Pressure and Its Measurement
Every pressure value has a zero reference. Absolute, gauge, vacuum, and differential pressure; the pressure of a liquid column; manometric units; density and specific gravity of liquids; atmospheric pressure and site elevation; and how a pressure indicator establishes its reference.
This is the first article in a series on the topics of pressure and head. This article covers:
- pressure as force per unit area,
- absolute, gauge, vacuum, and differential pressure,
- the pressure exerted by a liquid column,
- manometric units,
- density and specific gravity of liquids,
- atmospheric pressure and site elevation, and
- how a pressure indicator establishes its reference.
Pressure as Force per Unit Area
Pressure is force distributed over an area. A liquid or a gas at rest presses on every surface that contains it, and the pressure at a point is the force exerted per unit of area:
P = F / A
where:
- P = pressure (psi or kPa)
- F = force acting perpendicular to the surface (lbf or N)
- A = area over which the force acts (in² or m²)
In process plants the customary US unit is the pound per square inch, psi, and the metric units are the kilopascal and the bar, with 1 psi = 6.895 kPa and 1 bar = 100 kPa = 14.50 psi.
Absolute, Gauge, Vacuum, and Differential Pressure
A pressure value cannot be accurately interpreted until its zero reference is known.
Absolute pressure takes a perfect vacuum, the complete absence of pressure, as its zero reference. An absolute pressure is therefore never negative. In industry practice an absolute value is identified by the letter “a” or the word “absolute” after the unit, as in psia, bar(a), or kPa absolute.
Gauge pressure takes the local atmospheric pressure, at the place and time of the reading, as its zero reference. Positive values are pressures above atmospheric, and negative values are pressures below it. A gauge value is identified by the letter “g” or the word “gauge” after the unit, as in psig, bar(g), or kPa gauge.
Absolute and gauge pressure are related through the local atmospheric pressure:
Pabs = Pg + Patm
where:
- Pabs = absolute pressure (psia or kPa absolute)
- Pg = gauge pressure, negative below atmospheric (psig or kPa gauge)
- Patm = local atmospheric pressure at the site and at the time (psia or kPa absolute)
A pressure below the local atmospheric pressure is called a vacuum, and a vacuum reading states the amount by which the pressure lies below atmospheric. Expressed in the same unit, a vacuum reading is the gauge pressure with its sign reversed. A compound pressure indicator reads on both sides of zero gauge pressure, showing positive gauge pressure above zero, usually in psig, and vacuum below zero, usually in inches of mercury. The absolute pressure behind a vacuum reading is the local atmospheric pressure less the reading, in consistent units, and the largest vacuum any instrument can read equals the local atmospheric pressure.
A differential pressure is the difference between two pressures, such as the pressures upstream and downstream of a strainer. It is the same whether both pressures are expressed as gauge or both as absolute, because the atmospheric term cancels, provided both gauge values are referred to the same atmospheric pressure. A differential value is identified by the letter “d” or the word “differential” after the unit, as in psid or kPa differential. Gauge pressure is itself a differential pressure, the difference between the absolute pressure and the local atmospheric pressure.
The Pressure Exerted by a Liquid Column
A column of liquid exerts pressure on its base by the force of its own weight. Consider a column of height h and uniform cross-sectional area A, filled with a liquid of density ρ. The force at the base is the weight of the column, ρ·g·h·A, with g the gravitational acceleration. Dividing by the area gives the pressure the column exerts at its base, ρ·g·h, in which the area no longer appears. The pressure at a point depends on the density of the liquid and the depth of the point below the surface, not on the width or the shape of the vessel. A pressure indicator at the base of a narrow standpipe reads the same as one at the base of a wide tank when the liquid stands at the same height above each (Figure 1).
In consistent SI units the pressure the column exerts is:
P = ρ · g · h
where:
- P = pressure exerted by the column (Pa)
- ρ = density of the liquid (kg/m³)
- g = gravitational acceleration, 9.807 m/s²
- h = height of the liquid column above the point (m)
In US customary units, with the density in pounds per cubic foot and the height in feet, the pressure in psi follows directly:
P = ρ · h / 144
where:
- P = pressure exerted by the column (psi)
- ρ = density of the liquid (lb/ft³)
- h = height of the liquid column above the point (ft)
The two forms differ because of a unit convention. In US customary units the weight density of a liquid in pounds-force per cubic foot is numerically equal to its mass density in pounds-mass per cubic foot at standard gravity, so g disappears from the US form and the division by 144 converts square feet to square inches. For water at 39.2 °F (4 °C), whose density is 62.43 lb/ft³, a column 20 ft high exerts 62.43 × 20 / 144 = 8.67 psi.
