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Mathematical Aspects of Thermoacoustics Mathematical aspects of thermoacoustics Citation for published version (APA): Panhuis, in 't, P. H. M. W. (2009). Mathematical aspects of thermoacoustics. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR642908 DOI: 10.6100/IR642908 Document status and date: Published: 01/01/2009 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. 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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: 01. Oct. 2021 Mathematical Aspects of Thermoacoustics Copyright c 2009 by P.H.M.W. in ’t panhuis, Eindhoven, The Netherlands. All rights are reserved. No part of this publication may be reproduced, stored in a re- trieval 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 Jorrit van Rijt A catalogue record is available from the Eindhoven University of Technology Library ISBN 978-90-386-1862-3 NUR 919 Subject headings: thermoacoustics; acoustics; acoustic streaming; thermodynamics; per- turbation methods; numerical methods; boundary value problems; nonlinear analysis; shock waves. This research was financially supported by the Technology Foundation (STW), grant number ETTF.6668. Mathematical Aspects of Thermoacoustics 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 25 juni 2009 om 16.00 uur door Petrus Hendrikus Maria Wilhelmus in ’t panhuis geboren te Roermond Dit proefschrift is goedgekeurd door de promotor: prof.dr. J.J.M. Slot Copromotoren: dr. S.W. Rienstra en prof.dr. J. Molenaar To Ik Cuin Preface This project is part of a twin PhD program between the Departments of Mathemat- ics and Computer Science and Applied Physics and is sponsored by the Technology Foundation (STW), Royal Dutch Shell, the Energy Research Centre of the Netherlands (ECN), and Aster Thermoacoustics. I would like to express thanks to all people who participated in this project. First and foremost I would like to thank my mathematical supervisors dr. Sjoerd Rienstra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar for their expert guidance and stimulating support. I am also indebted to my physics colleagues Paul Aben, dr. Jos Zeegers, and prof.dr. Fons de Waele, who helped to broaden and deepen my understanding of the physics involved. I am also grateful for the many useful discussions I have had with the people from ECN, Aster Thermoacoustics, and Shell. My defense committee is formed by prof.dr. Anthony Atchley, prof.dr. Bendiks-Jan Boersma, and prof.dr. Mico Hirschberg, together with my supervisors dr. Sjoerd Rien- stra, prof.dr. Han Slot, and prof.dr. Jaap Molenaar. I would like to thank them for the time invested and their willingness to judge my work. I also want to thank prof.dr. Bob Mattheij for agreeing to be part of the extended defense committee. What made these four years especially enjoyable was the great working atmosphere within CASA and the Low Temperature group. My special thanks go out to my of- fice mate and partner-in-crime Erwin who started and finished his PhD (and Master) at the same time as I did. Many thanks also to all the current and former colleagues that I have had the pleasure to work with, in particular the PhD students and postdocs: Aga, Ali, Andriy, Bart, Berkan, Christina, Darcy, Davit, Dragan, Hans, Jurgen, Kakuba, Kamyar, Kundan, Laura, Marco, Maria ( 2), Mark, Matthias, Maxim, Michiel, Miguel, Mirela, Nico, Oleg, Patricio, Paul, Remko,× Remo, Roxana, Shruti, Sven, Tasnim, Temes- gen, Valeriu, Venkat, Wenqing, Yabin, Yan, Yixin, Yves, Zoran. I fondly think back to our daily lunches at the Kennispoort, the weekly poker games, the road-trips to Den- mark, the regular squash/tennis/football games, and the many nights in town that I have enjoyed with so many of you. Our two secretaries Mar`ese and Enna also deserve a special word of thanks, for making life of a PhD student so much easier by taking care of all administrative