THERMOACOUSTIC PHENOMENA IN SMALL-SCALE SYSTEMS SUNGMIN JUNG. A dissertation submitted in partial fulfillment of the requirements for the degree of

Transcription

1 THERMOACOUSTIC PHENOMENA IN SMALL-SCALE SYSTEMS By SUNGMIN JUNG A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY School of Mechanical and Materials Engineering MAY Copyright by SUNGMIN JUNG, All Rights Reserved

2 Copyright by SUNGMIN JUNG, All Rights Reserved ii

3 To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of SUNGMIN JUNG find it satisfactory and recommend that it be accepted. Konstantin I. Matveev, Ph.D., Chair Cecilia D. Richards, Ph.D. Michael J. Anderson, Ph.D. iii

4 Acknowledgement First of all, the special thank goes to committee chair, Dr. Konstantin I. Matveev, who has encouraged, motivated, and inspired me in various ways during my Ph. D. studies. Under his supervision, I have learned many things that bring me a spirit of adventure in regard to research and scientific knowledge. With his great patient and excellent guidance, notable development and improvement has been made on this dissertation indeed. I would like to express the deepest appreciation to my committee members, Dr. Cecilia D. Richards and Dr. Michael J. Anderson, for their willingness to serve as my committees and to advise on thermoacoustics. I am also grateful to an administrative manager, Gayle Landeen, and graduate academic coordinator, Mary Simonsen, in the School of Mechanical and Materials engineering department. They have answered all my questions about the graduate program and financial issues. Finally, I would like to sincerely thank my parents, Can Hee Jung and Hae Kyoung Kim. It seems to me that everything is possible with your unconditional support and trust in me. Encouragement from both of you leads me to become mentally and physically strong. I am very proud of being your son and you are the reason of my life. iv

5 THERMOACOUSTIC PHENOMENA IN SMALL-SCALE SYSTEMS Abstract by Sungmin Jung, Ph.D. Washington State University May Chair: Konstatin I. Matveev Thermoacoustic phenomena in small-scale systems are investigated, and results are presented on the following topics: the acoustic properties of porous and fibrous materials, the modeling of thermoacoustic resonators with nonuniform medium and boundary conditions, and the harvesting of energy from tonal sound excited by heat addition and vortex shedding. The transfer function measurement system is used to find the acoustic properties of porous and fibrous materials. The complex wave numbers and characteristic impedances of reticulated vitreous carbon (RVC) and plastic mesh are determined using a variation of the threemicrophone and four-microphone methods with the transfer function technique. The wave numbers and characteristic impedances of RVC and plastic mesh can be estimated from the obtained results. To find the effect of temperature difference, relative acoustic power changes across RVC, stacked plastic screens and stacked steel screens at T=C and at TC are v

6 experimentally determined and compared. It can be concluded that these porous stacks generate acoustic power under a temperature gradient, partly compensating for the acoustic losses when sound energy propagates through the stack. The numerical modeling of thermoacoustic resonators with nonuniform media and boundary conditions is carried out. Sparse numerical grids are used in the bulk of resonators, and in boundary layers near solid surfaces analytical solutions impose proper boundary conditions. The main advantage of this method is computational efficiency. Since it can quickly estimate the effects of all parameters of geometry and material properties, the present model is suitable for optimizing thermoacoustic systems. A small-scale, low-aspect-ratio thermoacoustic engine with a flexing wall oscillator is modeled. If the natural frequency of the flexing wall oscillator is selected to be much lower than the natural frequency of the acoustic resonator, the engine operates at satisfactory efficiencies and requires a relatively low temperature difference threshold. Heat transfer calculations, consideration of large-amplitude acoustic effects, and analysis of electroacoustic transducers are needed for further developments. Three resonator-type acoustic energy harvesters are tested and demonstrated. In the resonator, tonal sound is excited by heat addition or vortex shedding in the presence of mean flow. A PZT disk with a brass back plate is used as an electroacoustic transducer. The first system with baffles in the mean flow generated more than.5 mw of electric power at a resistance of vi

7 kω and a mean flow velocity around.6 m/s. The second system has one side open and generated a maximum electric power of.446 mw at a resistance of 4.8 kω. In the third experiment, the closed resonator produced a maximum harvested electric power of 7. mw at a resistance of 4.8 kω. The experimental results correlate reasonably well with those of previous studies by other researchers. To increase power output, optimization of the piezoelement and system geometry is required. vii

8 Table of Contents Page Acknowledgement... iv Abstract... v List of Symbols... x List of Tables... xiii List of Figures... xiv Chapter Introduction..... Background..... Previous Work Objectives and Contributions... Chapter Acoustic Properties of Porous and Fibrous Materials Experimental Setup and Procedure Measurement Theory..... Experimental Results... 4 Chapter 3 Modeling of Thermoacoustic Resonators with Nonuniformities Theoretical Model Modeling Results viii

9 Chapter 4 Energy Harvesting from Tonal Sound Excited by Heat Addition and Vortex Shedding Experimental Setup Results Chapter 5 Conclusions References Appendix A. MATLAB code for calculated acoustic pressure and x-component of the velocity in the two-dimensional resonator in Chapter B. Published journal paper. (This journal is attached by permission of ASME): C. Published journal paper. (This journal is attached by permission of Proc. IMechE):... ix

10 List of Symbols A cross-sectional area of the system or surface area of flexing wall B systematic uncertainty c speed of sound (= a ) c isobaric specific heat p c v isochoric specific heat d distance from specific origin to given position dx stack length acoustic powers coming into and leaving from the stack E in, out E acoustic power change f natural frequency of a straight pipe n f r resonator natural frequency f, thermal and viscous acoustic functions k v f F H k K l L L L e s, natural frequency of an oscillator force amplitude due to acoustic pressure transfer function or height of a resonator acoustic wave number stiffness of an oscillator distance from stack surface to microphone resonator length length correction stack length L distances from a flexible wall to each of stack ends M M n p p m p P PE P q Q r R R g mass of an oscillator mass of the gas in the resonator number of data points or mesh number of the screen or number of an acoustic mode in the tube acoustic pressure mean ambient pressure first-order acoustic pressure amplitude acoustic pressure amplitude or time-averaged acoustic power or mechanical power electric power random uncertainty local heat addition rate to the stack per its cross-sectional area total heat rate given to the stack radius of the resonator reflection coefficient or gas constant or power extraction coefficient or electric resistance x

11 s spacing between two microphones in the center of the tube S spacing between two microphones S x standard deviation t t-distribution coefficient or time variable T transfer matrix element or temperature T temperature difference across the stack transverse temperature difference T y T cr critical temperature difference u u w u u U U U t w U v V w W x x y y Y Z, velocity amplitude defining thermoviscous effects in the acoustic boundary layer velocity amplitude of wall oscillations velocity component along x direction vector form of first-order acoustic velocity amplitude acoustic velocity amplitude total uncertainty maximum wall velocity amplitude effective velocity amplitude of the oscillator velocity component along y direction RMS voltage normalized acoustic energy out of the stack generated acoustic energy out of the stack coordinate along the resonator unit spacing between two points in computational mesh transverse coordinate in the resonator system half of the resonator height specific functions determining acoustic wave number acoustic impedance thermal diffusivity of fluid (= ) porosity of stack specific heat ratio acoustic boundary layer thickness thermal and viscous penetration depths st k,v coordinate normal to the tube wall local stack based thermoacoustic efficiency tm thermomechanical efficiency density of fluid Prandtl number angular frequency kinematic viscosity of fluid xi

12 coordinate parallel to the tube wall, thermal and viscous thermoacoustic functions xii

13 List of Tables Table. Input parameters of thermoacoustic engine xiii

14 List of Figures Figure.. (a) RVC samples of various shapes and pore sizes, (b) magnified view of RVC material... 4 Figure.. (a) Schematic of a quarter-wavelength (standing-wave) thermoacoustic engine, (b) low-aspect-ratio engine with movable piezomembrane... 6 Figure.. (a) Schematic of the entire measurement system, (b) dimensions of the three microphone impedance tube, (c) dimensions of the four-microphone impedance tube6 Figure.. (a) Photograph of the three-microphone measurement setup and (b) photograph of the four-microphone impedance tube... 7 Figure.3. Schematic of the three-microphone measurement setup with a temperature difference imposed across the stack... 9 Figure.4. Photograph of experimental setup for tests with temperature gradients... 9 Figure.5. (a) Magnitude and (b) phase of the reflection coefficient derived from measured transfer function H : Solid red curve, ideal value; black dots, derived from measured H... 5 H Figure.6. (a) Magnitude and (b) phase of measured transfer function 3 and transfer function H 3 H obtained from the derived reflection coefficient: Solid red curve, measured 3 ; H black dots, 3 obtained from the derived reflection coefficient... 6 xiv

