# Alcohol Dilution & Proofing Calculation Methods

This article compares 3 methods for calculating alcohol dilution ratios. Adjusting the strength of the alcohol spirit to the correct value for bottling is known as "proofing". The advantages and shortcomings of each method are discussed and examples are given. The 3 methods covered are the AlcoDens Blending Calculator, The TTB Table No.6 Method and the Pearson's Square Method.

The methods discussed below are applicable to blends of alcohol and water only. If you are interested in calculations for spirits containing sugar please see the page on proofing and blending liqueurs.

In order to have a relevant "apples-with-apples" comparison between the methods the following typical situation, which occurs daily in hundreds of distilleries around the world, is used as the example to calculate in all cases.

Calculate the volume of water required to dilute 10 gallons of spirit at 95 % ABV to create a product at 40 % ABV. It can be assumed that all measurements and blending are performed at 60ºF and all quantities are measured volumetrically.

## 1. The AlcoDens Blending Calculator

The AlcoDens Blending Calculator is one of several functions available in the AlcoDens program. This calculator can calculate the strength of a blend of 2 sources of known quantity and strength, but here it is used in the alternate mode where it will calculate the quantity of the sources required to achieve a target blend strength, as would typically be done when proofing to bottling strength.

In the screenshot below it can be seen that all the temperature options are set to 60ºF and all quantities are set to volumetric. In this example Source 1 is the 95 % ABV spirit and Source 2 is the water (with a strength of zero). The only quantity that has been given is the 10 gallons for the volume of the 95 % spirit, so Source 1 is set as the "Known Quantity" in the bottom right block.

# AlcoDens Results for Standard Example This screenshot from the AlcoDens alcohol dilution calculator gives 2 results - the volume of dilution water that must be added (14.446 gallon) and the final volume of the total blend (23.75 gallon). It is easy to confirm that the total volume is correct. The initial 10 gallons of spirit had a strength of 95 % ABV so it must have contained 9.5 gallons of pure alcohol. This same quantity of alcohol constitutes 40 % by volume of the final blend so the volume of the blend is clearly 23.75 gallons (= 9.5 x 100/40).

The volume of dilution water required is not as easy to check. The results indicate that 10 gallons of 95 % ABV spirit blended with 14.446 gallons of pure water give a final blend volume of 23.75 gallons although the volumes of the raw materials add to 24.446 gallons. There seems to be 0.696 gallons "missing". Where did it go? The answer is the shrinkage or contraction that occurs when ethanol and water are mixed together.

Anyone who has mixed sand and stone to make concrete will know that 1 bucket of sand plus 1 bucket of stone make less than 2 buckets of mix because some of the sand fits into the spaces between the stones. This is not a perfect analogy for the ethanol-water behavior because the change in volume has more to do with the non-polar ethanol molecules interfering with the charges on the polar water molecules than it has to do with the sizes of the molecules. But the net effect is that ethanol and water molecules can pack together more tightly than expected and the volume shrinks.

At the heart of AlcoDens is a database that has very accurate data for the density of ethanol and water mixtures. This allows it to calculate alcohol dilution ratios very accurately. As you will see in the next section, the results obtained from AlcoDens agree very closely with the results obtained doing alcohol dilution calculations using the TTB Table No.6 Method, which has been in use for 100 years by American distillers proofing their products.

## 2. The TTB Table No.6 Method for alcohol dilution & proofing calculations

In the USA the manufacture and trade of alcoholic beverages are regulated by the Alcohol and Tobacco Tax and Trade Bureau - usually abbreviated as TTB. To facilitate alcohol gauging, blending and proofing calculations in the days before electronic calculators and computer spreadsheets the TTB issued a set of 7 Tables which allowed very accurate alcohol dilution calculations to be performed with a minimum of mathematics. The ingenuity of these tables is reflected in the fact that they are still in use today, 100 years after their original issue. These Tables can be freely downloaded from http://www.ttb.gov/foia/gauging_manual_toc.shtml

All the TTB Tables use Proof as the measure of strength. The Proof is simply twice the % ABV at 60ºF. For example, a blend that is 50 % ABV at 60ºF is 100 Proof.

It is TTB Table No.6 that is used for alcohol dilution calculations. This Table gives the parts of alcohol and the parts of water (both by volume) that make up a particular blend with a volume of 100 parts, all measured at 60ºF. Note that the parts of alcohol and the parts of water add up to more than 100 in all cases because of the shrinkage that was discussed above. For example, the 100 Proof blend mentioned above will contain 50.00 parts of alcohol and 53.73 parts of water.

# TTB Table No.6 Results for Standard Example These results are extremely close to those obtained in the AlcoDens proofing example, and since both of these alcohol dilution calculation methods have been used for years by distillers and bottlers, we can be very confident of the results obtained.

