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Thermoacoustic refrigerators : experiments and scaling analysis Citation for published version (APA): Li, Y. (2011). Thermoacoustic refrigerators : experiments and scaling analysis. Eindhoven: Technische Universiteit Eindhoven. https://doi.org/10.6100/IR716451 DOI: 10.6100/IR716451 Document status and date: Published: 01/01/2011 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 22. Jun. 2020

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Thermoacoustic refrigerators : experiments and scalinganalysisCitation for published version (APA):Li, Y. (2011). Thermoacoustic refrigerators : experiments and scaling analysis. Eindhoven: TechnischeUniversiteit Eindhoven. https://doi.org/10.6100/IR716451

DOI:10.6100/IR716451

Document status and date:Published: 01/01/2011

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 22. Jun. 2020

Thermoacoustic Refrigerators: Experiments and Scaling Analysis

Copyright © 2011 by Yan Li, 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 Li, Yan Thermoacoustic Refrigerators: Experiments and Scaling Analysis / by Yan Li.- Eindhoven: Technische Universiteit Eindhoven, 2011. Proefschrift.-ISBN 978-90-386-2670-3 NUR 929 This research was financially supported by MicroNed, grant number 1-B-7

Thermoacoustic Refrigerators: Experiments and Scaling Analysis

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

donderdag 27 oktober 2011 om 14.00 uur

door

Yan Li

geboren te LiaoNing, China

Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. H.J.M. ter Brake en prof.dr. A.T.A.M. de Waele Copromotor: dr.ir. J.C.H. Zeegers

Ten years living and dead have drawn apart I do nothing to remember But I cannot forget Your lonely grave a thousand miles away Nowhere can I talk of my sorrow Even if we met, how would you know me My face full of dust My hair like snow In the dark of night, a dream suddenly, I am home You by the window Doing your hair I look at you and cannot speak Your face is streaked by endless tears Year after year must they break my heart These moonlit nights? That low pine grave?

By Su Shi

谨以此文赠给我的弟弟

Contents 1 Introduction 1 1.1 Thermoacoustics · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 1

1.2 History of thermoacoustics· · · · · · · · · · · · · · · · · · · · · · · · ·· · ·· · 2

1.3 Objective of present work · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 6

1.4 The scope of this thesis · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 7

2 Basic theory of thermoacoustics 8 2.1 Wave equation and total energy flow · · · · · · · · · · · · · · · · · · · · 8

2.2 Acoustic energy · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 22

3 Standing-wave systems 26 3.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 26

3.2 Physical description of standing-wave systems · · · · · · · · · · · · · 29

3.3 Modeling standing-wave systems · · · · · · · · · · · · · · · · · · · · · · · 32

3.3.1 Zero viscosity · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 33

3.3.2 General analysis with viscosity included · · · · · · · · · · · 46

3.3.3 General analysis of “TAC” · · · · · · · · · · · · · · · · · · · · · 51

3.4 Experimental results · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 58

3.4.1 Experimental set-up · · · · · · · · · · · · · · · · · · · · · · · · · · · 58

3.4.2 Measurements · · · · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · 66

3.4.3 Theoretical computation · · · · · · · · · · ··· · · · · · · · · · · · 76

3.4.4 Conclusions· · · · · · · · · · · · · · · · · ·· · · · ·· · · · · · · · · · · · 84

4 Traveling-wave systems 85

4.1 Introduction · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 85

ii Contents

4.2 Physical description of traveling-wave systems · · · · · · · · · · · · 86

4.3 Modeling traveling-wave systems · · · · · · · · · · · · · · · · · · · · · · · 88

4.4 Optimizing regenerator material· · · · · · · · · · · · · · · · · · · · · ·· · · 114

4.4.1 Introduction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 114

4.4.2 Regenerator materials· · · · · · · · · · · · · · · · · · · · · · · · · · 114

4.4.3 Selection criteria· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· 120

4.4.4 Experimental set-up: a coaxial traveling-wave engine· 125

4.4.5 Energy balance in the experimental set-up· · · · · · · · ·· · 129

4.4.6 Measurement equipment and data handling· · · · · · · · ·· 130

4.4.7 Measuring procedure· · · · · · · · · · · · · · · · · · · · · · · · · · · 133

4.4.8 Results and discussion· · · · · · · · · · · · · · · · · · · · · · · · · · 135

4.4.9 Conclusion· · · · · · · · · · · · · · · · · · · · · · · · · · · ··· · · · ·· · 149

4.5 Experiments on a thermoacoustic refrigerator · · · · · · · · · · · · · 151

4.5.1 Introduction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 151

4.5.2 Design considerations· · · · · · · · · · · · · · · · · · · · · · · · · · 151

4.5.3 Experimental set-up· · · · · · · · · · · · · · · · · · · · · · · · · · · 152

4.5.4 Measurement equipment and data handling· · · · · · · · ·· 157

4.5.5 Losses· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · 160

4.5.6 Measuring procedure· · · · · · · · · · · · · · · · · · · · · · · · · · · 162

4.5.7 Results and discussion· · · · · · · · · · · · · · · · · · · · · · · · · · 163

4.5.8 Discussions· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· 183

4.5.9 Conclusion· · · · · · · · · · · · · · · · · · · · · · · · · · · ··· · · · ·· ·· 185

5 Scaling considerations 186

5.1 Introduction· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · · 186

5.2 Standing-wave systems · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 187

5.2.1 Constant temperature difference over the stack · · · · · 187

5.2.2 Constant time-averaged total energy flow · · · · · · · · · · 199

5.2.3 Constant time-averaged total energy flow density · · · · 201

5.3 Traveling-wave systems · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·· 202

5.4 Conclusions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 210

6 Conclusions and recommendations 211

6.1 Conclusions 211

Contents iii 6.2 Recommendations 214

Appendices 215

A Momentum equations derivation· · · · · · · · · · · · · · · · · · · · · · ·· · · · · · · 215

B Derivation of the temperature of the solid plate· · · · · · · · · · · · · · · · · · 217

C Derivation of the temperature oscillation of the fluid layer· · · · · · · · · 218

D Derivation of the time-averaged total energy flow· · · · · · · · · · · · · · · · 222

E Derivation of the decoupling the sound field into standing-wave and traveling-wave components· · · · · · · · · · · · · · · · · ·· · ·· · · · · · · · · · · · ·· 224

F Computation of loop section in a traveling-wave system· ·· ·· · · · · · ·· 226

G Transmission of acoustic impedance of a uniform pipe· · ·· · ·· · ·· · ·· 235

H Fortran code for computation of traveling-wave engine· · · ·· · ·· · · · · 237

I The design of the ambient heat exchanger in the traveling-wave refrigerator· · · · · · · · · · · · · · · · · · · · · ·· · · · ·· · · · · · · · · · · · · · · · · · · · 242

J Time evolution of two orientations: upward and downward in traveling-wave refrigerator measurement· · · · · · · · · · · · · · · · · · · · · ·· 246

K Acoustic field in the scaled-down standing-wave systems· · · · · · · · · 248

Nomenclature 254

Bibliography 258

Summary 264

Samenvatting 266

Dankwoord 269

Curriculum Vitae 271

Chapter 1

Introduction 1.1 Thermoacoustics Thermoacoustics is a subject which focuses on the interaction between solid walls and oscillating fluids from thermodynamic point of view. Under normal conditions, the periodical adiabatic compression and expansion make the temperature of the sound propagating medium oscillate with a tiny amplitude, which is hardly perceptible to human beings. For instance, a normal conversation, scaled as 60 dB, can produce an excess temperature of only a few hundredths of a degree Celsius. It is this almost invisibility of the thermodynamic effect which prevented thermoacoustics from being explored earlier. In 1980, Nikolaus Rott [1] first introduced the term “thermoacoustics” in a review of his previous work on a theory for this phenomenon. The theory, known as linear thermoacoustic theory, became the solid basis of nowadays thermoacoustic investigations and applications. In recent decades, investigations on the fundamental nature of the problems encountered in various thermoacoustic devices and explorations on industrial and household applications are widely carried out in many research groups world-wide. Many thermoacoustic devices were built and utilized. The performances and efficiencies are much enhanced. Thermoacoustic devices have the advantages over the conventional heat pumps and engines, that they have no mechanical moving parts, which brings high reliability and virtually maintenance-free to the customers. Moreover, they are environment friendly by using chemically inert working gases. It is a charming technology for today’s world, which is suffering all sorts of environmental problems: global warming, ozone depletion and others.

2 Chapter 1

1.2 History of thermoacoustics The first reported observation about the thermoacoustic effect in the community of physicists was in the year of 1802 by Bryan Higgins [2]. In 1777, about 10 years after the discovery of hydrogen, Higgins demonstrated that the burning of hydrogen produces water. He lowered a vertical glass tube, which was sealed at the far end, over the flame. What ensued is unexpected “singing”. Later, he also tried different glass tubes and produced “several sweet tones, according to the width, length and thickness of the glass jar or sealed tube”. This singing flame aroused the interest of many investigators. Many explanations for this interesting effect were proposed, but they were largely incorrect. Jones made a good discussion on all these theories and modified the Rayleigh’s theory [3]. Later, Putnam and Dennis gave a wide survey on all sorts of combustion oscillations related to this “singing flame” [4]. Another interesting thermoacoustical oscillation, namely the Rijke tube, was reported by Rijke in 1859 [5]. Rijke found that strong oscillations occurred when a heated wire screen was placed in the lower half of a vertical pipe with two open ends, as shown in Fig. 1.2.1a. It was also found that the oscillations would stop if the top of the pipe was closed, implying that the convective air current was necessary for this phenomenon. Oscillations became strongest when the heated screen was located one-fourth of the length of the pipe from the bottom end. Although Rijke gave some explanation, it was thought as inadequate to explain the detailed heat exchange mechanism causing the oscillations. Figure 1.2.1: (a) Rijke tube (b) Sondhauss tube.

Sound generated

Heated screen

Convection flow

(a)

Bulb

Tube stem

Heat addition Sound generated

(b)

Introduction 3

In the consequential years, many theoretical analysis and experimental work were provided to explain this phenomenon qualitatively and quantitatively as well. Feldman reviewed the literature [6]. The Rijke oscillations are observed in many industrial facilities, like gas furnaces, oil burners, gas-heated deep fat fryers, and rocket combustion chambers. This annoying, sometimes even destructive effect, are described as “screaming”, “screeching”, and “chugging”. An important difference between the Rijke effect and thermoacoustics is that in a Rijke tube an average velocity is present on top of the acoustic oscillations. As mentioned at the beginning in this section, before scientists worked on these thermoacoustical phenomena, the glass blowers had heard a lot of “glass singing”, when they blew bulbs on the ends of narrow tubes. Sondhauss was the first to study experimentally these “singing glasses”. He published his investigations [7] in 1850 on a tube which was open on one end and terminated in a bulb on the other end, with a steady gas flame applied to the closed bulb-end, as shown in Fig. 1.2.1b. Such a tube was therefore named as “Sondhauss tube”, which approximates best what we define today as thermoacoustic oscillations. Sondhauss discovered that a steady gas flame, applied to the closed bulb end, caused the air in the entire tube to oscillate and produce a clear sound which was characteristic of the dimension of the tube. He also observed that larger bulbs and longer tubes produced lower frequency sounds and that hotter flames produced more intense sounds. Knipp also observed that thermoacoustic oscillations occurred when a glass vapor trap was heated and suggested the apparatus could be used as a standard source of sound [8]. The first referred Sondhauss oscillation taking place in the cryogenic research is “Taconis oscillations”. Taconis observed spontaneous oscillations in a hollow tube with the upper end closed at room temperature and the lower end immersed in the liquid helium [9]. He explained how the large thermal gradient along the tube caused the oscillations. The Taconis oscillations have been investigated experimentally by Yazaki et al. [10] In 1878, Lord Rayleigh proposed his criterion on these related thermoacoustical oscillation phenomena [11]: “If heat be given to the air at the moment of greatest condensation or be taken from it at the moment of greatest rarefaction, the vibration is encouraged”. This qualitative explanation was proved to agree well with extensive experimental observations and widely accepted by thermoacoustic community. An important progress came in 1962, when Carter et al. experimentally investigated the Sondhauss oscillation to determine the feasibility of using the phenomenon to generate electricity [12]. They found that inserting a bundle of small glass capillaries at a suitable position inside the Sondhauss tube could greatly

4 Chapter 1

improve the performance. This bundle of small glass capillaries is the so-called “stack” in modern thermoacoustics. This discovery made the later applications, using thermoacoustic phenomena, feasible and practical. Extensive studies following this idea were performed by Feldman in his PhD work [13]. He also made a review on literature work about Sondhauss tube [14]. Compared with the history of heat-driven oscillations, which is rich and old, the reverse thermodynamic process, of generation of a temperature gradient by imposing acoustic oscillations is rather recent. The first work on thermoacoustic type cooling was carried out by Gifford and Longsworth in 1964 [15]. They invented the pulse-tube refrigerator driven by a low frequency acoustic wave, to cool down to a temperature of 150K. Due to the efforts of many researchers, the pulse-tube has become one of the most favored technologies for cryocooling. A complete history and review of pulse tube works is given by Radebaugh [16, 17]. More information about modeling and numerical analysis of pulse-tube refrigerators can be found in [18-19]. In 1975, P. Merkli and H. Thomann reported their observation of thermoacoustic effects in a resonance tube [20]. They found cooling in the section of the tube with maximum velocity amplitude and marked heating in the region of the velocity nodes. They also developed a theoretical model which agreed with experiment at low amplitudes. Although much experimental work had been done and after Rayleigh’s qualitative explanation, researchers got progress in theoretical exploration to quantitatively describe these thermoacoustic oscillations at a much later time. The formal study on the theoretical aspect was started by Kramers in 1949 [21]. He developed a theoretical model to explain “Taconis oscillations”, by employing the method of solution used previously by Kirchhoff to achieve an exact solution for gas vibrations in a tube of constant temperature throughout [22]. By confining the phenomenon to small amplitude wave, he could linearize hydrodynamic equations of mass, momentum, and energy. Although he successfully separated the wave components and solved the resulting linearized equations, he was unable to account for the spontaneous vibrations which were often observed in experiments. He attributed this unsatisfactory feature of his theory to some neglected terms in linearizing which were probably not negligible. Trilling did theoretical analysis on an induced sound field by applying a sudden temperature variation on the rest boundary of a viscous heat-conducting gas [23]. In his analysis, the temperature at the closed end of a semi-infinite gas-filled pipe suddenly raised, the gas near the hot wall expanded and moved outwards, function like a piston. He showed that the magnitude of the pressure pulse generated was proportional to that of the temperature increase and inversely proportional to the one-fourth root of the distance traveled.

Introduction 5

Chu published four theoretical papers about heat-generated pressure waves. In the first paper a modified wave equation with the heat addition as source term was derived to describe the pressure field generated by a moderate rate of heat release [24]. In the second paper, Chu analyzed the stability of systems containing a heat source [25]. In the third paper, Chu and Ying theoretically investigated non-linear oscillations produced by a sinusoidal heat release from a plane heater located at the midsection of a completely closed pipe [26]. In the fourth paper, he theoretically studied a self-sustained, thermally driven, non-linear oscillation in a closed pipe [14]. The breakthrough came in 1969 by a series of articles of Rott [27-32, 1]. Based on review of previous works, Rott re-examined the simplifying assumptions used in Kramers’ work and abandoned incorrect ones. His remarkable work has built a solid theoretical basis of thermoacoustics, and becomes one of the most refered papers in modern thermoacoustics. The review article, published by Rott in 1980 [1] on summary of his previous results, inaugurated an active and prolific era in thermoacoustics. Enormous related projects have been conducted and progresses have been achieved. The Condensed Matter and Thermal Physics group of Los Alamos National Laboratory started a research program to apply Rott’s theory to build functional devices. In 1988, G. Swift published a comprehensive article addressing important aspects of thermoacoustic devices [33]. In 2000, S. Backhaus and G. Swift [34] presented an efficient thermoacoustic traveling-wave engine which made this novel thermoacoustic technology competitive with present conventional thermal machines widely used commercially. This new technology has been now investigated and efforts have been put to applications in industrial and normal household facilities, in a world-wide scale: the US, Canada, France, Mexico, the Netherlands, China, Japan, and other countries. In the last few decades, thermoacoustics has gone through a prosperous time. Much progress and achievement have been collected and reviewed by Garrett [35].

6 Chapter 1

1.3 Objective of present work Section 1.2 describes the history of thermoacoustics, and from the developments of the work at Los Alamos by Swift and coworkers, the work at Penn State University by Garrett and coworkers, as well as developments at ECN in the Netherlands and many other laboratories in the world, traveling wave thermoacoustic engines are built, and sized often as large apparatus, having a length of 3 meters up till sizes of even 25 meters long. Also standing wave devices are generally 50 cm or longer. The apparatus that has been built at Penn State University in the group of Steven Garrett, although very compact is also of a size on the order of 0.50 m long and 0.25 m diameter. In space application as well as in laptop computers or even mobile phones there is a strong need of cooling devices to cool away the heat that is generated by the ultra fine IC components that have a high intensity local heat production. That motivated also this project out of the perspective of the MicroNed grant, where the focus is on cooling of small scale (space) devices. Now from thermoacoustic point of view, small scales will mean that high frequencies have to be used. Using high frequencies on the one hand pushes up the criteria on downsizing all the components that are needed to build such a device. But apart from that thermoacoustic heat transport is strongly related to temperature differences over a stack or regenerator of finite length. When downsizing the length it will mean that thermal gradients will increase and there must be a limit on scalability of such devices due to the laws of thermodynamics i.e. heat conduction or any other loss processes. This issue, the rules for scaling down thermoacoustic refrigerators to miniature size, and to discover the limitation of that is one of the main topics of this dissertation. The main goal is to provide some guidance for the design of small-scale thermoacoustic machines. The two types of thermoacoustic refrigerators, standing-wave and traveling-wave, are both investigated for scaling. The basis of this scaling forms an investigation by means of analysis using the thermoacoustic equations, and applying them to both types of devices. Cooling rates, heat conduction, and power production are investigated analytically and scaling rules can be derived to study the influence of scaling. Apart from that the modelling results are partially verified by comparing them with experimental apparatus as built by Swift, as well as in our own laboratory. It gives this work a solid foundation for future design work on scaling of thermoacoustic refrigerators.

Introduction 7

1.4 The scope of this thesis This thesis presents the following contents: Chapter 2 is dedicated to a brief review of the basic theory of thermoacoustics, which is the widely used linear thermoacoustic theory developed by Rott and implemented by Swift. Chapter 3 is concerned with standing-wave refrigerator systems. The working principles to generate cooling by an acoustic wave are described. The analytic expressions are applied into a model, that describes a 25 cm tubular standing wave resonator. In chapter 4 the theory of the traveling wave systems is derived and analytical expressions are found for the thermoacoustic equations describing the energy flows in this system. By applying these expressions into a numerical model a fast numerical design tool in Fortran has been written by which traveling wave systems can be studied efficiently. This model is applied to Swifts traveling wave engine described in reference [34] and shows good agreement with DeltaE computations. Apart from that chapter 4 describes two experiments with traveling wave apparatus, one co-axial type as developed by ECN, and a new concept namely a motor driven 1.3 meter long tubular traveling wave cooler, developed at TU/e. The results of the measurements are compared with the model to obtain insight concerning the validation. An important result that should already be mentioned here is that our numerical design tool for the traveling wave system indicated that for these smaller scale systems it is not necessary to contain a compliance to build a cooler. A best performance can be obtained with a single size diameter feedback tube. The experimental system of the TU/e cools very well by using such one diameter size feedback tube. Finally in chapter 5 the results of chapters 3 and 4 are combined into two analytical scaling models for standing wave as well as traveling wave systems. These models that start from a macroscopic known apparatus demonstrate that scaling towards millimeter size devices leads to a strong decrease of the performance. Chapter 6 concludes this thesis. In the appendixes some detailed experimental data and some mathematical derivations are presented.

Chapter 2 Basic theory of thermoacoustics 2.1 Wave equation and total energy flow Geometry The widely-used linear thermoacoustic theory as known today was first developed by Rott and reviewed by Swift [33]. First, the linearization of the Navier-Stokes and continuity equations gives us the wave equation of thermoacoustics. Next, energy conservation and heat transfer equations provide us the total energy flow expression. As shown in Figs. 2.1.1 and 2.1.2, we consider a stack of parallel plates in an acoustic field. The conditions are defined in Fig.2.1.2. The x axis is along the direction of sound propagation, the y axis normal to the fluid-solid boundary. y=0 is located in the center of the fluid. The thickness of the fluid layer between two adjacent stack plates is 02y as shown in Fig. 2.1.2. The y′ axis for the solid is

normal to the fluid-solid boundary, with 0=′y in the center of the solid and

ly =′ at the boundary, see Fig. 2.1.2. Axes y andy′ have opposite directions.

Figure 2.1.1: Geometry used for a multi-plate thermoacoustic system.

Stack of plates x

Sound wave

Basic theory of thermoacoustics 9

Figure 2.1.2: Geometry used for multi-plate stack.

Basic Equations

Assume that all variables oscillate at a single angular frequency ω and use an expansion up to first-order in the acoustic amplitude for all variables.

])(Re[ 1ti

m expp ωp+= (2.1.1)

]),(Re[)( 1ti

m eyxx ωρρ ρ+= (2.1.2)

]),(Re[]),(Re[ 11titi eyxyeyxxV ωω vu

vvr+= (2.1.3)

]),(Re[)( 1ti

m eyxxTT ωT+= (2.1.4)

]),(Re[)( 1ti

sms eyxxTT ω′+= T (2.1.5)

]),(Re[)( 1ti

m eyxxss ωs+= (2.1.6)

The subscripts “m” indicates mean value and “s” indicates solid. Throughout this study, complex quantities are represented by boldface type, with exceptions: I) the definition 1−=i and II) The Rott’s functions: vf , κf and sε . Thus, variables like )(),( 1 xx ρp1 and etc. are complex amplitudes. Note that we also make the following assumptions:

1. The theory is linear, zero-order and first-order terms are kept for equations other than energy equations. For energy equations, second–order terms are considered.

2. The zero order average fluid velocity 0=mVr

3. The solid is perfectly rigid. 4. The working fluid is considered to be an ideal gas. 5. Gravity is neglected. 6. Second viscosity is neglected [37].

Fluid Solid

Fluid Solid

Fluid

Solid

l2

02yx

x

y

y′

10 Chapter 2

7. The pressure dependency of the viscosity and thermal conductivity is neglected. In the geometry of Fig 2.1.2, we begin with deriving an expression for the x component of the fluid velocity. The momentum equation of a compressible, viscous fluid in two dimensions [36],

∂∂+

∂∂−

∂∂

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂

y

v

x

u

x

u

xx

p

y

uv

x

uu

t

u

3

22µρ

∂∂+

∂∂

∂∂+

x

v

y

u

yµ , (2.1.7)

where µ is the dynamic viscosity. The viscosity µ is a function of temperature.

Rearranging the terms in Eq. (2.1.7) yields:

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂

y

v

x

u

xy

u

x

u

x

p

y

uv

x

uu

t

u

32

2

2

2 µµρ

x

v

yy

u

yy

v

xx

u

x ∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂−

∂∂

∂∂+ µµµµ

32

34

. (2.1.8)

Substitute the variables expressed from Eq. (2.1.1) to Eq. (2.1.6), and keep the terms till first-order:

∂∂∂+

∂∂+

∂∂+

∂∂+−=

yxxyxdx

di m

12

21

2

21

2

21

21

1 3vuuup

uµµωρ

xyyyyxxx ∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂−

∂∂

∂∂+ 1111

32

34 vuvu µµµµ

. (2.1.9)

Since

ydT

d

y

T

dT

d

y ∂∂=

∂∂=

∂∂ 1Tµµµ

, (2.1.10)

Eq. (2.1.10) leads to the conclusion that the two terms in Eq. (2.1.9) yy ∂

∂∂∂ 1uµ

and

xy ∂∂

∂∂ 1vµ

are negligible second-order terms. So, now, Eq. (2.1.9) can be reduced to:

∂∂∂+

∂∂+

∂∂+

∂∂+−=

yxxyxdx

di m

12

21

2

21

2

21

21

1 3vuuup

uµµωρ

yxxx ∂∂

∂∂−

∂∂

∂∂+ 11

32

34 vu µµ

. (2.1.11)

Similar to Eq. (2.1.10),

∂∂+=

∂∂=

∂∂

xdx

dT

dT

d

x

T

dT

d

xm 1Tµµµ

(2.1.12)

Basic theory of thermoacoustics 11

The substitution of Eq. (2.1.12) in Eq. (2.1.11) and neglecting second-order variation terms, Eq. (2.1.11) can be finally expressed as:

∂∂∂+

∂∂+

∂∂+

∂∂+−=

yxxyxdx

di m

12

21

2

21

2

21

21

1 3vuuup

uµµωρ

ydx

dT

dT

d

xdx

dT

dT

d mm

∂∂−

∂∂+ 11

32

34 vu µµ

. (2.1.13)

The variations related to x, which is the axial wave propagation direction, are of the order of the radian wavelength πλ 2/=D , where λ is the wavelength. Those in the perpendicular y direction relate to the viscous penetration depth. Therefore, we

know that 11 vu is of the order of vδD , x∂∂ is of the order of D1 , y∂∂ is of

the order of vδ1 . Here,

)(2 ρωµδ =v (2.1.14)

is the viscous penetration depth. Since D<<vδ , the terms 21

2

x∂∂ uµ and

yx∂∂∂ 1

2vµ can

be neglected compared with 21

2 y∂∂ uµ .

Therefore, Eq. (2.1.13) reduces to

ydx

dT

dT

d

xdx

dT

dT

d

ydx

di mm

m ∂∂−

∂∂+

∂∂+−= 11

21

21

1 3

2

3

4 vuupu

µµµωρ . (2.1.15)

In normal working conditions, the temperature gradient is not extremely large. In general, the viscosity can be approximately described as [38]:

( ) µµµ bTT 00 /= (2.1.16)

For T0=300 K, values 0µ and µb for some gases, of common interest in thermoacoustics, are listed in the table 1.1.I.

T0=300 K µ

µµb

T

T

=

00

0µ (kg/m·s) µb

air 1.85E-5 0.76

nitrogen 1.82E-5 0.69

helium 1.99E-5 0.68

neon 3.2E-5 0.66

argon 2.3E-5 0.85

xenon 2.4E-5 0.85

Table 1.1.I Approximate values 0µ and µb for some gases

Therefore, the ratios of terms in Eq. (2.1.15) are

12 Chapter 2

xdx

dTy

T

bxdx

dT

dT

dy

mm

∂∂

∂∂

=

∂∂

∂∂

1

21

2

1

21

2

43

34 u

u

u

u

µµ

µ (2.1.17)

is of order 2

2

νδD

>>1, and

ydx

dTy

T

bydx

dT

dT

dy

mm

∂∂

∂∂

=

∂∂

∂∂

1

21

2

1

21

2

2

3

32 v

u

v

u

µµ

µ (2.1.18)

is of order 2

2

νδD

>>1.

So, for widely used working gases, in normal working conditions, (without extremely large temperature gradient dxdTm / ), the last two terms in Eq. (2.1.15) can be neglected. Therefore, the momentum equation can be reduced to:

21

21

1ydx

di m ∂

∂+−=

upu µωρ . (2.1.19)

This is the description of the oscillatory velocity profile as dependant on the oscillatory pressure gradient including viscous terms. With boundary conditions: at 0=y , because of the symmetry, 0/1 =∂∂ yu , and

at 0yy = , because of the solid wall, 01 =u , the solution of (2.1.19) follows (see

Appendix A)

( )[ ]( )[ ]

++

−=v

v

m yi

yi

dx

di

δδ

ωρ 0

11 1cosh

1cosh1

pu . (2.1.20)

As an illustration of the velocity profile, an example is plotted based on Eq. (2.1.20) in Fig 2.1.3. Here, the velocity variation along the y direction is shown with time as a parameter. In this example, the following parameters are adopted:

1. standing wave field inside the resonator tube: )/sin(1 Dxpp A=

2. 1.0=Ap bar, for helium in 300K and 1 bar of mean pressure.

3. νδ×= 0.20y , ωρµδν m/2= is the viscous penetration depth of the fluid at

(1000Hz, 1 bar, and 300K for helium). 4. The computed position is at the middle point of the resonator tube, i.e. x=λ/8,

where λ is the wavelength. The resonator tube length is λ/4.

Basic theory of thermoacoustics 13

0.0 0.5 1.0 1.5 2.0

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

u 1 (m

/s)

y/δν

ωt=0*(π/4)ωt=1*(π/4)ωt=2*(π/4)ωt=3*(π/4)ωt=4*(π/4)ωt=5*(π/4)ωt=6*(π/4)ωt=7*(π/4)

Figure 2.1.3: Relation between velocity in the x direction and position perpendicular to that (y direction) at different moments in time.

After deriving the axial flow velocity from the momentum equation, we now consider the temperature of the solid plate ),,( tyxTs . The following equation

holds:

sss Tt

T 2∇=∂

∂ κ , (2.1.21)

where ssss cK ρκ = is the thermal diffusivity of the solid, andsK , sρ , sc are the

thermal conductivity, density, and specific heat per unit mass, respectively. The solid’s thermal diffusivity is considered as constant. Substitution of Eq.(2.1.5) into Eq. (2.1.21) to first order yields:

′∂∂+

∂∂+=⋅⋅ tistism

sti

s ey

exdx

Tdie ωωω κω 2

12

21

2

2

2

1

TTT . (2.1.22)

Similar to the reduction of Eq. (2.1.13) and (2.1.15), it can be seen that

( ) ( ) 1)/(~/ 221

221

2 <<′∂∂∂∂ Dsss yx δTT , where ωκδ /2 ss = is the solid’s

thermal penetration depth, and

1)//()/(~)//()/( 122

1222 <<′∂∂ msssm TydxTd TT Dδ .

Thus, Eq. (2.1.22) reduces to

21

2

1y

i sss ′∂

∂=

TT κω . (2.1.23)

The temperature of the plate can be derived from this equation as (see Appendix B)

14 Chapter 2

( )[ ]( )[ ]s

sbs li

yi

δδ

+′+

=1cosh

1cosh11 TT . (2.1.24)

where 1bT is temperature amplitude at the boundary, and is given by Eq. (C.22) in

appendix C. The temperature in the fluid is found from the general equation of heat transfer [36].

( )+∇⋅∇=

∇⋅+∂∂

TKsVt

sT

vrvvρ (terms quadratic in velocity). (2.1.25)

Here, s is the fluid entropy per unit mass. From thermodynamics, it is known that

( ) ( )dpdTTcds p ρβ // −= , (2.1.26)

where β is its isobaric thermal expansion coefficient and equal to 1/T for ideal gas. Substitution of Eq. (2.1.1) to (2.1.6) into (2.1.25), using Eq. (2.1.26) and keeping the first order terms, Eq. (2.1.25) becomes

2

22

111 dx

TdK

dx

dT

dT

dKe

dx

dTceieic mmtim

pmtiti

pm +

=+⋅−⋅ ωωω ρωωρ upT

tititimtim ey

Kex

Keydx

dT

dT

dKe

xdx

dT

dT

dK ωωωω21

2

21

211

∂∂+

∂∂+

∂∂+

∂∂+ TTTT

. (2.1.27)

Compare the terms on the right-hand side of Eq. (2.1.27) with the very last one,

∂∂

TyK

dx

dT

dT

dK m 1

2

21

22

~TT

D

κδ<<1 (2.1.28a)

∂∂

TyK

dx

TdK m 1

2

21

2

2

2

~TT

D

κδ<<1 (2.1.28b)

2

21

21 ~

∂∂

∂∂

D

κδy

Kxdx

dT

dT

dK m TT<<1 (2.1.28c)

D

κδ~

21

21

yK

ydx

dT

dT

dK m

∂∂

∂∂ TT

<<1. (2.1.28d)

Thus, neglecting the relatively small terms compared in Eq. (2.1.28a) to (2.1.28d), Eq. (2.1.27) reduces to

21

2

111 yKi

dx

dTic m

pm ∂∂=−

+ TpuT ωωρ . (2.1.29)

Solving this second order differential equation, the temperature oscillation in the fluid layer can be obtained as (see Appendix C)

( )( ) ( )[ ] dx

dT

dx

d

yi

yi

cm

v

v

mpm

1

02

11 /1cosh1

]/1cosh[1

1 ppT

+−+−−=

δσδσ

ωρρ

Basic theory of thermoacoustics 15

( )( )( )

( )[ ]( ) ( )[ ]κ

κ

δεδε

ωρσρ /1cosh1/1cosh

11

/

02

11

yi

yi

f

fdxdTdxd

c sk

vs

m

m

pm +++

+

−+− pp

, (2.1.30)

where 22 // κδδµσ vp Kc == (2.1.31)

is the Prandtl number, and the Rott’s functions [ ]

ν

νν δ

δ/)1(

/)1(tanh

0

0

yi

yif

++

= (2.1.32)

[ ]κ

κκ δ

δ/)1(

/)1(tanh

0

0

yi

yif

++

= (2.1.33)

[ ][ ]ssss

pm

slicK

yicK

δρδρ

ε κ

/)1(tanh

/)1(tanh 0

+

+= . (2.1.34)

0 1 2 3

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

tanh

(1+

i)y0/δ

k

y0/δ

k

Real partImaginary part

Figure 2.1.4: The real and imaginary parts of ( )

+

κδ01tanh yi

In Fig.2.1.4, the real and imaginary parts of ]/)1tanh[( 0 κδyi+ are plotted. Note

that at κδ20 =y the function is almost unity.

16 Chapter 2

Figure 2.1.5: Geometry of the example case. As an illustration of the temperature oscillation for the 1T profile, an example is

plotted in Fig 2.1.6 and 2.1.7 based on Eq. (2.1.30). Here, the real part of the 1T

variation along the y direction at any position in the stack is shown. In this example, the same parameters as for the x component 1u in Fig.2.1.3 are adopted. The

geometrical schematic is shown in Fig. 2.1.5. The resonator tube is a quarter of one wave length (25cm) and the stack length is one fifth of the resonator tube length (5cm). The leading end of the stack is placed at 5 cm away from the pressure node.

The total energy flow 2E& along the stack is zero. At a fixed x position, the real and

imaginary part of 1T increases as approaching the center of the fluid layer. At a

fixed y position, 1T decreases as approaching the anti-node of the pressure wave.

The relative position in the stack is evalued as x/stack length in Fig 2.1.6 and 2.1.7. From Eq. (2.1.30), it is obvious that it consists of three groups of terms. For convenience, name them as:

the first term:pmcρ

11

pT = ; (2.1.35)

the second term:

( )( ) ( )[ ] dx

dT

dx

d

yi

yi m

v

v

m

1

021 /1cosh1

]/1cosh[1

1 pT

+−+−×−=

δσδσ

ωρ; (2.1.36)

and the third term:

( )( )( )

( )[ ]( ) ( )[ ]κ

κ

δεδε

ωρσρ /1cosh1/1cosh

11

/

02

111 yi

yi

f

fdxdTdxd

c sk

vs

m

m

pm +++×

+

−+−= pp

T .

(2.1.37) The first term comes from the adiabatic acoustic compressions and expansions. The second and the third terms come from the oscillatory movement of the fluid along the mean-temperature gradient in the fluid, with viscous effects included.

5 cm 5 cm 15 cm

x 0

Basic theory of thermoacoustics 17

0.1 0.2 0.3 0.40.5

0.60.7

0.80.9

1.0

-2

0

0.0

0.2

0.4

0.6

0.81.0

real

par

t of

T1 (

K)

y/y 0relative position in the stack

-2.000

-1.725

-1.450

-1.175

-0.9000

-0.6250

-0.3500

-0.07500

0.2000

Figure 2.1.6: Real part of 1T at various y position and x position in the stack.

0.1 0.2 0.3 0.40.5

0.60.7

0.80.9

1.0

-0.2

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0.81.0

imag

inar

y pa

rt o

f T

1 (K

)

relative position in the stack

y/y0

-0.2500

-0.1312

-0.01250

0.1063

0.2250

0.3438

0.4625

0.5813

0.7000

Figure 2.1.7: Imaginary part of 1T at various y position and x position in the stack.

18 Chapter 2

Wave equation Next, the wave equation for )(1 xp is derived. Starting with the continuity equation

( ) 0=⋅∇+∂∂

Vt

vρρ

. (2.1.38)

Substitution of the variables Eq. (2.1.1) to (2.1.6) into Eq. (2.1.38), and keeping the first order terms yields

( ) 0111 =

∂∂+

∂∂+

yxi mm

vuρ ρρω . (2.1.39)

Using Eq. (2.1.19), it can be written as:

21

21

1

1

yidx

d

im ∂∂+−= up

uωµ

ωρ . (2.1.40)

Substitution of Eq. (2.1.40) into Eq. (2.1.39) gives

012

12

21

2

12 =

∂∂+

∂∂

∂∂+−−

yi

yxdx

dm

vupρ ωρµω . (2.1.41)

Assuming ideal-gas behavior, we can write: RTp /=ρ (2.1.42)

where the specific gas constant is mRR univ /= ,universal gas constant

KmolJRuniv ⋅= /3.8 , and m molecular weight.

Thus, we have dTTRTpRTdpd ]/)/[(/ −=ρ . (2.1.43)

Substitute the adiabatic speed of sound RTa γ=2 [39], whereγ is the ratio of

isobaric to isochoric specific heats. We find

( ) 12

11 pTρ am γβρ +−= . (2.1.44)

Substitution of Eq. (2.1.44) in Eq. (2.1.41) yields

0121

2

21

2

12

2

12 =

∂∂+

∂∂

∂∂+−−

yi

yxdx

d

a mm

vuppT ωρµγωβρω . (2.1.45)

Integrating Eq. (2.1.45) with respect to y from 0 to 0y yields a wave equation for

the first-order acoustic pressure amplitude )(1 xp

∫∫∫∫

∂∂

∂∂+−− 0000

0 21

2

0 21

2

0 12

2

0 12 yyyy

m dyyx

dydx

ddy

ady

uppT µγωβρω

00

0

1 =∂∂+ ∫

y

m dyy

ivωρ . (2.1.46)

For the last term on the left hand side follows

00110

1

0

0 =−=∂∂

==∫ ymyym

y

m iidyy

i vvv ωρωρωρ . (2.1.47)

Basic theory of thermoacoustics 19

Because the boundary conditions: at 0yy = , due to the wall, 01 =v and at

0=y , 01 =v by symmetry.

Thus, Eq. (2.1.46) reduces to

00000

0 21

2

0 21

2

0 12

2

0 12 =

∂∂

∂∂+−− ∫∫∫∫

yyyy

m dyyx

dydx

ddy

ady

uppT µγωβρω . (2.1.48)

By substituting Eq. (2.1.30) for 1T , the first integration term is obtained

10 0

2

120

11 pT∫

+−=

y

spm

fy

cdy

εβωβρω κ

+−++

−−−

)1)(1(111

0s

vsvm fff

dx

dT

dx

dy

εσε

σσβ κp

. (2.1.49)

By substituting Eq. (2.1.20) for 1u , the last term in the left hand side of Eq.

(2.1.48) is obtained

∂∂

∂∂=

∂∂

∂∂=

∂∂

∂∂

=

=∫∫

0

00

0

1

0 21

2

0 21

2 yy

y

yy

yxdy

yxdy

yx

uuu µµµ

⋅= vfdx

d

dx

dy 1

0

p. (2.1.50)

Substitution of Eq. (2.1.49) and (2.1.50) into Eq. (2.1.48) yields

+−++

−−−

+−

)1)(1(11

11 1

010

2

s

vsvm

sp

fff

dx

dT

dx

dy

f

c

y

εσε

σσβ

εβω κκ p

p

0102

12

0102

2

=

⋅+−− vfdx

d

dx

dy

dx

dyy

a

pppγω

. (2.1.51)

By using the following relations,

1−=

γγ R

cp ; (2.1.52)

T

1=β ; (2.1.53)

RTa γ=2 . (2.1.54)

Eq. (2.1.51) can be rewritten as

( )

−+

+−+

dx

df

dx

daf

s

12

2

1 11

)1(1

pp ν

κ

ωεγ

( ) 01)1(1

112

2

=

+−++

−−+

s

svm fff

dx

d

dx

dTa

εσε

σσ

ωβ νκp

. (2.1.55)

20 Chapter 2

Using the state equation (2.1.42), the second term of Eq. (2.1.55) on the left hand side can be written as

( ) ( )dx

d

dx

dTf

a

dx

df

dx

da

dx

df

dx

da mv

m

vmv

12

21

2

21

2

2

11

1ppp −−

−=

−ω

βρω

ρω

.

(2.1.56) Substituting Eq. (2.1.56) into (2.1.55), the thermoacoustic wave equation is obtained

( )( ) 011

11

)1(1 1

2

21

2

2

1 =+−

−−

−+

+−+

dx

d

dx

dTffa

dx

df

dx

daf m

sm

m

s

ppp

εσωβ

ρωρ

εγ νκνκ .

(2.1.57) This equation describes a sound field modified by the interaction between fluid and solid plates. The coefficients, related to the acoustic pressure and its gradient, are complicated functions having a dependence on the temperature profile. If the temperature profile is known, the wave equation (2.1.57) can be solved. In chapter 4, this wave equation is reduced by some assumptions to describe the sound field inside the different components of a traveling-wave system. The time-averaged total energy flow We will proceed to derive an expression for the time-averaged energy flow. In steady state, and assuming that the system is ideally isolated from the surroundings, it can be deduced that the time-averaged energy flow must be independent of x. Start with conservation of energy [38]:

∑⋅−∇−

+⋅∇−=

+∂∂ vvvvvvv

VTKhVVeVt

22

21

21 ρρρ , (2.1.58)

where e and h are internal energy and enthalpy per unit mass, respectively, and ∑ is the viscous stress tensor, with components:

k

kij

k

kij

i

j

j

iij x

v

x

v

x

v

x

v

∂∂+

∂∂−

∂∂

+∂∂=∑ ξδδµ

32

. (2.1.59)

The first term on the left hand side in Eq. (2.1.58) is kinetic energy density of unit control volume. The second term on the left hand side is the internal energy density. The first term on the right hand side is from the enthalpy flow and kinetic energy. The second term is from thermal conduction. The last term on the right hand side results from the viscosity. The lowest-order variation in the energy is of second

order. All terms of higher order (e.g. 2

VVvv

are of third order) are neglected.

Basic theory of thermoacoustics 21

Integrating the remaining terms in Eq. (2.1.58) with respect to y from y=0 to 0=′y and time averaging, yields

( ) 00 00

0 0 00=

∑⋅−′

∂∂−

∂∂−∫ ∫ ∫∫

y l y

xs

s

ydyVyd

x

TKdy

x

TKdyuh

dx

d vvρ . (2.1.60)

The over bar denotes time averaging. The quantity within the square brackets is the time-averaged energy flow per unit of perimeter along x, defining this quantity as

∏E& , where Π is the perimeter of the stack plates:

∫ ∫ ∫∫ ∑⋅−′∂∂−

∂∂−=

∏0 00

0 0 00)(

y l y

xs

s

ydyVyd

x

TKdy

x

TKdyuh

E vr&ρ , (2.1.61)

where E& is the total energy flow through the stack and ∏ is the total perimeter of

the stack plates. Now hcan be expanded in the same way as in the Eqs. (2.1.1) to (2.1.6):

]),(Re[)( 1ti

m eyxxhh ωh+= (2.1.62)

Substitution of the equations from Eq.(2.1.1) to (2.1.6) and (2.1.62) into (2.1.61)

and expanding ∏E& to second order in the acoustic amplitude, (the variation

terms of third order and higher are again neglected) the first term in Eq. (2.1.61) becomes

(∫∫ ++= 00

0 112

210]Re[]Re[]Re[]Re[

y titim

timm

timm

yeehehehdyuh ωωωω ρρρ uρuu

)dyee titim ]Re[]Re[ 11

ωωρ hu+ . (2.1.63)

It is easy to see that the value of time averaging of the first term is also zero, i.e.

0]Re[ 1 =tieωu .

The integrals of the third and fourth terms in Eq. (2.1.63) sum to zero because the second-order time-averaged mass flow is zero:

0)]Re[]Re[]Re[(0

0 112

2 =+∫ dyeeey tititi

mωωωρ uρu . (2.1.64)

Hence, the Eq. (2.1.63) reduces to

dyeedyuhy y titi

m∫ ∫=0 0

0 0 11 ]Re[]Re[ ωωρρ hu . (2.1.65)

Using equation dpTdsdh )/1( ρ+= (2.1.66)

and Eq. (2.1.26) yields ( )dpTdTcdh p βρ −+= 1)/1( . (2.1.67)

By using Eq. (2.1.67), the Eq. (1.1.65) becomes

[ ]dyeecdyuhy titi

pm

y

∫∫ = 00

0 110]Re[]Re[ ωωρρ uT . (2.1.68)

22 Chapter 2

For the second and third integrals of Eq. (2.1.61), only the zero order terms are significant to be counted. The terms to second order or higher can be neglected. Therefore, the two integrals are

( )dx

dTlKKyyd

x

TKdy

x

TK m

l

ss

s

y

∫∫ +−≅′∂∂−

∂∂−

0 00

0

. (2.1.69)

Using arguments similar to those leading to Eq. (2.1.19), we find that the largest

terms in the last integral in Eq. (2.1.61) are of order D/210 uy µ , whereas uhρ has

the order of 2111 auup mρ≅ . Hence,

( ) 121

~ 2

2

00

00 <<=∑⋅ ∫∫DD

vvv

yy

x adyuhdyV

δνρ . (2.1.70)

So, the viscous term ( )xV ∑⋅vv

is negligible. Therefore, Eq. (2.1.61) becomes

[ ] ( )dx

dTlKKydyeec

E ms

y titipm +−=

∏ ∫ 00 112 0

]Re[]Re[ ωωρ uT&

. (2.1.71)

To remind us that it is an energy flow valid to second order in the acoustic quantities, here, the subscript “2” is added toE& .

Substitution of Eq. (2.1.30) for 1T and Eq. (2.1.20) for 1u , and integration yields (see Appendix D)

( )( )( )

++−−−∏=

σεωρνκ

ν 11

~~

1~

Im2 1

102

sm

fff

dx

dyE p

p&

( )( )( )

( )( )

+++−+×

−∏

+σε

εσρω

κννκν 11

/1~

~Im

~

1211

30

s

sm

m

p fffff

dx

d

dx

d

dx

dTcy pp

( )dx

dTlKKy m

s+∏− 0 , (2.1.72)

where the tilde denotes complex conjugation. This equation describes up till second order in the acoustic quantities energy flow in a thermoacoustic stack.

2.2 Acoustic energy Now, we will develop an expression for the time-averaged acoustic power W& used or produced in a segment of length x∆ in the stack. It is clear that the time-averaged products of first-order terms in pressure and acoustic particle velocity are the largest non-zero time-averaged power component. In acoustics, this is called the acoustic intensity, which describes the time-averaged “rate per unit area at which work is done by one element of fluid on an adjacent element” [39]. This

Basic theory of thermoacoustics 23

acoustic power is a second-order quantity. Therefore, the subscript “2” is added to the power symbol W& . According to the geometry in Fig. 2.1.1, no net acoustic power flows in the y direction. Thus, the difference in time-averaged acoustic intensity between the two positions along the stack must be the acoustic power per unit area generated or absorbed between those two positions. It is shown in Fig. 2.2.1.

Figure 2.2.1: Two faces in a multi-plate thermoacoustic system used for computation of the acoustic power flow. Consider a cross section of the stack at position x and one at position dxx + . Then the difference in acoustic power flow through these cross sections is given by

)()( 222 xWdxxWWd &&& −+=

( )[ ] ( )[ ]dyeedyeey

xtiti

y

dxxtiti

∫∫ ∏−∏= +

00

0)(11

0)(11 ]Re[]Re[]Re[]Re[ ωωωω upup .

(2.2.1) When dx is infinite small, we can write

( )∫∏=0

0

112 ]Re[]Re[y

titi dyeedx

ddxWd ωω up& ,

Or

∏= ∫

0

0

112 ]Re[]Re[

ytiti dyee

dx

d

dx

Wd ωω up&

. (2.2.2)

The velocity in the x-direction, averaged over the cross section is defined as:

dyy

y

∫=0

0

10

1

1uu (2.2.3)

Eq. (2.2.2) can be rewritten as:

( )]Re[]Re[ 1102 titi ee

dx

dy

dx

Wd ωω up∏=&

. (2.2.4)

By using Eq. (D.2), Eq. (2.2.4) can be expressed as

Left

x

Sound wave

Right

24 Chapter 2

[ ]

+∏=∏=

dx

d

dx

dy

dx

dy

dx

Wd 11

110110

2 ~~

Re21~Re

21 p

uu

pup&

. (2.2.5)

According to Eq. (2.1.20), the average velocity can be obtained as:

( )νωρf

dx

di

m

−⋅= 111

pu . (2.2.6)

The conjugate average velocity is

( )νωρf

dx

di

m

~1

~~ 1

1 −⋅−= pu . (2.2.7)

From Eq. (2.2.6), it can be obtained

ν

ωρf

i

dx

d m

−−

=1

11up

. (2.2.8)

From Eq. (2.2.7), dxd /~1u can be obtained as:

( )

−−=dx

df

dx

di

dx

d

m

11~~

1~ pu

ρων . (2.2.9)

According to Eq. (2.1.57):

( )( )( )

+−+−

+−−=

−1

12

2

2

21

1)1(

111

1p

pp

s

m

smm

f

dx

d

dx

dTffa

adx

df

dx

d

εγ

εσωβ

ρω

ρκνκν .

