Numerical Example: Breakage Function in Crushing

Numerical Example: Breakage Function in Crushing

Problem Statement

The breakage function for a crusher is given by:

$$B(x : y) = 0.5\left(\frac{x}{y}\right)^{0.6} + 0.5\left(\frac{x}{y}\right)^{3.0}$$

where

• $x$ = size of progeny particle

• $y$ = size of parent particle

Size class 3 has mesh size boundaries of 12.0 cm and 8.0 cm, and
Size class 6 has mesh size boundaries of 4.0 cm and 2.0 cm.

Calculate $b_{63}$, the fraction of material from size class 3 that reports to size class 6 after crushing.

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Solution

Step 1: Identify representative sizes

Representative size is taken as the geometric mean of class boundaries.

Size class 3 (parent):

$$y = \sqrt{12 \times 8} = \sqrt{96} = 9.8 \text{ cm}$$

Size class 6 (progeny):

$$x = \sqrt{4 \times 2} = \sqrt{8} = 2.83 \text{ cm}$$

Step 2: Write the breakage function

$$B(x:y) = 0.5\left(\frac{x}{y}\right)^{0.6} + 0.5\left(\frac{x}{y}\right)^{3.0}$$

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Step 3: Substitute values

$$\frac{x}{y} = \frac{2.83}{9.8} = 0.289$$
$$B = 0.5(0.289)^{0.6} + 0.5(0.289)^3$$

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Step 4: Calculate each term

$$(0.289)^{0.6} \approx 0.44$$
$$(0.289)^3 \approx 0.024$$
$$B = 0.5(0.44) + 0.5(0.024)$$
$$B = 0.22 + 0.012$$
$$B = 0.232$$

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Final Answer

$$\boxed{b_{63} = 0.23}$$

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Result Interpretation

• About 23% of material from size class 3 reports to size class 6 after crushing

• Lower fractions are expected because class 6 is much finer than class 3

• This result is physically realistic for a crushing operation

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