The temperature distribution in an internal combustion engine piston.

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1 Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection The temperature distribution in an internal combustion engine piston. Palacios, Daniel Hax Monterey, California. Naval Postgraduate School

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3 Library U. S. Naval Postgraduate School Monterey, California

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8 THE TEMPERATURE DISTRIBQTION IN AN INTERNAL COMBUSTION ENGINE PISTON D. H. PALACIOS

9 jajiflaridi HA MI MoiTiHiflT8ia aaotaha^ar Ear M0T8I1 a^icka KoiTaoaMOD

10 THE TEMPERATURE DISTRIBUTION IN AN INTERNAL COMFJSTION ENGINE PISTON by Daniel Hax Palaclos Lieutenant, Chilean Navy Submitted in partial fulfillment of the requirements for the degree of MASTER OP SCIENCE in MECHANICAL ENGINEERING United States Naval Postgraduate School Monterey, California 1952

11 jahharwi MA MI HoiToaiHTaia aaotaflaimat airr M0T8I1 awioma MoiTauaMoo eitnemsil up 9'i erict lo aomaioa '5o nataam MiflaaMiona JAOiMAiioaM loocfob 9^JBjjbj8i8;JBo*I IbvbM esijbd'fi becfjtau slniolllbo,^e'i9;jnom 2561

12 This work Is accepted as fulfilling the thesis requirements for the degree of MASTER OP SCIENCE in MECHANICAL ENGINEERING from the United States Naval Postgraduate School P. J. Kiefer Chairman Department of Mechanical Engineering Approved r R. S. Glasgow Academic Dean i

13 lo saisab sdi lo'i ad'aerasilup '! Blseil;? 3rict 30wax 08 "50 Hf^n'2AM OHIHasraiCWa JAOIHAHOaM srict moil Ioodo8 9;*Bir5Bis;tEo1 XbvbH BsctBctg bsijlau 19l9iX."G.<! nfimiibjc bsvoiqqa wogbblo.8.fl nboq olffidbbsa 1

14 PREFACE This Investigation was prepared during the period February - June 1962 at the United States Naval Postgraduate School, Monterey, California. The subject was suggested to the author by previous experience in maintenance of small high-speed diesel engines, during ^ ich burnt pistons had to be replaced on several occasions. The author wishes to extend his appreciation to Assistant Professor E. E. Drucker for his interest and advice throughout the development of the work. 11

15 B:tBubBi:gi Bol XbvjbM se^tbus oeilnu srij o*i ^661 enx/l - ^ UBj^rrcfe'^ BlniollIflO t^ei ;tnom,ioorfo8 Xaaelb bseqs-jzisjtff Iljeme lo ooniinojnljewi nt eonalnsqxs bsoblqei ecf o:f bbrf enocfeiq ^mud rfol ri<r gnlii/b ^Benlgna enolebooo Ijbisvsb no oct ctoliibloeiqqb Bid bnf);txe oct aedelw iodi:iub edt bnb d-esie^tnl Bid tol leafotng.3.3 iobbsioi^ cfnbitbleea.jjiow 9dcf lo ctneraqclevab 9d;t ;tuodbwoid:t eolvbb 11

16 TABIE OF CONTENTS Page number Preface List of Illustrations Table of Symbols Summary ii iv V 1 Chapter! Introduction 2 Chapter II Statement of Problem. 1. Description of Piston* 2. Temperature Scales. 3. Boundary Conditions Chapter III. Solution of Problem. 1. General Solution. 2. Solution for the disk. 3. Solution for the Barrel. 4. Conditions at annulus a-a. 5. Partial Solution using Tm Superposition of Sinusoidal Variation. 7. Conclusion Bibliography. Appendix I. General Solution. 23 iii

17 i9cfmun egfl*! vl B^ol;tB^c^e.c/III lo ctalj V exocrra-^a lo sxcfbt S nolijoirboictnl.1 leiqbrfo.ra9xcfoi:<i lo cfnsnre;tb;ta.ii lectqbiio J^.nod^eil lo noldqlioesq.x ^» 29X308 sijjctjbisqaiat.s b.bnol:tlbn^r^ -' Tsbnuoa,5 mexdoi*? lo csotiuloq.iii isctqarjo XX nol;tuxo8 XsisnaO *X SX.2^8 lb 9rf:t lol '"- ^f t/xos,2 ^X Xs'iisa ed;t lol noi:?i/xo8. 5X.B-B euxjjnnb db enol;tibnoo *^ 8X iiit gn.tet/ ar:l:ti/ioc! X^iliB*?.5 XBbloEijni2 io nouieoqiaqus,9 ax.nox:jblibv OS noleuxonoo.v ^rfqbiaoxxcfia C2,nol;tuIo8 XBisasD.1 xlbnsqqa til

18 LIST OF ILLUSTRATIONS Page number Figure! 5 Figure 2. 7 Figure 3, 8 iv

19 necfnir/a sfl*[ vl

20 TABLE OF SYMBOLS Symbol Q^antlty Units <^ Thermal Diffusivity. f t^ hr" -1 & Temperature of the Disk. &o Initial Temperature of Disk. ^ Temperature of Barrel. d> Initial Temperature of Barrel, p T Mass Density of Piston Material. Time b Thickness of Barrel Wall. Cp Specific Beat at Constant Pressure of Piston Ifeterial P Degrees Fahrenheit P P P F Ih^ f t-3 hr ft B Ib^-^ F- 1 P h^ Eeat-transfer Coefficient from Gas to Upper Surface of Disk B hr-1 ff -2 F'.1 hg Heat-transfer Coefficient from outside surface of barrel into media beyond the surface B hr-^ ft' '2 F" 1 1 Square Root of Minus One Jq Bessel function of the First Kind of Order Zero J", Bessel function of the First Kind of Order One. k Thermal Conductivity of Piston tit Material B hr"-"- ft"-^ P'"*- 1 Thickness of Disk ft

21 5.-j/elCI erf;* lo aii/cfsneqmst ^ 5.ifBia lo 8'ii/ctBi9qm9T Ijslctlnl oq "5.leiiBS lo aiucti3*t.9qra9t IjBJt^tinl ^<i) ^"^tl jgdl.ibjtiectam nocfel<i j.>j ^ctiensd bbam <^ id sralt T ctl.xlflw leiiaa lo eb9«>loiifr cf -. :ia&:i&aoo cts ctjash ol'ilosqs o -' "'5 ^" cfl a iblisctbm noiell lo sijjbeqi^l? ^IsrinSlrfB'? '^ moil :ta9lolll9oo i9lbabi:t-;tfl9n jcf X-.;r 2-5 "*"itl.^ I. -^"iri a :lbia lo eobliua isqqlj o;t eb^ - moil ^JneloilleoO i9lenbi:f-;tb9h grf OCtnl l911bcf lo 90BlliJ8 9bl8Ctt/0 jj 5 jtl '"id a GOBlijys 3ri;t bno-^9cf Blbsin 9a0 BunIM lo cfooh 9iBi/p8 1 ctbil*? 9rf;* lo nojt:tonul I9ee9a ql 0195; i9bio lo bnlx ;teil'5 ecii lo aotioai/l LB&eed. rl.eno i-^.5io lo bnlx '^"''5 no^tel^ lo -^ Ji via ojjdnoo IjaniiarfT a( -"^"ctl -^"iri a lblio:tflm ;tl a(bia lo BsaniioIdT I

