16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 1 of 19 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 2: Mission Analysis for Low Thrust 1. Constant Power and Thrust: Prescribed Mission Time Starting with a mass 0 M , and operating for a time t an electric thruster of jet speed c, such as to accomplish an equivalent (force-free) velocity change of V? , the final mass is dv dM M=-c dt dt dM dv = -c M if c=constant (consistent with constant power and thrust), then f M v=c ln M 0 -Vc f0 M=Me ? (1) and the propellant mass used V - c P0 M=M 1-e ? ?? ?? ?? (2) The structural mass is comprised of a part Mso which is independent of power level, plus a part Pα proportional to rated power P, whereα is the specific mass of the powerplant and thruster system. In turn, the power can be expressed as the rate of expenditure of jet kinetic energy, divided by the propulsive efficiency: 2 1 P= mc 2η i (3) and, since m i is also a constant in this case, p m=M t. i Altogether, then, 2P sso M M=M + c 2t α η (4) The payload mass is Lfs M=M-M. Combining the above expressions, 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 2 of 19 () 2 -Vc -VcsoL oo MM c =e - - 1-e MMt ?? α η (5) Stuhlinger [1] introduced a “characteristic velocity” ch 2t v= η α (6) whose meaning, from the definition of αis that, if the powerplant mass above were to be accelerated by converting all of the electrical energy generated during t, it would then reach the velocity ch v . Since other masses are also present, ch v must clearly represent an upper limit to the achievable mission V? and is in any case a convenient yardstick for both V? and c. Figure 1 shows the shape of the curves of Lso o M+M M versus ch c/v with ch Vv? as a parameter. The existence of an optimum c in each case is apparent from the figure. This optimum c is seen to be near ch v hence greater than V? . If V c ? is taken to be a small quantity, expansion of the exponentials in (5) allows an approximate analytical expression for the optimum c: 2 OPT ch ch 11V cv-V- 224v ? ?? (7) Figure 1 also shows that, as anticipated, the maximum V? for which a positive payload can be carried (with negligible so M ) is of the order of ch 0.8 v . Even at this high V? , Equation (7) is seen to still hold fairly well. To the same order of approximation, the mass breakdown for the optimum c is as shown in Figure 2. The effects of (constant) efficiency, powerplant specific mass and mission time are all lumped into the parameter ch v . Equation (7) then shows that a high specific impulse sp I = c gis indicated when the powerplant is light and/or the mission is allowed a long duration. Figure 2 then shows that, for a fixed V? , these same attributes tend to give a high payload fraction and small (and comparable) structural and fuel fractions. Of course the same breakdown trends can be realized by reducing V? for a fixed ch v . This regime was called quite graphically the “trucking” regime by Loh [2] . At the opposite end (short mission, heavy powerplant) we have a low ch v , hence low optimum specific impulse, and, from Figure 2, small payload and large fuel fractions. This is then the “sports car” regime [2] . References: Ref. [1]: Stuhlinger, E. Ion Propulsion For Space Flight. New York: Mc Graw-Hill Book Co., 1964. Ref. [2]: Loh, W. H. Jet, Rocket, Nuclear, Ion and Electric Propulsion Theory and Design. New York: Springer-Verlag, 1968. 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 3 of 19 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 (resistojets, arcjets, ion engines, MPD thrusters) and for each fuel and design, one can typically establish a connection between η and c alone. Thus, as we will see in detail later, η increases with c in both ion and MPD thrusters, whereas it typically decays with c for arcjets (beyond a certain c). In general, then, one needs to return to Equation (5) with ()=cηη in order to discover the best choice of c in each case. It is instructive to consider in some detail the particular case of the ion engine, both because of its own importance and because relatively simple and accurate laws can be obtained in that case. Ion engine losses can be fairly well characterized by a constant voltage drop per accelerated ion. If this is called? φ, and singly charged ions are assumed, the energy spent per ion is () 2 ii 1 2mc + m = ion mass; e = electron charge? φ , of which only 2 i 12m c is useful. The efficiency is then 2 2 i c = 2e c+ m η ?