FermCalc - Hydrometer Temperature Corrections and Alcohol Content Calculations

Introduction

The Alcohol Content panel has four functions:

  1. Correcting your hydrometer readings for temperature.
  2. Calculating the approximate alcohol content using the specific gravity (SG) drop method.
  3. Calculating the approximate alcohol and residual sugar contents using the hydrometer & refractometer method.
  4. Calculating the approximate alcohol content using the spirit-indication (boiling) method.

To calculate the alcohol content using the SG drop method, you need to know the initial and final hydrometer SG readings. These calculations are only valid if you have not added any additional water or sugar between the initial and final readings. To calculate the alcohol content using the hydrometer & refractometer method, you need simultaneous hydrometer and refractometer readings. To calculate the alcohol content using the boiling off method, you need two hydrometer readings obtained by the procedure described below. FermCalc can make temperature corrections to the hydrometer readings for all three methods.

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Making a Calculation

To make a calculation, follow these steps:

  1. Select Calculation > Alcohol Content from the menu, or select the Alcohol tab at the top of the main window.
  2. Select the type of calculation by selecting either the SG Drop Method tab, the Hydrometer & Refractometer Method tab, or the Boiling Off Method tab.
  3. Select the appropriate units for the input fields using the drop-down menus in the right-hand column.
  4. If the SG Drop Method was selected, enter the initial and final Hydrometer SG Readings of the wine.
  5. If the Hydrometer & Refractometer Method was selected, enter the Refractometer Reading of the wine and the Hydrometer SG Reading of the wine, both of which were measured on the same sample.
  6. If the Spirit Indication Method was selected, enter the initial and final Hydrometer SG Readings of the wine.  See the procedure below for instructions on obtaining the initial and final SG readings.
  7. Where applicable, enter the Reading Temperature at which the hydrometer SG readings were taken, and the Calibration Temperature of the hydrometer(s) used to measure each of the specific gravities.  (If you don't know what these are or if you don't want to do a temperature correction, just leave the reading temperatures and hydrometer calibration temperatures set at their default values.)
  8. If the OIML Calculator was selected, enter the Density and the Temperature to calculate alcohol content, or enter the Alcohol Content and the Temperature to calculate the density.

The temperature-corrected specific gravities and the approximate alcohol content of your wine will appear in the output fields as you type.

All gravity/density values are converted to SG when they are entered, and are subjected to a lower limit of 0.80 and an upper limit of 1.5545. Temperatures must range between 0ºC and 55ºC.  If any of the entered values are outside of these ranges, output fields are highlighted in red and an error message is displayed.

Calculation details are provided below.

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Calculation Details

Hydrometer Temperature Corrections

FermCalc allows hydrometer readings to be corrected for temperature in all three of the alcohol content calculation methods.  The corrections are functions of both the alcohol and solids content of the liquids.  The alcohol correction is determined from the OIML formula (OIML, 1973).  The solids correction is determined from a fit of tables published in the AOAC Official Methods of Analysis (Williams, 1984) for correcting hydrometer btix readings.  These tables have a reference temperature of 20°C, so first the readings need to be corrected for the hydrometer calibration temperature if it's different from 20°C.

The SG measured with a hydrometer is the ratio of the density of the liquid divided by the density of water at the hydrometer calibration temperature, or:

sgaTc = ρT/ρwTc (1)

where

sgaTc = apparent SG measured with a hydrometer with calibration
       temperature Tc
ρT = liquid density at temperature T, kg/L
ρwTc = water density at temperature Tc, kg/L

To correct the reading to a reference temperature of 20°C, we simply need to multiply the hydrometer reading by the ratio of the water density at the old calibration temperature to the water density at 20°C, or:

sga20 = sgTc (ρwTc/ρw20) (2)

where

sga20 = apparent SG corrected to a reference temperature of 20°C
ρw20 = water density at 20°C, kg/L

Water densities are calculated using the OIML formula assuming 0% alcohol.