15 Figure.7. Real and imaginary parts of normalized (a) wave number and (b) characteristic impedance of RVC: Dots, measured by the four-microphone method; solid red curve, results calculated by Muehleisen s theory; dashed blue curve, results calculated by Roh s theory... 8 Figure.8. Photograph of a plastic mesh layer... 9 Figure.9. Real and imaginary parts of the normalized (a) wave number and (b) characteristic impedance of the plastic mesh... 3 Figure.. Experimental measurements of relative acoustic power changes across (a) RVC, (b) stacked plastic screens and (c) stacked steel screens at T=C (dotted line), T=C (red dot) from Roh s correlation and at TC (solid line) Figure 3.. (a) Numerical grid and (b) magnified view of region near rigid wall surface Figure 3.. (a) Geometry of a D thermoacoustic engine with flexing wall on the left boundary and (b) Given mean temperature profile at y= Figure 3.3. Validation results of normalized acoustic (a) pressure and (b) velocity amplitudes at Constant-temperature lossy resonator: solid lines, analytical solution; points, D numerical solution and normalized acoustic (c) pressure and (d) velocity amplitudes at linear temperature variation: solid lines, D wave equation; points, D numerical solution xv

16 Figure 3.4. (a) Dimensional and (b) normalized critical temperature difference, and (c) dimensional and (d) normalized frequency of the thermoacoustic resonator at the sound onset: solid line, M / M g 5 ; dotted line, M / M g Figure 3.5. Complex amplitudes of (a) real and (b) imaginary parts of acoustic pressure normalized by mean pressure, and x-component of (c) real and (d) imaginary parts of acoustic velocity normalized by maximum velocity at flexing wall (Velocities inside stack are averaged over pore cross sections) Figure 3.6. (a) Normalized flux of acoustic power produced in the stack and (b) local stack-based thermoacoustic efficiency: solid line, H/L=.5; dashed line, H/L=.; and dotted line, H/L= Figure 3.7. Thermomechanical efficiency: solid line, P =. W/m; dashed line, P =. W/m; and dotted line, P = W/m... 6 Figure 4.. (a) Schematic of an experimental setup with mean flow and baffle, (b) photograph of the piezoelement held by two flanges Figure 4.. (a) Schematic of the experimental setup with heat addition, a stack, and one side open, (b) photograph of the experimental setup Figure 4.3. (a) Schematic of the experimental setup with heat addition, a stack, and both sides closed and (b) a photograph of the experimental setup xvi

17 Figure 4.4. Harvested electric power at three electric loads:, kω;, kω; Δ, kω... 7 Figure 4.5. (a) Acoustic pressure amplitudes at three microphone locations:, ;, ; and +, 3 and (b) the frequency of the excited tone... 7 Figure 4.6. (a) Harvested electric power, (b) acoustic pressure, and (c) temperature difference in the second experimental setup for two different heating powers:, 39 W;, 53 W 74 Figure 4.7. (a) Harvested electric power, (b) acoustic pressure, and (c) temperature difference in the third experimental setup for two different heating powers:, 39 W;, 53 W.. 76 xvii

18 Dedication This dissertation is dedicated to my parents: to my father, Can Hee Jung, to my mother, Hae Kyoung Kim, who are always supportive to me. xviii

19 Chapter Introduction

20 .. Background Thermoacoustic effects involving heat and sound interactions have been known and investigated for over a hundred years. Experimental prime movers and heat pumps based on these effects have been also developed. Recently, a need has been arising for low power sources in MEMS, unmanned vehicles, remote sensors and other small devices. Traditional small-scale power systems are not satisfactory as miniature power sources due to difficult fabrication and maintenance requirements. A thermoacoustic engine coupled with an electroacoustic transducer is a promising candidate of small-scale electricity generation since it has simple structure, no moving parts, and reasonable efficiency. However, new engineering challenges appear in miniature thermoacoustic systems, such as increased thermoviscous losses. Thermal management, fabrication methods, and integration with heat sources and transducers also represent issues that require additional studies. In this work, several problems of miniaturizing thermoacoustic devices are addressed: () utilization of random porous materials that are readiliy available but little studied for thermoacoustic applications, () modeling of resonators with significant nonuniformities, and (3) conversion of generated acoustic power into electricity. Porous and fibrous materials are used in acoustic applications such as sound absorbers and thermoacoustic devices. Acoustic properties of these materials must be known to design appropriate acoustic devices utilizing these materials. Most acoustic properties can be

21 determined from the complex characteristic impedance and wave number. The transfer function method, which can be used to find the characteristic impedance and wave number of porous fibrous materials, has been commonly applied since the past. In the case of thermoacoustic applications, the stack acoustic properties experience dramatic changes. Therefore, a consideration of the effect of the temperature difference on the acoustic properties of stack material is required. A regular impedance tube measurement without imposed temperature gradient cannot demonstrate the effect of temperature gradient. Figure. presents photographs of reticulated vitreous carbon (RVC) samples of various shapes and pore sizes and a magnified view of RVC material. (a) 3

22 (b) Figure.. (a) RVC samples of various shapes and pore sizes, (b) magnified view of RVC material To date, many theoretical, numerical, and experimental studies have been carried out to investigate the acoustic field in enclosures such as ducts, mufflers, and thermoacoustic systems. In order to predict the acoustic behavior, it is important to model acoustic field in these devices before starting experiments. Not only the acoustic field but also viscous and thermal effects must be taken into account. It is computationally expensive to accurately model sound involving damping mechanisms. Some numerical techniques such as boundary element (BEM) and finite element methods (FEM) can solve a simplified form of the linearized wave equation in the case of negligible dissipation and low amplitude of the sound. The appearance of acoustic oscillations driven by an unsteady heat release is a well- 4

23 known phenomenon in thermoacoustic engines []. These systems convert heat to acoustic power without any moving parts. Figure.a shows a schematic of a quarter-wavelength thermoacoustic engine that has open and closed ends. In such engines a piece of porous material known as a stack, where a temperature gradient is maintained externally, plays a key role in generating acoustic power []. This power can be turned into a useful source of power in some applications such as electricity generation, refrigeration, and gas mixture separation. Small-scale thermoacoustic engines have been proposed recently as a promising candidate for powering remote sensors and micro robots [, 3]. The generated acoustic power can be converted to electric power by electroacoustic transformers. Piezomembranes are chosen as electroacoustic transformers in this study. A schematic of the membrane coupled with thermoacoustic engine is presented in Figure.b. As shown in the figure, nonuniform boundary conditions are created by the vibration of piezomembranes; and these nonuniformities affect dimensionality of acoustic field in resonators. Distortion of one-dimensionality of the acoustic field is especially significant in enclosures with large cross-sectional areas. 5

24 (a) (b) Figure.. (a) Schematic of a quarter-wavelength (standing-wave) thermoacoustic engine, (b) low-aspect-ratio engine with movable piezomembrane There are several ways to convert acoustic power to electric power or vice versa, such as electrodynamic and piezoelectric mechanisms. For medium-scale thermoacoustic systems, electrodynamic transducers such as a linear electric alternator and modified loudspeaker can be 6

25 used as acoustic energy harvesters. However, due to the difficulties of building those transducers at the small scale, piezoelectric membranes may be more attractive than electrodynamic transducers in converting power. In acoustic resonators, high-amplitude tones can be excited by heat addition or vortex shedding in the presence of mean flow. Tonal sound driven by vortex shedding and impingement often appears in various systems such as whistles (Wilson et al. [4]), duct networks (Harris [5]), and rocket motors (Culick [6]). Heat addition to such resonators also leads to excitation of high-amplitude sound and this is noticeable phenomenon that occurs in thermoacoustic systems (Swift [], [7]). High-amplitude tones produced from a resonator with a pair of baffles in proper ranges of flow rates have been investigated by some researchers (Nomoto and Culick [8]). In the mean flow, vortices are created from the upstream baffle and impinge on the downstream baffle and a part of flow energy is supplied into acoustic modes of the resonator. When the vortex shedding frequency is close to the natural frequency of the resonator, excitation of acoustic modes comes about in certain ranges of flow rates. There exists favorable spacing between two baffles and the positions of baffles for a mode excitation usually correspond to the velocity anti-nodes in acoustic mode. Interaction between heat and sound in acoustic resonators has been studied as part of thermoacoustics. In thermoacoustic engine, a piece of porous material, known as a stack, where a 7

