High-amplitude thermoacoustic flow interacting with solid boundaries

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High-amplitude thermoacoustic flow interacting with solid

boundaries

Citation for published version (APA):

Aben, P. C. H. (2010). High-amplitude thermoacoustic flow interacting with solid boundaries. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR692950

DOI:

10.6100/IR692950

Document status and date: Published: 01/01/2010

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High-Amplitude Thermoacoustic

Flow Interacting with Solid

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Copyright c 2010 by P.C.H. Aben, Eindhoven, The Netherlands.

All rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author.

Printed by Print Service Technische Universiteit Eindhoven

Cover design by Paul Verspaget

A catalogue record is available from the Eindhoven University of Technology Library

Aben, Paul

High-Amplitude Thermoacoustic Flow Interacting with Solid Boundaries / by Paul Aben.

-Eindhoven: Technische Universiteit Eindhoven, 2010. Proefschrift. - ISBN 978-90-386-2399-3

NUR 928

This research was financially supported by the Technology Foundation (STW), grant number ETTF.6668.

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High-Amplitude Thermoacoustic

Flow Interacting with Solid

Boundaries

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen

op woensdag 8 december 2010 om 16.00 uur

door

Paul Cornelis Hubertus Aben

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Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A.T.A.M. de Waele

Copromotor: dr.ir. J.C.H. Zeegers

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Nomenclature

List of symbols with a short description and the page on which it is introduced for the first time.

Symbol Unit Description Page

α kinetic-energy correction factor 67 α W/(m2K) heat transfer coefficient 88 β momentum correction factor 66 β K−1 volume expansivity 131 γ ratio of specific heats 1 δ_ν m viscous penetration depth 16 δ_κ m thermal penetration depth 18

ζ m displacement 24

η efficiency 23

η_C Carnot efficiency 24

θ rad phase angle 25

κ m3/s thermal diffusivity 18

λ m wavelength 4

µ Pa·s dynamic viscosity 86 ν m3/s kinematic viscosity 14 ξ m−1 complex wave number 21

Π m wetted perimeter 13

ρ kg/m3 density 14

σ0 s−1 viscous stress tensor 14

ψ porosity 54

ω rad/s angular frequency 15

ω_z s−1 vorticity 108

A m2 cross-sectional area 13

A_c m2 cone area 36

bµ viscosity exponent 86

bk thermal conductivity exponent 86

Bl N/A motor force factor 36

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vi

c_p J/(kgK) specific isobaric heat capacity 14 cso J/(kgK) solid heat capacity 86

Cc contraction coefficient 68

C₊, D₊ Pa amplitude of rightward traveling wave 55 C₋, D₋ Pa amplitude of leftward traveling wave 55 COP coefficient of performance 24 COP_C Carnot coefficient of performance 24

d m pore wall thickness 53

D m diameter 38

D m pore size 53

D_p m plate separation 86

Dr drive ratio 15

˙E W total energy flow 19

∆ ˙E_ml W energy loss due to minor losses 66

f Hz frequency 41

f₀ Hz characteristic frequency 54 fν viscous Rott function 17

fκ thermal Rott function 18

F N force 36

g m/s2 acceleration due to gravity 131

h J/kg specific enthalpy 19

hν, hκ channel-geometry dependant function 17

I A current 36 k W/(Km) thermal conductivity 14 k N/m spring constant 36 k0 m−1 real component of ξ 55 k00 m−1 imaginary component of ξ 55 K minor-loss coefficient 67

K_c minor-loss coefficient for contraction 68 K_e minor-loss coefficient for expansion 67 KC_D Keulegan-Carpenter number based on D 113 KC_L Keulegan-Carpenter number based on L 111 l₀ m half the plate thickness 16 L_el Vs/A self-inductance 36

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vii ˙ m kg/(m2s) mass streaming 130 M kg mass 36 N_u Nusselt number 133 p Pa pressure 14 P_r Prandtl number 18 ˙ Q W cooling/heating power 23

R_s J/(kgK) specific gas constant 14

R_e Reynolds number 107

R_eδ Reynolds number based on δν 113

R_el Ω electrical resistance 36 R_me Ns/m friction constant 36

R_a Rayleigh number 133

R_H m hydraulic radius 13

Rr reflection coefficient 57

s J/(kgK) entropy per unit mass 14

St Strouhal number 107 S surface 65 t s time 14 T K or◦C temperature 14 T transfer matrix 21 u m/s x-component of v 14 u_c m/s coil velocity 37

U m3/s volume flow rate 15

v m/s velocity vector 14

v m/s y-component of v 14

|v0| m/s root-mean-square value of v 129

V V voltage 36

V_b m3 back volume 36

V_ind V induced voltage 36

w m/s z-component of v 14

˙

W W Work flow 23

˙

W W acoustic energy flow 60

˙

W_{i j} W acoustic energy flow at position(x_i+x_j)/2 62

Wp m plate width 86

x m position along tube axis 14

y m 14

y₀ m half the plate distance 16

z m 14

Z impedance 36

Zac N m−5s acoustical impedance 36

Zel V/A electrical impedance 37

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viii

List of sub- and superscripts and special operators.

Symbol Description Page

aν viscous 17 aκ thermal 18 aa ambient 2 aac acoustical 36 a_C cold 2 a_el electrical 36 a_H hot 2

a_in (power) into the device 24

a_L left-hand side 21

a_me mechanical 36

a_R right-hand side 21

a_out (power) out of the device 24

a_p stack plate 86

aso solid 19

a0 zeroth order (average over one period) 15

a1 first harmonic 15

a₂ second harmonic 15

hai cross-sectional average 17

a∗ complex conjugate 19

˙a per unit time 19

a_{i j} matrix element of a with row i and column j 38 a+ Moor-Penrose generalized inverse of K 57 aH Hermitian matrix of K 57

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Introduction

1.1 Thermoacoustics

Thermoacoustics concerns phenomena in which an interaction of acoustics with ther-modynamics takes place. In an acoustic wave the gas parcels always undergo tempera-ture variations, which is a consequence of their compression and expansion. When the acoustic wave is adiabatic, the temperature variations to first order T₁are given by

T₁ T0 = γ−1 γ p₁ p0 , (1.1)

in which T₀ is the average temperature, γ the ratio of specific heats, p₁ the pressure variation, and p₀ the average pressure. In adiabatic sound waves these temperature variations go unnoticed. However, when a solid is present near the acoustic wave, the wave interacts with the solid and can cause a transfer of heat from one location in the solid to another. This is called the thermoacoustic heat-pumping effect. This effect is the driving mechanism in stack-based coolers and heat pumps. Vice versa, when a sound wave interacts with a solid with a temperature gradient above a certain critical value, the temperature gradient enhances the sound wave.

