THERMAL ANALYSIS OF PULSE DETONATION ENGINES RAGHU RAMAN GHANDIKOTA. Presented to the Faculty of the Graduate School of

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1 THERMAL ANALYSIS OF PULSE DETONATION ENGINES by RAGHU RAMAN GHANDIKOTA Pesened he Faculy f he Gaduae Schl f The Unvesy f Texas a Alngn n Paal Fulfllen f he Requeens f he Degee f MASTER OF SCIENCE IN AEROSPACE ENGINEERING THE UNIVERSITY OF TEXAS AT ARLINGTON DECEMBER 008

2 ACKNOWLEDGEMENTS I wuld lke ake hs ppuny acknwledge he effs and supp f he fllwng peple wh have helped e n he successful cplen f y Mase s hess. Fs, I wuld lke hank y supevsng pfess, D. Dnald Wlsn, Pfess and cha, Depaen f Mechancal Engneeng, y en n he pusu f y Mase s degee. I a gaeful f hs eseeed effs and gudance whch have ulaely helped e each hs lesne n y lfe. He gave e he ppuny f wkng n he Aedynacs Reseach Cene and ulaely ade e fala wh he cncep f he pulse denan engne. The cnceps augh by h n a beahng ppulsn and hypesnc a beahng ppulsn paved he way bee undesand y hess pc. Secndly I wuld lke hank D Haj Shek f hs valuable suggesn n he aea f hea ansfe Inspe f beng busy, he was always avalable f help n gvng e valuable suggesns n y hess pc and D Zhen Xue Han f hs supp and encuageen all hugh y ases pga and wh wh I enjyed wkng wh as a Teachng Asssan. I wuld als lke hank D Fank Lu, Dec f Aedynacs Reseach Cene f pvdng e asssanshp all hugh y Mase s pga. I wuld als lke hank y fahe Ghandka Muhy, y he Vjayalaksh and y bhe Rav f he lve and affecn all hugh y lfe. Nvebe 0, 008

3 ABSTRACT THERMAL ANALYSIS OF PULSE DETONATION ENGINES RAGHU RAMAN GHANDIKOTA, MS The Unvesy f Texas a Alngn, 008 Supevsng Pfess: D. Dnald Wlsn The pulse denan engne PDE s a cpac syse ha can be deled hedynacally by he Huphey cycle and s capable f hgh effcency. The apd hea elease f schec fuel/xdze as esuls n develpng hgh epeaues. The hgh epeaues pduced dung s pean can cause he engne fal. Thus, peang hs engne f a lnge duan wuld bece dubus f he epeaues ae n dsspaed f. The cuen eseach as pef a pelnay heal analyss. Ths analyss cnsss f w seps. The fs sep s deene he hea dsspaed f he engne usng he unseady sae analyss, and he secnd sep s develp a pelnay hea exchange desgn. A desgn ehd s develped and an exaple desgn f a hea exchange s ppsed.

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS. ABSTRACT LIST OF ILLUSTRATIONS v LIST OF TABLES...x LIST OF SYMBOLS..x Chape. INTRODUCTION.... Pulse Denan Engne..... Fllng Pcess..... CJ Denan Pcess Tayl Raefacn Pcess Refleced Raefacn and Exhaus Pcess Pugng Pcess...7. Objecve Of Cuen Reseach...8. Checal Equlbu Applcans Enegy Analyss f Unseady-Flw Pcesses...9. DETERMINATION OF THERMODYNAMIC PROPERTIES USING CHEMICAL EQUILIBRIUM APPLICATIONS CODE.... Inpu he CEA Cde... v

5 . Oupu f he CEA Cde...4. AVERAGE THERMODYNAMIC PROPERTIES FOR ONE CYCLE...5. Fllng Pcess...5. CJ Denan Pcess...5. Tayl Raefacn Wave Pcess Fn Bunday Cndns f Tayl Raefacn Wave End Wall Cndns f Tayl Raefacn Wave Aveage Cndns f Flw Dung Denan Pcess....4 Refleced Raefacn Pcess Inal Cndns f Reflecn Pcess Fnal Cndns f Reflecn Pcess Aveage Cndns f Flw Dung Reflecn Pcess Blw Dwn Pcess Inal Cndns f Blw Dwn Pcess Fnal Cndns f Blw Dwn Pcess Aveage Cndns f Flw Dung Blw Dwn Pcess Pugng Pcess Inal Cndns f Pugng Pcess Fnal Cndns f Pugng Pcess CALCULATION OF HEAT DISSIPATED FOR ONE CYCLE OF OPERATION...5 v

6 4. Applcan f Enegy Expessn The Pulse Denan Engne Fllng Pcess CJ Denan Pcess Tayl Raefacn Wave Pcess Refleced Raefacn Wave Pcess Blw Dwn Pcess Pugng Pcess CALCULATION OF WALL TEMPERATURES USING GREEN S FUNCTIONS Tansen Analyss Slun f Hea Cnducn Thugh Hllw Cylndes Calculan f Wall Tepeaues f One Cycle f Opean Wall Tepeaues Dung Fllng Pcess Wall Tepeaues Dung CJ Denan Pcess Wall Tepeaues Dung Tayl Raefacn Wave Pcess Wall Tepeaues Dung Refleced Raefacn Wave Pcess Wall Tepeaues Dung Blw Dwn Pcess Wall Tepeaues Dung Pugng Pcess Cpasn f Cppe and Seel Meals Cpasn f Tansen Analyss wh Seady Sae Analyss...6 v

7 5.4. Seady Sae Hea Cnducn Relans Seady sae values f wall Tepeaues CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK Cnclusns Recendans f Fuue Wk...68 REFERENCES...70 BIOGRAPHICAL INFORMATION...7 v

8 FIGURE LIST OF ILLUSTRATIONS Page. Blck Daga f a Tubje Engne.... Blck Daga f a Pulse Denan Engne.... Scheac f a Pulse Denan Cycle....4 Css Secn f a Denan Tube Secnal Vew f he Denan Tube...6. Inpu Secn f he CEA Cde f Hydgen and Oxygen.... Fuel Selecn Secn f he CEA Cde.... Oxdze Selecn Secn f he Cde....4 Type f Denan Selecn Secn f he CEA Cde....5 Execun Secn f he CEA CODE.... Vaan f Tepeaue Dung Tayl Wave Pcess...0. Vaan f Pessue Dung Tayl Wave Pcess...0. Vaan f Densy Dung Tayl Wave Pcess....4 Tepeaue Vaan Afe Tayl Wave Pcess....5 Densy Vaan Afe Tayl Wave Pcess....6 Pessue Vaan Afe Tayl Wave Pcess....7 Vaan n Tepeaue Dung Reflecn Pcess...5 v

9 .8 Vaan n Pessue Dung Reflecn Pcess Vaan n Densy Dung Reflecn Pces Vaan n Tepeaue Dung Blw Dwn pcess...9. Vaan n Pessue Dung Blw Dwn Pcess...9. Vaan n Densy Dung Blw Dwn Pcess...0. Pessue Vaan f Tayl Wave and Reflecn Pcesses....4 Pessue Vaan f Tayl Wave and Reflecn Pcesses Denan Tube Dung Fllng Pcess Denan Tube Dung Denan Pcess Denan Tube Dung Tayl Wave Pcess Denan Tube Dung Refleced Raefacn Pcess Denan Tube Dung Blw Dwn Pcess Denan Tube Dung Pugng Pcess Css Secnal Vew f he Denan Tube Tepeaue Dsbun Alng he Inne Walls f Denan Tube Cppe Tepeaue Dsbun Alng he Oue Walls f Denan Tube Cppe...65 x

