Density Meters – Measuring Principle

This section contains some information about the basic principles of density determinations with digital density meters. The information is divided in several chapters. Please click on the desired section in the menu.



What is Density

density_formula_1.png

The density of a material is defined as its mass per volume.

The most common units for density are the SI unit kilogram per cubic meter (kg/m³) and the cgs unit gram per cubic centimeter (g/cm³).

 

The density depends on temperature. The tables below show the density of air and water in relation to the temperature:

Density of dry air at 1013.25 hPa

Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
0 0.00129 25 0.00118 50 0.00109 75 0.00101
1 0.00129 26 0.00118 51 0.00109 76 0.00101
2 0.00128 27 0.00118 52 0.00109 77 0.00101
3 0.00128 28 0.00117 53 0.00108 78 0.00101
4 0.00127 29 0.00117 54 0.00108 79 0.001
5 0.00127 30 0.00116 55 0.00108 80 0.001
6 0.00127 31 0.00116 56 0.00107 81 0.001
7 0.00126 32 0.00116 57 0.00107 82 0.00099
8 0.00126 33 0.00115 58 0.00107 83 0.00099
9 0.00125 34 0.00115 59 0.00106 84 0.00099
10 0.00125 35 0.00115 60 0.00106 85 0.00099
11 0.00124 36 0.00114 61 0.00106 86 0.00098
12 0.00124 37 0.00114 62 0.00105 87 0.00098
13 0.00123 38 0.00113 63 0.00105 88 0.00098
14 0.00123 39 0.00113 64 0.00105 89 0.00097
15 0.00123 40 0.00113 65 0.00104 90 0.00097
16 0.00122 41 0.00112 66 0.00104    
17 0.00122 42 0.00112 67 0.00104    
18 0.00121 43 0.00112 68 0.00103    
19 0.00121 44 0.00111 69 0.00103    
20 0.00120 45 0.00111 70 0.00103    
21 0.00120 46 0.00111 71 0.00103    
22 0.00120 47 0.00110 72 0.00102    
23 0.00119 48 0.00110 73 0.00102    
24 0.00119 49 0.00110 74 0.00102

Density of pure water

Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
Temp.
[°C]
Density
[g/cm³]
0 0.99984 25 0.99705 50 0.98804 75 0.97484
1 0.99990 26 0.99679 51 0.98758 76 0.97424
2 0.99994 27 0.99652 52 0.98712 77 0.97364
3 0.99997 28 0.99624 53 0.98665 78 0.97303
4 0.99997 29 0.99595 54 0.98617 79 0.97241
5 0.99997 30 0.99565 55 0.98569 80 0.97179
6 0.99994 31 0.99534 56 0.98521 81 0.97116
7 0.99990 32 0.99503 57 0.98471 82 0.97053
8 0.99985 33 0.99470 58 0.98421 83 0.96990
9 0.99978 34 0.99437 59 0.98371 84 0.96926
10 0.99970 35 0.99403 60 0.98320 85 0.96861
11 0.99961 36 0.99368 61 0.98268 86 0.96796
12 0.99950 37 0.99333 62 0.98216 87 0.96731
13 0.99938 38 0.99297 63 0.98163 88 0.96664
14 0.99925 39 0.99259 64 0.98109 89 0.96598
15 0.99910 40 0.99222 65 0.98055 90 0.96531
16 0.99895 41 0.99183 66 0.98000    
17 0.99878 42 0.99144 67 0.97945    
18 0.99860 43 0.99104 68 0.97890    
19 0.99841 44 0.99063 69 0.97833    
20 0.99821 45 0.99021 70 0.97776    
21 0.99800 46 0.98979 71 0.97719    
22 0.99777 47 0.98936 72 0.97661    
23 0.99754 48 0.98893 73 0.97603    
24 0.99730 49 0.98848 74 0.97544

What is Specific Gravity

density_formula_2.png

For historical reasons the density is often expressed as specific gravity (SG) or relative density (SR) which is a dimensionless quantity as it is the ratio of two densities.

If the reference is not exactly stated it is normally assumed to be be water. As the density depends on temperature, the specific gravity depends on the temperatures to which the densities of the substance and the reference relate. The two temperatures are specified by the notation (Ts/Tr) with Ts representing the temperature at which the sample's density has been determined and Tr the temperature at which the reference density is specified. Commonly used are for example SG(20°C/20°C) and SG(20°C/4°C).


The specific gravity was usually determined with so called Pycnometers. Pycnometers are flasks normally made of glass. The Pycnometer is weighed empty, full of a liquid with a known density (normally water) and full of the liquid whose specific gravity is to be determined. For the later two weighings the Pycnometer must be thermostatted precisely to the reference temperature.

The calculation of the specific gravity with the three masses is easy to perform:

 

density_formula_3.png

m0 = mass of the empty Pycnometer
m1 = mass of the Pycnometer filled with water
m2 = mass of the Pycnometer filled with the sample

If both weighings are performed at 20°C SG(20°C/20°C) is yielded. The actual volume of the Pycnometer has no influence on the result and the result does not depend on the calibration of the balance either. The only requirement is that the balance reads linearly with force. However it must be noted that a balance does not determine the exact masses required for the above calculation. This is due to the buoyancy effect of the air (Archimedes Principle). If we determine the specific gravity of air using the weights determined with the balance, the result would be 0 (which is of course not true).

 

density_formula_4.png

When calculating the specific gravity by means of the formula given above using the weights displayed by the balance, the so called apparent specific gravity SGA is yielded. In order to determine the true specific gravity SGV, the weighings would have to be performed in vacuo (for this reason the subscript V is used). SGV can be calculated based on SGA.