Lord Rayleigh was the first to give a qualitative description of thermoacoustic phe-nomena. In his work ”The Theory of Sound” [1], published in 1887, he discussed the ability to generate temperature differences using acoustic waves. The subject remained largely untouched for over eighty years, until in 1969 Nicholas Rott began a series of publications [2–7] that initiated a revival in thermoacoustic research. Rotts work forms the theoretical basis of most of modern standing-wave thermoacoustic research. Later, Rott’s publications have been reviewed by Swift [8]. Once the theory for parallel plates was established, the thermoacoustic theory for different pore shapes (triangular, rectan-gular, circular and pin arrays [9] instead of parallel plates) was developed by Arnott, Bass, and Raspet [10]. Also other authors contributed to the understanding of ther-moacoustics, including Wheatley and Atchley. Wheatley [11] introduced the thermoa-coustic couple and compared his measurement results quantitatively with Rott’s theory. Atchley worked on thermoacoustically generated temperature gradients in a thermoa-coustic couple [12], on the analysis of thermoathermoa-coustic prime movers and their onset of

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2 Introduction

self-oscillation [13–16], and together with Gaitan on the energy dissipation of higher harmonics and how to avoid this by detuning the resonance tube [17].

In ’t panhuis [18] rederived the thermoacoustic theory in a very systematic way. At the Los Alamos National laboratory a flexible code, called Design Environment for Low-amplitude ThermoAcoustic Engines (DeltaE) [19], was developed for the numeri-cal simulation of thermoacoustic devices. In 2007 a version with a better user interface was developed, called DeltaEC [20]. This code has been used by many groups to de-velop and test new devices.

1.2 History and applications

We distinguish heat-driven devices, called prime movers, and sound-driven devices, called refrigerators or heat pumps. In a prime mover the absorbed heat at a higher tem-perature T_His partially converted into work and the remaining energy is released at a lower temperature T_C < T_H. A refrigerator or heat pump absorbs heat at a lower tem-perature and requires input of work to release or pump heat to a higher temtem-perature, where the released heat is the sum of the absorbed heat and work. The difference be-tween a heat pump and a refrigerator is that in a heat pump, the heat is transferred from ambient temperature T_ato a higher temperature T_H >T_a, whereas in a refrigerator heat is transferred to ambient temperature Tafrom a lower temperature TC<Ta.

1.2.1 Standing-wave or stack-based devices

Already in the 19th century, glass blowers noticed that when a hot glass bulb was at-tached to a cool glass tube, it sometimes emitted sound, and Sondhauss [21] in 1850 quantitatively investigated the relation between the pitch of the sound and the dimen-sions of the apparatus.

Standing-wave devices are called this way, since their operation is based on a stand-ing wave that is created in a resonator tube. In case of a refrigerator one side of the resonator is closed and at the other side an acoustic power source is located, e.g. a loud-speaker, linear motor, or thermoacoustic prime mover. In case of a prime mover, one end of the resonator is closed and at the other end an acoustic load is located, e.g. a piston or a thermoacoustic refrigerator. For the operation of a standing-wave device an imperfect heat transfer of the gas with a solid is essential. To optimize the gas-wall interaction, a solid object, called a stack, is installed in the resonator. A stack consists of pores whose size is of the same order of magnitude as the thermal penetration depth in the gas. For a good performance the stack is located between a pressure node and a pressure anti-node, since the thermoacoustic effect requires both pressure and displace-ment oscillations. The waves in a thermoacoustic device are never pure standing waves, since standing waves cannot transport energy. The phase angle between pressure and velocity is close to 90◦, but never exactly 90◦.

In the Sondhauss tube the thermoacoustic effect occurs in a single pore. In standing-wave engines, however, the process occurs in a stack, which consists of many pores in parallel, all of which contribute to the acoustic power generation. Such a stack was not added to a Sondhauss tube until the 1960s. This important development allowed fill-ing a large-diameter tube with relatively small pores, creatfill-ing a large volume of strong

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1.2 History and applications 3

thermoacoustic power production, while leaving the rest of the resonator open and rela-tively low in dissipation. Heat exchangers spanning the ends of the stack are needed for efficient delivery and extraction of the large amounts of heat needed by a stack. Early use of such heat exchangers was described by Feldman and Carter [22] and by Wheatley et al. [23].

The first stranding-wave refrigerator was developed by Hofler [24] in 1986. He used a loudspeaker to drive a closed resonator tube with a stack positioned near the loud-speaker. At the other part of the tube a volume was attached to simulate an open end-ing. The tube is effectively a quarter-wave resonator. At both sides of the stack a heat exchanger is located, one at low temperature and the other at ambient temperature. Ti-jani et al. [25, 26] optimized this standing-wave refrigerator, using the back volume of the loudspeaker as a gas-spring system to adept the effective resonance frequency of the loudspeaker to its load. They managed to cool down to -70◦C. They also demonstrated the advantage of using gas mixtures, in order to get a low Prandtl number.

One application of stack-based devices is an ice-cream freezer that is developed by Garret [27] from the Pennsylvania State University in 2004 in cooperation with Ben & Jerrys Homemade. It is 25 cm in diameter, 48 cm tall, is filled with 10 bar helium, and is driven by a moving-magnet linear motor at around 100 Hz. The cooling capacity is 119 W at -24.6◦C and the coefficient of performance is 19% of Carnot. Although this refrigerator is more expensive than conventional refrigerators, no chemicals or gases are used that can be harmful to the environment. Another advantage of thermoacous-tic devices, including this refrigerator, is that they have only a few moving parts, and even more important, no moving parts in the cold, which makes them very reliable and relatively simple and flexible to build. This makes them also interesting for space applications [28].

When pumping natural gas through pipes to the surface, sensors are located far below the ground. To power these sensors a reliable electrical power supply is required. One way to provide this power is to create a side branch to the main flow, which was studied by Slaton [29–32]. At the edges of the side branch with the main pipe, vortices will be shed. When the vortex-shedding frequency corresponds with the resonance frequency of the side branch, a standing wave is created. By installing a stack in the side branch the acoustic power can be used to generate a temperature difference, that can produce electric power with thermoelectric elements.

1.2.2 Traveling-wave or regenerator-based devices

In Stirling engines and so-called traveling-wave engines, the conversion of heat to acous-tic power occurs in the regenerator, which smoothly spans the temperature difference between the hot heat exchanger and the ambient heat exchanger and contains small pores through which the gas oscillates. A regenerator differs from a stack by a much smaller pore size.

The pores in a regenerator are small enough that the gas in them is in excellent local thermal contact with their walls. A solid matrix such as a pile of fine-mesh metal screens or spherical particles is often used. Proper design causes the gas in the channels to move toward the hot heat exchanger while the pressure is high, and toward the ambient heat exchanger while the pressure is low. The time phasing described above is that of a traveling acoustic wave, which carries acoustic power from ambient to hot. In contrast

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4 Introduction

to standing-wave engines, acoustic power must be injected into the ambient end of a regenerator in order to create more acoustic power; the regenerator is an amplifier of acoustic power. Yazaki et al. [8] demonstrated a traveling-wave engine very similar to that first conceived by Ceperley [33–35], with the path length around the toroidal waveguide nearly equal to 2λ. At about the same time, De Blok [36] and the Los Alamos group [37, 38] invented a traveling-wave engine with the heat exchangers imbedded in a lumped-acoustic impedance torus much shorter than the wavelength λ.

