Basics of Fluid Mechanics, 2014a

78 CHAPTER 4. FLUIDS STATICS
If the light liquid volume in the two containers is known, it provides the relationship
between h 1 and h 2 . For example, if the volumes in two containers are equal then
−h 1 A 1 = h 2 A 2 −→ h 1 = − h 2 A 2
A 1
(4.21)
Liquid volumes do not necessarily have to be equal. Additional parameter, the volume
ratio, will be introduced when the volumes ratio isn’t equal. The calculations as results
of this additional parameter does not cause a significant complications. Here, this ratio
equals to one and it simplify the equation (4.21). But this ratio can be inserted easily
into the derivations. With the equation for height (4.21) equation (4.19) becomes
( (
P 1 − P 2 = gh 2 ρ 2 − ρ 1 1 − A ))
2
(4.22)
A 1
or the height is
P 1 − P 2
h 2 = [
] (4.23)
A
g (ρ 2 − ρ 1 )+ρ 2 1 A 1
For the small value of the area ratio, A 2 /A 1 ≪ 1, then equation (4.23) becomes
h 2 = P 1 − P 2
g (ρ 2 − ρ 1 )
(4.24)
Some refer to the density difference shown in equation (4.24) as “magnification factor”
since it replace the regular density, ρ 2 .
Inclined Manometer
One of the old methods of pressure
P
measurement is the inclined manometer.
outside
P 1
In this method, the tube leg is inclined relatively
to gravity (depicted in Figure 4.8).
dl
dy
This method is an attempt to increase the
θ
accuracy by “extending” length visible of
the tube. The equation (4.18) is then
Fig. -4.8. Inclined manometer.
P 1 − P outside = ρgdl (4.25)
If there is a insignificant change in volume (the area ratio between tube and inclined
leg is significant), a location can be calibrated on the inclined leg as zero 3 .
Inverted U-tube manometer
3 This author’s personal experience while working in a ship that use this manometer which is significantly
inaccurate (first thing to be replaced on the ship). Due to surface tension, caused air entrapment
especially in rapid change of the pressure or height.