View source for Gravity Classifier (Plitt-Flintoff)

== Description ==

This article describes a model of gravity classifiers proposed by Plitt and Flintoff (1985).{{Plitt and Flintoff (1985)}}{{Fuerstenau and Han (2003)}}

Gravity classifiers use the differential settling velocity of particles in a pool to separate feed pulp into coarse and fine product streams. 

Mechanical gravity classifiers use a mechanical action to both agitate pulp in the pool and convey the gravity-settled coarse particles away from the machine and the remaining fine particles in the overflow. Examples of mechanical gravity classifiers include ''spiral'', ''rake'' and ''bowl'' machines.

Non-mechanical gravity classifiers rely on free or hindered settling against a rising current and draw settled coarse particles through an underflow spigot. Finer particles report to the overflow. Examples of non-mechanical gravity classifiers include devices variously referred to as ''hydraulic classifiers'', ''hydrospearators'', ''hydrosizers'', or ''cone classifiers''.

== Model theory ==

[[File:GravityClassifierPlittFlintoff1.png|thumb|425px|Figure 1. Example partition curve predicted by the Plitt-Flintoff gravity classifier model.]]

Plitt and Flintoff proposed a model that describes the classification of particles based on a consideration of settling velocity and flow conditions in gravity classification devices.{{Plitt and Flintoff (1985)}}

=== Classification ===

Plitt and Flintoff's formulation applies the Reid-Plitt (1971) efficiency curve to describe the classification of particles by size.{{Reid (1971)}}{{Plitt (1971)}}

{{Model theory (Text, Reid-Plitt Efficiency Curve)}}

Determination of the corrected cut size (<math>d_{\rm 50c}</math>) and water recovery (<math>R_{\rm f}</math>) terms of the Reid-Plitt equation are described below.

The value of the sharpness parameter, <math>m</math>, can be specified for a given application, or estimated as described below.

=== Corrected cut size ===

Plitt and Flintoff assume the corrected cut size (<math>d_{\rm 50c}</math>) is the size of a particle which has a settling velocity equal to the mean upward rising velocity of pulp, <math>v</math> (m/h), i.e.:

:<math>v = \dfrac{Q_{\rm o}}{A} = \dfrac{48 g {d_{\rm 50c}}^2 (\rho_{\rm S} - \rho_{\rm Eff})}{\alpha \left ( 2 {\left(\dfrac{d_{\rm 50c}}{10}\right)}^{\frac{3}{2}} \sqrt{\dfrac{100g \rho_{\rm S} \rho_{\rm Eff}}{3}} + 10\sqrt{48} \beta \mu\right )}

</math>

where:
* <math>Q_{\rm o}</math> is the volumetric flow rate of pulp in the overflow (m<sup>3</sup>/h)
* <math>A</math> is the effective settling pool area of the classifier (m<sup>2</sup>)
* <math>\rho_{\rm S}</math> is the density of solids (t/m<sup>3</sup>)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)

The effective density of slurry in the classifier, <math>\rho_{\rm Eff}</math> (t/m<sup>3</sup>), is assumed to be the mean of the feed slurry density, <math>\rho_{\rm F}</math> (t/m<sup>3</sup>), and overflow slurry density, <math>\rho_{\rm OF}</math> (t/m<sup>3</sup>), i.e.:

:<math>\rho_{\rm Eff} = \dfrac{\rho_{F} + \rho_{OF}}{2}</math>

The apparent viscosity of slurry in the classifier, <math>\mu</math> (Pa.s), is estimated by the classical Krieger-Dougherty (1959) equation:{{Krieger and Dougherty (1959)}}

:<math>\mu = \mu_{\rm 0} \left ( 1 - \dfrac{\phi_{\rm Eff}}{\phi_{\rm m}} \right)^{-k \phi_{\rm m}}</math>

where:
* <math>\mu_0</math> is the viscosity of the carrier fluid, i.e. water (Pa.s)
* <math>\phi_{\rm m}</math> is maximum packing fraction of particles (v/v), assumed to be 0.64
* <math>k</math> is a particle shape constant, assumed to be 2.5

The effective volume solids fraction in the classier, <math>\phi_{\rm Eff}</math> (v/v), is assumed to be the mean of the feed solids volume fraction, <math>\phi_{\rm F}</math> (v/v), and the overflow solids volume fraction, <math>\phi_{\rm OF}</math> (v/v), i.e.:

:<math>\phi_{\rm Eff} = \dfrac{\phi_{F} + \phi_{\rm OF}}{2}</math>

The parameters <math>\alpha</math> and <math>\beta</math> are related to the particle shape, as shown in Table 1.

