Hydrocyclone (Narasimha-Mainza)

Description

This article describes the Narasimha-Mainza 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:

• A sharpness of separation equation
• A slurry viscosity term that includes the effects of very fine particles
• Terms for cyclone inclination, particle density, g-forces, flow regime (Reynolds Number) and turbulent diffusion

Narasimha et al. (2014b) subsequently presented a multi-component version of the same model, with revised equations for cut size and sharpness of separation per ore component:

Model theory

Figure 1. The dimensions of a hydrocyclone required by the Narasimha-Mainza model.

The Narasima-Mainza model comprises an equation for the efficiency curve (partition to overflow) and several sub-equations 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 Narasimha-Mainza model applies the Whiten-Beta efficiency curve to partition particles to the overflow stream:

$E_{\rm {oa}}({\bar {d}}_{i})=C\left[{\dfrac {\left(1+\beta \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)(\exp(\alpha )-1)}{\exp \left(\alpha \beta ^{*}{\dfrac {{\bar {d}}_{i}}{d_{\rm {50c}}}}\right)+\exp(\alpha )-2}}\right]$

where:

• $i$ is the index of the size interval, $i=\{1,2,\dots ,n\}$, $n$ is the number of size intervals
• $E_{\rm {oa}}({\bar {d}}_{i})$ is the fraction of particles of size interval $i$ in the feed reporting to the overflow stream (frac)
• ${\bar {d}}_{i}$ is the geometric mean size of particles in size interval $i$ (mm)
• $d_{\rm {50c}}$ is the corrected size at which 50% of the particle mass reports to underflow and 50% to overflow (mm)
• $C$ is the fraction of feed liquids (or fines) split to overflow (frac)
• $\alpha$ is a parameter representing the sharpness of separation
• $\beta$ is a term introduced to accommodate the so-called fish-hook effect, and controls the initial rise in the efficiency curve at finer sizes
• $\beta ^{*}$ is computed to ensure the Whiten-Beta function preserves the definition of $d_{\rm {50c}}$ in the presence of the fish-hook, i.e. $E=0.5C$ at $d_{\rm {50c}}$

Corrected cut size

The corrected cut size, $d_{\rm {50c}}$ (m), is computed from:

${\dfrac {d_{\rm {50c}}}{D_{\rm {c}}}}=K_{\rm {d0}}{\left({\dfrac {D_{\rm {o}}}{D_{\rm {c}}}}\right)}^{1.093}{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{-1.000}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{-0.703}({\text{Re}})^{-0.436}{\left({\dfrac {D_{\rm {i}}}{D_{\rm {c}}}}\right)}^{-0.936}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{0.187}{\left({\dfrac {1}{\tan \theta }}\right)}^{-0.1988}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{-1.034}{\left({\dfrac {\rho _{\rm {s}}-\rho _{\rm {f}}}{\rho _{\rm {f}}}}\right)}^{-0.217}$

where:

• $K_{\rm {d0}}$ is a calibration factor which should be fitted to operating data
• $D_{\rm {c}}$ is diameter of the cyclone (m)
• $D_{\rm {i}}$ is diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet (m)
• $D_{\rm {o}}$ is diameter of the vortex finder (overflow) (m)
• $D_{\rm {u}}$ is diameter of the apex/spigot (underflow) (m)
• $L_{\rm {c}}$ is length of the cylindrical section (m)
• $\theta$ is the cone full angle (deg.)
• $i$ is the angle of inclination from the vertical (rad)
• $f_{\rm {v}}$ is the volume fraction of solids in the feed (v/v)
• $\rho _{\rm {s}}$ is the density of solids in the feed (t/m³)
• $\rho _{\rm {f}}$ is the density of the fluid (liquids) in the feed (t/m³)
• $g$ is acceleration due to gravity (m/s²)

The Reynolds Number, ${\rm {Re}}$, is:

${\rm {Re}}={\dfrac {1000V_{i}D_{\rm {c}}\rho _{\rm {p}}}{0.001\mu _{\rm {r}}}}$

The feed inlet velocity, $V_{i}$ (m/s), is:

$V_{i}={\dfrac {Q_{\rm {f}}}{{\dfrac {\pi }{4}}{D_{\rm {i}}}^{2}}}$

where $Q_{\rm {f}}$ is the volumetric feed flow rate (m³/h), and $\rho _{\rm {p}}$ is the density of the feed pulp (t/m³).

