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 Relative to CS 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 Inertial Relative to CS 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, 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 Changing the total momentum instantaneously contained within the control volume, and/or Changing the net flow rate of momentum leaving the control volume

HOW A ROCKET WORKS F Chemical 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 Thermal Energy Kinetic Energy

HOW AN AIRCRAFT ENGINE WORKS Chemical Energy Thermal Energy Kinetic Energy 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