XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2087
compared the results of this method with the standard solv-
ers available in EIDORS, i.e., Gauss-Newton one step. The
solution can be obtained in a faster manner using the con-
cept of chordal sparsity (Zheng et al.,2020).
However, even after exploiting chordal sparsity, the
optimization problem is computationally costly when run
on a mesh containing 1600 elements. Therefore, we use a
relaxed version of Eq. (2)
∫, .∫ Find s t. Y Jl 0 1 1 ##h, -
3 (3)
We have observed that the solutions to Eq. 3 for central
disc-shaped phantoms are closer to the truth than the first-
order SDP relaxed solutions to Eq. 2. This optimization
problem is much faster as it can be posed as a convex prob-
lem. We solve it using the CVX solver for MATLAB.
Gauss-Newton One-step:
The mathematical formulation of Gauss-Newton (GN)
One step algorithm is (Adler, Dai, and Lionheart 2007) :
argmin Y Jl Rl
2
2
2
2 l =-+
l (4)
Here, Y is the difference voltage data ΔV for various current
patterns, J is the Jacobian/Sensitivity matrix, κ is the vector
of conductivity differences, and R is a fixed prior matrix. In
this paper, we use Noser prior available in EIDORS.
METHODOLOGY
Experimental data (Rakesh, Kumar Reddy, and Narasimha
2014 Reddy 2016) were collected from a laboratory hydro-
cyclone with a cylindrical diameter of 75mm. The primary
objective is to assess the efficiency of the RBO algorithm
in reconstructing the air core under the influence of design
conditions (spigot variations) and operating conditions
(feed inlet pressure) in both two-phase and three-phase sce-
narios. Four different pressure conditions and three spigot
variations were tested to understand the variation in air
core stability during experiments while maintaining a fixed
vortex finder diameter. Sixteen equidistant electrodes were
placed to facilitate the acquisition of reconstruction images.
A comprehensive list of all experimental variables is pro-
vided in Table 1.
ALGORITHM PERFORMANCE ANALYSIS:
In this section, we conducted a thorough analysis of algo-
rithm performance, assessing the reconstruction quality
and observing radial profiles to evaluate the effectiveness of
the RBO algorithm in two-phase and three phase experi-
mental data.
Two Phase (Air–Water) Flow Studies:
The relaxed binary optimization operates with two conduc-
tivity values, considering the air phase conductivity as 10–9
and the water phase conductivity as 1. For two-phase flows,
the algorithm precisely captures the air-core boundary. A
color bar is provided with the reconstructions to enhance
comprehension, where the blue color signifies the low con-
ductivity phase. In contrast, the red color represents higher
conductivity values. Furthermore, the study includes exper-
imental data for the solids phase at 20%. In the three-phase
flow experimental data, the solid phase consists of silica, a
non-conductive material hence, its conductivity is set to a
very low value of 10–9.
The Effect of Operating Pressure on Air-core Diameter:
Hydrocyclone operating at pressures ranging from 10 to
25 psi, with a spigot diameter of 12.5 mm and a vortex
finder diameter of 25 mm, was used in the study. As pres-
sure increases, air-core size increases, as observed in RBO
and GN one-step reconstructions Figure 3, which are also
aligned with the theoretical predictions (Diddi, Jampana,
and Mangadoddy 2022). As the feed pressure increases
inside the cyclone, the intensity of the swirl of feed flow
also increases, which leads to an increment in the discharge
rate of the cyclone’s products of both underflow and over-
flow. This creates a clear path for the atmosphere air to
flow inside, resulting in much lower pressure at the centre,
increasing the airflow and forming a larger air core. As the
pressure increases, the diameter of the air core also increases
(Diddi, Jampana, and Mangadoddy 2022). The ERT algo-
rithms also strengthened this understanding of physics by
their air-core diameter estimation.
The reconstruction profiles presented in Figure 3, show
notable differences between the two image reconstruction
algorithms. Specifically, the RBO algorithm demonstrates
improved conductivity distribution compared to the GN
one-step algorithm. The reconstructions from the RBO
Table 1. Summary of Experimental data
Experimental Variable Count
The conductivity of gas phase (mS/cm) 10–9
The conductivity of Solid phase (Silica)
(mS/cm)
10–9
The conductivity of water phase (mS/cm)
Number of reconstruction methods
Number of electrodes
Number of planes
Vortex finder diameter (mm)
Spigot diameter (mm)
Pressure variations (psi)
1
2 (RBO, GN)
16
1
25 mm
10,12.5,15 mm
10,15,20,25 psi
compared the results of this method with the standard solv-
ers available in EIDORS, i.e., Gauss-Newton one step. The
solution can be obtained in a faster manner using the con-
cept of chordal sparsity (Zheng et al.,2020).