10 LIST OF TABLES TABLE Page. Thedynac Ppees Afe CJ Denan Pcess CEA...4. Aveage Thedynac Ppees Dung Each Pcess Thedynac Ppees Dung he Fllng Pcess Thedynac Ppees Dung CJ Denan Pcess Thedynac Ppees Dung Tayl Raefacn Wave Pcess Thedynac Ppees Dung Refleced Raefacn Pcess Thedynac Ppees Dung Blw Dwn Pcess Thedynac Ppees Dung Pugng Pcess Te ped, Tepeaues and Hea Dsspaed Dung Each Pcess Wall Tepeaues Dung Each Pcess COPPER Wall Tepeaues dung Each Pcess STEEL...6 x

11 LIST OF SYMBOLS A Aea f he denan ube B C B nube Specfc hea f seel kj C p Specfc hea a cnsan pessue kg. K C Ceffcens f B nube f cylndes D Inne daee f he denan ube D Oue daee f he denan ube D Inne daee f he wae jacke D Oue daee f he wae jacke W h c Oveall hea ansfe ceffcen K J 0 Bessel Funcn. kg Mass flw ae f wae nsde he ube s. kg Mass flw ae a he nle s x

12 . kg s e Mass flw ae a he ex secn P Aveage pessue f he buned gas a P Pessue a he nle a Q Hea dsspaed by he PDE kj T Aveage ex epeaue f he gas K T cj CJ denan epeaue K T Tayl aefacn wave epeaue K T eflecn Aveage gas epeaue dung eflecn K T Blw dwn Aveage gas epeaue dung blw dwn K T Inle epeaue f he fuel and xdze xue K de Te aken f denan pcess s fll Te aken f fllng pcess s puge Te aken f pugng pcess s a Te aken f aefacn pcess s T, T w w Tepeaues f he nne and ue walls f he denan ube K x

13 T ν Aben epeaue 00 Kelvn Kneac vscsy v Inle velcy f he fuel xde xue s v e Ex velcy f he buned gas s V Inle and ex vlues f he denan ube,v e v fll Velcy a whch he denan ube s flled wh fuel v puge Velcy a whch a s blwn n he denan ube s x Refeence lcan nsde he denan ube x L Lengh f he denan ube x L Lengh f he denan ube kg ρ eavg Aveage ex densy f he buned gas ρ Inle densy f fuel and xde xue and kg H O γ * θ Ra f specfc heas. Nn-densnal paaee x

14 Subscps e Inle secn f he PDE Ex secn f he PDE Inal cndns nsde he denan ube Fnal cndns nsde he denan ube xv

15 CHAPTER INTRODUCTION. Pulse Denan Engne The pulse denan engnes ae a ype f engnes whch ae sple n cnsucn, lghe n wegh, have neglgble vng pas and pduce lage auns f hus. Much eseach has been caed u n hese ppulsn syses f any yeas due he nueus advanages hese engnes have ve adnal je engnes. Cpaed wh ubjes and ubfans, hese engnes ae sple n geey wh a lng ube, echncally eed he denan ube, whch s pen a ne end and clsed a he he end. The clsed end f he ube has valves cnl he flw f fuel and xdze xue Hydgen and Oxygen n hs case n he ube. These engnes nze he use f cpesss cpess he a befe cbusn, heeby educng he veall wegh and cplexy f he engne. Hweve a lw flgh velces a cpess s equed pup a n he chabe. The basc blck dagas f he ubje engne and he PDE ae shwn n he fllwng fgues.

16 COMPRESSOR COMBUSTOR TURBINE NOZZLE Fg. Blck Daga f a Tubje Engne CONTROL VALVES DETONATION TUBE NOZZLE Fg. Blck Daga f a Pulse Denan Engne The denan pcess, n addn pducng huge auns f hus, als ads n pducng hgh epeaues and pessues and hence hs engne has hgh effcences when cpaed wh he engnes. The fllwng fgue ulnes he vaus pcesses ha ae pa f he pulse denan cycle.

17 Fg. Scheac f a Pulse Denan Cycle The vaus pcesses ha ae llusaed n he abve fgue fg. ae explaned n deal n he fllwng secns. The Gas dynac analyss f he PDE cycle fllws he del develped by Takua End and Tsh Fujwaa Ref.... Fllng Pcess Ths s he s pcess ha akes place n he pulse denan cycle. Ths pcess s sply fllng he denan ube wh schec xue f fuel and xdze a a cean desgn velcy. The e aken f hs pcess shuld be nzed, because a slgh delay n fllng wuld have an pac n he ng f he eanng pcess and ulaely lead a delayed denan. The css secn f a denan ube s shwn belw and he fllng e s calculaed as fllws,

18 Fg.4 Css Secn f a Denan Tube If x L s he lengh f he denan ube, hen he fllng e s calculaed as, fll lengh f he ube x fllng Velcy v L fll The quany v fll s he velcy wh whch he xue f he fuel and xdze xue s puped n he denan ube.the value f v fll s assued be se paccally feasble value.e. 0/s, 0/s 00/s.The hghe he flw velcy, he lesse he fllng e and ulaely lesse cycle es. The value f fllng velcy shuld be caefully assued as f achevng hghe flw aes, a hghly effcen pupng syse shuld be ncpaed... CJ Denan Pcess Ths s he pcess akng place afe he fllng pcess. The vng xue s denaed a he clsed end f he denan ube, whch ceaes a wave whch ves a vey hgh velces wads he pen end. Cpaed wh he fllng pcess, hs pcess akes place a a facn f a llsecnd as hgh epeaues gases avel alng he ube wh velces ypcally a few husand ees pe secnd. The velcy wh 4

19 whch he denan wave avels f he pen end he clsed end s eed he denan velcy. The Denan e f hs wave s calculaed as lengh f he ube L I / [Fg.4] CJ wave Velcy Dcj x The value f D cj s baned f he NASA Checal Equlbu Applcans cde adaped f Ref. f a fuel and xdze xue when denaed... Tayl Raefacn Pcess Ths s he pcess whch s assued be akng place sulaneusly wh he denan wave. As he denan wave ves wads he pen end f he ube, he fuel and he xdze valves ae clsed and snce he gas aached he clsed end shuld be a es, he flw f gas beween he CJ denan wave and he clsed end s deceleaed. Ths decelean esuls n he genean f a aefacn wave whch fllws he denan wave. Snce he fn bunday f he Tayl wave cncdes wh ha f he CJ wave, he fn bunday cndns f hs wave ae sla he CJ cndns f he flw adaped f Ref.. The e aken by he alng edge f he aefacn wave each he end f he denan ube s Ref.. lengh f he ube Denan velcy 5 xl Dcj

20 ..4 Refleced Raefacn and Exhaus Pcess A e, he Tayl wave exs he denan ube. Dung hs pcess, anhe aefacn wave s geneaed whch sas ppagae back wads he clsed end. The fn bunday cndn avels a snc velcy and eaches he clsed end a II. The efleced aefacn wave s an expansn wave whch dps he pessue nsde he ube, bu he pessue s sll abve aben hus he flw cnnues evacuae he ube. Ths phase s called exhaus phase and lass f cean e ped III adaped f Ref.. Dung hs pcess, he buned gas s exhaused f he pen end f he ube. Als dung he e ped II III, he pessue s assued decay lnealy wh e unl eaches aben cndns. Hweve, he fnal epeaue f hs pcess s sll vey hgh. The e f aefacn pcess s calculaed as fllws Ref.. II lengh f he ube aefacn velcy xl Dcj 4 Fg.5 Secnal Vew f he Denan Tube III γ γ + γ xl Dcj

