Gas property determination
Natural gas has been utilized by human being as a source of energy for a long history. It is believed that the ancient people in Greek, Persia and India discovered natural gas many centuries ago. However, the Chinese has implemented the first piping system for gas transmission in Sichuan province (The NEED project, 2008). Initially, bamboo has been selected as the fabricating material.
However, for a long historical period, natural has not being used as a main energy resource due to the centralized reserve and lacking of efficient transport method. Things started to change after the long distance pipeline system has been invented, this type of system were firstly made by cast iron with socket and spigot joints which were packed with hemp and sealed with molten lead. Later, polyethylene and steel has been used as the constructing material.
Nowadays, gas trades are mostly carried out via pipeline transport. There have been lots of long distance pipeline gas transport systems working internationally or even intercontinentally. Many of the gases in Europe are originally exploited in Russia and transport all the way through to the destination.
The long distance transportation and variety of sources of global gas supplying have strongly improved the utilization of natural gas. However, they also cause some practical problems. One of the outstanding problems is the gas quality monitoring. Thus, many investigations have been carried out to measure different properties of natural gas. Schley et al (2004) has used a vibrating-wire viscometer to measure the viscosity of natural gas. Jaeschke et al (2002) has carried out a system to evaluate the energy contained in a gas by measure the calorific value. Avila et al (2006) has carried out a system to measure the dew-point curves of natural gas, etc.
Among all the thermal properties, density is a crucial one. It directly or indirectly related to the concentration, purity, Wobbe Index of the gas stream and some other gas quality criteria. The Wobbe Index is the most important characteristic for the classification of natural gases. It is determined by the following equation:
HS is the gross calorific value and d is the relative density (sometimes called specific gravity which is defined by the ratio of test gas density to the reference gas density) of the gas.
Many methods have been invented to carry out such a task. Some of the main method has been summarized by Civan(1989) and ASTM(American Standard of Test Method ) D-1070-85.
Single Inverted bell
The working mechanism of this type of equipment is simply the force balance. Test gas flow into a column which is suspended in oil, the weight of gas can be detected by adding counter weight at the end of a balance beam.
The simplified force balance can be written as:
Where D is the diameter of the bell, ρ is the density of test gas and h is the height of the gas column. For practical operation, correction must be applied for the surface tension and buoyancy effect of the bell edge in the oil.
Double inverted Bells- Arcco-Anubis Gravitometer
Compared to the single inverted bell method, the double inverted bells gravitometer has the advantage of the elimination of the surface tension effect by introducing an identical bell on the other end of the balance. The weights of bells and surface tension forces are equally balanced at the beginning. The weight difference between the reference gas (usually dry air) in one bell and the test gas in the other then can be determined, which can be led to the specific gravity.
Gas analysis using gas chromatograph
The density measurement by using gas chromatograph is achieved by the component analysis rather than direct measurement. As the gas chromatograph can be used to analyze the component data of the test gas, by applying a simple equation, the density of gas can be calculated. yi and ρi is the mole fraction and the pure component density of corresponding spices. Sometimes the pure component density ρi under certain condition is not available. However, for ideal gases under the same pressure, the density of a certain density is just:
M represents the Molecular weight and the subscript “a” refers to air. Thus as long as the density of air under certain pressure is known, the density of natural gas under the same pressure can be calculated
However, such a method takes time and the operation and maintenance cost is relatively high.
Ac-Me Gravity Balance
This type of instrument is an indirectly way to measure the density of gas, which means only the relative density of test gas to the reference gas can be measured.
The measurement is carried out with in a gas vessel. A balance beam with counter weight on one end and a buoy on the other end is integrated in the vessel. Gas is pumped into the vessel with a slight excess amount, then being released slowly by an outlet valve. The gas pressure will then be recorded when the balance reaches equilibrium. The same procedure then will be repeated using reference gas.
When the equilibrium is reached, the buoyancy act on the buoy for both cases should be the same, which means by ignoring the compressibility of the buoy and gases, the density of test gas and reference gas should be the same.
Thus a simple working equation can be easily derived to show that the relative density of test gas is:
Ac-Me Recording Gas Gravitometer
This method is also an indirect way of measuring gas density, since it only measures the relative density.
This system is similar to the single inverted bell method, which also composes a bell in sealing oil. The bell in this case is a large float; the weight of it is balanced by the counterweight on the other end of beam. Test gas passing through the float, the movement of the beam caused by buoyancy force will be recorded by the chart recorder which has been calibrated by known gases. The local temperature and barometric pressure need also to be recorded to process a correction for the specific gravity.
This correction sometimes is done through an automatic build-in mercury system. However, the correction mechanism is beyond the topic this review, details can be found in Morrison(1984) and Woomer (1987).
UGCI Recording Gravitometer
This kind of instrument can be treated as a promotion of the Ac-Me Recording Gas Gravitometer. The design of it is similar to the double inverted bells gravitometer which consists of two floats. The test gas and reference gas are in different float at different end both with equal distance from the fulcrum of the beam. The reference gas float is sealed. A special pressure regulator has been implemented in this system to regulate the pressure of sample gas to be identical to the reference gas. By doing this, the impact of ambient temperature variation on the relative density measurement can be eliminated. The movement of the bean is then recorded by a chart which has calibrated. The relative density can be read.
One typical example of the instrument using effusion method is the Schilling Apparatus. The working principle for this method is that providing the pressure and the temperature is the same, the time for flow the same volume of gas through a hole will be proportional to the square root of the molecular weight of gas.
The Schilling Apparatus is composited by one sealed container which has a vent screw at top for air flowing filled with water and a small tube with a three-way valve at the top. There are two reference points within the small tube which indicate the same volume of gas being specified. The test gas is first injected to the small tube by the three way valve to let the water reach the lower mark point of the tube. Then it is released through the top orifice which is connecting to the three-way valve until the water reaches the higher mark of the small tube. The same procedure will be repeated for the reference gas (normally, air). Both of the time for gas to flow through will be recorded as tg and ta respectively.
And the relative density is just:
Kimray gravitometer is a typical instrument working under this regime. Kimary gravitometer is such an instrument that a wheel which being divided into two part, each part is connected to a manometer. The gas and reference gas are filled into different part of the wheel independently. When the wheel is rotating, there will be a relation between the force balances which involves the term of gas density. Again if the sample side manometer has pre-calibrated using reference gas, the relative density of test gas can be read directly.
Kinetic Energy Method
The kinetic energy of flowing gas is defined as:
As shown above, the kinetic energy has the connection to gas density. If the same amount of kinetic energy can be applied to different gases, by measuring there speed or other relevant parameters, the relative density of gas can be obtained. A typical instrument is the Ranarex Gas Gravitometer. The test gas and reference gas are both sealed in a chamber respectively. Each gas chamber has an impulse wheel connected which driven by an impeller wheel. The impeller wheel and the impulse wheel are not connected. Both impeller wheels are driven by the same motor using a bell to achieve the same output. Then in this case the torques exerted on the impulse wheel rather than the speeds are measured. The difference between the opposite forces acts on the impulse wheel then can be measured to gain a relative density of the test gas to the reference gas.
Optics Method has also been applied on gas density measurement. A typical type of instrument is the Ashby-Jephcott laser interferometer (Burton & Slaybaugh, 1969). Turnbull et al (1993) has also used a similar instrument. This working principle of this type of equipment is that as the density of gas changes, the optical path length of laser beam also changes. As a result, the beam intensity in the test cavity also changes. This change of intensity is monitored by a photomultiplier which records a “fringe” or peak of maximum intensity each time the optical path length changes by one wavelength. Thus if the standard condition of a certain gas has been calibrated, the density changes can be determined by adapting such a principle. However, the problem is that, the increase of the same amount of wavelength will lead to a result in the photomultiplier exactly the same as decrease. So an additional component needs to be added to compose an interferometer schlieren system (LIS) in order to measure the density change of a gas with the sign. The whole system is shown below:
There are several types of different acoustical instrument which have been applied to measure the density of gas.
The Solartron Gas Specific Gravity Transducer represents a successful industrialized acoustic density sensor for measuring relative gas density. The development of such an instrument is based on that the natural frequency of a vibrating thin metal sheet will depend on the density of fluid flowing along it. As the resonating frequency can be measured electronically, known the relationship between natural resonating frequency and the gas density, the gas density can then be determined.
The resonating frequency is determined by:
where K is the stiffness of the vibrating element, M1 is the mass of element and M2 is the mass of fluid.
The only term which related to the fluid density in the equation is M2 , others remain constant, by doing a simply rearrangement, the working equation can be written as
K1 and K2 are constants which can be determined easily by calibration.
By adding a pressure regulator to achieve and a gas chamber component, the relative density of gas can also be measured.
Another method has been proposed by Haran(1988). His principle based on the phenomena that the pressure variation at some fixed point in a constant acoustic field will be proportional to the gas density.
His derivation is given below:
For a volume element dx, dy, dz, which is in an acoustic field along the x direction. The continuity equation should be
Where ρ is the gas density, u is the particle velocity.
For small-density excursions, the equation above becomes
The pressure p and the volume V vary adiabatically for excitation in the audio range. With
Where c is the sound speed and Cp and Cv is the specific heat at constant pressure and volume respectively.
The expressions for velocity and pressure in this acoustic field are
By using the boundary condition that at the transmitter which is x=0 and ua=u0exp(jt), rearranging and neglecting the small terms, the final approximate relation between pa and density of gas can be obtained
Based on this principle, Haran has built a prototype, with only two active elements: a transducer to generate the acoustic field and a receiver which detect the pressure change and produce a response voltage.
His initial prototype gave a divergence up to ±20% due to the reason which he thought was the gas cushion in confined pocket. So a revised design has been carried out.
Such a shell has claimed to have an accuracy within ±2% for densities below 0.1kg/m3 and ±0.2% for higher densities.
Later, Shakkottai et al(1990) revised Harran's work and gave another explanation for the reason why the prototype had a divergence up to ±20%, and a new design has been also carried out together with an American patent applied by Shakkottai et al(1992).
It has been stated in Shakkottai et al's work(1990) that under some circumstances that the pressure variation may depend on the sound speed c, which he treated as a drawback, however, if the sound speed c of the gas has already been determined, can the gas of density be determined?
When studying the electric circuits, a concept has been raised to descript the total resistance of a piece of electrical equipment which is called the electric impedance. This term of definition contains the resistance Re and reactance Xe of the equipment or the circuit, so it is defined as
Where V is the voltage across the circuit and i is the current.
It has been stated in some of the textbooks(Kinsler et at, 1982; Trusler, 1991;Hall,1987) that it is convenient to apply similar laws to an acoustic system.
A concept of acoustic impedance can be raised to show that how is the system's responding ability of a given acoustic pressure p. So the acoustic impedance is defined as:
Where U is the volume velocity.
However, it is worth noting that the term Za involves the term U, which is depend on the geometric natural of the test instrument. Thus another concept has also been raised to represent only the property of the wave-carrying medium. The characteristic acoustic impedance
Which shows how hard it is to drive the fluid move.
It can be shown that by solving wave equation, for the case of guided travelling plane wave, the relationship between the acoustic impedance and the specific impedance of wave-carrying medium is
Rearrange the equation above,
Thus if both the acoustic impedance and the speed of sound can be measured, the density of the wave-carrying medium which in this case, gas, can be determined
However, it is worth noting that this relation is just the simplest case for the guided travelling plane wave, for sound travel in a confined volume, different relation form and correction factor may need to be applied, which will be discussed later.
So now the problem becomes if both the speed of sound and the acoustic impedance of the gas can be determined at the same time in a same instrument, the density of may be able to be determined.
Speed of Sound measurement
The basic equation using to measure the speed of sound is
It is obviously that if both the travel distance and travel time of sound can be accurately measured the speed of sound can thus be determined.
Normally, there are two main methods to measure the speed of sound for fluid: Acoustic resonators and interferometers and pulse-echo methods. The acoustic resonators and interferometers are mainly applied in the gas speed of sound measurement; on the contrary the pulse-echo methods are mainly employed in the measurement of liquid (Meier and Kabelac, 2006).
As the main concern of this review is on gas measurement, so the pulse-echo methods of speed of sound measurement will not be covered.
There are several different criteria to characterize the resonators.
Variable volume V.S. Fixed volume cavity
Consider a plane wave travelling along in a confined tube which has stiff wall at the receiving side. Ignore any of the damping effect, and the wave is generating constant maximum amplitude P0 with a frequency ƒ in a sinusoidal way. Due to the refraction of the end wall, the total amplitude along the wave travelling direction will be the sum of the incident wave and the refraction wave. Thus the total amplitude can be written as:
For general sound wave equation
Differentiate the expression for ɛ with time twice, such an expression can be obtained
As for a plane wave which the pressure only varies in the x direction
Integral and use the boundary conditions that when x=0 and L that the maxima of the acoustic pressure is equal, thus we can obtain a
It is explicit from the expressions for ɛ and p that, the node of ɛ is always the antinodes of acoustic pressure p. As signal of the transducer is depend on pressure rather than the amplitude, and the transducers are conventionally installed on the end of the cavity tube. So it will be ideal that the antinode of pressure always occurs at the end of the tube. Which means the length of tube must satisfy kL=nπ, which lead to
So the length of L the cavity should be integral multiple of the have wave length.
There are two methods in order to achieve this result, either using a fixed frequency but variable volume resonator or a fixed volume but variable frequency resonator.
It is explicit that the design of variable path length resonator will involve some complexity since the introduction of the path length changing element. The typical design consists of a hollow tube fitted with a moveable piston and closed at one end by the active surface of a transducer. As we have stated before this method is normally operating under fixed frequency however, by adjusting the path length of the cavity to achieve the condition
The Kundt's tube is a typical design of variable path length resonator.
For the fixed path length resonator, the same condition is met by adjusting the frequency of acoustic source. As path length is fixed, increasing frequency of the wave generator from zero will meet the condition first when n is equal to 1 and as f doubled, n becomes 2. By adjusting the wave frequency, the similar effect of that changing path length will also occurs. Different mode of longitudinal resonance can be achieved.
Low frequency operation V.S. High frequency operation
As the resonators always operating under non-ideal condition, the basic theoretical working equations always have to be revised when operating. Correction factors will need to be implemented to obtain data with acceptable accuracy. The boundary layer effect is one of the factors which need to be considered when carrying out any acoustic measurement. This effect will be large if the resonator is operating under low frequencies. High- frequency method in some cases can avoid the large end effect, however, a high ƒ will result a small λ, as λ becomes small it is hard to distinguish adjacent normal modes. Although large boundary layer effect may occur under low frequency resonator, a specific single normal mode can be separated for investigation.
Spherical Resonators V.S. Cylindrical Resonators
Resonators can also be characterized by their geometric design. The typical geometric designs of resonators are Spherical Resonators and Cylindrical Resonators.
Normally, spherical designs are using as a type of fixed volume acoustic resonator for the obvious reason that it is not easy to change the volume of a stiff sphere in a controlled way. A series of papers has been published by Moldover et al (1986;1988) and Mehl et al(1981;1982;1986) to demonstrate the characters of such a geometric design.
As has been recognized, the spherical resonator provides the most accurate result over the other design of similar volume and operating frequency (Trusler, 1991).This is mainly due to the radially symmetric normal modes of spherical resonator, which eliminates the effect of viscous damping at the surface and the insensitive of the resonance frequencies caused by geometric imperfection. Both of the reasons lead to a higher quality factor.
An extra advantage of the spherical resonator is that the closed form solutions to the problem of coupling between fluid and shell motion. As gas density increase this coupling is tend to become the major factor that affecting the accuracy of the measuring result.
Moldover et al(1988) has measured the speed of sound for argon at 273.16K between 25 and 500kPa less than 1ppm from the lowest five radial modes at each pressure using a spherical resonator. The largest correction of Moldover's work account to 63ppm average over the (0,2)-(0,6) modes at 100kPa is much smaller than the corresponding correction(about 1600ppm) compare to Quinn's(1976) work of which a cylindrical design has been used.
However, the main drawback of this type of geometric design is the difficulty and cost for fabrication.
The main advantage of cylindrical design over spherical is the respectively simplicity of fabrication. Also by adapting a cylindrical geometry design, the Variable volume design can be simple achieved by adjust the path length of the cavity. This is the reason why most of the variable volume resonators chose to use a cylindrical design. When lacking wide-band transducers, a cylindrical resonator may become the only choice.
Younglove and Frederick(1990) have used a cylindrical resonator to measure the sound speed within an uncertainty about 0.05% while Hurly(1999) claimed his work with a uncertainty of ±0.01%.
Design of Our Rasonator
Ruffine and Trusler(2009) has used a cylindrical resonator to measure the speed of sound of argon, nitrogen and methane at temperatures between 293.15K and 333.15K under the pressure between 0.1MPa and 10MPa, which obtained an absolute average deviation of about 0.02% and a maximum absolute deviation of 0.06% to the most accurate equations of state. This is also the apparatus of which the following of this project will be used.
It is a fixed path length cylindrical resonator using two transducers with an inner diameter of 15mm, a cavity length of 20mm and a wall thickness of 3mm. The fabrication material is Type-316 stainless steel bar stock. Four axial tapped holes in each end have been dig in order to fit the transducer elements. Eight radial holes have also be dig for the purpose of filling and remove of gases. One of the advantages of cylindrical resonator is that it provides an extra degree of freedom which is the ratio of diameter to path length (Trusler, 1991). In this case the ratio has been selected as 3/4 with the purpose of isolate the second longitudinal mode in the frequency spectrum by avoiding the interference with the fundamental radial mode and compound modes of oscillation.
The two end plates which contain the transducer element are also made of Type-316 stainless steel. The diameter of each plate is 21mm and length is 20mm. They are counter-bored axially from the rear face of the resonator to produce a stiff diaphragm with a diameter of 15mm and a thickness of approximately 1.1 mm to detect the vibration. A piezoelectric disc (PZT, 10mm diameter and 0.4 mm thick, poled axially and nickel-plated on the major surfaces) has been attached to a shallow recess on the rear face of the diaphragm by using epoxy resin. The electrical connection between the piezoelectric disc and the rear electrode is achieved by using a small spring. The open end is closed by a cover plate fitted with a bulkhead SMA co-axial connector. The fundamental frequency ƒ1 of the transducer has been designed to be well above the operating frequency of the cavity in order in order to avoid the effect of over coupling. The acoustic perturbation theory has been applied to evaluate the shift of frequency of a longitudinal cavity resonance resulting from the non-zero compliance. The result shows that the frequency shift will only be 10-3 of the operating frequency even the maximum operating pressure has reached. Thus the design of diaphragm will provide an acceptable result.
In this case, the second longitudinal mode is studied. Under this mode, the wave length is the length of the cavity and the travel time of the sound wave to pass a wave length is the period in other word, the reciprocal of the operating frequency. However, correction has to be made. Thus the equation becomes:
ƒ200 is the operation frequency for the second longitudinal mode. L here is the cavity length.
∆ƒ is the sum of two correction terms which is showing below:
∆ƒend is the frequency shift which has been mentioned in the design of diaphragm. It is determined by
Where ρ is the gas density, u is the speed of sound in the gas, ƒ1 is the fundamental resonance frequency, which is determined by
Where ν1=1.015π is an eigenvalue, b is the radius, d is the thickness. E is Young's modulus, σ is Poisson's ratio, and ρs is the density of the fabrication material of the diaphragm.
C is the mean compliance in the limit of zero frequency, which is defined as:
Thus the first correction term can be determined. Problem now lie on the determination of ∆ƒv-t which is the viscous and thermal boundary layer effects factor. It is given by
Here Cp,m and CV,m is the isobaric molar heat capacity and isochoric molar heat capacity respectively. Dt is the thermal diffusivity, λ is the therma conductivity, M is the molar mass, ρ is the density, Dv is the viscous diffusivity and η is the shear viscosity.
However, it has also been examined that the as the total correction |∆ƒ|/ƒ200< 2×10-3 one can ignore all the corrections just allow an error of the order of 0.2%(Ruffine and Trusler, 2009). Since the purpose of the project is to measure the density of gas by measuring the speed of sound and the acoustic impedance, the density value of gas is assumed to be unknown, thus all these corrections above which involve the density of gas will not be feasible to carry out. An alternative way need to be raised, by assuming the gas to be perfect gas, all the density terms in the equations above can be change to the form of measurable quantities.
Acoustic impedance measurement
As the measurement of speed of sound has been achieved within an acceptable accuracy, the problem lies on the measurement of acoustic impedance. Such a task is usually done by an impedance sensor, Dalmond(2001) defined the impedance sensor as a system involving two transducers whose input signals are related to pressure and volume velocity. Benade and Ibisi(1987) has revised the instruments for impedance measurement in the musical wind instruments. A brief introduction will be summarized here.
1. The servo-controlled capillary-drive impedance head.
A servo-controlled loudspeaker is using in this type of instrument as the sound source, the sound then pass through a capillary into the measuring point. The response pressure at the measuring point then can be measured by a detecting microphone. As the volume flow is kept constant by the servo controlled loudspeaker, the impedance can then be determined by definition.
The capillary using in this type instrument varies. Lots of designs have been developed. The most common one was constructed by Johan Backus in 1974. However, the most complex but precise instrument is built by Rene Causse at IRCAM in Paris. However, this type of instrument is only used in the measurement of the magnitude of acoustical impedance due to the complex phase change involved in this system.
2. The velocity-servo magnetic-drive impedance head
The driving source is driven magnetically by a current flow through a voice coil. The quantity being controlled is the piston's velocity. This is achieved either by an accelerometer or an EMF set up in a second coil mounted on it. A volume velocity can be obtained by the known of cross sectional area. This together with the pressure measured by a microphone makes the acoustic impedance can be calculated by definition.
3. The Ionophone-driven impedance head
It has been first discovered by Frans Fransson and Erik Jansson between 1962 and 1975 a corona discharge phenomenon when two electrodes are close enough to each other. This phenomenon has later been developed as a true constant-volume flow source over a wide frequency range. By using another microphone to determine the acoustic pressure, the impedance can be calculated by definition. The time-differentiated signal from this microphone serves as an accurate measure of the volume velocity of the diaphragm.
4. Merhaut impedance head
It is originally developed by Josef Merhaut. The sound is generated by a loudspeaker horn driver connecting to a duct. The detector end is a diaphragm across the duct. There is an insulated planar grid placed behind the diaphragm together with it consists a condenser microphone. Both magnitude and the phase of the air column impedance can be measured by this type of head.
5. Hot-wire anemometer-based impedance head
The main difference to the other impedance heads is that the flow data is not directly measured. It is based on the fact that the cooling rate of a hot wire is depending on the speed of air flowing around it. However, as speed rather than velocity being measured, a dc flow needs to be used as a base to measure a sinusoidal flow variation. The value of the dc flow need to be selected carefully since both too large and too small will impact the measured value of impedance. The measuring result of the original design has an unexplained calibration shift of 50% in magnitude and several degrees of phase over 100-1000 Hz frequency range (Benada& Ibis, 1987).
6. Massive-piston, servo less impedance head
The involvement of servo-controller systems within all the instruments descripted above induced considerable complexity into the apparatus. Several systems have been proposed to achieve a servo less impedance head. The most promising one has used a small Alnico rod magnet placed along the axis of a solenoid and its surrounding magnetic return. It is elastically supported in a way that one end serves as the drive piston for the air column. An annular ring of RTV rubber provides both hermetic sealing and shear elasticity at the air column end of the magnet, while a similar ring at the other end preserves the alignment and provides an additional elastic restoring force for the vibrating piston. The natural frequency of oscillation of this of this driver piston turns out to be in the range of 500-600 Hz, with a Q of 4 or 5. Also it turns out that with proper design, the driving coil gives adequate excitation of the air column with the expenditure of only 1 or 2 W of power at a convenient impedance level.
As it has been stated in previous text, according to the definition of acoustic impedance, the two quantities required to calculate the acoustic impedance are acoustic pressure and the resulting volume velocity. As long as the transducer's response is related to the acoustic pressure or volume velocity by known relation, it can be used in the acoustic impedance measurement. As a result, the classification of acoustic impedance measuring method can also be classified by the quantity that the two transducers measuring.
1. The combination of a pressure and a volume velocity transducer
If conditions allows, the easiest way to measure acoustic impedance will be obviously to measure both the pressure and volume velocity simultaneously. Thus if the signal output from the microphone is e and the signal output of the volume velocity transducer is u, they can be treated as proportional to the pressure P and volume velocity U. In this case the acoustic impedance can be measured through
Proper determination of Re and Ru will lead to a measurement of the acoustic impedance in this case. Pratt et al(1977) and de Bree et al(1996) have developed different type of velocity sensors in their work. However, compare to measuring the volume velocity, it will be even much easier if a source of known volume velocity can be used. Many such sources have been developed under different principles (SINGH & SCHARY, 1978; FRANSSON and JANSSON, 1975; DALMONT,and BRUNEAU ,1992).
2. Two pressure transducers
a. Two pressure tranducers, one is proportional to the volume velocity
In such a design one transducer need to measure a pressure proportional to the volume velocity. A typical design can be descript that since a loudspeaker cannot really be treated as a constant volume velocity acoustic source due to the low acoustic impedance compare to the resonant loads, the measurement of its volume velocity can be achieved by allocating a pressure transducer at the back of the loudspeaker which has been enclosed by some sort of material, since in this case the pressure being measured under low frequency is proportional to the volume velocity. However, this kind of design suffers from a drawback that the driving frequency cannot beyond the first resonance of the cavity. So a promoted design has been developed by Webster(1947). The acoustic source consists of a loudspeaker and a capillary tube, which has high impedance. The pressure transducer is put in the volume between the loudspeaker and the capillary. If the impedance of the tube is greater than the measured one, volume velocity in is should be proportional to the pressure which has been measured.
b. Two transducers measuring pressure
The difference between the arrangements of the two pressure transducers of this method is that in this method, two transducers are allocated along a straight tube but in the former method, one transducer needs to be involved in the setup of the acoustic source. This method is based on the fact that the velocity is locally proportional to the pressure gradient. This method is sometimes called the two microphone method, which is the main concern of many papers. (Chu, 1986) (Gibiat & Laloě, 1990.)
3. Single pressure transducer
The single pressure transducer method is a modification of the two pressure transducers method with the purpose that of removing the errors of the calibration of two microphones with respect one to another. Thus, by using one pressure transducer but repeat the same measurement twice on different position can also achieve the same result(TARNOW, 1995). And if the source and reflect point are both far from the pressure transducer, a pulse can be used to determine the impedance by both measuring the incident and reflected signals (WATSON and BOSHWER,1988).
