Reference ID: MET-C9A0 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Powder blending is rarely perfect; particles of different components segregate because of size, density or electrostatic differences. The variance of composition among spot samples therefore carries two contributions: the random variance expected from a perfectly randomised mixture and the excess variance caused by segregation. By comparing the measured variance with these two limits one obtains a dimensionless mixing index \(M\) that quantifies how close the mixture is to the ideal random state. In process engineering the index is used to:
validate blenders after installation or maintenance,
set end-point criteria for batch mixing,
demonstrate compliance with infant-food or pharmaceutical homogeneity specifications.
Methodology & Formulas
Data reduction
Convert each analytical result \(w_i\) (mass fraction of the key component in increment \(i\)) to a fraction:
\[ x_i = \frac{w_i}{100} \]
Theoretical limits
Completely segregated (two-layer) limit:
\[ \sigma_0^2 = \bar{x}(1 - \bar{x}) \]
Completely random (binomial) limit for a sample containing \(m\) particles:
\[ \sigma_{\text{R}}^2 = \frac{\bar{x}(1 - \bar{x})}{m}, \qquad m = \frac{m_{\text{sample}}}{m_{\text{particle}}} \]
Mixing index
The fraction of the segregation gap that has been closed by mixing:
\[ M = \frac{\sigma_0^2 - s^2}{\sigma_0^2 - \sigma_{\text{R}}^2} \]
\(M = 0\) for a fully segregated blend, \(M = 1\) for a perfectly random blend.
Interpretation & Acceptance Regimes
Regime
Mixing Index Range
Typical Application
Segregated
\(0 \le M < 0.60\)
Re-mixing required
Intermediate
\(0.60 \le M < 0.90\)
Acceptable for most bulk foods
Random-like
\(0.90 \le M \le 1\)
Infant formula & pharmaceutical blends
The calculation assumes that each increment is obtained with a sample thief whose mass is large enough to contain at least ~1000 particles; otherwise the approximation for \(\sigma_{\text{R}}^2\) becomes unreliable. Similarly, at least ten increments are required for the \(t\)-based confidence on \(s^2\) to be meaningful.
The mean concentration only tells you the overall recipe balance; variance tells you how far individual samples deviate from that mean. A low variance means every future scoop, tablet, or capsule is close to the target potency, so downstream segregation or dose-uniformity failures are far less likely.
Take at least 30 increments across the entire batch, not just the discharge stream.
Use a thief or core sampler that removes the same fixed volume every time.
Collect while the blender is still rotating or immediately after it stops to avoid post-blend segregation.
Seal samples immediately and analyze in random order to remove analytical drift bias.
Subtract each assay from the batch mean, square the differences, sum them, and divide by n-1 to get the sample variance. If your spec is given as a relative standard deviation (RSD), divide the square root of the variance by the mean and multiply by 100 to obtain %RSD.
Most firms set the upper limit at 5 %RSD for the active pharmaceutical ingredient (API) across 10–20 locations. If the potency specification is ±5 % of label claim, the corresponding variance target is (5/3)² ≈ 2.8 %RSD, giving a 3-sigma cushion within the acceptance window.
Verify fill level: overfilling a blender can kill radial motion and leave dead zones.
Check if the new larger mixer has the same Froude number; if not, adjust rpm to match the pilot-scale tip speed.
Inspect for ingredient segregation during transfer; long free-fall chutes or vibratory feeders can de-mix powders by particle size.
Worked Example – Powder Mixer Qualification
A process engineer must verify that a new V-blender can produce a 10 % (w/w) premix of an active infant-formula ingredient. The acceptance criterion is that the relative standard deviation of the active fraction, measured on 1 g samples, must not exceed 0.5 %. The engineer withdraws 15 tablets of 1 g each and assays them; the mean assay is 9.93 % and the between-sample variance is reported as 8.81 × 10-6 (fraction)2. Does the blender meet the specification?
Convert the 1 g sample into number of particles:
\[ n = \frac{1.000\ \text{g}}{0.0005\ \text{g}} = 2000\ \text{particles} \]
Compute the theoretical random (Poisson) variance for a 10 % mixture:
\[ \sigma_0^2 = \frac{p(1-p)}{n} = \frac{0.10 \times 0.90}{2000} = 4.50 \times 10^{-5}\ \text{(fraction)}^2 \]
Convert the measured variance to percent relative variance:
\[ \sigma^2_{\text{rel}} = \frac{8.81 \times 10^{-6}}{(0.099)^2} \times 100\% = 0.088\% \]
Compare with the acceptance limit:
\[ 0.088\% < 0.25\% \quad \Rightarrow \quad \text{requirement satisfied} \]
Final Answer
The V-blender produces a 10 % active premix whose between-sample relative variance is 0.088 %2, well below the 0.25 %2 acceptance limit; the unit is therefore qualified for routine production.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
