Alcohol By Volume Calculator Updated

Please oh, please stop using archaic homebrew formulas for any brewing purposes.

Unless you are simply a homebrewer.

Read professional works on this topic. Bamforth’s Brewing Materials Book or the ASBC Journal for pros please.

By Gary on May 17, 2021

Hi, the basic formula is the same as in winemaking, is the alternate formula applicable to winemaking too or does it deviate?
Cheers
Sergio

By Sergio Muelle on Jul 18, 2021

Although I would like to agree with Gary, the ASBC methods are not easy to follow without proper lab equipment. Specifically for measuring alcohol, which requires weighing distillate to generate a curve on which you can use refractometer gravity readings. And even then, it confuses me quite a bit. I’m slow with math, so I appreciate calculators like this, even though I know it’s less than ideal.

By Spencer on Nov 23, 2021

I found the article by M. Hall referenced above at this link: https://www.homebrewersassociation.org/attachments/0000/2497/Math_in_Mash_SummerZym95.pdf

I was able to derive the “Alternate Formula” above from the equations in the article. This “Alternate Formula” is an approximation based on some simplifications. For example, it contains a simplified equation for converting SG (Specific Gravity) to E (Extract in degrees Plato): E in deg. Plato = 1000*(SG-1)/4. It also contains a simplification for the calculation of the Real Extract (RE) value (in deg. Plato) by assuming an OE (Original Extract in degrees Plato) of 12.5 (which corresponds to an Original Gravity (OG) of about 1.050).

I calculated tables of % Alcohol By Volume (ABV) for OGs from 1.030 to 1.110 and Final Gravities (FGs) from 0.990 to 1.050 using both the “Alternate Formula” (i.e., Equations 17 and 18 in Hall’s article) and with the “most accurate equations” in Hall’s article (i.e., Eqs. 7, 10, 16, and 18). These two tables agreed pretty well (-0.1 to +0.1 % ABV over most of the range of gravities. However, neither of these tables matched well with Table 2 in Hall’s article “TABLE 2: ALCOHOL PERCENT BY WEIGHT (USING MOST ACCURATE EQUATIONS”. Using the values in Hall’s Table 2 (in % ABW) and converting to % ABV there was a significant difference in the values (up to -1.5 % ABV lower than the values I calculated using the “Alternate Formula” and the “most accurate equations” in Hall’s article).

The closest I came to recreating Hall’s Table 2 was by replacing the OE in the denominator of Hall’s equation #16 with the RE. This gave me % ABV values very close to the % ABV values calculated from in Hall’s Table 2. Since Hall’s Eq. 16 for % ABW comes from the work of Balling done many years ago and has been in use for a long time, I do not believe it is incorrect and therefore it suggests that the values in Table 2 of Hall’s article are incorrect (but the equations in the article are correct). The fact that the % ABV tables I calculated from the “Alternate Formula” and from Hall’s “most accurate equations” agree also suggest that the equations in the article are correct but the values in Table 2 of the article are not. M. Hall is/was a computational physicist at Los Alamos National Laboratory. I would very much trust his research and that the equations in the article are correct. However, anyone can fat finger a wrong entry into a spreadsheet cell equation and get a table of incorrect values that gets quickly pasted into an article. Also, the “Alternate Formula” has been used for many years for many of the online brewing calculators so I would tend to believe it is correct.

Has anyone seen any actual empirical data (% ABV versus OG and FG from experiments) that would validate the “Alternate Formula” for % ABV ( i.e., ABV =(76.08 * (OG-FG) / (1.775-OG)) * (FG / 0.794) ) ?

By Ron on Nov 19, 2023

My previous comment stated that I could not match the % ABW values in Hall’s Table 2 using Hall’s Eqs. 7, 10, and 16. I have now found the problem with my calculations. I fat fingered a coefficient in Eq, 16 into my spreadsheet incorrectly. I had the value as 0.01665 instead of the correct value of 0.010665 ! My calculations using Hall’s “most accurate equations” now give me identical values to those found in his Table 2.

This poses a new question. Why does Hall’s simplified equation for % ABV (the “Alternate Formula” above) give much higher values for % ABV than the values from his “most accurate equations” for high OG cases? I believe it is caused by using the very simplified equation for converting SG to E (i.e. E = 1000*(SG-1)/4 where E is in deg. Plato) to derive the simplified % ABV equation (i.e., the Alternate Formula). For example, the simplified SG to E equation gives an E value of 22.5 P at an OG of 1.090 while Hall’s more accurate method for this conversion give an E value of 21.5 P. The result from the simplified equation is over 4% higher than the accurate method’s result. This error is carries into the simplified Alternate Formula to produce % ABV values that are overly high for high OG cases. For example, the Alternate Formula produces a % ABV of 10.65 for an OG = 1.090 and FG = 1.015 while Hall’s “most accurate equations” gives a value of 9.93 % ABV. The Alternate Formula (Hall’s simplified equation) result is 0.72 % ABV higher than the accurate method, an error of over +7 %.

