Derivation

Derivation and Explanation of the Brix-Based Calculator For Estimating %ABV

in Fermenting and Finished Beers

Overview

Stoichiometry of Ethanol Production From Maltose

Contribution of Ethanol to Apparent Brix

Measurement of %ABV Using Brix Refractometry

Derivation of Calculator Equations

Evaluation of The Method: Comparing Estimated and Measured %ABV

In this section, a derivation of equations used in the Brix-based calculator is presented.

During fermentation, the consumption of wort carbohydrates causes a decline in measured Brix. However, the produced EtOH, itself, contributes to apparent Brix; therefore the observed decline in Brix is less than would be expected from the mass of carbohydrate consumed. Knowledge of the stoichiometry that relates carbohydrate consumption to EtOH production (and wort mass losses from CO₂ evolution and settled yeast cells), coupled with knowledge of the contribution EtOH makes to apparent brix, allows resolution of the problem.

Stoichiometry

As considered in an earlier section, the stoichiometry used here is as follows:

1 g maltose ---> 0.484 g EtOH + 0.494 g CO₂ + 0.088 g TSS Eq [1]

Contribution of EtOH to apparent Brix

As considered in an earlier section, the contribution of EtOH to apparent Brix is as follows:

0.445 Brix (20˚C) per 1% alcohol by weight (ABW) Eq [2]

Definitions

W_i = initial (pre-fermentation) mass of wort = 100 g

W_f = final (post-fermentation) mass of wort (g per 100 g original wort)

B_i = initial (pre-fermentation) Brix as measured via refractometry (g equivalent sucrose per 100 g original wort)

B_f = final (post-fermentation) Brix as measured via refractometry before conditioning sugar is added (g equivalent sucrose per 100 g final wort)

C = g sugars consumed in fermentation per 100 g original wort

k = 0.445 Brix per 1% ABW (i.e., the factor relating %ABW to apparent Brix)

OG = original specific gravity (15˚)

-- estimated from initial real extract using an empirical equation adapted from Siebert, K.J. [ "Routine Use of a Programmable Calculator for Computing Alcohol, Real Extract, Original Gravity, and Calories in Beer," J. Amer. Soc. Brewing Chemists., Vol 38 (1), 27-33 (1980)].

OG = 1.000019 + 0.003865613*Pi + 0.00001296425*Pi^2 + 0.00000005701128*Pi^3
+ 0.001 (to correct from 20˚C to 15˚C)

Note that OG plays no role at all in the estimation of %ABW or %ABV. The calculator reports the estimated OG merely for those users who might wish it.

FG = final specific gravity (15˚C)

-- estimated from initial real extract and final Brix according to Bonham, L. K. [ "The Use of Handheld Refractometers by Homebrewers," Zymurgy, 43-45, January/February (2001)].

FG = 1.001843 – 0.002318474*Pi – 0.000007775*Pi^2 – 0.000000034*Pi^3 	 
  	                                              + 0.00574*Bf + 0.00003344*Bf^2 + 0.0000000*Bf^3

Note that the only role for FG in the calculator is its use in converting the primarily calculated %ABW to an estimate of %ABV. Numerically, that’s not a very important role. For example, suppose %ABW = 5.00%, density of EtOH at 15˚C is 0.794, and real FG is 1.000. Then %ABV_15˚C = 5.00%*1.000/0.794 = 6.30%. If a user mistakenly used a FG of 1.010 (a huge error of 10 points!), then the user would estimate %ABV_15˚C = 5.00%*(1.010)/0.794 = 6.36%. My point, here, is that it’s not worth fussing with refinements in estimating FG.

BSA = bottling sugar added for conditioning, after fermentation (ounces per 5 gal)

P_i = initial real extract (an estimate of the carbohydrate content of the wort, % by weight)

Note that P_i is generally less than B_i because there are non-carbohydrate components of wort that will contribute to apparent Brix. The issue is discussed later on this page.

P_f = final real extract (an estimate of the remaining carbohydrate content after fermentation, % by weight)

Note that P_f differs from B_f because the apparent Brix of fermented wort will include the aforementioned non-carbohydrate components originally present (and now concentrated in the fermented wort because total wort mass will have declined), as well as EtOH contributions.

