CE150 - CSU, Chico
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Transcript CE150 - CSU, Chico
CE 150
Fluid Mechanics
G.A. Kallio
Dept. of Mechanical Engineering,
Mechatronic Engineering &
Manufacturing Technology
California State University, Chico
CE 150
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Finite Control
Volume Analysis
Reading: Munson, et al.,
Chapter 5
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Conservation of Mass
• The conservation of mass principle
states that
Dmsys
Dt
0 , where msys dV
sys
• Applying the RTT with B = m and
b = 1:
dV
t cv
cs
(V nˆ )dA 0
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Special Cases
• Moving control volume - replace V
with relative velocity W = V-Vcv ,
where CV is moving at constant
velocity Vcv
• Deforming control volume – replace
V with relative velocity W + Vcs ,
where CS is moving at constant
velocity Vcs
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Newton’s 2nd Law:
Linear Momentum Eqn.
• Newton’s 2nd law for a system is
D
momentum Fsys
Dt
D
VdV Fsys
Dt sys
• Applying the RTT with B = mV
(linear momentum) and b = V :
VdV
cv
t
cs
V (V nˆ )dA
CE 150
Fcv
5
Newton’s 2nd Law:
Linear Momentum Eqn.
VdV
t cv
cs
V (V nˆ )dA
Fcv
• The first LHS term represents the
time rate of change of linear
momentum of the control volume
• The second LHS term represents the
net outflow of linear momentum
across the control surface
• The RHS term represents all body
and surface forces acting on the
control volume
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Newton’s 2nd Law:
Linear Momentum Eqn.
• Important notes regarding the linear
momentum equation:
– this is a vector equation; it can have
components in as many as three
orthogonal directions
– the first LHS term is zero for steady
flow through a nondeforming CV
– the V • n product determines whether
there is an inflow or outflow of linear
momentum
– the integral operations in the second
LHS term are simplified if the flow is
incompressible and the velocity is
uniform over the control surface
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Newton’s 2nd Law:
Linear Momentum Eqn.
• Important notes, continued:
– the CV is normally chosen to be
perpendicular to inflows and outflows;
this further simplifies the analysis
– if an anchoring force is sought, then the
CV should contain the fixture that
imposes the anchoring force; this
normally allows the atmospheric pressure
to cancel out over all surfaces
– the external forces in the RHS term
typically include anchoring or reaction
forces, pressure forces, and fluid weight
forces
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Moment-of-Momentum
Equation
• This material is omitted (sections
5.2.3, 5.2.4, and 5.3.5)
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The Energy Equation
• The conservation of energy equation,
or 1st Law of Thermodynamics, for a
(closed) system is
Esys Q W ,
or
Esys Qin Qout Win Wout
or
DE sys
Dt
Q in Q out Win Wout
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The Energy Equation
• The total energy stored in the system,
E, consists of internal, kinetic, and
potential energy:
E U 12 mV 2 mgz
or
1 2
e u 2 V gz
whe re
E edV
sys
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The Energy Equation
• Applying the RTT with B=E or b=e:
DE D
edV
Dt Dt sys
edV e (V nˆ )dA
cs
t cv
• The first RHS term represents the
time rate of change of the energy
stored within the CV
• The second term represents the net
loss of energy by fluid flow across
the control surface
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The Energy Equation
• Combining terms,
ˆ
e
d
V
e
(
V
n
)
dA
Q
W
net in
net in
cs
t cv
cv
• Work is transferred across the CV
boundary in several ways; when
analyzing fluid machinery such as
pumps, fans, compressors, turbines,
etc., rotating shaft work is common
• Heat transfer occurs by the modes of
of conduction, convection, and
radiation under the influence of a
temperature difference
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The Energy Equation
• Work also occurs where flow enters
and leaves the CV; this is associated
with the pressure and is often called
flow work; since it acts along the
control surface, it can be combined
with the flow of stored energy term:
e u p / 12 V 2 gz
or
1 2
e h 2 V gz
• h is the fluid enthalpy per unit mass
(J/kg or Btu/slug)
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The Energy Equation
• Combining terms once again, we
obtain a useful form of the energy
equation for a control volume
experiencing fluid flow:
edV
t cv
2
1
u p / 2 V gz (V nˆ )dA
cs
Qnet in Wshaft
net in cv
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