The relative slurry viscosity, $\mu _{\rm {r}}$, is the ratio of slurry and water viscosities, $\mu _{\rm {m}}$ and $\mu _{\rm {w}}$, which is approximated by:

$\mu _{\rm {r}}={\dfrac {\mu _{\rm {m}}}{\mu _{\rm {w}}}}=\left(1-{\dfrac {f_{\rm {v}}}{0.622}}\right)^{-1.55}({F_{-38\mu }})^{0.39}$

where $F_{-38\mu }$ is the cumulative fraction passing 38 μm in the feed (frac).

Liquids recovery

The fraction of feed liquid recovered to the underflow stream, $R_{\rm {f}}$, is related to $C$ (i.e. $C=1-R_{\rm {f}}$), and is computed as:

$R_{\rm {f}}=K_{\rm {w0}}{\left({\dfrac {D_{\rm {o}}}{D_{\rm {c}}}}\right)}^{-1.06787}{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{2.2062}{\left({\dfrac {V_{t}^{2}}{R_{\rm {max}}g}}\right)}^{-0.20472}{\left({\tan \left({\dfrac {\theta }{2}}\right)}\right)}^{-0.829}{\mu _{\rm {r}}}^{-0.7118}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{2.424}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{-0.8843}{\left({\dfrac {\rho _{\rm {s}}-\rho _{\rm {f}}}{\rho _{\rm {f}}}}\right)}^{0.523}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{1.793}$

where $K_{\rm {w0}}$ is a calibration factor which should be fitted to operating data.

$R_{\rm {max}}$ is the radius of the cyclone (m), i.e.:

$R_{\rm {max}}=0.5D_{\rm {c}}$

and the tangential velocity, $V_{\rm {t}}$ (m/s), is:

$V_{\rm {t}}=4.5V_{i}\left({\dfrac {D_{\rm {i}}}{D_{\rm {c}}}}\right)^{1.13}$

Sharpness of separation

The sharpness of separation parameter, $\alpha$, is:

$\alpha =K_{\alpha }{{{\left({\dfrac {D_{\rm {o}}}{D_{\rm {c}}}}\right)}^{0.27{\phantom {0}}}{\left({\dfrac {V_{\rm {t}}^{2}}{gR_{\rm {max}}}}\right)}^{0.016}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{0.868}{\left({\dfrac {{\left(1-fv\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{0.72{\phantom {0}}}}{{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{-0.567}{\left({\dfrac {\left(\rho _{\rm {s}}-\rho _{\rm {p}}\right)}{\rho _{\rm {s}}}}\right)}^{-1.837}{\mu _{\rm {r}}}^{-0.127}{\left({\tan \left({\dfrac {\theta }{2}}\right)}\right)}^{0.182{\phantom {-}}}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{-0.2}}}$

where $K_{\alpha }$ is a calibration factor which should be fitted to operating data.

