### Introduction

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.

FermCalc offers three options for this calculation:

- Specify Target Alcohol & Sugar > Calculate Additions and Volume – Specify the desired alcohol and sugar contents, and FermCalc will calculate the required additions of fortifier and sweetener to achieve the targets, as well as the resulting volume after the additions.
- Specify Target Alcohol, Sugar, & Volume > Calculate Additions – Specify the desired alcohol content, sugar content, and volume, and FermCalc will calculate the required amounts of wine, fortifier, and sweetener to achieve all three targets.
- Specify Additions > Calculate Resulting Alcohol, Sugar, & Volume – Specify the amounts of fortifier and sweetener to be added, and FermCalc will calculate the resulting alcohol content, sugar content, and volume.

All volumes and specific gravities have a temperature basis of 20°C (68°F). It is assumed that the fortifier contains only ethanol and water, and that the sweetener contains only sucrose and water. Specific gravities for all liquids containing both ethanol and sucrose are calculated using the Hackbarth model.

### Input Field Definitions

Sweetener – The type of sweetener to be added, either sugar, honey, or concentrate. The SG of the sweetener must be specified if honey or concentrate is selected.

Fortifier Alcohol Content – The alcohol content of the fortifier being added.

Target Alcohol Content – The desired alcohol content of the fortified wine.

Target Sugar Content – The desired total solids content of the fortified wine.

Wine Volume – The volume of the wine prior to fortification. Required only if either the first or third calculation option is selected.

Target Volume – The desired volume of the wine after fortification. Required only if the second calculation option is selected.

Wine Alcohol Content – The original alcohol content of the wine being fortified.

Wine Sugar Content – The total solids content (true Brix) of the wine being fortified.

Fortifier Added – The amount of fortifier added to the wine. Required only if the third calculation option is selected.

Sugar (or Honey or Concentrate) Added – The amount of sweetener added to the wine. Required only if the third calculation option is selected.

### Output Field Definitions

Fortifier Required – The calculated amount of fortifier required to yield the desired alcohol content, sugar content, and volume. Reported only if either the first or second calculation option is selected.

Sugar (or Honey or Concentrate) Required – The calculated amount of sweetener required to yield the desired alcohol content, sugar content, and volume. Reported only if either the first or second calculation option is selected.

Wine Required – The volume of wine required to yield the desired alcohol content, sugar content, and volume. Reported only if the second calculation option is selected.

Resulting Volume – The calculated volume of the fortified win after adding the specified amounts of fortifier and sweetener. Reported only if either the first or third calculation option is selected.

Resulting Alcohol Content – The alcohol content of the fortified wine after adding the specified amounts of fortifier and sweetener. Reported only if the third calculation option is selected.

Resulting Sugar Content – The total solids content of the fortified wine after adding the specified amounts of fortifier and sweetener. Reported only if the third calculation option is selected.

### Calculation Details

The fortification point calculations are based mainly on the previously developed mass balance equations (7-16), (7-17), and (7-18), repeated here for completeness:

v = _{f}sg_{f}ρ_{w}B_{f}v + _{i}sg_{i}ρ_{w}B_{i}m_{sa}B_{s} |
(7-16) |

v = _{f}sg_{f}ρ_{w}a_{wf}v + _{i}sg_{i}ρ_{w}a_{wi}v_{fa}sg_{fa}ρ_{w}a_{wfa} |
(7-17) |

v = _{f}sg_{f}ρ_{w}v + _{i}sg_{i}ρ_{w}v + _{fa}sg_{fa}ρ_{w}m_{sa} |
(7-18) |

where:

*v _{i}* = initial must volume, liters

*v _{f}* = final volume, liters

*sg _{i}* = initial specific gravity

*sg _{f}* = final specific gravity

*B _{i}* = initial Brix

*B _{f}* = final Brix

*B _{s}* = sugar Brix

*a _{wi}* = wine initial alcohol level, % by weight

*a _{wf}* = final alcohol level, % by weight

*a _{wfa}* = fortifier alcohol level, % by weight

*ρ _{w}* = density of water = 0.9982 kg/liter at 20ºC

*m _{sa}* = mass of sugar added, kg

*v _{wa}* = volume of water added, liters

These equations will be rearranged and solved differently depending on which variable we’re solving for.

