Introduction & Context

The tracer study for mixing time, denoted as t₉₅, is a critical diagnostic tool in process engineering used to quantify the performance of batch stirred tanks. Unlike continuous flow systems that rely on residence time distribution (RTD) analysis, this study focuses on the time required for a localized tracer pulse to achieve 95% homogeneity throughout the vessel volume. This metric is essential for ensuring product consistency, validating reaction kinetics, and optimizing impeller power consumption in industrial blending operations.

Methodology & Formulas

The calculation of mixing time relies on the hydrodynamic regime of the fluid and the experimental response of a tracer concentration probe. The following mathematical framework defines the transition from physical parameters to the final t₉₅ value.

First, the flow regime is characterized by the Reynolds number (Re) to ensure the validity of turbulent mixing correlations:

\[ Re = \frac{\rho \cdot N \cdot D^{2}}{\mu} \]

The theoretical mixing time (t_95,theory) is estimated based on the impeller pumping capacity and tank volume:

\[ t_{95,theory} = \frac{4 \cdot V}{N_{Q} \cdot N \cdot D^{3}} \]

When processing experimental data, the normalized response F(t) is calculated to determine the degree of homogeneity. To find the precise t₉₅ from discrete data points, linear interpolation is applied between two points that bracket the 0.95 threshold:

\[ t_{95,experimental} = T_{1} + \left( \frac{F_{target} - F_{1}}{F_{2} - F_{1}} \right) \cdot (T_{2} - T_{1}) \]

Parameter	Condition / Threshold	Implication
Flow Regime	Re < 10,000	Correlation invalid; transition or laminar flow requires specific empirical models.
Interpolation	F₁ ≥ 0.95 or F₂ ≤ 0.95	Data points do not bracket the target; select a wider time interval.
Mixing State	F(t) = 0.95	Definition of t₉₅; the system is considered effectively mixed.

To ensure accurate mixing time measurements, the injection point must be selected based on the following criteria:

Position the injection port as far as possible from the impeller discharge zone to avoid localized concentration spikes.
Ensure the tracer is introduced into the bulk flow region to represent the global mixing process.
Verify that the injection mechanism minimizes initial momentum effects that could artificially accelerate the mixing rate.

The selection of a tracer is critical for data integrity. Consider these factors:

The tracer must be chemically inert relative to the process fluid to prevent side reactions.
The physical properties, specifically density and viscosity, should closely match the process fluid to avoid buoyancy-driven stratification.
The detection method, such as conductivity or spectrophotometry, must have a linear response range that covers the expected concentration fluctuations.

Mixing time is typically defined as the duration required for the tracer concentration to reach and remain within a specified percentage of the final equilibrium value. Follow these steps:

Normalize the raw signal data against the steady-state concentration value.
Apply a 95% or 99% homogeneity criterion to the normalized signal.
Identify the time point where the signal fluctuations permanently enter the defined error band.
Perform multiple trials to calculate a statistically significant mean mixing time for the specific impeller speed and fluid viscosity.

Worked Example: Mixing Time Determination via Tracer Study

In a batch stirred tank (cylindrical, baffled, working volume 5 m³ water), a tracer pulse of concentrated salt solution is injected. The conductivity probe is placed at the tank wall, mid-baffle position. The normalized tracer concentration response F(t) is recorded until plateau. The goal is to determine the mixing time t₉₅ (time to reach 95% homogeneity).

Knowns

V = 5.0 m³
ρ = 1000.0 kg/m³
μ = 0.001 Pa·s
N = 2.0 rps
D = 0.4 m
N_Q = 0.5 (pumping number for 45° pitched blade turbine)
T₁ = 300.0 s, F₁ = 0.93
T₂ = 325.0 s, F₂ = 0.97
F_target = 0.95

Step-by-Step Calculation

Reynolds Number Check
Re = \(\frac{\rho N D^2}{\mu} = \frac{1000.0 \cdot 2.0 \cdot (0.4)^2}{0.001} = 320000.0\)
Since Re > 10⁴, the flow is fully turbulent and the theoretical mixing time correlation is valid.
Theoretical Mixing Time
t_95,theory = \(\frac{4 V}{N_Q N D^3} = \frac{4.0 \cdot 5.0}{0.5 \cdot 2.0 \cdot (0.4)^3} = 312.5 \, \text{s}\)
Interpolation from Experimental Data
The target F_target = 0.95 is bracketed by F₁ = 0.93 at T₁ = 300.0 s and F₂ = 0.97 at T₂ = 325.0 s.
ΔF = F₂ - F₁ = 0.97 - 0.93 = 0.04
t_95,exp = \(T_1 + \frac{F_{\text{target}} - F_1}{\Delta F} \cdot (T_2 - T_1) = 300.0 + \frac{0.95 - 0.93}{0.04} \cdot (325.0 - 300.0) = 312.5 \, \text{s}\)
Validity of Interpolation
F₁ = 0.93 < F_target = 0.95 < F₂ = 0.97, so the linear interpolation is valid and no extrapolation is required.

Final Answer

t₉₅ = 312.5 s (both theoretical prediction and experimental interpolation give the same value).

"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