Hydrocyclone (NarasimhaMainza)
Description
This article describes the NarasimhaMainza model for hydrocyclone size classification.
Narasimha et al. (2014a) describe an empirical hydrocyclone model that improves on the Plitt and Nageswararao approaches with the addition of several features:^{[1]}
 A sharpness of separation equation
 A slurry viscosity term that includes the effects of very fine particles
 Terms for cyclone inclination, particle density, gforces, flow regime (Reynolds Number) and turbulent diffusion
Narasimha et al. (2014b) subsequently presented a multicomponent version of the same model, with revised equations for cut size and sharpness of separation per ore component:^{[2]}
Model theory
The NarasimaMainza model comprises an equation for the efficiency curve (partition to overflow) and several subequations that describe its parameters:
 the corrected cut size
 the sharpness of separation
 liquid recovery to underflow
In addition, a relationship between throughput and operating pressure is presented.
Efficiency curve
The NarasimhaMainza model applies the WhitenBeta efficiency curve to partition particles to the overflow stream:
where:
 is the index of the size interval, , is the number of size intervals
 is the fraction of particles of size interval in the feed reporting to the overflow stream (frac)
 is the geometric mean size of particles in size interval (mm)
 is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
 is the fraction of feed liquids (or fines) split to overflow (frac)
 is a parameter representing the sharpness of separation
 is a term introduced to accommodate the socalled fishhook effect, and controls the initial rise in the efficiency curve at finer sizes
 is computed to ensure the WhitenBeta function preserves the definition of in the presence of the fishhook, i.e. at
Corrected cut size
The corrected cut size, (m), is computed from:
where:
 is a calibration factor which should be fitted to operating data
 is diameter of the cyclone (m)
 is diameter of a circular inlet or the diameter of a circle with the same area as a noncircular inlet (m)
 is diameter of the vortex finder (overflow) (m)
 is diameter of the apex/spigot (underflow) (m)
 is length of the cylindrical section (m)
 is the cone full angle (deg.)
 is the angle of inclination from the vertical (rad)
 is the volume fraction of solids in the feed (v/v)
 is the density of solids in the feed (t/m^{3})
 is the density of the fluid (liquids) in the feed (t/m^{3})
 is acceleration due to gravity (m/s^{2})
The Reynolds Number, , is:
The feed inlet velocity, (m/s), is:
where is the volumetric feed flow rate (m^{3}/h), and is the density of the feed pulp (t/m^{3}).
The relative slurry viscosity, , is the ratio of slurry and water viscosities, and , which is approximated by:
where is the cumulative fraction passing 38 μm in the feed (frac).
Liquids recovery
The fraction of feed liquid recovered to the underflow stream, , is related to (i.e. ), and is computed as:
where is a calibration factor which should be fitted to operating data.
is the radius of the cyclone (m), i.e.:
and the tangential velocity, (m/s), is:
Sharpness of separation
The sharpness of separation parameter, , is:
where is a calibration factor which should be fitted to operating data.
Cyclone capacity
The volumetric capacity of a cyclone at a given operating pressure, (m^{3}/h), is estimated from the pressurethroughput relationship:
where is a calibration factor which should be fitted to operating data, and is the pressure drop across the operating cyclone (kPa).
This expression may be used to estimate the number of cyclones required to accept a given total cluster feed flow rate at a fixed pressure, e.g. a process set point.
Alternatively, the pressure drop arising from a given feed flow rate may be calculated by rearranging the above equation:
Multicomponent modelling
Narasimha, Mainza and Holtham (2014)
The NarasimhaMainza model formulation only considers the classification of solid particles with a single average feed density, .
Narasimha et al. (2014b) explored the classification of multicomponent feeds, deriving modified equations for cut size and sharpness of separation per ore component:^{[2]}
and
where is the density of solid ore component (t/m^{3}).
A multicomponent version of the NarasimhaMainza model is available which replaces the average density cut size and sharpness of separation equations with their per ore equivalents above. The multicomponent version is named as NarasimhaMainza (Multi) to distinguish it from the primary NarasimhaMainza (2014) formulation.
Silveira, Delboni and Bergerman (2024)
More recently, Silveira et al. (2024) simulated multicomponent classification with the NarasimhaMainza model by assigning separate cut size and sharpness coefficients to the valuable mineral phase and overall ore partitions.^{[3]}
This concept is extended here to allow the specification of cut size and sharpness coefficients for each ore/mineral/class type in the feed, i.e. and replacing and .
Partition metrics
Several metrics are provided to characterise the partition curve.
The , also known as the cut or separation size, is defined as the size of a particle which has an even (50%) chance of appearing in either the underflow or overflow stream. The size is estimated via a loglinear interpolation of geometric mean size () against the uncorrected partition to underflow of all solids in the feed.
The Ecart Probable, or , is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for size classification as:^{[4]}
where and are the sizes of particles which have a 75% and 25% probability, respectively, of appearing in the underflow stream. The and sizes are estimated by loglinear interpolation of geometric mean size against the uncorrected partition to underflow of all solids in the feed.
The Imperfection, , is a normalised measure of the sharpness of separation, which is suggested to be independent of the magnitude of the , and is typically defined for size classification as:^{[4]}
Roping
Several methods are available to identify the potential for roping discharge from a hydrocyclone underflow.
