School of Aerospace Engineering
Conservation Equations
• Govern mechanics and thermodynamics of systems
• Control Mass Laws
– mass: can not create or destroy mass (e.g., neglect
nuclear reactions)
– momentum: Newton’s Law, F=ma
– energy: 1st Law of thermodynamics, dE=δQ- δW
– entropy: 2nd Law, dS=δQ/T+δPs
• Propulsion systems generally employ fluid flow
– need to write conservation laws in terms for
Control Volumes
Conservation Equations -1
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Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Reynolds Transport Theorem - RTT
• Provides general form for converting conservation
laws from control mass to control volumes
• Take arbitrary control volume (CV); can be moving
– mass and energy (“heat” Q and work W)
W&
can cross
n̂ m& A = ρur ⋅ nˆ
Q&
control surface
rel
r
u rel
(CS) boundaries
r
uCS
F
– forces also act
m&
on CS and
mass in CV Open CS
CV
ref. frame
Closed CS
r
u rel = velocity of material crossing CS relative to motion of CS
Conservation Equations -2
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
$(
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School of Aerospace Engineering
RTT Equation
• Take any extensive property B, that follows a
“conservation” law and its intensive version β (per
mass); can show
r
dB
d
=
ρβdV + ∫ ρβ(u rel ⋅ nˆ )dA
∫
dt CM dt CV
CS
Replace with appropriate Storage term Net flux of property leaving
Control Mass
(rate of increase
CV, carried by flow
Conservation Law
inside CV)
(outflow - inflow)
• Always leads to PICO relationship
Production + Input = Change (in time) + Output
Conservation Equations -3
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Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Mass Conservation
• If property of interest
is mass
• From RTT
B = m,
=
dB d( m)
=
=1
dm
dm
r
dB
d
=
ρβdV + ∫ ρβ(u rel ⋅ nˆ )dA
∫
dt CM dt CV
CS
d( m)
d
r
=
ρ1 dV + ∫ ρ1(u rel ⋅ nˆ )dA
∫
dt CM dt CV
CS
• CM:
d(m)
=0
dt CM
• CV:
Conservation Equations -4
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
0=
Integral Control
Volume Form of Mass
Conservation
r
d
ρdV + ∫ ρ(u rel ⋅ nˆ )dA
∫
dt CV
CS
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School of Aerospace Engineering
Simplified Mass Conservation
0=
r
d
ρdV + ∫ ρ(u rel ⋅ nˆ )dA
∫
dt CV
CS
• Uniform flow (at CS) - no variations across flow
r
∫ ρ(u rel ⋅ nˆ )dA = ∑ ρu rel A − ∑ ρu rel A
CS (t )
outlets
∑ m& − ∑ m&
=
outlets
• Add Steady-State
d
∫ ρdV = 0
dt CV
⇒
inlets
inlets
m& 1
∑ m& = ∑ m&
outlets
m& 3
m& 2
m& 1 + m& 2 = m& 3
inlets
PI=CO → Input = Output
Conservation Equations -5
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Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Conservation of Momentum
• Linear momentum
r
r
B = mu ,
r
r
=u
• RTT then gives
r
r
r r
d(mu)
d
=
ρudV + ∫ ρu (u rel ⋅ nˆ )dA
∫
dt CM dt CV
CS
“production”
• Use Newton’s Law
r
of
momentum
r
r
d(mu )
&
total force
= ∑ F = Pmomentum
acting on fluid
dt CM
r
r
r
r r
d
&
∑ FCV = Pmomentum = dt ∫ ρudV + ∫ ρu (u rel ⋅ nˆ )dA
CV
CS
Production =
Conservation Equations -6
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
Change
+
Out - In
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School of Aerospace Engineering
Force Terms
• Examine different forces that can act on matter in our
control volume
e.g., pressure,
e.g., gravity
r
r
r
shear,...
