Document
RESEARCHMEMORANDUM
ANALYTICAL INVESTIGATIONOF RAM-JET-ENGINE PERFORMANCE IN FLIGHT MACH NUMBER RANGE FROM 3 TO 7 By Philip J. Evans, Jr.
Lewis Flight PropulsionLaboratory
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W OFFICERMAILIRC(iHh#@E)
NATloNAL ADvIsORY COMMITTEE
FORAERONAUTICS
WASHINGTON
October 17, 1951
k
NACARM E51H02 l AERONAUTICS NATIONAL ADVISORY COWTTEE FOR .
RESEARCH MEMORANDUM ANALYTICAL INVESTIGATION OF RAM-JET-ENGINE F31RFORMANCE IN FLIGHT MACHNUMBER RANGE FROM3 TO 7 By Philip J. Evans, Jr.
k analytical investigation wasmadeof theperformance of isolated ran- jeten@nesin theflight Machnumber range from3 to 7. Calcula- ficiency diffuser tions weremadefortwotypes of tif~ser: a hfgh-ef .
of unspecified forin having a Hneti c-energy efficiency of 0.92, anda simple normal- shock diffuser. A combustion efficiency of 100percent anda fuelhaving a hydrogen-carbon ratio of 0.168 wereassumed.
Impmtantconclusions were: (1)a design altitude of 100,000 feet .
appears desirable fora high-efficiency, highMachnumber ram-jet eugine, and (2). gasoline provides sufficient energy release formaximum engine efficiency in thef13.ght Machnumber range investigated.
b Thecalculated maximum propulsive-thrust coefficient foran externally mounted ramjet. hating a high-efficiency diffuser decreased continuously fromapproximately 2.13at a flight Machnu&er of 3 to engine efficiency 0.57at a flight Machnumber of 7. Themaximum reached a peakof 0.47neara flight Machnumber of 5 andthen — decreased to 0.43at a Machnumber of 7.
INTRODUCTION Because of lackof ifiormation m-ram-jet design @iiperformance at highMachnumbers, primary consideration hasbeengiven therocket forvehicle propulsion above a flight Machnuniber of 4 within the earthc-s atmosphere.
In order to determine whether consideration should alsobe-given theramjetin thehighMachnumber range, an analysis wasmadeat /“” theNACALewis laboratory of ram-jet perfomnance at zero “angle of” ,~fi attack forflight Machnumbers between 3 and7. Thecomputed perform- $ i,~ antewasbased on “a hyih?ocarbon fuelhaving a hydrogen-c&rh-o’n ratio D of 0.168:’ However, because of thenarrow range of hydrog~-carbon ~ .- ~$ - ratio andthesimilarity of combustion products, thesameresults should be obtained withmostof theavailable hydrocarbons. In fact, , “ * Y 1.>./” ,.
it seems reasonable to expect that ”many nonhydrocarbohs btii~ at i thesanevalue of energy addition willyield approximately the same thrust andengine efficiency.
Consequently, results arepresented in terms of an ener~-addition theywillbe more generally applicable.
- T!, / NACARM E51H02 # .
In theanalysis, twotypes of diffuser wereconsidered: (1]a .
diffuser of unspecified formhaving a kinetic-energy efficiency of 0.92, and (2)a simple normal-shock diffuser, which provided a basisfor comparison. Theresults presented areforthedesign condition only andrepresent a different geometry foreachvalue of energy addition andflight Machnumber.Off-design performance wasnotconsidered in theanalysis.
SYMBOLS co
R
N Thefollowing symbols areusedin thisreprt: A cross-sectional area total engine-drag coefficient, De/~~ CD, e engine friction-drag coefficient, Df/~~ %,f propulsive-thrust coefficient, F/~~ + net-tmst coefficient, Fn/~~ . * c F,n skin-friction coefficient, (friction force/sq ft)/~ Cf =.
c speed of sound, (ft/see} D tiagof airplane minusdragof engine ___ — —.
De total engine drag engine friction drag Df Er energy-addition parameter defined as ratio between fuel energy actually added andenergy added by gasoline at its stoichiometric condition F propulsive thrust, Fn - De Fn net,or internal, thrust f/a fuel-air ratio gravitational constant, 32.17(ft/sec2) b ..—- — total, or reservoir, enthalpy static enthalpy .
h
,#&y!!*.@#f&q
->. , NACARM E51H02 J mechanical equivalent of heat, 778( ft-lb/Btu) .
L liftof airplane ‘ m massflow M Machnumber \’ DJ P reservoir, or total} pressure N ol m Pr relativ’e tatal pressure as usedin airtables of reference 6 (actual total pressuedivided by total pressure at reference temperature usedin table) static pressure P relative static pressure as usedin airtables of reference 6 Pr heating value of fuel,(Btu/lb fuel) Q dynsmic pressure, p@/2 gasconstant in characteristic equation of state R reservoir or total, temperature t static temperature velocity v gross weight at beginning of f~ght ‘o gross weight after fuelhasbeenburned
‘1
actual fuelflow,(lb/see) ‘f x range, (ft) ratio of specific heats T combustion efficiency FV engine efficiency, * kinetic-energy efficiencyof diffuser [equation (1}) efficiency parameter forcomplete engine
V’k (defined by equation (3))
l ..go :“-”-– , 4 NACARM E51H02 l ~,.<”: : -.
v?