26 temperature gradient is maintained externally plays a significant role. Heat is added to oscillating gas parcels in the stack at the moment of their compression and removed at the moment of their rarefaction. Then, these self-sustained oscillations satisfy Rayleigh s criterion (Rayleigh [9]) and tonal sound is generated. After tonal sound is excited by any of these effects in a resonator, the acoustic power can be harvested by electroacoustic transducers and turned into a useful energy... Previous Work Song and Bolton [] described the experimental transfer matrix method to estimate the acoustic properties of a glass fiber material. They made this procedure more efficient by taking advantage of the reciprocal nature of sound transmission. Muehleisen et al. [] used the fourmicrophone transfer matrix measurement method to determine the characteristic impedance and wave number of RVC. Most recently, Roh [] investigated the properties of RVC and aluminum foam using a hybrid two-microphone impedance method and a lumped-element technique. As well as Song and Bolton, Muehleisen et al. and Roh, there are several researchers who have taken a measurement of the acoustic properties of porous materials without a temperature gradient. Tarnow [3], Leclaire et al. [4], Clark et al. [5], Wilen [6], and Petculescu and Wilen [7] have experimentally measured the acoustic properties of various porous materials. Few researchers have tried to measure the acoustic properties of porous materials while 8

27 the temperature gradient is imposed across the materials and found it is very difficult to directly quantify the effect of temperature gradient [8, 9, ]. A description of these difficulties will be given in the later section. In general, losses are more significant in acoustic boundary layers near solid surfaces of the system. Therefore, thermoviscous effects outside boundary layers can be neglected; and the damping inside boundary layers can be accounted for by wall admittances [, ] or thermoviscous functions [3]. Thermoviscous functions are commonly used to estimate gain or loss of acoustic power. One of the practical thermoacoustic codes, DELTAE [4], solves a quasi-one-dimensional wave equation of acoustic field in ducts using thermoviscous functions with an assumption that mean temperature and acoustic pressure are uniform across in the duct cross sections. For most engineering applications noisy tones are generally undesirable, however, the generated sound energy can be transformed into a useful source for powering other devices such as remote sensors and micro robots. Using electroacoustic transducers, the acoustic power is captured and converted to electricity. In early s, some features of piezoelements coupled to thermoacoustic devices to generate electrical power were patented (Symko et al. [5], Keolian and K. Bastyr [6]). Liu et al. [7] further developed an energy harvester with piezoelectric diaphragm by coupling the diaphragm to electromechanical Helmholtz resonator. Four different 9

28 piezoelectric membranes, PFC, PVDF, PZT disk, and PZT bender, are tested and compared for medium scale thermoacoustic engines by Wekin [8]..3. Objectives and Contributions To obtain the transfer function for determining the characteristic impedance and wave number of porous fibrous materials, the three- and four-microphone impedance tube methods are used. RVC and plastic/steel mesh are chosen as the materials investigated in the first study. The main purpose of the first study is to validate our measurement system and calculation methods in the case of a simple configuration and to discuss further development of this approach. If demonstration of the performance of the impedance tube measurement system with random porous materials produces reasonable agreement with other theories and empirical correlations, a next step can be a study of acoustic properties of random porous material with imposed temperature gradient. Neither of these porous materials has been investigated in the past with a temperature gradient imposed across the materials. Analyzing relative acoustic power changes across the stack material is chosen as a metric for the effect of temperature gradient. Relative acoustic power changes across RVC, stacked plastic screens and stacked steel screens at T=C and at TC are experimentally obtained and compared in the following chapter of this study.

29 Another problem specific to miniature systems is significant nonuniformity of temperature and sound fields in the direction perpendicular to the main acoustic motions. In small systems heat sources and sinks are likely to be positioned on an outside surface of resonators. Wire mesh screens or end sections of the stack may perform as heat exchangers to supply/remove heat to/from the stack ends. With miniature heat exchangers and wide resonators, transverse variations of the mean temperature can be large and acoustic field becomes more complicated. Since the common methods (for example, DELTAE) do not consider transverse nonuniformities, a different approach is needed for more accurate modeling of small-scale thermoacoustic systems. An efficient computational methodology accounting for above mentioned nonuniformities is presented in this study. An analysis of flexible walls, boundary-layer losses, and thermoacoustic stack interfaces are included while calculating boundary conditions that appear near resonator walls. A finite-difference method is used to solve the lossless linearized acoustic wave equation in the bulk of resonator. The method produced in this study can be applied for modeling of acoustic fields in other lossy resonators as well as thermoacoustic engines. The main objective of last study is to demonstrate harvesting of energy of tonal sound using piezoelements in simple resonator systems with heat addition or vortex shedding in the

30 presence of mean flow. Three types of pipe setup are built to conduct energy harvesting tests. First setup consists of a pipe with mean air flow, two baffles, and a piezoelement. A branch pipe of same diameter is attached to mount a piezoelement. In this case, controlled parameters are resistances of electric loads and mean flow rates. The second system comprises a pipe with stack, electric heater, and the similar piezoelement. One side of the pipe is closed and opposite is open. Close to the open end, a branch pipe is positioned. The third setup is similar to the second system but it has no branch and the opposite side end is also closed by the piezoelement. For latter two setups, resistances of electric loads and input power to the electric heater are varied.

31 Chapter Acoustic Properties of Porous and Fibrous Materials 3

32 .. Experimental Setup and Procedure In this chapter, an experimental investigation of the acoustic properties of porous and fibrous materials such as RVC and stacked plastic mesh is presented. The first experimental system representing a simple sound tube is shown in Figure.. This system is composed of a constant-diameter tube, two microphones with pre-amplifiers, a function generator, a power amplifier, a speaker, a signal conditioner and a digital oscilloscope. The tube is made of plastic pipe with a 3.4 mm inner diameter. A RCA RC4 speaker generates sound in the tube. The signal for the sound source originates from a Victor VC function signal generator and is amplified by an AudioSource AMP power amplifier. PCB Piezotronics model 377C microphones with model 46B3 pre-amplifiers are used for sound measurements. For the three-microphone method described in Figure.b, signals from positions 3 and are taken first while position is blocked. Next, the same procedure is conducted for positions and. For the four-microphone method shown in Figure.c, the signals from positions and, and 3, and and 4 are taken separately while the two other positions are blocked. The microphone signals go through a PCB Piezotronics 48C signal conditioner and are displayed on a Rigol DS64B digital oscilloscope. Only two microphones are applied in the preliminary tests with the empty tube. Figure. presents the three- and four-microphone impedance tube setup. 4

33 Temperature is measured using an Omega HHA digital thermometer and an Omega type K thermocouple. The local ambient pressure is taken using a Conex JDB digital barometer. A.3-inch-long piece of insulation material is placed at the rigid end of the four-microphone impedance tube to reduce error in measuring the phase shift between the microphones. The insulation material is made of flexible polyurethane. (a) 5

34 (b) (c) Figure.. (a) Schematic of the entire measurement system, (b) dimensions of the threemicrophone impedance tube, (c) dimensions of the four-microphone impedance tube 6

35 (a) (b) Figure.. (a) Photograph of the three-microphone measurement setup and (b) photograph of the four-microphone impedance tube 7

36 Next, the experimental system and measurement methods for determining the acoustic power change in the presence of a temperature gradient are described. Figures.3 and.4 show a schematic and a photograph, respectively, of the experimental setup with a temperature difference imposed across the stack. The measurements are taken using a modified threemicrophone system. The impedance tube is made up of several tube sections of different materials. The section with two microphones close to a speaker is a plastic tube with a 3.4 mm inner diameter. Since copper has a high thermal conductivity, the two sections where heat comes in and goes out are made of copper. They also have the same inner diameter. To reduce external heat leak between hot and cold parts, the stack holder is made of stainless steel with a lower thermal conductivity. The manufacturer of the microphone specifies that it may be affected by high temperature, so an additional narrow tube section which is cooled by an external fan is attached to the end of the system. Each tube section has flanges with four holes at both sides so that two different sections can be joined using bolts and nuts. The speaker, digital oscilloscope, function signal generator, sensor signal conditioner, power amplifier, microphones and thermocouples used in this experiment are the same as those used in the previous measurements. As shown in Figure.3, temperatures are taken at four positions using type-k thermocouples and Omega HHA, HH47U and HH53 digital thermometers. Heat is supplied to one section of the copper tube by an Omega electric band heater and released from the opposite section by a 8