21 The e aken by he aefacn wave n avelng hugh he denan ube wll uch hghe han ha f he denan wave as he velces f he aefacn wave ae uch lwe han he denan wave...5 Pugng Pcess Ths s he fnal pcess n he pulse denan cycle. Ths akes place afe he aefacn pcess as shwn n fg.4. Ths pcess nvlves pupng a a hgh velces n he denan ube, n de clean he ube f any bun pues lef nsde he ube afe he denan and aefacn pcess. Ths pcess s vey val as n he absence f hs pcess, hgh epeaue pacles nsde he denan ube can cause he fuel-a xue au gne and ulaely lead deflagan. Ths pcess can be efeed as scavengng he denan ube befe he fllng pcess. The e aken f hs pcess s calculaed n he sae lnes as f he fllng pcess, he nly dffeence beng, n he lae, fuel and xdze xue s puped n he denan ube, whee as n he fe, nly a s puped fce he bun pacles u f he ube. Thus, pugng e s gven by lengh f he ube x puge pugng Velcy v L puge The value f v puge n he abve expessn, plays a val le as he pugng e s nvesely ppnal he pugng velcy. I s geneally pesued ha he value f 7

22 v puge s assued be se paccally feasble value.e. 0/s, 0/s 00/s.The hghe he puge velcy, he lwe he puge e and ulaely fase cycle es can be acheved.. Objecve Of Cuen Reseach Inspe f pssessng any advanages, f exaple; fase cycle es, neglgble vng pas, hgh hus pducn, he val fac affecng he paccal pleenan f PDEs s he hgh epeaues ha ae pduced dung he nal de f pean. The epeaues pduced dung ne cycle f pean, ay be vey hgh n he de f 000K.Thus n de ensue ha hese engnes peae f a lnge duan, ppe and effcen clng syse desgns shuld be ncpaed. The cuen eseach as cnduc a heal analyss f he engne and ppsng a pelnay hea exchange desgn.. Checal Equlbu Applcans Checal Equlbu Applcans Ref. s cpue pga whch calculaes he checal equlbu pduc cncenans and hedynac ppees f any se f eacans a a gven epeaue and pessue. The npu he cde s an equlbu xue f fuel and xdze whch ae aken be Hydgen and Oxygen a 00 Kelvn n he cuen analyss. The cde hen calculaes he hedynac ppees afe a 8

23 Chapan Jugue Denan, whch s ne ype f denan. The cde als calculaes all he hedynac ppees f he denaed xue..4 Enegy Analyss f Unseady-Flw Pcesses The pcess dung whch hee s a apd change n he flud ppees ve a cean e ped s geneally eed an unseady flw pcess. Whle analyzng hese knds f pcesses, s pan keep ack f he ass and enegy cnens f he cnl vlue as well as he enegy flux acss he bunday. The ass balance f any syse undegng any pcess can be expessed as u syse Whee s he change n he ass f he syse. F cnl syse fnal nal vlues, can be expessed as CV e Whee nle, e ex, nal sae, fnal sae. Ofen unseady flw pcesses can be deled as a unf sae, unf flw pcess, whch eques ha he flud ppees whn he cnl vlue vay nly wh e. The enegy balance elans f a unf flw syse ae expessed as adaped f Ref.. h h + u u Q W.4. e 9 Snce hee s n shaf wk paed he syse, neglecng W n he abve equan.4., he fnal enegy expessn f an unseady flw pcess s expessed as

24 h h + u u Q.4. e Whee h e ve C pte +.4. v h C pt u u v CvT v CvT

25 CHAPTER DETERMINATION OF THERMODYNAMIC PROPERTIES USING CHEMICAL EQUILIBRIUM APPLICATIONS CODE. Inpu he CEA Cde The npu he Checal Equlbu Applcans cde s he epeaue and pessue f a xue f hydgen and xygen. Inle Tepeaue: 00 K Inle Pessue: a The fllwng ages ulne a saple CEA applcan usng a xue f fuel/xdze. The ype f hedynac eacn desed s he Chapan Jugue denan as shwn hghlghed n he fllwng fgue. Fg. Inpu Secn f he CEA Cde f Hydgen and Oxygen

26 Fg. Fuel Selecn Secn f he CEA Cde Fg. Oxdze Selecn Secn f he Cde

27 Fg.4 Type f Denan Selecn Secn f he CEA Cde Fg.5 Execun Secn f he CEA CODE

28 . Oupu f he CEA Cde Afe eneng he npu values he cde, he cde calculaes he Chapan Jugue denan ppees f he xue. The ex values ae geneally dened wh a subscp CJ adaped f Ref. epesenng he values baned a Chapan Jugue denan. The values f hedynac ppees afe he Chapan Jugue denan ae abulaed belw. Table. Thedynac Ppees Afe CJ Denan Pcess CEA D cj p cj T cj ρ cj 85 /s 8.6 a 675 K 0.89 kg/ γ γ γ γ p c cj M R J/kg.K 4

29 CHAPTER AVERAGE THERMODYNAMIC PROPERTIES FOR ONE CYCLE. Fllng Pcess Inally, he ube s cpleely flled wh a unf denable xue f hydgen and xygen, he ppees f whch ae chaacezed by p a, T 00K and u 0 as he xue s a es. The e aken by hs pcess s expessed as xl 0. s.afe hs e ped, he denan ube s assued be v 50 / s fll 0 fll cpleely flled wh he fuel and xdze xue. The nal and fnal cndns f he flw dung hs pcess ae equal as hee ae n hedynac neacns akng place.. CJ Denan Pcess Afe he ube s cpleely flled wh he fuel and xdze xue, a shck wave s geneaed a he clsed end f he ube whch avels wh hgh velcy wads he pen end f he ube. Ths shck wave cpesses he fuel xue heeby esulng n enus ncease n he values f he hedynac ppees ahead f. The e aken by hs wave each he pen end f he ube s eed as he denan e. Ths e as explaned n he fs chape, s expessed as xl I /, whee x D L cj 5

30 and D cj 85 /s CJ wave velcy, hence I / s. The NASA 85 s CEA cde adaped f Ref. s ulzed calculae he ppees f he cbusn pducs afe CJ denan. The CJ denan ppees f he xue ae baned f Table. as T cj 675 K pcj 8.6 a kg ρ cj 0.89 The fllwng analyss s based n he End-Fujwaa del f analyzng he PDE Ref..The abve baned values f he CEA cde ae cpaed wh he fllwng Hugn elans f he CJ cndns deved f he End-Fujwaa del. ρ γ + ρ kg 0.86 γ.. p γ M γ + CJ p 8. a.. T p K.. ρ R The values f he hedynac ppees calculaed by he CEA cde ae appxaely equal he analycally calculaed values f he abve expessns 6

31 The Hugn elans ae used as a bass cpae wh he esuls f he CEA cde and als calculae he cndns f he flw dung he he pcesses.. Tayl Raefacn Wave Pcess As he denan wave avels f he clsed end he pen end f he ube, s fllwed by Tayl aefacn waves, caused by he end wall bunday cndns. Ths pcess lass f a cean e ped I / I, whee I / 0.5 s and I 0.7s.The leadng edge f he Tayl aefacn wave fllws edaely behnd he denan wave and hence he fn bunday cndns f hese waves ae equal ha f he CJ values... Fn Bunday Cndns f Tayl Raefacn Wave T K F equan.. p 8. a F equan.. kg ρ F equan.. The sae f he flw nsde he aefacn wave s wen by he elans f he self sla aefacn waves as adaped f Ref. γ γ + γ p x p.. γ γ x 7

32 γ γ ρ + x ρ.. γ γ x x x u u γ + The epeaue s calculaed f he equan f sae as.. T p..4 ρ R whee R 57 J kg. K f able. and p, ρ ae baned f... In he abve expessns, x s he psn f he CJ suface expessed as x Dcj and x s he vaable lengh f he alng edge he leadng edge f he Tayl wave. The x alng edge f he Tayl wave s assued be,hus when he leadng edge f he wave eaches he end f he ube x x L, he alng edge s a x xl. All he equed vaables ae subsued n he abve expessns and he ppees f he buned xue dung he Tayl aefacn wave ae calculaed a dffeen lcans nsde he denan chabe. The velcy a he alng edge f he Tayl wave s ze, and he hedynac ppees ae cnsan f he alng edge f he wave he end wall, and ae gven by he end wall values f T, p, ρ whch ae expessed by he fllwng elans. 8

