Hydrocyclone (Plitt)

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Description

This article describes the Plitt model for hydrocyclone classification, which is based on the original work of Plitt (1976) and updated by Flintoff (1987).[1][2]

Following a review of the Plitt model, Nageswararao et al. (2004) concluded:[3]

Because the model was published in full and thus enterered[sic] the public domain, the Plitt model saw widespread early use, particularly as a teaching tool. Following the addition of the Flintoff corrections, the Plitt model has also seen industrial use. With the proviso ... that, if at all feasible, the model should be fitted to data obtained under conditions as close as possible to those to be simulated, the model can be expected to give useful results.

Model theory

Figure 1. The dimensions of a hydrocyclone required by the Plitt model.

Plitt provides a method of computing the classification of particles by size, incorporating the so-called bypass of fines to the underflow stream.

Plitt's partition by size equation is:

where:

  • is the index of the size interval, , is the number of size intervals
  • is the uncorrected, i.e. actual, partition to underflow of particles in size interval (frac)
  • is the corrected partition to underflow of particles in size interval (frac)(frac), i.e. excluding the fines bypass effect
  • is the fraction of feed liquids recovered to underflow (frac)

Plitt suggests applying either the Rosin-Rammler or Lynch exponential sum partition curves to compute the uncorrected classification by size:

where:

  • is the geometric mean size of particles in size interval (mm)
  • is the size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
  • and are parameters which quantify the sharpness of separation of the Rosin-Rammler and Lynch partition curves, respectively

The corrected cut size, (μm), and partition curve shape parameters, and , are calculated with:[3]

where:

  • - are calibration factors which should be fitted from operating data
  • is the cyclone feed pressure (kPa)
  • is the ratio of underflow to overflow volumetric flow rates (frac)
  • is diameter of the cyclone (cm)
  • is diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet (cm)
  • is diameter of the vortex finder (overflow) (cm)
  • is diameter of the apex/spigot (underflow) (cm)
  • is the free vortex height (cm)
  • is the liquid viscosity (cP)
  • is the volume fraction of solids in the feed, in percentage units (%)
  • is the volumetric feed rate to the cyclone (L/min)
  • is density of solids in the feed (t/m3)
  • is density of liquids in the feed (t/m3)[4]
  • is density of the feed slurry (t/m3)
  • is the hydrodynamic exponent (typically between the values of 0.5 for laminar flow and 1.0 for turbulent flow)

The fraction of feed liquids recovered to underflow () is computed from:

where is the volumetric recovery of pulp to the underflow (frac):

The recovery of solids to underflow, (frac), is computed from the fractional distribution of mass by size in the feed, (w/w), and the partition , i.e.:

It can be seen from the above equations that and are interdependent via the partition . Therefore, an iterative procedure is used to calculate and refine the values of and . Few iterations are typically required for solution.

Partition metrics

Several metrics are provided to characterise the partition curve.

The , also known as the cut or separation size, is defined as the size of a particle which has an even (50%) chance of appearing in either the underflow or overflow stream. The size is estimated via a log-linear interpolation of geometric mean size () against the uncorrected partition to underflow of all solids in the feed.

The Ecart Probable, or , is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for size classification as:[4]

where and are the sizes of particles which have a 75% and 25% probability, respectively, of appearing in the underflow stream. The and sizes are estimated by log-linear interpolation of geometric mean size against the uncorrected partition to underflow of all solids in the feed.

The Imperfection, , is a normalised measure of the sharpness of separation, which is suggested to be independent of the magnitude of the , and is typically defined for size classification as:[4]

Roping

Several methods are available to identify the potential for roping discharge from a hydrocyclone underflow.

Plitt proposed that roping may occur when when the volumetric feed rate to the cyclone, (m3/h), exceeds a solids capacity limit:[5]

Plitt further proposed a limit to the volume fraction of solids in cyclone underflow, (% v/v), of:

where is the mass median particle size of the underflow, computed here as the P50 (μm).

The SPOC criterion indicates roping may occur when:[6]

where is the percentage volume fraction of solids in the underflow stream (% v/v). The SPOC criterion is only valid when .

Investigations by Bustamante (1991) and Concha et al. (1996) led to the limiting values of cyclone geometry in Table 1:[4]

Table 1. Transition from spray to roping discharge (after Gupta and Yan, 2016).[4]
Source Condition
Bustamante <0.34 Roping discharge
0.34 - 0.5 Roping or spray
>0.5 Spray discharge
Concha et al. <0.45 Roping discharge
0.45 - 0.56 Roping or spray
>0.56 Spray discharge

Excel

The Plitt hydrocyclone model may be invoked from the Excel formula bar with the following function call:

=mdUnit_Hydrocyclone_Plitt(Parameters as Range, Size as Range, Feed as Range, OreSG as Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

Inputs

The required inputs are defined below in matrix notation with elements corresponding to cells in Excel row () x column () format:


where:

  • is the partition curve function to apply, 0 = Rosin-Rammler, 1 = Lynch
  • is the number of cyclones operating in a cluster. Model calculations are scaled per cyclone.
  • is the number of ore types
  • is feed solids mass flow rate by size and ore type (t/h)
  • is the mass flow feed rate of liquids into the cyclone (t/h)

Results

The results are displayed in Excel as an array corresponding to the matrix notation below:


where:

  • is the number of iterations required to refine the interdependent values of and
  • is the mass flow rate of liquids to the overflow stream (t/h)
  • is the mass flow rate of liquids to the underflow stream (t/h)
  • is mass flow rate of solids to the underflow stream (t/h)
  • is mass flow rate of solids to the overflow stream (t/h)
  • is partition fraction of feed solids to the underflow stream (frac)
  • is partition fraction of feed solids to the overflow stream (frac)

Example

The images below show the selection of input arrays and output results in the Excel interface.

Figure 2. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 3. Example showing the selection of the Size (red frame), Feed (purple frame) and OreSG (green frame) arrays in Excel.
Figure 4. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

MD_Hydrocyclone page

The first tab page in the access window will have this name.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Tag Display This name tag may be modified with the change tag option.
Condition Display OK if no errors/warnings, otherwise lists errors/warnings.
ConditionCount Display The current number of errors/warnings. If condition is OK, returns 0.
GeneralDescription / GenDesc Display This is an automatically generated description for the unit. If the user has entered text in the 'EqpDesc' field on the Info tab (see below), this will be displayed here.

If this field is blank, then SysCAD will display the unit class ID.

Requirements
On CheckBox This enables the unit. If this box is not checked, then the MassFracToUF option appears below.
MassFracToUF Input Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the underflow stream when the model is off.
Method Partition (User) The partition to overflow for each size interval is defined by the user. Different values can be used for different solids.
Partition (Reid-Plitt) The partition to overflow for each size interval is defined by a Reid-Plitt efficiency curve. Different parameters can be used for different solids.
Partition (Whiten-Beta) The partition to overflow for each size interval is defined by a Whiten-Beta efficiency curve. Different parameters can be used for different solids.
Nageswararao The Nageswararao model is used to determine the partition of solids to underflow and overflow for each size interval.
Narasimha-Mainza (2014) The Narasimha-Mainza (2014) model is used to determine the partition of solids to underflow and overflow for each size interval.
Narasimha-Mainza (Multi) The Narasimha-Mainza (Multi) model is used to determine the partition of solids to underflow and overflow for each size interval.
Plitt The Plitt model is used to determine the partition of solids to underflow and overflow for each size interval.
RopingCalcs CheckBox Show addition calculations that predict the onset of cyclone underflow roping.
Options
ShowQFeed CheckBox QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQOF CheckBox QOF and associated tab pages (eg Sp) will become visible, showing the properties of the overflow stream.
ShowQUF CheckBox QUF and associated tab pages (eg Sp) will become visible, showing the properties of the underflow stream.
SizeForPassingFracCalc Input Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc Input Fraction passing for Size calculation. The fraction input here will be shown in the Stream Summary section.
Stream Summary
MassFlow / Qm Display The total mass flow in each stream.
SolidMassFlow / SQm Display The Solids mass flow in each stream.
LiquidMassFlow / LQm Display The Liquid mass flow in each stream.
VolFlow / Qv Display The total Volume flow in each stream.
Temperature / T Display The Temperature of each stream.
Density / Rho Display The Density of each stream.
SolidFrac / Sf Display The Solid Fraction in each stream.
LiquidFrac / Lf Display The Liquid Fraction in each stream.
Passing Display The mass fraction passing the user-specified size (in the field SizeForPassingFracCalc) in each stream.
Passes Display The user-specified (in the field FracForPassesSizeCalc) fraction of material in each stream will pass this size fraction.