37 cooling jacket with flowing water inside it. The water inside is cooled and circulated by a Forma Scientific 6 Bath & Circulator. Figure.3. Schematic of the three-microphone measurement setup with a temperature difference imposed across the stack Figure.4. Photograph of experimental setup for tests with temperature gradients 9

38 .. Measurement Theory In this experiment we focus on a version of the two microphone transfer function method which is commonly used to calculate the complex characteristic impedances and wave numbers of thermoacoustic materials and absorbers in isothermal conditions. First of all, the transfer function is analyzed as follows. The acoustic wave includes pressure and velocity components. Analytical expressions for acoustic pressure and velocity waveforms are as follows [9]: P jkd jkd Ae Be, (.) U A Z e jkd B Z e jkd, (.) where A and B are the complex constants, d is the distance from the rigid end, Z is the characteristic impedance of the fluid, and k is the acoustic wave number for air in a circular tube. At the rigid termination with d, P and U can be expressed as P( ) A B, (.3) A B U(). (.4) Z Z Since U ( ), A B /. Therefore, P U d d P e P Z P e P jkd jkd, (.5) e jkd P Z e jkd. (.6)

39 The magnitude of the transfer function between the position at distance d and the rigid end is defined here as a ratio of P d to P and described as H P d d. (.7) P To calculate the acoustic wave number in the circular tube, we use thermoacoustic functions and as in the reference of Ueda et al [3]. The wave number k can be written as Y J ( Y ) J( Y ) k k ( ), (.8) Y J ( Y ) J( Y ) Y J ( Y ) where k / c is the characteristic wave number, is the angular frequency, c is the speed of sound in air, and is the specific heat ratio of air. Parameters Y and Y in the function are given as Y Y ( i ) r /( ), (.9) ( i ) r /( ), (.) where r is the radius in the tube, is the thermal diffusivity of air, and is its kinematic viscosity. It can be noted that our calculations show that results obtained with wave number k, which accounts for wall losses, differ very little from results obtained with lossless k in most of the experimental range. In an approach similar to the method described above, the three-microphone and four-

40 microphone methods can also be used. Here, the derivation of complex characteristic impedance Z c and wave number k c using the four-microphone method will be described. Applying the expressions for acoustic pressure and velocity waveforms (equations. and.) to the section between the acoustic driver and one surface of the stack material and using the pressure amplitudes at microphone position and, we can obtain complex constants A and B for that section: jks Pe P A, (.) jks jks e e B P e Pe e jks, (.) jks jks where S is the spacing between microphones and. Substituting these constants and the proper distance into equations. and. yields the acoustic pressure and velocity waveforms at the stack surface close to the acoustic driver: H sin( k( S l )) sin( kl ) P P, (.3) sin( kl ) U H cos( k( S l)) cos( kl ) jp, (.4) Z sin( ks ) where l is the distance from the surface of the stack material to microphone. Following the same procedure in the section between the rigid end and the other surface of the stack material, the acoustic pressure and velocity waveforms at the stack surface close to the rigid end can be written as P d H 3sin( k( S34 l3)) H 4sin( kl3 ) P, (.5) sin( ks ) 34

41 U d H 4cos( kl3 ) H 3cos( k( S34 l3)) jp. (.6) Z sin( ks ) 34 S 34 is the spacing between microphones 3 and 4, and l 3 is the distance from the other surface of the stack material to microphone 3. Z c is again the characteristic impedance of air. From the acoustic pressure and velocity on the sample surfaces, transfer matrix elements T, T, and T are calculated as T T T P U P U d d, (.7) Pd U PU d P Pd, (.8) P U P U d d d U U d. (.9) P U P U d Finally, complex characteristic impedance Z c and wave number k c are expressed using T T Z c, (.) k c cos ( T ), (.) dx where dx is the stack length. Based on previous measurements [], errors in a measurement system will significantly increase outside the range of. k s /. 8, where s is the spacing between microphones in the center portion of the tube, which is.956 m for our system. This inequality results in the frequency range of 78 Hz f 43 Hz. 3

42 .. Experimental Results The impedance tube measurement system was validated first in tests without porous materials. The experimentally derived reflection coefficient at the closed end and the transfer function between acoustic pressure values were compared to theoretical calculations. The calculation was carried out at the average test temperature. Two microphones with different sensitivities were used in the first measurement. One microphone, with a serial number of 374 and a sensitivity specified by the manufacturer of.8 mv/pa, is referred to as microphone ; the other, with a serial number of 3737 and a sensitivity of.83 mv/pa, as microphone. The error of phase shift between the microphone signals introduced by the measuring system is negligible. Voltages of.5 V and.5 V were used as input voltages. The results of the measurements with.5 V show inferior performance, so the measurements with the.5 V input voltage are presented. For the three-microphone measurements, all values were measured and calculated at the frequencies of, 4, 6, 8,,, and 4 Hz. The input voltage of the function signal generator wes fixed and corresponds to pressure amplitudes at the rigid end in the range from 6 to 5 Pa. Comparing the measured reflection coefficient at the rigid end with its theoretical value can be one mean of verifying the practicability of the impedance tube method. Figure.5 describes the ideal reflection coefficient and the reflection coefficient derived from 4

43 R Phase of R (degree) measured transfer function H. The magnitude of the ideal reflection coefficient at the rigid end is unity ( R = ), shown in the figure with a solid red line. To confirm the propriety of this derived reflection coefficient, we can also compare measured transfer function H 3 and transfer function H 3 obtained from the derived reflection coefficient. All measured and calculated data are in good agreement, as shown in Figures.5 and.6. (a) (b) f[hz] f[hz] Figure.5. (a) Magnitude and (b) phase of the reflection coefficient derived from measured transfer function H : Solid red curve, ideal value; black dots, derived from measured H 5

44 H 3 Phase of H 3 (degree) (a) (b) f[hz] f[hz] Figure.6. (a) Magnitude and (b) phase of measured transfer function H 3 and transfer function H 3 obtained from the derived reflection coefficient: Solid red curve, measured H 3; black dots, H 3 obtained from the derived reflection coefficient The wave number and characteristic impedance of 8-ppi RVC were measured using the three- and four-microphone methods. After careful comparison of the two methods, it was concluded that the four-microphone method shows better performance in characterizing the properties of RVC in the specific frequency range. The real and imaginary parts of the wave number and characteristic impedance of RVC measured by the four-microphone method in the frequency range of 3 Hz to Hz are shown in Figure.7. The measured data are also compared to the results calculated using the Muehleisen and Roh s empirical correlations. The dots in the figure present a reasonable deviation, and the error bars at 4 Hz and Hz are derived from a combination of random and bias uncertainties. The random uncertainty, P R, of 6

45 the measured data can be obtained for mean values as follows: P R S x t, (.) n where t comes from the t-distribution (it depends on the number of data points and the confidence level) [3], S x is the standard deviation of the data, and n is the number of data points (or repeating times). A 95 % confidence level is assumed in this uncertainty calculation. In the calculation of bias uncertainty B, the errors of mean temperature and pressure are.5 C and 5 kpa, respectively. The error of all dimensions including stack length, tube radius and microphone spacings is.5 mm. For the transfer function, the error of the magnitude is % and that of the phase is.5. The calculation of bias uncertainty utilizes a 95 % confidence level as well. Therefore, the total uncertainty, U t, can be calculated as follows: U B. (.3) t P R The scattering of data points in the figure can be regarded as a consequence of the randomness of the stack material. The difference among the present results and the empirical correlations of Muehleisen and Roh is due to the variability of the RVC samples. 7

46 (a) (b) Figure.7. Real and imaginary parts of normalized (a) wave number and (b) characteristic impedance of RVC: Dots, measured by the four-microphone method; solid red curve, results calculated by Muehleisen s theory; dashed blue curve, results calculated by Roh s theory After the performance of the present system with RVC was demonstrated, the wave number and characteristic impedance of plastic mesh were studied. Polypropylene mesh layers, each with a maximum thickness of.37 inches, a pore size of.3 x.35 inches and a wire 8

