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.

Reading & Lecture

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.
Density tracers recovered form floats and sinks immediately show the partition curve.
Density tracers recovered from floats and sinks immediately show the partition curve. [image: (135-4-1)]

 


  • 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.
A diagram depicting perfect particle separation
A diagram depicting perfect particle separation
[image: (135-4-2)]

 


  • 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
A diagram illustrating less then perfect particle separation
Less then perfect particle separation
[image: (135-4-3)]

A diagram comparing two separation curves
A Comparison of separation curves
[image: (135-4-4)]
  • 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
Partition probability curve
Partition probability curve
[135-4-5]

  • 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.)
A diagram comparing separation efficiency
A comparison of separation efficiency
[image: (135-4-6)]

  • 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.
A diagram depicting bypass results
A diagram depicting bypass results
[image: 135-4-7)]

  • 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).
Bypass diagram
[image 135-4-8]

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)
Acceptable separation curves
Acceptable separation curves
[image: (135-4-9)]

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)
Undesirable separation curves
Undesirable separation curves
[image: (135-4-10)]
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).

Ball mills

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
A graph showing line slope partition
Observations? Slope in the first plot shows that 46% of feed reported to oversize.
[image: (135-4-11)]

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%

 

Observations?
Slope in first plot shows that 46 .9% of feed reported to oversize. Partition curve shows 0.30 mm cutsize, no oversize bypass and 9% undersize bypass. [image: (135-4-12)]

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' = \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.

A chart showing a mean particle size curve.


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 = \frac{d_{75} - d_{25}}{2d_{50}}

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

Probability of Underflow
Percent Probability of Underflow
[image: (135-4-14)]

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

X= d/d 50(c)

α = 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.
Chart showing increasing values of alpha
[image: (135-4-15)] Chart showing increasing values of alpha

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)
Total100.0061.57100.00
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

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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