Reference ID: MET-94FF | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Mixing‑time estimation predicts how long a mechanically agitated liquid batch needs to reach 95 % homogeneity, and by applying mixing endpoint detection you can verify the exact moment the desired uniformity is achieved, enabling you to size agitators, set batch cycle times, and guarantee product uniformity in reactors, blenders, and storage tanks across the chemical, pharmaceutical, food, and water‑treatment industries.
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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; this estimate can be validated through a tracer study for mixing time.
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 Tc = V / Q, where V is batch volume and Q is the primary flow rate generated by the impeller.
Estimate mixing time tm = Θ · Tc. Validate later with conductivity or color-change trials.
Constant mixing time requires constant Θ, which is only achieved if the dimensionless flow number (Fl = Q / N·D³) stays the same.
Keep Reynolds number in the same regime (turbulent or transitional).
Use N ∝ D–1 to hold Fl 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/Dax) 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/μapp; if Re < 200, mixing time scales roughly as tm ∝ μapp0.5.
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, NQ = 0.75 (dimensionless)
Mixing-time constant, Kmix = 4 (dimensionless)
Minimum Reynolds for fully turbulent regime, Remin = 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.
Estimate the bulk mixing time with the standard correlation: tmix = KmixV / 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