33 p γ γ + γ γ γ + γ M cj p..5 γ + γ + γ ρ γ ρ..6 T p..7 ρ R.. End Wall Cndns f Tayl Raefacn Wave T p K 6.47 a kg ρ 0.4 The abve expessns....4 ae slved usng Maheaca and he values ae baned f hs pcess ve he lengh f he denan ube. All he baned values ae pled a he e I /.The vaan n he cndns f he flw dung hs pcess s shwn n Fg.. 9

34 Vaan f Tepeaue dung Tayl Wave Pcess Sees 700 Tcj Tepeaue K Tavg T dsance x Fg. Vaan f Tepeaue Dung Tayl Wave Pcess Vaan f Pessue f Pcj End wall Pessue P 0 8 Pcj 6 4 Pessue a 0 8 Pavg 6 P dsance x Fg. Vaan f Pessue Dung Tayl Wave Pcess 0

35 Vaan f densy f ρcj end wall densy ρ 0.9 ρcj ρ kg/ 0.5 ρ dsance x Fg. Vaan f Densy Dung Tayl Wave Pcess.. Aveage Cndns f Flw Dung Denan Pcess kg ρ 0.40 Als a e I 0. 7s, he Tayl wave exs he denan ube and hence he cndns f he gas dung hs ped wll be cnsan hughu he ube and wll be equal he end wall cndnst, p, ρ.the pls f he abve case ae shwn belw f efeence. T 97 K p 7.9 a

36 Tepeaue vaan afe Tayl wave exs he denan ube T 00 Tepeaue K 50 Sees Dsance x Fg.4 Tepeaue Vaan Afe Tayl Wave Pcess Vaan f densy afe Tayl wave exs denan ube ρ Densy Dsance x Fg.5 Densy Vaan Afe Tayl Wave Pcess

37 Vaan n pessue afe Tayl wave 7 P 6 5 Pessue a dsance x Fg.6 Pessue Vaan Afe Tayl Wave Pcess.4 Refleced Raefacn Pcess As he denan wave alng wh he Tayl aefacn waves each he pen end f he ube, a efleced wave s geneaed ha avels f he pen end back wads he clsed end. The value f T, p, ρ whch ae he end wall cndns afe he Tayl aefacn pcess, ae he nal cndns f hs pcess..4. Inal Cndns f Reflecn Pcess T K, p 6.47 a,

38 4 ρ 0.4 kg. The cndns f he flw dung he eflecn pcess ae gven by he fllwng expessns f a self sla aefacn wave. ex L cj p x x D p + γ γ γ.4. ex L cj x x D ρ γ ρ γ +.4. R p T ρ.4. The ex cndns f gas afe he aefacn wave ae wen by he elan f he self sla aefacn waves as fllws. + P M γ γ γ γ γ p cj ex.4.4 ρ γ γ γ γ ρ ex R p T ex ex ex ρ.4.6

39 .4. Fnal Cndns f Reflecn Pcess T ex K p ex. 6a kg ρ ex 0. Sla., he aveage values dung hs pcess ae baned usng Maheaca. The nal cndns ae baned f.4. and he fnal cndns ae baned f secn.4.. The fllwng pfles ae baned f all he abve expessns and he vaan n cndns f he flw dung he eflecn pcess s shwn belw f efeence. Vaan n Tepeaue dung Reflecn Pcess T Tepeaue K s Tex Dsance Fg.7 Vaan n Tepeaue Dung Reflecn Pcess 5

40 Vaan f Pessue dung Reflecn pcess P Pessue a s Dsance Pex Fg.8 Vaan n Pessue Dung Reflecn Pcess Vaan f Densy dung Reflecn pcess ρ Densy s 0.5 ρex Dsance Fg.9 Vaan n Densy Dung Reflecn Pcess 6

41 .4. Aveage Cndns f Flw Dung Reflecn Pcess T K p a ρ 0. kg.5 Blw Dwn Pcess The efleced aefacn wave afe clldng wh he clsed end f he ube ges efleced he pen end nce agan and hence n hs pcess anhe aefacn wave s geneaed and sas ppagae hugh he ube f he clsed end he pen end.the e aken by he aefacn wave wll be uch hghe han he e aken by he denan wave whch s evden n he velcy values f he w waves. The e s hus gven as III γ + + γ + γ x D L cj.6 s..5. Inal Cndns f Blw Dwn Pcess T K, p 6.47 a & ρ kg. Dung he blw dwn pcess, he gas s assued be senpc and hence senpc elans ae used deene he end cndns f he flw. Dung hs pcess, he pessue s assued decay lnealy f an nal value f p aben cndns. The epeaue and densy decay ae calculaed f senpc elans. All he

42 baned values ae hen aveaged f he ene pcess. Hence he end cndns f hs pcess ae gven by he senpc elans expessed as p p ` ρ ρ γ T T γ γ.5. p.5. II p p ab III II γ p ρ ρ.5. p γ p γ T T p.5.4 The e vaable, n he abve expessns s vaed f II III, whee II.45 s and III.6 s. When pled wh espec e, he pessue decay ccus n a lnea anne wads he pen end f he ube and he pessue f he flw nsde he ube s aben a III.6 s..5. Fnal Cndns f Blw Dwn Pcess T 7. 9 K kg ρ 0. 8

43 p a The vaan n he ppees dung he aefacn e ped II III s pled. Vaan f Tepeaue dung Blw dwn pcess T Tepeaue K >.6 s e ped.45.6 ll secnds Tend Fg.0 Vaan n Tepeaue Dung Blw Dwn pcess Vaan f Pessue dung he Blw dwn pcess.45 ->.6 s 7 6 P 5 Pesue 4 Pavg Pab e ped.45s<<.6s Fg. Vaan n Pessue Dung Blw Dwn Pcess 9

44 Vaan f densy dung Blw dwn pcess 0.5 ρ 0. Densy s e ped llsecnds ρend Fg. Vaan n Densy Dung Blw Dwn Pcess F he pls dung he Blw dwn pcess, he values f pessue and densy afe he pcess ae als equal he nal aben cndns. Hweve, he epeaues sll ean a uch hghe value and d n each he aben cndns..5. Aveage Cndns f Flw Dung Blw Dwn Pcess T K p.65a kg ρ Pugng Pcess The values f he hedynac ppees a he end f he pugng pcess ae assued be aben as hgh velcy a s beng puped n he denan ube 0

45 scavenge he ube befe he fllng pcess and als cl he ube afe he hgh epeaue pcesses ha ake place befe puge..6. Inal Cndns f Pugng Pcess T 7.9 kg ρ 0. p a K The pugng e as explaned n chape s expessed as x L puge 0.0 s. I s V fll assued ha afe he pugng pcess, he cndns nsde he denan chabe ae aben as hgh velcy a cls he denan ube. Hence he fnal cndns nsde he ube ae aben gven by.6. Fnal Cndns f Pugng Pcess T 00 K p a kg ρ.6 a The fllwng fgues ulne he pfles f he cndns f he flw f all he ppees f he flw sang f denan pcess up pugng.the Tepeaue pessue and Densy pfles ae pled wh espec he dsance he lengh f he denan ube.

