XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 951
K K K
132 131 232 =[11]
to obtain the hydrophobic force constants (K132) between
mineral particles 1 and air bubble 2 in water. In using Eq.
[11], the values of K232 =10–17 J was used from Wang and
Yoon (2009).
One can then determine the disjoining pressure (Π)
acting in wetting films using the extended DLVO theory as
follows (Pan and Yoon, 2016),
(
sinh(lh) coth(lh)
h)
A132
K132
6rh
2
6rh
3
0
2
1
2
2
2
3
ff l }}
}2
P =-
-
+
-2}1
-
_icosech(lh)
H [12]
in which the first term represents the contribution from
the repulsive van der Waals (vdW) forces, the second term
representing the repulsive electrical double layer (EDL)
force, and the third term representing the attractive hydro-
phobic force. In Eq. [12], A132 is the Hamaker constant,
ε the permittivity of vacuum, ε0 the dielectric constant of
water, κ the reciprocal Debye length, and ψ1, and ψ2 are
the surface (or ζ-) potentials of the particles and bubbles,
respectively.
Substituting the Π(h) isotherm given in Eq. [12] into
the Derjaguin approximation (1934), one obtains the free
energy isotherm, which is a function of particle radius (R1),
bubble radius (R2), particle contact angle (θ) and surface
tension of water (cLV )as follows,
()2rrdr G h) R R
R R
h 2r
1
1h
r
r
LV
1 2
1 2
0
c
P( =
+
=-
3
=
=c
^cosi
m #
[13]
The value of Gibbs free energy at P (/2h) G 2 ==0 will
give the energy barrier (E1) as follows,
)E G( h 2rrdr
r
r
1 0 0
=3
P= =
=#[14]
The value of E1 determined using Eqs [14] can then be used
to determine kp using Eq. [5], which can be substituted
into Eq. [8] to obtain the overall flotation rate constant k.
Figure 3a shows the size-by-size flotation rate constants
(ki) and recoveries (Ri) plotted for the cases of using i) KAX
and ii) SC as collectors. As shown, the use of Super Collector
greatly increased both ki and Ri, demonstrating the benefits
of using stronger hydrophobizing agents. It is interesting
to note that the differences observed between KAX and SC
are much larger in plant simulations than observed in labo-
ratory flotation tests. In general, high energy dissipation
rates, typically f =10–15 kW/m3 are used to shorten the
flotation times. In plant operations, f =0.8–1.0 kW/m3
were used but at much longer flotation times. In the pres-
ent work, we used f =1 kW/m3 and 23.7 min of retention
time were used for plant simulations.
Figure 3b shows the grade vs. recovery curves obtained
using KAX and SC by plant simulations. Super Collectors
gave higher recoveries but at lower copper grades than
KAX, suggesting that the former can recover composite
particles and, hence, produced lower grade concentrates.
This finding also suggests that SC should help improve
the recovery of coarse particles. The size-by-size recovery
Figure 3. a) Size-by-size bank recoveries (blue) and flotation rate constant (red) as obtained using KAX and SC as collectors. b)
grade vs. recovery curves obtained from simulation of the rougher flotation bank with KAX (red) and SC (blue)
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Extracted Text (may have errors)

XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 951
K K K
132 131 232 =[11]
to obtain the hydrophobic force constants (K132) between
mineral particles 1 and air bubble 2 in water. In using Eq.
[11], the values of K232 =10–17 J was used from Wang and
Yoon (2009).
One can then determine the disjoining pressure (Π)
acting in wetting films using the extended DLVO theory as
follows (Pan and Yoon, 2016),
(
sinh(lh) coth(lh)
h)
A132
K132
6rh
2
6rh
3
0
2
1
2
2
2
3
ff l }}
}2
P =-
-
+
-2}1
-
_icosech(lh)
H [12]
in which the first term represents the contribution from
the repulsive van der Waals (vdW) forces, the second term
representing the repulsive electrical double layer (EDL)
force, and the third term representing the attractive hydro-
phobic force. In Eq. [12], A132 is the Hamaker constant,
ε the permittivity of vacuum, ε0 the dielectric constant of
water, κ the reciprocal Debye length, and ψ1, and ψ2 are
the surface (or ζ-) potentials of the particles and bubbles,
respectively.
Substituting the Π(h) isotherm given in Eq. [12] into
the Derjaguin approximation (1934), one obtains the free
energy isotherm, which is a function of particle radius (R1),
bubble radius (R2), particle contact angle (θ) and surface
tension of water (cLV )as follows,
()2rrdr G h) R R
R R
h 2r
1
1h
r
r
LV
1 2
1 2
0
c
P( =
+
=-
3
=
=c
^cosi
m #
[13]
The value of Gibbs free energy at P (/2h) G 2 ==0 will
give the energy barrier (E1) as follows,
)E G( h 2rrdr
r
r
1 0 0
=3
P= =
=#[14]
The value of E1 determined using Eqs [14] can then be used
to determine kp using Eq. [5], which can be substituted
into Eq. [8] to obtain the overall flotation rate constant k.
Figure 3a shows the size-by-size flotation rate constants
(ki) and recoveries (Ri) plotted for the cases of using i) KAX
and ii) SC as collectors. As shown, the use of Super Collector
greatly increased both ki and Ri, demonstrating the benefits
of using stronger hydrophobizing agents. It is interesting
to note that the differences observed between KAX and SC
are much larger in plant simulations than observed in labo-
ratory flotation tests. In general, high energy dissipation
rates, typically f =10–15 kW/m3 are used to shorten the
flotation times. In plant operations, f =0.8–1.0 kW/m3
were used but at much longer flotation times. In the pres-
ent work, we used f =1 kW/m3 and 23.7 min of retention
time were used for plant simulations.
Figure 3b shows the grade vs. recovery curves obtained
using KAX and SC by plant simulations. Super Collectors
gave higher recoveries but at lower copper grades than
KAX, suggesting that the former can recover composite
particles and, hence, produced lower grade concentrates.
This finding also suggests that SC should help improve
the recovery of coarse particles. The size-by-size recovery
Figure 3. a) Size-by-size bank recoveries (blue) and flotation rate constant (red) as obtained using KAX and SC as collectors. b)
grade vs. recovery curves obtained from simulation of the rougher flotation bank with KAX (red) and SC (blue)

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