22 Lq Length of Barrel. ft n Frequency of Gas Temperature _^ Fluctuation hr" r Radius ft Rq Outside Radius of Disk ft Ri Inside radius of Disk ft t Temperature, Standard Scale P tg Initial temperature of Piston P Ta Temperature of Cooling Medium Surrounding Barrel. F Tgas Gas Temperature P Tm Mean Temperature of Gases of Combustion. Tq Amplitude of Gas Temperature Fluctuation. P P Yq Bessel Function of the second Kind of Order Zero. Y^ Bessel Function of the second Kind of Order One. z Length Measured from Tom of Disk Downwards. ft vl

23 ;tl 2/Bia io BijJtbflfl ableitxai ^fl ;i1 3i82a lo BuJtbBi 9blenl Ifl q 9lBo8 biabniiis,3iucf si sqmat c^ 'i no^bl<i lo enjlrrffiieqffisct ibicmni Qit aix;lb9m gnllooo lo sijj^tbisqiiiet st q.isiiba snlbnuoiiifs 1 eit/d^bieqiest eao Bas*^ q lo B9 8B0 lo siir/;tb'i qm9t nb9m mt.noii^bjjcfnioo siuijb'isqraet bbc lo fci/ijtiqnia q't 1.noi-vtBLfcfouI*? bnooee dzt lo flol;tofli/5 leebes qy,otes lebio lo bnlx bnooob '='^" -'?-^'^at/^ I^BBsd x"^ lo bniii i(elcr lo raot moil hflti/ebsm rictgnej s i1 BbiBwnwoQ Iw

24 SUllMRY The objective of this work was to establish by annalytical means the temperature distribution in an internal combustion engine piston. For this purpose the piston was divided into a disk and a barrel, for which boundary conditions were expressed for the exterior surfaces and a common conical surface. The problem was solved in two parts t first, a solution was found for the case where the gas temperature was assumed representable by an average temperature; this solution was then modified to account for a sinusoidal fluctuation of gas temperature. The resulting temperature distribution, within the limits of the approximations made, is: for the disk: t -X. -^Tr. e.'^ "" /<u^ ( ^~^'^-j V^ J -^ for the barrel: 1. -r - --i ^Ltjf e^^'^^^^^^^lu^.q[n^)^r, YJ/,^[ 7** ij p»/

25 YflAM?5U2 Xd rfel. cfb.-tb9 o:t aew 3/iow elrfct lo svlctoatrfo edt flj8 nt aoi;tjjtfii:reio eiujbisqraa.-^... ziib&m l&oux^banb 9di B&oqttjq Bldi lo*!! *no;teiq anlgna nolcf^jjcfmoo Isniainl riolrfw io1,l9iiflcf B bns 2/8lb b octnl beblvlb ebw nocteiq xolie^xs 3ii:t 10 't be-r«*-mx9 919W 8rTol;tl5aoo ^ibbnuod bsviob 8BW raaxdoiq srit.sobiii/e ibolnoo nommoo b dhb b^obiiub eiedw eeao 9ri;t 10I bnirol gbw nol:m/ioe 8.cfsill :e^ibq ow^ nl e^biqvb ab ^d 9ldfl:tn9 8 9iq9i bgini/abb bbw BTutMteQmei ebs d;t ;tnijooob oi bsiutbora nod^ bbw nol;tijios Qldi ieiu:ibieqmei.9'ii/;tbi9qm9;t bbs lo aotibutoull ibblozualb & 10I d;* aldilw ^nouudtiibtb s^uiatba^^^ vntila&ei 9dT :al,9bbin Bnol^BmiA.'j'^iqqB adi lo BcflmlX :3f8lb 9dct 10I XsiiBd d;t nol 1

26 CHAPTER I INTROOTCTION Upon analyzing a piston of an Internal combustion engine with a viewpoint to understanding its behaviour as a heat dissipating member of the engine, it is readily realized that the piston absorbs a certain amount of heat through the upper surface of its crown. This heat is partly stored in the body of the piston until it reaches its steady state temperature; at the same time there is heat dissipation from the outside surface of the barrel and rings and also through its Inside surfaces. After the steady state is reached, all heat absorbed must be dissipated through these surfaces. According to 0. L. Adams (1), five percent of the heat liberated by the combustion of the fuel must pass through the piston crown surface into the piston. Of this heat, ten percent is dissipated by the lower surface of the crown of the piston, as stated by J. L. Hepworth (5)» Radiation from the gases into the piston only takes place through a very small portion of the time cycle and it amounts, according to B. Pinkel (9), to only ten percent of the total heat being absorbed by all the metallc partsforming the container for the hot gases; thus it may be seen that the heat absorbed by the piston Itself through radiation will be a part of this ten percent depending on the relative areas of the piston crown vi th respect to the area of the rest of the combustion chamber. Throughout this work it vl 11 be assumed that all heat absorption by the

27 cioiibudmoo IjBmsi'n.t na 'io no^^ajtq & ^nis^^lbrtfl aoqo' luojtvjbried BJl gnlbn6d"ei9bn/j o;t ;tnloqw9lv b rictiw eal^ae deed 1g ^tnijoms nib^iao b Bd''T.,oHR noiteiq sa'i isdi b^sllaei Xl:iiaq el ^aad BldT»mioio ac^i lo ojaltue isqqw 9ri:t dgjjond^t ^bbed-a adl cedobsi ctl ijtd^ajj flojsl-q 'ad;t to Y^x^cf sdct ni bsio^te osib baa esni-^ DnB loi'ijad aiij lo aobi'ijj-a sdiacri/o acij ihoii ei ctbcte Y^Jas^B edcf ie:tla.aaobliue sblenjt ail dgj^oidit 3B9il:i ri-guoid:} bscffo-iseib d.jsdth b&d'ro^c^ d^bsd IIb,b0rioB9i.BSOBiiue ;jb9d srid lo ;tn90i8q svll,(i) ambba.j.0 ocf gnlbioooa d^-'joidcj esbq rf-euni I9r.r1 sn'd lo aoicfancficoc sdrf' 7;cl bed'siecfll nsi,jb9d elaj lu.nojelq.sria- ocfnl aob'i'ina nv;oio nodelq sdd" lo nwoio eii lo goblni/e levrol ad* -^cf be^tsqleelb el ineoisq.(5) d;j'iowt9^.j.1. ^d bs^bcfs bb ^noielq gdcf 89j/Bcf Y-I^no noatalq edj ijoui asbbg 9ri;t moil nold'bxbbh bna 9lo^o ml;* eri;* lo aol;jr[oq XlBma ^-^isv b dsuoidrf soblq ctneorreq ne;t ^Ino o* ^(6) l93/nl<!.3 o;* gnjtbiooob,b;tnm)rab ;J1 actiflq ollb;:t9ra &d:i XIb t^"^ b9cfioedb gni:9d ^tasri ib^ocf sd* lo d x^^^ J-t 8ud:t ;e98b5 ;tod dct nol lenib^tnoo 9dd gnlmiol dsu^i'idi 1108*1 noctfilq d;* ^d badiofcdb ;tb9d d* ;tfldi neoe ao 3nlbn9q b irtsbneq nscf aldct lo dibq b d III./ nolitblbbi dj o* io q8 i d* iri 'nwoio no;j'blq 9d;t lo bbsib evict bisi adct eld;t d^ikidswoidt.ladraado nol;teijdnioo adi lo *b9i dct lo B9nB!^ ^ f vd noldqioadb ib9d 1 b ibd:t b nnueca E»d li bv d^l jiiow

