FermCalc - Sugar Requirements Calculations

Introduction

The Sugar Requirements panel calculates the amounts of sweetener and/or water to add to a given must or juice to achieve a target specific gravity and volume. The sweetener can be either sugar, honey, or concentrate.

FermCalc offers three options for this calculation:

1. Specify Target SG - Calculate H2O/Sugar Additions and Volume - Specify the target specific gravity only, and FermCalc will calculate the required additions of water or sugar as well as the resulting volume after the addition.
2. Specify Target SG and Volume - Calculate H2O/Sugar Additions - Specify the target specific gravity and volume, and FermCalc will calculate the required additions of water and/or sugar to achieve both targets.
3. Specify H2O/Sugar Additions - Calculate Resulting SG and Volume - Specify the amounts of water and/or sugar to be added, and FermCalc will calculate the resulting volume and specific gravity.

Making a Calculation

To make a calculation, follow these steps:

1. Select Calculation > Sugar Requirements from the menu, or select the Sugar tab at the top of the main window.
2. Select one of the three calculation types from the Calculation Type menu.
3. Select either Sugar, Honey, or Concentrate from the Sweetener drop-down list, and if Honey or Concentrate is selected enter its gravity.
4. Select the appropriate units for the input and output fields using the drop-down menus in the right-hand column.
5. If you've selected the first option in the Calculation Type menu, enter the Target SG, the Initial SG, and the Initial Volume of the must.
6. If you've selected the second option in the Calculation Type menu, enter the Target SG, the Target Volume, the Initial SG, and the Initial Volume of the must.
7. If you've selected the third option in the Calculation Type menu, enter the Initial SG and the Initial Volume of the must, and the amounts of Sugar Added and/or Water Added.

The calculated values will appear in the output fields as you type.

The entered specific gravity values must range between 1.5805 and 0.95.  If any values are outside of the valid range, red zeros will appear in the output fields and an error message will be displayed.  In the Specify Target SG and Volume mode, if you get a message that says "Target Volume must be increased or Initial Volume must be reduced.", your target volume is too small to allow the additions that are required to achieve the target specific gravity. In this case, change back to Specify Target SG to calculate the minimum target volume required to achieve the desired specific gravity.

The honey and concentrate calculations are very similar. The main difference is that honey additions are specified in mass units, and concentrate additions are specified in volume units.

Calculation details are provided below.

1. Mass Balance Equations
2. Specify Target SG - Calculate H2O/Sugar Additions and Volume
3. Specify Target SG and Volume - Calculate H2O/Sugar Additions
4. Specify H2O/Sugar Additions - Calculate Resulting SG and Volume

Calculation Details

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 mass of that component that is added.  For water and sugar the mass balances are:

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

where:

m_wf = final water mass, kg
m_wi = initial water mass, kg
m_wa = mass of pure water added, kg
m_wsa = mass of water in the sweetener (sugar or honey) added, kg
m_sf = final sugar mass, kg
m_si = initial sugar mass, kg
m_ssa = 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

(3)

where:

m_tf = final total mass, kg
m_ti = initial total mass, kg
m_sa = mass of sweetener added, kg

If we know the initial and final specific gravities of the must, we can relate the initial and final total mass to volume as follows:

mti = visgi	(4)
mtf = vfsgf	(5)

where:

v_i = initial volume, liters
v_f = final volume, liters
sg_i = initial specific gravity, kg/liter
sg_f = final specific gravity, kg/liter

And if we know the initial and final specific gravities, we can determine initial and final Brix, or percent sugar by weight, from the USDA tables as described on the Conversions page. Knowing Brix we can then relate the initial and final masses of the water and sugar to the initial and final total masses as follows:

msi = mtiBi/100	(6)
msf = mtfBf/100	(7)

where:

B_i = initial Brix
B_f = final Brix

Substituting equations (4) and (5) into equations (6) and (7) gives us:

msi = visgiBi/100	(8)
msf = vfsgfBf/100	(9)

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

mssa = msaBs/100

(10)

Where B_s is the Brix of the sweetener.

Substituting equations (8) through (10) into equation (2) we get:

vfsgfBf = visgiBi + msaBs

(11)

And substituting equations (4) and (5) into equation (3) we get:

vfsgf = visgi + msa + mwa

(12)

Equations (11) and (12) form the basis for all of these calculations.  They'll just be re-arranged and solved differently depending on what we're solving for.  There are three calculation options to consider.

1. Specify Target SG - Calculate H2O/Sugar Additions and Volume
2. Specify Target SG and Volume - Calculate H2O/Sugar Additions
3. Specify H2O/Sugar Additions - Calculate Resulting SG and Volume

Option 1: Specify Target SG - Calculate H2O/Sugar Additions and Volume

In this case the known values are v_i, sg_i, and sg_f.

If the target gravity is greater than the initial gravity, we know we'll be adding a sweetener and no water, so equation (12) reduces to:

vfsgf = visgi + msa

(13)

Substituting the right-hand side of equation (13) into equation (11) we get:

(visgi  + msa)Bf = visgiBi  + msaBs

(14)

Then we can re-arrange equation (14) to solve for the amount of sweetener to add:

msa = visgi(Bi - Bf) / (Bf - Bs)

(15)

Knowing how much sweetener we're adding we can then re-arrange equation (13) to solve for the final volume after the addition of the sweetener, which gives us:

vf = (visgi + msa) / sgf

(16)

The graphs below validate equations (15) and (16) by comparing their results to tabular data from various sources and to the results of my own experiments.

If the target gravity is less than the initial gravity, we know we'll need to add water to reduce the gravity. Since we know we'll be adding water only and no sugar, equation (11) reduces to:

vfsgfBf = visgiBi

(17)

Now we can solve equation (17) for the final volume:

vf = (visgiBi) / (sgfBf)

(18)

Since we know we'll only be adding water to the mixture, equation (12) reduces to:

vfsgf = visgi + mwa

(19)

Solving equation (19) for the mass of water to add we get:

mwa = vfsgf - visgi

(20)

The density (specific gravity) of water is 1.0 kg/liter, so the volume of water added in liters is equal to the mass in kilograms that we calculated from equation (20).

Option 2: Specify Target SG and Volume - Calculate H2O/Sugar Additions

In this case the known values are v_i, v_f, sg_i, and sg_f. To calculate the amount of sugar to add, we simply need to re-arrange equation (11) and solve for the amount of sweetener to add, or:

msa = (vfsgf Bf - visgiBi) / Bs

(21)

Knowing the amount of sweetener to add we can then re-arrange equation (12) to solve for the amount of water to add:

mwa = vfsgf - visgi - msa

(22)

Again, since the density (specific gravity) of water is 1.0 kg/liter, the volume of water added in liters is equal to the mass in kilograms that we calculated in equation (22).

The graph below validates equation (21) by comparing its results to tabular data from various sources and to the results of my own experiments.

Option 3: Specify H2O/Sugar Additions - Calculate Resulting SG and Volume

In this case the known values are v_i, sg_i, m_sa, and m_wa. To calculate the resulting specific gravity sg_f we first substitute the right-hand side of equations (12) into equation (11) to get:

(visgi + msa + mwa)Bf = visgiBi + msaBs

(23)

Then solving for B_f we get:

Bf = (visgiBi + msaBs) / (visgi + msa + mwa)

(24)

Knowing B_f we can then determine sg_f from the USDA tables as described on the Conversions page.

After we determine sg_f we can solve equation (12) for v_f to get:

vf = (visgi + msa + mwa) / sgf

(25)

© 2007-2010 Steve Gross
Last updated 08 May 2010.