(2.2.10) Substitution of Eq. (2.2.10) in (2.2.9) yields:

( )( ) 1211 ~

~1

~)1(

1~

~11

~~~p

pu

+−++

+−−−=

sm

m

sm

f

a

i

dx

d

dx

dTffi

dx

d

εγ

ρω

εσωρβ κνκ . (2.2.11)

Substituting Eq. (2.2.8) and Eq. (2.2.11) into Eq.(2.2.5), gives:

( ) ( )

+−

+−

∏=sm

m f

af

fy

dx

Wd

εργρ

ω κν

ν1

Im1

Im~12

12

2

12

2

10

2pu&

( ) ( )

−+−

−∏+ 110

~)

~1(~1

)~~

(Re

121

upν

νκ

εσβ

f

ff

dx

dTy

s

m . (2.2.12)

This is the acoustic power absorbed (or produced in the prime mover mode) in the stack per unit length. The first two terms in Eq. (2.2.12) are the viscous and thermal relaxation dissipation terms, respectively. These two terms have a dissipative effect in thermoacoustics and they will be present whenever a wave interacts with a solid surface. The third term can be either source or sink for acoustic power. It depends on the sign of the temperature gradient along the stack.

Basic theory of thermoacoustics 25

This chapter is dedicated to the review of Rott’s theory. The time-averaged total energy flow Eq.(2.1.72) is an important equation to be used often in the latter chapters.

Chapter 3 Standing-wave systems 3.1 Introduction Thermoacoustics employs the thermodynamic interaction between a solid wall and sound wave to realize heat transport (in heat pump mode, taking heat from cold end to the warm end) or energy conversion (in prime mover mode, changing the heat into mechanical energy—acoustic energy). In chapter 2, it is shown that these thermoacoustic effects occur in a thin gas layer, which is called the fluid’s thermal penetration depth. In a standing-wave system, these thermoacoustic effects are used by inserting a stack of parallel plates made from poorly conductive materials into a resonator tube. The plates are normally spaced by a distance about twice the working gas’ thermal penetration depth. Thus, most of the gas between the stack plates takes part in the heat transport. This is the basic idea behind the construction of a standing-wave thermoacoustic refrigerator. Normally, a standing-wave thermoacoustic refrigerator consists of an acoustic driver to generate a sound wave, a resonator tube filled with a working gas and a stack sandwiched between a cold-end heat exchanger and a hot-end heat exchanger. In practice, the resonator tubes are made in a quarter wave-length or in a half wave-length. These two types of standing-wave systems are illustrated schematically in the figures 3.1.1 a and b, respectively. In a quarter wave-length resonator, there is a large-volume gas reservoir at one end to create the right acoustic boundary condition at the end of the resonator. In a half wave-length resonator, one end is closed by the acoustic driver and the other one is just a closed end. The foundation of linear thermoacoustic theory by Rott [1] unveiled a very prolific era of thermoacoustics. This novel technology attracted much attention and interest from researchers. Among them, the Condensed Matter and Thermal Physics Group of Los Alamos National Laboratory (LANL) are the distinguished pioneers. In the

Standing-wave systems 27

early 1980s, they dedicated their efforts to the exploration of thermoacoustic concepts to create devices that would produce useful refrigeration or useful work [40-42].

Figure 3.1.1: (a) A simple illustration of a quarter wave-length standing-wave thermoacoustic refrigerator. Figure 3.1.1: (b) A simple illustration of a half wave-length standing-wave thermoacoustic refrigerator. Tom Hofler’s PhD work was part of these efforts, who built the first efficient thermoacoustic refrigerator [43]. Hofler’s standing-wave refrigerator was a quarter wave-length one, having a similar configuration as shown in Fig. 3.1.1a. A standing-wave type refrigerator was launched on the space shuttle Discovery in 1992 [44]. It was built in Naval Postgraduate School (NPS), designed to produce up to 80 K temperature difference over the stack, and to pump up to 4 W of heat. Another refrigerator built in NPS was the Shipboard Electronics Thermoacoustic Cooler to cool radar electronics on board of the warship USS Deyo in 1995 [45]. It was able to provide 400 W of cooling power for a small temperature span as designed. At Pennsylvania State University, a large refrigerator called TRITON was built for cooling of Navy ships, which was designed to generate a cooling power of 10 kW [45]. At LANL, a heat-driven thermoacoustic refrigerator “beer cooler” was built, which replaced the only moving part, the diaphragm of the loudspeaker, by thermoacoustic prime mover. It has no moving parts at all. Two of similar devices

Gas reservoir

Sound wave

Loudspeaker

Resonator tube

Cold HX

Ambient HX

Stack

Loudspeaker Resonator tube Stack

Sound wave

Ambient HX Cold HX

28 Chapter 3

were also built at NPS: one is called a Thermoacoustic Driven Thermoacoustic Refrigerator (TADTAR), having a cooling power of about 90 watt for a temperature span of 25°C; The other TADTAR is solar driven, having a cooling capacity of 2.5 watt for a temperature span of 17.7°C [47]. After this general introduction of standing-wave system, the following subsections are dedicated to: a simplified physical description of standing-wave system thermoacoustic cycle; modeling of standing-wave systems; and a standing-wave type experiment apparatus, which is similar to the so-called “TAC” (thermoacoustic couple), was built to validate the model developed in section of modeling. At the last part of this chapter, the measurement results are analyzed and compared with computation based on proposed modeling. A “TAC” was introduced for better understanding of heat transport inside a standing-wave system for its simply structure. With more work were carried out by researchers on similar apparatus at high amplitudes, large discrepancies between measurement and computation by using the linear thermoacoustic theory were found and reported. These deviations from linear thermoacoustic theory attracted much attention and a succession of works was conducted to explore the attributions. In this chapter, a new approach to evaluate the total energy flow along the stack and then to further compute other parameters about the system was proposed and tested by measurements.

Standing-wave systems 29

3.2 Physical description of standing-wave systems The basic principles of the thermoacoustic effect in a standing-wave type refrigerator will be given in this section. In chapter 2, the basic theory of thermoacoustics is developed from an Eulerian point of view of fluid mechanics. In this section, to make it easier to understand, a Lagrangian point of view is adopted [53]. A given typical parcel of the fluid will be traced as it moves. The sound wave that is sinusoidal in practice is simplified as a square wave. By these simplifications, the cycle is simplified to two reversible adiabatic steps and two irreversible constant-pressure steps, which is called the Brayton cycle. The animation of this thermal process is clearly illustrated at the Los Alamos National Laboratory web site [54]. This thermoacoustic cycle is schematically presented in Fig. 3.2.1. A given gas parcel oscillating along the stack plate at the distance of a fluid’s thermal penetration depth is traced. A longitudinal temperature gradient mT∇ is assumed to

exist along the stack. Additionally, it is supposed that the pressure antinode is to the right end of the plate and a node to the left. For simplicity, an inviscid ideal gas is assumed to be the working fluid. The four steps of the thermoacoustic cycle are shown separately in Fig. 3.2.1 (a). A starting point is set as the position where the traced gas parcel is at the left most position of its stroke in this oscillation. The temperature, pressure, and volume of the parcel are mm TxT ∇− 1 , 1ppm − , and

V respectively. In step 1, the gas parcel moves a distance 12x , where it reaches its

right most position. The given gas parcel is compressed by the periodic acoustic wave oscillation and the inside temperature is increased by an amount of 12T . The

pressure change 12p and the temperature change 12T are related by the

thermodynamic relationship

1121211

11p

c

Tp

s

T

Tp

sp

p

TT

pm

m

ppmpmsρ

βρρ

ρρ

=

∂∂

∂∂−=

∂∂−=

∂∂=

(3.2.1) where s is entropy per unit mass, ( ) mpT ρρβ // ∂∂−= is the ordinary thermal

expansion coefficient, and pc is the constant-pressure heat capacity per unit mass.

After step 1, the temperature, pressure and volume become

11 2TTxT mm +∇− , 1ppm + and 1VV − respectively. At the same time, the gas

parcel is located at the right most position, where the local temperature of the stack plate is mm TxT ∇+ 1 . At this moment, the temperature difference between the gas

parcel and the stack plate is

mTxTT ∇−= 11 22δ . (3.2.2)

30 Chapter 3

Figure 3.2.1: (a) Typical gas parcel in a thermoacoustic refrigerator mode, experiencing a four-step cycle with two adiabats (step 1 and 3) and two constant-pressure heat transfer (step 2 and 4). (b) An amount of heat is shuttled along the stack plate from one gas parcel to the next, as a result, heat Q is transported from the left end of the plate to the right end, using work W [53].

In a heat pump mode, there must be a positive Tδ . Therefore, in step 2, assuming that the gas cools down to the local temperature of the stack the heat that is transferred from the gas parcel to the stack plate is

TmcQ pδδ = , (3.2.3)

where m is the mass of the gas parcel. A schematic pV-diagram of the cycle is shown in Fig.3.2.2. Here, the adiabatic steps 1 and 3 are linearized which is allowed because of the small Vδ . The work used to drive this cycle is equal to the area ABCD, is given by

W

Plate mm TT ∇− 1x mm TT ∇+ 1x

κδ

12x

1WδGas Parcel

1ppm −

1ppm +

mm TT ∇− 1x 11 2TTT mm +∇− x

V 1VV −

2. Heat transfer

2Wδ

1ppm +mm TT ∇+ 1x

1Qδ

3. Adiabatic expansion

12x

3Wδ

1ppm −1ppm +

11 2TTT mm −∇+ x

mm TT ∇+ 1x

VV δ−

4. Heat transfer

4Wδ

1ppm −mm TT ∇− 1x

V

2Qδ

κδ

Qδ1x

SLCTHT

CQ HQPressure antinode

end

Pressure node end

Plate

1. Adiabatic compression a)

VVV δ−− 1

VVV δ−− 1

b)

Standing-wave systems 31

∫=ABCD

pdVWδ (3.2.4)

The used work can be seen from Fig. 3.2.2. Hence, the work used in this 4-step cycle is given by

VpW δδ 12−≈ . (3.2.5)

Figure 3.2.2: Schematic pV-diagram of the thermoacoustic cycle of Fig. 3.2.1. The four steps of the thermoacoustic cycle are illustrated: adiabatic compression 1, isobaric heat transfer 2, adiabatic expansion 3 and isobaric heat transfer 4. The area ABCD is the work used in the cycle [53]. After releasing the heat from the gas parcel to the stack plate, the gas parcel has the same temperature as the local stack plate and in this isobaric heat transfer process its volume is reduced. In step 3, the gas parcel moves back to the initial most left position via an adiabatic expansion. The pressure drops to 1ppm − by the

periodical sound movement. The accompanying temperature drop is 12T by this

adiabatic expansion. The volume expands by 1V to VV δ− . After this expansion,

the gas parcel becomes colder than the local stack plate. In step 4, the gas parcel absorbs heat from the local stack plate to eliminate the temperature difference

Tδ− by an isobaric expansion. In the two isobaric processes, step 2 and 4, the temperature difference between the gas parcel and the local temperature of the stack plate is crucial to the direction of the heat flow. If the Tδ is positive, then, in step 2, the gas parcel has a higher temperature than that of the stack plate, and in step 4, the gas parcel has a lower temperature. By this means, the gas parcel absorbs heat in the left most position and release heat at the extreme right position, realizing the heat transportation from left to right. If the Tδ is negative, the whole story reverses. The gas parcel absorbs heat from the stack plate when it arrives at the right most position and gives off heat as it is at the extreme left position. This

Pressure

Volume

A

B C

D

1

2

3

4

1ppm +

1ppm −

V VV δ− 1VV − VVV δ−− 1

32 Chapter 3

process heats up the gas parcel while compressed, and cools it down while expanded. This makes the sound amplified and is called “prime mover” mode. If the Tδ is zero, then no heat is exchanged between the gas parcel and the stack plate. Therefore, there exists a critical temperature gradient, which distinguishes the “heat pump” mode from the “prime mover” one. The critical temperature gradient is given by 0=Tδ , when the temperature change mTx ∇12 by the gas parcel

displacement from the sound wave transportation is just matched by the adiabatic temperature change 12T by the sound wave thermodynamic process. Eq. (3.2.2)

yields

11 / xTTT critcriticalm =∇=∇ (3.2.6)

Using Eq. (3.2.1) and ω/1 Aux = , where Au is the gas particle velocity amplitude

at the specific position along the stack and ω the angular frequency, it is obtained

Apm

Amcrit uc

pTT

ρβω

=∇ . (3.2.7)

Usually, the displacement of a given gas parcel is small with respect to the length of the plate. Thus there will be an entire train of adjacent gas parcels, each confined in its cycle motion. For the heat pump mode, during the first half cycle, the individual gas parcels will each move a distance of 12x towards the pressure

antinode and deposit heat Qδ locally on the stack plate. During the second half

cycle, each parcel moves back to its initial position and picks up heat Qδ from the

plate. But this heat was deposited there a half cycle earlier by an adjacent gas parcel. Eventually, the heat Qδ is passed along the plate from one gas parcel to the

next one in the direction of the pressure antinode. By means of these gas parcels in series, the heat Qδ is shuttled along the plate towards the pressure antinode end.

The ratio of the maximum acoustic pressure amplitude in the system and the mean pressure is called the “drive ratio”.

3.3 Modeling standing-wave systems In thermoacoustics, thermodynamics is strongly coupled with acoustics. To compute the thermoacoustic effects, it is necessary to find information of the acoustic field, which will be modified by the thermodynamic effects induced by the presence of a “stack”. The interaction between thermodynamics and acoustics is very complicated. In order to simplify the analysis, we, therefore, decouple the acoustics from the thermodynamics. For a standing-wave machine, the decoupling is realized by assuming that a standing wave acoustic field exists in the resonator,

Standing-wave systems 33

without being influenced by the presence of the stack. Normally, the sound field consists of two components: standing-wave and traveling-wave component. This can be seen in appendix E. In most practical cases, the traveling-wave component is much smaller than the standing-wave component. Therefore, to make a further simplification, we assume that the traveling-wave component is negligible.

In a standing-wave system, the total energy flow along the stack 2E& is an important

parameter, which can be deduced by the analysis of the energy balance of the

system. In the sections 3.3.1 and 3.3.2, the total energy flow along the stack 2E& is

assumed to be known. Both the total energy flow along the stack 2E& and the

viscosity of the working gas have much influence on the performance of the whole system. Therefore, the computations in sections 3.3.1 and 3.3.2 are arranged under

categories by using different combinations of viscosity and total energy flow 2E&

(zero viscosity and zero total energy flow 2E& =0; zero viscosity and non-zero total

energy flow 2E& ≠0; non-zero viscosity and zero total energy flow 2E& =0; non-zero

viscosity and non-zero total energy flow2E& ≠0 ). By doing so, the performance ∆T

and temperature gradient dTm/dx as a function of stack position are investigated in a manner of step by step from simple to complicated. 3.3.1 Zero viscosity First consider zero viscosity and the so-called “short-stack” approximation: assume that the stack plates do not disturb the sound field. The assumption of “zero viscosity” makes the axial flow velocity amplitude

1u independent on y. So, 1u is uniform between two adjacent stack plates.

Furthermore, we use the “boundary-layer” approximation which means κδ20 >>y

and sl δ2>> , so that the hyperbolic tangents ( ]/)1tanh[( 0 κδyi+ ) in Eqs. (2.1.31),

(2.1.32) and (2.1.33) can be set to unity. The acoustic field inside of the resonator is

sA pxp 11 )/sin( ≡= Dp , (3.3.1)

( )( ) smA iuxapyli 101 )/cos(//1 ≡+= Dρu . (3.3.2)

Here, Ap is the pressure amplitude in the pressure anti-node. The term )/1( 0yl+

accounts for the continuity of the volume flow at the ends of the stack. With all these assumptions applied to Eq. (2.1.72), the formula to compute the energy flow is as follows:

( ) ( )dx

dTlKKyupE m

sss

s

+∏−−Γ+

∏−= 0112 11

141

εδκ

& , (3.3.3)

34 Chapter 3

where

sss

pms c

c

δρδρ

ε κ= . (3.3.4)

In Eq. (3.3.3), the ratio between the actual temperature gradient and the critical gradient is represented by

crit

mT

T∇

∇=Γ . (3.3.5)

If the conductive loss is neglected (the second term on the right hand side in Eq.

(3.3.3)) and 02 =E& , then all available acoustic power is used to establish a

temperature gradient. This maximum attainable temperature gradient is the critical temperature gradient given by Eq. (3.2.7), determined by the cyclic compression and expansion of the gas. Substituting Eq. (3.3.1), (3.3.2) into (3.2.7) yields

( ) D

x

ylc

aT

pcrit tan

/1 0+=∇ ω

. (3.3.6)

In Eq. (3.3.6), the critical temperature gradient depends on the operating frequency and the geometrical properties of the system. Assuming that the working gas is helium, the working frequency is 1000 Hz at ambient temperature, and that the thickness of the stack plate is much smaller than the working gas layer i.e.

0/ 0 ≈yl . The critical temperature gradient plot in that case is shown in Fig.3.3.1.

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x/radian wavelength

Lo

g(c

riti

cal

tem

per

atu

re g

rad

ien

t)

Figure 3.3.1: Critical temperature gradient given by Eq. (3.3.6).

Standing-wave systems 35

Remark that in the position, x/radian wavelength=π/2, where a pressure anti-node is occurring the critical temperature gradient becomes infinity due to the tan function. At this position, steep gradient may occur. In the pressure node, at x=0, the temperature gradient is zero. Zero total energy flow Now, we focus on the situation when the total energy flow given by Eq. (3.3.3) is zero, which means that all the thermal power produced by the acoustic wave is consumed via conduction by the working gas itself and by the stack plates. So a temperature gradient is established without cooling power output. Also, we are assuming the working fluid is ideal gas. The length of the stack in all computations in this chapter is not so short that the temperature gradient can be considered as constant. Therefore, the temperature gradients depend on positions. Then from Eq. (3.3.3), the temperature gradient in that case can be expressed as:

( )( ) ( )2

10

11

14 spmss

ssm

uclKKy

up

dx

dT

ρδεωωδ

κ

κ

+++= . (3.3.7)

Substitution of the acoustic pressure Eq. (3.3.1) and the particle velocity Eq. (3.3.2) into Eq. (3.3.7) leads to

( )( )DD

/2cos/2sin

xDB

x

dx

dTm

⋅+= , (3.3.8)

Where

( )( )( )

( )ωδ

ρεωκ a

ylc

pylc

alKKyB p

Ap

mss 0

220

20

/11

/1

18 +⋅

+

+++= , (3.3.9)

and ( )

ωa

ylcD p 0/1+

= . (3.3.10)

The temperature difference between the starting point of the stack and the other end of the stack is represented as:

stackstack

stack

stackstack

stack

lx

x

lx

xp

mm

xDB

ylc

adx

dx

xdTxT

++

⋅++

−==∆D

2cosln

)/1(2)(

)(0

2

.

(3.3.11) In Eq. (3.3.11) xstack is the starting point of the stack and lstack is the length of the stack. Now considered as an example case, a quarter-wave-length standing wave machine, is shown in Fig. 3.3.2. The working frequency is assumed to be 1000 Hz. The working gas is helium. The stack length is taken as 1/20 wave length, which is short enough not to significantly disturb the sound field as assumed before (“short-

36 Chapter 3

stack” approximation). The plate thickness 2l is fixed to twice the thermal penetration depth of the solid material. All the parameters of this example model are listed in table 3.3.I.

Figure 3.3.2: Geometry of the standing-wave example case.

Table 3.3.I (a) Properties of working fluid and stack of the standing-wave example case

Working gas Helium Properties

Operating frequency 1000 Hz

Mean pressure 1 bar

Acoustic pressure amplitude

1 kPa (1% of mean pressure) or

10 kPa (10% of mean pressure)

Ambient temperature 300K

Density 0.16 kg/ 3m

Thermal conductivity 0.156 w/(m*K)

Cp 5193 J/(kg*K)

speed of sound 1019 m/s

Stack plate PVC

Properties

Thermal conductivity 0.2 W/(m*K)

Density 1300 kg/3m

Cp 1500 J/(kg*K)

4/λ x

0

Hot HX Cold HX Cold HX

Standing-wave systems 37

Geometry parameters data

Thermal penetration depth (gas) 2.44e-4 m

Thermal penetration depth (solid) 5.71e-6 m

Stack length (1/20 wave length) 0.051 m

Tube length ( ¼ wave length) 0.255 m

Diameter of tube (1/20 wave length) 0.051 m

Plate thickness 11.5 µm

Table 3.3.I (b) Geometrical parameters of the standing-wave example case

Figure 3.3.3: Illustration of the stack position. The heat exchangers are ideal. The left end of the resonator is where the pressure node locates. The right end of the resonator is closed by a solid wall or a speaker, where the pressure anti-node locates. The influence of the stack-plate spacing on the performance of the system is investigated, as well as that of the stack position in the resonator tube. Here the stack position is referred to the end of the stack nearest to the open end of the tube, see Fig. 3.3.3. The half distance between the stack plates 0y is varied and set to the

values of κδ2 κδ5 κδ10 κδ20 and κδ50 . Figs. 3.3.4 and 3.3.5 show the acoustic

pressure amplitude, and the acoustic particle velocity amplitude at different positions inside the resonator tube for the case of a drive ratio of 10%. Obviously, the pressure amplitude is maximum at the right end of the resonator (x=0.255m), whereas the velocity amplitude is maximum at the left end of the tube (x=0).

x Stack position xstack

38 Chapter 3

0.00 0.05 0.10 0.15 0.20 0.25 0.300

2000

4000

6000

8000

10000

Aco

ustic

pre

ssur

e am

plitu

de (

Pa)

x (m)

pm=1 bar, p

A=10 kPa (10%)

Figure 3.3.4: Acoustic pressure amplitude distribution inside the resonator, drive ratio=10%.

0.00 0.05 0.10 0.15 0.20 0.25 0.300

10

20

30

40

50

60

70

Aco

ustic

par

ticle

vel

ocity

am

plitu

de (

m/s

)

x (m)

pm=1 bar,p

A=10 kPa (10%)

Figure 3.3.5: Acoustic particle velocity amplitude distribution inside the resonator, drive ratio=10%

Standing-wave systems 39

0.00 0.05 0.10 0.15 0.20 0.25 0.300

2000

4000

6000

8000

10000

12000

14000

16000

dTm/d

x (K

/m)

x (m)

critical temperature gradient y

0/δ

κ=2.0

y0/δ

κ=5.0

y0/δ

κ=10.0

y0/δ

κ=20.0

y0/δ

κ=50.0

Figure 3.3.6: Calculated temperature gradient on the stack at various positions and for different stack plate spacing, drive ratio=10%.(Symbols are for indication of different ratios of y0/δκ.) Figure 3.3.6 shows that the temperature gradient decreases with increasing thickness of the gas layer. This is due to the increasing thermal conduction by the gas and plates while the amount of effective working gas is less. This effective amount of gas basically is in the layer of one thermal penetration depth away from the plate. This amount becomes less when the stack plate spacing increases because the number of plates in a given cross section reduces. For any gas-layer thickness in Fig. 3.3.6, the temperature gradient is always below the critical temperature gradient. This is because in the critical gradient thermal conduction is not taken into account. As the stack plate spacing decreases, the maximum local temperature gradient moves towards the right end of the resonator tube where the pressure anti-node lies. If the gas layer is thinner, the ratio of conductive heat loss to the power generated by the gas becomes smaller. So, the situation will be closer to that of the critical temperature gradient. In Fig.3.3.6, the curve of the thinnest gas layer approaches the critical temperature curve closest. The critical temperature gradient reaches infinity at the closed end of the resonator tube. This can be explained by Fig. 3.3.5: the acoustic particle velocity is zero at the closed end of the resonator tube, whereas the acoustic pressure amplitude is maximum (Fig. 3.3.4). Therefore,

40 Chapter 3

the temperature gradient goes to infinity at this position. In the case of conductive flow 1<Γ and dxdTm / becomes zero at the closed end since 01 =u (Eq. 3.3.3).

Since the temperature gradient in the stack material is equal to that in the gas, the conductive heat flow can be expressed as

dx

dTlKKy m

s ⋅+Π )( 0 . (3.3.12)

Per unit of cross-sectional area the conductive heat flow equals to

dx

dT

ly

lKKy ms

++

0

0 . (3.3.13)

The temperature difference that is established between the two ends of the stack can be determined by integrating the gradient as depicted in Fig. 3.3.6 along the length of the stack. The resulting temperature difference for the example case of table 3.3.I is shown in Fig. 3.3.7.

0.00 0.05 0.10 0.15 0.200

50

100

150

200

250

300

350

400

Tem

pera

ture

diff

eren

ce o

ver

the

stac

k (K

)

Stack position (m)

y0/δ

κ=2.0

y0/δ

κ=5.0

y0/δ

κ=10.0

y0/δ

κ=20.0

y0/δ

κ=50.0

Figure 3.3.7: Calculated temperature difference over the stack for different stack positions and different stack plates spacing, drive ratio=10%.(Symbols are for indication of different ratios of y0/δκ.) The temperature span over the stack reaches its maximum value at some position near the closed end of the resonator tube. As the gas layer becomes thinner, the

Standing-wave systems 41

stack starting position for maximum temperature difference over the stack moves towards the closed end of the resonator tube. The thinner the gas layer, the larger the temperature span. Although the temperature span is large at positions near the closed end, the properties of the gas and solid are assumed to be independent on temperature. So, all the physical parameters remain constant. These evaluations were also performed for a drive ratio of 1%. In that case, the temperature gradient developed in the stack obviously is much smaller as shown in Fig.3.3.8. In addition the peak becomes wider. This is due to the fact that as Ap

increases the parameter B in Eq. (3.3.8) becomes much smaller and the denominator in the gradient is largely determined by the cosine term. For the same reason, the temperature difference between the ends is smaller at smaller drive ratio, see Fig. 3.3.9.

0.00 0.05 0.10 0.15 0.20 0.25 0.300

200

400

600

800

1000

1200

1400

1600

dTm/d

x (K

/m)

x (m)

critical temperature gradient y

0/δ

κ=2.0

y0/δ

κ=5.0

y0/δ

κ=10.0

y0/δ

κ=20.0

y0/δ

κ=50.0

Figure 3.3.8: Calculated temperature gradient in the stack at various positions and for different stack plate spacing, drive ratio=1%.(Symbols are for indication of different ratios of y0/δκ.)

42 Chapter 3

0.00 0.05 0.10 0.15 0.200

10

20

30

40

50

60

Tem

pera

ture

diff

eren

ce o

ver

the

stac

k (K

)

Stack position (m)

y0/δ

k=2.0

y0/δ

k=5.0

y0/δ

k=10.0

y0/δ

k=20.0

y0/δ

k=50.0

Figure 3.3.9: Calculated temperature difference over the stack for different stack positions and different stack plates spacing, drive ratio=1%.

Non-Zero total energy flow In this section, the more general situation with non-zero total energy flow is investigated. Substitution of Eq. (3.3.1) and (3.3.2) into Eq. (3.3.3), the temperature gradient is expressed as:

( ) ( )( )

( )D

&D

/2cos/1

18/2sin 2

0

2

xDB

pyl

aEx

dx

dT A

ms

m

⋅+++⋅

∏−

=

ρεδκ .

(3.3.14)

Obviously, the temperature gradient reduces as 2E& increases. Part of the acoustic

power is “consumed” to establish the energy flow 2E& and is no longer available to

generate a temperature profile. Integration of the temperature gradient Eq. (3.3.14) along the stack plate yields the temperature distribution: If ( ) 0/2cos ≥+ DxBD

( )

⋅++

−==∆ ∫+

D

xDB

ylc

adx

dx

dTxT

p

Lx

x

mm

stackstack

stack

2cosln

/12)(

0

2

( )( )

( )( )

stackstack

stack

lx

xA

ms

xBD

xDBarctg

DBpyl

aE+

+−

−⋅

++⋅

∏−

D

DD&

/2cos

/2sin2/

/1

18 22

2220

2 ρεδκ

(3.3.15a)

If ( ) 0/2cos <+ DxBD

Standing-wave systems 43

( ) ( )[ ]D/2cosln/12

)(0

2

xDBylc

adx

dx

dTxT

p

x

x

mm

start

⋅++

−==∆ ∫

( )( )

( )( )

stackstack

stack

lx

xA

ms

xBD

xDBarctg

DBpyl

aE+

+−+⋅

−⋅

++⋅

∏−

D

DD&

/2cos/2sin2/

/118 22

2220

2 πρεδκ

.

(3.3.15b) Here, lstack is the stack length. The influence of different total energy flows on the system performance is investigated, for the example case of table 3.1.I, again at drive ratio of 1% and 10%. In this section, the influence of stack plate spacing is not further investigated. The stack plate spacing is fixed as four times the fluid’s thermal penetration depth, i.e. 2/0 =ky δ and so is the thickness of the solid stack

plate, i.e. 2/ =sl δ .

In order to choose a realistic value, the energy flow is related to the maximum conductive heat flow that may be present in the stack for the case of zero-total-energy-flow. If we denote the reference value as “Max”, the energy flow values used in the following computation is set to 0, 25%, 50% and 75% of “Max”.

( )max

0

+∏=dx

dTlKKyMax m

s (3.3.16)

in the case of 02 =E& , at the stack positions where the local temperature gradients

are maximum. In the case of 10% drive ratio and 2/0 =ky δ , this Max value can be obtained

from corresponding computation of the above “zero total energy flow” part, (they are obtained at positions near the closed end of the tube) which is 1.88 W for this tube geometry and diameter. In the case of 1% drive ratio, the Max value is 0.1684 W. According to the computation in the foregoing section, Max=1.88 W in the case of drive ratio 10%, and Max=0.1684 W for the case of 1%. The plots are shown below. Similar to Figs. 3.3.6 and 3.3.8: the local temperature gradient increases gradually and reaches its peak value at some position near the closed end, then drops dramatically as the stack moving towards the close end. The temperature gradient deteriorates only minor in the left part of the tube (0-0.20 m), but there is a fairly dramatic drop in the region at x=0.225 m and further. The reason is that due to the additional load, there must be a temperature drop at the end of the stack causing a large negative dT/dx at the last 1% of the stack end. When the total energy flow increases, the peak values drop and the corresponding stack position is moving away from the closed end, the pressure anti-node.

44 Chapter 3

0.00 0.05 0.10 0.15 0.20 0.25 0.30-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

12000

dTm/d

x (K

/m)

x (m)

E2=0.0

E2=Max*25%

E2=Max*50%

E2=Max*75%

Figure 3.3.10: Calculated temperature gradient in the stack at various stack positions and different total energy flows, drive ratio=10%.

0.00 0.05 0.10 0.15 0.20 0.25 0.30-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

dTm/d

x (K

/m)

x (m)

E2=0.0

E2=Max*25%

E2=Max*50%

E2=Max*75%

Figure 3.3.11: Calculated temperature gradient in the stack at various stack positions and different total energy flows, drive ratio=1%.

Standing-wave systems 45

0.00 0.05 0.10 0.15 0.200

50

100

150

200

250

300

350

400

Tem

pera

ture

diff

eren

ce o

ver

the

stac

k (K

)

Stack position (m)

E2=0.0

E2=Max*25%

E2=Max*50%

E2=Max*75%

Figure 3.3.12: Calculated temperature difference over the stack at different stack positions and different total energy flows, drive ratio=10%.

0.00 0.05 0.10 0.15 0.20-10

0

10

20

30

40

50

60

Tem

pera

ture

diff

eren

ce o

ver

the

stac

k (K

)

Stack position (m)

E2=0.0

E2=Max*25%

E2=Max*50%

E2=Max*75%

Figure 3.3.13: Calculated temperature difference over the stack at different stack position and different total energy flows, drive ratio=1%.

46 Chapter 3

In Fig. 3.3.11, at lower drive ratios, the effect can be seen more profoundly. There is a large effect of the energy load at lower pressure amplitudes. It is because the dT/dx|max approaches dT/dx|critical at high pressure amplitudes (see Fig.3.3.6 curves of 2/0 =ky δ and critical temperature gradient), but dT/dx|max approximates

1/10·dT/dx|critical at low pressure amplitudes (see Fig.3.3.8 curves of 2/0 =ky δ and

critical temperature gradient). But the energy grows at the square of pressure amplitude, meaning that the energy of 10% drive ratio case is 100 times larger than that of the 1% case. A similar trend can be found in the figures of temperature difference over the stack, Fig. 3.3.12 and 3.3.13. Again, the drive ratio has a big influence on the performance of the system. As shown in Fig.3.3.10 and 3.3.11, the local temperature gradient is negative at the

closed end of the resonator tube. That is due to the 2E& value we assigned. The

substitution of Eq. (3.2.7) and (3.3.5) into (3.3.3) yields:

( )( ) ( )

dx

dTlKKyup

T

dx

dTucE m

sss

s

mm

s

spm +∏−

+∏+

+∏−

= 011

2

12 14

114 ε

βδωε

ρδκ

κ&

(3.3.17) In the presented graphs, the energy flow was related to the reference value “Max” as

( ) ( )[ ]0max,

00max,

02

22 ==

+∏⋅=

+∏⋅=E

ms

E

ms dx

dTlKKyc

dx

dTlKKycE

&&

&

(3.3.18) in which cwas set to 0, 0.25, 0.5, and 0.75.

At the closed end of the resonator tube, the particle velocity is zero, i.e. 01 =su ,

and thus the first two terms on the right hand side in Eq. (3.3.17) are zero. Then, the gradient at the closed end can be expressed by substituting Eq. (3.3.18) into (3.3.17) as

0max, 2 =

−=E

m

closeend

m

dx

dTc

dx

dT

&

, (3.3.20)

which becomes negative for positive values of c . A similar explanation can be made for the negative values at x=0 in Fig.3.3.10 and 3.3.11. (which is hardly visible, because of the extended scale.) 3.3.2 General analysis with viscosity included In this section, many of the assumptions made before are eliminated. The assumption that a standing-wave is sustained inside the resonator tube without disturbance from the presence of the stack is maintained. The sound wave is

Standing-wave systems 47

described by Eq. (3.3.1) and (3.3.2) and now viscosity is included. The temperature dependence of the various parameters is also taken into account. Ideal gas property is assumed, i.e. 1=βmT . The ratio εs is neglected, i.e. εs≈0. As shown in the

definition of εs by Eq.(2.1.34), neglecting εs is realistic because the solid material heat capacitance is much larger than that of working gas. Substitution of Eq. (3.3.1) for the pressure wave and (3.3.2) for the velocity into the general total energy equation (2.1.72) yields

( )sA

m

p

A

mm

lKKyff

fxpcy

f

yl

f

ffxpy

y

lE

dx

dT

+−

+−+⋅

⋅−−

+

+−−

++

∏=

0

2

2

2

3

02

2

0

20

0

2

)1()

~(~

Im)cos()1(21

1

)1)(~

1(

)~

(Im)

2sin(

41

σσρω

σωρ

νκν

ν

ν

νκ

DD

DD

&

(3.3.21) This equation is solved numerically using the following procedure: 1. The stack plate is divided into elementary cells 2. The hot end of the stack is sustained at 300K 3. Computation starts from the hot end of the stack 4. For the first computational cell at the hot end of the stack, the temperature gradient is calculated with Eq. (3.3.21) based on gas properties 5. A new temperature for the immediate next computation cell is obtained via the temperature gradient. 6. This computation is iterated to the other end of the stack. The configuration that is considered is the same as discussed before and summarized in table 3.3.I a and b with a drive ratio of 10%. Only now, all parameters are considered temperature dependent. The thermal conductivity of helium is

Km

WTK 65.00038.0= . (3.3.22)

And the viscosity can be expressed as 64.0664.0 /1052.0 KsPaT ⋅×= −µ . (3.3.23)

Zero total energy flow The temperature difference across the stack ∆Tstack is calculated for different positions of the stack inside the tube. In contrast to the previous zero-viscosity case, we now will also investigate the effect of the operating frequency, or in other words of the tube length. In addition to the 1 kHz considered so far, we now will evaluate temperature profiles for 2 kHz and 3 kHz as well. When the operating

48 Chapter 3

frequency increases to 2 and 3 kHz from 1 kHz, the resonator tube length has to be decreased to 1/2 and 1/3 of the length of 1 kHz correspondingly.

0.00 0.05 0.10 0.15 0.20 0.250

50

100

150

200

250

300

∆Tst

ack (

K)

Stack position (m)

1 kHz 2 kHz 3 kHz

Figure 3.3.14: Temperature difference over the stack at various stack positions and zero total energy flow, drive ratio=10%. The temperature difference over the stack in these three computational cases is shown in Fig. 3.3.14. In the computations, the following parameters are fixed:

0.2/0 =ky δ and 0.2/ =sl δ ; stack length/wave length=1/20; resonator tube

diameter/wave length=1/20; the resonator tube length/wave length=1/4 (a quarter wave-length cooler). The temperature of the hot end of the stack is fixed at 300K in all the computation cases. Apparently, at higher operating frequency, the attainable temperature difference decreases. If the temperature difference over the stack is divided by the temperature of the hot end of the stack, and the stack position is divided by the length of resonator tube,

hot

coldhot

T

TTT

−=* and tube

stackL

XX =* (3.3.24)

then the dimensionless plots are shown in Fig.3.3.15. The maximum temperature difference across the stack decreases when the refrigerator is smaller or the operating frequency is higher.

Standing-wave systems 49

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.0

0.2

0.4

0.6

0.8

1.0

∆Tst

ack/T

Hot

Dimensionless stack position x*=xstack

/resonator length

1 kHz 2 kHz 3 kHz

Figure.3.3.15: Dimensionless temperature difference over the stack at different stack positions along x with zero total energy flow, drive ratio=10%. However, in the dimensionless figure, most part of the three curves overlap, whereas their peak parts diverge. Perhaps, it can be explained as follows:

1. According to Eq. (3.3.21), using 02 =E& , the temperature gradient can be

expressed as

+−

++

+−+⋅

⋅−

++−+

=

0

2

2

2

2

2

)1)(1(

)1)(~

(~

Im)cos()1(2

1

)1)(1()

~(~

Im)2

sin(4

1

y

lKK

f

fff

fxpc

fff

xp

dx

dT

s

s

sA

m

p

s

A

mm

σε

ε

σρωω

σεωρ

κ

ννκ

ν

νκν

DD

DD

(3.3.25) The imaginary parts of the complex combination of Rott’s functions are always negative in sign (to be shown in the next section). Rewriting it as:

)()(1

0

2

1

02

1

y

lKK

CC

y

lKKC

C

dx

dT

ss

m

++=

+−⋅−

−=

ωω

, (3.3.26)

where 1C and 2C are positive functions related to the numerator and denominator

in Eq. (3.3.25). They remain constant when changing the frequency. 2. The length of the stack scales with ω/1 .

50 Chapter 3

3. Using the results from 1 and 2, the total temperature difference across the stack scales with

)(

1

)(0

2

1

0

2

1

y

lKKC

C

y

lKK

CC

Tss

m

++=⋅

++∝∆

ωωω

. (3.3.27)

Thus, the temperature difference over the stack becomes less when the frequency increases. It is because the conductive loss becomes larger in the proportion of the total energy flow while scaling down. Non-Zero total energy flow An energy flow is assumed to be present along the stack that is related to the maximum conductive flow in case of zero energy flow—“Max”. As we did before, the energy flow is set to 0, 0.25, 0.5 and 0.75 of this “Max” value. The resulting temperature difference over that stack for 1 kHz operating frequency is plotted in Fig.3.3.16. Here, the reference value Max=1.732 W. From Fig. 3.3.16, we can see that the temperature difference over the stack drops if an energy load is applied to the stack at the same stack position. The temperature drop becomes obvious when the stack is closer to the close end. It is the same trend observed in the zero-viscosity and non-zero total energy flow computation in section 3.3.1.

0.00 0.05 0.10 0.15 0.20 0.250

50

100

150

200

250

300

∆Tst

ack (

K)

Stack position (m)

E2=0.0

E2=Max*25%

E2=Max*50%

E2=Max*75%

Figure 3.3.16: Calculated temperature difference over the stack at different stack positions along x with different total energy flows, drive ratio=10%.

Standing-wave systems 51

It is apparent that the temperature span across the stack has a dependency on the total energy flow along the stack. To see this effect, a plot of the temperature difference over the stack at different total energy flows is shown in Fig. 3.3.17. In this plot, the computation is made for the cases that the stack is fixed at positions near the closed end (xstack=0.153 m), and a position somewhere between the closed end and the pressure node (xstack=0.076 m).

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

∆Tst

ack (

K)

Total energy flow E2 (W)

xstack

=0.153 mx

stack=0.076 m

Figure 3.3.17: Temperature difference over the stack at a fixed stack position, where the maximum temperature difference over the stack is achieved, with various total energy flows along the stack, drive ratio=10%. Fig.3.3.17 shows that the temperature span across the stack decreases rapidly at increasing total energy flow. At some point, the maximum temperature span across the stack becomes zero, which means that no temperature drop at the cold end is available anymore with the specific amount of total energy flow along the stack. The curve for the stack position near the closed end decreases rapidly, while the other one for the position between the closed end and the pressure node decrease slowly. 3.3.3 General analysis of “TAC” Introduction A “thermoacoustic couple” (TAC) refers to the simplest class of thermoacoustic devices with a stack consisting of short plates without heat exchangers at any end, and was introduced by Wheatley et al [41]. The analysis of a TAC is helpful in

52 Chapter 3

better understanding the basic thermoacoustic heat transport and also the coupling between stack and its adjacent heat exchangers. Much work is done on TACs in experiments and numerical simulation. Wheatley et al [41] measured the temperature difference developed across a TAC as a function of its position in an acoustic standing wave, with low drive ratio. They also developed a theoretical expression to predict the temperature difference, stating from the point that the heat transferred via entropy flow in the gas is returned by diffusive conduction. It was assumed that the heat transfer between the couple and its surroundings was negligible, because thermal insulation was applied in their set-up. Later, Atchley et al [65] used similar TACs and extended the measurements to higher drive ratios up to 2%. By comparing the measurement results with calculations based on the theoretical expression developed by Wheatley et al [41], it was found that the agreement was good for drive ratios below approximately 0.4%. At higher drive ratios, the temperature difference along the stack was increasingly overestimated. In some later experimental investigations [66, 67], the predictions based on linear theory could not match the measured values either. Large discrepancies of up to 300% were reported. These deviations from linear thermoacoustic theory attracted much attention and a succession of works was conducted to explore the attributions. As stated by Piccolo and Cannistraro [67], possible causes can be: 1 non-linear effects due to the presence of harmonics higher than the fundamental; 2 turbulence and vortex generation; 3 heat leak to the surroundings; 4 additional thermal load to the cold end carried by acoustic streaming caused by heat generated in viscous losses along the resonator wall; 5 heat flow in transverse direction; 6 heat transferred to the ends of the stack in axial direction due to imperfect thermal isolation. Many numerical simulations [68, 69], in which perfect thermal isolation from the surroundings was assumed, were done to evaluate the heat flow in the transverse direction. In standard linear thermoacoustic theory, this heat flow is neglected by stating that the solid wall and adjacent gas have the same time-averaged temperature. In the numerical simulation in reference [68], a discrepancy up to 25% could be explained, which is only a small part of the large practical discrepancy between linear theory and experimental data mentioned above. Paul Aben experimentally made a detailed study about these nonlinear effects on a set-up similar to “TAC” in his PhD work [84]. He foused on the 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; and streamings taking place in a standing-wave device. His PIV visualization on the vorticity pattern, and jet streaming, natural convection also by PIV technique has showed a complicated situation at the ends of the stack.

Standing-wave systems 53

In the next section, an experimental apparatus similar to the so-called “TAC” was used, as shown in Fig. 3.3.18. It is driven by a loudspeaker, which generates acoustic power into the resonator tube at one end. The other end of the resonator is closed by an end plug. A stack is placed inside the resonator tube and its position can be adjusted by two steel tubes. The stack is housed in a stack cage and consists of many parallel plates separated by fishing lines. There are no heat exchangers at both ends of the stack. An analytical method for computing the temperature distribution along the stack is proposed. Although no thermal insulation for the system was employed, heat exchange with the environment is neglected in the proposed analytical model.

Figure 3.3.18: Schematic drawing of the apparatus similar to “TAC” and used for the experiment. Theory developed for the apparatus The acoustic field inside the resonator tube is described by

)/cos(1 DxpA=p and ( ) )/sin(/1 Dxapi mA ρ=u (3.3.28)

The acoustic field inside the stack is described by

)/cos(1 DxpA=p and ( )( ) )/sin(//1 01 Dxapyli mA ρ+=u (3.3.29)

where Ap is the maximum acoustic amplitude.

Ideal gas is assumed, i.e. 1=βmT , and sε is neglected, i.e. 0=sε . Substitution of

Eq. (3.3.29) into (2.1.72), in which the pressure gradient is replaced by the relation for the acoustic velocity, in that case leads to

+−−−

⋅−⋅Π

=

)1)(~

1(

~1Im

2sin

4

20

2 σωρ νf

ffxpy

A

AE vkA

mgas

res

DD

&

( ) dx

dTfff

xpcy

f

AAmvk

vA

m

pgasres ⋅

+−+⋅

⋅⋅−

Π⋅

−+

σσρων 1

~~

Imsin121

/ 22

2

30

2

DD

dx

dTK

AKy m

ssolid ⋅

∏+Π− 0 , (3.3.30)

Loudspeaker

Stack

End plug Resonator tube

54 Chapter 3

where Ares is the cross sectional area of resonator tube and Agas is the cross sectional area of gas in the stack. Asolid is the total cross sectional area of all solid material, including stack cage, stack plates, and fishing lines, is given by

fishfishcagesolid ANAlA ⋅++∏= , (3.3.31)

where Acage and Afish are the cross sectional area of the stack cage and single fishing line, respectively. П is the total effective plate length in cross sectional view. Hence, the temperature gradient is

+−−−

⋅⋅

+

Π=

)1)(~

1(

~Im

2sin

4

202

σωρ νf

ffxpy

A

AE

dx

dT vkA

mgas

resm

DD

&

( )

+−+⋅

⋅⋅−

− σσρων 1

~~

Imsin121

/ 22

2

30

2

vkv

A

m

pgasres fff

xpcy

f

AA

DD

1

00

+− s

gas

solid KyA

AKy . (3.3.32)

In order to use Eq. (3.3.32) to obtain the temperature gradient, it is necessary to

find the total energy flow along the stack 2E& . In the present situation of

experimental apparatus, it is difficult to obtain the exact number for 2E& . A

numerical simulation method could be time expensive and therefore is not an

option for the present work. In order to obtain an estimation for 2E& by using the

available measurement devices, we assumed that the energy exchange with the environment along the whole resonator tube is negligible. Since the thermometers were installed at the centers of both ends of the stack, they are far from the wall of the resonator tube, the temperatures are weakly influenced by the environment. Therefore, the assumption is acceptable. Thus the total energy flow into the resonator tube is the total acoustic power from the speaker. In the experiments, the total acoustic power from the speaker can be calculated by the measured acoustic

pressure and volume velocity. Thus, the total energy flow 2E& along the stack

without any loss is given by:

peakersspeakeracoustic UpPE 112 5.0 ⋅⋅==& , (3.3.33)

where Pacoustic is the total acoustic power from the speaker into the resonator tube. It is calculated by the acoustic pressure p1speaker from microphone and volume velocity U1speaker at the interface between the resonator tube and speaker diaphragm. U1speaker is obtained by using the measurement from the accelerometer mounted on the speaker diaphragm. As will shown later, here p1speaker and U1speaker are in phase guaranteed by equipment measurement. Therefore, the phase terms are not needed

in Eq.(3.3.33). As a comparison, that total energy flow 2E& is zero, the case of

Standing-wave systems 55

2E& =0, (3.3.34)

is also used for the later computation. We name the computation by using Eq. (3.3.33) as method 1 “M1”, and by Eq.(3.3.34) as method 2 “M2”, which were used in the figures later. Acoustic power losses in the “TAC” The “TAC” system is divided by the stack into three sections, as shown in Fig. 3.3.19.

Figure 3.3.19: Three loss sections of the resonator tube. The three sections are: zone 1, the resonant tube segment from the loudspeaker to the nearest end of the stack; zone 2, the stack zone; zone 3, the resonant tube segment from the other end of the stack to the plug. Losses in these three sections are computed by two methods in this work. (1) By solving the wave equation in section 1 and 3: The acoustic field can be determined by solving the acoustic wave equation in the tube sections 1 and 3 by using the pressure and velocity amplitudes at the two ends of the resonator tube. Using the wave equation (2.1.57) in chapter 2, and assuming that the temperature gradients in sections 1 and 3 are zero, and sε is zero.

The wave equation in section 1 and 3 becomes:

( ) 0)1()1(121

2

2

2

1 =−+−+dx

df

af

pp νκ ω

γ . (3.3.35)

The Rott’s functions are evaluated with the parameter of the resonator tube. The general solution of Eq. (3.3.35) is

xixi ee 21211

kk CCp ⋅+⋅= , (3.3.36)

where 1C and 2C are complex constants, only depending on the boundary

conditions, and 2,1k are two complex square roots

Stack Plug

Sound wave

1 Loss 1

2 3 Loss 2 Loss 3

56 Chapter 3

ν

κγωf

f

a −−+

±=1

)1(12,1k . (3.3.37)

Define real wave number akr ω= , (3.3.38)

and complex number

ν

κγξχf

fi

−−+

=+1

)1(1. (3.3.39)

Thus, the general solution can be rewritten as xikxkxkixk rrrr eeee χξχξ −− ⋅⋅+⋅⋅= 211 CCp . (3.3.40)

At x=0, where the speaker is located, the pressure and velocity are measured

speakerxp101 =

=p and speakerx

u101 ==

u (3.3.41)

By these boundary conditions, the complex constants are obtained

+⋅

−⋅

−=ξχ

ρ

ν if

uap peakersm

speaker

1

12

1 111C , (3.3.42a)

+⋅

−⋅

+=ξχ

ρ

ν if

uap peakersm

speaker

1

12

1 112C . (3.3.42b)

Thus, by substitution of Eq. (3.3.42a) and (3.3.42b) into Eq. (3.3.40), the pressure at any location in section 1 is known. Similarly, by using the boundary conditions at the end plug:

eLxp11 =

=p and 011 ==

= eLxuu (3.3.43)

the pressure distribution in section 3 is obtainable by the same method. By using Eq. (3.3.40), the pressures and velocities at both ends of the stack, i.e. xL

and xR in Fig.3.3.20, are obtainable, indicate them as L1p , L1u , R1p ,and R1u ,

respectively.

Figure 3.3.20: Acoustic pressures and velocities in stack section. The losses in the three sections with area A are

[ ]LLAupALoss 1111~Re

21

21

1 up⋅−⋅=speakerspeaker

, (3.3.44a)

stack

xL xR

p1L u1L

p1R u1R

Standing-wave systems 57

[ ] [ ]RRLL AALoss 1111~Re

21~Re

21

2 upup ⋅−⋅= , (3.3.44b)

[ ]RRALoss 11~Re

21

3 up⋅= . (3.3.44c)

In the figures for the loss computation in the next section, curves using this method are indicated by “(1)”. (2) By using the equation in chapter 2 The acoustic power variation in an elementary distance dx is given by Eq. (2.2.12) in chapter 2. With assumptions made before, which are 1=βmT and sε neglected,

i.e. 0=sε , Eq.(2.2.12) becomes:

( ) ( ) ( )

−−

+−−

∏−= κν

νρ

γρω f

af

fy

dx

Wd

m

m Im1

Im~12

12

2

12

2

10

2pu&

( )

−−

−∏+ 11

0 ~)

~1(

)~~

(Re

112

1up

ν

νκ

σ f

ff

dx

dT

T

y m

m

. (3.3.45)

In sections 1 and 3, the temperature gradients are assumed close to zero. Therefore, the acoustic power losses in sections 1 and 3 are computed with

( ) ( ) ( )

−−

+−−

−= κν

νρ

γρω f

af

fA

dx

Wd

m

mres Im

1Im~

121

2

2

12

2

12pu&

. (3.3.46)