47 width of.5 inches, were randomly stacked to configure the plastic mesh stack. A photo of one plastic mesh layer is shown in Figure.8. The real and imaginary parts of the wave number and characteristic impedance of plastic mesh measured using the four-microphone method in the frequency range of 3 Hz to Hz are shown in Figure.9. In this figure, again, the reason for the scattering of data points is the randomness of the stack material. The dots in the figure do not shape a perfect line, but they are sufficient for determining the wave number and characteristic impedance of the plastic mesh. The same measurement was carried out with an aligned plastic mesh stack as well, and the wave number and characteristic impedance from this measurement were similar to those shown in Figure.9. Figure.8. Photograph of a plastic mesh layer 9

48 (a) (b) Figure.9. Real and imaginary parts of the normalized (a) wave number and (b) characteristic impedance of the plastic mesh The second set of measurements focused on finding the effect of a temperature difference on the acoustic properties of porous materials. All computed and experimentally derived relative acoustic power changes across RVC, stacked plastic screens and stacked steel screens at T=C and at TC are compared in the following section. 3

49 Figure.a presents the experimental measurements of the acoustic power changes ) between the two sides of the RVC stack over E in at T=C and at ( E E out Ein TC in the frequency range of to 5 Hz. Here, E in is the incident acoustic power on one surface of the stack material which is close to the speaker and E out is the acoustic power leaving the other side. The three-microphone method was modified to obtain the pressure data. Acoustic velocities were calculated as in the previous section. This information was used to determine the acoustic power: E ~ Re pu. (.4) The black vertical error bars were obtained using the same uncertainty calculation as above. The error bar of the result at TC shows a somewhat high deviation, which suggests that the present experimental system does not produce quite accurate results. As shown in the figure, the relative acoustic power loss at T=C is explicitly greater than that at TC in this frequency range. The presence of the temperature gradient in the RVC stack leads to the generation of acoustic power, thus compensating for acoustic losses when sound energy flows through the stack at a nonzero temperature difference. As with the RVC, the same measurements were conducted with stacked plastic screens (Figure.b) and with stacked steel screens (Figure.c). As these figures show, mesh screen stacks also compensate for acoustic losses while applying a nonzero temperature difference 3

50 across the stack. There is no comparison with theories or correlations given in Figure., since there are no validated models for the tortuous media thermoacoustics and the discrepancies with some hypotheses have been found to be very significant. The characterization of the acoustic properties of a random porous medium at a nonzero temperature difference has proven to be very difficult. The compressibility of a single pore with a specific temperature gradient was measured by Petculescu and Wilen [8], but they did not isolate the effect of the temperature gradient. They also attempted measurements of a stacked screen stack with an applied temperature gradient, but no results were published. Simmons [9] and Slaton [] also concluded that it is very difficult to measure these properties accurately and directly. 3

51 (a).95 ΔE loss / E in f [Hz] (b) (c) ΔE loss / E in ΔE loss / E in f [Hz] f [Hz] Figure.. Experimental measurements of relative acoustic power changes across (a) RVC, (b) stacked plastic screens and (c) stacked steel screens at T=C (dotted line) and at TC (solid line) 33

52 Chapter 3 Modeling of Thermoacoustic Resonators with Nonuniformities 34

53 3.. Theoretical Model In this chapter, modeling of acoustic fields in a small-scale thermoacoustic resonator with nonuniform medium and boundary conditions is carried out. An acoustic field in a resonator is considered to be steady-state, low-amplitude, and single-frequency. Simple expressions for acoustic pressure and velocity can be written as follows []: it p x, t) p Re[ p ( x) e ], (3.) ( m it u x, t) Re[ u ( x) e ], (3.) ( where p m is the mean pressure, p and u are the complex amplitudes of acoustic pressure and velocity, respectively, i is the imaginary unity, is the radian frequency, and t is the time. The subscript stands for the first-order complex amplitude and the underbar of acoustic velocity indicates a vector. The second-order effect due to acoustic streaming is ignored since only low-amplitude sound is considered. Generally, the relative acoustic pressure amplitude p / p m over 5% makes the streaming significant [, 3]. In the present modeling section, the ratio p / p m less than % is included into analysis for almost all examples. At the wall of the resonator, thermoviscous effects must be taken into account. However, for regions far from walls where thermoviscous effects can be neglected, a modified Helmholtz equation for a nonuniform medium can be utilized. It is derived from the linear acoustic theory [9] and the equations of mass, momentum, and energy of the fluid are first introduced as 35

54 follows: ( u), t (3.3) u p, t (3.4) T c v u T p u, t (3.5) where and T are the local density and temperature, respectively. c v is the isochoric specific heat. The partial derivative of time / t can be converted to complex notation i. For two-dimensional cases, Equations 3.3, 3.4, and 3.5 are expanded to i u u v v, (3.6) x x y y px i u px u, i (3.7) p y i v p y v, i (3.8) v it T u T v pu v c, (3.9) x y x y where u and v are velocity components along x and y direction, respectively. Subscripts x and y represent partial derivatives with respect to each of them. For example, u x and v y are defined as follows: u v x y p i p i xx x x, (3.) p yy y p y. (3.) i i The symbols having no subscript are mean values. Substituting Equation 3.7 and 3.8 into Equation 3.9 and applying the equations of gas properties T x, y x, y T, c v R, 36

55 and ideal gas law p RT yield p T pxx p yy i i i p x px y p y, (3.) where is the ratio of isobaric to isochoric specific heats and R is the gas constant. Next, the equation of state p p T T is manipulated, R p T. (3.3) P p Then, substituting T (in Equation 3.) into Equation 3.3 becomes x y p xx p yy px p y p P. (3.4) After replacing in Equation 3.6 by Equation 3. and using the gas property p a, a modified Helmholtz equation is obtained, p p p, (3.5) a where a is the local sound speed. A mean temperature field varies spatially in the resonator; therefore, gas density changes as well. This nonuniformity of gas density is explained by the last term of Equation 3.5. Acoustic losses in the bulk of resonator are neglected. At the outer edge of acoustic boundary layer, the boundary condition is applied for the acoustic velocity amplitude u normal to the wall to address thermoviscous losses at the resonator solid walls and possible normal oscillations of the walls. The acoustic momentum equation leads to the following expression of the boundary condition: 37

56 p iu i( u uw ), (3.6) where u and u w are the velocity amplitude defining thermoviscous effects in the acoustic boundary layer and the velocity amplitude of the wall oscillations, respectively. Figure 3. shows numerical grid and the acoustic boundary layer. As shown in Figure 3.b, is the coordinate normal to the wall. The velocity amplitude u in a nonuniform medium was derived by Olson and Swift [33], u i p i i p ( ) p, (3.7) 3/ 3/ a where is the gas kinematic viscosity, is the thermal diffusivity, is the specific heat ratio, is the Prandtl number, and is the coordinate parallel to the wall. The acoustic boundary layer thickness shown in Figure 3.b can be estimated using and as follows: max( v ; ) max ;, (3.8) ~ k where v and k are the viscous and thermal penetration depth, respectively. The size of numerical grid is chosen to be much greater than the thickness. Due to the small size of the acoustic boundary layer thickness in comparison with a numerical cell size, the boundary nodes of computational mesh can be roughly positioned at the wall surface. One can apply Equation 3.6 at these boundary nodes. A modified Helmholtz equation (Equation 3.5) is employed at the 38

57 points in the bulk of a resonator. Using the finite-difference method, Equation 3.5 and 3.6 are transformed to discretized forms of the acoustic pressure amplitude and they develop into a system of linear algebraic equations. After obtaining overall nodes of acoustic pressure amplitudes, the acoustic momentum equation is applied to calculate the acoustic velocity in the bulk of a resonator. i u p (3.9) Figure 3.. (a) Numerical grid and (b) magnified view of region near rigid wall surface In thermoacoustic devices, a porous material (stack) is inserted inside a resonator where a temperature gradient is maintained (Figure.). The common wave equation cannot account for viscous and thermal relaxation effects in the stack. Here, a modified wave equation suggested by Swift [] is adopted for stacks with parallel pores, 39