The TTB Tables are designed to work at 60ºF and this example has been formulated to match that. There are correction factors included in the TTB Tables to allow working at different temperatures and although they do not change the basic method illustrated here they do complicate the math considerably. Also the Table No.6 data is given for whole numbers of Proof only and it is usually necessary to interpolate between the given data points to match actual measurements. AlcoDens automatically compensates for temperature differences and can calculate for any fractional strength, making it much easier to use than the TTB Tables for real life proofing operations.

## 3. The Pearson's Square Method for alcohol dilution calculations

You will surely remember the "ratio and proportion" problems we all did in grade school. These were problems like "If I share a bag of marbles amongst 5 boys they each get 8 marbles. How many marbles would each boy get if there were 20 boys?" While the answer to this problem is intuitively obvious, it becomes more complicated when we have fractional marbles and fractional boys. The Pearson's Square method is simply a way to lay out the alcohol dilution problem to aid solution.

Before describing the process, it must be pointed out that while there are instances where the Pearson's Square method for alcohol dilution calculation does give sufficiently accurate results, in general the method is not accurate because it does not take the shrinkage into account. Please understand the limitations of this method before using it.

The procedure is illustrated in the graphic below. The strengths of the two sources are written against the 2 left side corners of the square, as values A and B. The strengths can be in % ABV or Proof, but of course only one basis can be used per problem. The target strength is written in the center of the square as value C. By performing the illustrated subtractions diagonally the required number of parts of each of the 2 sources is calculated and written on the right hand corners, giving results D and E. Although the subtractions are done diagonally, the values for Source 1 are on the top corners and the values for Source 2 on the bottom corners.

The volume of at least one of Source 1, Source 2 or the Total must be known or assumed. The volumes for the other two can then be calculated by ratioing the parts to the known (or assumed) volume.

# Pearson's Square Results for Standard Example Unlike the previous 2 examples, the volumes of the two sources do add up to the total blend volume in this case. This is because there is no way for the shrinkage to be taken into account. If the calculated volume of water required to achieve the alcohol dilution for this example (i.e. 13.75 gallon) were used the actual final volume (taking shrinkage into account) would be 23.065 gallons and the strength would be 41.2 % ABV. As a generalization (but not absolute rule) the actual strength obtained will be higher than the target when the Pearson's Square method is used. This is why distillers who use this method for their proofing calculations sometimes talk of "creeping up on the target". After each dilution the blend strength has to be remeasured, the proofing dilution calculation redone, and more water added until the target is achieved. When using the AlcoDens or Table No.6 methods this can be achieved in a single step.

As an aside, it can be mentioned that if masses rather than volumes are used for the quantities of the 2 sources in the Pearson's Square method the results are even more inaccurate. If the strengths are measured in volumetric terms (eg % ABV or Proof) then the quantities must also be volumetric.

When the Pearson's Square is used to blend sources of similar strength, for example when blending wines, the result can be sufficiently accurate. However, when diluting distilled spirits with water or when fortifying wines with spirits the results will not be accurate and a more rigorous method should be employed for the blending calculations.

## 4. Comparison and discussion of alcohol dilution calculation methods

For the example used here it was found that the AlcoDens blending calculator and the TTB Table No.6 method gave identical answers, and it has been found in practice that these two methods are very accurate. The Pearson's Square method is not accurate for this type of dilution calculation and it results in additional physical work being required to achieve the target strength.

When alcohol and water are mixed the temperature rises slightly and it takes time for the entire blend to get to a uniform temperature once more. Also, the blending operation often results in micro-bubbles of air (which are invisible to the naked eye) being introduced to the spirit and lowering the density - and thus raising the apparent proof. For these reasons it is common for distillers to allow several hours of settling time after blending before re-measuring the strength. The blends are often left overnight to settle. This can result in significant delays when proofing using the "creeping up on the target" procedure necessitated by the Pearson's Square method.

As electronic scales and load cells have become cheaper and more accurate, there has been a tendency for distillers to do their blending on a mass basis rather than on the volumetric basis used in this example. The TTB Tables do include densities at 60ºF so it is possible to convert all volumetric quantities to masses. This requires a bit more work, but has the advantage that once the required mass has been calculated the temperature of the source is no longer an issue.

When working volumetrically it is necessary to either bring all the raw materials to 60ºF, or to apply correction factors at each step. If you use the TTB method these correction factors have to be applied manually after consulting the Tables, but AlcoDens has the correction factors built in and will automatically compensate for different temperatures.

Performing blending or proofing calculations on a mass basis with AlcoDens is no more work at all because all the corrections are automatically performed internally by the program. AlcoDens will even allow calculations to be done on a mixed basis, with the choice of mass or volume being made individually for each of the sources and the final blend, and any quantities given volumetrically will be automatically corrected for the actual temperatures.