Plitt Original Hydrocyclone

Summary

The Plitt Original Hydrocyclone model represents hydrocyclone classification using the empirical model proposed by L. R. Plitt in “A Mathematical Model of the Hydrocyclone Classifier”, CIM Bulletin, December 1976. The model estimates corrected cut size, flow split, pressure drop, sharpness of separation and the particle-size partition to underflow.

The model should be used when hydrocyclone performance must be estimated from cyclone geometry, feed flowrate, feed solids concentration, and calibration factors.

In this DPSIM implementation, the Plitt model is implemented using the metric equations from the original paper. The model includes calibration factors and an optional pressure-control extension that adjusts the number of hydrocyclones in operation based on calculated pressure.

DPSIM model key: DPSIM.Classification.PlittOriginal
Category: Classification
Subcategory: Hydrocyclones
Display name: Plitt Original Hydrocyclone

Parameters

Derived parameters

Derived parameters

Model Description

The Plitt Original Hydrocyclone model receives one feed stream and generates two product streams: underflow and overflow. In DPSIM, the product port represents the underflow and the tail port represents the overflow.

The feed volumetric flowrate is divided by the number of hydrocyclones in parallel:

Q_cyc=Q_F/N_parallel

The individual cyclone flowrate is converted to litres per minute:

Q=1000 Q_cyc/60

The geometric parameters are converted from inches to centimetres:

D_cm=2.54 D_in

The feed solids volume fraction is converted to percent:

φ=100 φ_v

Where:

The pressure drop is calculated as:

P=1.88 Q^1.78 exp(0.0055φ)/(D_c^0.37 D_i^0.94 h^0.28 (D_u^2+D_o^2)^0.87)

Where:

The hydraulic head is calculated from pressure and pulp density:

H=P/(9.81 ρ_p)

Where:

The flow split is calculated as:

S=1.9 (D_u/D_o)^3.31 h^0.54 (D_u^2+D_o^2)^0.36 exp(0.0054φ)/(H^0.24 D_c^1.11)

The volumetric recovery to underflow is:

R_v=S/(S+1)

Where:

For each component c, the corrected cut size is calculated as:

d_(50,c)=52.5 D_c^0.46 D_i^0.60 D_o^1.21 exp(0.063φ)/(D_u^0.71 h^0.38 Q^0.45 (ρ_(s,c)-ρ_l)^0.5)

Then calibration factors are applied:

d_(50,c)^*=d_(50,c) F_d50 F_(d50,c)

Where:

The sharpness of separation is calculated as:

m=1.94 exp(-1.58 R_v) (D_c^2 h/Q)^0.15

The calibrated sharpness for component c is:

m_c=m F_m F_(m,c)

Where:

The corrected partition curve is calculated as:

Y'(c,i)=1-exp(-0.693 (d_i/d(50,c)^*)^m_c)

Where:

The liquid bypass to underflow is calculated iteratively. Starting from R_f=R_v, the model calculates solids recovery:

R_s=sum_c sum_i M_(c,i)^F (R_f+(1-R_f)Y'_(c,i))/M_S^F

The liquid recovery to underflow is updated as:

R_f=(R_v-R_s φ_v)/(1-φ_v)

The iteration is repeated until convergence or until the maximum iteration count is reached.

After convergence, the liquid bypass is calibrated:

R_f^*=R_f F_b

For each component:

R_(f,c)=R_f^* F_(b,c)

Where:

The final class partition to underflow is:

Y_(c,i)=R_(f,c)+(1-R_(f,c))Y'_(c,i)

The partition is limited by the minimum and maximum class partition parameters:

Y_(c,i)^*=min(Y_max,max(Y_min,Y_(c,i)))

The underflow and overflow component masses are calculated as:

M_(c,i)^U=Y_(c,i)^* M_(c,i)^F

M_(c,i)^O=M_(c,i)^F-M_(c,i)^U

The stream size distributions and component-by-size matrices are rebuilt from these component masses.

The water split is calculated from the calibrated global liquid bypass:

M_W^U=R_f^* M_W^F

M_W^O=M_W^F-M_W^U

Pressure Control Extension

The pressure-control extension is not part of the original Plitt model. It is a DPSIM operational logic for adjusting the number of hydrocyclones in operation.

When pressure control is enabled, the model compares the calculated pressure with the high and low pressure thresholds. If pressure is above the high threshold, the number of hydrocyclones in parallel is increased by one, limited by the maximum available hydrocyclones. If pressure is below the low threshold, the number of hydrocyclones in parallel is decreased by one, limited by the minimum available hydrocyclones.

The model uses the metric Plitt equations internally. The user enters hydrocyclone geometry in inches, and the model converts these dimensions to centimetres before applying the equations.

The product stream corresponds to the underflow. The tail stream corresponds to the overflow.

The model preserves total solids and water mass by splitting the feed into underflow and overflow. Solids are split size-by-size and component-by-component according to the calculated partition curve. Water is split according to the calculated liquid bypass to underflow.

The d50, bypass and partition efficiency calibration parameters are DPSIM extensions. They allow calibration of the original Plitt equations against plant data, pilot data or vendor information.

The optional feed-size correction to d50 presented by Plitt is not implemented in this DPSIM version.

The model should be calibrated before being used for design conclusions, especially outside the operating range of the original Plitt data set.

References

Plitt, L. R. (1976). A Mathematical Model of the Hydrocyclone Classifier. CIM Bulletin, December 1976, 114–123.