. .
k~
l .- density P .
‘c total temperature after combustion Mvidedby total temperature J—— before combustion q Subscripts: — av 1,’ average
i?J
comp completely expanded gasoline g .- eff effective St Stoichicmletric — o free-stream conditions,, or inlet free-stresm tube — —.
2 combustor inlet downstream of flame holder nozzle throat ., 7 nozzle exit ASSUMPTIONS AND CALCULATIONS General Assumptions andRocedure .
“The following general assumptions weremade: 1. ‘The internal flowis one-dimensional. In applying theone- dimensional-flow e@ations, thevariations in y, theratio of specific heats of theworking fluid, dueto temperature andfuel-air ratio variations wereconsidered. Fortheflowinside theconibustor, an averagey before andafter combustion wasused. Theflowfromthe nozzle throat to thenozzle exitwas calculated as described in appendix A.
2. Theengine is operating at an altitude or 100,OOO feet. ~is assumption determines thefree-stresm static. temperature andpressure (table III05 reference 1) andhence thetotal temperature andtotal- temperature ratio % corresponding to a given flight Machnumber~ b“ andenergy addition. It alsode&rmines theReynolds number perunit .-: length usedto determine .the external friction-drag coefficient.
t@mQ9’Ei%!i
NACARM E51H02 3.
Theenergy lostby radiation andconduction is negligible.
.
theairentering theengine is equal 4. Thefree-stresm areaof to theinlet areaof theengine.
maintainable at thehightemperatures 5. Structural integrity is andpressures.
Diffusers Definition andsignificance of Mnetic-energy efficiency. - The ~netic-ener~ efficiency ~k of a diffuser is defined as thekinetic energy theairwould haveafter beingexpanded isentropically fromthe diffuser exitto thesaibient static pressure dividedby thefree-stream kinetic energy.Whenthisprocess is accomplished at constant y, the formula forthekinetic-ener~ efficiency is .
(1)
~k =1 y-l .
-z- %2
.
l assumption of a value for ~k determines thediffuser Consequently the total-pressure ratio P2/Po for each ~. Equation (1)is plotted in fiwe 1. In thisfigure thetotal-pressure ratio required fora
g~&n ~~ue of nk &creases rapidly with increasing flight Mach
number~. considered, verylittle error is In therange of ~ introduced forhigh values of Tk by considering T constant and equal to 1.4since, in general, equation (1)yields accurate results whenever the finaly after reexpansion is little different fromthe ambient staticT.
Thekinetic-enerm efficiency of a diffuser is significant for —.
ram-jet engines because of itssi&larity to a basicram-jet perform- anceparameter. Thenetinteml-thrust coefficient of an engine having constant y anda completely expanded exitis given by .
r where .
NACARM E51H02 and (3). ) Thequantity (See appendix B forderivation of equations (2) .
=Vlk represents thekinetic-energy efficiency of thecom@eteengine.
.— Because at highvalues of ~ al.must allthetatal-pressure losses .— occur in thediffuser, thediffuser kinetic-energy efficiency and equal.
~’k arenearly Theshape of thecurves of figure 1 (that is,thewell-defined shoulder at highvalues of ~) suggests thatthere willbe little advantage in operating at total-pressure ratios veryfarto theright of thisshoulder.
i?i N High-efficiency diffuser. - On thebasis of available highMach number data, thehigh-efficiency diffuser wasassumed to havea kinetic-energy efficiency ~k of 0.92at allflight Machnumbers.
An advantage of assuming Vk is thatlmowledge of theexact formof -— .— thehigh-efficiency diffuser is ~ecessary. Theassumption thatthe maximum~k is invariant with ~ wassuggested by thesmall variation of maximum~k in therange of ~ between 2 and4 and alsoby thefactthatthere is no excessive variation of the VW of theflowthrough a normal-shock wavein the ~ range from4 to 7.
.— .
Values of ~k fOrseveral existing andproposed diffuser designs (references 2 and3) areshown in figure 1. Thevalue of P2/Po * which follows fromtheassumption of ?l decreases quite rapidly as ~ increases (fig. 2) andsho@dbe at alnable witha refine”d * diffuser design.
Normal-shock diffuser. - In calculating thetotal-pressure ratio across theshock waveof thenormal-shock diffuser, imperfect-gas effects wereincluded by means of correction factors taken fromrefer- ence4. Thetotal-pressure ratio of thesubsonic diffuser wasarbi- trarily assumed to be 0.98. Theresultant over-all total-pressure ratio of thenormal-shock &Lffuser is considerably lower thanthat of thehigh-efficiency diffuser; however, itsvariation with K Is u quite similar (fig. 2].
