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 Mo, and operating for a time t an electric thruster of jet speed C, such as to accomplish an equivalent(force-free) velocity change of Av the final nass Is if c=constant(consistent with constant power and thrust), then V=CIn M4=M。e and the propellant mass used M=Mol-e c The structural mass is comprised of a part Mso which is independent of power level, plus a part a p proportional to rated power p, where a 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 (3) and since m is also a constant in this case m=M/t Altogether, then Ms (4) The payload mass is M=M Combining the above expressions, 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 1 of 19
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 M0 , 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 M =M e f 0 ∆ (1) and the propellant mass used V - c M =M 1-e P 0 ∆ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (2) The structural mass is comprised of a part Mso which is independent of power level, plus a part α P proportional to rated power P, where α 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: 1 2 P= mc 2η i (3) and, since m i is also a constant in this case, m=M t. p i Altogether, then, P 2 s so M M =M + c 2 t α η (4) The payload mass is M =M -M L fs . Combining the above expressions
(1 M M。znt Stuhlinger!ll introduced a"characteristic velocity. 2nt whose meaning from the definition of a is that, if the powerplant mass above were to be accelerated by converting all of the electrical energy generated during t, it ould then reach the velocity vch Since other masses are also present v must clearly represent an upper limit to the achievable mission Av and is in any case a convenient yardstick for both Av and Figure 1 shows the shape of the curves of M+M versus c/with△V/asa M parameter. The existence of an optimum c in each case is apparent from the figure. This optimum c is seen to be near v, hence greater than Av. If -is taken to be a small quantity, expansion of the exponentials in(5)allows an approximate analytical expression for the optimum c (7) Figure 1 also shows that as anticipated the maximum Av for which a positive payload can be carried(with negligible M)is of the order of 0.8V. Even at this high AV, 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 vh. Equation (7)then shows that a high specific impulse In =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 av, 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 Av for a fixed 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 vh, hence low optimum specific impulse, and from Figure 2, small payload and large fuel fractions. This is then the sports car"regime References Ref.[1]:Stuhlinger,E.lon Propulsion For Space Flight. New York: Mc Graw-Hill Book Co., 1964. Ref. [2 W.H. Jet, Rocket, Nuclear, Ion and Electric Propulsion Theory and Design. New York: g,1968 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 2 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 2 of 19 ( ) 2 L - Vc - Vc so o o M M c =e - - 1-e M M 2t ∆ ∆ α η (5) Stuhlinger[1] introduced a “characteristic velocity” ch 2 t v = η α (6) whose meaning, from the definition of α is 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 vch . Since other masses are also present, vchmust 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 L so o M +M M versus ch c/v with ∆V vch 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 vch 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 1 1V c v - V- 2 24 v ∆ ≅ ∆ (7) Figure 1 also shows that, as anticipated, the maximum ∆V for which a positive payload can be carried (with negligible Mso ) is of the order of 0.8 vch . 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 vch . 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 vch . This regime was called quite graphically the “trucking” regime by Loh [2]. At the opposite end (short mission, heavy powerplant) we have a low vch , 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
