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 1 st 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 2 nd Term: Rate of convection of mass into and out of CV through bounding surface, S 3 rd Term (=0): Production or source terms Last equation arises from vector equation: V intertial = V relative + V control surface Relative to CS Inertial

MOMENTUM EQUATION: NEWTONS 2 nd LAW This is a vector equation in 3 directions 1 st 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 2 nd 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) 3 rd 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 Relative to CS Inertial

HOW ALL ROCKETS WORKS 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 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”)? F Chemical Energy Thermal Energy Kinetic Energy

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?

SOLID ROCKET MOTOR CROSS-SECTION