Flotation Cell (Savassi): Difference between revisions

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Savassi's original model was further extended over many years by subsequent researchers, and the resulting methodology is frequently referred to as the "P9 model" in reference to the AMIRA P9 project that has driven much of the development since.{{Coleman et al. (2007)}}
Savassi's original model was further extended over many years by subsequent researchers, and the resulting methodology is frequently referred to as the "P9 model" in reference to the AMIRA P9 project that has driven much of the development since.{{Coleman et al. (2007)}}
The model described here is for '''''steady-state''''' simulation. For dynamic simulation, ''see [[Flotation Cell (Savassi, Dynamic)]]''.


== Model theory ==
== Model theory ==


{{Restricted content}}
<hide>
</hide><div class="user-show">
=== Floatability classes ===
=== Floatability classes ===
</div><hide>


Much of the original literature underpinning the modelling approach described here refers to the notion of a ''floatability class'', which groups all particles that exhibit similar floatability properties, without any particular reference to particle sizes or mineral compositions within the class.{{Runge et al. (1998)}} In order to align the flotation modelling approach with the comminution, classification and concentration processes described in other articles, a distinction is made between ''particle size'' and ''floatability component''.
Much of the original literature underpinning the modelling approach described here refers to the notion of a ''floatability class'', which groups all particles that exhibit similar floatability properties, without any particular reference to particle sizes or mineral compositions within the class.{{Runge et al. (1998)}} In order to align the flotation modelling approach with the comminution, classification and concentration processes described in other articles, a distinction is made between ''particle size'' and ''floatability component''.
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The mathematical relations below are developed on the basis of a particle size class which is further subdivided into floatability components. This allows the definition of floatability rate groups (e.g. fast, slow, non-floating) within mineral types (e.g. chalcopyrite, pyrite, non-sulphide gangue), by size fraction.
The mathematical relations below are developed on the basis of a particle size class which is further subdivided into floatability components. This allows the definition of floatability rate groups (e.g. fast, slow, non-floating) within mineral types (e.g. chalcopyrite, pyrite, non-sulphide gangue), by size fraction.


</hide><div class="user-show">
=== Recovery ===
=== Recovery ===
</div><hide>


Savassi's compartment model describes the recovery of particles by the mechanisms of true flotation, froth mass transfer and entrainment.
Savassi's compartment model describes the recovery of particles by the mechanisms of true flotation, froth mass transfer and entrainment.
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R_{ij} =  
R_{ij} =  
\begin{cases}
\begin{cases}
1 - \exp \big [ - P_{ij} \cdot (S_b \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau \cdot (1 - ENT_{ij} \cdot R_w) \big ] & \text{for batch or plug flow flotation}\\
1 - \exp \left ( - P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau \right ) \cdot (1 - {\rm ENT}_{ij} \cdot R_{\rm w}) & \text{for batch or plug flow flotation}\\
\\
\\
\dfrac{P_{ij} \cdot (S_b \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau \cdot (1 - R_w) + ENT_{ij} \cdot R_w}{(1 + P_{ij} \cdot (S_b \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau)(1 - R_w) + ENT_{ij} \cdot R_w} & \text{for continuous flotation}\\
\dfrac{P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau \cdot (1 - R_{\rm w}) + {\rm ENT}_{ij} \cdot R_{\rm w}}{(1 + P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau)(1 - R_{\rm w}) + {\rm ENT}_{ij} \cdot R_{\rm w}} & \text{for continuous flotation}\\
\end{cases}
\end{cases}
</math>
</math>
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* <math>j</math> is the index of the floatability component, <math>j = \{1,2,\dots,m\}</math>, <math>m</math> is the number of floatability components (e.g. mineral types, fast/slow types etc.)
* <math>j</math> is the index of the floatability component, <math>j = \{1,2,\dots,m\}</math>, <math>m</math> is the number of floatability components (e.g. mineral types, fast/slow types etc.)
* <math>R_{ij}</math> is the fraction of particles of size fraction <math>i</math> and floatability component type <math>j</math> recovered to concentrate (frac)
* <math>R_{ij}</math> is the fraction of particles of size fraction <math>i</math> and floatability component type <math>j</math> recovered to concentrate (frac)
* <math>P_{ij}</math> is the floatability of particles of size <math>i</math> and floatability component type <math>j</math> (min<sup>-1</sup>)
* <math>P_{ij}</math> is the floatability of particles of size <math>i</math> and floatability component type <math>j</math> (frac)
* <math>S_b</math> is the bubble surface area flux  (m<sup>2</sup>/s/m<sup>2</sup>)
* <math>S_{\rm b}</math> is the bubble surface area flux  (m<sup>2</sup>/s/m<sup>2</sup>)
* <math>Rf_{ij}</math> is the fraction of particles of size <math>i</math> and floatability component <math>j</math> recovered by froth (frac)
* <math>Rf_{ij}</math> is the fraction of particles of size <math>i</math> and floatability component <math>j</math> recovered by froth (frac)
* <math>C_{ij}</math> is a cell scale-up factor that relates flotation performance between cells of different sizes (e.g. laboratory versus industrial) (frac)
* <math>C_{ij}</math> is a cell scale-up factor that relates flotation performance between cells of different sizes (e.g. laboratory versus industrial) (frac)
* <math>\tau</math> is the residence time of slurry in the cell (min)
* <math>\tau</math> is the residence time of slurry in the cell (min)
* <math>ENT_{ij}</math> is the fraction of particles of size <math>i</math> and floatability component <math>j</math> recovered by entrainment (frac)
* <math>{\rm ENT}_{ij}</math> is the fraction of particles of size <math>i</math> and floatability component <math>j</math> recovered by entrainment (frac)
* <math>R_w</math> is the fraction of feed water recovered to concentrate (frac)
* <math>R_{\rm w}</math> is the fraction of feed water recovered to concentrate (frac)


The cell scale-up factor, <math>C_{ij}</math>, is suggested by Yianatos et al. (2010) to allow for the simulation of industrial-scale cells using kinetic rates derived from laboratory cells.{{Yianatos et al. (2010)}}
The cell scale-up factor, <math>C_{ij}</math>, allows for the simulation of industrial-scale cells using kinetic rates derived from laboratory cells.{{Runge et al. (2003)}}


</hide><div class="user-show">
=== Pulp volume ===
=== Pulp volume ===
</div><hide>


Pulp volume,  <math>V_{{effective}}</math> (m<sup>3</sup>), is related to cell paramters.
{{Model theory (Text, Flotation Cell, Savassi, Pulp Volume)}}
 
:<math>V_{{effective}} = (V_{{cell}} - V_{{mechanism}} - A_{{cell}} * H_{f} ) \cdot (1 - \epsilon_{{gas}})</math>
 
where:
* <math>V_{{cell}}</math> is the total volume of the cell
* <math>V_{{mechanism}}</math> is the volume of the impeller mechanism in the cell (m<sup>3</sup>)
* <math>A_{{cell}}</math> is the cross sectional area of the cell (m<sup>2</sup>)
* <math>H_{f}</math> is the froth depth from the top of the cell (m)
* <math>\epsilon_{{gas}}</math> is the gas hold-up in the pulp (v/v)


</hide><div class="user-show">
=== Gas hold-up ===
=== Gas hold-up ===
</div><hide>


Gas hold-up, <math>\epsilon_{gas}</math> (v/v), may be estimated from a linear relationship as suggested by the findings of Gorain et al. (1995):{{Gorain et al. (1995)}}
{{Model theory (Text, Flotation Cell, Savassi, Gas Hold-up)}}
 
:<math>\epsilon_{gas} = m_{gas} \cdot J_g + C_{gas}</math>
 
where <math>m_{gas}</math> and <math>C_{gas}</math> are the slope an intercept parameters, respectively, of the linear equation.
 
