-
7/27/2019 Sc in Engine AIAA-7349-822
1/6
and Chemica l Reac t ions / ' Pape r p resen ted a t T h i rd
Combus t ionand P ropu ls ion Co l loqu ium, NA T O -A GA RD , Pa
le rmo , S ic ily ,M a r c h 1 7 - 2 1 , 1958.28 L ees, L . , ' 'L
am ina r He a t T rans fe r Over Blun t -N osedBod ies a t Hyperson
ic F l igh t Speeds , " J E T P R OP UL S I O N , vol. 26,no. 4,
1956, pp. 259-269.29 Liepm ann, H. W., and Bleviss , Z . O. , "T he
Effects of Di ssoc ia t ion and Ion iza t ion on Compress ib le
Coue t te F low," Douglas Aircraft Co. , Report SM-19831, 1956.30 L
igh th i ll , M. J . , "Dy nam ics of a Dissoc ia t ing Gas , I -E
qu i -l ib r ium F low," Journal of Fluid M echanics, vol. 2, part
1, 1957,pp . 1-32.
31 L inne t t , J . W. , and Marsd en , D. G. H. , "T h e Kine t
ic s ofthe Recombina t ion o f Oxygen Atoms a t a Glass Sur face ,
" and"T he Recombina t ion o f Oxygen Atoms on Sa l t and Ox ide
Sur f aces , " Proceedings of the Royal Aeronautical Society,
Series A, no.1199, vo l . 234 , March 1956 , pp . 489- 515 .32 L
ogan , J . G . , J r . , "Re l axa t io n Pheno men a in Hyperson
icAerodynamics , " Pape r p resen ted a t IAS 25 th Annua l Mee t
ing ,J a n . 2 8 - 31 , 1957 ( IAS P rep r in t 728 ) .33 Meixner ,
J . , "Z ur T heor ie de r Warm ele i t fah igke i t R ea -g ie
render F lu ide r Mischungen , " Z. Naturforsch, vol. 7a, 1952,
pp.553-557.34 Metzdorf, H. J . , "F lows in Pa r t ly Dissoc ia ted
Gases , "Journal of The Aeronautical Sciences, vol. 25, no. 3 ,
1958, p . 200.35 Moore , L . L . , "A So lu t ion o f the L amina r
Bound ary L ay erE qua t ions for a Compress ib le F lu id wi th
Var iab le P roper t ie s , I n c lud ing Dissoc ia t ion , "
Journal of the Aeronautical Sciences, vol.no. 8, 1952, pp.
505-518.
36 Nerns t , W. , "Chemisches Gle ichgewich t und T empera -tu
rge fa l len , " Fes tsch r i f t L udw ig Bo l tzm ann G ewidmet ,
1904, pp .904-915 .37 Penner , S . S . , "Chem ica l Reac t ions in
F low Sys tem s , "N A T O - A G A R D , B u t t e r w o r t h s ,
L o n d o n , 1 9 5 5 .38 Penner , S . S . , Harshb arge r , F . ,
and Va l i , V . , "An In t rod uct ion to the U se of the Shock T
ub e for the D e te rm ina t ion ofPhys ico -Chemica l Pa ramete rs
, " Combus t ion Resea rches andR e v ie w s , N A T O - A G A R D
, B u t t e r w o r t h s , L o n d o n , 1 9 57 , p p .134-172.39
Prigogine, I . , and Buess, R. , Acad. Roy ale de Belgique, vol.38,
series 5, 1952, pp. 7 1 1 - 8 5 1 .
T h e p u r p o s e o f t h i s s t u d y i s t o d e t e r m i
n e t h e f e a s i b i l i tyo f a r e a c t io n e n g i n e e m
p l o y i n g a c o n t i n u o u s d e t o n a t i o np r o c e ss
a t t h e c o m b u s t i o n c h a m b e r . A r e a c t i o n - t
y p ee n g i n e e m p l o y i n g s t e a d y - s t a t e d e t o
n a t i v e c o m b u s t i o n i sc o n s i d e r e d . A s i m p
l i f i ed a n a l y s i s t r e a t s t h e s u p e r s o n i cm i
x i n g o f f u e l a n d a i r t o g e t h e r w i t h t h e r e q
u i r e m e n t sn e c e s s a r y t o a c h i e v e s t e a d y -
s t a t e d e t o n a t i v e c o m b u s t i o n .C a l c u l a t
i o n s o f s p e c i f i c t h r u s t a n d s p e c i f i c f u e
l c o n s u m p t i o n a s f u n c t i o n s o f f lig ht M a c h
n u m b e r a r e m a d e fo r h y d r o g e n a n d a c e t y l e
n e f u e l s . T h e r e s u l t s o f t h i s s t u d y i n d i c
a t e t h a t s o m e s u p e r s o n i c d i f f u s i o n o f t h
e a ir i s n e c e s s a r ye v e n t h o u g h s u p e r s o n i c
c o m b u s t i o n e x i s t s . I t i s c o n c l u d e dt h a t
t h e s p e e d r a n g e o f a i r - b r e a t h i n g e n g i n e
s m a y b em a t e r i a l l y e x t e n d e d .