The pressure exerted by the weight of the column adds to whatever pressure acts on the liquid’s surface. Pressure applied at the surface is transmitted undiminished to every point in the liquid, so the pressure at a depth h is the surface pressure plus ρ·g·h. In an open tank the surface is at atmospheric pressure, and a pressure indicator at the base reads ρ·g·h. In a closed vessel with a gas blanket, an indicator at the base reads the gauge pressure of the blanket plus ρ·g·h. In absolute terms the local atmospheric pressure is added in either case.
Manometric Units
The liquid-column equations show that a pressure can be expressed as the height of a column of liquid. Manometric units express a pressure in this way, as the height of a column of a reference liquid. The reference liquids are water at 39.2 °F (4 °C), with a density of 1,000 kg/m³ (62.43 lb/ft³), and mercury at 32 °F (0 °C), with a density of 13,595.1 kg/m³.* Table 1 lists the units in plant use.
| Unit | How many in 1 psi | How many in 1 kPa |
|---|---|---|
| Millimeter of water (mmH2O) | 703.1 | 102.0 |
| Millimeter of mercury (mmHg) | 51.71 | 7.501 |
| Inch of water (inH2O) | 27.68 | 4.015 |
| Foot of water (ftH2O) | 2.307 | 0.3346 |
| Inch of mercury (inHg) | 2.036 | 0.2953 |
| Meter of water (mH2O) | 0.7031 | 0.1020 |
Table 1. The number of each manometric unit in 1 psi and in 1 kPa.
* Some industry standards and conversion tables define the inch of water on water at 60 °F or 68 °F, which makes the unit approximately 0.1 or 0.2 percent smaller, respectively, than on water at 39.2 °F. Some also define the inch of mercury on mercury at 60 °F, which makes it approximately 0.3 percent smaller than on mercury at 32 °F. The differences are negligible against the accuracy of a typical dial pressure indicator but may exceed that of a pressure transmitter, so the reference temperature of a transmitter’s configured unit should be confirmed.
The units began with the instruments. The first instruments for measuring pressure were liquid columns, the barometer and the U-tube manometer, and the height of the column was the reading itself (Figure 2). Mercury is approximately 13.6 times as dense as water, so a barometer using mercury is approximately 1/13.6 the height of one using water. The standard atmosphere supports 29.92 in (760 mm) of mercury, against 33.90 ft (10.33 m) of water. Mercury’s vapor pressure is approximately 0.0012 mmHg at 68 °F, so the space above a barometer’s column is very nearly a perfect vacuum and the column reads absolute pressure. Inches and millimeters of mercury therefore became the customary units of atmospheric pressure and of vacuum, and atmospheric pressure measured at a site is often called barometric pressure, after the instrument.
Water suits small pressures, because a water column moves 13.6 times as far as a mercury column for the same pressure. A furnace draft of 0.10 inH2O is approximately 0.0036 psi and would move a mercury column only approximately 0.007 in. Inches of water therefore became a common unit for furnace and boiler draft, fan static pressure, and the pressure and vacuum settings of low-pressure storage tanks, with millimeters of water as the metric counterpart.
Density and Specific Gravity of Liquids
The pressure of a liquid column depends on the density of the liquid, and so does the size of a manometric unit. The density ρ of a substance is its mass per unit volume. A liquid generally expands as its temperature rises, so its density falls. Its density changes so little with pressure that a liquid is usually treated as incompressible in plant calculations. Water is densest at 39.2 °F (4 °C), at 62.43 lb/ft³ (1,000 kg/m³). Above that temperature its density falls as it warms, to 62.37 lb/ft³ at 60 °F and 60.12 lb/ft³ at 200 °F.
Specific gravity is the ratio of a liquid’s density to that of water at a stated reference temperature. Referred to water at 39.2 °F (4 °C), whose density is ρw, the specific gravity SG is:
SG = ρ / ρw
where:
- SG = specific gravity of the liquid at its temperature, relative to water at 39.2 °F (4 °C)
- ρ = density of the liquid at its temperature (lb/ft³, lb/US gal, kg/m³, or g/cm³)
- ρw = density of water at 39.2 °F (4 °C), which is 62.43 lb/ft³, 8.345 lb/US gal, 1,000 kg/m³, or 1.000 g/cm³
The two densities enter the formula only as a ratio, so any one unit serves for both. On the 39.2 °F basis, the specific gravity of mercury at 32 °F (0 °C) is 13.60, and that of water at 200 °F is 0.963. Although water at 39.2 °F (4 °C) is the reference used here, some industries state specific gravity at 60/60 °F, the density of the liquid at 60 °F relative to that of water at 60 °F.