Combustion Assumptions pertaining to thecombustion process weremadeas follows: Thefuelis anyhydrocarbon (such as gasoline) having a hydm~&-carbon ratio of 0.168.However, because of thesimilarity of u combustion products of hydrocarbons, theresulting total-temperature ratio. T across thecombustor at equal values of energy addition (based on datafromreference 5) should be verynearly e@al forall NACARM E51H02 b hydrocarbons~ andevenforsomenonhydrocarbons beingconsidered for .
ram-jet fuels, a given energy addition should produce at least approxi- .
Consequently, theresulting thrust mately thesamevalue of T.
coefficients andengine efficiencies willa~ly to mosthydrocarbon fuels andto a fewnonhydrocarbons as well. Thus, instead of presenting performance in terns of fuel-air ratio, as is frequently done, it is presented in terms of an energy-addition parameter Er defined as (f/a) ef f Q Er =
(f/a} g, st Qg
If thefuelis gasoline, Er is merely theeffective fi”el-air ratio relative to stoichiometric, thatis,theequivalence ratio.Forother fuels, however, thisenergy-addition parameter Er is theratio between theenergy actually added andtheenergy thatwould be added by gasoline at itsstoichiometric condition.
2. In thedetermination of T it is necessary thatthevalue of be consistent withtheassumed altitude combustor-inlet pressure P2 .
anddiffuser-pressure ratio, because theamount of molecular dissocia- tionincreases as thepressure decreases, causing theresultant % to a decrease. Thedifference in thepressure given by thetwotypes of” Uffuserat highvalues of ~ (noticeable fromfig.2) results in twosetsof curves of 7 against& forconstant value of energy- addition parameter (fig. 3). Thelarge decrease of T withincreasing ~ ismainlyaresult of theincrease of total temperature with ~.
3. Combustion is completed in theconstant-area combustor.
4. across theflame holder is equal to Thetotal-pressure drop twice thedynamic pressure ahead of theflame holder, that” is = 2q = p2v22 ‘2 - ‘3 2 . .
Thisvalue is probably somewhat highat large values of ~ because thehightemperatures encountered should permit theuseof a low-loss flame holder, but theresulting difference in thrust is small at the value of combustor-inlet Machnumber assumed herein.
5. Thedesia value of thecombustor-inlet ”Mach number~ was .
taken as 0.15.” Fortherange of ~ investigated, thisvalue of ~ is so lowthatthere is no choking in a straight pipeat thelower values of .%, andhence no limltition on T imposed fromthissource.
Ebmecalculations werealsomadefor ~ = 0.05to check theeffect of thisvariable on theengine geometry.
8 NACARM E51H02 provide background on combustion requirements, it In order to .
seemed advisable to include therange of combuslxm-inlet conditions .,..- encountered (fig. 4).
(Q T2J?ndvz) — — Thecombustor-inl.et pressure P2 (fig. 4(a)) correswnds ta M2 = o.15j at such low values of M2,however, p2 changes ve~ little — with M2. At highflight Machnumbers andaltitudes muchbelow 103,000 feet, thehigh-efficiency engine WY haveserious structural -.
difficulties because of thecombination of hightemperature andpressure ~ N the p2 encounter d. Because of thenormal-shock engine is much N — — lower, thestruct-1 diffic~lties areconsiderably lessenedj but at highaltitudesj. p2 maybe lowenough to cause ,combustion difficulties.
Thetotal temperature of theinlet air T2 thatwasusedherein wasdetermined fromtheairtables of Yeference 6 (see appendix A).
In order, to showthatsomerefinement is necessary, theexact value fromtheairtables is compared withthe~lue obtained by assuming constant y equal to 1.4 (fig. 4(b)).At ~ = 7, theerror is 470°R.
These values of-total temperature apply to theisothermal region @’ .
theatmosphere.between altitudes of 35,000 and105,000 feet. Above an altitqd. eof 105,000 fe,et, performance is reduced because the increased ambient airtemperature results in a reduced total- b.
temperatui?e ratio across thecombustor.
Since altitudes below -.
100,000 feetmaybe undesirable because ofthe highpressure, a reasonable compromise seems to be an altitude of 100,000 feetfor .
design of high-efficiency, highMachnumber r@n-jet engines. This altitude wastherefore chosen fortheanalysis.
Although,the values of V2 (fig. 4(c)) mayseemhighfor M2 = 0.15, theyarenotunreasonable andfallwithin therange cw- .=.
rently usedin afterburners.
,- External Drag Theexternal dragwascomputed fora_q~face thatis uniformly tapered fromtheinlet to theexit. Thelinearized supersonic-flow dataof reference 7 wereusedto estimate .ihe wavedrag.The wavedrag - _ .
above ~ . 4 wasobtained by extrapolation fromreference 7 accord- ingto thelinearized su~rsonic-flow. similarity rule.
Forsomeengines thecontour maybe curved, or it-may consist of broken straight.linesj theerror is small, however, because thewave dragis onlya small paitof thetotal dragand“the total dragi-s still .6: compared withthepropulsive thrust.
_.