Plitt proposed that roping may occur when when the volumetric feed rate to the cyclone, (m^{3}/h), exceeds a solids capacity limit:^{[5]}
Plitt further proposed a limit to the volume fraction of solids in cyclone underflow, (% v/v), of:
where is the mass median particle size of the underflow, computed here as the P_{50} (μm).
The SPOC criterion indicates roping may occur when:^{[6]}
where is the percentage volume fraction of solids in the underflow stream (% v/v). The SPOC criterion is only valid when .
Investigations by Bustamante (1991) and Concha et al. (1996) led to the limiting values of cyclone geometry in Table 1:^{[4]}
Table 1. Transition from spray to roping discharge (after Gupta and Yan, 2016).^{[4]} Source Condition Bustamante <0.34 Roping discharge 0.34  0.5 Roping or spray >0.5 Spray discharge Concha et al. <0.45 Roping discharge 0.45  0.56 Roping or spray >0.56 Spray discharge
Additional notes
Note that the equations presented by Narasimhsa et al. (2014) differ from those presented in Narasimha's original dissertation.^{[7]}
The user should be aware of which model formulation is being applied when adopting calibration parameters from external sources. Recalibration of model parameters via the Excel interface below is recommended in such cases.
Excel
The NarasimhaMainza hydrocyclone model may be invoked from the Excel formula bar with the following function call:
=mdUnit_Hydrocyclone_NarasimhaMainza(Parameters as Range, Size as Range, Feed as Range, OreSG Range, Optional Kd0 as Range, Optional Kalpha0 as Range)
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:
 is the set of equations to use (0 = Narasimha et al. (2014), 1 = Narasimha et al. (Multi))
 indicates whether the number of cyclones required at the given feed flow rate and pressure set point is returned (TRUE) or the operating pressure at the given feed flowrate and number of cyclones is returned (FALSE)
 is the number of cyclones operating in a cluster. The number of cyclones is ignored if is TRUE (the value is returned in the results instead)
 , the operating pressure, is ignored if is FALSE (the value is returned in the results instead)
 is the density of liquids (fluids) in the feed (t/m^{3})
 is the size of the square mesh interval that feed mass is retained on (mm)
 is the number of ore types
 is feed solids mass flow rate by size and ore type (t/h)
 is the mass flow feed rate of liquids into the cyclone (t/h)
 indicates optional arrays to specify cut size and sharpness parameters by ore/mineral/class type. If either array is omitted (default), the and values supplied in the array are used instead.
Results
The results are displayed in Excel as an array corresponding to the matrix notation below:
where:
 is the number of cyclones required at the given and operating pressure , if = TRUE
 is the volumetric feed flow rate per cyclone (m^{3}/h)
 is the total volumetric feed flow rate to the cluster of cyclones (m^{3}/h)
 is the pressure drop across the cyclone cluster with operating at feed rate , if = FALSE (kPa)
 is actual split of liquids to the overflow (frac)
 is the mass flow rate of liquids to the overflow stream (t/h)
 is the mass flow rate of liquids to the underflow stream (t/h)
 is the SPOC roping limit of volume fraction solids in the underflow stream (v/v)
 is the Plitt roping limit of volumetric flow rate of solids in the underflow stream (m^{3}/h)
 is the Plit roping limit of volume fraction solids in the underflow stream (v/v)
 is the underflow discharge type based on the Bustamante geometry limits (spray or roping discharge)
 is the underflow discharge type based on the Concha geometry limits (spray or roping discharge)
 is mass flow rate of solids to the underflow stream (t/h)
 is mass flow rate of solids to the overflow stream (t/h)
 is the actual partition of all particles of size to the overflow stream, computed as (frac)
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_Hydrocyclone 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 MassFracToUF option appears below. 
MassFracToUF  Input  Only appears if the On field above is not checked. Specifies the fraction of feed mass that reports to the underflow stream when the model is off. 
Method  Partition (User)  The partition to overflow for each size interval is defined by the user. Different values can be used for different solids. 
Partition (ReidPlitt)  The partition to overflow for each size interval is defined by a ReidPlitt efficiency curve. Different parameters can be used for different solids.  
Partition (WhitenBeta)  The partition to overflow for each size interval is defined by a WhitenBeta efficiency curve. Different parameters can be used for different solids.  
Nageswararao  The Nageswararao model is used to determine the partition of solids to underflow and overflow for each size interval.  
NarasimhaMainza (2014)  The NarasimhaMainza (2014) model is used to determine the partition of solids to underflow and overflow for each size interval.  
NarasimhaMainza (Multi)  The NarasimhaMainza (Multi) model is used to determine the partition of solids to underflow and overflow for each size interval.  
Plitt  The Plitt model is used to determine the partition of solids to underflow and overflow for each size interval.  
RopingCalcs  CheckBox  Show addition calculations that predict the onset of cyclone underflow roping. 
Options  
ShowQFeed  CheckBox  QFeed and associated tab pages (eg Sp) will become visible, showing the properties of the combined feed stream. 
ShowQOF  CheckBox  QOF and associated tab pages (eg Sp) will become visible, showing the properties of the overflow stream. 
ShowQUF  CheckBox  QUF 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. 
Cyclone page
The Cyclone page is used to specify the input parameters for the hydrocyclone model.
Partition page
The Partition page is used to specify or display the partition 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. 
Partition  
Method  Model/User  Select modelcalculated or userdefined partition 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 

