# Density of fluids

5 Rates

Density is a physical parameter that plays a vital and important role in all material states, whether solid, liquid, or gaseous. The following article focuses on the density of fluids, meaning the density of liquids and gases.

## Definition

Density (“True Density”) is the relation of mass and volume. As mass is independent of external conditions, such as buoyancy in air or gravity, it corresponds to weight in vacuo. The symbol of density is the Greek letter rho ρ:

$$\rho = {mass \over volume} \left[ {g \over cm^{3}} \right] or \left[ {kg \over m^{3}} \right]$$

Equation 1: The true density ρ in kg/m3 or g/cm3 of a liquid is defined as its mass m divided by its volume V. The mass m corresponds to the weight in vacuum and is independent of external conditions such as buoyancy in air or gravity.

The true density of liquids or gases can be measured with the oscillating U-tube principle. The unit of the true density is kg/m³ or g/cm³ (1 g/cm³ = 1000 kg/m³).

ρ [g/cm³]
Air 0.001
Water 0.998
Ethanol 0.79
Syrup 1.40
Plant Oil 0.91
Glycerin 1.26

Table 1: Densities of common fluids at 20 °C

A multi-layered cocktail (Figure 1): Fluids of higher density such as juices or syrup will sink; they are heavier and have less buoyancy. Fluids of lesser density such as alcohol or water have more buoyancy, they swim on top.

### Temperature dependence of density

A material’s volume and state changes with temperature. Temperature therefore has an important influence on the density. Consequently, an accurate density measurement requires accurate temperature determination and good temperature stability.[1]

An excellent example of the temperature dependence of density is the thermometer. With increasing temperature, the volume of mercury is expanded and rises. Same mass but more volume means less density.

Temperature is a vital factor for precise density measurement , in which you need precise temperature control or algorithms for compensation. A temperature difference of 0.1 °C might result in a density error of up to 0.0001 g/cm³.[2]

### Anomaly of water

Water is a unique liquid and reaches the density maximum at a temperature of 3.98 °C. Starting at 3.98 °C upwards, the volume of water increases and it becomes less dense. The same applies when water is cooled, just the other way around[3]. This anomaly causes lakes to freeze from the top down and water that is colder than 4 °C to freeze and swim on top.

## Common density units

A physical property like density is investigated for several reasons and is therefore reported in several units. The most frequent density unit is kilogram per cubic meter (kg/m³), used in petrochemistry, for example. In other industries, density is reported in gram per cubic centimeter (g/cm³). The conversion factor in this case is 1000 (1 g/cm³ = 1000 kg/m³). Density might also be reported in kilogram per liter (kg/L) or gram per liter (g/L).

True density ρ is often confused with apparent density ρapp. The apparent density of a sample is weight in air per volume:

$$\rho _{app} = {W \over V}$$

Equation 2: The apparent density ρapp of a sample is defined as the weight in air W divided by the sample’s volume V.

The values of apparent density and true density are different, even if their units are identical. The true density of air at 20 °C as measured in a density meter is 0.0012 g/cm³ whereas the apparent density of air at 20 °C is 0.0000 g/cm³ – air on a balance does not give a reading.

Apparent density can be calculated from true density considering the buoyancy of the sample in air and the weight and density of a reference weight in steel or brass. Nowadays, steel is defined as the material of choice for the weights. Earlier, brass was used.

$$\rho _{app} = {\rho _{true}- \rho _{air} \over 1- \cfrac{\rho _{air}}{\rho _{steel \hspace{1mm} or \hspace{1mm} brass}}}$$

Equation 3: Converting the true density of a sample ρtrue into apparent density ρapp. This conversion takes the true density of air (ρair≈ 0.0012 g/cm³) and true density of brass (ρbrass = 8.4 g/cm³) or steel (ρsteel = 8.0 g/cm³) into account.

For the determination of filling volumes with a balance from a density result, the apparent density is the required value.

Another vital unit for reporting density is Specific Gravity (SG) as found in several industries. Specific gravity is the measured density of a sample divided by the density of water at a certain temperature. For this reason, it is also called relative density (RD) because it is related to the density of water.

$$SG= {\rho _{true} \over \rho _{W}}$$

Equation 4: Specific Gravity (SG) is defined as sample density ρ divided by the density of water ρW at a specific temperature.

SG20/4 means sample density at 20 °C divided by water density at 4 °C. SG20/20 means sample density at 20 °C divided by water density at 20 °C. Comparing the results of the same sample SG20/20 and SG20/4 shows a difference because the water density is different at 4 °C and 20 °C.

The apparent specific gravity SGapp (sometimes referred to as apparent relative density Dapp)is dimensionless, which means it has no unit. It is calculated by dividing the apparent density of a sample ρapp by the apparent density of pure water ρapp,water at defined temperatures.

$${D_{app}{20\over20}} = SG_{app}{20\over20}={{{\rho _{app}}at20^{\circ}C}\over{{\rho _{app, water}}at20^{\circ}C}}$$

Equation 5: Apparent specific gravity DappT/T (or SGappT/T) is related to defined temperatures of the apparent density of the sample (ρapp) and pure water (ρapp, water).

