Modeling & Simulation of Mineral processing Systems

Explain the modeling of dense media separators, highlighting their advantages. Modeling of Heavy Media Separators Heavy media separators are gravity-based separation devices that use a dense medium (a liquid or suspension) to separate particles based on their specific gravity. The medium is typically a mixture of finely ground dense material, like magnetite or ferrosilicon, suspended in water, creating a liquid with a density higher than water but lower than the desired separation density. Principles of Modeling Partition Curve (Performance Curve) : The partition curve is a key tool used in modeling heavy media separators. It represents the probability of particles with a specific density reporting to the sink (or concentrate) stream. The critical cut density ( d 50 d_{50} ) is the density at which particles have an equal probability of reporting to the sink or float streams. Mathematical Representation : The probability P ( d ) P(d) of a particle reporting to the sink stre...

  Problem: The breakage function in a crusher machine is determined to be: B ( x : y ) = 0.4 ( x y ) 0.5 + 0.6 ( x y ) 3.0 B(x:y) = 0.4 \left(\frac{x}{y}\right)^{0.5} + 0.6 \left(\frac{x}{y}\right)^{3.0} If size class 2 has mesh size boundaries of 14.0 cm and 10.0 cm, and size class 5 has boundaries of 8.0 cm and 6.0 cm, calculate b 52 b_{52} . Explanation in Simple Terms: When rocks or particles are crushed in a crusher machine, they break into smaller pieces of various sizes. The  breakage function  is a mathematical way to describe how these larger particles break into smaller ones. It predicts the probability or fraction of material from a given size class breaking into smaller size classes. Key Terms Simplified: Size Classes : Particles are grouped into "size classes" based on their sizes. For example, particles between 14.0 cm and 10.0 cm belong to  size class 2 , and particles between 8.0 cm and 6.0 cm belong to  size class 5 . The goal is to underst...

  Problem Statement: The breakage function in a crusher machine is given by: B ( x : y ) = 0.3 ( x y ) 0.45 + 0.7 ( x y ) 3.2 B(x:y) = 0.3 \left(\frac{x}{y}\right)^{0.45} + 0.7 \left(\frac{x}{y}\right)^{3.2} If size class 2 has mesh sizes boundaries 13.4 cm and 9.5 cm and size class 5 has boundaries 7.50 cm and 5.60 cm, calculate b 52 Solution: Calculate b 52 b_{52} , the breakage distribution from size class 5 to size class 2, where: Size class 2 has mesh size boundaries D 2 = 13.4   cm D_2 = 13.4 \, \text{cm} and D 3 = 9.5   cm D_3 = 9.5 \, \text{cm} , Size class 5 has mesh size boundaries D 5 = 7.5   cm D_5 = 7.5 \, \text{cm} and D 6 = 5.6   cm D_6 = 5.6 \, \text{cm} . The formula for calculating b i j b_{ij} is: b i j = B ( D i − 1 : d p j ) − B ( D i : d p j ) b_{ij} = B(D_{i-1}: d_{pj}) - B(D_i: d_{pj}) Where: d p 2 = 13.4 + 9.5 2 = 11.28   cm d_{p2} = \frac{13.4 + 9.5}{2} = 11.28 \, \text{cm} (the average size in size class 2). Solution: Substitute the ...

  Illustrative Example 4.4: Terminal Settling Velocity of a Glass Cube Problem Statement : Calculate the terminal settling velocity of a glass cube with an edge dimension of 0.1 mm in a fluid with the following properties: Density of fluid ( ρ f \rho_f ) = 982 kg/m³ Viscosity of fluid ( μ \mu ) = 0.0013 kg/m·s Density of glass ( ρ s \rho_s ) = 2820 kg/m³ Additionally, calculate: The equivalent volume diameter ( d v d_v ) of the cube. The sphericity factor ( ϕ \phi ) of the cube. Solution Step 1: Calculate the Equivalent Volume Diameter ( d v d_v ) The equivalent volume diameter ( d v d_v ) is the diameter of a sphere with the same volume as the cube. Volume of the cube : V cube = a 3 = ( 0.0001 ) 3 = 1 × 1 0 − 12   m 3 V_{\text{cube}} = a^3 = (0.0001)^3 = 1 \times 10^{-12} \, \text{m}^3 Volume of a sphere : V sphere = π 6 d v 3 V_{\text{sphere}} = \frac{\pi}{6} d_v^3 Equating the two volumes : 1 × 1 0 − 12 = π 6 d v 3 1 \times 10^{-12} = \frac{\pi}...

