Introduction & Context

The calculation of the liquid-liquid interface radius is a fundamental procedure in process engineering, specifically within the design and operation of centrifugal separators. In a cylindrical bowl or disc stack centrifuge, two immiscible liquids of differing densities are subjected to a high-magnitude centrifugal force field. Under steady-state conditions, these fluids undergo rigid-body rotation, forming concentric layers where the heavier phase is forced toward the outer wall and the lighter phase migrates toward the axis of rotation.

Determining the precise location of the interface between these two phases is critical for ensuring product purity and preventing phase carry‑over. This calculation is essential for sizing overflow dams, density ring selection for separators, setting discharge weir radii, and optimizing the separation efficiency of industrial equipment such as cream separators, oil‑water decanters, and solvent extraction centrifuges.

Methodology & Formulas

The physical model relies on the principle of hydrostatic equilibrium within a rotating frame of reference. The pressure gradient generated by the centrifugal field must be balanced across the interface. For two incompressible, immiscible fluids, the pressure at the interface radius must be identical when approached from either the light phase or the heavy phase.

The governing equilibrium equation is defined as:

\[ \rho_{L} \cdot (r_{i}^{2} - r_{L}^{2}) = \rho_{H} \cdot (r_{H}^{2} - r_{i}^{2}) \]

To determine the interface radius r_i, the equation is rearranged algebraically. Notably, the angular velocity ω cancels out, demonstrating that for incompressible fluids, the equilibrium interface position is a function of geometry and density ratios rather than rotational speed.

The final expression for the interface radius is:

\[ r_{i} = \sqrt{\frac{\rho_{H} \cdot r_{H}^{2} + \rho_{L} \cdot r_{L}^{2}}{\rho_{L} + \rho_{H}}} \]

The following table outlines the operational constraints and physical regimes required for the validity of this model:

Constraint/Regime	Condition	Engineering Implication
Density Requirement	\(\rho_{H} > \rho_{L}\)	The heavy phase must occupy the outer radius to maintain stable stratification.
Geometric Validity	\(r_{L} < r_{i} < r_{H}\)	The interface must exist within the physical boundaries of the bowl; otherwise, phase breakthrough occurs.
Flow Regime	Rigid-Body Rotation	Assumes steady-state operation where fluid velocity matches bowl velocity; non-ideal flow may occur at low speeds.
Separation Stability	\(\Delta\rho > 50 \text{ kg/m}^{3}\)	A sufficient density difference is required to prevent a diffuse, unstable interface.

To calculate the interface radius, you must balance the gravitational and centrifugal forces acting on the two liquid phases. Follow these steps:

Identify the densities of the light and heavy liquid phases.
Determine the required residence time based on the droplet settling velocity.
Apply the force balance equation to solve for the radius where the phases separate.
Verify that the calculated radius falls within the mechanical constraints of the vessel geometry.

The accuracy of your model depends heavily on the precision of the following input parameters:

Density difference between the two liquid phases.
Dynamic viscosity of the continuous phase.
Interfacial tension between the liquids.
Operating temperature, as it significantly impacts both density and viscosity values.

The interface level control represents the setpoint maintained by the level transmitter. If this value fluctuates, the effective radius changes, which can lead to:

Carryover of the light phase into the heavy phase outlet.
Underflow of the heavy phase into the light phase outlet.
Instability in the separation zone, potentially causing emulsion formation.

Worked Example: Interface Radius in a Cream Separator

In a dairy processing plant, a tubular bowl centrifugal separator is used to separate cream (light phase) from skim milk (heavy phase). The process engineer must determine the equilibrium position of the liquid-liquid interface to ensure stable operation and correct product removal.

Known Parameters:

Density of heavy phase (skim milk), \(\rho_H = 1035\ \text{kg/m}^3\)
Density of light phase (cream), \(\rho_L = 950\ \text{kg/m}^3\)
Inner bowl wall radius, \(r_H = 0.15\ \text{m}\)
Light phase outlet (overflow dam) radius, \(r_L = 0.05\ \text{m}\)

Step-by-Step Calculation:

The governing equation for hydrostatic equilibrium in the centrifugal field is:
\[ \rho_L \left( r_i^{2} - r_L^{2} \right) = \rho_H \left( r_H^{2} - r_i^{2} \right) \]
This equation is rearranged to solve for the unknown interface radius \(r_i\):
\[ r_i = \sqrt{ \frac{\rho_H r_H^{2} + \rho_L r_L^{2}}{\rho_L + \rho_H} } \]

(Note: The rotational speed term cancels out, making the interface position independent of the centrifuge's speed under ideal conditions).
Substituting the known parameters into the equation:
\[ r_i = \sqrt{ \frac{ (1035) \cdot (0.15)^{2} + (950) \cdot (0.05)^{2} }{ 950 + 1035 } } \]
Performing the calculation yields the equilibrium interface radius.

Final Answer:

The equilibrium interface radius for this separator is \(r_i = 0.113\ \text{m}\).

"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