Air Engines -...

Air Engines

Downloaded From: http://ebooks.asmedigitalcollection.asme.org/ on 07/16/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use


Air Engines

The History, Science, and Reality of the PerfectEngine

Theodor Finkelstein

and

Allan J Organ

The American Society of Mechanical Engineers, New York


Published with corrections in 2009 by ASME Press. First Published 2001 in the United Kingdom by Professional Engineering Publishing Limited and in the United States by ASME Press. © 2001, Theodor Finkelstein and Allan J Organ © 2009, ASME, 3 Park Avenue, New York, NY 10016, USA (www.asme.org)

All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. INFORMATION CONTAINED IN THIS WORK HAS BEEN OBTAINED BY THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES BELIEVED TO BE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. ASME shall not be responsible for statements or opinions advanced in papers or . . . printed in its publications (B7.1.3). Statement from the Bylaws. For authorization to photocopy material for internal or personal use under those circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, tel: 978-750-8400, www.copyright.com. Requests for special permission or bulk reproduction should be addressed to the ASME Publishing Department, or submitted online at: http://www.asme.org/Publications/Books/Administration/Permissions.cfm ISBN-13: 978-07918-0171-0 ISBN-10: 0-7918-0171-3 ASME Order No. 801713 A CIP Catalogue record for this book is available from the British Library.


___________________________________

Cover illustration

___________________________________

Those familiar with the first printing of Air Engines will recall its distinctive jacket

illustration. The 6-cylinder concept under development by Dipl.-Ing Peter Feulner,

together with accompanying background information, was a valuable complement

to the material between the covers. The original art-work was not preserved,

however, and to the time-scale of this re-print it has not been possible to arrange

for re-use of the illustration.

The replacement jacket design makes an important symbolic statement: it is a

graphic generalization of the conversion algebra defining the equivalence between

thermodynamic volume ratio and kinematic displacement ratio , and between

thermodynamic phase angle and corresponding kinematic phase angle . Equally

it defines an essential condition for a coaxial ('beta') engine to be

thermodynamically equivalent to a given 2-cylinder ('alpha') engine - and vice-

versa. (The text has more on this radical unifying insight at Chap. 6.)

Ted Finkelstein casually knocked off the crucial algebra in the course of his tour de

force opus 'Optimization of phase angle and volume ratio for Stirling engines'.

With the companion paper 'Generalized thermodynamic analysis of Stirling

Engines', this material inaugurated the modern analytical base of Stirling engine

technology. To this extent, the jacket illustration symbolizes a defining step in the

progression from the air engine of the Industrial Revolution to the Stirling engine

of today - and from chapters 1 - 4 of the present account to the chapters which

follow.

The reader encountering Air Engines for the first time is encouraged to become

apprised of the Feulner multi-cylinder concept. A computer-generated solid-model

image and an informative link can be found on the Welcome page of http://

web.me.com/allan.j.o/Communicable_Insight


Related titles

For a full range of titles available from ASME International, please visit theASME Press website at http://www.asme.orgFor a full range of titles available from ASME International, please visit theASME Press website at http://www.asme.org/Publications/Books/



About the authorsTheodor Finkelstein studied mechanical engineering at Imperial College,London, obtaining his BSc with first-class honours. A Diploma of ImperialCollege followed. Working under Professor H Heywood he was awarded hisPhD in December 1952 for a thesis entitled Theory of Air Cycles with SpecialReference to the Stirling Cycle. He is a registered Professional Engineer.

He has worked mainly in the research departments of major corporations,but has also held academic appointments at the Universities of Wisconsin, SanFernando Valley State College, University of Calgary, and University ofCalifornia.

He pioneered the development of mathematical models of the Stirling cycleand the use of electronic computers, both analogue and digital, for obtainingsolutions. His analytical optimization of the Schmidt cycle model has neverbeen superseded.

Inventions, some patented, have included the internal heat supply andtransmission (1958) in the free piston machine (1960), balancedcompounding (1975), and the introduction of the Delta class of Stirlingengines (1998). He has contributed an average of one scientific paper peryear over a period of 50 years to the Stirling literature. In 1978, he and StigCarlqvist founded Stirling Associates International which continues to carryout consulting work in this field.

His later activities have concentrated exclusively on industrial Stirlingengine development. He lives with his wife, Hanna, in Beverly Hills,California.

