Theoretical Aspects of an Acoustical Possessing Hydrogen Specificity Daniel P. Lucero Electro-Analytical Transducer Corp., Fullerton, Gal$
Albert C . Krupnick George C. Marshall Space Flight Center, Huntsuille, Ala. The acoustical detection of hydrogen in a gas mixture can be achieved with specificity by utilizing the thermal relaxation properties of hydrogen. The thermal properties of hydrogen change from those of a diatomic to those of a monatomic gas below and above its relaxation frequency, respectively. There is a corresponding change in the velocity of sound through hydrogen gas. The detector consists of two ultrasonic transmitter-receiver systems that measure the acoustic velocity of a gas mixture or its absorption characteristics. One system operates at a frequency below and the other above the relaxation frequency. The acoustic velocity of the gas mixture is a function of frequency only when hydrogen is present. The difference in signal of the systems provides the readout signal. The design of the detector is outlined by an analysis which delineates the major problem areas. Also the interdependence of functional subsystems and the relative importance of the design parameters are established.
ACOUSTICAL DETECTORS have not experienced widespread utilization as analytical devices because of their lack of molecular specificity-ie., the detection of specific molecular species cannot be accomplished without some prior knowledge of the nature of the composition of the gas mixture. These devices, however, have been employed with success as detectors in gas chromatographs and in the control of chemical and petroleum process streams. The detection and assay of particular molecular species in a gas mixture by acoustic means is generally accomplished by variations of two distinct methods : measurement of the changes in the sonic velocity and measurement of the changes of the acoustic energy absorption characteristics of the gas mixture which can be ascribed to the presence of a particular molecular species. An acoustical detector which determines the velocity of sound through a gas mixture may be considered to be geometrically, for the purposes of mathematical representation, two parallel plane surfaces of infinite area separated by a known distance, One surface acts as the transmitter of acoustic energy or waves and the other as the receiver. The space between the surfaces is filled with the sample gas. Changes in the composition of the gas can be related to the changes in the time required for the sound wave to traverse the separation of the transmitter and receiver or to changes in the phase delay of the wave at the receiver surface with respect to the transmitter surface. The relationship between the phase delay and the properties of a binary gas mixture has been established (1) and is presented by Equation 1 below:
( 1 ) F. W. Noble, K. Abel, and P. W. Cook, ANAL.CHEM., 26, 1421 (1964).
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ANALYTICAL CHEMISTRY
(Ap) = difference in phase delay in pure gas and the gas mixture, degrees n = mole fraction of solute gas, dimensionless = distance between transmitter and receiver, cm S = frequency of wave, sec-I f = gravitational constant, 980 cm/sec2 g T = absolute temperature of gas, "K M = molecular weight of gas, g/mole = ratio of specific heat capacities of gas, dimensiony less Cp = specific heat capacity of gas at constant pressure, cal/g- OK 1 = subscript denoting solvent gas 2 = subscript denoting solute gas An inspection of Equation 1 shows that specificity is nonexistent for any molecular species in this system. The parameters in the equation that are representative of the molecular and equilibrium properties of the gas medium are average or effective parameters. Thus, molecular specificity can only be achieved by an acoustical device by observing changes in the acoustic characteristics of a gas mixture that are related to molecular properties and/or phenomena peculiar to a single molecular species. ACOUSTIC SPECIFICITY
Acoustic specificity to hydrogen molecules may be attained in a detector that employs the molecular thermal relaxation properties of hydrogen. Each diatomic molecule has six degrees of freedom; three external ( X , Y , and Z)translational energy modes and three internal energy modes, one vibrational and two rotational. The energy associated with the internal energy modes is quantized--i.e., of certain discrete values. When a system is thermodynamically disturbed, equilibrium is restored through a relaxation process of molecular collisions, which redistributes the energy among the internal and external degrees of freedom of the molecules until the average energy content of each degree of freedom is identical. When the energy contained in a translation mode is rapidly changed, as it is in the compression and rarefaction regions of the gas as a sound wave propagates through it, relaxation of the external modes will occur in a few collisions, but relaxation occurs more slowly in the internal energy modes, because of the smaller percentage of molecules available with sufficient energy to accomplish the quantum changes. If the period of the sound wave is much shorter (high frequency) than the time required for equilibrium t o be established with the internal modes, then the internal degrees of freedom are not involved in the energy changes of the gas during the compression and expansion cycles of the sound wave. For hydrogen, if the sound wave frequency is greater than 5 cycles per second, the vibrational degree of freedom does not
contribute to the specific heat of the molecule or to the mechanism of the wave propagation. The rotational energy modes of most molecules contribute fully to the specific heat capacity of that particular molecule for f < 1 Mc. At higher sound frequencies, there is a gradual lessening of the rotational contribution; for hydrogen there is no contribution for f > 100 Mc. The frequency at which the rotational contribution is changing at the most rapid rate is called the relaxation frequency. As a result of the reduction of the probability of the rotational energy exchange, there are two important acoustical effects that occur: increase in the velocity of propagation of acoustic energy and increase in the absorption of acoustic energy by the gas. The propagation of acoustic energy through a gaseous medium at a frequency near the thermal relaxation frequency of the hydrogen molecule will induce thermodynamic changes in the medium when hydrogen molecules are contained within the system. The changes occur because the number of degrees of freedom of the hydrogen molecules are reduced from those of a diatomic molecule to those of a monatomic molecule. Consequently, the specific heat capacity ratio and the velocity of a sound wave through the gaseous medium are changed when sound waves are propagated at the relaxation frequency of hydrogen. An acoustical hydrogen detector that is specific to hydrogen gas can be designed by utilizing the molecular thermal relaxation properties of the hydrogen molecule. The relaxation effect is detected by a measurement of the difference in the velocity of sound through the gas mixture at the hydrogen molecular relaxation frequency, and at a lower frequency where the relaxation effect is not present. Thus, there will be an additional change in the phase delay between the two frequencies due to the presence of hydrogen molecules in the system. The (Ap) will be expressed by Equation 1 and only those terms that are a function of frequency will changei.e., the ratio of specific heat capacities and the specific heat capacity of hydrogen at constant pressure. Therefore, if a detector is designed to measure and compare the phase delay of the gas mixture at a frequency, f,and at a frequency equal to or greater than the relaxation frequency, fR, of hydrogen, a specific quantitative detection of hydrogen gas is achieved. The specificity of the detector is illustrated by employing Equation 1 for a sample calculation. The detector frequencies are selected to be one and ten megacycles for this purpose. Hydrogen molecules in the gas mixture will be thermodynamically active as both diatomic and monatomic molecules at 1 and 10 Mc, respectively. There will be an expected phase delay change at 10 Mc equal to 10 times the phase delay change that would occur at 1 Mc due solely to the change in operational frequency. However, when hydrogen molecules are present in the gas mixture, an additional difference of the phase delay at 10 Mc appears because of the changes of the hydrogen molecules at the relaxation frequency. The relaxation effect contribution to the velocity of sound of the gas mixture will cause the phase delay change to decrease. This increase in velocity and decrease in phase delay change is expressed by the relationship: (A'P)~-IODIC = lOA(Ap)r=isfc
(2)
The factor A is a parametric grouping of the physical properties of the acoustical system and gas mixture. It is also representative of the changes induced by the parameters describing the hydrogen molecular thermal relaxation effect. It may be delineated by inspection of a sample calculation of the phase delay difference for a gas mixture consisting of
Table I. Thermodynamic and Molecular Properties of a Gas Mixture Containing Hydrogen at 1 and 10 Megacycles @ 1Mc @10Mc Parameter Specific heat capacity of hydrogen @ constant pressure ( C n ) ,cal/g-OK 3.47 2.48 Specific heat capacity of air @ concal/g-"K 0.24 0.24 stant pressure (CP,), Ratio of specific heat capacities of hydrogen (y2),dimensionless 1.4 1.66 Ratio of specific heat capacities of air (TI), dimensionless 1.4 1.4 Molecular weight of hydrogen, ( M ) , g/mole 2 2 Molecular weight of air, ( M I ) ,g/mole 28.8 28.8