:{| class="wikitable"
|+ Table 1. Particle shape types
|-
! Index !! Particle shape !! Example mineral !! <math>\boldsymbol{\alpha}</math> !! <math>\boldsymbol{\beta}</math>
|-
| 0|| Spheres|| || 0.942 || 3.03
|-
| 1|| Perfect cubes || Galena, sylvite || 1.038|| 3.01
|-
| 2|| Conchoidal fracture || Quartz|| 1.022|| 4.008
|-
| 3|| Dodecahedral || Sphalerite|| 1.05 || 5.9
|-
| 4|| Plates 10x thickness || || 2.43|| 4.7
|-
| 5|| Plates 100x thickness || Mica|| 7.69|| 12.29
|-
| 6|| Irregular fracture || Arsenopyrite|| 1.00|| 12.98
|}

The <math>d_{\rm 50c}</math> term is implicit in the settling velocity equation. Therefore a numerical root-finding algorithm is applied to approximate the value of <math>d_{\rm 50c}</math> for a given settling velocity and pulp conditions.

=== Water recovery ===

The Plitt and Flintoff model requires the volume fraction of solids in the underflow, <math>\phi_{\rm UF}</math> (v/v), to be specified as an input.

The recovery of feed water to the underflow, <math>R_{\rm f}</math>, is thus a function of the recovery of solids to the underflow, which is in turn determined by the Reid-Plitt classification equation.

The Reid-Plitt classification equation itself features the <math>R_{\rm f}</math> term, and therefore an iterative scheme is required to fully compute the Plitt and Flintoff model.

=== Sharpness estimation ===

Plitt and Flintoff suggest using the following relationship to estimate values of <math>m</math>, the separation sharpness:

:<math> m =
\begin{cases}
\dfrac{1.57}{\ln {\sqrt{\dfrac{\ln \left ( \dfrac{3}{S} \right )}{\ln \left (\dfrac{1}{3S} \right )}}}} & S < \dfrac{1}{3}\\
0 & S \geq \dfrac{1}{3}
\end{cases}
</math>

where <math>S</math> is the ratio of volumetric flowrates of pulp in the underflow (sands) and overflow streams, i.e. <math>S = \frac{Q_{\rm V,UF}}{Q_{\rm V,OF}}</math>.

Plitt (1971) found the value of <math>m</math> was in the range of 1 - 3.8 for classifiers. As such, the sharpness estimation above may not be appropriate for larger values of <math>S</math>, depending on the application. User caution is recommended.

=== Density-specific classification ===

Plitt and Flintoff's original model formulation purposely assumed that a mixture of particles of differing densities could be represented by a singular solid density value (<math>\rho_{\rm S}</math>).

This implementation can retain Plitt and Flintoff singular density value assumption, or compute classification curves separately for each density class, <math>(\rho_{\rm S})_j</math> (t/m<sup>3</sup>), where <math>j=\{1,2,\dots,p\}</math> and <math>p</math> is the number of density classes (or ore types, minerals etc).

=== Coarse solids removal capacity ===

If the computed mass flowrate of solids to the underflow stream, <math>(Q_{\rm M,UF})_{\rm S}</math> (t/h), exceeds the solids removal capacity of the classifier, <math>(Q_{\rm M,UF})_{\rm S}^{\rm Max}</math> (t/h), the <math>{d_{\rm 50c}}</math> value is increased by a small fraction. This process repeats until the underflow mass flowrate of solids is less than the removal capacity, i.e.

:<math>({d_{\rm 50c}})_{n+1} = 1.05 \cdot {d_{\rm 50c}}_{n}, \text{ while }(Q_{\rm M,UF})_{\rm S} > (Q_{\rm M,UF})_{\rm S}^{\rm Max}</math>

This action simulates a buildup of solids in the classifier which effectively reduces the settling pool area, driving up the cut size.

=== Sands solid content ===

Plitt and Flintoff provide the following table as a guide to the volume fraction of solids that may be expected from a classifier with a given cut size:

:{| class="wikitable"
|+ Table 2. Sands solids content (after Plitt and Flintoff, 1985){{Plitt and Flintoff (1985)}}
|-
! <math>\boldsymbol{d_{\rm 50c}}</math> (μm) !! Volume fraction solids (% v/v)
|-
| 600|| 57-60
|-
| 420|| 53-57
|-
| 300|| 52-54
|-
| 210|| 53
|-
| 150|| 52
|-
| 105|| 51
|-
| 74|| 49
|-
| Hydroseparators|| 20-40
|}

=== Partition metrics ===

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

== Excel ==

The Plitt-Flintoff gravity classifier model may be invoked from the Excel formula bar with the following function call:

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

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

=== Inputs ===

{{Excel (Text, Inputs)}}

:<math>\mathit{Parameters}=
\begin{bmatrix}
A\text{ (m}^2\text{)}\\
(Q_{\rm M,UF})_{\rm S}^{\rm Max}\text{ (t/h)}\\
\phi_{\rm UF}\text{ (v/v)}\\
\text{ Particle shape type (0-6)}\\
\text{ Density-specific (True/False)}\\
\text{ Calculate sharpness (True/False)}\\
\text{ User } m\text{ (-)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_{\rm L}\text{ (t/m}^{\text{3}}\text{)}\\
\mu_0\text{ (mPa.S)}\\
\end{bmatrix},\;\;\;\;\;\;

\mathit{Size} = \begin{bmatrix}
d_{1}\text{ (mm)}\\ 
\vdots\\ 
d_n\text{ (mm)}\\ 
\end{bmatrix},\;\;\;\;\;\;

\mathit{Feed}= \begin{bmatrix}
(Q_{\rm M,F})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{1p}\text{ (t/h)}\\ 
\vdots & \ddots & \vdots\\ 
(Q_{\rm M,F})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,F})_{np}\text{ (t/h)}\\ 
\end{bmatrix},\;\;\;\;\;\;

\mathit{OreSG}= \begin{bmatrix}
(\rho_{\rm S})_1\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm S})_p\text{ (t/m}^\text{3}\text{)}\\ 
\end{bmatrix}

</math>


where:
* <math>\text{Particle shape type}</math> is the index to the particle shape factors in Table 1 (0-6)
* <math>\text{Density-specific}</math> indicates whether to compute classification separately for each solids density class (''True'') or use a singular representative solids density (''False'')
* <math>\text{Calculate sharpness}</math> indicates whether to estimate the sharpness of separation (<math>m</math>) (''True'') or apply a user-specified value (''False'')
* <math>\text{User }m</math> is a user-specified value of the sharpness of separation (<math>m</math>) which is applied if <math>\text{Calculate sharpness}</math> is ''False'' and ignored otherwise (-) 
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>Q_{\rm M,F}</math> is feed solids mass flow rate by size and ore type (t/h)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the classifier (t/h)

=== Results ===

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

:<math>
\mathit{mdUnit\_GravityClassifier\_PlittFlintoff} = 
\begin{bmatrix}

\begin{bmatrix}
\text{Iterations}\\
\rho_{\rm S}\text{ (t/m}^3\text{)}\\
\rho_{\rm F}\text{ (t/m}^3\text{)}\\
\phi_{\rm F}\text{ (v/v)}\\
Q_{\rm o}\text{ (m}^3\text{/h)}\\
\phi_{\rm Eff}\text{ (v/v)}\\
\rho_{\rm Eff}\text{ (t/m}^3\text{)}\\
\mu\text{ (mPa.S)}\\
v\text{ (m/h)}\\
S\text{ (-)}\\
m\text{ (-)}\\
R_{\rm f}\text{ (frac)}\\
(Q_{\rm M,OF})_{\rm L}\text{ (t/h)}\\
(Q_{\rm M,UF})_{\rm L}\text{ (t/h)}\\
d_{50}\text{ (mm)}\\
E_{\rm p}\text{ (mm)}\\
I\\
\end{bmatrix}


\begin{array}{cccccc}

\begin{bmatrix}
\bar d_1\text{ (mm)}\\
\vdots\\
\bar d_n\text{ (mm)}
\end{bmatrix}


&

\begin{bmatrix}
(Q_{\rm M,UF})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,UF})_{1p}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,UF})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,UF})_{np}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(Q_{\rm M,OF})_{11}\text{ (t/h)} & \dots & (Q_{\rm M,OF})_{1p}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
(Q_{\rm M,OF})_{n1}\text{ (t/h)} & \dots & (Q_{\rm M,OF})_{np}\text{ (t/h)}\\
\end{bmatrix}

&

\begin{bmatrix}
(E_{\rm ua})_{11}\text{ (frac)} & \dots & (E_{\rm ua})_{1p}\text{ (frac)}\\
\vdots & \ddots & \vdots\\
(E_{\rm ua})_{n1}\text{ (frac)} & \dots & (E_{\rm ua})_{np}\text{ (frac)}\\
\end{bmatrix}

&

\begin{bmatrix}
(P_{\rm UF})_{1,{\rm All}}\\
\vdots\\
(P_{\rm UF})_{n,{\rm All}}\\
\end{bmatrix}