Contact details:PO Box 643, Beverly Hills, California, 90213, USAPhone: (USA) 310 472-2176 Fax: (USA) 310 476-2021 e-mail: [email protected]

Allan J Organ studied for his BSc in the Department of MechanicalEngineering at Birmingham University, where a final year project provided anirresistible introduction to the Stirling engine. Study of the regenerator beganat Toronto under Professor F C Hooper and led to the degree of MASc in 1964.He returned to Birmingham to take up a research position studying metal-forming at high rates of strain under Professor S A Tobias, for which he gaineda PhD in 1968. Metal-forming provided the bread and butter until 1970, whena British Council Visiting Professorship in Brazil gave the opportunity todivide the research interest between plasticity and Stirling engines.


Air Enginesx

The return from Brazil was to a position at King’s College London allowingfree rein of research activity, and it is not necessary to say which way thechoice went! A privilege of a further move to Cambridge in 1978 was alsocomplete freedom of research interest. The die being well-and-truly cast,attention turned to making Stirling engine analysis less restricted to a specificengine and more relevant to the genre as a whole. A first book, published byCambridge University Press in 1992, approached this task from the standpointof Dynamic Similarity, but had gone to press before the full benefits of thispowerful tool had been completely perceived and applied to best advantage.Attention reverted to the regenerator problem, which finally yielded in 1995.This afforded an opportunity for drawing together gas circuit design andregenerator theory using the full power of Energetic Similarity. The text waspublished in 1997 by Mechanical Engineering Publications (MEP).

He was awarded the post-doctoral degrees of DEng from the University ofBirmingham in 1993, and ScD by Cambridge in 2000. In addition to the twobooks, he has contributed 41 scientific and 10 technical papers to the literatureon Stirling engines and the regenerator, and 15 papers to metal-formingliterature.

He is a Fellow of the Institution of Mechanical Engineers and serves on theEditorial Board of the Journal of Mechanical Engineering Science(Proceedings Part C). He lives in the village of Dry Drayton near Cambridge.

Contact details: 32 Pettitt’s Lane, Dry Drayton, Cambridge CB3 8BTPhone: (UK) (0)1954 781815Fax: (UK) (0)1954 781253 e-mail: [email protected]


ContentsDedication xv

Foreword xvii

Preface xix

Notation xxiii

Chapter 1 Air engines 11.1 Introduction 11.2 Classification 11.3 The regenerator 21.4 Furnace gas engines 61.5 Ericsson engines 11

Chapter 2 The Stirling engine 212.1 The invention 212.2 Working principle 232.3 The patent 242.4 Degenerate Stirling engines – double-cylinder types 32

Chapter 3 Later single cylinder engines 413.1 German air engines 413.2 Heinrici, Bailey, and other variants 443.3 The Rider engine 49

Chapter 4 Philips engines 554.1 The rediscovery 554.2 Double-acting types 584.3 Future possibilities 604.4 Acknowledgements to the original four articles 64

Chapter 5 ‘Modern knowledge’ … and all that 655.1 Now, where were we? 655.2 Pre-Dark Ages 665.3 End of the Dark Ages 695.4 The ‘regenerator problem’ 705.5 A first physical model 71


5.6 Back to the (Philips) Laboratory 745.6.1 An early approach to regenerator design 745.6.2 Rebirth of the multi-cylinder concept 78

5.7 The SMF-Kroon engine 785.8 Some basic concepts 81

5.8.1 The ‘ideal’ gas 815.8.2 Reynolds number 815.8.3 Number of transfer units, NTU 81

5.9 Schumann’s solution to the initial blow 825.10 Interim summary 84

Chapter 6 Reassessment 876.1 Status quo 876.2 What is the Stirling engine design problem? 876.3 Fundamentals of thermal design 886.4 Equivalence conditions 906.5 Reappraisal of the 1818 engine 94

6.5.1 Basic dimensional data 946.5.2 Operating conditions 956.5.3 Kinematics and volume variations 976.5.4 Temperature ratio 100

6.6 Some essential basics 1016.6.1 Significance of temperature ratio 1016.6.2 Dead space ratio 1016.6.3 ‘Extra’ dead space 101

