Environmental Engineering Reference
In-Depth Information
4.4.2.1
Evaluation of the Static Permittivity through LCR Measurements
LCR measurements were performed using the same coaxial probe, but removing the
short circuit at the distal end. The probe (filled with the LUT) was considered as a
cylindrical capacitor whose capacitance (
C
liquid
) ideally depends on the liquid filling
the space between the cylindrical plates
6
, according to the following equation:
h
liquid
s
C
liquid
=
2
πε
0
ε
(4.7)
ln
(
b
/
a
)
liqui
s
is the static permittivity of the
LUT;
h
is the length of cylindrical plates (i.e., the length of the probe);
a
is the
external diameter of the inner conductor; and
b
is the inner diameter of the external
conductor.
The measured capacitance (
C
meas
) can be regarded as the parallel equivalent of
three separate contributions:
where
ε
0
is the permittivity of free space;
ε
C
meas
=
C
liquid
+
C
par
+
C
fr
(4.8)
where
C
par
is the parasitic capacitance (mostly due to the SMA adapter used to
connect the probe), and
C
fr
is the capacitance due to the fringing field at the probe
end [13]. LCR-based capacitive measurements with the probe in air at increasing
frequencies showed that, after a slight initial decrease, above 1 kHz the measured
capacitance was constant, thus indicating that the effect of parasitic and fringing
capacitances was stable; therefore, measurements at 1 kHz were used to estimate
the value of
C
par
+
C
fr
. This way, capacitance measurements on the materials under
test could be corrected, thus obtaining
C
liquid
and the
ε
s
of the liquid sample.
Successively, the LCR-based approach for the evaluation of
ε
s
was validated
through preliminary measurements on reference materials (i.e., ethyl acetate, 1,1,1-
trichloroethane, chloroform); measurements were repeated ten times for each con-
sidered material, thus compensating for random errors contributions. Results for the
evaluation of the values of
s
are summarized in Table 4.11 and are compared to
the literature values. The LCR-based method was used for the evaluation of the
ε
ε
s
values of oils; also in this case, measurements were repeated ten times for each type
of oil.
4.4.2.2
Evaluation of the Remaining Cole-Cole Parameters for Vegetable
Oils
The values of the static permittivity were considered as known (fixed) values in the
subsequent minimization routines carried out on FFT-transformed TDR data, em-
ploying the previously described approach: as expected, since these minimizations
6
LCR impedance measurements at low frequencies (i.e., at 1 kHz), confirmed the capacitor-
like behavior of the probe; in fact, the phase angle of the impedance was -90
◦
.
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