%RA = real attenuation (an estimate of the % conversion of wort carbohydrate by yeast during fermentation)

Final Wort Mass

Wort mass declines through fermentation because of mass transferred to gaseous phase (CO₂) and settled solids (yeast cells). Through stoichiometry,

W_f = W_i – mass CO₂ formed – mass yeast formed

W_f = 100 – 0.494*C – 0.088*C = 100 – 0.582*C Eq [3]

Final Brix

Final Brix should equal initial Brix, minus sugars consumed, but plus the contribution from produced EtOH:

B_f = [B_i – C + 0.484*C*k]*[100/W_f] Eq [4]

Note that the factor, [100/W_f] appears because Brix is based on mass percentage in solution, and the final mass of wort differs from the 100 g initial mass of wort. [If this point needs further explanation, consider this: imagine a situation where no sugar was consumed (C = 0), but water was evaporated. In that case, the final Brix would be greater than the initial Brix by virtue of the loss of non-sugar wort mass.]

Sugar Consumption

Eq [3] can be substituted into Eq [4], and the result solved for C, giving:

Eq [5]

%ABW

%ABW = (g EtOH produced per 100 g original wort)*[100/W_f]

= 0.484*C*[100/W_f]

= 48.4*C / [100 – 0.582*C] Eq [6]

with C given by Eq [5].

%ABV (before sugar added for bottle-conditioning)

%ABV_15˚C = %ABW*FG/0.794 Eq [7]

where 0.794 is the density of EtOH at 15˚C. Final specific gravity (FG) is adequately estimated using the empirical relationship earlier presented in Definitions.

%ABV (after bottle-conditioning)

The eventual, post-conditioning contribution to final, %ABV from added corn sugar at bottling is modeled using the same stoichiometry employed in the modeling of fermentation, Eq [1].

%ABV_after = %ABV_before+ 100*[BSA*(342/360)/(667.6*FG)]*0.484*(FG/0.794) Eq [8]

where 667.6 is the number of ounces (mass) in 5 gal of water; 667.6*FG is the number of ounces (mass) in 5 gallons of final wort; (342/360) is a factor to multiply by glucose to get the equivalent maltose [bottling sugar is presumed corn sugar, i.e. glucose]; 0.484 is the g EtOH produced per g maltose consumed. Note that FG actually cancels in the equation. It is shown only as an aid to understanding the construction of the equation.

Initial and Final Real Extracts

Brewers use the term “extract” to denote the carbohydrate content of wort. While the Brix scale is defined on the basis of sucrose solutions (1 Brix unit = 1 weight percent sucrose = 1 gram sucrose per 100 grams solution), the measurement of Brix using a refractometer will be influenced by non-carbohydrate constituents of wort that will affect refractive index. (It is also true that different forms of carbohydrate will affect Brix readings differently, though this factor is a lesser concern.) In the absence of a correction factor specifically determined for a particular wort, analysts commonly use a 1.04 correction factor. That is,

P_i = initial real extract = B_i/1.04 Eq [9]

It must be emphasized that P_i is not used at all in estimating %ABV. Its only use in the calculator is in the estimation of final extract, P_f, which is a measure of residual, non-fermented carbohydrates.

P_f = 100*(P_i – C)/W_f Eq [10]

The factor 100/W_f appears in this equation for the same reason it appeared in Eq [4]: “Extract,” like Brix, is defined on a % mass basis, and while the initial wort mass was 100 g, the final (W_f) is much lower and affects P_f.

Final, real extract is responsible for residual sweetness and “mouth feel.” It can serve as a quality-control parameter. If one of two intended, duplicate batches of beer has a significantly higher %ABV and lower P_f, the likely explanation is that mash conditions caused it to produce a wort with higher attenuation.

Some brewers use an experimental method to arrive at something akin to P_f. A sample of the final brew is boiled to remove EtOH, and then the lost volume is replaced with distilled water. A Brix measurement of the boiled-and-reconstituted sample is interpreted as remaining, non-fermentables. The difference between such a measure and P_f given by Eq [10], is that the Brix of the boiled-and-reconstituted sample will include the non-carbohydrate constituents in the original wort that contributed to Brix – i.e., the very constituents for which the 1.04 (or other) factor is intended to correct. In fact, such constituents will have been concentrated by a factor 100/W_f. For those users who might wish an estimate of this Brix value obtained after boiling/reconstituting, the calculator provides one:

Brix_reconst = 100*(B_i – C)/W_f Eq [11]

Real Attenuation

Real Attenuation is the percentage of the initial carbohydrate in the wort ("initial real extract") that has been consumed by yeast during fermenation. It is a measure of the completeness of conversion. Ale yeasts generally have lower real attentuations than do lager yeasts, because the former cannot fully ferment maltotriose, whereas the latter usually do. Aside from differences in yeast strain, real attention is also affected by pitching rate and mashing conditions.

%RA = 100*(P_i – P_f) / P_i Eq [12]

Contribution of EtOH to Brix; Measurement of %ABV Evaluation of Method