Friction drag. - A turbulent flat-plate boundhry layer wasassumed .— overtheentire-outer surface. Because there is a possibility thata d .?.
portion of theboundary layer maybe laminar, thedragusedherein is nearthemaximum Nssible. An effective engine.length of 20 feetand an altitude of 100,000 feetwereusedto determine theskin-friction coefficient.
.
NACARM E51H02 Theskin-friction coefficient (fig. 5) wasdetermined by using .
reference 8 andan unpublished reprt of Morris W. Rubesin of the Thecoefficient usedis from10 to 20 percent higher Ameslaboratory.
thantheexLended Frankl andVoishel value of reference 9, whichis shown therein to agreeclosely withtheavailable experimental data.
However, because theexperimenttitiwereallnear ~ = 2 ad 2.5) theaccuracy of theskinfriction usedat higher values of ~ is yetto be determined. In anyevent, theengine dragis small compared withthepropulsive thrust anderrors dueto lackof lmowledge of dragat thehi@ Machnymbers arenotexpected b change appreciably theresults obtained.
drag, consisting of thefriction drag Tbtal drag. - Thetotal plusthewavedrag, was calculated overa range of over-all arearatio &/A~ (fig. 6). Thevalues used. werefora fineness ratio(length dividedby meandiameter) of 8. Thisvalue was selected because it ram~et. It is clear thatin the ~A7 is a reasonable value for~ range of thisanalysis, the principal portion of thedragis friction hag .
.
Nozzle Themainassumption pertaining to thenozzle concerns themethod < of accounting fortheimperfect-gas effects as discussed in appendix A.
thethroat Theassumed nozzle total-pressure” ratio P7/P6 between andtheexitwas0.96forallamounts of expansion.
Nozzle expansion corres pending to maximum thrust. - Themaximum internal thrust occurs forthecompletely ~ded exit(P7= PO)if thenozzle losses do notvarywiththeamount of expansion; as &s assumed herein.However, forthehigh-efficiency engine theresultant propulsive thrust is maximum whenthenozzle is slightly underex@anded, thatis,when p7 is somewhat greater than PO,because when P7 is near PO thee?rbernal dragincreases mre rapidly thantheinternal (net) thrust as theamount of expansion is increased. In orderto findthemaximum propulsive thrust andthecorresponding amount of expansion, it is necessary to investigate netthrust anddragover a range of p7/PO.Thecalculations showthatthere is little difference between themaximum propulsive thrust andthepropulsive ““-.
, thrust of thecompletely expanded nozzle andt?mtthere is a fairly large range of P7/PO overwhich thepropulsive thrust is almost constsmt. .Ty_pical variations of and ~,n with p7/~ are c~ presented in figure 7. Thesecurves arefortheenergy-addition .
parameter Er eqpal to 0.4andhence do notnecessarily correspnd to thecondition of maximum engine efficiency.
Similar curves l (notshown) wereplotted forother values of ~ forthehigh- efficiency engine in order to obtain thevalue andldcation of the 10 NACARM E51H02 l Thevalues of p7/PO corresponding to themaximum thrusts msximums .
—.
.
obtainable at thevalue of Er corresponding ta maximum engine efficiency areshown in figure 8 forthehigh-efficiency engine.The amount of expansion is alsopresented in terms of theexitarea A7 correspcmlimg to maximum thrust divided by theexitareacorres~nding .— — to complete expansion On thisbasistheexpansion correspmd- A7,comp”” — tigto maximum thrust becomes considerably lessthancomplete at the higher values of ~.
Thisoptimum amount of expansion mustvary appreciably withtheassumed external dragbecause if there wereno external drag, complete expansion’would be optimum.
RESULTS ANDDISCUSSION Propulsive-Thrust Coefficient andEngine Efficiency Theperformance of thehigh-efficiency engine forvarious values of theflight Machnumber~ is shown in figure 9 in terms of the ratio of engine efficiency to combustion efficiency qe/~c as a function of propulsive-thrust coefficient ~. Themaximum values .
of propulsive-thrust coefficient and ‘qe/~c are plotted in figures 10 and1.1, respectively+ Themaximum propulsive-thrust coefficient (at Er= 1.0,which is stoichiometric forgasoline) decreases * continuously from2.13at ~ = 3 to 0.57at ~ = 7 (fig. 9 or 10), Therapid decrease of theproulsive-thrust M-efficient withincreasing ~ is mainly a result of thedecrease of total-temperature ratio ‘c.
Morethrust could be obtained by theuseof a fuelhating a higher energy release thangasoline (that is,EV > 1.0), butsuchan increase ,.
be. accompanied by a decr~ase’in engine efficiency.
in thrust would Theengine efficiency reaches itsmaximum value of approximately 0.47near ~ = 5 ‘and thendecreases gradually to approximately 0.43 at %=7. As canbe seenfromtheBreguet rangeformula Wo L log — x= QJve e~ D therange is proportional to qe andhence thehighvalue of qe thatmaybe attainable at thelarge values of ~ willbe an im@rtant factor in theconsideration of long-range, high-speed ram-jet-propelled vehicles.
