🎓 Lesson 6
D4
CSTR and PFR Derivations and Assumptions
A CSTR is like a well-stirred pot where incoming reactants instantly mix with everything inside, and a PFR is like a pipe where material flows through without mixing backward.
🎯 Learning Objectives
- ✓ Derive and apply the design equations for CSTR and PFR for first- and second-order irreversible reactions
- ✓ Analyze and compare reactor sizing requirements (volume) for identical conversion targets under identical conditions
- ✓ Explain how assumptions (e.g., perfect mixing, no backmixing, steady state) impact reactor performance and safety margins
- ✓ Design a two-reactor cascade (CSTR + PFR) to meet specified effluent concentration and selectivity constraints
- ✓ Apply residence time distribution concepts to diagnose deviations from ideal behavior in field-scale leaching or neutralization tanks
📖 Why This Matters
In mining, ideal reactor models underpin critical unit operations: heap leach pad design (approximated as PFR for percolating solution), agitated cyanide tanks (modeled as CSTRs), and acid mine drainage neutralization ponds. Misapplying assumptions—like assuming perfect mixing in a poorly agitated tank—can lead to incomplete metal recovery, hazardous reagent accumulation, or regulatory noncompliance. Mastering these models ensures robust, safe, and economically optimized process design.
📘 Core Principles
CSTR relies on three foundational assumptions: (1) perfect and instantaneous mixing → uniform composition/temperature everywhere; (2) steady-state operation → no accumulation; (3) constant density flow. This yields a simple algebraic design equation. In contrast, PFR assumes: (1) no radial dispersion and zero axial mixing → plug-like flow; (2) differential element analysis → leads to an ordinary differential equation; (3) reaction rate depends on local concentration. Real reactors fall between these ideals; the Damköhler number quantifies deviation. For mining applications, CSTR approximations dominate in agitated tanks (e.g., gold dissolution), while PFR better describes flow-through columns (e.g., resin adsorption beds or bio-oxidation channels).
📐 Design Equations for Single Reactions
The CSTR and PFR design equations derive from mole balances on species A. For a steady-state, constant-density system with rate –r_A = k·C_A^n, the required reactor volume depends on inlet concentration C_A0, target exit concentration C_A, and volumetric flow rate Q.
💡 Worked Example
Problem: A gold cyanidation tank treats 150 L/s of slurry containing 0.8 mM free cyanide (CN⁻). The oxidation of CN⁻ by oxygen follows pseudo-first-order kinetics: –r_CN = 0.025 s⁻¹ · C_CN. Design both a CSTR and a PFR to achieve 95% destruction (C_CN = 0.04 mM) at steady state.
1.
Step 1: Convert units — C_A0 = 0.8 mM = 0.0008 mol/L; C_A = 0.04 mM = 0.00004 mol/L; Q = 150 L/s = 0.150 m³/s; k = 0.025 s⁻¹.
2.
Step 2: Apply CSTR equation: V_CSTR = Q · (C_A0 − C_A) / (−r_A) = Q · (C_A0 − C_A) / (k·C_A) = 0.150 × (0.0008 − 0.00004) / (0.025 × 0.00004) = 0.150 × 0.00076 / 0.000001 = 114 m³.
3.
Step 3: Apply PFR equation (integrated first-order): V_PFR = Q ∫_{C_A0}^{C_A} dC_A / (−r_A) = Q/k · ln(C_A0/C_A) = 0.150 / 0.025 × ln(0.0008/0.00004) = 6 × ln(20) ≈ 6 × 2.9957 = 17.97 m³.
Answer:
The CSTR requires 114 m³, while the PFR requires only ~18 m³ — a 6.3× volume reduction — highlighting efficiency gains when flow approaches plug behavior, as in well-designed packed columns or laminar-flow leachate channels.
🏗️ Real-World Application
At Barrick Gold’s Cortez Mill (Nevada), the carbon-in-pulp (CIP) circuit uses six large agitated tanks in series. Each tank is modeled as a CSTR for design validation; kinetic data from pilot-scale tests confirmed pseudo-first-order adsorption of Au(CN)₂⁻ onto activated carbon. When unexpected cyanide breakthrough occurred, engineers diagnosed poor mixing (verified via tracer studies), violating the CSTR assumption — leading to retrofitting of high-efficiency impellers and baffle upgrades. Post-retrofit, residence time distribution narrowed, improving gold recovery by 1.8% and reducing cyanide consumption by 12% — directly linking ideal reactor assumptions to economic and environmental KPIs.
📋 Case Connection
📋 CO₂ Hydrogenation to Methanol in a Slurry Reactor (Carbon Recycling International, Iceland)
Low CO₂ solubility and slow surface reaction kinetics limiting productivity