XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 883
Recognizing that operating gap is a variable that
responds to the characteristics of the hydro-pneumatic sys-
tem and operating pressure, a mechanical-thermodynamic
model proposed elsewhere (Bauer, 2011) and already used
for lab-scale (Barrios and Tavares, 2016) and industrial-scale
(Campos et al., 2023b) HPGRs is presented below for the
prediction of such a variable, with its structure described
based on three states:
ii. State #0: The hydraulic system force is given by F0 =
0. The nitrogen pressure in the accumulator is equal
to the pre-charge pressure p0, which is defined at the
beginning of the operation of the HPGR. The gas
completely fills the internal volume V0 of the accu-
mulator. The relationship between the hydraulic sys-
tem force F and the operating pressure:
F a
p
p
m =(6)
where a
p is the piston area given as
a 4 p
p
2 rD
=and D
p
is piston diameter.
State #1: The static force F1 is loading the system (F1
F0). The force is sufficiently high to compress the gas in the
accumulator isothermally to volume V1, and the defined
initial pressure of the hydraulic system is p1. Based on the
isothermal state change from 0 to 1, we have:
p V p V0
1 1 0 =(7)
State #2: F is the hydraulic force and oscillates around
F1. Therefore, the gas volume is compressed (compres-
sion) and expanded (restitution) through a polytropic state
change to volume Vm and operational pressure pm:
p V p V
m m 1 1 =i i (8)
where θ and o adiabatic exponent given as 1.4 for diatomic
gases (Bauer, 2011).
The operating gap is then calculated as:
A
V Vm
g
p
1 ||0 =
-
+(9)
where
0 |is the initial gap, Ap is the effective area of the
pistons given by A a N
p p p #=,and Np is the number of
pistons.
The application of the presented model between
Equations (6) to (9) depends on a clear relationship between
the change in system pressure from the initial setpoint (p1)
to the hydraulic pressure (pm), which will rely on material
characteristics such as particle size, breakage characteristics
and particle bed stiffness. As such, as information from
the process relies mainly on the hydraulic pressure (pm), a
simple relationship can be established between this variable
and the initial pressure p1, given by:
log^p
log^p
m
1
1
2 j
j
=
+
h
h (10)
where
1 j and
2 j are fitting parameters (Campos et al.,
2023a).
The size reduction, on the other hand, should be pre-
dicted based on the analytical solution of the population
balance model by calculating the product size distribution
(wi,k) for Nb sections along the axial roll position (Torres
and Casali, 2009):
*o expe- w A U
S
z
,
,k
k
j
i
ij,k
g 1
=
=
/(11)
where *z is the distance between the beginning of the
compression region and the extrusion zone (Torres and
Casali, 2009), A
ij,k is Reid’s matrix (Reid, 1965) and S
,k is the breakage rate for material contained in each size class
j and section k along the roll length of the HPGR. The
analytical solution in Eq. (11) depends on the breakage
function and the specific selection function (Herbst and
Fuerstenau, 1980) to describe the material breakage behav-
ior. The cumulative non-normalizable breakage function is
given by (King, 2001):
for B x
x
x
x
x
ij
j
i
j
i
i $z zhc ~=+-
c b
c ^1 m m
for B
x
x
x
x
x
x
ij
i
j
i
j
i
i 1 za ~z ~=+-
h c b
c ^1 k m hc m (12)
where xi is the particle size, γ, β, ϕ, ω and η are fitting
parameters and bij is the distributed breakage function cal-
culate from b B B
ij i ij 1,j =-
-.The breakage rates for each
of the Nb sections are given by:
S s H
Pk
,k i
E
k
W= vv(13)
where Hk is a constant hold-up along the roll length (Torres
and Casali, 2009), Pk is the power profile and si is the spe-
cific selection function:
//xt /x ln_s ln^x s
i
E E
i i 1 1 1 2 1 p p =+i h h@2 6ln^x (14)
where s E
1 ,
1 p and
2 p are fitting parameters, x
i is the rep-
resentative size calculated from x x x
i i i 1 =
-and x
1 is a
reference size given by the top size class.
