Simulation & Modeling Comprehensive Notes

Mineral Processing Modeling & Simulation:
Comprehensive Study Notes

This guide covers the fundamental principles of modeling mineral processing units, including comminution, classification, and flotation, based on standard phenomenological models.

1. Fundamentals of Modeling

Q1. Define mathematical modeling and explain its significance in mineral processing systems with examples.

Definition: Mathematical modeling is the process of creating a set of equations that quantitatively describes the behavior of a physical system. In mineral processing, unlike fluid dynamics, the material is not a continuous medium but a "particulate system." Therefore, a model must describe how populations of particles (which vary in size, density, and composition) behave when subjected to the physical and chemical forces inside processing equipment.

Significance:

• Predictive Capability: Models allow engineers to calculate plant performance (grade, recovery, throughput) before the plant is built, reducing financial risk.
• Scale-Up: "Phenomenological" models—those based on physical laws rather than just curve-fitting—allow data from small laboratory tests to be used to design full-scale industrial machines.
• Optimization: Simulations allow for "what-if" scenarios (e.g., "What happens if we change the ball mill speed?") to optimize processes for maximum profit without disrupting actual production.

Examples:

• Comminution: Using the Population Balance Model to predict the particle size distribution exiting a mill.
• Classification: Using the Plitt Model to predict the cut-size of a hydrocyclone based on its geometry and feed pressure.

Q2. List and explain the basic concepts required to simulate an ore dressing plant.

To simulate an ore dressing plant, four fundamental building blocks must be integrated:

1. Flowsheet Structure: This is the "map" of the plant. It defines how Unit Operations (machines) are connected by Process Flow Streams (pipes/conveyors).
2. Unit Operation Models: Each machine requires a mathematical algorithm. This model acts as a transformation function: it accepts feed stream data and calculates product stream data based on machine settings.
3. Material Description: The simulator needs a numerical method to describe the ore. The material is described by classes. The simulator tracks the mass flow rate of particles in specific intervals of size (e.g., 10–12 mm) and composition (e.g., 0–10% mineral grade).
4. Simulation Executive (Solver): This is the computer logic that manages the calculation. It handles sequential calculation and the convergence of recycle loops (iterative mass balancing).

Q3. Explain the steps involved in developing a plant flowsheet simulation.

1. Data Collection: Gather ore characteristics (hardness, density) and machine parameters. For existing plants, sample streams to measure flow rates.
2. Flowsheet Construction: Draw the map of units and streams in the simulator.
3. Model Selection: Choose appropriate mathematical models for each unit (e.g., selecting the Karra model for screens).
4. Parameter Estimation: Calibrate the models using experimental data so they accurately mimic the specific ore and machines.
5. Execution: Run the simulation to solve mass balances and recycle loops.
6. Validation: Compare results with real-world plant data to ensure accuracy.

2. Particle Size Analysis

Q4. Describe the Rosin–Rammler equation, its parameters, and their physical meaning.

The Rosin–Rammler distribution is an empirical formula used to describe the particle size distribution of broken rock (crusher or mill products).

Equation:
P(D) = 1 - exp[ -(D / D63.2)λ ]

Parameters:

• P(D): The cumulative fraction of material passing size D.
• D_63.2 (Size Modulus): The size at which 63.2% of the material passes. It represents the overall coarseness of the sample.
• λ (Distribution Modulus): Indicates the "sharpness" or width of the distribution. High λ implies a narrow size range; low λ implies a wide spread of sizes.

Q5. Numerical Problem: Rosin–Rammler Calculation

Problem: A crusher product has Rosin–Rammler parameters D_63.2 = 40 mm and λ = 2.0. Calculate the percentage of material smaller than 20 mm.