φ (8) We should also include a factor of 0 1η ∠ to account for power processing and other losses. We then have 2 0 22 L c = c+v ηη (9) 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 4 of 19 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 5 of 19 were L v is a “loss velocity”, equal to the velocity to which one ion would be accelerated by the voltage drop .? φ Notice how this simple expression already indicates the importance of a high atomic mass propellant; ? φ is insensitive to propellant choice, and so L v can be reduced if i m is large. Equation (9) also shows the rapid loss of efficiency when c is reduced below L v . Using (9), we can rewrite (5) as 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 6 of 19 V22V -- soLL cc 2 ooch MMc+v =e - - 1-e MMv ?? ?? ?? ?? (10) where the definition of ch v (Equation (6)) is now made using 0 η instead of η. Once again, only approximate expressions for ch Vv? are feasible for the optimum c and mass fractions. Normalizing all velocities by ch v : L ch ch ch vcv x; v=; = vvv ? ≡δ (11) we obtain 2 2 OPT 2 vv x = 1+ - - +... 2 24 1+ δ δ (12) 3 22soL 2 oo MAX MM v +=1-21+ v+v- +. MM 12 1+ ?? δ ?? δ?? (13) 3 P 22 o M v1v = - +... M24 1+ 1+ ?? ?? δδ ?? (14) For =0,δ and neglecting the last term included in each case, we recover the simple expressions of Equation (7) and Figure 2. The main effect of the losses ()δ can be seen to be: (a) An increase of the optimum c, seeking to take advantage of the higher efficiency thus obtained. (b) A reduction of the maximum payload, (c) A reduction of the fuel fraction. Both these last effects indicate a higher structural 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 ch v , i.e. light engine or long duration). 2. The Optimum: Thrust Profile 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, m i and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the power: 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 7 of 19 () () () () 2 11 P= m t c t = F t c t 22ηη i (15) Consider the rate of change of the inverse mass with time: 22 1 d 1dM mM =- = dt dtMM ?? ?? ?? i (16a) Multiplying and dividing by 2 22 F=mc, i 22 22 1 d FaM == dt 2 P Mmc ?? ?? ?? η i (16b) where a=FM is the acceleration due to thrust. Integrating, t 2 f0 0 11 1 -= adt MM 2Pη ∫ (17) On the other hand, the mission V? is t 0 V= a dt? ∫ (18) and is a prescribed quantity. We wish to select the function a(t) which will give a maximum f M (Equation 17) while preserving this value of V? . 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, (17) and (18) integrate immediately. Eliminating a between these, we obtain f 2 0 0 M 1 = M MV 1+ 2tP ? η (19) 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. Using fLs M=M+M and 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 8 of 19 s M P= α and introducing the characteristic velocity (Equation 6), we rewrite (19) as ()() sL 2 00 s0 ch MM 1 =-1 MM MM + Vv ?? ?? ? ?? (20) and select the value of s 0 M M that will maximize L 0 M M . This is easily found to be s 0chch OPT M VV =1- Mvv ?? ? ??? ?? ? ? ?? ? ? (21) which, when used back in (21) gives 2 L 0ch MAX M V =1- Mv ?? ? ? ? ?? ? ? ?? ? ? (22) and then P 0ch OPT M V = Mv ?? ? ?? ?? (23) These are, within the assumptions, exact expressions. They are to be compared to the approximate expressions in Figure 2 or Equation (12)-(14) with s0 0 M =0, =0 M δ which were found to apply when c, and not a, was assumed constant. Clearly, the difference is noticeable only for ch v= Vv? near unity (its highest