Rocket Propulsion Fundamentals
Transcript
Rocket Propulsion Fundamentals
Rocket Propulsion Fundamentals ROCKET PROPULSION SYSTEMS Functions • Thrust generation by acceleration of an internally stored propellant for: – launch & orbit insertion – orbit maneuvering & maintenance – attitude control Basic Elements • Propellant(s) & Propellant Storage • Propellant Feed System • Energy Source • Energy Conversion • Accelerator Main Technologies & Energy Sources Cold Gas Rocket • Cold Gas Rocket Propulsion Systems (CGRPSs), thermal • Liquid Rocket Propulsion Systems (LRPSs ), chemical • Solid Rocket Propulsion Systems (SRPSs), chemical • Hybrid Rocket Propulsion Systems (HRPSs), chemical • Nuclear Rocket Propulsion Systems (NRPSs), nuclear • Electrical Rocket Propulsion Systems (ERPSs), electrical (photovoltaic, nuclear, etc.) Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals LIQUID ROCKET PROPULSION SYSTEMS System Architecture Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals SOLID & HYBRID ROCKET PROPULSION SYSTEMS System Architecture Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ELECTRIC ROCKET PROPULSION SYSTEMS System Architecture Arcjets Magnetoplasmadynamic (MPD) Thrusters Electrostatic Thrusters Hall Thrusters Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals KEY PROPULSION TECHNOLOGIES Operational Comparison Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ROCKET MASS BALANCE Rocket Continuity Equation • AS Notations and assumptions: – control volume V , the rocket itself, bounded by: AS the rocket external surface (impermeable, u ! dS = 0 ) Ae the nozzle exit area – inertial frame moving at the instantaneous rocket speed: rocket velocity v = 0 (but in general dv dt ! 0 ) flow velocity u – rocket mass m – propellant mass flow rate m! – nearly uniform flow at the nozzle exit (index e) • Then: ! " dV + # "u $ dS + Ae ! t #V where: dm ! = # " dV dt ! t V m! = # !u " dS Ae # AS "u $ dS = 0 ! v m Ae , ! e ue dm + m! = 0 dt is the rate of change of the rocket mass m is the propellant flow rate ( m! ! "eue Ae in the 1D approximation) Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ROCKET FORCES pa Rocket Momentum Equation • Notations and assumptions: AS v – uniform ambient pressure pa and gravitational body force g • D In an inertial frame where v = 0 (but in general dv dt ! 0 ): mg ! & $ "uu $ ( p $ pa ) 1 + % () * dS + # " g dV , "udV = # Ae + AS ' V ! t #V Ae , pe , !e d dv dm ue ! = F + FA + mg ( mv ) = m + v dt dt dt v=0 where the use of the relative pressure p ! pa is justified because a uniform pressure pa yields no net force on closed surfaces, and: " !g dV F ! " % #uu $ dS " % ( p " p ) dS !#"#$ !##"## $ is the rocket weight FA ! " # ( p " pa ) dS + # $ % dS AS AS ! ##"## $ ! # "# $ is the aerodynamic force ( u ! dS = 0 on AS ) mg = V Ae Ae momentum thrust A/D pressure force • a is the rocket thrust ( ! " 0 on Ae ) pressure thrust A/D viscous force Notice that, projecting in the forward direction, in the 1D approximation: ! e + ( pe " pa ) Ae F ! mu Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ROCKET POWERS AS Rocket Energy Equation • • m Notations and assumptions: – free rocket, adiabatic conditions ( g = pa = q! = Q! = 0 ) – hR , m R formation enthalpy and mass of reacting propellants – hP ! h!P + c pP ( T " T° ) formation enthalpy of combustion products v ue Ae , ! e , Te , pe From the energy equation for the usual control volume V : ! !p " " h dV + dS $ u " h = dV = 0 ! ( mRhR + mBOhBO ) + %Ae dS # u$ htP & 0 t t !