8.1 Crushing, Grinding & Bond Work Index Calculations

Crushing, Grinding & Bond Work Index Calculations

Comminution is the process of physical size reduction of ore particles through crushing and grinding. In many mineral-processing plants, comminution is a major consumer of energy and operating cost, but its share is orebody-, flowsheet-, tariff-, and accounting-dependent. Ranges such as 30%–50% are context examples, not a universal plant fraction. The primary objectives of comminution are to liberate valuable economic minerals from barren gangue matrix at the coarse economic liberation size and to prepare mineral surfaces for downstream physical or chemical separation processes.

Classical Comminution Theories & Energy-Size Reduction Laws

The fundamental relationship between energy input ($E$) and particle size reduction ($x$) is expressed by the general differential comminution equation:

$dE = -C \frac{dx}{x^n}$

Where $C$ is a material constant, $x$ is the characteristic particle size, and $n$ is an exponent determining the specific comminution regime.

1. Kick's Law ($n = 1$)

Kick's Law states that the energy required to crush a given quantity of material is directly proportional to the reduction ratio ($ \frac{F_{80}}{P_{80}}$), regardless of original particle size:

$E = C_K \ln\left( \frac{F_{80}}{P_{80}} \right)$

Where $F_{80}$ is the 80% passing size of the feed and $P_{80}$ is the 80% passing size of the product. Kick's Law is commonly used as a coarse-reduction approximation over a calibrated range; the model has no universal particle-size boundary.

2. Rittinger's Law ($n = 2$)

Rittinger's Law states that the energy consumed in comminution is directly proportional to the new surface area created during size reduction:

$E = C_R \left( \frac{1}{P_{80}} - \frac{1}{F_{80}} \right)$

Rittinger's Law relates energy to new surface area and is often used as a fine-size approximation over a calibrated range; it does not have a universal particle-size boundary.

3. Bond's Third Theory of Comminution ($n = 1.5$)

Fred C. Bond established an empirical relationship bridging the gap between Kick's and Rittinger's laws, asserting that the work input is proportional to the new crack length produced per unit volume. Bond's equation calculates specific power consumption ($W$, in $\text{kWh/tonne}$ or $\text{kWh/short ton}$):

$W = 10 W_i \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right)$

Where:

• $W$ = Specific energy required ($\text{kWh/tonne}$)
• $W_i$ = Bond Work Index ($\text{kWh/tonne}$), defined as the total specific energy required to reduce ore from infinite particle size to an 80% passing size of $100\ \mu\text{m}$
• $P_{80}$ = 80% passing size of the grinding circuit product ($\mu\text{m}$)
• $F_{80}$ = 80% passing size of the primary feed ($\mu\text{m}$)

If $P_{80}$ and $F_{80}$ are given in micrometers ($\mu\text{m}$), the constant $10$ converts units into standard specific energy. Bond's relation is empirical and must use a compatible test Work Index, size basis, units, and circuit correction factors.

Crushing Machinery & Equipment Selection

Crushing is performed dry in multi-stage sequences (primary, secondary, tertiary) using mechanical compression or impact forces.

Primary Crushers

• Blake Jaw Crusher: Heavy-duty primary crusher with a fixed jaw and a movable swing jaw pivoted at the top. This configuration provides maximum motion at the discharge throat, preventing clogging under coarse ROM (run-of-mine) feed up to $1.5\ \text{m}$.
• Dodge Jaw Crusher: Swing jaw pivoted at the bottom, creating a constant discharge throat opening but variable feed opening. Prone to choking; used mainly for small-scale sampling plants.
• Gyratory Crusher: Features a heavy conical crushing head mounted on a central vertical shaft that gyrates inside a concave shell. Delivers continuous crushing action with higher throughput capacity ($2,000\text{--}10,000\ \text{tph}$) than jaw crushers.

Secondary & Tertiary Crushers

• Cone Crushers: Modified gyratory crushers with a shortened shaft supported from below. Standard Cone Crushers feature steep crushing cavities for coarse secondary feed ($50\text{--}100\ \text{mm}$), while Short Head Cone Crushers feature flatter parallel crushing zones for fine tertiary feed ($10\text{--}25\ \text{mm}$).
• High-Pressure Grinding Rolls (HPGR): Consists of two counter-rotating rolls applying high hydraulic pressures ($50\text{--}150\ \text{MPa}$) to an ore bed. HPGR promotes interparticle crushing, generating micro-cracks in mineral grains that reduce downstream ball mill grinding energy by 15% to 25%.

Grinding Machinery & Mill Dynamics

Grinding utilizes tumbling mills loaded with grinding media (steel balls, steel rods, or coarse ore rocks) operating wet at high solids concentrations (65% to 80% solids by weight).

Tumbling Mill Types

1. Rod Mills: Utilize high-carbon steel rods as media. Rods exhibit line contact, selectively grinding coarse particles while shielding fine particles, producing a uniform product ($0.5\text{--}3.0\ \text{mm}$) with minimal slimes.
2. Ball Mills: Utilize forged steel balls ($20\text{--}100\ \text{mm}$ diameter). Point contact promotes intense impact and attrition grinding, achieving fine product sizes ($<74\ \mu\text{m}$ / -200 mesh).
3. Autogenous (AG) & Semi-Autogenous (SAG) Mills: AG mills grind ore using large coarse ore rocks as media without steel media. SAG mills add a light steel ball charge (4% to 15% mill volume) to assist ore breakage, achieving extreme reduction ratios ($F_{80} = 300\ \text{mm}$ to $P_{80} = 1\ \text{mm}$) in a single stage.

Mill Critical Speed Calculation

The Critical Speed ($N_c$) of a tumbling mill is the rotational speed at which centrifugal force acting on a grinding ball equals gravitational force, causing media to cling to the mill shell throughout the revolution:

$N_c = \frac{42.3}{\sqrt{D - d}} \quad (\text{rpm})$

Where $D$ is the internal mill diameter ($\text{m}$) and $d$ is the media diameter ($\text{m}$). If $d \ll D$, the equation simplifies to $N_c = \frac{42.3}{\sqrt{D}}$. Commercial ball mills operate at 65% to 80% of critical speed to create a cataracting regime (balls lift and impact the mill toe) combined with cascading (balls roll down the charge face producing attrition).

Closed-Circuit Grinding & Circulating Load Ratio

To prevent overgrinding of liberated mineral grains, tumbling mills operate in closed circuit with hydrocyclones or vibrating screens.

Circulating Load Ratio ($CLR$)

The Circulating Load Ratio evaluates the mass flow rate of coarse classifier underflow ($C$) returned to the mill relative to fresh plant feed ($F$):

$CLR = \frac{C}{F} \times 100\% = \frac{t_o - t_m}{t_m - t_u} \times 100\%$

Where $t_o$, $t_m$, and $t_u$ are cumulative mass fractions passing a specified sieve mesh in the classifier overflow, mill discharge, and classifier underflow, respectively. The suitable circulating load follows the ore, mill, classifier, water balance, power draw, product target, and stability; a quoted range is only a reference case and cannot ensure optimum grinding.

Summary of Comminution Machinery