We have, so far, regarded the efficiency n 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 cower loading(hence the jet velocity)is increased. For each thruster family can typically establish a connection between n and c alone. Thus, as we will see in detail later, n 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 is instructive to consider in some detail the particular case of the ion engine, both e to Equation(5)with n=n(c) in order to discover the best choice of c in each case 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 A o, and singly charged ions are assumed, the energy spent per ion is 1/2mc2+△中(m= Ion mass, e= electron charge), of which only 1/2m, c2 is useful The efficiency is the n- We should also include a factor of n z1 to account for power processing and other losses. We then have 16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Page 3 of 19
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 η η = c( ) 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 1 2m c + m = ion mass; e = electron charge i i ∆ φ , of which only 2 1 2m ci is useful. The efficiency is then 2 2 i c = 2e c + m η ∆φ (8) We should also include a factor of η0 ∠ 1to account for power processing and other losses. We then have 2 0 2 2 L c = c +v η η (9)
△V/Vch=0 .2 6 6811.2141.6182 Fig. 1 Payload Fraction vs Specific Impulse, Mission △ V and Characteristic Velocity For△v<<vch coP=vGch·△MV2:( M/MO)MA=(1-△WNa2 16.522, Space Propulsion ecture 2 Prof. Manuel Martinez-Sanchez Page 4 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 4 of 19
0zo-9<L00 5 Fig. 2 Optimized Mass Fractions were v, is a"loss velocity", equal to the velocity to which one ion would be accelerated by the voltage drop A o. Notice how this simple expression already indicates the importance of a high atomic mass propellant A o is insensitive to propellant choice, and so v, can be reduced if m is large. Equation(9) also shows the rapid loss of efficiency when c is reduced below v Using(9), we can rewrite(5)as 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 5 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 5 of 19 were vL 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 vL can be reduced if mi is large. Equation (9) also shows the rapid loss of efficiency when c is reduced below vL . Using (9), we can rewrite (5) as
△Mc2 here the definition of Vch(Equation(6) is now made using no instead of n. Once gain,only approximate expressions for Av/vch are feasible for the optimum c and mass fractions Normalizing all velocities by vch e obtain xn=√1+82-y (1 224√1+82 =1-21+82v+v2- 121+82 (13) M M-五+8224(1+ For 8=0, 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(8 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 Vch, 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 and c to vary in time if this is advantageous. Of course these variations are linked by the constancy of the 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 6 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 6 of 19 V 22 V - - L L c c so 2 o o ch M c +v M =e - - 1-e M M v ∆ ∆ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (10) where the definition of vch (Equation (6)) is now made using η0 instead of η. Once again, only approximate expressions for ∆V vch are feasible for the optimum c and mass fractions. Normalizing all velocities by vch : L ch ch ch c v v x ; v = ; = vvv ∆ ≡ δ (11) we obtain 2 2 OPT 2 v v x = 1+ - - + ... 2 24 1+ δ δ (12) 3 L so 2 2 2 o o MAX M M v + = 1 - 2 1+ v + v - + ... M M 12 1+ ⎛ ⎞ ⎜ ⎟ δ ⎝ ⎠ δ (13) 3 P 2 2 o M v 1v = - + ... M 24 1+ 1+ ⎛ ⎞ ⎜ ⎟ δ δ ⎝ ⎠ (14) For δ = 0,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 vch , 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:
m(t)c(t)=5F(t)c(t) Consider the rate of change of the inverse mass with time 1】) dM m (16a) Multiplying and dividing by F2=m c (16b) M2 mc2 where a= F/M is the acceleration due to thrust Inte 111 (17) P On the other hand, the mission Av is △V=adt (18) and is a prescribed quantity. We wish to select the function a(t) which will give a naximum M(Equation 17)while preserving this value of Av. 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 betweer these, we obtain M 1 (19) M M△V ntp 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 U M=M+M d 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 7 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 7 of 19 () () () () 1 1 2 P= m t c t = F t c t 2 2 η η i (15) Consider the rate of change of the inverse mass with time: 2 2 1 d M 1 dM m =- = dt dt M M ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ i (16a) Multiplying and dividing by 2 2 2 F =m c , i 2 2 2 2 1 d M F a = = dt 2 P M mc ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ η i (16b) where a=F M is the acceleration due to thrust. Integrating, t 2 f 0 0 11 1 - = a dt M M 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 Mf (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 M V 1+ 2 tP ∆ 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 M =M +M f Ls and