46 Vaan f Pessue alng he lengh f he denan ube f ne cycle f pean 0 8 Pcj 0.5 s 6 Pessue a P 0.7 s Pavg 4 Pex.45 s Dsance x Pcj P s P 0.7 s P Pex s Fg. Pessue Vaan f Tayl Wave and Reflecn Pcesses In he abve pl, can be clealy bseved ha he pessue afe CJ denan educes gadually up he d secn f he denan ube. Beynd he d secn, p cj he pessue beces he end wall pessue afe 0. 7s.Dung he eflecn pcess, he pessue decays f p pex whch s he end cndn f eflecn. Ulaely dung he blw dwn pcess as shwn Fg. he pessue f p decays lnealy p ab whch s appxaely equal a.

47 Tepeaue Vaan f Tayl wave and Reflecn pcesses Tcj 0.5 s T 0.7 s Tex.45 s Tcj 0.5 s 500 Tepeaue K T 0.7 s Tex.45 s dsance x Fg.4 Pessue Vaan f Tayl Wave and Reflecn Pcesses The epeaue dsbun fllws he sae vaan as n he case f pessue. The epeaue decays f a hgh value f Tcj up he d secn f he denan ube. Beynd he d secn f he ube, eaches he end wall epeaue T and eans cnsan unl 0. 7s.Dung eflecn; decays T ex as epesened n he gaph. Hweve afe he blw dwn pcess f fg.0, he epeaue nsde he denan chabe s sll vey hgh and des n each aben cndns lke n he case f pessue unl he ube s scavenged dung he puge pcess. The values f all pcesses ae abulaed as shwn n he fllwng able..

48 Table. Aveage Thedynac Ppees Dung Each Pcess Pcess Aveage Tepeaue K Aveage Pessue a Aveage kg Densy Fllng CJ Denan Tayl Raefacn Refleced Raefacn Blw Dwn Pugng Pcess a

49 CHAPTER 4 CALCULATION OF HEAT DISSIPATED FOR ONE CYCLE OF OPERATION The enegy expessn f an unseady flw pcess s gven by equan.4.6 be h h + u u Q 4. e 4. Applcan f Enegy Expessn The Pulse Denan Engne The vaus pcesses ha ake place n he pean f he pulsed denan cycle ae. Fllng Pcess. CJ Denan Pcess. Tayl Raefacn Wave Pcess 4. Refleced Raefacn Wave Pcess 5. Blw Dwn Pcess 6. Pugng Pcess The nle values f he hydgen and xygen xue ha s puped n he denan ube wh a desgn velcy f 50 / s ae pessue p a, kg Tepeaue T 00K and densy ρ The vlue π xl f a denan ube f 0.05 adus and lengh. Als he geey f he denan ube s assued be unf hughu and hence V Ve V The suface aea f he denan ube A πx L

50 R γ R AlsC v, C p whee R 0.57 γ γ kj kg. K and γ.8 f able. and hence C v kj kg. K and C p 5.48 kj kg. K.The values f Cv and C p calculaed n hs secn ae used n he hea analyss dung he denan pcess unl he pugng pcess. F calculans dung he fllng pcess, he values f baned f leaue f hydgen a 00 K. 4.. Fllng Pcess, h F/O C p and C v ae, ua, u F/O, h a e e Fg 4. Denan Tube Dung Fllng Pcess As depced n he abve fgue, dung hs pcess, he fuel a xue s puped n he denan ube whch n un fces he a pesen nsde he ube wads he pen end f he ube. The ass f he fuel-a xue puped n he ube s and he ass f a whch s fced u f he ube s. The nal cndns nsde he ube ae he ppees f aben a and he fnal cndns ae hse f he fuel-a xue snce a he end f he fllng pcess, he ube s cpleely flled wh he fuela xue. Hence Eq 4. can be ewen as e 6

51 Q h h +, whee e e u u v e ehe ρ av CPTe v h ρ F / OV CPT V a C V T a u ρ 4.. V F / O CVT F O C p and C v f Hydgen a 00 K 4.07 u ρ 4..4 / kj kj, 0.8.Slaly f kg. K kg. K a, C p kj kj.005, Cv 0.78.The nenal enegy n he abve expessn s kg. K kg. K calculaed usng he expessn u CVT. Als n hs case, as he ass f he fuel puped n he ube s he fnal ass ha s sed n he syse befe denan and e. The nle velcy v 50. I s assued ha he nle and he ex s velces ae equal v v. All he ppees f he fuel and a ae subsued n he e equans and he enegy ansfeed f he cnl vlue s calculaed. The fllwng able ulnes he ppees f he a and fuel a xue dung he fllng pcess and he hea dsspaed f he syse dung he fllng pcess. 7

52 Table 4. Thedynac Ppees Dung he Fllng Pcess F/O ρ F/O V V e v e a kg kg T T ρ e a / s kg T T e fllng Q kj kg K 00 K 0.0 s The hea ansfeed f he denan ube s calculaed be kj.the e aken by hs pcess s 0.0 s. Ulaely, he hea dsspan f he denan ube dung he fllng pcess s calculaed be s Q kj kj kj Q & KW 0.0 s s,whch s equal MW. fllng 4.. CJ Denan Pcess 0, 00K, u Buned T Denan e 0 Fg 4. Denan Tube Dung Denan Pcess Cnsdeng Eq 4., Q h h + u u e 8

53 As shwn n he abve fgue, dung hs phase,he nle and ex es ae cpleely negleced n equan 4. and hence he enegy equan can be ewen as u u Q.Als snce hee s n change n he ass nsde dung he denan phase, he dffeenal change n ass wh espec e s ze and hence. The abve enegy expessn can be ewen as Q u u, whee u v v CVT, whee T avg T cj T andu CVT + + hpr. In hese w +, expessns, v v 0 as he xue s assued be a es befe denan and hence knec eneges dung he pcess ae negleced and h PR f hydgen fuel s 4,800 kj.the ass nsde he denan ube s kg f fllng kg kj pcess. Inenal enegy u u fllng pcess + h PR and u C V T kg. kj The fllwng able ulnes he vaus ppees f he fuel hydgen kg xde xue befe and afe denan. The values f he aveage epeaues dung fllng and denan ae baned f Table.. 9

54 Table 4. Thedynac Ppees Dung CJ Denan Pcess kg e kg T K T K kg C C u kj kg u kj kg h PR kj kg Q kj Q & MW The hea ansfeed f he denan ube n hs pcess s calculaed be -5.9 kj. The denan e s secnds. Ulaely he hea dsspan ae dung hs pcess. Q s calculaed be 5.9kJ 6886 s s s Q kj kj Q & KW whch s equvalen - fllng 6 MW. 4.. Tayl Raefacn Wave Pcess 0, u@ T de Fg. Denan ube dung wave pce e Fg 4. Denan Tube Dung Tayl Wave Pcess 40

55 Cnsdeng Eq 4., Q h h + u u e Ths pcess s assued be akng place nsananeusly afe he denan pcess. Sla he abve pcess, he nle es n he abve expessn ae negleced and he abve expessn s ewen as Q h + u u. In hs equan, he ex e es ae hse f he buned gases whch leave he denan ube f he pen end. The sae pcess s epeaed as n he case f denan, he nly dffeence beng he pesence f he ex e n he enegy equan. The ex ass e. Dung hs pcess, e 4..5 ve h e C PTe de 4..7 Als u u denan, as hs pcess s assued be akng sulaneusly wh he CJ denan pcess, u v CVT,, whee + v 6. ρ A s. The fllwng able ulnes he vaus ppees f he fuel a xue dung hs pcess. 4

56 Table 4. Thedynac Ppees Dung Tayl Raefacn Wave Pcess de kg e kg C v, C p kj kg. K kg T e T K , T K T K kj u kj u kj ehe The hea ansfeed f he denan ube n hs pcess s calculaed be kj. The e s secnds. Ulaely he pwe dsspaed he hea dsspan ae dung hs pcess. Q s calculaed be Q kj Q& 0.65kJ s 0. s fllng 0007 kj KW, whch s equal MW. s 4..4 Refleced Raefacn Wave Pcess 0, T Tayl wave e 0 T eflecn Fg 4.4 Denan Tube Dung Refleced Raefacn Pcess 4