Rece ived Nov . 13 ,1957.1 T his re sea rch was suppor ted by
the U n i ted S ta te s Air Fo rcethrough the Air Force Office of
Scientif ic Research of the AirR e s e a r c h a n d D e v e l o p
m e n t C o m m a n d , u n d e r C o n t r a c t N o .AF 18(600
)-1199.2 R e s e a r c h A s s is t a n t , E n g i n e e r i n g R
e s e a r c h I n s t i t u t e .3 I n s t r u c t o r , D e p a r
t m e n t of A e r o n a u t ic a l E n g i n e e r in g . M e m
.ARS.JULY 1958
40 Rab inowicz , J . , "Aerodynamic S tud ies in the ShockT u b
e , " G A L C I T H y p e r s o n i c R e s e a r c h P r o j e c t
, M e m o r a n d u m38, J u n e 1 9 5 7 .41 Romig , M . F . , and
Dore , F . J . , "So lu t ions of the C ompress ib le L aminar
Boundary L ayer I nc lud ing the Case o f aDissoc ia ted F ree S t
ream," Conva i r , Repor t Z A-7-012, 1954 .42 Rose , P . H. , P
robs te in , R. F . , and Adam s , M. C , "T u rbu len t Hea t T
rans fe r T hrough a High ly -Coo led , Pa r t ia l ly Disso c ia
ted Bou ndary L aye r , " Avco Resea rch L abora to ry , Rese a
rchRep or t 14 , J an . 1958 .43 Rose , P . H. , and S ta rk , W. I
. , "S tagna t i on Po in t H ea tT rans fe r Measu remen ts in
Dissoc ia ted Air , " Journal of the Aero-nautical Sciences, vol.
25, no. 2, 1958, p . 86.44 Rosner , D. E . , "Bo und ary Cond i t
ions fo r the F low of aM u l t i - C o m p o n e n t G a s N e a r
a R e a c t i v e W a l l , " J E T PR O PUL S I O N( t o a p p e a
r ) .45 Rosner , D. E . , "Chem ica l ly F rozen Bou ndary L ayers
wi thSur face Reac t ion , " P r ince ton L Tn ivers i ty Aeronau t
ica l E ng inee r ing Rep or t 419 , Ma rch 1958 ; subm i t ted to
Journal of Aeronautical Sciences.46 Sca la , S . M. , "Hyperson ic
Hea t T rans fe r to Ca ta ly t icSur faces , " Journal of the
Aeronautical Sciences, vol. 25, no. 4 ,1958, p p . 2 7 3 - 2 7 5
.47 Sca la , S . M. , "Hype rson ic S tagna t ion Po in t Hea t T
rans fe rto Sur faces Hav ing F in i te Ca ta ly t ic E f f
iciency, " Pap er p resen teda t the Aero the rm ochemis t ry
Session , T h i rd U .S . Na t io na l Cong ress o f App l ied
Mechan ics , Brown U n ive rs i ty , J une 13 , 1958 .48 Scho t te
, W. , "H ea t T rans fe r to a Gas Phase C hemica lR e a c t i o n
, " Industrial Engineering and Chemistry, vol. 50, no. 4,1958, pp .
683-690 .49 U bbe lohde , A. R. , Journal of Chemical Physics, vol.
