Introduction & Context

Solid Distribution Uniformity (SDU) quantifies how evenly suspended particles—such as chocolate chips in cake batter—are dispersed throughout a stirred tank. In process engineering, SDU is a key performance indicator for mixing operations: poor uniformity leads to off-spec product, variable texture, or downstream separation issues. The metric is routinely used in food, pharmaceutical, and fine-chemical industries where batch integrity and regulatory compliance depend on consistent solids loading at every sampling location.

Methodology & Formulas

Convert dynamic viscosity
$ \mu_{\text{Pa·s}} = \mu_{\text{cP}} \times 0.001 $
Impeller rotational speed
$ N_{\text{rps}} = \dfrac{\text{rpm}}{60} $
Reynolds number for a stirred vessel
\[ Re = \dfrac{\rho N_{\text{rps}} D^{2}}{\mu_{\text{Pa·s}}} \]

Flow regime Re range

Laminar $ Re < 10 $

Transitional $ 10 \le Re < 10\,000 $

Fully turbulent $ Re \ge 10\,000 $
Mean measured concentration
\[ \bar{c} = \dfrac{1}{n}\sum_{i=1}^{n} c_{i} \]
Sample standard deviation
\[ \sigma = \sqrt{\dfrac{1}{n-1}\sum_{i=1}^{n}(c_{i}-\bar{c})^{2}} \]
Solid Distribution Uniformity
\[ \text{SDU} = 1 - \dfrac{\sigma}{\max(\bar{c},\;10^{-9})} \]
SDU ranges from 0 (completely non-uniform) to 1 (perfectly uniform).

Solid distribution uniformity quantifies how evenly particles are dispersed across a vessel, pipe, or reactor cross-section. Poor uniformity leads to:

Local hot spots or dead zones that reduce conversion or selectivity
Segregation that causes off-spec product or catalyst deactivation
Increased pressure drop and energy use due to maldistribution

Designing for uniformity minimizes these risks and improves overall plant profitability.

Focus on these measurable indicators:

Radial or axial concentration profiles from optical or capacitance probes
Pressure fluctuation amplitude and frequency in fluidized beds
Local solids flux using isokinetic sampling or laser Doppler methods
Electrical resistance tomography (ERT) or gamma-ray density maps

Trending these signals against baseline data quickly flags maldistribution before product quality drifts.

Low-cost operational tweaks often restore uniformity:

Increase superficial gas velocity within erosion limits to boost turbulence
Adjust distributor pressure drop to ≥30 % of bed pressure drop for even fluidization
Install or reposition internal baffles to break large bubbles or slugs
Optimize particle size distribution—narrow cuts reduce segregation

Pilot trials with these variables can yield 10–30 % improvement in residence-time uniformity.

Use the following rule set:

Empirical correlations suffice for similar vessel aspect ratios (<±20 %) and particle densities
Invoke CFD when internals, multiple feed points, or heat exchangers create 3-D effects that correlations ignore
CFD is justified for scale-ups beyond 5:1 volume ratio or when maldistribution costs exceed $1 M yr⁻¹ in lost yield

A hybrid approach—CFD to identify problem zones, then empirical tuning—often delivers the fastest ROI.

Worked Example – Checking Chocolate-Chip Uniformity in a Pilot-Scale Batter Mixer

A confectionery plant is scaling up a new muffin batter. To avoid “bottom-heavy” chocolate chips, the process engineer must verify that the Solid Distribution Uniformity (SDU) is ≥ 0.90 when the 30 cm pilot tank is run at 180 rpm. Samples are withdrawn at five elevations and three radial positions, and the chip mass fraction is measured.

Tank diameter: 0.3 m
Tank height: 0.3 m
Impeller diameter: 0.1 m
Impeller speed: 180 rpm (3.0 rps)
Batter density: 1100 kg m⁻³
Batter viscosity: 4.0 cP (0.004 Pa s)
Target chip mass fraction: 0.02 kg kg⁻¹
Measured chip mass fractions (15 samples): average 0.0199 kg kg⁻¹
Sum of squared deviations from target: 2.573 × 10⁻⁵

Calculate the Reynolds number to confirm turbulent regime: \[ Re = \frac{\rho N D^2}{\mu} = \frac{1100 \times 3.0 \times 0.1^2}{0.004} = 8250 \] $Re = 8250$ indicates fully turbulent flow.
Compute the standard deviation of the measured mass fractions: \[ \sigma = \sqrt{\frac{\sum (x_i - \mu_{\text{target}})^2}{n - 1}} = \sqrt{\frac{2.573 \times 10^{-5}}{14}} = 0.00136 \text{ kg kg}^{-1} \]
Evaluate Solid Distribution Uniformity: \[ SDU = 1 - \frac{\sigma}{\mu_{\text{target}}} = 1 - \frac{0.00136}{0.02} = 0.932 \]

Final Answer: SDU = 0.932 (dimensionless), exceeding the 0.90 specification; the mixer is acceptable for scale-up.

"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

Flow regime	Re range
Laminar	\( Re < 10 \)
Transitional	\( 10 \le Re < 10\,000 \)
Fully turbulent	\( Re \ge 10\,000 \)