value), 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 2 2 mc m a a= = Mc= Mc M2PM η ii where (16 a, b) have been used. Hence, 2P1 c= aM η 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 9 of 19 and since, by (16b), 1 M varies linearly with time, so will c. At the final time, when f M=M , 2 schs f fsL f 2M v M2P c= = = V aM V M +M M t ηαη ? ? and, from (21), (22), s sL ch M V = M+M v ? so that fch c=v (24) The rate of change of c follows from that of 1M(Equation 16) as 1 d dc 2 P M ==a dt a dt ?? ?? η ?? (25) so that, altogether, if t'represents some intermediate time, while t is the final time (used in ch v ), () ( ) ch ct'=v -at-t' (26) This varies between ch c=v - V? at t' = 0 and ch c=v at t' = t . The approximate result OPT ch 1 cv-V 2 ?? found when c was constrained to remain constant is therefore quite reasonable. Notice that (26) implies a constant absolute velocity of the exhaust gas, at the value () abs ch c=v-V-V0? Alternative Derivation with Variable Specific Impulse This is a more general treatment than that in pp. 15-19 of the Notes, and can be extended to more complicated situations, like non-constant efficiency. It also introduces some elements of Calculus of Variations, which is of general utility, and has many commonalities with Optimal Control theory. Decision #2 τ 2 1 Stage 1 v ch 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 10 of 19 We wish to minimize ft 0 F V= dt m ?? ? ∫ ?? ?? (1) subject to a given power (constant in time) 2 1mc 1Fc p= = 22ηη i (2) and to dm m=- dt i (3) write (1) as ft 0 2P V= dt mc η ? ∫ (4) which eliminates thrust. Next, treat (3) as a dynamic constraint, and append it to the cost through a time- dependent Lagrange multiplier (t).λ Define the Hamiltonian 2P dm H= - m+ mc dt η ?? λ ?? ?? i (5) or, using (2), 2 2P 2P dm H= - + mc dtc ηη?? λ ?? ?? (6) and minimize (unconstrained) the integral ft 0 Hdt ∫ .To do this, perturb about the optimum solution: fftt 00 HH Hdm Hdt = c + m+ dt = 0 dmcm dt dt ??? ???? δ δδδ ∫∫ ???? ?? ?? ???? ? ?? ?? () d m dt δ Integrate last term by parts: 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 11 of 19 f f t t 0 0 HHdH H c+ - m dt+ m =0 dm dmcmt dt dt ???? ?? ? ? ???? ?? ?? ?? ? ? δδδ ∫ ???? ?? ?? ?? ???? ?? ??? ? ???? (17) For optimality, we want to impose H =0 c ? ? (18) Hd H = dmmdt dt ?? ?? ?? ? ?? ? ?? (19) and at the ends, we say ( ) 0 mt=m (fixed), so () 0 m=0,δ and also ffm(t) = m (fixed, since we minimize V? between given masses), so, again, () ft m=0δ . Equation (18) gives in general (assuming ( )=cηη ) 23 2 2P 2 P 2P 2P -+2 +- =0 mc cmc c c ηη ?η?? λλ ?? ? ?? or 12 c ln -+ + - =0 mc m c λ?η?? λ ?? ? ?? (20) For example, if 2 0 22 2 c =, c+v ηη 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 12 of 19 () 2 2 22 2 2v ln = c cc+v ?η ? Here we take the simple case where η = constant, so (20) gives c=2 mλ (21) From (19), 2 2P d -+=0 dtmc ηλ (22) and using (21), 3 dP = dt m λη λ (24) We also can substitute (21) into (3) to get 22 dm P =- dt 2m η λ (25) Divide (24) by (25): d =-2 dm m λλ (26) which integrates to 2 m=Aλ (27) (A = undetermined constant). The value of A can be easily related to the optimized V? from Equation (4): fftt f2 00 2P P P V = dt = dt = t mc Am ηηη ? λ ∫∫ f Pt A= V η ∴ ? (28) To complete the time integration, go back to (24): 1 2 33 22 dPP == ; dt AA λη η λ ?? λ ?? λ ?? -1 2 3 2 P d= dt A η λλ 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 13 of 19 and integrating, 1 2 3 2 P 2= t+B 2A η λ 2 3 2 PB =t+ 2 2A ?? η λ ?? ?? (29) and from (27), 1 2 3 2 A m= PB t+ 2 2A η (30) The constants A and B can now be related to m 0 and m f : 1 2 0 2A m; B = 1 2 0 f ff f 32 2 2 0 mA1 m= = = Pt Pt m PtB1 ++1+ 2m2A 2A 2A ηη η (31) Using now (28), 2 0f 0f 0 22 f f mPt mPt mV == 2Pt2A Pt 2 V ηη? η η?? ?? ? ?? (32) We now introduce the specific mass of the power/propulsion system PP M , P α≡ and the Characteristic Velocity f ch 2t ν =, η α to rewrite (32) as 2 0f 0 2 ch pp mPt m V = m ν2A η ? ?? ?? ?? We also