16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 1 of 6 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 1b: Review of Rocket Propulsion The only practical way to accelerate something in free space is by reaction. The idea is the same as in air breathing propulsion (to push something backwards) but in rockets the “something” must be inside and is lost. Here is a revealing derivation of the thrust equation for vacuum: () () () () () () t 0 Mom.t=Mtvt+ mt' vt'-ct' dt'=Constant?? ??∫ i ()dMom =0 dt () dv dM M+v +mv-c=0 dt dt i But dM dv m=- M =mc dt dt → ii call this thrust, F. Notice () () dMv =F-vm=m c-v dt ii which can be (+) or (-) Using the same technique, the kinetic energy of the system rocket-jet is () () () t 2 2 0 11 KE = Mv + m t' v t' - c t' dt' 22 ?? ??∫ i So, () 222 2 dKE dv 1 dM 1 1 =Mv + v m v +c -2vc mc dt dt 2 dt 2 2 += ii mcv i m? i So, if thermal (or electrical) energy is expended internally at the rate E, i and converted to total kinetic energy with efficiency th e1. (or )η η ; i.e., no drag, no interaction of molecules with ambient air 16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 2 of 6 () th dKE =E, dt η i then 2 th 1 E= mc 2 η ii jet kinetic power Note that we counted both vehicle and wake KE as produced by E i , and this is unambiguous. If we try to define “useful Propulsive work” as Fv = mcv, i then we find that the “propulsive efficiency” prop 2 th F.v mvc 2v == 1 c E mc 2 =η η i i i is arbitrarily high! pr. c If v > , >1 . 2 ?? ?? ?? η For this reason, prop η is not used for rockets. But it is still true that thrusting at high speed increases kinetic energy more f 2 220 f0 0 f m11 ? mv = m v +c ln - v 22 ?? ?? ?? ?? ?? ?? ?? ?? ?? In the presence of external air, some modification is needed, leading to the well- known formula () eeea F=mu +A P -P mc≡ ii defines e cu=null Jet speed far from exhaust For finite P a , in thermal rockets, increasing A e increases u e (towards a limit max ep0 u2cT? ), but it eventually makes eae (P - P )A negative. The best A e is such as to make P e = P a . The thrust coefficient c F is used to quantify the performance of nozzles. Starting from () eeea F=mu +A P -P i (or finite P e ) 22 2 00 f0 ff mm1 =m c ln +v c ln 2m m ?? ?? ?? 16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 3 of 6 Isentropic flow r pp∝ r V p = constant and assuming ideal gas, 1 -1 tt t 0 0 t 22 m= uA = RT A +1 -1 γ ?? ?? ρρ ?? ?? γγ ?? ?? i 0 00 P RT 0t * PA m= c ? i with () 0* RT c= , Γγ () +1 2-1 2 = +1 γ γ ?? Γγ ?? γ ?? eeea 0t * t0 0 uAPP F=PA + - APPc ???? → ???? ???? non-dimensional c F So F0 t F=cPA Usually, F c 1.5-2～ 16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 4 of 6 Specific impulse (inverse of specific fuel consumption) is defined as * F sp ccFc I= = = gg mg i We want high I sp (high c), since 0 f mdv dm m=- c?V=c ln dt dt m → Using 0PSL m=m+m+m, () PSL m = propellant, m = structure, m = payload ?V P c LSL0S?V c m m = - m or m = m e - m e-1 ? So, for given m P , m S , the higher c (I sp ), the higher m L ; this dependence is at least linear (for small LP S ?Vc ,m m -m c ?V null ), but it becomes exponentially fast for high energy missions ( ?V 1 c null ). For chemical rockets, c is limited by chemical energy/mass in fuel max c2E.= In general, since 2 e th th 1 mu = E= mE 2 ηη iii e th u2E= η (Accounting for work of expansion leads to a replacement of E by H, the enthalpy.) For ideal expansion, Brayton cycle, so -1 e 0 th P =1- P γ γ ?? ?? ?? η and H=c P T 0 (ideal gas) -1 e eP0 0 P u= 2cT 1- P γ γ ?? ?? ?? ?? ?? ?? 16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 5 of 6 This shows we want high P 0 /P e . This means large area ratio e * A A , not necessarily high P 0 . In fact, for vacuum operation, higher P 0 simply means higher P e , with no increase of u e (or I sp ). In a closer look, higher P 0 can increase T 0 by inhibiting dissociation higher I sp . But the main reason to go to high P 0 is to reduce weight for a given thrust (also for boosting, where P e cannot be much lower than P a , so high P 0 /P e means high P 0 ). Roughly 2 2w 0 F0 t 4Rt M 2 Rt = R p FcPA πρ πσ πnull 0 R t=p 2σ 2 20w t2 F0 4RPRρM A = K R F 2cPK R π σ null 2w F0 F F0 M2 F R F = c P K R R = FKc cPK ρπ σ null () w 32 32 0 F M2 1 = PF Kc ρπ σ Thus, for a given thrust level, the engine mass scales like 0 1 P . For a given total impulse (not thrust), it may be better to reduce P 0 , reduce thrust, operate longer. In boosters, there is a complex tradeoff, involving gravity losses, drag penalties, improved I sp at high P 0 , etc. In general, for boosters it is found advantageous to go to high P 0 , limited mostly (in liquid rockets) by turbomachinery. For space engines the result is less clear, but they do tend to optimize at much lower P 0 . The power of a rocket can be extremely high. For the Shuttle 7 F3000 ton=3×10Nt , null c 3300 m/secnull 27310 111 mc Fc 3×10 3.3 10 =5 10 watt =50 GW!! 222 == ×× × i Thus, if one step in the power chain involves electrical power, the engine is likely to be very heavy. Why then electrical? Because it breaks the u e limit, allowing any I sp , or, in other words, it gives very fuel efficient rockets. With EM forces one can increase u e almost arbitrarily; however looking at the power requirements, 16.522, Space Propulsion Lecture 1b Prof. Manuel Martinez-Sanchez Page 6 of 6 cc 11 Fc Pmc 22 == ηη i cc PFc ac MM2 2 == ηη 0.5 watt/Kg 2 Kg/watt = 2000 Kg/KW→ -2 50 watt/Kg 2 10 20→× → So, with reasonable mass/power ratios for electric power one gets very low accelerations If one pushes I sp in an electric thruster, M P is reduced for given ?V, but M s increases due to the high P, unless a is very low (which may be impossible because of mission duration constraints). Thus, an optimum I sp is found to exist (~2000 sec, depending on conditions). Reference Martinez-Sanchez, M., and J. E. Pollard. “Spacecraft Electric Propulsion – An Overview.” Journal of Propulsion and Power 14, no. 5 (September–October 1998): 688-699. [A publication of the American Institute of Aeronautics and Astronautics, Inc.]