Cyclone capacity

The volumetric capacity of a cyclone at a given operating pressure, $Q$ (m³/h), is estimated from the pressure-throughput relationship:

$Q=K_{\rm {Q0}}{\left({\dfrac {D_{\rm {i}}}{D_{\rm {c}}}}\right)}^{0.45}{D_{\rm {c}}}^{2}{\sqrt {\dfrac {P}{\rho _{\rm {p}}}}}{\left({\dfrac {D_{\rm {o}}}{d_{c}}}\right)}^{1.099}{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{0.037}{\left({\tan \left({\dfrac {\theta }{2}}\right)}\right)}^{-0.405}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{0.30{\phantom {0}}}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{-0.048}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{-0.092}$

where $K_{\rm {Q0}}$ is a calibration factor which should be fitted to operating data, and $P$ 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:

$P=\rho _{\rm {p}}\left[Q{{K_{\rm {Q0}}}^{-1}{\left({\dfrac {D_{\rm {i}}}{D_{\rm {c}}}}\right)}^{-0.45}{D_{\rm {c}}}^{-2}{\left({\dfrac {D_{\rm {o}}}{d_{c}}}\right)}^{-1.099}{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{-0.037}{\left({\tan \left({\dfrac {\theta }{2}}\right)}\right)}^{0.405}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{-0.30{\phantom {0}}}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{0.048}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{0.092}}\right]^{2}$

Multi-component modelling

Narasimha, Mainza and Holtham (2014)

The Narasimha-Mainza model formulation only considers the classification of solid particles with a single average feed density, $\rho _{\rm {s}}$.

Narasimha et al. (2014b) explored the classification of multi-component feeds, deriving modified equations for cut size and sharpness of separation per ore component:

${\dfrac {(d_{\rm {50c}})_{j}}{D_{\rm {c}}}}=K_{\rm {d0}}{\left({\dfrac {D_{\rm {o}}}{D_{\rm {c}}}}\right)}^{1.093}{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{-1.000}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{-0.703}({\text{Re}})^{-0.436}{\left({\dfrac {D_{\rm {i}}}{D_{\rm {c}}}}\right)}^{-0.936}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{0.187}{\left({\dfrac {1}{\tan \theta }}\right)}^{-0.1988}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{-1.034}{\left({\dfrac {\rho _{{\rm {s}}j}-\rho _{\rm {f}}}{\rho _{\rm {f}}}}\right)}^{-1.37}$

and

$\alpha _{j}=K_{\alpha }{{{\left({\dfrac {D_{\rm {o}}}{D_{\rm {c}}}}\right)}^{0.27{\phantom {0}}}{\left({\dfrac {V_{\rm {t}}^{2}}{gR_{\rm {max}}}}\right)}^{0.016}{\left(\cos \left({\dfrac {i}{2}}\right)\right)}^{0.868}{\left({\dfrac {{\left(1-f_{\rm {v}}\right)}^{2}}{10^{1.82f_{\rm {v}}}}}\right)}^{0.72{\phantom {0}}}}{{\left({\dfrac {D_{\rm {u}}}{D_{\rm {c}}}}\right)}^{-0.567}{\left(\rho _{{\rm {s}}j}-\rho _{\rm {p}}\right)}^{-0.983}{\mu _{\rm {r}}}^{-0.127}{\left({\tan \left({\dfrac {\theta }{2}}\right)}\right)}^{0.182{\phantom {-}}}{\left({\dfrac {L_{\rm {c}}}{D_{\rm {c}}}}\right)}^{-0.2}}}$

where $\rho _{{\rm {s}}j}$ is the density of solid ore component $j$ (t/m³).

A multi-component version of the Narasimha-Mainza model is available which replaces the average density cut size and sharpness of separation equations with their per ore equivalents above. The multi-component version is named as Narasimha-Mainza (Multi) to distinguish it from the primary Narasimha-Mainza (2014) formulation.

Silveira, Delboni and Bergerman (2024)

More recently, Silveira et al. (2024) simulated multicomponent classification with the Narasimha-Mainza model by assigning separate cut size and sharpness coefficients to the valuable mineral phase and overall ore partitions.

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. $(K_{\rm {d0}})_{j}$ and $(K_{\alpha 0})_{j}$ replacing $K_{\rm {d0}}$ and $K_{\alpha 0}$.

Partition metrics

Several metrics are provided to characterise the partition curve.

