Cyclone

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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 three ways:

a) Define the cut point of the cyclone

b) Use the Krebs method to calculate the cut point

c) Use the Plitt method to calculate the cut point.

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 of the cyclone.

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

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	Input / Output	No. of Connections		Description
		Min	Max.
Feed	In	1	1	The slurry feed to the cyclone
OverFlow	Out	1	1	The over flow from the unit
UnderFlow	Out	1	1	The under flow from the unit

Model Theory

The model will simulate a cyclone using one of the three user defined methods. All of the models use the d₅₀ value 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 undersize.

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.

Define Cut Point

This is the simplest of the three methods. The user defines the cut point of the cyclone, the d₅₀, and the required percentage solids in the under flow from the unit.

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) _{File:Models-Cyclone-Image002.gif}

[math]\displaystyle{ \mathbf{\mathit{d_{50}=d_{50}(Base)*C_1*C_2*C_3*CorrectionFactor}} }[/math]

(2) _{File:Models-Cyclone-Image007.gif}

[math]\displaystyle{ \mathbf{\mathit{d_{50}(Base)=2.84*D_c^{0.66}}} }[/math]

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

(3) _{File:Models-Cyclone-Image006.gif}

[math]\displaystyle{ \mathbf{\mathit{C_1=\begin{pmatrix}\frac{53-V}{53}\end{pmatrix}^{1.43}}} }[/math]

where:	C1	= Correction for the influence of cyclone feed concentration
	V	= Percent solids by volume in cyclone feed

(4) _{File:Models-Cyclone-Image008.gif}

[math]\displaystyle{ \mathbf{\mathit{C_1=3.27*\boldsymbol{\Delta}P^{0.28}}} }[/math]

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

(5) _{File:Models-Cyclone-Image010.gif}

[math]\displaystyle{ \mathbf{\mathit{\boldsymbol{\Delta}P=\frac{1.88*Q^{1.78}*exp(0.005*\boldsymbol{\phi})}{D_c^{0.37}*D_i^{0.94}*h^{0.28}*(D_u^2+D_o^2)^{0.87}}}} }[/math]

where:	Q	= feed slurry volumetric flow-rate per cyclone in l/min
	f	= volumetric percent solids in cyclone feed
	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) _{File:Models-Cyclone-Image012.gif}

[math]\displaystyle{ \mathbf{\mathit{c_3=\begin{pmatrix}\frac{1.65}{G_S=G_L}\end{pmatrix}^0.5}} }[/math]

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 :

File:Models-Cyclone-Image014.gif

[math]\displaystyle{ \mathbf{\mathit{x=\frac{Particle_-diameter}{d_{50}}}} }[/math]

File:Models-Cyclone-Image016.gif

[math]\displaystyle{ \mathbf{\mathit{y^1=\frac{exp(4*x)-1}{exp(4*x)+exp(4)-2}}} }[/math]

where:

y

= recovery to underflow on a corrected basis

File:Models-Cyclone-Image018.gif

where:	y	= actual recovery to underflow
	Rf	= fraction of feed liquid reporting to the underflow product

Assumptions (Krebs)

The cyclone has 'optimum' dimensions, as given in equation 5 above.

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, 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:

NOTE The Plitt cyclone equations were updated with corrected data on March and July 2006, if users have tuned their cyclone models with existing data, they should then use the models with Oldcalcs tick box selected. For any new cyclone models, it is recommended to have the oldcalcs option unticked.

