Flotation Cell (Savassi)

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.

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.

Model theory

Floatability classes

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. 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.

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.

Recovery

Savassi's compartment model describes the recovery of particles by the mechanisms of true flotation, froth mass transfer and entrainment.

The compartment model may be formulated for both batch/plug flow and continuous (perfectly mixed) flow patterns.

Recovery is expressed as:

$R_{ij}={\begin{cases}1-\exp \left(-P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau \right)\cdot (1-{\rm {ENT}}_{ij}\cdot R_{\rm {w}})&{\text{for batch or plug flow flotation}}\\\\{\dfrac {P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau \cdot (1-R_{\rm {w}})+{\rm {ENT}}_{ij}\cdot R_{\rm {w}}}{(1+P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau )(1-R_{\rm {w}})+{\rm {ENT}}_{ij}\cdot R_{\rm {w}}}}&{\text{for continuous flotation}}\\\end{cases}}$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $j$ is the index of the floatability component, $j=\{1,2,\dots ,m\}$, $m$ is the number of floatability components (e.g. mineral types, fast/slow types etc.)
• $R_{ij}$ is the fraction of particles of size fraction $i$ and floatability component type $j$ recovered to concentrate (frac)
• $P_{ij}$ is the floatability of particles of size $i$ and floatability component type $j$ (frac)
• $S_{\rm {b}}$ is the bubble surface area flux (m²/s/m²)
• $Rf_{ij}$ is the fraction of particles of size $i$ and floatability component $j$ recovered by froth (frac)
• $C_{ij}$ is a cell scale-up factor that relates flotation performance between cells of different sizes (e.g. laboratory versus industrial) (frac)
• $\tau$ is the residence time of slurry in the cell (min)
• ${\rm {ENT}}_{ij}$ is the fraction of particles of size $i$ and floatability component $j$ recovered by entrainment (frac)
• $R_{\rm {w}}$ is the fraction of feed water recovered to concentrate (frac)

The cell scale-up factor, $C_{ij}$, allows for the simulation of industrial-scale cells using kinetic rates derived from laboratory cells.

Pulp volume

Pulp volume, $V_{\rm {effective}}$ (m³), is related to cell paramters.

$V_{\rm {effective}}=(V_{\rm {cell}}-V_{\rm {mechanism}}-A_{\rm {cell}}*H_{\rm {f}})\cdot (1-\varepsilon _{\rm {gas}})$

where:

• $V_{\rm {cell}}$ is the total volume of the cell
• $V_{\rm {mechanism}}$ is the volume of the impeller mechanism in the cell (m³)
• $A_{\rm {cell}}$ is the cross sectional area of the cell (m²)
• $H_{\rm {f}}$ is the froth depth from the top of the cell (m)
• $\varepsilon _{\rm {gas}}$ is the gas hold-up in the pulp (v/v)

Gas hold-up

Gas hold-up, $\varepsilon _{\rm {gas}}$ (v/v), may be estimated from a linear relationship as suggested by the findings of Gorain et al. (1995):

$\varepsilon _{\rm {gas}}=m_{\rm {gas}}\cdot J_{\rm {g}}+C_{\rm {gas}}$

where $m_{\rm {gas}}$ and $C_{\rm {gas}}$ are the slope an intercept parameters, respectively, of the linear equation.

The superficial gas rate, $J_{\rm {g}}$ (m/s), is approximated by:

$J_{\rm {g}}={\dfrac {Q_{\rm {air}}}{A_{\rm {cell}}\cdot 60\cdot 60}}$

where $Q_{\rm {air}}$ is the air rate (m³/h).

Residence time

Residence time, $\tau$ (min), may be computed on the basis of the cell feed or tailing stream volumetric flow rates.

$\tau ={\begin{cases}{\dfrac {V_{\rm {effective}}}{Q_{\rm {feed}}\cdot 60}}&{\text{for feed-based residence time}}\\\\{\dfrac {V_{\rm {effective}}}{Q_{\rm {tail}}\cdot 60}}&{\text{for tail-based residence time}}\\\end{cases}}$

where ${Q_{\rm {feed}}}$ and ${Q_{\rm {tail}}}$ are the volumetric flow rates of pulp in the feed and tail streams, respectively (m³/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.

