# Sugar Content Calculators

### Calculator Descriptions

FermCalc includes three calculators related to sugar content:

1. Chaptalization & Dilution – The Chaptalization & Dilution Calculator determines the amounts of sweetener and/or water to add to a given must to achieve a target specific gravity (SG) and volume.  Alternatively, the resulting SG and volume can be calculated from the specified sweetener and water additions.  The sweetener can be either sugar, honey, or concentrate.
2. Amelioration – Amelioration is the dilution of must with water in order to reduce the acidity or sugar content.  The Amelioration Calculator determines the required additions of water, sugar, and acid to a must to meet the desired acidity and SG levels given the initial acidity, initial SG, and initial must volume.  Alternatively, the target must volume can be specified, in which case FermCalc calculates the initial volume of must required, along with the required water, acid, and sugar additions. (Note: In the Java app, the Amelioration Calculator is included only under the Acid category.)
3. High Brix Measurement by Dilution – The High Brix Measurement by Dilution Calculator determines the specific gravity of a liquid from a specific gravity measurement after it has been diluted with a given volume of water.  This calculation is useful for determining the specific gravity of high-Brix liquids such as honey or concentrates with specific gravities that are outside of the range of standard hydrometers or refractometers.

All of the sugar content calculators are based on the mass balance equations developed below. Calculation details for each of the calculators are described on the individual calculator pages linked above.

### Mass Balance Equations

To derive the necessary equations for these calculations we need to perform a simple mass balance. In other words, we need to honor the constraint that the resulting or target mass of any component is equal to the initial mass of the component plus the added mass of that component.  For water and sugar the mass balances are:

 mwf = mwi + mwa + mwsa (2-1) msf = msi + mssa (2-2)

where:

mwf = final water mass, kg

mwi = initial water mass, kg

mwa = mass of pure water added, kg

mwsa = mass of water in the sweetener (sugar or honey) added, kg

msf = final sugar mass, kg

msi = initial sugar mass, kg

mssa = mass of sugar in the sweetener (sugar or honey) added, kg

We can also write a mass balance on the total mass as:

 mtf = mti + msa + mwa (2-3)

where:

mtf = final total mass, kg

mti = initial total mass, kg

msa = mass of sweetener added, kg

We can relate the masses of the liquids to their volumes and specific gravities as follows:

 mwa = vwaρw (2-4) mti = visgiρw (2-5) mtf = vfsgfρw (2-6)

where:

vwa = volume of water added, liters

vi = initial volume, liters

vf = final volume, liters

sgi = initial specific gravity

sgf = final specific gravity

ρw = density of water = 0.9982 kg/liter at 20ºC

If we know the initial and final specific gravities, we can determine initial and final Brix, or percent sugar by weight, from the Brix conversion equation. Knowing Brix we can then relate the initial and final sugar masses to the initial and final total masses as follows:

 msi = mtiBi/100 (2-7) msf = mtfBf/100 (2-8)

where:

Bi = initial Brix

Bf = final Brix

Substituting equations (2-5) and (2-6) into equations (2-7) and (2-8) gives us:

 msi = visgiρwBi/100 (2-9) msf = vfsgfρwBf/100 (2-10)

If we’re adding honey or concentrate as a sweetener, we’ll need to account for the fact that they contain both sugar and water.  We can express the amount of sugar added as:

 mssa = msaBs/100 (2-11)

Where Bs is the Brix of the sweetener.

Substituting equations (2-9) through (2-11) into equation (2-2) we get the sugar mass balance equation:

 vfsgfρwBf = visgiρwBi + msaBs (2-12)

And substituting equations (2-4) through (2-6) into equation (2-3) we get the total mass balance equation:

 vfsgfρw = visgiρw + msa + vwaρw (2-13)

Equations (2-12) and (2-13) form the basis for all of the sugar calculations.  They’ll just be re-arranged and solved differently depending on what we’re solving for.