.
As theener~-addition parauieter E$ increases (fig. 9),~ increases and qe~vc increases to a maximum andthendecreases.
u corres~nding to themaximum~e/vc increases Thevalue of Er NACARM E51H02 U J with ~ fromapproximately 0.35at ~ = 3 to 0.8at ~ = 7. !I!Ms .
trend indicates thatslightly above ~ = 7 gasoline mayno longer havesufficient energy release formaximum ~ssibleengine efficiency.
In thisevent a fuelpermitting greater energy addition thangasoline should be used. However, since at ~c near100percent gasoline doeshavesufficiently highenergy release in theMachnumber range of interest, theuseof fuels permitting a higher energy release maynotbe necessary fortheattainment of maximum efficiency. Other fuelcharacteristics are im~rt-tj therange is proportional to the IN N o-l heating value perpaund of thefuelas wellas ta theengine efficiency, andthedensity of thefuelis of great .im@rtance in anyflight vehicle design.8uchconsiderations, however, areoutside thescope of thisinvestigation.
A comparison of thepropulsive-thrust coefficients andtheengine efficiencies of boththehigh-efficiency andthenormal-shock engines is madein figures 10 and11,respectively. Themsximum propulsive-thrust coefficient of thenormal-shock engine is near1.57at ~ = 3, about three-fourths that.of thehigh-efficiency engine, andapproximately zeroat ~ = 7 (fig. 10).
Themaximum engine efficiency of thenormal-shock engine (about O. 27}is only57 percent of thatof thehigh-efficiency engine andoccurs at a lower flight Machnumber (~ = 4) (fig. 11].
AreaandVolume Relations It follows fromtheone-dimensional compressible-flow equations thatforanygiven combustor-inlet MachnumberM2 there exists a value of ~ above which theinlet flowarea ~ is greater thanthe combustor flowarea A2. Thevalues of ~ above whichtheinlet areaof thehigh-efficiency diffuser becom& greater thancombustor areaare5.7for M2 = 0.05, 3.6for M2 = 0.15, andabout 2.9for ~ = 0.25. Forthenormal-shock diffuser, unless thedesignM2 is .
extremely high(approximately 0.42at allvalues of ~}, thecombustor areais greater thantheinlet area. Thesignificance of this condition of equal areas is thatit is close to“the condition where thecombustor starts interfering withtheexternal contour; it is notespecially critical, tiwever, because thecombustor areamustbe somewhat greater thantheinlet areabefore interference starts owing to thetaper of theexternal contour, andalsobecause there canbe a sma~ mount ofinterference before thedragis increased appreciably.
A qualitative picture of thissituation is given in‘ figure 12,which presents sketches of engines withareasin correct .
proportion forvarious values of ~ and M2. It can be seen t~t engines designed formaximum engine efficiency haveapproximately the same @A7 forallvalues of ~ considered. Of interest is the 12 NACARM E51E02 .— , .- factthatfor a $iV@n amount of expansion, variations in M2 have .
little effect on thevalue of ~/A7. Onlyonesketch of thenormal- shock engine’(fig. 12(a}) is sho& because there areno significant variations of appearance with ~. The” following discussion will therefore pertain to thehigh-efficiency engine only(figs. 12(b), 12(c), and12(d)).Thespike diffuser shown In thesketches is purely schematic andis notintended to imply thediffuser design.
3, lowvalues of ~ At%= of theorder of 0.05willresult ~ in undesirable external dragincreases, @ereasforM2 ~ 0.15the — combustor interferes onlyslightly withtheasswned external contour.
thecombustor still interferes Slightlyj At ~= 5 and.M2.=0.05, whereas at M2 x 0.15 there is no interference at all. At ~=7, canbe madeevensmaller thamO.05 before theexternal contour M2 willbe affected. Figure 12 alsoillustrates howmuchtheusable volume increases with ~ and ~. Thisincreased volume &Lybe as highas goodcombustion ‘ a reason forselecting a value of M2 .
T permits .
,.
CONCLUDING REMARKS .
An analytical,investigation wasmadeof theperformance of an externally mounted ram-jet engine in theflight Mach number range from n“ 3 to 7 withdiffusers of twotypes, oneof undetermined formwitha ““ kinetic-energy efficiency of 0.92, andtheother, a simple normal- shock diffuser. A combustion efficiency of 100percent anda fuel having a hydrogen-qarbon ratio of 0.168 wereassumed. Theresults obtained were: 1. A design altitude of about 100,000 feetappears desirable fora high-efficiency highMachnumber ram-jet engine in order to avoid theexcessive pressures encountered at lower altitudes andthe increased ”anibient temperatures encountered athigher altitudes.
, 2.
Gasoline provides sufficient energy release formaximum engine efficiency in theflight Machnumbei.range investigated. At MEICh numbers above 7 andforvehicle designs requiring maxim~thrust evenat theexpense of efficiency, however, fuels permitting higher “- u- . .
.
energy release areindicated.