57 Cnsdeng Q h h + u u e Dung hs pcess, a efleced wave avels f he pen end he clsed end f he denan ube. The nle es n he enegy balance equan ae negleced as hee s n fuel nake n he syse. The enegy balance equan f hs pcess s hus gven as Q h + u u.dung hs pcess, hgh epeaue gases ex e he ube and hence he ex es cann be negleced. Sla he abve pcesses, he hedynac ppees a sae dung hs pcess ae he fnal ppees dung he Tayl aefacn pcess. The ex velces dung hs pcess, ae calculaed usng he ass flw paaee elan whch s expessed as. e ρ A v e e e. v e e,& ρ A e. e v. e whee e ρ eve 4..8 ρ A ρ 4..9 V u u v CvT u Tayl aefacn pcess 4

58 Table 4.4 Thedynac Ppees Dung Refleced Raefacn Pcess Tayl e kg C v, C p kj kg. K kg T e K pcess kg , T K T K h kj e e u kj u kj The hea ansfeed f he denan ube n hs pcess s calculaed be kj. The e s secnds. Ulaely he pwe dsspaed he hea dsspan ae dung hs pcess. Q s calculaed Q kj beq& s fllng.9kj s 889 kj s. 889 KW MW. 44

59 4..5 Blw Dwn Pcess 0, T eflecn, T blwdwn e >0 Fg 4.5 Denan Tube Dung Blw Dwn Pcess Cnsdeng Eq 4., Q h h + u u.the efleced aefacn e wave afe avesng f he pen end he clsed end f he denan ube eflecs back he pen end afe clldng wh he clsed end f he ube. Sla he abve pcess, he nle es n he abve expessn ae negleced and he abve expessn s ewen as Q h + u u. The nenal enegy e u v + CvT kj v and kj u CvT kg kg Table 4.5 Thedynac Ppees Dung Blw Dwn Pcess kg e kg C v, C p kj kg. K kg T e K , T K T K h kj e e u kj u kj

60 The hea ansfeed f he denan ube n hs pcess s calculaed be kj. The e s secnds. Ulaely he pwe dsspaed he hea dsspan ae dung hs pcess. Q s calculaed Q kj beq& s fllng.7 kj s 4..6 Pugng Pcess 56. kj s. 56 KW 0.56 MW. A a 00K,u T blwd e F / O,u T ab Fg 4.6 Denan Tube Dung Pugng Pcess The an a f pleenng hs pcess edaely afe he blw dwn pcess s fuhe cl he denan ube. Aben a s puped n he ube a cean desgn velces whch ads n bngng he epeaue nsde he denan chabe f a vey hgh value an aben value. Als, hs pcess can als be used scavenge he ube as he hgh velcy a fces all he buned cbusn pducs whch ae lef ve nsde he ube. In he absence f hs pcess, esdual hgh epeaue cbusn pducs wuld au-gne he fuel as enes he denan ube and ulaely lead deflagan. The enegy balance expessn n hs case s sla ha f he fllng pcess, he nly dffeence beng ha aben a s puped n he 46

61 ube eplacng fuel/xdze xue n he fllng pcess. The enegy balance expessn s expessed as Q h h +, whee e e u u v e ehe ρ F / OV CPTe v h ρ av CPT v u V a CVT ρ a u V ρ F / O v CVT + F / 4..4 The ppees a he nle ae he ppees f he aben a a 00 Kelvn epeaue and hence he ass a he nle s expessed as ρ V, whee kg ρ.64 and V and he nle ass s calculaed be kg. The nle enhalpy s calculaed usng he elan v h CPT + and kj calculaed be The ppees a sae ae baned dung he blw dwn kg pcess. 47

62 Table 4.6 Thedynac Ppees Dung Pugng Pcess a V V e v v e F/O kg e kg / s T T K T Te K s Q kj The hea ansfeed f he denan ube n hs pcess s calculaed be kj. The e s 0.0 secnds. Ulaely he pwe dsspaed he hea dsspan ae dung hs pcess. Q s calculaed be s Q kj Q& fllng 8.509kJ 0.0 s 407 kj s 407 KW MW. The al hea dsspaed f he denan ube f ne cycle f pean Q -5.6 kj.the hea dsspan ae f ne cycle f pean Q & Q cycle fll de Tayl Q eflecn Blwdwn puge MW. 48

63 CHAPTER 5 CALCULATION OF WALL TEMPERATURES USING GREEN S FUNCTIONS The bjecve f hs secn s pesen pcedues f deenan f epeaue whn he walls f he denan chabe. The denan ube has an nne adus f and an ue adus f ο, as shwn n he fllwng fgue 5.. The denan ube s assued be ade up f Cppe. The suface f hs enclsue, a ο lcan, s beng cled by a clan whch s wae n hs case. The nne walls f he denan ube face vayng epeaues dependng n he pcess akng place. y Denan zne x Clng flud Fg. 5. Css Secnal Vew f he Denan Tube 49

64 F a cylnde as shwn n he abve Fg. 5. wh an nne adus f whch s nally a epeaue and due a hedynac pcesses ccung nsde, apd changes n epeaue akes place acss s suface due bh cnducn hugh he walls and als due cnvecn due he flw f clan nsde. The ype f cnvecn eplyed n hs case s fced cnvecn as he clan s puped wh a cean velcy dsspae he hea f he walls f he cylnde. An esan f he epeaue slun ay be baned f he e dependence f he adal epeaue dsbun. Thee ae w sluns avalable f hs knd f pble. The fs ne beng he ansen analyss slun and he he beng he seady sae sluns. The fllwng ae he assupns n hs analyss.. The- physcal ppees ae ndependen f epeaue.. Axal cnducn n alng he walls f he denan chabe s negleced. Q& W. The avng hea flux q& A pcess and A 0.64., whee Q & s baned f Table 5. f each 4. Hea flux s cnsdeed be unf dung he denan ped. 5. Tansen Analyss Slun f Hea Cnducn Thugh Hllw Cylndes The slun f he epeaue dsbun f he abve enned case usng Geen s Funcns adaped f Ref. 5 f a fne bdy wh hgeneus bunday cndns as 50

65 5 + d d g G k d F G T 0,,,,0,, τ ο ο τ τ τ πα π 5.. In he abve equan, he nn ze nal epeaue dsbun x F s equaed ze as n he fllwng analyss, s assued ha he educed suface epeaue T T s a ze nally dung he begnnng f he pcess and hence he abve equan s ewen as d d g G k T x T 0,,,, τ τ τ τ πα 5.. In he abve equan, q a g & δ and,, τ G s he Geen s Funcn slun f a hllw cylnde. The geen s funcn sasfes he hgenus bunday cndns whch s descbed by he cndns 0 G a, 0 + hg G k a gven by + 4,, / V Y J S V Y J S V J B J e a G a β β β β β β β β π τ τ α β 5.. whee, Υ + Υ B S β β β, + BJ J V β β β, q& he hea flux dung each pcess and s he nne adus f he denan ube.

