View source for 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).{{Plitt (1976)}}{{Flintoff et al. (1987)}}

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

<blockquote>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.</blockquote>

== Model theory ==

[[File:HydrocyclonePlitt1.png|thumb|375px|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:

:<math>y_i = y'_i + R_f(1 - y'_i)</math>

where:
* <math>i</math> is the index of the size interval, <math>i = \{1,2,\dots,n\}</math>, <math>n</math> is the number of size intervals
* <math>y_i</math> is the ''uncorrected'', i.e. actual, partition to underflow of particles in size interval <math>i</math> (frac)
* <math>y'_i</math> is the ''corrected'' partition to underflow of particles in size interval <math>i</math> (frac)(frac), i.e. excluding the fines bypass effect
* <math>R_f</math> 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:

:<math>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}
</math>

where:
* <math>\bar d_{i}</math> is the [[Conversions|geometric mean size]] of particles in size interval <math>i</math> (mm)
* <math>d_{50}</math> is the size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
* <math>m</math> and <math>\alpha</math> are parameters which quantify the ''sharpness of separation'' of the Rosin-Rammler and Lynch partition curves, respectively

The corrected cut size, <math>d_{50c}</math> (μm), and partition curve shape parameters, <math>m</math> and <math>\alpha</math>, are calculated with:{{Nageswararao et al. (2004)}}

:<math>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}</math>

:<math>m = F_2 1.94 \left ( \dfrac{{D_c}^2h}{Q} \right )^{0.15} \exp \left (\dfrac{-1.58 S}{1 + S} \right )</math>

:<math>\alpha = 1.54 m - 0.47</math>

:<math>P = F_3 \dfrac{1.88 Q^{1.8} \exp (0.0055 C_V)}{{D_c}^{0.37} {D_i}^{0.94} h^{0.28} ({D_u}^2 + {D_o}^2)^{0.87}}</math>

:<math>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 (0.0054 C_V)}{{D_c}^{1.1} P^{0.24}}</math>

where:
* <math>F_1</math> - <math>F_4</math> are calibration factors which should be fitted from operating data
* <math>P</math> is the cyclone feed pressure (kPa)
* <math>S</math> is the ratio of underflow to overflow volumetric flow rates (frac)
* <math>D_c</math> is diameter of the cyclone (cm)
* <math>D_i</math> is diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet (cm)
* <math>D_o</math> is diameter of the vortex finder (overflow) (cm)
* <math>D_u</math> is diameter of the apex/spigot (underflow) (cm)
* <math>h</math> is the free vortex height (cm)
* <math>\eta</math> is the liquid viscosity (cP)
* <math>C_V</math> is the volume fraction of solids in the feed, in percentage units (%)
* <math>Q</math> is the volumetric feed rate to the cyclone (L/min)
* <math>\rho_s</math> is density of solids in the feed (t/m<sup>3</sup>)
* <math>\rho_L</math> is density of liquids in the feed (t/m<sup>3</sup>){{Gupta and Yan (2016)}}
* <math>\rho_p</math> is density of the feed slurry (t/m<sup>3</sup>)
* <math>k</math> 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 (<math>R_f</math>) is computed from:

:<math>R_f = \dfrac{R_v - C_V R_s}{1 - C_V R_s}</math>

where <math>R_v</math> is the volumetric recovery of pulp to the underflow (frac):

:<math>R_v = \dfrac{S}{1+S}</math>

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

:<math>R_s = \sum\limits_{i=1}^n{f_i y_i}</math>

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

=== Partition metrics ===

{{Model theory (Text, Hydrocyclone, Partition Metrics)}}

=== Roping ===

{{Model theory (Text, Hydrocyclone, Roping)}}

== Excel ==

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

<syntaxhighlight lang="vb">=mdUnit_Hydrocyclone_Plitt(Parameters as Range, Size as Range, Feed as Range, OreSG as Range)</syntaxhighlight>

{{Excel (Text, Help, No Arguments)}}

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>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^{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}

</math>


where:
* <math>\text{Partition method}</math> is the partition curve function to apply, ''0 = Rosin-Rammler, 1 = Lynch''
* <math>\text{Number of cyclones}</math> is the number of cyclones ''operating'' in a cluster. Model calculations are scaled per cyclone.
* <math>m</math> is the number of ore types
* <math>Q_m^F</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>Q_m^{Liquids}</math> 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:

:<math>
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^{LiqOF}\text{ (t/h)}\\
Q_m^{LiqUF}\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}
</math>


where:
* <math>Iterations</math> is the number of iterations required to refine the interdependent values of <math>R_f</math> and <math>y</math>
* <math>Q_m^{LiqOF}</math> is the mass flow rate of liquids to the overflow stream (t/h)
* <math>Q_m^{LiqUF}</math> is the mass flow rate of liquids to the underflow stream (t/h)
* <math>Q_m^{UF}</math> is mass flow rate of solids to the underflow stream (t/h)
* <math>Q_m^{OF}</math> is mass flow rate of solids to the overflow stream (t/h)
* <math>P^{UF}</math> is partition fraction of feed solids to the underflow stream (frac)
* <math>P^{OF}</math> 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.

{|
|- style="vertical-align:top;"
| [[File:HydrocyclonePlitt2.png|left|frame|Figure 2. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]|| [[File:HydrocyclonePlitt3.png|left|frame|Figure 3. Example showing the selection of the '''Size''' (red frame), '''Feed''' (purple frame) and '''OreSG''' (green frame) arrays in Excel.]] ||  [[File:HydrocyclonePlitt4.png|left|frame|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.

{{SysCAD (Page, Hydrocyclone, ScdMD*Hydrocyclone)}}

==== Cyclone page ====

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

{{SysCAD_Table_Header}}

|-
! colspan="3" style="text-align:left;" |''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.
|-
! colspan="3" style="text-align:left;" |''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.
|- style="vertical-align:top;"
|rowspan="2" | PartitionMethod
|Rosin-Rammler
|The Rosin-Rammler partition equation is applied to classification by size.
|- style="vertical-align:top;"
|Lynch
|The Lynch partition equation is applied to classification by size.

{{SysCAD (Text, Hydrocyclone, Liquids)}}

|-
! colspan="3" style="text-align:left;" |''Results''
|-
|Iterations
|style="background: #eaecf0" | Display
|Number of iterations required to stabilise partition calculation.
|-
|ClusterQv
|style="background: #eaecf0" | Display
|Volumetric flow rate of feed to the cyclone cluster (i.e. total flow).
|-
|CycloneQv
|style="background: #eaecf0" | Display
|Volumetric flow rate of feed to each cyclone in the cluster (i.e. per cyclone flow).
|-
|Cv
|style="background: #eaecf0" | Display
|Volume fraction of solids in feed stream.
|-
|eta
|style="background: #eaecf0" | Display
|Liquid viscosity (SysCAD property).
|-
|d50c
|style="background: #eaecf0" | Display
|Value of the corrected ''d<sub>50c</sub>''.
|-
|DeltaP
|style="background: #eaecf0" | Display
|Value of the pressure drop across the cyclone, ''P''.
|-
|S
|style="background: #eaecf0" | Display
|Value of the volumetric flow split ratio, ''S''.
|-
|Rv
|style="background: #eaecf0" | Display
|Fraction of feed slurry volume recovered to underflow stream.
|-
|m
|style="background: #eaecf0" | Display
|Value of the sharpness of separation, ''m'', used in the Rosin-Rammler partition function.
|-
|alpha
|style="background: #eaecf0" | Display
|Value of the sharpness of separation, <math>\alpha</math>, used in the Lynch partition function.
|-
|Rs
|style="background: #eaecf0" | Display
|The fraction of solids in the feed recovered to the underflow stream.
|-
|Rf
|style="background: #eaecf0" | Display
|The fraction of feed liquids recovered to the underflow, ''R<sub>f</sub>'', computed by the Plitt model.

{{SysCAD (Text, Hydrocyclone, Partition Metrics)}}

|}

{{SysCAD (Page, Hydrocyclone, Partition)}}

{{SysCAD (Page, Hydrocyclone, Roping)}}

{{SysCAD (Page, About)}}

== See also ==

* [[Hydrocyclone (Nageswararao)]]
* [[Hydrocyclone (Narasimha-Mainza)]]

== References ==

[[Category:Excel]]
[[Category:SysCAD]]