Study notes

Qwestrum Engineering360 · Chemical Engineering · Mechanical Operations

Particle Size Reduction

Size-reduction energy is estimated from comminution laws — Kick’s for coarse crushing, Rittinger’s for fine grinding, and Bond’s (via the work index) for the intermediate range that covers most industrial mills.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Specific energy increases as product particle size decreases
Crushers: jaw, gyratory; grinders: ball mill, hammer mill
Product size distribution is often log-normal

Topic details

Introduction

This McCabe-Smith topic quantifies the energy to crush and grind solids. You select the appropriate comminution law from the size range, apply Bond’s work index for practical mill sizing, and match equipment (jaw and gyratory crushers for coarse duty, ball and hammer mills for fine duty) to the required product size distribution.

Key relations & formulas

E = C (\frac{1}{\sqrt}{d_{p}} - \frac{1}{\sqrt}{D_{p}})

(Kick’s law region — coarse crushing uses Kick: E = C ln(D_p/d_p))

E = C (\frac{1}{d_{p}} - \frac{1}{D_{p}})

(Rittinger’s law, fine grinding)

W = 10 W_{i} (\frac{1}{\sqrt}{P_{80}} - \frac{1}{\sqrt}{F_{80}})

(Bond work index, kWh/ton)

Notation and sign conventions

Relation 1 —

E = C

E = C (\frac{1}{\sqrt}{d_{p}} - \frac{1}{\sqrt}{D_{p}})

(Kick’s law region — coarse crushing uses Kick: E = C ln(D_p/d_p))

Write this relation with symbols exactly as in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

E = C

E = C (\frac{1}{d_{p}} - \frac{1}{D_{p}})

(Rittinger’s law, fine grinding)

Relation 3 —

W = 10 W_{i}

W = 10 W_{i} (\frac{1}{\sqrt}{P_{80}} - \frac{1}{\sqrt}{F_{80}})

(Bond work index, kWh/ton)

Concept in depth

Breaking particles creates new surface, and the three classical laws differ in how they relate energy to size change. Kick’s law (energy ∝ log of size ratio) suits coarse crushing where volume-based fracture dominates; Rittinger’s law (energy ∝ new surface created) suits fine grinding where surface generation dominates; Bond’s law sits between, using an empirical work index and the 80%-passing sizes that plant engineers actually measure. Grinding is energy-intensive and inefficient — most input energy becomes heat — so the specific energy climbs steeply as the target size shrinks.

Assumptions and validity limits

State assumptions explicitly before using any relation for particle size reduction — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Mechanical Operations viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Mechanical Operations papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to particle size reduction.
4. Use equation 1:

E = C

.
5. Use equation 2:

E = C

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Particle Size Reduction appears in solids processing industries. In Indian chemical curricula this topic is tested because it connects theory to size reduction, filtration, and fluidization.

GATE and semester exams often combine particle size reduction with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use particle size reduction?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students apply Rittinger’s law to coarse crushing or Kick’s to fine grinding, mismatching the law to the size range. Others confuse F_80/P_80 (80% passing) with mean sizes in the Bond equation and drop the factor of 10 in the standard Bond form.

Quick revision checklist

Before attempting particle size reduction problems, confirm you can:

1. Specific energy increases as product particle size decreases
2. Crushers: jaw, gyratory; grinders: ball mill, hammer mill
3. Product size distribution is often log-normal

Revise the solved examples in Unit Operations of Chemical Engineering — McCabe, Smith & Harriott and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Bond work index energy

Problem

A mill reduces feed from F_80 = 10 mm to P_80 = 0.1 mm; the work index is W_i = 12 kWh/ton. Find the specific energy.

Solution

Convert to µm: F_80 = 10000, P_80 = 100. W = 10×12×(1/√100 − 1/√10000) = 120×(0.1 − 0.01) = 120×0.09 = 10.8 kWh/ton.

Conceptual check — Particle Size Reduction

Problem

In a Mechanical Operations semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of particle size reduction." What should a complete answer include?

Exams & GATE

McCabe-Smith Ch. 28 — pick the correct comminution law by size range.

📖 Standard books (India)

Unit Operations of Chemical Engineering — McCabe, Smith & Harriott
Read: Syllabus unit
Momentum, heat, and mass transfer operations

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📚Key formulas & points