Introduction & Context

Kick’s law is an empirical energy–size relationship used in comminution engineering. It postulates that the energy required to reduce the size of a brittle solid is proportional to the logarithm of the size reduction ratio. The law is most accurate for coarse grinding where the reduction ratio is modest and the product size is still relatively large. Typical applications include roller mills, jaw crushers, and hammer mills processing grains, ores, and cement clinker.

Methodology & Formulas

Define the size reduction ratio
\[ R = \frac{x_1}{x_2} \] where \(x_1\) is the mean feed size and \(x_2\) is the mean product size (both in consistent units).
Compute specific energy
\[ E = K_K \ln R \] with \(K_K\) the Kick coefficient, material-specific and expressed in kJ kg⁻¹.
Convert to power demand
\[ P = E \cdot \dot{m} \] where \(\dot{m}\) is the mass flow rate (kg s⁻¹) and \(P\) is the instantaneous power (kW).

Validity Regimes for Kick’s Law
Parameter	Range	Consequence if Outside
Reduction ratio \(R\)	\(1 < R \le 8\)	Model accuracy degrades; consider Bond or Rittinger laws.
Moisture content	\(\le 15\ \%\) w.b.	Empirical \(K_K\) may underestimate energy; re-calibrate coefficient.

Kick’s Law states that the energy required for particle breakage is proportional to the size reduction ratio, i.e. \(E = C_k \ln(D_f/D_p)\). Use it when you are dealing with coarse crushing (>50 mm feed) where new surface creation is minimal and deformation energy dominates. Switch to Bond when you need a more accurate prediction in the 0.1–50 mm range, and use Rittinger for very fine grinding where surface area creation becomes the major energy sink.

Run a drop-weight or pendulum test on 5–10 coarse particles:

Measure input energy \(E\) and initial/final sizes \(D_f\), \(D_p\).
Plot \(E\) vs \(\ln(D_f/D_p)\); the slope is \(C_k\).
Scale the laboratory value to motor power using the equipment efficiency (typically 20–30 % for jaw crushers).

Expect ±30 % accuracy for primary crushers. For definitive budgets, calibrate the constant with pilot-plant data at the target closed-side setting and include 15 % contingency for feed variability and liner wear.

No—Kick’s Law is purely a geometric model. Apply empirical factors:

Moisture >4 % w/w can raise \(C_k\) by 10–20 %.
Wide size distributions require weighted averaging of \(\ln(D_f/D_p)\) using the cumulative mass fraction.

Worked Example – Kick’s Law Energy Calculation

A small on-farm roller mill is being commissioned to grind tempered wheat from a coarse grist of 4 mm down to a finished flour granulation of 1 mm. The mill vendor quotes a Kick constant of 2.0 kJ·kg⁻¹ for wheat at 14 % moisture. With a nominal throughput of 0.5 kg·s⁻¹, the engineer needs to estimate the specific comminution energy and the motor power required.

Knowns

Kick constant, \(K_K\) = 2.0 kJ·kg⁻¹
Feed size, \(x_1\) = 4.0 mm
Product size, \(x_2\) = 1.0 mm
Moisture content = 14 % (by mass)
Mass flow rate, \(\dot{m}\) = 0.5 kg·s⁻¹

Step-by-Step Calculation

Compute the size reduction ratio: \[ \frac{x_1}{x_2} = \frac{4.0\ \text{mm}}{1.0\ \text{mm}} = 4.0 \]
Apply Kick’s Law for specific energy: \[ E = K_K \ln\left(\frac{x_1}{x_2}\right) = 2.0\ \text{kJ·kg}^{-1} \times \ln(4.0) \] \[ E = 2.0 \times 1.386 = 2.773\ \text{kJ·kg}^{-1} \]
Convert specific energy to power: \[ P = E \times \dot{m} = 2.773\ \text{kJ·kg}^{-1} \times 0.5\ \text{kg·s}^{-1} \] \[ P = 1.387\ \text{kJ·s}^{-1} = 1.387\ \text{kW} \approx 1.39\ \text{kW} \]

Final Answer

The specific comminution energy is 2.77 kJ per kilogram of wheat, and the continuous power draw on the mill shaft is approximately 1.39 kW.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

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