28 piston takes place by convection. Also all heat dissipated by the piston vi. 11 be assumed to do so by a process of convection. The temperature of the gases throughout the cycle varies widely; the temperature curves for an engine may be obtained either experimentally by direct measurement or annalytically from knowledge of the cycle under which the engine is operating. In this work the temperature of the gases will be assumed, for the sake of simplicity, to follow a sinusoidal variation about an average temperature. The main objective of this work will be to determine the temperature distribution throughout the piston. For this purpose a set of boundary conditions will be prescribed #iich resemble as closely as possible the actual conditions under which the piston operates.

29 be^iaqleeib :fsdri f.lb obia nou'oavrfoc vd ao^icr eeij/sd- nod"eiq lo sesoonq jb '^a oe ob oi bsmuaes ad IX iiv x^ocfelq sds x^.aol:toevnoo ^Bj.: c^nis^ ^^ '^o'i 89vitfo 9iirdBieqmecf e:id^ j^iebiw bbiibv TO i^nerasii/bbsra ^tosilb '^fcf ijilbinsraliaqxs isrictls benib^tcfo cf erf;* dolrfw lebna exo^t) Brict lo ssbsxwonij motl Y-C-^^^-f^^-^^^nnfl oj t^ctxoxxqmxe lo sj/be erict lol tbamueeb &d XXXw bqebs eijji^bieqraei sgbiavb hb ;tx;odb noxd^bxibv XabXoeunXa b woixol fllrat'» + --^^ oct 9d XXXw ifiowv aid:^ lo svxd-oat^^o nxaw sdt lo"?.noiexq dd^ ^tuodsiiondct noxdudxid'exb siuctbieqrasct edd^ b9dxi089iq 9d XXXw Enol;tXbnoo ij;ibbn0od lo ctee b eoqiuq eldit BnoX:tXbnoo XBi;:foB edi exdxse' ^ -^ r vjaecxo es Xdmseai dox chr.e^cfb'xsqo no^exq dd' doxdw lebau

30 CHAPTER II STATEMENT OP THE PROBLEM! Description of piston. A piston with the characteristics shown in figure 1 was selected for analysis. The crown will be a flat disk of uniform thickness 1. The barrel will be a cylindrical sleeve of constant thickness b. The boundary surface of disk and barrel will be surface a-a as shown in figure 1. This surface will have properties common to both disk and barrel and this will enable some of the unknown quantities entering the questions to be found 2. Temperature scales. Figure 2 shows a sketch of the temperature pattern that a particular point of the disk is expected to follow. This figure is given mainly to show graphically some of the values being used throughout the development. Figure 3 serves the same purpose as figure 2 with relation to the barrel. The temperature of the gases will be assumed to be; where n will be the frequency of the cycle. The temperature T^ surrounding the outside surface of the barrel will be assumed constant. The temperatures for the disk will be measured from the level designated as in figure 2 and will be expressed

31 II HSTIAHO MSJaOfl^ anr '50 TIiaM;'5TAT8.noiztq lo noliqtiozqa.1 I 8IU3II nl ftwodb eoldsjtiaiobijbdo srii ri:tlw noitsiq A 2/elb :t6ll B ecf XIlw rrwoio sfft.ela^lbnb lol bectoelse sbw iboiibnll^o 3 ad II tw leiibcf rpt.1 eearr^o'iii rano^inrj I0.d aaen^oldd' Jnjsjenoo 10 sv^yls d IIIw XsiiBd bns -sjelb I0 eobliije ^ibbnuod edt evfld LLl*f aoblix/b slrpt.1 Btug-^'i nl rrwode bb b-b eobl': ne II Iw Biri;t bnb Isiibcj dhb ^slb ^.l^ca ocj noisoioo eel^jisqo'iq Enoiiasjjp 9d^ snlia^ne BeI:tl:tnBjjp awonsinw sri^ lo amoa sldane bnijol 9d o:j bsibob 3i0;tfli9qra9T.2 aiei^aq aij^cfsisqmecf eri;t lo do:t9:5/8 b awode 2 siusl'^.wollol oi b9;j09qx9 si r-ialb adi lo (tnloq ibljjoi;tibq B *fid;t 9dct lo amob ^IlBoldqai^ wodb od- T^InlBra nevlb el eiugll sldt.ctnsmqolaveb edct iuod^uoici:i beau ^ntb(s aewlbv riilw S 9i!J7;Il SB eeoqiuq embz acii e9vi9e qiusi'^.leiibd edi o:t no.tibl9*i ;9d oct b9niueab ed illw bbq&s ^^^ "^o 9'i0;jBi9qm3;t dt slo^o 9dct lo ^onejupg-il 9d:t ad IIlw n 9i9dw lo eobliua sbla^uo 9d;t snlbnooiit/e jgt ijj^jbieqmac} 9dT cfnbienoo bgmuaeb ^d IIIw let-iscf 9d:J moil b9ijjeb9m 9d IIlw alalb ed;t lol 89iu^si9qm9;t Qf.'i baabsiqxe 9d HIn bna S giugll ni as b9;:rbnglb9b l9v9l 9d^

32 \X\\\X\\x\\ \ \ \ \ \ \ \ \ \ \ 2 % Tja* R, > \ \\ \ \ \ \ \ -^ A \ \ L_^ / \ \ \ b^ Fls-a."-^ 1

