Reference ID: MET-9862 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In liquid-liquid extraction processes, the formation of emulsions is a critical phenomenon that dictates mass transfer efficiency and phase separation kinetics. The Sauter mean diameter (d32) serves as a primary metric for characterizing the droplet size distribution within a dispersed phase. Understanding the balance between inertial forces and interfacial tension is essential for designing contactors, such as stirred tanks or extraction columns, where turbulent energy dissipation governs droplet breakup. This calculation is typically employed during the scale-up of chemical reactors to ensure that the interfacial area is sufficient for the required extraction rate while avoiding stable emulsions that complicate downstream separation.
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The droplet size is determined by evaluating the hydrodynamic regime through dimensionless numbers. The Reynolds number (Re) assesses the ratio of inertial forces to viscous forces, while the Weber number (We) evaluates the ratio of inertial forces to surface tension forces. The Sauter mean diameter is subsequently calculated using the Hinze-Kolmogorov correlation, which assumes that droplet breakup is driven by turbulent eddies in the inertial subrange. An empirical proportionality constant \(C\) is required to account for system-specific geometry and fluid properties; typical values for liquid-liquid dispersions range from 0.05 to 0.5, and the constant must be determined experimentally for each application.
The governing equations are defined as follows:
Reynolds Number:
\[ Re = \frac{\rho_{c} \cdot v \cdot D}{\mu_{c}} \]
Weber Number:
\[ We = \frac{\rho_{c} \cdot v^{2} \cdot D}{\sigma} \]
Sauter Mean Diameter:
\[ d_{32} = C \cdot D \cdot We^{-0.6} \]
Parameter
Condition
Regime/Requirement
Reynolds Number
\( Re \geq 1000 \)
Turbulent atomization regime
Weber Number
\( We \geq 10 \)
Stable droplet formation
Emulsions typically form when the interfacial tension between the organic and aqueous phases is significantly reduced, often due to the presence of surface-active impurities or high shear forces. Key factors include:
Presence of solid particulates that act as stabilizing agents at the interface.
Excessive mechanical agitation during the mixing stage.
Rapid changes in pH or ionic strength that alter the solubility of surfactants.
High viscosity ratios between the dispersed and continuous phases.
Mitigation strategies focus on destabilizing the interface and promoting droplet coalescence. Recommended approaches include:
Adjusting the impeller speed to reduce shear stress while maintaining mass transfer efficiency.
Implementing coalescer media, such as knitted wire mesh or packing, to encourage droplet growth.
Adding demulsifying agents to neutralize the charge on droplet surfaces.
Optimizing the phase ratio to ensure the desired phase remains continuous.
The appearance of a rag layer indicates the accumulation of interfacial contaminants. Engineers should perform the following:
Sample the interface to identify the chemical composition of the accumulated solids or surfactants.
Check upstream processes for potential contaminants like corrosion products or degradation of extractants.
Perform a bench-scale bottle test to determine if the emulsion is sensitive to temperature fluctuations or pH adjustments.
Evaluate the residence time in the settler to ensure it exceeds the critical coalescence time for the specific system.
Worked Example: Emulsion Formation in a Liquid-Liquid Extraction Column
A water-continuous liquid-liquid extraction column is being designed to process an organic phase. To ensure efficient mass transfer, the organic phase must be dispersed as fine droplets within the continuous aqueous phase. The following analysis evaluates whether the flow conditions in the column's inlet zone will produce a stable emulsion with a sufficiently small droplet size.
Knowns:
Density of the continuous (water) phase, \(\rho_c = 997.0\) kg/m³
Dynamic viscosity of the continuous phase, \(\mu_c = 0.00089\) Pa·s
Interfacial tension between phases, \(\sigma = 0.072\) N/m
Characteristic velocity of the flow, \(v = 0.5\) m/s
Characteristic length scale (impeller diameter or nozzle diameter), \(D = 0.05\) m
Empirical proportionality constant, \(C = 0.1\) (selected as a representative value for nozzle injection in extraction columns; must be validated experimentally for the specific system)
Step-by-Step Calculation:
Calculate the Reynolds number to determine the flow regime:
\[
Re = \frac{\rho_c \cdot v \cdot D}{\mu_c} = \frac{(997.0) \cdot (0.5) \cdot (0.05)}{0.00089} = 28006
\]
Since \(Re = 28006 > 1000\), the flow is fully turbulent, satisfying the regime requirement for droplet breakup.
Calculate the Weber number to assess droplet stability:
Since \(We = 173.1 > 10\), the inertial forces dominate over interfacial tension, ensuring stable droplet formation and preventing immediate coalescence.
Determine the Sauter mean diameter using the Hinze-Kolmogorov correlation for turbulent dispersions:
\[
d_{32} = C \cdot D \cdot We^{-0.6} = (0.1) \cdot (0.05) \cdot (173.1)^{-0.6} = 0.00023 \text{ m}
\]
This correlation predicts the volume-to-surface area mean droplet diameter produced by turbulent breakup in the inertial subrange. The empirical constant \(C\) must be calibrated for the specific equipment geometry and phase system.
Final Answer:
The predicted Sauter mean diameter of the dispersed phase droplets is \(d_{32} = 0.00023\) m (0.23 mm). The operating conditions (\(Re = 28006\), \(We = 173.1\)) confirm a turbulent, stable dispersion regime suitable for liquid-liquid extraction. Note that the empirical constant \(C\) must be validated experimentally for the specific system to refine this estimate.
"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