Backhaus and Swift [37–39], at the Los Alamos National Laboratory, have developed a heat-driven thermoacoustic refrigerator to liquify natural gas. It is a combination of a thermoacoustic engine that drives a orifice pulse tube. The engine is a regenator based on toroidal geometry.

In the process industry large quantities of waste heat are released to the environment that cannot be reused, mostly because the temperature level is too low. In order to reuse part of this waste heat, a heat pump is necessary that can provide a temperature lift to the required temperature levels. A thermoacoustic device can accomplish this goal. The energy research center of the Netherlands (ECN) [40, 41] develops traveling-wave devices to upgrade waste heat to either process heat or to generate cooling. Their thermoacoustic devices are a combination of an engine and a heat pump. The waste heat is used to create a temperature gradient and is converted into acoustic power. The acoustic power is used to drive a heat pump that upgrades part of the waste heat to a required temperature level.

The traditional Stirling engine has high efficiency, but it has moving parts. The thermoacoustic-Stirling hybrid engine has reasonably high efficiency and very high re-liability, but the toroidal topology needed is responsible for high fabrication costs. Fi-nally, the stack-based standing-wave thermoacoustic engine is reliable and costs little to fabricate, but its efficiency is only about 2/3 that of a regenerator-based system.

1.2.3 Special applications and phenomena

A Rijke tube is similar to the Sondhauss tube, in the sense that it turns heat into sound, by creating a self-amplifying standing wave. Rijke [42] discovered in 1859 a way of using heat to sustain a sound in a cylindrical tube, open at both ends. The tube was oriented vertically and a hot wire gauze was located at the lower half of the tube. The flow of air past the gauze is a combination of two motions. There is a uniform upwards motion of the air due to a convection current resulting from the gauze heating up the air. Superimposed on this is the motion due to the sound wave. Rijke oscillations only occur in a pipe having both ends open, since the upward motion, due to natural convec-tion past the hot gauze is an essential part of the operaconvec-tion of the Rijke tube. Feldman produced a literature overview of both the Rijke [43] and the Sondhauss [44] thermoa-coustic phenomena.

A completely different use of thermoacoustics is the separation of gas mixtures by thermoacoustic waves, which was discovered by Swift and Spoor [45–47]. The super-position of nonzero time-averaged mole flux on a thermoacoustic wave in a binary gas mixture in a tube produces continuous mixture separation.

A cascade thermoacoustic engine, developed by Gardner and Swift [48], is a device in which one standing-wave engine and two traveling-wave engines are cascaded in series. Most of the acoustic power is produced in the efficient traveling-wave stages.

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1.3 Objectives 5

The straight-line series configuration is easy to build and allows no Gedeon streaming. The engine delivers up to 2 kW of acoustic power, with an efficiency (the ratio of acoustic power to heater power) of up to 20%.

Luo et al. [49, 50] have been working on a novel cascade thermoacoustic prime mover. And currently Hu et al. [51–53] are still working on cascade thermoacoustic engines. They use much higher frequencies than Gardner and Swift did.

Bauwens et al. have worked on numerical simulations of the flow and heat transfer in a stack [54, 55] and on a transient theory [56].

1.3 Objectives

The thermoacoustic theory, as developed by Rott and Swift [57], is a linear theory. How-ever, in many situations the amplitudes are so high that the use of the linear theory is not justified [38, 58–61]. Therefore we decided to study the effects that appear at high amplitudes. The nonlinear effects that we are interested in, are:

• vortex shedding at the end of a parallel-plate stack,

• dissipation at the ends of a stack due to the sudden change in cross section, • transition to turbulence in-between plates,

• streaming.

In order to study these effects we have built a set-up to do detailed measurements of pressure, stack temperature, and local velocity fields. Since nonlinear effects that we want to study occur at high amplitudes (≈ 3% [62–67]), we need to build a set-up that can produce such high amplitudes. We want to emphasize that the goal is not to build a device that is optimized for efficiency, but rather to do measurements to gain a better understanding of different phenomena that are occurring in thermoacoustic devices.

The work is a dual PhD project, in which Peter in ’t panhuis focuses on the theoretical and mathematical aspects and we focus more on the practical and experimental part.

An additional goal of this work is to help several Dutch companies, including ECN, Aster Thermoacoustics, and Shell, that work on thermoacoustics and also partially funded this project. By getting more insight in thermoacoustics we hope that they will be able to improve their thermoacoustic devices.

1.4 Thesis outline

In this thesis we will first derive a well-known set of three differential equations that give a mathematical description of thermoacoustics for low amplitudes. These equa-tions can be applied to many different geometries, including parallel-plate stacks, pores, even various cross-section shapes, and resonator tubes. The equations include viscous and thermal dissipation, and thermal conduction. Losses due to entrance effects in a stack, changes in cross section, and streaming are not included. Since the time depen-dency is assumed to be harmonic, higher harmonics are also not included. The set of equations can only be solved analytically for special cases. For all the other cases we

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6 Introduction

wrote a code to solve the equations numerically. The thermoacoustic theory, the code, and the different types of thermoacoustic devices are discussed in chapter 2.

In chapter 3 we present the experimental set-up that is used, with a focus on the electroacoustics, i.e. the acoustic output of the loudspeaker with respect to the electrical input. By including the speaker equations in our code we can calculate the pressure and velocity in the resonator as a function of the speaker voltage and frequency. We have measured the current through the loudspeaker, its excursion, and also the pres-sure at different positions in the resonator, using microphones. The meapres-surements are compared with the model.

In chapter 4 we describe the use of a multi-microphone method to determine the transfer matrix of various stacks. From the transfer matrix elements the Rott functions can be determined as functions of the frequency. The Rott functions are fitted to an analytical solution with the pore size and porosity as fitting parameters. Furthermore, the multi-microphone method is used to determine energy fluxes at both ends of a stack. This way the dissipated energy in the stack is measured and compared with our model. The temperature profile in a parallel-plate is registered as a function of time, using 32 thermometers that are inserted in the center plate of the stack. Since the stack is not isolated from the environment, the stack temperature is not uniform over the cross sec-tion. The radial temperature gradients are included in the model, which is discussed in chapter 5. Temperature measurements in time are compared with our time-dependent model.

In chapter 6 we visualize the flow using Particle Imaging Velocimetry (PIV). A large part of the chapter discusses the vortex shedding that occurs behind the plate ends. The flow-visualization measurements are compared with a numerical simulation. In the center of the stack in-between two plates the velocity is measured. The velocity profile is compared with an analytical solution. At sufficiently high Reynolds numbers the profiles are expected to deviate from this analytical solution, due to turbulence. We will measure the transition of laminar to turbulent. The PIV technique is also used to measure jet streaming and natural convection.

Regenerator-based devices, consisting of a coaxial loop, are designed by Aster Ther-moacoustics and ECN. It is valuable to know how an oscillatory flow behaves in such a coaxial loop, which is studied using PIV in chapter 7. Especially, the flow at the sharp corners is of interest.

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Bibliography

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[2] N. Rott, “Damped and thermally driven acoustic oscillations in wide and narrow tubes,” Journal of Applied Mathematics and Physics, vol. 20, pp. 230–243, 1969.

