🎓 Lesson 4
D3
Mole Balances in Closed and Open Systems
A mole balance tracks how many 'chunks' of chemical substances (moles) enter, leave, stay, or change inside a system during a reaction—like counting bricks before, during, and after building a wall.
🎯 Learning Objectives
- ✓ Calculate molar accumulation in a closed blasting chamber during explosive decomposition
- ✓ Apply mole balances to design venting time for fume-laden blast holes in open-pit operations
- ✓ Analyze stoichiometric consumption of ANFO components (NH₄NO₃ + fuel oil) using balanced reaction equations
- ✓ Explain differences between transient and steady-state mole balances in underground vs. surface blasting scenarios
📖 Why This Matters
In mining and blasting engineering, understanding how explosives decompose—and how many moles of gases (e.g., N₂, CO₂, H₂O vapor) and solids (e.g., carbon soot, unreacted salts) are generated—is critical for safety, ventilation planning, and environmental compliance. A mole balance tells you *how much* toxic gas (like CO or NOₓ) forms per kg of explosive, guiding air requirements and post-blast re-entry protocols. Without it, engineers risk under-ventilation, delayed production, or regulatory noncompliance.
📘 Core Principles
Mole balances stem from the law of conservation of atoms: atoms rearrange but aren’t created or destroyed. In a closed system (e.g., sealed blast hole), the mole balance reduces to accumulation = generation (via reaction). In an open system (e.g., a vented stope), inflow, outflow, accumulation, and reaction generation all matter. Transient balances account for time-dependent changes (critical for rapid detonation events), while steady-state assumes no accumulation—valid for continuous leaching circuits feeding into blasting residue treatment. Stoichiometric coefficients from balanced reactions directly scale molar production/consumption, making accurate reaction modeling essential—especially when explosives deviate from ideal oxygen balance.
📐 General Mole Balance Equation
The general mole balance applies universally; its form simplifies depending on system type. For any species A: Input − Output + Generation = Accumulation. Generation is calculated via reaction rate multiplied by stoichiometric coefficient (ν_A) and reactor volume. In blasting contexts, generation dominates—input/output are often zero (closed) or approximated via diffusion models (open).
General Mole Balance
IN − OUT + GEN = ACCConservation-based accounting of moles for species A across a defined system boundary.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| IN | Molar inflow rate | mol/s | Moles of species A entering the system per unit time |
| OUT | Molar outflow rate | mol/s | Moles of species A exiting the system per unit time |
| GEN | Net molar generation | mol/s | r_A × V, where r_A is reaction rate (mol/m³·s) and V is system volume (m³) |
| ACC | Molar accumulation rate | mol/s | dN_A/dt — time rate of change of moles of A within the system |
Typical Ranges:
ANFO detonation (closed borehole): GEN ≈ 10⁴–10⁶ mol/s·m³
Post-blast ventilation (open stope): OUT ≈ 50–500 mol/s for CO removal
💡 Worked Example
Problem: A 10-kg charge of ANFO (94% NH₄NO₃ + 6% fuel oil, assumed as C₁₀H₂₀) detonates in a sealed 2-m³ blast chamber. Using the balanced reaction: 4 NH₄NO₃ + C₁₀H₂₀ → 4 CO₂ + 12 H₂O + 8 N₂ + 2 C (soot), calculate total moles of gaseous products formed.
1.
Step 1: Determine molar masses — NH₄NO₃ = 80.04 g/mol; C₁₀H₂₀ = 140.27 g/mol.
2.
Step 2: Compute moles in charge — NH₄NO₃ = 9400 g / 80.04 g/mol = 117.4 mol; C₁₀H₂₀ = 600 g / 140.27 g/mol = 4.28 mol.
3.
Step 3: Use limiting reactant — stoichiometric ratio requires 4 mol NH₄NO₃ per 1 mol C₁₀H₂₀ → 4.28 mol C₁₀H₂₀ needs 17.12 mol NH₄NO₃ (available: 117.4 mol ⇒ excess). So C₁₀H₂₀ limits.
4.
Step 4: Scale products — from reaction: 1 mol C₁₀H₂₀ → 4 mol CO₂ + 12 mol H₂O + 8 mol N₂ = 24 mol gas. So 4.28 mol × 24 = 102.7 mol total gas.
5.
Step 5: Verify — at STP, 102.7 mol ≈ 2.3 m³ gas volume, exceeding chamber volume ⇒ high overpressure expected.
Answer:
The result is 102.7 mol of gaseous products, which implies ~2.3 m³ at STP—well above the 2-m³ chamber volume, confirming need for controlled venting or pressure relief design.
🏗️ Real-World Application
At the Bingham Canyon Mine (Rio Tinto), engineers used mole balances to redesign fume management after detecting elevated CO levels post-blast in deep lift blasts. By modeling ANFO decomposition with oxygen-balanced stoichiometry and incorporating measured O₂ depletion and CO/CO₂ ratios from gas sampling, they recalculated expected CO generation per tonne of explosive. This led to installing timed blast-hole venting valves that opened only after peak gas generation subsided—reducing average CO exposure by 68% and cutting ventilation fan runtime by 22%, saving $1.4M/year in energy costs (2022 Operations Report, Rio Tinto Technical Bulletin No. 2022-07).