Finch-Dobby Column Flotation Advanced

The Finch-Dobby Column Flotation Advanced model represents column flotation using an empirical multi-mechanism formulation inspired by the modelling concepts presented by Finch and Dobby in Column Flotation.

The model calculates size- and component-dependent recovery through a collection zone with axial-dispersion correction. It also represents fast and slow floating populations, a maximum floatable fraction, transfer across the pulp-froth interface, recovery and detachment in the froth phase, hydraulic entrainment of fine particles, wash-water cleaning and an optional froth carrying-capacity limit.

The model should be used when column flotation performance must be estimated from column geometry, gas and liquid superficial velocities, gas holdup, bubble size, collection-zone mixing, wash water, froth behavior and component-specific kinetic parameters.

This is a calibratable DPSIM model inspired by Finch-Dobby column flotation theory. It is not a complete first-principles representation of every hydrodynamic and froth-phase phenomenon described in the reference.

DPSIM model key: DPSIM.Concentration.FinchDobbyColumnFlotationAdvanced
Category: Concentration
Subcategory: Flotation
Display name: Finch-Dobby Column Flotation Advanced

Parameters

Derived parameters

Model Description

The Finch-Dobby Column Flotation Advanced 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.

Column geometry

The cross-sectional area of one column is:

A_c=\pi D_c^2/4

The collection-zone height is calculated as:

H_c=min(f_c H_T,H_T-H_f)

The implementation imposes small positive numerical limits on the calculated collection-zone height.

Where:

Superficial velocities and collection-zone residence time

The feed-water flowrate per column is:

Q_(W,col)=Q_W^F/N_col

The feed-liquid superficial velocity including positive downward bias is:

J_l=Q_(W,col)/(3600A_c)+J_b

The liquid residence time in the collection zone is:

t_l=H_c(1-\epsilon_(g,c))/J_l

Where:

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

J_g=0.01J_(g,cm/s)

d_b=d_(b,mm)/1000

Axial mixing

The model represents axial mixing through an equivalent vessel number:

N_p=(a_N/0.6)(D_c/H_c)^0.63 J_g(1-\epsilon_(g,c))/J_l

Where:

Particle settling and residence time

The particle slip velocity is calculated iteratively for each component and size class:

u_(sp,c,i)=g d_i^2(\rho_c-\rho_l)(1-\phi_s)^2.7/[18\mu_l(1+0.15Re_(c,i)^0.687)]

The particle Reynolds number is:

Re_(c,i)=d_i u_(sp,c,i)\rho_c(1-\phi_s)/\mu_l

The implementation uses:

\rho_l=1000

\mu_l=0.0015

The particle residence time is corrected for downward particle slip:

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

Where:

Particle-bubble collection

For particles smaller than the component maximum floatable size, the collection factor is:

E_(c,i)=\alpha_c(d_i/d_b)^(2a_c)(1-(d_i/d_(max,c))^1.5)

For:

d_i>=d_(max,c)

the collection factor is:

E_(c,i)=0

The base collection rate is:

k_(c,i)=3J_gE_(c,i)/(2d_b)

Where:

The fast- and slow-population rates are:

k_(fast,c,i)=F_(fast,c)k_(c,i)

k_(slow,c,i)=F_(slow,c)k_(c,i)

Collection-zone recovery with axial dispersion

For either the fast or slow kinetic population:

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

The non-collected fraction is:

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

The collection-zone recovery is:

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

The component recovery from the combined kinetic populations is:

R_(kin,c,i)=f_(fast,c)R_(fast,c,i)+f_(slow,c)R_(slow,c,i)

with:

f_(slow,c)=R_(infinity,c)-f_(fast,c)

Where:

Material outside the maximum floatable fraction is treated as non-floating by true flotation, although it may still report to floated by entrainment.