Explain karra model for vibrating screens. Karra Model for Vibrating Screens The Karra model is an empirical model used to predict the performance of vibrating screens in mineral processing. It is particularly useful for determining screen efficiency and the partitioning of materials into oversize (retained) and undersize (passing) streams based on particle size distribution and operating conditions. Key Concepts of the Karra Model Screen Efficiency : The Karra model evaluates the efficiency of separation for each particle size fraction on a vibrating screen. Efficiency is expressed as a function of particle size, screen aperture, and operating parameters. Partition Function : The model uses a partition curve (or efficiency curve) to describe the probability ( P P ) that a particle of a given size will pass through the screen. The curve is sigmoidal, reflecting that particles smaller than the screen aperture pass through with high probability, while larger particles are r...

Illustrative example 5.2 Modeling and Simulation of Mineral Processing Systems The breakage function of a crusher machine is given by the following equation: B ( x , y ) = 0.3 + 0.7 ( x y ) 0.45 × 3.2 B(x, y) = 0.3 + 0.7 \left( \frac{x}{y} \right)^{0.45} \times 3.2 Where: Size class 2 has mesh size boundaries of 13.4 cm and 9.5 cm. Size class 5 has mesh size boundaries of 7.50 cm and 5.60 cm. To calculate b 52 b_{52} , use the formula: b i j = B ( D i − 1 , d p j ) − B ( D i , d p j ) b_{ij} = B(D_{i-1}, d_{pj}) - B(D_i, d_{pj}) Given: d p 2 = 13.4 + 9.4 2 = 11.28   cm d_p2 = \frac{13.4 + 9.4}{2} = 11.28 \, \text{cm} Now calculate b 52 b_{52} : b 52 = 0.3 + 0.7 ( 7.5 11.28 ) 0.45 × 3.2 − 0.3 + 0.7 ( 5.6 11.28 ) 0.45 × 3.2 b_{52} = 0.3 + 0.7 \left( \frac{7.5}{11.28} \right)^{0.45} \times 3.2 - 0.3 + 0.7 \left( \frac{5.6}{11.28} \right)^{0.45} \times 3.2 Breaking this down: First term calculation: 0.3 + 0.7 ( 7.5 11.28 ) 0.45 × 3.2 = 0.4393 0.3 + 0.7 \left( \frac{7.5}{1...

 Write a short note on energy requirements for comminution. Energy Requirements for Comminution The energy required for comminution is a critical consideration in mineral processing and is influenced by several factors, including the particle's properties and the comminution process itself. The key points are: Energy Absorption : Energy is absorbed during particle fracture and is determined by the distribution of flaws, the stress rate, and particle orientation in the stress field. Single-particle impact fracture data reveal that higher energy absorption correlates with a finer product size distribution. Energy Laws : The energy required for size reduction is modeled using empirical laws. Three major laws provide insights: Kick's Law ( E ∝ ln ⁡ D d E \propto \ln \frac{D}{d} ): Energy is proportional to the size reduction ratio. Rittinger's Law ( E ∝ 1 d − 1 D E \propto \frac{1}{d} - \frac{1}{D} ): Energy consumption is proportional to the increase in surfac...

Illustrative Example 4.5: Effect of Particle Density Problem: A hydrocyclone exhibits a corrected classification curve described by: e ( d p ) = 1 1 + x − λ e(dp) = \frac{1}{1 + x^{-\lambda}} where x = d p d 50 c x = \frac{dp}{d_{50c}} . When processing a quartz slurry: d 50 c = 16.5   μ m d_{50c} = 16.5 \, \mu\text{m} , Sharpness index ( S I SI ) = 0.72, Density of quartz ( ρ q \rho_q ) = 2670 kg/m 3 \text{kg/m}^3 . Determine the recovery of 10   μ m 10 \, \mu\text{m} particles to overflow when the hydrocyclone treats a magnetite slurry under comparable conditions, where: Density of magnetite ( ρ m \rho_m ) = 5010 kg/m 3 \text{kg/m}^3 , Recovery of water to underflow ( R u R_u ) = 18.5% (same in both cases). Solution: Step 1: Corrected cut size for magnetite slurry The cut size d 50 c , m d_{50c,m} for magnetite slurry can be calculated as: d 50 c , m = d 50 c ρ q − ρ f ρ m − ρ f , d_{50c,m} = d_{50c} \sqrt{\frac{\rho_q - \rho_f}{\rho_m - \rho_f}}, where: Fluid...

Short questions and answers   Write the Rosin Ramler Equation for particle distribution   Define throw of a crusher  Define progeny particle  Write the equation for screen angle factor   What do you mean by the term  Breakage' Write the expression for Reynolds number  Write the relationship between energy and breakage   What do you mean by classification function   Define the term through in Jaw crusher   Define D 50 for a Hydro cyclone Rosin-Rammler Equation for Particle Distribution : The Rosin-Rammler distribution function is given by: P ( D ) = 1 − exp ⁡ ( − ( D D 63.2 ) n ) P(D) = 1 - \exp{\left(-\left(\frac{D}{D_{63.2}}\right)^n\right)} Here, D 63.2 D_{63.2} is the size at which the function has a value of 0.632, and n n is a distribution parameter. (Page 8 of Modeling and Simulation of Mineral Processing Systems ) Throw of a Crusher : The "throw" of a crusher refers to the distance by which the movi...

