Density and Specific Gravity

9.3.1 Basic terms

The density, specific weight, and specific gravity were defined in Chap. 5 as follows:

Density rof a material is defined as the mass per unit volume. Units of den- sity are pounds (slug) per cubic foot [lb (slug)/ft3] or kilogram per cubic meter (kg/m3).

Specific weight g is defined as the weight per unit volume of a material, i.e., pounds per cubic foot (lb/ft3) or newton per cubic meter (N/m3).

Specific gravity (SG)of a liquid or solid is defined as the density of the mate- rial divided by the density of water or the specific weight of the material divided

by the specific weight of water at a specified temperature. The specific gravity of a gas is its density/specific weight divided by the density/specific weight of air at 60°F and 1 atmosphere pressure (14.7 psia).

The relation between density and specific weight is given by

g=rg (9.6)

where gis the acceleration of gravity 32.2 ft/s2or 9.8 m/s2depending on the units being used.

Example 9.7 What is the density of a material whose specific weight is 27 kN/m3?

r=g/g=27 kN/m3/9.8 m/s2=2.75 ×103kg/m3

Table 9.1 gives a list of the density and specific weight of some common materials.

9.3.2 Density measuring devices

Hydrometersare the simplest device for measuring the specific weight or den- sity of a liquid. The device consists of a graduated glass tube, with a weight at one end, which causes the device to float in an upright position. The device sinks in a liquid until an equilibrium point between its weight and buoyancy is reached. The specific weight or density can then be read directly from the grad- uations on the tube. Such a device is shown in Fig. 9.7a.

Thermohydrometeris a combination of hydrometer and thermometer, so that both the specific weight/density and temperature can be recorded and the spe- cific weight/density corrected from lookup tables for temperature variations to improve the accuracy of the readings.

Induction hydrometersare used to convert the specific weight or density of a liquid into an electrical signal. In this case, a fixed volume of liquid set by the overflow tube is used in the type of setup shown in Fig. 9.7b, the displacement device, or hydrometer, has a soft iron or similar metal core attached. The core is positioned in a coil which forms part of a bridge circuit. As the density/specific

TABLE 9.1 Density and Specific Weights

Specific weight Density

Material lb/ft3 kN/m3 slug/ft3 _×103kg/m3 Specific gravity

Acetone 49.4 7.74 1.53 0.79 0.79 Ammonia 40.9 6.42 1.27 0.655 0.655 Benzene 56.1 8.82 1.75 0.9 0.9 Gasoline 46.82 7.35 3.4 0.75 0.75 Glycerin 78.6 12.4 2.44 1.26 1.26 Mercury 847 133 26.29 13.55 13.55 Water 62.43 9.8 1.94 1.0 1.0

weight of the liquid changes, the buoyant force on the displacement device changes. This movement can be measured by the coil and converted into a den- sity reading.

Vibration sensorsare an alternate method of measuring the density of a fluid (see Fig. 9.8a). Fluid is passed through a U tube which has a flexible mount so that it can vibrate when driven from an outside source. The amplitude of the vibration decreases as the specific weight or density of the fluid increases, so that by measuring the vibration amplitude the specific weight/density can be calculated.

Pressure at the base of a column of liquid of known height (h) can be meas- ured to determine the density and specific gravity of a liquid. The density of the liquid is given by

(9.7)

ρ = p

gh

Figure 9.7 (a) A basic hydrometer, (b) An induction hydrometer.

Figure 9.8 Alternative methods for density measurement are (a) vibration sensor and (b) bubbler system.

The specific weight is given by

(9.8)

Example 9.8 What is the pressure at the base of a column of liquid if the height of the column is 298 cm and the density of the liquid is 1.26 ×103kg/m3?

p= rgh=1.26 ×9.8 ×298/100 =36.8 Pa

The weight of a known volume of the liquid can be used to determine density, i.e., a container of known volume can be filled with a liquid and weighted full and empty. The difference in weight gives the weight of liquid, from which the density can be calculated using the following equation:

(9.9)

where Wf=weight of container + liquid

Wc=weight of container

Vol =volume of the container

Differential bubblers can be used to measure liquid density or specific weight. Figure 9.8bshows the setup using a bubbler system. Two air supplies are used to supply two tubes whose ends are at different depths in a liquid, the differ- ence in air pressures between the two air supplies is directly related to the den- sity of the liquid by the following equation:

(9.10)

where ∆pis the difference in the pressures and ∆hthe difference in the height of the bottoms of the two tubes.

Example 9.9 What is the density of a liquid in a bubbler system if pressures of 500 Pa and 23 kPa are measured at depths of 15 cm and 6.5 m, respectively?

r_{= ×} 103 kg/m3

Radiation density sensors consist of a radiation source located on one side of a pipe or container and a sensing device on the other. The sensor is calibrated

23 500 1000 6 5 15 100 9 8 23 0 5 6 5 0 − −    × = − − . . . . . ( 115)×9 8. =0 36. ρ = ∆ ∆ p g h ρ = − × W W g f c Vol ρ = p h

with the pipe or container empty, and then filled. Any difference in the measured radiation is caused by the density of the liquid which can then be calculated.

Gas densities are normally measured by sensing the frequency of vibration of a vane in the gas, or by weighing a volume of the gas and comparing it to the weight of the same volume of air.

9.3.3 Density application considerations

Ideally, when measuring the density of a liquid, there should be some agitation to ensure uniform density throughout the liquid. This is to avoid density gra- dients due to temperature gradients in the liquid and incomplete mixing of liq- uids at different temperatures. Excessive agitation should be avoided.

Density measuring equipment is available for extreme temperatures and pressures, i.e., from 150 to 600°F and for pressures in excess of 1000 psi. When measuring corrosive, abrasive, volatile liquids, and the like, radiation devices should be considered.