33 I 4

34 :. for any point of the disk as: The initial temperature will necessarily he a constant and will he designated as ^o The temperatures for points of the barrel ^4 11 be measured in the j> scale of temperatures as shov/n in figure 3, where: The initial temperature <^^, will be a constant. It is realized that the value of Ta will initially be equal to t^, but it is impossible with any degree of ease to consider into the problem its initial rise from tq to Ta and thus it will be assumed constant and equal to T^* After a solution is reached, the results will be reverted to the standard level of measurement of temperatures in degrees F. which is designated as the t scale of temperatures. 3. Boundary conditions. The following set of boundary conditions has been selected (a). At the lower surface of the disk, the rate of heat rejection is ten percent of the rate of heat absorption at the upper surface. Since the rate of heat conduction is a function of the temperature gradient normal to the surface considered, and if we assume the coefficient of heat conduction to be constant throughout the piston, we may state: 6

35 . oq ej3 bsctjsrtsleeb ecf IIlw bcib &6 XI Jbv Isiijscf adct lo s;jnloq nol BS'in^tflisqme^ srft siijs-tl ^-t rr.tofie as aaii/ctbisqui ;* lo elaoe ^ 9ri:t nl bsit/ejseni :9i9riw».ctnfl^enoo b ed IIIt-^ ^^ ani/^biaqmect Xjsl;tlnl dt 9d ^IIjBlctlnl IIlw flt lo 9ulBV 9d:t :tbri:t b9sllb i el ctl BBS lo 99is9b -^nb rfilw IcflBBoqnil el ;*! :tjjcf,q;t od^ Isirp o:i Q^ moil 9b1i lbl:tlnl ait melcfoiq rf:t octnl igblenoo oi jgt o;t ibup bnb :fnb:fbaoo bsrax/bbb d IIlw ;tl Buri^ bnb ^T d litfr e^luesi ed:t,b9riob9i si nolcfuiob b i9:tla eaiucjaiaqrasct lo :tn9m9'ii/8b9m lo i&vai bisbnacfb dc^ o^ b9;ti979'x lo 9lBoe :f 9d* bb b9:tbngla b b1 doldw.'9 aeeigeb nl «8 9'ludB19qffl9;t.anolctlbnoo yiflbnuoa.5 n99d BBd Bnoldlbfloo ^ibbnuod lo jae ^ntfiiollol dt : b9:to I 8 lo 9;tsi 9dct <ifelb 9ri:t lo obiii/e 19^0! di :ta.(b) nol;tqnoedfl ;JB3d lo sjbi adj lo ;*n90'^i9q nscf al nol;to9(;9i ;jb d b1 flol^toybfloo ib9d lo o:fbi rf;t onls. obiiub isqqjj dit ia oaltim 9d:t o:t ibrrnon ^JnolbBig oiu^tbieqragct od;j lo nolctonol b nol;t )ijdnoo c'sgn lo :tnelolll9oo 9d:t arausbb w 11 bns»b9i9blenoo : ;Jb*b x^^ ^w,no;*elq ed:i cfx/odgwondct :tnb;tsnoo 9d o:t

36 Z-n > r Flea re 2

37 1

38 \ ^ 1 k. ^^^ I ^ ^ox y ^ ( ^- 1 t /l i - T^ tc. < -». r?lr;-are 3 8

39

40 r.ij^-'"'^^^'^^'^ There will be a difference in the areas of the upper and lower surfaces determined by the relative values of Rq and Ri» In this work we will neglect this difference, but it is recognized that it could be included by multiplying the value given as 10 by the ratio of Ri to Rq squared (b). At the upper surface of the disk the rate of heat absorption from the gases will equal the rate of heat conduction from the surface into the disk, or: i^ti'lr-'^'l'^--'^"^ 'i 'i (c). At the surface a-a the temperature of both the disk and the barrel will be equal. Since these temperatures will be measured from different levels, this relation may not be used directly without introducing undesirable constants. Therefore it is necessary to realize that at this boundary the isotherms from disk and barrel must coincide and furthermore they must be continuous functions of the variables involved. This may be stated symbolically along surface a-a: and:, d ^ ^ 9

41 >4rJ"'-Jr4 isqqjj erict lo bbsijs 9rI;J al onaislilb b 9cf IIlw eisrlt lo eoulbv 9vl:i&lBi edd -^d b3nlnii9:t3b bsobiiub lewol ba&,eon i6'lllb 8Jtd;t :toal3en IIlw 9.v s/iow aldct nl.ifl bnb qh -Ictljjra ic<^ bsbuionl ed bluoo :tje ^sdjt beslfl3o09i el dl ;tjjd qh o;t ih lo oid-bi 9d;t ^d CI bb nevlg 9x/Iflv 9rict gnl^iq baibjjpb lo 9crBi 9dct :il^tb Qdi lo 90Bliife I9qqjj 9riit cta.(d) :i&ed lo 9:iBi srict ibupe IIlw esbbg 9d:t moil nolctqioadb *B9rf :«io ^:it&lb 9rirf o;tnl eo&liue 9ri;t flioil nol^oubnoo L-^-.,aV'i-%,l5)i 9ri:t d;tod lo siu^biqqmsi sri* b^b sobiii/e 9d;t ia *(o) 8 9'iu:tsi9qffl9cf 9esrii sonia,lbjjp9 ed IIlw l9iibd edi bna :ilbtb XBm nol:tbl9i eldij»bl9v9l ctn9i9lllb moil bsiuebsm ed IIlw eldbilqqbnu sflloi/boi:tnl ;tjjod;tlw ^I;to9ilb b9bu 9d ioa. Bd^nB^tsnoo Bldc^ ib :ibdi 9s1Ib9i o;t TjiBBEgoen b1 il 9iol9'i9riT 9blonloo cfeimi l9iibd bnb ^&lb flioil 8rai9dctoel 9d;t ^labnuod 9ri;t lo 2flol;tonul auounld-noo 9d ieum z^^^ 9ionn9diii/l baa s^ilboliodra^e b9;tfl;*b 9d ^bth eldt &9vIovnl 89ldBliBV :B-B 90BlliJB s^o-t^ H & "t^ ^"^ L ^ : bnb

42 (d). The lower surface of the barrel will he assumed insulated, or: ^] ^ (e). The inner surface of the barrel will be assumed Insulated, or: 1+1 ^ (f ). At the outer surface of the barrel the rate of heat rejection from the barrel into the surrounding medium will be equal to the rate of heat conduction from the barrel into its outer surface, or: ^inu-^'(^-'-*^' (g At time ). * zero, a point : z - 0, r =. 0, will be at a temperature B(,, or: at 10

43 bemuzbb ecf IIlw leiibq adct "lo soblii/a lewol orfp.(b) :'io tbe;jblx;bnl Q * ( ii bsnu-seia ad IX Iw XdiTiJd erict lo aoblism i&anl erit.( ) to 9iBi 3di IsTiflcf cii.:f lo eoblijjb 'lecfoo 9d;t cta.,./ mutb»m scfl bn JJOTIJJ8 erict o^jnl IsiTsd edct raoil nolioet^i ^&Qci ed:i raonl jaolcfoubnoo cteerf lo ectjai rii o:t ibupe 9d Iliw :io teoeliue isd^uo zil ol*nl leiifid l>-,i^.i-=,,ii-tv ed II Iw to - 1 to-* s rrfaloq s,oiss * eml^ ;*A.(g) ;io t 0^ sii/ctbieqaisd- a ^fl JB -' e^ = f Cj r J. ^ T 01