The Rott functions fν and fκ are evaluated with the parameters of the resonater tube. In section 2, which contains the stack, the acoustic power losses is computed by using Eq. (3.3.45). The temperature gradient is computed by using Eq. (3.3.32) and (3.3.33). In the computation, the losses in three sections are numerically integrated by using Eq. (3.3.45) or (3.3.46). In the figures for the loss computation in the next section, curves using this method are indicated by “(2)”.

58 Chapter 3

3.4 Experimental results In the next sections, the model, used for theoretical analysis about scaling-down a standing-wave thermoacoustic refrigerator, is validated by experiments. An experimental set-up was built, consisting of a resonator tube with a movable plug at one end, a loudspeaker connected to the resonator by flanges, and stacks with plates spacings at 0.2mm, 0.4 mm, and 0.6 mm. The stack can also be displaced by steel tubes attached on it. These steel tubes contain the wiring of thermometers placed on the stack. Microphones, thermometers, accelerometer and lock-in amplifiers are used for necessary data acquisition. This set-up does not have heat exchangers at the ends of the stack. So it is similar to a thermoacoustic couple (TAC). The main function of this experiment is to validate the model, instead of achieving large temperature drop or cooling power. The performances at various stack positions are measured. Three stacks spacing at 0.2mm, 0.4 mm and 0.6 mm are used. The effects of different drive ratios, 0.5%, 1% and 2%, are investigated. 3.4.1 Experimental set-up A schematic illustration of the standing-wave TAC is shown in Fig.3.4.1 and photographs are given in Fig. 3.4.2 and 3.4.3. The set-up consists of a loud-speaker, a resonator tube and a parallel-plate stack installed in a thin walled cylinder. The length of the resonator tube can be adjusted by means of the plug.

Figure 3.4.1: Illustration of the standing-wave resonator tube. The loudspeaker is housed in a cylindrical plastic housing, which prevents acoustic energy from leaking out to the surroundings. It is a commercial moving-coil loudspeaker, Dynaudio loudspeaker type D54 AF. An accelerometer (Kistler 8614A1000M1, dev. is 1.67% in the frequency range of 300-600 Hz) is installed on the oscillatory diaphragm of the speaker. The voltage and the current to drive the speaker are also measured with lock-in amplifiers (PAR

Loudspeaker with an accelerometer

Stack in the stack holder

Plug

Microphone 1

Resonator tube

Sound wave

Thermometers Steel tubes

Microphone 2 Steel rod

Standing-wave systems 59

5210). The resonator tube is made of transparent perspex, and has an inner diameter of 25.7 mm and length of 546 mm. The loudspeaker diaphragm was adapted, so that acoustic energy is radiated into the resonator tube. The speaker is connected to one end of the resonator tube by flanges. The other end of the resonator tube is closed with a plug. The plug can be displaced by pulling a steel rod forth or backward. There are two locations for measuring the acoustic pressure. One microphone (microphone 2, ENDEVCO 8510B-5, the uncertainty is ±0.2% Full scale, ±69.6 Pa) is flush mounted at the end of the plug, facing the stack. The other one (microphone 1, ENDEVCO 8510B-2, the uncertainty is ±0.2% Full scale, ±27.6 Pa) is flush mounted at the wall of the resonator tube, close to the speaker diaphragm. So the acoustic input can be measured. The pressure signals of the two microphones are measured with lock-in amplifiers (PAR 5210). A stack of parallel plates is housed in a thin walled cylinder. The stack position can be changed by pulling steel tubes, which are attached to the stack holder. The temperatures at both ends of the stack are measured by two resistance thermometers (Jumo Pt 1000), and the resistance can be read from the “LakeShore” 218 temperature monitor. The working gas in the system is normal ambient air at fixed pressure. There are no heat exchangers installed at the ends of the stack. A schematic diagraph of the whole set-up is given in Fig. 3.4.2a. The measurement devices are labeled with numbers in Fig. 3.4.2b and listed as follows:

1. 15 MHz Function/Arbitrary waveform generator, Hewlett Packard 33120A 2. System DC power supply, Agilent 6614C 0-100V/0-0.5A 3. System DC power supply, Agilent 6614C 0-100V/0-0.5A 4. Power supply, Delta Elektronika E015-2 5. Lakeshore 218 temperature monitor 6. 224 programmable current source, Keithley 7. speaker amplifier 8. 2*150 watts linear precision power AMP, Dynacord L300 9. Model 5210 Lock-in Amplifier 0.5Hz-120kHz, EG&G Princeton Applied

Research 10. Two channel Digital Real-Time Oscilloscope, Tektronix TDS210

The photo of the part of standing-wave system is given in Fig. 3.4.3.

60 Chapter 3

Figure 3.4.2a: Schematic diagraph of the experimental setup and measurement devices.

PC

Lock-in Am

plifier

Lock-in Am

plifier

Lock-in Am

plifier

Lock-in Am

plifier

Lock-in Am

plifier

p p a V I

Plug

Microphone 1

Sound wave

Thermometers

Microphone 2

Lakeshore 218 temperature monitor

T

Power amplifier

Wave form generator

Standing-wave systems 61

Figure 3.4.2b: Photo of the experimental setup: on the table, resonator and loudspeaker set-up; on the left are electronics rack with lock-in amplifiers and other meters.

1 2,3

4 5 6 7

8

9

10

62 Chapter 3

Figure3.4.3: Photo of the resonator and loudspeaker.

Loudspeaker with an

accelerometer inside

Microphone1

Stack in the

stack cage R

esonator tube

Therm

ometers

Microphone 2

Plug

Standing-wave systems 63

The stacks The parallel-plate stacks consist of parallel plates made from Mylar material, spaced by fishing line spacers glued between the plates. In the experiments, the three spacing between the plates are 0.2 mm, 0.4 mm and 0.6 mm respectively. The stacks are 21 mm long, with diameter of 22.7 mm and housed in a stack holder. The stack holder is a 30 mm long, plastic cylinder, with a wall thickness of 1.5 mm. The cross section of the stack is shown in Fig. 3.4.4. The photographs of the stacks are given in Figs. 3.4.5 and 3.4.6, and all relevant parameters are listed in table 3.4.I.

Figure 3.4.4 Illustration of the stacks.

Figure 3.4.5: Photo of the 0.6 mm, 0.4 mm and 0.2 mm-spacing stacks in the stack holder (cross view).

Fishing line

Mylar plate

64 Chapter 3

Figure 3.4.6: Photo of the 0.4 mm-spacing stack in the stack holder (longitudinal view). Stack parameters 0.6 mm spacing 0.4 mm spacing 0.2 mm spacing

Half fluid layer thickness 0y 0.3 mm 0.2 mm 0.1 mm

Half solid plate thickness l

0.05 mm 0.05 mm 0.05 mm

Perimeter of the stack plates Π

1.08 m 1.55 m 2.65 m

Isobaric specific heat

pc 1.01 E3 J/(kg·K) 1.01 E3 J/(kg·K) 1.01 E3 J/(kg·K)

Diameter of the resonant tube

25.7 mm 25.7 mm 25.7 mm

Area of the speaker

7.1E-4 m² 7.1E-4 m² 7.1E-4 m²

Table3.4.I. Parameters used in computation of the experiment by utilizing the model.

Standing-wave systems 65

Resonance frequency Before conducting the thermoacoustic experiments on the stacks, introductory experiments were performed to investigate the resonance frequency. Two lock-in amplifiers are used to measure the acoustic pressure (from microphone 1) and the acceleration of the diaphragm of the speaker. The resonance frequency is defined as the frequency at which the phase difference between the acoustic pressure and the acoustic volume velocity at the position of diaphragm of the speaker is zero. Since the acceleration is ninety degrees out of phase with respect to the local acoustic volume velocity, a ninety-degree phase difference between the acceleration and the local acoustic pressure occurs at resonance. The judgement of resonance is by reading the phase measurements of the acoustic pressure and the acceleration from the two lock-in amplifiers. Besides that, Lissajous curves are also a good visual means of observing resonance on an oscilloscope. A ninety degree phase difference between the acceleration signal and the acoustic pressure signal gives an undistorted ellipsoid on the oscilloscope screen. Time for temperatures to reach steady state Prior to the actual experiments, the time required for thermal stabilization was investigated. When the loudspeaker is switched on, the acoustic energy is coupled into the resonator tube. As soon as the gas parcels around the stack begin to oscillate, the thermal interaction between gas and stack plates starts to build up a temperature gradient along these plates. There are no heat exchangers installed at both ends of the stack. In this situation, both ends of the stack begin to warm up and the temperature difference over the stack keeps increasing as well. After a short period of time, this thermoacoustic process reaches its equilibrium. The temperature difference between the two ends of the stack becomes stable. This time needed for reaching the steady state, is determined experimentally before conducting the series measurement, to make sure that all the data are taken in steady state. To determine this time period, a trial measurement is taken. First, the resonance frequency is found by the oscilloscope method mentioned above. The stack is at a fixed position, and the loudspeaker loaded at that resonance frequency. The moment when the loudspeaker is switched on is the “zero” point of time axis. A plot of the time history of the temperatures at both ends of the stack is given in Fig.3.4.7. It can be seen that the temperatures reach their steady states after increasing for some time. Around 1000 seconds seen from Fig.3.4.7, the temperatures stop increasing and stabilize at constant values. In the experiments of this chapter, the start-up time is taken as 30 minutes for every measurement. It can

66 Chapter 3

be seen from Fig. 3.4.7 that 30 minutes, equal to 1800 seconds, is enough for a specific operation to reach its steady state.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000295

296

297

298

299

300

301

302

303

304

305

306

Tem

pera

ture

(K

)

Time (second)

Figure 3.4.7: Time development of the temperatures at two ends of the stack. 3.4.2 Measurements Four kinds of measurements were conducted: I. Fixed resonator tube length, resulting in a distance between end plug and loudspeaker membrane of about 0.5m; fixed drive ratio (1% of the filling pressure) and three stacks (0.2 mm, 0.4 mm and 0.6 mm spacing). The filling pressure was always 1000 mbar. All measurements were done with a fixed resonator tube length of 438 mm (the distance from the flange surface, which is made from transparent perspex glued to the end of resonator tube, and facing the loudspeaker, to the end plug) and the pressure amplitude of the end plug was kept almost constant (the reading from the lock-in amplifier of Microphone 2 remained almost constant). For a specific stack, measurements were taken at a few stack positions. The stack position is specified as the distance Lstack as indicated in Fig.3.4.8, which is the distance between the diaphragm of the speaker and the nearest end of the stack. II. Fixed resonator tube length to 438 mm; fixed drive ratio (2% of the filling pressure); 0.6 mm-spacing stack and various stack positions. All the measurements were done with a fixed resonator tube length 438 mm and the stack of 0.6mm-spacing, and the pressure amplitude at the closed end was kept almost constant at 2% of the filling pressure. The measurements were taken at a

Standing-wave systems 67

few positions of the stack. This experiment is to investigate the influence of the driving pressure ratio on the performance of the system. The results are compared with those of 1% drive ratio for 0.6 mm-spacing stack in measurement I.

Figure 3.4.8: Illustration of the measurement position of the stack. III. Fixed resonator tube length to 121 mm; different drive ratio (0.5% and 1% of the filling pressure); 0.2 mm-spacing stack and various stack positions. The measurements were done with a fixed resonator tube length 121 mm and the stack of 0.2 mm-spacing, and the pressure amplitude at the closed end was kept almost constant at 0.5% of the filling pressure. The measurements were taken at a few positions of the stack. A second set of measurements was made at a drive ratio of 1%. This experiment was also made to investigate the influence of the driving pressure ratio on the performance of the system with different resonator tube length and different stack from measurement II. IV. Fixed resonator tube length to 121 mm; various drive ratio; 0.2 mm-spacing stack and fixed stack position. The measurements were done with a fixed resonator tube length 121 mm and the stack of 0.2 mm-spacing, and the stack was fixed at Lstack=137 mm. The measurements were taken at a few drive ratios. This experiment is to investigate the influence of different acoustic pressure at the end plug surface, i.e. different drive ratios, on the performance of the system. For every measurement, the steps are as follows: place the stack at a specific position and turn on the loudspeaker. Make a frequency sweep to search the resonance frequency by the Lissajous criterion. Fix the loudspeaker resonance frequency, and tune the amplitude of the acoustic wave generator such that the acoustic pressure at the surface of the end plug is constant at around a ratio (0.5%, 1% or 2%) of filling pressure (keep the reading of microphone 2 constant). Make the whole system run for 30 minutes. Record the readings from the lock-in amplifiers for microphone 1, current and voltage flowing through the loudspeaker, accelerometer, microphone 2 and temperatures of both ends of the stack from the temperature monitor. Then, after this recording, the measurement procedure is repeated for the next measurement point.

Loudspeaker

Lstack

Plug Stack

Sound wave

Resonator tube

68 Chapter 3

Discussion of the results I. From the measurement data, there are a lot of common features between the cases of three stacks under drive ratio of 1%. Under the prerequisites of fixed resonator tube and constant acoustic pressure at the surface of the end plug, with the stack moving toward the end plug, the input current and voltage into the loudspeaker decrease. That means the input total power to the whole system decreases when the stack is moved closer and closer to the plug end of the resonator tube, which is shown in Figs.3.4.9. The curves of 0.4 mm and 0.6 mm stacks have similar values, but the curve of 0.2 mm stack gives much larger power. The acoustic pressure in the vicinity of the loudspeaker does not vary much when the stack moves towards the plug end of the resonator tube. The acceleration rate of the diaphragm, however, decreases sharply. This shows that the total input acoustic power into the resonator tube decreases, which is shown in Fig.3.4.10. Again, the curves of 0.4 mm and 0.6 mm stacks have similar values, but the curve of 0.2 mm stack shows much larger values. By comparison of corresponding curves in Fig. 3.4.9 and 3.4.10, the rates of electrical power converting to acoustic power for 0.4 mm and 0.6 mm stacks are around 10%, but the rate for 0.2 mm stack is much lower, around 4%. According to the measurements of the cases of resonator tube length 438 mm and 1% drive ratio, the resonance frequencies of 0.6 mm stack varied in the range of 288 to 300 Hz; 288 to 300 Hz for 0.4 mm stack; and 259 to 300 Hz for 0.2 mm stack. The measurement also shows that the resonance frequency slightly increases as the stack is moved towards the end plug. In an ideal situation, a standing wave field exits inside the resonator tube. A pure standing wave does not transfer energy. So, in the ideal cases, no acoustic energy is needed. But, in reality, due to the viscous loss, thermodynamic loss and other factors, continuous acoustic energy input is needed to compensate these losses and thus maintain a standing wave. Figs. 3.4.10 shows that the acoustic energy input decreases when the stack moves closer to the closed end. Since the closed end is a velocity node, at a position closer to this end, the stack will exhibit lower viscosity losses and thus the acoustic energy input required to maintain the standing wave reduces. The measured temperature difference over the stack ∆Tstack varies against stack positions are given in Fig. 3.4.11. It shows that there is no much difference between 0.6 mm stack and 0.4 mm stack. The temperature difference over the stack ∆Tstack of 0.2 mm stack is much lower than those of 0.4 mm and 0.6 mm stacks. Among the three stacks, the 0.4 mm stack has the best performance. When the stack moves to the end plug from the middle of the resonator tube, ∆Tstack increases till a maximum, then decreases as the stack is close to the end plug.

Standing-wave systems 69

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500

1

2

3

4

5

6

7

Tot

al p

ower

into

the

spea

ker

(W)

Stack position (m)

0.6 mm stack0.4 mm stack0.2 mm stack

Figure 3.4.9: Total power input as a function of the stack position: 0.6 mm, 0.4 mm and 0.2 mm spacing stack, resonator tube length 438 mm and 1% of drive ratio.

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.00

0.05

0.10

0.15

0.20

0.25

0.30

Tot

al a

cous

tic p

ower

into

the

reso

nato

r tu

be (

W)

Stack position (m)

0.6mm stack0.4mm stack0.2mm stack

Figure 3.4.10: Total input acoustic power as a function of the stack position: 0.6 mm, 0.4 mm and 0.2 mm spacing stack, resonator tube length 438 mm and 1% of drive ratio.

70 Chapter 3

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

∆Tst

ack (

K)

Stack position (m)

0.6mm stack0.4mm stack 0.2mm stack

Figure 3.4.11: ∆Tstack as a function of the stack position: 0.6 mm, 0.4 mm and 0.2 mm spacing stack, resonator tube length 438 mm and 1% of drive ratio.

II . This 0.6 mm-spacing stack with 2% drive ratio case has similar features as those at 1% drive ratio. For both drive ratios, under the prerequisites of fixed resonator tube and constant acoustic pressure at the surface of the end plug, with the stack moving toward the end plug, the input current and voltage through the loudspeaker decrease. That means the total electrical power to the whole system decreases when the stack moving closer to the close end of the resonator tube, which is shown in Fig.3.4.12. The 2% drive ratio case consumed more than proportional electrical power than that of 1% case. For each case, the acoustic pressure of microphone 1 does not vary much when the stack is moved towards the closed end of the resonator tube. The acceleration rate of the diaphragm decreases sharply. This shows that the total input acoustic power into the resonator tube decreases, which is shown in Fig.3.4.13. The resonance frequency becomes higher as the stack is moved towards the end plug. The comparison of Fig.3.4.12 and 3.4.13 shows that both drive ratio cases have almost the same conversion rate, around 10%, from electrical power to acoustic power. Also, Figs. 3.4.12 and 3.4.13 show that the values of 2% drive ratio are nearly four times the corresponding ones of 1% drive ratio. The temperature difference over the stack ∆Tstack of two drive ratios is given in Fig.3.4.14. When the stack is placed away from the end plug, less than 0.39 seen from the figure, the difference of ∆Tstack between both drive ratios is not so large. When the stack is closer to the end plug, say stack position larger than 0.39, the

Standing-wave systems 71

∆Tstack of the 2% case becomes much larger than that of 1% case. When the stack position is further than 0.43, ∆Tstack of 2% is more than twice of that of 1% case.

0.30 0.35 0.40 0.45 0.500.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Tot

al p

ower

into

the

spea

ker

(W)

Stack position (m)

drive ratio 1%drive ratio 2%

Figure 3.4.12: Total power input as a function of the stack position: 0.6 mm spacing stack, resonator tube length 438 mm, 2% and 1% of drive ratio.

0.30 0.35 0.40 0.45 0.500.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

drive ratio 1%drive ratio 2%

Tot

al a

cous

tic p

ower

into

the

reso

nato

r tu

be (

W)

Stack position (m)

Figure 3.4.13: Total input acoustic power as a function of the stack position: 0.6 mm spacing stack, resonator tube length 438 mm, 2% and 1% of drive ratio.

72 Chapter 3

0.30 0.35 0.40 0.45 0.500

2

4

6

8

10

12

14

drive ratio 1%drive ratio 2%

∆Tst

ack (

K)

Stack position (m)

Figure 3.4.14: ∆Tstack as a function of the stack position: 0.6 mm spacing stack, resonator tube length 438 mm, 2% and 1% of drive ratio. III . Similar comparison to measurement II was carried out for a different resonator tube length of 121 mm and 0.2 mm stack. Two drive ratios 0.5% and 1% cases show similar trends. For both drive ratios, with the stack moving towards the end plug, the input current and voltage through the loudspeaker decrease. That means the total electrical power to the whole system decreases when the stack is moving closer to the closed end of the resonator tube, which is shown in Fig.3.4.15. The 1% drive ratio case consumed much more electrical power than that of 0.5% case. Similarly, Fig.3.4.16 shows that the total input acoustic power into the resonator tube decreases as the stack moves to the closed end for both cases. The 1% drive ratio case consumed much more acoustic power than that of 0.5% case. The comparison of Fig.3.4.15 and 3.4.16 shows that the energy conversion rate from electrical power to acoustic power in this case, around less than 1% to 2%, is much lower than case II. In case II, the resonance frequencies varied from 288 to 300 Hz. Taking the average frequency of 295 Hz, the gas thermal penetration depth is around 0.15 mm. For the 0.6 mm spacing stack, y0=0.3 mm=2δκ, which is in normal situation of a standing-wave type. In the 0.2 mm spacing case III, the resonance frequency varied from 687 Hz to 712 Hz. Taking the average value of 700 Hz, the gas thermal penetration depth is around 0.1 mm, which just equal to the y0 of 0.2 mm spacing stack. It means this stack has the spacing of a regenerator, but was inserted in a standing-wave type resonator tube.

Standing-wave systems 73

The temperature difference over the stack ∆Tstack of two drive ratios is given in Fig.3.4.17. For the case of drive ratio 1%, as the stack move to the end plug, ∆Tstack increases till a maximum and then decreases when the stack is close to the end plug. The ∆Tstack of the 1% case is much larger than that of 0.5% case.

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.160

2

4

6

8

10

12

14

16

18

20

Tot

al p

ower

into

the

spea

ker

(W)

Stack position (m)

drive ratio 0.5%drive ratio 1.0%

Figure 3.4.15: Total power input as a function of the stack position: 0.2 mm spacing stack, resonator tube length 121 mm, 0.5% and 1% of drive ratio.

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.160.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

drive ratio 0.5%drive ratio 1.0%

Tot

al a

cous

tic p

ower

into

the

reso

nato

r tu

be (

W)

Stack position (m)

Figure 3.4.16: Total input acoustic power as a function of the stack position: 0.2 mm spacing stack, resonator tube length 121 mm, 0.5% and 1% of drive ratio.

74 Chapter 3

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.160

1

2

3

4

5

6

7

8

drive ratio 0.5%drive ratio 1.0%

∆Tst

ack (

K)

Stack position (m)

Figure 3.4.17: ∆Tstack as a function of the stack position: 0.2 mm spacing stack, resonator tube length 121 mm, 0.5% and 1% of drive ratio.

0.0 5.0x105 1.0x106 1.5x106 2.0x106 2.5x106 3.0x106

0

2

4

6

8

100.0 5.0x10-5 1.0x10-4 1.5x10-4 2.0x10-4 2.5x10-4

0

2

4

6

8

10

(Drive ratio)2

Tot

al p

ower

into

the

spea

ker

(W)

(Acoustic amplitude)2 (Pa)2

Figure 3.4.18: Total power input to the speaker as a function of the acoustic pressure square at the end plug and drive ratio square: 0.2 mm spacing stack, resonator tube length 121 mm.

Standing-wave systems 75

0.0 5.0x105 1.0x106 1.5x106 2.0x106 2.5x106 3.0x106

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.200.0 5.0x10-5 1.0x10-4 1.5x10-4 2.0x10-4 2.5x10-4

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

(Drive ratio)2

(Acoustic amplitude)2 (Pa)2

Tot

al a

cous

tic p

ower

into

the

reso

nato

r tu

be (

W)

Figure 3.4.19: Total acoustic power into the resonator tube as a function of the acoustic pressure square at the end plug and drive ratio square: 0.2 mm spacing stack, resonator tube length 121 mm.

400 600 800 1000 1200 1400 1600 18002

4

6

8

10

12

14

∆Tst

ack (

K)

Acoustic amplitude (pa)

Figure 3.4.20: ∆Tstack as a function of the acoustic pressure at the end plug: 0.2 mm spacing stack, resonator tube length 121 mm.

76 Chapter 3

IV. When the acoustic pressure amplitude at the end plug surface increases, the power consumed by the system also increases, as shown in Fig.3.4.18. Similar trends are also shown for the total acoustic power into the resonator tube, as shown in Fig. 3.4.19. Figs. 3.4.18 and 3.4.19 also show that the power flows are linear functions of the acoustic pressure amplitude squared. The temperature difference over the stack ∆Tstack at different acoustic pressure is given in Fig.3.4.20. As expected, ∆Tstack increases almost linearly as the acoustic pressure at the end plug surface increases. 3.4.3 Theoretical computation The computation results by using section 3.3.3 are discussed below. (a) Computation of ∆Tstack as a function of stack positions or acoustic pressure at the end plug Experimental data for the cases of resonator tube length of 438 mm, along with computation results by using Eq.(3.3.33) and (3.3.32) as method 1 “M1”, and by Eq.(3.3.34) and (3.3.32) as method 2 “M2”, are given from Fig. 3.4.21 to Fig. 3.4.24. Computations and measurement data for the cases of resonator tube length of 121 mm are plotted from Fig. 3.4.25 to 3.4.27. In the computation, the tube length is set to half the wave length fa /=λ , where a is the speed of sound at

normal atmospheric pressure and ambient temperature 300K, and f is the

corresponding measured operating resonance frequency.

0.30 0.35 0.40 0.45 0.500

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.21: Comparisons between computation and experiment of case with 0.6 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

Standing-wave systems 77

0.30 0.35 0.40 0.45 0.500

2

4

6

8

10

12

14

16

18

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.22: Comparisons between computation and experiment of case with 0.4 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

-8

-6

-4

-2

0

2

4

6

8

10

12

14

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.23: Comparisons between computation and experiment of case with 0.2 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

78 Chapter 3

0.30 0.35 0.40 0.45 0.500

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.24: Comparisons between computation and experiment of case with 0.6 mm spacing stack, 2% drive ratio and resonator tube length 438 mm.

0.08 0.10 0.12 0.14 0.16

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.25: Comparisons between computation and experiment of case with 0.2 mm spacing stack, 1% drive ratio and resonator tube length 121 mm. As seen in Figs. 3.4.21 to 3.4.27, the computations based on the proposed analytical method by using Eq.(3.3.33) have better performance than those using method 2. In Figs.3.4.23, 3.4.25, 3.4.26 and 3.4.27, the computation results have

lower ∆Tstack than those of the experiment. In the computations, a larger 2E& leads

Standing-wave systems 79

to a lower ∆Tstack. Therefore, a lower ∆Tstack than experimentally measured is due to

the overestimation of 2E& by Eq. (3.3.33). That can be explained by the assumption

that no energy exchange with the environment takes place.

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15

-2

-1

0

1

2

3

4

5

6

7

∆Tst

ack (

K)

Stack position (m)

computation M1 computation M2 experiment

Figure 3.4.26: Comparisons between computation and experiment of case with 0.2 mm spacing stack, 0.5% drive ratio and resonator tube length 121 mm.

400 600 800 1000 1200 1400 1600 18000

2

4

6

8

10

12

14

16

18

20

22

24

26

28

∆Tst

ack (

K)

Acoustic amplitude (pa)

computation M1computation M2experiment

Figure 3.4.27: Comparisons between computation and experiment of the case with 0.2 mm spacing stack, stack position fixed at 137 mm and resonator tube length 121 mm with various acoustic pressures at the end plug.

80 Chapter 3

(b) The acoustic power losses The losses in the three sections of the resonator for all cases are given from Fig. 3.4.28 to 3.4.34. The losses computed by using “by solving the wave equation in section 1 and 3” (see section 3.3.3) are indicated as “(1)”, and those by “by using the equation in chapter 2” (see section 3.3.3) are indicated as “(2)”. As expected, the loss in section 1 increases, and the loss in section 3 decreases as the stack moves towards the end plug. As seen in the figures from Fig.3.4.28 to 3.4.33, the loss in section 2 decreases as the stack moves towards the end plug. In Fig.3.4.34, all losses in three sections increase as the acoustic pressure on the end plug surface increases. This can be explained by the stronger acoustic field leading to higher thermoviscous losses. Among three sections, the losses in section 2 increase more in scale than the losses in section 1 and 3. In these figures, two methods agree well with each other in section 1 and 3, but there are discrepancies in section 2. The discrepancy in section 2 is large for the cases with 0.2 mm-spacing stack, which are shown in Figs. 3.4.30, 3.4.32, 3.4.33 and 3.4.34. We consider the method (1) is more accurate than method (2).

0.30 0.35 0.40 0.45 0.500.000

0.005

0.010

0.015

0.020

0.025

0.030

Ene

rgy

(W)

Stack position (m)

loss before the stack (1) loss before the stack (2) loss in the stack (1) loss in the stack (2) loss behind the stack (1) loss behind the stack (2)

Figure 3.4.28: Comparisons of losses between computation methods (1) and (2) of case with 0.6 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

Standing-wave systems 81

0.30 0.35 0.40 0.45 0.500.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

Ene

rgy

(W)

Stack position (m)

loss before the stack (1) loss before the stack (2) loss in the stack (1) loss in the stack (2) loss behind the stack (1) loss behind the stack (2)

Figure 3.4.29: Comparisons of losses between computation methods (1) and (2) of case with 0.4 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500.00

0.05

0.10

0.15

0.20

0.25

Ene

rgy

(W)

Stack position (m)

loss before the stack (1) loss before the stack (2) loss in the stack (1) loss in the stack (2) loss behind the stack (1) loss behind the stack (2)

Figure 3.4.30: Comparisons of losses between computation methods (1) and (2) of case with 0.2 mm spacing stack, 1% drive ratio and resonator tube length 438 mm.

82 Chapter 3

0.30 0.35 0.40 0.45 0.500.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Ene

rgy

(W)

Stack position (m)

loss before the stack (1) loss before the stack (2) loss in the stack (1) loss in the stack (2) loss behind the stack (1) loss behind the stack (2)

Figure 3.4.31: Comparisons of losses between computation methods (1) and (2) of case with 0.6 mm spacing stack, 2% drive ratio and resonator tube length 438 mm.

0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.160.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Ene

rgy

(W)

Stack position (m)

loss before the stack (1)loss before the stack (2)loss in the stack (1)loss in the stack (2)loss behind the stack (1)loss behind the stack (2)

Figure 3.4.32: Comparisons of losses between computation methods (1) and (2) of case with 0.2 mm spacing stack, 1% drive ratio and resonator tube length 121 mm.

Standing-wave systems 83

0.08 0.09 0.10 0.11 0.12 0.13 0.140.00

0.01

0.02

0.03

0.04

Ene

rgy

(W)

Stack position (m)

loss before the stack (1)loss before the stack (2)loss in the stack (1)loss in the stack (2)loss behind the stack (1)loss behind the stack (2)

Figure 3.4.33: Comparisons of losses between computation methods (1) and (2) of case with 0.2 mm spacing stack, 0.5% drive ratio and resonator tube length 121 mm.

400 600 800 1000 1200 1400 1600 18000.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

Ene

rgy

(W)

Acoustic amplitude (pa)

loss before the stack (1)loss before the stack (2)loss in the stack (1)loss in the stack (2)loss behind the stack (1)loss behind the stack (2)

Figure 3.4.34: Comparisons of losses between computation methods (1) and (2) of case with 0.2 mm spacing stack, stack position fixed at 137 mm and resonator tube length 121 mm as a function of the acoustic pressure at the end plug.

84 Chapter 3

3.4.4 Conclusions An analytical method is presented for evaluating the temperature profile along a stack in a standing-wave “thermoacoustic couple”. A thermoacoustic resonator tube was built, and experiments with a “thermoacoustic couple” were made. These experiments have been compared with the model. The trend of the temperature differences over the stack between measurement and model is the same, however, a precise agreement depends much on the correct estimation of the total energy flow

along the stack 2E& . Better agreement about temperature differences along the stack

∆Tstack between experimental measurement and computation by using the proposed

method 1 than method 2 assuming 2E& =0 was obtained. If the total energy flow

along the stack 2E& is accurately given, the model discussed in section 3.3 is able to

describe and to explore the standing-wave system.

Chapter 4 Traveling-wave systems 4.1 Introduction Although a traveling-wave system has many similarities with a standing-wave system, it executes a different thermodynamic cycle compared to a standing-wave system. The intermaterial spacing in the regenerator of a traveling-wave system is much smaller than in the stack of a standing-wave system. The phasing between acoustic pressure and volume velocity in a traveling-wave system is nealy in phase, whereas a standing-wave type is nearly 90 degrees out of phase. This phasing relation makes a traveling-wave system work in a Stirling cycle. A Stirling cycle has no intrinsic irreversibilities so that a higher thermal efficiency occurs. Under ideal conditions, the ideal efficiency of the Stirling cycle is equal to the Carnot efficiency. This high efficiency makes a traveling-wave system more competitive in applications than its counterpart, the standing-wave system, which involves an intrinsically irreversible process. Stirling machines originated from the invention of Robert Stirling in 1816. His ingenious design went ahead of the development of thermodynamic theory. Afterwards, much work has been done fruitfully based on his patent [55, 56]. In 1964, an important discovery by W. Beale [85] made an inspiration to a piston-free Stirling machine. While teaching a course on thermal machinery at Ohio University, he found that the piston was not necessary in a Stirling β-configuration, once the machine started. Because the machine could continue to work only based on gas springs with the correct phasing relationship. The first device without any sliding seals was the Fluidyne engine, which used U-tube liquid pistons [57]. But its performance is limited to low frequencies due to the large mass of the liquid pistons. Ceperley discovered that a traveling acoustic wave propagating through a differentially heated regenerator executes a thermodynamic cycle similar to the Stirling cycle [58, 59]. As the wave travels up the temperature gradient through the

86 Chapter 4

regenerator, the wave is amplified and thermal energy converts into acoustical energy. The opposite direction results in acoustical energy being used to pump heat. Although Ceperley’s experimental engine was not able to amplify acoustic power, his remarkable idea stimulated interest in using sound to build piston-free Stirling type engines and refrigerators. Much later, Yazaki et al. first demonstrated a stack-based traveling-wave amplifer [60]. Unfortunately, the engine had a low efficiency. They realized that the low acoustic impedance of the working gas caused large viscous losses due to high acoustic velocities. In 1999, Backhaus and Swift presented a new configuration of a thermoacoustic Stirling heat engine [61] and later, in 2000, they made a detailed description of it [34]. Their ingenious invention solved the problem of low acoustic impedance, made a breakthrough in building a thermoacoustic engine to achieve the efficiency of a conventional gasoline-powered engine. In the next section, a physical description of traveling-wave systems is given. Section 4.3 presents an analytical model for a complete traveling-wave system of refrigerator. After that, in section 4.4, the regenerator material is considered. Experiments are performed on various regenerator materials in which also the model developed in section 4.3 is validated. A full traveling-wave refrigerator driven by a mechanical compressor was designed and built. This system is described in section 4.5 along with experiments again aiming at the validation of the model. 4.2 Physical description of traveling-wave systems A simplified physical description about how a traveling-wave thermal process develops is given in this section. A Stirling cycle for the refrigeration mode will be presented. A schematic description of an ideal Stirling cycle is shown in Fig. 4.2.1. From left to the right, it consists of a warm piston, a hot-end heat exchanger, a regenerator, a cold-end heat exchanger and a cold piston. In steady-state, starting at the moment when the warm piston is in its left most position, and the cold piston is in contact with the cold-end exchanger, a Stirling refrigeration cycle can be described in four steps: 1. From a to b. The warm piston moves to the right over some distance while

the cold piston is fixed. This isothermal process gives off heat flow HQ& to

the surroundings through the hot-end heat exchanger.

Traveling-wave systems 87

Figure 4.2.1: Ideal Stirling cycle with the P-V diagram. 2. From b to c. Both pistons move to the right over the same distance to ensure

the total volume enclosed remains constant. The gas enclosed passes through the regenerator. The gas enters the regenerator from the left end at room temperature and leaves it at the right end with low temperature. During this process, the heat is given off by the gas to the regenerator material.

3. From c to d. The cold piston moves to the right while the warm piston

remains fixed. This isothermal process takes up heat CQ& from the

application via the cold-end heat exchanger. So, here cooling takes place. 4. From d to a. Both pistons move to the left over the same distance to ensure

the total volume enclosed remains constant. The gas enclosed passes through the regenerator. The gas enters the regenerator from the right end at low temperature and leaves it at the right end at high temperature. During this process, the regenerator material is cooled by the cold gas. Then the whole system returns to its initial state and is ready for the next cycle.

So netto heat is given off to a warm heat exchanger, and cooling takes place at the cold heat exchanger. With this cooling power an application can be cooled. The main part of a thermoacoustic traveling-wave refrigerator, which contains the regenerator, retains all components of their mechanical equivalent, except for the pistons. The removal of the moving pistons of a Stirling refrigerator is realized by using an acoustic wave and a properly dimensionalized acoustic resonator.

p

V 1V 2V

a

b

c

d

1 2

3 4 HT

CT

HQ&

CQ&

b

c

d

a

a

HQ&

CQ&

HT CT

Regenerator Warm piston

Cold piston

88 Chapter 4

4.3 Modeling traveling-wave systems Figure 4.3.1: A traveling-wave refrigerator driven by a loudspeaker. In this section, a model for a traveling-wave system with a torus-shaped section is discussed, as depicted in Fig. 4.3.1. The system of Fig. 4.3.1 employs a thermoacoustic conventional configuration, using a large volume called compliance to connect the feedback inertance and the ambient heat exchanger. However, the author found that the compliance is not necessary for the performance of a traveling-wave system. Therefore, in the section 4.5, a traveling-wave refrigerator without a compliance was designed and used as the experimental apparatus. For the present section, in the consideration of generality, the theoretical model sticks to the conventional configuration. In Fig.4.3.1, the refrigerator system is driven by a driver, feeding acoustic power into the refrigerator part. A resonator tube is connected to the driver at one end and to the torus-shaped section at the other end. The length of the resonator tube depends on the wavelength of the working gas. The torus-shaped section contains a feedback inertance tube, a compliance volume, and a regenerator which is sandwiched between two heat exchangers. A traveling-wave system in such a configuration, which was first developed by Backhaus and Swift in the mode of an engine [34], has a favorable efficiency and thus is valuable for applications and research of the present work. Much work was focused on numerically modeling a traveling-wave system. Among them, SAGE and DeltaE are the most extensively used tools for modeling in the thermoacoustic community. Swift and his coworkers developed a computer program DeltaE which integrates the acoustic wave equations, and can be used for apparatus ranging from simple duct networks and resonators to thermoacoustic prime movers, refrigerators, and combinations at low-pressure amplitudes [71]. SAGE is developed by D. Gedeon, and utilizes finite difference methods to simulate the machine and predict the performances [72]. This commercially accessible software provides guidance in thermoacoustic system design. W.Dai et

Feedback inertance

Ambient HX

Compliance

Regenerator Cold end HX

Resonator tube Driver

Traveling-wave systems 89

al. [73] developed computer codes to simulate the fluid field inside a thermoacoustic system and make performance predictions. All of these computer programs numerically simulate the acoustic fields and predict the performances. It normally takes much time to obtain a converged result by using these programs. Although these numerical programs are powerful in simulation and prediction of a thermoacoustic system, they are expensive in computation time. The analytical method presented in this work has the advantage of predicting the performance in a fast manner. Therefore, it is a useful tool in parameter-optimization studies. This analytical expression is also helpful in obtaining a universal conclusion for scaling analysis of traveling-wave thermoacoustic systems.

Figure 4.3.2: Energy flow distribution of a torus traveling-wave refrigerator. In the regenerator of the looped configuration of a traveling-wave system, the energy flow arrangement, is shown in Fig.4.3.2. The dash-dot rectangular frame indicates the part containing the regenerator and two heat exchangers, where the main heat exchange between the environment and the system takes place. For the global system, the energy balance is

HCin QQW &&& =+ , (4.3.1)

where inW& is the acoustic input power into the refrigerator by the driver.

The total energy flow along the regenerator is given by

2EQW Hstup&&& +=− , (4.3.2)

Cstdn QEW &&& +=− 2 . (4.3.3)

In a regenerator, the total energy flow 2E& is very small. For an ideal regenerator,

and ideal gas as working gas, the heat capacity of the regenerator material and the good thermal contact between working gas and solid regenerator material maintain temporally isothermal conditions, so the temperature oscillation 01 =T everywhere.

If the conduction of heat through the regenerator can be neglected, Eq. (2.1.68) shows that 02 =E& [38].

CQ&

stupW −&

HT

CT

HQ& Control volumes

2E& stdnW −

& inW&

90 Chapter 4

Using Eq. (4.3.3), the cooling power is

2EWQ stdnC&&& −= − . (4.3.4)

To analyze the behavior of the acoustic power flows and total energy flow along the regenerator, the acoustic field has to be known. In order to develop an analytical model, here below all the sections of the traveling-wave system, as shown in Fig. 4.3.1, will be analyzed using the thermo-acoustic equations. First the general case for a section with temperature gradient and pressure gradient will be treated. From that the expressions for all the segments are derived. The most general wave equation that describes the behavior of the acoustic pressure 1p in an acoustical pipe section is given by Eq. (2.1.57),

( )( ) 011

11

)1(1 1

2

21

2

2

1 =+−

−−

−+

+−+

dx

d

dx

dTffa

dx

df

dx

daf m

sm

m

s

ppp

εσωβ

ρωρ

εγ νκνκ .

The Rott’s functions vf , κf and sε are evaluated locally and with corresponding

geometry. In order to solve Eq. (2.1.57) for the pressure, the temperature or temperature gradient must be known. For most of the components in Fig. 4.3.1, the temperature is constant. However, in the regenerator, the temperature varies in the axial direction x, and the parametervf , κf , viscous penetration depth νδ , thermal

penetration depth κδ , sound speed a , density mρ , gas conductivity κ and

viscosity µ are functions of x.

First the material properties like viscosity etc. are written as functions of temperature via substution of Eq. (2.1.16). Substitution of (2.1.16), (A.6), (2.1.38) and (2.1.31) into wave equation (2.1.57), after a long calculation, leads to

vv

b

mv

s

v

fyi

Tbfbff

dx

d

−⋅

++

++

−+−

−−+

+−

1]/)1[(cosh2

1

2

)3(

)1)(1(1

02

2

3

21

2 βδεσ

µ

µµκp

01

)1/()1(112

21 =⋅

−+−+⋅+⋅⋅ p

p

v

sm

f

f

adx

d

dx

dT εγω κ . (4.3.5)

This is a second order differential equation. Like in chapter 2, the solution has the following form

xeαCp =1 . (4.3.6)

Substitution of Eq. (4.3.6) into (4.3.5) yields

vv

b

mv

s

v

fyi

Tbfbff

−⋅

++

++

−+−

−−+

+−

1]/)1[(cosh2

1

2

)3(

)1)(1(1

02

2

3

2 βδεσ

µ

µµκα

Traveling-wave systems 91

01

)1/()1(12

2

=−

+−+⋅+⋅⋅v

sm

f

f

adx

dT εγω κα . (4.3.7)

For simplicity of writing, we define

dx

dT

fyi

Tbfbff m

vv

b

mv

s

v

++

++

−+−

−−=

+−

1]/)1[(cosh2

1

2

)3(

)1)(1(1

02

2

3

βδεσ

µ

µµκG

(4.3.8) and

v

s

f

f

a −+−+⋅=

1)1/()1(1

2

2 εγω κH . (4.3.9)

Thus Eq. (4.3.7) can be rewritten in terms of α

02 =+⋅+ HαGα . (4.3.10)

Therefore, the roots of Eq. (4.3.10) are

( )HGGα 421 2

2,1 −±−= . (4.3.11)

Thus the general solution of acoustic pressure oscillation is xx ee 21

211αα CCp ⋅+⋅= , (4.3.12)

where the complex coefficients 1C and 2C are determined by the boundary

conditions. The boundary conditions depend on the specific configuration of the refrigerator and also depend on local parameters. Therefore, it is difficult to obtain a universal expression of the solution, certainly not for a looped tube system with temperature gradients in it. However, in principle the solution (4.3.12) can be used to component-wise couple the tube sections and find a universal expression for the wave in the complete resonator. This will be explained below. In order to solve the equations, an assumption of zero temperature gradient on the solid wall, i.e. 0/ =dxdTm , is made for all components except the regenerator.

The regenerator, using the concept of differentiation, is divided into many differential elements to resolve the solution. Zero temperature gradient components For the zero-temperature-gradient components, the acoustic pressure equation is much simplified via Eq. (4.3.12) where 0/ =dxdTm has been used ( 0=G ) and

reduces to xixi ee k-k CCp ⋅+⋅= 211 , (4.3.13)

where the complex wave number is defined as:

92 Chapter 4

v

s

f

f

a −+−+⋅=

1)1/()1(1

2

22 εγω κk . (4.3.14)

Thus, it can be obtained that

[ ]xixi eeidxd k-k CCkp ⋅−⋅= 211 / . (4.3.15)

By using Eq. (2.2.6), the y-direction averaged velocity follows as

( )νωρf

dx

di

m

−⋅= 111

pu .

The volume velocity can be written as ( ) [ ]xixi

m

eeAf

A k-k CCk

uU ⋅−⋅−−== 2111

1ωρ

ν . (4.3.16)

In most practical cases, the components are tubes, at ambient temperature without an axial temperature gradient, with a radius much larger than the thermal and viscous penetration depths at ambient temperature. We consider sound propagating in the x direction in an ideal gas within a channel with cross-sectional area A and perimeterΠ , the hydraulic radius is defined as

Π= /Arh . (4.3.17)

Now that the hydraulic radius of a tube segment vhr δ>> and κδ can be sustained,

so Rott’s functions 0≈vf and 0≈κf .

The wave number can now be reduced to a real number from Eq. (4.3.14) ak /ω= , (4.3.18)

where the sound speed is at the working temperature of these tubes, usually, ambient temperature. The pressure and volume velocity for every tube segment are reduced to

ikxikx ee -CCp ⋅+⋅= 211 , (4.3.19)

[ ]ikxikx

m

eea

AA -CCuU ⋅−⋅−== 2111 ρ

. (4.3.20)

To describe the complete system, it is necessary to compute the acoustic impedance section-wise. In many general acoustic networks, a lumped compliance is used having a general impedance Z=1/iω·(V/ρma²). However the general method in this thesis is more accurate and is also used for the connection A→B. Therefore, the computation stations from A to E are distributed along the loop anti-clockwise, which is shown in Fig. 4.3.3 (c). The computation orientation is shown in Fig. 4.3.3 (a), where the original point is located at the interface of the resonator tube and the driver. Although in the tee section, acoustically generated flows and vortices may occur, it is possible to neglect these in the acoustic limit, and only apply mass conservation and equality of pressure in the tee. In that case, the joint tee is simplified as a point.

Traveling-wave systems 93

Figure 4.3.3: Illustration of computation stations for a traveling-wave refrigerator (a) the orientation of modeling computation: o indicates the origin (b) detailed sketch of the joint tee (c) the global distribution of computation stations. The computation stations are located at the interfaces between any two connected components within the loop. O is the origin of the coordinate system, where the driver is connected to the traveling-wave refrigerator part. The input acoustic flow from the driver, feeding into the refrigerator at original point of O, is indicated by in1p and in1U . The pressure and volume velocity at the end of the resonator tube

connecting the tee junction are indicated by input1p and input1U . The pressure and

volume velocity at one branch of the tee junction which is connected to the cold-end heat exchanger are indicated by out1p and out1U . The pressure and volume

velocity at another branch of the tee junction which is connected to the feedback inertance tube are indicated by fb1p and fb1U , see Fig. 4.3.3 (b) and (c). A locates

the interface of the feedback inertance tube and the compliance volume. B locates the interface of the compliance volume and the ambient heat exchanger. C locates the interface of the ambient heat exchanger and the regenerator. D locates the interface of the regenerator and the cold end heat exchanger. E locates the interface of the cold end heat exchanger and the tube section connecting the tee. As a prerequisite, the pressure at the interface of the driver and the resonator tube

in1p and the cold end temperature CT of the regenerator or cold end heat exchanger

should be given. The cooling power and efficiency of the refrigerator system can be computed by the model presented here. Resonator tube First, indicating the length of the resonator tube as resL and the diameter as resd , it

is known that at 0=x , by Eq. (4.3.19) and (4.3.20), we have

Resonator tube

o

o

in1p

x

in1U

A

C

D E

B (a)

out1p out1U input1p

fb1p

fb1U input1U (b)

(c)

94 Chapter 4

resresin 211 CCp += , (4.3.21)

[ ]resresm

resin a

d21

2

1 4CCU −−=

ρπ

. (4.3.22)

Thus the coefficients are

−= 2

111

421

res

mininres d

a

πρU

pC , (4.3.23a)

+= 2

112

421

res

mininres d

a

πρU

pC . (4.3.23b)

Therefore, the pressure and volume velocity at resLx = are resres ikL

resikL

resinput ee -CCp ⋅+⋅= 211 , (4.3.24)

[ ]resres ikLres

ikLres

m

resinput ee

a

d -CCU ⋅−⋅−= 21

2

1 4 ρπ

. (4.3.25)

Substitution of Eq. (4.3.23a) and (4.3.23b) into (4.3.24) and (4.3.25), and splitting the exponential factors into cosine and sine, yields

)sin(4

)cos( 21

11 resres

minresininput kL

d

aikL

πρU

pp −= , (4.3.26)

)sin(4

)cos( 12

11 resm

inresresininput kL

a

dikL

ρπ p

UU −= . (4.3.27)

Loop For the loop, the computation starts from the right side of the joint tee point, where the acoustic flow from the driver via resonator tube joins the acoustic flow out of the cold-end heat exchanger, and merges into the feedback flow, indicated as

fb1p and fb1U . The computation goes stepwise through the inertance tube, A to E

till the tee branch connected to the cold-end heat exchanger. For the derivation which is laborious, the reader is referred to appendix F. Finally the relation between fb1p fb1U and out1p out1U is found:

