Oblique detonation wave engine

Uni e sidade do Minho
Escola de Engenha ia
Tomás Pin o de Jesus Fe ei a
Oblique De ona ion Wa e Engine
Oc obe 2024
Uni e sidade do Minho
Escola de Engenha ia
Tomás Pin o de Jesus Fe ei a
Oblique De ona ion Wa e Engine
Specializa ion in Ae ospace Enginee ing Design
Wo k ca ied ou unde he guidance o he
P o esso D . Gus a o Rod igues Dias
P o esso D . F ancisco B ojo
Oc obe 2024
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B ojo, que me acompanha am e apoia am ao longo da ealização des e abalho. Pela sua
disponibilidade pa a discu i as minhas dú idas, mesmo em ho as a dias, pa a além das
c í icas que ajuda am a melho a a disse ação ealizada.
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c escimen o do in e esse já exis en e nes e cu so e nes a á ea. De salien a o P o esso Ped o
Dias, que não sendo o ien ado , ambém oi impo an e na ealização des a disse ação,
a a és da sua disponibilidade cons an e pa a esol e qualque ques ão.
A odos os meus amigos que iz ao longo des es 5 anos, que o na am es a expe iência
inesquecí el, a a és do seu apoio que a ní el académico, que a ní el pessoal.
E inalmen e, aos meus Pais e ao meu I mão, que a a és do seu apoio e amo incondicional,
o na am possí el es a caminhada, mesmo quando eu não acha a possí el. Sem ocês não
se ia possí el.
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INTEGRITY STATEMENT
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in o ma ion o esul s in any o he s ages leading o i s p epa a ion.
I u he decla e ha I know and ha I espec ed he Code o E hical Conduc o he Uni e si y
o Minho.

Mo o de de onação oblíqua
RESUMO
Nos úl imos anos a indús ia espacial e da a iação em p ocu ado soluções pa a esol e os
p incipais p oblemas, p incipalmen e ao ní el dos mo o es u ilizados. Alguns desses
p oblemas es ão ligados ao peso da ae ona e e ao impulso especí ico pa a ele ados núme os
de Mach. A hype sonic ai b ea hing p opulsion o e ece di e en es soluções pa a es e ipo de
p oblemas nomeadamen e a diminuição do peso, uma ez que não necessi a de anspo a
no in e io da ae ona e o oxidan e usado na combus ão pa a além de ambém o e ece um
impulso especí ico supe io pa a ele ados núme o de Mach. O p incipal ipo de mo o
u ilizado des e ipo de p opulsão oi du an e mui os anos, e ainda o é, o sc amje embo a cada
ez mais sejam ealizados es udos p ocu am descob i no os ipos de con igu ações, que
ap esen em melho desempenho, su gindo en ão os mo o es de de onação oblíqua.
Es a disse ação, ap esen a o desen ol imen o de um código, que p ocu a calcula e
compa a a pe o mance dos mo o es sc amje com mo o de de onação obliqua. Pa a
e e ua essa compa ação, é u ilizada uma con igu ação de á ea a iá el pa a o caso do mo o
sc amje , ao con á io do que é mais comum, á ea cons an e. A compa ação en e a á ea
a iá el e a á ea cons an e pe mi i á pe cebe se o mo o de de onação obliqua ap esen a
an agens a ní el de pe o mance, sob e as con igu ações de sc amje ap esen adas.
O es udo se á e e uado a a és da análise dos di e sos sis emas de um mo o ,
nomeadamen e a admissão e consequen e comp essão do a , a combus ão e inalmen e a
espe i a expansão, sendo ambém de inidos pa âme os que p ocu am de ini as condições
de ope ação des e mesmo mo o , p incipalmen e a al i ude de ope ação e o núme o de Mach.
PALAVRAS-CHAVE
De onações, mo o de de onação oblíqua, p opulsão hipe sónica, sc amje de á ea a iá el
i
Oblique de ona ion wa e engine
ABSTRACT
In ecen yea s, he space and a ia ion indus y has been looking o solu ions o sol e he
main p oblems, especially in e ms o he engines used. Some o hese p oblems a e linked o
he weigh o he ai c a and he speci ic h us o high Mach numbe s. Hype sonic
ai b ea hing p opulsion o e s di e en solu ions o his ype o p oblem, namely weigh
educ ion, since i does no need o ca y he oxidize used in combus ion inside he ai c a ,
in addi ion o also o e ing a highe speci ic impulse dus high Mach numbe . The main ype
o engine used in his ype o p opulsion was o many yea s, and s ill is, he sc amje , al hough
mo e and mo e s udies a e being ca ied ou o disco e new ypes o con igu a ions, which
p esen be e pe o mance, hen he ODWE (Oblique De ona ion Wa e Engine) eme ged.
This disse a ion p esen s he de elopmen o a code, which seeks o calcula e and compa e
he pe o mance o sc amje engines wi h oblique de ona ion engines. To make his
compa ison, a a iable a ea con igu a ion will be used o he case o he sc amje engine, as
opposed o wha is mo e common, cons an a ea. The compa ison be ween he a iable a ea
and he cons an a ea will allow us o unde s and i he oblique blas ing engine has
pe o mance ad an ages o e he sc amje con igu a ions p esen ed.
The s udy will be ca ied ou h ough he analysis o he a ious sys ems o an engine, namely
he in ake and consequen comp ession o he ai , he combus ion and inally he espec i e
expansion, and de ined pa ame e s ha seek o de ine he ope a ing condi ions o his same
engine, mainly he ope a ing al i ude and he numbe o Mach.
KEYWORDS
De ona ions, hype sonic p opulsion, oblique de ona ion wa e engine, a iable a ea sc amje
ii
INDEX
Ag adecimen os ........................................................................................................................ iii
Resumo .......................................................................................................................................