details. I am also thankful to the members of the football teams Pusphaira and Old Soccers, and hope they will have more success without me. On a more personal level, I would like to thank all my friends and family, for their continuous love and support. I especially want to show appreciation to my mom for her unbridled enthusiasm and my dad, who I wish could have been here today. I also want to thank my siblings Jos, Hellen, and Dorris and their significant others Marjanne, Joram, and Tonnie. Of course I should not forget to mention my little nephew Sep, who ii Preface is getting so big now. Last, but definitely not least, I would like to thank my girlfriend Jessey, to whom this thesis is dedicated, for her unlimited love and patience in these last few months. Peter in ’t panhuis Eindhoven, May 2009 Nomenclature General symbols and variables A [m2] cross-sectional area [m2] cross-section A 2 b [ms− ] specific body force field 1 c [m s− ] speed of sound 1 1 Cp [J kg− K− ] isobaric specific heat 1 1 Cs [J kg− K− ] specific heat of stack material 1 1 Cv [J kg− K− ] isochoric specific heat d [m] diameter f [Hz] frequency fν viscous Rott function fk thermal Rott function fs solid Rott function Fν viscous Arnott function Fk thermal Arnott function Fs solid Arnott function Fourier transform FG Green’s function 2 g [ms− ] gravitational acceleration H˙ [W] total power 1 h [J kg− ] specific enthalpy Im imaginary part 1 k [m− ] wave number 1 1 K [W K− m− ] thermal conductivity ℓ [m] displacement L [m] typical length 2 1 m˙ [kg m− s− ] time-averaged mass flux 2 1 M˙ [kg m− s− ] time-averaged volumetric mass flux 1 n˙ [mol s− ] molar flow rate P [W] power p [Pa] pressure pA [Pa] pressure oscillation amplitude pamb [Pa] ambient pressure iv Nomenclature Q˙ [W] heat flow per unit time 2 q [W m− ] heat flux r [m] radial coordinate Re realpart 1 1 Rspec [J kg− K− ] specific gas constant [m] radius R 1 1 s [J kg− K− ] specific entropy 1 S [J kg− ] entropy S surface Su [K] Sutherland’s constant t [s] time T [K] temperature 1 U [m s− ] typical fluid speed U˙ [W] internal energy 1 v =(u, v, w) [m s− ] velocity vector V [m3] volume W˙ [W] acoustic power x =(x, y, z) [m] spatial coordinate Z [N s m3] impedance 1 β [K− ] isobaric volumetric expansion coefficient Γ boundary, interface δ [m] penetration depth 1 ǫ [J kg− ] specific internal energy λ [m] wave length µ [Pa s] dynamic (shear) viscosity ζ [Pa s] second viscosity 3 ρ [kg m− ] density 2 τ [N m− ] viscous stress tensor θ [rad] angular coordinate 1 ω [rad s− ] angular frequency of the acoustic oscillations Dimensionless numbers A amplitude Br blockage ratio COP coefficient of performance COPR relative coefficient of performance COPC Carnot coefficient of performance Dr drive ratio Fr Froude number Ma acoustic Mach number NL Lautrec number gas Ns Lautrec number solid Pr Prandtl number R reflection coefficient Nomenclature v Sk Strouhal number based on δk Wo Womersley number Wζ second Womersley number β coefficient of nonlinearity γ ratio specific heats δ coefficient of stack dissipation ∆ deviation from resonance η coefficient of weak nonlinearity η efficiency ηR relative efficiency ηC Carnot efficiency ε aspect ratio ε driver Mach number εs stack heat capacity ratio κ Helmholtz number σ ratio thermal conductivities φ porosity µ dimensionless viscosity Sub- and superscripts and special operators a˜ dimensionless a time averaging a transverse averaging ha˙ i per unit time a∗ complex conjugate are f reference value a+ top plate a− bottom plate a0 steady zeroth order a1 first harmonic a2,0 steady second order (streaming) a2,2 second harmonic aC cold aH hot ag gas, fluid ak thermal aL thermal ap isobaric aR right as solid, source, stack center at outer aτ transverse components av isochoric aν viscous vi Nomenclature Contents Preface i Nomenclature iii 1 Introduction 1 1.1 Ahistoricalperspective .
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