The superficial gas rate, <math>J_g</math> (m/s), is approximated by:{{Gorain et al. (1999)}}
 
:<math>J_g = \dfrac{Q_{air}}{A_{{cell}} \cdot 60 \cdot 60}</math>
 
where <math>Q_{air}</math> is the air rate (m<sup>3</sup>/h).


</hide><div class="user-show">
=== Residence time ===
=== Residence time ===
</div><hide>


Residence time, <math>\tau</math> (min), may be computed on the basis of the cell '''feed''' or '''tailing''' stream volumetric flow rates.{{Schwarz et al. (2006b)}}
{{Model theory (Text, Flotation Cell, Savassi, Residence Time)}}
 
:<math>
\tau =
\begin{cases}
\dfrac{V_{effective}}{Q_{feed} \cdot 60} & \text{for feed-based residence time}\\
\\
\dfrac{V_{effective}}{Q_{tail} \cdot 60} & \text{for tail-based residence time}\\
\end{cases}
</math>
 
where <math>{Q_{feed}}</math> and <math>{Q_{tail}}</math> are the volumetric flow rates of pulp in the feed and tail streams, respectively (m<sup>3</sup>/h).


The volumetric flow rate of the tailing stream is a function of the recoveries of solids computed by the Savassi equation and water. As such, an iterative procedure is required to estimate the tail-based residence time.
The volumetric flow rate of the tailing stream is a function of the recoveries of solids computed by the Savassi equation and water. As such, an iterative procedure is required to estimate the tail-based residence time.


</hide><div class="user-show">
=== Bubble surface area flux ===
=== Bubble surface area flux ===
</div><hide>


The bubble surface area flux, <math>S_b</math> (m<sup>2</sup>/s/m<sup>2</sup>), is related to the superficial gas rate, <math>J_g</math>, and Sauter mean bubble diameter, <math>d_b</math> (m), by the equation:
{{Model theory (Text, Flotation Cell, Savassi, Bubble Surface Area Flux)}}
 
:<math>S_b = \dfrac{6 J_g}{d_b}</math>
 
The Sauter mean bubble diameter may be determined from gas dispersion measurements. Where a measurement of <math>d_b</math> is not available, <math>S_b</math> may be estimated by the empirical relation of Gorain et al. (1995):{{Gorain et al. (1995)}}
 
:<math>S_b = 123 {N_s}^{0.44} {J_g}^{0.75} {As}^{-0.10} {P_{80}}^{-0.42}</math>
 
where:
* <math>N_s</math> is impeller tip speed (m/s)
* <math>As</math> is impeller aspect ratio (m/m)
* <math>P_{80}</math> is the size that 80% of solid particles in the feed are finer than (µm)
 
The Gorain relation may be modified to suit alternative bubble surface area flux data sets with:
 
:<math>S_b = C_{Sb} {N_s}^{a} {J_g}^{b} {As}^{c} {P_{80}}^{d}</math>
 
where <math>C_{Sb}</math> and <math>a \dots d</math> are a user-defined coefficient  and set of exponents, respectively.


</hide><div class="user-show">
=== Froth recovery ===
=== Froth recovery ===
</div><hide>


Froth recovery, <math>Rf_{ij}</math>, may be estimated using the froth residence time model outlined by Harris et al. (2002).{{Harris et al. (2002)}}
{{Model theory (Text, Flotation Cell, Savassi, Froth Recovery)}}


:<math>Rf_{ij} = (1 - (P_d)_j) + (P_d)_j \cdot ENT_i \cdot R_w</math>
</hide><div class="user-show">
=== Entrainment ===
</div><hide>


where <math>(P_d)_j</math> is the probability of detachment of ore type <math>j</math> (frac), computed as:
{{Model theory (Text, Flotation Cell, Savassi, Entrainment)}}


:<math>(P_d)_j = 1 - \exp (- \beta_j \cdot FRT)</math>
</hide><div class="user-show">
==== No entrainment ====
</div><hide>


and <math>\beta_j</math> is the detachment rate constant (min<sup>-1</sup>) for ore type <math>j</math>.
When the effects of entrainment are ignored, the recovery equation reduces to:


The froth residence time, <math>FRT</math> (min), is computed based on either '''air''' or '''slurry''' flow rates:
:<math>
 
R_{ij} =  
:<math>FRT =  
\begin{cases}
\begin{cases}
\dfrac{H_f \cdot {\epsilon_{froth}}}{J_g} & \text{for air-based froth residence time}\\
1 - \exp \left ( - P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau \right )  & \text{for batch or plug flow flotation}\\
\dfrac{H_f \cdot A_{cell} \cdot (1 - \epsilon_{froth})}{Q_{con}} & \text{for slurry-based froth residence time}\\
\\
\dfrac{P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau }{(1 + P_{ij} \cdot (S_{\rm b} \cdot 60) \cdot Rf_{ij} \cdot C_{ij} \cdot \tau)} & \text{for continuous flotation}\\
\end{cases}
\end{cases}
</math>
</math>


where <math>\epsilon_{froth}</math> is the gas hold-up fraction in the froth phase (v/v), and <math>Q_{con}</math> is the volumetric flow rate of concentrate from the cell.
</hide><div class="user-show">
 
The concentrate flow rate, <math>Q_{con}</math>, is a function of the recoveries of solids computed by the Savassi equation and water. As with the tail-based residence time (see [[Flotation Cell (Savassi)#Residence_time|above]]), an iterative procedure is required to estimate froth recovery via the froth residence time method (even if a feed-basis is used to estimate pulp residence time).
 
=== Entrainment ===
 
Entrainment may be estimated by two methods, the hyperbolic equation approach by Savassi et al. (1998), and the froth residence time approach described by Vera et al (2002).{{Savassi et al. (1998)}}{{Vera et al. (2002)}}
 
The '''hyperbolic equation''' for entrainment is:
 
:<math>ENT_i = \dfrac{2}{\exp \left ( 2.292 \left ( \dfrac{\bar d_i}{E20} \right )^{adj} \right ) + \exp \left ( -2.292 \left ( \dfrac{\bar d_i}{E20} \right )^{adj} \right )}</math>
 
:<math>adj = 1 + \dfrac{\ln \delta}{\exp \left ( \dfrac{\bar d_i}{E20} \right )}</math>
 
where:
* <math>\bar d_i</math> is the geometric mean size of particles in size interval <math>i</math> (µm)
* <math>E20</math> is the particle size at which <math>ENT_i</math> is 0.2 (µm)
* <math>\delta</math> is the froth drainage parameter
 
The '''froth residence time''' approach first computes the froth residence time (<math>FRT</math>) according to the method descibed for froth recovery [[Flotation Cell (Savassi)#Froth_recovery|above]]. Entrainment is then estimated by applying the following equation:
 
:<math>ENT_i = \dfrac{1}{1 + \omega_i \cdot FRT}</math>
 
where <math>\omega_i</math> is the froth drainage factor for particles in size interval <math>i</math> (min<sup>-1</sup>).
 
=== Water recovery ===
=== Water recovery ===
</div><hide>


Methods for estimating the recovery of water are described by Harris et al. (2002).{{Harris et al. (2002)}}
{{Model theory (Text, Flotation Cell, Savassi, Water Recovery)}}
 
The '''fixed concentrate percent solids''' method assumes the concentrate is always recovered at a fixed pulp density. The recovery of water, <math>R_w</math>, is then computed to meet the fixed concentrate solids fraction requirement.
 
The '''solids dependent model''' assumes a power law relationship between the volumetric flow rate of water, <math>Q_w</math> (m<sup>3</sup>/h), and solids, <math>Q_s</math> (m<sup>3</sup>/h), in the concentrate stream:


:<math>Q_w = a \cdot {Q_s}^b</math>
</hide><div class="user-show">
==== Water rate function ====
</div><hide>


where <math>a</math> and <math>b</math> are the coefficient and exponent of the power law relationship, respectively.
The ''water rate function'' adopts a first order kinetic relationship between water recovery and pulp residence time:


Finally, the '''water rate function''' adopts a first order kinetic relationship between water recovery and pulp residence time:
:<math>R_{\rm w} = \dfrac{k_{\rm w} \tau}{1 + k_{\rm w} \tau}</math>


:<math>R_w = \dfrac{k_w \tau}{1 + k_w \tau}</math>
where <math>k_{\rm w}</math> is the kinetic rate parameter (min<sup>-1</sup>) for the recovery of water to the concentrate stream.
 
</hide>
where <math>k_w</math> is the kinetic rate parameter (min<sup>-1</sup>) for the recovery of water to the concentrate stream.


== Excel ==
== Excel ==
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The Savassi flotation cell model may be invoked from the Excel formula bar with the following function call:
The Savassi flotation cell model may be invoked from the Excel formula bar with the following function call:


<syntaxhighlight lang="vb">=mdUnit_Flotation_Savassi(Parameters as Range, Size as Range, Feed as Range, OreSG As Range, Floatability as Range, CellScaleUp As Range, DetachRateConst as Range, FrothRecovery as Range, DrainageFactor as Range, Entrainment as Range)</syntaxhighlight>
<syntaxhighlight lang="vb">=mdUnit_Flotation_Savassi(Parameters as Range, Size as Range, Feed as Range, OreSG As Range, Floatability as Range, Optional CellScaleUp As Range = Nothing, Optional FrothRecovery as Range = Nothing, Optional Entrainment as Range = Nothing, Optional DetachRateConst as Range = Nothing, Optional DrainageFactor as Range = Nothing)</syntaxhighlight>
 
Several of the function input parameters are marked as ''Optional'':
* If ''CellScaleUp'' is omitted, cell scale up ignored and <math>C_{ij}=1</math> for all <math>i</math> and <math>j</math>
* If ''FrothRecovery'' is omitted, froth recovery is ignored and <math>Rf_{ij}=1</math> for all <math>i</math> and <math>j</math>
* If ''Entrainment'' is omitted, recovery by entrainment is ignored and <math>{\rm ENT}_{ij}=0</math> for all <math>i</math> and <math>j</math>
* If ''DetachRateConst'' is omitted, <math>\beta_{j}=0</math> for all <math>j</math>
* If ''DrainageFactor'' is omitted, <math>\omega_{i}=0</math> for all <math>i</math>


{{Excel (Text, Help, No Arguments)}}
{{Excel (Text, Help, No Arguments)}}
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:<math>Parameters=
:<math>Parameters=
\begin{bmatrix}
\begin{bmatrix}
V_{cell}\text{ (m}^\text{3)}\\
V_{\rm cell}\text{ (m}^\text{3)}\\
V_{mechanism}\text{ (m}^\text{3)}\\
V_{\rm mechanism}\text{ (m}^\text{3)}\\
A_{cell}\text{ (m}^\text{2)}\\
A_{\rm cell}\text{ (m}^\text{2)}\\
H_f\text{ (m)}\\
H_{\rm f}\text{ (m)}\\
Q_{air}\text{ (m}^\text{3}\text{/h)}\\
Q_{\rm air}\text{ (m}^\text{3}\text{/h)}\\
\text{Gas holdup model}\\
\text{Gas holdup model}\\
\epsilon_{gas}\text{ (v/v)}\\
\varepsilon_{\rm gas}\text{ (v/v)}\\
m_{gas}\\
m_{\rm gas}\\
C_{gas}\\
C_{\rm gas}\\
\text{Residence time option}\\
\text{Residence time option}\\
\text{Flow pattern}\\
\text{Flow pattern}\\
\text{Residence time basis}\\
\text{Residence time basis}\\
\tau\text{ (min)}\\
\text{Sb option}\\
\text{Sb option}\\
S_b\text{ (m}^2\text{/s/m}^2\text{)}\\
S_{\rm b}\text{ (m}^2\text{/s/m}^2\text{)}\\
\text{Use calculated P80}\\
\text{Use calculated P80}\\
P_{80}\text{ (}\mu\text{m)}\\
P_{80}\text{ (}\mu\text{m)}\\
N_s\text{ (m/s)}\\
N_{\rm s}\text{ (m/s)}\\
A_s\text{ (m/m)}\\
A_{\rm s}\text{ (m/m)}\\
C_{Sb}\\
C_{\rm Sb}\\
a\\
a\\
b\\
b\\
Line 214: Line 172:
\text{Froth model}\\
\text{Froth model}\\
\text{Froth residence time basis}\\
\text{Froth residence time basis}\\
\epsilon_{froth}\text{ (v/v)}\\
\varepsilon_{\rm froth}\text{ (v/v)}\\
\text{Entrainment option}\\
\text{Entrainment option}\\
E20\text{ (}\mu\text{m)}\\
E20\text{ (}\mu\text{m)}\\
Line 221: Line 179:
\text{Water option}\\
\text{Water option}\\
\text{Concentrate fraction solids (w/w)}\\
\text{Concentrate fraction solids (w/w)}\\
a_w\\
a_{\rm w}\\
b_w\\
b_{\rm w}\\
k_w\\
k_{\rm w}\\
R_w\text{ (frac)}\\
R_{\rm w}\text{ (frac)}\\
(Q_{M,F})_{Liquids}\text{ (t/h)}\\
(Q_{\rm M,F})_{\rm L}\text{ (t/h)}\\
\rho_L\text{ (t/m}^\text{3}\text{)}\\
\rho_{\rm L}\text{ (t/m}^\text{3}\text{)}\\
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


Line 236: Line 194:


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


OreSG= \begin{bmatrix}
OreSG= \begin{bmatrix}
SG_{1}\text{ (t/m}^\text{3}\text{)} & \dots & SG_m\text{ (t/m}^\text{3}\text{)}\\  
(\rho_{\rm S})_{1}\text{ (t/m}^\text{3}\text{)} & \dots & (\rho_{\rm S})_m\text{ (t/m}^\text{3}\text{)}\\  
\end{bmatrix}
\end{bmatrix}
</math>
</math>
Line 249: Line 207:
:<math>
:<math>
Floatability = \begin{bmatrix}
Floatability = \begin{bmatrix}
P_{11}\text{ (t/h)} & \dots & P_{1m}\text{ (t/h)}\\  
P_{11}\text{ (frac)} & \dots & P_{1m}\text{ (frac)}\\  
\vdots & \ddots & \vdots\\  
\vdots & \ddots & \vdots\\  
P_{n1}\text{ (t/h)} & \dots & P_{nm}\text{ (t/h)}\\  
P_{n1}\text{ (frac)} & \dots & P_{nm}\text{ (frac)}\\  
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


CellScaleUp= \begin{bmatrix}
CellScaleUp= \begin{bmatrix}
C_{11}\text{ (t/h)} & \dots & C_{1m}\text{ (t/h)}\\  
C_{11} & \dots & C_{1m}\\  
\vdots & \ddots & \vdots\\  
\vdots & \ddots & \vdots\\  
C_{n1}\text{ (t/h)} & \dots & C_{nm}\text{ (t/h)}\\  
C_{n1} & \dots & C_{nm}\\  
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


DetachRateConst = \begin{bmatrix}
FrothRecovery = \begin{bmatrix}
(P_d)_{1}\text{ (frac)} & \cdots & (P_d)_m\text{ (frac)}\\  
Rf_{11}\text{ (frac)} & \dots & Rf_{1m}\text{ (frac)}\\
\vdots & \ddots & \vdots\\
Rf_{n1}\text{ (frac)} & \dots & Rf_{nm}\text{ (frac)}\\  
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;
</math>
</math>
Line 267: Line 227:


:<math>
:<math>
FrothRecovery = \begin{bmatrix}
Entrainment = \begin{bmatrix}
Rf_{11}\text{ (t/h)} & \dots & Rf_{1m}\text{ (t/h)}\\  
{\rm ENT}_{11}\text{ (frac)} & \dots & {\rm ENT}_{1m}\text{ (frac)}\\  
\vdots & \ddots & \vdots\\  
\vdots & \ddots & \vdots\\  
Rf_{n1}\text{ (t/h)} & \dots & Rf_{nm}\text{ (t/h)}\\  
{\rm ENT}_{n1}\text{ (frac)} & \dots & {\rm ENT}_{nm}\text{ (frac)}\\  
\end{bmatrix},\;\;\;\;\;\;
\end{bmatrix},\;\;\;\;\;\;


DetachRateConst = \begin{bmatrix}
\beta_{1}\text{ (min}^{\text{-1}}\text{)} & \cdots & \beta_m\text{ (min}^{\text{-1}}\text{)}\\
\end{bmatrix} \quad \text{or} \quad
\begin{bmatrix}
\beta_{1}\text{ (min}^{\text{-1}}\text{)} & \cdots & \beta_m\text{ (min}^{\text{-1}}\text{)}\\
(\eta_p)_{1}\text{ (min}^{\text{-1}}\text{)} & \cdots & (\eta_p)_m\text{ (min}^{\text{-1}}\text{)}\\
\end{bmatrix},
</math>
:<math>
DrainageFactor = \begin{bmatrix}
DrainageFactor = \begin{bmatrix}
\delta_{1}\\  
\omega_{1}\text{ (min}^{\text{-1}}\text{)}\\  
\vdots\\  
\vdots\\  
\delta_n\\
\omega_n\text{ (min}^{\text{-1}}\text{)}\\  
\end{bmatrix},\;\;\;\;\;\;
 
Entrainment = \begin{bmatrix}
ENT_{11}\text{ (t/h)} & \dots & ENT_{1m}\text{ (t/h)}\\
\vdots & \ddots & \vdots\\
ENT_{n1}\text{ (t/h)} & \dots & ENT_{nm}\text{ (t/h)}\\  
\end{bmatrix}
\end{bmatrix}
</math>
</math>
Line 288: Line 253:
where:
where:
* <math>\text{Gas holdup model}</math> specifies the method used to determine the gas-hold up, ''0 = No gas holdup, 1 = User defined, 2 = Jg dependent''
* <math>\text{Gas holdup model}</math> specifies the method used to determine the gas-hold up, ''0 = No gas holdup, 1 = User defined, 2 = Jg dependent''
* <math>\epsilon_{gas}</math> is the user-defined value of gas hold-up (v/v) if <math>\text{Gas holdup model} = 1</math>, ignored otherwise
* <math>\varepsilon_{\rm gas}</math> is the user-defined value of gas hold-up (v/v) if <math>\text{Gas holdup model} = 1</math>, ignored otherwise
* <math>m_{gas}</math> is the value of the slope coefficient if <math>\text{Gas holdup model} = 2</math>, ignored otherwise
* <math>m_{\rm gas}</math> is the value of the slope coefficient if <math>\text{Gas holdup model} = 2</math>, ignored otherwise
* <math>C_{gas}</math> is the value of the slope coefficient if <math>\text{Gas holdup model} = 2</math>, ignored otherwise
* <math>C_{\rm gas}</math> is the value of the slope coefficient if <math>\text{Gas holdup model} = 2</math>, ignored otherwise
* <math>\text{Residence time option}</math> specifies the method used to determine the residence time, ''0 = User defined, 1 = Calculated''
* <math>\text{Residence time option}</math> specifies the method used to determine the residence time, ''0 = User defined, 1 = Calculated''
* <math>\text{Flow pattern}</math> specifies the cell flow configuration, ''0 = Batch or Plug flow, 1 = Perfect Mixing (Continuous)''
* <math>\text{Flow pattern}</math> specifies the cell flow configuration, ''0 = Batch or Plug flow, 1 = Perfect Mixing (Continuous)''
* <math>\text{Residence time basis}</math> specifies the stream which is used to computed the residence time, ''0 = Feed, 1 = Tail''
* <math>\text{Residence time basis}</math> specifies the stream which is used to computed the residence time, ''0 = Feed, 1 = Tail''
* <math>\text{Sb option}</math> specifies the method used to determine the bubble surface area, ''0 = User defined, 1 = Gorain, 2 = User equation''
* <math>\text{Sb option}</math> specifies the method used to determine the bubble surface area, ''0 = User defined, 1 = Gorain, 2 = User equation''
* <math>S_b</math> is the user-defined value of bubble surface area flux (m<sup>2</sup>/s/m<sup>2</sup>) if <math>\text{Sb option} = 0</math>, ignored otherwise
* <math>S_{\rm b}</math> is the user-defined value of bubble surface area flux (m<sup>2</sup>/s/m<sup>2</sup>) if <math>\text{Sb option} = 0</math>, ignored otherwise
* <math>\text{Use calculated P80}</math> indicates whether the <math>P_{80}</math> for the Gorain equation (<math>\text{Sb option} = 1,2</math>) is user-defined or determined from the feed stream , ''0 = User defined, 1 = Calculated P80''
* <math>\text{Use calculated P80}</math> indicates whether the <math>P_{80}</math> for the Gorain equation (<math>\text{Sb option} = 1,2</math>) is user-defined or determined from the feed stream , ''0 = User defined, 1 = Calculated P80''
* <math>P_{80}</math> is the user-defined value of <math>P_{80}</math> (μm) if <math>\text{Sb option} = 1,2</math> and <math>\text{Use calculated P80} = 1</math>, ignored otherwise
* <math>P_{80}</math> is the user-defined value of <math>P_{80}</math> (μm) if <math>\text{Sb option} = 1,2</math> and <math>\text{Use calculated P80} = 1</math>, ignored otherwise
* <math>N_s</math> is the user-defined value of impeller speed (m/s) if <math>\text{Sb option} = 1,2</math>, ignored otherwise
* <math>N_{\rm s}</math> is the user-defined value of impeller speed (m/s) if <math>\text{Sb option} = 1,2</math>, ignored otherwise
* <math>A_s</math> is the user-defined value of impeller aspect ratio (m/m) if <math>\text{Sb option} = 1,2</math>, ignored otherwise
* <math>A_{\rm s}</math> is the user-defined value of impeller aspect ratio (m/m) if <math>\text{Sb option} = 1,2</math>, ignored otherwise
* <math>C_{Sb}</math> is a user-defined coefficient of the Gorain relation if <math>\text{Sb option} = 2</math>, ignored otherwise
* <math>C_{\rm Sb}</math> is a user-defined coefficient of the Gorain relation if <math>\text{Sb option} = 2</math>, ignored otherwise
* <math>a \dots d</math> are user-defined exponents of the Gorain relation  if <math>\text{Sb option} = 2</math>, ignored otherwise
* <math>a \dots d</math> are user-defined exponents of the Gorain relation  if <math>\text{Sb option} = 2</math>, ignored otherwise
* <math>\text{Froth model}</math> specifies the method used to determine froth recovery, ''0 = User defined, 1 = FRT''
* <math>\text{Froth model}</math> specifies the method used to determine froth recovery, ''0 = User defined, 1 = FRT Entrain, 2 = FRT User Np''
* <math>\text{Froth residence time basis}</math> specifies the phase which is used to computed the froth residence time, ''0 = Air, 1 = Slurry'', if <math>\text{Froth model} = 1</math>, ignored otherwise
* <math>\text{Froth residence time basis}</math> specifies the phase which is used to computed the froth residence time, ''0 = Air, 1 = Slurry'', if <math>\text{Froth model} = 1</math>, ignored otherwise
* <math>\epsilon_{froth}</math> is the value of the gas hold up fraction (v/v) if <math>\text{Froth model} = 1</math>, ignored otherwise
* <math>\varepsilon_{\rm froth}</math> is the value of the gas hold up fraction (v/v) if <math>\text{Froth model} = 1</math>, ignored otherwise
* <math>\text{Entrainment option}</math> specifies the method used to determine entrainment, ''0 = No entrainment, 1 = User defined, 2 = Hyperbolic, 3 = FRT dependent''
* <math>\text{Entrainment option}</math> specifies the method used to determine entrainment, ''0 = No entrainment, 1 = User defined, 2 = Hyperbolic, 3 = FRT dependent''
* <math>E20</math> is the value of the hyperbolic equation particle size term (v/v) if <math>\text{Entrainment option} = 2</math>, ignored otherwise
* <math>E20</math> is the value of the hyperbolic equation particle size term (μm) if <math>\text{Entrainment option} = 2</math>, ignored otherwise
* <math>\delta</math> is the value of the hyperbolic equation drainage factor (v/v) if <math>\text{Entrainment option} = 2</math>, ignored otherwise
* <math>\delta</math> is the value of the hyperbolic equation drainage factor (-) if <math>\text{Entrainment option} = 2</math>, ignored otherwise
* <math>\text{FRT entrainment basis}</math> specifies the phase which is used to computed the froth residence time for entrainment, ''0 = Air, 1 = Slurry'', if <math>\text{Entrainment option} = 3</math>, ignored otherwise
* <math>\text{FRT entrainment basis}</math> specifies the phase which is used to computed the froth residence time for entrainment, ''0 = Air, 1 = Slurry'', if <math>\text{Entrainment option} = 3</math>, ignored otherwise
* <math>\text{Water option}</math> specifies the method used to determine water recovery, ''0 = Fixed % solids, 1 = Solid dependent, 2 = rate function, 3 = User defined''
* <math>\text{Water option}</math> specifies the method used to determine water recovery, ''0 = Fixed % solids, 1 = Solid dependent, 2 = rate function, 3 = User defined''
* <math>\text{Concentrate fraction solids}</math> is the user-defined value of concentrate pulp density (w/w) if <math>\text{Water option} = 0</math>, ignored otherwise
* <math>\text{Concentrate fraction solids}</math> is the user-defined value of concentrate pulp density (w/w) if <math>\text{Water option} = 0</math>, ignored otherwise
* <math>a_w</math> is a user-defined coefficient if <math>\text{Water option} = 1</math>, ignored otherwise
* <math>a_{\rm w}</math> is a user-defined coefficient if <math>\text{Water option} = 1</math>, ignored otherwise
* <math>b_w</math> is a user-defined exponent if <math>\text{Water option} = 1</math>, ignored otherwise
* <math>b_{\rm w}</math> is a user-defined exponent if <math>\text{Water option} = 1</math>, ignored otherwise
* <math>k_w</math> is a user-defined kinetic rate parameter if <math>\text{Water option} = 2</math>, ignored otherwise
* <math>k_{\rm w}</math> is a user-defined kinetic rate parameter if <math>\text{Water option} = 2</math>, ignored otherwise
* <math>R_w</math> is a user-defined water split to concentrate if <math>\text{Water option} = 3</math>, ignored otherwise
* <math>R_{\rm w}</math> is a user-defined water split to concentrate if <math>\text{Water option} = 3</math>, ignored otherwise
* <math>(Q_{M,F})_{Liquids}</math> is the mass flow feed rate of liquids into the cell (t/h)
* <math>(Q_{\rm M,F})_{\rm L}</math> is the mass flow feed rate of liquids into the cell (t/h)
* <math>\rho_L</math> is the specific gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>\rho_{\rm L}</math> is the Specific Gravity or density of liquids in the feed (- or t/m<sup>3</sup>)
* <math>n</math> is the number of size intervals
* <math>n</math> is the number of size intervals
* <math>m</math> is the number of ore types (floatability classes)
* <math>m</math> is the number of ore types (floatability classes)
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>d_i</math> is the size of the square mesh interval that feed mass is retained on (mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{n}=0</math> mm)
* <math>d_{i+1}<d_i<d_{i-1}</math>, i.e. descending size order from top size (<math>d_{1}</math>) to sub mesh (<math>d_{n}=0</math> mm)
* <math>Q_{M,F}</math>  is the mass flow rate of particles in the feed (t/h)
* <math>Q_{\rm M,F}</math>  is the mass flow rate of particles in the feed (t/h)
* <math>SG</math> is the Specific Gravity or density of solids (- or t/m<sup>3</sup>)
* <math>\rho_{\rm S}</math> is the Specific Gravity or density of solids (- or t/m<sup>3</sup>)