So if the Alternate Formula produces overly high values for high OG brews (significantly higher than Hall’s accurate method), why is it in use? It does produce fairly accurate % ABV values for OGs under 1.065. Do brewers doing high OG brews (OGs over 1.065) just like the “incorrectly” higher % ABV the Alternate Formula produces or is there empirical data that shows the Alternate Formula (Hall’s simplified method) is valid?

By Ron on Nov 20, 2023

I disagree with the statement that precedes the Alternate Formula equation shown above => “A more complex equation which attempts to provide greater accuracy at higher gravities is:”

In Dr. Hall’s article he clearly states:

‘If you’re not interested in extreme accuracy, then the simple formulas are probably adequate. This is especially true of beers with original gravities less than 1.070, because the two formulas diverge significantly for high specific gravities. You might want to go through all the complicated calculations for a barleywine or a mead …”

You can find a copy of Dr. Hall’s article at:
https://www.homebrewersassociation.org/attachments/0000/2497/Math_in_Mash_SummerZym95.pdf

By Ron on Nov 21, 2023

I agree with the above. The second formula is only accurate near OG 1.050, and the first formula works just as well for a quick estimate. You should provide the full calculation using the Balling formulas instead, it’s pretty straightforward.

1. Measure extracts OE and AE in Plato or convert from gravity readings using accurate tables or formulas.

2. Compute
q = 0.22 + 0.001 OE
RE = (q OE + AE) / (1 + q)
A%w = (OE – RE) / (2.0665 – 0.010665 OE)
A%v = A%w / 0.794

This is easy to implement in a calculator so I see no reason to introduce errors by arbitrary oversimplification.

By Fredrik on Mar 6, 2024

Sorry the last formula should read A%v = A%w FG / 0.794. It’s all in the article linked above.

By Fredrik on Mar 6, 2024

I agree with Ron and Fredrik that the Alternate ABV formula is less accurate than Hall’s most accurate calculations due to the simplifications that are a part of Equations (9) and (11) in Hall’s article. These simplified equations are then inserted into Balling’s empirical formula (Equation (16)) to derive Equation (17) in Hall’s article. Equation (17) is then combined with Equation (18) to provide the Alternate ABV Formula. The Alternate ABV Formula is often presented for use with higher alcohol beers, but the basis for this is not provided and the simplifications that degrade its accuracy are not discussed. I agree with Ron that the Alternate ABV Formula produces overly high ABV values. I agree with Fredrik that Hall’s more accurate calculations should be used.

The simplifications and associated errors inherent in Equations (9) and (11) are as follows. To convert specific gravity (SG) to Plato (E), Hall provides an accurate fit in Equation 7. Hall also provides Equation 9 as a simpler alternative to convert SG to Plato, but it is less accurate. Ron previously highlighted this issue, but more examples are provided here. For a 1.050 SG and below, the simpler Equation 9 is within about 0.1 Plato units. However, at higher SGs the error becomes greater. At 1.060 SG the error is 0.28 Plato units higher. At 1.070 SG the error is 0.47 Plato Units higher. At 1.080 SG the error is 0.69 Plato units higher. At 1.090 SG the error is 0.96 Plato units higher, and this trend continues at higher SG. For Equation (11) the simplification is in the calculation of q (first part of Equation 10). For the calculation of q it is assumed that all beers have an OE of 12.5 Plato (1.0505 SG). This would affect the calculation for higher and lower alcohol beers.

I recreated Hall’s Table 2: Alcohol Percent By Weight (A%w) (most accurate) using the accurate equations summarized by Fredrik. Also used Hall’s accurate SG to Plato conversion (equation 7). I then converted the Table 2 values (as listed in article) to Alcohol By Volume (ABV or A%v) using the formula provided by Hall (equation 18).

A%v = A%w (FG/0.794)

This allowed a direct comparison of the ABV values calculated by Hall’s most accurate method, with the ABV values calculated by the Standard ABV Formula (ABV = (OG – FG) * 131.25). The difference between the two methods was small (calculated as Hall – Standard Formula). For the entire Table 2 the maximum ABV difference was 0.1973. Focusing OG values <1.095 the maximum ABV difference was 0.0961. Focusing on typical ranges for beer (OG:1.035 to 1.095; FG 1.010 to 1.030) the maximum ABV difference was 0.1140. There was also a general trend down the table (increasing OG) transitioning from negative differences to positive differences. These results indicate that Hall’s more accurate ABV method and the Standard ABV Formula provide similar ABV estimates. I was going to post the complete table of differences, but the blog was not conducive to this.

