Session 3: Hart, Selections from The Concept of Law

Session 3-4: Mission Analysis for Electric Propulsion Previously, we described the benefits of electric propulsion over chemical rockets, namely, propellant savings albeit at increased mass from the supply, delivery and processing of electric power. Given this tradeoff, it is expected that there will exist a value of specific impulse that will maximize payload mass for some particular mission parameters. In this Lecture we analyze a few cases amenable to analytical (or straightforward numerical) solutions. The fundamental idea behind mission analysis is to write an objective function and then optimize it with respect to some variables under specific constraints. We begin by writing the total (wet) mass of the spacecraft, m0 = mps + mp + ms + mpay (1) The mass of the propulsion/power system is, mps = αP = αF c 2η (2) If the mission time tm is fixed and we assume constant mass flow rate, ˙ m= mp = mt F tm c (3) Taking the structural and payload mass as constant (or at least independent of specific impulse), we apply the optimality condition (maximum payload), dmpay dc = 0 F tm αF − =0 2 copt 2η or F,P,t (4) Solving for the optimal specific impulse we obtain, � copt = 2ηtm α (5) Missions requiring thrusting for long periods of time, for example to counteract drag when orbiting in LEO, would benefit from high specific impulse. In finding the optimal specific impulse Eq. (5) we only specified the mission duration and assumed constant mass flow rate and thrust independent of Isp . If in addition there is now e −Δv/c a constraint in Δv then we should use mp = m0 1 − e for the propellant mass. Also, the propulsion/power system mass can be written as, mps � � �� αmc ˙ 2 αm0 c2 Δv = αP = = 1 − exp − c 2η 2ηtm (6) Here we notice the same group of parameters of Eq. (5). We define this group as a charac­ teristic velocity, 1 � r vch = 2ηtm α (Stuhlinger velocity) (7) with the understanding that this may not be the optimal value of the specific impulse given the Δv constraint. The meaning of this velocity, from the definition of α is that, if the propulsion/power mass were to be accelerated by converting all of the electrical energy generated during time tm , it would then reach the velocity vch . This is therefore the upper limit of Δv that can be achieved with an electric propulsion system. The balance of Eq. (1) can be written as, �  �  �   �2  � �  ��  Δv/vch c Δv/v v/vch mpay + ms H= = exp − − 1 − exp − c/vch vch c/vch m0 (8) where we have grouped the problem invariants into an objective function H (in this case non-dimensional, accounting for the payload and structural mass) to be maximized in terms of c/vch for a given Δv/vch . Fig. 1 shows the shape of H as a function of c/vch with Δv/vch as a parameter. The existence of an optimum specific impulse in each case is apparent in the figure. This optimum c is seen to be near vch . If Δv/c is taken to be relatively small, expansion of the exponentials in Eq. (8) allows an approximate analytical expression for the optimum c: 1 Δv 1 copt ≈ 1− − vch 2 vch 24 �  Δv vch  �2 + ... (9) 1 0.9 Δv/vch = 0 0.1 0.8 0.7 0.2 H 0.6 0.3 0.5 0.4 0.4 0.3 0.5 0.2 0.6 0.1 0.7 0.8 0 0 0.5 1 c/vch 1.5 2 Figure 1: Objective function H as a function of c/vch with Δv/vch as a parameter 2 Fig. 1 also shows that, as anticipated, the maximum Δv for which a positive payload can be carried (with negligible ms ) is of the order of 0.8vch . Even at this high Δv, Eq. (9) holds fairly well. 1 m pay + ms optimized mass fractions 0.9 exact m0 0.8 approximation 0.7 0.6 mp 0.5 m0 0.4 0.3 0.2 m ps 0.1 m0 0 0 0.1 0.2 0.3 0.4 !v/vch 0.5 0.6 0.7 0.8 Figure 2: Optimized mass fractions as a function of Δv/vch The effects of (constant) efficiency, propulsion/power specific mass and mission time are all lumped into the parameter vch . Eq. (9) then shows that a high specific impulse is indicated when the propulsion/power system is light and/or the mission has a long duration. Fig. 2 shows that, for a fixed Δv, these same attributes tend to give a high payload fraction and small (and comparable) propulsion/power and propellant fractions. Of course the same breakdown trends can be realized by reducing Δv for a fixed vch . This regime is appropriately known as the trucking regime. At the opposite end (short mission, heavy propulsion/power) we have a low vch , hence low optimum specific impulse, and from Fig. 2, small payload and large propellant fractions. Correspondingly, this is known as the sports-car regime. We have, so far, regarded the efficiency η as a constant, independent of the choice of specific impulse. This is not, in general, a good assumption for electric thrusters where the physics of the gas acceleration process can change significantly as the power loading (hence the jet velocity) is increased. For each thruster family and for each propellant and design, one can typically establish a connection between Δv and c alone. Thus, as we will see later in detail, η increases with c in ion, Hall, MPD and electrospray thrusters, whereas it typically decreases for arcjets (beyond a certain c). In general, then, one needs to return to Eq. (8) with η = η(c) in order to discover the best choice of specific impulse in each case. It is instructive to consider the particular case of ion and electrospray thrusters, both because relatively simple and