3 ,1915, p. 69.50 Z ieb land , H. , "Some E xper im en ta l Obse
rva t ions Rega rd ing the Convec t ive Hea t T rans fe r f rom
High ly Dissoc ia tedCom bus t ion Gases to Coo led Wal ls of Rock
e t En g ines , " Min is t ryof Supp ly ; E xp los ives Resea rch
and Deve l opme n t E s tab l i sh m e n t , G r e a t B r i t a i
n , T e c h n i c a l M e m o r a n d u m 1 3 / M / 5 6 , A u g
.1956.51 Z ig rang , D . J . , "N o te on Dissoc ia t ion E f fec
ts in Hyp er son ic Viscous F lows , " Journal of the Aeronautical
Sciences, vol.24, no. 12, 1957, p . 916.
N o m e n c l a t u r eaAU pFQh\J TK xK2mMPQRoSF CTVwx, yyp
======================
speed of soundareaspecific heat a t constant pressuret h r u s
tacceleration of gravityen tha lpy pe r un i t massspecific
thrustde fined by E qua t ion [20 ]defined by E qua t ion [21
]molecu la r we igh tM a c h n u m b e rp ressu reheat re lease per
unit massun ive rsa l gas cons tan tspecific fuel consumptiona b s
o l u t e t e m p e r a t u r evelocitymass f low ratecoord ina te
axesratio of specific heatsdens i ty
45 1
A P r e l i m i n a r y S t u d y o f t h e A p p l i c a t i o
n o f S t e a d y -S t a t e D e t o n a t i v e C o m b u s t i o
n to a R e a c t i o n E n g i n e
R . D U N L A P ,2 R. L. BREHM 2 a nd J . A . N I C HOLLS 8U n i
v e r s i t y o f M i c h i g a n , A n n A r b o r , M i c h .
-
7/27/2019 Sc in Engine AIAA-7349-822
2/6
xb{M) = defined b y E q u a t i o n [13]r}(M) = defined b y E q
u a t i o n [15]Subscriptsai f = gases a a n d / (also a = a i r a
n d / = fuel)D = Chapman- Jougue t de tona t ione = exit
conditionsig = ignit ion valuen = normal componen ts = stagnation
condit ionst = t angen t ia l componen t = und is tu rbed a ir
condit ions1, 2, 3, 4 = stat ions i n t h e engine
I n t r o d u c t i o nTO DATE there has been considerab le
exper imental andtheoretical work devoted to the study of
detonationwaves. I t has been found tha t upon ignition of various
combustible gas mixtures under certain conditions of
pressure,volume and com position, a flame front will propa gate
throug hthe mixture at speeds ranging from 3000 to 12,000 fps.
Thisfront (shock-initiated combustion) is called a detonation
waveand the speed at which it propagates is the detonation
velocity. So far this phenom enon has been observed mainly inshock
tubes in which the wave is in a transient sta te. Atpresent,
however, attempts are being made to accelerate acombustible gas at
high total temperature and pressure tothe local detonation velocity
and then cause i t to ignite, thusestablishing a standing
(steady-state) detonation wave (l). 4One is led to wonder if such a
burning process could beapplied to a propulsive system. I t seems
tha t a jet deviceutilizing this mode of combustion, as opposed to
deflagrationburning, would offer several adva ntage s. For example,
thesupersonic inlet diffuser could be greatly simplified since
theburning would occur at supersonic speeds. Th us the incoming air
need not be diffused subsonically and hence shock-swallowing
problems, as well as large total pressure losses,could be
eliminated. The static pressure rise, necessary forexpelling the
exhaust gases, would now be realized, at leastin part , across the
detonation wave rather than entirelythrough the diffuser. Furth erm
ore, since the deton ationprocess occurs at high velocities and
total temperatures, i tseems to be a natural means of extending the
speed range ofair-breathing vehicles. Other advan tages would
include ashortened combustion chamber with no need for an
ignitiondevice. An obvious disadv antag e is the lack of static th
rus t.In view of possible applications, a preliminary analysis
wasmade in an effort to determine some of the performance
characteristics of an engine in which heat is added by means of
astanding detonatio n. Special atten tion is given to the
supersonic mixing of fuel and air, and solutions are presented
forthe two cases when either the pressure or the area remaincons
tant through out the mixing zone. A general discussionof the
detonation process, together with the method in whichit could be
applied to a steady-flow engine, is also given.Finally, problems
associated with the matching of the various flow processes to produ
ce an efficient thru st-p rodu cingmechanism are discussed and some
over-all characteristics,such as specific thrust and specific fuel
consumption, arecalculated.I t should be pointed out th at the
analysis contained hereinis l imited to the following general assu
mptio ns: The w orkingfluid is assumed to be an in viscid perfect
gas mixture. Average values of the specific heats, applicable to
the temperature range and mixture considered, are used
throughout.Heat and frictional losses are assumed absent, and any
totalpressure losses due to the formation of oblique shock
waveswhile compressing a supersonic gas are neglected. Finally,i t
is supposed that there exists an "effective ignition temperature"
below which the fuel-air mixture must be kept prior todetonat ing
.