Abs ac ..................................................................................................................................... i
Index ......................................................................................................................................... ii
Lis o Figu es ............................................................................................................................. ix
Lis o Tables .............................................................................................................................. xi
Lis o Symbols, G eek le e s, Subsc ip s and Abb e ia ions .................................................. xii
1. In oduc ion ........................................................................................................................ 1
1.1 Mo i a ion ................................................................................................................... 1
1.2 Objec i es .................................................................................................................... 1
2. Bibliog aphic Re iew ........................................................................................................... 3
2.1 Hype sonic Ai b ea hing P opulsion ........................................................................... 3
2.2 Sc amje ....................................................................................................................... 3
2.3 Oblique De ona ion Wa e Engine ............................................................................... 4
2.3.1 Combus ion .......................................................................................................... 6
2.3.2 Chapman-Jougue Theo y .................................................................................... 8
2.4 Syn hesis ...................................................................................................................... 9
3. Me hodology ..................................................................................................................... 11
3.1 A mosphe e Model .................................................................................................... 11
3.2 Comp ession Sys em ................................................................................................. 13
3.2.1 Inle Type ............................................................................................................ 13
3.2.2 Isola o ................................................................................................................ 14
3.2.3 Numbe o Oblique Shocks ................................................................................. 16
3.3 Combus ion Sys em ................................................................................................... 19
3.3.1 Pos -combus ion ................................................................................................ 20
3.3.2 Oblique De ona ion Wa e Engine ...................................................................... 21
3.3.3 Pos -De ona ion P ope ies ............................................................................... 22
iii
3.4 Expansion Sys em ...................................................................................................... 23
3.4.1 Pe o mance ....................................................................................................... 24
4. Resul s ............................................................................................................................... 27
4.1 Model Valida ion ....................................................................................................... 27
4.1.1 Comp ession Sys em .......................................................................................... 27
4.1.2 Sc amje .............................................................................................................. 28
4.1.3 Oblique De ona ion Wa e Engine ...................................................................... 29
4.2 Pa ame ic S udies ..................................................................................................... 31
4.2.1 Comp ession Sys em .......................................................................................... 32
4.2.2 Oblique De ona ion Wa e .................................................................................. 33
4.2.3 Sc amje combus ion ......................................................................................... 35
4.3 Case S udy ................................................................................................................. 37
4.3.1 Inpu ................................................................................................................... 37
4.3.2 Pe o mance ....................................................................................................... 38
5. Conclusion ......................................................................................................................... 44
5.1 Fu u e Wo ks ............................................................................................................. 45
Re e ences ................................................................................................................................ 46
Annexes .................................................................................................................................... 48
Annex 1 – Code inpu ........................................................................................................... 48
Annex 2 – Code ..................................................................................................................... 49
x
Abb e ia ions
Desc ip ion
ODW
Oblique De ona ion Wa e
ODWE
Oblique De ona ion Wa e Engine

1
1. INTRODUCTION
In e es in hype sonic ai b ea hing p opulsion has been g owing in ecen yea s, since his
echnology allows se e al ad an ages o e adi ional space and a ia ion engines, namely he
possibili y o lying a speeds g ea e han i e imes he speed o sound, bu also he ac ha
i allows as e access o space and as e comme cial ligh s.
The eason why his wo k has been ca ied ou , as well as wha will be ca ied ou in i , will be
e e ed o in he Mo i a ion and Objec i es Chap e s.
1.1 Mo i a ion
The mos used ype o Hype sonic ai b ea hing p opulsion engines is he sc amje , which has
been he a ge o in ense s udies o se e al yea s, wi h he aim o enhancing hei
pe o mance. This ype o mo o s has he ad an age o ha ing a high speci ic impulse, in
addi ion o no ha ing o anspo he oxidize , compa ed o sc amje engines. Fo his
eason, i is possible o ca y mo e payload weigh , inc easing cos e iciency.
Al hough he sc amje has se e al ad an ages, i also has some disad an ages, namely he
complexi y o he combus ion sys em being high. Due o his eason, he concep o oblique
de ona ion engines eme ged, which eplaces he di usion p ocess in he combus ion o
sc amje s, by oblique de ona ions, educing he geome y and complexi y o he combus ion
sys em.
E en so, se e al s udies will s ill be needed o make he cons uc ion and use o his ype o
engine iable.
1.2 Objec i es
The objec i e o his disse a ion is o desc ibe he p ocess o de eloping a nume ical ool,
which aims o compa e he pe o mance o an oblique de ona ion engine wi h a sc amje
engine. Fo his, he esul s ob ained in his wo k has been compa ed wi h he esul s o o he
wo ks, namely wi h wo ks ha pe o m combus ion a cons an p essu e and cons an a ea,
since his p ojec will add ess, in he case o he sc amje , a iable a ea.
2
The nume ical ool should be able, conside ing he inpu s gi en, and he geome y o he
chosen inle , o calcula e he pe o mance o a gi en ange o Mach numbe s.
In he end, and a e he esul s a e ob ained by he nume ical ool, hey will be compa ed
wi h he esul s o o he s udies, o unde s and he ad an ages and disad an ages o i s use.
3
2. BIBLIOGRAPHIC REVIEW
In his chap e , he p elimina y concep s will be add essed, which will se e as a basis o he
concep s and o mulas, and he decisions made, in he me hodology chap e , and he
conclusions ha will be d awn om he esul s ob ained in he nume ical ool.
2.1 Hype sonic Ai b ea hing P opulsion
Wi h he aim o making hype sonic espi a o y p opulsion mo e accessible and e icien , his
echnology has been inc easingly s udied, al hough he e has been no p ac ical e olu ion in
he las 40 yea s in e ms o p opulsi e sys ems [1], a leas , in e ms o ehicles.
The e o e, he ai -b ea hing p opulsion sys ems mos used now a e amje and sc amje ,
which di e in he way in which he ai low en e s he combus o , causing combus ion in each
case o be subsonic and supe sonic, espec i ely [2].
In addi ion o hese con igu a ions, o he s ha e been p oposed, such as oblique wa e
de ona ion engines (ODWE), o a y de ona ion engines (RDE), and pulse de ona ion engines
(PDE) [2]. In he case o ODWE, i s main peculia i y is he ac ha combus ion occu s by
de ona ion and no by de lag a ion.
The main disad an age o ODWEs, and o sc amje s, is he ac ha hey ha e a high ini ial
Mach numbe , which makes hem dependen on o he engines o ake-o and accele a ion,
up o he desi ed ini ial Mach numbe , inc easing he complexi y o he p opulsi e sys em. Fo
his eason, much s udy is s ill needed in his a ea un il i s applica ion is iable.
2.2 Sc amje
To de ine wha a sc amje engine is, we i s mus unde s and he bases o a gas u bine and
a amje engine. The main di e ence be ween he wo is he ai comp ession p ocess be o e
combus ion, whe e in he gas u bine his is done h ough a comp esso , while in he amje ,
his is done h ough shock wa es, which a e no mally caused by he on body o he ehicle.
When looking mo e speci ically a he ope a ion o he amje engine, he ai low is slowed
down o subsonic speeds a e comp ession, making combus ion also subsonic. Acco ding o
Heise and P a e al. [3], he ope a ion o he amje engine is e icien o Mach numbe s in
he ange 3-6. As he Mach numbe exceeds his ange, he use o he amje engine is no
4
longe e icien , as i is no longe ad an ageous o educe he speed o he ai low o subsonic
speed. The e o e, i was necessa y o c ea e an engine ha did no equi e his decele a ion,
and could pe o m supe sonic combus ion, o igina ing he sc amje engine.
Al hough his con igu a ion o e s his ad an age, when compa ed o he amje engine, and
has been he subjec o nume ous s udies and e olu ions o e he yea s, he e a e s ill
p oblems ha need o be o e come o make his engine mo e e icien and iable, in addi ion
o he high ini ial Mach numbe . Since he ope a ing en i onmen has high empe a u es and
p essu es, which jeopa dizes he s uc u al in eg i y o he engine and ehicle, he need o
gua an ee s able and e icien mixing and combus ion in supe sonic egime, in addi ion o a
combus ion chambe o app op ia e dimensions, a e some o he sizing p oblems o his ype
o engine [3].
