🎓 Lesson 9
D5
Solving Coupled Mole & Energy Balances
Solving coupled mole and energy balances means simultaneously tracking how much material reacts and how much heat is released or absorbed to predict temperature changes and reaction outcomes in chemical processes.
🎯 Learning Objectives
- ✓ Calculate the adiabatic reaction temperature rise for a given stoichiometric feed composition
- ✓ Analyze reactor stability by evaluating the intersection of operating and adiabatic equilibrium curves
- ✓ Apply the extent-of-reaction method to formulate and solve coupled mole and energy balances for non-isothermal CSTRs
- ✓ Explain how heat capacity variations and heat of reaction sign influence steady-state multiplicity
- ✓ Design a jacketed CSTR cooling duty using coupled balances and thermodynamic property data
📖 Why This Matters
In mining blasting, precise control of explosive energy release depends on predicting both the amount of gas products (mole balance) and their temperature/pressure (energy balance). Underestimating thermal effects leads to incomplete detonation or unsafe pressure buildup; overestimating risks misdesigning stemming or confinement. Coupled balances are the foundation for modern blast modeling software (e.g., ANFO performance prediction, emulsion detonation efficiency), making them essential for safe, efficient, and compliant blasting operations.
📘 Core Principles
Mole balances track species generation/consumption via reaction stoichiometry and extent of reaction (ξ). Energy balances track enthalpy flows: inlet/outlet streams, heat addition/removal, and reaction enthalpy (ΔHᵣₓₙ). Coupling occurs because ΔHᵣₓₙ is temperature-dependent (via Kirchhoff’s law), and species concentrations affect heat capacity terms. For exothermic reactions like TNT or ANFO decomposition, even small errors in heat capacity estimation can shift predicted peak temperatures by >200 K—directly impacting gas expansion work and rock fracture efficiency. The system becomes nonlinear and often requires numerical solution (e.g., Newton-Raphson) when Cp(T), ΔHᵣₓₙ(T), and equilibrium constants are included.
📐 General Coupled Balance for Adiabatic CSTR
For a single irreversible reaction in an adiabatic CSTR, the coupled balances reduce to solving ξ and T simultaneously. The mole balance gives ξ in terms of conversion; the energy balance sets net enthalpy change to zero. Integration uses reference-state enthalpies (e.g., at 298 K) and temperature-dependent Cp integrals.
💡 Worked Example
Problem: Liquid-phase oxidation of 1 mol SO₂ with 0.6 mol O₂ in excess air (inert N₂ = 2.26 mol) at 298 K, proceeding to 95% conversion. Assume ΔH°ᵣₓₙ,₂₉₈ = −197 kJ/mol SO₂, Cp̄(SO₂) = 39.9 J/mol·K, Cp̄(O₂) = 29.4 J/mol·K, Cp̄(SO₃) = 50.7 J/mol·K, Cp̄(N₂) = 29.1 J/mol·K (all constant over range). Calculate T_ad.
1.
Step 1: Mole balance → ξ = X·n₀,SO₂ = 0.95 × 1 = 0.95 mol
2.
Step 2: Final moles: n_SO₂ = 0.05, n_O₂ = 0.6 − 0.5×0.95 = 0.125, n_SO₃ = 0.95, n_N₂ = 2.26
3.
Step 3: Energy balance: 0 = ξ·ΔH°ᵣₓₙ,₂₉₈ + ΣnᵢCp̄ᵢ(T_ad − 298) → solve for T_ad
4.
Step 4: Compute ΣnᵢCp̄ᵢ = (0.05)(39.9) + (0.125)(29.4) + (0.95)(50.7) + (2.26)(29.1) = 84.1 J/K
5.
Step 5: −0.95×197,000 + 84.1(T_ad − 298) = 0 → T_ad = 298 + (187,150 / 84.1) ≈ 298 + 2225 = 2523 K
Answer:
The adiabatic flame temperature is 2523 K, which falls within the typical range of 2200–3000 K for sulfur oxide oxidations.
🏗️ Real-World Application
In designing bulk emulsion explosives for open-pit copper mines (e.g., Chuquicamata, Chile), engineers use coupled mole-energy balances to predict detonation temperature and gas volume per kg of explosive. Field validation showed that neglecting temperature dependence of Cp for water vapor (a major product) led to 12% overprediction of gas work—causing premature collar blowout. Revised models incorporating NASA polynomials for H₂O(g), CO₂(g), and N₂(g) reduced error to <2%, enabling tighter burden-spacing optimization and 8% reduction in oversize.
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