NACA-ACR-E4E06 · Study of Jet-Propulsion System Comprising Blower, Burner, and Nozzle

NACA ACR No . E4E06 APPENDIX A TERTST H03SEPOWER OF J'ET In th8 follovling analysis} terms bav ing a negligible effect on the result "Were omitted from the equations.

The thrust horsepm'ler of the j et - propulsion device is given by applicat i on of the familiar mom e ntvm e~uation (1) It is expedient in the analysis to malm use of a fictitious jet veloci ty defined as follovls: Let V5a be the jet ve2.oci ty for the case where no fuel is burned in the c omIms tion chmnber but ' where the blower pO\ver} mass flo",} pressure at the combustion - chamber entrance} and the diffuser and b10,,:er efficie:a.cies r ema in the same as fo r the case 17here fuel is burned. This condition is mechanically achieved by adjusting the disch arge - nozzle opening.

It is further reql : ired in the definition of V5a that h ea t gene r- ated by turbulence resulting from energy losses in the diffuser} the blm-rer} and the burner is removed from the gas as it is fonned and therefore does not contribute to the exit velocity V5a' Let a, be defined by the relation ( 2) Then 550 thp (3 )

lw

2 0 When the theo r etical gain i n kinetic energy of the air for the cold condition is equated to the difference bet'l"een mechan i cal power added by the blower and the power losses at the diffuser and at the constr i ction in the combust i on chamber caused by the burner} there is obta i ned HACA ACR No. E4E06 19 An 8.lJproxilllation is apparent in equa.tion (4) wbere tbe d.iffuser efficiency is assllmed. to apply to t:he cl"ange in velocity :i.'rom Va to V2 ' The complication of a more exa.ct relation is not consid.- ered. viarranted. because of the sma.llne ss ai' the error involved..

Solution for VSa/Vo from equatton (4) gives (S) whero ( 6) or and.

Tj , Lot be d.efined. by d.

(8 ) Equation (6) becomes 5S011 Pe b (9 ) X = - (1 - Tjd. t ) ~2 2 0 Part of tho d.iffusing action bakes pJ.ace in the free-air stream ahead. of the nacelle . The d.iffusel' effiCiency includes losses in thls l'eeion as '-Tell as those occurrin,~ in the diffuser proper. When the velocity at the combustion ch...anber V becomes greater than the free-air velocity V ) a nozzle rather than a o d.iffuser is required. In such cases the sign ahead of the teI'!!lS N ~ CA ACR No. E4E06 and s h ov~d be ch onged in or der t hut these terms agai n app ea r as a loss in e ner gy . In al l cases th e efficie nc ;y TJd i s defined as the r a tio of the true chance in l~in('tic energy to the ideal change in kin e tic energy for the s ' 1me entra nce concH tlon and pl' eSSl,U'e cllRnge.

Ther ef o re) c on dtti o ns w here V 2> Vo Sh OlUd b e treated as spec i al c ases where) de pe ndin g on tile ove r- all efficiency of the venturi) th e second t erm on the ric.ht-hanci side of e'luat ions ( 4), (6)) and (7) c an be eit her corrected cr om itt ed .

Combi nin g e ( 3) a nd (5 ) 'lu at i ons I

550 thp ," --- '-

-- .-

(10) ya (1 + X)

2 LC

v

~l

1MV 2 2 0 The equation for the j e t thrust per unit mass flow is The qu a ntity . a, may be dertved in the follov in g manner. The dischar ge ve locity of the j et is given by the o'lu ation

1-- - ---· ---

V5 "C ) 2 Jc T [ -1 _ (P o \)7- '

V V P 4 L ,P4

wh e re T4 an d P4 are the totnl temper nture an d preSSlITe) r es pec - tiv el y.

For the cold con dit i on a similar e xpreseion fo r V 5a may be o bt ai ned with T4 replaced by T2 a ' where T2a i s a theoretical tot al t emperature at the burner entrance correspondin g to the case vlhere the jet dis charge vel ocit i s V5a· The va::" L lC of T2a J o bt a in ed. f r oLl the con ditions given in the definition of V5a is el l}) otnI' _t io..ll;y- NACA ACR No. E4E06 (12) The rema rk that fo llow s equation (4) applies also in the case of equation (12).

The j e t ve l o cit y fo r the cold conditi on is then g iv en by (1 3) Then Wh en the first tw o terms of the series expans i on for the

-r.:1

tlP 3 -4\ )' t e :nn taken are

- --p;)

~ I

V5 = C

v

--- ------ . ------ tlP3-4 T4 P3T 3 JC T p 2a

J

NACA AC... No , t:4.H.:06 1r111cre the assumpc; . on is made that. T2a/T3 may be taken equal to unity i n the term containin g 6P3-4· '1'hen (14) V 5 == Ve' a w hE- r e y- 1 C

(Po \ -y-

y ~ ~3-4 (1 5) 2 - _ ,

( 1 - - .- - ' --

-

\ )) •• I 2 • -0/ P3 V5 a Thus (16) But ~P 3-4 If t he a1?proximations are made that V4/V3 - T /'r2 a and Y3 = V2 , then When Y Sa is elimlna ted from tlJe e-luation for ( by mea.1S of equa- tion ( 5 ), there r esult s -- - -- - --~-- -- ~-~--- ----- -- NACA ACR ' ~o . E4E06 ?'-=.~, p '\ )' ( For the tliJe of s;:-s1.em under d i sC USSivD , the term ~) \P3, equation for ( ma;{ be r eplcced by unity without introducing a]!preci8.ble error in t:~e .:'ra ~ ,til,.,a 1 ra:.ll:,e of ope Tat J. (·n . 'i'l:us Com'binati :m 0_ . 0<luaU. ons (10) alto. (15) g . Lves the tYJrust YJorsepowe r equation as

r- - ----;-· - · -- l

S50 th:, { C 2 t 4 (l X)

- 1 I (1 8)

\1 V \' I

1 2 , T I 2a 2=MVC.

J Equat i ons ( 11 ) and (16) give the ti1rust equation as I - ---- - -- - _._-

/ V 2 T4 (2 2 \

(19)

F /}II = C ( ;;-- V 0 + VoX) - V 0

v

12a / The ra tio of tll? %P Can be oerived from eq uatjons ( 6) e and (18) or f:;:om equations ( 9) and (18) 24 NACA ACR NO. E4E06 i : I

~J ~cv~,~::_~

thp ( 20 )

x + 1 - Tld 1

In this analysis the thrust obtalned. from the rearward motj . on of tl1e fuel vTaS neglected . The follovTing relation vT ilJ. correct for this contribution to the tl1rl.lst : WfJ' \ W.V 210 - ( C orrected thrust horseIlO\' ler = thp 1 + + ....f.J~. __ ._ 360Cn,1 ) 1 . VOl, b / The thrust of the enGine exhaust vms also neGlected.

NACA ACR No. E4E06 25 APPEr-JDIX B FUEL COnSmfPTION Combustion of the jet fuel occurs at substantially constant pressur e j therefore or The term Mcp (T2 - T28,) 3600 represents the heat Bene~ated by turbulence :L n the diffuser and at the blol-rer. This heat i s usu- ally small compared with the l1eal; generated by combustion and in most cases can bo ne gl e cte d . I t will be neglected in the subse- quent equat10ns . For lov[ effj ci e ncies, to correct for this approximati on) it is merely llecessary to subtract from t he f uel rate obtain ed from the ensuinG equations a value equivalent to the heat generated by tu:cbulence . Then from equation (1 8 ) ( T4 c

3600 T2a .--- 1 ~ 550

p \,T2a TlcW fj = J . ---- (21) (lb)/(thp-hr) ., thp'

I

I 2 T4

hV 2

(1 + X) .. 11

\( C v £ I T2a J L If c 7.728 ( Btu)/(slug)(OF ) p and h 1 9 )000 ( Btu)/(lb f u el )

(T4 _ 1)

805 . 3 T2a \T / TlcW fj 2a (lb)/(th:. )-h1') thp = T4

1 --

V 2

(1 + X) -

_ yc/ '

I T2a

J

2G ~ffiCA ACR No. E4E06 The engi -::1 e fuel consumption can be assumed to be proportional to the eD g ine bra ce bo rse )m.re:(' . '1'11118) 'df s == BP ( lb )/ (hr) (22 ) e F ro m e q,u at':ons ( 18 ) and (22) Vl TIP 550 c J f e -- . ::.. -- x - .. ::- - --- .. --- .:.._- -.-_.- '- .. .- (lb) / ( t: Jp- ll r) thp (2 3 )

~N 0 J r' -;- .-- T : ~ ----- : ' I

2 I !I J C V E: - - ( 1 + •• ) 1 I I \; T2D.

...J From e q,ua tion ( 9) , El q,ua tloll ( 23 ) b ec o mGs - ----- · -- -- r· - ~ --- .- . .- == t tl }?

