Basics of Hydrostatic Level Measurement

 A vertical column of fluid generates a pressure at the bottom of the column owing to the action of gravity on that fluid. The greater the vertical height of the fluid, the greater the pressure, all other factors being equal. This principle allows us to infer the level (height) of liquid in a vessel by pressure measurement.

Pressure of a fluid column

A vertical column of fluid exerts a pressure due to the column’s weight. The relationship between column height and fluid pressure at the bottom of the column is constant for any particular fluid (density) regardless of vessel width or shape. This principle makes it possible to infer the height of liquid in a vessel by measuring the pressure generated at the bottom:

The mathematical relationship between liquid column height and pressure is as follows:

Where, P = Hydrostatic pressure ρ = Mass density of fluid in kilograms per cubic meter (metric) or slugs per cubic foot (British) g = Acceleration of gravity γ =Weight density of fluid in newtons per cubic meter (metric) or pounds per cubic foot (British) h = Height of vertical fluid column above point of pressure measurement

For example, the pressure generated by a column of oil 12 feet high (h) having a weight density of 40 pounds per cubic foot (γ) is:

Note the cancellation of units, resulting in a pressure value of 480 pounds per square foot (PSF). To convert into the more common pressure unit of pounds per square inch, we may multiply by the proportion of square feet to square inches, eliminating the unit of square feet by cancellation and leaving square inches in the denominator:

Thus, a pressure gauge attached to the bottom of the vessel holding a 12 foot column of this oil would register 3.33 PSI. It is possible to customize the scale on the gauge to read directly in feet of oil (height) instead of PSI, for convenience of the operator who must periodically read the gauge. Since the mathematical relationship between oil height and pressure is both linear and direct, the gauge’s indication will always be proportional to height.

An alternative method for calculating pressure generated by a liquid column is to relate it to the pressure generated by an equivalent column of water, resulting in a pressure expressed in units of water column (e.g. inches W.C.) which may then be converted into PSI or any other unit desired.

For our hypothetical 12-foot column of oil, we would begin this way by calculating the specific gravity (i.e. how dense the oil is compared to water). With a stated weight density of 40 pounds per cubic foot, the specific gravity calculation looks like this:

The hydrostatic pressure generated by a column of water 12 feet high, of course, would be 144 inches of water column (144 ”W.C.). Since we are dealing with an oil having a specific gravity of 0.641 instead of water, the pressure generated by the 12 foot column of oil will be only 0.641 times (64.1%) that of a 12 foot column of water, or:

As you can see, we arrive at the same result as when we applied the P = γh formula. Any difference in value between the two methods is due to imprecision of the conversion factors used (e.g. 27.68 ”W.C., 62.4 lb/ft3 density for water).