remember now that the final mass m f contains payload Lm , power/propulsion mass, and random dry mass so m , and write (31) as 0 Lsop 2 0 pp ch m m+m +m = m V 1+ m ν ?? ? ?? ?? (33) or 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 14 of 19 pp Lso 2 00 pp 0ch m m+m 1 =-1 mm m V + m ν ?? ?? ?? ? ?? ?? ???? (34) At this point we have the optimum time profiles, but we can do better by also selecting the optimum power level, which amounts to selecting the optimum specific impulse as well. We allow pp 0 m x= m to vary in (34) and maximize Lso 0 m+m =y m . Using ch V = ν ν ? (35) We had 2 x y= -x x+ν () 2 2 2 dy x +ν -x =-1=0 dx x+ν () 2 22 ν =x+ν ( ) 2 x=ν - ν = ν 1-ν or pp 0chch OPT m VV =1- m νν ?? ?? ?? ?? ?? ???? (36) Putting 2 x=ν - ν intoy , () 2 2 OPT ν - ν 1 y= -ν+ν = ν 1-ν -1 νν ?? ?? ?? () 2 OPT y=1-ν or 2 Lso 0ch OPT m+m V =1- m ν ????? ???? ???? (37) and then the propellant fraction follows from ppp LS0 000 mm m+m =1- m ? 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 15 of 19 p 0ch OPT m V = m ν ?? ? ?? ?? (38) Note these are exact formulas, no trailing terms. The specific impulse now follows from (21): 2 00ff 2 00 f mmVt2Pt 2Pt2A c=2 m= = = 1+ mmVmmV 2Pt ???ηη λ ?? ??η ?? f 0f 2Pt t c= + V mV t η ? ? and if we use optimum power, 2 pp ch ch 0f chf m ν tVt c= + V=ν 1- + V mVt ν t ??? ?? ?? ? ?? ch f t c=ν -V1- t ?? ? ?? ?? (39) so that c increases linearly from ch V at t=0 to ch V-V? at f t . Optimum Mission Time So far, t has been a free parameter, and we have found the mission performance (payload fraction) to improve with large t. But time has some costs associated with it, so increasing t is not necessarily desirable. These “costs” of time including capital immobilization, personnel costs during the long thrusting period, loss of opportunity, etc. There are several simple analytical ways to penalize long t choices. We select here one that maximizes the “Transportation Rate” M L /t. This makes most sense when there is a sequence of identical flights, each delivering payload M L in time t, but it can serve as a crude indicator even for a single mission. We also assume constant thrust, constant power (for simplicity of operation), and we include in the analysis the effect of a variable efficiency () 0 22 L = 1+c v η η , with LLOSi v=2ev m? , as in ion engines with constant ion cost LOSS ?v . We had for this case a payload ratio (optimized with respect to specific impulse) 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 16 of 19 22 LL ochchch Mvvv =1-2 1+ + vv ?????? ???? ???? (1) where 0 ch 2t v= η α (α=specific mass of the Power/Propulsion system) The thrusting time t is buried in v ch . To bring it out more explicitly, let us define a reference time 2 * V t= 2 0 α ? η (2) which depends on specified quantities only, and then a non-dimensional time * t t τ≡ (3) From the definition of v ch , then, 2 ch *2 2t vt == = VtV 0 η ?? τ ?? ?α ? ?? (4) and so (1) becomes 2 LL o Mv2z1 =1- 1+ + ; z MV ≡ ττ ? τ (5) A normalized transportation rate is now defined as LO * MM tt ψ≡ (6) or, from (5), 2 32 2 12 z 1 =- 1+ +ψ τττ τ (7) To maximize ,ψ set =0 1 ? ψ ?? ? ?? τ ?? . After some re-grouping, this leads to the equation for OPT τ : 22 (+2) +z =3+4zττ τ (8) 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 17 of 19 A convenient way to view this OPT (z)τ dependence is to define an intermediate parameter u= 2 +zτ ; we then obtain the parametric representation (for u≥2) () OPT 2 OPT 2u 2u-1 = 1+u z= u - ? τ ? ? ? τ ? (9) A good analytical approximation (valid for z=0, asymptotic for z >>1) is OPT 6 2+4z+ z+3 τ null (10) In particular, for a constant-efficiency model (v L =0, z=0) we see that OPT =4τ , or 2 OPT 0 2V t = α ? η . For other values of v L (or z), the results are shown in the attached graphs. 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 18 of 19 22 (+2) +z =3+4zττ τ 6 4z + 2 + +... z+3 τ null (z>>1) 2 4+5z +...τ null (z<<1) 16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 19 of 19