Cyclone page

The Cyclone page is used to specify the input parameters for the hydrocyclone model.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Plitt
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Dimensions
NumberCyclones / NumCyclones Input The number of operating cyclone units in the cluster.
CycloneDiameter / Dc Input Diameter of the cyclone.
InletDiameter / Di Input Diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet.
CycloneDiameter / Dc Input Diameter of the cyclone.
OverflowDiameter / Do Input Diameter of the vortex finder (overflow).
UnderflowDiameter / Du Input Diameter of the apex/spigot (underflow).
FreeVortexHeight / h Input The free vortex height.
Parameters
d50Factor / F1 Input Calibration factor for the d50c equation.
SharpnessFactor / F2 Input Calibration factor for the sharpness (m) equation.
PressureFactor / F3 Input Calibration factor for the pressure (P) equation.
SplitFactor / F4 Input Calibration factor for the volumetric split (S) equation.
HydrodynamicExponent / k Input Value of the hydrodynamic exponent (k) in the d50c equation.
PartitionMethod Rosin-Rammler The Rosin-Rammler partition equation is applied to classification by size.
Lynch The Lynch partition equation is applied to classification by size.
Liquids
LiquidsSeparMethod Split To UF (User) Liquids are split to underflow by a user-defined fraction of liquids in the feed.
UF Solids Fraction Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow solids mass fraction value (if possible).
UF Liquids Fraction Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow liquids mass fraction value (if possible).
UFSolidsFracReqd / UF.SfReqd Input Required value of the mass fraction of solids in the underflow stream. Only visible if UF Solids Fraction is selected.
UFLiquidsFracReqd / UF.LfReqd Input Required value of the mass fraction of liquids in the underflow stream. Only visible if UF Liquids Fraction is selected.
LiqSplitToUF / UF.LiqSplit Input/Display The fraction of liquids recovered to underflow.
Results
Iterations Display Number of iterations required to stabilise partition calculation.
ClusterQv Display Volumetric flow rate of feed to the cyclone cluster (i.e. total flow).
CycloneQv Display Volumetric flow rate of feed to each cyclone in the cluster (i.e. per cyclone flow).
Cv Display Volume fraction of solids in feed stream.
eta Display Liquid viscosity (SysCAD property).
d50c Display Value of the corrected d50c.
DeltaP Display Value of the pressure drop across the cyclone, P.
S Display Value of the volumetric flow split ratio, S.
Rv Display Fraction of feed slurry volume recovered to underflow stream.
m Display Value of the sharpness of separation, m, used in the Rosin-Rammler partition function.
alpha Display Value of the sharpness of separation, , used in the Lynch partition function.
Rs Display The fraction of solids in the feed recovered to the underflow stream.
Rf Display The fraction of feed liquids recovered to the underflow, Rf, computed by the Plitt model.
d50 Display The separation size, d50, of all solid particles in the feed.
EcartProbable / Ep Display The value of the Ecart Probable of the separation.
Imperfection / I Display The value of the Imperfection of the separation.

