# Cyclone

Navigation: Main Page -> Models -> Size Distribution Models

## General Description

This model is used to simulate either a single hydrocyclone, or a cluster of hydrocyclones. The main requirement for using this model is that the feed must contain solids with size distribution information. The model will calculate the split of solids between the under and over flows based on the cut point (either calculated or defined) and the size distribution of the feed stream.

The unit allows the user to define the cyclone in one of four ways:

1. Define the cut point of the cyclone
2. Use the Krebs method to calculate the cut point
3. Use the Plitt method to calculate the cut point
4. Define the partition curve of the cyclone

The first option allows the user to set the cut point without any further knowledge of the cyclone. The second and third options require the user to define the cyclone dimensions. These dimensions are then used to calculate the cut point and partition curve of the cyclone.

In the case of the Plitt method, the model will also calculate the liquid split between the over and under flows.

Notes:

1. In SysCAD 9.3 this model has been replaced by the Hydrocyclone model.
2. This unit cannot be used to model a gas cyclone.

## Diagram

The diagram shows the default drawing of the cyclone, with all of the streams that are available for operation of the unit. The physical location of the streams connecting to the cyclone is unimportant. The user may connect the streams to any position on the unit.

## Inputs and Outputs

 Label RequiredOptional InputOutput Number of Connections Description Min Max. Feed Required In 1 1 The slurry feed to the cyclone OverFlow Requuired Out 1 1 The over flow from the unit UnderFlow Required Out 1 1 The under flow from the unit

## Model Theory

1. The model will simulate a hydrocyclone using one of the four user defined methods - 'Define d50', 'Efficiency Curve', 'Krebbs' and 'Plitt' These are described below.
2. The 'Efficiency Curve' method allows the user to define the fraction of material in each size interval that will report to the cyclone overflow.
3. The other three methods use the d50 value, either calculated or defined by the user, to determine the solids separation. The value is defined as the particle size that has a 50% chance of reporting to the cyclone underflow. The majority of the particles finer than this size will report to the overflow, while the majority of those coarser will report to the cyclone underflow.
4. The user may also define the number of cyclones in the cluster. The default value, and the minimum number, is 1. This allows the model to calculate the pressure drop across each cyclone by dividing the flow to the model by the number of cyclones in the cluster. This allows the user to model a number of cyclones with a single drawing.

### Efficiency Curve

The user defines the Efficiency Curve of the cyclone by entering the fraction of material in each size interval that will report to the cyclone overflow. This will normally be the Actual efficiency curve, which takes into account the bypassing of fines to the underflow with the liquid. An example of an Efficiency curve is shown below:

The user also defines the fraction of solids in the Cyclone underflow, as this allows the model to calculate the liquid split across the cyclone.

### Define Cut Point

The user defines the cut point of the cyclone, the d50, the sharpness $\alpha$ and the required percentage solids in the under flow from the unit.

The model then uses the d50 defined by the user to calculate the fraction of solids in each size interval reporting to the underflow using the following equations:

$\mathbf{\mathit{x=\frac{Particle_-diameter}{d_{50}}}}$
where Particle_diameter - geometric mean of the size interval
$y^1=\frac{e^{\alpha x}-1}{e^{\alpha x} + e^\alpha-2}$
 where: y1 = recovery to underflow on a corrected basis
$\mathbf{\mathit{ y=y^1 +R_f (1-y^1)}}$
 where: y = actual recovery to underflow Rf = fraction of feed liquid reporting to the underflow product

### Krebs Model

The user defines the cyclone diameter, the percentage solids in the cyclone under flow and a correction factor for the cyclone geometry, if necessary.

The model calculates the d50 of the cyclone from the following equations1:

(1) $\mathbf{\mathit{d_{50}=d_{50}(Base)\times C_1\times C_2\times C_3\times Factor}}$

where Factor is a correction factor to account for differences in the cyclone geometry from the 'optimum' dimensions.