66 5 Subsung all he expessns n he abve equan, he epeaue dsbun s expessed as a d q G k T T 0, :, τ δ τ πα & 5..4 d q G k a T T 0, :, τ τ τ α π & 5..5 Nw subsung he value f he Geen s Funcns expessn as gven by equan 5.. n 5..5, τ β β β β β β β β π α π τ α β τ d q Y V J S Y V J S V J B J e a k T T & / 5..6 Tha can be wen as τ β β β β β β β β τ π α π τ α β ατ β d q Y V J S Y V J S V J B J d e e k T T / / 5..7 Ulaely, he epeaue expessn f ansen hea cnducn hugh hllw cylndes s expessed as + / Y V J S Y V J S V J B J e q T T k β β β β β β β β π α β 5..8

67 Ulzng he abve equan, he lcal values f he wall epeaues dung each pcess can be baned. In he abve expessn, k s he heal cnducvy f W cppe whch s 400, s he nne adus f he wae jacke, K q s he wall hea flux dung each pcess q Q& W, he e aken by each pcess. The A fllwng able 5. ulnes he e ped, he aveage epeaues and he hea dsspan ae dung each pcess. Table 5. Te ped, Tepeaues and Hea Dsspaed Dung Each Pcess Pcess Te ped s Tepeaues T g K Hea Dsspan ae MW Fllng CJ Denan Tayl aefacn Reflecn Blw dwn Pugng

68 The equan 5..8 s hen appled ban he value f wall epeaues dung vaus pcesses ha ake place dung he pulse denan cycle. Due he cplexy f calculans nvlved, he ene expessn s gven as an npu Maheaca, whch calculaes he wall epeaues dung each pcess.e. fllng, denan, aefacn and pugng. The npu daa he Maheaca cde n hs case s he fuel and xdze xue a aben cndns. Wae s assued be flwng aund he denan ube as shwn n fg 5. a an assued desgn velcy 5.4. s The naue f he wae flwng,.e. ehe ubulen lana s deened by ρvd calculang he Reynlds nube whch s expessed as R, Whee, v ed µ kg 5.4, ρ 000 s, D s he hydaulc daee expessed as D0 D N. s 0.4 andµ s he abslue vscsy f he flud 8.55e-4. The Reynlds nube s calculaed be.5e+06, whch s geae han he ypcally assued value f 0,000 equed f ubulen flw and hence he naue f wae flwng s cpleely ubulen. The Nussel nube N ud f ubulen flw n ccula ubes adaped f Ref.5 f a gven Pandl nube P f 5.8 f wae a 00 K and Reynlds nube R ed f.5e+06 s calculaed be appxaely Velcy V/s whch s assued be an epcal desgn value f 5.4 s 54

69 W. Theal cnducvy k, 6.e 0 f wae a 00 K K kg. Densy ρ 000 f wae whch s he clan n hs case, 4. Hydaulc Daee D h 0.4 The veall hea ansfe ceffcen s calculaed be h 90 W K. 5. Calculan f Wall Tepeaues f One Cycle f Opean The velcy f wae flwng hugh he ubes s kep cnsan hughu he ene cycle and hence a cnsan value f hea ansfe ceffcen s ananed. As shwn n fg 5., he ue wall f he denan ube s n cnac wh he clng wae whle he nne walls face he hgh epeaue gases and hence he ue walls ae uch cle han he nne walls. The values f Tw and T w ae calculaed f each pcess sang f fllng up pugng and hen all he baned values ae aveaged f he ene cycle e. The npu paaees ae. h c 90 W K,. T c Tg, Whee, T g s gven f he able 5. f each pcess and. p Te aken f a pcess baned f able 5. fllng, CJ Denan, Tayl aefacn, Refleced aefacn, Blw dwn and pugng, end s fll + de + Tayl + eflecn + Blwdwn + puge.the enegy eleased s Q & n was and s baned f Table 5. The pesen analyss s cnduced by selecng 55

70 w aeals naely Cppe and Seel AISI 04.The cea f selecng w dffeen aeals s he behav f hese eals dung hgh epeaues. The heal cnducvy f seel a 00 K s 4.9 W. K and ha f cppe s 40 W. K.The value f hc s ananed cnsan f all he pcesses. 5.. Wall Tepeaues Dung Fllng Pcess The npu paaees ae h c 90 W K, T g T 00K,, c pcess 0. 0 s, cyc s.the nube f pulses n p n hs case s.the heal cnducvy k f cppe a 00 K 40 W. K and Q& q& whee Q & s baned f Table 5. f A fllng pcess and A s he aea f he denan ube whch s equal 0.4. The dffeenal e ped s p whee p s he e aken by he fllng pcess. The 0 aveage epeaue dung hs pcess s baned usng he cuve fng echnques and hence he epeaues baned dung he fs half f he cycle ae added and ulpled by and he eanng epeaues ae added gehe and ulpled wh whch s equal e 0 p.the ange f epeaues befe and afe he dffeenal e ped ae negleced as he dffeenal change n epeaues f ne cycle s neglgble. The values baned f bh he cases ae added gehe and dvded by he cycle e s s ban he aveage nne wall epeaue 56

71 whch s K. The sae pcedue s caed u n he calculans f he ue wall epeaues, he nly dffeence beng n hs case s ha.the ue wall epeaue dung hs pcess s calculaed be 00 K. 5.. Wall Tepeaues Dung CJ Denan Pcess The nal wall epeaue f hs pcess s he fnal wall epeaue afe he fllng pcess. The npu paaees ae h c 90 W K, s pcess , cyc s and heal cnducvy k f cppe a 00 K 40 W. K Q& and q& A whee Q & s baned f Table 5. f CJ denan pcess and A s he aea f he denan ube whch s equal 0.4.The aveage wall epeaues baned dung hs pcess T w 6. 6 K.In spe f huge enegy elease dung hs pcess, he wall epeaues d n vay uch n hs pcess. Ths can be abued he fac ha he e aken by hs pcess s vey sall and als due he pesence f hgh velcy wae clng he denan ube n he he sde. The hgh heal cnducvy f cppe als plays a val le n he esul f lw epeaue se dung hs pcess.hweve f peaed f epeve cycles, he epeaues ncease lnealy wh e.the ue wall epeaue dung hs pcess s calculaed be 00.0 K. 57

72 5.. Wall Tepeaues Dung Tayl Raefacn Wave Pcess The nal wall epeaues ae baned f he fnal cndns f he denan pcess. The npu paaees ae h c 90 W K, s pcess , cyc s, Tc Tg 56 K and heal cnducvy k f cppe a 00 K 40 W. K and Q& q& whee Q & s baned f Table 5. f Tayl aefacn pcess and A s he A aea f he denan ube whch s equal 0.4.The sae pcedue s epeaed as n he case f Denan pcess and he aveage wall epeaues dung hs pcess ae Tw 6. K and T w 00K Wall Tepeaues Dung Refleced Raefacn Wave Pcess The nal epeaues f he walls ae he fnal epeaues dung he Tayl wave pcess. The npu paaees ae sla ha f he abve pcesses, he nly dffeence beng n he e ped and epeaues dung hs pcess T K, pcess s, cyc s and Q& q& whee Q & s baned A f Table 5. f eflecn pcess and A s he aea f he denan ube whch s equal 0.4. The aveage wall epeaues dung hs pcess ae T 6.8 K, T 00 K.The wall epeaues pduced dung hs pcess ae w w 58

73 hghe when cpaed wh he denan pcess due he fac ha he e aken by hs pcess s uch hghe han denan pcess Wall Tepeaues Dung Blw Dwn Pcess The npu paaees ae h c 90 W K, T g T K, c pcess s cyc s and heal cnducvy k f cppe a 00 K 40 W. K and Q& q& A whee Q & s baned f Table 5. f blw dwn pcess and A s he aea f he denan ube whch s equal 0.4 The aveage wall epeaues pduced dung hs pcess ae T w K and T 00. w K 5..6 Wall Tepeaues Dung Pugng Pcess The wall epeaues f hs pcess wll be equal ha f he fllng pcess as he cndns f he flw dung hs pcess ae aben and hence Tg Tavg 00 K.The npu paaees ae h c 90 W K, T g c 00 K, pcess 0. 0 s, cyc s and heal cnducvy k f cppe a 00 K 40 W. K and Q& q& whee Q & s A baned f Table 5. f pugng and A s he aea f he denan ube whch s equal 0.4. The wall epeaues dung hs pcess ae T w K and T w 00 K.All he aveaged wall epeaues f all he pcesses f a denan ube ade f cppe eal ae abulaed. 59