{{{ActionU}}} page

The {{{ActionU}}} page is used to specify or display the {{{ActionL}}} by species and size values.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Distribution
Name Display Shows the name of the SysCAD Size Distribution (PSD) quality associated with the feed stream.
IntervalCount Display Shows the number of size intervals in the SysCAD Size Distribution (PSD) quality associated with the feed stream.
SpWithPSDCount Display Shows the number of species in the feed stream assigned with the SysCAD Size Distribution (PSD) quality.
{{{ActionU}}}
Method Model/User Select model-calculated or user-defined {{{ActionL}}} to separate each solids species type.
Density Display Density of each solid species.
Size Display Size of each interval in mesh series.
MeanSize Display Geometric mean size of each interval in mesh series.
All (All column) Display
  • Actual overall {{{ActionL}}} to {{{DestinationL}}} of all solid species, for each size interval.
  • Excludes solid species not present in the {{{UnitL}}} feed.
{{{ActionU}}} Display
  • {{{ActionU}}} to {{{DestinationL}}} for each size interval, in each solid species, as determined by the selected model or user defined value.
  • Note: These values are displayed regardless of whether the solid species is present in the {{{UnitL}}} feed or not.
All (All row, All column) Display
  • Displays the actual, total, {{{ActionL}}} of all solids with a particle size distribution property in the feed to {{{DestinationL}}}.
  • Excludes solid species not present in the {{{UnitL}}} feed.
All (All row, per species) Display
  • Actual overall {{{ActionL}}} to {{{DestinationL}}} for each solid species, for all size intervals in that species.
  • Excludes solid species not present in the {{{UnitL}}} feed.

Roping page

This page displays the results for hydrocyclone roping limit calculations. The page is only visible if Roping is selected on the MD_Hydrocyclone page.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Roping
Underflow
SolidsVolFlow / SQv Display Volumetric flow rate of solids in cyclone underflow stream.
Plitt.MSu Display Plitt's volumetric flow rate of solids in cyclone underflow roping limit.
SolidsVolFrac / Svf Display Volume fraction of solids in the cyclone underflow stream.
SPOC Display SPOC volume fraction of solids in the cyclone underflow roping limit.
Plitt.phiL Display Plitt's volume fraction of solids in the cyclone underflow roping limit.
Geometry
BCondition Display Text string describing the spray/roping condition of the cyclone based on Bustamante's geometry limits.
CCondition Display Text string describing the spray/roping condition of the cyclone based on the Concha et al. geometry limits.

About page

This page is provides product and licensing information about the Met Dynamics Models SysCAD Add-On.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
About
HelpLink ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png Copies Product and License information to the Windows clipboard.
Product
Name Display Met Dynamics software product name
Version Display Met Dynamics software product version number.
BuildDate Display Build date and time of the Met Dynamics Models SysCAD Add-On.
License
File ButtonBrowse.png This is used to locate a Met Dynamics software license file.
Location Display Type of Met Dynamics software license or file name and path of license file.
SiteCode Display Unique machine identifier for license authorisation.
ReqdAuth Display Authorisation level required, MD-SysCAD Full or MD-SysCAD Runtime.
Status Display License status, LICENSE_OK indicates a valid license, other messages report licensing errors.
IssuedTo Display Only visible if Met Dynamics license file is used. Name of organisation/seat the license is authorised to.
ExpiryDate Display Only visible if Met Dynamics license file is used. License expiry date.
DaysLeft Display Only visible if Met Dynamics license file is used. Days left before the license expires.

Additional notes

  • Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
  • If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
  • Gas phase species report directly to the Overflow stream without split.

See also

References

  1. Plitt, L.R., 1976. A mathematical model of the hydrocyclone classifier. CIM bulletin, 79(776), pp.114-123.
  2. Flintoff, B.C., Plitt, L.R. and Turak, A.A., 1987. Cyclone modelling: a review of present technology. CIM Bull.;(Canada), 80(905).
  3. 3.0 3.1 Nageswararao, K., Wiseman, D.M. and Napier-Munn, T.J., 2004. Two empirical hydrocyclone models revisited. Minerals Engineering, 17(5), pp.671-687.
  4. 4.0 4.1 4.2 4.3 4.4 Gupta, A. and Yan, D.S., 2016. Mineral processing design and operations: an introduction. Elsevier.
  5. Dubey, R.K., Singh, G. and Majumder, A.K., 2017. Roping: Is it an optimum dewatering performance condition in a hydrocyclone?. Powder Technology, 321, pp.218-231.
  6. Napier-Munn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.