Introduction & Context

The calculation of Cost per Ton for size reduction is a fundamental requirement in process engineering, bridging the gap between theoretical particle mechanics and operational economics. By quantifying the energy intensity of comminution processes, engineers can optimize equipment selection, predict utility expenditures, and establish baseline performance metrics for industrial milling operations. This methodology is primarily utilized in the design and optimization of grinding circuits, such as hammer mills and roller mills, where energy consumption represents the most significant variable operating cost.

Methodology & Formulas

The calculation follows a structured approach, beginning with the determination of specific energy requirements based on Bond's Law, followed by the integration of mechanical efficiency and fixed operational expenditures.

The specific energy consumption is derived from the following relationship:

\[ E = 10 \cdot W_i \cdot \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right) \]

To account for real-world equipment performance, the actual specific energy is adjusted by the motor efficiency factor:

\[ E_{actual} = \frac{E}{Efficiency_{motor}} \]

The total hourly energy cost is calculated by multiplying the actual specific energy by the throughput and the unit cost of energy:

\[ Cost_{energy\_hourly} = E_{actual} \cdot Throughput \cdot Cost_{energy\_per\_kwh} \]

The total hourly cost incorporates fixed operational expenses:

\[ Cost_{total\_hourly} = Cost_{energy\_hourly} + Cost_{fixed\_hourly} \]

Finally, the cost per ton is determined by normalizing the total hourly cost against the system throughput:

\[ Cost_{per\_ton} = \frac{Cost_{total\_hourly}}{Throughput} \]

Parameter	Constraint / Threshold	Regime / Condition
Bond's Law Validity	50.0 ≤ Size ≤ 10000.0	Empirical Range
Moisture Content	< 0.15	Standard Application
Motor Efficiency	0.88	Standard Operating Efficiency

The cost per ton is a function of both capital expenditure and operational efficiency. Key variables include:

Specific energy consumption measured in kWh per ton.
Media wear rates and the cost of replacement liners or grinding elements.
Maintenance downtime and labor costs associated with equipment servicing.
Throughput capacity relative to the rated power draw of the mill or crusher.

Bond's Law provides a mathematical framework to estimate the energy required for size reduction based on the work index of the material. By calculating the energy input required to reduce the feed size to the target product size, engineers can:

Predict electricity costs per ton based on local utility rates.
Benchmark the efficiency of existing circuits against theoretical performance.
Optimize the feed size distribution to minimize total energy expenditure.

Distinguishing between these costs is essential for accurate Total Cost of Ownership (TCO) analysis. Direct costs are easily tracked, while indirect costs often hide inefficiencies:

Direct costs: Electricity, grinding media, and spare parts.
Indirect costs: Lost production revenue during unplanned downtime, inventory carrying costs for consumables, and environmental compliance fees related to dust or noise mitigation.

Worked Example: Cost per Ton for Dry Grain Size Reduction

A process engineer is evaluating the operating cost of a hammer mill for dry grain size reduction in a feed processing facility. The objective is to determine the cost per ton based on given material properties and economic parameters.

Knowns (Input Parameters):

Bond Work Index, $ W_i $: 12.0 kWh/ton
Feed size, $ F_{80} $: 5000.0 μm
Product size, $ P_{80} $: 500.0 μm
Throughput: 5.0 tons/hour
Energy cost: $0.10 per kWh
Fixed costs (labor, maintenance, depreciation): $25.00 per hour
Moisture content: 0.12 (fraction, or 12%)

Step-by-Step Calculation:

Calculate the theoretical specific energy using Bond's Law:
\[ E = 10 \cdot W_i \cdot \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right) \]
From the numerical results, $ E = 3.67 $ kWh/ton.
Adjust for motor efficiency (88% or 0.88): Actual specific energy = $ \frac{E}{\text{MOTOR_EFFICIENCY}} = \frac{3.67}{0.88} = 4.17 $ kWh/ton.
Calculate the hourly energy cost: Hourly energy cost = Actual specific energy × Throughput × Energy cost per kWh = $ 4.17 \text{ kWh/ton} \times 5.0 \text{ tons/h} \times \$0.10/\text{kWh} = \$2.085/\text{h} $.
Calculate the total hourly operating cost: Total hourly cost = Hourly energy cost + Fixed costs per hour = $ \$2.085 + \$25.00 = \$27.085/\text{h} $.
Calculate the cost per ton: Cost per ton = $ \frac{\text{Total hourly cost}}{\text{Throughput}} = \frac{\$27.085}{5.0 \text{ tons/h}} = \$5.417/\text{ton} $.

Final Answer: The cost per ton of size reduction is $5.417 per ton.

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

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle