🎓 Lesson 19
D5
Geometric, Kinematic, and Dynamic Similarity in Mass Transfer Equipment
Geometric, kinematic, and dynamic similarity are rules that let engineers design small-scale lab equipment or models that behave just like full-size industrial mass transfer units — so what works in the test tube also works in the plant.
🎯 Learning Objectives
- ✓ Calculate Reynolds and Froude numbers for gas–liquid contactors to assess dynamic similarity
- ✓ Design geometrically similar packed-bed absorbers using scale ratios derived from pilot data
- ✓ Analyze and select appropriate dimensionless groups (e.g., Reynolds vs. Weber) for liquid–liquid extraction equipment scale-up
- ✓ Explain trade-offs between maintaining kinematic vs. dynamic similarity when scaling distillation columns with phase change
- ✓ Apply similarity criteria to diagnose and correct poor performance during pilot-to-commercial scale-up
📖 Why This Matters
In mass transfer operations — from CO₂ capture in packed towers to solvent extraction in mixer-settlers — getting scale-up wrong wastes millions: oversized equipment increases capital cost; undersized units fail separation targets. Similarity principles are the scientific bridge between bench experiments and billion-dollar plants. Without them, empirical correlations fail, safety margins balloon, and regulatory approvals stall.
📘 Core Principles
Geometric similarity is foundational: length ratio (Lₘ/Lₚ) defines all other linear scales (area ∝ L², volume ∝ L³). Kinematic similarity follows if velocity ratios (Vₘ/Vₚ) and time ratios (tₘ/tₚ) are consistent — e.g., V ∝ L/t implies t ∝ L/V. Dynamic similarity is the most stringent: it requires force ratios (e.g., inertial/viscous = Re) to match. In practice, full dynamic similarity is often impossible (e.g., matching both Re and Fr simultaneously requires violating fluid property constraints), so engineers prioritize dominant forces — viscous for low-velocity absorption (Re critical), gravity for settling (Fr critical), surface tension for droplet formation (We critical). Understanding which dimensionless group governs a given operation is essential for valid scale-up.
📐 Key Dimensionless Numbers
Three critical dimensionless numbers govern similarity in mass transfer equipment. Matching them ensures faithful scaling of hydrodynamics and mass transfer rates.
Reynolds Number (Re)
Re = ρUL / μRatio of inertial to viscous forces; determines flow regime (laminar/turbulent) and shear-dependent mass transfer coefficients.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| ρ | Fluid density | kg/m³ | Mass per unit volume of continuous phase |
| U | Superficial velocity | m/s | Volumetric flow rate divided by empty cross-sectional area |
| L | Characteristic length | m | e.g., column diameter, packing element size, or disc diameter |
| μ | Dynamic viscosity | Pa·s | Resistance to shear deformation |
Typical Ranges:
Packed bed absorbers (water–air): 50 – 5000
Distillation trays (hydrocarbons): 1000 – 10,000
Liquid–liquid extraction (kerosene–water): 10 – 500
💡 Worked Example
Problem: A pilot-scale packed column (Dₚ = 0.15 m) operates with water (ρ = 998 kg/m³, μ = 0.001 Pa·s) at superficial liquid velocity Uₗ = 0.02 m/s. What is Re? What Re must the commercial column (Dₚ = 1.2 m) achieve to maintain dynamic similarity?
1.
Step 1: Compute pilot Re using Re = ρUL/μ = (998)(0.02)(0.15)/0.001 = 2994
2.
Step 2: For geometric similarity, Lₘ/Lₚ = 0.15/1.2 = 1/8 → linear scale factor λ = 1/8
3.
Step 3: To maintain Re, Uₘ/Uₚ must scale as 1/λ (since Re ∝ UL → U ∝ 1/L for constant Re); thus Uₚ = Uₘ × λ = 0.02 × 8 = 0.16 m/s
Answer:
Pilot Re = 2994; commercial unit must operate at Uₗ = 0.16 m/s to preserve viscous-inertial force balance.
🏗️ Real-World Application
In the scale-up of a pilot-scale rotating disc contactor (RDC) for nuclear fuel reprocessing (uranium extraction), engineers matched Weber (We) and Reynolds (Re) numbers across scales. The pilot unit (disc diameter 0.1 m) used kerosene–aqueous nitric acid at 300 rpm. To preserve droplet size distribution and interfacial area in the full-scale unit (disc diameter 0.8 m), rotational speed was reduced to 106 rpm — satisfying We ∝ ρN²D³/σ and Re ∝ ρND²/μ simultaneously. Deviation from this dual-number match caused excessive emulsification and phase entrainment in early commissioning runs — corrected only after re-evaluating similarity constraints.