Size reduction using Bond’s law (energy estimate) Limestone grinding: The energy required to grind from a very large size to 100 μm is 12.7 kWh/ton. Using Bond’s law, estimate the energy needed to grind from a very large size down to 50 μm.

Introduction / Context:
Bond’s law is widely used in mineral processing to estimate comminution energy. It relates specific energy consumption to the change in characteristic particle size and is especially useful for comparing energy needs at different target sizes for the same material and mill type.

Given Data / Assumptions:

• Material: limestone (same ore, same work index).
• Initial size is effectively “very large,” so its contribution in the Bond expression is negligible.
• Energy to reach 100 μm from very large size: 12.7 kWh/ton.
• Target: estimate energy to reach 50 μm from very large size.

Concept / Approach:
Bond’s law (for very large feed) implies energy E is proportional to 1/√P80, where P80 is the product size. Therefore E2/E1 = (1/√P2)/(1/√P1) = √(P1/P2). If the product size is halved, energy increases by √2 because the inverse square-root term grows accordingly.

Step-by-Step Solution:
Let E1 correspond to P1 = 100 μm: E1 = 12.7 kWh/ton.For P2 = 50 μm, E2/E1 = √(100/50) = √2 ≈ 1.414.Compute E2: E2 = 12.7 * 1.414 ≈ 17.96 ≈ 18 kWh/ton.

Verification / Alternative check:
If we write E = k*(1/√P), then k = E1 * √P1 = 12.7 * 10 = 127. For P2 = 50 μm, 1/√50 ≈ 0.1414, so E2 ≈ 127 * 0.1414 ≈ 17.96 kWh/ton. Same result confirms the calculation.

Why Other Options Are Wrong:

• 6.35 kWh/ton: would imply energy decreases when making a finer product, which contradicts Bond’s law.
• 9.0 kWh/ton: underestimates; still lower than the 100 μm case.
• 25.4 kWh/ton: corresponds to a doubling rather than √2 increase; too high for halving size.

Common Pitfalls:

• Using a linear or inverse relationship with size instead of inverse square-root.
• Forgetting that “very large feed” makes the feed-size term negligible.

Final Answer:
18 kWh/ton