After correcting for the hydrometer calibration temperature, we can correct for the sample temperature.  The coefficient of thermal expansion for aqueous ethanol-sucrose solutions, which determines the change in density for a given change in temperature, depends on both the ethanol and sucrose levels, but mainly on the ethanol level (Espejo and Armada, 2011).  To correct the measured SG, FermCalc determines a composite correction factor by first calculating three density correction factors as follows:

dw = ρw20 / ρwT (3)
da = ρa20 / ρaT (4)
ds = ρs20 / ρsT (5)

where

dw = density correction factor for pure water
da = density correction factor an ethanol-water solution with the same alcohol
        content as the liquid
ds = density correction factor a sucrose-water solution with the same sucrose
        content as the liquid
ρw20 = water density at 20°C, kg/L
ρa20 = ethanol-water solution density at 20°C, kg/L
ρs20 = sucrose-water solution density at 20°C, kg/L
ρwT = water density at the measurement temperature, kg/L
ρaT = ethanol-water solution density at the measurement temperature, kg/L
ρsT = sucrose-water solution density at the measurement temperature, kg/L

The corrected SG sgc20 is then calculated as:

sgc20 = sga20dads/dw (6)

The densites for water and ethanol-water solutions are based on the OIML formula.  The change in density of sucrose-water solutions with temperature is based on the following curve fit of the AOAC brix correction tables:

B = Ba + [a(T - 20)2 + b(T - 20)] (7)

where:

B = corrected Brix
Ba = apparent Brix at temperature T
T = measurement temperature, °C
a = 1.4525·10-7Ba2 - 2.5256·10-5Ba + 1.2495·10-3
b = -6.6927·10-6Ba2 + 9.6012·10-4Ba + 4.4174·10-2

The plot below compares the AOAC data with the calculated Brix corrections from equation (7) above.

temp_correct.gif

Some of the alcohol content calculation methods below are based on SG values at 60°F instead of 20°C, in which cases the appropriate densities are substituted into the above equations.  FermCalc only reports the SG values corrected to 20°C, so the actual SG values being used in the equations will be slightly different than the reported SG values in these cases.

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Alcohol Content by the Specific Gravity (SG) Drop Method

FermCalc includes four different methods for calculating alcohol content from the drop in SG that occurs during fermentation.  All four of these methods require hydrometer measurements of the initial and final specific gravities. (These methods do not work with refractometer measurements.)  Agreement between the methods is generally excellent.

Dr. William Honneyman (1966) compared alcohol levels calculated using the SG drop method to the distillation data of Thorpe & Brown (1914) and found that it gives results "reasonably comparable with tests by distillation, provided graded factors suited to each drop in gravity are used".  While he did not recommend a specific way to calculate the graded factors, he provided an important validation of the approach in general.

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Berry Method

This is the most commonly used SG drop method, and is described on pages 79-80 of First Steps in Winemaking by C. J. J. Berry (1987).  It estimates the alcohol content by dividing the drop in SG by the constant 0.00736, or:

av = (sgi - sgf) / 0.00736 (8)

where

av = alcohol content, % by volume
sgi = initial SG
sgf = final SG

Ritchie Products Co. (2004) claims to have compared the results of equation (8) to the results of gas chromatography for a wide range of wines, and found that the results were within 0.3% vol/vol.

This method has a temperature basis of 15.56°C (60°F).

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Duncan & Acton Method

This method is described on pages 64-66 of Progressive Winemaking by Peter Duncan and Bryan Acton (1967).  The Duncan & Acton method calculates the alcohol content from the initial and final specific gravities divided by a factor F that is a function of the corrected initial SG. The equations are as follows.

av = 1000(sgi - sgf) / F (9)
F = 7.75 - 3000(sgc - 1.0) / 800 (10)
sgc = sgi - 0.007 (11)

where

av = alcohol content, % by volume
sgi = initial SG
sgf = final SG
F = conversion factor
sgc = initial SG corrected for non-sugar solutes

Combining equations (9) through (11) above yields the following equation:

av = 1000(sgi - sgf) / [7.75 - 3.75(sgi - 1.007)] (12)

This method has a temperature basis of 15.56°C (60°F).

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Balling Method

The Balling method is normally used for beer but gives results that agree very well with the other methods. The equations used in FermCalc were taken from Michael Hall's article "Brew by the Numbers" in the Summer 1995 issue of Zymurgy magazine.  In the original equations, specific gravities are expressed as degrees Plato, which FermCalc treats as being the equivalent as degrees Brix. The method requires the calculation of a parameter called "Real Extract", which is an estimate of the residual solids content after fermentation has finished, as follows:

q = 0.22 + 0.001Bi (13)
RE = (q·Bi + Bf) / (1 + q) (14)

where

q = attenuation coefficient
RE = real extract
Bi = initial Brix
Bf = final Brix

The alcohol content (% by weight) is then calculated as:

aw = (Bi - RE) / (2.0665 - 0.010665Bi) (15)

where aw is the alcohol content in % by weight.

The result of equation (15) is then converted to % alcohol by volume as described here.

This method has a temperature basis of 17.5°C (63.5°F).