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8 BIBLIOGRAPHY

[13] A. Atchley, “Analysis of the initial buildup of oscillations in a thermoacoustic prime mover,” Journal of the Acoustical Society of America, vol. 95, no. 3, pp. 1661– 1664, 1994.

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[26] M. Tijani, Loudspeaker-driven Thermoacoustic Refrigeration. Eindhoven: Universiteits-drukkerij Eindhoven, 2001.

[27] M. E. Poese, R. W. M. Smith, S. L. Garrett, R. van Gerwen, and P. Gosselin, “Thermoacoustic refrigeration for ice cream sales,” in Proceedings of the 6th Gustav Lorentzen Natural Working Fluids Conference, 2004.

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BIBLIOGRAPHY 9

[28] S. L. Garrett, J. A. Adeff, and T. J. Hofler, “Thermoacoustic refrigeration for space application,” Journal of Thermophysics and Heat Transfer, vol. 7, no. 4, pp. 595–599, 1993.

[29] W. Slaton and J. Zeegers, “An aeroacoustically driven thermoacoustic heat pump,” Journal of the Acoustical Society of America, vol. 117, no. 6, pp. 3628–3635, 2005. [30] W. Slaton and J. Zeegers, “Acoustic power measurements of a damped

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[31] W. Slaton and J. Zeegers, “Thermoelectric power generation in a thermoacoustic refrigerator,” Applied Acoustics, vol. 67, no. 5, pp. 450–460, 2006.

[32] J. Zeegers and W. Slaton, “Aeroacoustic thermoacoustics,” in AIP Conference Pro-ceedings, vol. 838, pp. 416–423, 2006.

[33] P. H. Ceperley, “A pistonless stirling enginethe traveling wave heat engine,” Journal of the Acoustical Society of America, vol. 66, pp. 1508–1513, 1979.

[34] P. H. Ceperley, “Gresonant travelling wave heat engine,” US Patent 4, pp. 355,517, 1982.

[35] P. H. Ceperley, “Gain and efficiency of a short traveling wave heat engine,” Journal of the Acoustical Society of America, vol. 77, pp. 1239–1244, 1985.

[36] C. M. de Blok, “Thermoacoustic system,” Dutch Patent. International Application Number PCT/NL98/00515, 1998.

[37] S. Backhaus and G. W. Swift, “A thermoacoustic-stirling heat engine,” Nature, vol. 399, pp. 335–338, 1999.

[38] S. Backhaus and G. W. Swift, “A thermoacoustic-stirling heat engine: Detailed study,” Journal of the Acoustical Society of America, vol. 107, pp. 3148–3166, 2000. [39] B. Arman, J. Wollan, G. Swift, and S. Backhaus, “Thermoacoustic natural gas

liq-uefiers and recent developments,” in Proceedings of the 2003 International Conference on Cryogenics and Refrigeration, pp. 123–127, 2003.

[40] L. A. Nijeholt, T. J.A., and S. M.E.H., Spoelstra, “Simulation of a traveling-wave thermoacoustic engine using computational fluid dynamics,” Journal of the Acous-tical Society of America, vol. 118, no. 4, pp. 2265–2270, 2005.

[41] M. Tijani and S. Spoelstra, “Study of a coaxial thermoacoustic-stirling cooler,” Cryo-genics, vol. 48, no. 1-2, pp. 77–82, 2008.

[42] P. L. Rijke, “On the vibration of the air in a tube open at both ends,” Philosophical Magazine, vol. 17, pp. 419–422, 1859.

[43] K. T. Feldman, “Review of the literature on rijke thermoacoustic phenomena,” Jour-nal of Sound and Vibration, vol. 7, pp. 83–89, 1968.

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10 BIBLIOGRAPHY

[44] K. T. Feldman, “Review of the literature on sondhauss thermoacoustic phenom-ena,” Journal of Sound and Vibration, vol. 7, pp. 71–82, 1968.

[45] P. Spoor and G. Swift, “Mode-locking of acoustic resonators and its application to vibration cancellation in acoustic heat engines,” Journal of the Acoustical Society of America, vol. 106, no. 3 I, pp. 1353–1362, 1999.

[46] P. Spoor and G. Swift, “Thermoacoustic separation of a he-ar mixture,” Physical Review Letters, vol. 85, no. 8, pp. 1646–1649, 2000.

[47] G. Swift and P. Spoor, “Thermal diffusion and mixture separation in the acous-tic boundary layer,” Journal of the Acousacous-tical Society of America, vol. 106, no. 4 I, pp. 1794–1800, 1999.

[48] D. Gardner and G. Swift, “A cascade thermoacoustic engine,” Journal of the Acous-tical Society of America, vol. 114, no. 4 I, pp. 1905–1919, 2003.

[49] H. Ling, E. Luo, W. Dai, and X. Li, “Numerical simulation on a novel cascade ther-moacoustic prime mover,” in Proceedings of the Twentieth International Cryogenic En-gineering Conference, ICEC 20, pp. 333–336, 2005.

[50] E. Luo, E. Dai, and H. Ling, “Basic operation principle of a novel cascade thermoa-coustic prime mover,” in Proceedings of the Twentieth International Cryogenic Engi-neering Conference, ICEC 20, pp. 317–320, 2005.

[51] Z. Hu, Q. Li, X. Xie, G. Zhou, and Q. Li, “Design and experiment on a mini cascade thermoacoustic engine,” Ultrasonics, vol. 44, no. SUPPL., pp. e1515–e1517, 2006. [52] Z. Hu, Q. Li, Q. Li, and Z. Li, “A high frequency cascade thermoacoustic engine,”

Cryogenics, vol. 46, no. 11, pp. 771–777, 2006.

[53] Z. Hu, Q. Li, Z. Li, and Q. Li, “Orthogonal experimental study on high frequency cascade thermoacoustic engine,” Energy Conversion and Management, vol. 49, no. 5, pp. 1211–1217, 2008.

[54] L. Bauwens, “Oscillating flow of a heat-conducting fluid in a narrow tube,” Journal of Fluid Mechanics, vol. 324, pp. 135–161, 1996.

[55] O. Hireche, C. Weisman, D. Baltean-Carls, P. L. Quéré, M.-X. François, and L. Bauwens, “Numerical model of a thermoacoustic engine,” Comptes Rendus Mcanique, vol. 338, no. 1, pp. 18 – 23, 2010.

[56] L. Bauwens, “Thermoacoustics: Transient regimes and singular temperature pro-files,” Physics of Fluids, vol. 10, no. 4, pp. 807–818, 1998.

[57] G. W. Swift, “Thermoacoustic engines,” Journal of the Acoustical Society of America, vol. 84, no. 4, pp. 1145–1180, 1988.

[58] G. Swift, A Unifying Perspective for Some Engines and Refrigerators. Melville: Acous-tical Society of America, 2002.

[59] G. Swift, “Thermo acoustic engines and refrigerators,” Physics Today, vol. 48, no. 7, pp. 22–28, 1995.