∑ Fon CV = ∑ F body + ∑ Fsurface
• Body forces
r
Fbody =
on CV
r
∫ ρfdV
on CS
r
with f = body force/mass,
i.e, acceleration
CV
• Surface forces
– free surfaces: not connected to solid body crossing
control surface
– connected surfaces: solid boundaries where there
are reaction forces
Conservation Equations -7
$(
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Surface Forces
• Free surfaces normal stress shear stress σr
r
Fsurface = −
∫ pnˆdA +
free
r
∫ σ shear dA
free
n̂
CV
pA pressure acts in pA
0 if inviscid
opposite
direction to n
• Connected solid surfaces
– force on fluid is reaction force
(inverse) of force on solid body
r
• Combine
r
Fsolid body −
on fluid
r
Fon fluid
r
Fon fluid = − Fon solid body
n̂
r
Fon strut
Integral Control Volume Form of Momentum Conservation
r
r
r
r r
d
∫ pnˆdA + ∫ σ shear dA + ∫ ρfdV = dt ∫ ρudV + ∫ ρu (u rel ⋅ nˆ )dA
open
Conservation Equations -8
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
open
CV
CV
CS
$(
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School of Aerospace Engineering
Conservation of Energy
• Energy: microscopic + macroscopic forms of energy
internal energy
1
B = Eo = E + Ekinetic = E + mu 2
2
u2
energy per mass→ = eo = e +
2
• RTT then gives
r
d(Etot )
d
=
ρetot dV + ∫ ρetot (u rel ⋅ nˆ )dA
∫
dt CM dt CV
CS
• Use 1st Law Thermodynamics for energy conservation
of control mass
Conservation Equations -9
$(
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
1st Law of Thermodynamics
• Differential form dE = δQ − δW
CM
in
out
dE
dt
=
CM
δQin δWout &
−
= Qin − W& out
dt
dt
• Into RTT
Q& in − W&out =
W&
Q&
m&
CV
r
d
ρeo dV + ∫ ρeo (u rel ⋅ nˆ )dA
∫
dt CV
CS
• But work is related to forces (F·x) acting on CV
– already examined some kinds of forces
– let’s relate them to work terms
Conservation Equations -10
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
$(
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School of Aerospace Engineering
Work and Forces
• Relationship
• Body forces
W&
r
r
dx r r
= F ⋅ = F ⋅u
dt
r r
Since we let work be
positive when done BY fluid
W&body = − ∫ ρf ⋅ u dV
• Fluid forces (stresses)
CV
“Flow Work”
∫ p(u ⋅ nˆ )dA
r
W& press =
CS open
=−
W& shear
r r
∫ σ ⋅ udA
• Reaction Forces
CS open
– lump into “useful” work term, e.g., shaft work
W& shaft
Conservation Equations -11
$(
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
School of Aerospace Engineering
Energy Conservation
• Combine into RTT result (neglect shear forces)
Q& in − W& shaft +
r r
∫ ρf ⋅ udV − ∫ p(u ⋅ nˆ )dA = dt ∫ ρeodV + ∫ ρeo (u rel ⋅ nˆ )dA
r
CV
CS
open
d
r
CV
Similar Form
CS
Stagnation
Enthalpy 2
u
ho = h +
2
p
p
= ρ eo + = ρho
ρ
ρ
r r
d
r r
r
Q& in − W& shaft + ∫ ρf ⋅ u dV − ∫ p(u − u rel ) ⋅ nˆ dA =
∫ ρeodV + ∫ ρho (urel ⋅ nˆ )dA
d
t
CV
CS
CV
CS
Combine flow work
and energy flux
ρeo + ρ
open
• For reference frame moving with control volume at
constant velocity and no body forces
r
d
Q& in − W& shaft =
∫ ρeo dV + ∫ ρho (u ⋅ nˆ )dA
0
PI=CO
Conservation Equations -12
Copyright © 2001 by Jerry M. Seitzman. All rights reserved.
In - Out
dt CV
Change
CS
Out - In, of energy in mass
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