44 CHAPTER III SOLUTION OP PROBLEM 1. General solution. In solving the problem the disk and barrel will be treated separately and the solutions matched at the surface a-a. Both the disk and the barrel involve heat transfer in a cylinder; the disk may be considered as a flat solid cylinder, )ih ile the barrel may be considered as a hollow cylinder. Under these conditions the best solution of Fourier* 3 Law of Conduction of Heat will be that expressed in cylindrical coordinates: (1) Assuming the variables to be separable a solution as found in Appendix I ist From figure 2 it may be seen that the expected curves for temperatures will be of the form: for a particular point on the disk. At the same time a sinusoidal variation will be applied, which proceeding along the z-axis will be damped and out of phase; therefore our solution would have a form such as: 11

45 XX Ill RaT<!AHO «noirfi/io8 IsiensO! ecf LLItt l9^ia<s bcib ^&lb edi raalcfoiq srft gnlviob nl eoblii/b 9di Jfl berfoiara enolitulos erf;? bna 'cx9;tbij8q9b be*a9i;t nl lelbflflicf :ibed evlornl leiibd sdct bnb sfslb 9rf:t dctos blxoe ibil B EB beieblanoo sd ^Jsni ifelb srli jiabnllyo «woxiorf B EB beisblenoo ecf -^Bm laiibcf rfi sxx r*r ^lebnlx^o 1o cioliulob ctsed" 9r{it eaox:tibnoo aserf:f lebatj lebnlx^o beee9iqx itflrict 9cr IXIw isah "lo nolioubnoo lo wbj e»'i91'iuo'5 :e cfanxbt:ooo XBolibnlX^o nl l^ EB nolcfifxoe B exdbibqsb ed oi eaxctblibv eri;t snlmubea 3 8l I xlbaeqqa al basjol B9V1U0 be^osqxe od:t ;tbdct naab 9cr -^jam cm S 911/811 nioi'5 :nnol 9ri:J lo acf XIl-7 8 9'iu:fBi9qffl9;t lol,3f8lb 9d^ no itnloq ibluolctibq b io1,b lxqqb 9cf Lilvr nol;tblibv XBbloaanls b smli em&b edii ia lo cttro bnb beqnibb ed XXlw b1xb-s 9d:t gnoxb gnlbegooiq doldw :bb riojje miol b 9V3d bxwow nol;tuxoe nuo eiolsiedi t9ebdq

46 9 ^$a e^\ To ^^[kr- ^j^*'^* ^ ^ere cz is the phase angle and e accounts for damping* Examining the previous equation it may be seen that the variables z and T may not be separated as assumed in the solution found by equation (2), and thus this equation will not yield an exact solution. A direct solution of the problem was attempted by the use of Duhamel»s Theorem (3), but it was found impossible to match the result obtained for the disk with that for the barrel at the surface a-a. Under these conditions it was decided to divide the problem into one where the gas temperature was assumed to be the average temperature T^* and then proceed to superimpose on it the effect of the sinusoidal variation in gas temperature. 2. Solution for the disk. From equation (2): e. ^ e'-^^^'^'^^^c^t^. 2)^. J«^o(.0*3U-J Since 9 must be finite throughout the body: B - and therefore: 9 s F e'*^ * [C<-r>i)j ^ i>**- ij J ntou^'-j (3) From boundary condition (a): 'i^y- 12

47 SnlqniBb tol e^nwoocb ^^~9 bnfi el^b saadq sdct 3I so eiedw at bemubbb as bsjsisa?*?. ad ;fofi rbm T oftjs s eeldbliav adct nol:tbwp9 eld^f ai/rict bnfi a 2) nolcfbjjp? ^u bniwl cioi^sjiob qiu»not:iulor :tobxe nb blel^ ion IXlw eri:t -^d bs^qras^itb sbw piexdoiq srfd- to noj:;tctloa ;to9'xlb A 9ldl8«oqin.t bnuol bbw itl :tud «(e) ra?^ioeifr &HemBdt<l lo obu lol :*Bd:t ri:i Jm 5falb 9d:t 10I b9alb:tdo lilubqi 9ri:t doctara o* B-B eobliub edi is IsiiBd ad^ 9ri;t 9bivlD cj ud>jjuosb sb.v :tl eaojtctlbnoo eeed* i9bnu o;t b9mu8es bbw 9iir^B«i9qjBi9;t bbs ed:t e't9riw no o:tnl ffl9ldoiq -i9qu8 o;t b990oiq n^cii bab,^5? Qiif:iBi9qmBi 9sbi9vb 9dcr 9d BBS nl nol;jslibv ibblobunlb 3d:J lo ^ d;t il no aaoqoil.91mc^b'i9qai9;t 2f8lb 9d;J 10I aojt;ti/ioa.s :(S) aotibupe nioi'i :^bod 9d:t cfi/odsuoid^ 9;tlnll 9d :izum ^ 9onl3 u - a :9iol9i9ri:l bnb -1 T( ';-^'j»)>^ (C) :(b) nol^jlbnoo y^-iksunmj'o raoi'^ A.i^l'^)

48 13 Prom equation (3}t vi 3 th eref (Dre: ^ yu^i>t c C^^/ -0. / th en: 9 z Q t ^(a. c^^ '3 Jsi^a<L) (4) From boundary condition (b): or: then! coi,^^ <>. / k y The solution of this equation for b may only be done graphically; it will yield a series of values for b ^Ich will be designated as t^* Equation (4) now becomes: ^;^ue /c^a^^ t ^ '^^^.hj J ^»^*^^ (5) Prom boundary condition (g): 'T i a /L ^ o B * & Substituting this in equation (5): G s 9o

49 i- r i^) :(d) aolctioflod -^'i» un j^o u diu :io <*i /I..1 ^6,.*JkCv rnerfd nob ecf vtrro r^m d *iol nol,-tbtrps ejtri* to aoliisloq orit ffoliiw d ^G. 39WXBV lo selids JB blelx -^-f^w ^^ "^?T-C'' ' '''^sis,^d 8B bs^bnslbsb ed IIlw reewooecf won {h) nojt;j«ifp3 (e) e %.^ :(g) noi^lbnon ijibbnjjorf nio'i'? :(5) noictbt/pe flj alrf^ gnlctu^tlcjed^ja

