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## Hydrostatic Pressure

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Hydrostatic Pressure |

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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:

*P = ρgh P = γh*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 having a weight density (γ) of 40 pounds per cubic foot 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.

Any type of pressure-sensing instrument may be used as a liquid level transmitter by means of this principle. In the following photograph, you see a Rosemount model 1151 pressure transmitter being used to measure the height of colored water inside a clear plastic tube:

The critically important factor in liquid level measurement using hydrostatic pressure is liquid density. One must accurately know the liquid’s density in order to have any hope of measuring that liquid’s level using hydrostatic pressure, since density is an integral part of the height/pressure relationship (*P = **ρgh* and *P = γh*). Having an accurate assessment of liquid density also implies that density must remain relatively constant despite other changes in the process. If the liquid density is subject to random variation, the accuracy of any hydrostatic pressure-based level instrument will correspondingly vary.

It should be noted, though, that changes in liquid density will have absolutely no effect on hydrostatic measurement of liquid mass, so long as the vessel has a constant cross-sectional area throughout its entire height. A simple thought experiment proves this: imagine a vessel partially full of liquid, with a pressure transmitter attached to the bottom to measure hydrostatic pressure.

Now imagine the temperature of that liquid increasing, such that its volume expands and has a lower density than before. Assuming no addition or loss of liquid to or from the vessel, any increase in liquid level will be strictly due to volume expansion (density decrease). Liquid level inside this vessel will rise, but the transmitter will sense the exact same hydrostatic pressure as before, since the rise in level is precisely countered by the decrease in density (if h increases by the same factor that *γ* decreases, then *P = **γh* must remain the same!). In other words, hydrostatic pressure is seen to be a direct indication of the liquid mass contained within the vessel, regardless of changes in liquid density.

Differential pressure transmitters are the most common pressure-sensing device used in this capacity to infer liquid level within a vessel. In the hypothetical case of the oil vessel just considered, the transmitter would connect to the vessel in this manner (with the high side toward the process and the low side vented to atmosphere):

Connected as such, the differential pressure transmitter functions as a gauge pressure transmitter, responding to hydrostatic pressure exceeding ambient (atmospheric) pressure. As liquid level increases, the hydrostatic pressure applied to the “high” side of the differential pressure transmitter also increases, driving the transmitter’s output signal higher.

Some pressure-sensing instruments are built specifically for hydrostatic measurement of liquid level in vessels, doing away with impulse tubing altogether in favor of a special kind of sealing diaphragm extending slightly into the vessel through a flanged pipe entry (commonly called a *nozzle*). A Rosemount hydrostatic level transmitter with an extended diaphragm is shown here:

The calibration table for a transmitter close-coupled to the bottom of an oil storage tank would be as follows, assuming a zero to twelve foot measurement range for oil height, an oil density of 40 pounds per cubic foot, and a 4-20 mA transmitter output signal range:

Oil level |
Percent of range |
Hydrostatic pressure |
Transmitter output |

0 ft | 0 % | 0 PSI | 4 mA |

3 ft | 25 % | 0.833 PSI | 8 mA |

6 ft | 50 % | 1.67 PSI | 12 mA |

9 ft | 75 % | 2.50 PSI | 16 mA |

12 ft | 100 % | 3.33 PSI | 20 mA |