IV Rocket Propulsion Systems C Vehicle Acceleration by J M Seitzman for AE 4451 Jet and Rocket Propulsion Seitzman Vehicle Acceleration 1 Copyright 2008 2011 by Jerry M Seitzman All rights reserved AE4451 Propulsion Accelerating Rocket Rocket typically used to accelerate a vehicle du dt u constant Velocity Increment u ufinal uinitial net increase in vehicle speed Lots of propellant carried on board so mass of vehicle also changes with time dm dt Seitzman Vehicle Acceleration 2 Copyright 2008 2011 by Jerry M Seitzman All rights reserved m constant AE4451 Propulsion Rocket Acceleration Rocket in gravity field with thrust aligned with vehicle motion momentum conservation Fsolid body on fluid u D pa pn dA sheardA fdV open open g mgcos CV d u rel n dA u dV u dt CV CS ue u pe m y inertial ref frame x Steps Skipped du 1 m ue pe pa Ae g cos D dt m Seitzman Vehicle Acceleration 3 Copyright 2008 2011 by Jerry M Seitzman All rights reserved AE4451 Propulsion Rocket Acceleration du 1 m ue pe pa Ae g cos D m dt m m ueq du m ueq g cos D m dt m dm m dt du dm m ueq g cos D m dt u D pa g mgcos ue u pe m y inertial ref frame x Describes the velocity change of the rocket versus time Seitzman Vehicle Acceleration 4 Copyright 2008 2011 by Jerry M Seitzman All rights reserved AE4451 Propulsion Velocity Increment dm ueq g cos D m dt To find u integrate du m In general all the variables can change with time ueq g D m simplest case ueq constant and g D negligible u final m final dm 0 du u ueq uinitial u ueq ln minitial m final minitial minital ueq ln Ideal Rocket Eqn m final Mass R mo Initial mass mb Burnout mass mo mp Ratio ueq u I sp ge ln R I sp ge Seitzman Vehicle Acceleration 5 Copyright 2008 2011 by Jerry M Seitzman All rights reserved m Ideal case u fn mass change Isp not dependent on burn time AE4451 Propulsion Rocket Equation Gravity and Drag Losses In more general case du m ueq m dt g cos dt D m dt other terms u u propulsion u gravity udrag usteering if ueq still constant ueq ln R loss 1 2 comes from having to lift D CD u Aref mass against a gravity field 2 reduce by dropping m early want low u time where drop propellant fast density is high low alt short burn times lower CD CD M shape drop dead mass also want to minimize staging vehicle stresses Seitzman Vehicle Acceleration 6 Copyright 2008 2011 by Jerry M Seitzman All rights reserved u loss ue AE4451 Propulsion Drag Coefficient Example drag coefficient for a slender shaped rocket also depends on AOA 1 CD Wave drag increasing 0 5 CD nearly constant in hypersonic region 0 0 Adapted from Space Propulsion Analysis and Design 1995 Fig 2 13 Seitzman Vehicle Acceleration 7 Copyright 2008 2011 by Jerry M Seitzman All rights reserved 1 2 3 4 Mach Number AE4451 Propulsion LEO Velocity Budgets Vehicle Orbit hp ha km inclination deg Ariane A 44L 170 170 70 7802 1576 135 38 413 9138 Atlas l 149 607 27 4 7946 1395 110 167 375 9243 Delta 7925 175 319 33 9 7842 1150 136 33 347 8814 Space Shuttle 196 278 28 5 7794 1222 107 358 395 9086 Saturn V 176 176 28 5 7798 1534 40 243 348 9267 7896 1442 156 65 157 463 28 6 From Table 2 10 in Space Propulsion Analysis and Design Humble Henry and Larson McGraw Hill 1995 Ascent varies between 8 8 and 9 3 km s for these selected launch vehicles Negative sign indicates beneficial effect of rotation The third stage of Ariane 44L uses a continuous burn into a geosynch transfer orbit arbitrarily terminated burn at 170 km to give better comparison with other vehicles Injection occurs at 111 km An additional u 144 m s is required to circularize at apogee 352 9207 Titan IV Centaur Seitzman Vehicle Acceleration 8 Copyright 2008 2011 by Jerry M Seitzman All rights reserved uLEO ugrav udrag usteering urot u ui AE4451 Propulsion