Illustrative Example 4.5: Effect of Particle Density

Illustrative Example 4.5: Effect of Particle Density

Problem:

A hydrocyclone exhibits a corrected classification curve described by:

$$e(dp) = \frac{1}{1 + x^{-\lambda}}$$

where $x = \frac{dp}{d_{50c}}$.

When processing a quartz slurry:

• $d_{50c} = 16.5 \, \mu\text{m}$,
• Sharpness index ($SI$) = 0.72,
• Density of quartz ($\rho_q$) = 2670 $\text{kg/m}^3$.

Determine the recovery of $10 \, \mu\text{m}$ particles to overflow when the hydrocyclone treats a magnetite slurry under comparable conditions, where:

• Density of magnetite ($\rho_m$) = 5010 $\text{kg/m}^3$,
• Recovery of water to underflow ($R_u$) = 18.5% (same in both cases).

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Solution:

Step 1: Corrected cut size for magnetite slurry

The cut size $d_{50c,m}$ for magnetite slurry can be calculated as:

$$d_{50c,m} = d_{50c} \sqrt{\frac{\rho_q - \rho_f}{\rho_m - \rho_f}},$$

where:

• Fluid density ($\rho_f$) = $1000 \, \text{kg/m}^3$.

Substitute values:

$$d_{50c,m} = 16.5 \sqrt{\frac{2670 - 1000}{5010 - 1000}} = 16.5 \sqrt{\frac{1670}{4010}}.$$

Calculate:

$$d_{50c,m} = 10.65 \, \mu\text{m}.$$

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Step 2: Regimes of classification

Two limiting cases are considered:

1. Stokes regime ($\lambda = 0.5$): $$d_{50c,m} = 10.65 \, \mu\text{m}.$$
2. Fully turbulent regime ($\lambda = 1.0$): $$d_{50c,m} = 10.65 \times \sqrt{0.5} = 6.87 \, \mu\text{m}.$$

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Step 3: Classification function

The classification function $C(dp_i)$ is defined as:

$$C(dp_i) = R_u + (1 - R_u) \frac{1}{1 + x^{-\lambda}},$$

where $x = \frac{dp}{d_{50c,m}}$.

For $dp = 10 \, \mu\text{m}$, $x = \frac{dp}{d_{50c,m}} = \frac{10}{10.65}$.

• Logistic exponent: Using the sharpness index:

  $$\lambda = -\frac{\ln(SI)}{\ln(2)}.$$

  Substituting $SI = 0.72$:

  $$\lambda = -\frac{\ln(0.72)}{\ln(2)} = 0.685.$$

• Classification function in Stokes regime ($\lambda = 0.5$):

  $$C(10) = 0.185 + (1 - 0.185) \frac{1}{1 + \left(\frac{10}{10.65}\right)^{-6.686}}.$$

  Simplify:

  $$C(10) = 0.185 + 0.815 \times 0.396 = 0.5079.$$

• Recovery to overflow:

  $$R_o = 1 - C(10) = 1 - 0.5079 = 49.2\%.$$

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Final Answer:

The recovery of $10 \, \mu\text{m}$ particles to overflow in the magnetite slurry is 49.2%.