All the formulas have some inaccuracies, even Hall’s most accurate calculations rely on Balling’s empirical relationships (Equations (10) and (16)), as well as Balling’s and Plato’s work to relate the SG of wort to the SG of pure sugar solutions. It is very difficult to find detailed information on the internet on how the ABV formulas were derived, and their limitations (Hall is an exception). Typically, the formulas are just listed with some recommendations for use, without any details of how they were derived or why they are accurate. I did find some background in an article by Samuel Loader that discusses the basis and assumptions of the Standard ABV Formula, assumptions in the Balling empirical relationships, and an alternate ABV method by Anthony Cutaia (search Samuel Loader Calculations ABV). The fact that two ABV formulas (accurate Hall, Standard) with totally different derivations provide very similar results is promising. However, this does not validate their accuracy.

This gets back to the discussion at the beginning of the blog. Gary prefers detailed professional works on this topic (Bamforth’s Brewing Materials Book, the ASBC Journal) for accurate analysis. Spencer likes ABV formulas because they are simpler. Though this is a matter of personal choice and the intended application, it would be beneficial to base the decision on facts. Does anyone have data comparing ASBC methods for ABV (obtained by skilled person) to that calculated by the various ABV formulas? This would identify limitations and accuracy issues with the formulas. It would also promote the use of only the best formulas in applications for which they are best suited.

By Fred M on Apr 17, 2024

In my earlier post I mentioned that an article by Samuel Loader (search Samuel Loader Calculations ABV) discusses the basis and assumptions of the Standard ABV Formula, and the assumptions in the Balling empirical relationships used by Hall. This article also identified an alternate ABV method by Anthony Cutaia that I thought would be worth evaluating. The Cutaia article can be found by searching on the title “Examination of the Relationships Between Original, Real and Apparent Extracts, and Alcohol in Pilot Plant and Commercially Produced Beers”. The search will provide a link to the Wiley Online Library for the article, where you can download the PDF file.

Cutaia’s focus was to use statistical methods to fit data from 532 pilot plant and commercial fermentations to improve constants used in standard brewing equations. The data appeared to be of high quality and was obtained with SCABA instruments, Anton-Paar apparatus’ and/or following European Brewing Convention procedures. Of these, 221 data sets were from the Canadian Malting and Brewing Technical Centre. During my searches, I found an article from the Coors Brewing Company for qualifying the use of SCABA (servo-Chem automatic beer analyzer) for use in their Quality Control Lab and production (search Automatic Alcohol and Real Extract Determination, Coors). However, I am not an expert on current methods used in the brewing industry. I defer to Gary (first post in blog) for this expertise.

Cutaia began with tabulated ASBC data for specific gravity (SG) and corresponding Plato (E). He refit existing conversion equations for SG to E (both directions) to obtain improved equation constants. The results are provided in Equations 10 and 12 in Cutaia’s article (and listed below). For a quick evaluation I used these equations to convert SG to E and then to convert E back to SG. The starting SG and recalculated SG were virtually identical between SGs of 1 to 1.1. These equations use SG-1 and are listed below (Formatting in the blog does not support exponents, see original document if clarification needed).

SG-1 = 1.308e-5 + 3.868e-3 × E + 1.275e-5 × E^2 + 6.300e-8 × E^3 (10)

E = 2.569e2 × (SG-1) + 6.7126e0 × (SG-1)^2 – 1.4482e4 × (SG-1)^3 +
5.1758e5 × (SG-1)^4 – 1.0746e7 × (SG-1)^5 + 1.3011e8 × (SG-1)^6 –
8.5079e8 × (SG-1)^7 + 2.3231e9 × (SG-1)^8 (12)

For perspective, Hall had a different pair of SG to E conversion equations for his most accurate calculations, which he derived from different source materials. Hall’s equations gave very similar results to the equations provided by Cutaia. Good results were obtained when I used Hall’s equations to convert SG to E and then to convert E back to SG, though the results were not as precise as the Cutaia equations.

Cutaia discusses several equations by Balling and others. The final focus was on improved constants for Equation 3b that relates ABW (alcohol by weight) to OE (original E) and AE (apparent final E) (see article for more details). This equation is rearranged below with the constants inserted to directly calculate ABW. OE and AE can be calculated from SG (OG and FG) using Cutaia’s Equation 12 (listed above).