4 Numbers in parentheses indicate References a t en d of papers
.
2 3 4Fig. 1 E ngine configuration
U n reac ted m ix tu reof a combust i t le gas ->Gaseo u s r
eac t i o np ro d u c t s
s t a t e 3 ^ s t a t e kFig. 2 Normal detonation wave
In view of these assumptions, i t should be clear that
theresults presented in this paper can only indicate
approximatevalues of performance characteristics comparable to
otheridealized engines, which is all that was intended.Genera l C o
ns i dera t i o ns
The following is a brief discussion of the flow system assume d
in the ana lysis of this engine. Fig. 1 is a sketch of thisflow
system .The desirability of an engine requiring no diffusion of
theincoming air has already been mentioned . At first onemight th
ink t ha t no diffusion w ould be required in this enginesince both
mixing and combustion are occurring at supersonic velocities. How
ever, it will be shown tha t for maximum pressure recovery through
the engine there must besupersonic diffusion to some extent
previous to mixing thefuel and air. Th is diffusion occurs betw een
station s o and1. The fuel is injected into the supersonic stream
at station1 and is assumed fully mixed with the air at station 2.De
tona tion occurs at station 3. U nder certain conditions,namely, at
low flight speeds, the back pressure should be sufficient to init
iate the detonation normal to the flow at somestation 3. How ever,
this typ e of init iation may possibly beunstable, and in practice
some form of stabilization, such asa thin body placed in the
stream, will probably be required.At high flight speeds, certain
mixing requirements and efficiency considerations dictate that a
body must definitely beplaced in the stream. The resulting oblique
shock waveinitiates the chemical reaction and the body serves as
themeans of stabilization. For the oblique wave portion of
thisanalysis, the detonation is assumed to be stabilized on a
two-dimensional wed ge.Chemical equilibrium conditions are assumed
to exist atstation 4 imme diately downstream of the detonation.
Finally, frozen equilibrium flow is assumed in the isentropic
expansion to atmosphe ric pressure at the exit . Dow nstrea
mreflections of the stabilized wave are neglected and of
coursevariable geom etry is implied.It should be pointed out that
the question of stabili ty ofthe assumed mode of combustion is as
yet an unresolved one.In fact, this is the primary objective of the
Air Force contractsupporting the study reported here. This problem
is beingattac ked both theoretically and experime ntally. For
purposes of this paper, stabili ty has been implicit ly assumed.It
is felt that this is highly probable for wedge
stabilizationalthough it is admitted that a normal detonation wave
existing between stations 2 and 3 would in all l ikelihood be
unstable and some means of stabilization would be required.Such a
wave could not move all the way upstream to position1, however, due
to incomplete m ixing.In the over-all consideration of the
analysis, two areas of
4 52 J E T P R O P U L S I O N
-
7/27/2019 Sc in Engine AIAA-7349-822
3/6
special interest and im portance stand out the supersonicmixing
of fuel and air and the deton atio n process. E ach willbe
considered separately in the following two sections.C o n s i d e r
a t i o n s o f t h e D e t o n a t i v e P r o c e s s
For purposes of analysis a detonation wave is usuallytreated as
a discontinuity with heat addition. 5 First , consider the
one-dimensional model of a detonation in which thereference system
is attached to the wave and the gases moverelative to it (Fig.
2).The conservation equations for this system can be written
State 3
PzVz ~ piVi 7 3 M 3 P 3 T 4 M 4 P 40,3 Cli [ I ]
omentum: P 3 + PsJV = PA + PAV^ orP 8 ( l + 73M32) = P 4 ( l + 7
4 M 4 2 ) . . . [2 ]
CmT, + -j- + Q = CPiT< + - y o r
3 [ 3 ]
ext, consider an extension of the above model to the two-wedge
(Fig. 3). I n this model the incoming velocity is
ce there is no net pressure force in the tangen tial
direction,The normal components may be treate d as in the one-case.
