# AMIT 135: Lesson 4 Performance Modeling & Assessment

## Objectives

Upon completing this lesson students should be able to:

• Explain the method used to assess the performance of separators.
• Illustrate partition analysis for comminution circuit.
• Analyze different types of partition curves.
• Explain the methods to access separation efficiency from partition curve.

### Partition Analysis

• One of the most insightful methods for quantifying the performance of separators is a “partition analysis’.
• This detailed assessment is commonly performed using a “partition curve’.
• Partition curves show the probability of a particular particle having a given characteristics reporting to a given product stream.
• Can be achieved for any separation: particle size, density, magnetics, floatability, etc.

• To explain partition curves, let’s take a look at a “perfect’ particle size separation at 0.45 mm.
• As shown, 100% of particles > 0.45 mm in each in each size class in the feed report to oversize.

• If the separation is less than perfect, then particles can be “misplaced’ to the wrong streams.
• This includes:
• Misplacement of coarser particles to undersize
• Misplacement of finer particles to oversize

• If ideal, curve runs parallel to abscissa at the “cutsize’.
• More deviation from axis means more misplaced material.
• Shape is characteristic of separator type and operation.

• Cut — Point
= D50
= 0.45 mm
• Ecart Probability (Ep)
= [D75-D25]/2
=(0.5-0.4)/2=0.05 mmImperfection (I)
=EP/D50
=0.05/2 = 0.025

• Curve shape is an inherent characteristic of the type of separator employed.
• Commonly reported values for imperfection range from 0.005 to more than 0.50.
• Depends on:
• Type of equipment (e.g., screens, cyclones hydraulic sizers, etc)
• Characteristics of feed material (e.g., particle size, shape, density, etc.)
• Production demands (feed rate, water quality, etc.)

• Another important issue is “bypass’.
• Bypass can occur to both oversize and undersize.
• Oversize bypass is not unusual for screens
• Undersize bypass very common for classifiers.

• Bypass is the misplacement of fines via entrainment  into the oversize  product.
• Quantified by zero-size offset on the partition curve.
• Can sometimes >30 °/o  for fine sizing applications.
• Bypass typically has a large adverse downstream impact.
• Classifiers are often used in multiple stages  to  reduce bypass  (i.e., retreat oversize).

#### Acceptable Curves:

• Type 1- Ideal Symmetrical
• represents perfect  processes (e.g., laboratory sieve  data)
• Type 2 – Efficient Symmetrical
• OK for  efficient units (e.g., well  designed/operated  screen)
• Type 3 – Inefficient Symmetrical
• OK for less efficient units (e.g., fine hydraulic classifiers)

#### Undesirable Curves:

• Type 4- Oversize Nonsymmetrical
• shows loss of coarse to undersize (e.g., holes in screens)
• Type 5 – Undersize Nonsymmetrical
• shows loss of fines to oversize (e.g., overloaded screen)
Reasons:

Inherent unit characteristic, poor circuit design, excessive rates, mechanical failure, poor operating practices, others …

#### Step 1- Collect Samples

• Collect representative samples of the feed, oversize and undersize streams.
• Make sure that all streams have been taken into account.

#### Step 2 – Perform Size Analysis

• Perform a laboratory particle size analysis on each sample.
• Assess data to make sure that it is reliable (discussed later).

Size Class (Mesh)Mean Size (mm)Feed Mass (%)U/S (%)O/S (%)
+102.4044.020.007.85
10x201.2024.680.1211.78
20x280.71413.411.8826.50
28x350.50511.203.8218.10
35x480.3575.033.537.20
48x650.25210.8611.5210.08
65x1000.17812.5016.297.19
100x1500.12614.2720.965.72
150x2000.0898.4015.302.28
200x3250.05810.2517.522.16
-3250.0305.399.061.14
100.0100.0100.0

#### Step 3 – Conduct Calculations

• Plot (u-f) versus (u-o).
Data should form a line passing  through  zero .
• Line slope is the fraction of feed tonnage that reports to oversize.
• You may disregard unreliable points that do not appear to fall along the line.
Size Class (Mesh)Mean Size (mm)Feed Mass (%)U/S (%)O/S (%)X-axis (u-o)Y-axis (u-f)
+102.4044.020.007.85-7.85-4.02
10x201.2024.680.1211.78-11.66-4.56
20x280.71413.411.8826.50-24.62-11.53
28x350.50511.203.8218.10-14.28-7.38
35x480.3575.033.537.20-3.67-1.50
48x650.25210.8611.5210.081.440.66
65x1000.17812.5016.297.199.13.79
100x1500.12614.2720.965.7215.246.6
150x2000.0898.4015.302.2813.026.90
200x3250.05810.2517.522.1615.367.27
-3250.0305.399.061.147.923.67
100.0100.0100.0

#### Step 4 – Construct Partition Curve

• Calculate oversize partition for each size class using [(u-f)o]/[(u ­ o)f].
• Plot mean size versus partition factor .
• Compute performance indicators (cutsize, imperfection, bypass, etc.).

