Plitt model for hydrocyclone

Explain Plitt model for hydrocyclone.

The Plitt model for hydrocyclones, as detailed in the user's provided document, describes the relationship between hydrocyclone performance and geometric/operating parameters. Below is the explanation based on the referenced material:

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Key Aspects of the Plitt Model

1. Balancing Forces:

   • The classification in a hydrocyclone relies on the equilibrium between two opposing forces:
     • Centrifugal force: Moves particles outward toward the wall.
     • Hydrodynamic drag: Convects particles toward the axis.
   • The cut-size ($d_{50}$) corresponds to the particle size at which these two forces are balanced. This concept is discussed under the equilibrium orbit hypothesis in Section 4.5.5.

2. Flow Split Parameter ($S$):

   • The volumetric flow split ($S$) influences the separation and is primarily a function of:
     • Spigot diameter ($D_u$)
     • Vortex finder diameter ($D_o$)
     • Cyclone diameter ($D_c$)
     • Feed solids volume fraction ($\phi$)
     • Distance between vortex finder and spigot ($h$)
     • Slurry feed head ($H$)
   • The empirical formula is: $$S = a \left(\frac{D_u}{D_o}\right)^b \left(\frac{D_u}{D_c}\right)^c \left(\frac{D_o}{D_c}\right)^d \left(2 + h/D_c\right)^f \exp(-gH)$$ Recommended constants: $a = 3.79$, $b = 3.31$, $c = 0.36$, $d = 0.54$, $f = 1.11$, $g = 0.24$.

3. Recoveries:

   • Recovery metrics:
     • $R_v$: Volumetric flow rate to underflow/feed volumetric flow rate.
     • $R_f$: Fluid recovery to underflow.
     • $R_s$: Solids recovery to underflow.
   • $R_s$ is governed by particle-fluid interactions, described via Stokes' law.

4. Efficiency of Separation:

   • Efficiency depends on $R_v$ and cyclone size.
   • Empirical formula for selectivity index: $$\ln(\text{SI}) = -1.24 \exp\left(1.58 \frac{R_v}{\phi H^{0.15}D_c^2}\right)$$

5. Particle Size Distributions:

   • The overflow and underflow particle size distributions are characterized using cumulative distribution functions and exponential/logistic forms: $$p_o(i) = \frac{F_i(1-C(d_p))}{\sum F_i(1-C(d_p))}$$ $$p_u(i) = \frac{F_iC(d_p)}{\sum F_iC(d_p)}$$

6. Sources of Variability:

   • Practical uncertainty primarily arises from estimating the flow split ($S$).

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Section 4.5.7 of R.P. King's "Modeling and Simulation of Mineral Processing Systems" (page numbers approximately 102-125).