=== Results ===
=== Results ===
Line 334: Line 299:
\begin{bmatrix}
\begin{bmatrix}
\text{Iterations}\\
\text{Iterations}\\
V_{effective}\text{ (m}^3\text{)}\\
V_{{\rm effective}}\text{ (m}^3\text{)}\\
J_g\text{ (m/s)}\\
J_{\rm g}\text{ (m/s)}\\
\epsilon_{gas}\text{ (v/v)}\\
\varepsilon_{\rm gas}\text{ (v/v)}\\
\tau\text{ (min)}\\
\tau\text{ (min)}\\
P_{80}\text{ (}\mu \text{m)}\\
P_{80}\text{ (}\mu \text{m)}\\
S_b\text{ (m}^2\text{/s/m}^2\text{)}\\
S_{\rm b}\text{ (m}^2\text{/s/m}^2\text{)}\\
FRT\text{ (min)}\\
{\rm FRT}\text{ (min)}\\
FRT_{ENT}\text{ (min)}\\
{\rm FRT}_{{\rm ENT}}\text{ (min)}\\
R_w\text{ (frac)}\\
R_{\rm w}\text{ (frac)}\\
(Q_{\rm M,C})_{\rm L}\text{ (t/h)}\\
(Q_{\rm M,T})_{\rm L}\text{ (t/h)}\\
R\text{ (frac)}\\
R\text{ (frac)}\\
\end{bmatrix}
\end{bmatrix}
Line 349: Line 316:


\begin{bmatrix}
\begin{bmatrix}
- & \cdots & -\\
- & \cdots & -\\
- & \cdots & -\\
- & \cdots & -\\
- & \cdots & -\\
- & \cdots & -\\
Line 373: Line 342:


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


Line 389: Line 358:


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


Line 421: Line 390:


\begin{bmatrix}
\begin{bmatrix}
ENT_{11}\text{ (frac)} & \dots & ENT_{1m}\text{ (frac)}\\
{\rm ENT}_{11}\text{ (frac)} & \dots & {\rm ENT}_{1m}\text{ (frac)}\\
\vdots & \ddots & \vdots\\
\vdots & \ddots & \vdots\\
ENT_{n1}\text{ (frac)} & \dots & ENT_{nm}\text{ (frac)}\\
{\rm ENT}_{n1}\text{ (frac)} & \dots & {\rm ENT}_{nm}\text{ (frac)}\\
\end{bmatrix}
\end{bmatrix}