Hence the two-dimensional analysis re
onal normal wave model. No te that the cons for this model are
identical to the p revious3 and 7 4 are now
Note also that in the energy equation the tangentialOne finds
upon analyzing the above conservation equations
certain minimum . For all velocities above this6
I t is interesting to note that th e minimu m velocity is
theAssociated with this minim um velocity are down
relative to the wave is always unity. Also, the tot alloss
across such a detona tion is a minim um. ThisSince total pressure
is of prime importance in the efficient
uguet typ e wave. This proves to be a fortunate choicecan be
eliminated. Thu s, b}^ using experimental values
M i x i n g A n a l y s i sAn exact solution to the supersonic
mixing of fuel and air5 I t m a y b e thought of as a shock wave
with combustion.6 See (1) for a comprehensive discussion of the
oblique wave
A ir
Fig. 3 Oblique detonation wave
"T" / u 1 1 ( 1 1 1 f 1 1 1 u i i i i 1 ( t T i l . r.ial
L w i 1 i 1 1 ( / i ( rTi i i i J-LZ: r u c i Fuel Nozzles
Ti 1 1 1 1 / / ^/ / / / / / I I /T'zizi 1 1 1 1 1 nri ? 1 1 1 1
1 f^r.'Fig. 4 A method of fuel injection
utnn/L"^7777777777
t
7777777777777777,~Fig. 5 Mixing region
would be prohibitively complicated. Consider, for example,a
scheme for introducing gaseous fuel into a supersonic streamof air
as shown in Fig. 4. Air, at a high stagna tion te mpe rature and
pressure, would flow around the nozzles at supersonic speeds and
proceed to mix with th e fuel throug h molecular and turb ule nt
diffusion. Th e resulting shear flow wouldbe nonuniform and nonstea
dy. The flow would be furthercomplicated by the presence of shock
waves generated inthe nozzle region, as well as by possible local
burning in thebou nda ry layer in the vicinity of the nozzle exit.
Since itwould clearly be quite difficult to make a detailed
investigation, the following analysis will be directed toward a
simplified solution of the mixing problem.Consider the mixing of
two inviscid perfect gases (denotedby subscripts a an d/ ) as shown
in Fig . 5 . I t is assumed thatthe rate of change of any flow
parameter in the x or y direction is zero at s tations 1 and 2
(except at the init ial interfaceof the gases where a discontinuity
may exist in the y direction in the temperature and velocity), and
that the gases arefully mixed at sta tion 2.With these assumptions
the conservation equations may bewritten in difference form asm a s
s : piaVi aA la + PI/VI/AI/ = p2V* A2. [4 ]energy: Wi . ( , . + ^ )
+ w \f ( hi f + ?)
(-?) [51momentum in ^-direction:Px (A la + A lf) - P,A 2 + J ^ 2
PdA =
PlAiVf - PiaA laVi a2 ~ PlfAxjVif*. . . [6]U sing the equation
of state
P = P Tmand approximating the enthalpy by
h = C P T1958 4 5 3
-
7/27/2019 Sc in Engine AIAA-7349-822
4/6
w h e r e Cp is an a v e r a g e v a l u e of t h e s p e c i f
i c h e a t b e t w e e n T =0 a n d T = Th th e c o n s e r v a t
i o n e q u a t i o n s a r e r e w r i t t e n in
theformJiaMiaAiaP, ylfM uA lfP l y2M 2A2P,
aia aif U)
e n e r g y : Ts - * ( ,+i?,
m o m e n t u m :
w h e r e
) -Cpi/ Wif Tsiaj (r/i[ Cpia Wig 1CPlf Wif J
PiAi (l + yla ~ Mla* + 7 1 / Yx Mlf2)
Ts la...[S]
+x ^ 2 Pel A = PoA, (1 + 7 , M 2 2 ) . . . [9]4 i = ,l ia +
Alf
N o t i c e t h a t the m o m e n t u m e q u a t i o n c o n t
a i n s an i n t e g r a lt e r m t h a t is d e p e n d e n t u p
o n th e v a r i a t i o n of p r e s s u r e d u r i n gt h e m i
x i n g , w h i l e the o t h e r c o n s e r v a t i o n e q u a t
i o n s e x p r e s s ar e l a t i o n s h i p b e t w e e n the
end p o i n t s o n l y . It is a p p a r e n tt h a t a s o l u t
i o n can be f o u n d for c o n s t a n t p r e s s u r e m i x i
n g , inw hich cas eJ; PdA = P^Ao - Ar) = P2(A 2 - A,). [10]o r f o
r c o n s t a n t a r e a m i x i n g w h e r e
* AsL Pd A = 0. [11]I n g e n e r a l th e i n t e g r a l can
be a p p r o x i m a t e d by u s i n g an
a v e r a g e c o n s t a n t v a l u e of t h e p r e s s u re
d u r i n g m i x i n g a l t h o u g hth i s cas e w i l l no t be
cons idered .
