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# lecture2 - 16.522 Space Propulsion Prof Manuel...

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16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 1 of 1 9 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 P α 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: 2 1 P= mc 2 η i (3) and, since m i is also a constant in this case, p m=M t. i Altogether, then, 2 P ss o M M=M + c 2t α η (4) The payload mass is Lf s M=M-M . Combining the above expressions,

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16.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 2 of 19 () 2 -Vc so L oo M M c =e - - 1-e MM 2 t ∆∆ α η (5) Stuhlinger [1] introduced a “characteristic velocity” ch 2t 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 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 Ls o 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 11 V cv - V - 22 4 v ≅∆ (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.
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## This note was uploaded on 11/07/2011 for the course AERO 16.512 taught by Professor Manuelmartinez-sanchez during the Fall '05 term at MIT.

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lecture2 - 16.522 Space Propulsion Prof Manuel...

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