6.7 Summary of fundamentals to date 102

Chapter 7 Post-revival 1057.1 Synopsis 1057.2 The rhombic drive engines 1067.3 Sealing 1087.4 Multi-cylinder rhombic engines 1117.5 A widening of involvement 1127.6 Back to thermodynamic design – via an anomaly 112

Chapter 8 The ‘regenerator problem’ 1178.1 What regenerator problem? 1178.2 Early part-solutions 1198.3 The makings of cycle analysis 1208.4 The advent of computer simulation 123

Air Enginesxii


8.5 A first fluid particle trajectory map 1248.6 Lateral thinking 1258.7 Air versus helium versus hydrogen 127

Chapter 9 Two decades of optimism 1319.1 Summary 1319.2 The free-piston engine 1319.3 The fluidyne 1329.4 The Low ΔT variant 1349.5 The era of the computer 1379.6 Further advances by Philips 1399.7 Two UK initiatives 1419.8 More on similarity and scaling 143

Chapter 10 Thermodynamic design 14910.1 The thermodynamic design problem 14910.2 The task in perspective 14910.3 Pressure and flowrate 15210.4 Solution of the regenerator problem 15610.5 Gas circuit design by scaling 15810.6 Similarity principles and engine design 15910.7 A very un-scale model 16510.8 The study of the 1818 engine continued 166

Chapter 11 Completing the picture 16911.1 Regenerator analysis further simplified 16911.2 Some background 17011.3 Flush ratio 17011.4 Algebraic development 172

11.4.1 Temperature profile 17211.4.2 The ‘flush’ phase in perspective 17411.4.3 Temperature recovery ratio 17611.4.4 Matrix temperature swing 176

11.5 Common denominator for losses 17711.5.1 Heat transfer and flow friction correlations 17711.5.2 Heat transfer loss 179

11.6 Hydrodynamic pumping loss 17911.7 Matrix temperature variation again 18111.8 Optimum NTU 18111.9 Inference of NTU actually achieved 185

Contents xiii


11.9.1 From temperature recovery ratio, ηT 18511.9.2 NTU from mean cycle Nre 185

11.10 Evaluation of optimum NTU 18611.11 Implications 19111.12 Complete temperature solutions 19211.13 Thermodynamic study of the 1818 engine (continued) 19411.14 Interim deductions 197

Appendix to Chapter 11 198

Chapter 12 By intuition, or by design? 19912.1 An anomaly 19912.2 The 1818 engine and the regenerator 20012.3 Stirling’s regenerator design 205

12.3.1 A suitable expression for pumping loss 20512.3.2 The temperature solutions 207

12.4 The alternative 21012.5 Résumé 211

Chapter 13 ….. and the heyday to come 21313.1 Full circle? 21313.2 An air engine to challenge hydrogen and helium – the

Viebach CHP unit 21313.3 A bold initiative from New Zealand 21413.4 Future of the 1818 concept 21713.5 A gas-powered, cordless hair drier? 21813.6 A shot in the dark 225

Chapter 14 In praise of Robert Stirling 22914.1 Citation 22914.2 How might the unique genius of Robert Stirling by

celebrated? 23014.3 A task completed - or barely begun 230

Appendix Literary output of Theodor Finkelstein 233

References 237

Air Enginesxiv


This book is dedicated to the genius of Robert Stirling



Foreword

The title Air Engines is that of the classic series of articles that form the firstfour chapters of the book. But it is evident on reading that there are otherreasons for preferring this over the more obvious Hot-air Engines. The mostimportant is the dramatically improved performance of the modern, air-charged, external combustion engine relative to that of the hot-air toy or the19th century industrial engine. Indeed, the air engine promises to challenge thespecific power and efficiency of hydrogen- and helium-charged types.

Hot air (of the metaphorical variety) is an unfortunate feature of much of thegeneral literature on this particular technology, which carries a disproportionateburden of ideas that have never been tested, or of which test results have – forsome reason – been withheld. Be assured that this book contains no such hot-air. The fact is guaranteed by the well-known names of the authors, Allan JOrgan and Theodor Finkelstein, who together can account for nigh-on acentury of research and design work in this field. I spent two summers atCambridge as a student of Dr Organ before being truly satisfied with my owninsight. Perhaps I can offer the excuse that this book was not available at thattime.