B y c o m b i n in g E q u a t i o n s [7, 8, 9] an d [10 or 1 1
] , th e foll o w i n g s o l u t i o n s for th e d o w n s t r e
a m M a c h n u m b e r are f o u n df o r c o n s t a n t p r e s
s u r e or c o n s t a n t a r e a m i x i n gcons tan t p res s
ure mix ing : 4>(M 2) =
^Pia W\a1 +7 i ?% / Lp\f Wi f7 2 m-ia \ TS1/ CPla wla
^Tsxa Cpif wif/
LAaSlf Wif{M lf)asia Wia
L WiaJw h e r e
-
7/27/2019 Sc in Engine AIAA-7349-822
5/6
and that in the Maeh numbe r range between M = 2 and M =4 the
least loss in total pressure is obtained by injecting thefuel at
the low energy level. I t is physically plausible th atthere should
be a better total pressure recovery (referred tothe air total
pressure) when the fuel is at a low energy levelsince more thermal
energy from the air must be transferredto the fuel.7 As the Mach
number of the air increases, theair total temperature becomes so
large that the effect of thedifference in fuel-inlet co nditions
becom es less significant.The results shown in Fig. 6 indicate the
desirabili ty of diffusing the incoming air somewhat before
injecting the fuelbecause of the importance of total pressure
recovery in engineperformance.
D eta i led Ex a m i na t i o n o f the F l o w S y s temThe
integration of the previou s work into an over-all enginesystem
will now be considered.An important l imitation peculiar to this
type of engine isthat the static temperature of the unreacted
fuel-air mixturemust be kept below an "effective ignition
temperature."The "effective ignition temp era ture " is herein
defined as tha ttemperature at which a moving combustible gas
mixture willspontaneously ignite. Clearly, this tem pera ture will
depend on the dynamic and thermodynamic s tate of the gas
,composition, confining boundaries, the flow process involved,etc.
To the authors' knowledge, no studies involving all
these param eters have been ma de. To effect the calculations
for this engine, it was necessary to assume some value ofignition
temperature which appeared reasonable on the basisof known values
found in the l i terature for stagnant mixtures.In applying the
ignition-temperature condition, certainramifications mus t be
considered. Th e total pressure losscross the detonation can be
minimized by the attainmentof a Chapman-Jouguet detonation at the
lowest possible8 it isbvious that the detonation should occur at
the highest posle tem pera ture. This is adva ntage ous from sti l
l anoth er
ch number of mixing is decreased. Hence , for best perowed to
approac h its l imiting value, th e ignition tempe r. Also, from
this reasoning, it is concluded tha t supe r
In locating the detonation in this engine, the above dis
Recall , however, tha t to achieve deton ation, a
certainvelocity is required. At low flight speeds thisflow mu st be
expand ed. Th e detonation will then
pma n-J ougue t normal wave. As the fl ight speed is inl occur
further upst ream until finally the y will
jus t at the end of mixing . For any higher flight speeds ,f
veloci ty remains a min imum (Chapm an-Jou guet
With the general engine configuration thus determined, i t isble
to compute over-all performance. By specifying the
7 Recall that in a Motionless heat subtraction process the
totalof the gas will increase .8 The Chapman-Jouguet detonation
velocities used in this re
L Y 19 58
-
7/27/2019 Sc in Engine AIAA-7349-822
6/6
(a) Setting T2 = T ig, determine M 2 and T s2 f rom E quat
ion[8].(b) Knowing M 2 and T s2, compute M ia from E qua tion [12or
14].(c) Calculate Psi/Psia from E qua tion [16 or 17] .(d) Find V2
= M 2 (h and compare wi th VD .1. If V2 is less than VD , use the
normal wave solutionbelow (E quatio n [18]) to comp ute Ve.2. I f
V2 i s g reater than VD , use the oblique wave solut ion (E quat
ion [19 ]) .The following equations were derived by use of the
conservation equa tions between the end of mixing and the exit . Th
