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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

d
dV    U  nˆ dS  0

dt CV
S
Relative to CS
 
d
dV    U  U CS  nˆdS  0

dt CV
S
•
•
•
•
•
Inertial
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
MOMENTUM EQUATION: NEWTONS 2nd LAW

 

d
UdV   U U  nˆ dS   F

dt CV
Inertial
S
Relative to CS

  

d
UdV   U U  U CS  nˆdS   F

dt CV
S

•
•
•
•

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:



 F    pnˆ dS  t dS   gdV   F
ext
S
•
S
CV
–
Notice that sign on pressure, pressure always acts inward
–
Shear stress tensor, t, 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
HOW ALL ROCKETS WORKS
Chemical
Energy
F
Rocket Propulsion (class of jet propulsion) that
produces thrust by ejecting stored matter
•
Thermal
Energy
•
•
•
Kinetic
Energy
•
F  m eVe  Pe  Pa Ae
F  m eVe
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”)?
SOLID ROCKET MOTOR ANALYSIS: MASS CONSERVATION
http://www.fofweb.com/Subscription/Science/Sc/ffdsptech2530b.jpg
• 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?
SOLID ROCKET MOTOR CROSS-SECTION
http://www.aerospaceweb.org/question/propulsion/rocket/solid-rocket2.jpg