In he Figu e 2.1, i is possible o isualize a wo-dimensional diag am o he s uc u e o a
sc amje engine.
Figu e 2.1: Diag am o a sc amje engine [4]
2.3 Oblique De ona ion Wa e Engine
As seen p e iously, he engines mos used in he cons uc ion o hype sonic ai b ea hing
p opulsion ehicles a e he sc amje , which a e cha ac e ized by using de lag a ion as a
combus ion p ocess. Acco ding o P a e [5], he use o no mal de ona ion wa es was
p oposed by Roy in 1946 [6], and la e s udies p o ed i s easibili y, al hough i was ound
di icul o con ol he de ona ion wa e, as well as he complex geome y o he chambe ,
which his p ocess would equi e.
The use o oblique de ona ion wa es ins ead o no mal de ona ions was p oposed by Dunlap
e al [1], since as happens in a sc amje engine, i allows he low ha passes h ough he
bu ne o be supe sonic.
This mean ha i was no necessa y o implemen a a iable geome y o s abilize he wa e,
since his s abiliza ion would be ca ied ou by a wedge [7]. I we look a he geome ies o a
5
sc amje and an ODWE, we see ha he di e ence be ween hem is he exis ence o his
wedge in he combus o . As will be seen la e , he exis ence o a wedge is necessa y o he
s abiliza ion o shock wa es, unlike wha happens in sc amje engines. The posi ion o he
wedge depends on he ype o comp ession ha will be chosen o he wo k, ep esen ing
g ea in luence mainly on he le el o d ag gene a ed. The ypical model o an oblique
de ona ion engine can be obse ed in he Figu e 2.2.
Figu e 2.2: Diag am o an Oblique De ona ion Wa e Engine [8]
Ci ing Heise and P a e al. [3], “The immedia e ad an ages o he ODW p ocess a e ha he
d ag, con ec ion hea ing, leng h, weigh , cos , and main enance o he combus o a e almos
en i ely elimina ed”. In his way, di e en s udies we e ca ied ou o inc ease he e iciency
and iabili y o ODWEs. One o hem was he s udy ca ied ou by Valo ani e al. [9], which
sough o ind a solu ion o he high ini ial Mach numbe , wi h he possibili y o his being
educed. This would esul in a educ ion in ope a ing weigh , since less oxidan would be
needed o be anspo ed, since he lowe he Mach numbe , he smalle he amoun o
oxidan used, o sho e ime. Bu i would also allow he use o sho e bu ne s, when
compa ed o hose used in sc amje s. Ano he ad an age is he simplici y o he comp ession
sys em, especially when compa ed o Ramje engines. This is because, he empe a u e
a ia ion h oughou he comp ession p ocess is smalle , due o he exis ence o highe
empe a u es and p essu es in he bu ne due o de ona ion, since he exis ence o shock
wa es, causes an inc ease in p essu e and empe a u e, wi hou using mo e uel.
One o he mos impo an s udies was ca ied ou by Ash o d e al. [10], whe e he
pe o mance o an oblique wa e de ona ion engine was compa ed wi h ha o a di usi e
sc amje engine. Some conside a ions we e made, such as cons an a ea and cons an
p essu e, as well as he ac ha i is an ideal and pe ec gas. Fo Mach 10, a which he s udy
was ca ied ou , i was concluded ha he use o ODWEs can educe he leng h o he engine
up o 50%.

6
To eplace di usi e bu ning, Jiang e al [11], p oposed a p o o ype o a s anding oblique
de ona ion amje engine. The engine was success ully de eloped, concluding ha he model
wo ks s eadily, and con i ming ha oblique de ona ion can be s a iona y and con ollable in
he bu ne . This model can be iewed in Figu e 2.3
Figu e 2.3: Schema ic o he Sod amje p oposed by Jiang [11]
2.3.1 Combus ion
To unde s and he combus ion p ocess, i is impo an o de ine de lag a ion and de ona ion,
o unde s and hei di e ences.
S a ing wi h de lag a ion, a combus ion p ocess cha ac e is ic o sc amje engines, his can
be de ined as he passage o a lame on as a subsonic wa e h ough a uel mix u e,
acco ding o Diéguez e al [12].
Acco ding o Wolanski e al. [13], he de ona ion p ocess was i s desc ibed by Be helo ,
Vieille, Malla d and Le Cha elie in 1881. I was only wen y yea s la e ha Chapman and
Jougue p esen ed he ze o-dimensional heo y o de ona ion.
De ona ion is a combus ion wa e ha p opaga es h ough a gaseous mix u e o uel and
oxidize , causing a eac ion.
This can be classi ied acco ding o he Mach numbe ha occu s a e de ona ion. I he Mach
numbe is subsonic, we ha e a de ona ion called o e d i en, i i is sonic, we a e acing he
Chapman-Jougue de ona ion ( e e ed o in he ollowing chap e ), and inally, i he Mach
numbe is supe sonic, he de ona ion is called unde d i en.
As wi h oblique shocks, oblique de ona ions can ha e a weak o s ong shock solu ion, o he
same wedge angle. To e alua e he angle o he oblique de ona ion wa e as a unc ion o he
angle o de lec ion and hea addi ion, a diag am was c ea ed, based on he ups eam
condi ions, and conside ing he di e en possible condi ions downs eam. This diag am can
be seen in Figu e 2.4.
7
Figu e 2.4: Oblique de ona ion wa e angle as unc ion o de lec ion angle and hea addi ion [3]
Acco ding o he diag am, we can ha e s ong o weak o e d i en oblique de ona ion wa es.
In he case o s ong o e d i en de ona ions, associa ed inc eases in s a ic p essu e a e so
high ha hey cause de achmen and a e he e o e unna u al. In he case o weak o e d i en
de ona ion wa es, hese a e es ic ed by he Chapman-Jougue angle and he de achmen
angle, which is why his is he egion o in e es o ODW engines. The Chapman-Jougue angle
is connec ed o he Chapman-Jougue locus, which is he poin o minimum wa e angle a
each locus o s a es o a gi en 𝑞>0, as can be seen in Figu e 2.4. De ona ions a e no
conside ed o hese engines due o he downs eam Mach numbe being supe sonic, and i
needs o be subsonic o sonic, o ensu e ha he oblique de ona ion wa e is a ached and
s able.
Upon de ona ion, he on p opaga es a speeds in he o de o km/s in he ai - uel mix u e,
which p oduces a signi ican inc ease in p essu e. Acco ding o Wolanski e al [13], i he
de ona ion speed is abo e 1.8 km/s, he e is a p essu e inc ease o mo e han en imes. I he
de ona ion speed eaches 3km/s, a p essu e inc ease o up o wen y imes is ob ained.
