FermCalc - Blending Calculator - Introduction
- Input Field Definitions
- Calculation Details
- Standard Mode
- 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
- Advanced Mode
- Concentration
- pH
- Specific Gravity / Density
## IntroductionThe 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, and Specific Gravity/Density. ## Input Field DefinitionsStandard, 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. Back to top## Calculation Details## Standard ModeIn 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
where
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
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. Back to top## Case 1: Solve for the Property and Volume of the BlendFor 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. Back to top## Cases 2: Solve for the Volumes of Liquid 2 and the BlendFor 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:
The blend volume is then calculated from equation (4). Back to top## Case 3: Solve for the Volumes of Both LiquidsFor 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:
The volume of liquid 2 is then calculated by re-arranging equation (4) as
## Advanced Mode## ConcentrationThe 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. ## pHpH is not a concentration, but is the negative log of the hydronium ion concentration:
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:
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:
where: [H 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 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:
The sugar mass balance equation is:
where:
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
B:
_{b}
Now that we know sg using the
Brix conversion equation.
Then we just need to re-arrange equation (11) to solve for
_{b}v:
_{b}
For the second calculation option, where we're calculating the volumes of liquid 2 and the blend,
we again need to eliminate v:
_{2}
Then we can solve equation (14) for
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:
_{1}
Then we can solve equation (11) for Back to top
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