Subject: Turbojet engines (continued); Design parameters; Effect of mass flow on thrust.

Transcription

1 16.50 Leure 19 Subje: Turbje engines (ninued; Design parameers; Effe f mass flw n hrus. In his haper we examine he quesin f hw hse he key parameers f he engine bain sme speified perfrmane a he design ndiins, and hw he perfrmane varies if hese parameers are hanged, sill a he design ndiins. Laer we will lk a a mplemenary quesin, namely, hw he perfrmane f a pariular design hanges when ndiins are differen frm design ndiins. Wih he resuls ha we wrked u las ime fr he Turbje engine, le us lk a he. dependene f /m a n he prinipal parameers, M, τ and θ. We an view hem his way τ design hie (mpressr pressure rai M fligh speed θ τ r θ mbusr ule emperaure, an peraing variable, limied by urbine maerials sme maximum value. Assuming he exhaus is mahed, we an re-wrie Eq. (10 f he previus leure as 1. = [# (1 " " # ($ "1]" M m a " 1 # 0 $ we an see ha sine θ τ >1, belw fr a subsni ase: always inreases wih θ. This relainship is displayed 1

2 r a given θ, wha is he variain wih τ? By inspein we see here is a maximum a he maximum f he brakeed quaniy, s a he value f τ ha saisfies 1 # 1 [ # (1 $ $ # ( " $ 1] = $ # = 0 " # " # " This value is (τ = max This resul an be seen be equivalen T 3 = T 0 T 4, namely, he mpressr exhaus shuld be a he gemerial mean f he ambien and mbusr exhaus emperaures. If i were muh lwer r muh higher, he T-S diagram f he equivalen Brayn yle wuld be skinny, and enlse lile area ( lile wrk per uni mass: (a T lile mpressin (b Opimal mpr. ( T muh mpr. T T T T 4 T 4 T 4 T 3 T 0 T 0 T 0 T 3 S S S Wheher his pwer is uilized as je kinei energy, as in he urbje, r as shaf pwer in a urbprp, is immaerial. Als, as far as his argumen, he mpressin T 3 /T 0 an be arbirarily divided beween ram mpressin (θ 0 and mehanial mpressin (τ. Wha is he meaning f his fr he urbje? Puing his value in (10 and he rrespnding expressin fr he speifi impulse, we have he hrus and I fr engines pimized fr hrus per uni f airflw: T 3

3 ( = ( # max "1 + M "1 (Ι max = ( a h ( / 0 T g ( " " M (11 As an example ake: θ = 6.5 =.5, γ = 1.4 ( max = 5(.5 + M M (Ι = a ( h max T g 3.75 a h (83m /s(4.3x10 7 J /kg = = 6178s T g (1004J /kgk(00k(9.81m /s There is an upper limi n M reahed when he mpressr ule emperaure equals he urbine inle emperaure, s n fuel an be added. Tha is we mus have θ τ < θ Sine θ 0 inreases wih Mah number, he hereial limi is reahed when τ =1 and θ =θ, i.e. when 0 here is n mpressr and we have a ramje. Bu fr τ = he real limi is reahed when θ =, i.e., all he mpressin is due ne again he ram effe, bu we are allwing a gd margin fr hea addiin in he burner, s his ramje des prdue maximum hrus. r a urrenly praial value f θ = 9, θ < 3 r M < 3.9 Prpulsive Effiieny The Turbje engine is araive fr is simpliiy and is gd hrus behavir a high Mah numbers. Unfrunaely i is n very effiien a lw Mah numbers, beause is je veliy is high. T see his we nsider he Prpulsive Effiieny, defined as pwer # # airplane prpulsive " pwer # in # je u 0 m (u e # u 0 u 0 u = 0 = $ & u = e u 0 ' m ( u u # e # 0 (u e + u0 m (ue + u 0 % ( rm his we see ha here is a dire nfli beween he desire fr high je veliy give high hrus, and je veliy near he fligh veliy, maximize he prpulsive effiieny. 3

4 In erms f ur expressin fr hrus, sine u = M ( e 1 u 0 u e = ( / / M + 1 u 0 and we an wrie he expressin fr he prpulsive effiieny in erms f ur expressin fr hrus prpulsive = + M 0. Sine /m a ~ 3 fr lw M, η prp is n gd fr he urbje a lw Mah numbers. We will see laer hw his defiieny is remedied by adding a fan he engine prdue a Turbfan. Thermal and Overall Effiienies The Thermal Effiieny is define fr he Turbje Engine as $ & u u m e # 0 ' pwer # in # je % ( hermal " = pwer # in # fuel # flw m f h inally we an define an Overall Effiieny as pwer # # airplane verall " = pwer #in # fuel # flw u0 m f h We see ha verall = hermal prpulsive I is als impran ha he verall effiieny is direly relaed he speifi impulse: u0 gu0 gu verall = = = 0 I m f h m f g h h 4

5 MIT OpenCurseWare hp://w.mi.edu Inrduin Prpulsin Sysems Spring 01 r infrmain abu iing hese maerials r ur Terms f Use, visi: hp://w.mi.edu/erms.