The application of Helmholtz resonance to determination of the volume of solids, liquids and particulate matter : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Instrumentation and Process Engineering
The aim of this investigation was the creation of a high precision volume
measurement device using the Helmholtz resonator principle, the purpose of which
was to measure, without interference, liquids, solids and particulate samples. A
previous study by Nishizu et al. (2001) suggested they achieved an accuracy of about
+1% of full scale, where full scale is 100% fill of the resonator chamber. Theory
suggested that with careful design and measurement accuracy of approximately
+0.1% should be achievable.
A high precision resonator was designed using acoustic theory and drawn using
SolidWorksTM computer aided design software. This was then built using a
computerised numerical controlled milling machine. The resulting resonator was
coupled to a 16-bit high-speed data acquisition system driven by purpose-made
LabVIEWTM software. Using a resonant hunting method, repeatability was within
+1mL for a 3L chamber and the accuracy was better than +3mL, which is +0.1% of
full scale for liquid and solid samples.
Testing of particulate material gave results indicating complex behaviour occurring
within the resonator. Accuracy of sub-millimetre granular samples was restricted to
approximately +1%, and fill factors to about 50%. This reduction in accuracy was
caused by a combination of energy absorption and resonant peak broadening. Medium
sized particles, between 1mm and 15mm allowed measurement accuracies of
approximately +0.5%. Larger samples, greater than 15mm in diameter, gave results
with comparable accuracy to water and solids tested.
It was found that most materials required a post measurement curve fit to align
predictive volume calculations. All samples were observed to have a predictive
deviation curve with coefficients dependent on the material or general shape. This
curve appeared to be a function of sample regularity and/or whether the sample has
interstitial spaces. To achieve high measurement accuracy temperature compensation
was required to negate drifts in sample measurement.
Chamber mapping was conducted using a spherical solid moved to precise locations,
then making a three-dimensional frequency map of the inside of a dual port resonator.
This showed the length extension term for the moving mass of air in the port
penetrates roughly three times further than theory suggests. However, the influence of
this extra ‘tail’ was found to be negligible when calculating sample volumes.
A new method of measuring volume was developed using Q profile shifting and
ambient temperature information. Accuracies for this method were comparable to
those found using the resonant hunting method. A significant advantage of the new
method is a 2-3 second measurement time compared to approximately 40 seconds for
the resonant hunting method. The Q profile shifting method allowed volume
measurements on samples moving through a dual port resonator at speeds of up to
Free fall measurements proved unsuccessful using existing methods, but variations in
signal data for different sample sizes suggest the need for future investigation.
Follow-up studies may provide new interpretation models and methods for high-speed
acquisition and analysis required to solve freefall measurements.
Precise temperature (speed of sound) and flange factor (responsible for port length
extension) relationships were evaluated. The correction factor for the speed of sound
with temperature was found to be marginally different to established theory using the
Helmholtz equation due to temperature secondary effects in the port length extension
factor. The flange factor, which determines port length extension, for the
configurations used in this investigation was experimentally found to be
approximately 5% less than theoretical values.
It was established that the sample to be measured must be within a certain region of
the chamber for accurate volume measurements to be made. If the sample were larger
than the bounded region the resonant frequency would no longer obey the Helmholtz
relationship. This would thereby reduce the accuracy of the measurement. All samples
irrespective of cross sectional area were found to alter the resonant frequency when
they were over 85% of the chamber height.
An equalisation method termed environmental normalisation curve was developed to
prevent environmental and loudspeaker deficiencies from colouring Q profiles used in
Q profile shifting procedures. This was undertaken as Q profile shifting relies on
consistency in the Q profile. The environmental normalisation curve was able to
equalise external factors to within +0.4dB. The environmental normalisation method
could be used to post-process data or applied in real time to frequency generation.
The controlled decent Q profile shifting technique was refined further to be used in
continuous measurements in a single port resonator. Samples could theoretically be
measured up to 15% of full-scale fill before resonant peak predictability would
compromise accuracy. Measurement times were from one to three seconds, depending
on environmental temperature stability.
An alternative Helmholtz resonator was developed and investigated using an inverted
port. This variant has potential applications for a seal-less chamber and port with
rapid non-interference chamber access. Q factors for the inverted port resonator were
found to be significantly less than tradition [sic] Helmholtz resonators. It is believed this is
due to a larger boundary layer acoustic resistance occurring in the inverted port.
A variable chamber resonator was designed and built as a further development of the
Helmholtz resonator volume measurement system, as the uncertainty of measurement
is a function of resonant chamber size. Therefore, using the variable chamber
resonator the chamber size could be customised to the sample size. In this way the
uncertainty of measurement could be minimised. The variable chamber resonator was
used with both the resonant hunting method and the Q profile shifting method.
Volume measurements on produce and minerals using the variable chamber resonator
yielded results of similar accuracy to measurements on calibration samples. Each
sample type displayed characteristics that would make specific calibration necessary.
Both techniques were able to detect hidden void spaces, larger than 2% of the sample
volume, and in punctured samples. Therefore, both methods may be viable for rapid
sorting of produce and minerals.