When specific gravity is used in place of density, the density in the liquid-column equations becomes SG·ρw, the product of the specific gravity and the density of water at 39.2 °F (4 °C). In SI units, ρw·g = 1,000 × 9.807 = 9,807 Pa, or 9.807 kPa, for each meter of height, so the pressure the column exerts in kilopascals is:
P = 9.807 · SG · h
where:
- P = pressure exerted by the column (kPa)
- SG = specific gravity of the liquid at its temperature, relative to water at 39.2 °F (4 °C)
- h = height of the liquid column above the point (m)
In US customary units, ρw/144 = 62.43 / 144 = 0.4335 psi for each foot of height, so the pressure the column exerts in psi is:
P = 0.4335 · SG · h
where:
- P = pressure exerted by the column (psi)
- SG = specific gravity of the liquid at its temperature, relative to water at 39.2 °F (4 °C)
- h = height of the liquid column above the point (ft)
Atmospheric Pressure and Site Elevation
Atmospheric pressure is the weight of the column of air above the site, spread over the area of its base. Under the standard conditions used for reference, the atmosphere at sea level exerts 14.696 psia, which is 101.325 kPa, 1.01325 bar, 29.92 inHg, or 33.90 ftH2O (10.33 mH2O). In round numbers it is 14.7 psia.
The standard sea-level value also serves as a unit, the standard atmosphere (atm), defined as exactly 101.325 kPa. It was originally defined as the pressure of a column of mercury 760 mm high. This gave rise to the torr unit, defined as 1/760 of a standard atmosphere, equivalent to one millimeter of mercury for practical purposes. Units of torr are commonly applied when reporting deep vacuum as an absolute pressure.
Air is compressible, so its density falls with height, and the pressure falls faster near the ground than higher up. The decrease of atmospheric pressure with elevation is therefore not linear, though it is nearly so across the range of elevations at which plants are built. The U.S. Standard Atmosphere is a reference model of the atmosphere at a sea-level temperature of 59 °F (15 °C) with a fixed fall of temperature with altitude, and it gives the atmospheric pressure at a site from the site elevation above mean sea level. In US customary units, with the elevation in feet, the relation is:
Patm = 14.696 · (1 − 6.8756 × 10−6 · zsite)5.2559
where:
- Patm = standard atmospheric pressure at the site (psia)
- zsite = site elevation above mean sea level (ft)
In SI units, with the elevation in meters, the relation is:
Patm = 101.325 · (1 − 2.2558 × 10−5 · zsite)5.2559
where:
- Patm = standard atmospheric pressure at the site (kPa absolute)
- zsite = site elevation above mean sea level (m)
The relation describes a long-term average, not the pressure on a given day, and it holds through the whole range of plant elevations. In ordinary weather the pressure at a site stays within roughly ±0.4 psi (±3 kPa) of the standard value.
Because zero gauge pressure sits at the local atmospheric pressure, one gauge reading corresponds to different absolute pressures at sites of different elevation. The standard atmospheric pressure is 14.67 psia in Houston, about 50 ft above sea level, and 12.10 psia in Denver. An indicator at the base of the 20 ft water column of Figure 1 reads 8.67 psig at either site, which is 23.34 psia in Houston and 20.77 psia in Denver (Figure 3).
How a Pressure Indicator Establishes Its Reference
A Bourdon-tube pressure indicator reads gauge pressure by construction. The curved tube inside the case straightens as the pressure inside it rises, and its motion drives the pointer. The tube deflects by the difference between the process pressure inside it and the pressure in the case around it, and the case is open to the atmosphere through a vent, so the pointer reports pressure relative to the atmosphere.
A liquid-filled pressure indicator has a sealed case to retain the fill, with a vent plug that is opened after installation. A case left sealed may build up an internal pressure as its temperature changes or as it is moved to an elevation other than the one at which it was filled. A change in case pressure shifts the zero and is especially significant on a low-range indicator. The vent should therefore be confirmed open on any liquid-filled indicator relied on for low gauge pressures.
Where the absolute value itself is needed, an absolute-pressure transmitter is used. The reference side of its sensing element is evacuated and sealed, so it reads from zero absolute, and its reading does not change with the weather or the site elevation. Its data sheet and tag identify it as absolute.
Next in the series: Liquid Head: Pressure, Velocity, and Elevation Head — head as the mechanical energy per unit weight of a liquid, the static and stagnation pressures and how the orientation of a pressure tap or tube affects the reading, and the pressure, velocity, and elevation heads.