Basics of Fluid Mechanics, 2014a

82 CHAPTER 4. FLUIDS STATICS
This constant, B T /g ρ 0 , is a typical length
of the problem. Additional discussion
will be presented in the dimensionless issues
chapter (currently under construction).
The solution becomes
)
= z (4.44)
P −P
B T
gρ 0
(e
0
B T
− 1
Or in a dimensionless form
P −P0
B T
March 11, 2008
gρ 0 z
B T
(
e
Density in Liquids
)
P −P 0
B T
− 1
= zgρ 0
B T
(4.45)
Fig. -4.10. Hydrostatic pressure when there is
compressibility in the liquid phase.
The solution is presented in equation (4.44) and is plotted in Figure 4.10. The solution
is a reverse function (that is not P = f(z) but z = f (P)) it is a monotonous function
which is easy to solve for any numerical value (that is only one z corresponds to any
Pressure). Sometimes, the solution is presented as
P
P 0
= B T
P 0
B T
ln
( ) gρ0 z
+1
B T
+1 (4.46)
An approximation of equation (4.45) is presented for historical reasons and in
order to compare the constant density assumption. The exponent can be expanded as
⎛
⎞
piezometric
corrections
pressure
{ }} {
⎝ { }} { ( ) 2
B T P − P0
(P − P 0 ) +
+ B ( ) 3
T P − P0
⎠ = zgρ 0 (4.47)
+ ···
2
6 B T
It can be noticed that equation (4.47) is reduced to the standard equation when the
normalized pressure ratio, P/B T is small (