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BIBLIOGRAPHY 11

[60] M. E. Poese and S. L. Garrett, “Performance measurements on a thermoacoustic refrigerator driven at high amplitudes,” Journal of the Acoustical Society of America, vol. 107, no. 5, pp. 2480–2486, 2000.

[61] H. S. Roh, R. Raspet, and H. E. Bass, “Parallel capillary-tube-based extension of thermoacoustic theory for random porous media,” Journal of the Acoustical Society of America, vol. 121, no. 3, pp. 1413–1422, 2007.

[62] A. Berson and P. Blanc-Benon, “Nonperiodicity of the flow within the gap of a ther-moacoustic couple at high amplitudes,” Journal of the Acoustical Society of America, vol. 122, no. 4, pp. EL122–EL127, 2007.

[63] A. Berson, P. Blanc-Benon, and G. Comte-Bellot, “A strategy to eliminate all non-linear effects in constant-voltage hot-wire anemometry,” Review of Scientific Instru-ments, vol. 80, no. 4, 2009.

[64] A. Berson, G. Poignand, P. Blanc-Benon, and G. Comte-Bellot, “Capture of instan-taneous temperature in oscillating flows: Use of constant-voltage anemometry to correct the thermal lag of cold wires operated by constant-current anemometry,” Review of Scientific Instruments, vol. 81, no. 1, 2010.

[65] D. Marx and P. Blanc-Benon, “Numerical calculation of the temperature difference between the extremities of a thermoacoustic stack plate,” Cryogenics, vol. 45, no. 3, pp. 163–172, 2005.

[66] D. Marx and P. Blanc-Benon, “Computation of the mean velocity field above a stack plate in a thermoacoustic refrigerator,” Comptes Rendus M´ecanique, vol. 332, no. 11, pp. 867–874, 2004.

[67] D. Marx and P. Blanc-Benon, “Numerical simulation of stack-heat exchangers cou-pling in a thermoacoustic refrigerator,” AIAA Journal, vol. 42, no. 7, pp. 1338–1347, 2004.

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Chapter 2

Theory and model

2.1 General thermoacoustic theory

We derive three basic thermoacoustic differential equations. Two of them relate the pressure and volume-flow rate for a given temperature gradient and geometry. To-gether they form Rott’s wave equations. The third equation is the energy equation, from which a temperature profile can be determined for a given pressure, volume-flow rate, and geometry. The theory was developed in the seventies by Rott et al. [1–6] and was later reviewed by Swift [7], for the application of parallel plates. The theory is gen-eral in the sense that it applies to a whole range of different geometries. The size of the flow channels can vary from smaller than the thermal penetration depth, in case of regenerators, to a few times the thermal penetration depth, in case of a stack, to big resonator tubes. The hydraulic radius R_H, defined as the ratio of the cross-sectional area of the flow A to the wetted perimeter Π, is a measure of the flow channel size. The shape of the geometry can be parallel plates, circular cylinders, rectangular cylinders, or any other cylindrical regular shape. The generalization to arbitrary pore geometry was developed by Arnott et al. [8] using the assumption that the temperature at the bound-aries is constant. In ’t panhuis et al. [9] repeated this generalization, without using this assumption, by taking into account the heat equation in the solid, and also allowed for gradual changes of the hydraulic radius of the pore.

In the derivation of the three basic thermoacoustic equations we make the following major assumptions:

• The relevant parameters, pressure, density, velocity, and temperature can be lin-earized, i.e. can be written as the sum of a mean value and a first-order variation. The assumption that second-order effects are ignored yields to a major limitation of this ’linear’ theory to the amplitudes. For high-pressure amplitudes (higher than 10% of the mean pressure, according to literature) the second-order terms cannot be neglected.

• The acoustic wave has a single, fixed frequency and is propagating in only one direction that is along the tube axis. This implies that the hydraulic radius is re-quired to be smaller than the wavelength, to avoid cut-off modes.

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14 Theory and model

same state as in the beginning of the cycle.

• The local time-average mean velocity is equal to zero.

In addition to these major assumptions, some additional assumptions that are com-mon in fluid dynamics and thermodynamics, like perfect heat contact between the solid and gas, no-slip condition at wall, the gas is assumed to be a Newtonian fluid and a perfect gas, the only relevant viscous force to be shear viscosity, and the kinetic energy to be negligible in the energy equation.

2.1.1 Basic equations

Here we will derive the three basic equations that follow from conservation of mass, momentum, and energy respectively. The second law of thermodynamics and the ideal gas law are two additional basic equations. From the conservation of mass the well-known continuity equation follows

∂ρ

∂t +∇ · (ρv) =0, (2.1)

where ρ is the density, t the time, and v the velocity. From the conservation of mo-mentum and approximating the viscous term by ν∇2v, the well-known Navier-Stokes equation ∂v ∂t + (v· ∇)v=− ∇p ρ +ν∇ 2_v_, _(2.2)

can be derived. Here p is the pressure and ν the kinematic viscosity. By combining the first law of thermodynamics with the Eqs. 2.1 and 2.2, Landau and Lifshitz [10] derived a general equation of heat transfer for fluids

ρT ∂s ∂t +v· ∇s =∇ ·k∇T+ (σ0_{· ∇) ·}v, (2.3)

where s represents the entropy per unit mass, k the thermal conductivity, T the temper-ature, and σ0 the viscous stress tensor. Using ρTds = −dp+ρcpdT (for ideal gases),

with c_pthe specific isobaric heat capacity, we can rewrite the heat equation to

ρcp _∂T ∂t +v· ∇T − _∂p ∂t +v· ∇p =∇ ·k∇T+ (σ0_{· ∇) ·}v. (2.4)

The ideal gas law yields

p=ρR_sT, (2.5)

where Rsis the specific gas constant.

2.1.2 Linearization

The coordinate system for a flow channel with arbitrary cross section is shown in figure 2.1. The velocity in Cartesian coordinates v= (u, v, w), with u the x-component of v.

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2.1 General thermoacoustic theory 15

x

z

y

gas area solid

Figure 2.1: A geometry consisting of a homogeneous flow channel with an arbitrarily shaped

cross-section. The flow-channel length (in x direction) is much larger than its perimeter.

