Introduction & Context

In food-grade stirred-tank operations—such as keeping chocolate-chip-like solids uniformly suspended in batter—the target is to quantify “how evenly” the solids are distributed. The dimensionless degree of uniformity (mixing index) gives a single number that operators can benchmark against a minimum acceptable value (typically 0.95 for food suspensions). The calculation is used:

after any speed or time adjustment to verify that additional mixing is unnecessary;
during process qualification to document batch-to-batch reproducibility;
in scale-up to demonstrate that a larger tank delivers the same homogeneity as the pilot unit.

Methodology & Formulas

The workflow mirrors the sampling protocol in the Python module:

Collect n samples at different heights (and, if desired, radial positions); convert wt % readings to mass fractions c_i (kg solids kg^-1 slurry).
Compute the mean concentration in the tank: \[ \bar{c}= \frac{1}{n}\sum_{i=1}^{n}c_{i} \]
Compute the sample standard deviation: \[ \sigma = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(c_{i}-\bar{c})^2} \]
Calculate the mixing index using the design target concentration c_design (supplied as mass-loading) to avoid bias toward the accidentally richer or leaner batch: \[ I_{\text{uniform}} = 1 - \frac{\sigma}{\max(c_{\text{design}},10^{-9})} \]
Check hydrodynamic regime with the impeller Reynolds number based on impeller diameter D: \[ Re_{\text{impeller}} = \frac{\rho_{\text{f}}\,N_{\text{rad/s}}\,D^2}{\mu_{\text{f}}} \quad \text{with} \quad N_{\text{rad/s}} = N_{\text{rpm}}\cdot\frac{2\pi}{60} \]

Parameter	Validity range	Consequence if violated
Mean solids loading	0 < c_design < 0.15 kg kg^-1	Correlation invalid; use dense-slurry models
Reynolds number	Re_impeller > 10,000	Flow not fully turbulent; index may over-estimate uniformity

The most common metric is the Relative Standard Deviation (RSD) of sample concentrations taken across the vessel, pipe, or belt at steady state. Acceptable RSD values vary by industry:

Food & pharma: ≤ 5%
Mineral processing: ≤ 10%
Waste-water neutralization: ≤ 15%

Sampling frequency must exceed the theoretical blend-time, and each grab sample should represent ≤ 0.5% of the total batch mass.

Key variables are the dimensionless numbers:

Reynolds number (Re) – turbulent regime (Re > 10,000 for impeller) shortens blend time
Power number (Np) – sets mechanical power input; scale-up with constant power per unit volume
Suspension speed (Njs) – minimum shaft speed to keep solids completely off the bottom; use the Zwietering correlation
Circulation number – vessel turnovers per minute; target ≥ 5 for uniformity

Adjust impeller diameter, blade angle, and clearance to meet these targets without over-shearing the product.

Maintain three scale-up rules simultaneously:

Geometric similarity: keep H/T, D/T, and baffle width ratios constant
Kinematic similarity: keep tip speed ≤ 7 m s⁻¹ to avoid particle attrition
Dynamic similarity: maintain constant power per unit mass (P/m). If P/m is kept, RSD remains within ±1% of pilot plant value across scales from 1 m³ to 100 m³

Validate with tomography or inline NIR at both scales.

Choose based on slurry density and particle size:

Electrical Resistance Tomography (ERT): 0–30% solids, detects 1% concentration deviation every 2 s
Ultrasonic attenuation: suits 10–60 µm particles, 0–40% solids; signal drift <1% per month
Near-InfraRed (NIR) diffuse reflectance: 0–15% solids, fast for API content in pharma
Laser diffraction probes: gives chord length distribution every 4 s; requires dilution loop

All outputs can be trended in the DCS; set alarms at RSD thresholds derived from the specification.

Worked Example: Solid Distribution Uniformity in a Chocolate Chip Batter Mixer

A food processing plant mixes chocolate chips into a batter for cookie production. To quantify mixing uniformity, an engineer samples the slurry in a stirred tank after a fixed mixing time and calculates the degree of uniformity using the mixing index.

Knowns (Input Parameters):

Impeller diameter, D = 0.300 m
Liquid height, H_liq = 0.300 m
Impeller speed, N = 200 rpm
Impeller blade angle = 45.0°
Impeller depth below surface = 0.150 m
Solid (chocolate chip) density, ρ_s = 1200.0 kg m^-3
Particle diameter, d_p = 0.004 m
Design mass loading = 5.0 wt% (target concentration, \( c_{\text{design}} \) = 0.050 kg kg^-1)
Liquid (batter) density, ρ_f = 1100.0 kg m^-3
Liquid viscosity, μ_f = 0.003 Pa s
Number of samples taken, n = 6

Step-by-Step Calculation:

Sample the slurry at three heights (30%, 60%, 90% of H_liq) and two radial positions, yielding n=6 samples. The solids concentration for each sample is measured in wt% and converted to mass fraction c_i in kg kg^-1. One sample, for example, has c_i = 0.05187 kg kg^-1.
Compute the mean concentration: \( \bar{c} = \frac{\sum c_i}{n} = 0.050 \, \text{kg kg}^{-1} \).
Compute the sum of squared deviations: \( \sum (c_i - \bar{c})^2 = 2.098 \times 10^{-5} \, (\text{kg kg}^{-1})^2 \).
Compute the sample variance: \( \sigma^2 = \frac{\sum (c_i - \bar{c})^2}{n-1} = 4.196 \times 10^{-6} \, (\text{kg kg}^{-1})^2 \).
Compute the sample standard deviation: \( \sigma = \sqrt{\sigma^2} = 0.002 \, \text{kg kg}^{-1} \).
Compute the mixing index (degree of uniformity) using \( I = 1 - \frac{\sigma}{c_{\text{design}}} \). Substituting values: \( I = 1 - \frac{0.002}{0.050} = 0.959 \).
Verify turbulent flow regime by calculating the impeller Reynolds number. First, convert impeller speed: \( N = 200 \, \text{rpm} \times 0.104720 \, \text{rad s}^{-1} \, \text{per rpm} = 20.944 \, \text{rad s}^{-1} \). Then, \( Re = \frac{\rho_f N D^2}{\mu_f} = \frac{1100.0 \times 20.944 \times (0.300)^2}{0.003} = 691150 \). Since \( Re > 10,000 \), the flow is turbulent, validating the use of the mixing index correlation.
Benchmark the result. For food suspensions, \( I \geq 0.95 \) indicates "well mixed." Here, \( I = 0.959 \), so the mixture meets the uniformity standard.

Final Answer: The mixing index is \( I = 0.959 \), confirming that the chocolate chips are uniformly distributed in the batter (well mixed).

"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