Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 77
4.3.2.2 Pressure Measurement
The idea describes the atmospheric
measurement that can be
extended to measure the pressure
of the gas chambers. Consider a
chamber filled with gas needed to
be measured (see Figure 4.6). One
technique is to attached “U” tube
to the chamber and measure the
pressure. This way, the gas is prevented
from escaping and its pressure
can be measured with a minimal
interference to the gas (some
gas enters to the tube).
h 1
h 2
A 1 P 1
ρ 1
ρ 1
ρ 1
A 2
A 1 P 2
ρ 2
ρ 2
Fig. -4.7. Schematic of sensitive measurement device.
The gas density is significantly lower than the liquid density and therefore can be
neglected. The pressure at point “1” is
P 1 = P atmos + ρgh (4.18)
Since the atmospheric pressure was measured previously (the technique was shown
in the previous section) the pressure of the chamber can be measured.
4.3.2.3 Magnified Pressure Measurement
For situations where the pressure difference is very small, engineers invented more sensitive
measuring device. This device is build around the fact that the height is a function
of the densities difference. In the previous technique, the density of one side was neglected
(the gas side) compared to other side (liquid). This technique utilizes the
opposite range. The densities of the two sides are very close to each other, thus the
height become large. Figure 4.7 shows a typical and simple schematic of such an instrument.
If the pressure differences between P 1 and P 2 is small this instrument can
“magnified” height, h 1 and provide “better” accuracy reading. This device is based on
the following mathematical explanation.
In steady state, the pressure balance (only differences) is
P 1 + gρ 1 (h 1 + h 2 )=P 2 + gh 2 ρ 2 (4.19)
It can be noticed that the “missing height” is canceled between the two sides. It can
be noticed that h 1 can be positive or negative or zero and it depends on the ratio that
two containers filled with the light density liquid. Additionally, it can be observed that
h 1 is relatively small because A 1 >> A 2 . The densities of the liquid are chosen so that
they are close to each other but not equal. The densities of the liquids are chosen to
be much heavier than the measured gas density. Thus, in writing equation (4.19) the
gas density was neglected. The pressure difference can be expressed as
P 1 − P 2 = g [ρ 2 h 2 − ρ 1 (h 1 + h 2 )] (4.20)
ρ 2