−= )sin(

4)cos( 2

3111 tb

tb

mtbfbout kL

d

aikL

πρ D

Dpp

−+ )sin(

4)cos( 2

421 tb

tb

mtbfb kL

d

aikL

πρ D

DU , (4.3.28)

−= )sin(

4)cos( 1

2

311 tbm

tbtbfbout kL

a

dikL

ρπ D

DpU

Traveling-wave systems 95

−+ )sin(

4)cos( 2

2

41 tbm

tbtbfb kL

a

dikL

ρπ D

DU . (4.3.29)

where the involved complex coefficients of D are

( ) 30

2

1001 4

),(),(1 θ

ρτπ

θτω µµ a

bfRdibgRCi

m

fb+−=D , (F.38a)

( ) 4200202

4),(1),( θ

πρτωθτ µµ

fb

m

d

abgRCiibfR −−−=D , (F.38b)

3

2

10

3 41ln θ

τρπ

θτ

τωa

diCi

m

fb−−

=D , (F.38c)

420

24 )1(

ln4/ θ

τπτωρτθ

−+=

fb

m

d

CaD , (F.38d)

where temperature ratio τ (the ratio between the temperatures at both ends of the regenerator) is defined by Eq.(F.18), and four real coefficients, which only depend on the geometrical parameters, used in complex Ds are:

221 /)sin()sin()cos()cos( cplcplfbfbcplfb dkLkLdkLkL −=θ , (F.37a)

222 /)sin()sin()cos()cos( fbcplfbcplcplfb dkLkLdkLkL −=θ , (F.37b)

223 /)sin()cos()cos()sin( fbcplfbcplcplfb dkLkLdkLkL +=θ , (F.37c)

224 /)sin()cos()cos()sin( cplcplfbfbcplfb dkLkLdkLkL +=θ . (F.37d)

Tee junction The boundary conditions at the joint tee for the mode of thermoacoustic refrigerator are

inputoutfb 111 ppp == , (4.3.30)

inputoutfb 111 UUU += . (4.3.31)

Applying boundary condition (4.3.30) to Eq. (4.3.28), it follows that

+ )sin(

4)cos(1 2

311 tb

tb

mtbfb kL

d

aikL

πρ D

D-p

−= )sin(

4)cos( 2

421 tb

tb

mtbfb kL

d

aikL

πρ D

DU . (4.3.32)

Thus, the acoustic impedance on the right hand side of the joint tee is given by:

)sin(4

)cos(1

)sin(4

)cos(

23

1

24

2

1

1

tbtb

mtb

tbtb

mtb

fb

fbfb

kLd

aikL

kLd

aikL

πρ

πρ

DD-

DD

U

pZ

+

−== . (4.3.33)

Applying boundary condition (4.3.31) to Eq. (4.3.29), it follows that

96 Chapter 4

−−=−= )sin(

4)cos( 1

2

31111 tbm

tbtbfboutfbinput kL

a

dikL

ρπ D

DpUUU

++ )sin(

4)cos(1 2

2

41 tbm

tbtbfb kL

a

dikL

ρπ D

D-U . (4.3.34)

By using Eq. (4.3.30) and (4.3.34), the acoustic impedance of the joint tee is given by

=−

==outfb

fb

input

inputinput

11

1

1

1

UU

p

U

pZ

)sin(4

)cos()sin(4

)cos(11

1

12

32

2

4 tbm

tbtbtb

m

tbtb

fb

kLa

dikLkL

a

dikL

ρπ

ρπ D

DD

D-Z

+−

+

.

(4.3.35) Substitution of Eq. (4.3.33) into (4.3.35) yields

)sin(4

4)cos()(1

)sin(4

)cos(

232

2

413241

24

2

tbtb

m

m

tbtb

tbtb

mtb

input

kLd

a

a

dikL

kLd

aikL

+++−−+

−=

πρ

ρπ

πρ

DDDDDDDD

DD

Z .

(4.3.36) So this is the input impedance of the loop seen from the connection which connects the resonator tube and the tee junction. The complete system Using the transmission relation for the acoustic impedance, see appendix G, replacing cZ with inputZ in Eq. (G.10) yields the complete acoustic impedance of

the refrigerator

)sin()cos(4

)sin(4

)cos(4

2

2

2

resinputresres

m

resres

mresinput

res

mrfga

kLikLd

a

kLd

aikL

d

a

Z

ZZ

+

+=−

πρ

πρ

πρ

. (4.3.37)

It is noticeable that the complete acoustic impedance of the refrigerator rfga−Z only

depends on the geometrical configuration of the refrigerator and operational temperatures. Assume that the pressure at the interface between the driver and the resonator tube is known, i.e. in1p from the driver being known. Thus, the volume velocity in1U is

given by

rfgainin −= ZpU /11 . (4.3.38)

Traveling-wave systems 97

Substitution in1p and in1U into Eq. (4.3.26) and (4.3.27) yields

)sin(4

)cos( 21

11 resrfgares

minresininput kL

d

aikL

−=Z

ppp

πρ

, (4.3.39)

)sin(4

/)cos( 12

11 resm

inresrfgaresininput kL

a

dikL

ρπ p

ZpU −= − . (4.3.40)

By using the boundary condition (4.3.30) and Eq. (4.3.39), the pressure fb1p is

given by

−=

)sin(4

)cos( 211 resrfgares

mresinfb kL

d

aikL

Zpp

πρ

. (4.3.41)

By using Eq. (4.3.33), the volume velocity fb1U is given by

−=

)sin(4

)cos( 21

1 resrfgares

mres

fb

infb kL

d

aikL

ZZp

ρ. (4.3.42)

Now, all equations derived have to be reviewed to understand the result of the full analysis. If the output pressure at the driver in1p which forms the input to the

system is given, the key reference parameters fb1p and fb1U will be given by Eq.

(4.3.41) and (4.3.42). From this all the other acoustic pressures and volume velocities at all the computation stations from A to E can be obtained by the corresponding equations in appendix F. Energy flows Since the acoustic field is now known, the cooling power can be computed. As mentioned in the beginning of this section, the cooling power is given by Eq. (4.3.4)

2EWQ stdnC&&& −= − .

It can be seen from Fig.4.3.3, and using Eq. (D.2), the cooling power can be written as

211 ]~

Re[21

EQ DDC&& −= Up . (4.3.43)

Substitution of Eq. (F.34) and (F.35) into the first term of Eq. (4.3.43) on the right hand side yields

+++= ]~

~~Re[

]~

Re[]

~Re[

21

]~

Re[21 4132

242

31

2

111fbfbfb

fbDD ZDD

ZDD

Z

DDDDpUp .

(4.3.44) The term between brackets is defined as 3Θ

98 Chapter 4

]~

~~Re[

]~

Re[]

~Re[ 4132

242

313fbfbfb

ZDD

ZDD

Z

DDDD +++=Θ . (4.3.45)

Thus, Eq. (4.3.44) can be written as

3

2

111 21

]~

Re[21 Θ⋅= fbDD pUp . (4.3.46)

Using the distributed regenerator model Eq. (F.16) at position C, and noticing Eq.

(F.9) and 0TTpositionCm = , the pressure gradient at position C is given by

)(101 x

L

R

dx

dB

regpositionC

Up −= . (4.3.47)

By using Eq. (2.1.72) and (F.17), and assuming that the regenerator part is ideally thermally isolated from the ambient environment and the heat capacity of the solid material of the regenerator is large enough, therefore the constant total energy flow along the regenerator can be evaluated at position C

( )( )( )

++−−−−=

σεωρψ νκ

ν 11

~~

1~

Im2 11

02

sBB

regm

regreg fff

L

RAE Up&

( )( )( )

( )( )

+++−+−

−+

σεετ

σρωψ κννκ

ν 11/1

~~

Im1/1

122

12

20

30

s

sB

regregm

pregreg fffff

LL

RTcAU

[ ]reg

regsregreg LTAKK

1/1)0.1( 0

−−+− τψψ . (4.3.48)

Note that for an ideal regenerator, so with mesh size much below the thermal penetration depth of the acoustic wave, and without heat conduction, it follows that 02 =E& , as it should be.

Substitution of Eq. (4.3.46) and (4.3.48) into Eq. (4.3.43) yields

[ ]reg

regsregregfbC LTAKKQ

)/1(1)0.1(

2

103

2

1

τψψ −⋅−+−Θ⋅= p&

( )( )( )

++−−−+

σεωρψ νκ

ν 11

~~

1~

Im2 11

0

sBB

regm

regreg fff

L

RAUp

( )( )( )

( )( )

+++−+−

−−

σεετ

σρωψ κννκ

ν 11/1

~~

Im1/1

122

12

20

30

s

sB

regregm

pregreg fffff

LL

RTcAU .

(4.3.49)

Substitution of Eq. (F.28) and (F.29) into Eq. (4.3.49) makes CQ& be expressed by

fb1p and fbZ (eliminate B1p and B1U ). The cooling power is given in form of fb1p

and fbZ

Traveling-wave systems 99

[ ]reg

regsregregfbC LTAKKQ

)/1(1)0.1(

2

103

2

1

τψψ −⋅−+−Θ⋅= p&

( )( )( )

++−−−

++

σεβθθθθ

ωρψ νκ

ν 11

~~

1~Im2

43212

10

s

m

fbfbfb

regm

regreg ffTf

L

RA

ZZp

( )( )( )

++−−−

++

σεβ

πθθρ

ρθθπ

ωρψ νκ

ν 11

~~

1Re4

42 22

42312

2

10

s

m

fbfb

m

m

fbfb

regm

regreg ffTf

d

a

a

d

L

RA

Zp

( )

+

+−

−− 2

3222

32

2

22

2

12

20

3

0 ]Im[

241/1

12fb

fb

m

fb

m

fb

fb

fbregregm

pregreg

a

d

a

d

LL

RTcA

Z

Z

Zp

ρθθπ

ρθπθτ

σρωψ

( )( )( )( )

+++−+⋅

σεε κννκ

ν 11/1

~~

Ims

s fffff . (4.3.50)

Using Eq. (4.3.41), it is obtained

( )

+=

2

2

22

1

2

1

)sin(4)cos(

rfgares

resmresinfb

d

kLakL

Zpp

πρ

+

− rfgares

resm

d

kLa

Z1

Im)2sin(4

2πρ

. (4.3.51)

Substitution of Eq. (4.3.51) into (4.3.50), the cooling power can be rewritten as

{ } [ ]reg

regsregreginC LTAKK/Q

)/1(1)0.1(2 0231

2

1

τψψ −−+−Θ+ΘΘ= p& ,

(4.3.52) where the new functions 1Θ and 2Θ are defined as:

( )

+

+=Θ

−− rfgares

resm

rfgares

resmres d

kLa

d

kLakL

ZZ1

Im)2sin(4)sin(4

)cos( 2

2

2

21 π

ρπρ

(4.3.53)

( )( )( )

++−−−

+=Θ

σεβθθθθ

ωρψ νκ

ν 11

~~

1~Im2

432102

s

m

fbfbregm

regreg ffTf

L

RA

ZZ

( )( )( )

++−−−

++

σεβ

πθθρ

ρθθπ

ωρψ νκ

ν 11

~~

1Re4

42 22

42312

0

s

m

fbfb

m

m

fb

regm

regreg ffTf

d

a

a

d

L

RA

Z

100 Chapter 4

( )

+

+−

−− 2

3222

32

2

22

2

20

3

0 ]Im[

241/1

12fb

fb

m

fb

m

fb

fbregregm

pregreg

a

d

a

d

LL

RTcA

Z

Z

Z ρθθπ

ρθπθτ

σρωψ

( )( )( )( )

+++−+⋅

σεε κννκ

ν 11/1

~~

Ims

s fffff . (4.3.54)

The functions 1Θ , 2Θ and 3Θ only depend on the specific configuration and working

temperature conditions. It should be emphasized that this final result is obtained under some assumptions, which are stated during derivation: no blockage ideal heat exchangers; no losses outward; no mean fluid flow; no temperature gradient along the pipes except the regenerator. Coefficient of performance (COP) From Eqs. (4.3.26) and (4.3.27), it can be obtained that the input acoustic work to the system from the driver is

[ ] [ ]inputinputinininW 1111

~Re

21~

Re21

UpUp ==& . (4.3.55)

Using Eqs. (4.3.33) and (4.3.34), the volume velocity can be rewritten as

−−=−= )sin(

4)cos( 1

2

31111 tbm

tbtbfboutfbinput kL

a

dikL

ρπ D

DpUUU

++ )sin(

4)cos(1 2

2

41

tbm

tbtb

fb

fb kLa

dikL

ρπ D

D-Z

p. (4.3.56)

Substitution of Eq. (4.3.30), (4.3.33) and (4.3.56) into (4.3.55) yields

4

2

121 Θ⋅= fbinW p& , (4.3.57)

where 4Θ is defined as

( ) ( )

+

−=Θ )cos(

~)sin(

~4

1)cos(1Re 4123

44 tbtbtb

mtb kLkL

d

aikL -DD

DD-

πρ

+

− )sin(4

)cos(~

/)sin(4

~

24

22

2

tbtb

mtbtb

m

tb kLd

aikLkL

a

di

πρ

ρπ D

DD

. (4.3.58)

Using Eq. (4.3.51), the input acoustic power can be rewritten as

41

2

121 Θ⋅Θ⋅= ininW p& . (4.3.59)

Thus, by the definition of coefficient of performance for a refrigerator, it is given by

[ ]regin

regsregreg

in

C

L

TAKK

W

QCOP

)/1(1)0.1(22

41

2

1

0

4

23 τψψ −ΘΘ⋅

−+−

ΘΘ+Θ==

p&

&. (4.3.60)

Traveling-wave systems 101

Despite the laborious amount of computation these expressions give an analytical means of studying a traveling-wave system. So if the configuration and the temperatures at both ends of the regenerator are known, the efficiencies and powers are readily computed. Computational validation of the model To test the analytical model, a comparison between results, computed by the proposed analytical equations, and by DeltaE [38] is made for the thermoacoustic engine described in S. Backhaus and G. W. Swift’s paper in 2000 [34] and described in the appendix of his book [38], as shown in Fig. 4.3.6. The system consists of a ¼-wavelength resonator filled with 30-bar helium, a torus-shaped section containing a regenerator that is sandwiched by two heat exchangers, and a variable acoustic load. It will be described in detail below. The main components are as listed: 1→2: The main cold heat exchanger is of shell-and-tube construction consisting of 299 stainless-steel tubes with 2.5-mm inside-diameter with wall thickness of 0.7 mm, and length of 20-mm-long tubes welded into two 1.6-mm-thick stainless-steel plates. The diameter of the heat exchanger is 9.5 cm. It is cooled by chilled water of 15°C. 2→3: The regenerator is made from a 7.3-cm-tall stack of stainless-steel wire screens machined to a diameter of 8.89 cm. The wire screen has mesh 120 and wire diameter of 65 µm. These wire screens are contained by a thin-wall stainless-steel can. The detailed description about this kind of regenerator material, stainless wire screen, and its relative parameters will be explained later in section 4.4.2.2. The volume porosity of the regenerator is given as 0.72 and the hydraulic radius is given as 42 µm in the paper [34]. 3→4: The hot heat exchanger consists of a 0.64-cm-wide by 3.5-m-long Ni-Cr ribbon wound zigzag on an alumina frame. The ribbon is divided into 3 equal-length segments and driven with 208-V three-phase power in a delta configuration. 4→5: The thermal buffer tube (TBT) is a tapered, 24-cm-long open cylinder tube of Inconel 625. The wall thickness is 4.0 mm. The upper 8.0 cm of the TBT is a straight cylinder while the lower 16.0 cm is flared in diameter from 8.9 cm to 9.6 cm with a 1.35° half-angle taper. 5→6: The secondary heat exchanger is water-cooled and of shell-and-tube, used to maintain the lower end of TBT at room temperature. It contains 109 4.6-mm inside-diameter, 10-mm-long stainless-steel tubes welded into two 1.6-mm-thick stainless-steel plates. 6→7→8: The junction is a standard-wall, 3 ½-in. nominal, stainless-steel tee. The inside diameter is around 9.0 cm.

102 Chapter 4

Figure 4.3.6: A schematic drawing of the engine, resonator and variable acoustic load. a and b are figures from reference paper [34]. c is schetch of the whole engine.

1 2 3 4 5 6

7 8

9

10

11 12

13 14 15 16

c

Traveling-wave systems 103

8→11: The feedback inertance is composed of three separate sections. The first section (8→9) is a 3 ½-in. to 3-in. nominal. long-radius reducing elbow. The centerline length of the elbow is 20.9 cm, and the final inside diameter of the elbow is 7.8 cm. The second section (9→10) is a 3-in. nominal, stainless-steel pipe of 25.6 cm long. The third section (9→10) is a machined cone that adapts the 3-in. nominal pipe to the compliance. Its initial inside diameter is 7.8 cm and end is enlarged to 10.2 cm by a taper angle of 13.5°. Its length is 10.2 cm. 11→12: The compliance consists of two 4-in. nominal, short-radius 90° elbows made from carbon steel. Its internal volume is 2830 cm³. 12→1: Between the compliance and the main cold heat exchanger is a device termed as “jet pump”, used to prevent Gedeon streaming. 7→16: The resonator consists of three sections. The first section (7→13) is a machined cone that adapts the 3 ½-in. nominal tee to a 4-in. nominal, carbon-steel pipe. The initial inside diameter 9.0 cm is enlarged to 10.2 cm over a length of 10.2 cm, giving a 6.8° taper angle. The main section of the resonator (13→14) is a 1.9-m length of 4-in. nominal, carbon-steel pipe. The inside diameter is 10.2 cm. The last section of the resonator includes a 7° cone (14→15) which enlarges the inside diameter of the resonator from 10.2 cm to 25.5 cm over a length of 1.22 m. The large end of the cone is closed with a 25.5-cm-diameter pipe with an approximate length of 52 cm, terminated by a 2:1 ellipsoidal cap (15→16). The operation condition described in Swift’s book [38], page 272-280 of appendix B.4, is used as comparison example. As the acoustic load is not clear for the author in the computation case in the book [38], the acoustic pressure at the junction tee is designated as the input parameter in1p for the computation, which is given as

057231.2 += Ep Pa in the book [38]. The temperatures at the main cold heat

exchanger and hot heat exchanger are given as ambient temperature 300 K and 900 K, respectively. The working frequency and mean pressure are given in the book: f=84.12 Hz and Pm=3103 kPa. The necessary geometrical parameters are used as listed in the above paragraph. Gas model and properties like viscosity is described in the model in section 4.3. The code is built in Fortran (appendix H) and follows the sequence: 1. Input of the geometrical parameters, ambient temperature 300 K and hot heat

exchanger temperature 900 K, using Eq. (F.38 a b c d) and (F.39 a b c d) to compute the real and complex coefficients θ1 θ2 θ3 θ4 D1 D2 D3 D4 .

2. Use Eq. (4.3.33) and (4.3.36) to obtain Zfb and Zinput . 3. The volume velocities are given by

inputininput ZU /11 p= and fbinfb ZU /11 p= .

4. Use Eq. (4.3.28) and (4.3.29) to compute P1out and U1out .

104 Chapter 4

The results using the analytical lumped-element model discribed by Backhaus and Swift [34] are also presented in the following table as comparison.

DeltaE

Results from the

related analytical

equations of this

chapter

Results from

lumped-element

model [ 34]

Heating power

into hot heat

exchanger [W]

3037 3108 11042

inputU1

(amplitude,

phase) [m³/s,

degree]

0.29, -89.5 0.26, -83.7 0.17, -62.2

fbU1

(amplitude,

phase) [m³/s,

degree]

0.20, 86.4 0.18, 84.1 0.08, 69.3

outU1

(amplitude,

phase) [m³/s,

degree]

8.63E-2, -77.6 9.21E-2, -59.9 0.96E-2, -20.7

21locationU

(amplitude,

phase) [m³/s,

degree]

8.56E-3, 12.2 8.65E-3, 12.7 2.79E-2, -19.9

31locationU

(amplitude,

phase) [m³/s,

degree]

3.02E-2, -46.6 3.02E-2, -46.4 0.96E-2, -20.7

Table 4.3.I: Comparison of computation from DeltaE, analytical model developed in this chapter, and lumped-element model [34]. The volume velocities at the three branches of the tee, shown in figure 4.3.7, and the input heating power at the hot heat exchanger are computed by the analytical model described in the previous section and also by the lumped-element in reference [34]. The flows at location 2 and 3 are given and compared as well.

Traveling-wave systems 105

Figure 4.3.7: A schematic drawing of the tee and pressures and velocities in three branches. The results are listed in table 4.3.I. The results from the proposed analytical model and those from DeltaE provided by the book [38] agree well with each other. But the results from the lumped-element model differ much from those of DeltaE. Therefore, the proposed analytical model is more accurate than the lumped-element model. Prediction of influence of geometrical parameters on the performance of an example refrigerator The thermoacoustic engine of Backhaus and Swift’s [34], as shown in Fig. 4.3.6, was assumed to operate in the cooling mode, used as an example refrigerator in this computation. The geometry can be found in the description following Fig. 4.3.6. In the refrigerator mode, the main cold heat exchanger (1→2) works as an ambient heat exchanger and the hot heat exchanger (3→4) becomes cold heat exchanger. The resonator tube (7→16) was a quarter-wave-length tube is Swift’s engine. In this computation, the resonator tube (7→16) was extended to a half-wave-length tube. It is assumed that input acoustic pressure in1p at one end of the resonator

connecting the driver is constant 057231.2 += Ep Pa. The working frequency and

mean pressure for the engine mode given in the book [38]: f=84.12 Hz and Pm=3103 kPa remain the same. The working gas is helium like in the engine mode. Assume that this refrigerator is driven by a driver and the ambient heat exchanger is maintained at environmental temperature around 300 K. From Eq. (4.3.37), it can be concluded that the complete acoustic impedance of the refrigerator only depends on the geometry and working condition parameters (TC, Pm and working gas). If the cold-end temperature is given, the influence of any of the relevant geometrical parameters on the impedance can be investigated by using Eq. (4.3.37). We assume that the cold-end temperature is 200 K. For every computation, only one geometrical parameter is varied at constant further settings. The geometrical parameters used in Eq. (4.3.37) (see appendix F) which are varied individually at each time in computation are listed in the table 4.3.II. In table 4.3.II, the parameters having a big influence on the total acoustic impedance are marked by “(!!)”; less important parameters for impedance are marked with “(!)”. Those parameters

in1p

out1p out1U input1p

fb1p

fb1U input1U

106 Chapter 4

having negligible influence are not marked. The important parameters are discussed below as well as the operating frequency.

component length diameter hydraulic radius rh

porosity

resonator tube Lres (!!) dres × ×

feedback inertance tube Lfb (!!) dfb (!) × ×

compliance tube Lcpl (!) dcpl (!) × ×

regenerator Lreg (!!) dreg (!) rh-reg (!!) ψreg (!!)

tube segment from the cold end heat exchanger to the center of tee junction

L tb dtb × ×

Table 4.3.II: Geometrical parameters in Eq. (4.3.37) and appendix F. 1. Dependency on the resonator tube length If only the resonator tube length is varied, the complete acoustic impedance changes in amplitude and phase as shown in Figs. 4.3.8 and 4.3.9. The effect on COP and relative COP is given in Fig. 4.3.10. As seen in Figs. 4.3.8 and 4.3.9, both the amplitude and phase of the complete acoustic impedance have a large variation with the resonator tube length. So tuning of this length in experimental set-ups will be important.

0 1 2 3 4 5 6 7 84.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Log(

ampl

itude

of c

ompl

ete

acou

stic

impe

danc

e of

the

refr

iger

ator

Za_

rfg)

Resonator tube length (m)

Figure 4.3.8: The amplitude of the complete acoustic impedance of the example refrigerator as a function of the resonator tube length.

Traveling-wave systems 107

0 1 2 3 4 5 6 7 8

-100

-80

-60

-40

-20

0

20

40

60

80

100

Pha

se o

f com

plet

e ac

oust

ic im

peda

nce

of th

e re

frig

erat

or Z

a-rf

g

Resonator tube length (m)

Figure 4.3.9: The phase of the complete acoustic impedance of the example refrigerator as a function of the resonator tube length.

0 1 2 3 4 5 6 7 8

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

CO

P

Resonator tube length (m)

COP Relative COP to Carnot's

Figure 4.3.10: COP and relative COP as a function of the resonator tube length. In Fig.4.3.10, the COP and relative COP do not show a strong dependency on the resonator tube length. With respect to the acoustic coupling between driver and thermoacoustic device, it is vital to choose the suitable resonator length in the design.

108 Chapter 4

2. Dependency on the feedback inertance tube Now the feedback inertance tube length is varied. The original geometrical parameters except the length of the feedback inertance tube in Eq. (4.3.37) and related other equations are used. The corresponding complete acoustic impedances at various lengths of the feedback inertance tube are obtained. Although the variation of the amplitude of the complete acoustic impedance is not so large in Fig. 4.3.8, the phase shows a strong dependency. The variation of COP and relative COP is given in Fig. 4.3.11. As seen in Fig. 4.3.11, the COP and relative COP have strong dependency on the feedback tube length. Although Fig. 4.3.11 shows a high COP at zero feedback inertance tube length, it leads to a zero cooling power and a zero input acoustic power in practical operation. This effect is also shown by the computation results given by Fig. 4.3.12. Therefore, zero-feedback-tube-length is not a realistic option for the design of a traveling-wave refrigerator.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-4

-3

-2

-1

0

1

2COPRelative COP to Carnot's

CO

P

Feedback inertance tube length (m)

Figure 4.3.11: COP and relative COP as a function of the feedback inertance tube length. If only the diameter of the feedback inertance tube is varied, the results show that the complete acoustic impedance has a small variation of both the amplitude and the phase (about 4 degrees difference if the diameter varies from 0.04 m to 0.16 m).

Traveling-wave systems 109

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

-50000

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

cooling powerinput acoustic power

Pow

er (

W)

Feedback inertance tube length (m)

Figure 4.3.12: Cooling power and input acoustic power as a function of the feedback inertance tube length. 3. Dependency on the compliance tube Now the length and the diameter of the compliance tube are varied each at one time, the results show that the complete acoustic impedance has small variations in the amplitude and variation in the order of around 10-15 degrees in the phase on both the length (vary in the range of 0.1 m to 1.2 m) and the diameter of the compliance tube (vary in the range of 0.04 m to 0.2 m). 4. Dependency on the regenerator The computation has shown that the complete acoustic impedance is slightly dependent on the length, diameter and porosity of the regenerator, and the hydraulic radius of the wire gauzes. The results show small variation in the amplitude and the phase (around 4-5 degrees difference). That might be attributed to the small variation ranges for these parameters in computation by considering the practice. The regenerator is still a key component and important for further investigation. The influence of two important parameters of the regenerator material, porosity and hydraulic radius, on the COP and relative COP are given in Figs. 4.3.13 and 4.3.14. In Fig. 4.3.13, the efficiency increases with increasing porosity. In Fig. 4.3.14, the curve shows that there exists an optimum value of hydraulic radius, which makes a maximum efficiency. Fig. 4.3.14 shows a flat peak over a broad range of the hydraulic radius. These two important parameters will be discussed more in detail theoretically and experimentally in the following section 4.4.

110 Chapter 4

0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.4

0.5

0.6

0.7

0.8

0.9

COPRelative COP to Carnot's C

OP

Porosity of regenerator

Figure 4.3.13: COP and relative COP as a function of the porosity of the regenerator.

0 10 20 30 40 50 60

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

COPRelative COP to Carnot's

CO

P

Hydraulic radius of regenerator (µm)

Figure 4.3.14: COP and relative COP as a function of the hydraulic radius of the regenerator material. The influence of the regenerator length on the COP and relative COP are given in Figs. 4.3.15. The influence on cooling power and input acoustic power is given in Fig. 4.3.16. Fig. 4.3.15 shows that the regenerator has to be longer than a certain value to gain positive COP, and the COP increases with increasing regenerator length. When the regenerator length is larger than a certain value, the COPs almost

Traveling-wave systems 111

stay constant. In Fig. 4.3.16, the cooling power and input acoustic power curves first increase with increasing regenerator length, then decrease slowly.

0.00 0.05 0.10 0.15 0.20 0.25

-2

-1

0

1

2

COPRelative COP to Carnot's

CO

P

Regenerator length (m)

Figure 4.3.15: COP and relative COP as a function of the regenerator length.

0.00 0.05 0.10 0.15 0.20 0.25

-1000

0

1000

2000

3000

4000

5000

cooling powerinput acoustic power

Pow

er (

W)

Regenerator length (m)

Figure 4.3.16: Cooling power and input acoustic power as a function of the regenerator length. 5. Dependency on the tube segment from the cold end heat exchanger to the center of tee junction The results have shown that the complete acoustic impedance is slightly dependent on the length and diameter of this tube segment.

112 Chapter 4

6. Dependency on operating frequency The original geometrical parameters were used and the cold end temperature was set as 200 K. Helium gas at filling pressure of Pm=3103 kPa is used. The operating frequencies were varied as input into Eq. (4.3.37) and the complete acoustic impedance changed in amplitude and phase as shown in Fig. 4.3.17 and 4.3.18. In Figs. 4.3.17 and 4.3.18, the amplitude and phase of the complete acoustic impedance have a large variation with operating frequency, which are analogous to Figs.4.3.8 and 4.3.9. This shows that there is strong coupling between the resonator tube length and operating frequency. The influence of operating frequency on the efficiency and relative efficiency is given in Fig.4.3.19. Fig.4.3.19 shows that an optimum operating frequency exists.

0 20 40 60 80 100 120 140 160 1803.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Log(

ampl

itude

of c

ompl

ete

acou

stic

impe

danc

e of

the

refr

iger

ator

Za_

rfg)

Frequency (Hz)

Figure 4.3.17: The amplitude of the complete acoustic impedance of the example refrigerator as a function of operating frequency.

Traveling-wave systems 113

0 20 40 60 80 100 120 140 160 180

-100

-80

-60

-40

-20

0

20

40

60

80

100

Pha

se o

f com

plet

e ac

oust

ic im

peda

nce

of th

e re

frig

erat

or Z

a-rf

gFrequency (Hz)

Figure 4.3.18: The phase of the complete acoustic impedance of the example refrigerator as a function of operating frequency.

0 20 40 60 80 100 120 140 160 180

-3

-2

-1

0

1

2

3

COPRelative COP to Carnot's

CO

P

Frequency (Hz)

Figure 4.3.19: COP and relative COP as a function of operating frequency.

114 Chapter 4

4.4 Optimizing regenerator material 4.4.1 Introduction The regenerator is a key component in a traveling-wave system, where the heat exchange between the working gas and the solid filling material takes place to realize the thermal cycle. A regenerator consists of a porous medium. A variety of porous materials can be chosen: ceramic honeycomb with different channel shapes, metal honeycomb, stainless steel wire screens, for instance. Therefore, the thermal and geometrical properties of the regenerator materials are of great interest in optimizing the performance of a traveling-wave system. In this section, the optimization of regenerator material is investigated experimentally. Samples of ceramic and metal honeycomb in different channel sizes, and stainless steel wire screens in different hydraulic radii and porosities, were applied in a coaxial traveling-wave engine to characterize their performance. In practice, stainless steel wire screens are widely used as filling material for regenerators. Therefore, stainless steel screens are the main concern of this experiment. This work is an extension of the TASOR project (Thermo Acoustic Systems to Upgrade Waste heat) which is supported by Senter Novem. The TASOR project is cooperation with the Energy research Center of the Netherlands (ECN) who coordinated the project, Huisman Elektrotechniek, Aster Thermoacoustics, NRG who makes the computational fluid dynamic calculations and the Eindhoven University of Technology (TUE) who is responsible for the research to regenerators of a thermoacoustic engine. 4.4.2 Regenerator materials 4.4.2.1 Ceramic and metal honeycomb regenerator Ceramic and metal honeycomb regenerator materials can differ widely in the shapes and sizes of channels. Photos of two samples of ceramic and metal honeycomb regenerators are given in Fig. 4.4.1. In this study, the tested honeycomb samples all have the same geometry. They are cylinders consisting of many parallel channels with square-shaped cross-section, as sketched in Fig.4.4.2. Based on periodicity the geometry can be reduced to one single cell for analysis. The length of the cells is indicated asregL . For one cell, the inner size is denoted

asd , and the wall-thickness is denoted asδ . CPSI CPSI, “cells per square inch” in full, is an important quantity to characterize the amount of cells per unit area of cross section. For square shaped cells, it can be calculated as

Traveling-wave systems 115

24.25

+=

δdCPSI , (4.4.1)

where d and δ are specified in mm. Figure 4.4.1: Photos of a ceramic and a metal honeycomb regenerators.

Figure 4.4.2: Sketch of a honeycomb regenerator. Hydraulic radius and hydraulic diameter According to the definition of hydraulic radius for flow through a channel with arbitrary geometry, which is the ratio between the volume occupied by fluid to the “wetted area”, the hydraulic radius of square shape channel honeycomb regenerator is evaluated as

44

2d

Ld

Ld

A

Vr

reg

reg

wetted

fluidh =

⋅⋅== . (4.4.2)

The hydraulic diameter is equal to four times the hydraulic radius:

regL

One cell

δ

d

116 Chapter 4

dA

VD

wetted

fluidh == 4 . (4.4.3)

Porosity The porosity is defined as the ratio between the volume, occupied by fluid, to the total volume of the regenerator. It is sufficient to consider only one cell. Thus, the porosity can be expressed as

( ) ( )2

2

2

2

δδψ

+=

⋅+⋅

==d

d

Ld

Ld

V

V

reg

reg

total

fluidreg . (4.4.4)

Sometimes this is also named as OFA, open fluid area. 4.4.2.2 Regenerator of stainless steel wire screens For wire screen regenerators, the situation is more complex than for square shaped channel honeycombs. The wires are curved, and one sheet overlaps another one in a random way when they are packed into a regenerator. A photo of regenerator screens and the top view and side view of one screen are given in Fig.4.4.3. Some assumptions have to be made before calculating the parameters hydraulic radius and porosity. The diameter of the wires is indicated as δ and the inner size of one cell is noted asd . A sketch of one cell is given in Fig.4.4.4 (a). We assume that the wires of a screen are not curved, but perfect cylinders. Thus, a cell is reduced to a structure confined by four half-cylinders (the wires) perpendicular to each other at the corners. The length of one single half-cylinder is δ+d , as shown in Fig.4.4.4 (b). A cell in principle has a thickness of twice the wire diameter because of the crossing of the wires. In order to find a realistic estimation of one cell thickness, the pattern how several screens are packed has to be investigated. For randomly packed screens, they can be shifted or rotated with respect to each other, so that the squared cells do not lie just above each other, as shown in Fig. 4.4.5 (a). In the following, it is assumed that the squared spacings can be shifted in only two directions, shown in Fig. 4.4.5 (b), not allowed to rotate. This shifting means that the square spacings can partially be seated into each other, if more screens are stacked up. Thus, the effective thickness of one screen layer is smaller thanδ2 , as shown in the lower line of Fig. 4.4.5. The effective thickness is evaluated by pfγ , which is defined as the screen packing density. So 2=pfγ

corresponds with the non-shifted situation. The normal value of pfγ is expected to

be around 2. By using the above modeling for wire screen regenerators, the hydraulic radius, hydraulic diameter and porosity can be obtainable as shown below.

Traveling-wave systems 117

Figure 4.4.3: (a) Photo of regenerator screens and its top (b) and side views (c) of one screen.

Figure 4.4.4: A sketch of one cell in a wire screen regenerator and the simplified unit cell. Mesh Similar to CPSI used in honeycomb regenerators, mesh, M is frequently used to count the number of aligned cells per inch in a screen. It is obtained by

δ+=

dM

4.25, (4.4.5)

where d and δ are in mm.

(a) One cell and its characteristic lengths

(b) One cell under assumptions and its characteristic lengths

(a)

(b)

(c)

118 Chapter 4

Figure 4.4.5: Sketches of the packing of the wire screens. The upper line describes the case where the screens are perfectly aligned so that every screen layer occupies twice the wire thickness. The lower line describes the case where the screens are shifted in only two directions so that one screen layer occupies less than twice the wire thickness. Hydraulic radius and diameter The fact that the cells are not aligned with each other due to screens shifting is not taken into account, but the effective thickness regarding to the factor γ caused by

shifting is considered. By this assumption, the complex structure is reduced to the repetition of one cell so that one cell is sufficient to be considered instead of the whole regenerator. Then, the total volume occupied by one cell is

( )2δδγ +⋅= dV pftotal . (4.4.6)

The volume of gas equals the total volume minus the volume of wires:

( ) ( )δπδδδγ +⋅⋅−+⋅=−= ddVVV pfwirestotalfluid 421

42

2. (4.4.7)

The wetted area is

( )δδπ +⋅= dAwetted 21

4 . (4.4.8)

By using the definition of hydraulic radius and diameter, they are expressed as:

δγ pf δγ pf

Traveling-wave systems 119

( )π

δπδγ2

2/−+==

d

A

Vr pf

wetted

fluidh , (4.4.9)

( )π

δπδγ2

244

−+==

d

A

VD pf

wetted

fluidh . (4.4.10)

Porosity By similar calculations, the porosity is given by

( )δγδπψ

+−==

dV

V

pftotal

fluidreg 2

1 . (4.4.11)

4.4.2.3 Thermal conductivity By using Fourier’s law, the heat flux in the axial direction is

dx

dTKq m

eff−=& , (4.4.12)

where effK is effective thermal conductivity of the porous regenerator.

If the regenerator has a cross sectional areaA , as shown in Fig.4.4.6 (a), the total

heat flow condQ& through the regenerator is

dx

dTKAQ m

effcond −=& . (4.4.13)

Figure 4.4.6: Sketches of a regenerator and the analogous model as two parallel electrical resistors. In practical situation, the heat flow is complicated. Because the solid part is not a massive rod and the fluid is not homogeneous tube flow. In this work, the heat flow is modeled analogous to a current in a resistor. The fluxes through the thermal

(a) A regeneraor

regL

HotT ColdT

Solid

Fluid

(b) The regenerator modeled as two parallel electrical resistors.

A

120 Chapter 4

resistance of solid part and working gas are analogous to current in parallel resistors, as shown in Fig. 4.4.6 (b). Assume that the solid part and the working gas have the same temperature gradient in the x direction. Thus, the total heat flow consists of heat flow through the solid part and working gas:

dx

dTKA

dx

dTKAQQQ m

fluidm

ssolidfluidsolidcond −−=+= &&& . (4.4.14)

By using the porosity, the areas of solid part and working gas are given by ( )regsolid AA ψ−= 1 and regfluid AA ψ= (4.4.15)

Substitution of Eq. (4.4.15) into (4.4.14), it yields

( )[ ]dx

dTKKAQ m

regsregcond ⋅+⋅−−= ψψ1& . (4.4.16)

By comparison of Eq. (4.4.13) and (4.4.16), the effective thermal conductivity is ( ) KKK regsregeff ⋅+⋅−= ψψ1 . (4.4.17)

In this way an estimation of the thermal losses can be made, neglecting any convective effects and contact to contact thermal resistivities. 4.4.3 Selection criteria There are three main criteria to choose a regenerator material: First criterion The hydraulic radius may not be too large, otherwise the oscillating gas parcels in the regenerator will not be locally isothermal, nor too small, which could result in losses due to large fluidic resistance. For the best performance, there should be an optimum ratio between the hydraulic radius and the working gas’ thermal penetration depth. Here, the first guess is 1/3 of a reference thermal penetration depth, i.e. gashr δ⋅≈ 3/1 (the reference thermal penetration depth gasδ is evaluated

with the operation frequency of the system, operation mean pressure and the working gas property parameters at the average value of the hot and cold-end temperatures of the regenerator). The specific value of the ratio will be found later by comparison of the performance of different regenerators. Considering the correlation between the hydraulic radius and porosity, an optimum value for porosity is investigated as well. First, this criterion for hydraulic radius and porosity can be checked by the theory described in section 4.3. Considering that the experimental set-up for measurements is a coaxial traveling-wave engine (the details are given in the next section 4.4.4), the formula for engine mode is used in the following computation. All the stainless steel screen regenerators, which are measured in this study, are computed and plotted in the figures. In all computations, the configuration of the engine, the hot and cold-end temperatures, and the given input acoustic pressure at

Traveling-wave systems 121

some position are the same for all computations of different stainless steel screen regenerator samples. In other words, the working conditions remain the same for all computations. Since the experimental set-up is a coaxial traveling-wave engine, meaning that the geometry is not a clearly defined looped configuration which is used in section 4.3, the geometrical input parameters are approximated and adapted to a looped configuration. The thermal penetration depth used as the reference

gasδ for the hydraulic radius in Figs. 4.4.7, 4.4.8 and 4.4.9 is 166 µm. This value is

for a temperature of 490 K of argon, mean pressure of 10 bar and frequency of 59 Hz being the average setting of the traveling-wave system. The efficiency of the engine as a function of porosity and of the ratio of hydraulic radius and the reference penetration depth is plotted in Fig. 4.4.7. The graph shows clearly that there is an optimum in hydraulic radius, and that there is a trend that larger porosity leads to larger efficiencies.

0.0 0.1 0.2 0.3 0.40.5

0.60.7

0.80.9

0.00.51.01.52.02.53.03.54.04.55.05.5

6.0

6.5

7.0

0.00.1

0.20.3

0.40.5

0.60.7

0.80.9

Effi

cien

cy [%

]

Hydra

ulic

radi

us/

Ther

mal

pen

etra

tion

dept

h

Porosity

Figure 4.4.7: Theoretical predictions for stainless steel screen regenerators: efficiency as a function of porosity and the ratio of hydraulic radius and the reference thermal penetration depth (i.e.166µm).

122 Chapter 4

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

Effi

cien

cy [%

]

Hydraulic radius/Thermal penetration depth

Porosity 40% Porosity 60% Porosity 80%

Figure 4.4.8: Theoretical predictions for stainless steel screen regenerators: efficiency as function of the ratio of hydraulic radius and the reference thermal penetration depth at fixed porosity.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

Eff

icie

ncy

[%]

Porosity

rh/δ

gas=0.1

rh/δ

gas=0.2

rh/δ

gas=0.3

rh/δ

gas=0.5

rh/δ

gas=0.7

Figure 4.4.9: Theoretical predictions for stainless steel screen regenerators: efficiency as function of porosity at fixed ratio of hydraulic radius and the reference thermal penetration depth. The efficiency at a fixed porosity as function of ratio of hydraulic radius and the reference penetration depth is given in Fig. 4.4.8. The computation shows that

Traveling-wave systems 123

indeed there exists an optimum value of the ratio for a fixed porosity. It is also seen that the optimum value for the ratio moves to smaller values when the porosity becomes larger. The efficiency at a fixed ratio of hydraulic radius and the reference penetration depth as function of porosity is given in Fig. 4.4.9. Generally, the performance increases with porosity. Due to the absence of influence of heat capacity of the solid part of a regenerator in the model described in section 4.3, the performance deterioration that must occur at porosities above 90% is not seen in Fig. 4.4.9. Therefore, the prediction based on the model is not reliable when the porosity is above 90%. Second criterion During the acoustic cycle, heat will penetrate into the regenerator material over a distance characterized by the solid’s thermal penetration depth. Therefore, it has no advantage to employ relatively thick wires. The efficiency will decrease if useful space is wasted by using relatively thick solid material. Therefore, the diameter of wire δ should be less than twice the solid material’s thermal penetration depth,

sδδ 2< . (4.4.18)

This criterion will determine the porosity of the regenerator. Tijani et al. showed that 2yo ≥ 8δs for a plate of stack and maybe also for a wire [88]. Third criterion The ratio between the heat capacity of the regenerator material and the heat capacity of the working gas should be large. A large heat capacity means that a large amount of energy can be stored per unit volume, which ensures no temperature oscillation in the solid material to guarantee a stable local temperature. In the present study, a required ratio of 10 is assumed:

10>gaspm

sss

Vc

Vc

ρρ

. (4.4.19)

sV and gasV are the volumes of solid material and the working gas.

These criteria are plotted for the honeycomb materials in Figs. 4.4.10 and 4.4.11, and for the screen materials in Figs. 4.4.12 and 4.4.13. By the three criteria, the good choices for a regenerator material should be in the range which is below the dash-dot line (selecton criterion 3), left to the dash line (selecton criterion 2), and close to the solid line (selecton criterion 1).

124 Chapter 4

0 0.05 0.1 0.15 0.2

10-1

100

101

Wall thickness [mm]

Ope

n ch

anne

l dia

met

er [

mm

]

selection criterion 1selection criterion 2selection criterion 3test samples

Figure 4.4.10: Selection criteria for ceramic honeycomb regenerators (logarithmic plot).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

Wall thickness (delta) [mm]

Ope

n ch

anne

l dia

met

er (

d) [m

m]

selection criterion 1

selection criterion 2selection criterion 3

measured honeycombs

Figure 4.4.11: Selection criteria for ceramic honeycomb regenerators (linear scale).

Traveling-wave systems 125

0 0.1 0.2 0.3 0.4 0.5

10-1

100

101

Wire thickness [mm]

Ope

n w

ire d

ista

nce

[mm

]

selection criterion 1selection criterion 2selection criterion 3test samples

Figure 4.4.12: Selection criteria for stainless steel screen regenerators (logarithmic plot).

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Wire thickness (delta) [mm]

Ope

n w

ire d

ista

nce

(d)

[mm

]

selection criterion 1selection criterion 2selection criterion 3measured screens

Figure 4.4.13: Selection criteria for stainless steel screen regenerators (linear scale). 4.4.4 Experimental set-up: a coaxial traveling-wave engine The experimental set-up is shown in Fig.4.4.14. It consists of a coaxial half-wave-length thermoacoustic engine, a resonator tube, a variable acoustic load, measuring devices, and data acquisition equipment.

126 Chapter 4

Figure 4.4.14: Schematic view of the coaxial traveling-wave engine test set-up.The regenerator part is shown in the left-below corner, marked out by an ellipse. The components are explained below. The Resonator tube The resonator tube is made of a symmetric tube of stainless steel which is able to withstand a pressure up to 20 bar. The dimensions in mm and geometry of the resonator tube are given in Fig.4.4.15. It is mounted on dampers to minimize the transfer of vibration. One end is closed by an end flange which is fixed with bolts and nuts. The thermoacoustic engine is mounted inside of this end. The other end is empty and connected to a variable acoustic load by a flange.

Figure 4.4.15: Dimensions of the resonator tube (in mm). Thermoacoustic engine Fig. 4.4.16 (a) and (b) show the so-called insert of the thermoacoustic engine. From bottom to top, the engine consists of several elements: membrane, cold-end heat exchanger (cooler), regenerator, hot-end heat exchanger (heater) and two flow

440 440 920

95.5 54.8 95.5

Traveling-wave systems 127

straighteners. Two cooling liquid flow tubes connect the heat exchanger to a thermal Julabo cooler, as seen in Fig. 4.4.16 (a). A thermal insulation ring of Vespel surrounds the cooler, regenerator and heater. It is placed between the holder tube and the part consisting of cooler, regenerator and heater. The insulation ring has a thickness of 0.52 mm and is able to withstand a temperature up to 400 °C. The holder tubes are supported by three identical threaded rods. The rim of the holder of the thermoacoustic engine is mounted at a distance of 130 mm from the end flange of the resonator tube. The elements are described in detail below. Membrane: This is a latex diaphragm closing the lower part of the engine to prevent Gedeon streaming. This streaming refers to a non-zero time average mass flow in the toroidal space (around the cylinder formed by the regenerator and the heat exchangers in this experiment set-up) shuttling heat from hot end to the cold end. The membrane allows the gas to oscillate, but blocks the net mass flow at this position.

Figure 4.4.16: The coaxial thermoacoustic engine.

(a) Photo of the engine. The upper part points to the load side of the resonator tube.

(b) Sketch of the engine and all the components with their positions in the tube.

128 Chapter 4

Figure 4.4.17: The heater and cooler of the thermoacoustic engine. Cold-end heat exchanger (cooler): The temperature of the cold end of the regenerator is fixed by a heat exchanger with cooling water. The photo is given in Fig. 4.4.17 (a). A Julabo HL-4F32 cooler is used to maintain the temperature of the water at a constant temperature of 22 °C. Regenerator: The regenerator is a wire-screen filled cylinder with a height of 11.2 mm and a diameter of 55.0 mm. Wire screens are stacked up to fill the available space. All the tested samples used in this experiment fit in this space. Hot-end heat exchanger (heater): The engine is driven by heat generated with an electric heater. A spiral shaped wire heater is mounted on a perforated plate. The heater is connected to an electric power supply from Delta Elektronica type SM3004-D. The power is varied to heat the hot end of the regenerator to a maximum temperature of 450°C. The heater is shown in Fig. 4.4.17 (b). First flow straightener: A first flow straightener is positioned directly above the heater. It is made from porous metallic material and used to make the acoustic flow uniform. Second flow straightener: A second flow straightener is mounted 10 cm further in the heater tube. It functions similarly as the previous one, but with larger pores. The variable acoustic load A variable acoustic load is connected to the end of the resonator tube. It consists of a throttling valve and a buffer vessel with a volume of 1 dm³, as schematically given in Fig. 4.4.18. The dissipated acoustic power can be measured by comparing the phase and amplitude of the acoustic wave at the end of the resonator with those in the buffer vessel.

(a) Photo of the cold-end heat exchanger (cooler)

(b) Photo of the hot-end heat exchanger (heater)

Traveling-wave systems 129

Figure 4.4.18: The variable acoustic load with pressure sensors. 4.4.5 Energy balance in the experimental set-up

Figure 4.4.19: The energy balance for the thermoacoustic engine. The energy flow out of the box 1 equals to the flow into box 2. The energy balance of the thermoacoustic engine is shown in Fig. 4.4.19. In this experimental setup, the heat to drive the engine is supplied by an electrical heater. This gross energy is partially lost to the environment through the walls of the resonator tube, and only a fraction is used by the engine to convert heat to acoustic energy. The whole system is approximately symmetric. Thus, if the system is divided into two parts at the middle of the resonator tube, the amount of acoustic power dissipated in one part approximately equals that in the other part, as seen in Fig. 4.4.19. The dissipated acoustic power,dissipatedP , is due to thermoviscous losses

by motion of gas near the walls of the tube and radiation out of the tube. The

acoustic energy −acousticP that leaves the left confined imaginary box 1 is:

dissipatedlossescoolerheateracoustic PPPPP21−−−=− . (4.4.20)

The acoustic energy +acousticP into the right part confined imaginarily by box 2 is

dissipatedloadacoustic PPP21+=+ . (4.4.21)

AC load

130 Chapter 4

acousticP is the acoustic power measured at the center of the resonator tube.

According to energy conservation, the energy that flows out of box 1 equals that entering box 2:

acousticacousticacoustic PPP == −+ . (4.4.22)

The acoustic energy dissipated in the variable loadloadP is in principle useful energy

that can be applied as power for instance to convert into mechanical energy. All energy losses in the engine section are counted into lossesP . They are heat

convection from the hot end of the regenerator to the cold end without any acoustic power production, thermal radiation losses, as well as Gedeon streaming (“DC” flow). Performance of the engine The performance of an engine can be evaluated by the raw efficiency η or the

fraction of the Carnot efficiency relativeη ( Carnotrelative ηηη /= ). The raw efficiency

η is the ratio of useful work, produced by the engine, to the total heat added to the

system. In this case, the energy loadP is taken as useful work, produced by the

system, in the consideration that the dissipated acoustic power along the walls, dissipatedP , is unavailable for practical use. Thus, the raw efficiency is

calculated by %100/ ×= heaterload PPη . (4.4.23)

4.4.6 Measurement equipment and data handling The main concerns of this measurement are the energy flows. To determine those, pressure sensors and temperature sensors are mounted. All the measuring devices and the connections to the computer are schematically shown in Fig. 4.4.20. There are six temperature sensors in the system. Two Pt1000 thermometers are used to measure the temperatures of the cooling water flowing in and out of the cooler, respectively. Three thermocouples are placed in the regenerator, one at the hot end, the second at the middle, and the third at the cold end, as the enlarged section of Fig. 4.4.20 shows. One thermocouple is placed between two flow straightners outside the regenerator in the thermal buffer tube zone. Six pressure sensors are mounted in the system. Three microphones are mounted in the wall around the middle part of the resonator tube to acquire the information for calculatingacousticP .

Two microphones are mounted around the variable acoustic load to calculate the acoustic power dissipated inside. One pressure sensor is mounted on the surface of the end flange connecting to the variable acoustic load, which is for the static mean pressure measurement.

Traveling-wave systems 131

Figure 4.4.20: Overview of the measuring devices and data acquisition. During the measurement, all data are collected with a National Instruments PXI-system and handled by a LabView program. The parameters are monitered via a LabView data acquisition system. The system composes two files for every measuring point. These data files are analyzed with a Matlab data processor. In the following, the measurement of every unit of the thermoacoustic system is discussed, as well as the calculation of energy flows. Cooling and heater power The supplied heat from the electrical heater is calculated by

IUPheater ⋅= , (4.4.24)

where U and I are the voltage and current of the power supply, respectively. The waste heat dumped at the cold end of the regenerator is removed by the cooling water through the cooler. The amount of heat coolerP is calculated by

( )inoutpwaterwatercooler TTcVP −⋅⋅⋅= ρ& , (4.4.25)

where V& is the flow rate of the cooling water. inT and outT are the temperatures of

the cooling water flowing in and out from the cooler, and are measured by two thermocouples. waterρ and pwaterc are density and specific heat of water,

respectively. Acoustic energy The acoustic intensity is determined from the signals of two adjacent pressure sensors, by using the average of the two signals to obtain the pressure and the difference between the two signals to obtain velocity. It is called the “Two-microphone method”, which is described in reference [62]. Three sensors are

132 Chapter 4

applied for generating three acoustic flow measurements. The signals of the three pressure sensors mounted halfway of the resonator tube are used to obtain the acoustic power following:

( )CBCBgas

acoustic ppx

AP φφ

ωρ−⋅⋅⋅

∆= sin

2, (4.4.26)

where A is the cross-sectional area of the resonator tube, and CB, are two points

with a distance of x∆ along the tube. Bp and Cp are the pressure amplitudes at

points B and C respectively. Bφ and Cφ are the phase angles of the pressure at

these points. Power to the acoustic load The power delivered to the variable acoustic load is determined by using a similar method as in reference [62]. Information from two microphones, one upstream of the throttling valve and the other in the buffer, is used. It is schematically illustrated in Fig.4.4.18. The energy dissipated by the throttling valve is given by

( )TETEload ppP

VP φφ

γω −⋅⋅⋅⋅= sin

21

0

0 , (4.4.27)

where γ is the ratio between the isobaric and isochoric specific heats and 0V is the

volume of the buffer vessel. 0P is the mean pressure in the tube, measured by a

static pressure sensor at the end of the tube in form of relative pressure compared to the atmospheric pressure. A dynamic pressure sensor upstream of the throttling valve measures the pressure amplitude Ep and phase angle Eφ . Similarly, another

dynamic pressure sensor in the buffer vessel measures the pressure amplitude Tp

and phase angle Tφ .

Dissipated energy

dissipatedP is the amount of acoustic energy that is dissipated along the walls of the

resonator tube. If the acoustic energy acousticP and the energy to the load loadP are

determined as described above, the dissipated energy can be calculated by Eq. (4.4.21)

( )loadacousticdissipated PPP −⋅= 2 . (4.4.28)

Energy losses in TA-engine The lost energy in the engine is mostly due to convection and radiation to the surroundings, as shown in Fig.4.4.19. With all the energy terms known by above calculations, the energy lossesP can be determined by using Eq. (4.4.20)

dissipatedacousticcoolerheaterlosses PPPPP21−−−= . (4.4.29)

Traveling-wave systems 133

4.4.7 Measuring procedure Preparation of the set-up For every measurement run, only one regenerator sample is tested. After the regenerator is placed in the holder and the system is closed, the system is pumped off by a rotation pump and an oil-diffusion pump. Then, the system is filled with the desired gas, helium or argon at an absolute pressure of 10 bar. While filling, the pressure can be read on the Labview window. Starting up The system produces stable acoustic resonance only if the temperature gradient is large enough, which means that the heat, fed by the heater, must exceed some threshold. In this set-up, the heater power can be varied by changing the voltage or current of the electric power supply. When the engine starts to generate sound and a stable resonance is produced, it can be seen on the oscilloscope visualization of the pressure signals. Then, the XY-visualization of the two outer microphones at the center of the resonator tube shows a stable ellipse and all the five dynamic pressure sensors show harmonic oscillation. At the point of threshold it is often observed that the sound intensity increases and, while the thermoacoustic heat pumping starts, the thermal balance in the regenerator is disturbed. This leads to an unstable mode with a periodicity of several minutes. So, when onset is reached, sound generation occurs, heat pumping starts, and because of the thermal balance in the regenerator being disturbed, the sound production decays to zero. Over a window of several watts, this unstable behavior occurs, but finally at a sufficiently large heat input a stable sound intensity is maintained. In the starting-up stage, no net acoustic energy is produced and the wave is not stable. Temperature oscillation occurs during this stage. De Waele [70] gave a detailed theoretical analysis on this onset temperature and oscillation stability of a traveling-wave engine with a torus-shaped section. Similar results were obtained in LeMans at the laboratoire acoustique, see Penelet [87]. Measuring To start the measurements of the performance of the engine, normally, a power of 10 W is fed to the heater. The temperature evolution can be followed on the Labview program window. After all the temperatures have reached stable values, data from all the sensors and thermocouples, as well as information for the heater and cooling water flow are stored in two files. For about thirty seconds, all the data are stored. Then, heater power is stepwise increased to start the next measurement. If the engine starts to produce sound, the throttling valve for acoustic load has to be opened to maintain a fixed drive ratio of 2.7%. The drive ratio is the ratio between the amplitude of the sound wave at the end of the resonator tube and the mean pressure inside the system. When more and more heat is added to the heater, the

134 Chapter 4

throtting valve must be opened more and more correspondingly to maintain a constant drive ratio. The steps are repeated for every energy input to the heater till the maximum temperature of the hot end of the regenerator reaches around 450 °C, which is the maximum temperature allowed. All the files are processed using a Matlab program to compute all the parameters of interest as a function of increasing heaterP , or as a function of the corresponding temperature difference

across the regenerator. The data of one measurement of 30 seconds are processed by the Matlab program to one average value as the value for analysis. List of standard measurement settings The measurements are conducted under standardized measurement settings needed for calculations, as listed below: Working gas: Argon or helium; Mean pressure: 10 bar; Temperature of cooling water flowing in the cold end heat exchanger: 22 °C; Drive ratio: 2.7%; Maximum temperature at hot end of the regenerator: 720 K; Mean regenerator gas temperature used for calculations: 490 K, which is an estimate of the mean temperature during a measurement. This temperature, 490 K, is also used in the tables below as average temperature to determine the properties of the gas, and related dimension(less) numbers. Regenerator samples The ceramic honeycomb regenerators were supplied by Corning and the metal honeycomb samples were manufactured at the University of Liverpool. The dimensions of all honeycomb regenerators were measured using a microscope. The wire screens are from Metaalgaasweverij Dinxperlo. The mesh number M and diameter of wires δ are listed by the manufacturer and used for calculations. Values of hydraulic radius hr , hydraulic diameter hD , and porosity regψ are

calculated by corresponding equations in section 4.4.2. The basic average gas’

thermal penetration depth averageκδ , which works as reference in later

computations is calculated by using the properties of the working gas at 10 bar and

490 K. For argon, averageκδ is 166 µm. Some important parameters of all the

samples, which are tested in this study, are listed in the table 4.4.I and 4.4.II.

Traveling-wave systems 135

ceramic honeycomb Ar 0.08 (±0.01) 0.55 (±0.01) 1600 40 137.5 550 0.76 1.2 0.5ceramic honeycomb Ar 0.06 (±0.02) 0.43 (±0.02) 2700 52 107.5 430 0.77 1.6 0.48ceramic honeycomb Ar 0.07 (±0.01) 0.34 (±0.01) 3800 62 85 340 0.69 2.0 0.64ceramic honeycomb Ar 0.06 (±0.01) 0.28 (±0.01) 7800 75 70 280 0.68 2.4 0.66ceramic honeycomb Ar 0.03 (±0.01) 0.09 (±0.01) 43300 208 22.5 90 0.56 7.4 0.89ceramic honeycomb Ar 0.03 (±0.02) 0.04 (±0.02) 106000 326 10 40 0.33 16.7 1.36ceramic honeycomb Ar 0.02 (±0.01) 0.08 (±0.01) 46300 215 20 80 0.64 8.3 0.74ceramic honeycomb Ar 0.02 (±0.01) 0.13 (±0.01) 30300 174 32.5 130 0.75 5.1 0.52metal honeycomb Ar 0.12 (±0.01) 0.63 (±0.01) 1100 33 157.5 630 0.71 1.1 18metal honeycomb Ar 0.12 (±0.01) 0.38 (±0.01) 2600 51 95 380 0.58 1.8 25metal honeycomb Ar 0.12 (±0.01) 0.28 (±0.01) 4000 63 70 280 0.49 2.4 31metal honeycomb He 0.12 (±0.01) 0.63 (±0.01) 1100 33 157.5 630 0.71 1.7 18metal honeycomb He 0.12 (±0.01) 0.28 (±0.01) 4000 63 70 280 0.49 3.9 31

[W/mK]

CPSI

[#cells/inch2]

M [#cells/inch] [µm] [µm]

regenerator type gas [mm] [mm]

δ d hr hD regψh

Tg

r

δeffλ

Table 4.4.I Ceramic and metal honeycomb regenerator properties.

stainless steel wire screeens Ar 0.035-0.036 0.066 62500 250 1.8 20.3 81.2 0.70 8.17 8.23stainless steel wire screeens Ar 0.035-0.036 0.059 72900 270 2.0 20.8 83.1 0.70 7.99 8.10stainless steel wire screeens Ar 0.035-0.036 0.075 52900 230 1.9 25.0 99.8 0.74 6.65 7.10stainless steel wire screeens Ar 0.035-0.036 0.049 90000 300 1.6 12.5 50.1 0.59 13.26 11.22stainless steel wire screeens Ar 0.05 0.104 27225 165 2.1 38.6 154.3 0.76 4.30 6.63stainless steel wire screeens Ar 0.03 0.224 10000 100 1.6 58.2 232.8 0.89 2.85 3.11stainless steel wire screeens Ar 0.03 0.288 6400 80 1.6 72.8 291.3 0.91 2.28 2.55stainless steel wire screeens Ar 0.03 0.478 2500 50 1.3 93.6 374.3 0.93 1.77 2.03stainless steel wire screeens Ar 0.1 0.218 6400 80 1.8 65.5 261.8 0.72 2.54 7.48stainless steel wire screeens Ar 0.1 0.147 10609 103 1.7 39.8 159.0 0.61 4.18 10.44stainless steel wire screeens Ar 0.06 0.115 21025 145 2.1 44.4 177.5 0.75 3.74 6.84stainless steel wire screeens Ar 0.05 0.077 40000 200 1.9 25.1 100.4 0.67 6.61 8.99stainless steel wire screeens Ar 0.039-0.040 0.102 32400 180 1.9 32.9 132.0 0.77 5.06 6.26stainless steel wire screeens Ar 0.039-0.040 0.087 40000 200 1.9 28.4 114.0 0.74 5.87 6.99stainless steel wire screeens Ar 0.039-0.040 0.062 62500 250 1.9 20.8 83.3 0.68 8.00 8.70stainless steel wire screeens Ar 0.035 0.110 32400 180 2.0 37.2 149.0 0.81 4.48 5.17

regenerator type gas [mm] [mm]

CPSI [#cells/i

nch2]

M [#cells/inch] [µm] [µm] [W/mK]

δ dpfγ hr hD regψ

h

Tg

r

δeffλ

Table 4.4.II Stainless steel wire screens regenerator properties. 4.4.8 Results and discussion The results of measurements on the performance of different regenerator samples are presented in this subsection. All data are obtained at operation under the standard measurement settings described in the last subsection, except for some samples. One exception is the sample of ceramic honeycomb regenerator with a CPSI of 43300. It was measured twice: one time without the outer flow straightner for comparison with standard condition, which is indicated by (1) in Fig. 4.4.26, and the other one with both flow straightners as in standard condition, which is indicated by (2). Another exception is about the stainless steel wire screen regenerator with mesh 180 and wire diameter 0.035-0.036 mm. It was also measured twice: one time, indicated as (1), was measured at a lower drive ratio of 2.3% than the standard one, and the second time, indicated as (2), was measured at standard drive ratio of 2.7%. Energy balance According to the energy equations (4.4.20) (4.4.21) and (4.4.22), the ratios of four energy consumptions to total heater power have to sum up to 1:

136 Chapter 4

12/

=+++heater

dissipated

heater

acoustic

heater

losses

heater

cooler

P

P

P

P

P

P

P

P. (4.4.30)

In Fig.4.4.21-4.4.23, the relative fraction of the power contribution to the following four quantities is plotted: heat power transported to the cold end of the regenerator Pcooler (blue), heat losses to the environment Plosses (green), acoustic power in the valve Pacoustic (yellow), and acoustic dissipation Pdissipated/2 (pink). It is clear that the blue zone in all the graphs is dominant. Except for the metal honeycombs, it becomes in general smaller at increasing heat power. But at the threshold when acoustic power starts to be generated there is an increase in cooler losses, which is due to increase of thermal transport in the regenerator due to the acoustics. At increasing heat power the green zone increases, which is due to increased convective losses and thermal radiation. Both effects are nonlinear and although the fractional losses are plotted, these losses contribute more and more at the expense of the cooler losses. The yellow and pink zones are present only after the onset of the acoustic wave. The fraction of the acoustic power and losses is small with respect to the other two losses. This is mainly due to two aspects, the coaxial geometry is not insulated and there is a large thermal gradient over a thin regenerator (about 10 mm thick) leading to significant thermal radiation and conduction losses. In case of the metal honeycomb, see Fig. 4.4.21, the production of acoustic power remains very small due to the relatively large size of the channels compared to the penetration depth as well as to the huge axial heat conduction. Note that the x axis scale extends here to 270 W still with negligible acoustic power production. The poor performance of metallic honeycombs is also clearly shown in Figs. 4.4.24 and 4.4.25. The production of Pacoustic as a function of ∆T (∆T=TH-TC, where TH and TC are the temperatures of hot and cold-end of the regenerator, respectively) for metallic honeycombs is given in Fig. 4.4.26. From the measurement of the temperature difference over the regenerator, we conclude that in principle at a ∆T of 80 K, the engine already produces acoustic power. In most of the measurements with metal honeycombs, this ∆T is attained. However, the cell size of these honeycombs is too large (too small CPSI number). Only for CPSI number of 4000 there is acoustic power generated. Of course, acoustic power can be generated with metal honeycombs of CPSI=1100 or 2600, however only at powers and temperatures far above the available power that we have. The main reason for not generating acoustic power is the disagreement between cell size and thermal penetration depth.

Traveling-wave systems 137

0 25 50 75 100 125 150 175 200 225 250 2750

20

40

60

80

100

Pow

er d

istr

ibut

ion

[%]

Pheater

[W]

Pcooler/Pheater [%] Plosses/Pheater [%] Pacoustic/Pheater [%] (Pdissipated/2)/Pheater [%]

Figure 4.4.21: Relative power distribution for the metal honeycomb regenerator of CPSI 4000.

0 10 20 30 40 50 60 70 80 90 100 1100

20

40

60

80

100

Pow

er d

istr

ibut

ion

[%]

Pheater

[W]

Pcooler/Pheater [%] Plosses/Pheater [%] Pacoustic/Pheater [%] (Pdissipated/2)/Pheater [%]

Figure 4.4.22: Relative power distribution for the ceramic honeycomb regenerator of CPSI 46300.

138 Chapter 4

0 10 20 30 40 50 60 70 80 900

20

40

60

80

100

Pow

er d

istr

ibut

ion

[%]

Pheater

[W]

Pcooler/Pheater [%] Plosses/Pheater [%] Pacoustic/Pheater [%] (Pdissipated/2)/Pheater [%]

Figure 4.4.23: Relative power distribution for the stainless steel screen regenerator with mesh 180 and wire diameter 0.035.

0 50 100 150 200 250 300 350 400-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Pac

oust

ic (

W)

Pheater

(W)

CPSI 1100, Ar CPSI 1100, He CPSI 2600, Ar CPSI 4000, Ar CPSI 4000, He(1) CPSI 4000, He(2)

Figure 4.4.24: acousticP as a function of heaterP for metal honeycomb regenerators.

Traveling-wave systems 139

0 50 100 150 200 250 300 350 400-0.05

0.00

0.05

0.10

0.15

0.20 CPSI 1100, Ar CPSI 1100, He CPSI 2600, Ar CPSI 4000, Ar CPSI 4000, He(1) CPSI 4000, He(2)

Effi

cien

cy [%

]

Pheater

(W)

Figure 4.4.25: Efficiency as a function of heaterP for metal honeycomb regenerators.

0 20 40 60 80 100 120 140 160 1800.0

0.5

1.0

1.5

2.0

2.5

Pac

oust

ic (

W)

CPSI 1100, Ar CPSI 1100, He CPSI 2600, Ar CPSI 4000, Ar CPSI 4000, He(1) CPSI 4000, He(2)

∆T (K)

Figure 4.4.26: acousticP as a function of T∆ for metal honeycomb regenerators.

Power dissipated in the variable acoustic load The energy, dissipated in the acoustic load, is determined via Eq. (4.4.27). The energy of loadP as a function of the heating powerheaterP for ceramic honeycomb

samples and stainless steel screen regenerators is plotted in Figs. 4.4.27 and 4.4.28.

140 Chapter 4

The production of Pacoustic as a function of ∆T for ceramic honeycombs and stainless steel screen are given in Fig. 4.4.29 and 4.4.30. For an ideal engine that operates under ideal circumstances with no losses, it can be loaded as soon as any heater energy is applied, soloadP increasing fundamentally

linear with heaterP . However, in a real engine, there exists a threshold value for

heaterP below which there is no acoustic energy production. From Eq. (4.4.21),

power can only be added to the acoustic load if 2/dissipatedacoustic PP > . Below that,

the efficiency is zero and no useful energy is provided by the engine. Under those heat loads, all power produced by the engine is used up by losses and dissipation along the walls. For most of the ceramic samples, this threshold point is at

WPheater 5040−≈ . For stainless steel screen samples, the point is reached at 20-

30 W. The three ceramic samples, with CPSI lower than 7800, do not give any significant loadP . Here the thermal penetration depth of the wave is much smaller

than the characteristic cell size of the material. For the ceramic honeycomb regenerators, the one with CPSI 30300 gives the maximum of WPload 5.3= at

WPheater 130= . It is clear from Fig. 4.4.27 that the threshold and performance of

the honeycombs is related to the CPSI number or effective cell width. In general, the stainless steel screen samples generate more power and perform better than the ceramic honeycombs. For the stainless steel screen samples, the one with mesh 80 and wire diameter of 0.03 mm gives the maximum of

WPload 7.6= at WPheater 140= . But the best performance in terms of efficiency is

with mesh 100/0.03mm. In these two figures, the curves are almost linear after the threshold points. The sample of mesh 80 and wire diameter of 0.03 mm in Fig.4.4.28, for instance, shows this typical trend. As stated above, in ideal situation, loadP increases linearly

with heaterP , heaterloadidealload PP ⋅= α, . In real situation, the acoustic power is

generated after heaterP exceeds the threshold value thresholdP . As seen in Fig. 4.4.29,

at this threshold power, the ∆T for these materials is approximately 85 (±10) °C. Physically speaking this could be defined as the lowest threshold ∆T at optimum CPSI number for this engine. As seen in Figs. 4.4.27 and 4.4.28, loadP increases

linearly with heaterP after reaching the threshold value:

( )thresholdheaterloadload PPP −⋅= α .

In Fig.4.4.30, similar to Fig.4.4.28, Pacoustic increases almost linearly with ∆T after reaching the threshold ∆T for a specific stainless steel screen material (distinguished by mesh number and wire diameter).

Traveling-wave systems 141

0 20 40 60 80 100 120 140 160 180 200 220 240-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

CPSI 1600 CPSI 2700 CPSI 3800 CPSI 7800 CPSI 43300(1) CPSI 106000 CPSI 46300 CPSI 30300 CPSI 43300(2)

Plo

ad (

W)

Pheater

(W)

Figure 4.4.27: loadP as a function of heaterP for ceramic honeycomb regenerators.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

0

1

2

3

4

5

6

7

mesh180,wire0.035-0.036(1) mesh180,wire0.039-0.040 mesh200,wire0.039-0.040 mesh250,wire0.039-0.040 mesh180,wire0.035-0.036(2) mesh250,wire0.035-0.036 mesh230,wire0.035-0.036 mesh165,wire0.05 mesh270,wire0.035-0.036 mesh300,wire0.035-0.036 mesh100,wire0.03 mesh80,wire0.03 mesh50,wire0.03

Plo

ad (

W)

Pheater

(W)

Figure 4.4.28: loadP as a function of heaterP for stainless steel screen regenerators.

142 Chapter 4

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6CPSI 1500CPSI 3000 CPSI 3900CPSI 7300CPSI 43300(1)CPSI 106000CPSI 46300CPSI 30300CPSI 43300(2)

Pac

oust

ic (

W)

∆T (K)

Figure 4.4.29: acousticP as a function of T∆ for ceramic honeycomb regenerators.

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

7

8

9

10 mesh180,wire0.035-0.036(1) mesh180,wire0.039-0.040 mesh200,wire0.039-0.040 mesh250,wire0.039-0.040 mesh180,wire0.035-0.036(2) mesh250,wire0.035-0.036 mesh230,wire0.035-0.036 mesh165,wire0.05 mesh270,wire0.035-0.036 mesh300,wire0.035-0.036 mesh100,wire0.03 mesh80,wire0.03 mesh50,wire0.03P

acou

stic (

W)

∆T (K)

Figure 4.4.30: acousticP as a function of T∆ for stainless steel screen regenerators.

Performance of the engine Efficiency By using Eq. (4.4.23), the raw efficiencies of ceramic honeycomb and stainless steel screen regenerators are determined and plotted against heaterP , as shown in Figs.

Traveling-wave systems 143

4.4.31 and 4.4.32. As seen in the discussion of loadP , there is a threshold value

below which there is no energy production. After that, the efficiency increases as the heater power increases. For some samples, the efficiency increases to a maximum value and then slightly decreases. The decrease after reaching a maximum could be possibly attributed to the increasing nonlinear convection losses. The effect of a second flow straightner can be found in the comparison between two measurements of the ceramic honeycomb sample with CPSI 43300, as seen in Fig. 4.4.31. Apparently, the performance using two flow straightners, marked with (2) in the figures, is better than the one without second flow straightner, marked with (1). It can be concluded that the second flow straightner prevents convection losses. Similarly, two working conditions were applied to the stainless steel screen sample with mesh 180 and wire diameter of 0.035 mm. One time, the regenerator worked at drive ratio of 2.3%, marked by (1), and the second time, at standard drive ratio of 2.7%, marked by (2). The results show that the higher drive ratio gives a much better performance.

0 20 40 60 80 100 120 140 160 180 200 220 2400

1

2

3

Eff

icie

ncy

[%]

Pheater

(W)

CPSI 1600 CPSI 2700 CPSI 3800 CPSI 7800 CPSI 43300(1) CPSI 106000 CPSI 46300 CPSI 30300 CPSI 43300(2)

Figure 4.4.31: Efficiency as a function of heaterP for ceramic honeycomb

regenerators.

144 Chapter 4

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

1

2

3

4

5

6

7

Effi

cien

cy [%

]

Pheater

(W)

mesh180,wire0.035-0.036(1) mesh180,wire0.039-0.040 mesh200,wire0.039-0.040 mesh250,wire0.039-0.040 mesh180,wire0.035-0.036(2) mesh250,wire0.035-0.036 mesh230,wire0.035-0.036 mesh165,wire0.05 mesh270,wire0.035-0.036 mesh300,wire0.035-0.036 mesh100,wire0.03 mesh80,wire0.03 mesh50,wire0.03

Figure 4.4.32: Efficiency as a function of heaterP for stainless steel screen

regenerators. For ceramic honeycomb regenerators, the highest efficiency reached by the samples is 2.6%, whereas for stainless steel screen regenerators, the maximum is 5.8% (sample with mesh 100 and wire diameter of 0.03 mm). Generally, the stainless steel screen regenerators give higher efficiency than the ceramic honeycombs. Efficiency relative to the Carnot efficiency The relative efficiency is evaluated as the ratio of raw efficiency to the Carnot effeciency relativeη ( Carnotrelative ηηη /= ). The Carnot efficiency is the theoretical

maximum that an engine is able to reach and that is bound by the second law of thermodynamics:

H

CHCarnot T

TT −=η , (4.4.31)

where TH and TC are temperatures at the hot and cold ends.When the heating power

heaterP increases, the Carnot efficiency increases because of the increasing

temperature difference across the regenerator. The relative efficiency against heaterP for both kinds of samples are plotted in Figs.

4.4.33 and 4.4.34. The maximum relative efficiency for the ceramic honeycomb samples is around 7%, (sample with a CPSI of 30300 and 46300). Again, the stainless steel screen samples give better performance than the ceramic honeycombs. The maximum relative efficiency for the stainless steel screen samples is 11.7% (sample with mesh 100 and wire diameter of 0.03 mm).

Traveling-wave systems 145

0 20 40 60 80 100 120 140 160 180 200 220 2400

1

2

3

4

5

6

7

8 CPSI 1600 CPSI 2700 CPSI 3800 CPSI 7800 CPSI 43300(1) CPSI 106000 CPSI 46300 CPSI 30300 CPSI 43300(2)

Pheater

(W)

Eff

icie

ncy/

Car

not e

ffic

ienc

y [%

]

Figure 4.4.33: Relative efficiency as a function of heaterP for ceramic honeycomb

regenerators.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

1

2

3

4

5

6

7

8

9

10

11

12

13 mesh180,wire0.035-0.036(1) mesh180,wire0.039-0.040 mesh200,wire0.039-0.040 mesh250,wire0.039-0.040 mesh180,wire0.035-0.036(2) mesh250,wire0.035-0.036 mesh230,wire0.035-0.036 mesh165,wire0.05 mesh270,wire0.035-0.036 mesh300,wire0.035-0.036 mesh100,wire0.03 mesh80,wire0.03 mesh50,wire0.03E

ffici

ency

/Car

not e

ffici

ency

[%]

Pheater

(W)

Figure 4.4.34: Relative efficiency as a function of heaterP for stainless steel screen

regenerators. The maximum relative efficiency points out the optimum value for heaterP . For the

ceramic samples, the optimum heaterP is between 70-90 W, and 60-90 W is optimum

for the stainless steel screen samples.

146 Chapter 4

Maximum efficiency as a measure for the performance The overall machine efficiency is low and less than 10%, and even the relative efficiency is not over 13%. For the main goal of this work, namely to compare different regenerator materials, this is not a problem. This study aimed at finding the optimum geometrical properties for a regenerator material in a thermoacoustic system with respect to the cell size or mesh number related to the thermal penetration depth. Based on measurements, stainless steel wire screen regenerators give the best performance among the three kinds of samples tested in this study. For a regenerator, many geometrical parameters, such as CPSI number, hydraulic radius, porosity, mesh number as described in section 4.4.2 are used for characterizing them, and they are inter-related. Among these parameters, the porosity and hydraulic radius are chosen for this study, which are direct for analysis and include inherently other parameters by mutual dependency. The maximum raw efficiency is plotted based on experimental measurements as functions of porosity and the dimensionless hydraulic radius (ratio of hydraulic radius and thermal penetration depth at 490 K of argon, mean pressure of 10 bar and frequency of 59 Hz ,which is 166 µm), respectively. From the measurements, there exists an optimum value for dimensionless hydraulic radius (the ratio of hydraulic radius to thermal penetration depth at average temperature 490 K), which is around 0.3 for stainless steel screens, as seen in Fig. 4.4.35. The trend for stainless steel screens agrees with the computation in Fig. 4.4.8. Although there are some apparently unexpected low values, for instance at 0.4 (mesh 80 and wire diameter 0.1 mm), but the trend is clear. The optimum range is obvious. Those unexpected points diverging from the trend can possibly be attributed to the effect of porosity. This becomes clear when interpreting Fig. 4.4.36 where porosity is plotted as a function of dimensionless hydraulic radius. The low efficiency results in Fig. 4.4.35 correspond to the low porosity samples in Fig. 4.4.36 as discussed. So this shows experimentally that both porosity and hydraulic radius are vital parameters. If both porosity and dimensionless hydraulic radius are used as inputs in the model in section 4.4.3, the computation of stainless steel wire screen regenerators also gives similar distribution as in the experiment. This is shown in Fig. 4.4.37. For ceramic honeycomb samples, the optimum value is much lower than that of stainless steel screens, which is in the range 0.14-0.2 as shown in Fig. 4.4.35.

Traveling-wave systems 147

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Hydraulic radius/Thermal penetration depth

Max

imu

m e

ffic

ien

cy [

%]

stainless steel wire screens

ceramic honeycomb

Figure 4.4.35: Measured maximum efficiency as a function of dimensionless hydraulic radius for stainless steel screen regenerators and ceramic honeycombs.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Hydraulic radius/Thermal penetration depth

Po

rosi

ty

Figure 4.4.36: Porosity as a function of dimensionless hydraulic radius for stainless steel screen regenerators.

148 Chapter 4

0

1

2

3

4

5

6

7

0 0.1 0.2 0.3 0.4 0.5 0.6

Hydraulic radius/Thermal penetration depth

Max

imu

m e

ffic

ien

cy [

%]

experiments

computation

Figure 4.4.37: Computed maximum efficiency as a function of dimensionless hydraulic radius for stainless steel screen regenerators.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

Porosity

Max

imu

m e

ffic

ien

cy [

%]

stainless steel wire screens

ceramic honeycomb

Figure 4.4.38: Measured maximum efficiency as a function of porosity for stainless steel screen regenerators and ceramic honeycomb.

Traveling-wave systems 149

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1

Porosity

Max

imu

m e

ffic

ien

cy [

%]

experiments

computation

Figure 4.4.39: Computed maximal efficiency as a function of porosity for stainless steel screen regenerators. The measured maximum efficiency against porosity is given in Fig. 4.4.38. The same trend was already explained in Fig. 4.4.9. The efficiency monotonically increases with porosity, except for very high porosity when the heat capacity of the solid part of the regenerator is not big enough to sustain a stable local temperature. The dip at around porosity 0.7 can be explained by the variation of dimensionless hydraulic radius. In the experiments, the variation of porosity is accompanied by the variation of hydraulic radius. Therefore, the diverging points from the general trend emerge as a result of the combined variation of porosity and hydraulic radius. If both porosity and dimensionless hydraulic radius are used as inputs in the computation as discussed in section 4.4.3, the computation also gives similar distribution as in the experiment, which is shown in Fig. 4.4.39. 4.4.9 Conclusion The performance of a traveling-wave thermoacoustic system is affected by the geometrical properties of the regenerator in a profound manner. From this study, it can be concluded that there is an optimum value for the dimensionless hydraulic radius, which is around 0.3 for stainless steel wire screen regenerators and 0.14-0.2 for square cell honeycombs. The efficiency goes up with increasing porosity, to a maximum value above which the heat capacity of the solid becomes the limiting factor.

150 Chapter 4

The maximum efficiency that could be established with regenerator in experiments corresponded well with computations based on the model described in section 4.3.

Traveling-wave systems 151

4.5 Experiments on a thermoacoustic refrigerator 4.5.1 Introduction A thermoacoustic-Stirling refrigerator, which utilizes a traveling wave to realize the Stirling cycle within the regenerator, was built to explore the performance at high frequency and relatively small size. A further objective of this experimental set-up was validation of the model described in section 4.3. Helium was chosen as the working gas because of its suitable properties to produce more cooling power than the other gases under the same working conditions. However, the sound speed in helium is much larger than in the other gases leading to larger size at the same operation frequency. The refrigerator is driven by a pressure wave generator (PWG) provided by CFIC Q-Drive. The design considerations are discussed in more detail in the next section. After that, in section 4.5.3, the resulting experimental set-up is described and section 4.5.4 focuses on the measuring equipment and data handling. Losses in the set-up are discussed in section 4.5.5. The measuring procedure is considered in section 4.5.6 followed by measuring results and discussion in section 4.5.7. This section ends with conclusions in 4.5.8. 4.5.2 Design considerations 1. The diameters of tube sections were chosen from commercially avalaible

products. The coupling in size between tubes and tee was also taken into account. With all these considerations, the tubes in the loop and tee were chosen to be of stainless steel inner diameter 29.7 mm and 2 mm wall thickness.

2. The operation frequency is determinded by the operation frequency of the PWG provided by CFIC Q-Drive. Originally, two theoretical designs of the set-up were made, one operating at 400 Hz and the other one at 1000 Hz. However, the maximum that Q-Drive could provide us with was 320 Hz PWG. Therefore, the design was adapted to 320 Hz.

3. The system was designed as a half-wave length system. Therefore, the resonator tube length was roughly determined by the operation frequency of 320 Hz.

4. The rough length of the loop was determined by using the model of section 4.3, and refined later by using a numerical simulation solver from the research group of E. Luo and W. Dai [73]. This also determines the length of the regenerator and the tube segment connecting the cold-end heat exchanger and the tee. As stated before, the author found that a lumped volumetric compliance is not necessary for a good performance. Therefore, in the design,

152 Chapter 4

there was no compliance. A single tube diameter also allows for simpler manufacturing and assembly.

5. The position of regenerator was determined by using the above-mentioned numerical simulation solver to place it at the position with a large acoustic pressure and a small volume velocity position.

6. The regenerator material, dimensions of stainless steel wire screen were chosen based on the operation frequency and the conclusive selection criteria in section 4.4.

7. All the dimensions were refined by using the numerical simulation solver mentioned under point 4.

4.5.3 Experimental set-up The structure of the system and its 3-D schematic drawing are given in Figs. 4.5.1 a and b, respectively. Essentially, it consists of a half-wavelength resonator filled with helium, with one end connected to the pressure wave generator and the other connected to a torus-shaped section. The torus-shaped section contains three heat exchangers, a regenerator and a duct forming a loop. Six microphones and five thermometers are mounted along the tubes and connected to the data-acquisition equipement. A photograph of the complete system is given in Fig.4.5.1 c.

Figure 4.5.1(a): Structure of the thermoacoustic-Stirling refrigerator.

Ambient heat exchanger

Electrical heater

Secondary heat exchanger

Regenerator

Thermal buffer

Pressure wave generator

Traveling-wave systems 153

Figure 4.5.1(b): Experimental set-up of the thermoacoustic-Stirling refrigerator (Two separate tube sections are included for tuning the resonator length).

Figure 4.5.1(c): Photo of the thermoacoustic-Stirling refrigerator. Pressure wave generator (PWG) The acoustic power is generated by a powerful pressure wave generator, model 2s102W-X by Q-Drive. It has a piston area of 91.5 cm² and motor stiffness of 34.3kN/m. The wave generator is fixed by bolts and nuts on an aluminium profile. Two microphones (P7 and P8) are installed in two chambers to monitor the acoustic pressure inside the PWG. The resonator tube The resonator tube is made of a stainless steel tube with uniform inner diameter of 38.4 mm and wall thickness of 2 mm. It consists of three segments (1, 2, and 3) and the total length is 1.2 m as designed. A schematic drawing is given by Fig.4.5.2. Segment 1, which is 800 mm long, is surrounded by a larger diameter tube of 44.3 mm inner diameter and wall thickness of 2 mm. Water cooling circulation is

Pressure wave generator

Resonator tube

The loop

Microphones

P8

Average pressure sensor P9

Pressure wave generator

P7

154 Chapter 4

applied in the gap between the two tubes. By doing so, the heat due to acoustic dissipation along the wall can be removed and the amount of heat can be determined by measuring the water temperatures in and out. Segment 2, which is 200 mm long, can be replaced by another tube of different length or their combination to make different resonator lengths. The two optional tubes are 100 mm and 50 mm long. The segment tube 3 is 200 mm long. Along segment 3, two microphones are installed to measure the acoustic power passing through, and one static pressure sensor is for measuring the filling pressure inside the system. One end of the resonator tube is coupled to the PWG by a flange and the other end is connected to a reduced tube by a flange. The reduced tube has diameters of 38.4 mm and 29.7 mm, length of 26 mm.

Figure 4.5.2: Schematic drawing of the thermoacoustic-Stirling refrigerator. The loop The reduced tube is connected to one branch of the tee junction. The junction is a commercial available, stainless-steel tee with inner diameter of 29.7 mm. One branch of the tee is connected to the feedback tube, which is a torus tube section of 261 mm long with uniform inner diameter of 29.7 mm. One end of the feedback tube is bent to connect to the tee and the other end connects to the tube segment that contains the regenerator and the heat exchangers. The regenerator and the thermal buffer tube are surrounded by a vacuum chamber filled with super insulation to be thermally isolated from the environment (see also Fig.4.5.5 (a)). Along the bent tube, three microphones are installed and a fourth microphone is installed in the tee junction. The ambient heat exchanger is 6 mm thick. A diaphragm is installed close to the ambient heat exchanger to prevent DC flow. The photo of the ambient heat exchanger is given by Fig. 4.5.3. The ambient heat exchanger was a delicate manufacturing product, involving much precision machining work using spark discharging technique in copper. The proper functioning of the ambient heat exchanger plays a key role in the performance of the whole system. The high working frequency requests thin fins and a small

Segment 2 Segment 1 Segment 3 Reduced tube

Jacket water cooling

Loop section

PWG

Traveling-wave systems 155

distance between any two fins. The thickness of a fin is 0.11 mm and the distance between two adjacent fins is 0.17 mm, as shown by the photo of a prepared test plate in Fig. 4.5.3. Keolian and Hofler have conducted a series of studies on heat exchangers [74-77]. In the work of 1994 [74], Hofler concluded that a heat exchanger with ξhx/Lhx in the range of 3 to 8 can be thermally effective as a source or sink for thermoacoustic heat transport if y0/δκ is in the range of 0.75 to 0.5. Here, the peak displacement amplitude of a gas parcel in the heat exchanger is given by ξhx, and the length of the heat exchanger is given by Lhx. The separation between adjacent fins is 2y0 and δκ is the thermal penetration depth of the working fluid. The conclusion from Hofler was employed in the design of the heat exchangers in this setup. The details of designing the ambient heat exchanger are given in appendix I.

Figure 4.5.3: Photo of the ambient heat exchanger.

Figure 4.5.4: Photo of the electrical heater at the cold end of the regenerator.

156 Chapter 4

Figure 4.5.5(a): Schematic drawing of the loop section of the thermoacoustic-Stirling refrigerator. The dimensions of the components inside the vacuum chamber are given in the right-side drawing.

Figure 4.5.5(b): Photo of the loop section of the thermoacoustic-Stirling refrigerator. The regenerator has a thickness of 23 mm and diameter of 29.7 mm. The regenerator is made of a stack of stainless steel wire screens of Mesh 180 and wire diameter of 0.039-0.04 mm. The regenerator is sandwiched between an electrical

Ambient heat exchanger

Secondary heat exchanger

Regenerator

Thermal buffer

Vacuum chamber

Tee

29.7 mm

29.7 mm

29.7 mm

Electrical heater

6 mm

23 mm

29.7 mm

6 mm

25 mm

Feedback tube chamber 4 mm

Diaphragm

Vacuum chamber

Tee

Feedback tube

Traveling-wave systems 157

heater and the ambient heat exchanger. In between two perforated plates are used to fix the regenerator screens. The electrical heater forms a heat load to the refrigerator and represents the cooling load. The heater is depicted in Fig. 4.5.4, with the fin thickness of 0.11 mm and interfin distance of 0.17 mm. The thermal buffer tube is 25 mm long, and is lined with a thermal isolation ring made of Peek. One end of the thermal buffer tube is connected to a secondary heat exchanger of 6 mm thick. The second heat exchanger has the fin thickness of 0.12 mm and interfin distance of 0.27 mm. The dimensions are given in Fig. 4.5.5 (a) and a photograph of the complete loop is given in Fig. 4.5.5 (b). 4.5.4 Measuring equipment and data handling Thermometers are mounted on the heat exchangers as well as in the water channels of the cooling water. Pressure sensors are mounted flush with the wall at various parts of the system. In this way the acoustic signal information as well as the heat fluxes can be determined. Fig. 4.5.6 shows the distribution of all measuring devices. A HP 33120A function generator generates a sinus wave that is used as input for a Dynacord L2400 amplifier in bridged operation. The frequency and amplitude of the sinus signal can be read from the control panel.

Figure 4.5.6: Overview of the measuring devices. The amplified sinus signal passes through the Hioki power sensor and goes to the PWG (model 2s102W-X by Q-Drive). Two microphones Endevco (P7, P8) are mounted on the walls of the two back volumes of the pistons to monitor the acoustic pressure inside the two chambers. The two microphones, P7 and P8, are measured by an EG&G lockin type amplifier. Microphones P1 to P6, PCB Piezotronics are mounted along the resonator tube and the loop. P9 is to measure

Function Generator sinus

V1

Amplifier HIOKI power sensor

PWG

P7, P8 via Endevco

EG&G Lockin

Q V2 P7

P8 P2 P3 P1

P5

P6

P4

P9 T1 T2

T3 T4

T5 T6 T7

T8 T9

T10 T11

Julabo cooler

Grundfos pump Flow

meter

DC pressure

Heater V, I

158 Chapter 4

the static mean pressure inside of the system. When the heater is on, the power is provided by a power supply type E030-5 of Delta Elektronika. T1 to T5 are Pt1000 temperature sensors that are used to monitor the inside temperatures of the heat exchangers. T2 is attached to the solid part of the secondary heat exchanger for measuring the temperature of the metal material. T1 is attached to a support, which is attached to the secondary heat exchanger, for measuring the oscillating gas temperature flowing back and forth. In a similar way, T4 is to measure the temperature of the solid part of the cold-end heat exchanger and T3 is to measure the temperature of the gas. T5 is for measuring the temperature of the gas oscillating through the ambient heat exchanger. T6 to T11 are to measure the temperature of the water circulating inside the heat exchangers. Water is pumped by a Grundfos pump and the temperature of the circulating water is maintained at 25°C by a Julabo F32 cooler. The water flow rate is measured by flowmeters installed downstream of the Grundfos pump. T6 is the temperature of water flowing in and T7 is the temperature of water flowing out of the ambient heat exchanger. T8 and T9 are the temperatures of water flowing in and out of the secondary heat exchanger. T10 and T11 are the in and out flow temperatures of the water cooling pipes that cool the resonator tube wall. The thermometers T6-T11 have an accuracy of 0.01 K and are very relevant as these are needed to determine the thermal fluxes of the apparatus. There is no displacement sensor (LVDT) mounted on the piston of PWG. And due to time problem, the acoustic power at the PWG was not measured in this experiment. During a measurement, all data (except P7 and P8) are collected with a PXI system by National Instruments and handled by a LabView program. The parameters are shown on the LabView window and a text file can be generated for every measuring point. The data written in the file can be read in a Matlab-file to compute the desired quantities, which are discussed below. Energy exchanged in heat exchangers The heat released by oscillating gas through the ambient heat exchanger is absorbed by the water circulation. The absorbed heat PambHX can be obtained by

( )67 TTcVPwaterpwaterwaterambHX −⋅⋅⋅= ρ& , (4.5.1)

where waterV& is the volume flow rate of the water through the heat exchanger.

Similarly, the energy exchange taking place at the secondary heat exchanger and the water cooling of the resonator tube can be obtained by

( )89sec TTcVPwaterpwaterwaterHX −⋅⋅⋅= ρ& , (4.5.2)

( )1011 TTcVPwaterpwaterwaterresHX −⋅⋅⋅= ρ& . (4.5.3)

Cooling power

Traveling-wave systems 159

The cooling power at the cold end of the regenerator is dissipated by an electrical heater. Therefore, the cooling power is given by

IUPcooling ⋅= , (4.5.4)

where U and I are the voltage and current of the power supply, respectively. The microphones were mounted to determine the acoustic power flowing through by the “two-microphone method” [62], but the results could not comply with analysis of energy balance. In this configuration, phase differences were so small that errors in the microphone readings (e.g. resulting from vibrations) deteriorated the overall accuracy to such an extent that they could not be used for reliable evaluations of efficiency. Therefore, the acoustic power computed from parameters of these microphones is not used in this work. Acoustic power The acoustic power is obtained by analyzing the energy balance of the system. This is shown by the control volume in the dashed rectangle in Fig. 4.5.7. Acoustic power enters the control volume to drive the refrigerator. The control volume has energy exchange with the external heat sources or sinks at the ambient heat exchanger, secondary heat exchanger and the electrical heater at the cold-end heat exchanger.

Figure 4.5.7: Energy balance of the loop. The regenerator and thermal buffer tube are thermally insulated by a vacuum chamber (see Fig.4.5.5 (a)). Therefore, the heat exchange of this part with the environment is considered to be zero. Besides the heat exchange mentioned above, the loop exchanges heat with the environment along the torus tube, as denoted as Penvironment in Fig. 4.5.7. Considering the practical situation of the operation of the setup, Penvironment is small enough to be neglected in energy balance analysis. In the steady state, the energy conservation of the control volume gives

Penvironment

PambHX

PsecHX

Pacoustic

Pcooling

160 Chapter 4

HXambHXcoolingacoustic PPPP sec+=+ . (4.5.5)

The acoustic power flowing into the control volume can be given by

coolingHXambHXacoustic PPPP −+= sec . (4.5.6)

4.5.5 Losses There are many possible losses in this system. Two of these can be derived from the measurement results and will be extensively discussed here. 1. Viscous loss along the tube wall The viscous loss converts part of the acoustic power into heat, which is absorbed by the solid tube wall and will warm up the wall. Part of the resonator tube wall is cooled down by water circulation between temperature measurement of T10 and T11 as shown in Fig. 4.5.6. Based on the heat taken away by the water circulation PresHX and the surface area of the tube wall between T10 and T11, the total viscous loss of the complete refrigerator along the wall, denoted as Ptotal wall loss, can be approximated by:

totalresHXlosswalltotal AAPP ⋅= − )/( 1110 (4.5.7)

A10-11 is the inner surface area of the tube wall between T10 and T11. Atotal is the total inner surface area of the tube wall of the complete refrigerator. Later in the thesis the energy balance as depicted in Fig.4.5.7 is considered with the dashed box (system boundary starting at microphone P2, see Fig.4.5.6). The viscous loss along the tube wall of the part downstream the location of microphone 2 can be approximated by:

21110 )/( MicresHXlosswall AAPP ⋅= − . (4.5.8)

AMic2 is the inner surface area of the tube wall downstream microphone P2. 2. Convection streaming losses When the driver is switched on, the temperature at the cold end of the regenerator goes down rapidly. A temperature gradient builds up between the cold end of the regenerator and the secondary heat exchanger. Flows inside the system cause convective heat exchange, as shown in Fig. 4.5.8. Note that this convective heat exchange is not only natural convection but may be caused by any streaming in the system as well. The zone where the convection takes place is indicated. Since the temperature at the secondary heat exchanger T2 is anchored at environmental temperature, we indicate it as the thermally grounded case. If the secondary heat exchanger is thermally floating, the temperature T2 will fall with the temperature T4 of the cold end of the regenerator. In that case, T2 is lower than the environmental temperature and higher than T4.

Traveling-wave systems 161

Ambient HX

P2

Secondary HX

Pconvection

T2

T4

Figure 4.5.8: Convection loss in the thermal buffer tube in thermal grounded case caused by streaming. Figure 4.5.9: Convection loss in the thermal buffer tube in thermal floating case.

Figure 4.5.10: Thermal resistance model for convection loss in the loop part of the refrigerator system. Due to the temperature gradients between T2 and T4, and between T2 and its surrounding environment, heat exchange by convection arises, as shown in Fig. 4.5.9. As an assumption, the convective heat exchange is proportional to the temperature difference between two surfaces, using Newton’s law of heat transfer

( )BA TTQ −⋅= α& . Therefore, the convection loss can be modeled and computed

for the thermally floating case, in which it is not measured.

R24 R02

T0 T4 T2

T5

P02 P24 Regenerator Secondary heat exchanger

Ambient environment

Ambient HX

Secondary HX

Pconvection

T2

T4

P2

T2

162 Chapter 4

Assume that the cold end of the regenerator with temperature T4 is connected with the secondary heat exchanger by a thermal resistance R24, and that the secondary heat exchanger is connected with the environment with temperature T0 by thermal resistance R02, as shown in Fig. 4.5.10. In Fig. 4.5.10, the energy flow between the environment and the secondary heat exchanger is P02. Similarly, the energy flow between the secondary heat exchanger and the cold end of the regenerator is P24. Concerning the secondary heat exchanger, there are two cases given below. (A) Thermally grounded condition When the secondary heat exchanger is maintained at environmental temperature by water circulation through it, there is no heat exchange between the environment and the secondary heat exchanger. That means the energy exchange shown in the dashed oval in Fig. 4.5.9 does not take place, i.e. P02=0. The energy flow between the secondary heat exchanger and the cold end of the regenerator P24 is due to internal convection and is balanced by water circulation through the secondary heat exchanger PsecHX. Therefore, it can be written as:

( )241

2424sec TT

RPP HX −⋅== . (4.5.9)

By using the corresponding data from measurement in the thermal grounded case, the thermal conductance 1/R24 can be obtained, which is then considered to be constant. (B) Thermally floating condition When the secondary heat exchanger is floating, the temperature T2 of the secondary heat exchanger will drop when the cold end of the regenerator T4 is cooling down. There is no thermal insulation applied at the tee junction part, the energy flow P02 takes place as shown in the dashed oval in Fig. 4.5.9. For this case holds P02=P24. The convection loss can be obtained by:

( )241

2424 TT

RPPconvection −⋅== , (4.5.10)

where 1/R24 was obtained from the thermally grounded case as discussed above. 4.5.6 Measuring procedure Filling the system Every time when the system is opened, the system has to be refilled with pure helium gas before conducting a new measurement. To refill it by pure helium gas, the system was flushed three times. During filling and releasing gas, the mean pressure (filling pressure) is monitored by the static pressure sensor P9. Two kinds of measurement were carried out, indicated as operation A and B:

Traveling-wave systems 163