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Cutaia, Reid, & Speers Method

Cutaia, Reid & Spears (2009) analyzed data from 532 beers to develop equation (12) below relating alcohol content to the initial and final specific gravities.

aw = (Bi - Bf)(0.372 + 0.00357Bi) (16)

The alcohol content of the beers ranged from 3% to 7% by weight (approx. 3.8% to 8.7% by vol.).  As with the Balling equation above, the specific gravities for the analyzed beers were expressed in degrees Plato, which is assumed by FermCalc to be the same as degrees Brix.

The result of equation (16) is then converted to % alcohol by volume as described here.

This method has a temperature basis of 20°C (68°F).

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Estimation of Solids Content (True Brix)

After we know the alcohol content, we can estimate true Brix, which represents solids content in % by weight, by using the model developed by James Hackbarth (2011), which is described below. This is done by treating the specific gravity SG and alcohol content A as known values and iteratively solving equations (38) through (44) below for the true Brix E.  FermCalc uses the alcohol content calculated by the Duncan & Acton method for this calculation.

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Alcohol Content by the Hydrometer & Refractometer Method

FermCalc includes four methods of estimating alcohol content and residual solids (true Brix) from simultaneous refractometer and hydrometer readings.  All of the methods are designed to be used after fermentation, but they should be able to yield reasonable estimates of alcohol content during fermentation as long as there is enough alcohol to affect the measurements and the sample is degassed enough that the measurements are not affected by dissolved CO2.

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Rogerson & Symington Method

Rogerson & Symington (2006) developed a method to estimate alcohol content and residual solids (true Brix) from based on refractometer and hydrometer readings on 35 port wines. In the words of the authors, "It is not applicable for the analysis of dry wines, whether fortified or not, which contain insufficient soluble solids for Baumé determination by hydrometer, and is yet to be evaluated for sweet table wines, such as sauternes."  However it is included in FermCalc because many home winemakers seem to find it useful for monitoring fermentation progress and calculating alcohol content.

FermCalc first converts the hydrometer reading sg to degrees Baumé using the following equation.

= 145 - 145/sg (17)

where is degrees Baumé.

Alcohol content is then calculated as:

av = 1.646Ba - 2.703 - 1.794 (18)

where

av = alcohol content, % by volume
Ba = refractometer Brix reading (apparent Brix)

True Brix, Bt, which represents the estimated residual solids content in % by weight, is then calculated as:

Bt = Ba - 0.358av (19)

This method has a temperature basis of 20°C (68°F).

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Son et al. Method

H. S. Son et al. (2009) developed the following six empirical equations based on refractometer, hydrometer, and alcohol content measurements on 30 wines before and during fermentation.

Bt = -0.352Bi + 1.264Ba + 2.006 (20)
Bt = 0.201Bi + 0.782Bh - 0.921 (21)
av = 0.967Bi - 0.766Ba - 5.793 (22)
av = 0.625Bi - 0.457Bh - 3.814 (23)
Bt = 0.529Ba + 0.457Bh - 0.344 (24)
av = 0.833Ba - 0.996Bh + 3.927 (25)

where

av = alcohol content, % by volume
Bi = initial Brix reading
Ba = refractometer Brix reading (apparent Brix)
Bh = hydrometer Brix reading
Bt = true Brix (% solids by weight)

Equations (24) and (25) allow calculation of alcohol content and true Brix directly from hydrometer and refractometer readings.  However, I found these equations to be inaccurate, yielding estimates of alcohol content that appear too high in the lower-alcohol range and too low in the upper range.  Instead of using equations (24) and (25), I developed alternative equations from equations (20) through (23) that appear much more accurate. Combining equations (22) and (23) to eliminate Bi gives:

av = 1.400Ba - 1.292Bh + 0.197 (26)

Combining equations (20) and (21) to eliminate Bi gives:

Bt = 0.459Ba + 0.498Bh + 0.143 (27)

Equations (26) and (27) are used by FermCalc to calculate alcohol content and true Brix.

This method has a temperature basis of 20°C (68°F).

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Roesener Method

This method was published online by Werner Roesener (2001) and is very popular among home winemakers, but there is no documentation regarding the derivation of the equations.  My testing indicates that it yields results that are very similar to the other methods.

Simplifying the original equations we get:

av = 1.5184Ba + 365(1.0 - sg) (28)
s = 2520(sg - 1.0) + 3.1853av (29)

where s is the dissolved solids content in g/L. FermCalc converts the solids content in g/L to true Brix in percent by weight using equation (30) below.

Bt = s/sg/10 (30)

This method has a temperature basis of 15.56°C (60°F).

