Introduction & Context
In process engineering, the filter centrifuge is a critical unit operation used for the solid-liquid separation of slurries, particularly in the dewatering of crystalline products. The performance of this equipment is governed by the intensity of the centrifugal force field and the resulting pressure drop across the filter cake. Understanding these parameters is essential for optimizing cycle times, ensuring product purity, and maintaining mechanical integrity.
The Separation Factor (G) quantifies the magnitude of the centrifugal force relative to gravity, serving as a primary indicator of the potential for particle settling and liquid drainage. The Centrifugal Pressure Drop (ΔPc) represents the driving force that pushes the mother liquor through the porous cake structure. These calculations are standard practice during the scale-up and operational monitoring of batch, vertical-axis, perforated-basket centrifuges.
Methodology & Formulas
The calculation process follows a sequential derivation starting from the rotational speed of the basket to determine the kinetic energy and pressure distribution within the liquid layer.
First, the rotational speed in revolutions per minute (N) must be converted to angular velocity (ω) in radians per second:
\[ \omega = \frac{2 \cdot \pi \cdot N}{60} \]
The Separation Factor (G) is calculated at the basket wall (R2) to determine the maximum force field experienced by the solids:
\[ G = \frac{\omega^{2} \cdot R_{2}}{g} \]
The driving force for filtration is the centrifugal pressure drop (ΔPc), which is a function of the liquid density (ρ), the angular velocity, and the radial boundaries of the liquid layer (R1 and R2):
\[ \Delta P_{c} = \frac{1}{2} \cdot \rho \cdot \omega^{2} \cdot (R_{2}^{2} - R_{1}^{2}) \]
Finally, the cake thickness is determined by the difference between the basket radius and the inner radius of the liquid/cake layer:
\[ \delta_{cake} = R_{2} - R_{1} \]
| Parameter |
Operational Regime / Threshold |
| Separation Factor (G) |
300 to 1500 (Dimensionless) |
| Rotational Speed (N) |
800 to 1500 RPM |
| Centrifugal Pressure (ΔPc) |
0.5 to 5.0 bar |
| Cake Thickness (δcake) |
0.05 to 0.15 m |
The separation factor, often denoted as \( G \), represents the ratio of centrifugal acceleration to gravitational acceleration. It is calculated using the following parameters:
- The angular velocity of the centrifuge basket, \( \omega \), in radians per second.
- The inner radius of the filter basket, \( R_{2} \).
- The standard acceleration due to gravity, \( g = 9.81 \, \text{m/s}^2 \).
The formula is defined as \( G = \frac{\omega^{2} \cdot R_{2}}{g} \).
Worked Example: Filter Centrifuge Separation Factor
A vertical-axis perforated-basket centrifuge is used to dewater washed sugar crystals in a batch process. The engineer needs to calculate the separation factor to assess the centrifugal force field and the centrifugal pressure to evaluate the driving force for filtration through the cake.
Known Parameters (Inputs):
- Rotational speed, \( N = 1000.0 \, \text{RPM} \)
- Basket inner radius, \( R_{2} = 0.35 \, \text{m} \)
- Cake inner radius (after formation), \( R_{1} = 0.25 \, \text{m} \)
- Mother liquor density (assumed water), \( \rho = 1000.0 \, \text{kg/m}^3 \)
- Gravitational acceleration, \( g = 9.81 \, \text{m/s}^2 \)
Step-by-Step Calculation:
-
Convert rotational speed to angular velocity:
The angular velocity \( \omega \) is calculated from \( N \) using \( \omega = \frac{2 \cdot \pi \cdot N}{60} \).
Calculating: \( \omega = \frac{2 \cdot \pi \cdot 1000.0}{60} = 104.72 \, \text{rad/s} \).
-
Calculate the separation factor (G-force) at the basket wall:
The separation factor \( G \) at radius \( R_{2} \) is given by \( G = \frac{\omega^2 \cdot R_{2}}{g} \).
Substituting the values: \( \omega = 104.72 \, \text{rad/s} \), \( R_{2} = 0.35 \, \text{m} \), \( g = 9.81 \, \text{m/s}^2 \).
Calculating: \( G = \frac{(104.72)^2 \cdot 0.35}{9.81} = \frac{10966.2784 \cdot 0.35}{9.81} = \frac{3838.19744}{9.81} = 391.252 \) (dimensionless).
-
Calculate the centrifugal pressure drop across the cake:
The centrifugal pressure \( \Delta P_{c} \) is calculated using \( \Delta P_{c} = \frac{1}{2} \cdot \rho \cdot \omega^2 \cdot (R_{2}^2 - R_{1}^2) \).
First, calculate \( R_{2}^2 - R_{1}^2 = (0.35)^2 - (0.25)^2 = 0.1225 - 0.0625 = 0.06 \, \text{m}^2 \).
Then: \( \Delta P_{c} = \frac{1}{2} \cdot 1000.0 \cdot (104.72)^2 \cdot 0.06 = 328986.813 \, \text{Pa} \).
Convert to bar using \( 1 \, \text{bar} = 10^5 \, \text{Pa} \): \( \Delta P_{c} = \frac{328986.813}{10^5} = 3.28986813 \approx 3.29 \, \text{bar} \).
-
Determine the cake thickness:
Cake thickness is \( \delta_{cake} = R_{2} - R_{1} = 0.35 \, \text{m} - 0.25 \, \text{m} = 0.1 \, \text{m} \).
Final Answer:
- Separation factor at the basket wall: \( G = 391.252 \)
- Centrifugal pressure drop: \( \Delta P_{c} = 328987 \, \text{Pa} \) or \( 3.29 \, \text{bar} \)
- Cake thickness: \( \delta_{cake} = 0.1 \, \text{m} \)
These values are within empirical ranges for industrial basket centrifuges, indicating effective dewatering potential.
