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.^{[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]}
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.^{[3]} 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, nonfloating) within mineral types (e.g. chalcopyrite, pyrite, nonsulphide 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.^{[3]}^{[4]}
Recovery is expressed as:
where:
 is the index of the size interval, , is the number of size intervals
 is the index of the floatability component, , is the number of floatability components (e.g. mineral types, fast/slow types etc.)
 is the fraction of particles of size fraction and floatability component type recovered to concentrate (frac)
 is the floatability of particles of size and floatability component type (frac)
 is the bubble surface area flux (m^{2}/s/m^{2})
 is the fraction of particles of size and floatability component recovered by froth (frac)
 is a cell scaleup factor that relates flotation performance between cells of different sizes (e.g. laboratory versus industrial) (frac)
 is the residence time of slurry in the cell (min)
 is the fraction of particles of size and floatability component recovered by entrainment (frac)
 is the fraction of feed water recovered to concentrate (frac)
The cell scaleup factor, , allows for the simulation of industrialscale cells using kinetic rates derived from laboratory cells.^{[5]}
Pulp volume
Pulp volume, (m^{3}), is related to cell paramters.
where:
 is the total volume of the cell
 is the volume of the impeller mechanism in the cell (m^{3})
 is the cross sectional area of the cell (m^{2})
 is the froth depth from the top of the cell (m)
 is the gas holdup in the pulp (v/v)
Gas holdup
Gas holdup, (v/v), may be estimated from a linear relationship as suggested by the findings of Gorain et al. (1995):^{[6]}
where and are the slope an intercept parameters, respectively, of the linear equation.
The superficial gas rate, (m/s), is approximated by:^{[7]}
where is the air rate (m^{3}/h).
Residence time
Residence time, (min), may be computed on the basis of the cell feed or tailing stream volumetric flow rates.^{[8]}
where and are the volumetric flow rates of pulp in the feed and tail streams, respectively (m^{3}/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 tailbased residence time.
Bubble surface area flux
The bubble surface area flux, (m^{2}/s/m^{2}), is related to the superficial gas rate, , and Sauter mean bubble diameter, (m), by the equation:
The Sauter mean bubble diameter may be determined from gas dispersion measurements. Where a measurement of is not available, may be estimated by the empirical relation of Gorain et al. (1999):^{[7]}
where:
 is impeller tip speed (m/s)
 is impeller aspect ratio (m/m), the ratio of impeller diameter (m) to impeller height (m)
 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:
where and are a userdefined coefficient and set of exponents, respectively.
Froth recovery
Froth recovery, , may be estimated using the froth residence time model outlined by Harris et al. (2002).^{[9]}
where is the probability of detachment of ore type (frac), computed as:
and is the detachment rate constant (min^{1}) for ore type .
The froth residence time, (min), is computed based on either air or slurry flow rates:
where is the gas holdup fraction in the froth phase (v/v), and is the volumetric flow rate of concentrate from the cell.
The concentrate flow rate, , is a function of the recoveries of solids computed by the Savassi equation and water. As with the tailbased residence time (see above), an iterative procedure is required to estimate froth recovery via the froth residence time method (even if a feedbasis 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).^{[10]}^{[11]}
The hyperbolic equation for entrainment is:
where:
 is the geometric mean size of particles in size interval (µm)
 is the particle size at which is 0.2 (µm)
 is the froth drainage parameter
The froth residence time approach first computes the froth residence time () according to the method descibed for froth recovery above. Entrainment is then estimated by applying the following equation:
where is the froth drainage factor for particles in size interval (min^{1}).
When the effects of entrainment are ignored, the recovery equation reduces to:
Water recovery
Methods for estimating the recovery of water are described by Harris et al. (2002).^{[9]}
The fixed concentrate percent solids method assumes the concentrate is always recovered at a fixed pulp density. The recovery of water, , 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, (m^{3}/h), and solids, (m^{3}/h), in the concentrate stream:
where and 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:
where 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 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 gashold up, 0 = No gas holdup, 1 = User defined, 2 = Jg dependent
 is the userdefined value of gas holdup (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 userdefined value of bubble surface area flux (m^{2}/s/m^{2}) if , ignored otherwise
 indicates whether the for the Gorain equation () is userdefined or determined from the feed stream , 0 = User defined, 1 = Calculated P80
 is the userdefined value of (μm) if and , ignored otherwise
 is the userdefined value of impeller speed (m/s) if , ignored otherwise
 is the userdefined value of impeller aspect ratio (m/m) if , ignored otherwise
 is a userdefined coefficient of the Gorain relation if , ignored otherwise
 are userdefined exponents of the Gorain relation if , ignored otherwise
 specifies the method used to determine froth recovery, 0 = User defined, 1 = FRT
 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 userdefined value of concentrate pulp density (w/w) if , ignored otherwise
 is a userdefined coefficient if , ignored otherwise
 is a userdefined exponent if , ignored otherwise
 is a userdefined kinetic rate parameter if , ignored otherwise
 is a userdefined 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/m^{3})
 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/m^{3})
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 tailsbased 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.
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 userspecified size (in the field SizeForPassingFracCalc) in each stream. 
Passes  Display  The userspecified (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.
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.  
SpeciesSize  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 scaleup 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.  
SpeciesSize  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.  
SpeciesSize  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. 
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.  
SpeciesSize  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 modelcalculated or userdefined 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 

