🎓 Lesson 7
D4
Sizing Reactors Using Design Equations
Sizing a reactor means figuring out how big it needs to be so that a chemical reaction completes safely and efficiently.
🎯 Learning Objectives
- ✓ Calculate reactor volume for a first-order irreversible reaction in a PFR using the design equation
- ✓ Design a CSTR cascade to achieve 95% conversion for a second-order liquid-phase reaction
- ✓ Analyze the effect of temperature and activation energy on required reactor size using Arrhenius-based rate laws
- ✓ Explain the physical and operational differences between CSTR and PFR sizing outcomes for identical reaction systems
- ✓ Apply dimensionless numbers (e.g., Damköhler number) to assess kinetic vs. transport limitations in reactor scale-up
📖 Why This Matters
In mining and metallurgical processing—such as leaching of copper ores or cyanidation of gold concentrates—reactor sizing directly impacts metal recovery, reagent consumption, and tailings stability. An undersized reactor causes incomplete reaction and lost product; an oversized one wastes capital, energy, and space. For blasting engineers, understanding reactor sizing principles transfers directly to modeling explosive reaction zones, detonation wave propagation, and gas expansion dynamics in blastholes—making this foundational for both chemical process safety and explosive efficiency.
📘 Core Principles
Ideal reactors are simplified models used to derive design equations: the Batch reactor (constant volume, no flow), Continuous Stirred-Tank Reactor (CSTR, perfect mixing, steady-state), and Plug Flow Reactor (PFR, no axial mixing, differential element analysis). Each obeys a general mole balance: Input – Output + Generation = Accumulation. For steady-state flow reactors, accumulation = 0. The design equation emerges by substituting the rate law (–rₐ) into the mole balance and solving for volume (V). Key assumptions include constant density (for liquid-phase), isothermal operation (unless explicitly modeled), and well-defined kinetics. As complexity increases—e.g., variable density, non-isothermal behavior, or autocatalysis—the design equation expands to include energy balances and compressibility corrections.
📐 Key Calculation
The PFR design equation integrates reaction rate over conversion: V = Fₐ₀ ∫₀ˣ dX / –rₐ. For a first-order irreversible reaction A → products, this yields V = (Fₐ₀ / k) ln(1 / (1 – X)). This formula assumes constant density, isothermal conditions, and no pressure drop.
💡 Worked Example
Problem: A gold cyanidation leach reactor processes 50 L/min of ore slurry containing 0.8 mol/m³ dissolved Au(CN)₂⁻. The first-order rate constant k = 0.025 min⁻¹ at 25°C. What PFR volume is required to achieve 90% conversion?
1.
Step 1: Identify knowns — Fₐ₀ = (50 L/min) × (0.8 mol/m³) × (1 m³/1000 L) = 0.04 mol/min; X = 0.90; k = 0.025 min⁻¹
2.
Step 2: Apply V = (Fₐ₀ / k) ln(1 / (1 – X)) = (0.04 / 0.025) × ln(1 / 0.10) = 1.6 × ln(10) ≈ 1.6 × 2.3026 = 3.684 L
3.
Step 3: Verify — 3.7 L is physically feasible for lab-scale leaching; industrial units would scale linearly with flow rate and incorporate safety factor (1.2–1.5×)
Answer:
The required PFR volume is 3.68 L, which falls within the typical range of 3–5 L for pilot-scale cyanide leach reactors targeting >85% Au extraction.
🏗️ Real-World Application
At the Boddington Gold Mine (Western Australia), a two-stage CSTR system was sized to treat 1,200 m³/h of carbon-in-leach (CIL) slurry. Using measured kinetics (–rₐ = k·Cₐ², k = 0.0042 m³/mol·h), engineers calculated individual reactor volumes of 850 m³ each to achieve 92% cyanide-mediated gold dissolution. The design incorporated online pH and ORP monitoring to adjust residence time dynamically—demonstrating how theoretical sizing anchors real-time control strategies in hydrometallurgical operations.
🔧 Interactive Calculator
🔧 Open Reaction Engineering and Kinetics Calculator📋 Case Connection
📋 Bioethanol Fermentation Bioreactor Scale-Up with Inhibition Kinetics
Ethanol inhibition caused premature cessation at large scale despite matching nominal conditions
📋 Nitric Acid Absorption Tower Design for Tail-Gas Treatment
Incomplete absorption of NO and NO₂ due to slow liquid-phase oxidation kinetics and poor gas distribution