## 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:

m + _{wf} = m_{wi}m + _{wfa}m_{wsa} |
(1) |

m + _{sf} = m_{si}m _{ssa} |
(2) |

m + _{af} = m_{ai}m _{afa} |
(3) |

where:

m = final water mass, kg_{wf}m = wine initial water mass, kg_{wi}m = mass of water in the fortifier added, kg_{wfa}m = mass of water in the sweetener (sugar, honey, or concentrate) added, kg_{wsa}m = final sugar mass, kg_{sf}m = wine initial sugar mass, kg_{si}m = mass of sugar in the sweetener (sugar, honey, or concentrate) added, kg_{ssa}m = final alcohol mass, kg_{af}m = wine initial alcohol mass, kg_{ai}m = mass of alcohol in the fortifier added, kg _{afa} |

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

m + _{tf} = m_{ti}m + _{fa}m_{sa} |
(4) |

where:

m = final total mass, kg_{tf}m = wine initial total mass, kg_{ti}m = mass of fortifier added, kg_{fa}m = mass of sweetener added, kg _{sa} |

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

m = _{fa}v_{fa}sg_{fa}ρ_{w} |
(5) |

m = _{ti}v_{i}sg_{i}ρ_{w} |
(6) |

m = _{tf}v_{f}sg_{f}ρ_{w} |
(7) |

where:

v = volume of fortifier added, liters_{fa}v = wine initial volume, liters_{i}v = final volume, liters_{f}sg = fortifier specific gravity_{fa}sg = wine initial specific gravity_{i}sg = final specific gravity_{f}ρ = density of water = 0.9982 kg/liter at 20ºC _{w} |

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:

m = _{si}m/100 _{ti}B_{i} |
(8) |

m = _{sf}m/100 _{tf}B_{f} |
(9) |

where:

B = initial wine Brix_{i}B = final Brix _{f} |

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

m = _{si}v/100 _{i}sg_{i}ρ_{w}B_{i} |
(10) |

m = _{sf}v/100 _{f}sg_{f}ρ_{w}B_{f} |
(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:

m = _{ssa}m/100 _{sa}B_{s} |
(12) |

Where *B _{s}* is the Brix of the sweetener.

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

m = _{ai}v/100_{i}sg_{i}ρ_{w}a_{wi} |
(13) |

m = _{af}v/100_{f}sg_{f}ρ_{w}a_{wf} |
(14) |

m = _{afa}v/100_{fa}sg_{fa}ρ_{w}a_{wfa} |
(15) |

where:

a = wine initial alcohol level, % by weight_{wi}a = final alcohol level, % by weight_{wf}a = fortifier alcohol level, % by weight _{wfa} |

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

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

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

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

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

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

Equations (16) through (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.

## Post-Fermentation Fortification Calculator

### 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

#### 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 (18) for *m _{sa}* and solve equation (17) for

*v*and then substitute them into equation (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} |
(19) |

Then we can re-arrange equation (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} |
(20) |

Knowing how much fortifier we’re adding we can then re-arrange equation (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} |
(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 (18) for *m _{sa}* and solve equation (17) for

*v*and then substitute them into equation equation (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} |
(22) |

Then we can solve equations (20) and (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 (16) by equation (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} |
(23) |

We can solve for *a _{wf}* by dividing equation(17) by equation (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} |
(24) |

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

*B*and

_{f}*a*using the Hackbarth model and solve equation (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} |
(25) |

## Fortification Point Calculator

### Introduction

Port wine is traditionally made by adding a fortifier to a fermenting grape must at a specific point during the fermentation. The addition of the fortifier stops the fermentation by elevating the alcohol content of the must above the alcohol tolerance ot the yeast, thereby leaving the desired amount of residual sugar. The fortifier is traditionally a distilled wine spirit, known as aguardente or brandy, with an alcohol content of 76-78% v/v.

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.

After this “ideal” fortification point is calculated, the user has the option to override the calculated value of SG at the fortification point in order to model the impact of fortifying at a different SG. If the entered SG is lower than the ideal value, then there is not enough residual sugar to achieve the target sugar content, so the sweetener addition required to achieve the target sugar content is calculated. If the entered SG is higher than the ideal value, the sugar content will exceed the target sugar content after fortification, so the resulting sugar content after the addition of fortifier is calculated.

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.

### 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. This is only used if the user enters a value of wine SG at fortification that is less than the “ideal” value, in which case the addition of a sweetener is required to achieve the target sugar content.

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

Wine Volume – The volume of the fermenting wine at the time of fortification. If this volume option is selected, the resulting volume after fortification is calculated.

Target Volume – The volume of the fortified wine after the addition of the calculated fortifier and sweetener additions. If this volume option is selected, the required wine volume at the fortification point is calculated.

Initial Must SG – The SG of the must prior to the start of fermentation.

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

Target Sugar Content – The desired total solids content of the fortified wine after the addition of fortifier and sweetener.

Wine SG at Fortification – The SG of the fermenting wine at the time of fortification. This field can serve either as an input field or an output field. Initially the “ideal” fortification point SG is calculated based on the inputs above. If the calculated value is overridden by the user, then either the target sugar content or the required sweetener addition are calculated, depending on whether the entered SG value is greater than or less than the ideal value.