50 : Therefore: (6) 3. Solution for the barrel. Prom Equation (2): (j^zhe^^^"^ '/L^33 ^L^^3][^X^/.J^/i/K.(Aj] (7) From boundary condition (d)r J - therefore then? f.jp^-^cf%f^ ^^ (j.ljpj-.raj * VK(AjJ (8) Prom boundary condition (e): (^] ^ s/ ^: (/<?.) therefore: Prom boundary condition (f ): 14

51 M O) :(S) nolcfbjjp^" raoi*? J (5) nol^tlbaoo ^ibbauocf raoi'il - ili ( a* A : edit (8) [(J\)^^* i^)^^^\{.^.^).^ ^^^^^^^"".i'i. ^ :(e) floi-ilbnoo ^isbni/otf raoi*? :8iol3i9rict (6) :(1) floldibnoo ijiabnuod raoi'5 c5l*iv\^^^

52 : 15, Solving this equation graphically for the value of f a set of values fp are found that satisfy this equation, and equation (9) becomes: 4, Conditions at the surface a-a» Recapitulating and: Prom boundary condition (c) at points along the surface a-a: J^^ ^ r^ ^3" ^3^ (11) 0^" % (12) For equation (11) it is easily seen that for n-1 and for a particular point along the surface a-a: 1 T^ ^ ^ vs'^ (13) then for n = 5: (14) this suggests that as a method of approach to the evaluation of a, g, and P we assume: g = ^m

53 r>r ^1 lo 8x/Iflv ed:t not -^XlBolriqBiB nol^tbwpe Bidet anivios ^^olibjjpe eldct ^lelctbe itjbr{;t bnuol sib gl esuxbv 1o ^98 b -Bsniooed (6) nolitbilfps bnfl (01) l^-^v^^ ;A.U^^i^-^-^Jf^cv.^^^^^^^^^39 3. ^.B-B oblijje edct :i& anoj:;tlbnoo.j^ : anl^bluitlqbosh : 5aB 9d;* gnolfl e;^rrloq cfs (o) nolcjlbnoo ^ibbnuod irrot^ (XI) -^TT -^ ^'^^^ (21) ^s^l ilk, ilt ' -^U I«n Tol ^Bdi creee ^Xlesa e.f j-f CIX^ noictbups «io'5 jb-a eoblni/e adi gnois jaioq isinoliiaq a io1 baa (51) 1)* : 2 - n 'lol n8d;t (M) \: M ed;j oi rioiio'iq'it ly uod:t6m b sb ctad:^ sjaosaub eldct rsmijbbb ew «I bna ^-g,b lo aol^awxavs

54 : Acceptance of this assumption involves the necessity, in order to maintain the equality of the above equation, that: '^m " Sp where: m = p Also, in order that the effect of time on the temperature be equal for the same point on the surface a-a as approached by either equation (6) or (10), that: fp = a Applying this condition to equation (13), we find: where m = p This equation is valid for points along the surface a-a, ^ ere: i yfi^^~^) and should yield the values of Pp. Now, the replacement of equation (15) into equation (14) will yield an equation in terms of trigonometric and Bessel functions of r, valid for values of r ranging from Rl to Rq. Since this equation may not be simplified to any appreciable extent, it has been considered more advisable to replace the values of z and r of a particular point along the surface a-a, such that: z =i/2 ria r = (Ri Ro)/2 =- Ra 16

55 ar,"^;tieb90 n arfij eevlovnl nol;*qfflt/b8jb slri^ to onactqsooa q-» m"^ q = m leierfir -Bieqras^t sdct no 9mt^ lo cfoalla dd' iad^i isfoio ni tosla Bfi fl-b sobliub edd no dnloq ambs edd lol ibupe sd eiud :;*Bddf,(0I) lo (6) not;"?^up3 laddla ^cf bsriobo^qqb B = qi : ball 9w,(5I) nojt^bup o:t nojt;*ibnoo BJtdrf gnl-^iqqa q s ffl eoblii/e 9d:t gnolb acfnloq lol bllsv zl aotiaupb eldt : eiodw.q*i lo eenlbv eciii blatx blwode 5nB nolctbwpe o;tnjt (51) noldsjjps lo ^nepieoblqei add,woli bnb olid 9010 no 1*1 j- lo emned nl nolctbupe hb blal^ IIlw (^X) moil snlgnai i lo seulbv lol btlav,n lo enotioar/t Iseeaa XCiB oi bellllqfflle sd ;Jon \:Bm noldsjups etdi aonls.qh oi Ifl QL6BBtvb eiom baieblanoo need esd -^?,;tne;txe exdbl oaiqqa ;tnloq ibxjjol;tibq b lo i bnb s lo cssjjxbv add obxqsi oi : :^Bdd doue ^b-b aoblni/e add snoxa br - S\(o^ IH) = 1

56 then we find r (16> where: m s p If now in equation (12) for n = 1 this same point in the surface a-a is investigated: Equations (16) and (17) should give the same value for Pp. There is no way of proving that this will be so except by working out a particular problem. If in so doing it is found that the values of Pp satisfy simultaneously both equations, this will then mean that the assumption made that: S = ^m and the derived expressions: Tm - Sp and : fp «a are Justified. In case that the values of Pp found by means of one of the equations mentioned do not satisfy the other equation, there is still the possibility of introducing this value in the second equation and determining a relationship between Ri, b, Lq, and 1 such that both equations are satisfied and for which the assumption made is valid. This would introduce a limitation to the physical proportions of the pistons for which this development is useful. The relationship just mentioned would take the form? 17

57 : ball ew neri:t OX) q a r: terr,eirf7:r nl ;*n.toq emae elri* X» n lol (SI) ciolibirpe al won II (VI) 9uXbv sflibb ad:t 9vl8 bxfiorie (VX) bna (dx) enolctai/pa o8 96 IX Iw Blri;t ^Brict gnlvoiq to tjbw oa ai siorft.q*i lol Snio^ OB nl 'il.motdotiq isxi/oi.:t«tbq b ii:;o g-^^^'-i^.v ^r^f ;Jq9oxe -i^xbi/oanbitliimle 'ilalitse gl lo sst/iav 9ri:J isrf;* bnuol el ctl 9bjani notiqmubzb eri;t ;Jsdi nb9ra nsritf ILLw elri^ tsnol^bup rfijocf : Bnoi.aa6«tqx9 bevlieb Qcii bcib q3 r mt B z ql :bnb.bslllcteijt 91JB eno lo gnbqra -^cf bnijo*! q*i lo 89wIbv ed;j :JBrf:t 9aB0 nl isdcto 9d:t ^leldbb ion ofo bsnolitnain enol^fli/p an'ct lo SnXoijboi;tnl lo ^ilxlcfiaeoq sai LlltB el 9i9ri;t ^aoliaupe B 3nlnlxa'i9;t9b bns aolibupe bnooae sdct nl wxbv aldi ditod ;JBdi rioue X bne ^^J,d ^ih n99w:t9d qldbnolctsx9'x ebam noi^jqnioeeb ecii doldn lol bna bsl'lalvfae 91B enol^fa^pa ibole^dq 9d;t o* nolcts^jlicli a 9ouboiJal blijow eldt.bllav b1 el :tn-imaox9v9b ztd:i r{2t&?r rro^ srfoctelq odct lo anoliioqcnq iffliol 9dJ 9iiB:l bi.uow bsnoune-m jbi/^ 4J.ri8nolcfal9i 9dT.luleeu VI