The basic equations can be simplified by linearizing the relevant variables:

p(x, t) = p₀+Re{p₁(x)eiωt}, (2.6a) ρ(x, y, z, t) = ρ₀(x) +Re{ρ₁(x, y, z)eiωt}, (2.6b) T(x, y, z, t) = T0(x) +Re{T1(x, y, z)eiωt}, (2.6c)

u(x, y, z, t) = Re{u₁(x, y, z)eiωt}, (2.6d) U(x, t) = Re{U₁(x)eiωt}, (2.6e) where ω is the angular frequency and U₁ is the volume flow rate, i.e. the sectional integrated velocity over a surface ⊥x. The indexes denote the order. In case of the pressure, for instance, p₀ is the zeroth-order term, which is the time mean value, and p₁is the first-order term, also called acoustic velocity. The drive ratio, Dr, is defined as

the ratio of|p1|and p0. Note that p1is not a function of y and z and that p0is constant,

as was shown with small-parameter asymptotics by In ’t panhuis [11]. He also showed that T0and ρ0are only x dependent. We will now substitute the linearized variables

into the three basic equations and will then gather all first-order terms. Substituting the linearized density and velocity into Eq. (2.1) yields

iωρ1+ ∂ ∂x(ρ0u1) +ρ0 ∂v ∂y +ρ0 ∂w ∂z =0. (2.7) Using a similar concept for the x-component of the momentum equation results in

iωρ₀u₁=−dp_dx1+ρ₀ν ∂ 2 u1 ∂y2 +∂ 2 u1 ∂z2 ! . (2.8)

A linearization of the heat equation, Eq. 2.4, yields

ρ₀cp iωT1+u1 dT0 dx −iωp1=k " ∂2T₁ ∂y2 +∂ 2 T1 ∂z2 # , (2.9)

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2.1.3 Velocity profile

First we will derive the velocity profile for the special case of a parallel-plate channel and then for the general case of a flow channel with an arbitrary cross section.

Parallel-plate channel:

x

z

y

2

y

0 l0 gas solid center of plate center of plate

Figure 2.2: Parallel-plate geometry. The center of the gas channel is at y=0 and the boundaries are at y=±y₀. At the centers of the plates a periodic boundary condition in the y-direction is applied. The plate thickness is 2l₀.

We will solve the momentum equation, Eq. 2.8, first for a parallel-plate geometry (figure 2.2), and then for an arbitrary geometry. For parallel plates, separated by 2y₀ with y=0 in the center, the following boundary conditions apply:

u₁(x,−y₀) = 0, (2.10a) u1(x, y0) = 0. (2.10b)

Since the problem is two-dimensional, it is independent of the z-position: u1=u1(x, y).

We assume to be sufficiently far away, i.e. more than twice the displacement length, from the plate ends. Solving the differential equation, Eq. 2.8, leads to

u₁= i

ωρ₀ dp1

dx +C1cosh[(1+i)y/δν] +C2sinh[(1+i)y/δν], (2.11) where C1and C2are complex constants and δνthe viscous penetration depth, defined

as

δ_ν =

r 2ν

ω. (2.12)

From the boundary conditions, Eqs. 2.10a and 2.10b, it follows that

C1 = i ωρ₀ dp1 dx 1 cosh[(1+i)y₀/δ_ν], (2.13a) C₂ = 0. (2.13b)

This makes the solution for parallel plates

u₁= i ωρ₀ 1− cosh[(1+i)y/δν] cosh[(1+i)y0/δν] dp₁ dx . (2.14)

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General solution for a flow channel:

Other channel geometries (figure 2.1) lead to different solutions for u₁, but all of them can be written as a general solution

u₁= i

ωρ₀[1−hν(y, z)] dp1

dx , (2.15)

with hν(y, z)a complex function, depending on the channel geometry, the frequency,

and weakly on x due to the temperature dependence of the viscosity. The volume flow rate is the cross-sectional integrated velocity

U₁= Z Z Au1dA= iA ωρ₀[1−fν] dp1 dx , (2.16)

where A is the cross-sectional area and fνis the spatial average of hνand is called the

viscous Rott function

fν=

1 A

Z Z

Ahν(y, z)dA. (2.17)

Eq. 2.16 is an important equation, as it is one of the three equations that is used by our model that will be discussed in section 2.2. For parallel plates

hν = cosh[(1+i)y/δ_ν] cosh[(1+i)y0/δν] , (2.18a) fν = tanh[(1+i)y₀/δ_ν] (1+i)y0/δν . (2.18b)

The hνand fνfunctions of other geometries, cylindrical, triangular, rectangular, and pin

arrays, are derived by Arnott et al. [8].

2.1.4 Thermoacoustic continuity equation

Here we will derive a second equation, the thermoacoustic continuity equation, for a general pore geometry. T1is derived from the linearized heat equation and is used to

determine ρ1, using the ideal gas law. ρ1 is substituted into the linearized continuity

equation, which is integrated over the cross section.

By using the divergence theorem and that v = 0 at the pore boundaries, it can be shown thatRR

A∂v/∂ydA= RR

A∂w/∂zdA=0. This equation can be used to integrate Eq.

2.7 over the cross section, resulting in

iAωhρ₁_{i +} d

dx(ρ0U1) =0, (2.19) wherehidenotes the cross-sectional average. Solving the heat equation (Eq. 2.9) in the gas, with the boundary condition that the temperature at the solid is constant (T1=0),

results in T₁= (1−hκ)p1 ρ₀c_p − 1 iωA dT0 dx (1−hκ)−Pr(1−hν) (1− fν)(1−Pr) U₁, (2.20)

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where P_r=c_pµ/k is the Prandtl number and hκand fκare channel-geometry dependant

functions that are very similar to hνand fν, with the only difference that δνis replaced

with δκ. For instance, for parallel plates

hκ = cosh[(1+i)y/δ_κ] cosh[(1+i)y₀/δ_κ], (2.21a) fκ = tanh[(1+i)y₀/δ_κ] (1+i)y0/δκ . (2.21b)

The thermal penetration depth δκ is given by

δ_κ =

r 2κ

ω, (2.22)

with κ = k/ρ₀c_pthe thermal diffusivity. The viscous and thermal penetration depths are related by the Prandtl number

Pr=

δ2_ν

δ_κ2. (2.23)

If we take the cross-sectional average of Eq. 2.20

hT₁i = (1−fκ)p1 ρ₀c_p − 1 iωA dT0 dx (1−fκ)−Pr(1− fν) (1−fν)(1−Pr) U₁. (2.24)

For flow channels with a hydraulic radius much larger than both the thermal and viscous penetration depth (RH δν, RH δκ) the Rott functions approach zero fν→

0, fκ→0. In case of a parallel-plate stack this can be easily seen by taking y0→∞. For

gas parcels sufficiently far away from the flow-channel wall hν →0, hκ → 0, Eq. 2.20

can be simplified to T₁= p1 ρ₀c_p − 1 iωA dT₀ dxU1. (2.25)

From this equation it is easily seen that T1=0 if,

dT₀ dx = iωA ρ₀cp p₁ U1 . (2.26)

This equation can be fulfilled in case of a standing wave with a critical temperature gradient (∇T)crit= ωA|p₁| ρ₀cp|U1| . (2.27) If|dT0

dx| = (∇T)crit, the gas parcels are at the same temperature as the plate during the

whole cycle, and consequently no heat is exchanged between the plates and the gas. This would be the case for an inviscid thermoacoustic couple in steady state. Inviscid standing-wave engines have|dT0

dx| > (∇T)critand inviscid standing-wave refrigerators

or heat pumps have|dT0

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To solve Eq. 2.24 we need boundary conditions for T₁at the gas channel walls. For parallel plates Swift [7] takes into account the heat equation in both the gas and the solid. For flow channels with arbitrary cross sections this is more difficult. Arnott et al. [8] assume that the temperature on the boundary of the flow channel is constant. When the thermal capacity and conductivity of the solid is much larger than that of the gas, this is a reasonable assumption. In ’t panhuis et al. [9] showed how to solve differential equation for arbitrary pores without using this assumption. Just like Swift did for parallel plates, they take into account the heat equation in both the gas and the solid.