Partition  Display 

All (All row, All column)  Display 

All (All row, per species)  Display 

Roping page
This page displays the results for hydrocyclone roping limit calculations. The page is only visible if Roping is selected on the MD_Hydrocyclone page.
Tag (Long/Short)  Input / Display  Description/Calculated Variables/Options 

Roping  
Underflow  
SolidsVolFlow / SQv  Display  Volumetric flow rate of solids in cyclone underflow stream. 
Plitt.MSu  Display  Plitt's volumetric flow rate of solids in cyclone underflow roping limit. 
SolidsVolFrac / Svf  Display  Volume fraction of solids in the cyclone underflow stream. 
SPOC  Display  SPOC volume fraction of solids in the cyclone underflow roping limit. 
Plitt.phiL  Display  Plitt's volume fraction of solids in the cyclone underflow roping limit. 
Geometry  
BCondition  Display  Text string describing the spray/roping condition of the cyclone based on Bustamante's geometry limits. 
CCondition  Display  Text string describing the spray/roping condition of the cyclone based on the Concha et al. geometry limits. 
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 overflow stream without split.
See also
References
 ↑ Narasimha, M., Mainza, A.N., Holtham, P.N., Powell, M.S. and Brennan, M.S., 2014. A semimechanistic model of hydrocyclones—Developed from industrial data and inputs from CFD. International Journal of Mineral Processing, 133, pp.112.
 ↑ ^{2.0} ^{2.1} Narasimha, M., Crasta, J, Sreenivas, T. and Mainza, A. N., 2014. Performance of hydrocyclone separating bicomponent mixture. In Proceedings of the XXVII International Mineral Processing Congress, Santiago, Chile, 2014.
 ↑ Silveira Jr, A., Delboni Jr, H. and Bergerman, M.G., 2024. Modeling and Simulation of Hydroxyapatite Recovery in the Desliming Circuit of the Tapira Industrial Plant, Brazil. Minerals, 14(3), p.272.
 ↑ ^{4.0} ^{4.1} ^{4.2} ^{4.3} Gupta, A. and Yan, D.S., 2016. Mineral processing design and operations: an introduction. Elsevier.
 ↑ Dubey, R.K., Singh, G. and Majumder, A.K., 2017. Roping: Is it an optimum dewatering performance condition in a hydrocyclone?. Powder Technology, 321, pp.218231.
 ↑ NapierMunn, T.J., Morrell, S., Morrison, R.D. and Kojovic, T., 1996. Mineral comminution circuits: their operation and optimisation. Julius Kruttschnitt Mineral Research Centre, Indooroopilly, QLD.
 ↑ Narasimha, M., 2009. Improved Computational and Empirical Models of Hydrocyclones. PhD Thesis, University of Queensland (unpublished).