Lynch and Rao models

 Apply Lynch and Rao models to solve classification problems. 

Applying Lynch and Rao Models to Solve Classification Problems

The Lynch and Rao models are widely used to describe and analyze the performance of size classifiers, such as hydrocyclones and screens, in mineral processing. These models use mathematical expressions to define the relationship between particle size and the partition value, providing insights into the efficiency and separation characteristics of classifiers.

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Lynch Model

The Lynch model describes the partition curve using a logistic function. It accounts for the cut size ($d_{50}$) and the sharpness of separation, allowing engineers to evaluate classification efficiency.

Partition Curve Equation:

$$P(d) = \frac{1}{1 + \left( \frac{d}{d_{50}} \right)^n}$$

Where:

• $P(d)$: Partition value for particle size $d$.
• $d_{50}$: Cut size, the particle size with a 50% probability of reporting to the product.
• $n$: Sharpness index, a higher value indicates sharper separation.

Steps to Apply the Lynch Model:

1. Determine $d_{50}$ and $n$**:

   • Conduct experiments to measure the partition curve and estimate these parameters.

2. Predict Partition Values:

   • Use the above equation to calculate the partition values for different particle sizes.

3. Analyze Performance:

   • Plot the partition curve and evaluate the sharpness of separation and bypass fraction.

Example:

For a classifier with $d_{50} = 50 \, \mu\text{m}$ and $n = 3$:

$$P(30) = \frac{1}{1 + \left( \frac{30}{50} \right)^3} = 0.865$$

This means 86.5% of 30 µm particles report to the product.

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Rao Model

The Rao model provides an alternative approach to defining the partition curve and emphasizes the bypass fraction ($B$) and cut size ($d_{50}$). It is particularly useful when there is significant material bypassing the classification process.

Partition Curve Equation:

$$P(d) = B + (1 - B) \cdot \frac{1}{1 + \left( \frac{d}{d_{50}} \right)^n}$$

Where:

• $B$: Bypass fraction, the portion of feed material directly reporting to the product, independent of particle size.

Steps to Apply the Rao Model:

1. Determine $B$, $d_{50}$, and $n$:

   • Measure the bypass fraction and partition curve data to estimate these parameters.

2. Predict Partition Values:

   • Use the equation to calculate partition values for various particle sizes.

3. Interpret Results:

   • Assess how the bypass fraction impacts the classifier's performance and sharpness of separation.

Example:

For a classifier with $B = 0.1$, $d_{50} = 40 \, \mu\text{m}$, and $n = 4$:

$$P(20) = 0.1 + (1 - 0.1) \cdot \frac{1}{1 + \left( \frac{20}{40} \right)^4} = 0.1 + 0.9 \cdot 0.941 = 0.947$$

This indicates that 94.7% of 20 µm particles report to the product.

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Comparison of Lynch and Rao Models

Feature	Lynch Model	Rao Model
Key Parameters	$d_{50}$, $n$	$d_{50}$, $n$, $B$
Bypass Fraction	Not explicitly included	Explicitly included
Use Case	General classification problems	Classifiers with significant bypass

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Applications in Classification Problems

1. Hydrocyclones:

   • Use the Lynch model to calculate cut size and sharpness index for optimizing cyclone performance.

2. Screens:

   • Apply the Rao model to account for bypass material and predict screen efficiency.

3. Flow Sheet Optimization:

   • Incorporate these models in simulation software (e.g., MODSIM, JKSimMet) to optimize classification circuits.

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Conclusion

The Lynch and Rao models are powerful tools for analyzing and solving classification problems in mineral processing. By accurately modeling the partition curve, these methods enable engineers to design, optimize, and control classifiers to achieve desired performance.

Reference: R.P. King, Modeling and Simulation of Mineral Processing Systems, p. 98–125.