When does height equal pressure? Well the simple answer is it doesn’t, but more on that later. A less simple answer is that height based pressure unit standards for manometers have evolved significantly over time. This has left an interesting legacy that you need to be aware of if you calibrate pressure instruments using height based units.
The SI unit for fluid pressure is the pascal (Pa), named after Blaise Pascal, a French mathematician and physicist who lived in the late 15th century. Pascal is very familiar to pressure metrologist’s because of Pascal’s principal which states that a change in pressure in one part of a pressure vessel will be experienced equally in other parts of the pressure vessel without loss and is applied at right angles to the walls of the vessel [1]. He invented the Pascal press, a way of generating force using pressure. And as a mathematician Pascal introduced probability theory which is the foundation of measurement uncertainty analysis. A Pa is defined as a newton (N) per square meter (m2) and has the fundamental units of kg/ms2.
How much pressure is a pascal? It’s pretty small. If you add a tenth of a millimeter of water in a glass, the difference in pressure from the bottom of the water to the top created by the hydrostatic head is approximately one pascal. The hydrostatic head pressure is easily calculated as the product of the liquid density, gravity and the height of the column of liquid (ρglh). Since a Pa is so small the most widely used units are kPa and MPa. Pa itself is commonly used for very low draft range gauge or differential pressure, or for very low absolute pressure, such as for an evacuated pressure vessel.
It would make many pressure metrologists happy if we could just stop the discussion there. There is no technical need for any other pressure units with the exception of meters or feet used for altitude. Units other than SI include bar and mbar which are widely used, and are just another multiple of the Pa. 1 bar = 100000 Pa (0.1 MPa) and 1 mbar = 100Pa (hPa). There is pound-force per square inch (psi) which is founded on imperial units and is very popular in the United States. Another very popular unit is kg-force or gram-force per square cm or meters. These units simulate mass being accelerated by standard gravity, 9.80665 m/s2 applied to the area defined. Even though these last few mentioned are not SI units their conversions are well defined in publications such as NIST Special Publication 811 [2] and do not present any problems other than occasional issues with resolution in software or just not being easy to convert in your head.
The last units of measure to talk about, and the subject of this article, are the height based pressure units. These pressure units are different because they are expressed in length of a substance such as water or mercury, which is obviously not pressure. Why are they described like that? In the second paragraph I presented an equation to calculate pressure based on a hydrostatic head. A device used to measure pressure that uses the hydrostatic head pressure effect to measure pressure is called a manometer. There are different types of manometers but they are fundamentally a vertical U-shaped tube that measures the difference between two pressures by the measuring the difference in the height of the fluid created by the difference in pressure on each end of the tube. Manometers can be as rudimentary as a table top water manometer where the difference in height is measured by visual graduations on an attached scale or as sophisticated as NIST’s (National Institute of Science and Technology) ultrasonic interferometer manometers (UIM) that measure low pressures with extraordinary low uncertainties.
Historically, scientists and engineers who measured pressures with manometers in a process developed the practice of specifying pressure in terms of height. They could do this because the pressure measurements were crude enough that they could consider gravity and density of the liquid to be constant and therefore use the difference in height in the column as the pressure measurement. The most common liquids used were water and mercury so the pressure units of inches of water or inches of mercury were widely used. To fill the need of processes defined by these units process instrument manufacturers built instruments with scales reading in the height based units.
As technology improved, and the necessity for lower uncertainty pressure measurements evolved, defining gravity and the conditions of the liquid became more important. Gravity, almost universally, became standard gravity, and since the density of the liquid is very dependent on its temperature, the temperature of the liquid was defined. The unit would normally have the defined temperature in its name, for example inches of water at 20˚C. These were then by definition different from the actual measurements made on manometers since those were most likely not performed at standard gravity or some pre-defined liquid temperature. Now, ironically, if a manometer is used to perform a calibration on a process instrument that is ranged in height units, it is calculated using the equation for a manometer to get Pa then converted to the height based unit for comparison.
Over the years the height based unit conversions have been a source of error. This is primarily due to the inadequacy of the definition of the units. Even as late as the 80’s and 90’s manufacturers of process instruments did not define height based units relative to a temperature. The difference in inches of water at 4˚C and 20˚C, two very popular reference temperatures, is 0.18%. In 1995 I was given the task to define software conversions from a Pa to the selectable units being offered in our products. At that time we did not have NIST Special Publication 811 so other sources were used for the conversions. When it came to defining height based pressure units we used the handbook of chemistry and physics to create a function of water density with temperature based on a table of pure water. At some time later we acquired a copy of SP 811 and found we were different by ~0.002% and 0.003% at 4˚C and 60˚F respectively and that there was not a conversion for 20˚C, the most frequent reference used. Upon investigation it was found that the height based units for water in SP 811 were based on “normal” water, not pure water. So it was not only gravity and temperature of the water that needed to be specified but the purity of the water. To help with this we included conventional height based units defined in ISO 31-3 [3] for both water and mercury.
In 2010 Fluke Calibration acquired Ruska and Pressurements product lines from GE Sensing. They of course had their own story on height based units and theirs was in much better agreement with SP 811. They also included a reference to 25˚C. But it was impossible to go back and revise the embedded software for so many old DHI products. PC software platforms needed to agree with product. It was decided to leave all legacy Ruska and DHI products the way they were and support both in our PC software and differentiate by the naming convention of the unit. We had always used the name xxWa (Wa being for water) for DHI so the units defined by the Ruska legacy product we used xxH2O, the xx being the length unit. Thankfully there was not an issue with height units in mercury. xxHg is used for all situations.
In conclusion there are a couple suggestions I would make. When evaluating or calibrating in height based pressure units, be sure you have the correct conversion to Pa. It is advisable to show the conversion used in your documentation. If you are using Fluke Calibration calibrators or software, if the unit is described as xxWa
References
1. “Isaac Asimov’s Biographical Encyclopedia of Science and Technology”, Isaac Asimov, 1972, pp 118 and 119.
2. “Guide for the Use of the International System of Units (SI)”, NIST Special Publication 811 2008 Edition.
3. “Quantities and units - Part 4: Mechanics”, ISO 80000-4:2006(E), March 1, 2006. (replaced ISO 31-3)
If you are interested in pressure calibration, we've got a lot of great information on our website. For example, check out this page on pressure calibration basics. Or this page on things to consider when selecting a pressure calibration solution.