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Barth & Race Method

This method was originally developed by Georg Barth (1905) in Germany for analyzing beers. The original equations are:

aw = 759.8(ri - 1.3330) - 292.3(sg - 1.0) (31)
Bt = 336.6(ri - 1.3330) + 130.3(sg - 1.0) (32)

Where ri is the measured refractive index.  The equations were later modified by J. Race (1908) to yield more accurate results for beers with alcohol contents greater than 4.5% by weight.

aw = 778(ri - 1.3330) - 290(sg - 1.0) (33)
Bt = 350(ri - 1.3330) + 130(sg - 1.0) (34)

FermCalc uses equations (33) and (34) to calculate alcohol content and true Brix because they were intended for higher alcohol concentrations and might be more applicable for winemaking calculations.  The result of equation (32) is converted to % alcohol by volume as described here.

This method has a temperature basis of 15.56°C (60°F).

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Alcohol Content by the Boiling (Spirit-Indication) Method

This method was first proposed by M. E. Tabarié in 1830 as a simplified alternative to the distillation procedure.  It is based on the principle that alcohol causes the same depression in SG in wine as it does in pure water.

The method involves evaporating (boiling off) a portion of the wine sample until all of the alcohol is evaporated, and then replacing the evaporated volume with distilled water.  The difference between the specific gravities of the wine and the volume-corrected residue are then used to estimate the SG of the distillate, which represents the SG of a pure water/ethanol mixture, from which the alcohol content can be estimated. The experimental procedure is summarized below.

  1. Measure the SG (sgw) of the wine to be tested.
  2. Take a sample of about 250-500 mL (1-2 cups) of the wine and boil the sample down to approximately half of its original volume to drive off all of the alcohol.
  3. Allow the boiled residue to cool to room temperature.
  4. Add distilled water to the residue until the total volume is restored to the original sample volume.
  5. Measure the SG of this volume-corrected residue sgr, which will be greater than sgw because the alcohol has been replaced by water.

It is recommended that a narrow-range hydrometer be used for the SG measurements since small errors in these measurements can result in large errors in the results.

In addition to estimating the alcohol content, we can also calculate the solids content (true Brix) of the wine from the SG of the volume-corrected residue.  First we just need to convert the residue SG measurement to a Brix value by using the Brix conversion equation. This conversion yields the solids content in % by weight of the residue. We then need to convert this value to the solids content in % by weight of the wine by multiplying by the ratio of specific gravities, or:

Bt = Btr·sgr/sgw (35)

where

Bt = true Brix (solids content) of wine, % by weight
Btr = true Brix (solids content) of the volume-corrected residue, % by weight
sgw = SG of wine
sgr = SG of the volume-corrected residue

The four methods that FermCalc uses to calculate the alcohol content from the SG measurements are described below.

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Tabarié (Division) Method

Tabarié originally proposed estimating the SG of the distillate from the ratio of the wine and residue specific gravities, or:

sgd = sgw/sgr (36)

where

sgd = SG of distillate
sgw = SG of wine
sgr = SG of the volume-corrected residue

FermCalc determines the alcohol content in % by volume from sgd by using the OIML formula (OIML, 1973).

This method has a temperature basis of 20°C (68°F).

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Blunt (Subtraction) Method

T. P. Blunt (1891) suggested that the Tabarié division formula always underestimates alcohol content, and suggested calculating sgd from the difference between sgr and sgw instead of the ratio, or.

sgd = 1.0 - (sgr - sgw) (37)

FermCalc uses the OIML formula to estimate alcohol content in % by volume from the results of equation (37).

S. Harvey (1892) presented experimental results suggesting that Blunt's formula is more accurate than Tabarié's.  However, a few pages later in the same issue of The Analyst, A. H. Allen presented data for sugar and alcohol solutions ranging from 24% to 53% alcohol by weight suggesting that the Tabarié formula was more accurate.  While Blunt is generally credited with showing that the subtraction formula is more accurate than the division formula, the subtraction formula was clearly in use well before Blunt wrote his paper (Mulder, 1857).

This method has a temperature basis of 20°C (68°F).

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Honneyman Method

This method is described on pages 124-126 of The Art of Making Wine (Anderson & Hull, 1970), and is attributed to the researches of Dr. William Honneyman (1966).  This method is really identical to the Blunt subtraction method described above, so it might seem redundant to include it here.  The main difference is that this method uses Dr. Honneyman's table below to estimate the alcohol content, and in the Blunt method we're using the OIML equation. 