Du ing de lag a ion, he speed o he lame is in he o de o dozens o m/s, so he combus ion
will ha e o be o ganized in a s oichiome ic a io, a a highe bu ning speed, which esul s in
high combus ion empe a u es and p oduc ion o high NOx concen a ion. Due o hese
phenomena, in common a ia ion engines, which ha e a u bine, as is he case wi h u bo ans,
u boje s, among o he s, i is necessa y o mix mo e ai be o e he u bine, due o he high
empe a u e, inc easing he complexi y o he sys em.
8
In de ona ion engines his complexi y does no exis , since he combus ion empe a u e is
lowe , in addi ion o no using igni ion de ices [1].
Acco ding o Rosa o e al. [14], de ona ion can e ec i ely inc ease he e iciency o he
he modynamic cycle by up o 20%, when compa ed o ypical cycles ha a e based on
di usi e bu ning, since he g ea e he speed o bu ning o con e sion o he "ma e ial",
ypically ens o housands o imes as e han engines ha use de lag a ion, esul ing in mo e
compac and e icien sys ems.
2.3.2 Chapman-Jougue Theo y
As we saw p e iously, Chapman and Jougue es ablished he i s heo y abou de ona ions,
he ze o-dimensional heo y o de ona ion. This desc ibed he de ona ion wa e as a
hyd odynamic discon inui y, whe e ene gy would be eleased, assuming ha i was s able,
plana , and one-dimensional.
Ano he impo an opic o his wo k is he Rankine-Hugonio condi ions, which desc ibe he
ela ionship be ween he s a es on bo h sides o a shock wa e o combus ion wa e
(de lag a ion o de ona ion), o one-dimensional lows in luids o o one-dimensional
de o ma ions in solids.
In he equa ion (2.1), Rankine-Hugonio condi ions a e desc ibed, whe e he p e ixes 1 and 2
e e o he s a e be o e and a e de ona ion, espec i ely. The e o e, ℎ1 and ℎ2 ep esen ,
espec i ely, he speci ic en halpies be o e and a e de ona ion, 𝑝1 and 𝑝2, he p essu es
be o e and a e de ona ion and, inally, 𝜌1 and 𝜌2, he low densi ies be o e and a e
de ona ion.
ℎ2−ℎ1=12(𝑝2−𝑝1)(1
𝜌1+1
𝜌2)
(2.1)
Using hese condi ions, i is possible o plo he Rankine Hugonio cu e, whe e i will be
possible o de e mine he condi ions o s a e 2, conside ing he condi ions o s a e 1 ha a e
p o ided. One o hese cu es can be isualized in Figu e 2.5, whe e egions l and ll co espond
o supe sonic wa es, hence de ona ions, while egions lV and V co espond o subsonic wa es,
also called de lag a ions.
9
Figu e 2.5: Rankine-Hugonio cu e [15]
In he case o de ona ion, conside a ions o gas dynamics a e su icien o p edic he
p opaga ion speed o he wa e, h ough he shock wa e, while in de lag a ion i is necessa y
o know he s uc u e o he wa e, in addi ion o he anspo p ocesses, whe he u bulen
o di usi e.
Chapman and Jougue sugges ed ha de ona ions a el a he lowes speed o all solu ions
in he de ona ion b anch. Obse ing Figu e 2.5, we can see ha he e a e wo b anches, which
co espond o he Chapman-Jougue poin s, he uppe poin , 𝐶𝐽𝑢, and he lowe poin , 𝐶𝐽𝑙.
The lowe one is in he de lag a ion b anch, while he uppe one is in he de ona ion b anch,
which co esponds o he poin used o ODWEs, since his ype o engine uses de ona ions
and no de lag a ions, in addi ion o co esponding o he poin o lowes en opy and o al
p essu e los .
2.4 Syn hesis
Acco ding o he poin s add essed du ing he bibliog aphic e iew, i is pe cei ed ha he e
is a lack o knowledge o oblique de ona ion engines, especially in he possibili y o hem
eplacing he mos used hype sonic engines oday, he sc amje .
These easons led o his wo k, wi h a iable a ea con igu a ion, bu also he wo k ca ied ou
by Pe ei inha e al. [16], wi h cons an a ea con igu a ion, o be ca ied ou . In hem, a
compa ison is made be ween oblique de ona ion engines and di e en con igu a ions o
16
whe e 𝛾1 and 𝛾2 a e he speci ic hea a ios o he gas a he co esponding c oss-sec ion, and
R is he gas cons an .
Then, he a ea a he isola o exi /combus o inle can be calcula ed using he equa ion (3.14).
𝐴2=𝐴1
√𝛾2𝑅𝑇2[1
𝜋2(1+𝛾𝑀12)−1]
(3.14)
Finally, he o al p essu e a he exi o he isola o can be calcula ed, as well as he eco e y
coe icien o he o al p essu e o he isola o . These can be calcula ed using he equa ions
(3.15) and (3.16), espec i ely.
𝑃2=𝑃1
[
𝜋2(1+𝛾2−1
2𝑀22)𝛾2
𝛾2−1
(1+𝛾1−1
2𝑀12)𝛾1
𝛾1−1
]
(3.15)
𝜎2=𝑃2
𝑃1
(3.16)
3.2.3 Numbe o Oblique Shocks
A e choosing he ype o inle o use, i is necessa y o de e mine he numbe o oblique
shocks ha will be gene a ed by he inle , acco ding o he desi ed condi ions, namely he
desi ed cycle empe a u e a io. The numbe o shocks mus gua an ee ha his a io is
e icien and he inc ease in en opy is kep o a minimum, in addi ion o p o iding an
accep able pe o mance.
In his p ojec , he sizing o he comp ession sys em has been based on he cycle empe a u e
a io 𝑇3/𝑇0, ep esen ing he empe a u e a io be ween he bu ne inle and he ou side, as
shown by he Figu e 3.1.
I is also impo an o ensu e ha all oblique shock wa es ansmi an equal amoun o
geome ic o a ion o he low, in addi ion o ensu ing ha he shock-on-lip condi ion occu s
[3].
Acco ding o Sma e al [21], he g ea e he numbe o shocks, he mo e he op imal o al
p essu e eco e y will inc ease, al hough his dec eases wi h he inc ease in Mach numbe .
This e ec can be seen in Figu e 3.3, ha desc ibes how he op imal o al p essu e eco e y
a ies wi h he Mach numbe , o a di e en numbe o shocks.

17
Figu e 3.3: Maximum o al p essu e eco e y o wo-dimensional sc amje inle s wi h up o i e shocks [21]
In Figu e 3.4, i is analysed how he comp essi e e iciency a ies wi h he numbe o Mach,
o a di e en numbe o oblique shock wa es.