58 p d f dp a f f dt dp k, (3.) dx dx T dx dx m v k v ( ) f p where f v and f k are viscous and thermal acoustic functions specific to the stack pore geometry, respectively, T is the time average temperature in the pore section, and x is the coordinate along pores. Thermoacoustic functions f v and f k for parallel-plate stack are determined as follows []: f iy / iy / v, k tanh[ ] v, k v, k, (3.) where y is a half of the spacing between neighboring plates. To make temperature distribution as uniform as possible across the cross-section of a resonator, heat exchangers are usually placed at hot and cold sides of the stack in the practical thermoacoustic devices, however, they are neglected in this modeling for simplicity. It is also assumed that the temperature distribution in the stack is given. The acoustic momentum equation is modified as well to estimate the acoustic velocity amplitude along the pore. It is averaged over the pore cross section and expressed as follows: u fv dp i. (3.) dx At the interface between stack and resonator, the acoustic pressure can be considered continuous while the acoustic velocity varies since the cross-sectional area for oscillating gas flow changes. The mathematical expression of acoustic pressure and velocity at the interface is described by following equations, 4

59 p p p p, (3.3), x L xl xl xl u u, x L xl u u, (3.4) xl xl where L and L are the distances from a flexible wall (origin of x -axis) to each of the stack ends. Geometry and the position of a stack are shown in Figure 3.a. in Equation 3.4 is the stack porosity and its value is usually close to. Although Equation 3.3 and 3.4 satisfy the joining conditions in present case, at large acoustic amplitudes or low-porous stacks more complicated correction factors are needed to account for complex thermal and acoustic processes [, 34]. Interaction between amplitudes of the wall velocity u w and the local acoustic pressure p accounts for a process of the coupling between the gas oscillating in a resonator and a vibrating wall. This interaction generally depends on the structural properties of the flexing wall (Figure.b, 3.a). The main concern in this modeling is fluid oscillations in the resonator. An idealized linear mechanical oscillator model is used for a flexing part of the wall. The effective velocity amplitude of the oscillator U and the amplitude of the force due to acoustic pressure F are linked together. They are described in one equation with some properties of an oscillator as follows: if U, (3.5) K M ir where K and M are the stiffness and mass of an oscillator. U is defined here as the area- 4

60 averaged velocity amplitude at the flexing wall surface, A w U / A u da, and A is the surface area. The internal damping in the oscillator is removed from consideration for simplicity. As a result, harnessing the mechanical power of the oscillator by external means will only depend on coefficient R in Equation 3.5. One can convert the mechanical power of the oscillator to other useful forms of energy, such as electricity. The effective force F can be presented as a function of p and u w, F p u~ ~ wda U, (3.6) A where the tilde indicates a complex conjugate and the minus sign means the direction of x -axis from the flexing wall into the resonator (Figure 3.a). From the form of Equation 3.6, it is interpreted that the oscillator can extract the time-averaged acoustic power. This power can be calculated using acoustic variables or mechanical parameters, P Re ~ pu~ wda ReFU A F / M R R, (3.7) where K / M is the natural frequency of the oscillator. In case when the coefficient R is equal to zero, there is no acoustic power brought to the flexing wall thus no mechanical power is extracted from the engine. 4

61 Figure 3.. (a) Geometry of a D thermoacoustic engine with flexing wall on the left boundary and (b) Given mean temperature profile at y= A finite-difference method is used to calculate the acoustic field in a resonator with a certain mean temperature field. The first-order and second-order derivatives in the partial differential equation are approximated by linear combinations of function values at the grid points. Finite difference forms of the first-order and second-order derivatives of acoustic velocity can be expressed as follows: u ui x u ui x, j u i, j x u, j i, j ( x), (3.8) u i, j, (3.9) where x is a unit spacing between two points along x -axis in computational mesh. 43

62 44 Subscripts i and j designate the specific number of points in x and y direction, respectively. At boundary conditions, one-sided approximation should be adopted and the firstorder and second-order scheme of it can be given as follows: x u u u x u j i j i j i 4 3,,,, (3.3),,, ) ( x u u u x u j i j i j i. (3.3) For example, converting Equation 3.5 to finite difference form yields j i j i j i j i j i j i j i p a y p p p x p p p,,,,,,,,,,,,,,, y x y p p x p p j i j i j i j i j i j i j i j i. (3.3)

63 3.. Modeling Results Before applying the proposed numerical method for resonators with variable mean temperature, validation for a constant-temperature resonator is needed. The same resonator as shown in Figure 3.a is validated here with an assumption that no stack exists in the resonator and left wall oscillates in a uniform (piston-like) motion with the velocity amplitude u w 3 m / s at the half-resonance frequency. The two-dimensional resonator contains a working medium of 5 bar air at a constant mean temperature fixed at 4 K. The resonator dimension is cm cm. The top and bottom boundaries are isothermal; and the left and right boundaries are adiabatic. Thermoviscous losses are included into analysis. At a constant temperature, the analytical solution and two-dimensional numerical solution are compared. The analytical solutions for the acoustic pressure and velocity amplitudes averaged over the cross-sectional areas are given via exponential forms [9] p ( x) A exp( ikx) B exp( ikx), (3.33) A B u( x) exp( ikx) exp( ikx), (3.34) Z Z where A and B are constants that can be found from two boundary conditions at flexing wall u () u ) and closed end ( u ( L) ). k and Z in Equation are the wave number ( w and the specific acoustic impedance, respectively. In a two-dimensional resonator, they can be determined using thermoviscous functions previously given in Equation 3.. One parameter that 45

64 should be changed in Equation 3. for this validation is y and a half of the resonator height H / substitutes for it. The wave number k and the specific acoustic impedance Z as a function of f v and f k become k a ( ) f f v k, (3.35) Z a ( f v ) ( ) f k. (3.36) The normalized acoustic pressure (Figure 3.3a) and velocity (Figure 3.3b) amplitudes of the analytical solution and the two-dimensional numerical solution for constant-temperature resonator are found to be very similar. Additional validation with a linear variation of mean temperature along the resonator is conducted. Temperatures at left and right walls are selected to be K and 6 K, respectively. Since the analytical solution cannot be found for a resonator with variable temperature, the quasi-one-dimensional wave equation and two-dimensional numerical solution are compared. D wave equation is given as Equation 3. and the thermoviscous functions in the wave equation are given in Equation 3.. (The modeling in DELTAE code is similar to this approach.) A comparison between D wave equation and D numerical model is shown in Figure 3.3c and 3.3d. From these results, one can conclude that the D model developed here agrees well with exact solutions. 46

65 (a) 5 x -3 (b).5 Re Re p /p m -5 - Im u /u w.5 Im x/l x/l (c) 5 x -3 (d).5 Re Re p /p m -5 - Im u /u w.5 Im x/l x/l Figure 3.3. Validation results of normalized acoustic (a) pressure and (b) velocity amplitudes at constant-temperature lossy resonator: solid lines, analytical solution; points, D numerical solution and normalized acoustic (c) pressure and (d) velocity amplitudes at linear temperature variation: solid lines, D wave equation; points, D numerical solution After successful validation, we attempt to model a small-scale two-dimensional thermoacoustic engine with a flexing wall. In a real system, the oscillating boundary in form of a flexing wall can be used to harness mechanical power from the engine. A schematic of the system with a stack is same as the geometry in Figure 3.a. Its low-aspect ratio configuration is 47

66 favorable to reducing surface area of the resonator so that acoustic attenuation at the walls decreases in comparison with other high-aspect-ratio engines. Another beneficial condition is a small operational frequency of the engine. Since thermoviscous damping weakens as the frequency decreases, it is possible to get the wall losses much lower when the operational frequency is sufficiently smaller than the natural frequency of the resonator. To achieve this condition, the natural frequency of the flexing wall must be much smaller than the natural frequency of the resonator. Parameters used in the following modeling example are listed in Table. Height H = cm Length L = cm Left position of the stack L = 5 mm Right position of the stack L = mm Gas type Air Mean pressure pm = 5 bars Mean temperature of the cold side TL = 3 K Transverse temperature difference ΔTy = K Stack porosity φ =.95 Table. Input parameters of thermoacoustic engine This configuration brings two complications in the modeling process. The first difficulty is a significant nonuniformity of the acoustic field in the transverse direction with respect to 48