(2) $\mathbf{\mathit{d_{50}(Base)=2.84\times D_c^{0.66}}}$

 where: d50(Base) = the cut size a "standard cyclone" can achieve, in microns. Dc = cyclone diameter, in cm

(3) $\mathbf{\mathit{C_1=\begin{pmatrix}\frac{53-C_v}{53}\end{pmatrix}^{-1.43}}}$

 where: C1 = Correction for the influence of cyclone feed concentration Cv = Feed solids volume concentration (%)

(4) $\mathbf{\mathit{C_2=3.27\times \boldsymbol{\Delta}P^{-0.28}}}$

 where: C2 = Correction for the influence of pressure drop DP = Pressure drop across the cyclone in kPa - calculated in (5) using Plitt method

(5) $\mathbf{\mathit{\boldsymbol{\Delta}P=\frac{1.88\times Q^{1.78}\times exp(0.0055\times C_v)}{D_c^{0.37}\times D_i^{0.94}\times h^{0.28}\times (D_u^2+D_o^2)^{0.87}}}}$

 where: Q = feed slurry volumetric flow-rate per cyclone in l/min Cv = Feed solids volume concentration (%) DC = Diameter of cyclone DI = Diameter of cyclone inlet = 0.2 * DC DU = Diameter of cyclone underflow, or apex = 0.15 * DC DO = Diameter of cyclone overflow, or vortex finder = 0.3 * DC h = Free vortex height of cyclone = 1.5 * Dc
 All the cyclone dimensions are in cm. These values are for an 'Optimum Cyclone' 2 .

(6) $\mathbf{\mathit{C_3=\begin{pmatrix}\frac{1.65}{G_S-G_L}\end{pmatrix}^{0.5}}}$

 where: C3 = Correction for the influence of SG GS = SG of solids GL = SG of liquids
 The model then uses the d50, either defined as in 4.1 or calculated in 4.2, to calculate the fraction of solids in each size range using the following equations 2,3:
$\mathbf{\mathit{x=\frac{Particle_-diameter}{d_{50}}}}$
where Particle_diameter - geometric mean of the size interval
$\mathbf{\mathit{y^1=\frac{exp(4x)-1}{exp(4x)+exp(4)-2}}}$
 where: y = recovery to underflow on a corrected basis
$\mathbf{\mathit{ y=y^1 +R_f (1-y^1)}}$
 where: y = actual recovery to underflow Rf = fraction of feed liquid reporting to the underflow product

#### Assumptions (Krebs)

1. The cyclone has 'optimum' dimensions, as given in equation 5 above.
2. The mass weighted mean of the solids density is used to determine the cut point.

### Plitt Model

The user defines all of the cyclone dimensions and the model then calculates the d50 in microns, the pressure drop across the cyclone and the liquid distribution between the over and under flows.

The following equations are used to determine the cyclone operation3:

(7) d50 Calculation

$\mathbf{\mathit{d_{50}=Factor*\left\{\frac{50.5*D_c^{0.46}*D_i^{0.6}*D_o^{1.21}*exp(0.063*C_v)}{D_u^{0.71}*h^{0.38}*Q^{0.45}*(\rho_s-\rho_l)^{0.5}}\right\}}}$

where d50 is in microns

Dc- Cyclone diameter, cm
Di - Cyclone feed inlet diameter, cm
Do- Cyclone overflow or Vortex finder diameter, cm
Du- Cyclone underflow or Apex diameter, cm
h - Free vortex height in cyclone, cm
Q - Volumetric flow rate of each cyclone feed at temperature,l/min
Cv - Volumetric percent of solids in feed slurry at temperature
$\mathbf{\mathit{\rho_s}}$ - Solids density at Temperature (t/m3)
$\mathbf{\mathit{\rho_l}}$ - Liquid density at Temperature (t/m3)
Factor - d50 correction factor.