Obse ing he Figu e 3.3 and Figu e 3.4, i is concluded ha a sys em o 4 oblique shocks is
he mos e icien choice, since a highe numbe o shocks leads o complica ions in sys em
sizing, while a lowe numbe leads o a educ ion in he op imum o al p essu e eco e y.
Figu e 3.4: Adiaba ic comp ession e iciency as a unc ion o ees eam Mach numbe , s a ic empe a u e a io, and
numbe o oblique shock wa es [3]
To calcula e he low p ope ies a e each shock, he oblique shock equa ions has been
applied. In his way, equa ions (3.17), (3.18), (3.19), (3.20) and (3.21), aken om Pe ei inha
e al. [16], we e implemen ed in he p ojec code.
18
𝑝𝑟𝑎𝑡𝑖𝑜=1+ 2𝛾0
𝛾0+1(𝑀2sin2𝛽−1)
(3.17)
𝑝𝑟𝑎𝑡𝑖𝑜=(𝛾0+1)𝑀2sin2𝛽
2+(𝛾0−1)𝑀2sin2𝛽
(3.18)
𝑇𝑟𝑎𝑡𝑖𝑜=𝑝𝑟𝑎𝑡𝑖𝑜
𝜌𝑟𝑎𝑡𝑖𝑜
(3.19)
an𝜃=2co 𝛽(𝑀2sin2𝛽−1)
𝑀2(𝛾0+cos(2𝛽))+2
(3.20)
𝑀𝑟𝑎𝑡𝑖𝑜=1
𝑀(2
𝛾0−1+𝑀2sin2𝛽)1/2(2𝛾0𝑀2sin2𝛽
𝛾0−1 −1)1/2
(3.21)
In o de o de e mine comp essi e e iciency and he kine ic ene gy (ƞ𝑐 and ƞ𝐾𝐸, espec i ely),
he equa ions (3.22), (3.23) and (3.24), aken om [3], a e applied.
𝜋𝑐=𝑝𝑡3
𝑝𝑡0=𝑝3
𝑝0(1φ)𝛾𝑐/(𝛾𝑐−1)
(3.22)
ƞ𝑐=φ−(1
𝜋𝑐)(𝛾𝑐−1)/𝛾𝑐
φ−1
(3.23)
ƞ𝐾𝐸=1− 2
(𝛾𝑐−1)𝑀02[(1
𝜋𝑐)(𝛾𝑐−1)/𝛾𝑐−1]
(3.24)
whe e φ= 𝑇3/𝑇0.
19
3.3 Combus ion Sys em
The nex hing o pay a en ion o is he combus ion s udy, whe e i is conside ed ha i s a s
wi h he ai and uel al eady homogeneously mixed.
To unde s and how combus ion akes place, i can be desc ibed using a chemical equa ion
(3.38), called he comple e s oichiome ic equa ion, desc ibed in he equa ion (3.25).
𝐶𝑥𝐻𝑦+(𝑥+𝑦4)(𝑂2+79
21𝑁2)→𝑥𝐶𝑂2+𝑦2𝐻2𝑂+79
21(𝑥+𝑦4)𝑁2
(3.25)
In his case i ep esen s he comple e combus ion o ai and hyd oca bon uel.
Nex , i is necessa y o calcula e he s oichiome ic uel/ai a io. This can be ob ained h ough
he equa ion (3.26).
𝑓𝑠𝑡=36𝑥+3𝑦
103(4𝑥+𝑦)
(3.26)
Fo he case o non-s oichiome ic mix u es, i is impo an o de ine ano he p ope y, called
equi alence a io, which ep esen s he a io be ween he eal uel/ai a io and he
s oichiome ic uel/ai a io. The ep esen a ion o he equi alence a io is made in he
equa ion (3.27).
𝜙=𝑓
𝑓𝑠𝑡
(3.27)
The equi alence a io, acco ding o Heise and P a [3], mus be in he ange o 0.2 o 2, so
ha combus ion occu s wi hin a easonable ime scale, which is he ange conside ed in his
wo k.
Ano he impo an poin o he p ojec is he selec ion o uel. Fo his pu pose, se e al uels
we e compa ed, in di e en aspec s, as can be seen in he Table 3.1, whe e ρ is o s anda d
condi ions and 𝑇𝑖𝑔𝑛 o s anda d condi ions and 1 a m.
The choice ended up being 𝐻2, o ob ain a mo e eliable compa ison be ween he esul s o
he wo ks. When obse ing he Table 3.1, i can be seen ha 𝐻2 has he highes hea o
combus ion alue, in addi ion o a highe au o-igni ion empe a u e. The main d awback o
20
using 𝐻2 is he need o a lo o s o age space due o i s e y low densi y, al hough se e al
s udies and ad ances ha e been ca ied ou o coun e his p oblem.
Table 3.1: P ope ies o se e al uels [22]
Fuel
ℎ𝑝𝑟[𝑀𝐽/𝑘𝑔]
𝜌[𝑘𝑔/𝑚3]
𝑇𝑖𝑔𝑛[𝐾]
𝑓𝑠𝑡
𝐻2
119.96
0.08
845.15
0.02913
𝐶𝐻4
50.01
0.65
810.15
0.05825
𝐶2𝐻6
47.49
1.22
745.15
0.06241
𝐶3𝐻3
46.3
1.79
743.15
0.06408
𝐶4𝐻10
45.74
2.36
693.15
0.06497
3.3.1 Pos -combus ion
I is now necessa y o calcula e he p ope ies o he low a e combus ion. Since we a e
dealing wi h a ea sec ions, and since he adius o he sec ion a e combus ion is di e en
om he inle en ance adius, his di e ence mus be conside ed. Fo his p ojec , a sec ion
adius o 101.8 mm will be conside ed.
To calcula e hese p ope ies, i is necessa y o apply he equa ions (3.28), (3.29) and (3.30),
aken om he wo k o Gu e al. [20]. The a iables ep esen ed wi h poin 2 a e calcula ed a
he exi o he isola o .
𝑀4=𝑀2
√𝜏(𝑟)(1+𝛾4(𝑟)−1
2𝑀22)−(𝛾4(𝑟)−1
2𝑀22)
(3.28)
𝑃3=𝑃2
[1+𝛾4(𝑟)−1
2𝑀22(1− 1
𝜏(𝑟))]𝛾4(𝑟)
𝛾4(𝑟)−1
(3.29)
𝑇3(𝑟)=𝑇2𝜏(𝑟)
(3.30)
whe e,
21
𝜏(𝑟)=Ҩ(𝑟)ℎ𝑝𝑟
34.32𝑐𝑝2𝑇2
(3.31)
To simpli y he calcula ions, ce ain alues aken om o he wo ks we e assumed. In case o
Ҩ(𝑅), a alue o 0.2 was assumed, acco ding o Gu e al.[20], and o 𝛾4(𝑅) a alue o 1.286
was assumed, acco ding o Yang e al. [23].
