A briefer version of this article appears in the Dec, 2010-Jan. 2011 issue of The Home Wine Press, our club newsletter. The version below provides a more thorough explanation of the subject.

PA Revisited—The Final Word

Rethinking Our Approach to Calculating Potential Alcohol

By Dale Ims

What follows is essentially a collection of work disproving an equation I had found which relates the potential alcohol (PA) content of a juice to its sugar content (measured in Brix). The proofs are in what amounts to the chronological order in which I developed them, and in which I presented them to Roger Boulton, the first-named author of the book in which the equation first appeared. It was only after I collected all the proofs into the document which appears in slightly modified form below and emailed it to Boulton that he acknowledged that the equation was incorrect.

Why PA Revisited

Last year I submitted an article to the RAHW newsletter (The Home Wine Press Feb.-March 2010) in which I related how different books, pamphlets and on-line references predicted different levels of alcohol from the complete fermentation of a juice with a given Brix or specific gravity (SG). In that article I showed how variable those predictions of potential alcohol were, and presented what seemed to me to be a totally plausible relationship between the sugar content of a juice and the alcohol content of the post-fermentation liquor. This seemingly plausible relationship was not something that I came up with, but something I found in a book by Jean Jacobsen (Introduction to Wine Laboratory Practices and Procedures), and which that author extracted from an earlier, well-respected book (Principals and Practices of Winemaking, by Roger B Boulton, et al). The relationship of interest is:

Equation 1.

Potential Alcohol = 0.59 * (Brix – 3).

As described in the previous article, the (Brix – 3) term in the above equation recognizes the fact that some of the components in the juice which effect the SG/Brix reading done by hydrometer or refractometer are non-fermentable and cannot be converted to alcohol in spite of the fact that they affect the SG of the juice. Also as described in the previous article, the value of the (0.59) coefficient seemed to have a theory-based origin, with the value of 0.59 resulting from the product of the 51.1% efficiency of the sucrose/fructose conversion to ethanol, multiplied by the 92% efficiency of the yeast and divided by the density of ethanol (0.789 gm/ml) to convert the weight% to volume%.

Figure 1. Differences between potential alcohol values calculated via the use of Equation 1 (“form”) and those calculated via my spreadsheet process (“calc vol%” in magenta).

This all seemed very reasonable and attractive, but I've come to realize and prove that it's not correct! I discovered this problem while trying to do some spreadsheet calculations of the fermentation process to see how the specific gravity (SG) of the must changes as the sugar is converted to alcohol. Those calculations utilized the incremental conversion of simple sugars to alcohol with a 47% efficiency (the 51.1% chemical efficiency above times the 92% yeast efficiency, also above). In each step of this multi-step process, a quantity of sugar is removed from the juice/must and replaced by a volume of ethanol which has a mass equal to 47% of that of the sugar removed. The process requires that you keep track of the amounts (masses and volumes) of all of the components, but if you do it all right, you get the densities and SG's at all of the intermediate conditions.

I found that the final alcohol contents computed in this step-wise process were consistently higher than those predicted by the above equation; the graph in Figure 1 illustrates the differences between potential alcohol values calculated via the use of Equation 1 and those calculated via my spreadsheet process. Note in Figure 1 that the differences between the two methods of calculation (those done with the use of the equation labeled “form” and those done via the spreadsheet labeled “calc vol%”) are larger at the higher Brix levels.

After exhaustively checking my calculations without finding an error and thinking long and hard about what the difference could be, it occurred to me that in Equation 1 above, it is implicitly and incorrectly assumed that the mass of the fermenting mixture remains constant throughout the fermentation.

Let's look at that assertion for a moment. As stated above, the 0.59 coefficient is the result of multiplying a chemical-conversion efficiency (51.1%) times a yeast efficiency (92%) to get an overall conversion efficiency of 47% and then dividing by the density of ethanol to convert the resulting weight% to volume%. The significance of the 47% is that the total weight of ethanol resulting from fermentation of the (fermentable) sugars in a given mass of juice is equal to that fraction (47%) of the weight of those fermentable sugars. Clearly, if the total mass of the wine after fermentation were equal to that before fermentation, then the weight% ethanol in the wine would be equal to 47% of the weight% of the fermentable sugar in the starting liquid. However, there is a considerable loss of mass —as much as 10% to 12%—during the fermentation of a given juice due to the evolution of CO₂. Therefore, given the loss of mass during fermentation, any weight% calculation which is based upon the initial mass of the juice will be too low; this is because weight% is calculated by dividing the weight of the selected component (ethanol here) by the total weight of liquid, and a reduction in the total weight results in an increase in the weight%.

For some reason, the alcohol contents in beverages have been specified in terms of the volume% of ethanol. As indicated previously, the 0.59 coefficient in Equation 1 includes a division by the density of ethanol for the purpose of converting the weight% (which we assert is incorrectly calculated) to volume%. Now, strictly speaking, the conversion of weight% to volume% is not as simple as dividing by the density of the ethanol—although it's not a bad approximation. In addition, water and ethanol have an anomalous mixing characteristic, with mixtures of the two liquids always having a combined volume less than the sum of the volumes of the two components.

