Introduction & Context

Mixing-time estimation predicts how long a mechanically agitated liquid batch needs to reach 95 % homogeneity. The result is used to size agitators, set batch cycle times, and guarantee product uniformity in reactors, blenders, and storage tanks throughout the chemical, pharmaceutical, food, and water-treatment industries. A short, reliable mixing time minimises off-spec material and energy waste.

Methodology & Formulas

Convert rotational speed
\(N_{\text{rps}}=\dfrac{N_{\text{rpm}}}{60}\)
Convert dynamic viscosity
\(\mu_{\text{Pa·s}}=\mu_{\text{cP}}\;0.001\)
Pumping capacity (impeller)
\(Q=N_{Q}\;N_{\text{rps}}\;D^{3}\)
where
\(N_{Q}\) flow number (dimensionless)
\(D\) impeller diameter, m
Mixing-time constant
\(t_{\text{mix}}=K_{\text{mix}}\;\dfrac{V}{Q}\)
where
\(K_{\text{mix}}\) dimensionless constant (≈4 for turbulent stirred tanks)
\(V\) batch volume, m³
Reynolds number
\(\text{Re}=\dfrac{\rho\;N_{\text{rps}}\;D^{2}}{\mu_{\text{Pa·s}}}\)
with
\(\rho\) fluid density, kg m^-3

Correlation validity regime
Regime	Reynolds number	Applicability
Turbulent	\(\text{Re}\geq 10\,000\)	Correlation valid
Transitional/Laminar	\(\text{Re}< 10\,000\)	Correlation not recommended; mixing time will be underestimated

The predicted \(t_{\text{mix}}\) is the time required to achieve 95 % of the final concentration uniformity, assuming the vessel is geometrically similar to standard stirred-tank configurations and that the impeller operates in the turbulent regime.

A first-pass estimate can be obtained with the dimensionless mixing time (Θ) correlation for your impeller type.

Look up Θ for your impeller (e.g., Θ ≈ 4 for a pitched-blade turbine in turbulent flow).
Measure or calculate the average circulation time T_c = V / Q, where V is batch volume and Q is the primary flow rate generated by the impeller.
Estimate mixing time t_m = Θ · T_c. Validate later with conductivity or color-change trials.

Constant mixing time requires constant Θ, which is only achieved if the dimensionless flow number (F_l = Q / N·D³) stays the same.

Maintain geometric similarity (D/T, C/T, Z/T ratios).
Keep Reynolds number in the same regime (turbulent or transitional).
Use N ∝ D^–1 to hold F_l constant; power per volume will rise as P/V ∝ D^–1, so confirm motor sizing.

Once the turbulent macro-mixing time drops below the time required for the final micro-mixing or diffusion-controlled step, further speed increases give diminishing returns.

Check if the Bodenstein number (Bo = uL/D_ax) indicates convection-limited rather than diffusion-limited regime.
Consider feed pipe or dip-tube placement; poor meso-mixing can mask gains from faster impeller speed.

Shear-thinning fluids create a viscosity field that varies with radius; the effective Reynolds number changes locally.

Use the apparent viscosity at the representative shear rate γ ≈ k·N, where k is 10–30 for turbines.
Recalculate Re = ρND²/μ_app; if Re < 200, mixing time scales roughly as t_m ∝ μ_app^0.5.
For yield-stress fluids, ensure impeller tip speed exceeds the critical speed to eliminate stagnant caverns; otherwise mixing time becomes indeterminate.

Worked Example – Estimating Mixing Time in a Water Storage Tank

A small municipal water-treatment plant needs to verify that a rapid-mix tank will achieve the target blend time before chemical dosing. The vessel is a vertical cylinder, 1.5 m in diameter and 1.13 m straight side, giving a working volume of 2 m³. A single pitched-blade impeller (D = 0.45 m) is driven at 180 rpm. Water at 20 °C is the only phase.

Knowns

Tank volume, V = 2.0 m³
Impeller diameter, D = 0.45 m
Rotational speed, N = 180 rpm = 3.0 rps
Fluid density, ρ = 1000 kg/m³
Fluid viscosity, μ = 1 cP = 0.001 Pa·s
Impeller flow number, N_Q = 0.75 (dimensionless)
Mixing-time constant, K_mix = 4 (dimensionless)
Minimum Reynolds for fully turbulent regime, Re_min = 10,000

Step-by-step calculation

Check the Reynolds number to confirm turbulent regime:
\[ \text{Re} = \frac{\rho N D^{2}}{\mu} = \frac{1000 \times 3.0 \times 0.45^{2}}{0.001} = 607\,500 \]
Since 607,500 ≫ 10,000, the flow is fully turbulent.
Calculate the impeller pumping capacity:
Q = N_Q N D³ = 0.75 × 3.0 × 0.45³ = 0.205 m³/s
Estimate the bulk mixing time with the standard correlation:
t_mix = K_mix V / Q = 4 × 2.0 / 0.205 = 39.0 s

Final Answer

The predicted mixing time for the tank is 39 s. This satisfies the plant requirement of achieving uniformity well within the 60 s window allowed for flash mixing.

"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