esolution across the detonation wave was determined so thata
Chapman-Jouguet (normal or ob l ique) wave alwaysoccurred.normal
detonation:
F e =[w+*-'r-]
VD V 7 V 1 X
(74+
L 2 a s2 2J L 2 a s2 * _ \[18]
74(72-1) I
oblique detonation:V e = where * -"a22 + 72 Fz)2)2(7*4-1) I V -
VL
.[19]
[20]
V 7 2 / Fz> 2(74 -i) \ I + 7 4 / W 2 / L JPerformance
characteristics such as specific impulse and
specific fuel consumption can now be computed by the
usualformulas
CT =Ve ( l + ^ )\ Wig/ va
Wl a [ 2 2 ]
[231FC - 3600^-', HR-
R es u l t s a nd C o nc l us i o nsFigs. 7, 8 and 9 show the
results of calculations of specificthrust and specific fuel
consumption for hydrogen and acetylene fuels. I n each case the
fuel-air mixture was stoichiom et
ric and the effective ignition temp erat ure was assumed tobe
2000 R. The Cha pman-J ouguet detonat ion veloci tieswere approx
imately equ al at a value of 5900 fps. The constant pressure mixing
solution was used throughout.From these graphs, i t can be seen
that there is no thrustbelow a fl ight Mach number of 4 and that
the specific thrustreaches a maximum between Mach 6 and 7, then
decreases.The shape of the specific thrust curve can be explained
asfollows: Firs t, below a certa in flight speed the tota l energ
yof the fuel-air mixtur e is insufficient to ac hieve ste ady de
tonation . At slightly higher fl ight speeds a detonation willoccur
if the gases are expanded to a very high Mach number.However , for
very h igh detonat ion Mach numbers , the to talpressure loss is
excessive and no thru st is realized. As theflight speed is
increased, the Mach number of detonationdecreases, and hence the
specific thrust begins to increase.Finally the specific thrust
reaches a maximum when the detonation is stabilized at its limiting
position (the end ofmixing). For sti ll higher fl ight speeds the C
hapm an-J ougu etoblique wave solution exists at a constant Mach
number ofdetonat ion at T ig, and hence the total pressure ratio
across thewave remains cons tant. Th us the specific thru st
decreasesbecause of the rising total pressure loss associated with
themixing process.The importance of fuel inlet conditions can be
seen fromFig. 7. Th e uppe r curve corresponds to M if = 2 and T8if
=2000 R, while the other curve represents the lower l imitingvalues
for these para met ers. The fuel-inlet conditions areseen to be
quite important in determining the operating rangefor this type of
engine.Th e curve s in Figs. 8 and 9 com pare two different
fuels.Since the detonation velocities were the same for both
mixtures and the effective ignition temperatures assumed equal,the
differences in these curves are due to the unequal molecular
weights, fuel-air ratios, and specific heat ratios.In deciding upon
the feasibili ty of the proposed engine,the specific thrust and
specific fuel consumption may be compared to an ideal ramje t. The
prese nt engine offers comparable performance at much higher fl
ight Mach numbers,and indicates a possible means of extending the
speed rangeof air-breathing vehicles. Finally, i t mu st be
stressed tha tthese performance calculations are highly dependent
on theeffective ignition temperature assumed and are thus l
imitedby the accuracy of this assum ption. Furth erm ore, the
results have not been optimized from the standpoint of
mixingprocess and fuel-inlet conditions.
R eferences1 Rutkowski, J., and Nicholls, J . A.,
"Considerations for theAttainment of a Standing Detonation Wave,"
Proceedings ofthe Gas Dynamics Symposium, Northwe stern U
niversity, 1956.Also issued as OSR-TN-55-216.2 Lewis, B., and Von E
lbe, G., "Combustion, Flames andE xplosions of Gases," Academic
Press, New York, 1951.3 Morrison, R. B., "A Shock Tube
Investigation of Deto-native Com bustion," U niversity of Michigan,
E ngineering Research I nsti tute R eport U MM -97, Ann Arbor,
1952.
4 5 6 J E T P R O PU L S I O N