(7) d50 Calculation

_{File:Models-Cyclone-Image028.gif}

[math]\displaystyle{ \mathbf{\mathit{d_{50}=Factor*\begin{Bmatrix}\frac{50.5*D_c^{0.46}*D_i^{0.6}*D_o^{1.21}*exp(0.063*\boldsymbol{\phi})}{D_n^{0.71}*h^{0.38}*(\boldsymbol{\rho}_s-\boldsymbol{\rho}_l)^{0.5}}\end{Bmatrix}}} }[/math]

where D_c- Cyclone diameter, cm

D_i - Cyclone feed inlet diameter, cm

D_o- Cyclone overflow or Vortex finder diameter, cm

D_u- 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

F - Volumetric percent of solids in feed slurry at temperature

r_s - Solids density at Temperature (t/m³)

r_l - Liquid density at Temperature (t/m³)

Factor - d₅₀ correction factor (This is also named d50_Factor1 in SysCAD)

(8) Pressure Drop across the Cyclone in kPa

_{File:Models-Cyclone-Image005.gif}

where PressFactor Pressure drop correction factor (This is also named Pressure_Factor3 in SysCAD)

(9) Recovery of feed volume to the underflow product

R_v = S/(S+1) - Recovery of feed volume to the underflow product

where SFactor - S correction factor (This is also named Sharp_Factor4 in SysCAD)

File:Models-Cyclone-Image025.gif - Pressure drop across cyclone, in metres of feed slurry

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

(10) Sharpness Separation

_{File:Models-Cyclone-Image009.gif}

where: R_v - 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

Lynch

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

_{File:Models-Cyclone-Image053.gif}

where: d_i - geometric mean of the size interval

a = 1.54 * m - 0.47

m - measure of the sharpness of separation

Rosin-Rammler

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

_{File:Models-Cyclone-Image056.gif}

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:

_{File:Models-Cyclone-Image018.gif}

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

OldCalcs Option:

This option is available to users only for backward compatibility purposes. We have listed below the old equations and old definitions of the variables to let users know what has been changed / corrected. As a recommendation, we suggest users to use the corrected equations for any new cyclone models and only use the oldcalcs for any tuned existing models.

The following equations and variable definitions are different when OldCalcs option is selected:

(a) Equations used for d50 and m are:

_{File:Models-Cyclone-Image059.gif}

_{File:Models-Cyclone-Image061.gif}

For descriptions of the variables used in the equations, please refer to equation (7) and (11). Note that

Q in the sharpness equation is the total volumetric flow into the cyclones.

(b) F is Volumetric percent of solids in feed slurry at reference temperature which is normally 25dC.

(c) di used in the Lynch and Rosin-Rammler equation is the top size of the size interval.

Assumptions (Plitt)

The pressure drop equation assumes free discharge from both the under and over flows from the cyclone.

The mass weighted mean of the solids density is used to determine the cut point.

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.

Data Sections

The default sections and variable names are described in detail in the following tables. The default cyclone access window consists of 2 section. This section, which has the same name as the cyclone tag, contains all of the information relating to the unit. The last tab Audit is fully described in Audit Section. See Model Examples for enthalpy calculation Examples.

First Section- Tag Name

Tag / Symbol	Input or Calc	Description
Common First Data Section
Method?	Input	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 underflow solids. 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.

The parameters in the access window change depending on the cyclone model chosen by the user. The parameters for each model will be detailed separately below:

Define Cut Size

Tag / Symbol	Input / Calc	Description
CutSize	Input	The required d50 of the cyclone.
UFSolids	Input	The percentage of solids in the cyclone underflow.
DP	Input	The pressure drop across the cyclone. This parameter is only visible for dynamic simulation.
Z_Rqd	Input	The required relative datum of the unit.
Z	Output	The actual relative datum of the unit. These two parameters are only visible in the dynamic mode.
OFLiqSplit	Output	The percent of feed liquids reporting to overflow.
UFLiqSplit	Output	The percent of feed liquids reporting to underflow.
OFDens	Output	The slurry density of the overflow
UFDens	Output	The slurry density of the underflow

Krebs

Tag / Symbol	Input / Calc	Description
Diam	Input	The diameter of the cyclone.
Factor	Input	A factor, which can be used to adjust the calculated, cut point of the cyclone. The default value is 1. (See equation (1) in the Krebs theory)
UFSolids	Input	The percentage of solids in the cyclone underflow.
OFLiqCorrFac	Input	Overflow Liquid Correction Factor
Z_Rqd	Input	The required relative datum of the unit.
Z	Output	The actual relative datum of the unit. These two parameters are only visible in the dynamic mode.
Results:
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.
CalcDP	Output	The calculated pressure drop across the cyclone.
OFLiqSplit	Output	The percent of feed liquids reporting to overflow.
UFLiqSplit	Output	The percent of feed liquids reporting to underflow.
OFDens	Output	The slurry density of the overflow
UFDens	Output	The slurry density of the underflow

