Introduction & Context
In process engineering, the classification of separation units is a critical first step in equipment design and selection. This calculation framework distinguishes between Solid-Liquid Separation (clarifiers, settlers, and thickeners) and Liquid-Liquid Separation (separators and decanters). Proper classification ensures that the correct physical models—such as gravity settling or centrifugal separation—are applied to the feed stream.
These calculations are essential for determining the required surface area for gravity-based separation or for evaluating the necessity of high-G centrifugal force when particle or droplet sizes are too small for efficient natural settling. This methodology is foundational in industries ranging from wastewater treatment to dairy processing and chemical refining.
Methodology & Formulas
The sizing basis relies on the terminal settling velocity of a dispersed phase within a continuous phase. The following formulas define the physical behavior of the system:
The dynamic viscosity of the continuous phase is converted from centipoise to Pascal-seconds:
\[ \mu = \mu_{cP} \cdot 0.001 \]
The density difference between the dispersed phase and the continuous phase determines the driving force for separation:
\[ \Delta\rho = |\rho_{d} - \rho_{c}| \]
The terminal settling velocity is calculated using Stokes' Law, which assumes a laminar flow regime around the particle or droplet:
\[ v_{t} = \frac{g \cdot d^2 \cdot \Delta\rho}{18 \cdot \mu} \]
To validate the use of Stokes' Law, the particle Reynolds number must be calculated to ensure the flow remains in the laminar regime:
\[ Re_{p} = \frac{\rho_{c} \cdot v_{t} \cdot d}{\mu} \]
Finally, the required settling area for a gravity-based unit is determined by the volumetric flow rate and the terminal velocity:
\[ A = \frac{Q}{v_{t}} \]
| Regime/Condition |
Criteria |
| Stokes' Law Validity |
\( Re_{p} < 1.0 \) |
| Gravity Separation Feasibility (Density) |
\( \Delta\rho \geq 50 \text{ kg/m}^3 \) |
| Gravity Separation Feasibility (Size) |
\( d \geq 1 \cdot 10^{-6} \text{ m} \) |
Worked Example: Milk Separator Classification and Sizing Basis
A dairy plant is evaluating a gravity settler for preliminary separation of cream from skim milk. The goal is to classify the unit operation and estimate the required settling area for a specified throughput, using idealized Stokes' law as a baseline.
- Known Input Parameters:
- Continuous phase (skim milk) density, \( \rho_c = 1035.0 \, \text{kg/m}^3 \)
- Dispersed phase (cream droplet) density, \( \rho_d = 920.0 \, \text{kg/m}^3 \)
- Continuous phase viscosity, \( \mu_{c,cp} = 1.5 \, \text{cP} \)
- Characteristic droplet diameter, \( d = 5.0 \times 10^{-6} \, \text{m} \) (5 µm)
- Volumetric feed flow rate, \( Q = 0.001 \, \text{m}^3/\text{s} \)
- Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \)
- System Classification: The feed is milk, an emulsion of fat droplets (dispersed liquid phase) in an aqueous continuous phase. This constitutes a liquid-liquid separation, specifically a cream separator scenario.
- Property Conversion and Density Difference:
- Convert viscosity to SI units: \( \mu_c = \mu_{c,cp} \times 0.001 = 1.5 \times 0.001 = 0.0015 \, \text{Pa·s} \).
- Calculate density difference: \( \Delta\rho = | \rho_d - \rho_c | = | 920.0 - 1035.0 | = 115.0 \, \text{kg/m}^3 \). The absolute value ensures a positive driving force; the negative sign would indicate buoyant rise of the less dense cream droplets.
- Terminal Settling Velocity (Stokes' Law): Apply the idealized formula for a rising droplet:
\[
v_t = \frac{g \cdot d^2 \cdot \Delta\rho}{18 \cdot \mu_c} = \frac{9.81 \cdot (5.0 \times 10^{-6})^2 \cdot 115.0}{18 \cdot 0.0015} = 1.045 \times 10^{-6} \, \text{m/s}.
\]
This very low velocity underscores why centrifugal force is typically used in practice.
- Validity Check (Reynolds Number): Verify Stokes' law applicability:
\[
Re_p = \frac{\rho_c \cdot v_t \cdot d}{\mu_c} = \frac{1035.0 \cdot 1.045 \times 10^{-6} \cdot 5.0 \times 10^{-6}}{0.0015} \approx 3.60 \times 10^{-6}.
\]
Since \( Re_p \ll 1 \), the laminar flow assumption holds for this initial estimate.
- Required Settling Area (Gravity Basis): For a gravity clarifier, the area is based on the overflow rate principle:
\[
A = \frac{Q}{v_t} = \frac{0.001}{1.045 \times 10^{-6}} = 957 \, \text{m}^2.
\]
This area represents the theoretical surface area needed for gravity separation at the given flow.
Final Answer: The system is classified as a liquid-liquid separation (milk cream separator). For the idealized Stokes' law gravity settling model, the terminal rising velocity of cream droplets is approximately \( 1.045 \times 10^{-6} \, \text{m/s} \), and the required settling area for the flow rate of \( 0.001 \, \text{m}^3/\text{s} \) is approximately \( 957 \, \text{m}^2 \). This large area highlights the inefficiency of gravity alone, justifying the use of centrifugal separators in industrial applications.
