Mixture Properties (RX) Derivations

Mixture properties (RX) are calculated using the ratio matrix and ingredient properties, and can be calculated using a weighted average or additively.

Each formulation’s mixture property \(x_{mix,p}\) is calculated using the following equation, in summation form, for each property \(p\) in the set of all properties \(P\):

\[x_{mix,p} = \sum_{i}^{I} r_i x_{i,p} \;, \qquad \forall \; p \in P\]
Where,
\(x_{mix,p}\) = mixture property for property \(p\)
\(r_i\) = fractional ratio of ingredient \(i\)
\(x_{i,p}\) = property \(p\) value for ingredient \(i\)

The generalized equation, in matrix form, for the entire mixture property block is then as follows:

\[X_{mix} = R \cdot X\]
Where,
\(X_{mix}\) = all mixture properties
\(R\) = ratio matrix
\(X\) = ingredient properties matrix

In matrix form, a visual representation of the mixture properties calculation is displayed below:

rx matrix form image here

As shown, the matrix multiplication results in a final matrix with dimensions of Formulations (N rows) x Properties (K columns) with each cell representing the final mixture property value for a formulation.

The example shown above is the additive case for calculating mixtures for ingredients with the same properties (e.g. within the same class of ingredients). However, in many cases, each formulation uses a mix of ingredients from different classes, each of which contains its own set of properties. The RX framework can be applied to the multi-class case, where RX blocks are calculated for each class with the relevant ingredients in each formulation. In the case where the formulation does not use any ingredients of a class, the properties for that class will be set to missing (NaN).

Additionally, if an ingredient is used in a formulation and contains missing property values, then the RX for that property will be set to missing.

Configuration Options

RX Type: Additive or Weighted Average

  • Additive: Calculation as shown previously – sum product of the ingredients in the current class by their respective property values.

  • Weighted Average: Additive calculation but divided by the sum of ratios of the current class.

\[x_{mix,p} = \frac{\displaystyle\sum_{i}^{I} r_i x_{i,p}}{\displaystyle\sum_{i}^{I} r_i},\qquad \forall \; p \in P\]

Mixing Rule: Linear, Natural Logarithm, or Logarithmic

  • Linear: No transformation of RX calculation, as shown previously.

  • Natural Logarithm: The RX calculation is carried out with the natural logarithm \(\ln\) applied to the selected property for all ingredients. Then, the exponential is applied to the resulting RX. Below is the mathematical formulation for the additive case.

\[x_{mix,p} = \large e^{\sum\nolimits_{i}(r_i)(\ln x_{i,p})}\]

  • Logarithmic: The RX calculation is carried out with the base 10 logarithm \(\log_{10}\) applied to the selected property for all ingredients. Then, the exponential is applied to the resulting RX. Below is the mathematical formulation for the additive case.

\[x_{mix,p} = \large 10^{\sum\nolimits_{i}(r_i)(\log_{10} x_{i,p})}\]

Global RX

Global properties may be added for additional analysis. For global RX calculations:

  • One property from each class is used and all ingredients are used in the calculation.

  • All classes must be used.

  • Global calculations are only carried out in an additive fashion (sum of ingredients in a formulation is 1, and thus, weighted average would produce the same result).

  • A mixing rule (linear, logarithmic, natural logarithm) may be selected for the global property.

  • Similar to the prior property calculations, if a used ingredient contains missing property data, then the RX for the property will be set to missing.

  • The mathematical formulation is shown as the following for the linear case:

\[x_{mix,p}^{GLOBAL} = \displaystyle \sum_{i}^{I} r_{i} x_{i,p}^{GLOBAL}\]