mae 4262: rockets and mission analysis conservation equations and examples mechanical and aerospace...
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MAE 4262: ROCKETS AND MISSION ANALYSIS
Conservation Equations and Examples
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
CONSERVATION OF MASS
• This is a single scalar equation
– Velocity doted with normal unit vector results in a scalar
• 1st Term: Rate of change of mass inside CV
– If steady d/dt( ) = 0
– Velocity, density, etc. at any point in space do not change with time, but may vary from point to point
• 2nd Term: Rate of convection of mass into and out of CV through bounding surface, S
• 3rd Term (=0): Production or source terms
• Last equation arises from vector equation: Vintertial = Vrelative + Vcontrol surface
0ˆ dSnUdVdt
d
CV S
0ˆ dSnUUdVdt
d
CV S
CS
Relative to CS
Inertial
MOMENTUM EQUATION: NEWTONS 2nd LAW
FdSnUUdVUdt
d
CV S
ˆ
FdSnUUUdVUdt
d
CV S
CS
ˆ
• This is a vector equation in 3 directions
• 1st Term: Rate of change of momentum inside CV or Total (vector sum) of the momentum of all parts of the CV at any one instant of time
– If steady d/dt( ) = 0
– Velocity, density, etc. at any point in space do not change with time, but may vary from point to point
• 2nd Term: Rate of convection of momentum into and out of CV through bounding surface, S or Net rate of flow of momentum out of the control surface (outflow minus inflow)
• 3rd Term:
– Notice that sign on pressure, pressure always acts inward
– Shear stress tensor, , drag
– Body forces, gravity, are volumetric phenomena
– External forces, for example reaction force on an engine test stand
• Application of a set of forces to a control volume has two possible consequences
1. Changing the total momentum instantaneously contained within the control volume, and/or
2. Changing the net flow rate of momentum leaving the control volume
ext
CVSS
FdVgdSdSnpF
ˆ
Relative to CSInertial
HOW ALL ROCKETS WORKSRocket Propulsion (class of jet propulsion) that produces thrust by ejecting stored matter
• Propellants are combined in a combustion chamber where chemically react to form high T&P gases
• Gases accelerated and ejected at high velocity through nozzle, imparting momentum to engine
• Thrust force of rocket motor is reaction experienced by structure due to ejection of high velocity matter
• Same phenomenon which pushes a garden hose backward as water flows from nozzle, gun recoil
• Examples to come in next lecture: mass, momentum and derivation of Rocket Equation
QUESTION (Hill and Peterson, Chapter 1, p.3):
Could a jet or rocket engine exert thrust while discharging into a vacuum (with not atmosphere to “push against”)?
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VmF
APPVmF
FChemicalEnergy
ThermalEnergy
KineticEnergy
SOLID ROCKET MOTOR ANALYSIS: MASS CONSERVATION
• How does the exhaust velocity vary with,– Changes in density as the solid propellant burns?– Regression velocity of the solid grain?– Cross-sectional area of the grain relative to the exit area?
http://www.fofweb.com/Subscription/Science/Sc/ffdsptech2530b.jpg
SOLID ROCKET MOTOR CROSS-SECTION
http://www.aerospaceweb.org/question/propulsion/rocket/solid-rocket2.jpg
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