About This Tool

This Energy Requirement Calculator for comminution provides a comparative analysis of the energy needed to reduce the size of solid materials. It implements three foundational empirical laws in mineral processing and chemical engineering: Kick’s Law, Rittinger’s Law, and Bond’s Law. By inputting particle sizes and material-specific constants, users can estimate and compare the specific energy (e.g., kWh/t) and total power (kW) required for a crushing or grinding operation.

What This Calculator Does

The primary function of this calculator is to quantify the energy consumption for particle size reduction (comminution). It simultaneously computes the energy requirements based on three different theoretical models:

• Kick’s Law: Best for coarse crushing, where energy is proportional to the reduction ratio.
• Rittinger’s Law: Ideal for fine grinding, linking energy to the creation of new surface area.
• Bond’s Law: A widely used intermediate model, which proposes energy is proportional to the new crack length produced, making it versatile for both crushing and grinding applications.

The tool presents results in a table and a visual bar chart, allowing for easy comparison of the three models’ predictions for a given scenario.

When to Use It

This calculator is a valuable tool for process engineers, metallurgists, and students in the fields of mineral processing, cement production, and chemical engineering. Common applications include:

• Preliminary Equipment Sizing: Estimating the power draw for selecting mills and crushers.
• Process Optimization: Evaluating the energy efficiency of existing comminution circuits.
• Feasibility Studies: Predicting operational costs related to energy consumption in new projects.
• Academic and Educational Purposes: Demonstrating the differences and application ranges of the three fundamental comminution laws.

Inputs Explained

Results Explained

The calculator provides two key outputs for each of the three laws:

• Specific Energy: This is the energy consumed per unit mass of material processed (e.g., kWh/t). It is a measure of the energy efficiency of the size reduction process. The tool allows you to view this in different units like kWh/t, kJ/kg, or hp·h/short ton.
• Total Power: If a mass flow rate is provided, this is the calculated continuous power draw required by the equipment (e.g., kW or hp). It is calculated by multiplying the specific energy by the mass flow rate.

Comparing the three results helps in understanding which model might be most applicable. Typically, for the same reduction ratio, Kick’s law predicts the lowest energy, Rittinger’s the highest, and Bond’s falls in between.

Formula / Method

The calculations are based on the standard forms of the three comminution laws. The energy (E) is calculated as follows:

Kick’s Law

Energy is proportional to the logarithm of the reduction ratio.

E = Kk * ln(df / dp)

Rittinger’s Law

Energy is proportional to the new surface area created.

E = Kr * (1/dp - 1/df)

Bond’s Law

Energy is proportional to the new crack length formed, representing an intermediate case.

E = 10 * Wi * (1/√dp - 1/√df)

Note: In Bond’s formula, d_p and d_f must be in micrometers (μm), and the Work Index (W_i) is in kWh/short ton. The calculator handles all unit conversions internally to provide a consistent output in kWh per metric tonne (kWh/t).

Step-by-Step Example

Let’s calculate the energy required to crush granite from a feed size of 80 mm to a product size of 2 mm, with a throughput of 250 t/h.

Given Parameters:

• d_f = 80 mm = 80,000 µm
• d_p = 2 mm = 2,000 µm
• m = 250 t/h
• K_k = 1.1 kWh/t (typical for hard rock)
• K_r = 150 kWh·µm/t (typical for hard rock)
• W_i = 15.13 kWh/short ton (for Granite)

Calculations:

1. Kick’s Law:
   E = 1.1 * ln(80000 / 2000) = 1.1 * ln(40) ≈ 1.1 * 3.689 = 4.06 kWh/t
2. Rittinger’s Law:
   E = 150 * (1/2000 - 1/80000) = 150 * (0.0005 - 0.0000125) = 150 * 0.0004875 = 73.13 kWh/t
3. Bond’s Law:
   First, calculate in kWh/short ton:
   E = 10 * 15.13 * (1/√2000 - 1/√80000) ≈ 151.3 * (0.02236 - 0.00354) = 151.3 * 0.01882 = 2.85 kWh/short ton
   Then, convert to kWh per metric tonne (1 short ton = 0.907185 metric tonnes):
   E = 2.85 / 0.907185 ≈ 3.14 kWh/t
4. Total Power (Bond’s Law Example):
   Power = Specific Energy * Mass Flow Rate = 3.14 kWh/t * 250 t/h = 785 kW

Tips + Common Errors

• Unit Consistency: The most common error is inconsistent units. This tool automatically converts common units, but always double-check that your inputs (e.g., mm, µm, in) are correctly selected.
• Size Convention (F80/P80): Remember that d_f and d_p typically represent the 80% passing size, not the maximum or average particle size.
• Applicability of Laws: Do not treat the laws as universally correct. Kick’s law often underestimates energy for fine grinding, while Rittinger’s can overestimate for coarse crushing. Bond’s law is generally the most reliable for mill design.
• Work Index Source: The Bond Work Index (W_i) is material-specific and should ideally be from laboratory tests on a representative sample. Using values from a lookup table provides an estimate only.
• Feed vs. Product Size: Ensure the initial (feed) particle size d_f is always larger than the final (product) particle size d_p. The calculator will flag an error if this is not the case.

Frequently Asked Questions

References

1. Wills, B. A., & Finch, J. A. (2015). Wills’ Mineral Processing Technology: An Introduction to the Practical Aspects of Ore Treatment and Mineral Recovery. Elsevier. (Chapter 6: Comminution).
2. Perry, R. H., & Green, D. W. (2008). Perry’s Chemical Engineers’ Handbook (8th ed.). McGraw-Hill. (Section 21: Solid-Solid Operations and Processing).
3. Bond, F. C. (1952). The third theory of comminution. Transactions of the American Institute of Mining, Metallurgical, and Petroleum Engineers, 193, 484-494.
4. Napier-Munn, T. J., et al. (1996). Mineral Comminution Circuits: Their Operation and Optimisation. Julius Kruttschnitt Mineral Research Centre (JKMRC), The University of Queensland.

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