Chemistry Reference
In-Depth Information
3.7
RHEOMETRY
3.7.1
Averaging effects at the pipe wall
Often, profiles show an unrealistic sigmoidal shape close to the pipe
wall, while in theory one would expect the maximum shear rate at the
wall. This effect was already explained in the early 1970s (Jorgensen
et al.
, 1973; Jorgensen and Garbini, 1974) by the convolution of flow
velocity profile and the sample volume. The sample volume is gener-
ated by the ultrasound transducer and has approximately a drop-like
shape. The axial length and shape are determined by the pulse length
and the dynamic characteristics of the piezoelectric element. The di-
ameter of the sample volume depends on the pressure field. Thus, the
echo signal of one measurement channel is, simplified, an intensity-
weighted sum of all flow velocities covered by the pulse. In the men-
tioned articles, Jorgensen and later Hughes and How (1993, 1994) also
show a way to make a deconvolution of the two signals involving a
transfer to the frequency domain and back to the time domain using
(inverse) FFT. An alternative method for the deconvolution would be
a single value decomposition. Another approach to obtain a more re-
alistic profile at the pipe wall is shown by Tortoli
et al.
(1995). The
method, based on the maximum frequency
f
max
of the Doppler spec-
trum, involves a correction factor
k
for the sample volume shape, which
also depends on the transducer characteristics. To calculate the flow
velocities
v
, the following equation (Tortoli
et al.
, 2006) which is a
variation of the classical Doppler equation is used (
α
is a Doppler
angle):
f
max
c
v
=
(3.24)
2
f
0
(cos(
α
)
+
k
sin(
α
))
3.7.2
Fitting
When using the measured data and a model such as power law (Equation
3.14) or Herschel-Bulkley (Equation 3.21), there are a few critical points
to be considered. Usually, a non-linear optimisation is used to obtain
the parameters
n
,
K
and
τ
0
(Herschel-Bulkley only). The result of the
fitting can be quite sensitive to several parameters such as the section of
the start and end point of the profile half used for the fitting (Birkhofer,
2007) or details of the profile shape which can depend on measurement
parameters such as the pulse length or even the emission voltage or
amplification.
