Rocket Equation and Losses

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					MAE 4262: ROCKETS AND MISSION ANALYSIS

                Rocket Equation and Losses



                     January 22, 2008

      Mechanical and Aerospace Engineering Department
               Florida Institute of Technology

                        D. R. Kirk
ROCKET EQUATION: IMPORTANT TRENDS
TYPICAL DV MISSION REQUIREMENTS




                      http://www.strout.net/info/science/delta-v/intro.html
DV CAPABILITY FOR VARIOUS ROCKETS




 REF: Space Propulsion Analysis and Design, by Humble, Henry and Larson
TYPICAL PERFORMANCE PARAMETERS (T and Isp)
                                       GRAVITY
•   Remember that gravity on Earth (~ 9.81 m/s2) may be calculated fundamentally
     – Average radius of the Earth ~ 6,378 km or 3,963 miles
     – Mass of the Earth ~ 5.9742x1024 kg
                                                 GMm
                                      F  mg 
                                                  R2
                                                       2
                                              Re 
                                              R h
                                      g  ge      
•   Some typical values for Earth:            e   
     – High power amateur model rocket ~ 100,000 ft, 30.5 km, 19 miles
         • g/ge = 99%
     – Shuttle in LEO (altitude of 300 km, 186 miles)
         • g/ge = 91%
     – Satellite in GEO (altitude of 42,000 km, 26,000 miles)
         • g/ge = 1.7%

•   Note that the radius of the moon is about 1,737 km and mass is 7.36x1022 kg
     – So g on the surface of the moon is about 1.62 m/s2
                        COMPARISON OF GRAVITY, DENSITY AND PRESSURE
                                     VERSUS ALTITUDE
                                    1.00
                                                                      Gravity                                 Gravity
                                    0.90                                                                      Density
                                                                                                              Pressure

                                    0.80


                                    0.70
Percent of Sea Level Static Value




                                    0.60


                                    0.50


                                    0.40
                                               Pressure
                                    0.30
                                               Density
                                    0.20


                                    0.10


                                    0.00
                                           0     100      200   300   400       500        600   700   800   900     1000
                                                                            Altitude, km
                         COMPARISON OF GRAVITY, DENSITY AND PRESSURE
                                      VERSUS ALTITUDE
                                    1.00


                                    0.90                                          Gravity

                                    0.80            Density

                                    0.70
Percent of Sea Level Static Value




                                                 Pressure
                                    0.60
                                                                                             Shuttle
                                    0.50


                                    0.40


                                    0.30
                                                                                            LEO
                                    0.20
                                               Gravity
                                    0.10       Density
                                               Pressure
                                    0.00
                                           1                  10                     100               1000
                                                                   Altitude, km
TYPICAL DRAG VARIATION FOR ROCKETS
DRAG: SUPERSONIC MISSILE EXAMPLE
       COMMENTS: LAUNCH FROM SURFACE OF EARTH
•   To get to orbit (or to escape), direction of travel must be parallel to Earth’s surface (not
    perpendicular)

•   We launch vertically off the surface of the Earth, WHY?
     – Gravity
         • When rocket is vertical, gravity is acting against T and V
     – Drag
         • V2 dependence: Drag ↑ as rocket accelerates
              – Large effect in lower atmosphere
              – Acceleration of vehicle is almost constant even though mass is changing
         • Density dependence: r ↓ very rapidly in atmosphere (r/rS.L. ~ 1% at 100,000 ft)

•   All rocket pass through condition of maximum dynamic pressure (MAX Q)
     – Many rockets stay vertical through this part
     – Get through atmosphere as quickly as possible
     – BUT before rocket really starts to speed up
                            COMMENTS: LAUNCHERS
•   Need certain velocities to get to space (and stay in space), escape, insertion, transition
    velocities, etc. → give DV requirements

•   Don’t want to carry fuel (heavy fuel is working against you)
     – Burn fuel early in flight → high accelerations, V2 ↑
     – Atmosphere is counter argument: drag, dynamic pressure

•   Why not launch horizontal?
     – Less gravity loss
     – Drag loss is high, more time in atmosphere
     – Lots of structural stress
     – Launch might look different on moon

•   Vertical launch segment:
     – Get out of dense atmosphere quickly, but still at relatively low speed
     – Don’t spend too much time here (vertical segment contributes nothing to eventual
        vertical orbital velocity)
     – Highest gravity losses, but sustain them to get lower density then really increase DV
  Variation in air density (r), velocity (V), altitude (h),
and dynamic pressure (q) during a Space Shuttle launch

				
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