Kick’s law in comminution: if specific energy is expressed as energy per unit mass, what are the appropriate units of Kick’s constant Ck in the relationship E/m = Ck * ln(d1/d2)?

Introduction / Context:
Energy–size relationships (Kick, Bond, Rittinger) help estimate power needs for crushers and mills. Dimensional analysis ensures constants carry the correct units for consistent design calculations and scale-up comparisons.

Given Data / Assumptions:

• Kick’s form: E/m = Ck * ln(d1/d2), where ln(d1/d2) is dimensionless.
• E/m is specific energy (energy per unit mass), often reported as kWh/ton or kWh/kg.
• We use SI-derived practical units common in industry.

Concept / Approach:
Because the logarithmic term is dimensionless, Ck must carry the same units as E/m. If E/m is stated as kWh/kg, then Ck also has units of kWh/kg. Any extra time factor in the denominator would be incorrect for a specific energy constant.

Step-by-Step Solution:
Note that ln(d1/d2) has no units.Therefore Ck units = units of E/m = energy/mass.Common reporting unit in practice: kWh/kg (or kWh/ton by scaling).

Verification / Alternative check:
Textbook tables listing Ck values present them in energy per mass units; converting between J/kg and kWh/kg is straightforward (1 kWh = 3.6 * 10^6 J).

Why Other Options Are Wrong:
kWh/(s·kg): introduces an extra 1/s, which would correspond to power per mass, not specific energy.kg/s: has no energy dimension at all.kW·s/kg equals J/kg numerically and is convertible to kWh/kg; however, the standard plant unit for comminution constants and specific energy is kWh/kg (or kWh/ton), making option (b) the correct and conventional choice.

Common Pitfalls:
Confusing specific energy with specific power (kW per ton). Always keep track of time in the numerator for power and in the energy term for specific energy.

Final Answer:
kWh/kg