74 Table 5. Wall Tepeaues Dung Each Pcess COPPER Pcess w T T w Fllng CJ Denan Tayl Raefacn Reflecn Blw Dwn Pugng The pesen analyss s caed u f ne cycle f pean and he values f wall epeaues f Table 5.4 clealy sugges ha he epeaues f he wall vay vey slwly wh e due he fllwng cndns. The e aken f ne cycle f pean s s and he e aken by he denan pcess whch geneaes he axu hea s s. Ths suggess ha he denan e s less han % f he al cycle e. The eanng pcesses geneae hea whch s /0 h f he facn f he hea dsspaed dung he denan pcess.. The velcy f he CJ wave s 85 s, whch suggess ha he wave avels and exs he ube even befe he wall aeal can sense a change n epeaues. 60

75 . The clng edu whch s wae n hs case s assued be flwng hugh he wae jacke suundng he denan ube a hgh velces, W 5.4 and hus hgh value f h c 90. s. K 4. The eal assued n hs case s cppe whch has hgh heal cnducvy f W K I s evden f he abve cases ha a epeaues change acss he walls s n sgnfcan f ne cycle f pean. The wall epeaue espnds vey slwly wh e and afe a few cycles sas ncease lnealy wh e. In de acheve geae epeaue vaans, he nube f pulses can be nceened a uch geae value say 000 n de ban a seady sae slun. The aveaged values f he wall epeaues f a denan ube ade up f cppe and f cycle f pean ae T w K and T w K 5. Cpasn f Cppe and Seel Meals The pesen analyss als as n cpang dffeen aeals used n he cnsucn f he denan ube. The eal such seleced shuld be capable f whsandng such huge epeaues ha ae eleased dung he pulsed denan cycle. In de whsand such huge epeaues, he eal seleced shuld have hgh heal cnducvy n de allw he hea flw hugh. Hence all he abve baned values f wall epeaues f cppe ae cpaed wh hse f Seel AISI04.The nly dffeence be ade n he analyss wh seel ae ha f he ppees f seel a 6

76 00 K.The heal cnducvy f seel s 4.9 W. K kg, he densy ρ 7900 and C J 477 p kg.. I s pey evden ha he wall epeaues pduced whle usng K seel wuld be hgh when cpaed cppe as he heal cnducvy f seel s uch lwe han cppe. The fllwng able 5.5 ulnes he aveaged wall epeaues dung each pcess f a denan ube ade f seel f ne pulse. Table 5. Wall Tepeaues Dung Each Pcess STEEL Pcess w T T w Fllng CJ Denan Tayl Raefacn Reflecn Blw Dwn Pugng Upn cpang he values f ables 5. and 5., he wall epeaues ae uch hghe n he case f seel f ne cycle f pean. Ths suggess ha epeaues sgnfcanly se f seel afe few cycles and hence seel cann whsand such huge epeaues pduced f lnge duan f e. Hence f he abve analyss, cppe s pefeed ve seel due s hgh heal capacy whch s evden f he 6

77 lwe epeaues. The aveaged wall epeaues n hs case ae T w K and T w 00.0 K. 5.4 Cpasn f Tansen Analyss wh Seady Sae Analyss 5.4. Seady Sae Hea Cnducn Relans Gven he naue f he pulse denan cycle, wh evey pcess akng place n a facn f llsecnds, hee wuld n be any vaan n he wall epeaues nally f a few cycles. Hweve as he value f s nceened.e. hee ae a epeve nube f pulses akng place ve a lng duan f e hee wuld be a subsanal ncease n he wall epeaue. Ths pcess f epeve pulses wll ulaely cause hgh wall epeaues afe a cean e ped. The wall epeaues ncease expnenally wh e and afe a cean e ped bece cnsan and d n vay wh e. These epeaues ae knw as seady sae epeaues and ae baned usng he seady sae cnducn elans. The classcal cnducn slun unde he seady sae cndn yelds Q& k T T + ln / + k h 6.. Afe seng, ne bans T k T Q& + ln / k h Ths s a anscendenal equan ha pduces T and hen he epeaue feld a he lcans s banable usng he abve equans as 6

78 T T ln / + k / h Q& 6.. T T ln / + k / h The sae cndns exs nsde he denan ube when he e ped. Ulzng he abve equan 6.., he seady sae values f he wall epeaues can be baned. The aveage hea dsspaed s calculaed as fllws. Q& avg Q& fll fll + Q& de de + Q& fll ayl de ayl + Q& Tayl eflecn eflecn eflecn + Q& Blwdwn Blwdwn blwdwn puge + Q& puge puge Usng he abve value f Q &, he wall epeaues f he denan ube unde seady avg sae cndns ae calculaed usng Maheaca. The epeaue dsbun alng he nne and ue walls f he denan ube vaes asypcally unl hey each seady sae values. Vaan f Inne Wall Tepeaue seady sae value Tepeaue K 90 Sees Te, secnds Fg 5. Tepeaue Dsbun Alng he Inne Walls f Denan Tube Cppe 64

79 Vaan f Oue Wall Tepeaue Seady Sae Cndns Tepeaue K 900 Sees Te, secnds Fg 5. Tepeaue Dsbun Alng he Oue Walls f Denan Tube Cppe I can be nfeed f he abve gaphs ha he wall epeaues nally ean aben f a few cycles and hen ncease gadually unl hey aan seady sae cndns. The wall epeaues aan seady sae a abu 00 secnds and ean cnsan afe hs e ped. The seady sae values f wall epeaues ae gven belw Seady sae values f wall Tepeaues T T w w 87.6 K 6.9 K The calculans n he ansen analyss have been pefed f ne cycle f pean. Hweve, as he nube f cycles f pean s nceased, he wall 65

80 epeaues ncease gadually and each he seady sae cndns afe a cean aun f e and hence shuld be ned ha he wall epeaues d n vay a lage exen nally. Theefe, he ansen analyss sluns agee wh he seady sae analyss f he nube f cycles s nceased and als huge nceenal changes n he wall epeaues ake place when he hea ansfe ceffcen h c s lweed a uch lesse value. The abve analyss has been epeaed wh seel and has been fund u ha he seady sae epeaues n he case f seel ae f he agnude f 4000 K whch s pey uch abve he elng pn f seel. I s evden ha seel cann whsand such hgh epeaues whch ae geneaed n he walls f he denan ube f a seel denan ube and hence as he nube f cycles f pean s nceased, due he expnenal ncease n he epeaues, he eal eaches elng pn heeby causng sevee sucual daage. Hence seel has been excluded f hs analyss and nly cppe s cnsdeed. 66

81 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 6. Cnclusns The pesen heal analyss as n develpng a pelnay hea exchange desgn n de dsspae he huge auns f hea geneaed f he denan ube and ake he pulse denan engne peanal ve lng peds f e. The pupse f he eseach has been accplshed and he fllwng paaees f he pulse denan engne f ne cycle f pean wee calculaed and he behav suded.. The aveage values f Tepeaue, Pessue and Densy f each pcess n he pulse denan cycle.. The ne hea dsspan ae f he pulse denan engne f ne cycle f pean.. The asypc appach f he wall epeauest w and T w he seady sae cndns f a denan ube ade f cppe was suded and analyzed. All f he abve enned paaees wee calculaed, by eplyng he unseady sae analyss and he ansen hea cnducn elans, he aveage values f epeaue, he ne hea dsspaed and he nne and he ue wall epeaues wee successfully baned. These values wee used calculae he ass flw ae f he clng wae equed cl he ube and he nal szng f he hea exchange was caed u. Ths analyss als caes u he aeals selecn f he denan ube. 67