Using Eq. 2.5 in combination with Eq. 2.24, substituting it in Eq. 2.19, and using Eq. 2.16 results in iAω(ρ0 p0 p₁−ρ0 T0h T₁i) +U₁ d dx( p₀ RT0 ) +ρ₀dU1 dx =0. (2.28) Substituting Eq. 2.24 forhT1ileads to

iAω(ρ0 p₀p1− (1−fκ)p1 c_pT₀ ) + dT₀ dx (1−fκ)−Pr(1− fν) (1−fν)(1−Pr) ρ₀U₁ T₀ − ρ₀U₁ T₀ dT₀ dx +ρ0 dU₁ dx =0. (2.29) Using cp= γγ₋₁ p₀

ρ₀T₀, combining terms, and dividing by ρ0leads to the final differential

equation dU1 dx =− iωA[1+ (γ₋1)fκ] γp₀ p1+ fκ−fν (1−fν)(1−Pr) dT0 dx U1 T₀. (2.30)

2.1.5 Total energy flow

The total energy flow ˙E is the sum of the enthalpy flow and heat flow:

˙E= 1 2ρ0 Z Z ARe[h1u ∗ 1]dA− (Ak+Asokso) dT0 dx , (2.31) with h the specific enthalpy, A_sothe solid area, k and k_so, the thermal conduction coeffi-cient of the gas and the solid, and∗denoting the complex conjugate. For ideal gases

h1=cpT1. (2.32)

A substitution of this into Eq. 2.31 leads to

˙E= 1 2ρ0cp Z Z ARe[T1u ∗ 1]dA− (Ak+Asokso) dT0 dx. (2.33)

Substituting Eq. 2.20 and u∗₁ = 1−hν∗

1− fν∗

U₁∗

A into this equation and performing the

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20 Theory and model ˙E = 1 2Re p1U1∗ 1− fκ−fν∗ (1+Pr)(1−fν∗) + ρ0cp|U1| 2 2Aω(1−Pr2)|1−fν| 2Im(fκ+Prfν∗) dT₀ dx − (Ak+Asokso) dT₀ dx . (2.34)

This is the third important equation that is used in the model.

2.2 Model

In section 2.1 three 1-D differential equations were derived (Eq. 2.16, 2.30, and 2.34), with three variables (U₁(x), p₁(x), and T₀(x)), and one unknown constant ( ˙E). When the boundary conditions are known, the system can be solved. In general, the set of three differential equations cannot be solved analytically. For this reason a numerical code is developed in Matlab. A number of thermoacoustic numerical codes exist, among which the famous DeltaE, which was developed by Swift [12]. We have chosen to write our own code, in collaboration with Wei Dai from the Chinese Academy of Sciences, for the following reasons:

Flexibility: Having a code of our own provides flexibility. Measured data can be used as input for the code.

Better insight: We want to gain more insight in the equations and algorithms behind the code. This way we also learn about the limitations of the code.

Independence: Not being dependent on others.

Easy to use: In Matlab it is easy to write scripts for making good plots and connect them to the model.

2.2.1 Transfer matrix

A stack with a fixed temperature gradient can be described by the first two equations, Eqs. 2.16 and 2.30,

dp₁

dx +R2U1 = 0, (2.35a) dU1

dx +R1p1−R3U1 = 0. (2.35b) Note that p1and U1are explicit functions of x, whereas the parameters

R1 = iωA(1+ (γ−1)fκ)/γp0, (2.36a) R₂ = iωρ₀/(1−fν)A, (2.36b) R3 = (fκ− fν)dT0/dx (1− fν)(1−Pr)T0 , (2.36c)

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2.2 Model 21

depend only implicitly on x by the T dependency. A substitution of (2.35a) into (2.35b) leads to the ’wave’ equation of Rott:

d2p₁ dx2 −R3

dp₁

dx −R1R2p1=0. (2.37) For constant R₁, R₂, and R₃this differential equation has as general solution

p₁= (C₁cosh(R₄x) +C₂sinh(R₄x))exp(R₃x/2), (2.38a) U₁=− 2R4C2+R3C1 2R₂ cosh(R4x) + 2R₄C₁+R₃C₂ 2R₂ sinh(R4x) exp(R₃x/2), (2.38b) where R4= q

R2₃/4+R₁R₂and C1and C2are complex constants. The values of C1and

C₂depend on the boundary conditions:

p₁=p_L, U₁=U_L, for x=x_L=0, (2.39) where pL and UL are the pressure and volume flow rate at the left side of the

compo-nent. For C1 = pLand C2 =−

R₃p_L+2R₂U_L

2R₄ both boundary conditions are fulfilled. The

pressure and volume flow rate at the right-hand side of the component, at x = x_R, p_R and U_R, are linear combinations of p_Land U_Land can be written as

p_R U_R =T p_L U_L . (2.40)

Here T is a transfer matrix given by

T=exp(R₃x_R/2)   cosh(R₄x_R)−R3sinh(R4xR) 2R₄ − R₂ R₄ sinh(R4xR) −R1 R₄sinh(R4xR) cosh(R4xR) + R₃sinh(R₄x_R) 2R₄  . (2.41) When R3xR1 the transfer matrix simplifies to

T=   cosh(p R1R2xR) − q_R 2 R₁ sinh(pR1R2xR) −qR1 R₂ sinh(p R1R2xR) cosh(p R1R2xR)  . (2.42)

Since in this case cosh(R4xR) =cos(ωc

q_1+(γ

−1) fk

1− fν xR), the pressure waves (using 2.38a)

can be written as

p1(x) =C+e−iξx+C−eiξ x, (2.43)

with ξ the complex wave number given by

ξ= ω c s 1+ (γ₋1)f_k 1−fν . (2.44)

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2.2.2 Energy equation and temperature

When p₁, U₁, and two boundary conditions for the temperature are known, T₀(x)can be determined from the energy equation. This equation is solved using a discretization, as is shown in figure 2.3. The geometry is equidistant divided in n sections with size ∆x. In the section i the temperature is T_0,i, the pressure is p_1,i, and so on. When considering the energy flow equation at the dashed contour, it follows that

˙E(xL) =Q˙i+ ˙E(xR), (2.45)

where ˙Qiis the heat flow from the environment. ˙E(xL)and ˙E(xR)are computed from

Eq. 2.30. When ∆x is sufficiently small, we can use the approximations dT0/dx|x_L =

(T_0,i−T_0,i−1)/∆x and dT₀/dx|x_R = (T0,i+1−T0,i)/∆x. We also use the approximations

T_0,x

L = (T0,i−1+T0,i)/2 and T0,xR = (T0,i+T0,i+1)/2 for the temperature, and the

pressure and volume flow rate at positions xLand xRare approximated similarly. Four

T

0,i-1

T

0,i

T

0,i+1

x

L

x

R

T

0,i+2

T

0,i-2

∆

x

Q

i

Figure 2.3: An overview of the discretization for the x-dependence that is used for the energy

flow equation. ˙Qiis the heat exchange with the environment and T0,iis the discretized

tempera-ture.

different boundary conditions are possible: the energy flow at the left hand side of a component; the energy flow at the right hand side; the temperature at the left hand side; and the temperature at the right hand side. Out of these four different possible conditions, two are required to solve the energy flow equation. Eq. 2.45 for 2≤i≤n−1 in addition to the two boundary conditions, give us n equations for n unknown T₀’s.

