APPENDIX 2: Physics of Hydrostatic Pressure
A liquid manometer is a simple and reliable means of monitoring
pressures that do not change rapidly. It simply uses the weight of a measured vertical
column of liquid to balance the pressure exerted against the bottom of the column.
We have defined weight as the force exerted by gravity on a mass (m): Fg
= mg. To determine the weight of a column of liquid of known dimensions (the manometer
in Fig. 30-11
), we must
first know the density (mass per unit volume) of the liquid. Density has dimensions
of m/l3
, and the SI units are kilograms per cubic meter (kg/m3
).
Because liquids are almost incompressible, their density is little influenced by
pressure (but affected by temperature). The density of water at room temperature
is 997.8 kg/m3
, or 1.0 g/cm3
.
The pressure (p) exerted by the bottom of the vertical column
of liquid in a manometer (see Fig.
30-11
) is determined as follows. If the cross-sectional area of the liquid
cylinder is A and the height of the cylinder is z, its volume is V = Az. If the
liquid has a density of ρ, the mass of the column is
m = ρV = ρAz (1)
and its weight is
W = mg = ρAzg (2)
The liquid column exerts a force equal to its weight on its base, whose surface area
is A, thus creating the following pressure on the surface:
p = Force/Area = ρAzg/A = ρgz (3)
The pressure exerted by the manometer is therefore independent of its cross-sectional
area (A); it depends on only the density of the working fluid and the vertical height
of the column. If we know the liquid density, measurement of the column height (z)
allows us to calculate the pressure (p). For example, if the working fluid is mercury
(e.g., sphygmomanometers), the density is 13,680 kg/m3
(13.68 g/cm3
,
or 13.68 times the density of water). The relationship of pressure to height of
the column is then
p = ρgz = (13,600 kg/m3
)(9.8 N/kg)(z)
p [N/m2
] = 133,300(z) (4)
The unit of pressure newton/meter2
(N/m2
) is called the pascal
(Pa). Because a pascal is a small unit of pressure, we usually use kilopascals (103
Pa) or kPa. If we express p in kPa and z in millimeters of mercury (rather than
meters), Equation 4 becomes
p [kPa] = 0.1333(z) [mm] (5)