### Output Field Definitions

Wine Alcohol Content – The calculated alcohol content of the fermenting wine at the fortification point.

Wine Sugar Content – The calculated total solids content (true Brix) of the fermenting wine at the fortification point.

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

Fortifier Required – The calculated amount of fortifier required to yield the desired alcohol content, sugar content, and volume.

Sugar (or Honey or Concentrate) Required – The calculated amount of sweetener required to yield the desired alcohol content, sugar content, and volume. This will be zero unless the user enters a wine SG at fortification that is less than the “ideal” value.

Resulting Volume – The calculated volume of the fortified wine after adding the specified amounts of fortifier and sweetener. Reported only if wine volume is selected as the volume option above.

### Calculation Details

We’ll need to specify the initial specific gravity of the must *sg _{m}* in order to calculate the alcohol produced during the fermentation from the drop in specific gravity. We’ll use the Balling Method to calculate the amount of alcohol produced because it’s simple to apply and seems to be just as accurate as the other SG-drop methods.

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

*v*),

_{f}*sg*,

_{m}*a*,

_{wf}*B*,

_{f}*sg*,

_{f}*sg*,

_{fa}*a*, and

_{wfa}*m*. Since we’re not adding a sweetener in this case we can eliminate the terms

_{sa}*m*and

_{sa}*B*. We need to solve for

_{s}*v*(or

_{f}*v*),

_{i}*sg*,

_{i}*a*,

_{wi}*B*, and

_{i}*v*. Since

_{fa}*sg*,

_{i}*a*, and

_{wi}*B*are all interdependent, we’ll need to iterate to find a solution.

_{i}
We start by substituting equation (18) into equation (17) to eliminate *v _{fa}* and solve for

*v*:

_{f}v = [_{f}v(_{i}sg_{i}a – _{wfa}a)] / [_{wi}sg(_{f}a – _{wfa}a)] _{wf} |
(26) |

Or, if we’re solving for *v _{i}*:

v = [_{i}v(_{f}sg_{f}a – _{wfa}a)] / [_{wf}sg(_{i}a – _{wfa}a)] _{wi} |
(27) |

In these equations the subscript *i* denotes values at the fortification point. For the first iteration we’ll assume a value of 1.0 for *sg _{i}*. After solving equation (26) or (27) we can solve equation (16) for

*B*:

_{i}B = _{i}v / _{f}sg_{f}B_{f}v_{i}sg_{i} |
(28) |

Next we update the estimate of *sg _{i}* using Hackbarth model with the calculated values of

*B*and

_{i}*a*, and then repeat the calculations in equations (26) through (28) until we converge on a solution. Then we can solve equation (18) for the volume of fortifier to add:

_{wi}v = (_{fa}v – _{f}sg_{f}v) / (_{i}sg_{i}sg)_{fa} |
(29) |

After the “ideal” fortification point is calculated, the user has the option to override the calculated value of *sg _{i}* in order to model the impact of fortifying at a different SG. When this is done, we first calculate the resulting value of

*a*using the Balling method and the value of

_{wi}*B*using the Hackbarth model.

_{i}
If the value of *sg _{i}* entered by the user is lower than the ideal value, then there is not enough residual sugar to achieve the target sugar content. In this case we can then solve equation (19) or (22) above, depending on whether the initial volume or the target volume were specified, and then solve equations (20) and (21) to determine the required additions of fortifier and sugar.

If the value of *sg _{i}* entered by the user is higher than the “ideal” fortification point, then the sugar content of the wine will be too high to yield the target sugar content. In this case, no sweetener will be added and we’ll need to calculate the resulting sugar content

*B*. To do this we’ll follow an iterative procedure similar to the one described above utilizing equations (26) through (29), but we’ll use a starting value of 1.0 for

_{f}*sg*, and instead of equation (28) we’ll solve equation (16) for

_{f}*B*:

_{f}B = _{f}v / _{i}sg_{i}B_{i}v_{f}sg_{f} |
(30) |

Below the results of the FermCalc Fortification Point calculator are compared to the classic work of Pato & Miranda (1938) and to the fortification calculator found on the Australian Wine Research Institute (AWRI) web site. The density at fortification and the amount of fortifier to add were calculated to yield a fortified wine with an alcohol content of 20% v/v and a density of 1028.51 g/L using a fortifying spirit with an alcohol content of 77% v/v. The FermCalc results generally fall between the two other methods, and show better agreeement with the Pato & Miranda results. (The forumulas used by the AWRI calculator aren’t documented on their site, but it apparently uses the formulas found on the Monash Scientific site because they give identical results.)

## Hackbarth SG Calculator

### Introduction

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.

### Input Field Definitions

Alcohol Content – Alcohol content of the solution. Range: 0% to 100% vol/vol.

Sugar Content – Sugar content of the solution. Range: 0% to 100% by wt.

### Output Field Definitions

Calculated SG – The calculated specific gravity (20°C/20°C) of the solution.

### Calculation Details

Specific gravity is calculated using the Hackbarth model for calculating the specific gravity of ethanol-sucrose solutions. The model has a temperature basis of 20°C (68°F).