58 :,, /4Aai\, 0/im <.<- -^ tra b^ tc. l «, r ; ;; On,, ^>. t ' i-.^mdjldk^ (18) 5. Partial solution using T^. now On the light of previous conclusions, the solution is / / * / M Q^fiZL^'^ ij^^^y ^^^^JJ^oUr'') (19) >M -( f-' j> ^ _Z, Vp e ^ ^'' ^^^ '^ ' ^ lo^iyl.)!^^^.) -i- F^VoC/roJ (20) and since: ^ c and : d> - t - To. we find, for the disk: im - f ^ - I and for the barrel: (22) 6. Superposition of sinusoidal variation. In order to superimpose the effect of the sinusoidal fluctuation of the gas temperature about a mean temperature Tqi, the above equation (21) in a simplified form: 18

59 81 (81) L:Hi^ii^w^)i _ ' - -;; - rw> el nalijnioe ai^-t jbnolbtflorroo ei/olvaiq lo (tdgll dd ao swoq (ex) *- «k I* Kf ( 02) ^U^')oN ^^ ^ U ^Vol.\(* -^ - ^) ' -«^ c^"^ " ' ^3 <* J^^^ < > jot - :j - ^ : bns :?ialb 3ci^ nol,&nll 9W (12) rlsiibcf 9d;J lol bns (SS).noldBiiBv fbb^oeimle 1o nol:tlboq'i9qi; ^d labloeunie edit lo :to. xx-= r.nj ^c;i--..im.ti9qx;6 o;t 'isbio nl etudjbieqfliact nbew b ^twocfb eturrfbiaqms^ eeg rf:f lo no l;t Bt/;Jojjil traiol bsllllqinie a nl (IS) nolcfaupe evocfs orirf t^^t

60 s after a quarter of a cycle, or at T*^ we find that: ~r<\as ' '^ -^ 'o />t^ J T r, T" therefore: Tc^s - T. -^ ^ If this were put into equation (21) it would give an answer corresponding to the situation where this new temperature would have been imposed on the piston since time zero. To avoid this, instead of X> '<u^ J~>i'^ (23) use the following expression given by M. Jakob and G. A. Hawkins (7): To e.<u/wv (-en-nr -J y ir / (24) where: Tq is the amplitude of the oscillation at the surface of the disk; e accounts for damping effects along z, and _. y^ accounts for the change in phase along z. The above expression v;as given for a' thick plate subjected to conditions similar to those of this problem. Even if in this case the disk may perhaps not be considered a thick plate, it is felt that this equation will yield a fairly good approximation if it is considered that it is common knowledge that the effects of fluctuating surface temperatures do not penetrate to any great extent beyond the surface exposed. In this particular case this effect is further minimized by the fact that the fluctuation of temperatures is very rapid. 19

61 u i:isidi bciti 9w J^* "T" ^b io ^^loxo b to leiibup b *i9ilb ^ rf'i ^ /«>**, O *'' «' ' * >. jft pi 2eiol9ieri:t o"^ -^ -ct 5^ J, ^. i aa 9vJs bliiow -tl (IS) nol;j jjp9 odnl ;Jijq aiew elricf II wen eiri;* a*x9d»r nolrffiwctle r^:* * Snlbnoqee'r - -^ 'T9wena eonle noielq ari;;^ no beeoqoil nsecf &vbd bluoir e-iud Bi9qnie;t.oiss 9 c:l:t ^lo 5B9ieni tsirict blovb ot.a.0 bnb cfoiljbl.m xd nevlg nol8e9iqx9 sniwollol rfit 98 :(V) Bnl:jfwBH (*2) 3 : eiedw OAlitm 9d^ i& noljtbliioeo ed^ lo ebuillqnib sdcf el qt f-na ^5 snolb a:ice11e ^ntqmbb lol ectrrt'oonb "** ^ a.5 gnols 9EBdq nl 9snBdo srid- lol ectnx/oooij '^^ j_ -di/a e^sxq ^otdi a lol nevls bbw noleesiqx vodb 9riT nev3.mexcfotj ejtd:? lo seodi oct ibllmtz Bnotilbcioo oi beioel B bei&blenoo ed ion sqa:ri3q ^fltir ^tl.b ed^ eeao tldi nl 11 a blai'^ IIlw aolctbwps ztd^ ctbdtf itlsl el ;*!,eifllq aioldi el ctl :Jsd;t beiqblenoo el ;tl Iri nol;jam.txoiqqb boog '^C'^ilal OBlijje snld-birc^ojjil lo e.-toslia :j:.j dadj e^beiv/on.'«norcmoo baox^d ia&ixe ctaaig ^na o:t d^aicf sneq cton ob eeiirctaieqmed ^tool*): eldi bbbo iblrrojtdiaq eld;* nl *b9eoqxe 90B*lijje odd lo nojt^jbudoui JL :t.o oj^i^j ctoa'l odd y^' bq&lmlntm leridiul e1. blqai TJT9V bi ssnwctaisqmed

62 Prom equations (23) and (24) It may be seen that what happens at the surface at time zero will have Its effect at a depth z with a time lag of: The addition of the term expressed In equation (24) will affect in equal form both the equations (21) and (22), Since for the barrel the values of z will, with the exception of its topmost part, be far greater than those for the disk, this effect will only be perceptible In Its upper section. Including equation (24) into equations (21) and (22) we have as a final result: For the disk:- >H~I fit Per the barrel: Vvts. ( f> Z I 7» Conclusions* It is realized that the results obtained are not rigoroua but it is felt that they constitute a fairly good approximation to the actual temperature distribution in the piston. Rirther work might be done on the subject by solving an actual problem in order to check the practicability of 20

