Illustrative example 5.2

Illustrative example 5.2

Modeling and Simulation of Mineral Processing Systems

The breakage function of a crusher machine is given by the following equation:

$$B(x, y) = 0.3 + 0.7 \left( \frac{x}{y} \right)^{0.45} \times 3.2$$

Where:

• Size class 2 has mesh size boundaries of 13.4 cm and 9.5 cm.
• Size class 5 has mesh size boundaries of 7.50 cm and 5.60 cm.

To calculate $b_{52}$, use the formula:

$$b_{ij} = B(D_{i-1}, d_{pj}) - B(D_i, d_{pj})$$

Given:

$$d_p2 = \frac{13.4 + 9.4}{2} = 11.28 \, \text{cm}$$

Now calculate $b_{52}$:

$$b_{52} = 0.3 + 0.7 \left( \frac{7.5}{11.28} \right)^{0.45} \times 3.2 - 0.3 + 0.7 \left( \frac{5.6}{11.28} \right)^{0.45} \times 3.2$$

Breaking this down:

1. First term calculation:

$$0.3 + 0.7 \left( \frac{7.5}{11.28} \right)^{0.45} \times 3.2 = 0.4393$$

2. Second term calculation:

$$0.3 + 0.7 \left( \frac{5.6}{11.28} \right)^{0.45} \times 3.2 = 0.2934$$

Now, subtract the second term from the first term:

$$b_{52} = 0.4393 - 0.2934 = 0.1459$$

Thus, $b_{52} = 0.1459$.

This example and its detailed computation align with Illustrative Example 5.2 in R.P. King's text, found on pages 136-138.