As suming .,.., ' 0. 90 " e and B 0. 6 ( l b f'uel )/(b bT)<lr) then (2 5) (lb)/(tl1 ~ -hr )

- _. - -- - - -- . - .~ -- NACA ACR No. E4E06 27 APPENDIX C RELATIONS BE'l11EEJ.~ ATMOSPIlERIC CONDITIONS AND CONDI'l 'IOlm AT COMBUS'l'IOH -C HAMBER EWi'RA1~CE Accllrate values of V and T2 may be found by the followine metho ,d . The conditions after the diffuser and. before th e blO'loTer (V J D , an d. Tl ) can be 0 b La incd from the follm.,ring three formulas: l l From the conservation of ener g;y

k

J 2 .

+ T (26)

JC 1

From the definition of diffuser e.(ftciency

, L-l

'1

/ ) 'l

'r f ~l

:= T)d (T - '1'0) (27)

l o 1 \--

- 1 J

I Pol L and MRTl Vl (28 ) A "9 D 1 The valu es or' V ) '9 2) and T2 may b e found by the follo w lng fOI'mul as:

Z.:l

, ,, )' (2 9)

(~~)

\P

J . I (3 0) and I

I

J

-~- " - -_.

NACA AC IJ. ·, No . E4EOG

(31)

'TI For a n a ppr o::-: i 11lb te va .J.l'e of V , values fOl' and P?

-'-2 ....

j.n can be l 'e ple,cso . T ancl equation (31 ) . Equations ( 26 ) b Y' Po a 110_ ('~O ) g j.ve 1 2

l-v 2

'J 50P e

-:zy 0

2 2 -_.+ (32) 'I'

T9 ;- ~~Jc- =

Jc+ 2

JC p p P waich c an then be use~ to fied T .

If M, P T O J lJ 8:1.0. Vo a1'e blown 8!ld V2 is a dRumed eJ OJ to l'e equal t o V as an a p.Y.' 0x i: ,' at-':'oa, t"Ol1 .l·or any va~ne 0 ... ' V l l the values of Tl an,l T2 calJ. be aota '.De ::' f : .'om equa t' ions (26) and (30), re2'!. ~ ec.tiyel: '· . By u:..:'<) at -th::'s u.:e G:(_~ (J th e valn (; s of PI and. P2 can be obtaille ct from 0q.uat. i 0ns (27) O,Ld (29), r es!lectively , and the v alue of AD from equation ( 31) .

1. P i nk el , Benjamin , Tur :'-t.~: , L. Richard, and Vo es, Fl'erl ; Des ':' gn of N , zz :;' es f a .:. ' t :le In r Lv: . J.ual C: r linder Exhaust Jet Propulsion Syst om. NACA .A 8H, A" ) l'il L 9 4~, 2 . Die hl , Ha1..ter S . : Standa:r 'c< Atmos-pllere - Ta bles a nd Dat a .

N~CA Rep. No . 21 8, 1825 .

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s Po

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X (b) Jet t h rust horsepower per blower power.

Figure 2.- Curves for obtaining jet thrust horsepower.

550., P [ 2] IIp 2 X ~ ~ - (1 - TJ ) 1 - (V IV) - ~ (V Iv ) d 2 liV 2 0 'Ia 2 0 o Fig.3a,b

MACA ACR No. E~E06

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Figure 3.- Specific fuel consumption factor of jet-propulsion d evice.

X c ~ - (l""'fld' [1 - (V /V )21 _ ~ (V Iv )2 2 o ~Vo J q2 2 0

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o 2000 4000 6000 8000 10,000

thp Figure 10 .- Variation of specific fuel consumption with jet thrust horsepower for various combustion-chamber

temperatures and altitudes (P = 1200 hpj

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