# S #V ! t ! t #V "t and, since mBO hBO is constant and by continuity ! m R ! t = " m! = " % #u $ dS : Ae 1 % ( !hR $ "u # dS + $ ' h!P + c pP ( T ! T° ) + u # u * "u # dS + 0 ! Pc = Q! j + W! j Ae Ae 2 & ) where: P!c = m! (hR ! h"P ) (combustion power) ! pP ( Te " T° ) (exhaust thermal power) Q! e ! c pP ( T " T° ) #u $ dS ! mc % Ae 1 1 2 1 2F 1 ! u ! u # u ! dS $ u m = ue $ Fc e "Ae 2 2 2 c 2 $ # 1) Me2 2 W! e ue2 2 ( != " " "1 P!c c pP ( T # T° ) + ue2 2 1 + ($ # 1) Me2 2 W! e = (exhaust mechanical power) (rocket engine efficiency) Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals FREE ROCKET PERFORMANCE Thrust Performance Parameters • Effective exhaust velocity: ! e + ( pe " pa ) Ae ( p " pa ) Ae F mu c= ! = ue + e m! m! m! Notice: – c is a function of the ambient pressure pa – usually ( pe ! pa ) Ae m! << ue and therefore c ! ue (exhaust velocity) • Specific impulse and volume specific impulse (or density impulse): F F F and Isp = ! c = g0 Isp Id = = = ! g0 Isp ! 0 mg V! m! ! Notice: – Isp – – – • Id g0 V! mostly relevant to mass optimization mostly relevant to volumetric and aerodynamic optimization is the gravity acceleration at sea level is the volumetric flow rate of the stored propellant Problem: Show that for rockets operating with two propellants (fuel F and oxidizer O ): m! O + m! F Id = ! Isp g0 ! "= m! O "O + m! F " F Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals FREE ROCKET PERFORMANCE Tsiolkovsky’s Free Rocket Equation • • • • Assumptions: – straight trajectory – no body forces, no atmosphere ( g = pa = 0 ) – constant effective exhaust velocity c F v m From the rocket mass and axial momentum balances: Dm Dv and = ! m! m =F Dt Dt ṁ ! and eliminating m! with the continuity equation: Using F = mc ue Dv Dm dm ! = !c m = mc ! dv = "c Dt Dt m Integrating between the initial and burn-out masses m0 , mBO the rocket velocity change is: m (Tsiolkovsky’s free rocket equation) ! v = "c ln BO m0 Compare with Breguet eq’n for leveled cruise with lift-to-drag ratio L D and velocity v0 : L m ! x = "v0 Isp ln BO D m0 Problem: Why are both equations logarithmic in the vehicle mass and linear in the propulsive (and aerodynamic) performance parameters Isp (and L D )? Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ROCKET THRUST PERFORMANCE Equations and Parameters • Total impulse: I= ! tb 0 where tb is the thrust duration or burn time Fdt For operation at constant specific impulse: tb ! = cmP I = Isp g0 ! mdt where mP is the propellant mass 0 • Thrust profile: " progressive dF dt > 0 (SRPS's) $ F ( t ) ! #regressive dF dt < 0 (SRPS's, HRPS's) $neutral dF dt = 0 (LRPS's, ERPS's, NRPS's, SRPS's) % • Thrust-to-weight ratio: – for free flight: v! F (acceleration in g-No.) = g0 mg0 – for vertical take-off: v! F = !1 " g0 mg0 F mg0 >1 take-off Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals PERFORMANCE OF KEY PROPULSIVE TECHNOLOGIES Performance Comparison Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals PERFORMANCE OF KEY PROPULSIVE TECHNOLOGIES Exhaust Velocity v/s Vehicle Acceleration Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals PERFORMANCE OF KEY PROPULSIVE TECHNOLOGIES Specific Impulse v/s Thrust Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals MAJOR LAUNCH VEHICLE FAMILIES Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals SINGLE-STAGE ROCKETS Rocket Masses • Let: mP = propellant mass mT = tank mass mE = engine & propellant management mass m L = pay-load mass (all other masses) • • • Then: mS = mE + mT (structural mass) m0 = mS + mP + m L (initial mass) m0 ! mP = mS + m L (burn-out mass) Define: (tankage fraction, dependent on propellant type and feed system) ! = mT mP ! = mS ( mS + mP ) (structural coefficient, ! constant for equal rocket technology) (payload ratio) ! = m L m0 Then, using the free rocket equation mP m0 = 1 ! e ! " v c , obtain: ( ) m E m0 + " 1 # e # $ v c mS mE + mT != = = mS + mP mE + mT + mP mE m0 + " 1 # e # $ v c + 1 # e # $ v c ( m L m0 ! m P mS m0 ! m L = ! m0 m0 mS + m P m0 ) e! $v c ! % $v " #= > 0 " c > cmin = ! 