I obtained a copy of Dr. Honneyman's book and determined that his table is based on the alcoholometric tables of Thorpe (1915), which have a temperature basis of 60°F.  With this information I was able to extend his table to higher alcohol levels and increase the resolution.  Now with the appropriate temperature correction the Honneyman method gives results that are nearly identical to the other boiling methods.

sgr - sgw Alcohol
Content
(% vol/vol)
sgr - sgw Alcohol
Content
(% vol/vol)
sgr - sgw Alcohol
Content
(% vol/vol)
0.0000 0.0 0.0090 6.4 0.0180 14.1
0.0015 1.0 0.0100 7.2 0.0190 15.1
0.0020 1.3 0.0110 8.0 0.0200 16.0
0.0030 2.0 0.0120 8.8 0.0210 17.0
0.0040 2.7 0.0130 9.7 0.0220 18.0
0.0050 3.4 0.0140 10.5 0.0230 19.0
0.0060 4.1 0.0150 11.4 0.0240 20.0
0.0070 4.9 0.0160 12.3 0.0250 21.0
0.0080 5.6 0.0170 13.2 0.0260 22.0

This method has a temperature basis of 15.56°C (60°F).

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Hackbarth Method

James Hackbarth (2009) showed that the Tabarié approach works reasonably well for dry wines and low-extract beers, but is inaccurate for beverages with higher alcohol and sugar concentrations due to solute-solute interactions that take place at the higher concentrations, an idea that was first proposed by Leonard (1897).  Based on extensive laboratory experimentation and detailed analysis of the results, Hackbarth (2011) developed a new model for estimating the SG of a mixture from its sucrose (extract) and alcohol concentrations.  Since the spirit-indication procedure gives us measurements of the wine SG and the extract concentration (true Brix), we can treat these as known quantities and use the Hackbarth model to solve for the alcohol content.

The Hackbarth model utilizes the following equations.

Z = -1.020733575·10-2E1A0.5 + 6.223951696·10-4E2A0.5
      - 3.463023825·10-6E3A0.5 + 7.234029153·10-3E1A1
      - 4.496851490·10-4E1A2 + 9.045618812·10-6E1A3
      - 5.427265684·10-8E1A4 - 1.719663278·10-4E2A1
      + 2.302760700·10-9E3A3
(38)
Eb = E·100/(100 - A) + Z (39)
Ab = A·100/(100 - E) + Z (40)
SGWE = (100 - Eb)/[100/fe(Eb) - Eb/fe(100)] (41)
SGWA = (100 - Ab)/[100/fa(Ab) - Ab/fa(100)] (42)
SGW = SGWE · SGWA (43)
SG = 100/[E/fe(100) + A/fa(100) + (100 - E - A)/SGW] (44)

where

Z = correction for solute interactions
A = alcohol concentration, % by weight
E = sucrose concentration (true Brix), % by weight
SG = SG of the ternary solution (wine)
Ab = alcohol concentration in the binary solution, % by weight
Eb = sucrose concentration (true Brix) in the binary solution, % by weight
fa(A) = 11th order polynomial for calculating SG from alcohol
        concentration based on the OIML general formula (OIML, 1973)
fe(E) = 10th order polynomial for calculating SG from sucrose
        concentration based on AOAC Plato tables
SGWA = SG of water in the binary solution of alcohol
SGWE = SG of water in the binary solution of sucrose
SGW = SG of water in the ternary solution (wine)

FermCalc solves equations (38) through (44) iteratively using the alcohol content calculated from the Blunt model as the initial estimate.  The iteration loop is repeated until the difference between the value of SG calculated by equation (44) and the wine SG sgw is less than 10-8.

This method has a temperature basis of 20°C (68°F).

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OIML Calculator

The OIML Calculator uses the general formula for calculating the densities of mixtures of ethanol and water found in International Recommendation 22: International Alcoholometric Tables by the International Organisation of Legal Metrology (OIML, 1973).  The general formula has the form:

ρ = A1 + ΣiAipi-1 + ΣjBj(t-20)j + ΣmΣnCm,npn(t-20)m (45)

where

ρ = density of ethanol/water mixture, kg/m3
p = ethanol concentration by weight, fraction
t = temperature, ºC
A, B, C = coefficients
i = 2 to 12
j = 1 to 6
m = 1 to 5
n = 1 to 11

Given an alcohol content and a temperature, equation (45) can be solved directly for the density of the mixture.  Given a density and a temperature, FermCalc calculates the associated alcohol content using an iterative technique.

© 2007-2014 Steve Gross
Last updated 20 December 2014.