Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 79
The difference in the pressure of two different
liquids is measured by this manometer. This idea is
similar to “magnified” manometer but in reversed.
The pressure line are the same for both legs on line
ZZ. Thus, it can be written as the pressure on left
is equal to pressure on the right legs (see Figure
4.9).
right leg
left leg
{ }} { { }} {
P 2 − ρ 2 (b + h) g = P 1 − ρ 1 a − ρh) g (4.26)
Rearranging equation (4.26) leads to
1
Z
a
Z
h
b
2
P 2 − P 1 = ρ 2 (b + h) g − ρ 1 ag− ρhg (4.27)
For the similar density of ρ 1 = ρ 2 and for a = b
equation (4.27) becomes
P 2 − P 1 =(ρ 1 − ρ) gh (4.28)
Fig. -4.9.
manometer.
Schematic of inverted
As in the previous “magnified” manometer if the density difference is very small the
height become very sensitive to the change of pressure.
4.3.3 Varying Density in a Gravity Field
There are several cases that will be discussed here which are categorized as gases,
liquids and other. In the gas phase, the equation of state is simply the ideal gas model
or the ideal gas with the compressibility factor (sometime referred to as real gas).
The equation of state for liquid can be approximated or replaced by utilizing the bulk
modulus. These relationships will be used to find the functionality between pressure,
density and location.
4.3.3.1 Gas Phase under Hydrostatic Pressure
Ideal Gas under Hydrostatic Pressure
The gas density vary gradually with the pressure. As first approximation, the ideal gas
model can be employed to describe the density. Thus equation (4.11) becomes
∂P
∂z = − gP
RT
(4.29)
Separating the variables and changing the partial derivatives to full derivative (just a
notation for this case) results in
dP
P
= −gdz RT
(4.30)