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