4. Various transducers
Some other types of transducer have been also used in impedance measurement. However, the restriction is that the quantities being measured should be related to the pressure and volume velocity by known equations. An example is that electrical impedance has been used to be measured in order to infer the acoustic impedance (BRUNEAU and BRUNEAU, 1986).
After the brief discussion of the instruments and methods which has been used in acoustic impedance measurement, the possibility of using the cylindrical resonator which has been applied on sound speed measurement need to be stated here since this is the key issue at the moment blocking the way to infer the gas density.
The general formula for the spatial distribution of the acoustic pressure in the driven cavity has been derived by Trusler(1991) in the form of
The term is the Green function for driven cavity which expanded in terms of the normal modes. Where is the wavefunction, is the complex conjugate of , ω is the angular velocity, ρ is the density of gas, ΛN is the normalization constant of Nth normal node, KN(ω) is perturbed eigenvalue, k is propagation constant, V is the volume of the cavity, Sω is the source point and represents the source within the cavity which surface vibrates with a single angular frequency ω. Under the design of our resonator, the driven frequency is equal to the resonance frequency based on the equation
Where gN is the resonance half with of Nth normal mode. ƒ is the driven frequency, ƒN is the resonance frequency of Nth normal mode.
Also as it is a cylindrical cavity it is convenient to transform it to a polar coordinates, the expression of acoustic can be derived in the form of
At a certain mode of resonance
Also because of the resonator is working at second longitudinal mode of resonance, and it is design in a symmetrical way , and , to remove the subscript N as all the values are under 2nd longitudinal mode, lead to
The expression of Sω is
Where b is the radii of the source and x0 is the mean displacement amplitude.
Substitute the equation above back to ()
The velocity amplitude in this case is iωx0, and the πb2 is the cross sectional area of the cavity. The mean amplitude of volume velocity is the mean velocity amplitude times the cross sectional area. Thus,
This expression imply that as the acoustic impedance can be measured, and all the quantities at the right hand side except the gas density is known , the density of gas is theoretically obtainable.
Huang's paper has derived the expression to be,
being the swept volume.
At initial spot may will give that there should be some relation between the input voltage signal amplitude to the volume velocity amplitude of the transducer generated, also there must be some relation between the detect voltage signal amplitude to the acoustic amplitude.
It can be assumed
However, base on the non-linear response of transducer, the two response factor may not be constant. A initial assumption would be it is frequency dependant, for the output signal of pressure the response factor can be assumed to be
as it is a reverse process with respect to the pressure detecting transducer.
Thus by plotting the data, a linear behavior will be expected. And the gas density can be calculated by known the calibration factor and all the other known quantities.
Huang (2009) derived the expression in the same way, however, due to the different equation before the assumption of him was
A here is the ratio of u to e, however, he seemed to forget that the response factor has been involved twice, thus what he may intend to conclude should be
Conclusion and brief plan
From the theoretical and experiment method review, a brief conclusion can be drawn that it may be theoretically possible to measure gas density by knowing the sound speed and the acoustic impedance.
As the cavity we have has already been proven to be successful in speed of sound measuring, the problem lies on the acoustic impedance measurement.
Professor Trusler(1991) and Huang(2009)'s work has indicated a possible way to solve the current problem by using a response factor which is related to the frequency rather than being constant. Though the linearization Huang(2009) has done was not so satisfying , this may due to the problem of which he has forgotten the response factor need had involved twice in the measuring since the signals of both transducers has been used. Or, possibly this may be caused by his process of working equation derivation.
Fortunately, the data which measured by Huang may still be useful. Another linearization of the experiment data obtained by Huang can be carried out before taking any measurement to see whether the issue can be solved or new assumption need to be made for the relation for detecting and output signals of transducer.
A further reading regarding to the response of transducers may be necessary.
After these, some extra measurement can be performed if necessary.
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