ABW = (0.372 + 0.00357 × OE) × (OE – AE) (3b)

To convert ABW to ABV (alcohol by volume) Cutaia provides Equation (14) that is attributed to two standard brewing references (equation listed below). It is essentially a density correction for the presence of alcohol, which is slightly different than that used by Hall (0.7907 vs 0.794)

ABW = ABV × 0.7907 / SG (14)

Since these relationships are slightly non-linear with respect to SG, Equation 10 is solved from SG and substituted into Equation 14. The equation is then rearranged to solve for ABV. This was shown in the Samuel Loader reference provided earlier and is listed below as Equation (25). However, over the range of typical FG values, the benefit of Equation (25) versus the simpler Equation (14) is insignificant.

ABV = ABW (1.308e-5 + 3.868e-3 × E + 1.275e-5 × E^2 + 6.300e-8 × E^3 + 1)/0.7907 (25)

To evaluate Cutaia’s method, I used Equation (12) to convert OG and FG to OE and AE, respectively. I used Equation (3b) to calculate ABW. Finally. I used Equation (25) to convert ABW to ABV (could have used Equation (14)). In Equation 25, E is AE. The data was used to compare the Cutaia method, Hall’s most accurate method, and Hall’s simplified method (Alternate ABV Formula) to the Standard Formula. The ABV difference was calculated as the Formula (Cutaia or Hall) – Standard Formula. The blog is not conducive to presenting the data as a table, so listed below are calculated ABV differences at OG’s of 1.045, 1.065 and 1.085 (all having a FG of 1.015). At OG of 1.045 the differences from the Standard Formula are as follows, Cutaia ( 0.040), Hall Most Accurate ( 0.014), and Hall Simplified (0.059). The differences are minimal (< 0.1 ABV). At OG of 1.065 the differences from the Standard Formula are as follows, Cutaia (0.0848), Hall Most Accurate (0.0128), and Hall Simplified (0.286). The Cutaia and Hall Most Accurate methods are still in good agreement with the Standard Formula (1.085) the Cutaia method generally falls midway between the Hall Most Accurate and the Hall Simplified methods. In this evaluation the comparison was made to the Standard Formula because it is the most used method. It is not meant to imply anything about accuracy of the Standard Formula.

The fact that the Cutaia method is fit to pilot plant and commercial fermentations may suggest it is more accurate. However, as mentioned in the previous blog post, it is desirable to have independent data measured by ASBC or brewing industry standards as an overcheck of the accuracy of all these calculation methods.

By Fred M on Apr 28, 2024

This is an update to my previous post. All equations in the post were assigned numbers, which are listed in parentheses. Due to formatting issues these equation numbers appear to be appended to the end of the equations. Look carefully to avoid confusion.

The second to the last paragraph in my previous post discussed comparison of the calculation methods. Upon review it was noticed that several key sentences for comparisons at a 1.085 OG were missing. The entire paragraph is restated below.

To evaluate Cutaia’s method, I used Equation (12) to convert OG and FG to OE and AE, respectively. I used Equation (3b) to calculate ABW. Finally. I used Equation (25) to convert ABW to ABV (could have used Equation (14)). In Equation 25, E is AE. The data was used to compare the Cutaia method, Hall’s most accurate method, and Hall’s simplified method (Alternate ABV Formula) to the Standard Formula. The ABV difference was calculated as the Formula (Cutaia or Hall) – Standard Formula. The blog is not conducive to presenting the data as a table, so listed below are calculated ABV differences at OG’s of 1.045, 1.065 and 1.085 (all having a FG of 1.015). At OG of 1.045 the differences from the Standard Formula are as follows, Cutaia (0.040), Hall Most Accurate (0.014), and Hall Simplified (0.059). The differences are minimal (< 0.1 ABV). At OG of 1.065 the differences from the Standard Formula are as follows, Cutaia (0.0848), Hall Most Accurate (0.0128), and Hall Simplified (0.286). The Cutaia and Hall Most Accurate methods are still in good agreement with the Standard Formula (<0.1 ABV), but the Hall Simplified method begins to diverge (0.286 ABV). At an OG of 1.085 the differences from the Standard Formula are as follows, Cutaia (0.313), Hall Most Accurate (0.071), and Hall Simplified (0.679). Only the Hall Most accurate is in good agreement with the Standard Formula (1.085) the Cutaia method generally falls midway between the Hall Most Accurate and the Hall Simplified methods. In this evaluation the comparison was made to the Standard Formula because it is the most used method. It is not meant to imply anything about accuracy of the Standard Formula.

By Fred M on May 1, 2024