1. Formula: P(D) = 1 - exp[ -(D / D_63.2)^λ ]
2. Substitute: D = 20, D_63.2 = 40, λ = 2.0
3. Ratio: 20 / 40 = 0.5
4. Power: (0.5)² = 0.25
5. Exponential: e^-0.25 ≈ 0.7788
6. Result: P(20) = 1 - 0.7788 = 0.2212

Answer: 22.12% of the material is smaller than 20 mm.

Q6. Draw and explain a typical Particle Size Distribution (PSD) curve.

A PSD curve plots Particle Size (x-axis, usually logarithmic) against Cumulative % Passing (y-axis).

Key Features:

• S-Shape: Most comminution products follow a sigmoidal (S-shaped) curve.
• D₈₀: The size at which 80% of material passes; a standard control point for grinding circuits.
• Slope: A steep slope indicates a narrow distribution (particles are mostly the same size); a flat slope indicates a wide distribution.

3. Screening & Classification

Q7. Explain the Karra model for vibrating screens and the factors influencing efficiency.

The Karra Model predicts screening performance using a "Capacity Factor" approach. It calculates the theoretical tonnage of undersize material a screen can transmit based on its aperture area, modified by real-world factors.

Formula Concept: Capacity = Basic Capacity (A) × B × C × D × E × F × G

Factors:

• Basic Capacity (A): Dependent on aperture size and open area.
• Oversize Factor (B): More oversize material helps push fines through the mesh.
• Half-size Factor (C): High amounts of very fine material increase capacity.
• Near-size Factor (G): Particles close to the aperture size cause "pegging" (plugging), reducing capacity significantly.
• Wet Screening (E): Water sprays wash fines through, increasing capacity.

Q8. Numerical Problem: Screen Efficiency

Problem: A screen with 5 mm aperture has a Karra predicted cut size (d₅₀) of 3.2 mm and sharpness m = 4. Calculate the probability of a 4 mm particle passing.

1. Efficiency to Oversize (Retained) Formula:
Eo(d) = 1 - exp[ -0.693 * (d / d₅₀)^m ]

2. Substitute values:
Ratio = 4 / 3.2 = 1.25
Power = 1.25⁴ ≈ 2.441

3. Calculate Exponential:
exp[ -0.693 * 2.441 ] = exp[-1.69] ≈ 0.184

4. Efficiency to Oversize:
Eo = 1 - 0.184 = 0.816

5. Probability of Passing (Undersize):
Prob = 1 - Eo = 1 - 0.816 = 0.184

Answer: The particle has an 18.4% chance of passing.

4. Hydrocyclones

Q9. Explain the Plitt model for hydrocyclone performance.

The Plitt Model consists of four empirical equations linking cyclone geometry and operation to performance:

1. Cut Size (d_50c): The size with a 50% chance of separation. Increases with cyclone diameter and feed solids; decreases with pressure.
2. Flow Split (S): Ratio of underflow to overflow volume, controlled by the spigot/vortex finder ratio.
3. Pressure Drop (P): Energy required to push slurry through the unit.
4. Sharpness (m): Describes the steepness of the separation curve.

Q10. Numerical Problem: Hydrocyclone Density Shift

Problem: A cyclone has d_50c = 20 μm for Feldspar (ρ=2550). Calculate the recovery of 12 μm Hematite particles (ρ=5200) to the overflow. Water recovery R_f = 20%. Sharpness SI = 0.8 (assume exponent in efficiency equation). Efficiency eqn provided: e = 1 / (1 + x^SI).

1. Calculate new cut size (d₅₀) for Hematite:
d₅₀ ∝ 1 / √(ρ_s - ρ_l)
Ratio = √[(2550 - 1000) / (5200 - 1000)] = √[1550/4200] = 0.607
New d₅₀ = 20 * 0.607 = 12.15 μm

2. Calculate Efficiency to Underflow (e) for 12 μm:
x = d / d₅₀ = 12 / 12.15 = 0.988
e = 1 / (1 + 0.988^0.8) = 1 / (1 + 0.99) = 0.502

3. Actual Recovery to Underflow (Ru):
Ru = Rf + (1 - Rf) * e
Ru = 0.20 + (0.80 * 0.502) = 0.6016

4. Recovery to Overflow:
Rec = 1 - Ru = 1 - 0.6016 = 0.398

Answer: 39.8% recovery to overflow.