3. Forthehigh-efficiency engine themaximum propulsive-thrust coefficient decreased continuously from2.13-”at a flight Machnumber — of 3 to 0.57at a flight Machnumber of.7; whereas, forthenormal- .
shock engine, thissamecoefficient decreased from1.57to zeroin . .
thessme Machnumber range.Themimum engipe efficiency of the * NACARM E51H02 13 high-efficiency engine increased to a maximum of 0.47neara flight Machn+er of 5 andthendecreased gradually to about 0.43at a Machnumber of 7;whereas, themaximum forthenormal-shock engine wasonly0.27andoccurred neara Machnumber of 4.
Lewis Flight Propulsion Laboratory National Advisory Committee forAeronautics Cleveland, Ohio t NACARM E51H02 AEPENDIX A .
PROBLEMS ASSOCIATED WITHVARIATIONS IN y Since at highflight Machnumbers~ large variations of temperature, and.hence large variations in theratio of specific heats y, areunavoidable, theusual formulas andtables forconstant “y areno longer strictly applicable. In order to obtain thetotal tem- perature when@yen ~ andtheambient st,atic temperature to)use ismadeof theairtable(reference 6) andtheenergy equation g Theambient static enthalpy ~ andtheambient speed of sound co depend on to only, andhence thetotal enthalpy ~ is readily calculated; thenthecorresponding value Ot total ternp~twe an be — obtained fromtheairtable.Thecorresponding ratio of static to — total pressure canbe easily foumd because it equals therelative- .
pressure ratio pr,o/Pr,o, where pr and Pr aretabulated with .
thestatic andto&l temperatures, respectively, in theairtable.
.
.
Although it would havebeendesirable, sucha procedure wasnot passible forexhaust-nozzle calculations be~ause of thelackof corres- — ponding tables for combustion yroducts at thehightemperatures encountered.
Therefore, in order b calculate exitconditions, a constant value of T wasusedequal to Ta,7+ T7 Tav= where Ta,7 corresponds to thetotal temperature after combustion — Since’ and Y7 wasdetermined fromthestatic exittemperatm?e ‘7“ depends on yav,an iteration procedure hadto be used. It ‘7 converged veryrapidly, however. Thismethod permits thecalculation of thetemperature ratio t7/T7 andhence M7 when p7/P7 is known.
canthenbe calculated fromtheusual one- T%earearatio ~/A7 dimensional ener~ andmass-flow considerations.
of thethrust equation) ._.
It is veryimportant to use”the proper form namely, ~ 2~,n=(l+f/a)~=-l r NACARM E51H02 .
instead of .
(A2)
Equation (Al) follows directly fromequation (B2)(see awendix B) because Equation (A2) follows mosteasily from equation (Ill) because @42~ mV = @v2 = Equations (Al) and (A2) areequivalent andwould bothyield the same answer if thecorrect value of ~/A7 wereused. However, equation (IQ)is verysensitive to errors in ~/A7. Equation (A2)canbe usedto advantage in calculating ~/A7 once ~,n hasbeenfound .
fromequation (Al) because thenthesituation is reversed and ~/A7 willbe insensitive to ~,n. An iteration procedure forcalculating l ~/A7 is s~gestedj namely, useof a veryapproximate value for @A7 in equation (Al) andthenuseof theresulting value of CF,n in equation (A2) to obtain a moreaccurate value of @A7. Sucha procedure worksverywellfornozzles in therange of expansion encountered.
16 NACARME51H02 APPENDIX B ‘ . . .
DERIVATION OF THRUST EQUATIONS Fromtheequation forthenet (internal) t~st (Bl)
I?n = m7v7 - ~vo+ [3?7 - ~)+
g thenet-thrust coefficient is found to be (B’) Equation (2)canbe derived fromequation (B2)in thefold.owing manner: Fromtheenergy equation, thevelocity ratio canbe expressed as V72 H7 - h7 V02 ‘=%-% “ l which becomesj forcymstant y, ~g-l-~.+l) V72~T7 - t7 (B3) To V02 To - to —- ‘o By useof thedefinition of ~, theenergy relation between tem- perature andMachnumber, theisentroplc relation between temperature andpressure, andthedefinition of ~’ ~ given by’equation (3), equation (B3)canbe reduced to .
Iz=Tvtk
(B4)
V02 m7 when p7 = po. Hence, since— = 1 + f/a,equation (2)follows from % equations (B2) and (B4).
.
NACARM E51H02 17 REEERENCFS .
1. ~entative Tables fortheProperties of the Warf ield, Calvin N.: Upper Atmosphere. NACATN 1200, 1947.
l 2. Ferri, andExperimental Antonio, andNucci, huis M.: Theoretical Analysis of Low-Drag Supersonic Inlets Having a Circular Cross Section anda Central I!ady at MachNunibers of 3.30,2.75, and 2.45. NACARM L8H13, 1948.
E % 3. Dsmkhoff, W. F.: A Performmce Analysis of theHermes B-1 Supersonic Ramjet.Rep.Nb.55267, Thermal FbwerE@temsDiv., Gen.Elec.Co.,Aug.1948. (Project Hermes, U.S.ArmyOrdnance.)