Line 458: Line 427:


where:
where:
* <math>Iterations</math> is the number of iterations required to compute the tails-based residence time
* <math>\text{Iterations}</math> is the number of iterations required to compute the tails-based residence time
* <math>FRT_{ENT}</math> is the froth residence time (min) calculated for the entrainment method, which may be different to that calculated for froth recovery based on user air/slurry selections
* <math>{\rm FRT}_{{\rm ENT}}</math> is the froth residence time (min) calculated for the entrainment method, which may be different to that calculated for froth recovery based on user air/slurry selections
* <math>(Q_{\rm M,C})_{\rm L}</math> is the mass flow rate of liquids in the cell concentrate (t/h)
* <math>(Q_{\rm M,T})_{\rm L}</math> is the mass flow rate of liquids in the cell tailing (t/h)
* <math>R</math> is the overall recovery of all solids mass to concentrate (frac), i.e. the ''mass pull'', computed as <math>\textstyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{R_{ij}}}</math>
* <math>R</math> is the overall recovery of all solids mass to concentrate (frac), i.e. the ''mass pull'', computed as <math>\textstyle \sum_{i=1}^{n}{\sum_{j=1}^{m}{R_{ij}}}</math>
* <math>Q_{M,C}</math> is the mass flow rate of particles in the concentrate stream (t/h)
* <math>Q_{\rm M,C}</math> is the mass flow rate of particles in the concentrate stream (t/h)
* <math>Q_{M,T}</math> is the mass flow rate of particles in the tail stream (t/h)
* <math>Q_{\rm M,T}</math> is the mass flow rate of particles in the tail stream (t/h)
* <math>\bar{d}_i</math> is the [[Conversions|geometric mean size]] of the internal mesh series interval that mass is retained on (mm)
* <math>\bar{d}_i</math> is the [[Conversions|geometric mean size]] of the internal mesh series interval that mass is retained on (mm)
* <math>R_j</math> is the recovery of all solids of ore type <math>j</math> (frac), i.e. the ''unsized'' recovery of floatability components
* <math>R_j</math> is the recovery of all solids of ore type <math>j</math> (frac), i.e. the ''unsized'' recovery of floatability components
Line 477: Line 448:
| [[File:FlotationSavassi2.png|left|frame|Figure 2. Example showing the selection of the '''Feed''' (upper purple frame), '''OreSG ''' (green frame), '''Floatability''' (pink frame), '''CellScaleUp''' (brown frame), '''DetachRateConst''' (teal frame), '''FrothRecovery''' (blue frame), '''DrainageFactor''' (red frame), and '''Entrainment''' (lower purple frame) arrays in Excel.]]
| [[File:FlotationSavassi2.png|left|frame|Figure 2. Example showing the selection of the '''Feed''' (upper purple frame), '''OreSG ''' (green frame), '''Floatability''' (pink frame), '''CellScaleUp''' (brown frame), '''DetachRateConst''' (teal frame), '''FrothRecovery''' (blue frame), '''DrainageFactor''' (red frame), and '''Entrainment''' (lower purple frame) arrays in Excel.]]
|-
|-
| [[File:FlotationSavassi3.png|left|thumb|x1300px|Figure 3. Example showing the outline of the Results (blue shaded area) array in Excel.]]
| [[File:FlotationSavassi3.png|left|thumb|x1350px|Figure 3. Example showing the outline of the Results (blue shaded area) array in Excel.]]
|}
|}


== SysCAD ==
== SysCAD ==


{{Under construction|section}}
The sections and variable names used in the SysCAD interface are described in detail in the following tables.
 
{{SysCAD (Page, Flotation, DLL*Flotation)}}
 
{{SysCAD (Page, Flotation Cell, Savassi, Cell)|method=0}}
 
{{SysCAD (Page, Flotation Cell, Savassi, CellScaleUp)}}
 
{{SysCAD (Page, Flotation Cell, Savassi, Froth)}}
 
{{SysCAD (Page, Flotation Cell, Savassi, Entrainment)}}
 
{{SysCAD (Page, Flotation Cell, Savassi, Floatability)}}
 
{{SysCAD (Page, Hydrocyclone, Partition)|ActionU=Recovery|ActionL=recovery|DestinationU=Concentrate|DestinationL=concentrate|UnitL=cell}}
 
{{SysCAD (Page, About)}}
 
==== Additional notes ====
 
{{SysCAD (Text, No PSD Splits)|gasstream=tailings}}
 
== External links ==
 
* [https://help.syscad.net/Met_Dynamics_-_Flotation Met Dynamics - Flotation (help.syscad.net)]
* [https://help.syscad.net/Example_-_09_Met_Dynamics_Projects#Crushing,_Grinding_and_Flotation_Example Crushing, Grinding and Flotation Example project (help.syscad.net)]


== References ==
== References ==

Latest revision as of 13:32, 1 May 2025

Description

This article describes a model that estimates the recovery of ore and gangue from a mechanical flotation cell.

The approach is outlined by Savassi (2005), who proposed a compartment model that separates the mechanisms of true flotation, froth recovery, and entrainment.[1]

Savassi's original model was further extended over many years by subsequent researchers, and the resulting methodology is frequently referred to as the "P9 model" in reference to the AMIRA P9 project that has driven much of the development since.[2]

The model described here is for steady-state simulation. For dynamic simulation, see Flotation Cell (Savassi, Dynamic).

Model theory

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Floatability classes

Recovery

Pulp volume

Gas hold-up

Residence time

Bubble surface area flux

Froth recovery

Entrainment

No entrainment

Water recovery

Water rate function

Excel

The Savassi flotation cell model may be invoked from the Excel formula bar with the following function call: 

=mdUnit_Flotation_Savassi(Parameters as Range, Size as Range, Feed as Range, OreSG As Range, Floatability as Range, Optional CellScaleUp As Range = Nothing, Optional FrothRecovery as Range = Nothing, Optional Entrainment as Range = Nothing, Optional DetachRateConst as Range = Nothing, Optional DrainageFactor as Range = Nothing)

Several of the function input parameters are marked as Optional:

  • If CellScaleUp is omitted, cell scale up ignored and for all and
  • If FrothRecovery is omitted, froth recovery is ignored and for all and
  • If Entrainment is omitted, recovery by entrainment is ignored and for all and
  • If DetachRateConst is omitted, for all
  • If DrainageFactor is omitted, for all

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 () x column () format:



where:

  • specifies the method used to determine the gas-hold up, 0 = No gas holdup, 1 = User defined, 2 = Jg dependent
  • is the user-defined value of gas hold-up (v/v) if , ignored otherwise
  • is the value of the slope coefficient if , ignored otherwise
  • is the value of the slope coefficient if , ignored otherwise
  • specifies the method used to determine the residence time, 0 = User defined, 1 = Calculated
  • specifies the cell flow configuration, 0 = Batch or Plug flow, 1 = Perfect Mixing (Continuous)
  • specifies the stream which is used to computed the residence time, 0 = Feed, 1 = Tail
  • specifies the method used to determine the bubble surface area, 0 = User defined, 1 = Gorain, 2 = User equation
  • is the user-defined value of bubble surface area flux (m2/s/m2) if , ignored otherwise
  • indicates whether the for the Gorain equation () is user-defined or determined from the feed stream , 0 = User defined, 1 = Calculated P80
  • is the user-defined value of (μm) if and , ignored otherwise
  • is the user-defined value of impeller speed (m/s) if , ignored otherwise
  • is the user-defined value of impeller aspect ratio (m/m) if , ignored otherwise
  • is a user-defined coefficient of the Gorain relation if , ignored otherwise
  • are user-defined exponents of the Gorain relation if , ignored otherwise
  • specifies the method used to determine froth recovery, 0 = User defined, 1 = FRT Entrain, 2 = FRT User Np
  • specifies the phase which is used to computed the froth residence time, 0 = Air, 1 = Slurry, if , ignored otherwise
  • is the value of the gas hold up fraction (v/v) if , ignored otherwise
  • specifies the method used to determine entrainment, 0 = No entrainment, 1 = User defined, 2 = Hyperbolic, 3 = FRT dependent
  • is the value of the hyperbolic equation particle size term (μm) if , ignored otherwise
  • is the value of the hyperbolic equation drainage factor (-) if , ignored otherwise
  • specifies the phase which is used to computed the froth residence time for entrainment, 0 = Air, 1 = Slurry, if , ignored otherwise
  • specifies the method used to determine water recovery, 0 = Fixed % solids, 1 = Solid dependent, 2 = rate function, 3 = User defined
  • is the user-defined value of concentrate pulp density (w/w) if , ignored otherwise
  • is a user-defined coefficient if , ignored otherwise
  • is a user-defined exponent if , ignored otherwise
  • is a user-defined kinetic rate parameter if , ignored otherwise
  • is a user-defined water split to concentrate if , ignored otherwise
  • is the mass flow feed rate of liquids into the cell (t/h)
  • is the Specific Gravity or density of liquids in the feed (- or t/m3)
  • is the number of size intervals
  • is the number of ore types (floatability classes)
  • is the size of the square mesh interval that feed mass is retained on (mm)
  • , i.e. descending size order from top size () to sub mesh ( mm)
  • is the mass flow rate of particles in the feed (t/h)
  • is the Specific Gravity or density of solids (- or t/m3)

