2890 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
Mass transfer of water to the concentrate
the ith size interval to concentrate ENT
i =
Mass transfer of entrained particles of
(1)
where the mass transfer of water to the concentrate can be
quantified based on pulp, feed, or tailings.
Entrainment is considered a two-step process. First,
particles are transported from the pulp into the bottom
of the froth layer. This occurs in the region just below the
pulp-froth interface. Hydrodynamic conditions such as
impeller speed and superficial gas velocity play a key role
in entrainment. Next, the entrained particles report to
the concentrate with water drainage through the froth –
a process affected by froth structure (Wang et al., 2015).
The main mechanisms proposed for entrainment are liq-
uid convection in the hydrodynamic boundary layer under
bubbles, water transport in bubble wake and swarm effects,
and turbulence from cell agitation. Finer particles are more
prone to entrainment due to their lower settling velocities
and additionally for hydrophobic particles, the true recov-
ery reduces significantly at finer sizes (Wang et al., 2016
Yang et al., 2019).
Particle settling velocity plays a significant role in its
tendency for entrainment and is a function of particle size
and density (Smith &Warren, 1989 Maachar &Dobby,
1992 Schubert, 1999). Among the operating variables,
impeller speed, airflow rate and froth height play a major
role in controlling entrainment rate. Higher impeller speeds
provide greater turbulence leading to increased particle sus-
pension and higher entrainment (Akdemir and Sonmez,
2003 Cilek, 2009 Wang et al., 2016). Increasing the air-
flow rate recovers more water and solids across the pulpfroth
interface but also enhances drainage in the froth. Lower
froth heights increase entrainment as the froth is wetter
and has a shorter residence time for drainage (Szatkowski,
1987 Zheng et al., 2006 Wang et al., 2016). Minimizing
entrainment through a careful control of process conditions
is key to producing high quality concentrates by having
more control on the true flotation (Cutting, 1989 Hoang
et al., 2019).
Quantifying entrainment and incorporating its impact
on flotation performance is key for process modelling and
simulation. Entrainment of target valuable mineral must be
considered along with true flotation to properly quantify
flotation rate constant and true recovery. To characterize
the entrainment phenomenon, researchers have developed
numerous mathematical models to predict entrainment
flow rates and recovery.
Entrainment models can be divided into two main cat-
egories: (1) models that directly calculate entrainment flow,
and (2) models that estimate the degree of entrainment,
from which entrainment flows are determined (Wang et
al., 2015). A fundamental model was developed by Moys
(1978) which related entrainment to suspended solid con-
centration, drainage rate, gas rate, froth height, and water
recovery. While the large number of parameters makes
application impractical for industrial use, the model pro-
vides qualitative insights into factors influencing entrain-
ment. Maachar and Dobby (1992) proposed a direct
entrainment model based on laboratory column flotation
experiments. Their results showed feed water recovery and
entrainment were functions of wash water rate, feed rate,
particle size, and density. Their proposed empirical model
for the entrainment was:
ES =exp (−0.0325 ∙ ∆ρ) ∙ exp (−0.063 ∙ dp) ∙ RFW (2)
RFW =2.58JF−1 exp(−13.1 ∙ JB) (3)
where ES is the solids recovery by entrainment, ∆ρ is the
specific gravity difference between the solids and the water,
dp is the particle size in µm, RFW is the water recovery from
the feed, JF is the superficial feed rate (the ratio between
the volumetric flowrate and the cell cross-sectional area),
and JB is the bias water flowrate in the froth (the differ-
ence between the superficial velocities for the wash water
flow and the concentrate water flow). This model does not
incorporate all the parameters influencing entrainment,
thus, further validation is required before this model can be
reliably applied to industrial mechanical and column cells.
Neethling and Cilliers (2002) on the other hand, mod-
eled bubble motion, coalescence, drainage, and particle
transport. Their model assumed attached particles follow
bubble motion, while unattached particles move with liq-
uid affected by hindered settling and dispersion. The sim-
plified model proposed to predict gangue entrainment was:
F
v
CS
,Entrained
,
y Air P
F S
Settlin i F
i
i
##
##
2y
2C
f
f
f fP
=-f^Q,Q
+-^Q
h
h
(4)
where Fy,Entrained is the vertical flux of the gangue at a point
y, f(Q,QAir) is the dispersion function, dependent on the
air and water flux, among other variables, εF is the frac-
tional slurry content of the froth, εP is the fractional water
content of the Plateau borders, Csi is the concentration of
solids of class i within the Plateau borders, Q is water flux,
Mass transfer of water to the concentrate
the ith size interval to concentrate ENT
i =
Mass transfer of entrained particles of
(1)
where the mass transfer of water to the concentrate can be
quantified based on pulp, feed, or tailings.