Mean Size (mm)Feed Mass (%)U/S (%)O/S (%)Y-axis (u-f)X-axis (u-o)Percent Feed Weight
U/S
Percent Feed Weight
O/S
Reconstituted Feed WeightPartition Number
2.4044.020.007.854.027.850.003.683.68100.00
1.2024.680.1211.784.5611.660.065.535.5998.86
0.71413.411.8826.5011.5324.621.0012.4313.4392.57
0.50511.203.8218.107.3814.282.038.4910.5280.73
0.3575.033.537.201.503.671.873.385.2564.32
0.25210.8611.5210.08-0.66-1.446.114.7310.8443.61
0.17812.5016.297.19-3.79-9.18.653.3712.0228.07
0.12614.2720.965.72-6.6-15.2411.132.6813.8119.43
0.0898.4015.302.28-6.90-13.028.121.079.1911.64
0.05810.2517.522.16-7.27-15.369.301.0110.319.83
0.0305.399.061.14-3.67-7.924.810.535.3410.01
100.0100.0100.053.0846.92100.00

Yield to Oversize Stream – 46.92%

### Partition Curve Observations

• The partition curve completed for a particle size separation utilized the weight distribution of all process streams including the feed stream.
• Depending on the  breakage characteristics of the material and the  location of the sample points, particle breakage could occur  which  affects the component balance around the process.
• As such, it is sometimes preferred to use component assays (e.g., solid concentration or assays such as iron content) and the two-product equation to determine mass yield.
• Using the mass yield, the feed  is then  reconstituted to determine  the  partition  numbers.

### Corrected Partition Number

The equation for determining the corrected Partition Number is:

 $Y'&space;=&space;\frac{Y-R_{1}}{1-R_{1}-R_{2}}$ (5-10) (5-10)

Y’ is the corrected partition number, Y the actual partition number, R1 , the fractional amount of ultrafines by-passing to the underflow stream and R2 , the fractional amount of coarsest particles by-passing to the flow stream.

• The by-pass of coarse material to the overflow stream is rare but may occur due to a worn vortex finder.
• R2 =0 can be assumed in most cases.

### Separation Efficiency

Separation efficiency should always be measured from the corrected partition numbers.

•  Bypassed particles were not subjected to the separation forces .

For particle size separations, the imperfection value (I) is the preferred measurement:

$I&space;=&space;\frac{d_{75}&space;-&space;d_{25}}{2d_{50}}$

d75, d50 , and d25= the particle size having 75 %, 50% and 25% probabilities, respectively,  of reporting to the underflow stream.

### Separation Performance Projection

• Many equations are available that model typical performance curves associated with
• Lynch and Rao found that the Reduces Efficiency curve can be modeled by the following expression:

$Y'&space;-&space;\frac{exp(\alpha&space;\chi&space;)-1)}{exp(\alpha&space;\chi&space;)&space;+&space;exp(\alpha&space;)-2))}&space;\;&space;\;&space;\;&space;\;&space;(5-11)$

X= d/d 50(c)

Î± = the curve slope and the value is indicative of the classification efficiency.

• The Lynch model can be used to predict the performance of a classifying cyclones.

#### Example: Performance Prediction

Given a feed particle size distribution and an alpha value of 2.5, predict the performance of a cyclone. The amount of ultrafine by-pass is assumed to be 20% by weight.

1234=Eq.[5-10]5=Eq.[5-10]6=2*57=6/(∑6)
Mean Particle Size (microns)Weight (%)d/d50(c)Corrected Partition NumberActual Partition NumberUnderflow Weight (%)Normalized Underflow Weight (%
12002.40121.001.002.403.90
8507.508.51.001.007.5012.18
6008.9061.001.008.9014.45
4256.404.251.001.006.4010.39
3006.9031.001.006.8911.19
2124.302.120.970.974.196.80
1504.501.50.820.863.866.28
1064.001.060.550.642.554.14
753.400.750.310.451.522.46
5351.700.530.170.3417.3628.20
Total100.0061.57100.00

Mass Yield to Underflow = 61.57%

#### Summary

• “Partition factor” represents the probability that a given particle size in the feed stream will report to the oversize  product.
• “Partition analysis” makes it possible to determine key performance indicators.
• Cutsize (D50)
• Imperfection (I)
• Bypass
• Plant personnel should monitor and strive to maintain sizing “efficiencies ” since this greatly impacts other plant operations.

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