This document is intended as an introduction to a problem that we would like to submit to the maths Study Group with Industry at University of Lancaster 2nd – 5th April 2002.

The problem involves a transducer supplied to Nan Gall by Solartron.  It is used for measuring fluid density and viscosity.  Nan Gall would like to apply this transducer for down oil-well applications.  The transducer has not been used down hole before though it has been used in petroleum processing plants.  Nan Gall buy only the transducer without Solartron’s electronics and software.  This is because Solartron’s electronics will not fit in our pressure rated housing, is not rated to down-hole temperatures and draws too much current for battery operation.  See Solartron’s website:

We do not have a confidentiality agreement with Solartron.  However, they are aware that we are doing our own electronics development and research into down-hole applications of the transducer.  They have told me that they have performed some simulation of the system in the past but have not released the results to us.

The transducer is based on the principle that the resonant frequency of an element is dependant on the density of the fluid in which it is immersed.  This is presumably because some of the fluid is dragged with the vibrating element altering the effective mass.  The viscosity of the fluid applies a damping force to the system.  The Q of the resonance therefore decreases with increasing viscosity.

A tuning fork design is used because it is immune to external sources of vibration.  The tuning fork is excited by a driver piezoelectric element.  The resulting motion of the tuning fork is sensed by a pick-up piezoelectric element.  The voltage applied to the driving piezo is proportional to the stress applied to the tuning fork.  If the pick-up piezo is open circuit, the voltage obtained from the pick-up piezo is proportional to the strain.  Alternatively, if the pick-up piezo is short circuit, then the current output is proportional to rate of strain.

Nan Gall’s electronic circuit applies a phase shift to the signal from the pick-up piezo and amplifies it to a constant peak-to-peak level.  This voltage is then applied to the driver piezo.  Depending on the phase shift applied it is possible to vibrate the tuning fork “on-resonance” or to either side of the resonance.  For example, if the pick-up is short circuit then:

0° phase shift causes vibration on resonance.

+45° phase shift causes vibration at 3dB down point with higher frequency (driver is 45° in front of pickup).

–45° phase shift causes vibration at 3dB down point with lower frequency (driver is 45° behind pickup).

The transducer manufacturers recommended operation at the upper 3dB point for measuring density.  Our experiments have verified that this is the case at least for fluids up to about 100cP viscosity, with a very good linear fit between frequency and density.

Note that a simple model based on the equation of a simple damped harmonic oscillator:

(where m is the inertia (mass of tuning fork plus dragged fluid), v is the damping constant related to viscosity, k is the spring constant and F is the applied force) predicts that the resonant frequency is independent of viscosity.  The frequency of the upper 3dB point would appear to depend on viscosity.  Maybe this model is invalid because the volume of fluid dragged by the tuning fork is dependent on viscosity.

The questions that we would like to be addressed by the study group are the following:

1.       What is the best way to operate the transducer to determine fluid density.  Why and to what degree is the frequency of the upper 3dB point independent of viscosity?

2.       If the transducer is mounted in a cylindrical housing, how will this affect its operation?  The transducer will be mounted inside a housing that is smaller than what is recommended by the transducer manufacturer due to restrictions of running in an oil well.  This affects the resonant frequency.

Alan Fleming  15/03/2002