Figure 2. Three different processes for the conversion of weight% ethanol to volume%: simple volume addition (“App Vol%”), division by the density of ethanol (“1/p”), and  the excess-volume-corrected process (“act vol%”).

It is worthwhile at this point to compare the approximate methods of converting weight% to volume% to an accurate technique utilizing the excess-volume concept. The graph in Figure 2 compares three different processes for the conversion of weight% ethanol to volume%: via simple volume addition (“App Vol%”), via the approximation to simple volume addition accomplished by division by the density of ethanol (“1/p”), and via the excess-volume-corrected process (“act vol%”). Note that the differences among the converted values are not large, and that the results obtained from the simple addition of the components are low compared to the actual values while those obtained by dividing by the density of ethanol are high.

A Correction Factor

Since I had concluded that the error in Equation 1 is due to the equation’s failure to account for the fact that there is a significant loss of weight or mass during fermentation, it seemed that we might be able to develop a correction factor based on the ratio of the starting weight or mass of the juice divided by the final weight or mass of the wine. If we start with an initial amount of juice with a given Brix reading, from the information in the first page of this document, we should expect that the weight or mass of alcohol produced during fermentation will be 47% of the weight or mass of the fermentable sugar in the juice. We should expect to lose the remaining 53% of the weight or mass of the fermentable sugar to CO₂ evolution and the feeding of the yeast. If we do a little algebra on the simple equation which says in mathematical terms that the sum of the weight or mass of the wine produced plus the weight or mass of the components lost during the fermentation process should equal the initial weight or mass of the starting juice, we can arrive at a simple expression for the ratio of the initial weight or mass to the final weight or mass:

Equation 2.

m_i / m_f = {1 / [1 – 0.0053 * (Brix – 3)] }

Figure 3. Differences between potential alcohol levels calculated via Equation 1 (“PA  eqn” in dark data points and line) and the multiplication of the results of Equation 1 by the correction factor value resulting from Equation 2 (“corr PA” in magenta).

In the graph in Figure 3, we have plotted the potential alcohol levels resulting from juices with a range of Brix values; the levels of potential alcohol were determined via Equation 1 (the dark data points and line) and via the multiplication of the results of Equation 1 by the correction factor value resulting from Equation 2 (the magenta data points and line). The similarity between Figure 3 and Figure 1 is obvious, and in fact, the data set of the values from the potential alcohol equation (Equation 1) are common to both graphs. Any differences which might exist between the values generating the upper curve in Figure 3 and the upper curve in Figure 1 are due to the differing ways in which the conversions of weight% to volume% were made: the calculations here incorporate the (1/density) conversion while those for Figure 1 utilized the excess-volume correction.

Another Continent Heard From

The book Principles and Practices of Winemaking (Boulton, et al) includes several other formulae for estimating potential alcohol in addition to that shown as Equation 1. In particular, the book states that a formula widely used in Europe is:

Equation 3.

Potential ethanol in vol% = 0.059 * [2.66 * Oe – 30]

where Oe is the Oechsle of the juice and is defined as:

Oe = 1000 * (SG_20/20 – 1.000).

Thus, to use this equation to calculate potential alcohol of a juice, one must measure the specific gravity of the juice at a temperature of 20°C with a hydrometer standardized at 20°C, subtract 1.000 from the hydrometer reading and multiply the difference by 1000 to first calculate the Oechsle of the juice and then enter that value into Equation 3. In Equation 3, the Oechsle of the juice is multiplied by 2.66 and then 30 is subtracted, with the subtraction done to compensate for non-fermentable components in the juice.

Figure 4. Results from Equation 3 (“PA-Oe”), from my spreadsheet calculations including the excess-volume correction (“Act Vol%”), from Equation 1 with the correction factor applied (“corr PA-B”), and from Equation 1 alone (“PA-Brix”).

I found a table of data for sugar/water solutions which conveniently included SG values and °Brix in addition to a number of other specifications for the same solutions, and was able to calculate the potential alcohol value for a 24 °Brix (SG = 1.099) juice using Equations 1 and 3. The results were 12.4% and 13.8% respectively for Equations 1 & 3! That was an interesting result, so I did the PA calculations for the full range of Brix readings as in the graphs above. The graph in Figure 4 shows the results of those calculations; there are four sets of data plotted on this graph: the results from Equation 3 (“PA-Oe”), those from my spreadsheet calculations including the excess-volume correction (“Act Vol%”), those from Equation 1 with the correction factor applied (“corr PA-B”), and those from Equation 1 alone (“PA-Brix”).

It is apparent from Figure 4 that the outlier data set is the one produced by Equation 1, while the other three curves are tightly clustered. It would seem that equations or formulas used in different countries to calculate the same result should give essentially equal values, and the rather significant difference between the data set resulting from use of Equation 1 (“PA-Brix”) and Equation 3 (“PA-Oe”) is troubling. On the other hand, the good agreement between the predictions of Equation 3 and my spreadsheet calculations as well as my corrected version of Equation 1 lends credence to my calculations and the conclusions upon which they are based.

