Some of our Canadian customers have pointed out that they find differences when they compare the examples included in the Canadian documentation with the results given by AlcoDens for the same input numbers. This page explains that the Canadian tables and AlcoDens are entirely consistent with each other and discusses how to avoid any discrepancies.

The root of the problem is that while the OIML tables are based on absolute or "in vacuum" masses and densities and the American TTB tables are based on "in air" values, the Canadian tables are a hybrid of both "in air" and "in vacuum" values. This mixing of the two systems is not well described in the documentation accompanying the Canadian tables but it is clear from an examination the examples. The difference between an "in air" and "in vacuum" mass is roughly 0.11%, so it is often just ignored as a rounding error. The Canadian documentation seems to refer only to the OIML formulations so it comes as a bit of a surprise to find the "in air" basis used for some of the published factors.

In summary, the Canadian tables use "in vacuum" values for converting density readings from hydrometers to %ABV values and use "in air" values when converting between masses and volumes. This distinction is probably justifiable because density and SG hydrometers are calibrated with "in vacuum" values and of course when you weigh your spirit you can only do it "in air". The authors of the Canadian tables might have felt that highlighting this distinction would cause unnecessary confusion, but in fact it causes extreme confusion when digging deeply into the numbers or when comparing the Canadian values with other tables or with AlcoDens.

A point worth bearing in mind when considering all these factors is that the buoyancy due to measuring "in air" only affects masses and NOT volumes. The impact of "in air" measurements on density is via the mass component of the density. Also, %ABV values are not affected by the "in air" vs "in vacuum" distinction because they are based entirely on the volumes of the pure alcohol and the spirit. This is seen in Example 4 below.

There is a secondary reason for some discrepancies between the AlcoDens results and the Canadian tables. This is that the Canadian tables give only 1 digit after the decimal for the density in kg/m³ and for the %ABV while the OIML tables give 2 digits and AlcoDens follows the OIML example. This is a minor effect but is mentioned here because it does explain some of the differences.

The examples discussed below are taken from the "Canadian Alcoholometric Tables 1980" available on the internet. This document includes 4 examples. The first example will be examined in detail to make the reasoning absolutely clear, and the remaining 3 will be shown in outline because they illustrate different types of calculation and to reinforce our claim of consistency between the Canadian tables and AlcoDens.

## Mass/Density Procedure - Example 1

Scale tank indication = 20 135 kg (kilograms)
Hydrometer indication = 922.6 (kg/m³)
Temperature = 20°C
A = 1.0851 (Litres spirit per kg at 20°C)
B = 53.7 (% ABV at 20°C)
20 135 kg x 1.0851 = 21 848.4885 L (litres of spirits at 20°C)
21 848.4885 L x 53.7% = 11 732.638 L (litres of absolute ethyl alcohol at 20°C)

It should be pointed out here that the "Hydrometer indication" is what the TTB calls an "apparent density" reading. i.e. it is the value that is read off the hydrometer at the temperature of the sample. In this example it makes no difference because the sample temperature and the hydrometer calibration temperature are both 20°C. Note that in the graphic above the Mass Basis is set to OIML (in vacuum). We see that the AlcoDens value for the %ABV is very close to the value given (i.e. Factor B) in the example, and the difference is fully explained by the Canadian tables rounding the value to only 1 digit. A manual interpolation of the printed OIML tables confirms the AlcoDens value.

Now we need to look at the calculated volume. In AlcoDens the volume is calculated using the Alcohol/Water Volume Temperature Correction calculator as shown below. The gross volume at 20°C is calculated by AlcoDens as 21824 litres, but Example 1 gives 21848.4885 litres. The difference is therefore 24.5 litres (disregarding the excessive number of decimals given by the tables in this instance).

The reason for this difference is the value of Factor A. If the density is 922.6 kg/m³ then it could also be expressed as 0.9226 kg/litre. In Example 1 the Factor A is given in the units Litres per kg so its value should be the reciprocal of the density in kg/litre. But 1/0.9226 = 1.0839 and not 1.0851 as given in the example. What gives?

The best way to illustrate the cause of this difference is to use the AlcoDens Alcohol/Water Volume Temperature Correction calculator to determine the volume of 1 kg of spirit because this volume will have the value of Factor A. Here the calculation is done for both the OIML and TTB cases.  The second graphic above shows that when the Mass Basis is set to TTB the volume of 1 kg of spirit is calculated to be 1.0851 litre, which is exactly the value given in Example 1 for Factor A. This confirms that when dealing with masses the Canadian tables use the "in air" assumption that is also used by the TTB tables.

Another discrepancy comes to light when we look at the OIML version of the calculation in the above graphic. Earlier we said that the value of Factor A under the OIML or "in vacuum" assumption should be the reciprocal of the density and should have a value of 1.0839, but the graphic above shows a value of 1.0838. The reason for this is that the %ABV value of 53.7 (as per Example 1) has been used. If this value is changed to the more accurate 53.74 %ABV the volume of 1 kg would be calculated as 1.0839 (using the OIML option) as shown below. Now that we have verified that the value of 1.0851 is correct for Factor A (Litres spirit per kg at 20°C) the measured mass of 20 135 kg can be converted to the volume of 21 848.5 litres in agreement with Example 1.

The LAA (litres of absolute alcohol) can similarly be calculated, but the value obtained will be slightly different depending on whether the %ABV is taken as 53.7 or 53.74. The LAA would therefore be either the 11 732.6 litres in the example or 11 741.4 litres using a %ABV of 53.74. This would make a tax difference of less than 0.1 cent on the dollar, and would average out to zero over time because the rounding would sometimes be up and sometimes be down.

