Finch Column Flotation

Summary

The Finch Column Flotation model represents column flotation using a simplified collection-zone formulation based on the concepts presented by Finch and Dobby in Column Flotation. The model estimates particle collection in the collection zone from superficial gas velocity, gas holdup, bubble size, column geometry, liquid velocity, axial dispersion and component-specific collection parameters.

The model should be used for column flotation simulations where component and size recoveries must depend on particle size, maximum floatable size, bubble size, gas velocity, gas holdup, collection-zone height, bias velocity and dispersion intensity.

In this DPSIM implementation, the column product stream represents the floated stream, while the tail stream represents the bottom stream.

DPSIM model key: DPSIM.Concentration.FinchColumnFlotation
Category: Concentration
Subcategory: Flotation
Display name: Finch Column Flotation

Parameters

Derived parameters

This model does not create derived parameters.

Model Description

The Finch Column Flotation model receives one feed stream and generates two product streams. In DPSIM, the product port represents the floated stream and the tail port represents the bottom stream.

The column cross-sectional area is calculated as:

A_c=πD_c^2/4

Where:

The collection-zone height is calculated from the total column height:

H_c=f_cH_T

Where:

The feed is divided equally among the columns in parallel:

M_(F,col,c,i)=M_(F,c,i)/N_col

W_(F,col)=W_F/N_col

Where:

The liquid superficial velocity in the collection zone is calculated as:

u_l=W_(F,col)/(3600A_c)+u_bias

Where:

The gas superficial velocity and bubble diameter are converted to SI units:

u_g=0.01u_(g,cm/s)

d_b=d_(b,mm)/1000

Where:

The gas holdup is:

ε_g=E_g/100

Where:

The liquid residence time in the collection zone is calculated as:

t_l=H_c(1-ε_g)/u_l

Where:

The vessel number used to represent axial dispersion is calculated as:

N_p=(a_N/0.6)(D_c/H_c)^0.63 (u_g(1-ε_g)/u_l)

Where:

For each component c and size class i, the particle size is converted from micrometres to metres:

d_i=d_(i,µm)/1000000

The component-specific collection probability term is calculated as:

P_(c,i)=α_c(d_i/d_b)^(2a_c)(1-(d_i/d_(max,c))^1.5), d_i<d_(max,c)

P_(c,i)=0, d_i≥d_(max,c)

Where:

The collection rate is calculated as:

k_(c,i)=3/2 (u_g/d_b) P_(c,i)

Where:

The particle slip velocity is calculated iteratively using a hindered-settling correction:

u_(sp,c,i)=g d_i^2(ρ_c-ρ_l)(1-φ_s)^2.7/(18µ_l(1+0.15Re_(p,c,i)^0.687))

Re_(p,c,i)=d_i u_(sp,c,i)ρ_c(1-φ_s)/µ_l

Where:

The particle residence time is corrected for slip velocity:

t_(p,c,i)=t_l u_l/(u_l+u_(sp,c,i))

Where:

The axial-dispersion collection response is calculated using:

a_(c,i)=sqrt(1+4k_(c,i)t_(p,c,i)N_p)

The non-collected fraction leaving the collection zone is:

R_(nc,c,i)=4a_(c,i)exp(1/(2N_p))/((1+a_(c,i))^2 exp(a_(c,i)/(2N_p))-(1-a_(c,i))^2 exp(-a_(c,i)/(2N_p)))

The recovery to the floated stream is:

R_(c,i)=1-R_(nc,c,i)

The recovery is limited internally to the interval from 0 to 1.

Where:

The floated mass from one column is:

M_(Flt,col,c,i)=R_(c,i)M_(F,col,c,i)

The total floated mass is obtained by multiplying by the number of columns:

M_(Flt,c,i)=N_col M_(Flt,col,c,i)

The bottom mass is calculated by difference from the original feed mass:

M_(Bot,c,i)=M_(F,c,i)-M_(Flt,c,i)

Where:

The total floated and bottom retained masses by size class are:

M_i^Flt=sum_c M_(Flt,c,i)

M_i^Bot=sum_c M_(Bot,c,i)

The retained size distributions are normalized as:

p_i^Flt=M_i^Flt/sum_i M_i^Flt

p_i^Bot=M_i^Bot/sum_i M_i^Bot

The component fractions in each size interval are:

z_(Flt,c,i)=M_(Flt,c,i)/M_i^Flt

z_(Bot,c,i)=M_(Bot,c,i)/M_i^Bot

Where:

The water split is calculated from the feed water recovery to the floated stream:

W_Flt=R_W W_F

W_Bot=W_F-W_Flt

with:

R_W=R_(W,%)/100

Where:

The model preserves total solids and water by splitting the feed into floated and bottom streams. The model is a simplified Finch-Dobby column flotation representation focused on collection-zone recovery. It does not explicitly model froth-zone cleaning, wash-water entrainment, froth carry rate limits, interface level, air-sparger hydraulics, bubble-size population, reagent chemistry or detailed column control.

References

Finch, J. A.; Dobby, G. S. (1990). Column Flotation. Pergamon Press.