Recovery  Display 

All (All row, All column)  Display 

All (All row, per species)  Display 

About page
This page is provides product and licensing information about the Met Dynamics Models SysCAD AddOn.
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 AddOn. 
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, MDSysCAD Full or MDSysCAD 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.
References
 ↑ Savassi, O.N., 2005. A compartment model for the mass transfer inside a conventional flotation cell. International Journal of Mineral Processing, 77(2), pp.6579.
 ↑ 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.
 ↑ ^{3.0} ^{3.1} Runge, K.C., Alexander, D.J., Franzidis, J.P., 'Morrison, R.D. and Manlapig, E., 1998. JKSimFloata tool for flotation modelling', AusIMM ‘98  The Mining Cycle, Mount Isa, 19  23 April 1998, pp 361  370.
 ↑ Schwarz, S., Alexander, D., Whiten, W.J., Franzidis, J.P. and Harris, M.C., 2006, January. JKSimFloat V6: improving flotation circuit performance and understanding. In XXIII International Mineral Processing Congress Proceedings, Istanbul, Turkey.
 ↑ Runge, K. C., Franzidis, J., and Manlapig, E. (2003). Structuring a flotation model for robust prediction of flotation circuit performance. IMPC 2003, Cape Town, South Africa, 29 Sept  3 Oct, 2003. Marshalltown, South Africa: SAIMM.
 ↑ Gorain, B.K., Franzidis, J.P. and Manlapig, E.V., 1995. Studies on impeller type, impeller speed and air flow rate in an industrial scale flotation cell part 2: Effect on gas holdup. Minerals Engineering, 8(12), pp.15571570.
 ↑ ^{7.0} ^{7.1} Gorain, B.K., Franzidis, J.P. and Manlapig, E.V., 1999. The empirical prediction of bubble surface area flux in mechanical flotation cells from cell design and operating data. Minerals Engineering, 12(3), pp.309322.
 ↑ Schwarz, S. and Alexander, D., 2006. JKSimFloat V6.1 Plus: Improving flotation circuit performance by simulation. In Mineral Process Modelling, Simulation and Control  Conference Proceedings, Laurentian University, Sudbury, Ontario, Canada, June 67, 2006.
 ↑ ^{9.0} ^{9.1} Harris, M., Runge, K., Whiten, W. and Morrison, R., 2002. JK–SimFloat as a practical tool for flotation process design and optimization. Mineral processing plant design, practice and control: Proceedings, Society for mining, metallurgy and exploration. Society of Mining Engineers.
 ↑ Savassi, O.N., Alexander, D.J., Franzidis, J.P. and Manlapig, E.V., 1998. An empirical model for entrainment in industrial flotation plants. Minerals Engineering, 11(3), pp.243256.
 ↑ Vera, M.A., Mathe, Z.T., Franzidis, J.P., Harris, M.C., Manlapig, E.V. and O'Connor, C.T., 2002. The modelling of froth zone recovery in batch and continuously operated laboratory flotation cells. International Journal of Mineral Processing, 64(23), pp.135151.