Reference ID: MET-2330 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In process engineering, the separation of solid particles from a liquid phase is a fundamental unit operation. The selection between filtration and centrifugation is primarily governed by the settling characteristics of the particles within the fluid medium. This calculation determines the terminal settling velocity of particles under gravitational influence and validates the applicability of Stokes' Law. Understanding these parameters is critical for designing sedimentation tanks, hydrocyclones, and centrifuge equipment, ensuring that the chosen separation technology aligns with the physical behavior of the slurry.
Methodology & Formulas
The methodology relies on the application of Stokes' Law to calculate the terminal velocity of a spherical particle falling through a viscous fluid. The validity of this model is constrained by the Reynolds number, which must remain within the laminar flow regime to ensure the drag force is accurately represented by the Stokes' drag coefficient.
The terminal velocity v under gravity is calculated as follows:
For centrifugal separations, such as in centrifuges or hydrocyclones, the gravitational acceleration \( g \) is replaced by the centrifugal acceleration \( a_c = \omega^2 \cdot r \), where \( \omega \) is the angular velocity and \( r \) is the radius of rotation. The modified formula becomes:
The flow regime is characterized by the particle Reynolds number Re, defined as:
\[ Re = \frac{\rho_{f} \cdot v \cdot D}{\mu} \]
This definition applies to both gravitational and centrifugal settling, with \( v \) being the terminal velocity under the respective acceleration.
Where:
g is the acceleration due to gravity (9.81 m/s²)
a_c is the centrifugal acceleration (m/s²)
D is the particle diameter (m)
ρs is the density of the solid particle (kg/m³)
ρf is the density of the fluid (kg/m³)
μ is the dynamic viscosity of the fluid (kg/m·s)
ω is the angular velocity (rad/s)
r is the radius of rotation (m)
Regime
Condition
Applicability
Laminar (Stokes' Law)
Re < 1.0
Valid for sedimentation calculations under gravity or centrifugal force, provided the acceleration term is adjusted accordingly. Ensures accurate drag force estimation.
Transition/Turbulent
Re ≥ 1.0
Stokes' Law invalid; requires empirical drag correction (e.g., using the drag coefficient \( C_D \) for higher Reynolds numbers).
To determine the optimal separation method, process engineers should evaluate the following factors:
Particle size distribution and concentration of the solids.
Required purity levels for the final product.
Physical properties of the slurry, such as viscosity and density differences, which affect settling velocities as calculated by Stokes' law or modified versions for centrifugation.
Economic considerations, including capital expenditure and operational maintenance costs.
Settling characteristics under gravity versus centrifugal force, based on terminal velocity calculations.
Centrifugation is generally the superior choice under these conditions:
The solids are fine or colloidal, which would likely blind a filter medium, and centrifugal force can enhance separation efficiency.
The process requires continuous operation with minimal downtime for cleaning, as many centrifuges operate continuously.
The density difference between the solid and liquid phases is significant, leading to higher terminal velocities under centrifugal acceleration.
The slurry is highly viscous, making pressure-driven filtration inefficient, whereas centrifugation can overcome viscous drag with increased force.
When gravity settling velocities (as calculated) are too slow for practical separation times, necessitating higher acceleration.
Filtration faces specific operational challenges that engineers must mitigate:
Filter media fouling or blinding, which necessitates frequent backwashing or replacement, increasing downtime and costs.
High pressure drops across the cake, which can lead to increased energy consumption and potential equipment wear.
Difficulty in handling compressible cakes that deform under pressure, reducing filtration efficiency.
Batch-wise operation cycles that may limit overall throughput compared to continuous centrifuges, affecting scalability.
Dependence on particle size; for very small particles, filtration may be ineffective without pre-treatment, whereas centrifugation can handle a wider range via adjusted acceleration.
Worked Example: Settling Velocity of Aluminum Particles in Glycerin
A process engineer is evaluating gravity sedimentation for separating aluminum particles from a glycerin solution. Three particle sizes are analyzed to determine their settling velocities and verify the applicability of Stokes' law, which informs the selection between filtration and centrifugation. This example focuses on gravity settling as a basis for comparison.
Knowns (Input Parameters):
Acceleration due to gravity, g = 9.81 m/s2
Density of aluminum solid particles, ρs = 2600.0 kg/m3
Density of glycerin fluid, ρf = 1274.0 kg/m3
Dynamic viscosity of glycerin, μ = 1.0 kg/m·s
Particle diameters: D1 = 0.002 m, D2 = 0.004 m, D3 = 0.010 m
Step-by-Step Calculation:
Calculate the terminal settling velocity for each diameter using Stokes' law under gravity:
\[ v = \frac{g \cdot D^2 \cdot (\rho_s - \rho_f)}{18 \cdot \mu} \]
For D1 = 0.002 m: v1 = \(\frac{9.81 \cdot (0.002)^2 \cdot (2600.0 - 1274.0)}{18 \cdot 1.0}\) = 0.002891 m/s
For D2 = 0.004 m: v2 = \(\frac{9.81 \cdot (0.004)^2 \cdot 1326.0}{18 \cdot 1.0}\) = 0.011563 m/s
For D3 = 0.010 m: v3 = \(\frac{9.81 \cdot (0.010)^2 \cdot 1326.0}{18 \cdot 1.0}\) = 0.072267 m/s
Convert velocities to mm/s for practical interpretation (1 m/s = 1000 mm/s):
v1,mm/s = 2.891 mm/s
v2,mm/s = 11.563 mm/s
v3,mm/s = 72.267 mm/s
Calculate the Reynolds number to check Stokes' law validity:
\[ Re = \frac{\rho_f \cdot v \cdot D}{\mu} \]
Verify Stokes' law condition (Re < 1.0): All calculated Re values are less than 1.0, so Stokes' law is valid for all cases under gravity.
Final Answer:
The terminal settling velocities under gravity are v1 = 2.891 mm/s, v2 = 11.563 mm/s, and v3 = 72.267 mm/s for particle diameters of 2 mm, 4 mm, and 10 mm, respectively. Since all Reynolds numbers are below 1.0, Stokes' law applies, indicating that gravity sedimentation may be feasible for smaller particles, but centrifugation could be considered for faster separation of larger particles by increasing the effective acceleration beyond gravity.
"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
