PSol3 - .5'4. After verifying the derivation of Eq. (5.27)...

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Unformatted text preview: .5'4. After verifying the derivation of Eq. (5.27) for the ideal ramicr. find the Mach number M that maximizes Slim. for Tm = 2500 K, T, = 223K. Evaluate the Specific thrust at that Mach number and compare your results with Fig. 5.9. Both Fig. 5.9 (for the ideal rarnjet) and Fig. 5.12 (which makes allowance for losses) indicate that at a given flight Mach number the specific fuel con- sumption rises with Tm“. Looking for an explanation of this effect, deter- mine the variation of propulsion. thermal, and overall efficiencies with Tm. for the ideal ramjet for M = 3 and 2000 < T"... < 3000 K. Shaw for the limiting case Eff/n}. -> 0 that 2 Tu. file") and TSFC—> 2 M\/fil;£€_ {1+T_lui\ (a) T 50:: T %:= 121° =n.ae \l—%=3.37 _ Ll-EOOO a GP 0- 100021205 293'? 4:: [0,36—M2/5 =._ ng—M" 192"!- “STIT- Jb’R‘Te 5" JIM-(237) zze J]: = M 2.37.3 [(H—FDSJ'; 1 kN.5 Vila l°°° W H "3 M -F Wm a e: °‘ 2.0 amt-97 0.973 N 2.5 0.0473 Lace <%)max _ 1'01 3.0 0.0444 0.98%! e M-z: 2.6 (thP = {fiwa -—u]u = 2 the—1%; (H4399??- "’%—1 I+1= flat): u 1.. 41H“ =. 6+2??— J-g‘.‘ =, H“? "($32. "- zFQg ‘- -For [Vi-=3 Z$-='I.Ll- = “FEM g {1331+ 531m) .5 [$73 a: °i+ 134 99 77: +1: %—(l+z-'ITIMZ) get—2.60 Ci? ..._. o '— 4—7 Cf 4 I759.- 20‘7"??? E's-Lg" R _13 _ cg???“ $733.. [92.14- (D -= retain/s ® . _cd_ 1‘3; 7% 4 ue. 77? 1+»! m l+Eo 01000) %= wj-L—tc'z 7.3) -.= 56.5 M '5 20cm 9.03 0.032.; 0.555 0.75: 0.643 0.'f7o 250° 11.36 0.04444- all-96 0.68! 0.646 0.4% 3000 £3.64 0.0570 0.453 0.64?- o.643 0.423 A: Tau. increases 7!], deda'n es sulns‘l'anflally while Iii-He. change. in 77.”! 5 “Haas “#0 drops. C" 5%: NWENOJT—a 4] CD To.» - Ton—1' ® "*F _. .fi cp-ru. 73‘ @ can be wfl‘H’cn E... Jfl in which <1=C%E; @ T60» "' 1+ .F 55 (D b ECOMS %= M‘lxfi'ra [.IZJr-FXHW —1] (9 when 1}". =0 6+4Xl+v€4) =1— m but" since {>0 and F410 ’ F=O 119* 50 From ® 1%:1: m M = an—n TSFC. -: . 4‘ MWEW —17 Hm RFC = = ...—2‘_._— ++0 Mm [4,: WW7: 3+ m ] Mm: (mo = Mfib‘R'fi [1+ agfcf-‘T‘C'i 2;:- M1)” 5‘10. For a bypass engine it may be shewn that if the specific heat to; is constant, the fuel—air ratio f <<1. and all losses are negligible. the static engine thrust can be determined from Jaeweseet in which T denotes stagnation temperature and subscript 1 denotes the compressor and fan inlet. 2 denotes compressor outlet, f denotes fan outlet, 3 denotes turbine inlet. andfiis the bypass ratio. a. Prove or disprove that this expression is valid. b. For a given ratio T3/T., what compressor pressure ratio will maximize the static thrust? e. For a given bypass ratio-3: what fan pressure ratio will maximize the static thrust? d. With optimum compressor and fan pressure ratios, how will the thrust vary with WT. andfl? T 3 $4“ 5—: “hue. + Emue-F ® 77+ -T5 = Tau—775 ’03 —m) 1 Turbine world. 5 13_T+= “13:17 +B(T'.§:—'TT) us=xl ZCPC Ta-Ts -—(Ti*"‘t7-—B('1?t=—TT§® No Pressure losses T T (9 i— l9 _ 'l" 4:; " 13¢ 25.1. P. " P: "6%) "1 Combining 6) C2) @ ® ._.2- _ ..-- .2: BE:- % xii-i 1%“ B(%')—nv?+iw '@ Given TT%JB,1_':_E -Hq|'.s is MMMIZCd when I) .1 50 lees-l: Com Pf 395': on rat'i'io ifi :95 == 3‘ a." :5 and “Vein R T_ T’ a}?— filF-‘rshr ‘3 4“ Bless @ 1.3....2‘113 1 Given 3—3 9 maximize'J-when 11; __-E —-.‘—_—-+ 6,) TI ’ T:— +. 3%“ ‘ HS Lee. when gfiL‘EL + Ewi“ Use o-C' Ecr.® in yields @3 fig? ., @-4$+.)(i+s) ' -For CompreSSOr and ~Fan pressure rat'fos which will maximize fakefl-F thrust: 5-11. Compare the takeoff thrusts of two engines operating with the same core en- gine airflow rate, compressor exit pressure (12 atm}. and turbine inlet tem— perature (1500 K), and at the same ambient pressure and temperature. One of them is a turbojet (bypass ratio equals zero) and one has bypass ratio equal to 10 and fan pressure ratio 1.6. For convenience we here neglect the effect of Iossas in the compressor, burner, and turbine and make approximations associated with fuel-air ratio f << 1 and constant specific heat of the working fluid {with 1r = L4). Find the ratio of thrusts at takeoff, with flight velocity u << u... the exhaust velocity. Rat-tic 0.? "turboje-t and "Fan TthI'tS: 9 ‘1' :EE .. (U¢+Bu4)++ (D 1 “Jim 20.35;; us. "Titrboje ' E}, Lie: zap 13—1-5- uc- chPtiTérfl %)—1;_'-+l let i" fi—fi: “a \lzc-‘rrf saw—r?) —r¥*—-1 ® Tart-sown a”: .lch'fi [$191-1] 6) U2- =iiZcpi TaflT'é‘CrZ'TD—BCTz-F') Ue» '-' 261”? E "7.1(1-r-‘55—r‘?“ "50$ -=‘-) (“'3 (2}) t3 .549. A lurbojet engine (with a = 0] has a pressure ratio of 30 and a maximum temperature of 1700 K. The component efficiencies and ambient conditions are the same as those given in Table 5.1. For a flight Mach number of 0.85 determine a. The specific thrust, b. The thrust specific fuel consumption. c. The engine thermal efficiency. It. The propulsion efficiency. 9. The overall efficiency. 115.315 dot-Ea. e-F Table 5.1 and catcula-i-fon procedure a-F' 99170 “£75,111: resut-i-s are: (wi'f'h u =- 250.8 m/s an d 54/155; = 096) f3; [<31 23. 67’ 131 K 248.0 F35 KFa 763 K 0.02.60 Pi; 1431 954-5 Tag. K I 700. 155 K 1260 .7 P05 KR: Zia-'4' ue. m/s Hams 37m «Ms/55 0310' 1'ch Ks/K .s 0.0256 77th 0. 535 ’KF 0,365 77a a. 1.95- ...
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PSol3 - .5'4. After verifying the derivation of Eq. (5.27)...

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