# Filling volume control: Why, when, and how?

Whereas some products are quantified by their weight – for example some pre-packed goods in Europe –, others are filled, paid, and even taxed by their volume. Volume measurement is a commonly used practice, but frequently it is easier to measure the weight of a fluid product with a balance, followed by calculating the filling volume by using the density of the sample.

It is a quite straightforward procedure to convert the weight of a liquid sample into volume and vice versa if the density of the sample is known. But – does this always apply without any prerequisite?

### Question 1: Are there any influencing factors to be considered that could affect the density result?

There certainly are – just think about how frozen water can crack a water bottle that was filled to the rim! The density of liquids and gases is highly temperature-dependent. As a consequence, precise density measurements either have to be temperature-controlled or they require an accurate temperature measurement.

### Question 2: Is the temperature coefficient really that much of an issue and how can it be determined?

If the density of a sample is measured at a certain temperature and needs to be referenced to a standard temperature, the temperature coefficient has to be known.

To determine the temperature coefficient, measure the true densities ρ1 and ρ2 of the same sample at two different temperatures T1 and T2, and divide the density difference by the difference in temperatures.

The result is the temperature coefficient. It always carries a positive sign.

### Question 3: Does “density” always mean the same?

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

The values of true and apparent density are different, even if their units are identical.

Just imagine an empty vessel on a balance: 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!

### Question 4: How to determine the filling volume?

The filling volume V of a certain sample weight W (W = weight in air) can be calculated based on the sample’s apparent density.

The volume can be shown directly on a DMA display if the required coefficients for the calculation are provided.

### Question 5: What density result is needed to convert the weight of e.g. a tank full of liquid merchandise into its volume?

Let us say that an empty and a full truck are weighed and the tank’s volume is known. With this information plus the sample’s density and temperature, the calculation is possible, well-proven and straightforward and therefore frequently applied wherever the measurement of volumes is not possible.

The apparent density is smaller than the true density and can be calculated from the true density considering the buoyancy of the sample in air and the weight and density of a reference weight in steel or brass.

### Question 6: How to convert true into apparent density?

The apparent density ρapp is defined as

where

ρapp = apparent density of the sample

ρsteel = 8.0 g/cm³; ρbrass = 8.4 g/cm³

ρair = true density of air (≈ 0.0012 g/cm³)

ρtrue, sample = true density of the sample

Today, steel is generally considered to be the reference.

### Question 7: How to determine the accuracy of the calculated filling volume?

The accuracy of the calculated filling volume ΔV depends on the density of the solution and the accuracy of the instrument Δρ.

Let us tackle this question with an example:

The accuracy of the density meter is 0.001 g/cm³ in the viscosity range <100 mPa·s and the density range 0 g/cm³ to 2 g/cm³. Thus, the accuracy of the calculated filling volume ΔV for a density value of 0.7 g/cm³ is

### Question 8: Looking at instrument specifications, is it possible to predict which density meter version will fulfill your demands on accuracy for the weight-to-volume conversion result?

A selection of density meters is available. First of all, the specified accuracies of the different models are required as listed in Table 1.

Table 1: Accuracies of various DMA density meters

 Instrument Specified instrument accuracy Δρ DMA 35 ∓0.001 g/cm³ DMA 500 ∓0.001 g/cm³ DMA 4100 M ∓0.0001 g/cm³ DMA 4500 M ∓0.00005 g/cm³ DMA 5000 M ∓0.000005 g/cm³

Second, the accuracy has to be calculated for the respective density as explained above (compare Question 7). Table 2 summarizes the obtained results for samples with various densities with different density meters.

Table 2: Accuracies of calculated filling volumes

 Density [g/cm³] DMA 35, DMA 500 [%] DMA 4100 M [%] DMA 4500 M [%] DMA 5000 M [%] 0.7 ∓0.143 ∓0.0143 ∓0.00714 ∓0.000714 1 ∓0.100 ∓0.0100 ∓0.00500 ∓0.000500 1.3 ∓0.077 ∓0.0077 ∓0.00385 ∓0.000385 1.6 ∓0.062 ∓0.0062 ∓0.00312 ∓0.000312 2 ∓0.050 ∓0.0050 ∓0.00250 ∓0.00250 2.3 ∓0.043 ∓0.0043 ∓0.00217 ∓0.000217

Comparing the results in Table 2 to the required accuracy for the respective application tells you which density meter is the most suitable one.

### Comparing the accuracies DENSures that you choose the most suitable density meter for your requirements!

Anton Paar offers a wide selection of digital density meters.