Entrainment is considered a two-step process. First,
particles are transported from the pulp into the bottom
of the froth layer. This occurs in the region just below the
pulp-froth interface. Hydrodynamic conditions such as
impeller speed and superficial gas velocity play a key role
in entrainment. Next, the entrained particles report to
the concentrate with water drainage through the froth –
a process affected by froth structure (Wang et al., 2015).
The main mechanisms proposed for entrainment are liq-
uid convection in the hydrodynamic boundary layer under
bubbles, water transport in bubble wake and swarm effects,
and turbulence from cell agitation. Finer particles are more
prone to entrainment due to their lower settling velocities
and additionally for hydrophobic particles, the true recov-
ery reduces significantly at finer sizes (Wang et al., 2016
Yang et al., 2019).
Particle settling velocity plays a significant role in its
tendency for entrainment and is a function of particle size
and density (Smith &Warren, 1989 Maachar &Dobby,
1992 Schubert, 1999). Among the operating variables,
impeller speed, airflow rate and froth height play a major
role in controlling entrainment rate. Higher impeller speeds
provide greater turbulence leading to increased particle sus-
pension and higher entrainment (Akdemir and Sonmez,
2003 Cilek, 2009 Wang et al., 2016). Increasing the air-
flow rate recovers more water and solids across the pulpfroth
interface but also enhances drainage in the froth. Lower
froth heights increase entrainment as the froth is wetter
and has a shorter residence time for drainage (Szatkowski,
1987 Zheng et al., 2006 Wang et al., 2016). Minimizing
entrainment through a careful control of process conditions
is key to producing high quality concentrates by having
more control on the true flotation (Cutting, 1989 Hoang
et al., 2019).
Quantifying entrainment and incorporating its impact
on flotation performance is key for process modelling and
simulation. Entrainment of target valuable mineral must be
considered along with true flotation to properly quantify
flotation rate constant and true recovery. To characterize
the entrainment phenomenon, researchers have developed
numerous mathematical models to predict entrainment
flow rates and recovery.
Entrainment models can be divided into two main cat-
egories: (1) models that directly calculate entrainment flow,
and (2) models that estimate the degree of entrainment,
from which entrainment flows are determined (Wang et
al., 2015). A fundamental model was developed by Moys
(1978) which related entrainment to suspended solid con-
centration, drainage rate, gas rate, froth height, and water
recovery. While the large number of parameters makes
application impractical for industrial use, the model pro-
vides qualitative insights into factors influencing entrain-
ment. Maachar and Dobby (1992) proposed a direct
entrainment model based on laboratory column flotation
experiments. Their results showed feed water recovery and
entrainment were functions of wash water rate, feed rate,
particle size, and density. Their proposed empirical model
for the entrainment was:
ES =exp (−0.0325 ∙ ∆ρ) ∙ exp (−0.063 ∙ dp) ∙ RFW (2)
RFW =2.58JF−1 exp(−13.1 ∙ JB) (3)
where ES is the solids recovery by entrainment, ∆ρ is the
specific gravity difference between the solids and the water,
dp is the particle size in µm, RFW is the water recovery from
the feed, JF is the superficial feed rate (the ratio between
the volumetric flowrate and the cell cross-sectional area),
and JB is the bias water flowrate in the froth (the differ-
ence between the superficial velocities for the wash water
flow and the concentrate water flow). This model does not
incorporate all the parameters influencing entrainment,
thus, further validation is required before this model can be
reliably applied to industrial mechanical and column cells.
Neethling and Cilliers (2002) on the other hand, mod-
eled bubble motion, coalescence, drainage, and particle
transport. Their model assumed attached particles follow
bubble motion, while unattached particles move with liq-
uid affected by hindered settling and dispersion. The sim-
plified model proposed to predict gangue entrainment was:
F
v
CS
,Entrained
,
y Air P
F S
Settlin i F
i
i
##
##
2y
2C
f
f
f fP
=-f^Q,Q
+-^Q
h
h
(4)
where Fy,Entrained is the vertical flux of the gangue at a point
y, f(Q,QAir) is the dispersion function, dependent on the
air and water flux, among other variables, εF is the frac-
tional slurry content of the froth, εP is the fractional water
content of the Plateau borders, Csi is the concentration of
solids of class i within the Plateau borders, Q is water flux,