3.3.2 Oblique De ona ion Wa e Engine
P e iously, in Chap e 2, i was said ha he ypes o de ona ions ha could be applied in
hype sonic ai b ea hing p opulsion we e weak o e d i en de ona ions and Chapman-Jougue
de ona ions, al hough he la e co espond o hose o minimum o al p essu e loss and he
poin o minimum en opy inc ease. Conside ing he ypes o de ona ions ha can be used in
his p ojec , he de eloped p og am mus be able o de e mine he wedge angle ha
gene a es he Chapman-Jougue de ona ion, while o he emaining Mach numbe s, he
weak o e d i en de ona ion mus be gua an eed.
To ensu e ha p e-igni ion does no occu , in he speci ic case o bu ning 𝐻2, below
s oichiome ic condi ions, he empe a u e eco ded be o e de ona ion mus be below
1000 K, while o gua an ee igni ion, he empe a u e a e de ona ion mus be g ea e han
1000 K. I hese condi ions a e no me , he p og am will wa n he use o he e o ha has
occu ed.
To de e mine he Chapman-Jougue de ona ion angle o he Mach numbe conside ed, he
equa ion (3.32) mus be applied.
(𝑀1𝑛
2−1)2−2(𝛾+1)𝑀1𝑛
2𝑄=0
(3.32)
whe e,
𝑀1𝑛=𝑀1sin𝛽
(3.33)
𝑄=𝑄𝑛𝑏
𝑐𝑝𝑇
(3.34)

22
To ind he wedge angle ha gene a es he Chapman-Jougue de ona ion unde he desi ed
condi ions, he equa ion (3.35) mus be sol ed.
𝜃𝐶𝐻.𝐽.=𝛽𝐶𝐻.𝐽.− an−1
[
1+𝛾𝑀1𝑛𝐶𝐻.𝐽.
2
(𝛾+1)𝑀1𝑛𝐶𝐻.𝐽.
2√(𝑀1/𝑀1𝑛𝐶𝐻.𝐽.
2)2−1
]
(3.35)
Fo he emaining Mach numbe s o he ange unde s udy, and in o de o gua an ee weak
o e d i en de ona ion, he de ona ion angle will be calcula ed using he equa ion (3.36), since
he wedge angle has al eady been de ined in he p e ious equa ion (3.35).
𝑄=−𝛾+1
2𝑋2𝑀12sin2𝛽+(1+𝛾𝑀12sin2𝛽)𝑋−(1+𝛾−1
2𝑀12sin2𝛽)
(3.36)
The equa ions (3.32) and (3.35) we e aken om Mu hy e al.[24], and X has been ob ained
h ough he equa ion (3.39).
3.3.3 Pos -De ona ion P ope ies
A e calcula ing he condi ions be o e de ona ion, mainly he de ona ion angles o each
Mach numbe in he desi ed ange, i is necessa y o de e mine he condi ions a e
de ona ion. Fo his i is assumed ha he ai and uel a e homogeneously mixed [5], al hough
his is no comple ely eal.
Since he in luence o he uel empe a u e is neglec ed, he ai empe a u e a e he
comp ession sys em is equal o he empe a u e o he ai / uel mix u e.
To calcula e he p essu e a e de ona ion, he equa ion (3.37) is used,
𝑝2=𝑝1+𝜌1𝑢1𝑛
2(1−𝑋)
(3.37)
and he de ona ion empe a u e a io can be calcula ed using equa ion (3.38).
𝑇2
𝑇1=1+ 𝑢1𝑛
2
2𝐶𝑝𝑇1(1−𝑋)2+𝑄ƞ𝑏
𝐶𝑝𝑇1
(3.38)
while X is calcula ed by
23
𝑋≡𝜌1
𝜌2=𝑢2𝑛
𝑢1𝑛= an(𝛽−𝜃)
an𝛽
(3.39)
whe e,
𝑢1𝑛=𝑢1sin𝛽
(3.40)
𝑢1𝑡=𝑢1cos𝛽
(3.41)
𝑢2𝑛=𝑢2sin(𝛽−𝜃)
(3.42)
𝑢2𝑡=𝑢2cos(𝛽−𝜃)
(3.43)
3.4 Expansion Sys em
The las sys em o be assessed is he expansion sys em, whose main objec i e is o accele a e
he low o p oduce he maximum amoun o h us . I can ope a e in di e en ways, whe he
o e -expanded, unde -expanded o ideally, depending on he a io be ween he engine inle
and ou le p essu es. I his a io is 1, we a e acing an ideal con igu a ion; while i i is g ea e
han 1, we will ha e an o e -expanded nozzle con igu a ion, allowing a lowe weigh in he
nozzle con igu a ion; Finally, i he a io is less han 1, we a e dealing wi h an unde -expanded
nozzle, which causes a dec ease in he h us p oduced and an inc ease in he nozzle geome y
and consequen ly in i s weigh .
In his wo k an ideal nozzle will be conside ed, whe e he p essu e a inle o he engine and
on he ou le is he same, making he a io be ween hem 1. The expansion p ocess will be
adiaba ic and isen opic, in addi ion o being one-dimensional and calo ically pe ec low.
To calcula e he p ope ies o he expansion sys em, he equa ions (3.44), (3.45) and (3.46)
will ha e o be applied.
𝑇5=𝑇4[1−ƞ𝑒(1−(𝑝5
𝑝0𝑝0
𝑝4))𝑅
𝑐𝑝𝑒]
(3.44)
24
𝑉5=√𝑉42+2𝑐𝑝𝑒(𝑇4−𝑇5)
(3.45)
𝐴5
𝐴0=(1+𝑓)∗𝑝0
𝑝5𝑇5
𝑇0𝑉0
𝑉5
(3.46)
3.4.1 Pe o mance
Finally, he las s ep is he de ini ion o he equa ions ha will e alua e he pe o mance o
he designed engine, which we e aken om he wo k o P a e al. [3]. As p e iously
men ioned, assump ions we e made ha a e a om wha happens in eali y, a leas i is
necessa y o unde s and and know how o in e p e he esul s ob ained in acco dance wi h
he assump ions made p e iously.
The i s pe o mance ac o o be e alua ed is speci ic h us , which is de ined by he equa ion
(3.47).
Unins alled h us
Inle ai mass low a e=F
m󰇗0
(3.47)
When using he s eam h us unc ion, he e is ano he way o de e mine he speci ic h us ,
om eloci ies and empe a u es ob ain a he en ance and exi o he engine, de ined in he
equa ion (3.48).
𝐹
𝑚󰇗0=(1+𝑓)𝑆𝑎5−𝑆𝑎0−𝑅0𝑇0
𝑉0(𝐴5
𝐴0−1)
(3.48)
whe e,
𝑆𝑎5=𝑉5(1+𝑅𝑇5
𝑉52)
(3.49)
𝑆𝑎0=𝑉0(1+𝑅𝑇0
𝑉02)
(3.50)
25
𝑆𝑎 ep esen s he s eam h us unc ion a inle and exi o he engine (Poin 0 and 5,
espec i ely), depending on he low eloci ies and empe a u es eco ded in hese sec ions.
