Laws of size reduction: Rittinger, Kick and Bond

Energy for size reduction: the general law dE/dD = −C/Dⁿ, Rittinger, Kick and Bond (work index) with ratio-method and power numericals.

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Why it matters

Size reduction is one of the most energy-hungry operations in the process industries: crushing and grinding consume a large share of the power in cement, mineral and coal plants, and most of that energy ends up as heat. The three classical laws — Rittinger, Kick and Bond — let you estimate the power for a new duty from test data, compare crushing stages, and see why grinding the last few microns costs so much.

Key ideas

What the energy does. Breaking a solid creates new surface, but only a tiny fraction (often quoted as about 1% or less) of the power input is stored as new surface energy; the rest goes into elastic deformation that does not lead to fracture, friction, noise and heat. The crushing efficiency η_c is the ratio of surface energy created to energy absorbed. Because η_c is so small, the laws below are empirical correlations, with constants fitted to test data for each material and machine.

One differential law, three exponents. All three laws follow from dE/dD = −C / Dⁿ where E is the energy per unit mass and D a representative particle size.

  • n = 2, Rittinger (1867): energy is proportional to the new surface created, i.e. to (1/D_b − 1/D_a). Best for fine grinding, where large new surface is produced.
  • n = 1, Kick (1885): energy depends only on the reduction ratio D_a/D_b; reducing 100 → 10 mm costs the same as 10 → 1 mm. Best for coarse crushing of large lumps, where deformation energy dominates.
  • n = 1.5, Bond (1952): energy is proportional to the length of new cracks, giving (1/√D_b − 1/√D_a). Bond's law sits between the other two and is the one used in practice for crushing and grinding design across a wide size range.

Bond work index W_i. The energy (kWh per short ton or per tonne, as defined in your data book) needed to reduce a material from a very large size to 80% passing 100 µm. It is a measured material property — take values from tables or a Bond mill test, never invent them. In Bond's law the sizes are taken as the 80% passing sizes of feed and product (F₈₀ and P₈₀).

Reduction ratio. D_a/D_b. A single crusher usually achieves a limited ratio (roughly 3–8 for coarse crushers, much higher for fine mills), so large total reductions are done in stages.

Limits. Rittinger fails for coarse crushing (it under-predicts) and Kick fails for fine grinding (it badly under-predicts, because the surface created per tonne rises sharply as size falls). All three ignore machine efficiency, moisture and the size distribution shape; use them for estimates and scale-up, not for exact power.

Formulas

General: dE/dD = −C / Dⁿ (n = 2 Rittinger, 1 Kick, 1.5 Bond)

Rittinger: P/ṁ = K_R·(1/D_b − 1/D_a) Kick: P/ṁ = K_K·ln(D_a / D_b) Bond: P/ṁ = 0.3162·W_i·(1/√D_b − 1/√D_a) with D in mm, P/ṁ and W_i in kWh/t Equivalent Bond form: W = 10·W_i·(1/√P₈₀ − 1/√F₈₀) with P₈₀, F₈₀ in µm

  • P: power (kW); ṁ: feed rate (t/h), so P/ṁ is energy per tonne (kWh/t). In SI form, P/ṁ in J/kg with K_R in J·m/kg and K_K in J/kg.
  • D_a, D_b: feed and product sizes (Bond: 80% passing sizes).
  • K_R, K_K: empirical constants for the material and machine; W_i: Bond work index (kWh/t).

Worked examples

Example 1 (standard): comparing the laws. Crushing a rock from 50 mm to 10 mm needs 2 kWh/t. Estimate the energy to crush the same rock from 10 mm to 2 mm by each law.

  1. Kick: both steps have reduction ratio 5, so E = 2 kWh/t.
  2. Rittinger: E₂/E₁ = (1/2 − 1/10)/(1/10 − 1/50) = 0.40/0.08 = 5, so E = 5 × 2 = 10 kWh/t.
  3. Bond: E₂/E₁ = (1/√2 − 1/√10)/(1/√10 − 1/√50) = (0.7071 − 0.3162)/(0.3162 − 0.1414) = 2.236, so E = 4.47 kWh/t. Bond lies between Kick and Rittinger, as expected.

Example 2 (GATE level): power from the work index. A crusher handles 50 t/h of an ore with W_i = 12.7 kWh/t (from a data book). Feed is 80% passing 20 mm and product 80% passing 1 mm. Find the power.

  1. P/ṁ = 0.3162·W_i·(1/√D_b − 1/√D_a) with D in mm.
  2. 1/√1 − 1/√20 = 1 − 0.2236 = 0.7764 mm^(−1/2).
  3. P/ṁ = 0.3162 × 12.7 × 0.7764 = 3.118 kWh/t.
  4. P = 3.118 kWh/t × 50 t/h = 156 kW. Check with the µm form: 10 × 12.7 × (1/√1000 − 1/√20000) = 127 × (0.03162 − 0.00707) = 3.118 kWh/t. Same.

Common mistakes

  • Treating E from these laws as total joules. They give energy per unit mass; multiply by feed rate for power.
  • Mixing units in Bond's law: 0.3162 goes with D in mm; 10 goes with µm. Using metres with either constant gives nonsense.
  • Using Kick for fine grinding or Rittinger for primary crushing.
  • Forgetting that Bond uses 80% passing sizes, not mean sizes.
  • Assuming a large share of power goes into new surface; the crushing efficiency is tiny.

