Hydrocyclone (Plitt)

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).

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

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:

$y_{i}=y'_{i}+R_{f}(1-y'_{i})$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $y_{i}$ is the uncorrected, i.e. actual, partition to underflow of particles in size interval $i$ (frac)
• $y'_{i}$ is the corrected partition to underflow of particles in size interval $i$ (frac)(frac), i.e. excluding the fines bypass effect
• $R_{f}$ 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:

$y'={\begin{cases}1-\exp \left(-\ln 2\left({\dfrac {{\bar {d}}_{i}}{d_{50}}}\right)^{m}\right)&&{\text{Rosin-Rammler}}\\\\{\dfrac {\exp \left(\alpha {\dfrac {{\bar {d}}_{i}}{d_{50}}}\right)-1}{\exp \left(\alpha {\dfrac {{\bar {d}}_{i}}{d_{50}}}\right)+\exp(\alpha )-2}}&&{\text{Lynch}}\\\end{cases}}$

where:

• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $d_{50}$ is the size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
• $m$ and $\alpha$ are parameters which quantify the sharpness of separation of the Rosin-Rammler and Lynch partition curves, respectively

The corrected cut size, $d_{50c}$ (μm), and partition curve shape parameters, $m$ and $\alpha$, are calculated with:

$d_{50c}=F_{1}{\dfrac {39.7{D_{c}}^{0.46}{D_{i}}^{0.6}{D_{o}}^{1.21}\eta ^{0.5}\exp(0.063C_{V})}{{D_{u}}^{0.71}h^{0.38}Q^{0.45}\left[{\dfrac {\rho _{s}-\rho _{l}}{1.6}}\right]^{k}}}$

$m=F_{2}1.94\left({\dfrac {{D_{c}}^{2}h}{Q}}\right)^{0.15}\exp \left({\dfrac {-1.58S}{1+S}}\right)$

$\alpha =1.54m-0.47$

$P=F_{3}{\dfrac {1.88Q^{1.8}\exp(0.0055C_{V})}{{D_{c}}^{0.37}{D_{i}}^{0.94}h^{0.28}({D_{u}}^{2}+{D_{o}}^{2})^{0.87}}}$

$S=F_{4}{\dfrac {18.62{\rho _{p}}^{0.24}\left({\dfrac {D_{u}}{D_{o}}}\right)^{3.31}h^{0.54}({D_{u}}^{2}+{D_{o}}^{2})^{0.36}\exp(0.0054C_{V})}{{D_{c}}^{1.1}P^{0.24}}}$

where:

• $F_{1}$ - $F_{4}$ are calibration factors which should be fitted from operating data
• $P$ is the cyclone feed pressure (kPa)
• $S$ is the ratio of underflow to overflow volumetric flow rates (frac)
• $D_{c}$ is diameter of the cyclone (cm)
• $D_{i}$ is diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet (cm)
• $D_{o}$ is diameter of the vortex finder (overflow) (cm)
• $D_{u}$ is diameter of the apex/spigot (underflow) (cm)
• $h$ is the free vortex height (cm)
• $\eta$ is the liquid viscosity (cP)
• $C_{V}$ is the volume fraction of solids in the feed, in percentage units (%)
• $Q$ is the volumetric feed rate to the cyclone (L/min)
• $\rho _{s}$ is density of solids in the feed (t/m³)
• $\rho _{L}$ is density of liquids in the feed (t/m³)
• $\rho _{p}$ is density of the feed slurry (t/m³)
• $k$ 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 ($R_{f}$) is computed from:

$R_{f}={\dfrac {R_{v}-C_{V}R_{s}}{1-C_{V}R_{s}}}$

where $R_{v}$ is the volumetric recovery of pulp to the underflow (frac):

$R_{v}={\dfrac {S}{1+S}}$

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

$R_{s}=\sum \limits _{i=1}^{n}{f_{i}y_{i}}$

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

Partition metrics

Several metrics are provided to characterise the partition curve.

The $d_{50}$, 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 $d_{50}$ size is estimated via a log-linear interpolation of geometric mean size ($\ln {\bar {d}}$) against the uncorrected partition to underflow of all solids in the feed.

The Ecart Probable, or $E_{\rm {p}}$, is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for size classification as:

$E_{\rm {p}}={\dfrac {d_{75}-d_{25}}{2}}$

where $d_{75}$ and $d_{25}$ are the sizes of particles which have a 75% and 25% probability, respectively, of appearing in the underflow stream. The $d_{75}$ and $d_{25}$ 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, $I$, is a normalised measure of the sharpness of separation, which is suggested to be independent of the magnitude of the $d_{50}$, and is typically defined for size classification as:

$I={\dfrac {E_{\rm {p}}}{d_{50}}}={\dfrac {d_{75}-d_{25}}{2d_{50}}}$

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, $M_{\rm {SU}}$ (m³/h), exceeds a solids capacity limit:

$M_{\rm {SU}}>0.35{D_{\rm {u}}}^{2.35}$

Plitt further proposed a limit to the volume fraction of solids in cyclone underflow, $\Phi _{\rm {L}}$ (% v/v), of:

$\Phi _{\rm {L}}=62.3\left[1-\exp \left(-{\dfrac {d_{\rm {u}}}{60}}\right)\right]$

where $d_{\rm {u}}$ is the mass median particle size of the underflow, computed here as the P₅₀ (μm).