## Mass/Density Procedure - Example 2

Scale tank indication = 23 876 kg (kilograms)
Hydrometer indication = 937.4 (kg/m³)
Temperature = 10°C
A = 1.0762 (Litres spirit per kg at 20°C)
B = 50.0 (% ABV at 20°C)
23 876 kg x 1.0762 = 25 695.3512 L (litres of spirits at 20°C)
25 695.3512 L x 50.0% = 12 847.676 L (litres of absolute ethyl alcohol at 20°C)

The important difference between this example and the previous one is that the temperature of the sample is now 10°C. This means that we have to correct the indicated (or apparent) density to the true density at 20°C. To emphasize this point, the data from Example 2 can be compared with the original OIML data. At 10°C the OIML tables give the density at 50.0 %ABV as 937.56 kg/m³. The reason for this difference from the Example 2 density of 937.4 kg/m³ is that the OIML value is the true density at 10°C and the Canadian tables value is the apparent density that would be indicated by a hydrometer calibrated at 20°C but used in a sample at 10°C.

The practical consequence of this is that when using the Canadian tables it must be remembered that the tables include the hydrometer temperature correction and no correction factor should be applied to the hydrometer reading before using the density to determine the %ABV.

In AlcoDens the true density and the %ABV can be found using the Alcohol/Water Hydrometer Temperature Correction calculator, as shown below. Note that because we are working with a density measured by a hydrometer the Mass Basis is set to OIML ("in vacuum"). As in Example 1, the AlcoDens and Canadian values for the %ABV (Factor B in Example 2) differ slightly because of the additional rounding in the Canadian tables. Now that we have the %ABV we can use the AlcoDens Alcohol/Water Volume Temperature Correction calculator to calculate the volume of 23 876 kg of spirit at 50.0 %ABV and 20°C. Note that the Mass Basis must now be switched to TTB. The volume from AlcoDens is 25699 litres, giving a difference of 3.6 litres from the Example 2 value. The Factor A value from AlcoDens can be calculated from the mass and volume, giving 25699 / 23876 = 1.0764. This very small difference is again largely due to the rounding in the Canadian tables. The Factor A converts a mass (which is not temperature dependent) to a volume at 20°C (and therefore also not temperature dependent) and should therefore not vary with temperature. However, in the Canadian tables for a strength of 50.0 %ABV between 10°C and 20°C the value of Factor A varies between 1.0762 and 1.0764. This is because the value of 50.0 %ABV is rounded to 3 significant digits while Factor A is given to 5 significant digits.

The calculated Volume of Absolute Alcohol (LAA) from AlcoDens is 12849 litres. This is 1.3 litres or 0.01% higher than the Canadian value and would make a tax difference of about 0.01 cents in the dollar. As before these differences would average out to zero over time. Even if the AlcoDens %ABV value of 49.96 were to be used it would make a difference of less than 0.1 cents in the dollar.

## Volume/Density Procedure - Example 3

Flowmeter reading = 24 615.0 L (litres at sample temperature)
Hydrometer indication = 905.8 (kg/m³)
Temperature = 20°C
B = 61.5 (% ABV at 20°C)
C = 1.0000 (Litres of spirits at 20°C per litre at sample temperature)
24 615.0 L x 1.00000 = 24 615.0 L (litres of spirits at 20°C)
24 615.0 L x 61.5% = 15 138.225 L (litres of absolute ethyl alcohol at 20°C)

Examples 1 and 2 were based on the quantity of spirits being measured by mass, but Examples 3 and 4 involve the spirits being measured by volume. In Example 3 the volume of the spirits is measured at 20°C so the Factor C (which converts the volume at the measured temperature to a volume at 20°C) will be exactly 1.0000 so we don't need to worry about it in this example.

Since the density is also measured at 20°C it does not need to be corrected and the Alcohol/Water Strength Converter can be used to convert the density to the %ABV directly. Note that again the Mass Basis has been set to OIML because we are working with hydrometer readings. As before, the small difference is explained by the extra rounding in the Canadian tables. The volume is unchanged so no additional explanation is required for this example.

## Volume/Density Procedure - Example 4

Flowmeter reading = 21 643.0 L (litres at sample temperature)
Hydrometer indication = 897.4 (kg/m³)
Temperature = 30°C
B = 61.7 (% ABV at 20°C)
C = 0.9909 (Litres of spirits at 20°C per litre at sample temperature)
21 643.0 L x 0.9909 = 21 446.0487 L (litres of spirits at 20°C)
21 446.0487 L x 61.7% = 13 232.212 L (litres of absolute ethyl alcohol at 20°C)

In Example 4 the sample temperature is different from the calibration temperature of the hydrometer and in AlcoDens the indicated density must be corrected to the true density at 20°C using the Alcohol/Water Hydrometer Temperature Correction calculator, as shown below. Note that because we are working with a density measured by a hydrometer the Mass Basis is set to OIML ("in vacuum"). The difference between the AlcoDens and Example 4 values for the %ABV (Factor B) is again simply the extra rounding of the value in Example 4. In order to calculate Factor C we must use the AlcoDens Alcohol/Water Volume Temperature Correction calculator to calculate the volume at 20°C. This is shown below. Note that this is done for both of the TTB and OIML options and the values are the same in both cases. This confirms the statement made in the fourth paragraph at the top of this page where it was stated that the %ABV and volumes are not affected by the buoyancy of the air.  The Factor C is calculated as 21447 / 21643 giving a value of 0.9909 in agreement with Example 4.

The differences between the AlcoDens and Example 4 volumes, and between the calculated litres of absolute alcohol, only occur in the fifth digits and are therefore negligible since the %ABV was rounded to only 3 significant digits.