However, even after exploiting chordal sparsity, the
optimization problem is computationally costly when run
on a mesh containing 1600 elements. Therefore, we use a
relaxed version of Eq. (2)
∫, .∫ Find s t. Y Jl 0 1 1 ##h, -
3 (3)
We have observed that the solutions to Eq. 3 for central
disc-shaped phantoms are closer to the truth than the first-
order SDP relaxed solutions to Eq. 2. This optimization
problem is much faster as it can be posed as a convex prob-
lem. We solve it using the CVX solver for MATLAB.
Gauss-Newton One-step:
The mathematical formulation of Gauss-Newton (GN)
One step algorithm is (Adler, Dai, and Lionheart 2007) :
argmin Y Jl Rl
2
2
2
2 l =-+
l (4)
Here, Y is the difference voltage data ΔV for various current
patterns, J is the Jacobian/Sensitivity matrix, κ is the vector
of conductivity differences, and R is a fixed prior matrix. In
this paper, we use Noser prior available in EIDORS.
METHODOLOGY
Experimental data (Rakesh, Kumar Reddy, and Narasimha
2014 Reddy 2016) were collected from a laboratory hydro-
cyclone with a cylindrical diameter of 75mm. The primary
objective is to assess the efficiency of the RBO algorithm
in reconstructing the air core under the influence of design
conditions (spigot variations) and operating conditions
(feed inlet pressure) in both two-phase and three-phase sce-
narios. Four different pressure conditions and three spigot
variations were tested to understand the variation in air
core stability during experiments while maintaining a fixed
vortex finder diameter. Sixteen equidistant electrodes were
placed to facilitate the acquisition of reconstruction images.
A comprehensive list of all experimental variables is pro-
vided in Table 1.
ALGORITHM PERFORMANCE ANALYSIS:
In this section, we conducted a thorough analysis of algo-
rithm performance, assessing the reconstruction quality
and observing radial profiles to evaluate the effectiveness of
the RBO algorithm in two-phase and three phase experi-
mental data.
Two Phase (Air–Water) Flow Studies:
The relaxed binary optimization operates with two conduc-
tivity values, considering the air phase conductivity as 10–9
and the water phase conductivity as 1. For two-phase flows,
the algorithm precisely captures the air-core boundary. A
color bar is provided with the reconstructions to enhance
comprehension, where the blue color signifies the low con-
ductivity phase. In contrast, the red color represents higher
conductivity values. Furthermore, the study includes exper-
imental data for the solids phase at 20%. In the three-phase
flow experimental data, the solid phase consists of silica, a
non-conductive material hence, its conductivity is set to a
very low value of 10–9.
The Effect of Operating Pressure on Air-core Diameter:
Hydrocyclone operating at pressures ranging from 10 to
25 psi, with a spigot diameter of 12.5 mm and a vortex
finder diameter of 25 mm, was used in the study. As pres-
sure increases, air-core size increases, as observed in RBO
and GN one-step reconstructions Figure 3, which are also
aligned with the theoretical predictions (Diddi, Jampana,
and Mangadoddy 2022). As the feed pressure increases
inside the cyclone, the intensity of the swirl of feed flow
also increases, which leads to an increment in the discharge
rate of the cyclone’s products of both underflow and over-
flow. This creates a clear path for the atmosphere air to
flow inside, resulting in much lower pressure at the centre,
increasing the airflow and forming a larger air core. As the
pressure increases, the diameter of the air core also increases
(Diddi, Jampana, and Mangadoddy 2022). The ERT algo-
rithms also strengthened this understanding of physics by
their air-core diameter estimation.
The reconstruction profiles presented in Figure 3, show
notable differences between the two image reconstruction
algorithms. Specifically, the RBO algorithm demonstrates
improved conductivity distribution compared to the GN
one-step algorithm. The reconstructions from the RBO
Table 1. Summary of Experimental data
Experimental Variable Count
The conductivity of gas phase (mS/cm) 10–9
The conductivity of Solid phase (Silica)
(mS/cm)
10–9
The conductivity of water phase (mS/cm)
Number of reconstruction methods
Number of electrodes
Number of planes
Vortex finder diameter (mm)
Spigot diameter (mm)
Pressure variations (psi)
1
2 (RBO, GN)
16
1
25 mm
10,12.5,15 mm
10,15,20,25 psi