2684 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
and
in the underflow zone.
,z,th
,th:=-^1
,th:=
,,th:=
,th:=
F
f
f
f
f
f
in zone 3,
in zone 2,
in zone 1,
B
E B B E
B B b B
B B
B B b b B B
U B
b B 3
2
1
2
1
3
z
z z
{zB z zBhq
z z
z
=
-
-+--
-+--
-+--
-
in the effluent zone,
^{,z
^{,z
^{,z
^{,z
^{,zB,th:=-^1
^^1
^1
^1
^{h
^{h
^{h
^th{
^th{
^jb
^j
^j
^z
^z
^z
^1
^1
^1
^thh{
^thh{
^thh{
hfb
hf
hf
hq
hq1
h
h
h
hq
hq
Z
[
\
]
]
]
]
]
]]
(0.4)
with fb(φ):= ν∞φ(1 – φ)nRZ the classical sedimentation Kynch flux density function, ν∞ denotes the terminal
settling velocity of an individual solid particle and nRZ the hindered settling exponent (we refer to Bürger et
al. 2005 and Betancourt et al. 2014 for a full description of the mathematical issues regarding to the sedi-
mentation problem). For the bubbles (aggregates) we have jb(fB) as the bubble flux density function given by
(Bürger et al. 2022):
,
jb
0
1. for
for
B
B B
n
B n n
B c
c
1
1 term
term
b
S b
##
#
y z z
z zch2
zBh2nS
z z
z z
-
-
-
+-
+^zBh:=
^1
^1
^1 h Z
[
\
]
]y ]
(0.5)
Where vterm is the constant velocity of a single bubble in liquid and nb a dimensionless constant. ϕC refers to
a critical bubble concentration that is when the bubbles are in contact with each other forming a froth. The
expression in the second case of the definition of jb is derived from a compatibility condition which makes it
possible to express the drainage velocity of liquid in the froth relative to the bubbles with respect to gravity
and dissipation in terms of vterm and another dimensionless constant nS .The latter emerges from empirical
Figure 1. Schematic drawing of the model of the pilot flotation column