Introduction & Context
The decanter centrifuge is a critical unit operation in process engineering for the solid-liquid separation of slurries. By applying high centrifugal forces, these machines accelerate the sedimentation of particles that would otherwise settle too slowly under gravity alone. A decanter consists of a horizontal cylindrical-conical bowl and an internal screw conveyor. The primary clarification occurs in the cylindrical section of the bowl, while the conical beach dewaters the solids. This reference sheet presents the fundamental calculations for the cylindrical clarification zone, which governs the overall separation capacity, using Ambler’s theory of sedimentation.
Methodology & Formulas
The following equations define the mechanical and hydrodynamic performance of the cylindrical clarification section. The particle settling velocity assumes Stokes’ law regime (particle Reynolds number \(Re_{p} = \rho_{f} v_{s} d_{p} / \mu < 0.1\)).
The centrifugal acceleration at the bowl wall:
\[ a_{c} = \omega^{2} \cdot R_{o} \]
The relative centrifugal force (G-force) is the ratio of centrifugal acceleration to standard gravity:
\[ G = \frac{a_{c}}{g} \]
The terminal settling velocity of a particle at the bowl wall, assuming Stokes’ law regime, is:
\[ v_{s} = \frac{d_{p}^{2} \cdot (\rho_{p} - \rho_{f}) \cdot a_{c}}{18 \cdot \mu} \]
or equivalently
\[ v_{s} = \frac{d_{p}^{2} \cdot (\rho_{p} - \rho_{f}) \cdot \omega^{2} \cdot R_{o}}{18 \cdot \mu} \]
The volume of the liquid pool in the cylindrical section is:
\[ V = \pi \cdot L \cdot (R_{o}^{2} - R_{i}^{2}) \]
The Sigma factor, representing the equivalent area of a gravity settling tank with the same clarification capacity, is derived from the pool volume and the logarithmic ratio of the radii:
\[ \Sigma = \frac{V \cdot \omega^{2}}{g \cdot \ln(R_{o} / R_{i})} \]
| Parameter |
Condition/Constraint |
Requirement |
| Geometric Validity |
\(R_{i}\) vs \(R_{o}\) |
\(R_{i} < R_{o}\) |
| Physicality |
Radii |
\(R_{i}, R_{o} > 0\) |
| Flow Regime |
Particle Reynolds Number |
\(Re_{p} = \frac{\rho_{f} \cdot v_{s} \cdot d_{p}}{\mu} < 0.1\) |
| Mathematical Validity |
Logarithmic Ratio |
\(\ln(R_{o} / R_{i}) > 0\) |
Worked Example: Decanter Centrifuge Application Criteria
Scenario: A wastewater treatment plant employs a decanter centrifuge to remove fine solid particles (density 1200 kg/m³) from water. The process requires evaluating the centrifuge's separation capability using Ambler's Sigma theory for the cylindrical section.
Knowns
- Outer bowl radius, \( R_{o} = 0.25 \) m
- Inner liquid pool radius, \( R_{i} = 0.15 \) m
- Angular velocity, \( \omega = 300.0 \) rad/s
- Particle diameter, \( d_{p} = 1 \times 10^{-5} \) m (10 µm)
- Particle density, \( \rho_{p} = 1200.0 \) kg/m³
- Fluid density, \( \rho_{f} = 1000.0 \) kg/m³
- Fluid viscosity, \( \mu = 0.001 \) Pa·s (water at 20°C)
- Cylindrical section length, \( L = 1.0 \) m
- Gravitational acceleration, \( g = 9.81 \) m/s²
Step-by-Step Calculation
-
Centrifugal acceleration at the bowl wall:
\[ a_{c} = \omega^{2} \cdot R_{o} = (300.0)^{2} \times 0.25 = 22\,500.0\; \text{m/s}^2 \]
-
G-force (relative centrifugal force):
\[ G = \frac{a_{c}}{g} = \frac{22\,500.0}{9.81} = 2\,293.6 \]
-
Density difference between particle and fluid:
\[ \Delta\rho = \rho_{p} - \rho_{f} = 1200.0 - 1000.0 = 200.0\; \text{kg/m}^3 \]
-
Terminal settling velocity at the bowl wall (Stokes’ law):
\[ v_{s} = \frac{d_{p}^{2} \cdot \Delta\rho \cdot a_{c}}{18 \cdot \mu} = \frac{(1 \times 10^{-5})^{2} \times 200.0 \times 22\,500.0}{18 \times 0.001} = 0.025\; \text{m/s} \]
Check Stokes’ regime: \( Re_{p} = \frac{1000 \times 0.025 \times 10^{-5}}{0.001} = 0.25 \times 10^{-3} = 0.00025 \ll 0.1\ \checkmark \)
-
Liquid pool volume in the cylindrical section:
\[ V = \pi \cdot L \cdot (R_{o}^{2} - R_{i}^{2}) = \pi \times 1.0 \times (0.0625 - 0.0225) = \pi \times 0.04 = 0.1257\; \text{m}^3 \]
-
Logarithmic radius ratio:
\[ \ln\left(\frac{R_{o}}{R_{i}}\right) = \ln\left(\frac{0.25}{0.15}\right) = 0.5108 \]
-
Sigma factor (Ambler's theory):
\[ \Sigma = \frac{V \cdot \omega^{2}}{g \cdot \ln(R_{o} / R_{i})} = \frac{0.1257 \times 90\,000}{9.81 \times 0.5108} = 2\,256.9\; \text{m}^2 \]
Final Results
G-force: 2 293.6
Settling velocity: 0.025 m/s
Sigma factor: 2 256.9 m²
These values confirm that the centrifuge operating at 300 rad/s provides a Sigma factor of approximately 2 260 m², indicating its equivalent settling area for the given particle–fluid system. The particle Reynolds number is well within the Stokes’ regime, validating the use of the settling velocity formula.