5 % hydrogen in dry air. The factor A may be isolated by substituting Equation 1 at 10 Mc into Equation 2:
When hydrogen is absent from the gas mixture, the rat io o the molecular and thermodynamic terms-Le., the factor Ais equal to 10, since the molecular weight, specific heat capacity, and specific heat capacity ratio are not frequency dependent, However, when hydrogen molecules are present in the gas mixture, the thermodynamic properties of the gas mixture are a function of frequency, due to the hydrogen thermal relaxation effects at 10 Mc. The factor A is equal to 1.13 as evaluated from the individual terms at 1 and 10 Mc given by Table I. The calculation shows that there is a 13 increase in the total phase shift at 10 Mc when hydrogen is present at a concentration level of 5 in dry air and the entire relaxation effect is exerted. A detector with a I-cm transmitter-receiver separation at 1 Mc and 1-mm transmitter-receiver separation at 10 Mc yields a phase delay change due only to the presence of hydrogen of 31". In practice, however, it may be necessary to increase the transmitter-receiver separation of the low frequency system to approximately 10 cm to avoid phase distortion and error from reflected waves, It is possible that gases other than hydrogen with a thermal relaxation frequency close to that of hydrogen (10 Mc) may be present in the gas mixture. This could cause a signal interference or a false reading. Oxygen and nitrogen possess vibrational and rotational relaxation frequencies of 50 cps and 53 Mc and 4 cps and 240 Mc, respectively. Interference from monatomic molecules is not possible since thermal relaxation does not occur; thus, helium will not interfere with the signal. Carbon dioxide and carbon monoxide possess relaxation frequencies lower than 1 Mc. DETECTOR CONFIGURATION
The interrelationships in the transport of acoustic energy through the solid and gaseous media and at gas-solid interfaces of the detector are described by an analytical model. The critical parameters are related to two fundamental constraints imposed upon the detector: the transmitter and receiver surfaces must be structurally isolated; and the acoustical impedance of the solid and gaseous media at the transmitter and receiver surfaces must be as closely matched as possible. The configuration of the detector is described by two sets of flat parallel surfaces separated by a known disVOL. 40, NO. 8, JULY 1968
1223
Figure 1. Transmitter/receiversystem configuration tance. Their lateral dimensions are infinite relative to the separation. Figure 1 illustrates the transmitter-receiver systems and identifies the important parameters. The space separating the transmitter and receiver is completely open and exposed to the gaseous environment to permit rapid sampling. The separation is relatively small yet large enough to preclude mechanical interferences and vibrations that could be conducted through the structure supporting the transmitter and receiver. Large plane surface areas are required for the transmitter and receiver to provide an acoustical radiation field which is relatively free of edge distortions and which will not restrict the flow of gas into the sampling volume. Supporting Structure. The design constraints of the detector supporting structure can be functionally grouped into three categories : the transmitter and receiver and each transmitter-receiver set must be structurally isolated from one another; a firm and rigid mechanical attachment of the transmitting and receiving devices to the structure must be maintained; and the space separating the transmitter-receiver surfaces must be dimensionally stable. Structural isolation of the transmitter from the receiver is mandatory to minimize the amount of energy transported to the receiver at the detector frequency by any medium other than the sample gas. Complete isolation, of course, cannot be achieved since some degree of mechanical coupling is necessary to ensure dimensional stability. Semi-isolation, however, can be accomplished by coupling the transmitter and receiver through a long structural path length possessing a large effective acoustic energy absorption coefficient. The resonant frequency of the structure should be relatively small to induce high frequency energy transmission losses by an equivalent viscous damping process. The resonant frequency is reduced by increasing the path length of the structure and selecting a construction material possessing a low modulus of elasticity to material density ratio. Other energy transmission losses can be introduced by purposely adding acoustic wave scattering sites such as mechanical joints, interfaces, etc. Piezoelectric transducers are ordinarily used as the transmitting and receiving devices for purposes similar to this 1224
ANALYTICAL CHEMISTRY
application. The efficiency and general operating characteristics of the transducers are related to their geometrical and dimensional properties which are primarily established by the mode of their mechanical attachment to the structure. There are basically two modes by which the transducers can be attached: a rigidly backed and peripheral attachment. The rigidly-backed attachment primarily differs from the peripheral attachment in that the back side of the transducer is not free to move. The thickness dimension of the transducer which determines its resonant frequency is dependent on this condition. It is essential that the back surface of the transducer not move, otherwise an inefficient and off-resonant operation will result. However, it has a great advantage in that a large heat sink is provided and the problems introduced by the large temperature coefficients of piezoelectric transducers are diminished. A peripheral mode of attachment allows the back side of the transducer to move freely. The resonant thickness dimension of the transducer is twice that of a rigidly-backed transducer and thus its mechanical strength is increased. This mode of attachment is less difficult to achieve than the rigidly-backed mode. However, a relatively poor heat sink is provided and the problems attendant with changes in temperature are increased. The dimensional stability of the detector or the space separating the transmitter and receiver must be within relatively small tolerance limits to prevent false and spurious signals. Thus, differences in the mechanical rigidity, unsymmetrical forces, thermal expansion, etc. of corresponding transmitter-receiver sets must be minimized. The design of the structure must incorporate techniques whereby unsymmetrical movement of the transmitter-receiver set is cancelled out by compensating movements. Transmitter-Receiver Transducers. The transmitting and receiving surfaces of the detector may be comprised of piezoelectric transducers oscillating in their thickness mode. A uniform sinusoidal voltage is applied across the transmitting transducer which acts as the driving force to vibrate the surface in a controlled manner at its resonant frequency. The transducer must be fabricated with extremely small tolerances in its thickness dimension to minimize its internal
ADHESIVE LAYERS OF NEAR ZERO THICKNESS AND ZERO ENERGY ABSORPTION /--,
Figure 2. Multilayer acoustic transducer 2; = Acoustic impedance of driven piezoelectric disk Zl = Acoustic impedance of layer 1 2, = Acoustic impedance of layer 2 2, = Acoustic impedance of air or other gaseous media Z , = Effective acoustic impedance of multilayered transducer Z , = 2,% ~
2 1
acoustic impedance. Its thickness will be either one quarter or one half a wavelength of the acoustic wave propagating through the material and will depend upon the method of attachment to the supporting structure. The geometrical configuration of the transducers should approach that of a perfectly flat circular disk to ease the problems of mechanical attachment. Curved focusing surfaces are not applicable in the design because of the close proximity of the transmitter and receiver which limits the mode of transporting the acoustic wave to straight line propagation. The effective acoustic impedance of the transducer at its interface with the sample gas must be reduced by attaching additional one-quarter wavelength layers fabricated of materials with successively decreasing impedance to obtain better matching at the gas interface. The ratio of impedance of a material such as quartz to that of air may be as high as 0.0358 X 106. Multilayered transducers can reduce the impedance ratio from 3.5 x l o 4 to at least 3.5 X lo2,which is a significant improvement. The driving voltage is not applied to the secondary layers of the transducer, but only to the piezoelectric disc. Oscillations of the secondary layers occur in their thickness mode. Their thickness is one quarter of the wavelength of the acoustic wave propagating through the material. The most serious problem in the fabrication of the multilayered transducer is that of bonding the layers to each other. An adhesive should be employed which possesses the largest possible bonding strength and is relatively stable over the operating temperature limits. The adhesive should approach zero thickness at the interfaces so as not to act as an additional layer of material which may shift the resonant frequency of the multilayered system. These design constraints are necessary to minimize the losses in the transducer. Figure 2 illustrates the fundamental aspects of the multilayered transducer. Rate of Response Characteristics. Rate of response characteristics of the detector are established by the geometry