 Sample Air at T = 20 °C, p = 1013 mbar Water at T = 20 °C True Density ρ [g/cm³] 0.00120 0.99820 Specific gravity SG²⁰/₂₀ 0.00120 1 Specific gravity SG²⁰/₄ 0.00120 0.99823 Apparent specific gravity SGₐₚₚ²⁰/₂₀ 0 1

Table 2: Comparison of air and water density units at 20 °C

## History of density measurement

For more than 50 years now modern density meters based on the oscillating U-tube principle have been used to measure the density of fluids. However, the history of density measurement started in the third century BC, when Archimedes, a Greek mathematician, scientist, and philosopher, was asked to determine whether the king’s crown was made of pure gold. As tradition tells he took a bath and found a solution: He took the crown and a bar of pure gold of the exact same weight. Then he immersed both objects in a water container. The crown displaced more water than the gold bar – the water went up higher – so the crown obviously had more volume. The same mass but more volume means lower average density, so the crown was lighter than pure gold and was not pure.[4][5]

In the 4th century Hypatia of Alexandria, a philosopher, mathematician, teacher, and first female contributor to mathematics in ancient times, developed the hydrometer (hydroscope).[6][7]

Weighing precious metals in air and then in water was a common practice among jewelers in Europe when Galileo Galilei, the famous Italian mathematician, astronomer, and physicist, described an instrument in the 16th century that is still used for high-precision density measurements today: The hydrostatic balance . Again, an object is immersed in the fluid, but here the object is attached to a highly sensitive balance, and the density values are read from the movement of the counterweight.[8]

## Density measurement methods for fluids

### Hydrometers

A hydrometer is a floating glass body with a bulb filled with a metal weight and a cylindrical stem with a scale. The hydrometer is immersed in the sample and the density of the sample can be read directly from the scale: The deeper it sinks, the less dense is the sample. If a hydrometer is immersed in a glass of water, it would sink deeper than in a glass of syrup, because syrup is denser than water.[9][10]

There are many different hydrometers available depending on the use. The number read off the scale is not always density but also derived quantities . A lactometer is used for measuring the density (creaminess) of milk, a saccharometer for measuring the concentration of sugar in a liquid, or an alcoholometer for measuring the ethanol content in spirits.[11][12] Hydrometers are probably the most basic and inexpensive density measurement tools, but they require good temperature control, which might be quite complicated and a large sample volume (up to 100 mL). Due to the small size of a hydrometer’s scale, results can easily be misread.[13]

### Pycnometers

A pycnometer consists of a glass flask and a stopper (sometimes with integrated thermometer). It is placed on a balance and after determining the weight of the empty pycnometer you can calculate its volume by filling in a calibration liquid of known density (e.g. water) using the corresponding definition of density (volume = weight / density). Afterwards, by weighing the pycnometer filled with sample the density of the sample can be determined (density = weight / volume).[14]

Using a pycnometer can yield accurate and reliable results if the temperature control is as precise as the balance used. They are affordable, but can break easily. The method is rather slow and time-consuming and a skilled operator is needed. Another drawback is the large sample volume that is required, usually 10 mL to 100 mL.[15]

### Hydrostatic balances

The hydrostatic balance is based on the Archimedes principle.[16] It consists of a very precise balance and a sinker (e.g. a sphere) of exactly known volume that is attached to one scale pan. The sinker is immersed completely in the sample and the apparent weight loss of the sinker is determined by weighing out. The apparent weight loss of the sinker equals the weight of the fluid it displaces, so the precise volume and weight are known.[17]

Hydrostatic balances are reliable and precise. However they are expensive and very time-consuming. Another disadvantage is that installation (e.g. insulation on a concrete foundation) is challenging and an accurate temperature control is essential.[18]

### Digital density meters

Modern digital density meters are based on the oscillating U-tube principle. The tube, usually a U shaped glass tube, is excited and starts to oscillate at a certain frequency depending on the filled-in sample. Through determination of the corresponding frequency the density of the sample can be calculated.

Digital density meters based on the oscillating U-tube principle are very effective instruments that allow fast and precise measurements of fluid densities over a wide range of temperature and pressure. They measure the true density (density in vacuo), so there is no influence of air buoyancy or gravity.

In contrast to traditional static methods (such as hydrometers, pycnometers, or hydrostatic weighing ) only a small amount of sample is needed, approx. 1 mL to 2 mL. Digital density meters are easy to operate and there are no special requirements regarding ambient conditions or temperature control.[15][19]

Modern high-precision density meters additionally provide a viscosity correction and a reference oscillator to enable accurate results over a large range of densities, temperatures, and viscosities.

Figure 5: Oscillating U-tube principle (U-tube filled with air or water)

The oscillation of the cell is mechanically or electronically induced. The instrument constants to adjust the density meter are used to calculate the density of a sample from its oscillation frequency or oscillation period.