&

\begin{bmatrix}
(d_{\rm 50c})_1 & \dots & (d_{\rm 50c})_p\\
\end{bmatrix}\\

\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\\

\end{array}

\end{bmatrix}
</math>


where:
* <math>\text{Iterations}</math> is the number of iterations required to converge the Plitt-Flintoff model
* <math>(Q_{\rm M,OF})_{\rm L}</math> is the mass flow rate of liquids to the overflow stream (t/h)
* <math>(Q_{\rm M,UF})_{\rm L}</math> is the mass flow rate of liquids to the underflow stream (t/h)
* <math>Q_{\rm M,UF}</math> is mass flow rate of solids to the underflow stream (t/h)
* <math>Q_{\rm M,OF}</math> is mass flow rate of solids to the overflow stream (t/h)
* <math>(P_{\rm UF})_{i,{\rm All}}</math> is the actual partition of all particles of size <math>i</math> to the underflow stream, computed as <math>\frac{\sum_{j=1}^p(Q_{\rm M,UF})_{ij}}{\sum_{j=1}^p(Q_{\rm M,F})_{ij}}</math> (frac)

=== Example ===

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

{|
|- style="vertical-align:top;"
| [[File:GravityClassifierPlittFlintoff2.png|left|frame|Figure 2. Example showing the selection of the '''Parameters''' (blue frame) array in Excel.]]
| rowspan="2"| [[File:GravityClassifierPlittFlintoff3.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:GravityClassifierPlittFlintoff4.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, Classifier, DLL*Classifier)}}

==== Classifier page ====

The Classifier page is used to specify the input parameters for the classifier model.

{{SysCAD (Text, Table Header)}}

|-
! colspan="3" style="text-align:left;" |''PlittFlintoff''

{{SysCAD (Text, Help Link)}}

|-
|EffPoolArea / A
|Input
|Effective pool area
|-
|SolidsRemovalCapacity
|Input
|Maximum coarse solids removal capacity of the classifier
|-
|SandsSolVolFrac / UF.Svf
|Input
|Volume fraction of solids in the sands (underflow) stream
|- style="vertical-align:top;"
|rowspan="7" | ParticleShape
|Spheres
|Particles are spherical.
|- style="vertical-align:top;"
|Perfect cubes
|Particles are perfect cubes.
|- style="vertical-align:top;"
|Conchoidal fracture
|Particles shapes are due to conchoidal fracture.
|- style="vertical-align:top;"
|Dodecahedral
|Particles are dodecahedral
|- style="vertical-align:top;"
|Plates 10x
|Particles are plates with length 10x width.
|- style="vertical-align:top;"
|Plates 100x
|Particles are plates with length 100x width.
|- style="vertical-align:top;"
|Irregular fracture
|Particles shapes are due to irregular fracture.
|-
|DensitySpecific
|CheckBox
|Indicates whether to determine classification per species.
|-
|CalculateSharpness
|CheckBox
|Indicates whether to estimate sharpness based on volume split or apply user-specific value.
|-
|UserSharpness / User_m
|Input
|User-specified value of sharpness. Only visible if ''CalculateSharpness'' is not selected. 

{{SysCAD (Text, Hydrocyclone, Liquids)|method=0}}

|-
! colspan="3" style="text-align:left;" |''Results''
|-
|Iterations
|style="background: #eaecf0" | Display
|Number of internal iterations required to converge the Plitt-Flintoff model.
|-
|Feed.SRho
|style="background: #eaecf0" | Display
|Density of solids in the feed stream
|-
|Feed.Rho
|style="background: #eaecf0" | Display
|Density of slurry in the feed stream
|-
|Feed.Svf
|style="background: #eaecf0" | Display
|Volume fraction of solids in the feed stream
|-
|Overflow.Qv / OF.Qv
|style="background: #eaecf0" | Display
|Volumetric flowrate of slurry in the overflow stream.
|-
|EffSolVolFrac / Eff.Svf
|style="background: #eaecf0" | Display
|Effective volume fraction of solids in the classifier.
|-
|EffDensity / Eff.SRho
|style="background: #eaecf0" | Display
|Effective density of solids in the classifier.
|-
|SlurryViscosity / mu
|style="background: #eaecf0" | Display
|Apparent viscosity of slurry in the classifier.
|-
|SettlingVelocity / v
|style="background: #eaecf0" | Display
|Settling velocity in the classifier.
|-
|VolFlowSplit / S
|style="background: #eaecf0" | Display
|Ratio of volumetric flow rate of slurry in the underflow to the volumetric flow rate of slurry in the overflow.
|-
|Sharpness / m
|style="background: #eaecf0" | Display
|Sharpness of separation in the Reid-Plitt expression.
|-
|Rf
|style="background: #eaecf0" | Display
|Fraction of feed liquids recovered to underflow stream.

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

|-
|d50c
|style="background: #eaecf0" | Display
|Value of the corrected ''d<sub>50c</sub>'', per species.
|}

{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Partition|ActionL=partition|DestinationU=Underflow|DestinationL=underflow|UnitL=classifier}}

{{SysCAD (Page, About)}}

==== Additional notes ====

{{SysCAD (Text, No PSD Splits)|gasstream=overflow}}

== See also ==

* [[Partition (Size, Reid-Plitt)]]

== References ==

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