student handout 12 2014
TRANSCRIPT
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Macroscopic energy balance 2: Bernoulli equation
CHEE 3363Spring 2014Handout 12
Reading: Fox 4.4 (Bernoulli section)
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Learning objectives for lecture
1. State the Bernoulli equation and give the conditions under which it can be used.
2. Apply the Bernoulli equation to solve problems.
2
- Particularly simple case: streamline CVFluid is steady, incompressible, frictionless077;-;8-+1)4+7674=5-*7=6,-,*AE7?;
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t
CV
dV +
CS
v dA = 0
VsA+ ((Vs + ds)(A+ dA)) = 0
(Vs + ds)(A+ dA) = VsA
VsdA+AdVs + dAdVs = 0
VsdA+AdVs = 0
Continuity equation
dsVs + dVs
A+ dA
Vs
p
p+ dp
x
y
z
g
streamline
!64AE7?1;
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Momentum equation along s 1
dsVs + dVs
A+ dA
Vs
p
p+ dp
x
y
z
g
streamline
Equation:
Surface force:
5
Fsurf,s + Fbody,s =
t
CV
us dV +
CS
usv dA
Fsurf,s = pA (p+ dp)(A+ dA) +
(p+
dp
2
)dA
= Adp1
2dpdA
pressure forces on end faces
pressure force acting in s direction on surface
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Momentum equation along s 2Body force:
75-6
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Momentum equation along s 3
7
Divide by A and neglect 2nd order terms:
dp
g dz = Vs dVs = d
(V
2s
2
)
d
(V
2s
2
)+
dp
+ g dz = 0
integrate
Adp1
2dp dA gAdz
1
2g dAdz = VsAdVs
Put terms together:
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v2
2+
p
+ gz = const
Bernoulli equation
$01;-9=)
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