Plitt Hydrocyclone Bravo

Summary

The Plitt Hydrocyclone Bravo model represents hydrocyclone classification using an alternative Plitt-type empirical formulation. It calculates pressure, hydraulic head, volumetric split, corrected cut size, sharpness of separation, water bypass and size-by-size partition to underflow.

This model is intended to provide an alternative empirical hydrocyclone calculation structure while retaining the general classification framework of Plitt-type hydrocyclone models.

DPSIM model key: DPSIM.Classification.PlittHydrocycloneBravo
Category: Classification
Subcategory: Hydrocyclones
Display name: Plitt Hydrocyclone Bravo

Parameters

Derived parameters

Model Description

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

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

Q_i=Q_F/N_parallel

Where:

The feed volumetric solids fraction is:

φ_v=V_S^F/V_P^F

Where:

The pressure correlation is calculated from the implemented pressure term:

P_int=129.72875 Q_i^1.78 exp(0.55φ_v)/(D_c^0.37 D_i^0.94 h^0.28 (D_u^2+D_o^2)^0.87)

The reported pressure is:

P=0.0980665 P_int

The hydraulic head used by the split equation is:

H=P_int/(9.8100004196 ρ_p)

Where:

Difference from Plitt Original: the pressure equation retains the general Plitt-type structure but uses constants transformed for the implemented unit basis. The model uses inch geometry and m3/h per cyclone rather than directly using the original metric units.

The base volumetric split ratio is:

S=3.3411661493 (D_u/D_o)^3.31 h^0.54 (D_u^2+D_o^2)^0.36 exp(-0.8884871132φ_v) F_S/(H^0.24 D_c^1.11)

The calculated value is limited between the minimum and maximum flow split parameters:

S^*=min(S_max,max(S_min,S))

Where:

For each component c, the component-adjusted split is:

S_c=S^* F_(S,c)

Where:

Difference from Plitt Original: Bravo uses an alternative split coefficient and solids-concentration term, and it imposes explicit user-defined limits on S. These limits are not part of the original Plitt equation.

The water bypass to underflow is calculated as the volumetric recovery to underflow:

B_pw=S_c/(S_c+1)

The feed bypass term used in the partition curve is:

R_f=B_pw

Where:

Difference from Plitt Original: the original Plitt formulation relates liquid recovery to volumetric recovery and solids recovery. Bravo does not perform an iterative Plitt liquid-recovery calculation; it uses R_f=B_pw=S/(S+1).

The sharpness of separation is calculated as:

m=2.963 exp(-1.58 S_c/(S_c+1))(D_c^2 h/Q_i)^0.15/F_m

The result is limited by the maximum sharpness parameter:

m^*=min(m,m_max)

The component-adjusted sharpness is:

m_c=m^* F_(m,c)

Where:

Difference from Plitt Original: Bravo uses an alternative coefficient for the sharpness equation, applies a user correction divisor, and limits the maximum value of m.

The corrected cut size has two calculation branches.

For cut size mode 0, the exponential branch is:

d_50=F_cut 4418.82577186 D_c^0.46 D_i^0.60 D_o^1.21 exp(3.9300207955φ_v)/(D_u^0.71 h^0.38 Q_i^0.45 31.6227766017)

For cut size modes 1 and 2, the alternative branch is:

d_50=F_cut 4418.82577186 D_c^0.46 D_i^0.60 D_o^1.21 φ_v^0.41 A_alt^0.35/(D_u^0.71 h^0.38 Q_i^0.45 31.6227766017)

The model then applies a component density correction, a solids-load correction and the component d50 factor:

d_(50,c)^*=d_50 F_(ρ,c) F_load F_(d50,c)

with:

F_(ρ,c)=sqrt((ρ_p-1)/(ρ_(s,c)-1))

and:

F_load=(M_S^F/(100 N_parallel))^(-0.0465008346)

Where:

Difference from Plitt Original: Bravo includes alternative cut-size modes, a solids-load correction, a component density correction based on pulp density and component SG, and a global cut-size correction factor. These are not part of the original Plitt equation.

The corrected Rosin-Rammler partition curve is calculated as:

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

Where:

If “Include water bypass in solids partition” is enabled, the final partition is:

E_(c,i)=R_f+(1-R_f)Y'_(c,i)

If the option is disabled, the final partition is:

E_(c,i)=Y'_(c,i)

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

E_(c,i)^*=min(E_max,max(E_min,E_(c,i)))

Where:

Difference from Plitt Original: Bravo provides an explicit switch to include or exclude bypass in the solids partition curve and applies minimum and maximum class partition limits. These are DPSIM compatibility and numerical robustness extensions.

The underflow and overflow component-size masses are calculated as:

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

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

The total retained size distributions are calculated by summing over components:

M_(U,i)=sum_c M_(U,c,i)

M_(O,i)=sum_c M_(O,c,i)

p_(U,i)=M_(U,i)/sum_j M_(U,j)

p_(O,i)=M_(O,i)/sum_j M_(O,j)

The component fractions in each size interval are:

z_(U,c,i)=M_(U,c,i)/M_(U,i)

z_(O,c,i)=M_(O,c,i)/M_(O,i)

Where:

The water split is calculated from the base hydraulic water bypass:

W_U=B_pw,base W_F

W_O=W_F-W_U

Where:

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

The pressure-control extension is not part of the original Plitt model.

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 operating hydrocyclones is increased by one, limited by the maximum number available. If pressure is below the low threshold, the number of operating hydrocyclones is decreased by one, limited by the minimum number available.

The pressure-control action is applied after the current model calculation. Therefore, the new number of hydrocyclones affects the following calculation cycle.

The alternative cut-size modes, load correction, density correction, split limits, maximum sharpness and class-partition limits are DPSIM calibration and robustness extensions.

The use of R_f=B_pw=S/(S+1) is a simplification relative to the original Plitt liquid-recovery treatment.

The model should be calibrated against plant surveys, pilot data or vendor data before being used for design conclusions.

References

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