4. Meyerhoff, Leonard, andReissner, Hsm.s J.: The Standing, One- Dimensional Shock WaveUndertheInfluence of Temperature- Dependent Viscosity, HeatConduction andSpecific Heat. PIBAL Rep.No.150,Aero. Eng.smdAppl. Mech.D@., Polytechnic Inst.
of Brooklyn, June1949. (contract No.N60nr-206, TaskOrderl.)
“5. McCann, W. J.: Thermodynamic Charts forInternal -Ccmibustion-lhugine .
.
FluidE .
NACA TN 1883, 1949. (Retised by L. R. Turner and RnoryA.Bauer.)
l 6. Keenan, “Thermodynamic Properties of Joseph H.,and-ye, Joseph: Air. JohnWiley& Sons, Inc., 1945.
7.
Jack, JohnR.: Theoretical WaveDrags and%essure Distributions forAxially metric Operi-Nose Bodies.KACATN 2115, 1950.
8. Schl.icting, H.: Lectuxe Series “I!oundary Layer T&on.” PartII - Turbulent Flows.NACATM1218,X349. - 9. Rubesin, Kn?ris W.,Maydew, Randall C.,and Varga, Steven A.: An Analytical andExperimental Investigation of theSkinFriction of theTurbulent Boundary Layeron a Flat Plate at Supersonic 1951.
Speeds.NACATN2305, P co r — Equation (1) — -- High-efficiency Mffuc.., .mmd kreln —-— RO1’531-6hc4k mffMer (blghm?hmubr V81w * rig.2) TU&slmckSpib dfnlmr, ‘Erpmlwnt@J.
Imnt.mpic spike dlffwerj etihnted (reference 5) mght Wch UJJaer % 1.03 .
la .95 — c1 .65 / #’ 2 .m ., i / : / !.
/ > / /’ / E 2 } .6s /
/
/II .ea
/
1.5 ) .55 --
)
/1
I
.52 .-
/
-
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II /’ !2
.45*
.9 .7 .8 1.0
. 1 9 .5 .4 .5 .8
.
.
c l * WZz .
1: b NACARM E51H02 .7 Diffuser High-efficiency .6 ——— ‘Normal-shock \
*“
.5
(f
.
.
.
.4 I .3 \ \ \ \ \ .2 \ \ \ \ \ .1 .
\ \ \, -N’% -- --- --- y 7.
3 4 5 6, Flight Mach number, M.
Figure 2.- Variation oftotal-pressure ratio across diffuser with flight Mach nuuiber.
NACARM E5iEt02 Euergy-sddit ian psramet er m I Engine —— — — Normal-shock .
\ Y \ 1.4 1.0 — 5 4 5 6. 7 .- Flight Mach number, ~ Figure 3. - Vsrlation of’ tatal-temperature .ratio across combustion zone with flight 14ach number forseveral values of” ener~-addition parameter ‘- _. -;’ for both high-eff iciency and nornial-shock engines. E@rocarbon fuel; .- .
altitude ~ 160,000 feet.
NACARM E51HOZ .
1o,ooo I 1 I I I I plltituae, ft+ ~ Ei,oco I I A I t I I I I •D Diffuser 6,o0J 4,003 2,000
“2 1,000
N .U .
.
Im AN .
.
m a .8 .6 .4 .2 .1 Flight Machnuniher, ~ (a) Static pFessure at comlmstor inletforvarious altitudes.
* High-efficiency andnorml-shock diffusers.
Figure 4.-Variationof flight Mach number.’ ccmlmstor-inlet conditions tith .
NACARME51H02 .
.
Ratioof specific / heats I /~ “ — Variable (airtables, Z(YJO reference 6) / —— Constant at 1.4 A t f 4 5 6 7 Flight Machnumber, ~ -, .— (b) Total temperature of inlet airfortwomethckis of computation.
-- Ambient static temperature, 392.0 R.
.
m Combustor-inlet Machnumber % 0.15 : 4m $ ‘ ‘ + 1 1 I 1 -1 1 1 t / —— Flight Machnumber, ~ .- (c) Velocity at combus.tor inlet forvarious values of combustor-inlet 8- Machnwsber.Speed of sound basedon temperature obtained fromair tables.Asibient static temperature, 392° R. “ Figure 4.” - Concluded. Variation of combuetor-inlet condition8 withflight .- 0 , .0028
!2
l?’
N .
; ; % $+ g .0020 u d \ : * : ~ , ,, .“ ~ .0016 ..
,.
g ,.
~
I
# .0012 .0008 3 4 5 6 7 Flight Mach number, ~ Flwre 5. - Vartatioll of external Bkin-fFiction coefficient Turbulent boundary layer; effec- tith flight Mach number.
tive engine length, 20 feet; altitude, lCK),000feet.
.15 \ 1 \ \ \ I \ \ Engine-drag \ coefficient \ .L3 .
Total.~,e \ \ ——.
— Friction ~,f \ \ \ \ \ \ \ \ \ > ‘% \ .
