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Problem 3.44
[Difficulty: 3]
Given:
Atmospheric conditions at ground level (
z
= 0) in Denver, Colorado are
p
0
= 83.2 kPa,
T
0
= 25°C.
Pike's peak is at elevation z = 2690 m.
Find:
p
/
p
0
vs
z
for both cases.
Solution:
Governing Equations:
dp
dz
ρ
−
g
⋅
=
p
ρ
R
⋅
T
⋅
=
Assumptions:
(1) Static fluid
(2) Ideal gas behavior
(a) For an incompressible atmosphere:
dp
dz
ρ
−
g
⋅
=
becomes
pp
0
−
0
z
z
ρ
g
⋅
⌠
⎮
⌡
d
−
=
or
0
ρ
0
g
⋅
z
⋅
−
=
p
0
1
gz
⋅
RT
0
⋅
−
⎛
⎜
⎝
⎞
⎟
⎠
⋅
=
(1)
At
z
2690 m
⋅
=
p
83.2 kPa
⋅
1
9.81
m
s
2
⋅
2690
×
m
⋅
kg K
⋅
287 N
⋅
m
⋅
×
1
298 K
⋅
×
Ns
2
⋅
kg m
⋅
×
−
⎛
⎜
⎝
⎞
⎟
⎠
×
=
p
57.5 kPa
⋅
=
(b) For an adiabatic atmosphere:
p
ρ
k
const
=
ρρ
0
p
p
0
⎛
⎜
⎝
⎞
⎟
⎠
1
k
⋅
=
dp
dz
ρ
−
g
⋅
=
becomes
dp
ρ
0
−
p
p
0
⎛
⎜
⎝
⎞
⎟
⎠
1
k
⋅
g
⋅
dz
⋅
=
or
1
p
1
k
dp
ρ
0
g
⋅
p
0
1
k
−
dz
⋅
=
But
p
0
p
p
1
p
1
k
⌠
⎮
⎮
⎮
⎮
⌡
d
k
k1
−
0
−
()
−
k
⋅
=
hence
k
−
p
−
k
p
0
−
k
−
⎛
⎜
⎝
⎞
⎟
⎠
⋅
ρ
0
g
⋅
p
0
1
k
−
g
⋅
z
⋅
=
Solving for the pressure ratio
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 Fall '07
 Lear
 Fluid Mechanics

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