of the space separating the transmitter-receiver surfaces and the mechanism transporting hydrogen molecules into the space. The transport mechanism which defines the upper limit of the detector time constant is molecular diffusion of hydrogen molecules through air. When the sample space of the detector is approximated by a flat right circular cylinder, the time constant is described by the equation tc = R 2 / 4 D , where tc = detector time constant, seconds, R = radius of the cylinder, cm, and D = molecular diffusion coefficient of hydrogen through air, cm2/sec. When R = 1 cm and D 0.2 cm*/sec, tc = 1.25 seconds. The time constant of a system where gas is flowing into the sample space is much smaller and is given by the relationship: tc = aR2S/5Q, where Q = sample gas flow rate into the detector, cma/sec. Thus, tc = 0.189 sec, when R = 1 cm, S = 0.5 cm, and Q = 1.66 cma/sec. DETECTOR READOUT SYSTEM Two distinct systems are available to the detector to perform the readout task: the phase delay readout system and the energy absorption system. Each system is independently activated by different effects of the relaxation process. Phase Delay Readout System. The detector consists of two sets of parallel transmitting and receiving surfaces. One transmitter-receiver set is operating at a frequency f b and the second transmitter-receiver set is operating at a frequency f a , which are less than and greater than the relaxation frequency, respectively. The phase change difference between the detector sets can be attributed solely to the presence of hydrogen when the frequency and dimensional conditions of [he detector sets are related by the equation fb/h = SbiSa. Ideal conditions are assumed to prevail in this relationship such that signal distortions introduced by single and twice reflected waves reaching the receiver surface are absent. VOL 40, NO. 8, JULY 1968
1225
The two sets of surfaces are identical in all respects, except for their operating frequencies and transmitter-receiver separation. The phase measuring system that may be employed has been described in the literature (2). It essentially measures the phase shift between the transmitter driving voltage and the receiver output voltage. The difference in the phase delay at the different frequencies is : ((Oa)MIX - ((Ob)MIX = ( P
+ Ap)a + Ap)b (9
=
(A(O>a- (A(O>b (4)
where (A (o) is expressed by Equation 1. Gas mixtures of air containing less than 10% hydrogen will yield (Ap) that are negative at both frequencies because the velocity of sound in the gas mixture is increased over that where hydrogen is absent. When the relaxation process does not occur in the gas mixture, the velocity of sound and the corresponding phase delay will not change with frequency. However, when the relaxation process occurs, the velocity of sound and the corresponding phase delay will change with frequency. A computational analysis illustrated the phase delay change for the condition where thermal relaxation does not occur and where relaxation occurs to be 13z of the phase delay change at the lower frequency. Thus, Equation 4 is equal to 0.13(Ap). An estimate of the magnitude of the lower limit of the phase delay that the readout system must measure can be determined by assumingf, large enough to reduce the hydrogen molecules t o a thermally monatomic state. Therefore, if V, V
= =
velocity of sound in gas mixture when f > fR, cmjsec velocity of sound in gas mixture when f < fR, cmjsec
then the velocity of sound at 10 Mc approximately equals 4V,/9. The estimate in the difference in phase delays at fa and f b is given by: [((OMIXh - ((DMIX)blla=lOMo 0 . 1 3 ( d~ b
=
"9
(5)
Equation 5 can be reduced to: ((OM1X)u - ((OY1X)b
2.8'/%Hz
An accurate reading in the phase delay difference of this magnitude will require a reading of the phase delay at each frequency t o approximately 0.01 %. The noise, signal attenuation, and other possible signal interferences may influence the design of the readout system and impose additional design constraints. The thermal noise of the associated electronics will be -157 dbm for a band width of 50 cycles and a temperature of 298 OK. Therefore, the signal power at the input of the receiver must be approximately -64 dbm. The attenuation of the amplitude of the transmitted wave by the sample gas may be estimated by an examination of the linear absorption coefficient which is a function of the frequency of the transmitted wave and the total gas pressure. The calculation of the power attenuation with a reduction in pressure indicates that the amplitude of the transmitted wave is small when the total gas pressure is below 0.2 atmosphere. The detector may be required to operate in the environmental temperature range from -100 "C to f 8 O "C. Variation from the extremes of this temperature range will intro-~ ~~
(2) F. W. Noble, ZSA J . , 8, 54(1961).
