Figure 1. Pressure at a vertical column of fluid.
The pressure at the base of the fluid column, P, is expressed as follows:
Pressure
A pressure is defined as a force divided by the area on which the force exerts.
The units of pressure are N/m2 or Pa, lbf/in2 or psi, and
so on. Consider a vertical column of fluid with the height of h (m), the cross-sectional
area of A (m2), and the density of r (kg/m3).
Figure 1. Pressure at a vertical column of fluid.
The pressure at the base of the fluid column, P, is expressed as follows:
A pressure is also expressed as a head of a fluid. A head of a fluid is defined as the length such as 33.9 ft H2O or 76 cm Hg. The relationship between a pressure P (force/area) and the head Ph (height of a fluid) is represented by the following equation:
Example 1 Calculation of a Pressure as a Head of Fluid (adapted from Felder and Rousseau, 2000)
Express a pressure of 2.00 ´ 105Pa in terms of mm Hg.
Solution: Solve Equation 3.2 for Ph (mm), assuming that g = 9.807 N/kg and noting that the density of mercury is 13.6 ´ 1000 kg/ m3 = 13,600 kg/m3.
Example 2. Pressure Below the Surface of a Fluid (adapted from felder and Rousseau, 2000)
What is the pressure 30.0 m below the surface of a lake? Atmospheric pressure (the pressure at the surface) is 10.4 m H2O, and the density of water is 1000.0 kg/m3. Assume that g is 9.807 m/s2.
Solution: First, the hard way, using Equation 3.1:
= 3.96 ´ 105 N/m2 (Pa)
Next, the easy way, using Equation 3.2:
Atmospheric pressure at sea level is 760 mmHg or 101.325 kPa. The atmospheric pressure is measured by using mercury manometer (Figure 2).
Pure vacuum
Figure 2. Mercury Manometer.
The relationship between an absolute pressure and the gauge pressure is given in the following formula:
This relationship is expressed in the following diagram (Figure 3):
Figure 3. The relationship among absolute, gauge, and vacuum pressure.
The fluid pressure must be same at any two points at the same height in a continuous fluid. This principle is applied to the manometer to measure the fluid pressure (Figure 4).
Figure 4. Manometer variables.
In the above figure, since the pressure at the height of the lower surface of the manometer fluid is the same in both arms of the manometer, we can write the following equation:
If
r 1 and r 2 are very small compared with r f, r 1 and r 2 are neglected and then,