What is the Rosin Ramler Equation for particle size   distribution. Explain in detail.  The Rosin-Rammler Equation for particle size distribution is expressed as: P ( D ) = 1 − exp ⁡ [ − ( D D 63.2 ) n ] P(D) = 1 - \exp\left[-\left(\frac{D}{D_{63.2}}\right)^n\right] Where: P ( D ) P(D) = Cumulative fraction of the total mass of particles with sizes less than or equal to D D . D D = Particle size. D 63.2 D_{63.2} = Particle size at which 63.2% of the material passes (a characteristic parameter of the distribution). n n = Distribution modulus, indicating the spread of the size distribution. A larger n n corresponds to a narrower distribution. Explanation Mathematical Representation : P ( D ) P(D) ranges from 0 to 1 as D D increases. At D = D 63.2 D = D_{63.2} , P ( D ) = 0.632 P(D) = 0.632 , which is why D 63.2 D_{63.2} is used as the reference size. Log-Log Transformation : The Rosin-Rammler distribution is often analyzed in its linearized for...

What are joint distribution functions? Explain in detail. Joint distribution functions, as detailed in R.P. King's "Modeling and Simulation of Mineral Processing Systems," describe the combined distribution of two or more properties of particles in a population. These functions are particularly significant when multiple characteristics of particles influence their performance in mineral processing operations. Definition and Utility The joint distribution function , P ( D , g ) P(D, g) , represents the fraction of particles that possess a specific size D D and mineralogical composition g g . This function is crucial for modeling processes that depend simultaneously on size and composition, such as comminution or separation. Discrete Representation For practical applications, joint distribution functions are often discretized. The particle population is divided into size classes and grade (composition) classes, creating a grid in the size-composition space. The mass fr...

Kick’s energy equation Kick's energy equation is one of the empirical laws for estimating energy requirements in comminution processes, particularly for coarse particle size reduction. It is expressed as: E = k ⋅ ln ⁡ ( D 1 D 2 ) E = k \cdot \ln \left(\frac{D_1}{D_2}\right) Variables: E E : Energy required per unit mass of material (J/kg or other energy units). k k : Kick's constant, which depends on the material properties and machine characteristics. D 1 D_1 : Initial particle size (before size reduction). D 2 D_2 : Final particle size (after size reduction). Key Assumptions: Kick's equation assumes that the energy required for size reduction is proportional to the size reduction ratio ( D 1 / D 2 D_1/D_2 ). It is most accurate for coarse crushing where the reduction in particle size is relatively small. Usage: This equation is often used in engineering applications to estimate power requirements for crushing operations in mineral processing systems. You c...

 Explain different phases in Flotation. Phases of Flotation: Pulp Phase (Collection Phase) : Objective : In this phase, valuable minerals are selectively attached to air bubbles while undesirable gangue minerals remain in suspension. Mechanisms : Particle attachment occurs when the hydrophobic mineral particles collide with air bubbles. Hydrodynamic forces in the flotation cell govern the probability of collision and attachment. Kinetics : The attachment and detachment kinetics are described using rate constants, which are influenced by bubble size, particle size, and turbulence intensity. (Refer to pages 290–307 for detailed kinetic modeling and equations). Froth Phase : Objective : This phase involves the separation of mineral-laden bubbles from the pulp. Features : Stability and drainage of the froth layer are crucial. Gangue particles might be entrapped or entrained in the froth, which affects the purity of the concentrate. Dynamics : Froth stability is...

Derive equation of critical speed of a ball mill  The critical speed of a ball mill is the speed at which the centrifugal force exerted by the rotating mill shell on the balls is equal to the gravitational force acting on the balls. At this speed, the balls are held against the mill shell due to centrifugal force and do not fall, which results in no grinding action. Mathematically, the critical speed is expressed as: N c = 1 2 π g R − r N_c = \frac{1}{2\pi} \sqrt{\frac{g}{R - r}} Where: N c N_c = critical speed (in revolutions per second or minute), g g = gravitational acceleration, R R = internal radius of the mill, r r = radius of the grinding ball. In practical terms, the critical speed is often described in terms of the percentage of this theoretical speed (e.g., 75–90% of the critical speed) to achieve optimal grinding efficiency. ---------------------- Critical speed is derived based on the balance of forces acting on a ball in a rotating mill. The critical sp...