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ANALYTICAL CHEMISTRY
duce changes in the dimensional tolerance of the system and perhaps affect the stability of the detector. Assuming an average temperature coefficient for the detector system structure to be 24 X 10+ "C-', the percentage change in the separation of the transmitting and receiving surfaces over this temperature range could be as much as 0.2%. A change in the separation which corresponds to a phase delay change of 0.01 " at 10 Mc is approximately lo-' cm or a per cent dimensional change of 0.001. This change means that the thermal expansion for each cell must agree within 0.005%, to cancel out the dimensional phase delay change due to temperature variations and temperature gradients in the detector system, Absorption Readout System. In the frequency region where the acoustic velocity of hydrogen gas is a function of frequency, there is an attendant molecular process that is manifested by an increase in the absorption of energy above that which is predicted by classical considerations (3). The maximum relaxation absorption occurs at the relaxation frequency of the hydrogen molecule. Therefore, when two detectors are utilized operating at two distinct frequencies, as with the phase delay system, a specific detection and measurement of the concentration of hydrogen gas can be achieved by the identical comparison procedure. The absorption readout method is described by a simple analytical technique which relates the absorption of acoustic energy of a gas mixture when hydrogen is present and when it is absent. The energy of the received signals may be expressed as a function of the energy of the source signals, the separation of the transmitter-receiver surfaces, and the amount of energy absorbed by classical processes as shown below:
energy of the received signal in the transmitterreceiver system operating at f b , ergs E, = energy of the received signal in the transmitterreceiver system operating at fa, ergs Eo = energy of the transmitted signals, ergs f f = acoustic absorption coefficient, atm-sec2/cm P = total pressure of the gas mixture, atm
Eb
=
The magnitude of the received signal of both detectors is identical when hydrogen molecules are not present in the gas sample and when the relationship between the transmitter-receiver separation and frequency described by the equation: = fa2/fb2. The reduction of the energy of the transmitted signal due to the relaxation absorption process may be estimated on the basis of the assumption that the partial absorption coefficients of a gas mixture is proportional to the absorption coefficients of the gases comprising the mixture and the composition of the gas. Proportionment of the absorption coefficients of the gas constituents in this manner will yield the effective or total absorption coefficient of a binary gas mixture as a function of the composition : ffT
n
=
= ffc f n f f A
mole fraction of hydrogen in the gas mixture
aA = excess absorption coefficient above classical for
pure hydrogen gas, atm-sec2/cm (3) J. J. Markham, R. T. Beyer, and R. B. Lindsay, Reu. Mod. Phys., 23, 353 (1951).
cyc
= classical absorption coefficient of mixture not in-
cluding relaxation (approximately the same as pure air for dilute solution) atm-sec2/cm aT = the absorption coefficient of the mixture, atm-sec2/cm The detector readout signal would be comprised of the ratio of the difference of the acoustic energy absorbed at each frequency to the acoustic energy absorbed at the frequency below the relaxation frequency of hydrogen: (E, - Eb)/Eb. The absorption coefficients at each frequency are related by the expression: ac,S, = c t c b s b . Substitution of this expression into the signal equation yields:
The exponential term may be expanded into the series function : e-x
=
1 - x, x
-P
0, x = m A S a
Therefore, signal equation can be reduced to : Ea - Eb _-
=
naJ,, ncuAS,
