chap4_57_67 - AAE 439 Velocity Definitions Exit Velocity ve...

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Unformatted text preview: AAE 439 Velocity Definitions Exit Velocity ve The physical velocity of the exhaust flow at the nozzle exit plane. Effective Exhaust Velocity c A theoretical exhaust velocity that takes into account the pressure differences at the exit plane. T = mv e + (p e - p a )A e = mc = m I sp g 0 (p e - p a )A e T c = I sp g 0 = = v e + m m Characteristic Velocity c* Another theoretical velocity that indicates the efficiency and energy released by a combustor/propellant combination. c* = p0 A* m Ch4 48a 57 AAE 439 Velocity Definitions Characteristic Velocity This form of characteristic velocity is useful because we can use it and the definition of the thrust coefficient CF to give an alternative definition for the effective exhaust velocity c as follows: c = c* CF With this formula the effective exhaust velocity, which helps calculate missionlevel parameters like thrust and specific impulse, can be calculated from a pair of system-level parameters that only depend on your choice of propellant and the physical details of your rocket engine. Ch4 58 48b AAE 439 ROCKET PROPULSION SYSTEM Analysis of Thrust Equation: Thrust depends on altitude due to pa. For launch vehicles, pressure thrust can change a large amount over the firing duration due to large atmospheric pressure changes. Maximum thrust occurs in vacuum, since pa=0: For optimum expansion ratio (pa= pe), thrust is due to momentum thrust only: Specific Impulse: Effective Exhaust Velocity: Tvac = m v e + p e A e (max) Topt (design ) = m ve v e (p e - p a )A e F I sp = = + g 0m g 0 g 0m (p e - p a )A e c = I sp g 0 = v e + m This relation indicates that the effective exhaust velocity increases above the exit velocity if the pressure thrust is positive. Ch4 59 AAE 439 Exhaust Plume Pattern Perfectly Expanded Flow: pe = pa Underexpanded Flow: pe > pa We obtain no pressure thrust. We obtain positive pressure thrust. Overexpanded Flow: pe < pa We obtain negative pressure thrust. Vulcain engine, hot-firing at DLR P5 ground test facility RL-10 engine w/o NE Saturn 1B with 8 H1 engines, Apollo 7 mission Ch4 60 AAE 439 EXAMPLE Ch4 61 AAE 439 4.6 OPERATION OF NOZZLES Ch4 62 AAE 439 Effect of Back Pressure Ch4 63 AAE 439 Effect of Back Pressure Region I Condition 2 is dividing line between regime I ad II, where Mach Number at throat is unity. Region II Normal shock appears downstream of throat. Downstream of shock is subsonic deceleration. As back pressure is lowered, shock moves down nozzle. Condition 4 marks shock at the exit plane of nozzle. Region III Flow within entire nozzle is supersonic. Compression occurs outside the nozzle in for of oblique shocks, pe<pa. Region IV Condition 6 is termed the design condition for a nozzle under supersonic conditions, pe=pa. Expansion occurs outside the nozzle in form of oblique expansion waves, pe>pa. Flow rate depends on the back pressure only in Regime I. Flow rate is independent of back pressure for Regimes II, III and IV. Ch4 64 AAE 439 Exhaust Plume Pattern Perfectly Expanded Flow: pe = pa Underexpanded Flow: pe > pa We obtain no pressure thrust. We obtain positive pressure thrust. Overexpanded Flow: pe < pa We obtain negative pressure thrust. Vulcain engine, hot-firing at DLR P5 ground test facility RL-10 engine w/o NE Saturn 1B with 8 H1 engines, Apollo 7 mission Ch4 65 AAE 439 Effect of Back Pressure Ch4 66 AAE 439 NOZZLE FLOW SEPERATION T = m v e + pe - pa A e Momentum-thrust (MT) term largest for pe=pa Thrust Equation ( ) Qualitative Analysis: Pressure-thrust (PT) term largest for pa=0 Practical Considerations: pe=0 requires nozzle of Infinite Expansion Ratio (maximizing MT in vacuum), Rocket propulsion of launch vehicles is performed in atmosphere ( pa 0). Condition for Maximum Thrust: ! dT =0 d p e p0 ( ) pe = pa Thus, max. thrust for a given altitude is achieved by a nozzle that expands flow to ambient pressure at that altitude. T = A * p0 2 2 - 1 + 1 +1 -1 -1 pe 1 - + pe - pa A e p 0 ( ) Ch4 67 ...
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This document was uploaded on 01/15/2012.

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