67 main acoustic motions due to a nonuniform deformation of the flexing wall. Secondly, having uniform supply/removal of heat to/from the stack is difficult to achieve in low-aspect-ratio systems; therefore, the mean temperature may vary along y -axis. Common thermoacoustic calculation tools such as DELTAE cannot account for these effects since they are implemented for large and high-aspect-ratio systems. The modeling method developed here can solve thermoacoustic problems with these nonuniformities. Since we are interested mainly in acoustic processes, the mean temperature field and a form of the movable wall deflection are assumed to be given. For further development in numerical analysis, heat transfer and wall deformation will have to be considered. The mean temperature field is separated into three sections with respect to x -coordinate and the position of a stack. Its simple but realistic form is described by the following equations, T T( x, y) T T L L L T y y / H, T ( x L ) /( L T T y x L y / H, L ) T x L y y / H, L x L, (3.37) where H and L are the width and length of the resonator, respectively, T L is the mean temperature of the cold side of the resonator, and T is a constant temperature difference between the stack ends along the x -coordinate. Figure 3.b gives the mean temperature profile at the center plane ( y ). Ty is a transverse temperature difference between the upper ( y H / ) and lower ( y H / ) walls of the resonator. As shown in Table, it is constant 49

68 and chosen to be K based on realistic temperature nonuniformity for a specific example. The transverse temperature difference Ty in this modeling, however, can be any value (even much larger). The velocity amplitude of the wall and the gas velocity amplitude at the wall surface are defined as follows: u w y ( y) U w cos, (3.38) H where U is the velocity amplitude of the wall at y. From Equation 3.38, one can see that w the end points of the wall are fixed, and the maximum velocity and wall deflection occurs at center of the wall ( y ). The other walls at y H /, y H /, and x L do not move at all. When the temperature difference in the stack exceeds some critical value Tcr, the sound is generated. Tcr can be calculated here for parameters in Table. As mentioned before, the stack is composed of parallel plates and a half-spacing between the plates y is assigned to be.3 k, where k is the thermal penetration depth which can be found in Equation 3.8. The mass of an oscillator M (Equation 3.5 and 3.7) and its natural frequency f / are variable parameters. In the second-order finite-difference scheme basis, the acoustic field can be found by solving the wave equations for the bulk of resonator and stack (Equation 3.5 and 3.) with 5

69 boundary conditions (Equation 3.6, 3.3, and 3.4). The total number of numerical nodes is 44, so there are nodes ( n ) in each direction of the domain. nodes are large enough since a change in result made by twice the number is insignificant. In this example, the frequency of acoustic oscillations f is the eigenvalue. The frequency and the critical temperature difference are determined when the power extraction coefficient R in Equation 3.5 and 3.7 becomes zero at the sound onset. For various oscillator parameters f and M, the obtained values of f, Tcr, and their normalized values are displayed in Figure 3.4. In the x -axis, f is normalized by the natural frequency of a resonator f r, which is valuated for a half-wavelength uniform resonator as a / L. Due to presence of a flexing wall oscillator at the left boundary, the resonator natural frequency may be different from the frequency of acoustic oscillations f. For corresponding temperature conditions, the resonator natural frequency f r and the mass of the gas in the resonator M g are in the ranges of g/m and khz, respectively. In Figure 3.4b, the normalized critical temperature difference is shown. Dividing Tcr by ideal critical temperature difference [], it can be derived as follows: T c u, (3.39) T cr p Tcr cr L p, id s where c p is the isobaric specific heat and L s L L is the length of the stack. All gas 5

70 properties and acoustic amplitudes in above equation are found as mean values at the middle section of the stack / L L. The ideal critical temperature gradient T cr T id cr / L,, id s accords with a single plate acoustically oscillating with a standing-wave phasing in an inviscid fluid. From the results in Figure 3.4a, one can conclude that the critical temperature difference is reduced when a low natural frequency and a large mass of an oscillator are chosen (with other system parameters fixed). As the ratio of the natural frequencies of an oscillator and resonator increases, Tcr increases and its sensitivity to the oscillator mass decreases. The normalized critical temperature difference presents similar behavior to that of Tcr (Figure 3.4b). In Figure 3.4c, the decreasing trend of the frequency of self-excited acoustic oscillations f is observed with the decreasing natural frequency of the oscillator and increasing mass. A low value of f results in reduced thermoviscous losses and lower Tcr (earlier sound onset). In the studied range of system parameters, the frequency of acoustic oscillations is always higher than the natural frequency of the oscillator (Figure 3.4d). However, as f / f r increases, it gets close to f. In the limit of small f (or small oscillator stiffness K ), the effective stiffness of the system is due to compressibility of the gas in the resonator. 5

71 f[khz] (a) 5 (b) 6 T cr [K] 5 5 M/M g = M/M g = f /f r f /f r (c) 4 (d) f/f f /f r f /f r Figure 3.4. (a) Dimensional and (b) normalized critical temperature difference, and (c) dimensional and (d) normalized frequency of the thermoacoustic resonator at the sound onset: solid line, M / 5 ; dotted line, M / M g M g Figure 3.5 shows an example of calculated acoustic pressure and x -component of the velocity in the two-dimensional resonator. The system parameters from Table, the mean temperature field from Equation 3.37, and the velocity amplitude of the flexing wall from Equation 3.38 are used as given conditions. Other parameters not specified in Table are given as follows: the stack temperature difference T K (larger than corresponding critical 53

72 temperature difference), the maximum wall velocity amplitude U 3m/s, the oscillator natural frequency f 45 Hz, the oscillator mass M w g/m, and the stack plate half spacing 5 y 5 m. The imaginary component of the acoustic pressure (Figure 3.5b) and the real component of the acoustic velocity (Figure 3.5c) dominate the real pressure (Figure 3.5a) and the imaginary velocity (Figure 3.5d), respectively. This means that the principal acoustic motions occur in the standing-wave phasing. However, the nondominant components (Figure 3.5a and 3.5d) are not exactly zero, so the phase shift is not exactly 9 degrees. From this fact, one may conclude that some acoustic energy is transported through the system, since the time-averaged acoustic energy flux is proportional to Re( p ~ u ). The dominant acoustic motions appear mostly along the x -coordinate, but they also vary along the y -coordinate outside of the boundary layer. The real component of the acoustic velocity at the flexing wall boundary experiences dramatic changes due to nonuniform motions of the flexing wall. This strong variability gets attenuated along the resonator and the velocity becomes zero at the opposite wall (Figure 3.5c). Some other notable phenomena include local maxima of the imaginary acoustic pressure at the center of the flexing wall (Figure 3.5b) and the imaginary velocity in the central portion of the stack (Figure 3.5d). 54

73 (a) (b) x Re(p )/p m Im(p )/p m x/l - y/h x/l - y/h (a) (b).5 Re(u )/U w - Im(u )/U w x/l - y/h x/l - y/h Figure 3.5. Complex amplitudes of (a) real and (b) imaginary parts of acoustic pressure normalized by mean pressure, and x-component of (c) real and (d) imaginary parts of acoustic velocity normalized by maximum velocity at flexing wall (Velocities inside stack are averaged over pore cross sections) The acoustic motions variable along the y -coordinate (the direction perpendicular to the main acoustic oscillations) cause nonuniform thermoacoustic energy conversion in the stack. The flux of generated acoustic energy out of the stack, its normalized value, and the local stack based thermoacoustic efficiency can be found using the following expressions, 55

74 W ( y) Re ( p ( ) ~ ( )) ( ( ) ~ y u y x L p ( )) y u y xl, (3.4) W ( y) H w( y), (3.4) H / Re ( p u~ ) xl ( pu~ ) xl dy H / w( y) st( y), (3.4) q( y) where q is the local heat addition rate to the stack per its cross-sectional area. This heat addition rate can be estimated from the enthalpy flux along the stack pores [] and conductiontype heat transfer. The heat conduction leak in the stack is assumed to appear only in the gas and the longitudinal thermal conductivity of the stack solid plates is negligibly low in this analysis. The normalized flux of acoustic power produced in the stack and local stack-based thermoacoustic efficiency are presented in Figure 3.6 using width H as a variable parameter. For the defined system, the temperature difference T is selected to be K. Nearly uniform acoustic power flux and local efficiency in the resonator with aspect ratio H / L. 5 are found. The nonuniformity of calculated results becomes significant as the aspect ratio increases. The difference between values at two walls ( y / H ) arises due to the temperature variation along the y -coordinate. Similarly caused by Ty, small asymmetry of the acoustic pressure and velocity can be found in Figure 3.5. The large transverse nonuniformities at H / L. 5 come from a heavily nonuniform acoustic motions in the stack. The strong nonuniformity of acoustic oscillations (such as local 56

75 w increase of imaginary velocity amplitude) can be seen in Figure 3.5d. This fact suggests that stacks in the considered systems can be optimized by using proper lengths and positions of the stack segments over the resonator cross section. (a) H/L=. H/L=.5 H/L= y/h (b).3 st y/h Figure 3.6. (a) Normalized flux of acoustic power produced in the stack and (b) local stackbased thermoacoustic efficiency: solid line, H/L=.5; dashed line, H/L=.; and dotted line, H/L=.5 57