The power profile used in Eq. (13) allows the model to
predict the variation of the product size distribution along
Recognizing that operating gap is a variable that
responds to the characteristics of the hydro-pneumatic sys-
tem and operating pressure, a mechanical-thermodynamic
model proposed elsewhere (Bauer, 2011) and already used
for lab-scale (Barrios and Tavares, 2016) and industrial-scale
(Campos et al., 2023b) HPGRs is presented below for the
prediction of such a variable, with its structure described
based on three states:
ii. State #0: The hydraulic system force is given by F0 =
0. The nitrogen pressure in the accumulator is equal
to the pre-charge pressure p0, which is defined at the
beginning of the operation of the HPGR. The gas
completely fills the internal volume V0 of the accu-
mulator. The relationship between the hydraulic sys-
tem force F and the operating pressure:
F a
p
p
m =(6)
where a
p is the piston area given as
a 4 p
p
2 rD
=and D
p
is piston diameter.
State #1: The static force F1 is loading the system (F1
F0). The force is sufficiently high to compress the gas in the
accumulator isothermally to volume V1, and the defined
initial pressure of the hydraulic system is p1. Based on the
isothermal state change from 0 to 1, we have:
p V p V0
1 1 0 =(7)
State #2: F is the hydraulic force and oscillates around
F1. Therefore, the gas volume is compressed (compres-
sion) and expanded (restitution) through a polytropic state
change to volume Vm and operational pressure pm:
p V p V
m m 1 1 =i i (8)
where θ and o adiabatic exponent given as 1.4 for diatomic
gases (Bauer, 2011).
The operating gap is then calculated as:
A
V Vm
g
p
1 ||0 =
-
+(9)
where
0 |is the initial gap, Ap is the effective area of the
pistons given by A a N
p p p #=,and Np is the number of
pistons.
The application of the presented model between
Equations (6) to (9) depends on a clear relationship between
the change in system pressure from the initial setpoint (p1)
to the hydraulic pressure (pm), which will rely on material
characteristics such as particle size, breakage characteristics
and particle bed stiffness. As such, as information from
the process relies mainly on the hydraulic pressure (pm), a
simple relationship can be established between this variable
and the initial pressure p1, given by:
log^p
log^p
m
1
1
2 j
j
=
+
h
h (10)
where
1 j and
2 j are fitting parameters (Campos et al.,
2023a).
The size reduction, on the other hand, should be pre-
dicted based on the analytical solution of the population
balance model by calculating the product size distribution
(wi,k) for Nb sections along the axial roll position (Torres
and Casali, 2009):
*o expe- w A U
S
z
,
,k
k
j
i
ij,k
g 1
=
=
/(11)
where *z is the distance between the beginning of the
compression region and the extrusion zone (Torres and
Casali, 2009), A
ij,k is Reid’s matrix (Reid, 1965) and S
,k is the breakage rate for material contained in each size class
j and section k along the roll length of the HPGR. The
analytical solution in Eq. (11) depends on the breakage
function and the specific selection function (Herbst and
Fuerstenau, 1980) to describe the material breakage behav-
ior. The cumulative non-normalizable breakage function is
given by (King, 2001):
for B x
x
x
x
x
ij
j
i
j
i
i $z zhc ~=+-
c b
c ^1 m m
for B
x
x
x
x
x
x
ij
i
j
i
j
i
i 1 za ~z ~=+-
h c b
c ^1 k m hc m (12)
where xi is the particle size, γ, β, ϕ, ω and η are fitting
parameters and bij is the distributed breakage function cal-
culate from b B B
ij i ij 1,j =-
-.The breakage rates for each
of the Nb sections are given by:
S s H
Pk
,k i
E
k
W= vv(13)
where Hk is a constant hold-up along the roll length (Torres
and Casali, 2009), Pk is the power profile and si is the spe-
cific selection function:
//xt /x ln_s ln^x s
i
E E
i i 1 1 1 2 1 p p =+i h h@2 6ln^x (14)
where s E
1 ,
1 p and
2 p are fitting parameters, x
i is the rep-
resentative size calculated from x x x
i i i 1 =
-and x
1 is a
reference size given by the top size class.
The power profile used in Eq. (13) allows the model to
predict the variation of the product size distribution along