Question 8.4: Particle Separation by Elutriation Particles of ore suspend......
Particle Separation by Elutriation
Particles of ore suspended in water are to be separated by elutriation. A small test rig comprising a vertically mounted cylinder with an internal diameter of 5 cm is to be used to determine the effectiveness of separation in which a flow of particle-bearing water is continuously fed. For a rate of flow to the elutriator of 300 mL min ^{–1} , determine the effectiveness of the elutriator in terms of particle sizes that can be separated and retained.
Elutriation is the process of separating suspended particles in a liquid by the upward flow of liquid such that the smaller particles with insufficient buoyancy are washed out or elutriated. It is typically used for separating particles into different size fractions. The particles are assumed to obey Stokes’ law (Equation 8.1) for which the terminal velocity of the descending particles is equal to the upward velocity of the water fed to the separation device. The Reynolds number is less than 2000 and is expressed in terms of the flow rate to confirm that the upward flow is laminar:
U_{t}={\frac{g\left(\rho_{p}-\rho\right)d_{p}^{2}}{18\mu}} (8.1)
\mathrm{Re}={\frac{4\rho\dot{{Q}}}{\pi d\mu}}={\frac{4\times1000\times300\times10^{-6}/60}{\pi\times0.05\times0.001}}=127 (8.16)
Within the circular cross section of the cylindrical elutriator, the maximum velocity of the upward flow occurs along the centreline, which for laminar flow corresponds to twice the average velocity. This is
U_{\mathrm{max}}={\frac{2{\dot{Q}}}{a}}={\frac{8{\dot{Q}}}{\pi d^{2}}}={\frac{8\times300\times10^{-6}/60}{\pi\times0.05^{2}}}=5.09\times10^{-3} \mathrm{ms}^{-1} (8.17)
This corresponds to the terminal velocity of the particles, which based on Stokes’ law corresponds to a critical size of
d_{p}={\sqrt{\frac{18\mu U_{t}}{g(\mathbf{\rho}_{p}-\mathbf{\rho})}}}={\sqrt{\frac{18\times0.001\times5.09\times10^{-3}}{9.81\times(1800-1000)}}}=1.08\times10^{-4}{\mathrm{~m}} (8.18)
That is a diameter of 108 μm. This is the largest size of particle that can be elutriated corresponding to the highest terminal velocity when it is located at the centre of the elutriation tube. That is, it is furthest from the walls of the tube.