Introduction

The Sugar Requirements panel calculates the amounts of sugar and/or water to add to a given must or juice to achieve a target specific gravity.

FermCalc offers three options for this calculation:

1. Specify Target Gravity - 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 Gravity 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 Gravity 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 the appropriate units for the input and output fields using the drop-down menus in the right-hand column.
4. Enter the Initial Volume and Initial Specific Gravity of the must in the appropriate fields.
5. If you've selected the first option in the Calculation Type menu, enter the Target Specific Gravity of the must.
6. If you've selected the second option in the Calculation Type menu, enter the Target Specific Gravity and the Target Volume of the must.
7. If you've selected the third option in the Calculation Type menu, enter 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 1.55 and 0.99.  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 Gravity and Volume mode, if red zeros appear in the output fields, your target volume is too small to achieve the target specific gravity. In this case, change back to Specify Target Gravity to determine the minimum target volume required to achieve the desired specific gravity.

Calculation Details

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	(1)
msf = msi + msa	(2)

where:

m_wi = initial water mass (kg)
m_wf = final water mass (kg)
m_wa = mass of water added (kg)
m_si = initial sugar mass (kg)
m_sf = final sugar mass (kg)
m_sa = mass of sugar added (kg)

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

mtf = mti + msa + mwa

(3)

where:

m_ti = initial total mass (kg)
m_tf = final total mass (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 calculate initial and final Brix, or percent sugar by weight, from equation (1) 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)
mwi = mti(1.0 - Bi/100)	(8)
mwf = mtf(1.0 - Bf/100)	(9)

where:

B_i = initial Brix
B_f = final Brix

Substituting equations (4) and (5) into equations (6) through (9) gives us:

msi = visgiBi/100	(10)
msf = vfsgfBf/100	(11)
mwi = visgi(1.0 - Bi/100)	(12)
mwf = vfsgf(1.0 - Bf/100)	(13)

Equations (1) through (13) form the basis of these calculations.  They'll just be re-arranged and solved differently depending on what we're solving for.  There are three scenarios to consider.

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

Case 1: Specify Target Gravity - 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 sugar and no water. So, equation (1) reduces to:

mwi = mwf

(14)

Substituting equations (12) and (13) into equation (14) we get:

visgi(1.0 - Bi/100) = vfsgf(1.0 - Bf/100)

(15)

Now all we need to do is re-arranged equation (15) to solve for the final volume after the addition of the sugar, which gives us:

vf = [visgi(1.0 - Bi/100)] / [sgf(1.0 - Bf/100)]

(16)

Now we can calculate the mass of the sugar addition. We start by re-arranging equation (2) as:

msa = mtf - mti

(17)

Substituting equations (4) and (5) into equation (17) we get:

msa = vfsgf - visgi

(18)

The graphs below validate the results of equations (16) and (18) by comparing their results to tabular data from various sources.

If the target gravity is greater 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 (2) reduces to:

msf = msi

(19)

Substituting equations (10) and (11) into equation (19) we get:

vfsgfBf/100 = visgiBi/100

(20)

Now we can solve equation (20) for the final volume:

vf = (visgiBi) / (sgfBf)

(21)

Since we know we'll add only water to the mixture, we can reduce and re-arrange equation (3) to solve for the mass of water added:

mwa = mtf - mti

(22)

Then substituting equations (4) and (5) into equation (22) gives us:

mwa = vfsgf - visgi

(23)

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 (23).

Case 2: Specify Target Gravity 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 start by re-arranging equation (2) as:

msa = msf - msi

(24)

Substituting equations (10) and (11) into equation (24) gives us:

msa = vfsgf Bf/100 - visgiBi/100

(25)

To calculate the mass of water to add, we need to first re-arrange equation (3) as:

mwa = mtf - mti - msa

(26)

Substituting equations (4) and (5) into equation (26) gives us:

mwa = vfsgf - visgi - msa

(27)

Again, since the density (specific gravity) of water is 1.0 kg/l, the volume of water added in liters is equal to the mass in kilograms that we calculated in equation (27).

The graph below validates the results of equation (25) by comparing its results to tabular data from various sources.

Case 3: Specify H2O/Sugar Additions - Calculate Gravity 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 substitute equations (3), (4), (6), and (7) into equation (2) and solve for B_f to get:

Bf = 100(visgiBi/100 + msa)/(visgi + msa + mwa)

(28)

Knowing B_f we can calculate sg_f from equation (1) on the Conversions page.

Then, after we know sg_f we can combine equations (3) through (5) and solve for v_f as:

vf = (visgi + msa + mwa)/sgf

(29)