Plitt

Tag/Symbol	Input / Calc	Description
Dc	Input	The diameter of the cyclone.
Di	Input	The inside diameter of the cyclone inlet.
Do	Input	The inside diameter of the overflow or the vortex finder.
Du	Input	The inside diameter of the underflow or the spigot diameter.
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.
Scale	Input	This is used as a "once off" factor to adjust "Dc", "Di", "Do", "Du" and "h" as a group. If user prefers to specify "Dc", "Di", "Do", "Du" and "h" separately, simply type in a number in the appropriate field and "Scale" will be reset to "*" denoting that it is not in use.
OldCalcs	tickbox	Some corrections were made to the Plitt model equations in March 2006. This option is mainly for backward compatibility purposes. For changes please refer to the model theory section. Users already using the Plitt model might have tuned their model using tuning factors to fit plant data, in that case, they should select the oldcalcs tick box to keep the tuned model. For any new models, it is best to have this option unticked.
Factor / d50_Factor1	Input	Factor is used to adjust "d50". [d50 used = Factor * d50 calc]
SharpFactor / SharpFactor2	Input	Sharpness Factor is used to adjust "m". [m used = SharpFactor * m calc]
PressFactor / Pressure_Factor3	Input	Press Factor is used to adjust "dP". [dP used = PressFactor * dP calc]
SFactor / S_Factor4	Input	SFactor is used to adjust "S". [S used = SFactor * S calc]
OFLiqCorrFac / OFLiq_Factor5	Input	Overflow liquid correction factor, used to adjust the liquid split to the overflow by the specified factor.
SharpEqn	Input	Rosin-Rammler - The method by which the model calculates the sharpness of separation. See equations 11 and 12 in the Plitt theory.
		Lynch - The minimum display point on the graph
Z_Rqd	Input	The required relative datum of the unit.
Z	Output	The actual relative datum of the unit. These two parameters are only visible in the dynamic mode.
Results:
RefTemp	Output	The reference temperature used.
RefPress	Output	The reference pressure used.
QvPerCyclone	Output	Volumetric flowrate across each cyclone.
VolFrac	Output	Volumetric percent of solids in feed slurry
d50	Output	The calculated d50 of the cyclone
PressDrop	Output	The calculated pressure drop across the cyclone.
S	Output	Used to calculate the recovery of feed volume to the underflow.
Rv	Output	Recovery of feed volume to the underflow.
m	Output	This is a measure of the sharpness of separation of the cyclone.
Alpha	Output	The a term calculated using the Lynch sharpness equation.
Rs	Output	Recovery of feed solids to the underflow.
Rf	Output	Recovery of feed liquid to the underflow.
OFLiqSplit	Output	The percent of feed liquids reporting to overflow.
UFLiqSplit	Output	The percent of feed liquids reporting to underflow.
OFDens	Output	The slurry density of the overflow
UFDens	Output	The slurry density of the underflow

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 user can configure the 'Information' data to display the discharge partition curve. 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.
Mode	Passing	The partition curve is displayed as the fraction passing each size interval.
	Retained	The partition curve is displayed as the fraction retained in each size interval.
Size Intervals	Button	> Ascending - The numbers are displayed with the large sizes at the top.
		> Descending - The numbers are displayed with the smaller sizes at the top.
If the user has chosen to define the screening application using a partition curve, then they can enter both the size Fractions and the size distribution (Manual) or the size Fractions only, for the Discharge Partition Curve. The product from the Screen will have the same size distribution as the one defined here. If the user defines the size distribution manually, using only a few points, the model will use linear interpolation between the points to find the missing values. If the model is using the Karra method to calculate the products, then this page will display partition curves for each size distribution defined for the project.