Reference ID: MET-E4C7 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The “Mixing Index” (M) is a dimensionless figure‑of‑merit that quantifies how close a binary particulate solid is to the theoretical “random‑mixed” state. In process engineering, it is the standard metric for documenting blend uniformity of pharmaceutical premixes, food fortification blends, and powder detergents. A value of 0 indicates complete segregation, 1 indicates a theoretical random variance, i.e., a statistically random mixture, while intermediate numbers flag a partially segregated bed. Because the mixing index is closely related to the statistical spread of component concentrations, understanding the variance calculation for powder mixtures can provide deeper insight into blend quality. The index is accepted by ICH‑Q8/Q9 guidance, FDA ANDA submissions, and is routinely printed on batch records for 100 kg – 1 t ribbon or paddle mixers; for detailed requirements see the mixing uniformity specification.
Methodology & Formulas
Mean measured mass fraction
\[ \bar{x} = \frac{\Sigma x_i}{k} \]
where \(k\) is the number of withdrawn samples.
End-point variances
Completely segregated (un-mixed) variance:
\[ \sigma_{0}^{2} = p\,q \]
Random-mixed (theoretical) variance:
\[ \sigma_{r}^{2} = \frac{p\,q}{n} \]
with \(p + q = 1\) and \(n\) the number of particles in the analytical sample.
Mixing index (robust to near-zero denominator):
\[ M = \frac{\sigma_{0}^{2} - \sigma^{2}}{\max\!\bigl(\sigma_{0}^{2} - \sigma_{r}^{2},\,1 \times 10^{-12}\bigr)} \]
Validity regime
Requirement
M range
\(0 \le M \le 1\)
Particle size ratio
0.3 – 3
Minimum particle count per sample
\(n \ge 10^{4}\)
The Mixing Index quantifies how uniformly components are distributed in a mixture. Calculating it helps process engineers verify blend quality, minimize waste, and ensure product consistency without relying on visual inspection.
Collect at least 30 samples across the vessel to capture spatial variance.
Ensure each sample has sufficient mass to contain at least 104 particles (n ≥ 104) for statistical validity, as per the validity regime.
Increase sample count for cohesive powders or high aspect-ratio mixers.
For binary systems, use \( M = \frac{\sigma_0^2 - \sigma^2}{\sigma_0^2 - \sigma_r^2} \), where \( \sigma_0^2 \) is the variance of a fully segregated blend, \( \sigma^2 \) is the measured sample variance, and \( \sigma_r^2 \) is the variance of a random mixture. Values near 1 indicate excellent mixing. For robustness in computation, use \( M = \frac{\sigma_0^2 - \sigma^2}{\max(\sigma_0^2 - \sigma_r^2, 1 \times 10^{-12})} \) to avoid division by zero.
Review mixing time: extend or reduce depending on over-mixing risk.
Check fill level and impeller speed against equipment vendor charts.
Verify raw material moisture, particle size, and density differences; address segregation variables.
Worked Example: Mixing Index Calculation for Vitamin-B12 Premix
A process engineer assesses the homogeneity of a vitamin-B12 premix in wheat flour after batch mixing. The target is 2% vitamin by mass. Ten 25-gram samples are analyzed post-mixing to compute the mixing index.
Knowns (Input Parameters):
Mass fraction of vitamin-B12, \( p = 0.02 \) (dimensionless)
Mass fraction of wheat flour, \( q = 0.98 \) (dimensionless)
Number of particles per sample, \( n = 5,000,000 \)
Number of samples, \( k = 10 \)
Measured vitamin mass fractions from samples: 0.0198, 0.0204, 0.0195, 0.0201, 0.0202, 0.0199, 0.0203, 0.0197, 0.0200, 0.0201
Step-by-Step Calculation:
The mean measured mass fraction is given directly: \( \bar{x} = 0.02 \).
The sample variance is computed from the data: \( \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{k-1} = 7.8 \times 10^{-8} \).