The $d_{50}$, 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 $d_{50}$ size is estimated via a log-linear interpolation of geometric mean size ($\ln {\bar {d}}$) against the uncorrected partition to underflow of all solids in the feed.

The Ecart Probable, or $E_{\rm {p}}$, is a measure of the deviation of a partition curve from a perfect separation, and is typically defined for size classification as:

$E_{\rm {p}}={\dfrac {d_{75}-d_{25}}{2}}$

where $d_{75}$ and $d_{25}$ are the sizes of particles which have a 75% and 25% probability, respectively, of appearing in the underflow stream. The $d_{75}$ and $d_{25}$ sizes are estimated by log-linear interpolation of geometric mean size against the uncorrected partition to underflow of all solids in the feed.

The Imperfection, $I$, is a normalised measure of the sharpness of separation, which is suggested to be independent of the magnitude of the $d_{50}$, and is typically defined for size classification as:

$I={\dfrac {E_{\rm {p}}}{d_{50}}}={\dfrac {d_{75}-d_{25}}{2d_{50}}}$

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, $M_{\rm {SU}}$ (m³/h), exceeds a solids capacity limit:

$M_{\rm {SU}}>0.35{D_{\rm {u}}}^{2.35}$

Plitt further proposed a limit to the volume fraction of solids in cyclone underflow, $\Phi _{\rm {L}}$ (% v/v), of:

$\Phi _{\rm {L}}=62.3\left[1-\exp \left(-{\dfrac {d_{\rm {u}}}{60}}\right)\right]$

where $d_{\rm {u}}$ is the mass median particle size of the underflow, computed here as the P₅₀ (μm).

The SPOC criterion indicates roping may occur when:

$C_{\rm {VU}}>56+0.2(C_{\rm {V}}-20)$

where $C_{\rm {VU}}$ is the percentage volume fraction of solids in the underflow stream (% v/v). The SPOC criterion is only valid when $C_{\rm {V}}<35{\text{ }}\%{\text{ v/v}}$.

Investigations by Bustamante (1991) and Concha et al. (1996) led to the limiting values of cyclone geometry in Table 1:

Table 1. Transition from spray to roping discharge (after Gupta and Yan, 2016).

Source	$D_{\rm {u}}/D_{\rm {o}}$	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.

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 Narasimha-Mainza 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 ($i$) x column ($j$) format:

${\mathit {Parameters}}={\begin{bmatrix}{\text{Method}}\\{\text{Pressure control}}\\{\text{Number of cyclones}}\\D_{\rm {c}}{\text{ (m)}}\\D_{\rm {i}}{\text{ (m)}}\\D_{\rm {o}}{\text{ (m)}}\\D_{\rm {u}}{\text{ (m)}}\\L_{\rm {c}}{\text{ (m)}}\\\theta {\text{ (deg.)}}\\i{\text{ (deg.)}}\\P{\text{ (kPa)}}\\K_{\rm {d0}}\\K_{\rm {Q0}}\\K_{\rm {w0}}\\K_{\alpha }\\\beta \\(Q_{\rm {M,F}})_{\rm {L}}{\text{ (t/h)}}\\\rho _{\rm {f}}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\quad {\begin{array}{l}Size={\begin{bmatrix}d_{1}{\text{ (mm)}}\\\vdots \\d_{n}{\text{ (mm)}}\\\end{bmatrix}},\quad {\mathit {Feed}}={\begin{bmatrix}(Q_{\rm {M,F}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,F}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,F}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}},\quad {\mathit {OreSG}}={\begin{bmatrix}(\rho _{\rm {s}})_{1}{\text{ (t/m}}^{\text{3}}{\text{)}}&\dots &(\rho _{\rm {s}})_{m}{\text{ (t/m}}^{\text{3}}{\text{)}}\\\end{bmatrix}},\quad {\mathit {Kd0}}={\begin{bmatrix}(K_{\rm {d0}})_{1}&\dots &(K_{\rm {d0}})_{m}\\\end{bmatrix}}^{*},\quad {\mathit {Kalpha0}}={\begin{bmatrix}(K_{\alpha 0})_{1}&\dots &(K_{\alpha 0})_{m}\\\end{bmatrix}}^{*}\end{array}}$

where:

• ${\text{Method}}$ is the set of equations to use (0 = Narasimha et al. (2014), 1 = Narasimha et al. (Multi))
• ${\text{Pressure control}}$ 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)
• ${\text{Number of cyclones}}$ is the number of cyclones operating in a cluster. The number of cyclones is ignored if ${\text{Pressure control}}$ is TRUE (the value is returned in the results instead)
• $P$, the operating pressure, is ignored if ${\text{Pressure control}}$ is FALSE (the value is returned in the results instead)
• $\rho _{\rm {f}}$ is the density of liquids (fluids) in the feed (t/m³)
• $d_{i}$ is the size of the square mesh interval that feed mass is retained on (mm)
• $m$ is the number of ore types
• $Q_{\rm {M,F}}$ is feed solids mass flow rate by size and ore type (t/h)
• $(Q_{\rm {M,F}})_{\rm {L}}$ 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 $K_{\rm {d0}}$ and $K_{\alpha 0}$values supplied in the ${\mathit {Parameters}}$ array are used instead.

Results

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

${\mathit {mdUnit\_Hydrocyclone\_NarasimhaMainza}}={\begin{bmatrix}{\begin{bmatrix}{\text{Number of cyclones}}\\{\text{Qv per cyclone (m}}^{\text{3}}{\text{/h)}}\\{\text{Qv per cluster (m}}^{\text{3}}{\text{/h)}}\\C_{\rm {V}}{\text{ (}}\%{\text{ v/v)}}\\P{\text{ (kPa)}}\\R_{\rm {v}}{\text{ (frac)}}\\R_{\rm {f}}{\text{ (frac)}}\\{\text{Water split to OF (frac)}}\\d_{\rm {50c}}{\text{ (mm)}}\\(Q_{\rm {M,OF}})_{\rm {L}}{\text{ (t/h)}}\\(Q_{\rm {M,UF}})_{\rm {L}}{\text{ (t/h)}}\\d_{50}{\text{ (mm)}}\\E_{\rm {p}}{\text{ (mm)}}\\I\\{\text{SPOC }}C_{\rm {VU}}{\text{ (v/v)}}\\{\text{Plitt }}M_{\rm {SU}}{\text{ (m}}^{3}{\text{/h)}}\\{\text{Plitt }}\Phi _{\rm {L}}{\text{ (v/v)}}\\{\text{Bustamante condition}}\\{\text{Concha condition}}\\\end{bmatrix}}{\begin{array}{cccccc}{\begin{bmatrix}{\bar {d}}_{1}{\text{ (mm)}}\\\vdots \\{\bar {d}}_{n}{\text{ (mm)}}\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,UF}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,UF}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,UF}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,UF}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(Q_{\rm {M,OF}})_{11}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OF}})_{1m}{\text{ (t/h)}}\\\vdots &\ddots &\vdots \\(Q_{\rm {M,OF}})_{n1}{\text{ (t/h)}}&\dots &(Q_{\rm {M,OF}})_{nm}{\text{ (t/h)}}\\\end{bmatrix}}&{\begin{bmatrix}(E_{\rm {oa}})_{11}{\text{ (frac)}}&\dots &(E_{\rm {oa}})_{1m}{\text{ (frac)}}\\\vdots &\ddots &\vdots \\(E_{\rm {oa}})_{n1}{\text{ (frac)}}&\dots &(E_{\rm {oa}})_{nm}{\text{ (frac)}}\\\end{bmatrix}}&{\begin{bmatrix}(P_{\rm {OF}})_{1,{\rm {All}}}\\\vdots \\(P_{\rm {OF}})_{n,{\rm {All}}}\\\end{bmatrix}}&{\begin{bmatrix}\alpha _{1}&\dots &\alpha _{m}\\(d_{\rm {50c}})_{1}&\dots &(d_{\rm {50c}})_{m}\\\beta _{1}^{*}&\dots &\beta _{m}^{*}\\\end{bmatrix}}\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\end{array}}\end{bmatrix}}$

where:

• ${\text{Number of cyclones}}$ is the number of cyclones required at the given ${\text{Qv per cluster}}$ and operating pressure ${\text{P}}$, if ${\text{Pressure control}}$ = TRUE
• ${\text{Qv per cyclone}}$ is the volumetric feed flow rate per cyclone (m³/h)
• ${\text{Qv per cluster}}$ is the total volumetric feed flow rate to the cluster of cyclones (m³/h)
• ${\text{P}}$ is the pressure drop across the cyclone cluster with ${\text{Number of cyclones}}$ operating at feed rate ${\text{Qv per cluster}}$, if ${\text{Pressure control}}$ = FALSE (kPa)
• ${\text{Water split to OF}}$ is actual split of liquids to the overflow (frac)
• $(Q_{\rm {M,OF}})_{\rm {L}}$ is the mass flow rate of liquids to the overflow stream (t/h)
• $(Q_{\rm {M,UF}})_{\rm {L}}$ is the mass flow rate of liquids to the underflow stream (t/h)
• ${\text{SPOC }}C_{\rm {VU}}$ is the SPOC roping limit of volume fraction solids in the underflow stream (v/v)
• ${\text{Plitt }}M_{\rm {SU}}$ is the Plitt roping limit of volumetric flow rate of solids in the underflow stream (m³/h)
• ${\text{Plitt }}\Phi _{\rm {L}}$ is the Plit roping limit of volume fraction solids in the underflow stream (v/v)
• ${\text{Bustamante condition}}$ is the underflow discharge type based on the Bustamante geometry limits (spray or roping discharge)
• ${\text{Concha condition}}$ is the underflow discharge type based on the Concha geometry limits (spray or roping discharge)
• $Q_{\rm {M,UF}}$ is mass flow rate of solids to the underflow stream (t/h)
• $Q_{\rm {M,OF}}$ is mass flow rate of solids to the overflow stream (t/h)
• $(P_{\rm {OF}})_{i,{\rm {All}}}$ is the actual partition of all particles of size $i$ to the overflow stream, computed as ${\frac {\sum _{j=1}^{m}(Q_{\rm {M,OF}})_{ij}}{\sum _{j=1}^{m}(Q_{\rm {M,F}})_{ij}}}$ (frac)

Example

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

Figure 2. Example showing the selection of the Parameters (blue frame) array in Excel.

Figure 3. Example showing the selection of the Size (red frame), Feed (purple frame) and OreSG (green frame) arrays in Excel.

Figure 4. Example showing the outline of the Results (light blue frame) array in Excel.

SysCAD

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

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 (Reid-Plitt)	The partition to overflow for each size interval is defined by a Reid-Plitt efficiency curve. Different parameters can be used for different solids.
	Partition (Whiten-Beta)	The partition to overflow for each size interval is defined by a Whiten-Beta 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.
	Narasimha-Mainza (2014)	The Narasimha-Mainza (2014) model is used to determine the partition of solids to underflow and overflow for each size interval.
	Narasimha-Mainza (Multi)	The Narasimha-Mainza (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 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.

Cyclone page

The Cyclone page is used to specify the input parameters for the hydrocyclone model.