Results

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

where:

  • is the number of iterations required to compute the tails-based residence time
  • is the froth residence time (min) calculated for the entrainment method, which may be different to that calculated for froth recovery based on user air/slurry selections
  • is the mass flow rate of liquids in the cell concentrate (t/h)
  • is the mass flow rate of liquids in the cell tailing (t/h)
  • is the overall recovery of all solids mass to concentrate (frac), i.e. the mass pull, computed as
  • is the mass flow rate of particles in the concentrate stream (t/h)
  • is the mass flow rate of particles in the tail stream (t/h)
  • is the geometric mean size of the internal mesh series interval that mass is retained on (mm)
  • is the recovery of all solids of ore type (frac), i.e. the unsized recovery of floatability components

Example

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

Figure 1. Example showing the selection of the Parameters (blue frame) array in Excel.
Figure 2. Example showing the selection of the Feed (upper purple frame), OreSG (green frame), Floatability (pink frame), CellScaleUp (brown frame), DetachRateConst (teal frame), FrothRecovery (blue frame), DrainageFactor (red frame), and Entrainment (lower purple frame) arrays in Excel.
Figure 3. Example showing the outline of the Results (blue shaded area) array in Excel.

SysCAD

The sections and variable names used in the SysCAD interface are described in detail in the following tables.

MD_Flotation 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 MassFracToCon option appears below.
MassFracToCon Input Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the concentrate stream when the model is off.
Method User defined The recovery to concentrate for each size interval is defined by the user. Different values can be used for different solids.
Savassi The Savassi model is used to determine the recovery of solids and liquids to concentrate.
Options
ShowQFeed CheckBox QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream.
ShowQCon CheckBox QCon and associated tab pages (eg Sp) will become visible, showing the properties of the concentrate stream.
ShowQTail CheckBox QTail and associated tab pages (eg Sp) will become visible, showing the properties of the tailings stream.
SizeForPassingFracCalc Input Size fraction for % Passing calculation. The size fraction input here will be shown in the Stream Summary section.
FracForPassingSizeCalc Input 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.

Cell page

The Cell page is used to specify the input parameters for the flotation model.

Tag (Long/Short) Input / Display Description/Calculated Variables/Options
Savassi
HelpLink ButtonModelHelp.png Opens a link to this page using the system default web browser. Note: Internet access is required.
Iterations Display Shows the number of internal model iterations (per SysCAD step) required to converge the cell model.
Cell
CellVolume/ VCell Input The total volume of the cell.
MechanismVolume / VMechansim Input Volume of the impeller mechanism in the cell.
CellArea / ACell Input Cross sectional area of the cell.
FrothDepth / Hf Input Froth depth from the top of the cell.
AirRate/ QAir Input Flow rate of air into the cell.
SuperficialGasRate / Jg Display The superficial velocity of air through the cell.
EffCellPulpVolume / VEffective Display Volume of cell available for pulp.
GasHoldUp
GasHoldUpModel No Gas Hold Up No gas is held up in the pulp, i.e. EGas = 0
User Defined A fixed value of EGas is entered by the user,
Jg Dependent Gas hold up, EGas is computed from the value of Jg, mGas and CGas
mGas Input Only displayed if Jg Dependent is selected. Value of slope parameter of EGas relationship.
CGas Input Only displayed if Jg Dependent is selected. Value of intercept parameter of EGas relationship.
EGas Input / Display Fraction of pulp volume occupied by gas.
ResidenceTime
ResTimeOption User Defined The user specifies a fixed cell residence time.
Calculated Cell residence time is calculated from pulp volume and flow rate.
FlowPattern Plug Flow Recovery is calculated using the plug flow equation.
Perfect Mixing Recovery is calculated using the perfect mixing equation.
ResTimeBasis Feed Residence time is calculated using the volumetric flow rate of the feed stream.
Tailing Residence time is calculated using the volumetric flow rate of the tailing stream.
CellResidenceTime / tau Input / Display Residence time of pulp in the cell.
BubbleFlux
SbOption User Defined The user specifies a fixed value of Sb.
Gorain Sb is calculated by the Gorain equation.
User Equation Sb is calculated by the an equation with user-specified parameters.
Mean Diameter Sb is calculated using the mean bubble diameter.
ImpellerTipSpeed / Ns Input Only visible if Gorain or User Equation is selected. Speed of the impeller tip.
ImpellerAspectRatio / As Input Only visible if Gorain or User Equation is selected. Aspect ratio of the impeller.
UseCalculatedP80 CheckBox Only visible if Gorain or User Equation is selected. Specifies whether the feed stream P80 value should be used in the Sb equation. If not, P80 is user defined.
C Input Only visible if User Equation is selected. Coefficient of the user equation.
a Input Only visible if User Equation is selected. Exponent of the user equation.
b Input Only visible if User Equation is selected. Exponent of the user equation.
c Input Only visible if User Equation is selected. Exponent of the user equation.
d Input Only visible if User Equation is selected. Exponent of the user equation.
e Input Only visible if User Equation is selected. Exponent of the user equation.
MeanBubbleDiameter / db Input Only visible if Mean Diameter is selected. Sauter mean bubble diameter.
Sb Display Value of the bubble surface area flux.
WaterRecovery
WaterOption Fixed % Solids The user specifies the required concentrate solids mass fraction.
Solids Dependent Water is recovered to concentrate according to the solids dependent power law function.
Rate Function Water is recovered to concentrate according to the rate function kinetic equation.
User Defined The fraction of feed water recovered to concentrate is specified by the user.
QCon.SfReqd Input Only visible if Fixed % Solids is selected. Required concentrate solids mass fraction.
aW Input Only visible if Solids Dependent is selected. Parameter of the power law function.
bW Input Only visible if Solids Dependent is selected. Parameter of the power law function.
kW Input Only visible if Rate Function' is selected. Kinetic parameter of the rate function.
Rw Input / Display The fraction of feed water recovered to concentrate.

CellScaleUp page

This page is used to specify the cell scale up input parameters.

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.
CellScaleUp
Selection Stream A single CellScaleUp value is specified for all species and sizes.
Species CellScaleUp values are specified per species, for all sizes of that species.
Size CellScaleUp values are specified per size interval, for all species in that interval.
Species-Size Individual CellScaleUp values are specified for each size interval of each species.
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
CellScaleUp Input / Display Cell scale-up factor (C) for each size interval, in each solid species.

FlotationSavassi4.png FlotationSavassi5.png

FlotationSavassi6.png FlotationSavassi7.png

Froth page

This page is used to specify the froth recovery input parameters.