Operation A: Maximum temperature difference across the regenerator (zero cooling load) This is also the measurement of the lowest temperature that the cold end of the regenerator can reach. In this case, the electrical heater at the cold end of the regenerator was off, so no added cooling load on the cold end. The PWG is set at 10 W, which can be read off from the Hioki power meter. It generates direct input to the PWG, and this fixed value is maintained. After thirty minutes of operation, the steady state has established. The data were stored in file. Then the input power was stepwise increased to 30, 50 and 70 W. At each input power, the same operation was repeated. Operation B: Performance as a function of cooling power In this measurement, the electrical heater at the cold end of the regenerator was powered to generate a heat load. Measurements were made at fixed inputs (30 W, 50 W, 70 W) to the PWG. At every input, the heater power was varied, and the system parameters were stored in file as steady state was reached. All the data files are processed using a Matlab program. In every file, around 30 values are recorded for each parameter. Therefore, the mean value for each parameter is calculated and used for related energy computation in the Matlab program. 4.5.7 Results and discussion In the experiments, two orientations of the loop section are investigated: upward and downward with varied input acoustic power under thermally grounded condition. The results showed that the performances of two orientations are close but the downward configuration behaved more stable than the upward. The time evolutions of the two orientations are given in appendix J. Therefore, all the measurements presented later in this work were conducted with the loop section in downward configuration. In the downward configuration, any unstability caused by natural convection between the secondary heat exchanger and the cold end of the regenerator is weakened. I. Comparison of performance of different resonator tube lengths In the design phase, the resonator tube length was optimized at filling pressures of 11 bar and 15 bar. The resonator tube introduced in section 4.5.3 is shown in Fig. 4.5.11. There are three microphones installed on segment 3. So neither segment 3 nor segment 1 can be removed from the system. Only segment 2 can be replaced by another tube of different length or their combinations. There are four options: (a) 200 mm, (b) 100 mm, (c) 50 mm and (d) no tube (0 mm). The length containing only tube a (200mm) as segment 2 is the design value. The lengths larger and

164 Chapter 4

smaller than the designed length were measured by conducting operation A in thermally grounded condition. The measured combination of the optional tubes is listed in Table 4.5.I. Some cases did not show resonance and are indicated with “No” in Table 4.5.I. Figure 4.5.11: Three segments of the resonator tube.

Resonant frequency

Tube a 200 mm

Tube b 100 mm

Tube c 50 mm

Tube d 0 mm

Resonator length (mm)

Yes √ √ √ × 1350

No √ × √ × 1250

Yes √ × × × 1200

Yes × √ × × 1100

No × × × √ 1000

Table 4.5.I: Measured resonator tube segment 2 of optional tubes and their combinations. “√” indicates “used”, and “×” indicates “not used”. The cases without a resonant frequency are not discussed here. The comparison of three lengths of resonator tube at filling pressure of 11 bar and 15 bar are given in Fig. 4.5.12 and 4.5.13. ∆T is the temperature difference across the regenerator, i.e. ∆T=T5-T4. From the comparison in Figs. 4.5.12 and 4.5.13, the designed length (1200 mm) has the best performance. In section 4.3, the strong dependency of the complete acoustic impedance of the refrigerator on the resonator tube length is shown in Figs. 4.3.8 and 4.3.9. Subsequently, the complete acoustic impedance of the refrigerator has strong influence on the final performance through the coupling between the refrigerator and the PWG. It can be understood that the resonance of the complete system (refrigerator and the PWG) only takes place when the impedances of the refrigerator and the PWG are matched (reactance part of the impedance of the complete system, refrigerator and the PWG, is zero). In the design phase, the length of the resonator tube was optimized by numerical computation at the designed working condition. In Fig. 4.5.13, the performance of 1350 mm is close to that of the designed length 1200 mm. Because Pmean=15 bar is not the designed

Segment 1 Segment 2 Segment 3

800 mm 200 mm

Traveling-wave systems 165

working condition. The length of 1200 mm might not be the optimum value in that specific case.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

45

∆ T

(K

)

Pdriver

(W)

1200 mm 1350 mm 1100 mm

Pmean

=11 bar

Figure 4.5.12: Temperature difference across the regenerator for three different lengths of resonator tube at filling pressure of 11 bar.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40 1200 mm 1350 mm 1100 mm

Pmean

=15 bar

∆ T

(K

)

Pdriver

(W)

Figure 4.5.13: Temperature difference across the regenerator for three different lengths of resonator tube at filling pressure of 15 bar.

166 Chapter 4

II. Comparison of performance of thermally grounded and floating cases on the maximum temperature difference across the regenerator with zero cooling load In order to investigate the temperature characteristics of the apparatus with the resonator at its designed length, the so called zero-load measurements (operation A) were made. In that case, no power is supplied to the heater and the thermoacoustic cooler reaches its lowest temperature. Zero-load measurements (operation A) were made for thermally grounded and floating cases at filling pressures of 11 bar, 12 bar and 15 bar. The results are given in Fig. 4.5.14. The thermally floating cases show a better performance than the thermally grounded ones. The temperature difference ∆T in the thermally grounded cases is smaller due to the heat exchange between cold-end heat exchanger and the secondary heat exchanger as discussed in section 4.5.5. In the thermally floating cases, the secondary heat exchanger cools down along with the cold-end heat exchanger, leading to less heat loss from T2 to T4, and thus a larger ∆T can result. Computation by using the model described in section 4.3 and corresponding measurement of thermally floating cases is given in Fig. 4.5.15. In the model, there is no secondary heat exchanger. Therefore, the thermally floating case is the closest to the situation in the model.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40

45

50

55

60

∆T (

K)

Pdriver

(W)

Pmean

=11 bar (thermally grounded)P

mean=12 bar (thermally grounded)

Pmean

=15 bar (thermally grounded)P

mean=11 bar (thermally floating)

Pmean

=12 bar (thermally floating)P

mean=15 bar (thermally floating)

Figure 4.5.14: Zero-load temperature difference across the regenerator in thermally grounded and floating cases at various filling pressure.

Traveling-wave systems 167

In all computations, the temperature at the hot end of the regenerator, TH, is assumed to be at ambient temperature. The filling pressure (mean pressure Pm) is set at a constant value corresponding with the measurement. The operating frequency is set at the corresponding measured value. The working gas is helium and the regenerator material is stainless steel wire screens. The properties of the working gas and the regenerator material, such as thermal conductivity, ratio of specific heat, Prandtl number, used to compute Eqs. (4.3.37) and (4.3.52) can be found in Refs. [38,39]. The geometrical parameters (L-fb d-fb L-cpl d-cpl L-reg d-reg rh ψreg L-tb d-tb L-res d-res) are given in section 4.5.3. The acoustic pressure measured at P2, as shown in Fig. 4.5.6, is given as input acoustic pressure p1in and increased stepwise. The distance between microphone P2 and the center of the tee junction is set at Lres. For every p1in, an iteration was executed to obtain the lowest temperature for this p1in. In one iteration, the cold-end temperature of the regenerator TC was decreasingly varied in a range of values. Therefore, for one calculation in an iteration, a specific p1in and TC are inputed into Eq. (4.3.52) and related equations, and the cooling power can be obtained. In the next calculation in the same iteration, a different TC and the same p1in are inputed into Eq. (4.3.52) and related equations, and the cooling power for this TC and p1in can be obtained. After calculations for many TC and the same p1in in one iteration, many cooling power values are obtained. Since TC is given in a decreasing manner, the cooling power values are obtained in a decreasing manner as well. When the cooling power is almost zero, the cold-end temperature of the regenerator TC is stored as the lowest temperature Tlowest at this working condition. Thus the temperature difference across the regenerator ∆T (∆T=TH –Tlowest) at this input acoustic pressure p1in is obtained. Then another iteration for the next p1in is repeated. In Fig. 4.5.15, the trends from computation and the corresponding measurement are the same. The computation results overpredict the experimental values with a factor of about 2. The discrepancy between computation and experimental measurement can be explained by the highly idealized assumptions made to build the analytical model. All the losses, except dissipation and heat conduction within the regenerator, are neglected in the model. Some of these losses are analyzed in more detail below. Both the computation and the measuremts show that the performance decreases with increasing filling pressure Pmean. This can be explained by the fact that the configuration is optimized at a filling pressure of 11 bar in the design. When Pmean

becomes larger, the gas thermal penetration depth δκ becomes smaller. With the same regenerator material, the ratio of hydraulic radius rh to thermal penetration depth δκ becomes larger. This means that the heat exchange becomes worse.

168 Chapter 4

1.0x104 1.5x104 2.0x104 2.5x104 3.0x104

0

10

20

30

40

50

60

70

80

90

100∆T

(K

)

P2 (Pa)

computation Pmean

=11 bar experiment P

mean=11 bar

computation Pmean

=12 bar experiment P

mean=12 bar

computation Pmean

=15 bar experiment P

mean=15 bar

Figure 4.5.15: Temperature difference across the regenerator of thermal floating cases and computation at various filling pressure. Some energy flows and losses As stated above, losses play an important role in the performance of the system. From the many possible loss mechanisms, we limit the following discussion to the losses considered in section 4.5.5. In addition to the losses, some energy flows in this set-up, which are important for understanding the energy balance of the whole system, are presented. Comparison between the energy flows and the losses can help to understand the discrepancy between computation results and the measurements. Both thermally grounded and floating cases are considered for further comparison. The heat, exchanged at the ambient heat exchanger PambHX , is obtained by using Eq. (4.5.1). The heat exchange at the secondary heat exchanger PsecHX is given by Eq. (4.5.2) in the thermally grounded case, and is zero in the thermally floating case. In the measurement of the largest attainable temperature difference across the regenerator (operation A), the electrical heater is off. Therefore the cooling power is zero in using Eq. (4.5.6) to obtain Pacoustic. The viscous losses along the tube wall Ptotal wall loss and Pwall loss are given by Eqs. (4.5.7) and (4.5.8). By using the corresponding data (PsecHX, T4 and T2) from measurements in the thermally grounded condition and Eq. (4.5.9), the thermal

Traveling-wave systems 169

conductance 1/R24 can be obtained by linear fit of PsecHX and temperature difference between T4 and T2 by linear fit tool in software Origin. Eq.(4.5.9) shows that the linear relationship between PsecHX and the difference of T4 and T2 (T4-T2) has to pass the original point (0,0) in the linear fit curve. The linear fit of case Pmean=11 bar is taken as an example as shown in Fig. 4.5.16. It seems that there exists a better linear fit, which will pass through more measurement points than the line illustrated in Fig. 4.5.16. But that line would not pass through the original point (0, 0). This method of linear fit was repeated for the other two cases of Pmean=12 and 15 bar. The resulting conductance 1/R24 for these filling pressures is listed in the table 4.5.II. The convection loss in the thermally floating cases is obtained by using Eq. (4.5.10) and corresponding value of thermal conductance 1/R24 in table 4.5.II.

Pmean 11 bar 12 bar 15 bar

1/R24 (W/K) 0.366 0.321 0.409

Table 4.5.II: Conductance 1/R24 derived from Eq.(4.5.9) at various filling pressures.

-32 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12-14

-12

-10

-8

-6

-4

-2

Pse

cHX

(W)

T4-T2 (K)

measured value linear fit (by Eq.(4.5.9))

Figure 4.5.16: Linear fit of temperature difference T4-T2 and PsecHX by Origin for the case of Pmean=11 bar in thermally grounded condition. The energy flows and losses for a filling pressure of 11 bar are given in Figs. 4.5.17 to 4.5.19. The Figs. 4.5.20 to 4.5.22 are for a filling pressure of 12 bar and from 4.5.23 to 4.5.25 are for a filling pressure of 15 bar. Figs. 4.5.17, 4.5.20, and 4.5.23 are plots for energy flows and viscous losses with respect to the complete

170 Chapter 4

refrigerator. Since these were zero-load measurements (operation A), the cooling power is zero, i.e. Pcooling=0. For the thermally grounded cases, the energy flows taking place at the ambient heat exchanger and the secondary heat exchanger, and the total wall loss defined in Eq. (4.5.7) for viscous loss of the complete inner surface of the refrigerator are shown in the figures. For the thermally floating cases, the heat exchange at the secondary heat exchanger is zero. Therefore, only the energy flows at the ambient heat exchanger and the total wall losses are shown in the figures. Energy flowing from a heat exchanger to the outside world is defined positive. The energy flows at the secondary heat exchanger PsecHX in the three figures are negative, because heat flows from the outside to that heat exchanger (since that is cooled along with the cold heat exchanger). Figs. 4.5.18, 4.5.21, and 4.5.24 are plots for energy flows and viscous losses in the part downstream of the microphone 2 (P2 in Figs.4.5.6 and 4.5.9), which is the part in the dashed rectangle in Fig. 4.5.7. The acoustic power going into the loop section Pacoustic is calculated by using Eq. (4.5.6) and the viscous loss Pwall loss is computed by using Eq. (4.5.8). The convection loss Pconvection is obtained by using Eq. (4.5.10). In these three figures, the values of the viscous loss Pwall loss for thermally grounded case are close to their corresponding points in thermally floating case. The convection loss Pconvection is negative because of the thermal coupling between cold heat exchanger and secondary heat exchanger (R24 in Fig.4.5.10). The convection loss in the thermally grounded case is not separately depicted since it equals the heat flowing to the secondary heat exchanger (PsecHX). The acoustic power Pacoustic in the thermally floating cases is larger than the corresponding one in thermally grounded cases. In the latter cases, more acoustic energy is needed to compensate for the loss from the secondary heat exchanger to the environment, which is shown by P02 in Fig. 4.5.10. Figs. 4.5.19, 4.5.22 and 4.5.25 show that the sum of the losses (Pconvection and Pwall

loss) is comparable to the input acoustic power. Therefore, neglecting these losses in the analytical model of section 4.3 will be the main cause of the discrepancy between computation results and measurements as shown in Fig. 4.5.15. Furthermore, note that the losses are about half the acoustic power, these also explaning the factor of two difference between computation and measurements.

Traveling-wave systems 171

0 10 20 30 40 50 60 70

-15

-10

-5

0

5

10

15

20

25

30

Pow

er (

W)

Pdriver

(W)

PambHX

(thermally grounded)P

secHX (thermally grounded)

Ptotal wall loss

(thermally grounded)P

ambHX (thermally floating)

Ptotal wall loss

(thermally floating)P

mean=11 bar

Figure 4.5.17: Energy flows and viscous loss along the wall of thermally grounded and thermally floating cases at filling pressure of 11 bar.

0 10 20 30 40 50 60 70

-14-12-10

-8-6-4-202468

10121416182022

Pow

er (

W)

Pdriver

(W)

Pacoustic

(thermally grounded) P

secHX (thermally grounded)

Pwall loss

(thermally grounded) P

acoustic (thermally floating)

Pconvection

(thermally floating) P

wall loss (thermally floating)

Pmean

=11 bar

Figure 4.5.18: Energy flows and losses in the part downstream of microphone 2 of thermally grounded and thermally floating cases at filling pressure of 11 bar.

172 Chapter 4

0 10 20 30 40 50 60 70

2

4

6

8

10

12

14

16

18

20

22

Pow

er (

W)

Pdriver

(W)

Pacoustic

(thermally floating) P

convection+P

wall loss

(thermally floating)P

mean=11 bar

Figure 4.5.19: Input acoustic power at microphone 2 and sum of losses in the part downstream of microphone 2 of thermally floating case at filling pressure of 11 bar.

0 10 20 30 40 50 60 70

-10

-5

0

5

10

15

20

25

PambHX

(thermally grounded)P

secHX (thermally grounded)

Ptotal wall loss

(thermally grounded)P

ambHX (thermally floating)

Ptotal wall loss

(thermally floating)P

mean=12 bar

Pow

er (

W)

Pdriver

(W)

Figure 4.5.20: Energy flows and viscous loss along the wall of thermally grounded and thermally floating cases at filling pressure of 12 bar.

Traveling-wave systems 173

0 10 20 30 40 50 60 70

-12-10-8-6-4-202468

1012141618202224

Pacoustic

(thermally grounded) P

secHX (thermally grounded)

Pwall loss

(thermally grounded) P

acoustic (thermally floating)

Pconvection

(thermally floating) P

wall loss (thermally floating)

Pmean

=12 barP

ower

(W

)

Pdriver

(W)

Figure 4.5.21: Energy flows and losses in the part downstream of microphone 2 of thermally grounded and thermally floating cases at filling pressure of 12 bar.

0 10 20 30 40 50 60 700

5

10

15

20

25

Pacoustic

(thermally floating) P

convection+P

wall loss

(thermally floating)P

mean=12 bar

Pow

er (

W)

Pdriver

(W)

Figure 4.5.22: Input acoustic power at microphone 2 and sum of losses in the part downstream of microphone 2 of thermally floating case at filling pressure of 12 bar.

174 Chapter 4

0 10 20 30 40 50 60 70

-15

-10

-5

0

5

10

15

20

25

30

35

PambHX

(thermally grounded)P

secHX (thermally grounded)

Ptotal wall loss

(thermally grounded)P

ambHX (thermally floating)

Ptotal wall loss

(thermally floating)P

mean=15 bar

Pow

er (

W)

Pdriver

(W)

Figure 4.5.23: Energy flows and viscous loss along the wall of thermally grounded and thermally floating cases at filling pressure of 15 bar.

0 10 20 30 40 50 60 70

-15

-10

-5

0

5

10

15

20

25

Pacoustic

(thermally grounded) P

secHX (thermally grounded)

Pwall loss

(thermally grounded) P

acoustic (thermally floating)

Pconvection

(thermally floating) P

wall loss (thermally floating)

Pmean

=15 bar

Pow

er (

W)

Pdriver

(W)

Figure 4.5.24: Energy flows and losses in the part downstream of microphone 2 of thermally grounded and thermally floating cases at filling pressure of 15 bar.

Traveling-wave systems 175

0 10 20 30 40 50 60 700

5

10

15

20

25

30 Pacoustic

(thermally floating) P

convection+P

wall loss

(thermally floating)P

mean=15 bar

Pow

er (

W)

Pdriver

(W)

Figure 4.5.25: Input acoustic power at microphone 2 and sum of losses in the part downstream of microphone 2 of thermally floating case at filling pressure of 15 bar. III. Comparison of performance at varying cooling power for thermal grounded and floating cases In this measurement, the electrical heater is switched on to provide a heat load at the cold end of the regenerator (operation B). The cooling power is obtained by Eq. (4.5.4). The input power into the PWG was fixed at 30, 50 and 70 W for every measurement. The filling pressure Pmean was fixed at 11 and 15 bar. The temperature difference across the regenerator at various cooling powers is plotted for the thermally grounded, and the thermally floating cases. The COP, given by Eq. (4.5.11), at various cooling power is also plotted for both cases.

acousticcooling PPCOP /= . (4.5.11)

The Carnot COP is the maximal theoretical performance that a refrigerator can achieve, and it is given by:

CH

CC TT

TCOP

−= . (4.5.12)

The relative COP, the ratio of COP to Carnot COP, is calculated as:

CR COPCOPCOP /= . (4.5.13)

176 Chapter 4

0 1 2 3 4 5 6 7 8 9 10 11 126

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

∆T (

K)

Cooling power (W)

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally grounded

Figure 4.5.26: Temperature difference across the regenerator as a function of cooling power at filling pressure 11 bar for thermally grounded case.

0 1 2 3 4 5 6 7 8 9 10 11 12

12

14

16

18

20

22

24

26

28

30

32

34

36

38 Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=15 bar

thermally grounded

∆T (

K)

Cooling power (W)

Figure 4.5.27: Temperature difference across the regenerator as a function of cooling power at filling pressure 15 bar for thermally grounded case.

Traveling-wave systems 177

0 1 2 3 4 5 6 7 8 9 10 11 12 13 1410

15

20

25

30

35

40

45

50

55

∆T (

K)

Cooling power (W)

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally floating

Figure 4.5.28: Temperature difference across the regenerator as a function of cooling power at filling pressure 11 bar for thermally floating case.

0 1 2 3 4 5 6 7 8 9 10 11 1215

20

25

30

35

40

45

50

55 P

driver=30 W

Pdriver

=50 W P

driver=70 W

Pmean

=15 barthermally floating

∆T (

K)

Cooling power (W)

Figure 4.5.29: Temperature difference across the regenerator as a function of cooling power at filling pressure 15 bar for thermally floating case. As shown in the Figs. 4.5.26 and 4.5.27 for thermally grounded, and Figs. 4.5.28 and 4.5.29 for thermally floating cases, for all total input powers, the ∆T decreases nearly linearly with increasing cooling power. Therefore, ∆T as a function of cooling power Pcooling can be given as:

178 Chapter 4

maxmax

1cooling

cooling

P

P

T

T −=∆∆

. (4.5.14)

Here, ∆Tmax is the maximum ∆T for a specific input power, which is the temperature difference at zero cooling load. Pcooling max is the maximum cooling power that the system achieves for a specific input power, which is obtained at ∆T=0. At relatively small levels of cooling power, the acoustic input power is merely required to compensate for the losses. Therefore, the required input power will hardly change as the cooling power is increased. Thus it can be expected that at small levels of cooling power, the COP is proportional to that cooling power:

coolingPCOP∝ . (4.5.15)

This is confirmed by Figs. 4.5.30-4.5.33.

0 1 2 3 4 5 6 7 8 9 10 11 120.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally grounded

Cooling power (W)

CO

P

Figure 4.5.30: COP as a function of cooling power at filling pressure 11 bar for thermally grounded case.

Traveling-wave systems 179

0 1 2 3 4 5 6 7 8 9 10 11 120.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8 Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=15 bar

thermally grounded

CO

P

Cooling power (W)

Figure 4.5.31: COP as a function of cooling power at filling pressure 15 bar for thermally grounded case.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally floating

Cooling power (W)

CO

P

Figure 4.5.32: COP as a function of cooling power at filling pressure 11 bar for thermally floating case.

180 Chapter 4

0 1 2 3 4 5 6 7 8 9 10 11 120.0

0.2

0.4

0.6

0.8

1.0 Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=15 bar

thermally floating

Cooling power (W)

CO

P

Figure 4.5.33: COP as a function of cooling power at filling pressure 15 bar for thermally floating case. The relative COPs are given in Figs. 4.5.34 to 4.5.37 for both thermally grounded and floating cases. In these figures, the maximum values for most of the curves are not yet reached in the range of cooling powers which were tested. By using Eq. (4.5.12), the Carnot COP can be written as:

1)( −

∆=

−−−=

−=

T

T

TT

TTT

TT

TCOP H

CH

CHH

CH

CC . (4.5.16)

By using Eqs. (4.5.15) and (4.5.16), the relative COP is given as

TT

PT

TT

P

COP

COPCOP

H

cooling

H

cooling

CR ∆−

∆=

−∆∝=

1/. (4.5.17)

By employing Eq. (4.5.14), Eq. (4.5.17) can be rewritten as:

−−

maxmax

max

1

1

cooling

coolingH

cooling

coolingcooling

R

P

P

T

T

P

PP

COP . (4.5.18)

A maximum COPR is achieved in Eq.(4.5.18) if

∆−

∆+

∆= 111

maxmax

2

maxmax T

T

T

T

T

T

P

PHHH

cooling

cooling (4.5.19)

In the measurements, TH>>∆Tmax, thus Eq.(4.5.19) reduces to:

Traveling-wave systems 181

max21

coolingcooling PP = . (4.5.20)

In our measurements data ∆Tmax was in all experiments smaller than 50 K. Substituting TH=300 K in that case, Eq.(4.5.19) would yield a ratio of 0.48 instead of 0.5 which is obtained by assuming TH/∆Tmax→∞. So, a maximum in the relative efficiency is expected at a value of Pcooling that is around half Pcooling max. When extrapolating the curves in Figs. 4.5.26, we find for the grounded case at 30 W, Pcooling max=7.5 W; at 50 W Pcooling max=13.7 W; and at 70 W Pcooling max=17W. Maxima in the relative COP could thus be expected at 3.8 W, 6.9 W and 8.5 W, respectively. Fig. 4.5.34 shows that the actual maxima are at slightly higher values of cooling power. This is due to the fact that the COP increases more than proportional to the cooling power as shown in Fig. 4.5.30. The same analysis can be given for the other cases.

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

9

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally grounded

Cooling power (W)

Rel

ativ

e C

OP

[%]

Figure 4.5.34: Relative COP as a function of cooling power at filling pressure 11 bar for thermally grounded case.

182 Chapter 4

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

9

Rel

ativ

e C

OP

[%]

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=15 bar

thermally grounded

Cooling power (W)

Figure 4.5.35: Relative COP as a function of cooling power at filling pressure 15 bar for thermally grounded case.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

8

9

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=11 bar

thermally floating

Cooling power (W)

Rel

ativ

e C

OP

[%]

Figure 4.5.36: Relative COP as a function of cooling power at filling pressure 11 bar for thermally floating case.

Traveling-wave systems 183

0 1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

Pdriver

=30 W P

driver=50 W

Pdriver

=70 WP

mean=15 bar

thermally floating

Rel

ativ

e C

OP

[%]

Cooling power (W)

Figure 4.5.37: Relative COP as a function of cooling power at filling pressure 15 bar for thermally floating case. 4.5.8 Discussion Measurement data obtained from the experimental set-up described in section 4.5.3 were compared to computation results based on the analytical model described in section 4.3. The model correctly predicts the trends but overestimates the cooling performance by about a factor of 2. This is due to the fact that losses due to viscous effects and convection arise in practice that are not included in the model. Many other research groups found similar losses in thermoacoustic-Stirling type refrigerators. Tijani and Spoelstra developed and measured a coaxial thermoacoustic-Stirling cooler [78]. This cooler has a different feedback structure from the toroidal one and thus the loop section is more compact. Argon at a mean pressure of 15 bar is the working gas and it operates at 60 Hz. The cooler has achieved a relative efficiency of 25% of Carnot and a low temperature of -54°C without heat load. Tijani and Spoelstia also measured the acoustic power dissipated in the resonator by thermo-viscous processes at the wall as a function of the drive ratio. They found that the viscous losses in the resonator at the operating point of high performance (drive ratio=4.3%) reached as high as about half the power input. Therefore, they proposed to improve the design by using a quarter-wavelength resonator instead of a half-wavelength resonator and thus reducing the resonator losses.

184 Chapter 4

Luo and Dai have been working on the thermoacoustic-Stirling refrigerator with toroidal structure and driven by a thermoacoustic-Stirling engine [73, 79, 80]. They built a traveling-wave thermoacoustic refrigerator driven by a traveling-wave thermoacoustic engine and carried out numerical simulations about this set-up [73]. Their set-up achieved a lowest temperature of -64.4 °C and 250 W cooling power at -22.1 °C when the system was filled with 3.0 MPa helium gas, working at 57.7 Hz and 2.2 kW of heat input into the thermoacoustic engine. They investigated the performance at different heating powers and mean pressures. They found that their numerical simulation program worked well in the frequency prediction and reasonably well in the prediction of refrigerator inlet pressure amplitudes. But they saw a large discrepancy between computation and measurement in the prediction of the refrigerator’s cooling power as a function of cold-end temperature. They attributed this large discrepancy to the serious underestimation of some loss mechanisms strongly related to the thermoacoustic refrigerator’s cold-end temperature, such as Rayleigh streamings inside the refrigerator’s thermal buffer tube. This loss has similarity with the loss indicated as Pconvection in section 4.5. In Figs.4.5.18, 4.5.21 and 4.5.24, the loss Pconvection is much larger than the viscous loss along the tube wall Pwall loss and it is comparable to the acoustic power Pacoustic in the thermally floating cases. Luo and Dai explained that neglecting the influence of nonlinear behavior, especially the turbulence, in the model could be another cause of the large discrepancy. This has similarities with the presented comparison between computation and experimental measurements as shown in Fig.4.5.15. In the work in [80], Luo and Dai theoretically investigated a thermoacoustically driven Stirling cryocooler working at a high frequency of 500 Hz. By numerical simulations, they concluded that the high operating frequency significantly decreases the refrigeration efficiency. The relative efficiency (COP/Carnot efficiency) of this 500 Hz refrigerator was predicted theoretically as about 8%-15%. As discussed in the following chapter 5, the efficiency of a scaled-down system, which will work at high frequencies as a result of scaling-down, decreases with scaling factor. Ueda et al. experimentally investigated pressure p and velocity U in the acoustic field of a thermoacoustic Stirling engine with pressure sensors and a laser Doppler velocimeter for flow visualization [81, 82]. They found that the negative phase lead (around -20°) of U relative to p, rather than a pure traveling-wave phase (0°), made the engine achieve a high efficiency. They inserted a second regenerator into the looped tube to work as a cooler. Thus, the cooler used the work flow generated by the engine to produce cooling [81, 83]. The cooler could reach a low temperature of -25°C without heat load. They measured the lowest temperature at the cold end TC of the cooler regenerator under different filling pressures when the total input

Traveling-wave systems 185

power into the hot end of the engine regenerator QH remained constant at 210 W. The measurement showed an optimum filling pressure for each kind of working gas. The cold-end temperature TC went higher when the filling pressure Pm was larger or smaller than the optimum value. This has clear similarity to the trend in Fig.4.5.15, where the filling pressure of 11 bar has the best performance in comparison with 12 and 15 bar. 4.5.9 Conclusion The analytical model, developed in section 4.3, is able to correctly predict the trends, but overestimates the cooling performance by comparison with the measurements results in section 4.5. The discrepancy is due to the highly idealized assumptions in the model which exclude many losses taking place in practice. This analytical model is acceptable for the next scaling analysis.

Chapter 5 Scaling considerations 5.1 Introduction Many large-scale refrigerators were developed in the past decades. Small-scale refrigerators also attracted some researchers’ attention. Many of these investigations can be found in patents. In 2002, Chen reported on two miniature standing-wave type refrigerators [48]. The maximum temperature difference was 10°C using an in-house MEMS stack material, under the operation conditions of 1 atm at around 4 kHz and 10 kHz. Symko has been devoted to the exploration of miniature thermoacoustic refrigerators [49-51]. His miniature standing-wave type coolers can be as small as a few centimeters long, and operating at thousands of hertz. These impressively small thermoacoustic coolers have promising applications on various industrial and electronics utilization. Olson and Swift investigated similitude in thermoacoustics [86]. They proposed a group of dimensionless parameters and rewrote the basic thermoacoustic equations in dimensionless form. These dimensionless parameters are helpful to predict the performance of a thermoacoustic device by measuring a scale model which satisfies the similitude principles. Benavides analyzed the minimum reachable size for thermoacoustic devices related to the working gas [52]. He used the properties of the working gas to find the limitation for scaling-down. In conclusion, he found that the reachable minimum size for different gases could be the order of micrometer. This chapter focuses on what during scaling-down could be the major problem that limits miniaturization from the point of geometrical configuration and stack-material construction. By using individual scaling factor in every dimension, the performance can be predicted when a reference system is designed to shrink in any or any combination of the three dimensions: x, y, z. Only geometry change is considered, therefore, the changes don’t have to satisfy the similitude principles.

Scaling considerations 187

5.2 Standing-wave systems In this chapter, the scaling of standing-wave systems is analyzed. It consists of three subsections, referring to three different boundary conditions: constant temperature difference over the stack, constant total energy flow and constant total energy flow density. 5.2.1 Constant temperature difference over the stack In this section, it is supposed that the temperature difference across the stack remains unchanged while scaling down. We are going to focus on the variation of the cooling power when scaling down.

Figure 5.2.1: (a) An original standing-wave refrigerator in its original coordinate system (b) a scaled-down standing-wave refrigerator in a scaled-down coordinate system.

x

y

T∆

L

z

xxx ϕ/=′

yyy ϕ/=′

T∆

xLL ϕ/=′

zzz ϕ/=′

(a)

(b)

188 Chapter 5

The analysis is made through the following approach: starting point is from an original refrigerator in original coordinates and a scaled-down refrigerator in scaled-down coordinates. To make a general analysis, the three axes of the scaled-down coordinate system are scaled with three different scaling factors: xϕ , yϕ , zϕ ,

as shown in Fig.5.2.1.

Figure 5.2.2: Two types of energy flow in a standing-wave: A and B.

Figure 5.2.2(a): Type A: a quarter-wave-length standing-wave cooler system.

Figure 5.2.2(b): Type B: a half-wave-length standing-wave cooler system. The temperature span across the stack in the original refrigerator T∆ remains the same in the scaled version. First, we assume that the stack and the two heat

Driver TH TC

Hot HX

Cold HX

Stack

Pressure node

Pressure anti-node

Pressure anti-node

TH TC

Hot HX

Cold HX

Stack

Driver

Pressure node

Pressure anti-node

(A) (B) CQ&

inW2&

HT

CT HQ&

Control volume

2E&

CQ&

inW2&

HT

CT HQ& Control

volume

2E&

Scaling considerations 189

exchangers are contained in a perfect resonator tube, which means that no dissipation and no power flow is needed to maintain the standing wave going. So, we can assume that all the dissipation of work flow takes place in the stack. Generally speaking, there are two types of energy flow, as shown in Fig.5.2.2. We name them as energy flow types A and B. In practical situations, type A and B are illustrated in Figs. 5.2.2(a) and (b), respectively. The total energy flows are differently composed in the two types. In type A, a control volume is applied to the cold heat exchanger and part of the stack, as shown in Fig. 5.2.2 (A). The total energy flow along the stack in type A is given by

CQE && =2 . (5.2.1)

So, the cooling power in the cold heat exchanger is given by the total energy flow from cold to hot heat exchanger. Similarly, a control volume is applied in type B, containing the cold heat exchanger and part of the stack. The energy balance in the control volumes is given by

Cin QWE &&& += 22 . (5.2.2)

inW2& is the work fed by the driver. In this case the cooling power equals the

difference between the total energy flow and the acoustic power input from the driver. It can be seen that the cooling power depends on the total energy flow along the stack in both types. So, for type A, the cooling power is given by

2EQC&& = , (5.2.3)

and for type B:

inC WEQ 22&&& −= , (5.2.4)

In order to investigate the variation of the cooling power after scaling down, it is

necessary to look at the variation of the total energy flow 2E& along the stack.

Now the parameters of the scaled-down refrigerator system are applied to the equations in chapter 2. The following equations are obtained:

resonator tube length x

LL ϕ=′ , (5.2.5)

angular frequency ωϕω x=′ , (5.2.6)

radian wave length xϕ

DD =′ . (5.2.7)

In the stack region, now the following equations hold for the thermal penetration depth of the fluid

κκ δϕωϕρωρ

δxxpmpm c

K

c

K 122 ==′′

′=′ , (5.2.8)

the viscous penetration depth of the fluid

190 Chapter 5

νν δϕωϕρ

µωρ

µδxxmm

122 ==′′

′=′ , (5.2.9)

and the thermal penetration depth of the solid

s

xxss

s

ss

ss c

K

c

K δϕωϕρωρ

δ 122 ==′′

′=′ . (5.2.10)

While scaling, the ratio of stack spacing to thermal penetration depth is assumed to be kept constant, because it is by definition an optimized parameter for a fixed ratio

κκ δδ00 yy

=′′

, (5.2.11a)

it leads to

00

1yy

xϕ=′ , (5.2.11b)

Also, the ratio between stack plate thickness and material thermal penetration depth is assumed to be fixed while scaling:

ss

ll

δδ=

′′

, (5.2.12a)

it leads to

llxϕ

1=′ . (5.2.12b)

In the original system, the perimeter is approximately calculated as follows:

ly

dydz

ly

A Stube

+=

+=Π

∫∫

00

, (5.2.13)

where Atube is the cross-sectional area of the tube. In the scaled-down system, the perimeter is given by

( )Π=

+=

′+′

′′=

′+′′

=Π′−

′∫∫∫∫

zy

x

x

S

zy

Stube

ly

dydz

ly

zdyd

ly

A

ϕϕϕ

ϕ

ϕϕ

02

1

1

00

)(

. (5.2.14)

The Prandtl number is a material property of the working fluid, and is not affected by scaling:

σµµ

σ ==′

′=′

K

C

K

C pp. (5.2.15)

The Rott’s functions vf ′ , κf ′ and sε ′ for the scaled-down system are given by

substitution of equations (5.2.8) to (5.2.12ab) into equations (2.1.32) to (2.1.34). These are also not affected by scaling:

Scaling considerations 191

ν

ν

ν

ν

νν

δ

δ

δ

δf

yi

yi

yi

yif =

+

+

=

′′+

′′+

=′0

0

0

0

)1(

)1(tanh

)1(

)1(tanh, (5.2.16)

κ

κ

κ

κ

κκ

δ

δ

δ

δf

yi

yi

yi

yif =

+

+

=

′′+

′′+

=′0

0

0

0

)1(

)1(tanh

)1(

)1(tanh, (5.2.17)

( )

( )

( )

( )s

ssss

pm

ssss

pm

slic

yic

lic

yicε

δδρ

δδρ

δδρ

δδρε κ

κκ

κ=

+

+

=

′′+′′

′′+′′

=′1tanh

1tanh

1tanh

1tanh 00

. (5.2.18)

If the input acoustic pressure at the interface between the driver and the resonator tube is constant while scaling, then, at the same positions relative to the resonator-tube length, the acoustic pressure is also not affected (see appendix K)

11 pp =′ . (5.2.19)

Therefore, the pressure and temperature gradient become

dx

ddxd

xd

dx

x

111 ppp ϕϕ

==′′

, (5.2.20)

dx

dTdxdT

xd

Td mx

x

mm ϕϕ

==′′

. (5.2.21)

Substitution of the equations (5.2.6) to (5.2.21) into the total energy flow equation (2.1.72) yields

( ) ( )( )( )

++−−−∏⋅=′ −

σεωρϕϕ νκ

ν 11

~~

1~

Im2 1

1012

smzy

fff

dx

dyE p

p&

( )( )( )

( )( )

+++−+×

−∏

+σε

εσρω

κννκν 11

/1~

~Im

~

1211

30

s

sm

m

p fffff

dx

d

dx

d

dx

dTcy pp

( ) ( )dx

dTlKKy m

szyx +Π− −0

1ϕϕϕ . (5.2.22)

The two terms scaling with different factors are defined as

( )( )( )

++−−−∏=−

σεωρνκ

ν 11

~~

1~

Im2 1

10

sm

fff

dx

dyacousticsE p

p

( )( )( )

( )( )

+++−+×

−∏

+σε

εσρω

κννκν 11

/1~

~Im

~

1211

30

s

sm

m

p fffff

dx

d

dx

d

dx

dTcy pp, (5.2.23)

192 Chapter 5

and

( )dx

dTlKKyconductionE m

s+∏=− 0 . (5.2.24)

Eq. (5.2.22) can now be written as

{ } { }conductionEacousticsEEzy

x

zy

−⋅−−⋅=′ϕϕ

ϕϕϕ1

2& . (5.2.25)

For energy flow type A, the cooling power equals the total energy flow along the stack, as seen in Eq. (5.2.3)

{ } { }conductionEacousticsEEQzy

x

zyC −⋅−−⋅=′=′

ϕϕϕ

ϕϕ1

2&& . (5.2.26)

For energy flow type B, more work needs to be done. We assume that the input acoustic pressure at the interface between the driver and the resonator tube p1in is constant in scaling, i.e. inin 11 pp =′ , it is shown in appendix K that the volume

velocity at the interface U1in is obtained as:

( ) ( ) inzyzy

ininin 1

1

0

1

0

11 U

Zp

Zp

U ⋅==′

′=′ −ϕϕ

ϕϕ. (K.30)

The total input acoustic power at the interface of the driver and the resonator tube into the resonator tube after scaling is given by:

[ ] ( ) [ ] ( ) inzyininzyininin WW 21

111

112

~Re

21~

Re21 && ⋅=⋅⋅⋅=′⋅′⋅=′ −− ϕϕϕϕ UpUp (5.2.27)

Therefore, for energy flow type B, according to Eq. (5.2.4), the cooling power in the scaled-down system can be expressed as

{ } { }conductionEWacousticsEWEQzy

xin

zyinC −⋅−−−⋅=′−′=′

ϕϕϕ

ϕϕ 222 )(1 &&&& .

(5.2.28) From Eq. (5.2.26) and (5.2.28), it is obvious that the cooling power in the scaled-down system consists of two groups of energy flow scaling with different factors.

The first group of energy flow scales with a factor( ) 1−zyϕϕ , whereas the

conduction term scales as( ) 1−zyx ϕϕϕ . This means that the thermal conduction will

finally dominate the losses during scaling down. The cooling capacity of the system will decrease after scaling down. So, there must be a limitation for scaling down. If the system scales down uniformly in all directions, i.e. ϕϕϕϕ === zyx , the

resultant conclusion is even more clear. Then Eq. (5.2.26) and (5.2.28) become

{ } { }conductionEacousticsEQC −⋅−−⋅=′ −− 12 ϕϕ& (5.2.29)

and

Scaling considerations 193

{ } { }conductionEWacousticsEQ inC −⋅−−−⋅=′ −− 12

2 )( ϕϕ && . (5.2.30)

So the cooling power decreases rapidly with 2−ϕ , but the conduction losses with 1−ϕ showing that for large ϕ these will tend to dominate, making CQ& negative. So

for large ϕ and fixed ∆T the losses via conduction will be larger than the available

acoustic power. Total energy flow density The cross sectional area of the tube section where the stack is located

)()22(2

100 lylyA +∏=+∏= . (5.2.31)

After scaling, the area is given by

( ) Aly

lyA zy

xzy

x 100

)()( −=

+Π=′+′∏′=′ ϕϕ

ϕϕϕϕ

. (5.2.32)

The total energy flow per unit of cross-sectional area after scaling is obtained as:

( )( )( )

++−−−

+=

′′

σεβ

ωρνκ

ν 11

~~

1~

Im2

1

1

11

1

0

2

s

m

m

ffTf

dx

d

ylA

Ep

p&

( )( )( )

( )( )

+++−+×

−++

σεε

σρωκννκ

ν 11/1

~~

Im~

121

1 113

0s

sm

m

p fffff

dx

d

dx

d

dx

dTc

yl

pp

dx

dTK

y

lK

yl

msx

+

+⋅−

00

1

1ϕ . (5.2.33)

It can be rewritten as:

−⋅−

−=

′′

A

conductionE

A

acousticsE

A

Exϕ2

&. (5.2.34)

The total energy flow per unit area decreases when the system scales down. The energy flow density in the first term group remains the same, whereas the energy flow per unit area due to gas and plate thermal conduction increases when scaling down. Eq.(5.2.34) shows that the maximum scaling factor in x-direction is given by the ratio of acoustic energy flow and conductive flow in the original case. Coefficient of performance (COP) With the assumption that the input acoustic pressure at the interface between the driver and the resonator tube p1in is constant in scaling, i.e. inin 11 pp =′ , it is shown

in appendix K that the volume velocity at the interface U1in is obtained as:

194 Chapter 5

( ) ( ) inzyzy

ininin 1

1

0

1

0

11 U

Zp

Zp

U ⋅==′

′=′ −ϕϕ

ϕϕ. (K.30)

The total input acoustic power at the interface of the driver and the resonator tube into the resonator tube after scaling is given in Eq.(5.2.27). According to the definition, the COP of a standing-wave thermoacoustic refrigerator after scaling is given as:

in

C

W

QPCO

2′′

=′&

&. (5.2.35)

Therefore, substituting Eq. (5.2.26) and (5.2.27) into (5.2.35), the COP of the scaled-down system of type A becomes

( ) { } ( ) { }( ) inzy

zyxzy

in

C

W

conductionEacousticsE

W

QPCO

21

11

2&&

&

⋅−⋅−−⋅

=′′

=′ −

−−

ϕϕϕϕϕϕϕ

( )in

x W

conductionECOP

2

1&

−−−= ϕ . (5.2.36a)

For type B, substitution of Eq. (5.2.28) and (5.2.27) into (5.2.35), the COP after scaling becomes

( ) { } ( ) { }( ) inzy

zyxzy

W

conductionEWacousticsEPCO

21

12

1 )(&

&

⋅−⋅−∆−−⋅

=′ −

−−

ϕϕϕϕϕϕϕ

( )in

x W

conductionECOP

2

1&

−−−= ϕ . (5.2.36b)

From Eq. (5.2.36a) and (5.2.36b), it can be seen that the COPs for both types A and B are described by exactly the same equation. The last term in Eq. (5.2.36ab) is the product of the scaling factor and the original ratio of energy losses due to thermal conduction of working gas and plates to acoustic work. This term causes the reduction of COP after scaling. That means that the energy loss due to thermal conduction of the working gas and plates will become more and more dominant, when the system scales down. For the cases of constant total energy flow 2E& and constant total energy flow per

unit area AE /2& , explicit analytical expressions can not be obtained. These two

cases will be discussed later by numerical computation. To make this theoretical analysis visible, a computation is made for uniform scaling and for energy flow type A. Type B can be treated fully analogously. A sketch of the reference system is given in Fig.5.2.3. It is assumed that the left end is connected to a large buffer volume to work as an open end. The right end is driven by a loudspeaker. Therefore, the acoustic power is fed into the resonator from the end facing to the hot-end heat exchanger, as in type A of Fig. 5.2.2. The cold and

Scaling considerations 195

hot heat exchangers are assumed to be ideal. The configuration of the standing-wave refrigerator described in Fig.3.3.2, Table 3.3.I (a) and (b) is used as the reference system. It operates at 1000 Hz. The scaling performance of the original configuration under different working conditions: mean pressure of 1 bar, 10 bar and 100 bar is compared, while the drive ratio remains 10% all the time. Comparison between different temperature span across the stack under the same operational condition is also investigated. All these comparisons are made to study how to expand the space for scaling down a specific system.

Figure 5.2.3: A sketch of the reference standing-wave refrigerator. Comparison a) Different temperature difference over the stack (1) The stack position is fixed at x/Lres=0.63 (in the reference system that is x=0.16

m), the output cooling power CQ& is 13 W. The temperature difference over the

stack ∆Tstack is 100 K, mean pressure of 1 bar and drive ratio of 10%. Under the condition that ∆Tstack is kept at 100 K, scaling down of the system leads to a change of the energy group related to productive acoustic power Eq. (5.2.23) and the thermal conduction of the working gas and stack plates Eq. (5.2.24). The energy changes taking place at the leading end of the stack are shown in Fig. 5.2.4. The net cooling power is the difference between the two curves in Fig.5.2.4 and that decreases rapidly when the system size is reduced. In Fig.5.2.4, the energy flow containing acoustic power, which is the first term group in Eq. (5.2.26) and defined by Eq. (5.2.23), decreases faster than the consumptive energy flow due to thermal conduction of the working gas and stack plates. At some point, the two curves cross each other. If the scaling-down goes further than that value, no cooling power is available anymore. Here in this specific case, the value is approximately 45. That means that the model refrigerator can be scaled down to around 1/45 of its original size, not smaller, if non-zero cooling power is required. The decrease of cooling power at the same ∆Tstack after scaling-down implies the reduction of cooling capability.

0 x xstack=0.63Lres

Lres

Cold HX

Hot HX

Pressure anti-node

Pressure node inW2

&

196 Chapter 5

1 10 100 1000

-5

-4

-3

-2

-1

0

1

2

Log

(ene

rgy

flow

) (W

)

Scaling factor

E-acousticsE-conduction

Figure 5.2.4: Energy flows containing acoustic power and heat conduction through working gas and stack plates as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (100 K) mean pressure is 1 bar, drive ratio=10%, x/Lres=0.63. (2) If the system operates at another temperature span across the stack ∆Tstack, the scaling performance will behave differently. Using the same configuration, and placing the stack at the same position x/Lres=0.63 (x=0.16 m in reference system),

the output cooling power CQ& is increased to 39 W at a temperature difference over

the stack of 50 K. The mean pressure is 1 bar and drive ratio remains 10%. The energy flows are plotted in Fig. 5.2.5. In Fig.5.2.5, the two curves are similar to those in Fig.5.2.4. The energy flow containing acoustic power decreases faster than the energy flow due to thermal conduction of the working gas and stack plates. Again, at the crossing point of the two curves the net cooling power decreases to zero. Here in this case, the maximum scaling factor is around 177. That means that the reference refrigerator can be scaled down to 1/177 of its original size, if we keep ∆Tstack at 50 K with non-zero cooling power. Compared with the previous case, the factor to scaling-down is greatly enlarged, however, under a lower performance condition. Further scaling-down can be realized by the reduction of temperature difference over the stack. This is of course not an attractive option because of the smaller and smaller ∆T. Another option is to increase the mean pressure of the system and keeping the same drive ratio.

Scaling considerations 197

1 10 100 1000-5

-4

-3

-2

-1

0

1

2

E-acousticsE-conduction

Log

(ene

rgy

flow

) (W

)

Scaling factor

Figure 5.2.5: Energy flows containing acoustic power and heat conduction through working gas and stack plates as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (50 K) mean pressure is 1bar, drive ratio=10%, x/Lres=0.63.

1 10 100 1000

-4

-3

-2

-1

0

1

2

3

E-acousticsE-conduction

Log

(ene

rgy

flow

) (W

)

Scaling factor

Figure 5.2.6: Energy flows containing acoustic power and heat conduction through working gas and stack plates as a function of the down-scaling factor. Temperature difference over the stack remains constant (100K), mean pressure is 10 bar, drive ratio=10%, x/Lres=0.63.

198 Chapter 5

Comparison b) Different mean pressure at the same drive ratio Computations are made for the same system configuration working at mean pressures 10 bar and 100 bar. The drive ratio is constant at 10%. (3) If in the reference system, the mean pressure is 10 bar, and the stack position is

fixed at x/Lres=0.63 (x=0.16 m), the output cooling power CQ& is 121 W at a

temperature difference over the stack ∆Tstack of 100 K. The temperature over the stack remains constant (100 K), when scaling down the system. The change of the energy flows is shown in Fig.5.2.6. Based on the intersection point in Fig. 5.2.6, the increase of the mean pressure leads to a much larger window for scaling-down. In this case, it can be scaled down to nearly 1/387 of the original size, which is about a factor of 10 more than in the 1 bar case of Fig.5.2.4. (4) A much higher mean pressure 100 bar is applied to the same system configuration for a further comparison. The drive ratio is 10%. The stack position is fixed at x/Lres=0.63 (x=0.16 m in reference system) and the output cooling power

CQ& is 612 W in the reference system at the temperature difference over the stack

∆Tstack of 100 K. Keeping ∆Tstack constant (100 K), The change of the two groups of energy flow by the scaling-down of the system is shown in Fig. 5.2.7.

1 10 100 1000

-4

-3

-2

-1

0

1

2

3

E-acousticsE-conduction

Log

(ene

rgy

flow

) (W

)

Scaling factor

Figure 5.2.7: Energy flows containing acoustic power and heat conduction through working gas and stack plates as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (100K), mean pressure is100 bar, drive ratio=10%, x/Lres=0.63.

Scaling considerations 199

The intersection point in Fig 5.2.7 indicates that the system can be scaled to 1/1807 of the original size. Compared with case of mean pressure 10 bar, the window to scale-down is increased again by increasing mean pressure. 5.2.2 Constant time-averaged total energy flow As mentioned in section 5.2.1, for the analysis of ∆Tstack at constant energy flow

