Control volume analysis - My FIT (my.fit.edu)
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Transcript Control volume analysis - My FIT (my.fit.edu)
MAE 5360: Hypersonic Airbreathing Engines
Integral Forms of Mass and Momentum Equations
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
Kinematic Properties: Two ‘Views’ of Motion
1. Lagrangian Description
– Follow individual particle trajectories
– Choice in solid mechanics
– Control mass analyses
– Mass, momentum, and energy usually formulated for particles or systems of fixed
identity
• ex., F=d/dt(mV) is Lagrangian in nature
2. Eulerian Description
– Study field as a function of position and time; not follow any specific particle paths
– Usually choice in fluid mechanics
– Control volume analyses
– Eulerian velocity vector field:
V r , t V x, y, z, t u x, y, z, t iˆ vx, y, z, t ˆj wx, y, z, t kˆ
–
Knowing scalars u, v, w as f(x,y,z,t) is a solution
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
Integral vs. Differential Form
• Integral form of mass conservation
d
dV V nˆ dS 0
dt V
S
AdV A nˆdS
V
• Apply Divergence (Gauss’) Theorem
S
V nˆdS V dV
S
• Transform both terms to volume integrals
V
dV 0
V
t
V
V 0
t
• Results in continuity equation in the form of
a partial differential equation
• Applies to a fixed point in the flow
• Only assumption is that fluid is a continuum
– Steady vs. unsteady
– Viscous vs. inviscid
– Compressible vs. incompressible
Summary: Incompressible vs. Constant Density
D
V
0
Dt
V 0
t
• Two equivalent statements of conservation
of mass in differential form
• In an incompressible flow
u v w
D
Dt
x
y
z
• Says particles are constant volume, but not
V 0
necessarily constant shape
D
• Density of a fluid particle does not change
0
as it moves through the flow field
Dt
• Incompressible: Density may change within the flow field but may not change
along a particle path
• Constant Density: Density is the same everywhere in the flow field
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
Application to Rocket Engines
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
Application to Airbreathing Engines
Chemical
Energy
Thermal
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
Kinetic
Energy