(8) Pressure Drop across the Cyclone in kPa

$\mathbf{\mathit{ \Delta P =PressFactor*\frac{1.88*Q^{1.78}*exp(0.0055*C_v)} {D_{c}^{0.37}*D_{i}^{0.94}*h^{0.28}*(D_u^2+D_o^2)^{0.87} }}}$
where PressFactor - Pressure drop correction factor (This is also named Pressure_Factor3 in SysCAD)

(9) Pressure drop across cyclone, in metres of feed slurry

$\mathbf{\mathit{H=\frac{\Delta P}{gravity*\rho_{Feed}}}}$

(10) Recovery of feed volume to the underflow product

$\mathbf{\mathit{R_v=\frac{S}{S+1}}}$
where
$\mathbf{\mathit{S = SFactor* \frac{1.9*(\frac{D_u}{D_o})^{3.31}*h^{0.54}*(D_u^2+D_o^2)^{0.36}*exp(0.0054*C_v)} {H^{0.24}*D_c^{1.11}}}}$
where SFactor - S correction factor (This is also named Sharp_Factor4 in SysCAD)

(11) Sharpness Separation

$\mathbf{\mathit{m=SharpFactor*1.94*exp\left[(-1.58*R_v)*\left(\frac{D_c^2*h}{Q} \right)^{0.15}\right]}}$

where: Rv - Recovery of feed volume to the underflow product

SharpFactor - Sharpness correction factor (This is also named Sharp_Factor2 in SysCAD)
Q - Volumetric flow rate of each cyclone feed at temperature, l/min

The user may choose either the Lynch or the Rosin-Rammler equation to calculate the solids size distribution across the cyclone.

Lynch

(12) Recovery to underflow on a corrected basis for the size interval

$\mathbf{\mathit{y_i'=\frac{exp\left(a\frac{d_i}{d_{50}}\right)-1}{exp\left(a\frac{d_i}{d_{50}}\right)+exp(a)-2}}}$

where: di - geometric mean of the size interval

a = 1.54 * m - 0.47
m - measure of the sharpness of separation

Rosin-Rammler

(13) Recovery to underflow on a corrected basis for the size interval

$\mathbf{\mathit{y_i'=1-\exp\left(-0.693147\times\left(\frac{d_i}{d_{50}} \right)^m\right)}}$

where di - geometric mean of the size interval

m - measure of the sharpness of separation.

The actual recovery to the underflow, y, is then calculated using the same equation for both methods:

$\mathbf{\mathit{y=y'+R_f(1-y')}}$

where Rf = fraction of feed liquid reporting to the underflow product

#### Assumptions (Plitt)

1. The pressure drop equation assumes free discharge from both the under and over flows from the cyclone.
2. The mass weighted mean of the solids density is used to determine the cut point.
3. Q in the sharpness equation is the total volumetric flow into the cyclones.

### References

R.A.Arterburn, The Sizing and Selection of Hydrocyclones, Metallurgical Handbook.

D.T.Tarr, Practical application of liquid cyclones in mineral dressing problems, Krebs Engineers documentation. October 1965

L.R.Plitt, A mathematical model of the hydrocyclone classifier, CIM Bulletin, December 1976

T.P.Napier-Munn, Mineral Comminution Circuits: Their Operation and Optimisation, 1999

## Data Sections

The default sections and variable names are described in detail in the following tables. The default Cyclone access window consists of 4 sections. This number may increase or decrease, based on user configuration.

### Summary of Data Sections

1. Cyclone-1 tab - Contains general information relating to the unit.
2. PartCrv tab - This second tab displays the partition curve data for the cyclone.
3. Info tab - Contains general settings for the unit and allows the user to include documentation about the unit and create Hyperlinks to external documents.
4. Links tab, contains a summary table for all the input and output streams.
5. Audit tab - Contains summary information required for Mass and Energy balance. See Model Examples for enthalpy calculation Examples.

### Cyclone Page

Class: Cyclone-1 - The first tab page in the access window will have this name.