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.
Froth
Method User Defined The user specifies fractional froth recovery.
FRT Entrain The froth residence time model is used to determine fractional froth recovery values. The non-draining fraction is equal to calculated entrainment.
FRT User Np The froth residence time model is used to determine fractional froth recovery values. The non-draining fraction can be specified by the user.
Selection Stream Only visible if User Defined Method is selected. A single froth recovery value is specified for all species and sizes.
Species Only visible if User Defined Method is selected. Froth recovery values are specified per species, for all sizes of that species.
Size Only visible if User Defined Method is selected. Froth recovery values are specified per size interval, for all species in that interval.
Species-Size Only visible if User Defined Method is selected. Individual froth recovery values are specified for each size interval of each species.
FrothVoidage / Efroth Input Only visible if FRT Entrain or FRT User Np methods are selected. Volumetric fraction of void space in the froth.
FrothResTimeBasis Air Only visible if FRT Entrain or FRT User Np methods are selected. Froth residence time is determined based on the superficial gas velocity (Jg).
Slurry Only visible if FRT Entrain or FRT User Np methods are selected. Froth residence time is determined based on the volumetric flow rate of concentrate.
FrothResidenceTime / FRT Input / Display Only visible if FRT Entrain or FRT User Np methods are selected. Residence time in froth.
DetachRateConst / Beta Input Only visible if FRT Entrain or FRT User Np methods are selected. Detachment rate constant per solid species.
NonDrainingFrac / Np Input Only visible if FRT Entrain or FRT User Np methods are selected. Non-draining fraction per solid species.
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
FrothRecovery Input / Display Fractional recovery in froth (Rf) for each size interval, in each solid species.

FlotationSavassi4.png FlotationSavassi5.png

FlotationSavassi6.png FlotationSavassi7.png

Entrainment page

This page is used to specify the entrainment input parameters.

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.
Entrainment
Method No Entrainment. No entrainment, all values of ENTp are zero.
User Defined The user specifies fractional recovery by entrainment.
Hyperbolic The Hyperbolic model is used to determine fractional recovery by entrainment values.
FRT Dependent The froth residence time approach is used to determine fractional recovery by entrainment values.
Selection Stream Only visible if User Defined Method is selected. A single recovery by entrainment value is specified for all species and sizes.
Species Only visible if User Defined Method is selected. Recovery by entrainment values are specified per species, for all sizes of that species.
Size Only visible if User Defined Method is selected. Recovery by entrainment values are specified per size interval, for all species in that interval.
Species-Size Only visible if User Defined Method is selected. Individual recovery by entrainment values are specified for each size interval of each species.
E20 Input Only visible if Hyperbolic Method is selected. Parameter of the Hyperbolic entrainment equation.
DrainageParameter / Delta Input Only visible if Hyperbolic Method is selected. Parameter of the Hyperbolic entrainment equation.
FrothVoidage / Efroth Input / Display Only visible if FRT Dependent Method is selected. Volumetric fraction of void space in the froth.

Equal to the FrothVoidage input on the Froth page if the Froth Residence Time froth recovery method is selected, otherwise a user defined value.

FrothResTimeBasis Air Only visible if FRT Dependent Method is selected. Froth residence time for recovery by entrainment is determined based on the superficial gas velocity (Jg).
Slurry Only visible if FRT Dependent Method is selected. Froth residence time for recovery by entrainment is determined based on the volumetric flow rate of concentrate.
FrothResidenceTime / FRT Input / Display Only visible if FRT Dependent Method is selected. Residence time in froth for recovery by entrainment determination.

May be different to FrothResidenceTime on Froth page if different FrothResTimeBasis (Air/Slurry) is selected for each.

Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
DrainageFactor Input Only visible if FRT Dependent Method is selected. Drainage factors () for the FRT entrainment equation, for each size interval.
Entrainment Input / Display Recovery by entrainment (ENTp) values for each size interval, in each solid species.

FlotationSavassi4.png FlotationSavassi5.png

FlotationSavassi6.png FlotationSavassi7.png

Floatability page

This page is used to specify the floatability input parameters.

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.
Floatability
Selection Stream A single floatability value is specified for all species and sizes.
Species Floatability values are specified per species, for all sizes of that species.
Size Floatability values are specified per size interval, for all species in that interval.
Species-Size Individual floatability values are specified for each size interval of each species.
Size Display Size of each interval in internal mesh series.
MeanSize Display Geometric mean size of each interval in internal mesh series.
Floatability Input Floatability parameters for each size interval, in each solid species.

Recovery page

The Recovery page is used to display (or specify) the recovery by species/component/element/individual phase 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.
Recovery
Method Model / User Select model-calculated or user-defined recovery 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 recovery to concentrate of all solid species, for each size interval.
  • Excludes solid species not present in the cell feed.
Recovery Display
  • Recovery to concentrate 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 cell feed or not.
All (All row, All column) Display
  • Displays the actual, total, recovery of all solids with a particle size distribution property in the feed to concentrate.
  • Excludes solid species not present in the cell feed.
All (All row, per species) Display
  • Actual overall recovery to concentrate for each solid species, for all size intervals in that species.
  • Excludes solid species not present in the cell feed.
CmpRecovery
Components
ButtonHideComponents.png
ButtonShowComponents.png
 Hides or shows component recovery table.
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 recovery to concentrate of all solid components, for each size interval.
  • Excludes solid components not present in the cell feed.
CmpRecovery Display
  • Recovery to concentrate for each size interval, in each solid component, as determined by the selected model or user defined value.
  • Excludes solid components not present in the cell feed.
All (All row, All column) Display
  • Displays the actual, total, recovery of all solid components with a particle size distribution property in the feed to concentrate.
  • Excludes solid components not present in the cell feed.
All (All row, per component) Display
  • Actual overall recovery to concentrate for each solid component, for all size intervals in that component.
  • Excludes solid components not present in the cell feed.
EleRecovery
Elements
ButtonHideElements.png
ButtonShowElements.png
 Hides or shows element recovery table.
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 recovery to concentrate of all solid elements, for each size interval.
  • Excludes solid elements not present in the cell feed.
EleRecovery Display
  • Recovery to concentrate for each size interval, in each solid element, as determined by the selected model or user defined value.
  • Excludes solid elements not present in the cell feed.
All (All row, All column) Display
  • Displays the actual, total, recovery of all solid elements with a particle size distribution property in the feed to concentrate.
  • Excludes solid elements not present in the cell feed.
All (All row, per element) Display
  • Actual overall recovery to concentrate for each solid element, for all size intervals in that component.
  • Excludes solid elements not present in the cell feed.
IPhRecovery
IPhases
ButtonHideIPhases.png
ButtonShowIPhases.png
 Hides or shows individual phases recovery table.
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 recovery to concentrate of all solid individual phases, for each size interval.
  • Excludes solid individual phases not present in the cell feed.
IPhRecovery Display
  • Recovery to concentrate for each size interval, in each solid individual phase, as determined by the selected model or user defined value.
  • Excludes solid individual phases not present in the cell feed.
All (All row, All column) Display
  • Displays the actual, total, recovery of all solid individual phases with a particle size distribution property in the feed to concentrate.
  • Excludes solid individual phases not present in the cell feed.
All (All row, per individual phase) Display
  • Actual overall recovery to concentrate for each solid individual phase, for all size intervals in that component.
  • Excludes solid individual phases not present in the cell feed.

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 ButtonLicensingHelp.png Opens a link to the Installation and Licensing page using the system default web browser. Note: Internet access is required.
Information ButtonCopyToClipboard.png 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 ButtonBrowse.png 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.

Additional notes

  • Solid species that do not possess a particle size distribution property are split according to the overall mass split of the default particle size distribution species selected in the SysCAD Project Configuration.
  • If the default particle size distribution species is not present in the unit feed, the overall split of all other species with particle size distributions combined is used, as determined by the model.
  • Gas phase species report directly to the tailings stream without split.

External links

References

  1. ↑ Savassi, O.N., 2005. A compartment model for the mass transfer inside a conventional flotation cell. International Journal of Mineral Processing, 77(2), pp.65-79.
  2. ↑ Coleman, R.G., Franzidis, J.P. and Manlapig, E., 2007. Validation of the AMIRA P9 flotation model using the floatability characterisation test rig (FCTR). Ninth Mill Operators’ Conference, Fremantle, WA, 19 - 21 March 2007.
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