#### Option 1: Specify Target Alcohol & Sugar > Calculate Additions and Volume

In this case the known values are *v _{i}*,

*sg*,

_{i}*a*,

_{wi}*B*,

_{i}*sg*,

_{f}*a*,

_{wf}*B*,

_{f}*sg*, and

_{fa}*a*. We need to solve for

_{wfa}*v*,

_{fa}*m*, and

_{sa}*v*.

_{f} If we solve equation (7-18) for *m _{sa}* and solve equation (7-17) for

*v*and then substitute them into equation (7-16) we can solve for

_{fa}*v*:

_{f}v = [_{f}v(_{i}sg_{i}B – _{i}B + _{s}B)] / [_{s}a_{wi}/a_{wfa}sg(_{f}B – _{f}B + _{s}B)]_{s}a_{wf}/a_{wfa} |
(7-19) |

Then we can re-arrange equation (7-17) to solve for the amount of fortifier to add:

v = (_{fa}v – _{f}sg_{f}a_{wf}v) / (_{i}sg_{i}a_{wi}sg)_{fa}a_{wfa} |
(7-20) |

Knowing how much fortifier we’re adding we can then re-arrange equation (7-18) to solve for the amount of sweetener to add:

m = (_{sa}v – _{f}sg_{f}v – _{i}sg_{i}v)_{fa}sg_{fa}ρ_{w} |
(7-21) |

#### Option 2: Specify Target Alcohol, Sugar, & Volume – Calculate Additions

In this case the known values are *sg _{i}*,

*a*,

_{wi}*B*,

_{i}*v*,

_{f}*sg*,

_{f}*a*,

_{wf}*B*,

_{f}*sg*, and

_{fa}*a*. We need to solve for

_{wfa}*v*,

_{fa}*m*, and

_{sa}*v*.

_{i} Again we solve equation (7-18) for *m _{sa}* and solve equation (7-17) for

*v*and then substitute them into equation equation (7-16), but this time we solve for

_{fa}*v*:

_{i}v = [_{i}v(_{f}sg_{f}B – _{f}B + _{s}B)] / [_{s}a_{wf}/a_{wfa}sg(_{i}B – _{i}B + _{s}B)]_{s}a_{wi}/a_{wfa} |
(7-22) |

Then we can solve equations (7-20) and (7-21) for the amounts of fortifier and sweetener to add.

#### Option 3: Specify Additions – Calculate Resulting Alcohol, Sugar, & Volume

In this case the known values are *v _{i}*,

*sg*,

_{i}*a*,

_{wi}*B*,

_{i}*v*,

_{fa}*sg*,

_{fa}*a*, and

_{wfa}*m*. We need to solve for

_{sa}*v*,

_{f}*sg*,

_{f}*a*, and

_{wf}*B*.

_{f} We can easily solve for *B _{f}* by dividing equation (7-16) by equation (7-18):

B = (_{f}v + _{i}sg_{i}ρ_{w}B_{i}m) / (_{sa}B_{s}v + _{i}sg_{i}ρ_{w}v + _{fa}sg_{fa}ρ_{w}m)_{sa} |
(7-23) |

We can solve for *a _{wf}* by dividing equation (7-17) by equation (7-18):

a = (_{wf}v + _{i}sg_{i}ρ_{w}a_{wi}v) / (_{fa}sg_{fa}ρ_{w}a_{wfa}v + _{i}sg_{i}ρ_{w}v + _{fa}sg_{fa}ρ_{w}m)_{sa} |
(7-24) |

We can then determine *sg _{f}* from

*B*and

_{f}*a*using the Hackbarth model and solve equation (7-18) for

_{wf}*v*to get:

_{f}v = (_{f}v + _{i}sg_{i}ρ_{w}v + _{fa}sg_{fa}ρ_{w}m) / (_{sa}sg)_{f}ρ_{w} |
(7-25) |