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”)?

ee

eaeee

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