Based on he equa ions desc ibed, i is possible o ob ain he low a e o he ai mass, which
can be ob ained h ough he de i a i e o equa ion (3.48), which indica es no only he
unins alled h us equi ed o he ope a ion, bu also he speci ic h us o he de eloped
engine.
The nex ac o o conside is he speci ic uel consump ion, which can be de ined by he
equa ion (3.51).
Fuel mass low a e
Unins alled h us =S=m󰇗
F
(3.51)
The speci ic uel consump ion can also be de ined h ough he speci ic h us , using he
equa ion (3.52).
𝑆= 𝑓𝐹
𝑚󰇗0
(3.52)
Nex , is one o he mos impo an pe o mance pa ame e s o e alua e he pe o mance o
an engine, he speci ic impulse, ep esen ing he e iciency wi h which he engine p oduces
h us .
This is de ined by he equa ion (3.53).
Unins alled h us
Fuel weigh low a e=Isp=F
g0m󰇗
(3.53)
Nex , i will be he p opulsi e e iciency, which ep esen s he a io o h us powe o he
mechanical powe o he engine, being de ined by he equa ion (3.54), which can be applied
due o he ac ha an ideal nozzle has been selec ed, whe e he inle and ou le p essu e o
he engine a e he same.
32
4.2.1 Comp ession Sys em
Once again, he s udy begins wi h he comp ession sys em. The i s s udy ca ied ou seeks
o demons a e how he 𝑇𝑟𝑎𝑡𝑖𝑜, in luences he e iciency o he comp ession sys em and
kine ic ene gy. This in luence can be seen in he Figu e 4.3 and Figu e 4.4, espec i ely.
Figu e 4.3 : Inle comp ession sys em e iciency as unc ion o 𝑀0 [16]
S a ing wi h Figu e 4.3, we no iced ha as he 𝑇𝑟𝑎𝑡𝑖𝑜 inc eases, he e is a dec ease in he
e iciency o he comp ession sys em.
On Figu e 4.4, we no iced ha he beha iou o he e iciency o kine ic ene gy is e y simila
o he beha iou o he e iciency o he comp ession sys em, since when 𝑇𝑟𝑎𝑡𝑖𝑜 inc eases,
he e is a dec ease in he e iciency o kine ic ene gy.
The e o e, when choosing he 𝑇𝑟𝑎𝑡𝑖𝑜, he nega i e impac ha i has on bo h e iciencies
should be conside ed, al hough a low alue o 𝑇𝑟𝑎𝑡𝑖𝑜, nega i ely in luences he o e all
pe o mance o he engine, so i is necessa y o choose he alue o 𝑇𝑟𝑎𝑡𝑖𝑜.
Figu e 4.4: Kine ic ene gy e iciency as unc ion o 𝑀0 [16]

33
Obse ing he Figu e 4.5, i can be seen how he Mach numbe a he bu ne inpu a ies wi h
he ees eam Mach numbe , o di e en alues o 𝑇𝑟𝑎𝑡𝑖𝑜. As he 𝑇𝑟𝑎𝑡𝑖𝑜 inc eases he M3
dec eases.
This cha allows us o choose he alue o 𝑇𝑟𝑎𝑡𝑖𝑜, which allows a supe sonic low a he bu ne
inle , al hough low Mach numbe s a he bu ne inle , ha e some ad an ages, namely in e ms
o combus ion. In his case, low Mach numbe s mean high alues o 𝑇𝑟𝑎𝑡𝑖𝑜, his implies high
empe a u es inside he engine, implying speci ic ma e ials in his egion, inc easing he cos
o he de eloped engine.
Figu e 4.5: 𝑀3 as a unc ion o 𝑀0 [16]
4.2.2 Oblique De ona ion Wa e
The nex s udy o be ca ied ou was ha o oblique de ona ion wa es. This s udy seeks o
unde s and how 𝑄 in luences he wedge and de ona ion angles o a Chapman-Jougue
de ona ion. Bo h in luences can be seen in Figu e 4.6 and Figu e 4.7, espec i ely.
Figu e 4.6: Wedge angle as a unc ion o ees eam Mach numbe [16]
34
Figu e 4.7: De ona ion angle as a unc ion o ees eam Mach numbe [16]
As he alue o 𝑄 inc eases, he wedge angle equi ed o gene a e a Chapman-Jougue
de ona ion will also inc ease, also leading o an inc ease in he de ona ion angle.
Nex , i was s udied how he de ona ion p essu e and empe a u e a io a e in luenced by he
hea lux. This in luence can be isualized in he Figu e 4.8 and Figu e 4.9.
Figu e 4.8: P essu e a io as unc ion o ODW hea lux [16]
35
Figu e 4.9: Tempe a u e a io as unc ion o ODW hea lux [16]
I can hen be concluded ha as he hea lux o ODW inc eases, he e is also a linea inc ease
in he 𝑝𝑟𝑎𝑡𝑖𝑜 and 𝑇𝑟𝑎𝑡𝑖𝑜.
4.2.3 Sc amje combus ion
In his sec ion, he in luence o φ o a cons an p essu e bu ne will be s udied. To ca y ou
his s udy, he p ope ies isible in Table 4.7.
Table 4.7: Inpu s o he s udy o in luence o φ [16]
P ope y
Assigned
Value
𝑂𝑛−𝑑𝑒𝑠𝑖𝑔𝑛 𝑀0
8
𝑂𝑛−𝑑𝑒𝑠𝑖𝑔𝑛 𝑇𝑟𝑎𝑡𝑖𝑜
2
𝑉𝑓𝑥
𝑉3
0.5
𝑉𝑓
𝑉3
0.5
𝐶𝑓𝐴𝑤
𝐴3
0
𝐶𝑃𝐵
1510
J/kgK
𝑇°
222 K
ℎ𝑝𝑟
119.96
MJ/kg
ƞ𝑏
1
36
The esul s ob ained o he bu ne a cons an p essu e can be seen in he Figu e 4.10, Figu e
4.11 and Figu e 4.12.
As expec ed, as he equi alence a io inc eases, he e is an inc ease in he empe a u e a he
ou pu o he bu ne .
Figu e 4.10: Tempe a u e a he exi o he bu ne as a unc ion o 𝑀0 [16]
A di e en beha iou is obse ed a he bu ne ou pu speed, whe e low equi alence a io
alues lead o highe bu ne exi speed alues.
A he le el o he a ea along he bu ne , i is necessa y ha i inc eases, so ha he p essu e
is main ained. The e o e, an inc ease in he equi alence a io leads o highe a ia ions in he
a ea along he bu ne , making i easie o main ain p essu e.