and introducing the characteristic velocity(Equation 6), we rewrite(19)as 1 (20) MM[(M,M5)+(△V/ M and select the value of s that will maximize L. This is easily found to be (21) hich, when used back in(21) gives and then These are, within the assumptions, exact expressions. they are to be compared to the approximate expressions in Figure 2 or Equation(12)-(14)with 8=0, 50=0 which were found to apply when C, and not a, was assumed constant. Clearly, the difference is noticeable only for v= Av/vch 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 mc m MM where(16 a, b)have been used. Hence, c=2n1 16.522, Space Propulsion ecture 2 Prof. manuel martinez-Sanchez Page 8 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 8 of 19 Ms P = α and introducing the characteristic velocity (Equation 6), we rewrite (19) as ( ) ( ) L s 2 0 0 s 0 ch M M 1 = -1 M M M M + 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 0 ch ch OPT M V V = 1- M vv ⎛⎞ ⎛ ⎞ ∆ ∆ ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ (21) which, when used back in (21) gives 2 L 0 ch MAX M V = 1- M v ⎛⎞ ⎛ ⎞ ∆ ⎜⎟ ⎜ ⎟ ⎝⎠ ⎝ ⎠ (22) and then P 0 ch OPT M V = M v ⎛ ⎞ ∆ ⎜ ⎟ ⎝ ⎠ (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 v= V v ∆ ch 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 M 2P M η i i where (16 a, b) have been used. Hence, 2 P1 c = a M η
and since by(16b),. varies linearly with time, so will C At the final time, when M=M C, ==2n M /a=Ahm, Ms+M and,from(21),(22), M+M so that The rate of change of c follows from that of 1/M(Equation 16)as ac so that altogether, if t' represents some intermediate time, while t is the final time (used in vh), c(t)=vah-a(t-t) This varies between c=Vs-△Vatt=0 an c=vat t= t The approximate result copr =Vch-5AV 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 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 16.522, Space Propulsion Prof. manuel martinez-Sanchez Page 9 of 19
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 M=Mf , 2 s ch s f f sL f 2 P 2M v M c= = = aM V M + M V M t η η α ∆ ∆ and, from (21), (22), s s L ch M V = M +M v ∆ so that f ch c =v (24) The rate of change of c follows from that of 1 M (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 vch ), () ( ) ch c t' = v - a t - 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 c v- 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 c =v - V-V 0 abs ch ∆ ( ) 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 2 1Stage 1 vτ ch
We wish to minimize subject to a given power(constant in time) 1 mc 1 Fc nd te write(1)as which eliminates thrust Next, treat (3)as a dynamic constraint, and append it to the cost through a time dependent Lagrange multiplier i(t) Define the hamiltonian H=2nP m dt or, using(2) H 2nP( 2nP, dm (6) and minimize(unconstrained )the integral J Hdt To do this, perturb about the optimum solution 8+2a+2m+nmj=0 dt d Integrate last term by parts: 16.522, Space Propulsion Lecture 2 Prof. manuel martinez-Sanchez Page 10 of 19
16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 10 of 19 We wish to minimize tf 0 F V = dt m ⎛ ⎞ ∆ ∫ ⎜ ⎟ ⎝ ⎠ (1) subject to a given power (constant in time) 2 1 m c 1 Fc p= = 2 2 η η i (2) and to dm m=- dt i (3) write (1) as tf 0 2 P V = dt mc η ∆ ∫ (4) which eliminates thrust. Next, treat (3) as a dynamic constraint, and append it to the cost through a timedependent Lagrange multiplier λ (t). Define the Hamiltonian 2 P dm H = - m+ mc dt η ⎛ ⎞ λ ⎜ ⎟ ⎝ ⎠ i (5) or, using (2), 2 2 P 2 P dm H= - + mc dt c η η ⎛ ⎞ λ ⎜ ⎟ ⎝ ⎠ (6) and minimize (unconstrained) the integral tf ∫0 Hdt .To do this, perturb about the optimum solution: t t f f 0 0 H H H dm Hdt = c + m + dt = 0 c m dt dm dt ⎡ ⎛⎞ ∂∂ ∂ ⎤ δ δ δ δ ∫ ∫ ⎢ ⎜ ⎟⎥ ⎣ ⎝⎠ ∂ ∂ ⎛ ⎞ ⎦ ∂ ⎜ ⎟ ⎝ ⎠ ( ) d m dt δ Integrate last term by parts: