sound onset can be estimated by Equations (4, 29, 30, 3, 32), using experimentally measured acoustic pressure magnitude and temperatures at the ends of the stack. Additionally, it must be assumed that functions f k and f in Equations (29, 30) are independent of oscillation amplitudes. This assumption is reasonable for regular-geometry stacks with longitudinal pores, but it oersimplifies the real flow in tortuous stacks at high-amplitude elocities. Therefore, this calculation gies only a rough estimation. The results for acoustic power produced in the stack are shown in Figure 7 (a). The power increases with the temperature difference and the oscillation frequency, which is higher in shorter engines. W [mw] η s % (a) T- T [deg C] c (b) T- T [deg C] c Figure 7: Estimations for (a) acoustic power produced in the stack and (b) stack thermoacoustic efficiency. Corresponding engine length: solid cure, 67 mm; dashed cure, 00 mm; dotted cure, 24 mm. 39

51 The thermoacoustic energy conersion efficiency in the stack can be estimated as follows, W& η s =, (42) H& where H & is the thermoacoustic enthalpy flow through the stack, which equals to the heat supplied to the hot side of the stack. It does not account for any external heat leaks, such as ia the ceramic stack holder. The total enthalpy flow along the stack is gien by Swift [9] as follows, ~ 2 ~ f k f ρ c p U H& = Re p U + ~ k ( )( f ) Im σ σ 2Aω( σ ) f, (43) ~ ( f + f ) A( k+ k ) s T xs where k is the gas heat conductiity and k s = W/m K is the effectie heat conductiity of RVC matrix specified by the manufacturer. The estimated stack efficiency is shown in Figure 7 (b). The efficiency increases with T, partly due to smaller heat condition contribution to H & at larger pressure amplitudes. It should be noted that only a fraction of produced acoustic power can be utilized for practical purposes, such as generating electricity. A significant fraction of W & will be consumed by losses inside the resonator. 4.3 Optimized design System optimization for thermoacoustic engine is also conducted. For critical temperature difference, it is considered as the highest performance if the difference reaches the lowest alue. To find out more optimal conditions for the lowest critical temperature difference, the dependence of Tc on the radius of resonator, stack position, mean pressure, and pore radius 40

52 of stack in 67-mm engine is obtained with other parameters fixed. The results of these ariables are shown in Figure 8. The optimum conditions for the lowest critical temperature difference are obtained when the radius of resonator, stack position, mean pressure, and pore radius of the stack are 8 mm, 2 cm, 0 kpa, and 0.26 mm respectiely in 67-mm engine T c [deg C] T c [deg C] (a) Radius [m] (b) Stack position [m] T c [deg C] T c [deg C] (c) Mean pressure [kpa] (d) Pore radius [m] x 0-4 Figure 8: Critical temperature difference for ariable (a) radius of resonator, (b) stack position, (c) mean pressure, and (d) pore radius of stack in 67-mm engine. 4

53 Appropriate working fluid for the lowest critical temperature difference in the system can be determined in Figure 9. Nitrogen yields highest performance and using argon results in worst performance. All specifications for the lowest critical temperature difference results are arranged in Table Critical temp. dif Nitrogen Air Helium Argon Gas type Figure 9: Critical temperature difference for arious gases in 67-mm engine. 42

54 Experimental data Optimized design Gas Air Nitrogen Resonator radius 7 mm 8 mm Stack position (from closed-end).85 cm 2.0 cm Mean press kpa 0 kpa Pore radius 0.32 mm 0.26 mm T c Table 4: Optimization results for the lowest critical temperature difference in 67-mm engine. For acoustic power flow and efficiency, optimization for the highest performance is obtained by arying relatie pressure amplitude, p / P m, which is the ratio of acoustic pressure amplitude inside the engine and mean pressure. As shown in Table 5, the optimized design deliers higher performance in comparison with experimental data of the existing engine. The ariations of acoustic power flow and efficiency for p / P m are shown in Figure 20 as well. 43

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