Fo he engine in gene al, by inc easing he c oss-sec ion a ea, you a e inc easing he d ag
gene a ed by he engine, which is a nega i e aspec .
Figu e 4.11: Veloci y a he exi o he bu ne as a unc ion o 𝑀0 [16]
37
Figu e 4.12: A ea a io as a unc ion o 𝑀0 [16]
4.3 Case S udy
A e he explana ion, o he pa ame ic s udies ob ained by Pe ei inha e al. [16], and a e
ca ying hem ou , o p o e he esul s ob ained by Pe ei inha e al. [16], he case s udy
in ended in his wo k, can be de eloped.
Based on he pa ame ic s udies, he inpu s o he case s udy ha e been ca e ully selec ed,
so ha he case s udy is as accu a e and eal as possible.
4.3.1 Inpu
Some conside a ions we e made du ing he choice o inpu s, namely he ac ha he
ees eam numbe o Mach is 8, al hough in his wo k an isola o is implemen ed. The ange
o he Mach numbe implemen ed in he s udy should be small since some p ope ies a e
conside ed cons an along his ange.
O he poin s o be conside ed a e he choice o he 𝑇𝑟𝑎𝑡𝑖𝑜, as seen in he chap e on
pa ame ic s udies, in addi ion o he empe a u e be o e de ona ion being a leas 1000 K,
o p e en p e-igni ion. I hese condi ions a e no egis e ed, he code will wa n he use .
In he case o he alues, especially he speci ic hea s, hese mus be di e en in he sc amje
and in he ODWE, since he empe a u e and p essu e eco ded in he bu ne o he ODWE's
is much highe when compa ed o hose o he sc amje .
The alues used as inpu o he case s udy can be isualized in Table 4.8.

38
Table 4.8: Case s udy inpu s
P ope y
Assigned Value
h [m]
0
𝑀0𝐷
10
𝑀𝑢𝑏
15
𝑇𝑟𝑎𝑡𝑖𝑜
2
𝛾𝑐
1.362
Sc amje 𝛾𝑏
1.238
𝑂𝐷𝑊𝐸 𝛾𝑏
1.170
Sc amje 𝛾𝐶
1.238
𝑂𝐷𝑊𝐸 𝛾𝐶
1.170
𝐶𝑝0
1005
𝐶𝑝𝑐
1090
𝑆𝑐𝑟𝑎𝑚𝑗𝑒𝑡 𝐶𝑝𝑏
1510
𝑂𝐷𝑊𝐸 𝐶𝑝𝑏
2000
𝑆𝑐𝑟𝑎𝑚𝑗𝑒𝑡 𝐶𝑝𝑒
1510
𝑂𝐷𝑊𝐸 𝐶𝑝𝑒
2000
𝑉𝑓𝑥
𝑉3
0.5
𝑉𝑓
𝑉3
0.5
Sc amje ƞ𝑏
0.9
ODWE ƞ𝑏
0.9
Sc amje ƞ𝑒
0.9
ODWE ƞ𝑒
0.9
x o CxHy
0
y o CxHy
2
ℎ𝑝𝑟 [𝑀𝐽/𝑘𝑔]
119.96
4.3.2 Pe o mance
In his sec ion, he esul s ob ained will be analysed, a e he implemen a ion o he case
s udy in he nume ical ool de eloped.
P ima ily, he analysis o he speci ic h us will be made as a unc ion o he Mach numbe a
he inle , as shown in Figu e 4.13. As expec ed, o all h ee cases, he maximum speci ic h us
is eco ded o a Mach numbe o 10, and as he Mach numbe inc eases, he e is a dec ease
39
in speci ic h us , due o he mo e exp essi e inc ease in he eloci y o he low ou side in
ela ion o he escape eloci y. Fo sc amje engines, we ound ha he a iable a ea
con igu a ion has a speci ic h us highe han he cons an p essu e con igu a ion. E en so,
his alue is much lowe when compa ed o he speci ic h us o he case o ODWE's. Ano he
impo an poin is he ac ha he slope o he cu e o he a iable a ea con igu a ion is
qui e p onounced, being in ODWE, ligh e , and he cons an p essu e almos non-exis en .
Figu e 4.13: Speci ic Th us as unc ion o ees eam Mach numbe
Acco ding o hese esul s, i is expec ed ha he engine wi h he highes speci ic h us will
ha e he lowes speci ic uel consump ion. This is p o en by he Figu e 4.14. I is also no ed
ha he lowes speci ic uel consump ion is eco ded o lowe Mach numbe s, and as he
Mach numbe inc eases, so does he speci ic uel consump ion. Once again, he ODWE is he
engine ha has he lowes speci ic uel consump ion, being in e io o he a iable a ea
engine, and e en mo e so o he cons an p essu e engine.
40
Figu e 4.14: Speci ic uel consump ion as unc ion o ees eam Mach numbe
Nex , he speci ic impulse will be analysed, which can be obse ed in Figu e 4.15. The mo o
ha has he highes speci ic h us is he ODWE, ollowed by he a iable a ea and inally he
cons an p essu e mo o . I is also no ed ha he highes speci ic impulse alues eco ded a e
o lowe Mach numbe s.
Figu e 4.15: Speci ic impulse as unc ion o ees eam Mach numbe
Obse ing he Figu e 4.16, we ealized ha he nex ac o o be s udied was p opulsi e
e iciency. We ealized ha in his case he ODWE has he lowes p opulsi e e iciency, since
i has he highes exhaus eloci y, which can be p o en by he equa ion (3.55). On he
con a y, he cons an p essu e mo o has he highes p opulsi e e iciency, since i has he
lowes exhaus speed. An impo an poin is he ac ha he cons an -p essu e mo o is he
only mo o ha con e s powe , mechanical ene gy in o h us powe mo e e icien ly.
41
Figu e 4.16: P opulsi e e iciency as unc ion o ees eam Mach numbe
In e ms o he mal e iciency and obse ing he Figu e 4.17, we ealize ha ODWE has he
highes he mal e iciency compa ed o sc amje engines, since o dina y he mal machines do
no ha e shock wa es, o he han bu ning. The shock wa es allow an inc ease in p essu e and
empe a u e, wi hou using mo e uel, imp o ing he mal e iciency. This alue inc eases as
he numbe o Mach in he inle also inc eases. In he case o sc amje engines, he a iable
a ea engine has be e he mal e iciency when compa ed o he cons an p essu e engine,
al hough i dec eases as he Mach numbe inc eases.
Figu e 4.17: The mal e iciency as unc ion o ees eam Mach numbe
The nex s udy o be ca ied ou is he o e all e iciency s udy, as shown in Figu e 4.18. This
pa ame e demons a es how e icien ly he engine uses he ene gy ha was ini ially s o ed
48
ANNEXES
Annex 1 – Code inpu

49
Annex 2 – Code
50
51
52
53

54
55
56
57
64

65