#### Sensor shape and material

Density sensors are mostly straight or U-shaped tubes. They can be made of glass, e.g. borosilicate glass 3.3, metals, metal alloys, or plastics depending on the application and resistance towards the sample and cleaning agents[20].

#### Countermass

A countermass is linked to the measuring tube to reduce parasitic resonances (“external oscillations”) from other components, e.g. electronic parts. It is linked to the housing of the density meter by elastic supports and acts like a mechanical filter for external oscillations. The countermass has a resonance frequency that lies far below the frequencies used for density measurement. The countermass also ensures that the nodal points of the tube are constantly in position. The sample volume is set by the nodal points and therefore only the mass changes depending on the filled fluid while the volume remains stable.[21]

#### Reference oscillator

In case of sensors made of glass a built-in reference oscillator eliminates not only long-term drifts due to aging effects of the material, but also temperature changes that influence the elasticity. A reference oscillator therefore enables that only one single adjustment is used to cover the whole temperature range and provides the possibility to perform temperature scans of a sample.[21]

#### Temperature regulation

Temperature regulation of the cell is mostly performed with Peltier elements, which have displaced water baths. Peltier elements allow precise and fast temperature regulation and provide both effective heating and cooling of the measurement sensor.

#### Excitation and evaluation of the oscillation

A system of magnets and coils can be used to provide electronic excitation and to keep the system oscillating continuously at the characteristic frequency. The drawback of magnets is that they put additional weight on the oscillating sensor, which has a negative influence on the achievable accuracy. The most precise method to excite a sensor is using a Piezo element, a crystal or ceramic material that changes its dimension upon applied electrical voltage.

Optical pick-ups can detect a light beam that is interrupted by a minute coating on the oscillating tube. The pick-ups then record the oscillation period. On the other hand, Piezo elements can also be used to represent the period of oscillation very accurately if the usable effect of the element is inverted: it is pressurized by the moving sensor unit and generates electric voltage that represents the period of oscillation.

Although analog processing of the oscillation pattern is robust and affordable the precision is limited. Nowadays digital signal processors (DSP) are state of the art and provide great advantages over the analog technology, even allowing the recognition of energy loss connected to sample viscosity.

#### Adjustment and instrument constants (apparatus constants)

The measuring principle of an oscillation-type density meter is based on the correlation between the density r of a fluid filled and the corresponding oscillation period t (1 divided by frequency of oscillation f) according to the formula:

$$\rho = A \tau^2+ B$$

Equation 6: The density of a sample ρ can be calculated using the instrument constants (A, B) and the measured oscillation period τ

To obtain the instrument constants A, B from the corresponding frequency values at least two reference liquids with known densities have to be filled into the cell. The instrument constants comprise the cell volume and its mass as well as the spring constant.[15]

Setting the instrument constants of a density meter is called adjustment . An adjustment is an operation to bring the instrument (density meter) to a state of a performance suitable for its use, by setting or adjusting the instrument constants. Systematic measuring deviations are removed to an extent which is necessary for the provided application and permanently modifying the instrument.[20]

During an adjustment two standards are usually measured (Figure 8), e.g. dry air and pure (e.g. bi-distilled) degassed water. Knowing the density reference values of the standards allows linking the density values with a specific oscillation period and building up a linear relation between density and oscillation period. Based on this relation, unknown densities of different samples can be defined vice versa by measuring their period of oscillation.

#### Calibration

A calibration is a set of operations to establish a relationship between the reference density of a density standard and the corresponding density reading of the instrument. No intervention is made which permanently modifies the instrument.[20]

A calibration is performed to validate the quality of measurements and adjustments.

### How viscosity affects the oscillating U-tube density reading

The oscillation frequency measured not only depends on the density of the filled sample but also on its viscosity. Due to the oscillation of the tube, shear forces (a sort of friction) occur between the fluid and the tube wall and result in damping. Damping is promoted with increasing viscosity of the sample and that results in a density over-reading (the density value shown is too high).[22] Modern density meters compensate this effect and automatically perform a viscosity correction using a special technique in which two different oscillation modes are applied.[21]

#### Advantages of digital density measurement

There are several advantages if density is used for concentration determination: for example, it covers an extremely wide field of applications. A chemical company can use the same density meter for several acids, for caustics , for the quality control of incoming raw material as well as final products, and for process control at the production line. Further, the instruments are easy to use and easy to adjust.

The results are highly reproducible and show good repeatability over a wide range. Other advantages of a density meter are that generally no dilution or sample preparation is needed. Sample recovery is therefore possible as the sample remains unchanged during the measurement. There are no manual calculations needed and the method allows a quick measurement that is completely independent of the user.

In contrast to other methods for concentration determination, such as titration, photometry, or gas chromatography, the handling of chemicals is reduced to a minimum; only 1 mL to 2 mL of sample have to be filled into the measuring cell, e.g. with a syringe or an automatic filling unit.

## Conclusion

Density measurement of fluids is now used all over the world in quality control, for product characterization, and to monitor production processes. This article gives an insight into fluid density and common units as well as the historical background and measurement technologies.

## References

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