Symbol / Tag

Input / Calc

Description/Calculated Variables / Options

### Requirements

SplitMethod / Method DefineCutSize The user only needs to specify the actual d50 of the cyclone and the percent solids in the underflow.
Krebs The unit uses the Krebs model to calculate the operating parameters of the cyclone. The user specifies the cyclone diameter and the percent solids in the underflow. The model calculates the d50 of the cyclone.
Plitt The unit uses the Plitt model to calculate the operating parameters of the cyclone. The user specifies the all of the physical parameters of the cyclone. The model calculates the d50 of the cyclone and the liquid split to the under and over flows.
Efficiency Curve The unit uses a user specified Efficiency Curve (or partition curve) to calculate the separation of the solids. The user also specifies the fraction of solids in the Cyclone underflow. The model calculates the Actual d50 of the cyclone and the liquid split to the under and over flows.

No_Of_cyclones / NumCyclones

Input

The number of cyclones in the bank. This allows the user to emulate a bank of cyclones in a single unit.

The parameters in the access window change depending on the cyclone model chosen by the user. The input parameters for each model will be detailed separately below. Where a model has output parameters that are only displayed for that particular model, these are also shown individually.

### Define Cut Size

Cut_Size / CutSize Input The required d50 of the cyclone.
UnderFlowSolids / UFSolids Input The percentage of solids in the cyclone underflow.
Sharpness / Alpha Input The required sharpness (default 4.0).

### Krebs

CycloneDiam / Diam Input The diameter of the cyclone.
CorrectionFactor / Factor Input This is the correction factor to account for differences in the cyclone geometry from the 'optimum' dimensions. This is used to adjust the calculated cut point of the cyclone. The default value is 1. (See Equation (1) in the Krebs theory)
UnderFlowSolids / UFSolids Input The percentage of solids in the cyclone underflow.
OldFeedConcCalc Input Cv (Feed solids volume concentration) in calculation of correction factor C1 was incorrect by a factor of 100. Calculation has been corrected, but for backward compatibility of tuned existing projects the old (incorrect) calculation can be used if this option is selected. C1 in turn effects d50 calculation which drives the separation. Option is new to Build 135.19957 and Build 136.19935.

Results:

Calc_d50 Output The calculated d50 of the cyclone.
InterSectPt Output The calculated intersection point of the cyclone. This is the size at which the cumulative percentage oversize of the underflow is equal to the cumulative percentage undersize in the overflow.
Pressure_Drop / CalcDP Output The calculated pressure drop across the cyclone.

### Plitt

Cyclone_Diameter / Dc Input The diameter of the cyclone.
Inlet_Diameter / Di Input The inside diameter of the cyclone feed inlet.
Vortex_Diameter / Do Input The inside diameter of the overflow or the vortex finder.
Spigot_Diameter / Du Input The inside diameter of the underflow or the spigot diameter.
Height / h Input The free vortex height of the cyclone, defined as the distance from the bottom of the vortex finder to the top of the underflow orifice.

h = Total length of the cyclone - vortex length

ApplyScale Input This is used as a "once off" factor to scale all of the values in "Dc", "Di", "Do", "Du" and "h" as a group. The user types a value into ApplyScale and SysCAD will scale the values in "Dc", "Di", "Do", "Du" and "h" by that value and then reset ApplyScale to "*".
SharpEqn Rosin-Rammler The method by which the model calculates the sharpness of separation. Please see Equation (13) in the Plitt Model Theory.
Lynch The method by which the model calculates the sharpness of separation. The Lynch method will give a slightly more sharp separation than the Rosin-Rammler method. Please see Equation (12) in the Plitt Model Theory.
OldLiqSplitCalc Tick Box This is only available in builds 134.11986 and later. Prior to this build the liquid split calculation in the Plitt model was incorrect. This has now been corrected. However, for backward compatibility, users may enable the old (incorrect) liquid split calculation.
Note: In new projects this will be disabled by default. In old projects it will be enabled, and hence users will not see a change in their results, unless they un-tick this box.
Factors
d50_Factor1 / Factor Input This Factor is used to adjust the d50 of the cyclone. Increasing this value will increase the d50 used in the calculations, and hence increase the amount of solids reporting to the cyclone Overflow. Please see Equation (7) in the Plitt Model Theory.
Sharp_Factor2 / SharpFactor Input Sharpness Factor is used to adjust "m". Increasing this value will increase the sharpness of separation and more of the coarse particles will report to the underflow. Please see Equation (11) in the Plitt Model Theory.
Pressure_Factor3 / PressFactor Input Press Factor is used to adjust "dP". Increasing this value will increase the pressure drop across the cyclone and hence change the size recovery of the cyclone. Please see Equation (8) in the Plitt Model Theory.
S_Factor4 / SFactor Input SFactor is used to adjust "S", which influences the recovery of feed volume to the cyclone underflow. Increasing this value will increase the recovery to the cyclone underflow. Please see Equation (10) in the Plitt Model Theory.
OFLiq_Factor5 / OFLiqCorrFac Input Overflow liquid correction factor, used to adjust the liquid split to the overflow.

