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Fluid Mechanics - Fluid Density Expressions

Fluid density is commonly expressed as a ratio in comparison to pure water at standard temperature1. This ratio is known as specific gravity. For example, the specific gravity of glycerin may be determined by dividing the density of glycerin by the density of water:

 

 

The density of gases may also be expressed in ratio form, except the standard of comparison is ambient air instead of water. Chlorine gas, for example, has a specific gravity of 2.47 (each volumetric unit of chlorine having 2.47 times the mass of the same volume of air under identical temperature and pressure conditions). Specific gravity values for gases are sometimes called relative gas densities to avoid confusion with “specific gravity” values for liquids.

As with all ratios, specific gravity is a unitless quantity. In our example with glycerine, we see how the identical units of pounds per cubic foot cancel out of both numerator and denominator, to leave a quotient with no unit at all.

An alternative to expressing fluid density as a ratio of mass (or weight) to volume, or to compare it against the density of a standard fluid such as pure water or air, is to express it as the ratio of volume to mass. This is most commonly applied to vapors such as steam, and it is called specific volume. The relationship between specific volume and density is one of mathematical reciprocation: the reciprocal of density (e.g. pounds per cubic foot) is specific volume (e.g. cubic feet per pound). For example, consulting a table of saturated steam properties, we see that saturated steam at a pressure of 60 PSIA has a specific volume of 7.175 cubic feet per pound. Translating this into units of pounds per cubic feet, we reciprocate the value 7.175 to arrive at 0.1394 pounds per cubic foot. Industry-specific units of measurement do exist for expressing the relative density of a fluid. These units of measurement all begin with the word “degree” much the same as for units of temperature measurement, for example:

  • Degrees API (used in the petroleum industries)

  • Degrees Baum´e (used in a variety of industries including paper manufacture and alcohol production)

  • Degrees Twaddell

The mathematical relationships between each of these “degree” units of density versus specific gravity2 is as follows:

 

 

Two different formulae exist for the calculation of degrees Baum´e, depending on whether the liquid in question is heavier or lighter than water. For lighter-than-water liquids:

 

Note that pure water would measure 10o Baum´e on the light scale. As liquid density decreases, the light Baum´e value increases. For heavier-than-water liquids:

 

Note that pure water would measure 0o Baum´e on the heavy scale. As liquid density increases, the heavy Baum´e value increases.

Just to make things confusing, there are different standards for the heavy Baum´e scale. Instead of the constant value 145 shown in the above equation (used throughout the United States of America), an older Dutch standard used the same formula with a constant value of 144. The Gerlach heavy Baum´e scale uses a constant value of 146.78:

 

There exists a seemingly endless array of “degree” scales used to express liquid density, scattered throughout the pages of history. For the measurement of sugar concentrations in the food industries, the unit of degrees Balling was invented. This scale was later revised to become the unit of degrees Brix, which directly corresponds to the percent concentration of sugar in the liquid. The density of tanning liquor may be measured in degrees Bark. Milk density may be measured in degrees Soxhlet. Vegetable oil density (and in older times, the density of oil extracted from sperm whales) may be measured in degrees Oleo.

 

1Usually, this standard temperature is 4 degrees Celsius, the point of maximum density for water. However, sometimes the specific gravity of a liquid will be expressed in relation to the density of water at some other temperature.

2For each of these calculations, specific gravity is defined as the ratio of the liquid’s density at 60 degrees Fahrenheit to the density of pure water, also at 60 degrees Fahrenheit.

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