Bubble surface area flux

The bubble surface area flux, $S_{\rm {b}}$ (m²/s/m²), is related to the superficial gas rate, $J_{\rm {g}}$, and Sauter mean bubble diameter, $d_{\rm {b}}$ (m), by the equation:

$S_{\rm {b}}={\dfrac {6J_{\rm {g}}}{d_{\rm {b}}}}$

The Sauter mean bubble diameter may be determined from gas dispersion measurements. Where a measurement of $d_{\rm {b}}$ is not available, $S_{\rm {b}}$ may be estimated by the empirical relation of Gorain et al. (1999):

$S_{\rm {b}}=123{N_{\rm {s}}}^{0.44}{J_{\rm {g}}}^{0.75}{A_{\rm {s}}}^{-0.10}{P_{80}}^{-0.42}$

where:

• $N_{\rm {s}}$ is impeller tip speed (m/s)
• $A_{\rm {s}}$ is impeller aspect ratio (m/m), the ratio of impeller diameter (m) to impeller height (m)
• $P_{80}$ 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:

$S_{\rm {b}}=C_{\rm {Sb}}{N_{\rm {s}}}^{a}{J_{\rm {g}}}^{b}{As}^{c}{P_{80}}^{d}$

where $C_{\rm {Sb}}$ and $a\dots d$ are a user-defined coefficient and set of exponents, respectively.

Froth recovery

Froth recovery, $Rf_{ij}$, may be estimated using the froth residence time model outlined by Harris et al. (2002).

$Rf_{ij}=(1-(P_{\rm {d}})_{j})+(P_{\rm {d}})_{j}\cdot {\rm {ENT}}_{i}\cdot R_{\rm {w}}$

where $(P_{\rm {d}})_{j}$ is the probability of detachment of ore type $j$ (frac), computed as:

$(P_{\rm {d}})_{j}=1-\exp(-\beta _{j}\cdot {\rm {FRT}})$

and $\beta _{j}$ is the detachment rate constant (min^-1) for ore type $j$.

The froth residence time, $FRT$ (min), is computed based on either air or slurry flow rates:

${\rm {FRT}}={\begin{cases}{\dfrac {H_{\rm {f}}\cdot {\varepsilon _{\rm {froth}}}}{J_{\rm {g}}}}&{\text{for air-based froth residence time}}\\{\dfrac {H_{\rm {f}}\cdot A_{\rm {cell}}\cdot (1-\varepsilon _{\rm {froth}})}{Q_{\rm {con}}}}&{\text{for slurry-based froth residence time}}\\\end{cases}}$

where $\varepsilon _{\rm {froth}}$ is the gas hold-up fraction in the froth phase (v/v), and $Q_{\rm {con}}$ is the volumetric flow rate of concentrate from the cell.

The concentrate flow rate, $Q_{\rm {con}}$, is a function of the recoveries of solids computed by the Savassi equation and water. As with the tail-based residence time (see 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).

The hyperbolic equation for entrainment is:

${\rm {ENT}}_{i}={\dfrac {2}{\exp \left(2.292\left({\dfrac {{\bar {d}}_{i}}{E20}}\right)^{adj}\right)+\exp \left(-2.292\left({\dfrac {{\bar {d}}_{i}}{E20}}\right)^{adj}\right)}}$

$adj=1+{\dfrac {\ln \delta }{\exp \left({\dfrac {{\bar {d}}_{i}}{E20}}\right)}}$

where:

• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (µm)
• $E20$ is the particle size at which ${\rm {ENT}}_{i}$ is 0.2 (µm)
• $\delta$ is the froth drainage parameter

The froth residence time approach first computes the froth residence time (${\rm {FRT}}$) according to the method descibed for froth recovery above. Entrainment is then estimated by applying the following equation:

${\rm {ENT}}_{i}={\dfrac {1}{1+\omega _{i}\cdot {\rm {FRT}}}}$

where $\omega _{i}$ is the froth drainage factor for particles in size interval $i$ (min^-1).

When the effects of entrainment are ignored, the recovery equation reduces to:

$R_{ij}={\begin{cases}1-\exp \left(-P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau \right)&{\text{for batch or plug flow flotation}}\\\\{\dfrac {P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau }{(1+P_{ij}\cdot (S_{\rm {b}}\cdot 60)\cdot Rf_{ij}\cdot C_{ij}\cdot \tau )}}&{\text{for continuous flotation}}\\\end{cases}}$

Water recovery

Methods for estimating the recovery of water are described by Harris et al. (2002).