Liquid to Overflow = OFLiqCorrFac * Calculated Liquid to Overflow.

Results:

d50 Output The calculated d50 of the cyclone.
PressDrop Output The calculated pressure drop across the cyclone.
S Output The calculated ratio of the underflow volumetric flow to the overflow volumetric flow. Please see Equation (10) in the Plitt Model Theory.
Rv Output Recovery of feed volume to the underflow.
Sharpness / m Output The efficiency exponent in efficiency curve. This is a measure of the sharpness of separation of the cyclone.
Alpha Output The alpha term calculated when using the Lynch sharpness equation.
Rs Output Recovery of feed solids to the underflow.
Rf Output Recovery of feed liquid to the underflow.

### Efficiency Curve

The user will also see a tab 'EffCrv' where they must enter the fraction of each size interval reporting to the Cyclone overflow. An example of an Efficiency Curve is shown in Efficiency Curve Theory.

UnderFlowSolids / UFSolids Input The percentage of solids in the cyclone underflow.

### Common Data Fields

QvPerCyclone Output The feed volume flow per cyclone (equal to total feed volume flow if NumCyclones is equal to 1).
SolidsVolFrac / VolFrac Output The volumetric fraction of solids in the cyclone feed at feed conditions.
UnderFlow_Solids / UFSolids Output The fraction of solids in the cyclone underflow stream.
OverFlow_Solids / OFSolids Output The fraction of solids in the cyclone overflow stream.
UnderFlow_SolidsConc@25 / UFSolConc25 Output The solids concentration at 25&degC in the cyclone underflow stream.
OverFlow_SolidsConc@25 / OFSolConc25 Output The solids concentration at 25&degC in the cyclone overflow stream.
UnderFlow_SolidsRecovery / UFSolRec Output The fraction of feed solids that reports to the cyclone underflow stream.
OF_Liq_Split / OFLiqSplit Output The fraction of feed liquids that reports to the cyclone overflow stream.
UF_Liq_Split / UFLiqSplit Output The fraction of feed liquids that reports to the cyclone underflow stream.
OF_Density / OFDens Output The overflow slurry density
UF_Density / UFDens Output The underflow slurry density
Options
ShowQFeed Tickbox Show the total feed to the cyclone.

### Partition Curve Section

The Cyclone has a section showing the partition curve. The model will calculate the partition curve for the Cyclone configuration and display it on this page.

 Tag / Symbol Input or Calc Description The black rectangle at the top of the page is for display purposes only. The two white boxes below the window allow the user to configure the lower and upper display limits of the curve. Log > Lin Button This button toggles the x display between log and ln. Auto > Fixed Scale Button This button toggles the display between auto scaling and user defined fixed scale. Stepped > Smooth Button Toggles the display between discrete points (Stepped) and a smooth curve. Size Intervals Button > Ascending - Clicking on this displays the values from small to large. > Descending - Clicking on this displays the values from large to small. Graph : On > Off Button Toggles the graphical display on and off.

## Adding this Model to a Project

Insert into Configuration file

Sort either by DLL or Group.

 DLL: Separ2.dll → Units/Links → Size Separation: HydroCyclone (old) OR Group: Size Distribution → Units/Links → Size Separation: HydroCyclone (old)