2694 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
Summing over the all excitation frequencies f, where VLf
is the magnitude of the voltage at each f, giving the total
stress:
S
V
R f
f f
2rdC
1
f L
L
f
0
0 =
+_i2 /(7)
The force acting on the sensor is calculated by multiplying
the stress and the exposed piezoelectric film area:
F =S · Ap (8)
According to Meng et al. (2014), the PVS film measures the
drag force which they suggest is proportional to the kinetic
energy fluctuation. In fact, a pressure difference Δp occurs
near the edge of the piezoelectric film when the incoming
vortices from the surrounding fluid with velocity v deceler-
ate. In this work, it is posited that the force, F, calculated
in Eq. (8) is a function of this pressure difference Δp which
in turn is proportional to (ρ/2)v2. And the drag force on
the thin film is considered negligible. The list of parameters
needed to calculate F are listed in Table 2.
RESULTS AND DISCUSSION
Piezoelectric Force Measurements and Turbulent
Kinetic Energy (TKE)
The force measurements from the piezoelectric vibration
sensor were compared with TKE using two techniques
(experimental: PIV, dimensional analysis: TKEDR) that
were performed in cells with two different volumes (1L and
35L, respectively). The two validations allow to assess the
variability in the signal as well as the relationship between
force and TKE across a different geometry.
(i) Particle Image Velocimetry (PIV)
The single-phase PIV experiments were performed in a lab-
scale stirred tank (D =90 mm), which was equipped with
a Rushton impeller. TKE measurements were made at a
radial distance of 7.75 mm and axial distance of 30 mm
using PIV. Further details on the experimental methods are
explained in Sommer et al. (2021). From the 2D-PIV, the
TKE from two-component velocity fluctuations was calcu-
lated by:
TKE ul vl 4
3
PIV
2 2h =+^(9)
Force measurements at the same positions were also con-
ducted using the piezoelectric sensor and the results of both
techniques are shown in Figure 3 (a). It clearly indicates
a similar trend between the two methods, showing higher
turbulence with increasing tip speed. Turbulence is higher
Figure 2. Piezoelectric sensor measurement workflow
Table 2. List of piezoelectric sensor variables
Variable Magnitude Units
g
33 330 Vm/N
t 2.80e-05 m
d 9.24e-03 Vm2/N
C 5.00e-10 Farads
C
0 5.00e-11 Farads
R
L 1.3e+06 Ohms
Ap 2.55e-04 m2
Summing over the all excitation frequencies f, where VLf
is the magnitude of the voltage at each f, giving the total
stress:
S
V
R f
f f
2rdC
1
f L
L
f
0
0 =
+_i2 /(7)
The force acting on the sensor is calculated by multiplying
the stress and the exposed piezoelectric film area:
F =S · Ap (8)
According to Meng et al. (2014), the PVS film measures the
drag force which they suggest is proportional to the kinetic
energy fluctuation. In fact, a pressure difference Δp occurs
near the edge of the piezoelectric film when the incoming
vortices from the surrounding fluid with velocity v deceler-
ate. In this work, it is posited that the force, F, calculated
in Eq. (8) is a function of this pressure difference Δp which
in turn is proportional to (ρ/2)v2. And the drag force on
the thin film is considered negligible. The list of parameters
needed to calculate F are listed in Table 2.
RESULTS AND DISCUSSION
Piezoelectric Force Measurements and Turbulent
Kinetic Energy (TKE)
The force measurements from the piezoelectric vibration
sensor were compared with TKE using two techniques
(experimental: PIV, dimensional analysis: TKEDR) that
were performed in cells with two different volumes (1L and
35L, respectively). The two validations allow to assess the
variability in the signal as well as the relationship between
force and TKE across a different geometry.
(i) Particle Image Velocimetry (PIV)
The single-phase PIV experiments were performed in a lab-
scale stirred tank (D =90 mm), which was equipped with
a Rushton impeller. TKE measurements were made at a
radial distance of 7.75 mm and axial distance of 30 mm
using PIV. Further details on the experimental methods are
explained in Sommer et al. (2021). From the 2D-PIV, the
TKE from two-component velocity fluctuations was calcu-
lated by:
TKE ul vl 4
3
PIV
2 2h =+^(9)
Force measurements at the same positions were also con-
ducted using the piezoelectric sensor and the results of both
techniques are shown in Figure 3 (a). It clearly indicates
a similar trend between the two methods, showing higher
turbulence with increasing tip speed. Turbulence is higher
Figure 2. Piezoelectric sensor measurement workflow
Table 2. List of piezoelectric sensor variables
Variable Magnitude Units
g
33 330 Vm/N
t 2.80e-05 m
d 9.24e-03 Vm2/N
C 5.00e-10 Farads
C
0 5.00e-11 Farads
R
L 1.3e+06 Ohms
Ap 2.55e-04 m2