Mineral processing involves the separation of valuable minerals from gangue, requiring various unit operations such as comminution, classification, and separation. Modeling these processes mathematically is crucial for process optimization and control. Differential equations form the backbone of these mathematical models, capturing the dynamic behavior of key variables like particle size, composition, and flow rates. Below, we detail the governing differential equations for some critical unit operations in mineral processing.
Comminution
Comminution, including crushing and grinding, reduces the size of ore particles. Models often use population balance equations (PBE) to describe the particle size distribution (PSD).
Governing Equations:
: Number density of particles of size at time .
: Breakage velocity.
: Selection function (fraction of particles selected for breakage).
: Breakage function (distribution of progeny sizes).
Energy Balance:
The energy consumed during grinding can be modeled as:
: Total energy.
: Energy efficiency factor.
: Power input.
Classification (Cyclones and Screens)
Classification separates particles based on size or density. For hydrocyclones, the dynamics are governed by fluid-particle interactions.
Mass Balance for Solids:
: Concentration of solids in the product.
: Concentrations in feed and overflow streams.
: Volumetric flow rates of feed and overflow.
Partition Curve (Efficiency Curve):
: Probability of particle size reporting to the underflow.
: Cut size.
: Sharpness index.
Flotation
Flotation separates hydrophobic and hydrophilic particles in a slurry using air bubbles. The kinetics of flotation are modeled as:
Mass Balance for Valuable Mineral:
: Concentration of valuable mineral in the pulp.
: Rate constant of mineral recovery.
: Bubble rise rate.
: Concentration of mineral in froth.
Froth Phase Dynamics:
: Froth height.
: Cross-sectional area of the cell.
: Air and liquid flow rates.
Thickening
Thickening separates solids from liquids through gravity sedimentation.
Solids Flux Equation:
: Solids concentration.
: Depth in the thickener.
: Settling velocity, typically a nonlinear function of .
Material Balance:
: Mass of solids in the thickener.
: Feed and underflow flow rates.
: Solids concentrations in feed and underflow.
Leaching
Leaching dissolves valuable minerals using chemical solutions. The rate of dissolution follows kinetics equations.
Rate Equation:
: Mineral concentration.
: Rate constant.
: Surface area of particles.
: Equilibrium and current concentrations of leachant.
Mass Balance for Leachant:
: Feed flow rate.
: Input leachant concentration.
: Reactor volume.
Magnetic and Gravity Separation
These processes rely on differences in magnetic susceptibility or density.
Gravity Separation:
: Particle velocity.
: Gravitational acceleration.
: Particle and fluid densities.
: Drag coefficient.
: Particle diameter.
Magnetic Force Balance:
: Magnetic force.
: Magnetic susceptibility.
: Magnetic field strength.
Conclusion
Differential equations provide an essential framework for describing and simulating the dynamics of mineral processing unit operations. Accurate parameter estimation and validation using experimental or plant data are crucial for their practical application in process control and optimization.