2.2.3 Iteration

We have a method to determine p1and U1when T0is known and a method to determine

T₀when p1and U1are known. But in practice none of them are known. For this reason

an iteration technique is used (figure 2.4).

step 1 We make a first guess of the temperature T₀(x)

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2.3 Thermoacoustic devices 23

step 3 From the calculated p₁(x)and U₁(x)we compute T₀(x)as explained in section 2.2.2

step 4 We compare the computed T₀(x)with the profile from the previous calculation step. If the difference is sufficiently small, the iteration process is completed, oth-erwise we repeat steps 2 and 3.

3. calculate new T0(x) 2. calculate p1(x), U1(x) 1. initial guess of T0(x)

4. is

|T

0new

-T

0old

|

<

ε

?

no

yes

end

Figure 2.4: The schematic of the iteration algorithm.

2.3 Thermoacoustic devices

2.3.1 Introduction

Two types of thermoacoustic devices that make use of the thermoacoustic effect can be distinguished:

Engines A temperature difference, TH−TC, is used to generate power ˙Wout, as is shown

in figure 2.5(a).

Refrigerators or heat pumps Power ˙W_inis used to extract heat from one location and release heat at another location at a higher temperature, as is shown in figure 2.5(b). In case of a refrigerator the objective is to create cooling power ˙Q_C by keeping T_H at the environmental temperature and in case of a heat pump the objective is to generate heating power ˙Q_H by keeping T_C at the environmental temperature.

From the first law of thermodynamics it follows that in steady state ˙W_in,out=Q˙_H−_Q˙_C_. For engines the efficiency is defined as η= _W˙_out_/_Q˙_H_{. The maximum efficiency, which} can only be reached in an ideal device, is called Carnot efficiency ηC, and ηC = (TH−

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T_C)/T_H. For refrigerators the coefficient of performance is defined as COP = _Q˙_C_/_W˙_in and the Carnot coefficient of performance is defined as COP_C=T_C/(T_H−T_C).

TH TC engine Wout QC QH (a) engine TH TC refrigerator or heat pump Win QC QH (b) refrigerator or heat pump

Figure 2.5: Schematic drawing of the working principle of two different types of thermodynamic

devices. The squares represent thermal reservoirs. The two top reservoirs are at temperature TH

and the two bottom reservoirs at TC. The arrows show the energy-flow directions.

a) An engine converts the heating power ˙Q_Hat temperature T_H partially into power ˙W_outand transfer the waste heat ˙Q_Lto the cold reservoir.

b) The power ˙W_in is used to generate a cooling power ˙Q_Lin case of a refrigerator and a heating power ˙Q_Hin case of a heat pump.

The devices can also be categorized in a different way: stack-based devices (section 2.3.2) and regenerator-based devices (2.3.3). We define a stack as a geometry for which the hydraulic radius of the pores is of similar size as the thermal penetration depth, RH 'δκ, and a regenerator as a geometry for which the hydraulic radius of the pores

is much smaller than the thermal penetration depth, RH δκ. In literature these

cat-egories are often referred to as standing-wave and traveling-wave devices. Since, in practice, the waves are never purely standing nor purely traveling, this nomenclature can lead to confusion.

2.3.2 Stack-based devices

We consider a standing wave in a straight resonator, closed at both ends, at the first resonance frequency (figure 2.6). The pressure and velocity are out of phase. If we neglect the interaction of the wave with the wall, the wave is a perfect standing wave. When we follow a gas parcel during one cycle, we can draw a pressure-displacement, p−ζ, plot of it (figure 2.7(a)). The displacement ζ is the position of one gas parcel during one cycle with respect to its average position x0. We write the pressure as

p=p₀+|p₁|cos(ωt+θ_p), (2.46) and the displacement as

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2.3 Thermoacoustic devices 25 ωt = 0 ωt = π p-p0 x ωt = π/2; 3π/2 (a) pressure ωt = 3π/2 ωt = π/2 u x ωt = 0; π (b) velocity

Figure 2.6: Acoustic pressure and velocity as functions of the position in a straight resonator at

the first resonance frequency, at four different phase angles ωt.

with θ a phase angle. A substitution from 2.47 into 2.46 results in two equations for the pressure as a function of the displacement

p=p₀+|p₁|cos arccos(ζ ζ₁)± (θp−θζ) , −1≤ ζ |ζ₁_| ≤1. (2.48) The two equations in (2.48) together describe an ellipse. If we normalize the pressure p0 = (p−p₀)/|p₁|and displacement ζ0 =ζ_/|ζ₁_|, the ellipse axis is at an angle of 45◦ with the ζ-axis for |θ_p₋θ_ζ_{| <} π/2 and -45◦ for π/2 < |θ_p₋θ_ζ_{| <} π. In case of

|θ_p₋θ_ζ_{| =}π/2 the ellipse is a circle and in case of|θ_p₋θ_ζ_{| =}π or|θ_p₋θ_ζ_{| =}0 it is a straight line. The eccentricity of the ellipse yields e=cos(θ_p₋θ_ζ).

p

0 p

0 (a) p-ζ

p

x

0 (b) p-ζ at various positions

Figure 2.7: Pressure-displacement plots at different positions, x, in the resonator.

In figure 2.7(b) the p−ζplots of parcels at different positions are shown for a stand-ing wave. Since the pressure and displacement are in phase, i.e. θp = θζ, it follows

that

p=p₀+ |p1|

High-amplitude thermoacoustic flow interacting with solid boundaries

High-amplitude thermoacoustic flow interacting with solid

boundaries

High-Amplitude Thermoacoustic

Flow Interacting with Solid

High-Amplitude Thermoacoustic

Flow Interacting with Solid

Boundaries

PROEFSCHRIFT

Paul Cornelis Hubertus Aben

Nomenclature

Contents

Chapter 1

Introduction

1.1

Thermoacoustics

1.2

History and applications

1.2.1

Standing-wave or stack-based devices

1.2.2

Traveling-wave or regenerator-based devices

1.2.3

Special applications and phenomena

1.3

Objectives

1.4

Thesis outline

Bibliography

Chapter 2

Theory and model

2.1

General thermoacoustic theory

2.1.1

Basic equations

2.1.2

Linearization

x

z

y

2.1.3

Velocity profile

x

z

y

y

2.1.4

Thermoacoustic continuity equation

2.1.5

Total energy flow

2.2

Model

2.2.1

Transfer matrix

2.2.2

Energy equation and temperature

T

T

T

x

x

T

T

∆

x

Q

2.2.3

Iteration

4. is

|T

-T

|

<

ε

?

no

yes

end

2.3

Thermoacoustic devices