accurate laws can be obtained in those cases. Losses for these thrusters can be fairly well characterized by a constant voltage drop per accelerated ion or charged droplet. If this is called Δφ, the energy spent per particle of mass mi and charge q is, 3 1 mi c2 + q∆φ 2 (10) of which only mi c2 /2 is useful. The efficiency is then, η = η0 1 + (vL /c)2 (11) Eq. (11) includes a factor η0 < 1 to account for power processing and other losses and where vL is a loss velocity, equal to the velocity to which charged particles would be accelerated by the voltage drop ∆φ, r vL = 2 q ∆φ mi (12) This expression indicates that losses are reduced if the specific charge q/mi is low. This shows the importance of high atomic mass in ion engines and electrosprays working in the pure ionic regime, since ∆φ is not very sensitive to propellant choice, and vL can be reduced if mi is large. Eq. (11) also shows the rapid loss of efficiency when c is reduced below vL . Using Eq. (11), we can rewrite Eq. (8) as, −∆v/c H = e
c2 + vL2 −∆v/c − 1 e − 2 vch (13) where the definition of vch in Eq. (7) is now made using η0 instead of η. Once again, only approximate expressions in ∆v/vch are feasible for the optimum specific impulse and mass fractions. Normalizing all velocities by vch : x= c , vch vL vch (14) √ ν ν2 1 + δ2 − − √ + ... 2 24 1 + δ 2 (15) ν= ∆v vch and δ= we obtain, xopt = √ Hmax = 1 − 2ν 1 + δ 2 + ν 2 − mp m0 opt ν 1 = √ − 2 24 1+δ  ν3 √ + ... 12 1 + δ 2 ν √ 1 + δ2 (16) 3 + ... (17) The main effect of the losses characterized by δ can be seen to be: (a) An increase of the optimum specific impulse, seeking to take advantage of the higher efficiency thus obtained. (b) A reduction of the maximum payload, 4 (c) A reduction of the fuel fraction. Both these last effects indicate a higher propulsion/power mass fraction, due to the need to raise rated power to compensate for the efficiency loss. It is worth noting also that the losses are felt least in the trucking mode (high vch , i.e. light engine or long duration operation). As was mentioned, there is no a priori reason to operate an electric thruster at a constant thrust or specific impulse, even if the power is indeed fixed. We examine here a simple case to illustrate this point, namely, one with a constant efficiency as in the classical Stuhlinger optimization, but allowing F , ṁ and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the power, P = ṁ(t)c2 (t) F (t)c(t) = 2η 2η (18) Consider the rate of change of the inverse mass with time: F2 a2 d(1/m) 1 dm ṁ ṁ2 = − 2 = 2 = 2 = 2 2 = dt m dt m mm ˙ m mc ˙ 2ηP (19) where a = F/m is the acceleration due to thrust. Integrating, 1 1 1 − = m0 2ηP mf tm a2 dt (20) 0 On the other hand, the mission Δv is, tm Δv = adt (21) 0 and is a prescribed quantity. We wish to select the function a(t) which will give a maximum mf while preserving this value of Δv. The problem reduces to finding the shape of a(t), whose square integrates to a minimum while its own value has a fixed integral. The solution (which can be found by various mathematical techniques, but is intuitively clear) is that a should be a constant. Using this condition, Eq. (20) and (21) integrate immediately. Eliminating a between these, we obtain, mf = m0  �  �−1 m0 Δv 2 1+ 2ηtm P (22) The level of power is yet to be selected. It will determine the average specific impulse, and it is to be expected that an optimum will also exist. The final mass consists of the payload and propulsion/power system, mf = mpay + mps . Using the definition of vch and P = mps /α, our objective function is now, �  �  mpay mps 1 H = = −1 m0 (mps /m0 ) + (Δv/vch )2 m0 5 (23) and select the value of mps /m0 that will maximize H. This is found to be, mps m0 opt ∆v = vch   ∆v 1− vch (24)  2 ∆v = 1− vch (25) which, when used back in Eq. (23) gives, Hmax mpay = m0 opt and then, mp m0 = opt ∆v vch (26) These are, within the assumptions, exact expressions. They could be compared to the results in Fig. 2 which were found to apply when c, and not a, was assumed constant. The difference is noticeable only for the highest values of ∆v/vch and is negligible for smaller values. It is of some interest to inquire at this point how the jet velocity c should vary with time in order to keep the acceleration constant. We have, mc ˙ c a= = m m  2ηP c2   then c= 2ηP a  1 m (27) and since 1/m varies linearly with time (because a is constant), so will c. At the final time, when m = mf ,  cf = 2ηP a     2   2  1 2η(mps /α) 1 vch mps /m0 vch ∆v = = = = vch mf ∆v/tm mf ∆v 1 − mp /m0 ∆v vch (28) From Eqs. (19) and (27), the rate of change of c becomes, dc = dt  2ηP a  d(1/m) = dt  2ηP a  a2 =a 2ηP (29) so that, altogether, for some intermediate time t, c(t) = vch − a(tm − t) (30) Thus c varies between vch − ∆v at t = 0 and vch at t = tm . The approximate result copt ≈ vch − ∆v/2 found when c was constrained to remain constant is therefore quite reasonable. Notice that Eq. (30), in the ideal case with no external forces acting on the vehicle, implies a constant absolute velocity of the exhaust, cabs = c − v = c − (at + v0 ) = vch − ∆v − v0 (31) This means that in the optimal case, exhaust particles move at constant velocity with respect to an inertial observer. Even though some energy was spent in realizing this absolute motion, in time, no more energy is effectively added to the exhaust, thus minimizing energy losses. 6