5. Comminution (Crushing & Grinding)

Q11. Explain "progeny particles" and breakage mechanisms.

Progeny Particles: The daughter fragments produced when a parent particle breaks.

Mechanisms:

• Shatter: High-impact fracture (e.g., hammer blow). Produces a wide range of fragment sizes.
• Cleavage: Slow compression. Particle splits along weak veins into a few large pieces.
• Attrition: Surface wear/rubbing. Produces fine dust and leaves the parent mostly intact.

Q12. Numerical Problem: Breakage Function

Problem: Breakage function B(x,y) = 0.5(x/y)^0.6 + 0.5(x/y)^3.0. Calculate fraction breaking from Class 3 (12-8cm) into Class 6 (4-2cm).

1. Parent Size (y): Geometric Mean(12, 8) = √96 ≈ 9.8 cm
2. Progeny Limits: Top = 4 cm, Bottom = 2 cm.
3. Cumulative B at Top (4 cm):
Ratio = 4 / 9.8 = 0.408
B(4) = 0.5(0.408)^0.6 + 0.5(0.408)³
B(4) = 0.292 + 0.034 = 0.326
4. Cumulative B at Bottom (2 cm):
Ratio = 2 / 9.8 = 0.204
B(2) = 0.5(0.204)^0.6 + 0.5(0.204)³
B(2) = 0.194 + 0.004 = 0.198
5. Discrete Fraction (b63):
b63 = B(4) - B(2) = 0.326 - 0.198 = 0.128

Answer: 0.128 (12.8%).

Q13. Explain the Population Balance Model (PBM).

PBM is an accounting method for mill simulation. It tracks mass moving between size classes.
Equation: Accumulation = In - Out + Birth - Death

• Selection Function (S): The rate at which particles of a specific size are selected to break.
• Breakage Function (B): Describes the size distribution of the progeny particles produced when a parent breaks.

Q14. Numerical Problem: Kick's Law

Problem: Calculate energy to reduce size from 50 mm to 12.5 mm using Kick's law (K = 3.2 kJ/kg).

Formula: E = K * ln(d_in / d_out)
Ratio = 50 / 12.5 = 4
E = 3.2 * ln(4) = 3.2 * 1.386 = 4.44

Answer: 4.44 kJ/kg.

6. Flotation & Gravity Separation

Q15. Explain the phases in a flotation cell.

1. Pulp Phase: The agitated zone where particles are suspended and bubbles collide with them.
2. Bubble Phase: Rising bubbles carrying attached hydrophobic particles.
3. Froth Phase: The stable foam layer on top where concentrate is recovered and water drains.
4. Entrained Phase: Particles trapped in water between bubbles (Plateau borders) in the froth.

Q16. Numerical Problem: Stokes' Law Settling

Problem: Calculate settling velocity of 0.12 mm particle (ρ=2750) in water (ρ=998, μ=0.001).

Formula: v = [g * d² * (ρ_s - ρ_l)] / 18μ
d = 1.2 x 10^-4 m
d² = 1.44 x 10^-8
Δρ = 1752
v = [9.81 * 1.44e-8 * 1752] / 0.018
v = 0.0137 m/s

Answer: 1.37 cm/s.

Q17. Explain Dense Media Separation (DMS) modeling.

DMS separates particles strictly by density. Models use a Partition Curve (Tromp curve) to calculate the probability of a particle sinking.

• Cut Point (ρ₅₀): Density where a particle has a 50% chance of sinking/floating.
• EPM (Ecart Probable Moyen): Measures the inefficiency/slope of the curve.