2E& and constant energy flow per unit area AE /2& , explicit analytical expressions

are not available. Therefore, only results via numerical computation are presented here. (1) The stack position is again fixed at x/Lres=0.63 (x=0.16 m in reference system),

and the output cooling power CQ& is 13 W (6593 W/m²) at the temperature

difference over the stack ∆Tstack of 100 K in the reference system. The mean pressure is 1 bar and the drive ratio is 10%. At a constant total energy flow

2E& (13W), the scaling-down of the system leads to a decrease of the temperature

difference over the stack. It is shown in Fig 5.2.8. The curves for variation of temperature difference over the stack ∆Tstack at other values of total energy flow are also computed for comparison with the one of 13 W. The trend in Fig.5.2.8 is that ∆Tstack drops when the size of the system decreases. For each of the energy flows, at some scaling factor, the temperature difference over the stack is zero. It indicates the minimum size to scale down at a specific constant total energy flow along the

stack plates at which ∆T vanishes. For instance, the curve of 2E& =1.3 W, the

system can be decreased by a factor of 7.2 maximum. If the total energy flow along the stack becomes larger, the decrease in ∆Tstack becomes steeper. Looking at it from another point of view, a smaller cooling power output enlarges the scaling-down window for the same system. (2) The same computation is carried out for the case of 10 bar mean pressure. The stack position is fixed at x/Lres=0.63 (x=0.16 m in reference system), with an output

cooling power CQ& of 121 W at the temperature difference over the stack ∆Tstack of

100 K in the reference system. Keeping the energy flow 2E& constant (121 W), the

scaling-down of the system leads to change of ∆Tstack. It is shown in Fig.5.2.9. As seen from Fig.5.2.8 and 5.2.9, the performances of scaling-down for both cases are similar.

200 Chapter 5

0 3 6 9 12 15 18 21 240

20

40

60

80

100

120

140

∆Tst

ack (

K)

Scaling factor

E2=13 W

E2=1.3 W

E2=0.13 W

Figure 5.2.8: Temperature difference over the stack as a function of scaling factor. Total energy flow remains constant. Mean pressure is1bar, drive ratio=10%, x/Lres=0.63.

0 3 6 9 12 15 18 21 240

20

40

60

80

100

120

140

∆Tst

ack (

K)

Scaling factor

E2=121 W

E2=12.1 W

E2=1.21 W

Figure 5.2.9: Temperature difference over the stack as a function of scaling factor. Total energy flow remains constant, mean pressure is 10 bar, drive ratio=10%, x/Lres=0.63.

Scaling considerations 201

5.2.3 Constant time-averaged total energy flow density Using the same system configuration, the stack position is again fixed at

x/Lres=0.63 (x=0.16 m in reference system), and the output cooling power CQ& is 13

W at a temperature difference over the stack ∆Tstack of 100 K in the reference system. The mean pressure of the system is 1 bar and drive ratio is 10%. Keeping

the energy flow density resAE /2& constant (6593 W/m²), the scaling-down of the

system leads to a change of temperature difference over the stack. The behavior of ∆Tstack is shown in Fig.5.2.10. The curves for variation of the temperature difference over the stack ∆Tstack at other values of energy flow density are also plotted for comparison with the case of 6593 W/m². For comparison with 1 bar, a mean pressure of 10 bar for the same configuration is also considered and the resultant curves are plotted in Fig.5.2.11. As shown in Fig. 5.2.10, ∆Tstack decreases as the system is scaled down, under the prerequisite that the total energy flow density remains constant. By increasing the

constant energy flow density along the stack resAE /2& , the maximum scaling factor

decreases. Furthermore, comparison of Figs.5.2.10 and 5.2.11 shows that increasing the mean pressure leads to a better performance of the scaled-down system.

1 10 100 10000

20

40

60

80

100

∆Tst

ack (

K)

Scaling factor

E2/A

res=6593 W/m2

E2/A

res=13186 W/m2

E2/A

res=19780 W/m2

Figure 5.2.10: Temperature difference over the stack as a function of scaling factor. Total energy flow density remains constant. Mean pressure is 1 bar, and drive ratio=10%, x/Lres=0.63.

202 Chapter 5

1 10 100 1000 100000

20

40

60

80

100

∆Tst

ack (

K)

Scaling factor

E2/A

res=59218 W/m2

E2/A

res=118436 W/m2

E2/A

res=177655 W/m2

Figure 5.2.11: Temperature difference over the stack as a function of scaling factor. Total energy flow density remains constant. Mean pressure is 10 bar, and drive ratio=10%, x/Lres=0.63. It is important to note that in this case the scaling factor can be increased till infinity without reaching a negative temperature difference. The reason for this is obvious because on each curve the energy flow per surface area remains constant. The decrease is purely due to thermal conduction caused by an increased thermal gradient matching the enthalpy flow.

5.3 Traveling-wave systems Like in the discussion of the standing-wave system, it is supposed that the temperature difference across the regenerator remains unchanged while scaling down. We focus on the variation of the cooling power. The reference traveling-wave system has the configuration described in section 4.3, as shown in Fig.4.3.1. Again, two coordinate systems are considered: an original refrigerator in an original coordinate system and a scaled-down refrigerator in a scaled-down coordinate system. As shown in Fig.5.2.1, the three axes of the scale-down coordinate are scaled down with three different scaling factors: xϕ , yϕ , zϕ .

Assuming that the diameters of the tube components are much larger than the working-gas penetration depth, and that the mean pressure mP remains unchanged,

Scaling considerations 203

the analysis for all the dimensions as made in section 5.2 from Eq. (5.2.5) to (5.2.21) is the same for traveling-wave systems. The traveling-wave system has the following parameters that are different from the standing-wave system: The low-Reynolds-number-limit flow resistance of the regenerator

[ ]210

200 )/()(/6)(/6 xreghzyregxregreghregreg rALrALR ϕϕϕϕµµ −

−− =′′′′≈′

[ ] 02

0 )()(/6)( RrAL zyreghregregzy ϕϕµϕϕ == − , (5.3.1)

The porosity of the regenerator

regreg ψψ =′ , (5.3.2)

The isothermal compliance

011

0 )(/)(/ CpLApLAC zyxmregregregzyxmregregreg−− ==′′′′=′ ϕϕϕψϕϕϕψ . (5.3.3)

Since the temperatures at the cold end and hot end remain the same, we have ττ =′′=′ CH TT / . (5.3.4)

Therefore, we have for the earlier defined functions Eq.(F.21) and (F.22) in appendix F:

( ) ),()2)(1/1(

1)/1(,

2

µµ

µ ττ

ττµ

bfb

bfb

=+−′−′

=′+

, (5.3.5)

( ) ( )),(

2/1ln)/1(

)2(1)/1(

)1/1(1

,2

2

2

2 µµµ

µ τττττ

τµµ

bgbb

bgbb

=

+′′

−+

−′−′

=′++

. (5.3.6)

The real wave number scales as follows: kaak xx ϕωϕω ==′′=′ // . (5.3.7)

In combination with the length in the x direction xL , we have

xxxxx kLLkLk ==′′ )/( ϕϕ . (5.3.8)

Now substituting these in the complex coefficients from 1D to 4D (Eqs.(F.36a) to

(F.36d) in appendix F), we find that the scaled coefficients are:

( )[ ]22001 /)sin()sin()cos()cos(),(1 cplcplfbfbcplfb dLkLkdLkLkbgRCi ′′′′′′−′′′′′′′′′−=′ µτωD

[ ]2202

/)sin()cos()cos()sin(4

),(fbcplfbcplcplfb

m

fb dLkLkdLkLka

bfRdi ′′′′′′+′′′′

′′′′′′

τπ µ

( )[ ]2200 /)sin()sin()cos()cos(),(1 cplcplfbfbcplfb dkLkLdkLkLbgRCi −−= µτω

[ ] 1220

2

/)sin()cos()cos()sin(4

),(D=++ fbcplfbcplcplfb

m

fb dkLkLdkLkLa

bfRdi

ρτπ µ ,

(5.3.9)

[ ]2202 /)sin()sin()cos()cos(),( fbcplfbcplcplfb dLkLkdLkLkbfR ′′′′′′−′′′′′′′−=′ µτD

204 Chapter 5

( ) [ )cos()sin(4

),(1 200 cplfbfb

m LkLkd

abgRCii ′′′′

′′′′′′−−

πρτω µ

] ( ) 222 /)sin()cos( Dzycplcplfbfb dLkLkd ϕϕ=′′′′′′+ , (5.3.10)

[ ]2203 /)sin()sin()cos()cos(

1ln

cplcplfbfbcplfb dLkLkdLkLkCi ′′′′′′−′′′′

′−′′′

=′τ

τωD

[ ]222

/)sin()cos()cos()sin(4 fbcplfbcplcplfb

m

fb dLkLkdLkLka

di′′′′′′+′′′′

′′

−τρ

π

( ) 31D−= zyϕϕ (5.3.11)

[ ] τ ′′′′′′′−′′′′=′ //)sin()sin()cos()cos( 224 fbcplfbcplcplfb dLkLkdLkLkD

[ ]222

0 /)sin()cos()cos()sin()1(

ln4cplcplfbfbcplfb

fb

m dLkLkdLkLkd

Ca ′′′′′′+′′′′′−′

′′′+

τπτωρ

4D= . (5.3.12)

Substitution of these scaled coefficients into Eq. (4.3.33) yields:

( ) fbzy

tbtb

mtb

tbtb

mtb

fb

Lkd

aiLk

Lkd

aiLk

ZD

D-

DD

Z ϕϕ

πρ

πρ

=′′

′′

+′′′

′′′

′−′′′

=′)sin(

4)cos(1

)sin(4

)cos(

23

1

24

2

. (5.3.13)

Substitution of the coefficients into Eq. (4.3.36) yields:

[ −′′−′′+

′′′

′−′′′=′ 32412

42 1)sin(

4)cos( DDDD

DDZ tb

tb

mtbinput Lk

d

aiLk

πρ

( ) inputzytbtb

m

m

tbtb Lk

d

a

a

diLk Z

DDDD ϕϕ

πρ

ρπ =

′′

′′

+′′

+′′′+′−1

232

2

41 )sin(4

4)cos()( .

(5.3.14) Via substitution of Eq. (5.3.14) into (4.3.37), the scaled-down acoustic impedance of the refrigerator system becomes:

( ) rfgazy

resinputresres

m

resres

mresinput

res

mrfga

LkiLkd

a

Lkd

aiLk

d

a−− =

′′′+′′′

′′′

+′′′=′ Z

Z

ZZ ϕϕ

πρ

πρ

πρ

)sin()cos(4

)sin(4

)cos(4

2

2

2 . (5.3.15)

Substitution of these scaled-down parameters into the equations for real coefficients 1θ to 4θ (Eqs.(F.37a) to (F.37d) in appendix F) leads to:

122

1 /)sin()sin()cos()cos( θθ =′′′′′′−′′′′=′ cplcplfbfbcplfb dLkLkdLkLk , (5.3.16a)

222

2 /)sin()sin()cos()cos( θθ =′′′′′′−′′′′=′ fbcplfbcplcplfb dLkLkdLkLk , (5.3.16b)

322

3 /)sin()cos()cos()sin( θθ =′′′′′′+′′′′=′ fbcplfbcplcplfb dLkLkdLkLk , (5.3.16c)

Scaling considerations 205

422

4 /)sin()cos()cos()sin( θθ =′′′′′′+′′′′=′ cplcplfbfbcplfb dLkLkdLkLk . (5.3.16d)

Substitution of the above scaled-down parameters into the Eqs.(4.3.53), (4.3.54) and (4.3.45), for coefficients of 1Θ , 2Θ and 3Θ yields

( ) 12

2

2

21

1Im

)2sin(4)sin(4)cos( Θ=

′′′′

+

′′′′

+′′=Θ′−− rfgares

resm

rfgares

resmres d

Lka

d

LkaLk

ZZ πρ

πρ

,

(5.3.17)

( )( )( )

′+′+′−′

−′−

′′′

+′

′′

′′

′′′

=Θ′σε

βθθθθρω

ψ νκν 11

~~

1~Im2

432102

s

m

fbfbregm

regreg ffTf

L

RA

ZZ

( )( )( )

′+′+′−′

−′−

′′

′′+

′′′′′

′′′

+σε

βπ

θθρρ

θθπρω

ψ νκν 11

~~

1Re4

42 22

42312

0

s

m

fbfb

m

m

fb

regm

regreg ffTf

d

a

a

d

L

RA

Z

( )

′′′′+

′′+

′′

−′′′

′−′′′

− 232

22

32

2

22

2

20

3

0 ]Im[

241/1

12fb

fb

m

fb

m

fb

fbregregm

pregreg

a

d

a

d

LL

RTcA

Z

Z

Z ρθθπ

ρθπθτ

σρωψ

( )( )( )( ) ( ) 2

1

11/1

~~

Im Θ=

′+′+′′′+′−′

+′⋅ −zy

s

s fffff ϕϕ

σεε κννκ

ν , (5.3.18)

( ) 314132

242

313 ]~

~~Re[

]~

Re[]

~Re[ Θ=

′′′

+′

′′+

′′+′′=Θ′ −

zyfbfbfb

ϕϕZ

DDZ

DD

Z

DDDD . (5.3.19)

After substituting these scaled coefficients into Eq. (4.3.52), the scaled-down cooling power is given by:

{ } [ ]reg

regsregreginC LTAKKQ

′′−′′−+′−Θ′+Θ′Θ′′=′ )/1(1

)1(2 0231

2

1

τψψ/p&

{ } [ ]sregregzy

xin

zyin

in KK )1(2/1

231

2

12

1

2

1 ψψϕϕ

ϕϕϕ

−+−Θ+Θ⋅Θ′

= pp

p

regreg L

TA)/1(1

0

τ−⋅ (5.3.20)

If the input acoustic pressure from the driver remains the same, i.e. inin 11 pp =′ , then

Eq. (5.3.20) becomes

{ } [ ]reg

regsregregzy

xin

zyC L

TAKK/Q)/1(1

)1(21

0231

2

1

τψψϕϕ

ϕϕϕ

−−+−Θ+ΘΘ=′ p& .

(5.3.21)

206 Chapter 5

As seen from Eq. (5.3.21), the cooling power after scaling down consists of two groups of terms scaling with different factors. Similar to the standing-wave system in section 5.2 for, the two groups of terms are defined as

{ }231

2

1 2 Θ+Θ⋅Θ=− /acousticsE inp , (5.3.22)

[ ]reg

regsregreg LTAKKconductionE

)/1(1)1( 0

τψψ −−+=− . (5.3.23)

Eq. (5.3.21) can be rewritten as:

{ } { }conductionEacousticsEQzy

x

zyC −⋅−−⋅=′

ϕϕϕ

ϕϕ1& . (5.3.24)

The cooling power of the original system is given by:

{ } { }conductionEacousticsEQC −−−=& (5.3.25)

From Eq. (5.3.24), it is obvious that the cooling power in the scaled-down system consists of two groups of energy flow scaling with different factors. The first group

of energy flow scales with a factor( ) 1−zyϕϕ , while the conduction term scales

as ( ) 1−zyx ϕϕϕ . This is fully analogous to the standing-wave system Eq.(5.2.26).

The scaling behaviour is also fully analogous to that of the standing-wave system. Coefficient of performance (COP) For the computation of the COP the expression is needed for 4Θ′ (Eq.(4.3.58)) that

is in the scaled version given by:

( ) ( )

′′′′+

′′

′′

−′′′=Θ′ )cos(~

)sin(~

41)cos(1Re 412

344 tbtb

tb

mtb LkLk

d

aiLk -DD

DD-

πρ

( ) 41

24

22

2

)sin(4

)cos(~

/)sin(4

~Θ=

′′′

′+′′′

′′′′− −

zytbtb

mtbtb

m

tb Lkd

aiLkLk

a

di ϕϕ

πρ

ρπ D

DD

(5.3.26) Assuming inin 11 pp =′ , the scaled-down COP is obtained by substitution of all the

scaled-down parameters into Eq. (4.3.60) [ ]

regin

regsregreg

L

TAKKPCO

′′−⋅

Θ′Θ′′

′′−+′−

Θ′Θ′+Θ′

=′ )/1(1)0.1(22

41

2

1

0

4

23 τψψp

[ ]regin

regsregregx L

TAKK )/1(1)0.1(22

41

2

1

0

4

23 τψψϕ −⋅

ΘΘ−+

−Θ

Θ+Θ=p

[ ]regin

regsregregx L

TAKKCOP

)/1(1)0.1(2)1(

41

2

1

0 τψψϕ −⋅

ΘΘ−+

−−=p

. (5.3.27)

Scaling considerations 207

By substituting Eq. (5.3.24), (4.3.59), (5.3.26), (5.3.17) into definition that

inC WQPCO && ′′=′ / , the COP of the scaled-down system becomes:

( ) { } ( ) { }( ) inzy

zyxzy

in

C

W

conductionEacousticsE

W

QPCO

&&

&

⋅−⋅−−⋅

=′′

=′ −

−−

1

11

ϕϕϕϕϕϕϕ

( )in

x W

conductionECOP

&

−−−= 1ϕ . (5.3.28)

Eq. (5.3.27) or (5.3.28) shows that the COP for the scaled system decreases. The last term in Eq. (5.3.27) or (5.3.28) is the product of the scaling factor and the original ratio of energy losses due to thermal conduction of working gas and regenerator material to acoustic work. This term causes the reduction of COP after scaling. This is also fully analogous to the standing-wave system Eq.(5.2.36). As an example, scaling is considered of the traveling-wave refrigerator designed and used in section 4.5. The configuration is described in section 4.5.3 and shown in Fig.4.5.1, i.e. the original system. For simplicity, the computation is made for a uniform scaling, i.e. a universal scaling factor for all three dimensions:

ϕϕϕϕ === zyx . The scaling performance of the original configuration under

different working conditions: different temperature spans across the regenerator at the same mean pressure of 11 bar are compared. Comparison between different mean pressures, 11 bar and 15 bar, with constant temperature span across the regenerator (40 K) under the same operational condition (the drive ratio remains 2.5% all the time) is also investigated. These comparisons are made to study how to expand the space for scaling down a specific system. Comparison a) Different temperature difference over the regenerator (1) The reference-case traveling-wave refrigerator works at 320 Hz and the output

cooling power CQ& is 33 W. The temperature difference over the regenerator is 40 K,

and the mean pressure is 11 bar. Under the condition that the temperature difference over the regenerator remains constant at 40 K, scaling of the system leads to a change of the energy flow defined by Eq. (5.3.22) and the heat conduction of the working gas and regenerator material Eq. (5.3.23). This is shown in Fig. 5.3.1. The net cooling power, i.e. difference between the curves in Fig.5.3.1, decreases fast when the system decreases in size. As seen in Fig.5.3.1, the scaling behaviour is fully analogous to that of standing-wave system in section 5.2: the energy flow defined by Eq. (5.3.22), decreases faster than the consumptive energy flow due to thermal conduction of the working gas and regenerator material. At a scaling factor of about 4, the two curves cross.

208 Chapter 5

That means that the model refrigerator can be scaled down to around one fourth of its original size, if non-zero cooling power is required.

1 10 100

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0Lo

g(en

ergy

flow

) (W

)

Scaling factor

E-acoustic E-conduction

Pmean

=11 bar, ∆T=40 K

Figure 5.3.1: Energy flows containing acoustic power and heat conduction through working gas and regenerator material as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (40 K) mean pressure is 11 bar. (2) If the system operates at a smaller temperature span across the regenerator, the scaling performance will behave similarly but the scale factor can be larger. When the temperature difference across the regenerator is 23 K, the output cooling power

CQ& is increased to 37 W. It still works at 320 Hz and the mean pressure remains 11

bar. The energy flows with scaling are plotted in Fig. 5.3.2. In Fig.5.3.2, the two curves decrease differently and cross at around 9. That indicates that the reference refrigerator can be scaled down to 1/9 of its original size, if we keep the temperature difference over the stack at 23 K with non-zero cooling power. Compared with the last case, the space for scaling-down is enlarged under a lower temperature difference over the regenerator. Analogous to the discussion of standing-wave systems in section 5.2, further scaling-down can be realized only at the cost of a reduced temperature difference over the regenerator.

Scaling considerations 209

1 10 100-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

E-acoustic E-conduction

Pmean

=11 bar, ∆T=23 K

Log(

ener

gy fl

ow)

(W)

Scaling factor

Figure 5.3.2: Energy flows containing acoustic power and heat conduction through working gas and regenerator material as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (23 K) mean pressure is 11 bar.

1 10 100-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

E-acoustic E-conduction

Pmean

=15 bar, ∆T=40 K

Log(

ener

gy fl

ow)

(W)

Scaling factor

Figure 5.3.3: Energy flows containing acoustic power and heat conduction through working gas and regenerator material as a function of the linear down-scaling factor. Temperature difference over the stack remains constant (40 K) mean pressure is 15 bar.

210 Chapter 5

Comparison b) Different mean pressure at the same drive ratio Computations are made for the system working at mean pressures of 11 bar and 15 bar. The drive ratio is constant at 2.5%. (3) In reference system, at a mean pressure of 15 bar, the output cooling power

CQ& is 45 W at the temperature difference over the regenerator of 40 K. The change

of the energy flows under scaling at constant temperature difference is shown in Fig.5.3.3. Based on the intersection point in Fig. 5.3.3, the increase of mean pressure (drive ratio constant) leads to a slightly bigger window for scaling-down in comparison with case (1) in Fig.5.3.1. In this case, it can be scaled down to nearly 1/5 of the original size.

5.4 Conclusions In the case of constant temperature difference over the stack or the regenerator, the comparison between Eq. (5.2.26) or (5.2.28) for standing-wave systems and Eq. (5.3.24) for traveling-wave shows that the scaling behaviour of a standing-wave system is the same as that of a traveling-wave system. The cooling power in the scaled-down system consists of two groups of energy flow scaling with different factors. The first group of energy flow is related to the acoustic power and scales

with a factor 1)( −zyϕϕ , whereas the conduction term scales as 1)( −

zyx ϕϕϕ . This

means that the thermal conduction will finally dominate the losses during scaling down. The cooling capability of the system will decrease when scaling down. Therefore, there is a limitation for scaling down, when the temperature difference of the scaled-down system becomes zero. The COP decreases rapidly under scaling, as can be seen from Eq. (5.2.36) for a standing-wave system and Eq. (5.3.28) for a traveling-wave system. The term, which is the product of the scaling factor and the original ratio of energy losses due to thermal conduction to acoustic work, causes the reduction of COP after scaling. That means that the energy loss due to thermal conduction will become more and more dominant, when the system scales down.

Chapter 6 Conclusions and recommendations 6.1 Conclusions Chapter 2 In chapter 2 the general analytical expressions are derived for thermoacoustic systems. Here the main definitions of the functions are made that are used in

chapters 3 and 4. The function 2E& that describes the time-averaged total energy

flow in a stack plays an important role in the scalebility of the thermoacoustic systems. Chapter 3: In chapter 3 a standing wave system is modeled using analytical expressions for cooling power and COP. This model describes clearly the temperature distribution in a stack of a tubular standing wave device driven by a loudspeaker. The influence of the stack plate spacing is studied in detail, as well as the position dependency of the stack. It is clear that the stack plate spacing must be sufficiently small to have highest performance of heat transport. The model is compared with experimental data of a standing-wave “thermoacoustic couple” that could be positioned in the resonator. The model predicts correctly the trends, although additional losses (that are not modeled) lead to discrepancy between the model results and measurements.

It is clear that a correct choice of the total energy flow 2E& and loss function plays

an important role in matching experiment and model. Chapter 4 With the analytic model that is derived in chapter 4 traveling wave systems can be modeled. The great advantage of this model is that it is a fast solver because a full analytic expression is obtained to model the complete traveling wave acoustic feedback phenomenon of the apparatus. Therefore it is easy to change sizes and

212 Chapter 6

lengths of the geometry in the model and run the model to obtain performance curves. There is a good agreement between the model derived and developed in this thesis, and the Los Alamos DeltaE solver, when the simulation results of both models applied to Swift's traveling-wave engine are compared. Therefore the model developed in this thesis was used to simulate, design and optimize a new device built at TU/e. The model has made clear that optimization of an around 1.3 meter size traveling-wave cooler, driven by an external driver (loudspeaker) is possible without using a compliance. The optimum system design was attained for a one-diameter-size feedback tube containing heat exchangers and regenerator. Measurements on different regenerator materials in a thermally driven coaxial traveling-wave apparatus have made clear that the MESH number (for wire gauzes) and CPSI number (for honeycombs) are very sensitive parameters. The maximum performance of the system that was tested in this thesis occurred at a (hydraulic radius)/(thermal penetration depth) ratio of 0.30 for stainless steel wire gauzes and 0.16 for honeycombs (the reference thermal penetration depth taken at average regenerator temperature of 490 K). Besides that the performance depends also on the porosity of the regenerator. Maximum performances will generally occur at regenerator porosities between 80-90%. These sensitivities are in agreement with predictions by the model developed in this thesis. The different regenerators with different porosity and hydraulic radius in the measurements result in different acoustic impedances globally and locally, which finally result in different phasing between the pressure and the velocity in the regenerator and thus lead to different thermodynamic cycles. That makes a difference in the final performance of the engine. The experiments have shown that wire gauze materials exceed the performance of honeycombs with a factor 2. Probably this is due to two effects. One effect is the difference in thermal capacity between the light ceramic honeycombs and the metal wire gauze. A second reason might be the advantage of the randomized structure of wire gauze regenerators. The randomized nature of these wire gauzes leads to a more efficient thermal contact with the oscillating gas even though at higher porosity of the wire gauze regenerator itself in comparison with the honeycombs. The experiments with the traveling wave cooler have demonstrated that a 1.3-meter-long (small) traveling wave system cools without compliance. So thermoacoustic systems can be designed without the special precaution of compliance. Further scaling down seems possible. The thermal performance that was measured in the traveling wave cooler is about a factor 2 lower in comparison with the model results. The trends of the model

Conclusions and recommendations 213

compare very well with the measurements. The difference between experiment and design model is due to the neglect of most of losses, which take place inside the system in practical operations, in the analytical model. This is probably caused by streaming, which can only be removed by inserting additional flow straightners. The experiments have demonstrated that it is indeed possible to manufacture efficient heat exchangers from copper fins, using high end spark cutting technology. This manufacturing is a tedious task of high tech machining, and is very expensive. Downsizing to smaller coolers will obviously push up the limits of making even smaller heat exchange components and stacks for which probably lithography technology is needed. Chapter 5 The findings of chapter 5 show that the cooling capacity, and the COP decrease rapidly when scaling down. Therefore, there is a limitation for scaling down. It is possible to make a mini-thermoacoustic standing wave machine, although there is always a thermal conduction balance limitation for scaling down. For a standing-wave system, the effective cooling power decreases rapidly when scaling down with the prerequisite that the temperature span over the stack remains constant (see Fig. 5.2.4). Although the reference system can be only scaled down to nearly 1/45th of the original size before this thermal conduction loss balance occurs, the further scaling down can be realized by taking some methods. The methods are: (1) reducing the required constant temperature span over the stack (see Fig. 5.2.5); (2) increasing filling pressures and applying the same relative pressure ratio (see Figs. 5.2.6 and 5.2.7). If the filling pressure is as high as 100 bar, the scaling down system can be as small as 1/1800th of the original size (see Fig.5.2.7), which is around 0.142 mm long system and small enough for many applications. If the prerequisite is keeping constant time-averaged total energy flow or constant time-averaged total energy flow density, the temperature span over the stack ∆T decreases when scaling down (see Figs. 5.2.8, 5.2.9, 5.2.10 and 5.2.11). For traveling wave systems approximately the same rules apply as for standing wave systems. The reference system in chapter 5 having a length in the order of one meter was taken as our own model system which is already a relatively small traveling wave system. The scaling analysis shown with figures 5.3.1 and further makes clear that a factor 10 for a 11 bar system is the maximum scale down factor. This means that it is impossible to scale these systems down below a size of 10 cm. Again using very high average pressures of 100 bar or more, but in that case extremely high power drivers will be needed. This seems not compatible or at least a big challenge.

214 Chapter 6

The findings of chapter 5 tell us in general that the COP will definitely deteriorate at smaller devices. This limitation of scaling down is caused by the conduction losses. Therefore, the exploration on effective thermoacoustic materials for stack, and regenerator with less conduction losses is recommended by the author.

6.2 Recommendations In order to validate better, and improve the standing wave model, a standing wave system is needed with warm and cold heat exchanger. This assures a fixed thermal boundary condition at the end of the stack, whereas in the tested system in this thesis (thermoacoustic couple) the boundary conditions are floating. The scalability should be tested by building a small scale standing wave device with a length of about 10 cm, this will prove experimentally the statements in this thesis. The performance of honeycombs should be explored more than was done in this thesis with the coaxial traveling wave system. Emphasis should be placed on thin walled glass stack materials with porosities of 90%. Here even care could be taken of cell size versus temperature, meaning a tapered regenerator where penetration depth is always matched with the correct cell size at variable temperature along the regenerator. It is possible to manufacture ultra thin walled honeycombs from Pyrex glass. The mechanically driven traveling wave cooler that was tested in chapter 4 can be improved by placing additional thermometers in the regenerator in order to investigate the temperature gradient, and to find out if streaming takes place in the regenerator itself. Furthermore the effect of flow straightners should be explored in order to diminish the heat leak between the cold side of the regenerator and the room temperature heat exchanger. The performance of the multi-microphone system should be investigated, and improved. This can partially be done by isolating the cooler with a flexible bellows from the driver, or by isolating the driver itself so that the effect of mechanical vibrations can be diminished. With a well working multi-microphone measurement system the acoustical power can be compared with the power measured from the heat exchangers. Another way to improve the multi-microphone measurement is to increase the distance between the microphones so that a larger phase difference will occur resulting in more accurate measurements. The Fortran design model should be used to make optimization studies of traveling wave systems. When the current experimental mechanically driven cooler is improved the new results should be compared again with the model. It is expected that model and experiment will then be in better agreement with each other.

Appendices

A: momentum equation derivation In chapter 2, the momentum equation was derived as

21

21

1ydx

di m ∂

∂+−=

upu µωρ . (2.1.19)

Rewriting Eq. (2.1.19) into a standard form as an ordinary second order differential equation gives

dx

di

ym 1

121

2 1 pu

uµµ

ωρ=−

∂∂

. (A.1)

The solution 1u of this equation can be written as the sum of the solution of the

homogenous equation and a particular solution. The homogenous equation is

0121

2

=−∂∂

uu

µωρ mi

y. (A.2)

Assuming a solution of the form yeαCu =1 , (A.3)

and substitution of Eq.(A.3) into (A.2) yields

02 =−µ

ωρ miα . (A.4)

Thus, the two roots of Eq. (A.4) are

v

i

δ+±= 1

2,1α . (A.5)

The viscous penetration depth is defined in Eq. (2.1.14) as ωρµδ mv 2= .

Therefore, the solution of the homogenous equation (A.2) is obtained ( ) ( ) vv yiyi ee δδ /1

2/1

11+−+ ⋅+⋅= CCu . (A.6)

To obtain the particular solution, it can be assumed that ( ) ( ) vv yiyi eyey δδ /1

2/1

1*1 )()( +−+ ⋅+⋅= CCu . (A.7)

The unknown coefficient functions )(1 yC and )(2 yC for the particular solution

can be found by solving the following joint equations

216 Appendix

dx

die

dy

ydie

dy

yd

edy

yde

dy

yd

v

yi

v

yi

yiyi

vv

vv

1/)1(2/)1(1

/)1(2/)1(1

11)(1)(

0)()(

pCC

CC

⋅=

+−⋅⋅++⋅⋅

=⋅+⋅

+−+

+−+

µδδδδ

δδ

(A.8)

Solving the two joint equations, the solutions are

vyiv eidx

d

dy

yd δδµ

/)1(11

121)( +−

+⋅⋅= pC

, (A.9)

vyiv eidx

d

dy

yd δδµ

/)1(1

121)( +

+⋅⋅−= pC2 . (A.10)

Integration of Eq. (A.9) and (A.10) yields

vyi

m

edx

diy δ

ωρ/)1(1

1 2)( +−⋅⋅= p

C , (A.11)

vyi

m

edx

diy δ

ωρ/)1(1

2)( +⋅⋅= p

C2 . (A.12)

Substitution of Eq. (A.11) and (A.12) into Eq. (A.7) yields the particular solution

dx

di

m

1*1

pu ⋅=

ωρ. (A.13)

The summation of solutions of the homogenous equation and the particular solution yields the final general solution:

( ) ( )

dx

diee

m

yiyi vv 1/12

/111

pCCu ⋅+⋅+⋅= +−+

ωρδδ . (A.14)

Next, the boundary conditions are applied to obtain the coefficients 1C and 2C .

Boundary conditions:

a) at 0=y , because of the symmetry, 01 =∂∂

y

u

b) at 0yy = , because of the solid wall, 01 =u .

By using boundary condition a, we obtain

21 CC = (A.15)

By using boundary condition b, the coefficients are obtained

]/)1cosh[(2

1

0

11

vm yidx

di

δωρ +⋅⋅−=

pC . (A.16)

This is the description of the oscillatory velocity profile as dependant on the oscillatory pressure gradient including viscous terms.

( )[ ]( )[ ]

++

−=v

v

m yi

yi

dx

di

δδ

ωρ 0

11 1cosh

1cosh1

pu . (A.17)

Appendix 217

B: derivation of the temperature of the solid plate In chapter 2, the equation of the temperature of the solid plate was derived as

21

2

1 yi s

ss ′∂∂

=T

T κω . (2.1.23)

Rewrite it as

0121

2

=−′∂

∂s

s

s i

yT

T

κω

. (B.1)

It is a homogenous equation. Assume a solution of the form y

s e ′= αCT 1 . (B.2)

Substitution of Eq. (B.2) into (B.1) yields

02 =−s

i

κω

α . (B.3)

The two roots of Eq. (B.3) are

s

i

δ+±= 1

2,1α . (B.4)

Therefore, the solution of the homogenous equation (B.1) is obtained ( ) ( ) ss yiyi

s ee δδ /12

/111

′+−′+ ⋅+⋅= CCT . (B.5)

Next, the boundary conditions are applied to get the coefficients 1C and 2C .

Boundary conditions:

c) at 0=′y , because of the symmetry, 01 =′∂

∂y

sT;

d) at ly =′ , because of the solid wall, 11 bs TT = , where 1bT is temperature

amplitude at the boundary, and yet undetermined with the temperature of the fluid together.

By using boundary condition c, we obtain

21 CC = (B.6)

By using boundary condition d, the coefficients are obtained

]/)1cosh[(21

1s

b

li δ+=

TC . (B.7)

Substitution of Eq. (B.6) and (B.7) into Eq. (B.5) yields the final solution of Eq. (B.1)

( )[ ]( )[ ]s

sbs li

yi

δδ

+′+

=1cosh

1cosh11 TT . (B.8)

218 Appendix

C: derivation of the temperature oscillation of the fluid layer In chapter 2, the heat transfer equation was reduced to

21

2

111 yKi

dx

dTic m

pm ∂∂=−

+ TpuT ωωρ . (2.1.29)

Rewrite Eq. (2.1.29) into a standard form as an ordinary second order differential equation

11121

2

puTT

K

i

dx

dT

K

c

K

ci

ympmpm ωρωρ

−=−∂∂

. (C.1)

As did in the above part, the solution 1T of this equation can be written as the sum

of the solution of the homogenous equation and a particular solution. First, we start with solving the homogenous equation of Eq. (C.1)

0121

2

=−∂∂

TT

K

ci

ypmωρ

. (C.2)

Assume a solution of the form as yeαCT =1 . (C.3)

Substitution of Eq. (C.3) into (C.2) yields

02 =−K

ci pmωρα . (C.4)

The two roots of Eq. (C.4) are

κδi+±= 1

2,1α , (C.5)

where the fluid’s thermal penetration depth is

ωρδκ

pmc

K2= . (C.6)

Therefore, the solution of the homogenous equation (C.2) is obtained ( ) ( ) κκ δδ /1

2/1

11yiyi ee +−+ ⋅+⋅= CCT . (C.7)

To obtain the particular solution, assume the particular solution is of the expression as

( ) ( ) κκ δδ /12

/11

*1 )()( yiyi eyey +−+ ⋅+⋅= CCT . (C.8)

The unknown coefficient functions )(1 yC and )(2 yC for the particular solution

can be found by solving the following joint equations

0)()( /)1(2/)1(1 =⋅+⋅ +−+ κκ δδ yiyi e

dy

yde

dy

yd CC

Appendix 219

K

iie

dy

ydie

dy

yd yiyi 1/)1(2/)1(1 1)(1)( pCC ωδδ κ

δ

κ

δ κκ −=

+−⋅⋅++⋅⋅ +−+

++−+

]/)1cosh[(]/)1cosh[(

10

1

v

vmp

yi

yi

dx

d

dx

dT

K

ic

δδ

ωp

(C.9)

Solving the two joint equations, the solutions are

++−

+= +−

10

1/)1(1

]/)1cosh[(]/)1cosh[(

112

1)(p

pCK

i

yi

yi

dx

d

dx

dT

K

ice

idy

yd

v

vmpyi ωδδ

ωδ

κδκ

(C.10)

++−

+−= +

10

1/)1(

]/)1cosh[(]/)1cosh[(

112

1)(p

pC2

K

i

yi

yi

dx

d

dx

dT

K

ice

idy

yd

v

vmpyi ωδδ

ωδ

κδκ

(C.11) Integration of Eq. (C.10) and (C.11) yields

( )κδκ δω

ωδ

κ −⋅

+= +−

K

i

dx

d

dx

dT

K

ice

iy mpyi 11/)1(

21 )1(21

)(pp

C

220

1 1]/)1cosh[(

1−− −

⋅+

−κδδδω vv

mp

yidx

d

dx

dT

K

ic p

+++⋅v

vv yiyi

δδ

δδ

κ

]/)1sinh[(]/)1cosh[(, (C.12)

( )κδκ δω

ωδ

κ −⋅

+= +

K

i

dx

d

dx

dT

K

ice

iy mpyi 11/)1(

22 )1(21

)(pp

C

220

1 1]/)1cosh[(

1−− −

⋅+

−κδδδω vv

mp

yidx

d

dx

dT

K

ic p

+−+⋅v

vv yiyi

δδ

δδ

κ

]/)1sinh[(]/)1cosh[(. (C.13)

Substitution of Eq. (C.12) and (C.13) into Eq. (C.8) yields the particular solution

]/)1cosh[(

]/)1cosh[(

1

11

0

12

12

1*1

v

vm

m

m

mpm yi

yi

dx

d

dx

dT

dx

d

dx

dT

c δδ

σσ

ωρωρρ ++

−−−= ppp

T ,

(C.14)

where 22 // κδδµσ vp Kc == is the Prandtl number.

The summation of solutions of the homogenous equation and the particular solution yields the final general solution

( ) ( ) *1

/12

/111 TCCT +⋅+⋅= +−+ κκ δδ yiyi ee

220 Appendix

( ) ( )dx

d

dx

dT

cee m

mpm

yiyi 12

1/12

/11

1 ppCC

ωρρκκ δδ −+⋅+⋅= +−+

]/)1cosh[(]/)1cosh[(

11

0

12

v

vm

m yi

yi

dx

d

dx

dT

δδ

σσ

ωρ ++

−− p

. (C.15)

Next, the boundary conditions are applied to get the coefficients 1C and 2C .

Boundary conditions:

e) at 0=y , because of the symmetry, 01 =∂∂

y

T

f) at 0yy = , because of the solid wall, 11 bTT =

By using boundary condition e, we obtain

21 CC = . (C.16)

By using boundary condition f, the coefficients are obtained

−−−

+⋅=

111

]/)1cosh[(21 1

21

10

1 σωρρδκ dx

d

dx

dT

cyim

mpmb

ppTC . (C.17)

Substitution of Eq. (C.16) and (C.17) into Eq. (C.15) yields the final solution of Eq. (C.1)

pm

m

mpmb cdx

d

dx

dT

cyi

yi

ρσωρρδδ

κ

κ 112

11

01 1

11]/)1cosh[(]/)1cosh[( ppp

TT +

−−−

++=

]/)1cosh[(]/)1cosh[(

111

0

12

12

v

vm

m

m

m yi

yi

dx

d

dx

dT

dx

d

dx

dT

δδ

σσ

ωρωρ ++

−−− pp

. (C.18)

Now, the only unknown parameter is the temperature 1bT at the surface where the

fluid and the solid wall have contact, i.e. 0yy = and ly =′ . As the heat flow into

the fluid and the solid wall at the boundary has the same amount but opposite in direction. It means the following boundary condition is true

lyssyyyKyK

=′=′∂∂−=∂∂ )/()/( 11

0TT . (C.19)

From Eq. (2.1.24), it can be obtained that

]/)1tanh[(1

11

ss

b

ly

s lii

δ+⋅+⋅=

′∂∂

=′

TT

. (C.20)

From Eq. (C.18), it can be obtained that

]/)1tanh[(1

111

01

21

11

0

κκ

δδσωρρ

yii

dx

d

dx

dT

cym

mpmb

yy

+⋅+⋅

−−−=

∂∂

=

ppT

T

]/)1tanh[(1

11

01

2 vv

m

m

yii

dx

d

dx

dT δδσ

σωρ

+⋅+⋅−

+ p. (C.21)

Substitution of Eq. (C.20) and (C.21) into (C.19) yields

Appendix 221

1

01

]/)1tanh[(]/)1tanh[(−

+++=s

ssb

li

K

Kyi

δδ

δδ

κ

κT

+⋅

−+⋅

κ

κ

δδ

σωρρ]/)1tanh[(

111 01

21 yi

dx

d

dx

dT

cm

mpm

pp

+⋅

−−

v

vm

m

yi

dx

d

dx

dT

δδ

σσ

ωρ]/)1tanh[(

11 01

2

p. (C.22)

Substitution of Eq. (C.22) into (C.18) yields the final solution for the temperature oscillation of the fluid layer:

( )( ) ( )[ ] dx

dT

dx

d

yi

yi

cm

v

v

mpm

1

02

11 /1cosh1

]/1cosh[1

1 ppT

+−+−×−=

δσδσ

ωρρ

( )( )( )

( )[ ]( ) ( )[ ]κ

κ

δεδε

ωρσρ /1cosh1/1cosh

11

/

02

11

yi

yi

f

fdxdTdxd

c sk

vs

m

m

pm +++×

+

−+− pp

, (C.23)

where the Rott’s functions are as defined in Eqs. (2.1.32), (2.2.33), and (2.1.34): [ ]

ν

νν δ

δ/)1(

/)1(tanh

0

0

yi

yif

++

= , (2.1.32)

[ ]κ

κκ δ

δ/)1(

/)1(tanh

0

0

yi

yif

++

= , (2.1.33)

[ ][ ]ssss

pm

slicK

yicK

δρδρ

ε κ

/)1(tanh

/)1(tanh 0

+

+= . (2.1.34)

222 Appendix

D: derivation of the time-averaged total energy flow The time-averaged product of two oscillatory quantities with the same frequency, for example, a current )cos( IA tII φω += and voltage )cos( VA tVV φω += , is

given by

∫∫ ++==ττ

φωφωττ 00

)cos()cos(11

dttVtIIVdtIV VAIA

)cos(21

VIAAVI φφ −= . (D.1)

with ωπτ /2= is the period of the oscillation. If write the oscillatory current and voltage in complex notation, then, they are:

tiA

tiA eeI I ωφω II == + )( and ti

Ati

A eeV V ωφω VV == + )( , respectively, where Ii

AA eI φ=I and ViAA eV φ=V are the complex amplitudes.

The time-averaged value can be computed by

[ ] [ ] )cos(2

1Re

2

1~Re

2

1VIAA

iA

iAAA VIeVeIIV VI φφφφ −=⋅== −VI . (D.2)

where the tilde denotes complex conjugation. By using Eq. (D.2), the Eq. (2.1.71) can be rewritten as

[ ][ ] ( )dx

dTlKKydyc

E ms

y

pm +−=∏ ∫ 00 11

2 0 ~Re21

uTρ&

. (D.3)

Rewrite Eq. (D.3) as

[ ] ( )dx

dTlKKydyc

E ms

y

pm +−

=

∏ ∫ 00 112 0 ~Re

21

uTρ&

. (D.4)

First, we focus on the term in the outer square bracket. For simplicity of writing, this part is defined as

[ ]dycy

pm∫= 0

0 11~uTF ρ . (D.5)

Substituting Eq. (2.1.30) for 1T and Eq. (2.1.20) for1u , Eq. (D.5) can be rewritten

as

( )[ ]( ) ( )[ ]∫

+++−−= 0

00

11

/1cosh1/1cosh

1~y

sm yi

yi

dx

di

κ

κ

δεδ

ωρp

pF

( )[ ]( )[ ] ∫+

++−⋅ 0

0

113

0

~

1cosh1cosh

1y

m

m

p

v

v

dx

dT

dx

d

dx

dicdy

yi

yiconj

ppρωδ

δ

( )( ) ( )[ ] ( )( )

( )[ ]( )[ ]

++

+−++

+−+−⋅

κ

κκ

δδ

εσε

δσδσ

/1cosh/1cosh

11/1

/1cosh1]/1cosh[

100 yi

yiff

yi

yi

s

vs

v

v

Appendix 223

( )[ ]( )[ ] dy

yi

yiconj

v

v

++−⋅

δδ

01cosh1cosh

1 . (D.6)

Here, ( )functionconj denotes taking complex conjugation of the function.

Eq. (D.6) consists of two integrations, denoting them as 1F and 2F , respectively,

for the sake of easy writing. Therefore, Eq. (D.4) can be written as

[ ] [ ]{ } ( )dx

dTlKKy

E ms+−+=

∏ 0212 ReRe

21

FF&

. (D.7)

For the first integration1F , it can be obtained

[ ]

+−

++−−=

σεωρκ

1

~

11~

1~

ReRe 110

1

fff

dx

diy v

sv

m

pp

F . (D.8)

After a long calculation and keeping in mind that only real terms are needed, the term related to the second integration is

[ ]

+−⋅

+++

−=

σεε

σρωκκ

1

~

1/1~

11~

ReRe 113

02

v

s

vsv

m

m

p fffff

dx

dT

dx

d

dx

dyic ppF (D.9)

Note that it is true for complex computation ( )[ ] ( )[ ]biabiai +−=+ ImRe . (D.10)

Using Eq. (D.10) and substituting Eq. (D.8) and (D.9) into Eq. (D.7) yields

( )dx

dTlKKy

fff

dx

dyE ms

v

sv

m

+−

+−

++−=

∏ 01102

1

~

11~

1~

Im2 σεωρ

κpp&

+−⋅

+++

−+

σεε

σρωκκ

1

~

1/1~

Im1

1~

211

30 v

s

vsv

m

m

p fffff

dx

dT

dx

d

dx

dcy pp. (D.11)

Therefore, the total energy flow along the stack is given by

( )( )( )

++−−−∏=

σεωρνκ

ν 11

~~

1~

Im2 1

102

sm

fff

dx

dyE p

p&

( )( )( )

( )( )

+++−+×

−∏

+σε

εσρω

κννκν 11

/1~

~Im

~

1211

30

s

sm

m

p fffff

dx

d

dx

d

dx

dTcy pp

( )dx

dTlKKy m

s+∏− 0 . (D.12)

224 Appendix

E: derivation of the decoupling the sound field into standing-wave and traveling-wave components The following is dedicated to the assumption that there exist standing-wave and traveling-wave in the sound field inside a thermoacoustic device. Let’s start with the wave equation, i.e. Eq. (2.1.57): In the experiment, it is obvious that the stack plate material and the resonator tube have a much larger thermal capacity compared with the working gas, i.e. 0≈sε .

The temperature span over the stack is small enough that the density and the Rott’s functions κffv , do not have a dramatic change everywhere including the stack

zone. Therefore, the wave equation can be simplified by realistic situation as

( ) ( ) 01

1)1(1 1

2

21

2

2

1 =−−−

−+−+dx

d

dx

dTffa

dx

d

dx

dfaf m

m

vmk

ppp

σωβ

ρωργ νκ .

(E.1) Using the assumption of ideal gas, i.e mT/1=β , rewrite it as

( )( )( )

01

11111 12

21

21

2

=−

−++⋅⋅⋅−−

−− ppp

v

km

mv

vk

f

f

adx

d

dx

dT

Tf

ff

dx

d γωσ

. (E.2)

In this experiment, the temperatures are all around ambient room temperature 300 K, and the temperature gradient is small.

The value of function ( )( )v

vk

f

ff

−−−11 σ

is also small, roughly in the order of 1. So,

the second term in Eq. (E.2) can be neglected. After all these simplification, the wave equation now is written as

( )0

111

12

2

21

2

=−

−++ pp

v

k

f

f

adx

d γω. (E.3)

It is a second order of differential equation. The general solution of this differential equation is:

xx ee 211

kk BAp ⋅+⋅= . (E.4)

2,1k are the roots of the eigen equation 022 =+ ky , where

( )v

k

f

f

a −−+=

111

2

22 γω

k (E.5)

Therefore, we obtain

Appendix 225

( )v

k

f

f

a −−+±=

111

2,1

γωk . (E.6)

Coefficients A and B depend on the boundary conditions. Define a complex number

( )v

k

f

fi

−−+=+

111 γξχ , (E.7)

and a real wave number akr /ω= . (E.8)

Rewrite 2,1k as

( ) ( )ξχξχ rrr ikkik +±=+±=2,1k . (E.9)

Hence the acoustic pressure field is xikxkxikxk rrrr eeee ξχξχ −− ⋅⋅+⋅⋅= BAp1 . (E.10)

Coefficients A and B depend only on the local boundary conditions. xkre χ and xkre χ− are real numbers depend on local conditions. The wave consists of incident

wave and reflected wave, if there is a change of acoustic impedance in the way of

wave propagating. The pressure reflection coefficient is the ratio of xkre χ⋅A and xkre χ−⋅B . Set a reference pressure in such a way that makes xkre χ⋅A have phase

of “0”, becoming a real number. Define two coefficients LC and RC as xk

LreC χ⋅= A and xk

Rre χ−⋅= BC (E.11)

Rewrite equation (E.10) as xik

Rxik

Lrr eeC ξξ −⋅+⋅= Cp1 , (E.12)

Or ( ) ( ))sin()cos()sin()cos(1 xkixkxkixkC rrRrrL χχχχ −⋅++⋅= Cp . (E.13)

After some mathematical rearrangement, Eq. (E.13) can be written as xik

LRrLreCxkC χχ −⋅−+⋅= )()cos(21 Cp . (E.14)

The coefficient of the second term of Eq. (E.14), LR C−C , is a constant complex

number, writing it as ri

ALR eRC φ=−C , (E.15)

where AR and rφ are amplitude and phase of the complex number LR C−C .

Substitution of Eq. (E.15) into (E.14), yields )(

1 )cos(2 rr xkiArL eRxkC φχχ −−⋅+⋅=p . (E.16)

From equation (E.16), it is clear that the acoustic wave consists of two components locally, standing-wave component )cos(2 xkC rL χ⋅ and a right-going traveling-

wave component )( rr xkiA eR φχ −−⋅ .

226 Appendix

F: computation of loop section in a traveling-wave system Computation station A: The length of the feedback inertance tube is indicated as

fbL and the diameter as fbd . By means of the same method as did to obtain Eq.

(4.3.26) and Eq. (4.3.27), the pressure and volume velocity at computation station A are found to be given by

)sin(4

)cos( 21

11 fbfb

mfbfbfbA kL

d

aikL

πρU

pp −= , (F.1)

)sin(4

)cos( 12

11 fbm

fbfbfbfbA kL

a

dikL

ρπ p

UU −= . (F.2)

If we use the concept of transfer matrix, it is helpful to understand the method in this appendix. For the acoustic pressure and volume velocity at the two ends of any component in the loop section can be related by a transfer matrix:

ba

=

1

1

2221

1211

1

1

U

p

AA

AA

U

p, (F.tm1)

where the subscription, a and b, denote the two ends of a component. For easy writing, here the transfer matrix is denoted as:

[ ]

=−

2221

1211

AA

AATM ba . (F.tm2)

If write (F.1) and (F.2) in the form of transfer matrix, the transfer matrix is given as:

[ ]

−=−

)cos()sin(4

)sin(4

)cos(

2

2

fbfbm

fb

fbfb

mfb

fbA

kLkLa

di

kLd

aikL

ρπ

πρ

TM . (F.tm3)

Computation station B: Between A and B, is a volume having a character of a compliance. Normally, the geometry depends on specific machine. If the geometry parameters are known, the pressure and volume velocity at B can be obtained by the same method used for proceeding results by splitting the volume into elementary computation cells. Here, for simplicity, this compliance volume is considered as a tube segment of length cplL with a large diameter cpld . Its diameter

is larger than those of feedback tube and the ambient heat exchanger. Repeating the same calculation, the pressure and volume velocity at B are

Appendix 227

)sin(4

)cos( 21

11 cplcpl

mAcplAB kL

d

aikL

πρU

pp −= , (F.3)

)sin(4

)cos( 12

11 cplm

AcplcplAB kL

a

dikL

ρπ p

UU −= . (F.4)

The transfer matrix is given as:

[ ]

−=−

)cos()sin(4

)sin(4

)cos(

2

2

cplcplm

cpl

cplcpl

mcpl

AB

kLkLa

di

kLd

aikL

ρπ

πρ

TM . (F.tm4)

Computation station C: The assumption of a zero-temperature gradient does hold for the heat exchangers. Therefore, Eqs. (4.3.13) to (4.3.16) can be used to the computation of heat exchangers. In practice, to get good thermal contact with the gas, the hydraulic radius of the heat exchangers, especially the ambient heat changer, is of the order of working gas’ thermal penetration depth, i.e. vf and

κf having order of 1. Thus, the simplifications used for big channel are not valid

any more. The ambient heat exchanger has a porosity aHXψ , diameter aHXd , and

length aHXL . Start with Eq. (4.3.13) and (4.3.16), applying the boundary conditions

and using the same method, the pressure and volume velocity at C can be obtained

( ) )sin(1

4)cos( 2

111 aHXaHX

aHXaHXaHXaHXv

mBaHXaHXBC L

dfiL k

kU

kppπψ

ωρ−

−= , (F.5)

( ))sin(

4

1)cos(

21

11 aHXaHXm

aHXaHXaHXaHXvBaHXaHXBC L

dfiL k

kpkUU

ωρπψ−

−= ,

(F.6) where the complex wave number is evaluated at temperature of ambient heat exchanger and the Rott’s functions computed with hydraulic radius of the ambient heat exchanger

v

saHX f

f

a −+−+⋅=

1)1/()1(1

2

22 εγω κk , (F.7)

and the hydraulic radius is

aHX

aHXaHXaHXh

dr

Π=− 4

2πψ. (F.8)

In practical situations, the lengths of the heat exchangers are short compared with the complete system. Here, for simplicity, it is assumed that all the heat exchangers are ideal ones (no blockage, porosity=100%). Therefore, we have

228 Appendix

BC 11 pp = and BC 11 UU = . (F.9)

The transfer matrix is given as:

[ ]

=− 10

01BCTM (F.tm5)

Regenerator C-D: Between C and D is the regenerator. There is a strong temperature gradient along the regenerator. The assumption of zero-temperature gradient is not valid for the regenerator. Therefore, Eqs. (4.3.13) to (4.3.16) are not valid for regenerator. In this work, a distributed model for the regenerator proposed by Backhaus and Swift [34] is employed.

Figure F.1: Distributed model of the regenerator. In this distributed model, the regenerator of length regL is split into xLN reg ∆= /

segments. Each segment is of length x∆ and has a temperature span of nT∆ , as

shown in Fig. F.1. With an ideal gas as the working fluid, the compliance of each segment of the regenerator is the isothermal form

regLxCC /0∆=∆ , (F.10)

where mregregreg pLAC /0 ψ= . regψ and regA are the volume porosity and cross-

sectional area of the regenerator. mT is the local mean (average) temperature in the

regenerator. Due to the temperature distribution along the regenerator and the inside compliance, the volumetric velocity changes across each regenerator segment, i.e.

nnmnnnn CiTT ,1,,1,11,1 / pUUU ∆−∆=−+ ω . (F.11)

Dividing both sides of the equation by x∆ and letting 0→∆x yields the differential equation for 1U :

1011 p

UU

reg

m

m L

Ci

dx

dT

Tdx

d ω−= . (F.12)

2,1U

2R∆

2,

22,1

mT

T∆U C∆

2,1p 1,1U

1R∆

1,

11,1

mT

T∆U

C∆

1,1p N,1U

NR∆

Nm

NN

T

T

,

,1 ∆U C∆

N,1p ,down1U

down,1p

stdnT −

up,1p

up,1U

stupT −

Appendix 229

The fluid resistance of each segment of the regenerator is given by µb

nm

regn T

T

L

xRR

⋅∆=∆

0

,0 , (F.13)

where

)/(3 200 reghregregreg rALR −≈ ψµ (F.14)

is the low-Reynolds-number-limit flow resistance of the regenerator when its entire length is at ambient temperature 0T . Over a wide range in temperature, the

viscosity of gases can be described as ( ) µµµ bTT 00 /= , where µb is a constant

depending on the gas. The viscosity temperature dependence is counted in the

resistance by the factor of ( ) µbnm TT 0, / The pressure drop across each regenerator

segment is mainly caused by the fluid resistance, i.e.

nnnn R ,11,1,1 Upp ∆−=− − . (F.15)

Similarly, the differential equation for the pressure is obtained by dividing through with x∆ and letting 0→∆x .

10

01 Up

µb

m

reg T

T

L

R

dx

d

−= . (F.16)

Assume that the temperature has a linear distribution axially along the regenerator, which is a good approximation to the real case. Therefore, the temperature is given by

−+=−

+= −−−

− xL

TxL

TTTxT

regstup

reg

stupstdnstupm

1/11)(

τ, (F.17)

where the temperature ratio

stdnstup TT −−= /τ . (F.18)

In this cooler mode, stupT − is the ambient temperature0T and the stdnT − is the given

cold end temperatureCT . In the mode of engine, stupT − is the ambient

temperature0T and the stdnT − is the given hot end temperature HT .

Under typical operation conditions, the pressure drop across the regenerator is

small, less than 10% [34], i.e. upregen ,1,1 1.0 pp ≈∆ . Thus, upx ,11 )( pp = is assumed

and substituted into Eq. (F.12). The integration of Eq. (F.12) yields

−+−

−+= xL

Cix

L reg

upup

reg

1/11ln

1/11/1

1 ,10,11

ττ

ωτ pUU . (F.19)

Now Eq. (F.19) is substituted into Eq. (F.16) and integrated it along the regenerator. The linear temperature distribution is also assumed.

230 Appendix

The pressure drop is given by: ),(),( ,1000,1111 µµ τωτ bgRCibfR upupDCreg pUppp +=−=∆ , (F.20)

where the functions ( )µτ bf , and ( )µτ bg , are defined as:

( ))2)(1/1(

1)/1(,

2

+−−=

+

µµ τ

ττµ

bbf

b

, (F.21)

( ) ( )

+−

+−

−=

++

2/1ln)/1(

)2(1)/1(

)1/1(1

,2

2

2

2µµ

µτττ

ττ

µµ

bbbg

bb

. (F.22)

Substitution of regLx = , Cup 1,1 UU = and Cup 1,1 pp = in Eq. (F.19), the volume

velocity at the cold end of the regenerator is obtained in terms of the inlet conditions,

ττω

τ −+=

1ln101

1CC

D

Ci pUU . (F.23)

Substitution of Cup 1,1 UU = and Cup 1,1 pp = into Eq. (F.20) yields an expression for

the pressure in terms of inlet pressure and volume velocity: ( ) ),(),(1 010011 µµ ττω bfRbgRCi CCD Upp −−= . (F.24)

The transfer matrix is given as:

