The breakage function in a crusher machine: Numerical Example

 

Problem:

The breakage function in a crusher machine is determined to be:

$$B(x:y) = 0.4 \left(\frac{x}{y}\right)^{0.5} + 0.6 \left(\frac{x}{y}\right)^{3.0}$$

If size class 2 has mesh size boundaries of 14.0 cm and 10.0 cm, and size class 5 has boundaries of 8.0 cm and 6.0 cm, calculate $b_{52}$.

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Explanation in Simple Terms:

When rocks or particles are crushed in a crusher machine, they break into smaller pieces of various sizes. The breakage function is a mathematical way to describe how these larger particles break into smaller ones. It predicts the probability or fraction of material from a given size class breaking into smaller size classes.

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Key Terms Simplified:

1. Size Classes:

   • Particles are grouped into "size classes" based on their sizes. For example, particles between 14.0 cm and 10.0 cm belong to size class 2, and particles between 8.0 cm and 6.0 cm belong to size class 5.
   • The goal is to understand how particles from size class 2 end up as smaller particles in size class 5 after being crushed.

2. Breakage Function ($B(x:y)$):

   • This is a formula that tells us how likely it is for a particle of size $x$ to break into smaller particles of size $y$.
   • For example, if $x$ is the starting size (14.0 cm or 10.0 cm), and $y$ is the size after breaking, the formula computes this probability.

3. What is $b_{52}$?

   • $b_{52}$ measures how much material from size class 2 ends up in size class 5 after being crushed.
   • To calculate $b_{52}$, we use the breakage function for both the upper and lower limits of size class 5 (8.0 cm and 6.0 cm). Then, we find the difference between these values to get the final result.

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Real-Life Analogy:

Imagine you’re smashing stones with a hammer, and you want to know how many of the larger stones (size class 2) break into smaller stones (size class 5). The breakage function is like a rulebook that helps predict how the stones will break. $b_{52}$ tells you the fraction of stones from size class 2 that have now become part of size class 5 after smashing.

In this problem, the calculations use the formula to find the specific fraction ($b_{52}$) based on the sizes and properties of the stones and crusher.

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Solution:

The formula for calculating $b_{ij}$ is:

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

where $D_{i-1}$ and $D_i$ are the boundaries of size class $i$, and $d_{pj}$ is the representative size of size class $j$.

Step 1: Calculate the representative size for size class 2

$$d_{p2} = \sqrt{D_{2-1} \cdot D_2} = \sqrt{14.0 \cdot 10.0} = 11.83 \, \text{cm}$$

Step 2: Breakage function for class 5 boundaries

The upper boundary ($D_{5-1}$) and lower boundary ($D_5$) are 8.0 cm and 6.0 cm, respectively.
For $B(D_{i-1}; d_{pj})$:

$$B(8.0; 11.83) = 0.4 \left(\frac{8.0}{11.83}\right)^{0.5} + 0.6 \left(\frac{8.0}{11.83}\right)^{3.0}$$

For $B(6.0; 11.83)$:

$$B(6.0; 11.83) = 0.4 \left(\frac{6.0}{11.83}\right)^{0.5} + 0.6 \left(\frac{6.0}{11.83}\right)^{3.0}$$

Step 3: Calculate $b_{52}$

$$b_{52} = B(8.0; 11.83) - B(6.0; 11.83)$$

Let me calculate this numerically.

Solution Results:

1. The representative size for size class 2 is:

   $$d_{p2} = 11.83 \, \text{cm}$$

2. The breakage function values are:

   $$B(8.0; 11.83) = 0.5144$$ $$B(6.0; 11.83) = 0.3631$$

3. The value of $b_{52}$ is:

   $$b_{52} = 0.5144 - 0.3631 = 0.1513$$

Thus, the calculated breakage function value $b_{52}$ is 0.1513