Pulp-froth interface and froth recovery

The mean froth residence time used by the implementation is:

t_f=H_f(1-\epsilon_(g,f))/J_g

The froth recovery after detachment is:

R_(froth,c)=F_(froth,c)exp(-k_(det,c)t_f)

The true-flotation recovery is calculated sequentially as:

R_(true,c,i)=R_(kin,c,i)R_(int,c)R_(froth,c)

The true-flotation mass before carrying-capacity correction is:

M_(true,c,i)=M_(F,c,i)R_(true,c,i)

Where:

Froth carrying capacity

When the carrying-capacity limit is enabled, the capacity of all columns is calculated as:

C_f=F_C A_c J_(g,cm/s)N_col

If true-flotation solids exceed this capacity, the entire true-flotation matrix is reduced proportionally:

F_(carry)=min(1,C_f/sum_c sum_i M_(true,c,i))

M_(true,c,i)^*=F_(carry)M_(true,c,i)

Where:

The carrying-capacity limit is applied to true-flotation solids before entrainment is calculated.

Feed-water recovery and wash-water correction

In user-recovery mode:

R_W=R_(W,0)

In wash-water mode, the preliminary dilution-wash number is:

N_(W,0)=Q_(wash)/(Q_W^F R_(W,0))

The wash factor is:

F_W=exp(-\eta_W N_(W,0))

The positive-bias correction is:

F_B=1/(1+10J_b)

The corrected feed-water recovery is:

R_W=R_(W,0)F_WF_B

Where:

The reported froth dilution washes are calculated after the water-recovery correction:

N_W=Q_(wash)/(Q_W^F R_W)

The wash-water attenuation applied to entrainment is:

F_(wash,ent)=exp(-\eta_W N_W)

Entrainment

For each component and size class, the entrainment size factor is:

G_(ent,c,i)=1/[1+(d_i/d_(50,ent,c))^(n_(ent,c))]

The entrainment recovery of material not already recovered by true flotation is:

R_(ent,c,i)=min(1,max(0,F_(ent,c)G_(ent,c,i)R_WF_(wash,ent)))

The entrained mass is:

M_(ent,c,i)=[M_(F,c,i)-M_(true,c,i)^*]R_(ent,c,i)

Where:

Final solids products

The floated mass is:

M_(Flt,c,i)=min(M_(F,c,i),M_(true,c,i)^*+M_(ent,c,i))

The bottom mass is:

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

The total 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 product retained fractions are:

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 product 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

Water balance

In user-recovery mode, wash water is not added to the water balance:

W_(total)=W_F

In wash-water mode:

W_(total)=W_F+W_(wash)

The floated water is calculated only from feed-water recovery:

W_Flt=R_WW_F

The bottom water is:

W_Bot=W_(total)-W_Flt

Therefore, in wash-water mode the implementation assigns the added wash water to the bottom stream.

Recovery calculations

The overall solids recovery to floated is:

R_M=100 sum_c sum_i M_(Flt,c,i)/M_S^F

The true-flotation solids recovery is:

R_(true)=100 sum_c sum_i M_(true,c,i)^*/M_S^F

The entrainment solids recovery is:

R_(ent)=100 sum_c sum_i M_(ent,c,i)/M_S^F

For each component:

R_(true,c)=100 sum_i M_(true,c,i)^*/sum_i M_(F,c,i)

R_(ent,c)=100 sum_i M_(ent,c,i)/sum_i M_(F,c,i)

R_(total,c)=100 sum_i M_(Flt,c,i)/sum_i M_(F,c,i)

The model uses one representative bubble diameter and uniform mean gas holdups for the collection and froth zones.

Axial mixing is represented by an equivalent vessel-number correlation rather than by explicit physical compartments.

The pulp-froth interface and froth phase are represented by sequential recovery factors. The model does not solve a full froth population balance or internal recycle between zones.

The wash-water and bias effects are empirical. In wash-water mode, all added wash water is assigned to the bottom stream after calculating feed-water recovery.

The carrying-capacity limit is applied to true-flotation solids before entrainment is added.

The implementation uses a fixed liquid density of 1000 kg/m³ and a fixed viscosity of 0.0015 Pa·s in the particle-slip calculation.

The kinetic, froth, entrainment, wash-water and carrying-capacity parameters should be calibrated against laboratory, pilot-column or plant survey data.

References

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