## Introduction

The Blending Calculator determines the resulting property and volume of a blend of two liquids given the properties and volumes of the liquids being blended. It can also determine the volumes of the two liquids required to achieve desired values of the blend’s property and volume. Three calculation options are offered:

- Solve for Property and Volume of Blend – Given the properties and volumes of liquids 1 and 2, the property and volume of the blend will be calculated.
- Solve for Volumes of Liquid 2 and Blend – Given the property and volume of liquid 1, the property of liquid 2, and the desired property of the blend, the volumes of liquid 2 and the blend will be calculated.
- Solve for Volumes of Both Liquids – Given the properties of liquids 1 and 2, and the desired property and volume of the blend, the volumes of liquids 1 and 2 will be calculated.

In Standard mode, the Blending Calculator employs an ideal, volume-weighted mixing rule to calculate the resulting properties and volume of a blend from the properties of the constituent liquids being blended. This is equivalent to a Pearson’s Square calculation. This calculation will yield accurate results for just about any concentration that is expressed in terms of mass per unit volume or volume per unit volume, such as specific gravity, alcohol content, acidity, residual sugar, etc. However, it does not work for pH because pH is the negative log of a concentration, and it must be converted to a concentration prior to making the calculation. It will also not work well for properties expressed as mass per unit mass, such as Brix, in which case mass balance equations must be used.

In Advanced mode, the calculation depends on the type of property being calculated. The calculator currently offers three property types: Concentration, pH, or Specific Gravity/Density.

## Input Field Definitions

Standard, Advanced – Select between Standard and Advanced mode. In Standard mode, a standard “Pearson’s Square” calculation is performed regardless of the property type. In Advanced mode, the calculation depends on the property type: concentration calculations are the same as those used in Standard mode; pH is first converted to a concentration before performing the calculation; and SG / Density calculations are performed using a separate set of mass balance equations.

Property Type – Select from Concentration, pH, or Specific Gravity / Density. Required only in Advanced mode.

SG Units – Select the units for which SG will be entered and output. Required only in Advanced mode when Specific Gravity / Density is chosen as the property type.

Solve For – Select which two values will be calculated.

Liquid 1 & 2 Property – The properties (SG, acidity, etc.) of the two liquids being blended.

Liquid 1 & 2 Volume – The volumes of the two liquids being blended. These fields can be either input fields or output fields depending on the calculation option selected.

Blend Property – The property of the blend of the two liquids. This field can be either an input field or an output field depending on the calculation option selected.

Blend Volume – The volume of the blend of the two liquids. This field can be either an input field or an output field depending on the calculation option selected.

## Calculation Details

### Standard Mode

In Standard mode, the Blending Calculator employs what amounts to an ideal, volume-weighted mixing rule to calculate the resulting properties and volume of a blend from the properties of the constituent liquids being blended, or

p = Σ_{b}p / Σ_{i}v_{i}v_{i} |
(1) |

v = Σ_{b}v_{i} |
(2) |

where

p = property of blend_{b}p = property of liquid i_{i}v = volume of blend, liters_{b}v = volume of liquid i, liters _{i} |

This approach works reasonably well for properties such as concentrations or properties that are expressed in terms of mass per unit volume or volume per unit volume, such as titratable acidity (g/L) and alcohol content (vol/vol). Other properties require special consideration as will be discussed later.

For a system of two liquids, equations (1) and (2) can be re-written as

p = (_{b}p + _{1}v_{1}p) / (_{2}v_{2}v + _{1}v) _{2} |
(3) |

v = _{b}v + _{1}v _{2} |
(4) |

Equations (3) and (4) are equivalent to the Pearson’s Square approach. Given equations (3) and (4) above, we can do some simple algebra and solve for any two unknowns provided that all of the other quantities are known. FermCalc provides three options:

- Solve for the Property and Volume of the Blend.
- Solve for the Volumes of Liquid 2 and the Blend.
- Solve for the Volumes of Both Liquids.

For cases 2 and 3, the property value of the blend must be between the property values of the two liquids. If this is not the case, output fields are highlighted in red and an error message is displayed.

#### Case 1: Solve for the Property and Volume of the Blend

For this case we specify the properties and volumes of both liquids, and we simply need to apply equations (3) and (4) above to calculate the property and volume of the blend.

#### Cases 2: Solve for the Volumes of Liquid 2 and the Blend

For this case, we specify the properties of both liquids, the desired property of the blend, and the volume of one of the liquids. We first need to re-arrange equation (3) to solve for the volume of liquid 2. The resulting equation is:

v = _{2}v(_{1}p – _{b}p) / (_{1}p – _{2}p) _{b} |
(5) |

The blend volume is then calculated from equation (4).