The treatment opens with a classification of air engines as a basis for anaccount of their conception and historical evolution. Inevitably, attention soonfocusses on Robert Stirling’s remarkable double invention – the thermalregenerator and the famous engine of 1816 – as being the only viable basis fora modern prime mover. The story of the air engine is essentially that of thedevelopment of modern thermodynamics from its origins in Caloric theory.Finkelstein masterfully combines the two aspects so that, when Chapter 4concludes with the re-birth of the modern Stirling engine at the hands ofPhilips, you have a comprehensive insight of the thermodynamic andmechanical design aspects without use of a single mathematical symbol!

Structurally the book may be unique in consisting of two sections, the earlychapters written 40 years before the rest. It is a tribute to his handling of thematerial that Dr Finkelstein’s contribution is as pertinent now as when written,and appears in essentially unmodified form. Only a brief regression is calledfor in the bridging chapter to draw attention to early developments that havecome to light since 1959. Subsequent chapters bring the account up-to-the-minute, continuing in the process to highlight the relationship betweentechnical progress and the evolution of theoretical understanding.

Superimposed on the chronological account is an explanation ofthermodynamic design tools which can be applied to your own engine. There


Air Enginesxviii

are details of air engines currently under development for applicationsespecially suited to their particular characteristics.

The book offers an easy-to-read, coherent, and scientific account based onthe authors’ pioneering development of (dimensionless) scaling parameters.This amounts to the distillation of practical and theoretical know how andexperience that has previously appeared over many years in high-level papersand books and which is confirmed by the latest research results.

The slow progress of the air engine towards widespread commercialexploitation is due, more than anything else, to a lack of communicable insight.An evident priority of the book is to make good this lack. Thus it provides along-overdue basis for communication among air engine experts and hobbyists,serving for practical design or as a reference text in research and academicwork. A ‘forensic’ enquiry into the original engine of 1818 opens up a newdimension in the technical history of air engines.

I am deeply impressed by the work of the authors. But my sympathy is alsowith a vision urgently required by my generation, and upon which this topicbears: there is a pressing requirement for an environmental friendly, renewableglobal energy conversion system having limited risk. My current view is thatthe air engine has a key role in CHP (Combined Heat and Power) systems,including solar dish and biomass-fuelled types. The technology is now in place,but successful implementation awaits the formation of a critical mass offunding, manpower and, above all, of communicable insight. This book takes astep in this direction.

If you think the air engine just functions by moving a gas between a hot anda cold space you will find there is more to explore than you ever expectedbefore opening this book. Whatever your previous technical background youwill watch an air engine with greater fascination the next time around – even ifit is, indeed, a toy of the ‘hot-air’ variety.

Dipl.-Ing. Peter MäckelKassel

[email protected]


Preface to the original Air Engines

In these articles a survey of the development of air engines is given, startingwith the primitive ‘furnace gas engine’, built at the beginning of the nineteenthcentury, and leading up to the modern single-stage Philips’ air liquefiers. Thedifferent types of air engines are classified into three groups: open-cycle internalcombustion, open-cycle external combustion, and closed-cycle engines,invented respectively by Cayley in 1807, Ericsson in 1840, and Stirling in 1816.The principle of operation for these is explained and typical examples of eachare given. Engines ranging from a small toy motor to a monstrosity with four14 ft diameter cylinders, running at 9 r/min, are described.

To the writer’s knowledge this is the first time that a comprehensive set ofillustrations and descriptions of air engines has been assembled – in fact, apicture of Stirling’s original engine of 1816 has previously appeared in print ononly one occasion. Recent new developments are dealt with and someinteresting proposals for future applications of these basic scientific principlesare made.

Theodor Finkelstein1959

Editorial NoteThe aim has been to retain Air Engines in a form as close to the original aspossible. Nevertheless, some minor departures will be noted. There has, forexample, been difficulty in securing copyright permission for certain of theoriginal diagrams. Similar figures from known sources have been substitutedwhere available, and any explanatory notes adjusted accordingly. The note‘source unknown’ against a figure indicates that no substitute could be found.(Should the reader know the origin of any of these figures the authors wouldwelcome notification so that due acknowledgement can be included in anyfuture edition.)