The SPOC criterion indicates roping may occur when:

$C_{\rm {VU}}>56+0.2(C_{\rm {V}}-20)$

where $C_{\rm {VU}}$ is the percentage volume fraction of solids in the underflow stream (% v/v). The SPOC criterion is only valid when $C_{\rm {V}}<35{\text{ }}\%{\text{ v/v}}$.

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

Table 1. Transition from spray to roping discharge (after Gupta and Yan, 2016).

Source	$D_{\rm {u}}/D_{\rm {o}}$	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 ($i$) x column ($j$) format:

$Parameters={\begin{bmatrix}{\text{Partition method}}\\{\text{Number of cyclones}}\\D_{c}{\text{ (m)}}\\D_{i}{\text{ (m)}}\\D_{o}{\text{ (m)}}\\D_{u}{\text{ (m)}}\\h{\text{ (m)}}\\F_{1}\\F_{2}\\F_{3}\\F_{4}\\(Q_{M,F})_{Liquids}{\text{ (t/h)}}\\\rho _{L}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\eta {\text{ (cP)}}\\k\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Feed={\begin{bmatrix}(Q_{M,F})_{11}{\text{ (t/h)}}&\dots &(Q_{M,F})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{M,F})_{n1}{\text{ (t/h)}}&\dots &(Q_{M,F})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;OreSG={\begin{bmatrix}(\rho _{s})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{s})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}$

where:

• ${\text{Partition method}}$ is the partition curve function to apply, 0 = Rosin-Rammler, 1 = Lynch
• ${\text{Number of cyclones}}$ is the number of cyclones operating in a cluster. Model calculations are scaled per cyclone.
• $m$ is the number of ore types
• $Q_{M,F}$ is feed solids mass flow rate by size and ore type (t/h)
• $(Q_{M,F})_{Liquids}$ 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:

$mdUnit\_Hydrocyclone\_Plitt={\begin{bmatrix}{\begin{array}{c}{\text{Iterations}}\\\rho _{s}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\rho _{p}{\text{ (t/m}}^{\text{3}}{\text{)}}\\C_{V}{\text{ (}}\%{\text{ v/v)}}\\Q{\text{ (m}}^{\text{3}}{\text{/h)}}\\d_{50c}{\text{ (mm)}}\\P{\text{ (kPa)}}\\S{\text{ (frac)}}\\R_{v}{\text{ (frac)}}\\m\\\alpha \\R_{s}{\text{ (frac)}}\\R_{f}{\text{ (frac)}}\\(Q_{M,OF})_{Liquids}{\text{ (t/h)}}\\(Q_{M,UF})_{Liquids}{\text{ (t/h)}}\\d_{50}{\text{ (mm)}}\\E_{p}{\text{ (mm)}}\\I\\\end{array}}{\begin{array}{ccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{M,UF})_{11}{\text{ (t/h)}}&\dots &(Q_{M,UF})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{M,UF})_{n1}{\text{ (t/h)}}&\dots &(Q_{M,UF})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{M,OF})_{11}{\text{ (t/h)}}&\dots &(Q_{M,OF})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{M,OF})_{n1}{\text{ (t/h)}}&\dots &(Q_{M,OF})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(P_{UF})_{1}{\text{ (mm)}}\\\vdots \\(P_{UF})_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(P_{OF})_{1}{\text{ (mm)}}\\\vdots \\(P_{OF})_{n}{\text{ (mm)}}\end{bmatrix}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• $Iterations$ is the number of iterations required to refine the interdependent values of $R_{f}$ and $y$
• $(Q_{M,OF})_{Liquids}$ is the mass flow rate of liquids to the overflow stream (t/h)
• $(Q_{M,UF})_{Liquids}$ is the mass flow rate of liquids to the underflow stream (t/h)
• $Q_{M,UF}$ is mass flow rate of solids to the underflow stream (t/h)
• $Q_{M,OF}$ is mass flow rate of solids to the overflow stream (t/h)
• $P_{UF}$ is partition fraction of feed solids to the underflow stream (frac)
• $P_{OF}$ 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.

ScdMD*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 feed material will pass directly to the overflow stream.
Method	User defined	The partition to underflow for each size interval is defined by the user. Different values can be used for different solids.
	Whiten-Beta	The partition to underflow 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.
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.
RopingCalcs	CheckBox	Show addition calculations that predict the onset of cyclone underflow roping.
SizeForPassingFracCalc	Input (global)	Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc	Input (global)	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		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, $\alpha$, 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		Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information		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		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.

See also

• Hydrocyclone (Nageswararao)
• Hydrocyclone (Narasimha-Mainza)