# Fortification Calculators

### Calculator Descriptions

FermCalc contains three calculators that are related to wine fortification:

1. Post-Fermentation Fortification – The Post-Fermentation Fortification Calculator determines the amounts of fortifier and sweetener to add to a given wine to yield a fortified wine with the desired alcohol content, sugar content, and volume.  Alternatively, the resulting alcohol content, sugar content, and volume can be calculated from the specified fortifier and sweetener additions.  The sweetener can be either sugar, honey, or concentrate.
2. Fortification Point – The Fortification Point Calculator determines 1) the specific gravity (SG) at which to stop an active fermentation to yield the desired level of residual sugar in a fortified wine, and 2) the amount of fortifier required yield the desired alcohol level.  It accounts for the alcohol produced during fermentation, and for the dilution of the residual sugar by the addition of the fortifier.
3. Hackbarth SG – The Hackbarth SG Calculator utilizes the Hackbarth model to estimate SG from given values of alcohol content and sugar content.  Its purpose is mainly to test and validate the Hackbarth calculation in FermCalc, but it is also useful in the preparation of fortified wines and liqueurs.

The fortification 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 that component plus the added mass of that component.  It is assumed that the fortifier contains only alcohol and water, and that the sweetener contains only sucrose and water.  For water, sugar, and alcohol the mass balances are:

 mwf = mwi + mwfa + mwsa (7-1) msf = msi + mssa (7-2) maf = mai + mafa (7-3)

where:

mwf = final water mass, kg

mwi = wine initial water mass, kg

mwfa = mass of water in the fortifier added, kg

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

msf = final sugar mass, kg

msi = wine initial sugar mass, kg

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

maf = final alcohol mass, kg

mai = wine initial alcohol mass, kg

mafa = mass of alcohol in the fortifier added, kg

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

 mtf = mti + mfa + msa (7-4)

where:

mtf = final total mass, kg

mti = wine initial total mass, kg

mfa = mass of fortifier added, kg

msa = mass of sweetener added, kg

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

 mfa = vfasgfaρw (7-5) mti = visgiρw (7-6) mtf = vfsgfρw (7-7)

where:

vfa = volume of fortifier added, liters

vi = wine initial volume, liters

vf = final volume, liters

sgfa = fortifier specific gravity

sgi = wine initial specific gravity

sgf = final specific gravity

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

Knowing the initial and target sugar (true Brix) levels we can then relate the initial and final sugar masses to the initial and final total masses as follows:

 msi = mtiBi/100 (7-8) msf = mtfBf/100 (7-9)

where:

Bi = initial wine Brix

Bf = final Brix

Substituting equations (7-6) and (7-7) into equations (7-8) and (7-9) gives us:

 msi = visgiρwBi/100 (7-10) msf = vfsgfρwBf/100 (7-11)

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 (7-12)

Where Bs is the Brix of the sweetener.

Alcohol masses can be related to the volumes, specific gravities, and alcohol levels of the liquids as follows:

 mai = visgiρwawi/100 (7-13) maf = vfsgfρwawf/100 (7-14) mafa = vfasgfaρwawfa/100 (7-15)

where:

awi = wine initial alcohol level, % by weight

awf = final alcohol level, % by weight

awfa = fortifier alcohol level, % by weight

Substituting equations (7-10) through (7-12) into equation (7-2) we get the sugar mass balance equation:

 vfsgfρwBf = visgiρwBi + msaBs (7-16)

Substituting equations (7-13) through (7-15) into equation (7-3) we get the alcohol mass balance equation:

 vfsgfρwawf = visgiρwawi + vfasgfaρwawfa (7-17)

And substituting equations (7-5) through (7-7) into equation (7-4) we get the total mass balance equation:

 vfsgfρw = visgiρw + vfasgfaρw + msa (7-18)

Equations (7-16) through (7-18) form the basis for all of these calculations.  They’ll just be re-arranged and solved differently depending on what we’re solving for.