The original account did not have access to the illustration of Cayley’s airengine of 1807. This has since come to light, and now complements thediagram of the ‘gradual combustion’ engine which earlier deputized for it. Moreis now known about Robert Stirling’s original patent specifications (of whichthere was a Scottish and an English version), calling for adjustment to a fewwords of the narrative. A change to the Harvard system of referencing (Author,


Air Enginesxx

date) has required reallocation of those titles originally grouped under a singlenumerical (65) reference.

Otherwise, the text is as originally published. It is now reproduced by kindpermission of the Editor of The Engineer.


Preface to the joint editionWhy another book on Stirling engines?

There are, no doubt, authors who can cite one clear motive. Most of us,however, find book writing a soul-wrenching experience, and need to beinduced by a combination of incentives.

First, I had for decades been trying to persuade Ted Finkelstein to bring hisunique series of articles in The Engineer of 1959 up to date, put it betweencovers, and thereby perhaps achieve a measure of long-overdue credit for hisimaginative and independent contributions to the field. Persuasion failed, buthe kindly encouraged me tackle the task on our joint behalf. Whether the resulthas the charisma it would have carried at the hands of the original masterremains to be judged, but at least there is a result.

Second, an intriguing mystery offers itself for investigation: accounts of themodern technological development of the Stirling engine contain repeatedallusion to the crucial role played by the science of thermodynamics, heattransfer, and fluid flow. Perversely, details have been consistently withheld.Even Hargreaves’ otherwise definitive account* is studiously silent on thispoint. A potential legacy of thermodynamic design guidelines is missing,together with any accompanying insight this might have afforded into thethermodynamic personality of the Stirling engine. A narrative chronology oftwentieth-century developments would shed little light on this unsatisfactorysituation which, therefore, provides the considerably more interesting exerciseof reading between the lines. This being the case, the opportunity is taken ofprobing a further, perplexing aspect of recent history, namely, the role ofregenerator theory. The nineteenth century saw the regenerator itself – thatcrucial, central component – omitted or relegated to a token function. Thetwentieth century produced a huge corpus of analytical work on the regeneratormassively outweighing Stirling cycle analysis per se, and continued the earliertradition by assiduously ignoring it for engine simulation and design.

Examining these events permits reconstruction of the thermodynamic designmethodology that might have been. This in turn offers the third incentive (ifone were needed) for undertaking the present account: to reconstruct thethermodynamic personality of the celebrated engine of 1818, and thus toexamine claims for the 2 h.p. output.

Finally, there is the matter of air engines – not hydrogen- or helium-chargedengines, but air engines – their heyday past, and their heyday to come.

*Hargreaves, C. M. (1991) The Philips Stirling Engine Elsevier, Amsterdam.


Air Enginesxxii

Gathering together the various threads leads to the conclusion that the air- ornitrogen-charged engine has been wrongly set aside in the rush to exploit theapparent superiority of machines charged with hydrogen or helium. A furtheraspect of Stirling’s original vision which has gone by the board is its elegantsimplicity. Reverting to the earlier priorities, but this time with the aid ofappropriate thermodynamic design tools, has shown how an air- (or, equally,nitrogen-) charged engine can out-perform the helium-based counterpart in therapidly evolving technology of domestic Combined Heat and Power (CHP) – arole to which the operating characteristics of the Stirling engine are uniquelysuited. Apparently the economics are well-matched also.

Some two centuries later, the commercial future appears to lie, after all, withan engine little changed from the original concept. Hence retention for thisbook of the title of the pioneering articles which form its foundation andinspiration: Air Engines.

Allan J Organ


Notation(Note: The notation for the kinematics of Stirling’s 1818 engine used in Table6.1 is defined in Fig. 6.5, and is not duplicated here.)

a Acoustic speed √{γRT} m/sa, b, c, d Dimensionless parameters of heat transfer and

flow friction correlationsa/f air-to-fuel ratio (by mass)Aff Free-flow cross-sectional area m2

b/s bore-to-stroke ratioc specific heat J/kg Kcp, cv specific heat at constant pressure and at

constant volume respectively J/kg Kcw specific heat of matrix material J/kg KCf friction factor ratio of shear stress at wall

to ½ρu2

dFL readjustment interval of fluid temperature profile, defined in text m

dw diameter of individual wire of regenerator gauze m

dwarp, dweft diameters of wires in warp and weft directions respectively m

D operator − substantial derivative (D/dt = ∂/∂t + u∂/∂x) s-1

D internal diameter of cylinder mf cyclic frequency s-1

fNT function of temperature ratio, NT, defined in text

fNTCR function of ¶v, γ, NTCR defined in texth coefficient of convective heat transfer W/m2 KHb enthalpy per blow based on temperature

difference ΔT Jj Colburn j-factor Nst Npr

2/3

k thermal conductivity W/m KLr passage length in regenerator packing mm variable mass kgm' mass rate kg/smw mesh number of gauze (wires/m) m-1