The fixed concentrate percent solids method assumes the concentrate is always recovered at a fixed pulp density. The recovery of water, $R_{\rm {w}}$, 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, $Q_{\rm {w}}$ (m³/h), and solids, $Q_{\rm {s}}$ (m³/h), in the concentrate stream:

$Q_{\rm {w}}=a_{\rm {w}}\cdot {Q_{\rm {s}}}^{b_{\rm {w}}}$

where $a$ and $b$ are the coefficient and exponent of the power law relationship, respectively.

Finally, the water rate function adopts a first order kinetic relationship between water recovery and pulp residence time:

$R_{\rm {w}}={\dfrac {k_{\rm {w}}\tau }{1+k_{\rm {w}}\tau }}$

where $k_{\rm {w}}$ is the kinetic rate parameter (min^-1) for the recovery of water to the concentrate stream.

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 $C_{ij}=1$ for all $i$ and $j$
• If FrothRecovery is omitted, froth recovery is ignored and $Rf_{ij}=1$ for all $i$ and $j$
• If Entrainment is omitted, recovery by entrainment is ignored and ${\rm {ENT}}_{ij}=0$ for all $i$ and $j$
• If DetachRateConst is omitted, $\beta _{j}=0$ for all $j$
• If DrainageFactor is omitted, $\omega _{i}=0$ for all $i$

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 ($i$) x column ($j$) format:

$Parameters={\begin{bmatrix}V_{\rm {cell}}{\text{ (m}}^{\text{3)}}\\V_{\rm {mechanism}}{\text{ (m}}^{\text{3)}}\\A_{\rm {cell}}{\text{ (m}}^{\text{2)}}\\H_{\rm {f}}{\text{ (m)}}\\Q_{\rm {air}}{\text{ (m}}^{\text{3}}{\text{/h)}}\\{\text{Gas holdup model}}\\\varepsilon _{\rm {gas}}{\text{ (v/v)}}\\m_{\rm {gas}}\\C_{\rm {gas}}\\{\text{Residence time option}}\\{\text{Flow pattern}}\\{\text{Residence time basis}}\\\tau {\text{ (min)}}\\{\text{Sb option}}\\S_{\rm {b}}{\text{ (m}}^{2}{\text{/s/m}}^{2}{\text{)}}\\{\text{Use calculated P80}}\\P_{80}{\text{ (}}\mu {\text{m)}}\\N_{\rm {s}}{\text{ (m/s)}}\\A_{\rm {s}}{\text{ (m/m)}}\\C_{\rm {Sb}}\\a\\b\\d\\e\\{\text{Froth model}}\\{\text{Froth residence time basis}}\\\varepsilon _{\rm {froth}}{\text{ (v/v)}}\\{\text{Entrainment option}}\\E20{\text{ (}}\mu {\text{m)}}\\\delta \\{\text{FRT entrainment basis}}\\{\text{Water option}}\\{\text{Concentrate fraction solids (w/w)}}\\a_{\rm {w}}\\b_{\rm {w}}\\k_{\rm {w}}\\R_{\rm {w}}{\text{ (frac)}}\\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\rho _{\rm {L}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\;\;\;\;\;\;Feed={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\;\;\;\;\;\;OreSG={\begin{bmatrix}(\rho _{\rm {S}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {S}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}}$

$Floatability={\begin{bmatrix}P_{11}{\text{ (frac)}}&\dots &P_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\P_{n1}{\text{ (frac)}}&\dots &P_{nm}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;CellScaleUp={\begin{bmatrix}C_{11}&\dots &C_{1m}\\\vdots &\ddots &\vdots \\C_{n1}&\dots &C_{nm}\\\end{bmatrix}},\;\;\;\;\;\;FrothRecovery={\begin{bmatrix}Rf_{11}{\text{ (frac)}}&\dots &Rf_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\Rf_{n1}{\text{ (frac)}}&\dots &Rf_{nm}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;$

$Entrainment={\begin{bmatrix}{\rm {ENT}}_{11}{\text{ (frac)}}&\dots &{\rm {ENT}}_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\{\rm {ENT}}_{n1}{\text{ (frac)}}&\dots &{\rm {ENT}}_{nm}{\text{ (frac)}}\\\end{bmatrix}},\;\;\;\;\;\;DetachRateConst={\begin{bmatrix}\beta _{1}{\text{ (min}}^{\text{-1}}{\text{)}}&\cdots &\beta _{m}{\text{ (min}}^{\text{-1}}{\text{)}}\\\end{bmatrix}},\;\;\;\;\;\;DrainageFactor={\begin{bmatrix}\omega _{1}{\text{ (min}}^{\text{-1}}{\text{)}}\\\vdots \\\omega _{n}{\text{ (min}}^{\text{-1}}{\text{)}}\\\end{bmatrix}}$

where:

• ${\text{Gas holdup model}}$ specifies the method used to determine the gas-hold up, 0 = No gas holdup, 1 = User defined, 2 = Jg dependent
• $\varepsilon _{\rm {gas}}$ is the user-defined value of gas hold-up (v/v) if ${\text{Gas holdup model}}=1$, ignored otherwise
• $m_{\rm {gas}}$ is the value of the slope coefficient if ${\text{Gas holdup model}}=2$, ignored otherwise
• $C_{\rm {gas}}$ is the value of the slope coefficient if ${\text{Gas holdup model}}=2$, ignored otherwise
• ${\text{Residence time option}}$ specifies the method used to determine the residence time, 0 = User defined, 1 = Calculated
• ${\text{Flow pattern}}$ specifies the cell flow configuration, 0 = Batch or Plug flow, 1 = Perfect Mixing (Continuous)
• ${\text{Residence time basis}}$ specifies the stream which is used to computed the residence time, 0 = Feed, 1 = Tail
• ${\text{Sb option}}$ specifies the method used to determine the bubble surface area, 0 = User defined, 1 = Gorain, 2 = User equation
• $S_{\rm {b}}$ is the user-defined value of bubble surface area flux (m²/s/m²) if ${\text{Sb option}}=0$, ignored otherwise
• ${\text{Use calculated P80}}$ indicates whether the $P_{80}$ for the Gorain equation (${\text{Sb option}}=1,2$) is user-defined or determined from the feed stream , 0 = User defined, 1 = Calculated P80
• $P_{80}$ is the user-defined value of $P_{80}$ (μm) if ${\text{Sb option}}=1,2$ and ${\text{Use calculated P80}}=1$, ignored otherwise
• $N_{\rm {s}}$ is the user-defined value of impeller speed (m/s) if ${\text{Sb option}}=1,2$, ignored otherwise
• $A_{\rm {s}}$ is the user-defined value of impeller aspect ratio (m/m) if ${\text{Sb option}}=1,2$, ignored otherwise
• $C_{\rm {Sb}}$ is a user-defined coefficient of the Gorain relation if ${\text{Sb option}}=2$, ignored otherwise
• $a\dots d$ are user-defined exponents of the Gorain relation if ${\text{Sb option}}=2$, ignored otherwise
• ${\text{Froth model}}$ specifies the method used to determine froth recovery, 0 = User defined, 1 = FRT
• ${\text{Froth residence time basis}}$ specifies the phase which is used to computed the froth residence time, 0 = Air, 1 = Slurry, if ${\text{Froth model}}=1$, ignored otherwise
• $\varepsilon _{\rm {froth}}$ is the value of the gas hold up fraction (v/v) if ${\text{Froth model}}=1$, ignored otherwise
• ${\text{Entrainment option}}$ specifies the method used to determine entrainment, 0 = No entrainment, 1 = User defined, 2 = Hyperbolic, 3 = FRT dependent
• $E20$ is the value of the hyperbolic equation particle size term (μm) if ${\text{Entrainment option}}=2$, ignored otherwise
• $\delta$ is the value of the hyperbolic equation drainage factor (-) if ${\text{Entrainment option}}=2$, ignored otherwise
• ${\text{FRT entrainment basis}}$ specifies the phase which is used to computed the froth residence time for entrainment, 0 = Air, 1 = Slurry, if ${\text{Entrainment option}}=3$, ignored otherwise
• ${\text{Water option}}$ specifies the method used to determine water recovery, 0 = Fixed % solids, 1 = Solid dependent, 2 = rate function, 3 = User defined
• ${\text{Concentrate fraction solids}}$ is the user-defined value of concentrate pulp density (w/w) if ${\text{Water option}}=0$, ignored otherwise
• $a_{\rm {w}}$ is a user-defined coefficient if ${\text{Water option}}=1$, ignored otherwise
• $b_{\rm {w}}$ is a user-defined exponent if ${\text{Water option}}=1$, ignored otherwise
• $k_{\rm {w}}$ is a user-defined kinetic rate parameter if ${\text{Water option}}=2$, ignored otherwise
• $R_{\rm {w}}$ is a user-defined water split to concentrate if ${\text{Water option}}=3$, ignored otherwise
• $(Q_{\rm {M,F}})_{\rm {L}}$ is the mass flow feed rate of liquids into the cell (t/h)
• $\rho _{\rm {L}}$ is the Specific Gravity or density of liquids in the feed (- or t/m³)
• $n$ is the number of size intervals
• $m$ is the number of ore types (floatability classes)
• $d_{i}$ is the size of the square mesh interval that feed mass is retained on (mm)
• $d_{i+1}<d_{i}<d_{i-1}$, i.e. descending size order from top size ($d_{1}$) to sub mesh ($d_{n}=0$ mm)
• $Q_{\rm {M,F}}$ is the mass flow rate of particles in the feed (t/h)
• $\rho _{\rm {S}}$ 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:

$mdUnit\_Flotation\_Savassi={\begin{bmatrix}{\begin{bmatrix}{\text{Iterations}}\\V_{\rm {effective}}{\text{ (m}}^{3}{\text{)}}\\J_{\rm {g}}{\text{ (m/s)}}\\\varepsilon _{\rm {gas}}{\text{ (v/v)}}\\\tau {\text{ (min)}}\\P_{80}{\text{ (}}\mu {\text{m)}}\\S_{\rm {b}}{\text{ (m}}^{2}{\text{/s/m}}^{2}{\text{)}}\\{\rm {FRT}}{\text{ (min)}}\\{\rm {FRT}}_{\rm {ENT}}{\text{ (min)}}\\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)}}\\\end{bmatrix}}&{\begin{bmatrix}-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\-&\cdots &-\\\end{bmatrix}}\\{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,C}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,C}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,C}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}\\{\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,T}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,T}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,T}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}\\{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}Rf_{11}{\text{ (frac)}}&\dots &Rf_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\Rf_{n1}{\text{ (frac)}}&\dots &Rf_{nm}{\text{ (frac)}}\\\end{bmatrix}}\\{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}{\rm {ENT}}_{11}{\text{ (frac)}}&\dots &{\rm {ENT}}_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\{\rm {ENT}}_{n1}{\text{ (frac)}}&\dots &{\rm {ENT}}_{nm}{\text{ (frac)}}\\\end{bmatrix}}\\{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}R_{11}{\text{ (frac)}}&\dots &R_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\R_{n1}{\text{ (frac)}}&\dots &R_{nm}{\text{ (frac)}}\\\end{bmatrix}}\\{\begin{bmatrix}-\end{bmatrix}}&{\begin{bmatrix}R_{1}{\text{ (frac)}}&\dots &R_{m}{\text{ (frac)}}\end{bmatrix}}\end{bmatrix}}$

where:

• ${\text{Iterations}}$ is the number of iterations required to compute the tails-based residence time
• ${\rm {FRT}}_{\rm {ENT}}$ 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
• $(Q_{\rm {M,C}})_{\rm {L}}$ is the mass flow rate of liquids in the cell concentrate (t/h)
• $(Q_{\rm {M,T}})_{\rm {L}}$ is the mass flow rate of liquids in the cell tailing (t/h)
• $R$ is the overall recovery of all solids mass to concentrate (frac), i.e. the mass pull, computed as $\textstyle \sum _{i=1}^{n}{\sum _{j=1}^{m}{R_{ij}}}$
• $Q_{\rm {M,C}}$ is the mass flow rate of particles in the concentrate stream (t/h)
• $Q_{\rm {M,T}}$ is the mass flow rate of particles in the tail stream (t/h)
• ${\bar {d}}_{i}$ is the geometric mean size of the internal mesh series interval that mass is retained on (mm)
• $R_{j}$ is the recovery of all solids of ore type $j$ (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 overflow stream.
ShowQTail	CheckBox	QTail and associated tab pages (eg Sp) will become visible, showing the properties of the underflow 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		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.
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.
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.

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.
	Froth Residence Time	The froth residence time model is used to determine fractional froth recovery values.
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 Froth Residence Time Method is selected. Volumetric fraction of void space in the froth.
FrothResTimeBasis	Air	Only visible if Froth Residence Time Method is selected. Froth residence time is determined based on the superficial gas velocity (Jg).
	Slurry	Only visible if Froth Residence Time Method is selected. Froth residence time is determined based on the volumetric flow rate of concentrate.
FrothResidenceTime / FRT	Input / Display	Only visible if Froth Residence Time Method is selected. Residence time in froth.
Beta	Input	Only visible if Froth Residence Time Method is selected. Detachment rate constant 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.

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 ($\omega$) 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.

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 specify or display the recovery by species 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.

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