🎓 Lesson 18
D5
Ideal Reactor Models: CSTR, PFR, and LFR
An ideal reactor model is a simplified way to picture how chemicals mix and react inside a vessel—like imagining water flowing smoothly through a pipe (PFR), stirring perfectly in a tank (CSTR), or moving like a plug with no mixing (LFR).
🎯 Learning Objectives
- ✓ Calculate residence time distribution (RTD) functions E(t), F(t), and I(t) for CSTR, PFR, and LFR
- ✓ Analyze RTD curves to infer dominant flow patterns (e.g., bypassing, dead zones) in real blast-hole or slurry transport systems
- ✓ Explain how deviations from ideal models affect reagent utilization efficiency in leaching columns or explosive slurry mixing tanks
- ✓ Apply tracer test data to estimate dispersion numbers and quantify deviation from PFR/CSTR behavior
- ✓ Design a two-stage CSTR system to achieve target conversion for a first-order reaction in heap leach solution circulation
📖 Why This Matters
In mining, reactor hydrodynamics governs critical processes—from explosive slurry homogenization in bulk emulsion plants to acid distribution in heap leaching and reagent contact time in flotation circuits. Misjudging flow patterns can lead to incomplete reactions (e.g., low copper recovery), hazardous accumulation of unreacted explosives, or premature detonation due to poor mixing. Understanding ideal models lets engineers diagnose real-system flaws—not just calculate—but *interpret* why a leach pad underperforms or why a mixing tank produces inconsistent emulsion density.
📘 Core Principles
All ideal reactors are defined by their assumptions about mixing and flow: CSTR assumes infinite back-mixing → outlet concentration equals reactor concentration; PFR assumes zero radial/axial dispersion → fluid elements retain identity and age distribution is sharp; LFR assumes fully developed laminar flow (Re < 2000) → velocity profile is parabolic, causing broad RTD due to differential transit times. The Residence Time Distribution (RTD) function E(t) is the cornerstone metric—it describes the probability density that a fluid element spends time t in the reactor. Real systems are hybrids: a blast-hole saturated with air-entrained slurry may behave like a dispersed PFR; a large agitated tank for cyanide mixing approximates a CSTR only if baffles and impeller design suppress short-circuiting. Deviation from ideality is quantified using dimensionless numbers: the Dispersion Number (D/uL), Damköhler number (Da = kτ), and segregation index (I = σ²_θ / σ²_CSTR).
📐 RTD Functions & Mean Residence Time
The mean residence time τ = V/Q is common to all models, but E(t) differs fundamentally. For design and diagnosis, E(t) enables prediction of conversion, selectivity, and tracer breakthrough. Use E(t) to size reactors, interpret pulse/challenge tests, and benchmark against field RTD data from conductivity or radioisotope tracers.
💡 Worked Example
Problem: A leach solution holding tank has volume V = 450 m³ and inflow rate Q = 75 m³/h. A pulse of NaCl tracer is injected. Calculate E(t) at t = 6 h for CSTR, PFR, and LFR models.
1.
Step 1: Compute τ = V/Q = 450 / 75 = 6 h.
2.
Step 2: For CSTR: E(t) = (1/τ) exp(−t/τ) = (1/6) exp(−6/6) = 0.1667 × 0.3679 = 0.0613 h⁻¹.
3.
Step 3: For PFR: E(t) = δ(t − τ) → infinite spike at t = 6 h; value undefined pointwise but integral = 1.
4.
Step 4: For LFR: E(t) = τ/(2t²) for t ≥ τ/2 → E(6) = 6/(2×36) = 6/72 = 0.0833 h⁻¹.
Answer:
E(t=6 h): CSTR = 0.061 h⁻¹, PFR = δ-function (instantaneous exit), LFR = 0.083 h⁻¹ — illustrating broader spread for LFR and exponential decay for CSTR.
🏗️ Real-World Application
At Barrick’s Cortez Gold Mine (Nevada), RTD studies using bromide tracer in carbon-in-leach (CIL) tanks revealed severe short-circuiting: measured E(t) peaked at ~1.8 h instead of the design τ = 4.2 h, indicating 57% of flow bypassed active reaction zones. Engineers retrofitted baffles and adjusted impeller pitch, shifting E(t) toward CSTR behavior and increasing gold recovery by 2.3%. This case demonstrates how ideal models serve as diagnostic baselines—not targets—to quantify and correct hydrodynamic inefficiencies in mineral processing reactors.
📋 Case Connection
📋 CFD-Guided Mixer Redesign in Pharmaceutical Bioreactor
Oxygen mass transfer limitation causing cell viability drop at large scale