Air Enginesxxiv

mwarp, mweft mesh number in warp or weft direction, inverse of pitch distance m-1

M mass of working fluid taking part in cycle kgnv number of valves/cylinder of four-cycle

internal combustion engineNB Beale number, Wbrake/prefVsw = powerbrake/prefVswfNF Fourier modulus, αw/ωdw

2

NFL flush ratio: ratio of mass of fluid throughregenerator per blow to mass of fluid contained in matrix

NMA characteristic Mach number, or dimensionless speed, ωVsw

1/3/√RTrefNotto powerotto/prefVswfNpr Prandtl number μcp/kNre Reynolds number, 4ρ|u|rh/μNst Stanton number, h/ρ|u|cpNT characteristic temperature ratio, TE/TCNTCR thermal capacity ratio ρwcw/ρc for

incompressible case or ρwcwTref/pref for compressible case. Respective numerical values differ by the factor (γ −1)/γ

NTF Finkelstein number, Wcomputed /prefVsw = powercomputed/prefVswf

NTr regenerator heat transfer scaling parameter (Lr/rhr)3/2NSG

-½

NSG characteristic Stirling number or dimensionless pressure pref/ωμref

NTU number of transfer units, NstLreg/rhp absolute pressure Papw, perim. wetted perimeter mq' heat rate WQ' volume flowrate m3/sQr heat stored in regenerator per cycle Jr compression ratio Vmax/Vminrh hydraulic radius, free-flow area/wetted

perimeter mR gas constant for specified gas J/kg KS stroke (of piston or displacer) mS entropy J/K t time s


Notation xxv

T absolute temperature KTg, Tw local, instantaneous absolute temperature of

fluid and wall respectively Ku particle velocity m/su mean mass velocity or bulk velocity, m'/ρAff m/s Vd dead or unswept volume m3

VE modulus of expansion space volume variation m3

Vsw swept volume m3

W work per cycle JW'f friction power, or fan power according to

context Wα thermodynamic phase angle, the angle by

which events in expansion space lead those in compression space

α coefficient of thermal diffusivity m2/sαff dimensionless free-flow area, Aff/Vsw

2/3

β kinematic phase angle, angle by which motion of displacer leads that of piston in displacer-type machine

γ isentropic index, ratio of specific heats cp/cvδ dimensionless dead space, Vd/VswΔQr regenerator heat deficit, difference between

heat deposited and retrieved JΔT Tg − Tw Kεw wall temperature excursion as fraction of TCζf, ζsf specific work per blow lost to pumpingζq, ζsq specific work per blow lost to heat transfer

in regeneratorηT regenerator temperature recovery ratio,

e.g. (Tgexit − TE)/(TC − TE)

κ thermodynamic volume ratio, VC/VEλ kinematic displacement ratio, ratio of

displacement of work piston to that of displacer in displacer-type (beta) machine

Λ Hausen’s ‘reduced length’, equivalent to NTUμ coefficient of dynamic viscosity Pa sν dead space parameter δr loge NT/(NT − 1)Π Hausen’s ‘reduced period’, equivalent to

NFL NTU¶v/[(1-¶v) NTCR]ρ density kg/m3


Air Enginesxxvi

τ T/Tref = T/TCφ crank angle or dimensionless time, ωtω angular frequency rad/s¶v volumetric porosity, (volume wetted by fluid)/

(envelope volume)

Subscriptsb value per blowE, e, C, c expansion, compression f friction or fan as per contextg, w gas (or fluid), wall (or wire)p, d piston, displacerr regeneratorotto relating to Otto cyclexe, xc expansion/compression exchangerI, II forward, reverse blow∝ environment or free-stream

Superscripts+ additional (as in dead space)' (prime) per unit time


Date post:	08-Jul-2018
Category:	Documents
Upload:	phungkhue
View:	213 times
Download:	0 times

Air Engines -...

Documents