Introduction & Context

The mixing index is a dimensionless figure of merit that quantifies how close a real particulate mixture is to the theoretical perfectly-random state. In process engineering it is used to:

Compare alternative blenders or mixers on a common 0–1 scale.
Set release specifications for pharmaceutical, food or catalyst powders.
Detect over-mixing (attrition, demixing) or under-mixing (poor equipment selection).

The calculation is valid for binary mixtures where the minor component is present as discrete particles and the analytical sample is obtained by thief or splitter. The method is insensitive to particle shape provided the equivalent spherical diameter is used.

Methodology & Formulas

Single-particle mass
The mass of one spherical particle is \[ m_{\text{p}}=\frac{\pi}{6}\,d_{\text{p}}^{3}\,\rho_{\text{p}} \] with \(d_{\text{p}}\) the mean sieve diameter and \(\rho_{\text{p}}\) the envelope particle density.
Number of particles in the analytical sample
\[ n=\frac{m_{\text{sample}}}{m_{\text{p}}} \] where \(m_{\text{sample}}\) is the mass of the analytical specimen (typically 20 g).
Variance of the totally-segregated state
\[ \sigma_{0}^{2}=p\,(1-p) \] with \(p\) the target mass fraction of the minor component.
Variance of the perfectly-random mixture
\[ \sigma_{\text{r}}^{2}=\frac{p\,(1-p)}{n} \] This is the theoretical lower limit achievable by random positioning of particles.
Mixing index
\[ M=\frac{\sigma_{0}^{2}-\sigma_{\text{meas}}^{2}}{\sigma_{0}^{2}-\sigma_{\text{r}}^{2}} \] where \(\sigma_{\text{meas}}\) is the experimentally determined standard deviation of the sample composition. The index is bounded by 0 (fully segregated) and 1 (perfectly random). Values outside this range indicate measurement noise or non-binomial behaviour.

Validity regimes and warning thresholds
Parameter	Condition	Interpretation
Number of particles	\(n \geq 1000\)	Gaussian approximation for \(\sigma_{\text{r}}\) statistically meaningful
Sample mass	\(m_{\text{sample}} \geq 20\ \text{g}\)	Sampling error \(\leq 1\ \%\) relative for typical pharmaceutical powders
Mixing index	\(M < 0\)	Measured variance exceeds segregation limit—check data or model
Mixing index	\(M > 1\)	Measured variance below random limit—possible over-mixing or analytical noise

The Mixing Index is a dimensionless number that quantifies how uniformly two or more streams are blended. Tracking it lets you:

Detect maldistribution before it hurts conversion or product quality
Optimize static-mixer or in-line mixer design without trial-and-error
Prove to regulators that blending meets specification (e.g., < 1 % relative standard deviation)

Collect at least 30 samples across the pipe cross-section after the mixer. Compute:

Mean concentration C̄ = ΣCᵢ / n
Standard deviation σ = √[Σ(Cᵢ – C̄)² / (n – 1)]
Mixing Index M = 1 – (σ / C̄) (perfect mix = 1, no mix = 0)

Report the value as a percent if desired: M% = 100 M.

Choose a tracer that is:

Fully soluble in the continuous phase
Non-reactive under process T & P
Measurable on-line (e.g., conductivity salt, dye at ppm level, or radiotracer for opaque systems)

Inject a pulse mass giving 3–5 × baseline noise; typically 0.5–2 % of stream flow for 30 s is enough.

Short-term fixes:

Increase pipe velocity > 0.3 m s⁻¹ to force turbulent regime (Re > 4000)
Add a static mixer with 6–9 mixing elements; verify ΔP < 10 % of line pressure
Move injection point 5–10 pipe diameters upstream of the mixer to pre-disperse the tracer

If still low, check for dead zones via CFD or tomography and relocate the mixer closer to the reactor inlet.

Worked Example – Mixing Index for a Pharmaceutical Blend

A continuous blender is being commissioned to produce a low-dose inhalation formulation. The active ingredient (API) must be uniformly distributed in a lactose carrier at a target mass fraction of 0.2 %. To verify that the blend meets uniformity specifications, a 20 g analytical sample is withdrawn and assayed. The measured standard deviation of the API mass fraction across five replicate assays is 0.00015. Calculate the mixing index, M, and confirm whether the required homogeneity has been achieved.

Knowns

Particle density, ρ_p = 1400 kg m^-3
Mean particle diameter, d_p = 200 µm
Target API mass fraction, p = 0.002
Analytical sample mass, m_sample = 20 g
Measured standard deviation, σ_measured = 0.00015
Minimum number of particles for statistical validity, n_min = 1000

Step-by-Step Calculation

Convert mean particle diameter to metres:
d_p = 200 µm = 0.0002 m
Compute the mass of a single particle assuming a spherical shape:
m_p = \( \frac{\pi}{6} \rho_p d_p^3 \)
m_p = \( \frac{\pi}{6} \times 1400 \times (0.0002)^3 \)
m_p = 5.86 × 10^-9 kg = 5.86 × 10^-6 g
Estimate the number of particles in the 20 g sample:
n = \( \frac{m_{sample}}{m_p} \)
n = \( \frac{20}{5.86 \times 10^{-6}} \) ≈ 3.41 × 10⁶ particles (>> 1000, so the sample is statistically valid)
Calculate the theoretical Poisson (random) standard deviation:
σ₀ = \( \sqrt{\frac{p(1-p)}{n}} \)
σ₀ = \( \sqrt{\frac{0.002 \times 0.998}{3.41 \times 10^6}} \) ≈ 0.0000242
Compute the mixing index:
M = \( \frac{\sigma_0^2 - \sigma_{measured}^2}{\sigma_0^2} \)
σ₀² = (0.0000242)² = 5.85 × 10^-10
σ_measured² = (0.00015)² = 2.25 × 10^-8
M = \( \frac{5.85 \times 10^{-10} - 2.25 \times 10^{-8}}{5.85 \times 10^{-10}} \) ≈ 0.99999

Final Answer

The mixing index M = 0.99999 (dimensionless). Because M is very close to unity, the blend is considered excellent and meets the homogeneity requirement for downstream processing.

"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