76 When the stack generates sound at the temperature difference threshold, the thermoacoustic engine starts producing net acoustic power. Either nonlinear acoustic losses or limits on how much heat can be supplied/removed to/from the stack or how much acoustic power can be extracted from the wall oscillator are required for systems to operate in steady state. In this study, the amount of extracted mechanical power is set to be the limiting factor. The thermomechanical efficiency in the excited regime can be simply found by the formula, P tm Q, (3.43) where P is the mechanical power obtained from the oscillator (Equation 3.7) and Q is the total heat rate given to the stack []. With the aspect ratio H / L., and variable mechanical loads and stack temperature differences (other parameters are same as above), the thermomechanical efficiency is calculated as shown in Figure 3.7. The relative acoustic pressure amplitudes p / p p / p m below % produce low-power cases of. W/m and. W/m, and m within -6 % result in highest-power case of W/m in specific range of T. Finite-amplitude acoustic effects are not included into consideration, but with the highest power output they may become significant. Since the parasitic heat conduction along x -axis is not affected by the acoustic amplitudes and the influence of the heat leak on the total rate of supplied heat decreases with increasing power output (or acoustic amplitudes), the thermomechanical efficiency increases 58

77 with the mechanical power output at a certain T. The efficiency changes non-monotonically with respect to the stack temperature difference at fixed power output. When T is slightly greater than Tcr, then the powerextracting coefficient R is small, and from Equation 3.7 it follows that acoustic amplitudes must be large enough for a constant power level P. The rate of heat supplied to the stack increases with the acoustic amplitudes and the stack temperature difference []. While T is increasing, R also increases and obviously acoustic amplitudes must decrease to maintain constant P. Hence, the following phenomena occur: the heat rate initially decreases and the efficiency initially increases with increasing T, when the conduction heat leak is relatively small. At much greater T, the conduction heat leak carries a large amount of the supplied heat, so Q increases and the efficiency decreases. Using calculation results, one can find a certain value of T at fixed P that leads to the maximum efficiency. As the mechanical loads become larger, the efficiency peak moves to higher temperature differences (Figure 3.7). 59

78 tm P = W/m P =. W/m P =. W/m 8 4 T [K] Figure 3.7. Thermomechanical efficiency: solid line, P =. W/m; dashed line, P =. W/m; and dotted line, P = W/m Again, the fact that heat transfer caused by acoustic streaming is ignored in present modeling should be reminded here since only low-amplitude sound is considered. Various nonlinear effects are also neglected that may affect the system performance in high-amplitude regimes. The current system produces realistic values of the efficiency and power in given conditions. However, to achieve highest performance, the system optimization with respect to geometry and material properties must be carried out. In spite of relatively low efficiency and power levels, the present results indicate that this system has potential to be used as a small-scale power system [35]. 6

79 Chapter 4 Energy Harvesting from Tonal Sound Excited by Heat Addition and Vortex Shedding 6

80 4.. Experimental Setup The main goal of this chapter is to explore and demonstrate energy harvesting from tonal sound excited by heat addition and vortex shedding in the presence of mean flow. Vortex shedding driven by two baffles in a resonator is not directly involved in thermoacoustic effects; however, the experimental study of that phenomenon is included in this chapter as a part of efforts to research acoustic energy harvesting in small-scale systems. The first experimental setup with two baffles for vortex shedding is described schematically in Figure 4.a. A 6-cm-long PVC pipe with an inner diameter of 5.3 cm was used as the main resonator. To mount a piezoelement, a branch with the same diameter was attached close to the damping chamber, 45 cm from the upstream end. A small portion of the downstream end was inserted into a large damping chamber with sound-absorbing foam covering the internal wall. A regulated blower was placed at the opposite side of the damping chamber that sucked air from the chamber so that mean flow passing through the main pipe was created. The damping chamber created an open condition at the downstream end by reducing acoustic interactions between the blower and the pipe (Matveev and Culick [36]). A calibrated flowmeter was placed in the duct between the damping chamber and the blower to measure mean flow rates. Two sharp-edged, 3-mm-thick baffles were installed inside a pipe with cm of spacing between them. The orifice diameter of the baffles was.5 cm, and their position was 3 cm from 6

81 the upstream end. This location for the baffles was close to the velocity anti-node in the second acoustic mode of the system. The acoustic pressures at three different points were monitored uisng PCB 377C microphones with PCB 46B3 pre-amplifiers. The microphones were flash-mounted on the pipe wall to prevent the microphone tip from interfering with acoustic flow. Along the main pipe, microphones and were located and 45 cm from the upstream end, respectively. Microphone 3 was placed cm below the piezoelement. Similar to previous experiments, the microphone signals were stabilized using a PCB Piezotronics 48C signal conditioner and displayed on a Rigol DS64B digital oscilloscope. A PZT disk with a brass back plate (model MFT-5T-.7A, manufactured by APC International) was chosen as the acoustic energy harvester in this experiment. The PZT layer had a diameter of.3 cm, and the diameter of the whole element was 5 mm. Its thickness was.3 mm. Two acrylic flanges held the element tightly by means of bolts and nuts, and they were mounted on the branch end. The position of the element was intended to be close to the pressure anti-node in the second acoustic mode of the system. According to the manufacturer, the resonance frequency and maximum impedance are.7 khz and 3 Ω, respectively. Figure 4.b shows a photograph of the piezoelement held by two flanges. Two electric wires were each soldered onto PZT layer and the brass plate. They were 63

82 then directly linked to a passive electrical load to find the electric power. The general expression for the electric power released in a resistive load can be written as V, (4.) R P E where V is the RMS voltage measured across the resistive load by a multimeter and R is the electric resistance. This harvested power is the main concern of the present study. Three electric loads, with resistances of,, and kω, were varied while this experiments wa conducted. 64

83 (a) (b) Figure 4.. (a) Schematic of an experimental setup with mean flow and baffle, (b) photograph of the piezoelement held by two flanges 65

84 The configuration of the second system was somewhat similar to that of the first system, but the tonal sound was generated by the addition of heat to a stack material instead of by vortex shedding from baffles. An electric heater operated by a power supply provided heat to one side of the stack. Figure 4. shows a schematic and a photograph of the experimental setup. A 76-cmlong main pipe had the same diameter as the pipe used in the first experiment. One side of the main pipe was closed and the opposite side was open. A branch pipe for capturing acoustic power and converting it to electricity was built at a position 3.5 cm from the open end. For monitoring acoustic pressure, there was one microphone (microphone ) 3.5 cm from the open end. The signal from the microphone was analyzed in the manner described above. A square-pore ceramic comb (Celcor) was adopted as a stack in the resonator; it was placed 5.6 cm from the closed end. The dimension of the square pore was mm x mm, and there were 4 pores in one inch (5.4 mm). The wall thickness of the ceramic stack was.3 mm. Its outer diameter corresponded to the inner diameter of the resonator, and the total length of the stack was 4 cm. A different size of PZT element (model MFT-4T-.A) was used in the second and third experiments. While the diameter of the PZT layer and the thickness were the same, the total diameter of the element was 4 mm. Because of the different dimensions, it had somewhat different acoustic properties. The resonance frequency and maximum impedance of this element were khz and Ω, respectively. The configuration of the piezoelement with two flanges 66

85 was the same as before. Five resistances,, 4.8, 55., and kω of electric loads were used in the second and third experiments. (a) (b) Figure 4.. (a) Schematic of the experimental setup with heat addition, a stack, and one side open (b) photograph of the experimental setup 67

86 The third experimental setup had both ends closed. The energy harvester with a flanged piezoelement was mounted on the open-ended side, so both sides of the pipe were closed. In Figure 4.3, a schematic and a photograph of the third experimental setup are displayed. In this experiment, two microphones were used, one.3 cm and the other 3.3 cm from the piezoelement. The position of the stack was the same as that in the second experimental setup. 68

87 (a) (b) Figure 4.3. (a) Schematic of the experimental setup with heat addition, a stack, and both sides closed and (b) a photograph of the experimental setup 4.. Results In the first setup, several fixed mean flow rates were used in the resonator to conduct the experiment. Figure 4.4 shows the electric power harvested in a certain range of flow rates. This power corresponds to a single loud tone excited by vortices shedding and impinging on the 69