Hydrostatic weighing

Hydrostatic weighing, also referred to as underwater weighing, hydrostatic body composition analysis and hydrodensitometry, is a technique for measuring the density of a living person's body. It is a direct application of Archimedes' principle, that an object displaces its own volume of water.

Measurements of sea ice density using hydrostatic weighing in kerosene

Method

The procedure, pioneered by Behnke, Feen and Welham as means to later quantify the relation between specific gravity and the fat content,^[1] is based on Archimedes' principle, which states that: The buoyant force which water exerts on an immersed object is equal to the weight of water that the object displaces.

Example 1: If a block of solid stone weighs 3 kilograms on dry land and 2 kilogram when immersed in a tub of water, then it has displaced 1 kilogram of water. Since 1 liter of water weighs 1 kilogram (at 4 °C), it follows that the volume of the block is 1 liter and the density (mass/volume) of the stone is 3 kilograms/liter.

Example 2: Consider a larger block of the same stone material as in Example 1 but with a 1-liter cavity inside of the same amount of stone. The block would still weigh 3 kilograms on dry land (ignoring the weight of air in the cavity) but it would now displace 2 liters of water so its immersed weight would be only 1 kilogram (at 4 °C).

In either of the examples above, the correct density can be calculated by the following equation:^[2]

${\displaystyle D_{b}={\frac {M_{a}}{{\frac {M_{a}-M_{w}}{D_{w}}}-RV}}}$

Where:

• D_b = Density of the body;
• M_a = "Mass in air" (i.e. dry weight);
• M_w = "Mass in water" (i.e. underwater weight);
• D_w = Density of water (based on water temperature);
• RV = Residual volume (the unfilled space enclosed by the body- e.g. volume of air in the lungs + respiratory passages after a maximum exhalation).

The residual volume in the lungs can add error if not measured directly or estimated accurately. Residual volume can be measured by gas dilution procedures or estimated from a person's age and height:^[3]

• RV-Est(liters, Men) = 1.310 × Ht. (meters) + 0.022 × Age (yrs., take as 25 for 18-25) − 1.232
• RV-Est(liters, Women) = 1.812 × Ht. (meters) + 0.016 × Age (yrs., take as 25 for 18-25) − 2.003

These estimates are for adults aged 18-70, have standard deviation of about 0.4 litres and have dependence on ethnicity, environmental factors, etc.^[4] Residual volume may also be estimated as a proportion of vital capacity (0.24 for men and 0.28 for women).^[5]

Application

Once body density has been calculated from the data obtained by hydrostatic/underwater weighing, body composition can be estimated. The most commonly used equations for estimating the percent of body fat from density are those of Siri^[6] and Brozek et al.:^[7]

Siri (1956): Fat % = [4.950 /Density - 4.500]×100

Brozek et al. (1963): Fat % = [4.570 /Density - 4.142]×100

Hydrostatic weighing is also used to estimate the density of sea ice, as it is considered the most precise method.^[8] Typically, a sample of sea ice is weighed in air and in kerosene, as kerosene has a lower density than sea ice, can be cooled to sub-zero temperatures, and does not melt the ice. Sea ice density experiences substantial seasonality, with larger values during winter and lower values during summer.^[9] Due to the small difference between the density of seawater and sea ice, such seasonal changes in sea ice density affect its freeboard and introduce large uncertaintes of sea ice thickness estimates using satellite altimeters.^[10]