Go to the Source

The book Principles and Practices of Winemaking (Boulton, et al) cites a paper by George Marsh (1958) in the American Journal of Oenology and Viticulture as a reference for Equation 1, and I managed to get a copy of that paper. That paper is largely a review of previous work on the topic of alcohol yield and seems to be primarily focused on the origin of the 47 weight% yield of alcohol from sugar via fermentation. The paper does seem to me to be more than a little confusing, but it is clear that nowhere within that paper is there an equation of the form of Equation 1 explicitly presented. It is apparent that in at least one portion of the Marsh paper an equation of the form of Equation 1—if not identical—was used to calculate some potential alcohol levels from sugar concentrations. However, in other parts of the paper it is implied and indicated that an equation of the form of Equation 1 wherein the sugar concentration is specified in gm/L rather than Brix should be used to calculate potential alcohol levels.

Having recognized the (admittedly non-exclusive) use of gm/L in the calculation of potential alcohol levels in the Marsh paper, it occurred to me that one could utilize dimensional analysis to determine what kind of sugar specification should be used in an equation of the form of Equation 1. As indicated at the beginning of this document, the 0.59 coefficient in Equation 1 is the result of the multiplication of a chemical-conversion efficiency (51.1% by weight) times a yeast efficiency (92% by weight), with a subsequent division by the density of ethanol to (approximately) convert the answer to volume%. If we look at the units on each side of the equal sign in Equation 1, we see that on the left side we have just Potential Alcohol—supposedly in units of ml/ml or L/L. In fact, the left side of Equation 1 is unitless, and for the equality indicated by the equal sign in the equation to be true, the right side of the equation must likewise be unitless.

The right side of Equation 1 has all the units that went into the 0.59 coefficient times the units of Brix. In the above paragraph we stated that the chemical-conversion efficiency is an efficiency and thus unitless. Likewise, the yeast efficiency is unitless. Therefore, the units of the coefficient are solely determined by those of the density of the ethanol—which is the divisor in the calculation, so the units are ml/gm. Now we can’t cancel out the units of the coefficient (ml/gm) by multiplying by a unitless specifier like Brix. We can, however, make the units cancel out if we specify the sugar concentration in terms of gm/L—as might be inferred from the Marsh paper!

We can test how well the values of potential alcohol predicted by the modified version of Equation 1 (using gm/L for sugar concentrations) compare to those predicted by the unmodified Equation 1—and to those predicted by Equation 3 if we have a set of data for sugar solutions with SG, Brix and gm/L specifications. I have such a data set, and so I was able to do that comparison. In addition, from the data set I was able to determine that the subtraction of 3 from the Brix reading of the juice in Equation 1 can be carried over to the modified version of Equation 1 by noting that a sugar/water solution of 3° Brix is equivalent to one of 30.3gm/L. Thus, the modified form of Equation 1 is:

Equation 4.

Potential Alcohol = 0.059 * (gm/L – 30.3)

where the factor of 10 difference in the coefficient value is due to the ml to L conversion and the desire for the result to be specified as a percent.

Figure 5. Potential Alcohol results obtained using Equation 1 (“PA-Brix”), Equation 3 (“PA-Oe”) and Equation 4 (“PA-gm/L”).

The graph in Figure 5 compares Potential Alcohol vs. Brix results obtained using Equations 1, 3 and 4. Note here that the results from use of Equation 4 (“PA-gm/L”) are very similar to those from Equation 3 (“PA-Oe”) while those from Equation 1 are standing alone. It is clear that Equation 4—with its use of gm/L to specify the sugar content—produces results which are more consistent with those from Equation 3.

It has not escaped my notice that Equation 3 and Equation 4 are virtually identical. In fact, it can be shown that the sugar concentration of a sugar/water solution—specified in gm/L—is equal to the product of a proportionality factor of value 2.62 times the Oechsle of the solution! I do not know the origin of the discrepancy between this 2.62 and the 2.66 in Equation 3, but consider the difference to be of little importance; however, the differences seen between the “PA-gm/L” and the “PA-Oe” curves in Figure 5 are due to this 2.66 vs 2.62 discrepancy.

Conclusions

We have shown in several different ways that an equation relating potential alcohol levels to the degrees Brix of a juice, presented in the widely respected book Principles and Practices of Winemaking, is invalid. Ultimately it was perhaps the comparison of the potential alcohol levels predicted by the invalid equation to those predicted by a formula widely used in Europe for the same purpose that proved the point. Or maybe it was the careful reading of the reference from which the subject equation arose, aided by the dimensional analysis and the recognition that use of a different specifier for sugar concentration was required. I was convinced when I recognized that the equation of interest here calculates a percentage based on the mass of the starting juice, and that there is a change in the mass during fermentation.

What To Do?

This is not a national emergency! The errors in the predicted levels of potential alcohol—and the effects that might result in wines produced from this erroneous basis—are not large, although they may be as high as 1.4% (12.4 vs. 13.8%) for a 24°Brix juice. Regardless, it seems to me that this aspect of this important book should be corrected; there is more than enough misinformation out there already—especially for the home winemaker!

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