Transcript Conservation of mass - Florida Institute of Technology

MAE 4261: AIR-BREATHING ENGINES
Integral Forms of Mass and Momentum Equations
January 17, 2012
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
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 A ROCKET WORKS
Chemical
Energy
F
Thermal
Energy
Rocket 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
Kinetic
Energy
F  m eVe  Pe  Pa Ae
F  m eVe
HOW AN AIRCRAFT ENGINE WORKS
Chemical
Energy
Thermal
Energy
Kinetic
Energy
F  m eVe  m oVo  Pe  Pa Ae
F  m Ve  Vo 
• Flow through engine is conventionally called THRUST
– Composed of net change in momentum of inlet and exit air
• Fluid that passes around engine is conventionally called DRAG
Excellent website for how internal components work: http://www.geae.com/education/index.html