For GATE CH

Typical questions: given the energy for one size reduction, predict another by a named law (ratio method, so the constant cancels); compute power from Bond's work index and a feed rate; identify which law applies to coarse or fine reduction; recognise the general dE/dD law and its exponents. Practise the ratio method for all three laws and keep the Bond unit pairing straight.

Quick check

  1. Which law says 100 → 10 mm costs the same energy as 10 → 1 mm?
  2. What is the exponent n in dE/dD = −C/Dⁿ for Bond's law?
  3. Define the Bond work index.
  4. If Rittinger's law holds, how does the energy for 4 mm → 1 mm compare with 8 mm → 2 mm? Answers: 1. Kick's law. 2. 1.5. 3. Energy (kWh/t) to reduce the material from a very large size to 80% passing 100 µm. 4. It is twice as large (0.75 versus 0.375 mm⁻¹).

Interview questions

All Mechanical Operations interview questions

Try answering each one aloud before you open it.

  1. 1.What is Rittinger's Law in size reduction?Concept

    Rittinger's Law states that the energy required for size reduction is proportional to the new surface area generated. It is most applicable to fine grinding where the increase in surface area is significant. The law is mathematically expressed as E = K_R (1/D2 - 1/D1), where E is the energy required, K_R is Rittinger's constant, and D1 and D2 are the initial and final diameters of the particles.

  2. 2.Explain Kick's Law in the context of size reduction.Concept

    Kick's Law suggests that the energy required for size reduction is proportional to the size reduction ratio, meaning it depends on the relative size change rather than the absolute size. It is expressed as E = K_K ln(D1/D2), where E is the energy required, K_K is Kick's constant, and D1 and D2 are the initial and final diameters of the particles. This law is more applicable to coarse crushing where the size reduction ratio is more relevant.

  3. 3.Describe Bond's Law and its significance in size reduction.Concept

    Bond's law states that the work per unit mass is proportional to the new crack length created, which gives P/ṁ = 0.3162·W_i·(1/√D_b − 1/√D_a) with D in mm (or W = 10·W_i·(1/√P₈₀ − 1/√F₈₀) with sizes in µm). It corresponds to n = 1.5 in the general law dE/dD = −C/Dⁿ, between Kick (n = 1) and Rittinger (n = 2). The sizes are 80% passing sizes and W_i, the Bond work index, is a measured material property: the kWh/t needed to reduce the material from very large size to 80% passing 100 µm. It is the standard basis for sizing crushers and ball mills.

  4. 4.Why is Rittinger's Law more applicable to fine grinding processes?Application

    Rittinger's Law is more applicable to fine grinding because it assumes that the energy required is directly related to the increase in surface area. In fine grinding, the surface area increases significantly as particles are reduced to smaller sizes, making Rittinger's assumption more valid. This contrasts with coarse grinding, where the surface area increase is less pronounced.

  5. 5.What happens if you apply Kick's Law to a fine grinding process?Application

    If Kick's Law is applied to a fine grinding process, it may underestimate the energy required. Kick's Law is based on the size reduction ratio, which is more relevant to coarse crushing. In fine grinding, the increase in surface area is significant, and Rittinger's Law would provide a more accurate estimation of the energy required.

  6. 6.How does Bond's Law provide a balance between Rittinger's and Kick's Laws?Application

    Bond's Law provides a balance by considering both the surface area and the size reduction ratio. It is based on the square root of the surface-to-volume ratio, which makes it applicable to intermediate grinding processes. This approach allows Bond's Law to be used in situations where neither Rittinger's nor Kick's Laws are fully applicable, providing a more generalized estimation of energy requirements.

  7. 7.Calculate the energy required per kg to reduce particles from 10 mm to 1 mm using Rittinger's Law, with K_R = 1.5 J·mm/kg.Numerical

    Rittinger's law gives energy per unit mass: E/m = K_R(1/D_b − 1/D_a). With D_a = 10 mm and D_b = 1 mm, E/m = 1.5 × (1/1 − 1/10) = 1.5 × 0.9 = 1.35 J/kg. The units of K_R must be energy × length per mass so that the result comes out per kilogram; multiply by the feed rate to get power.

  8. 8.Using Kick's Law, calculate the energy required per kg to reduce particles from 100 mm to 10 mm, with K_K = 0.5 J/kg.Numerical

    Kick's law is E/m = K_K·ln(D_a/D_b) = 0.5 × ln(100/10) = 0.5 × 2.303 = 1.15 J/kg. Because only the reduction ratio appears, Kick's law predicts the same 1.15 J/kg for 10 mm to 1 mm — which is why it is used for coarse crushing and fails for fine grinding.

  9. 9.In what scenarios would Bond's Law be preferred over Rittinger's and Kick's Laws?Application

    Bond's Law is preferred in scenarios involving intermediate grinding processes where neither the surface area increase nor the size reduction ratio alone can accurately predict the energy requirement. It is particularly useful when dealing with materials that do not fit neatly into the categories of fine or coarse grinding, providing a more balanced approach.

  10. 10.Explain the significance of the constants in Rittinger's, Kick's, and Bond's Laws.Concept

    The constants in Rittinger's, Kick's, and Bond's Laws (K_R, K_K, and K_B respectively) are empirical values that depend on the material being processed and the equipment used. They are determined experimentally and reflect the efficiency of the size reduction process for specific conditions. These constants are crucial for accurately calculating the energy requirements for size reduction.

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