1! % ln % Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals MULTISTAGE ROCKETS Multistage Rocket Performance • For missions with high propellant masses ( mP comparable to m0 ): – large tank and engine masses need to be accelerated – final thrust and acceleration become excessive Hence staging can be advantageous • For each stage i = 1, 2,... n : !i = e " # vi ci " $ i (1 " !i ) % # vi = "ci ln &'$ i + !i (1 " $ i ) () where: ci i th exhaust velocity ! i i th structural coefficient !i i th payload ratio • For n stages: – total velocity change: ! v = " ! vi i – overall payload ratio: ! = " !i i Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals MULTISTAGE ROCKETS Multistage Rocket Optimization • Alternative approaches: – minimum m0 for given ! v and m L ! max {!} for given ! v !i – maximum ! v for given m0 and m L ! max {" v} for given ! !i • (new design) (new mission) Overall velocity change and payload ratio constraint for a rocket with i = 1, 2,... n stages: ! v = " i ! vi = # " i ci ln &'$ i + %i (1 # $ i ) () ! = " i !i and # ln ! = $ i ln !i For the constrained optimization of ! v consider the augmented objective function: ( F = ! ( i ci ln $%" i + #i (1 ! " i ) &' ! k ln # ! ( i ln #i ) ( k is a Lagrangian multiplier) Hence, for F to be extremum for some values of the !i ’s: (1 # $ i ) ci + k = 0 !F =# !"i $ i + "i (1 # $ i ) "i ! "i = # i (1 $ # i ) ci k $ 1 where k is determined using the constraint on the payload ratio: # (1 $ # i ) ( n th order polynomial for 1 k ! k ) ! = " !i = " i c k $ 1 i i i In general need to choose the best maximum corresponding to the n solutions for k = k j : * ! v = max +" ) ci ln %&# i + $ij (1 " # i ) '( . j , i / where !ij = " i (1 # " i ) ; ci k j # 1 j = 1, 2,... n Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals MULTISTAGE ROCKETS Multistage Rocket Optimization (continued) • For equal exhaust velocities ( c1 , c2 ,... cn = c ): %1 c #i ( !1= ' $ k & " i 1 ! # i *) 1n "i % 1 # "i ( !i = ! $ 1 # " i '& i " i *) 1n 1n , & 1 " #i ) / ! v = "c2 ln .# i + # i ( $ % +* 1 # ' .i 10 i i • For similar stages ( c1 , c2 ,... cn = c and ! 1 , ! 2 ,... ! n = ! ): (equal payload ratios) !i = ! 1 n ! v = "nc ln %&# + $ 1 n (1 " # ) '( (equal velocity changes) and: % e " # v nc " $ ( !=' & 1 " $ *) n In particular, for an infinite number if similar stages: lim # v = $c lim n ln '(% + & 1 n (1 $ % ) )* = $c (1 $ % ) ln & n!" n!" Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals MULTISTAGE ROCKETS Multistage Free Rockets Problem • Carry out the payload fraction optimization (1st approach) of multistage rockets. Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ELECTRIC ROCKET OPTIMIZATION Optimum Specific Impulse • Assumptions: – electric thruster effective beam velocity c – mission requirement ! v << c (not too unrealistic) – propellant mass flow rate m! = mP t BO – nearly uniform (1D) beam properties – electric thruster efficiency ! = W! e P!el – spacecraft specific mass ! = mS P!el • In the above assumptions: 1 2 1 mP 2 ! = W! e ! mc c 2 2 t BO P!el ! c2 ! mS = ! Wel = ! = mP " 2"t BO mP "v for = 1 ! e! "v c # ! v << c m0 c # ! c2 & m0 = m P + mS + m L = % 1 + mP + m L 2"t BO (' $ exhaust beam mechanical power spacecraft structural mass propellant mass ratio spacecraft initial mass Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals ELECTRIC ROCKET OPTIMIZATION Optimum Specific Impulse (continued) • Pay-load ratio: % %1 mL # c 2 ( mP #c ( != = 1 " '1 + + 1 " + '& c 2$t *) , v - ". m0 2$t BO *) m0 & BO as #%0 c! $ %&" ( mP ! " ) ( P! , m el S ! ") Hence, differentiating w.r.t. c , the pay-load ratio is maximum for: d! & 1 $ ) "( 2 # ,v = 0 + dc ' c 2%t * BO corresponding to: copt ! g0 Ispopt = 2"t BO # From this result: !opt " 1 # $ v 2% &t BO or: t BOopt • 2! & # v ) = ( + " '1$ %* 2 Notice that t BOopt is a strong function of ! " 1 , of ! v and, to a lesser extent, of ! and ! . Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals PROPULSION SYSTEM DESIGN PROCESS Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11. Rocket Propulsion Fundamentals PROPULSION SYSTEM DESIGN PROCESS (continued) Luca d’Agostino, Dipartimento di Ingegneria Aerospaziale, Università degli Studi di Pisa, 2010/11.