[ ]

−−=−

τττω

ττω µµ

11

ln),(),(1

0

000

CibfRbgRCi

CDTM (F.tm6)

Computation station E: Notice that we assume all the heat exchangers are ideal ones in the above part. Therefore, the pressure and volume velocity across the cold end heat exchanger are considered unchanged:

DE 11 pp = , DE 11 UU = , (F.25)

The transfer matrix is given as:

[ ]

=− 10

01DETM (F.tm7)

Computation station E-Tee: The length from the surface E to the joint tee is indicated as tbL and the diameter as tbd . Using the same method as before and

considering Eq.(F.25), the pressure and volume velocity at the tee branch connected to the cold-end heat exchanger are given by:

)sin(4

)cos( 21

11 tbtb

mDtbDout kL

d

aikL

πρU

pp −= , (F.26)

Appendix 231

)sin(4

)cos( 12

11 tbm

DtbtbDout kL

a

dikL

ρπ p

UU −= . (F.27)

The transfer matrix is given as:

[ ] [ ]

−== −−

)cos()sin(4

)sin(4

)cos(

2

2

tbtbm

tb

tbtb

mtb

DoutEout

kLkLa

di

kLd

aikL

ρπ

πρ

TMTM (F.tm8)

Direct relation between out1p , out1U and fb1p , fb1U :

Now the relations of acoustic pressures and volume velocities between all the subsequent computation stations from A to E are obtained. It can be seen that all the acoustic pressures and volume velocities can be expressed in terms of

fb1p and fb1U . The following analysis to find the relation between fb1p , fb1U and

out1p , out1U is carried out. The end result is looking for the impedance that the

loudspeaker “sees” when looking into the resonator. Substitution of Eq. (F.1) and (F.2) into Eq. (F.3) and (F.4) yields

−= )sin()sin()cos()cos( 2

2

11 cplfbcpl

fbcplfbfbB kLkL

d

dkLkLpp

[ ] fbcplcplfbfbcplfbm dkLkLdkLkLa

i 122 /)sin()cos(/)cos()sin(

4U+−

πρ

, (F.28)

−= )sin()sin()cos()cos( 2

2

11 cplfbfb

cplcplfbfbB kLkL

d

dkLkLUU

[ ] fbcplfbcplcplfbfbm

kLkLdkLkLda

i 122 )sin()cos()cos()sin(

4p+−

ρπ

. (F.29)

By substituting Eq. (F.9) into Eq. (F.23) and (F.24), the acoustic pressure and volume velocity are given by

BB

D

Ci1

011 1

lnp

UU

ττω

τ −+= , (F.30)

( ) ),(),(1 010011 µµ ττω bfRbgRCi BBD Upp −−= . (F.31)

Substitution of Eq. (F.28) and (F.29) into Eq. (F.30) and (F.31) yields

[ ]{ )/()sin()sin(/)cos()cos( 2211 fbcplfbcplcplfbfbD dkLkLdkLkL ττ −= UU

[ ]+

−+ 220 /)sin()cos(/)cos()sin(

)1(ln4

cplcplfbfbcplfbm dkLkLdkLkL

Ca

τπτωρ

[ ] −

−+ 220

1 /)sin()sin()cos()cos(1

lncplcplfbfbcplfbfb dkLkLdkLkL

Ci

ττω

p

232 Appendix

[ ]

+− )sin()cos()cos()sin(4

22cplfbcplcplfbfb

m

kLkLdkLkLda

i

τρπ

, (F.32)

( )[{ )sin()sin()cos()cos(),(1 20011 cplfbfbcplfbfbD kLkLdkLkLbgRCi −−= µτωpp

] [ ]})sin()cos()cos()sin(4

),(/ 2202

cplfbcplcplfbfbm

cpl kLkLdkLkLda

bfRid ++

ρτπ µ

[ ]{ 2201 /)sin()sin()cos()cos(),( fbcplfbcplcplfbfb dkLkLdkLkLbfR- −µτU

( ) [ 200 /)cos()sin(

4),(1 fbcplfb

m dkLkLa

bgRCiiπρτω µ−+

]}2/)sin()cos( cplcplfb dkLkL+ . (F.33)

To make D1U and D1p easy to write, they are rewritten as

fbfbD 12111 UDpDp ⋅+⋅= , (F.34)

fbfbD 14131 UDpDU ⋅+⋅= . (F.35)

The complex coefficients from 1D to 4D only depend on the geometry of the

refrigerator and operational conditions. They are given by

( )[ ]22001 /)sin()sin()cos()cos(),(1 cplcplfbfbcplfb dkLkLdkLkLbgRCi −−= µτωD

[ ]2202

/)sin()cos()cos()sin(4

),(fbcplfbcplcplfb

m

fb dkLkLdkLkLa

bfRdi ++

ρτπ µ (F.36a)

[ ]2202 /)sin()sin()cos()cos(),( fbcplfbcplcplfb dkLkLdkLkLbfR −−= µτD

( ) [ )cos()sin(4

),(1 200 cplfbfb

m kLkLd

abgRCii

πρτω µ−−

]22 /)sin()cos( cplcplfbfb dkLkLd+ (F.36b)

[ ]2203 /)sin()sin()cos()cos(

1ln

cplcplfbfbcplfb dkLkLdkLkLCi −−

τωD

[ ]222

/)sin()cos()cos()sin(4 fbcplfbcplcplfb

m

fb dkLkLdkLkLa

di+−

τρπ

(F.36c)

[ ] τ//)sin()sin()cos()cos( 224 fbcplfbcplcplfb dkLkLdkLkL −=D

[ ]222

0 /)sin()cos()cos()sin()1(

ln4cplcplfbfbcplfb

fb

m dkLkLdkLkLd

Ca +−

+τπ

τωρ. (F.36d)

For the simplicity of writing, four real functions are defined, which only depend on the geometrical configurations:

221 /)sin()sin()cos()cos( cplcplfbfbcplfb dkLkLdkLkL −=θ (F.37a)

222 /)sin()sin()cos()cos( fbcplfbcplcplfb dkLkLdkLkL −=θ (F.37b)

Appendix 233

223 /)sin()cos()cos()sin( fbcplfbcplcplfb dkLkLdkLkL +=θ (F.37c)

224 /)sin()cos()cos()sin( cplcplfbfbcplfb dkLkLdkLkL +=θ . (F.37d)

Therefore, the complex coefficients of D can be rewritten as

( ) 30

2

1001 4

),(),(1 θ

ρτπ

θτω µµ a

bfRdibgRCi

m

fb+−=D (F.38a)

( ) 4200202

4),(1),( θ

πρτωθτ µµ

fb

m

d

abgRCiibfR −−−=D (F.38b)

3

2

10

3 41ln θ

τρπ

θτ

τωa

diCi

m

fb−−

=D (F.38c)

420

24 )1(

ln4/ θ

τπτωρτθ

−+=

fb

m

d

CaD . (F.38d)

Substituting Eq. (F.34) and (F.35) into Eq. (F.26) and (F.27), the acoustic pressure

out1p and volume velocity out1U can be written in terms of fb1p and fb1U :

−= )sin(

4)cos( 2

3111 tb

tb

mtbfbout kL

d

aikL

πρ D

Dpp

−+ )sin(

4)cos( 2

421 tb

tb

mtbfb kL

d

aikL

πρ D

DU , (4.3.28)

−= )sin(

4)cos( 1

2

311 tbm

tbtbfbout kL

a

dikL

ρπ D

DpU

−+ )sin(

4)cos( 2

2

41 tbm

tbtbfb kL

a

dikL

ρπ D

DU . (4.3.29)

If we use the form of transfer matrix, computation stations from A to B in the form of Eq.(F.tm1) are given as:

[ ]fb

fbA

A

=

1

1

1

1

U

pTM

U

p, (F.tm8)

[ ]A

AB

B

=

1

1

1

1

U

pTM

U

p. (F.tm9)

By substituting Eq.(F.tm8) into (F.tm9), the transfer matrix between B and fb is obtained as: [ ] [ ] [ ] fbAABfbB −−− = TMTMTM . (F.tm10)

Repeating this computation for stations C, D, E to “out” (the tee branch connected to the cold-end heat exchanger), the transfer matrix between “out” and “fb” is obtained as:

234 Appendix

[ ] [ ] [ ] [ ] [ ] [ ] [ ] fbAABBCCDDEEoutfbout −−−−−−− = TMTMTMTMTMTMTM .

(F.tm11) If write Eqs.(4.3.28) and (4.3.29) in the form of transfer matrix, the transfer matrix is given as: [ ] =− fboutTM

−−

−−

)sin(4

)cos()sin(4

)cos(

)sin(4

)cos()sin(4

)cos(

22

41

2

3

24

223

1

tbm

tbtbtb

m

tbtb

tbtb

mtbtb

tb

mtb

kLa

dikLkL

a

dikL

kLd

aikLkL

d

aikL

ρπ

ρπ

πρ

πρ

DD

DD

DD

DD

.

(F.tm12) Therefore, the direct relation between “out” and “fb” in the form of transfer matrix is given as:

[ ]fb

fbout

out

=

1

1

1

1

U

pTM

U

p, (F.tm13)

Appendix 235

G: transmission of acoustic impedance of a uniform pipe Next, the transmission of acoustic impedance of a pipe is derived. Because the set-up we use is connected to the looped tube via a long straight pipe element. Two random positions in a tube are indicated by bx and cx in the direction of acoustic

wave propagation, showed in Fig. G.1. By using the Eq. (4.3.19) and (4.3.20), the acoustic impedance is given by

( ) xixi

xixim

ee

ee

Af kk-

kk-

CCCC

kUp

Z⋅−⋅⋅+⋅

−==

12

12

1

1

1 ν

ωρ (G.1)

Figure G.1: Two random positions along a tube. Splitting the exponential factors into cosine and sine and combining terms, Eq. (G.1) becomes

( ) )sin()cos(

)sin()cos(

1

12

12

12

12

xix

xix

Afm

kkCCCC

kCCCC

k

kZ

−+−

+−−

−=

ν

ωρ. (G.2)

Eq. (G.2) is the acoustic impedance at x. Replacing x with bx and cx . The acoustic

impedances of them are respectively

( ) )sin()cos(

)sin()cos(

1

12

12

12

12

bb

bbm

b

xix

xix

Af kkCCCC

kCCCC

k

kZ

−+−

+−−

−=

ν

ωρ, (G.3)

( ) )sin()cos(

)sin()cos(

1

12

12

12

12

cc

ccm

c

xix

xix

Af kkCCCC

kCCCC

k

kZ

−+−

+−−

−=

ν

ωρ. (G.4)

Now, the impedance conZ is defined as:

o x bx cx

236 Appendix

( ) kZ

Afm

conν

ωρ−

=1

. (G.5)

Eliminating )/()( 1212 CCCC +− , and using (G.3) (G.4), one finds the relation of

acoustic impedance between these two positions:

)](sin[)](cos[

)](sin[)](cos[

bccon

bbc

bcbccon

b

con

c

xxixx

xxixx

−−−

−−−=

kZZ

k

kkZZ

ZZ

(G.6)

Or

)](sin[)](cos[

)](sin[)](cos[

bccon

cbc

bcbccon

c

con

b

xxixx

xxixx

−+−

−+−=

kZZ

k

kkZZ

ZZ

. (G.7)

This is the transmission relation for the acoustic impedance. For open channel tube, i.e. the hydraulic radius vhr δ>> and κδ can be sustained,

thus Rott’s functions 0≈vf and 0≈κf , Eq. (G.5) can be reduced to

A

aZ m

con

ρ= . (G.8)

In that case, the transmission equations become much simpler,

)](sin[)](cos[

)](sin[)](cos[

bcbbcm

bcm

bcbm

c

xxkixxkA

a

xxkA

aixxk

A

a

−−−

−−−=

Z

ZZ ρ

ρρ

(G.9)

Or

)](sin[)](cos[

)](sin[)](cos[

bccbcm

bcm

bccm

b

xxkixxkA

a

xxkA

aixxk

A

a

−+−

−+−=

Z

ZZ ρ

ρρ

. (G.10)

Appendix 237

H: Fortran code for computation of traveling-wave engine * simple program for traveling-wave machine * edited on 13 Feb., 2009 *-------------------------------------------------- ----------------- Real R,gama,b_miu,pi,f,round_f,Pm,P_in, T_cold,T_hot,density, & conduct_fluid,conduct_solid,Cp_fluid,sound_speed,miu,Pr, & k,L_fb,d_fb,L_cpl,d_cpl,porosity,d_reg,L_reg,r_h,area_gas, & area_solid,L_tb,d_tb,C0,R0,tau,f_reg,g_reg,sita1,sita2, & sita3,sita4,pen_f_d,v_pen_fluid,ratio_v,ratio_k,term1,F3, & term2,term3,F2,Q_c,U_mu,U_angle,U_mu1,U_angle1,U_mui,U_anglei & ,power,inertance,compliance,U_mu2,U_angle2,U_mu3,U_angle3, & U_mu4,U_angle4 complex i,f_v,t_v,f_k,t_k,D1,D2,D3,D4,coef_i1,coef_i2,p_D,U_D, & coef_i3,Z_fb,Z_1,Z_input,U_input,U_fb,Z_i,U_i,U_1fb,U_1c, & p_out,U_out,U_1out open(21, file='out_dw.dat') R=8.3145/(4.003e-3) gama=5.0/3.0 b_miu=0.68 pi=3.1416 i=cmplx(0.0,1.0) f=84.12 Pm=3103000.0 p_in=2.723e5 T_hot=300.0 T_cold=900.0 density=Pm/(R*T_hot) conduct_fluid=0.156084 Cp_fluid=5.19312*1000.0 sound_speed=1019.56 miu=1.9937e-5 Pr=Cp_fluid*miu/conduct_fluid conduct_solid=40.0 round_f=2.0*3.14*f k=round_f/sound_speed L_fb=(10.2+25.6+20.9)*0.01 d_fb=7.8*0.01

238 Appendix

L_cpl=0.348 d_cpl=10.2*0.01 porosity=0.72 d_reg=8.89*0.01 L_reg=7.3*0.01 r_h=42.0e-6 area_gas=pi*d_reg**2*porosity/4.0 area_solid=pi*d_reg**2*(1.0-porosity)/4.0 L_tb=(24.0+1.0+7.0)*0.01 d_tb=8.89*0.01 C0=porosity*pi*d_reg**2*L_reg/4.0/Pm R0=4.5*miu*L_reg/(porosity*0.25*pi*d_reg**2*r_h**2) tau=T_hot/T_cold f_reg=((1.0/tau)**(b_miu+2.0)-1.0)/(1.0/tau-1.0)/(b_miu+2.0) g_reg=(((1.0/tau)**(b_miu+2.0)-1.0)/(b_miu+2.0)**2-(1.0/tau)** & (b_miu+2.0)*log(1.0/tau)/(b_miu+2.0))/(1.0/tau-1.0)**2 sita1=cos(k*L_fb)*cos(k*L_cpl)- & d_fb**2*sin(k*L_fb)*sin(k*L_cpl)/d_cpl**2 sita2=cos(k*L_fb)*cos(k*L_cpl)- & d_cpl**2*sin(k*L_fb)*sin(k*L_cpl)/d_fb**2 sita3=sin(k*L_fb)*cos(k*L_cpl)+ & d_cpl**2*cos(k*L_fb)*sin(k*L_cpl)/d_fb**2 sita4=sin(k*L_fb)*cos(k*L_cpl)+ & d_fb**2*cos(k*L_fb)*sin(k*L_cpl)/d_cpl**2 D1=(1.0-i*round_f*C0*R0*g_reg)*sita1+i*pi*d_fb**2*R0*f_reg*sita3 & /(4.0*density*sound_speed) D2=-R0*f_reg*sita2-i*(1.0-i*round_f*C0*R0*g_reg)*sita4* & 4.0*density*sound_speed/(pi*d_fb**2) D3=i*round_f*C0*log(tau)*sita1/(1.0-tau)-i*pi*d_fb**2*sita3/ & (4.0*density*sound_speed*tau) D4=sita2/tau+4.0*density*sound_speed*round_f*C0*log(tau)*sita4/ & (pi*d_fb**2*(1.0-tau)) Z_fb=(D2*cos(k*L_tb)-i*4.0*density*sound_speed*D4*sin(k*L_tb)/ & (pi*d_tb**2))/(1.0-D1*cos(k*L_tb)+i*4.0*density*sound_speed* & D3*sin(k*L_tb)/(pi*d_tb**2)) Z_1=pi*d_tb**2*D2/(4.0*density*sound_speed)+ & 4.0*density*sound_speed*D3/(pi*d_tb**2)

Appendix 239

Z_input=1.0/((D4*cos(k*L_tb)-i*pi*d_tb**2*D2*sin(k*L_tb)/ & (4.0*density*sound_speed)-1.0)/Z_fb+D3*cos(k*L_tb)-i*pi* & d_tb**2*D1*sin(k*L_tb)/(4.0*density*sound_speed)) F3=real(D1*conjg(D3))+real(D2*conjg(D4))/(abs(Z_fb))**2 & +real(D2*conjg(D3)/Z_fb+D1*conjg(D4)/conjg(Z_fb)) U_input=p_in/Z_input U_mu=cabs(U_input) U_angle=180.0*atan(aimag(U_input)/real(U_input))/pi U_fb=p_in/Z_fb U_mu1=cabs(U_fb) U_angle1=180.0*atan(aimag(U_fb)/real(U_fb))/pi p_D=D1*p_in+D2*U_fb U_D=D3*p_in+D4*U_fb p_out=p_in*(D1*cos(k*L_tb)-i*4.0*density*sound_speed*D3* & sin(k*L_tb)/(pi*d_tb**2))+U_fb*(D2*cos(k*L_tb)-i*4.0 & *density*sound_speed*D4*sin(k*L_tb)/(pi*d_tb**2)) U_out=p_in*(D3*cos(k*L_tb)-i*pi*d_tb**2*D1*sin(k*L_tb)/(4.0* & density*sound_speed))+U_fb*(D4*cos(k*L_tb)-i*pi*d_tb**2*D2* & sin(k*L_tb)/(4.0*density*sound_speed)) U_mu3=cabs(U_out) U_angle3=180.0*atan(aimag(U_out)/real(U_out))/pi pen_f_d=sqrt(2.0*conduct_fluid/ & (density*Cp_fluid*round_f)) v_pen_fluid=sqrt(2.0*miu/(density*round_f)) ratio_v=r_h/v_pen_fluid call tanh(ratio_v,t_v) f_v=t_v/(cmplx(1.0,1.0)*cmplx(ratio_v,0.0)) ratio_k=r_h/pen_f_d call tanh(ratio_k,t_k) f_k=t_k/(cmplx(1.0,1.0)*cmplx(ratio_k,0.0))

240 Appendix

coef_i3=conjg(f_v)+(f_k-conjg(f_v))/(1.0+Pr) coef_i2=1.0-conjg(f_v)-(f_k-conjg(f_v))/(1.0+Pr) coef_i1=(sita1*sita2/conjg(Z_fb)+sita3*sita4/Z_fb)*coef_i2 term1=area_gas*R0*aimag(coef_i1)/(2.0*round_f*density*L_reg) term2=area_gas*R0*(pi*d_fb**2*sita1*sita3/(4.0*density* & sound_speed)+4.0*density*sound_speed*sita2*sita4/ & pi/(d_fb*cabs(Z_fb))**2)*real(coef_i2)/ & (2.0*round_f*density*L_reg) term3=area_gas*Cp_fluid*T_hot*R0**2*(1.0/tau-1.0)/L_reg**3 & /(2.0*round_f**3*density)/(1.0-Pr)*aimag(coef_i3) & *((sita2/cabs(Z_fb))**2+(pi*d_fb**2*sita3/(4.0* & density*sound_speed))**2+pi*d_fb**2*sita2*sita3 & *aimag(Z_fb)/(cabs(Z_fb))**2/(2.0*density*sound_speed)) Q_c=(abs(p_in))**2*(F3/2.0-F2) & +(area_gas*conduct_fluid+area_solid*conduct_solid) & *(1.0-1.0/tau)*T_hot/L_reg Z_i=cmplx(1.09574e6,2.074e6) U_i=p_in/Z_i U_mui=cabs(U_i) U_anglei=180.0*atan(aimag(U_i)/real(U_i))/pi inertance=0.57e3 compliance=56.87e-11 U_1c=round_f**2*inertance*compliance/R0*p_in/ & (1.0+i*round_f*inertance/R0) U_1fb=-i*round_f*compliance*p_in-U_1c U_mu2=cabs(U_1fb) U_angle2=180.0*atan(aimag(U_1fb)/real(U_1fb))/pi U_1out=tau*U_1c U_mu4=cabs(U_1out) U_angle4=180.0*atan(aimag(U_1out)/real(U_1out))/pi power=0.5*real(p_in*conjg(U_1c))*3.0

Appendix 241

write(21,*) 'DeltaE',' Qh=',3037.000, & 'U_input= ',0.2870,' ',-89.452, ' U_fb= ',0.2029, & ' ',86.355,' U_out= ',8.6321e-2,' ',-77.571 write(21,*) 'yan model',' Qh= ',Q_c, 'U_input=' & ,U_mu,' ',U_angle,' U_fb=',U_mu1,' ',U_angle1, & ' U_out=', & U_mu3,' ',U_angle3,' p_out=', p_out write(21,*) 'swift smplified lump model', ' Qh=', power, & 'U_input= ',U_mui,' ',U_anglei,' U_fb=',U_mu2,' ', & U_angle2,' U_out=',U_mu4,' ',U_angle4 end *========================================================== * Subroutine tanh is used for calculate function tanh((1+i)y0/peneration depth) * edit on 12 April, 2006 *========================================================== subroutine tanh(alfa,f_tanh) real alfa,r_part,i_part,B,C,D,E complex f_tanh B=(exp(alfa)-exp(-1.0*alfa))*cos(alfa) C=(exp(alfa)+exp(-1.0*alfa))*sin(alfa) D=(exp(alfa)+exp(-1.0*alfa))*cos(alfa) E=(exp(alfa)-exp(-1.0*alfa))*sin(alfa) r_part=(B*D+C*E)/(D**2+E**2) i_part=(C*D-B*E)/(D**2+E**2) f_tanh=cmplx(r_part,i_part) end

242 Appendix

I: The design of the ambient heat exchanger in the traveling-wave refrigerator By the conclusion in reference [74], heat exchanger with ξhx/Lhx in the range of 3 to 8 can be thermally effective as a source or sink thermoacoustic heat transport if y0/δκ is in the range of 0.75 to 0.5. The peak displacement amplitude of a gas parcel in the heat exchanger is given by ξhx, and the length of the heat exchanger is given by Lhx. The half separation between adjacent fins is y0 and δκ is the gas thermal penetration depth. Hofler also pointed out that the acoustic loss or dissipation of the heat exchanger is relatively less for y0/δκ=0.5. First, the gas thermal penetration depth δκ is calculated for the designed working condition. By the definition, the fluid’s thermal penetration depth is given as

fc

K

pm πρδκ 2

2⋅

= . (I.1)

Under the designed working condition: helium at mean pressure of 11 bar, temperature of 300 K, and operating frequency of 320 Hz, the related gas properties are given: К=0.1567 W/(m·K), ρm=1.7571 kg/m³, and Cp=5.1925 kJ/(kg·K) Substitution of the gas parameters into Eq. (I.1) gives the thermal penetration depth: δκ =1.31×10-4 m. Considering the criterion y0/δκ=0.5, the separation between adjacent fins is designed as: 2y0=δκ . (I.2) The interfin distance should be 0.131 mm. By taking capability of manufacturing into account, the distance between adjacent fins is designed as 0.17 mm. By numerical simulation, the volume velocity in the ambient heat exchanger is U1=5.95×10-3 m³/s. By using the relation between volume velocity and peak displacement amplitude of a gas parcel

fA

U

hxhx π

ξ21

⋅= , (I.3)

where Ahx is the cross sectional area of the heat exchanger. The substitution of U1 and Ahx into (I.3) gives ξhx=4.275 mm. By the criterion “ξhx/Lhx in the range of 3 to 8”, the length of heat exchanger should be in the range of 0.534 mm to 1.425 mm. Further consideration on the criterion that small temperature difference between the top and the root of a fin shows that such small length Lhx will result in a large temperature span over a sigle fin. A fin of the heat exchanger is shown in Fig. I.1. The length of a fin is lhx, and the temperature of the root of a fin is Troot.

Appendix 243

Figure I.1: Two adjacent fins of the heat exchanger. The total amount of heat exchanged at the ambient heat exchanger is indicated as Φ. The heat of unit cross-sectional area of the heat exchanger is given by:

2

4

hxhx dA πφ Φ=Φ= . (I.4)

The heat taken away by each fin is obtained as: ( ) finfin ly ⋅+⋅= δφφ 01 2 . (I.5)

Figure I.2: A fin and the cross-sectional view with an element for energy analysis. An element in a fin, shown in Fig.I.2, is analyzed and the heat loaded from the contact surface with gas is given by:

Troot

l fin

Lhx

δfin

dx

dx x 0

heat

qx qx+dx

x

Troot y0

l fin

Lhx

y0 δfin

244 Appendix

dxl

LdxLl fin

hxhxfin

⋅=⋅⋅⋅

11 φφ (I.6)

Considering Eq. (I.2), the heat flows in the element of the fin, which is dashed out in Fig. I.2, follow:

01 =⋅+⋅⋅+⋅⋅−+

dxl

LqLqfin

hxfindxxxhxfinxx

φδδ , (I.7)

where the qx is the heat flow in the x direction. When dx→0, Eq. (I.7) becomes:

dxl

dxdx

dqL

fin

xfinhx ⋅−=⋅⋅⋅ 1φδ . (I.8)

Rewrite Eq.(I.8) as:

finhxfin

x

Lldx

dq

δφ

⋅−= 1 . (I.9)

By Fouriers law, the conduction within the fin is expressed as:

dx

xdTq x

sx

)(Κ−= , (I.10)

where Кs is the conductivity of the solid material of the fin and Tx(x) is the temperature distribution along x. Substitution of Eq. (I.10) into Eq. (I.9) yields:

shxfinfin

x

Lldx

xTd

Κ=

δφ1

2

2 )(. (I.11)

Therefore, it can be written as:

11)(

CLl

x

dx

xdT

shxfinfin

x +Κ

⋅=δφ

. (I.12)

By using the boundary condition:

at finlx = , 0)( =

dx

xdTx (I.13)

the coefficient in Eq. (I.12) is obtained as:

shxfinLC

Κ−=

δφ1

1 . (I.14)

By using Eq.(I.14) and (I.12), the temperature distribution can be written as:

21

21

2)( C

L

x

Ll

xxT

shxfinshxfinfinx +

Κ⋅−

Κ⋅=

δφ

δφ

. (I.15)

By using the second boundary condition: at 0=x , rootx TT = , (I.16)

the coefficient in Eq. (I.15) is obtained as:

Appendix 245

rootTC =2 . (I.17)

The temperature distribution is given as:

rootshxfinshxfinfin

x TL

x

Ll

xxT +

Κ⋅−

Κ⋅=

δφ

δφ 1

21

2)( . (I.18)

The temperature at the top of the fin is expressed as:

rootshxfin

fin

shxfinfin

fin

lxx TL

l

Ll

lT

fin+

Κ⋅

−Κ

⋅=

= δφ

δφ 1

21

2. (I.19)

The temperature difference between the top and the root of the fin is given as:

shxfin

finrootlxxfin L

lTTT

fin Κ⋅

=−=∆= δ

φ2

1 . (I.20)

Substitution of Eq. (I.4) and (I.5) into (I.20) yields:

+⋅

ΚΦ=∆ 1

22 02

2finhx

fin

shxfin

y

L

l

dT

δπ. (I.21)

By numerical simulation for design, the heat exchanged at the ambient heat exchanger is around 243 W, i.e. Φ=243. The designed diameter of the heat exchanger is 29.7 mm. The solid material is choosen as copper, whose conductivity is taken as 400 W/(m·K). Substitution of these properties into Eq. (I.21) yields:

hx

finfin L

lT

2

1116⋅=∆ . (I.22)

The length of fins is 4 mm, i.e. lfin=4 mm. Therefore, by Eq. (I.22), the temperature span across the fin would be in the range of 12.5 to 33 K, if the length of the heat exchanger is in the range of 0.534 mm to 1.425 mm as calculated at the beginning. If so, the temperature span would be relatively large for a fin. A heat exchanger with length of 0.534 mm to 1.425 mm would be a hard task for manufacturing, and be fragile in use. By all these considerations, the length of the ambient heat exchanger was designed to be 6 mm, which makes the temperature span across the fin as small as 2.9 K.

246 Appendix

J: Time evolution of two orientations: upward and downward in traveling-wave refrigerator measurement

Here, the time evolutions for two orientations of the loop section—upward and downward, are given to show that the loop downward configuration is more stable. The temperature distribution of T1 to T11 is given in Fig. 4.5.6. During the operations, the power into the PWG (P_driver in the Figs.J.1 and J.2) was increased step-wise. For every input power to the PWG, the temperatures from T1 to T11 are traced and recorded. For Figs. J.1 and J.2, the left vertical axis is for temperature (°C) and the right vertical axis is for power (W). In Figs J.1 and J.2, the temperature T1, T2, and T5-T11 are stable, or “flat” for a constant P_driver. Temperatures T3 and T4 are measurement of gas temperature through and the solid temperature on the cold end heat exchanger. In Fig.J.1, T3 and T4 fluctuate slightly when P_driver is 30W. When P_driver is 40 and 50 W, T3 and T4 show a difference between each other. When P_driver is more than 60 W, T3 and T4 start to fluctuate much.

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

0 500 1000 1500 2000

Time (s)

Tem

per

atu

re (C

)

0

10

20

30

40

50

60

70

80

90

100

Po

wer

(W)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

P_driver

Figure J.1: Time evolution of the loop upward configuration.

Appendix 247

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

0 400 800 1200 1600 2000 2400 2800 3200

Time (s)

Tem

per

atu

re (

C)

0

10

20

30

40

50

60

70

80

90

100

Po

wer

(W

)

T1

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

P_driver

Figure J.2: Time evolution of the loop downward configuration. In Fig.J.2, T3 and T4 show stable values, and the values of T3 are consistent with those of T4 when P_driver is 30, 40 and 50 W. When P_driver is more than 60 W, T3 and T4 start to fluctuate much. By comparison of Figs. J.1 and J.2, it can be concluded that the loop downward configuration is more stable than the loop upward. Therefore, in the measurements of section 4.5, the loop downward configuration was employed.

248 Appendix

K: Acoustic field in the scaled-down standing-wave systems The resonator tube of a standing-wave system containing a parallel-plate stack is schematically given in Fig. K.1. The resonator tube is divided into three zones by the stack: Zone 1: from the interface between the driver and the resonator tube to the left end of the stack, it has a length of L1. The acoustic pressure and volume velocity at the interface with the driver are p1in and U1in. Zone 2: stack is placed between xL and xR, and having a length of Lstack. The stack is sandwiched by cold and hot heat exchangers. The effects of two heat exchangers on the acoustic field are neglected, assuming that they are ideal. Zone 3: between the end of the resonator tube and the right end of the stack, it has a length of L3. The acoustic pressure and volume velocity confined by the boundary condition of the end of the resonator tube are p1end and U1end. The position where the boundary condition applies is xend.

Figure K.1: A sketch of the resonator tube of a standing-wave system. Boundary conditions at xend: Closed end: if the resonator tube is closed at xend, the solid wall imposes the volume velocity at this position zero, i.e. U1end=0. Then the acoustic impedance at this position is given by

∞==end

endend

1

1

Up

Z . (K.1)

Open end: if the resonator tube is connected to a big gas reservoir, the open end imposes the acoustic pressure at this position zero, i.e. p1end=0. Then the acoustic impedance at this position is given by

01

1 ==end

endend U

pZ . (K.2)

o x Lx Rx endx

p1end U1end

p1in U1in

L1 Lstack L3

Appendix 249

In zone 1 and 3, the tube segments are assumed as wide-open channel tube, i.e. the hydraulic radius vhr δ>> and κδ , thus Rott’s functions 0≈vf and 0≈κf . Using

Eq. (G.10) in appendix G, substitution of Zc=Zend and xc-xb=L3 into Eq. (G.10) yields:

]sin[]cos[

]sin[]cos[

33

33

kLikLA

a

kLA

aikL

A

a

endres

m

res

mend

res

mR

Z

ZZ

+

+= ρ

ρρ

. (K.3)

Ares is the cross sectional area of the resonator tube. Wave number k is defined as Eq. (E.8) in appendix E:

akk r

ω== . (K.4)

Substitution of Eq. (K.1) into (K.3) gives the acoustic impedance at xR when xend is a closed end:

]sin[]cos[

3

3

kL

kL

A

ai

res

mR

ρ−=Z . (K.5)

Substitution of Eq. (K.2) into (K.3) gives the acoustic impedance at xR when xend is an open end:

]cos[]sin[

3

3

kL

kL

A

ai

res

mR

ρ=Z . (K.6)

Similarly, the acoustic impedance at a random position x3 between xend and xR in zone 3 is given by:

)](sin[)](cos[

)](sin[)](cos[

33

33

3

xxkixxkA

a

xxkA

aixxk

A

a

endendendres

m

endres

mendend

res

m

−+−

−+−=

Z

ZZ ρ

ρρ

. (K.7)

In the zone 2, where the stack places, assumption of wide-open channel is not valid anymore. The hydraulic radius of the stack channel has the same order of gas penetration depth, i.e. κδ2~hr and νδ2 , thus Rott’s functions vf and κf are

computed and taken into account. In the zone 2, the transmission equation (G.7) in appendix G is employed. Substitution of Zc=ZR and xc-xb=Lstack into Eq. (G.7) gives the acoustic impedance at xL:

]sin[]cos[

]sin[]cos[

stackcon

Rstack

stackstackcon

R

con

L

LiL

LiL

kZZ

k

kkZZ

ZZ

+

+= . (K.8)

250 Appendix

The constant impedance conZ is defined as Eq.(G.5) in appendix G:

( ) kZ

gas

mcon Afν

ωρ−

=1

. (K.9)

Here Agas is the cross-sectional area of the gas in the stack region. The complex wave number in stack zone k is given in appendix E as:

( )v

k

f

f

a −−+=

111

2

22 γω

k , (E.5)

and ( )

v

k

f

f

a −−+±=

111

2,1

γωk . (E.6)

Similarly, the acoustic impedance at a random position x2 between xL and xR in zone 2 is given by:

)](sin[)](cos[

)](sin[)](cos[

22

222

xxixx

xxixx

Rcon

RR

RRcon

R

con −+−

−+−=

kZZ

k

kkZZ

ZZ

. (K.10)

Using transmission equation (G.10) in appendix G, and substitution of Zc=ZL and xc-xb=L1 into Eq. (G.10) gives the acoustic impedance at x=0:

]sin[]cos[

]sin[]cos[

11

11

0

kLikLA

a

kLA

aikL

A

a

Lres

m

res

mL

res

m

Z

ZZ

+

+= ρ

ρρ

. (K.11)

Similarly, the acoustic impedance at a random position x1 between xL and 0 in zone 1 is given by:

)](sin[)](cos[

)](sin[)](cos[

11

11

1

xxkixxkA

a

xxkA

aixxk

A

a

LLLres

m

Lres

mLL

res

m

−+−

−+−=

Z

ZZ ρ

ρρ

. (K.12)

In the scaled-down system, as shown in section 5.1, we have:

Tube segments length: x

LL ϕ1

1 =′ , (K.13)

x

stackstack

LL ϕ=′ , (K.14)

and x

LL ϕ3

3 =′ . (K.15)

Similarly, for the random positions in three zones in scaled-down system become:

( )11

1 xxxx LxL −=′−′ −ϕ , (K.13a)

Appendix 251

( )21

2 xxxx RxR −=′−′ −ϕ , (K.14a)

and ( )31

3 xxxx endxend −=′−′ −ϕ . (K.15a)

With Eq.(5.1.6), the wave number k′ of the scaled-down system becomes:

kaa

k xx ϕωϕω ==′′

=′ . (K.16)

The cross-sectional area of the resonator tube in scaled-down system becomes:

( ) reszyres AA 1−=′ ϕϕ . (K.17)

The assumption that the ratio of stack spacing to thermal penetration depth is kept constant while scaling, stated in Eq. (5.1.11a), holds here. So is the assumption that the ratio between stack plate thickness and solid thermal penetration depth is fixed while scaling, i.e. Eq. (5.1.12a). Thus the equations (5.1.8) to (5.1.18) are valid here. Therefore, the Rott’s functionsvf ′ and κf ′ for the stack zone in scaled-down

system are given as stated in section 5.1:

ν

ν

ν

ν

νν

δ

δ

δ

δf

yi

yi

yi

yif =

+

+

=

′′+

′′+

=′0

0

0

0

)1(

)1(tanh

)1(

)1(tanh, (5.1.16)

κ

κ

κ

κ

κκ

δ

δ

δ

δf

yi

yi

yi

yif =

+

+

=

′′+

′′+

=′0

0

0

0

)1(

)1(tanh

)1(

)1(tanh. (5.1.17)

With Eq.(5.1.11ab) and (5.1.12ab), the porosity of the stack in scaled-down system is given:

stackstack ly

y

ly

y ψψ =+∏

∏=′+′∏′

′∏′=′

)()( 0

0

0

0 . (K.18)

The gas area Agas in scaled-down system is given:

( ) ( ) gaszystackreszyresstackgas AAAA ⋅==′′=′ −− 11 ϕϕψϕϕψ . (K.19)

Substitution of equations (K.16), (5.1.16) and (5.1.17) into (E.6) yields the complex wave number in the stack zone of the scaled-down system:

( )2,12,1 1

11kk x

v

k

f

f

aϕγω =

′−′−+

′′

±=′ . (K.20)

Therefore, the substitution of equations (K.13) to (K.17), (5.1.16) and (5.1.17) into the equations for acoustic impedance at xR (K.5) or (K.6) yields:

( )[ ][ ] ( ) Rxy

xx

xx

reszy

m

res

mR Lk

Lk

A

ai

Lk

Lk

A

ai ZZ ϕϕ

ϕϕϕϕ

ϕϕρρ =

⋅⋅−=

′′′′

′′′

−=′ − /sin

/cos

]sin[

]cos[

3

31

3

3 ,

(K.21)

252 Appendix

Or

( )[ ][ ] ( ) Rxy

xx

xx

reszy

m

res

mR Lk

Lk

A

ai

Lk

Lk

A

ai ZZ ϕϕ

ϕϕϕϕ

ϕϕρρ =

⋅⋅=

′′′′

′′′

=′ − /cos/sin

]cos[]sin[

3

31

3

3 .

(K.22) In the stack zone, the substitution of equations (K.13) to (K.22) into the equations for acoustic impedance (K.8) and (K.9) yields the acoustic impedance of scaled-down system at xL:

( ) ( )( ) ( ) conzy

xgaszy

mx

gas

mcon

AfAfZ

kkZ ϕϕ

ϕϕϕωρϕρω

νν

=−

=′′′−

′′=′ −111

, (K.23)

]sin[]cos[

]sin[]cos[

stackcon

Rstack

stackstackcon

R

con

L

LiL

LiL

′′′′

+′′

′′+′′′′

=′′

kZZ

k

kkZZ

ZZ

( )( )

( )( )

con

L

xstackxconzy

Rzyxstackx

xstackxxstackxconzy

Rzy

LiL

LiL

ZZ

kZ

Zk

kkZ

Z

=+

+=

]/sin[]/cos[

]/sin[]/cos[

ϕϕϕϕϕϕ

ϕϕ

ϕϕϕϕϕϕϕϕ

. (K.24)

Rewrite Eq.(K.24) as:

( ) Lzycon

conLL Z

ZZ

ZZ ϕϕ=′

⋅=′ . (K.25)

Similarly, in zone 1, substitution of equations (K.12) to (K.24) into the equations for acoustic impedance (K.10) yields the acoustic impedance of scaled-down system at x=0:

]sin[]cos[

]sin[]cos[

11

11

0

LkiLkA

a

LkA

aiLk

A

a

Lres

m

res

mL

res

m

′′′+′′′

′′

′′′

′′+′′′

′′′

=′Z

ZZ ρ

ρρ

( )( ) ( )

( ) ( ) ]/sin[]/cos[

]/sin[]/cos[

111

111

1

xxLzyxx

reszy

m

xx

reszy

mxxLzy

reszy

m

kLikLA

a

kLA

aikL

A

a

ϕϕϕϕϕϕϕϕρ

ϕϕϕϕρϕϕϕϕ

ϕϕρ

Z

Z

+

+=

( ) 0Zzyϕϕ= . (K.26)

Similarly, the acoustic impedances at random positions x1, x2 and x3 in scaled-down system become:

Appendix 253

( ) 1

11

11

1

)](sin[)](cos[

)](sin[)](cos[Z

Z

ZZ zy

LLLres

m

Lres

mLL

res

m

xxkixxkA

a

xxkA

aixxk

A

a ϕϕρ

ρρ =

′−′′′+′−′′′

′′

′−′′′

′′+′−′′′

′′′

=′ , (K.27)

( ) 2

22

22

2

)](sin[)](cos[

)](sin[)](cos[Z

kZZ

k

kkZZ

ZZ zy

Rcon

RR

RRcon

R

con

xxixx

xxixx

ϕϕ=′−′′

′′

+′−′′

′−′′+′−′′′′

′=′ , (K.28)

( ) 3

33

33

3

)](sin[)](cos[

)](sin[)](cos[Z

Z

ZZ zy

endendendres

m

endres

mendend

res

m

xxkixxkA

a

xxkA

aixxk

A

a ϕϕρ

ρρ =

′−′′′+′−′′′

′′

′−′′′

′′+′−′′′

′′′

=′ .

(K.29) When the input acoustic pressure at the interface between the driver and the resonator tube p1in is constant in scaling, i.e.

inin 11 pp =′ , (K.30)

the volume velocity at the interface U1in is obtained as:

( ) ( ) inzyzy

ininin 1

1

0

1

0

11 U

Zp

Zp

U ⋅==′

′=′ −ϕϕ

ϕϕ. (K.31)

Therefore, the substitution of Eq. (K.31) into Eq. (K.27) gives the acoustic pressure at a random position 1x′ in the scaled-down system:

( ) ( )11

1111

111 xxzyinzyinxx =−

′=′=⋅=′⋅′=′ pZUZUp ϕϕϕϕ . (K.32)

Similarly, the substitution of Eq. (K.31) into Eq. (K.28) and (K.29) gives the acoustic pressures at random positions 2x′ and 3x′ in the scaled-down system:

( ) ( )22

1211

211 xxzyinzyinxx =−

′=′=⋅=′⋅′=′ pZUZUp ϕϕϕϕ , (K.33)

( ) ( )331

1311

311 xxzyinzyinxx =−

′=′=⋅=′⋅′=′ pZUZUp ϕϕϕϕ . (K.34)

Therefore, the acoustic pressure remains unchanged locally at the same relative position, when p1in is constant in scaling, i.e. inin 11 pp =′ . In another word, the

statement 11 pp =′ is valid everywhere in the scaled-down system.

Nomenclature

Lower case

p Pressure [Pa]

x Position along sound propagation [m]

y Position perpendicular to sound propagation [m]

u xcomponent of velocity [m/s]

v y component of velocity [m/s]

s entropy per unit mass [J/(kg·K)]

a Sound speed [m/s]

pc Isobaric heat capacity per unit mass [J/(kg·K)]

sc Specific heat of the stack material [J/(kg·K)]

Vc Isochoric specific heat [J/(kg·K)]

0y Plate half-gap [m]

l Plate half-thickness [m]

e Energy per unit volume [J/m³]

f frequency [Hz]

vf Viscous Rott function —

κf Thermal Rott function —

h Enthalpy per unit mass [J/kg]

i Imaginary unit —

m Molecular weight [kg/mol]

r Radius of tube [m]

d Diameter of tube [m]

t time [s]

k Wave number [m¯¹]

),( µτ bf Function given by Eq. (F.21) —

Nomenclature 255

),( µτ bg Function given by Eq. (F.22) —

Upper case

A Area [m²]

B Function given by Eq. (3.3.9) —

D Function given by Eq. (3.3.10) or complex

coefficients given by Eq. (F.38a) to (F.38d) —

C Coefficients of the solution of second order

differential equation —

T Temperature [K]

Vv

Velocity [m/s]

E& Total energy flow [J/s]

W& Acoustic power flow [J/s]

Q& Heat flow [J/s]

K Thermal conductivity [W/(K·m)]

L length [m]

U Volumetric velocity [m³/s]

Re Real part of a complex function or number —

Im Imaginary part of a complex function or number —

I current [A]

V Voltage or volume [V] or [m³]

Z Acoustic impedance [Pa·s/m³]

F Force exerted on piston of driver [N]

0R Low-Reynolds-number-limit flow resistance of

regenerator at ambient temperature [Pa·s/m³]

0C Basic compliance of regenerator [m³/Pa]

256 Nomenclature

Lower case Greek

β Thermal expansion coefficient [K¯¹]

γ Ratio, isobaric to isochoric specific heats —

ρ density [kg/m³]

µ Dynamic viscosity [Pa·s]

ξ the second viscosity [Pa·s]

ν Kinematic viscosity [m²/s]

sε Stack heat capacity ratio —

κδ Fluid’s thermal penetration depth [m]

vδ Fluid’s viscous penetration depth [m]

sδ Solid’s thermal penetration depth [m]

δ Wire diameter of stainless steel wire screen [m]

λ Wave length [m]

D Radian wavelength [m]

η efficiency —

κ Thermal diffusity [m²/s]

φ Phase angle —

ϕ Scaling factor —

σ Prandtl number —

ω Angular frequency [rad/s]

α Root of an equation —

θ Real coefficients given by Eq. (F.37a-d) —

τ Temperature ratio CH TT / —

ψ Porosity —

Upper case Greek

Γ Normalized temperature gradient —

Π perimeter [m]

Nomenclature 257

Σ Viscous stress tensor [N/m²]

ℜ Specific gas constant [J/(kg·K)]

univℜ Universal gas constant [J/(mol·K)]

Θ Real coefficients given by Eq. (4.3.53), (4.3.54),

(4.3.45) and (4.3.58) —

Sub- and superscripts

crit Critical —

m Mean —

s Solid or standing —

κ thermal —

v viscous —

A amplitude —

h hydraulic —

H hot —

C cold —

1 First-order —

2 Second-order —

Others

~ Take conjugation —

‹ › Take space averaging —

¯ ¯ ¯ Take time averaging —

′ Scaled-down coordinate or y direction of solid —

→ vectors —

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[81] Y.Ueda, T.Biwa, U.Mizutani, and T.Yazaki, “Experimental studies of a thermoacoustic Stirling prime mover and its application to a cooler”, J. Acoust. Soc. Am. 115 (3), 1134-1141 (2004).

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[84] Paul Aben, “High-Amplitude thermoacoustic flow interacting with solid boundaries”, Ph.D. dissertation, Department of Applied Physics, Eindhoven University of Technology, (2010).

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[88] M.E.H.Tijani, S.Spoelstra, and P.W.Bach, “Thermal-relaxation dissipation in thermoacoustic systems”, J. Applied Acoustics, 65, 1-13 (2004)

Summary Thermoacoustics, as the word spelling indicates, is an interdisciplinary field in physics. Both acoustics and thermodynamics are involved in the description of this interesting phenomenon. When a solid wall is present in direction of the wave vector of an acoustic field, the interaction between the wall and the acoustic wave generates a transfer of heat from one location of the wall to another. This mechanism can be used to drive so called standing- or traveling-wave type of refrigerators and heat pumps. This is possible provided that instead of a solid wall a heat transfer medium which essentially is a porous plug, is applied to increase the effective surface area for the heat transport. Such a porous plug is known as stack for standing wave type of apparatus, and regenerator in the case of traveling wave type of apparatus. When the mechanism is used in the reverse way - an applied temperature gradient along a wall - , an acoustic wave is generated spontaneousley by the interaction between the sound wave and a solid. This is possible provided that the temperature gradient exceeds a certain critical value, and by properly designing a resonator and stack geometry to amplify one specific frequency. With a mechanic-electric converter a so called, prime mover power generator can be made. When a regenerator, which has a much smaller pore size than the stack, is employed in an acoustic network, the Stirling cycle can be realized. The so-called traveling-wave type refrigerators or engines are machines that use thermoacoustic effects to realize the Stirling cycle. The advantages of thermoacoustic machines over conventional refrigerators and engines, such as no moving parts, environment-friendly, and high reliability, make thermoacoustic apparatus of interest for future applications. Much work has been done on larger-scale (1 - 25 meter) thermoacoustic machines in the past decades. With the growing demand for small scale cooling devices in for instance laptop computers, mobile phones, and satellites, development of miniature refrigerators draws more and more attention. This PhD work is therefore dedicated to the analysis of miniaturization of thermoacoustic refrigerators. Both types: standing-wave and traveling-wave are investigated. An introduction and historical review on thermoacoustics is given in chapter 1. In chapter 2, the linear thermoacoustic theory is reviewed. The main content of chapter 3 is about modeling of standing-wave systems and validation of the modeling by investigation of a standing-wave type apparatus, which is similar to the so-called “TAC” (thermoacoustic couple). At the end of chapter 3, the measurements are analyzed and compared with computations based on proposed modeling.

Summary 265

Chapter 4 is devoted to modeling traveling-wave systems and model validation by experiments. An analytical model for a complete traveling-wave system of refrigerator is developed. After that, the optimization of regenerator material is considered. Various regenerator materials: metal honeycomb and ceramic honeycomb with different channel shapes, stainless steel wire screens in different hydraulic radii and porosities, were applied in a coaxial traveling-wave engine to characterize their performance. The metal honeycombs gave partially because of their larger cell size a poor performance, the ceramic honeycoms have an improved performance, and the best performance was measured with the stainless steel wire screens. The measurements of ceramic honeycombs and stainless steel wire screens showed that there is an optimum value for the dimensionless hydraulic radius (the ratio of hydraulic radius to a reference thermal penetration depth), which is around 0.3 for stainless steel wire screen regenerators and 0.16-0.2 for square cell honeycombs. The efficiency goes up with increasing porosity, to a maximum value above which the heat capacity of the solid becomes the limiting factor. The conclusion for stainless steel wire screen regenerators is also in agreement with the theoretical prediction by using the proposed analytical model. A full traveling-wave refrigerator driven by a mechanical compressor was designed and built. This system is again aiming at the validation of the model with measurements. The theoretical computation showed an acceptable agreement with measurements. In chapter 5, the analytical models for standing-wave systems (developed in chapter 3) and traveling-wave systems (developed in chapter 4) are utilized to characterize the performance in scaling down. By inserting three different scaling factors: xϕ , yϕ , zϕ , into the analytical models, the cooling power and efficiency

of scaling down systems are obtained. The results show that the scaling behaviour of a standing-wave system is the same as that of a traveling-wave system. The cooling power in the scaled-down system consists of two groups of energy flow scaling with different factors: one group of energy flow scales with a

factor 1)( −zyϕϕ , whereas the conduction loss scales as 1)( −

zyx ϕϕϕ . Apparently, the

conduction loss term decreases less than the other one and results in a reduced cooling power. So the efficiency decreases rapidly in scaling down. The term, which is the product of the scaling factor and the original ratio of energy losses due to thermal conduction to acoustic work, causes the reduction of efficiency after scaling down. The unified scaling behaviour of standing-wave and traveling-wave systems points out that the thermal conduction loss will finally dominate the losses and becomes the limitation factor for scaling down. We hope that the future design of mini-thermoacoustic-machine will benefit from this finding to reduce the conduction loss, finally to develop smaller (< 5 cm) and more powerful thermoacoustic refrigerators.

Samenvatting Thermoakoestiek is zoals het woord al aangeeft een interdisciplinair veld van onderzoek. Zowel akoestiek als thermodynamica spelen een rol in dit gecompliceerde en interessante fenomeen. Als een wand aanwezig is in een akoestisch geluidsveld treedt tussen de wand en het gas waar het geluidsveld heerst een thermisch warmtepompproces op, waardoor warmte wordt verplaatst van een plaats in de wand naar de andere. Met dit warmtetransportmechanisme kan een staande of lopende golf type koelmachine of wamtepomp worden aangedreven. Dit is mogelijk door gebruik te maken van een resonator. De randvoorwaarde daarbij is dat voldoende wandoppervlak aanwezig is en dit wordt bereikt door de wand uit te voeren als een poreuze plug. Deze poreuze plug wordt ook wel een stack of regenerator genoemd, al naar gelang de poriegrootte en het type thermoakoestisch apparaat. Als het mechanisme wordt omgekeerd - dus een temperatuurgradient aangebracht langs een wand - kan een zwak akoestische golf in een resonator versterkt worden, mits deze gradient een kritische waarde overschrijdt, dit wordt een prime mover geneomd. Met een mechanisch-electrische omvormer kan de akoestiek worden omgezet in electriciteit. Als een regenerator met een kleine poriegrootte wordt ingezet als deel van een akoestische netwerk in een resonator is het zelfs mogelijk om een Stirling cyclus te realizeren. Het is daarmee mogelijk om een zogeheten "lopende golf" koelmachine te realiseren of een warmtemotor. Al deze apparaten maken gebruik van een (benadering) van de Stirling cyclus. De voordelen van thermoakoestische machines boven conventionele koelmachines en motoren, zoals geen bewegende delen, milieuvriendelijk, en hoge betrouwbaarheid, maken thermoakoestische apparaten van interesse voor toekomstige toepassingen. In de afgelopen 20 jaar is veel onderzoek uitgevoerd aan grootschalige thermoakoestische machines met lengteschalen van 2 tot 25 meter. Met de groeiende vraag naar koelers in kleinschalige apparaten zoals satellieten, mobiele telefoons, of laptop computers, komt er ook meer interesse voor de ontwikkeling van miniatuur koelers waarbij thermoakoestiek een mogelijke kandidaat kan zijn. Dit promotieonderzoek is daarom gericht op de analyse van miniaturisatie van thermoakoestische koelapparaten. Zowel staande golf als lopende golf types zijn onderzocht. Een inleiding en historisch overzicht van thermoakoestiek wordt gegeven in hoofdstuk 1. In hoofdstuk 2 wordt de lineaire theorie van thermoakoestiek besproken. Dit vormt de noodzakelijke basis die nodig is om later in de hoofdstukken 3, 4 en 5 de schaling te onderzoeken. Hoofdstuk 3 handelt over het modelleren van staande golf systemen en het valideren van deze modellen met een experimentele opstelling die bekend staat als een thermoakoestisch koppel. In het

Samenvatting 267

laatste gedeelte van hoofdstuk 3 worden de meetresultaten geanalyseerd en vergeleken met modelberekeningen. Hoofdstuk 4 gaat in op het modelleren van lopende golf systemen en model validatie met experimenten. Een analytisch akoestisch model voor een complete lopende golf machine is ontwikkeld. Voorts is de optimalisatie van regenerator materiaal beschouwd. Verschillende honingraat regenerator materialen: metaal, keramisch en met verschillende poriegrootte, alsook RVS metaalgaas met verschillende hydraulische radius en porositeit zijn toegepast in een co-axiale lopende golf machine om hun prestatie te karakteriseren. De metalen honingraten gaven voornamelijk door hun te grote celgrootte de slechtste prestatie, de keramische honingraten een gemiddelde prestatie en de beste prestatie wordt geleverd door de metaalgaas regeneratoren. Uit de metingen is naar voren gekomen dat voor zowel honingraat als gaasmateriaal er een optimale waarde is van de hydraulische radius, welk 0.3 is voor gaasmateriaal en 0.16-0.2 voor honingraten met vierkante celdoorsnede. Verder is gebleken dat de efficientie oploopt bij toenemende porositeit tot een maximum dat wordt bereikt bij een porositeit van ongeveer 90%, waarna de efficientie sterk afneemt vanwege de afnemende warmtecapaciteit van het regeneratormateriaal in de limiet van een hoog poreuze regenerator. Het analytische model laat een goede overeenkomst zien met de resultaten van de metingen. Een lopende golf koelmachine die wordt aangedreven door een mechanische compressor is ontworpen en gebouwd en metingen aan dit systeem zijn uitgevoerd. Het is gebleken dat de metingen in redelijke overeenstemming zijn met de modelvoorspellingen. De verschillen ontstaan door thermische verliezen die niet zijn meegenomen in het model. In hoofdstuk 5 zijn de analytische modellen voor staande en lopende golf systemen gebruikt om de prestaties van thermoakoestische koelmachines te karakteriseren voor miniaturisatie met schaalfactoren. Door een drietal schaalfactoren ϕx , ϕy , ϕz , (ϕ is de verhouding tussen het origineel en het miniatuurmodel) in de thermoakoestische vergelijkingen in te voeren kan het gedrag van koelvermogen en de efficientie van geschaalde systemen berekend worden. Het gedrag van lopende en staande golf systemen is identiek. Het koelvermogen in het geschaalde model bestaat uit twee groepen van energietermen met verschillende factoren.: een groep die het akoestische vermogen levert schaalt met de factor , 1/(ϕy ,ϕz), terwijl de andere groep van de warmtegeleiding schaalt met , ϕx/(ϕy ,ϕz). Bij een zuiver axiale schaling nemen de verliezen van de warmtegeleiding naar verhouding dus toe, met als gevolg een verminderd koelvermogen. De efficientie vermindert dus snel bij miniaturiseren. Het universele schaalgedrag van thermoakoestische systemen toont aan dat uiteindelijk de warmtegeleiding domineert en de limietfactor is bij miniaturiseren. Toekomstige ontwerpen van thermoakoestische koelers op kleine

268 Samenvatting

schaal (< 5 cm) zijn daarom alleen mogelijk als de warmtegeleidingsverliezen gereduceerd kunnen worden door andere regeneratormaterialen te kiezen.

Dankwoord I would like to thank all people who participated in this project and my colleagues whom I ever worked, and shared the enjoyable working atmosphere with. I am very grateful to my daily advisor and co-promotor, Jos Zeegers, who coached and supported me continuously during these five years of my PhD research. He also helped me much in the starting phase of the PhD period to settle quickly in Eindhoven. Furtermore I owe many thanks to my promotor prof. Marcel ter Brake who initiated, and steered this project, and my second promotor prof. Fons de Waele, both for all their expert guidance and support. Especially I like to mention here the vast amount of time, and efforts they have spent in the phase of the reading, correcting, and all support during the writing, of this thesis. It should be mentioned here that this project was only possible through the kind financial support of MicroNed, and this was made possible by Marcel's initiatives, as well as the possibility to extend my contract with 8 months to finalize the research work. This work would not have been possible without the large technical support team, who has realized all equipment. I am very thankful to our mechanical engineers Leo van Hout, Paul Niël, and Henny Manders, who designed, and developed the set-ups that were used in this research. Furthermore I have received much support from our electronics engineers Peter Helfferich and Freek van Uittert, who designed and built the electronics components often on day to day basis. Jørgen van der Veen kindly helped with the photographs of parts of the set-up. Much daily support in the actual manufacturing of the standing wave systems came from the faculty workshop in cooperation with the support from team leader Marius Bogers. I am thankfull to Henk van Helvoirt, Ginny Fransen, and Han den Dekker who did the actual manufacturing. Then the two large set-ups of the traveling wave systems have been built by the University GTD workshops. Many technicians were involved in this. I like to thank Mariëlle Dirks Smit who did the actual manufacturing via discharge machining of the copper heat exchangers, Lucien Cleven and Jeroen Baijens turned, machined, welded, and built the complex set-up. Jovita Moerel, and Jos van Kruijsdijk designed, and built the electronical assembly of the set-up. Hans Wijtvliet together with Freek van Uittert wrote all Labview codes to control the measurements. Erwin Dekkers performed the strength computations, this all in support by team leader Harrie de Laat. My special thanks go to Paul Aben who shared the office with me for four years, and helped me in translations of many letters I received in Dutch to English. I also

270 Dankwoord

want to thank my later office mate Christian Berendsen, who shared my room in the last year in the office in Cascade. It was an enjoyable time for me to have lunch, play poker games and chat with my former colleagues, Paul Aben, Wenqing Liang, Paul Niël, in the old coffee room during lunch break. For the stack materials we received much support from a number of people. I am very thankful to prof. Chris Sutcliffe, Kaj Berggreen and Adam Clare, from the Center of Materials of the School of Engineering of the University of Liverpool for the supply of the stainless steel and plastic honeycombs. Furthermore I like to thank John Wight of Corning USA, and Thierry Dannoux of Corning France for the production of the large number of ceramic samples with high cell density, which are in general not commercially available. Then I like to thank Stan Lam from the Metaalgaasweverij Dinxperlo for his help on supplying special wire gauze samples to be tested in the traveling wave set-up. The work and design of the coaxial thermally driven traveling wave system was possible thanks to the kind help of dr. Hassan Tijani, and I am very grateful for his advice. Furthermore I like to thank Elise Moers who helped me with the experiments on this engine. In 2007 I could visit the Chinese Academy of Sciences in Beijing for a period of three months. I am very thankful to dr. Wei Dai for his kind support, introducing and helping me to learn the numerical thermoacoustic simulation tools developed by him at CAS. I would like to thank all my colleagues in Mesoscopic Transport Phenomena group, whom I spent much joyful time with in playing bowling and dinners in prof. Mico Hirschberg’s house. I would like to express my thanks to our secretary, Brigitte van de Wijdeven, for taking care of all administrative details. On a personal level, I want to show appreciation and gratitude to my husband and my parents for their love and patience in these PhD years.

Curriculum Vitae

Yan Li was born in Liaoning, China, on February 14th 1977. After finishing her pre-university education in hometown (Huludao in Liaoning province) in 1995, she started her study, majoring in the speciality of aerospace engine in the department of jet propulsion in Beijing University of Aeronautics and Astronautics (BUAA, Beihang University, China) that same year. In the fourth year of her undergraduate study, the year of 1999, she worked on project “Numerical Investigation of Rotor/Stator Interaction Noise in an Aeroengine”. During the period of September 1999 to March 2002, she studied in field of aerospace propulsion theory and engineering in the department of jet propulsion in BUAA and obtained her master’s degree with thesis entitled “Sound Radiation Generated by Ducted Fan with Supersonic Blade Tip Speed”. From September 2003 till November 2005, she studied in McMaster University (Canada) in the department of mechanical engineering and obtained master of applied science with thesis entitled “Flow-Acoustics of T-Junctions: Effect of T-Junction Geometry”. From November 2005 till July 2010, she had been working as a PhD student at Eindhoven University of Technology (the Netherlands) in the department of applied physics. The project was mainly about scaling analysis of thermoacoustic refrigerators, which was sponsored by MicroNed, and performed under the supervision of prof.dr.ir. H.J.M.ter Brake, prof.dr.A.T.A.M.de Waele, and dr.ir.J.C.H.Zeegers. She had the defense on her thesis, entitled “Thermoacoustic Refrigerators: Experiments and Scaling analysis”, on the 27th of October 2011.