#### Case 3: Solve for the Volumes of Both Liquids

For this case, we specify the properties of both liquids and the desired property and volume of the blend. Substituting and re-arranging equations (3) and (4) again we get:

v = (_{1}v – _{b}p_{b}v) / (_{b}p_{2}p – _{1}p) _{2} |
(6) |

The volume of liquid 2 is then calculated by re-arranging equation (4) as

v = _{2}v – _{b}v_{1} |
(7) |

### Advanced Mode

#### Concentration

The standard Pearson’s Square equations are valid for concentrations, which are properties expressed in terms of mass per unit volume or volume per unit volume, such as specific gravity, alcohol content, acidity, and residual sugar. Therefore, when Concentration is selected as the property type, the calculations are performed using the equations above for Standard Mode.

#### pH

pH is not a concentration, but is the negative log of the hydronium ion concentration:

pH = -log[H_{3}O^{+}] |
(8) |

Pambianchi (2010) showed that considerable error can result from using pH values directly in Pearson’s Square calculations. Because of this, pH values should not be used directly for blending calculations. Instead we need to first convert pH to a concentration. Calculating the hydronium ion concentration from pH we get:

[H_{3}O^{+}] = 10^{-pH} |
(9) |

Now that we’ve converted pH to a concentration, we can apply the equations developed above in the Standard Mode section. For example, equation (3) above becomes:

[H_{3}O^{+}] = ([H_{b}_{3}O^{+}] + [H_{1}v_{1}_{3}O^{+}]) / (_{2}v_{2}v + _{1}v) _{2} |
(10) |

where:

[H_{3}O^{+}] = hydronium ion concentration of blend_{b}[H _{3}O^{+}] = hydronium ion concentration of liquid 1_{1}[H _{3}O^{+}] = hydronium ion concentration of liquid 2_{2} |

After calculating the hydronium ion concentration of the blend, we can then convert it back to pH using equation (8).

#### Specific Gravity / Density

Under certain circumstances, the standard Pearson’s Square formulas can result in significant error if used for chaptalization and dilution calculations. This is particularly true when units of

°Brix are used directly in the calculation, and when the difference is large between the Brix levels of the liquids being blended. For example, let’s say we want to dilute honey at 80°Brix with water (0°Brix) to yield 10 liters of 24°Brix must. A standard Pearson’s Square calculation would say you need 3 liters of honey and 7 liters of water. However, if we use more rigorous mass balance equations we would calculate volumes of 2.3 liters of honey and 7.7 liters of water. Quite a difference! If the Brix levels were expressed as SG or density, the results would agree much more closely. Even more so if they were expressed as g/L sugar.

The mass balance equations used for the Specific Gravity / Density blending calculations are based on those used in the Chaptalization & Dilution Calculator and yield the same results regardless of which SG unit is chosen. The total mass balance equation is:

v = _{b}sg_{b}v + _{1}sg_{1}v_{2}sg_{2} |
(11) |

The sugar mass balance equation is:

v = _{b}sg_{b}B_{b}v + _{1}sg_{1}B_{1}v_{2}sg_{2}B_{2} |
(12) |

where:

sg = specific gravity of blend_{b}sg = specific gravity of liquid 1_{1}sg = specific gravity of liquid 2_{2}B = Brix of blend_{b}B = Brix of liquid 1_{1}B = Brix of liquid 2 _{2} |

For the first calculation option, where we’re calculating the SG and volume of the blend, we need to substitute equation (11) into equation (12) to eliminate *v _{b}*, and then rearrange to solve for

*B*:

_{b}B = (_{b}v + _{1}sg_{1}B_{1}v) / (_{2}sg_{2}B_{2}v + _{1}sg_{1}v)_{2}sg_{2} |
(13) |

Now that we know *B _{b}*, we can determine

*sg*using the Brix conversion equation. Then we just need to re-arrange equation (11) to solve for

_{b}*v*:

_{b}v = (_{b}v + _{1}sg_{1}v) / _{2}sg_{2}sg_{b} |
(14) |

For the second calculation option, where we’re calculating the volumes of liquid 2 and the blend, we again need to eliminate *v _{b}*, but this time solve for

*v*:

_{2}v = [_{2}v(_{1}sg_{1}B – _{1}B)] / [_{b}sg(_{2}B – _{b}B)]_{2} |
(15) |

Then we can solve equation (14) for *v _{b}*.

For the third calculation option, where we’re calculating the volumes of both liquids, we need to substitute equation (11) into equation (12) to eliminate *v _{2}*, and then solve for

*v*:

_{1}v = [_{1}v(_{b}sg_{b}B – _{b}B)] / [_{2}sg(_{1}B – _{1}B)]_{2} |
(16) |

Then we can solve equation (11) for *v _{2}*:

v = (_{2}v – _{b}sg_{b}v) / _{1}sg_{1}sg_{2} |
(17) |