Problem 3.12
[Difficulty: 4]
Given:
Model behavior of seawater by assuming constant bulk modulus
Find:
(a) Expression for density as a function of depth h.
(b) Show that result may be written as
ρ
=
ρ
o
+ bh
(c) Evaluate the constant b
(d) Use results of (b) to obtain equation for p(h)
(e) Determine depth at which error in predicted pressure is 0.01%
Solution:
From Table A.2, App. A:
SG
o
1.025
=
E
v
2.42 GPa
⋅
3.51
10
5
×
psi
⋅
=
=
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure  h is positive downwards)
(Definition of Bulk Modulus)
E
v
dp
d
ρ
ρ
=
Then
dp
ρ
g
⋅
dh
⋅
=
E
v
d
ρ
ρ
⋅
=
or
d
ρ
ρ
2
g
E
v
dh
=
Now if we integrate:
ρ
o
ρ
ρ
1
ρ
2
⌠
⎮
⎮
⎮
⌡
d
0
h
h
g
E
v
⌠
⎮
⎮
⌡
d
=
After integrating:
ρ
ρ
o
−
ρ ρ
o
⋅
g h
⋅
E
v
=
Therefore:
ρ
E
v
ρ
o
⋅
E
v
g h
⋅
ρ
o
⋅
−
=
and
ρ
ρ
o
1
1
ρ
o
g
⋅
h
⋅
E
v
−
=
(Binomial expansion may
be found in a host of
sources, e.g.
CRC
Handbook of
Mathematics
)
Now for
ρ
o
g
⋅
h
⋅
E
v
<<1, the binomial expansion may be used to approximate the density:
ρ
ρ
o
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Lear
 Fluid Mechanics

Click to edit the document details