‘\ \ \ + Fli.@ WA : value s encounter ed L g .09— b d \ nulioer \\ * % \ % % - ‘\\ ‘ 3 %’ v w ,- \ w --.
- I .05 .
.
-..
--- -
I
I I I I
I
.03 .4 .5 .6 .7 .8 .9 1.0 .3 Over-all area ratio,AJ~ V.wiatloo d exkmcl drag coefficientwith over-all nea ratiofor varioua7alue8 of fllght Figwe 6. - Wch nmskr. Conicalcontiur;lengthove.r, mean d3mmter, 8. Coefficient ba.6ed on Ioletarea %.
.
, . s Thrust coefficient propulsive ~ 1.5 ——. — Net or internal ~,n F-l@t Mach nwnber ‘o F MM M9ch --- ._ --- -- nmimr .6 1.2 ‘o -—. ——- —_ —-- ___ e 1.1 ~ . — g .5- .8 m’ % s 1.0 ij” – – ‘– – – – ‘- ‘-- “-– ‘–- u J 4; ‘- - ‘- - ‘- ‘- .9 $. 3 s — -. _ n __ ._ _ ___ ____ __ - $ 4 .
— .0 . 2 — —- “Lo 1.2 1.4 “1 .6 1.8 2.0 2:2 “7i.o 1.2 1.4 1-6 !iozzle-~mion premsterp P7/Po Figure 7. - Typicalvariationsof propiU.sive- and net-tim+etcnefficientn with nozzle-expwsionPme.msterin high-efficiency N m engine. Energy-addition parameter,0.4.
.
IlACARME51H02 1.0 2.0 Nozzle-expansion M“, parameter / g +7,coItTp / ——— -’ & P+o &l. i3 “9 $: / .; / =?
/ % / $ / i / .8 i # a / / g 3“ / g.
/ & y .7 1$ / : L4 I / w N / S!
~ \ : & .6 1.2 .— .7 4 5 6 Flight Machnumber, ~ Figure8. - Vari@lonwithfligkt Machnuniber of valuesof twu .* ..
nozzle-e~ansion paramet=.s corresponding to ~i~ prowl- — slvethrust at valueof energy-addition paametercorrespond- ingto maximum engine efficiency. High-efficiency engine.
.
.5 A / 9 ‘6 Flight Mach 7 5 .4.
\ \ * “ - .3 Energy-addftion parameter % / o 0.2 q .4 .2 A .6 .8 v 1.0 ) c! I I I ~~ A /
~
I —.
.1 -.
0 .4 .8 1.2 1.6 2.0 2 .
prop~sfve-thrust coefficient? ~F Figure 9. - Variation ofratio of engine efficiency to combustion efficiency withpropulsive-thrust coefficient for various flight Machnumbers.High- M efficiency engine; nozzle expansion corresponding to maximum thrust; combuator-inlet Mach number, 0.15.
ti E51H02 “ .
J \ \ \ \ Propulsive-thrust \ coefficient \ l Corresponding tomaximum ‘\ \ \ engine efficiency \ — —— \ \ ——Maximum Possible with k \ gasoline \ \ \ \ \ \ \ L \ \ \ \ \ \ \ Engi~e ‘\ \ ‘ ~ \, I \ .High-efficiency \ .
\ \ T \ — 5 4 5 “6 7 Flight Mach number,’ ~ Figure 10.- Variation ofpropulsive-thrust coefficient with flight Mach number fortwotypes ofengine.
Combustor-inlet Mach number, 0.15.
NACARM E51.E02 .. -.’” # .
.
.
.5 - c.)
< / ‘ Go \ . -— .- . Engine .
/- % Eigh-efficiency /’ /0 ! ‘4 / / % ~ / s /’ Engine efficiency .2 / ~ .3 , Maximum .—— — Corresponding to g \ maxhlum thrust / CJ / $ --- .
/ // -N.
k / ‘ \\ \ 2 .2 h.
: \ .rl % No rmal-shock % al \ \ \.
g“.1 \ W \ o \ \ $ \ - T .
‘o” s.
4 6 7 Flight Machnumber, ~ ..
.
Figure 11.- Variation of ratio of engine efficiency to combustion efficiency wl.th flight Mach number fortwotypes ofengine.
~ -0.15 l+.= 0.15 M? = 0.05 (b) ~gh-efficiency engine; flightMach n~er, 3; (a)Wrmal-shockengine; flight~ch nomk, 5; typicalof any ~ ratioof Inleth exitarea,0.53.
M2 = 0.15 ~.ols 1.$-0.05 $.005 (a) ~gh+fficiency engine; flight ktmh Utier, T; (C) .Mgh-efficiency engine; flight ~ch -her, 5; math of inlet.toexitarea,0.51.
ratio of inlet to exit area, 0.57.
Figure M. - Elketches showingvariationsof gecmtry with flightMach numberand corbustor-inlet Mach numberwith areas to Bcale. Valueof energy-addition puwneter comespmds to mw.imumengineefflcleney.