Tag (Long/Short)	Input / Display	Description/Calculated Variables/Options
NarasimhaMainza
HelpLink		Opens a link to this page using the system default web browser. Note: Internet access is required.
PressureControl	CheckBox	If enabled, the number of cyclones is adjusted to maintain operating pressure at the feed volumetric flow rate.
OreSpecific	CheckBox	Only appears if the Narasimha-Mazina (2014) model method is selected. If enabled, allows Kd0 and Kalpha parameters to be specified per species.
Dimensions
NumberCyclones / NumCyclones	Input	The number of operating cyclone units in the cluster.
CycloneDiameter / Dc	Input	Diameter of the cyclone.
InletDiameter / Di	Input	Diameter of a circular inlet or the diameter of a circle with the same area as a non-circular inlet.
CycloneDiameter / Dc	Input	Diameter of the cyclone.
OverflowDiameter / Do	Input	Diameter of the vortex finder (overflow).
UnderflowDiameter / Du	Input	Diameter of the apex/spigot (underflow).
CylSectLength / Lc	Input	Length of the cylindrical section of the cyclone.
ConeFullAngle / Theta	Input	Full angle of the cone section of the cyclone.
InclinationAngle/ i	Input	Angle of inclination of cyclone from the vertical.
Parameters
Kd0	Input	Calibration factor for the d50c equation.
KQ0	Input	Calibration factor for the pressure-flowrate equation
Kw0	Input	Calibration factor for the liquids recovery to underflow equation.
Kalpha0	Input	Calibration factor for the sharpness of separation eqaution.
Beta	Input	'Fish hook' parameter of the Whiten-Beta efficiency curve equation.
Liquids
LiquidsSeparMethod	Split To UF (User)	Liquids are split to underflow by a user-defined fraction of liquids in the feed.
	Split To UF (Model)	Liquids are split to underflow by a fraction computed by the associated model.
	UF Solids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow solids mass fraction value (if possible).
	UF Liquids Fraction	Sufficient liquids mass is recovered to the underflow stream to yield the user-defined underflow liquids mass fraction value (if possible).
UFSolidsFracReqd / UF.SfReqd	Input	Required value of the mass fraction of solids in the underflow stream. Only visible if UF Solids Fraction is selected.
UFLiquidsFracReqd / UF.LfReqd	Input	Required value of the mass fraction of liquids in the underflow stream. Only visible if UF Liquids Fraction is selected.
LiqSplitToUF / UF.LiqSplit	Input/Display	The fraction of liquids recovered to underflow.
Results
ClusterQv / Cluster.Qv	Display	Volumetric flow rate of feed to the cyclone cluster (i.e. total flow).
CycloneQv / Cyclone.Qv	Display	Volumetric flow rate of feed to each cyclone in the cluster (i.e. per cyclone flow).
OperatingPressure / P	Display	Pressure drop across the cyclone.
VolFracSolids / fv	Display	Volume fraction of solids in feed stream.
RelSlurryVisc / mur	Display	Relative slurry viscosity.
Rf	Display	Fraction of feed liquids recovered to underflow stream.
d50	Display	The separation size, d50, of all solid particles in the feed.
EcartProbable / Ep	Display	The value of the Ecart Probable of the separation.
Imperfection / I	Display	The value of the Imperfection of the separation.
Alpha	Display	Value of $\alpha$, per species.
d50c	Display	Value of the corrected d50c, per species.
Beta*	Display	Value of $\beta ^{*}$, per species.

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 model-calculated or user-defined 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	Actual overall partition to overflow of all solid species, for each size interval. Excludes solid species not present in the cyclone feed.
Partition	Display	Partition to overflow 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 cyclone feed or not.
All (All row, All column)	Display	Displays the actual, total, partition of all solids with a particle size distribution property in the feed to overflow. Excludes solid species not present in the cyclone feed.
All (All row, per species)	Display	Actual overall partition to overflow for each solid species, for all size intervals in that species. Excludes solid species not present in the cyclone feed.

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 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 overflow stream without split.

See also

• Hydrocyclone (Plitt)
• Hydrocyclone (Nageswararao)