 

The specific gravity is often measured to determine concentrations of substances in aqueous solutions: The specific gravity of the solution is determined and converted to a concentration with a table SG vs. concentration. It is extremely important to use the correct form of the specific gravity when using such tables!

How does a Digital Density Meter work?

Measuring principle

KEM manufactures digital density meters based on the oscillating U-tube technique: The sample to be measured is filled into a U-shaped tube which is induced to vibrate. The eigenfrequency of the oscillation of the U-tube is influenced by the mass and therefore by the density of the sample.

 

The animations below illustrate this measuring principle:


density_oscillation_1.gifdensity_oscillation_2.gif

density_oscillation_3.png

A ball attached to leaf spring which is mounted on a wall starts to oscillate when it is pressed down with a finger and released. The frequency of the oscillation depends on the weight of the ball: A heavier ball (blue) oscillates with a lower frequency than a lighter ball (orange). If both balls have exactly the same size, the frequency of oscillation depends thus on the density of material the ball is made of: The higher the density of the material, the lower the frequency of oscillation.

 

The same happens with the U-tube of the density meter: The higher the density of the sample inside the cell, the lower the oscillation frequency.  The animations below illustrate this: The cell filled with water (high density, blue) oscillates with a higher frequency than the cell filled with air (low density, grey).

density_oscillation_4.gifdensity_oscillation_5.gif

 

The oscillation frequency of the U-tube filled with a sample can be expressed as follows:

 

density_formula_5.png

⍴ = density of the sample in the cell
Vc= internal volume of the U-tube
mc = mass of the empty U-tube
K = U-tube specific constant

 

With this calibration factor (F) it is then possible to calculate the density of any sample:

 

density_formula_8.png

Viscosity error

In practice, the oscillation frequency of the U-tube does not only depend on the density of the sample but also on it's viscosity: If the U-tube is filled with a highly viscous sample it's oscillation is dampened due to the shear forces which occur in the liquid. The observed oscillation frequency is too low and the density value yielded too high. The picture below illustrates the magnitude of this error in relation to the viscosity of the sample.

 

viscosity_error.png

The graph illustrates that the measuring error caused by the sample's viscosity is always less than 0.001 g/cm³.

  • Digital density meters which offer only a 3 place accuracy like the DA-100 and the DA-130 density meters from KEM do thus not require any correction of this error.
  • For samples with a viscosity below 10 mPa·s the shear forces do not cause any measurable error.

 

The digital density meters DA-640, DA-645 and DA-650 from KEM feature an automatic correction of the error caused by the samples viscosities. In order to evaluate the cell specific data required to correct this error, several measurements of viscous density standards must be performed with each instrument in our production plant. Many applications as for example alcohol content determinations in spirits do not require any correction of the viscosity error and it does not make sense for the corresponding customers to pay for the extra work required to evaluate the cell specific data required for the correction of the viscosity error. For this reason, KEM offers the three models mentioned above as well without automatic correction of the viscosity error at a considerably lower price (DA-640B, DA-645B and DA-650B).

 

In order to ensure traceability to recognized national standards of the results yielded with a digital density meter, system performance checks should be performed with certified density standard material. If the samples measured require a correction of the measuring error caused by their viscosity, viscous density standard material should be used for these system performance checks. For this reason, KEM offers certified standard reference material with different viscosities (see section 'Accessories' of the corresponding instruments).

Temperature control

Highly accurate density measurements require a highly accurate temperature control of the measuring cell. KEM's laboratory density meters are equipped with a solid state thermostat a so called Peltier element for this purpose. Peltier elements create a heat flux between the junction of two different materials, they transfer heat from one side of the device to the other side against the temperature gradient (from cold to hot) with the consumption of electrical energy. Peltier elements are compact, have no moving parts and can be used for heating and cooling. When used for cooling, Peltier elements are only 5-10 % as efficient compared to vapor-compression devices used in normal refrigerators. Due to their low efficiency they thus have a relatively high power consumption.

In order to minimize energy consumption, the DA-640, DA-645 and DA-650 density meters from KEM

  • are equipped with strong fans to cool the hot side of the Peltier elements. In order to prevent the accumulation of fine dust on the surface of the cooling fins (which would reduce the cooling efficiency), the density meters are equipped with a fine dust filter, which is easily accessible from outside (see picture below).
  • feature a freely programmable time switch which allows to automatically deactivate the thermostat when the instrument is not in use.

 

density_air_filter.jpg

 

To measure and control the temperature inside the cell, KEM's density meters are equipped with several NTC temperature sensors. NTC temperature sensors offer one important advantage compared to Pt100 and Pt1000 temperature sensors: Under ideal conditions, good quality Pt100 and Pt1000 sensors have a drift of approximately 0.005 °C per year. To ensure a temperature measurement accuracy of 0.02 °C density meters equipped with this type of temperature sensors must be checked and readjusted if required every 5 years at least. NTC temperature sensors have a drift of less than 0.002 °C per year only. To ensure a temperature measurement accuracy of 0.02 °C KEM density meters must thus be checked every 10 – 15 years only. NTC temperature sensors show - contrary to Pt100 and Pt100 sensors - a non linear dependence between resistance and temperature. For this reason, KEM's certified temperature sensors come with a so called thermistor table which is stored in the density meter and serves to accurately convert the measured resistance to a temperature by Lagrange Interpolation.

 

Sometimes is not possible to ensure a 100% homogeneous temperature distribution in the measuring cell as there is always a (small) heat flow from the outside. If a density meter is used in environments with high temperature fluctuations this non stable heat flow can cause measurement errors. To avoid such errors KEM's density meters are equipped with a reference temperature sensor which enables the instrument to automatically correct errors caused by fluctuations of the ambient temperature.

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