63 ctbriw jbd^ a998 90, ' < ' * '^S) bp.ii iinoljb^jpfj mu'i'i ioelle s;tl svbri IXiw oiss 6flii;t ;fa eoalixre erii Jb eneqqari :lo SBl sifllcf JB ri^iw s dcfqeb a ;tb (*2) nol;tbjjp9 al beeeeiqxe rpi9.-t erlct to aoi.:tib5b erft.(22) bcib (12) enolcfbups e-f ol Lsupe ai.toella IXlw -qeoxe erf^ rfijlw tlxlw s lo &QuiBv eri^t Xaiisd erfct lol 9onl8 xol abod.1 nbd:j is^jasig n&l ed,:tibq ijeofflqod" ectl lo nol.i e*x n2 excfl^fqeoieq 9cf ^Xno XIIw itoell eld* ^a/elfa ri* nold^oee leqqu (22) bna (X2) enolctbi/pe ocfax (i^2) nolctaups snlbi/xonl :;txi/«et XBnlt b eb evbri ew 4- ( "^ ^ \ ^ ^^^ -^ tifelb d;?- lo*? :X iibcf eci:i lo'^i enolsjjxonoo.v i/oiosjti ioa IB benxb^jdo e^txwee'i 9ri;t itacii b sixb i tl il -IxcrqqB boog -^XilBl b ^u:tt:tbnoo ^ cl;t ^bch d-x! b2 ;ti ;taai.noiexii.'j^f nl noxtfi/cil'id'exb ^uujb'iaqtns^t ibuiob ecii oi noxitam SnivXoe '^cf ioeldue ric^ no aob scf ^txiglm siiovr lediii/i lo \;;tixicfbcslvtob'tq rf;t ioedo oct «i bio nl msxcfoiq XauctoB hb OS

64 . the method used to determine Pp. By actual experiment on an engine the overall results of this work could he tested for accuracy and the percentage of error, if any could be found 21

65 be:tb9ct ecf bluoo liiom eldj 'io ecriusei IlBievo 3ii.-t eniana hjb d biwoo '^njb 'il «ioii9 lo agbitrreoieq arf* bnis "^jofiijjoojb io1 bni/ol 12

66 BIBLIOGRAPHY 1, Adams, O.L. Elements of Diesel Engineering. New York, Henley Publishing Co., # Boelter, L.M.K., and others. Heat Transfer Notes. Berkeley, University of California Press, Carslaw, H.S. The Conduction of Heat. London, Macmillan and Co., Gray, A., G.B. Mathews, and T.M. Macrobert. A Treatise on Bessel Functions. London, Macmillan and Co., Eepvjorth, J.L. Piston Assemblies for Road Transport Oil Engines. Engineering 168: London, December 9th., Ingersoll, L.R., and O.J. Zobel. Heat Conduction. New York, McGraw Hill, Jakob, M., and G.A. Hawkins, Elements of Heat Transfer and Insulation. New York, John Y/iley and Sons, McAdams, V/.E., Heat Transmission. New York, McGraw Hill, Pinkel, B. Heat Transfer Processes in Air-cooled Engine Cylinders. Washington, National Advisory Committee for Aeronautics, Report No. 612, Sokolnikoff, I.S., and E.S. Higher Mathematics for Engineers and Physicists. New York, McGraw Hill,

67 YIHAHCOUaie.ei-SI,6891^ BlrriolilfiO to \^^lsi vinu,^el93/i9a»nobfloj.;tb H lo aotioubnod erit S.H ^wjaleibc,»xsei foo bnfl njblilffioam,lz'til foo bfifl njalllmoj^,nobnoj zciol^oaifi leesaa flo 110 <tioqenbt[t bsofl lol eslicfniesba no;*el1.j.l ^d:i'lowq9e.3 iscfmaosq,nobnoj SC8-G29:89I snii3anisna[.eenlsna wew *nol:toubcjoo i&bh *l&dior.l.o bnb,.h.j ^Iloenssnl.8,81^GI,IIiH wbioom,3/ioy lolenbit ctbsh lo sinemels.»eni:?iwbh A.O bns»»m,cfo2ibl.v.0561,enos bnb Y -f-t^ nriol <:j/ioy w H.noIiJBli/eal bnb ^irttt wctrom,7/"ioy W9IT.nolea JtmBnBiT issh,.h.w,em3baom.8 enlgns belooo-ila nl eeeesooi*! i&lenbit ;tbeh.5,l9i[nl1.6 seid^lraraoo -^loblvba lbnol;tb}! ^no^tanldebw.eisbnll^o 8Cei,SI3.oH ^loqbfi,eol;tubnot:ea io1 lol soictamari.tbm lariglh.2.3 bas, t.a.i tlloj/lnloi/oa.01 I^ei,II1H wsisom,3jioy wslf.biblole^cfl bnb zieenl^as. SS

68 APPENDIX I GENERAL SOLUTION To find a general solution to Fourier^ s Law of Heat Conduction for the case where the temperature should be analyzed in the unsteady state for a body best described in terms of cylindrical coordinates, we use: Assume: ^._ ^ ('^ ) * /^ (/t) - f (^ J then; 0^^ <Xt ' a ckn?- a,/? Ji^ i. d^^ (2) Since r, z, and T are independent variables, we may say : -*- d^ 5 - a (3) 1 = t (4) Equation (3) may be rearranged as: which is Bessel»s equation of the first kind of order zero. 23

69 K0ITUIG2 JAH3;iH0 b&cflioseb ieecf Y^ocf jb lol e;t :J^B icbba^tenu aiict njt bes-^ijaaa (I) (^ ^ Ik I, in.\. li : i9dw inedct I ^BDi ew tealdslibv ^tnsbnsqabal eib T bns,s,1 eofxls (5) "^-, ^^ L ^^ \ J'.^ 5^^, ^A > (5) «1 X ^^ I :ea begnbiibei dcf '^jara (5) aol3&up^ ^ib ^ Via oi 5 labno lo bnlj/ ;tbill 9cii to not:ibup& e'leeeaa el rfolrfw p,^.

70 ) Then: 1^ ^ R CT^U'v) f 3 KU^) (6) Equation (4) may be rearranged as; which yields :?^Ccr>4-i + «^ '**-» ^v (7 Equation (5) may be rearranged as: which yields: ^ = E e (8) But, since: t = & ^ ^ * 2 we find using equations (6), (7), and (8), that: In this equation the constants a, b, A, B, C, D, and E must be found so as to satisfy the boundary conditions of the specific problem involved.

71 O) U*^«N +U*)«i>f\= ST :ne fp (V),^,.:*ivCL4.^^<^;^.^ : zblstx dotrtk ^k res begnbiibsi 9cf ^bki (3) nolibjjpa (8) 3 B - 9 teblei-^ riolriw S» ^ -. ^ = j :eonia,iija :;jbdct 4(8) bnb,(v) ^(o) enol^tsupa gnleu bnll ew ^Ui»).V S* c^^^ox^^'^^^^^^cl ^ ^Acv^d\ 3 5 '. j bnb ^a,0»a ta <a,b b jirii^'baoc ari^t aoictjejjps elrid" nl enol;tlbnoo -^Bbnuocf erf* ^TelctBe o;j eb ob bni^ol scf ;tejjnr 3.bevlovni melcfoiq olllosqe adct to

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76 OCT BIMOCRV Lacios The temperature distribution in an internal comi^ bustion engine piston. "^ Jg OCT 2 BINDERY Thesis P paiscios The temperature distribution in an internal combustion engine piston. U. 5. Ndvul P'.«t,c,'r«(iuale School JVIxiiUii-Av, California

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