Locating the inflection point of frequency-dependent velocity dispersion by acoustic relaxation to identify gas mixtures

Measuring adiabatic sound speed is an effective method to characterize gases with different molecular weights because sound speed mainly depends on molecular weight at a given temperature. However, it is still a challenge to apply this method to different gas mixtures with similar or even the same sound speeds. Acoustic relaxation in gases may overcome this challenge because sound speed becomes dispersive due to frequency-dependent heat capacity. Based on our previous work on reconstructing acoustic velocity dispersion with a simple measurement method, in this paper, we propose capturing the inflection point of velocity dispersion to identify gas mixtures. Standard detection areas are constructed using the theoretical location of the inflection point scaled by the acoustic velocity and relaxation frequency with different temperatures for target gases. The captured inflection point is located in the detection areas to obtain gas compositions. Thus, gas mixtures with the same molecular weights, such as 86.9% CO2–13.1% N2, 95% CO2–5% H2 and 95% CO2–5% pH2, can be differentiated using our method from only their acoustic velocities. The results show that the maximum absolute error of the compositions for CO2 can be effectively reduced from 3.8% to 0.2% by our temperature correction function. Therefore, the proposed method can identify gas mixtures qualitatively and quantitatively by only measuring acoustic velocity.


Introduction
The measurement of adiabatic sound speed is an effective method to determine the composition of gaseous mixtures. Thus, gas sensing by sound speed has several advantages, such as low-cost [1], high reliability [2,3], fast response [4], and non-necessity of calibration [5], compared to other gas sensing methods, such as electrochemistry [6] and laser sensing [7,8]. To determine the compositions of gas mixtures using sound speed, different schemes and techniques have been successfully implemented. Witschi and Haber [9] first detected the presence of hydrogen or methane in underground mine air through the change of sound speed. Based on Haber's method, Garrett [10] suggested an acoustical gas analyzer for hydrogen and methane with high accuracy and precision by electronic temperature compensation. For complete automation, Hallewell et al [11] developed an online monitor to reveal small fluctuations in the compositions of binary gas mixtures from ultrasonic measurement. In summary, the above methods that used sound speed can identify only binary gas with low selectivity. Some researchers aimed to detect ranges of sound speed for more than two components. Lueptow and Phillips [5] determined the combustion quality of natural gas using the fact that the sound speed of CH 4 is higher than that of C 2 H 6 , C 3 H 8 , CO 2 and N 2 ; Zipser and Wächter [12] compared the ratio of anisentropic sound speed to isentropic speed to distinguish ternary gas mixtures and improve selectivity. All of the above methods have achieved positive results for identifying gas mixtures. However, they only sensed the sound speed of gas mixtures from different molecular weights and failed to acquire more molecular information. Thus, for different gas mixtures with the same molecular weights, such as 95% CO 2 -5% pH 2 , 95% CO 2 -5% H 2 , and 86.9% CO 2 -13.1% N 2 , gas sensing by sound speed has been a challenge for almost a century.
According to acoustic relaxation theory, acoustic velocity changes not only with molecular weight but also with the heat capacity ratio of gas mixtures, and it becomes dispersive from frequency-dependent heat capacity [13]. Different gas mixtures with the same molecular weights may have the same acoustic velocities but different relaxation frequencies, at which acoustic relaxation happens most notably. Because each molecule has a unique relaxation mode, molecular relaxation information can be employed to identify gas compositions [14,15]. For example, Hu et al [15] located absorption spectral peaks to detect gas mixtures by jointly measuring the attenuation coefficient and sound speed, which is based on Zhang et al's [16] two-frequency acoustic measurements. Liu et al [17,18] verified Hu et al's simplified theory by extracting molecular internal effective specific heat. We provide a mixed rotational and vibrational relaxation model of hydrogen gas mixtures and decouple the multiple rotational relaxation into a group of singe relaxation processes to improve the accuracy of ternary gas mixture sensing based on acoustic velocity and absorption [19,20]. However, compared with acoustic velocity, the absorption coefficient has a large measurement error in the order of 5% and results in low detection sensitivity [21]. Recently, we proposed a simple measurement method to reconstruct acoustic velocity dispersion by only measuring acoustic velocities at several fixed frequencies [22]. On the Sshaped curve of frequency-dependent velocity dispersion due to acoustic relaxation processes, we find the inflection point of the dispersion curve to determine the molecular weight and the relaxation frequency at the same time. Thus, by combining our simple measurement of acoustic dispersion [22] and the mixed rotational and vibrational relaxation model [19], we propose a method to characterize gas compositions by locating the inflection point in this paper.
The remainder of this paper is organized as follows. In section 2, we discuss the location of the inflection point of the velocity dispersion in gas scaled by acoustic velocity, acoustic frequency, and temperature to demonstrate that it can be used to identify gas mixtures. In section 3, we obtain the inflection point from measured velocities by reconstructing the velocity dispersion to distinguish gas mixtures with similar, or even the same, acoustic velocities. In section 4, we analyze the detection uncertainty from the velocity measurement error and provide a temperature correction approach. In section 5, we present some concluding remarks.

Gas composition sensing theory based on the location of the inflection point of acoustic velocity dispersion
Acoustic propagation in gases can be characterized by a complex-valued effective wave number k eff (ω) and the effective thermodynamic sound speed c e ; both of them are expressed as [23] where α r (ω) is the relaxation absorption and V (ω) is acoustic velocity. V (ω) is defined as where M and T are the molar mass and temperature of the gas, respectively; R is the universal gas constant.
is the heat capacity ratio and ω = 2πf is the acoustic angular frequency. Under adiabatic conditions, C eff V (ω) has a fixed value for each kind of gas molecule, meaning γ (ω) has traditionally been divided into three categories based on whether the gas is monatomic, or composed by diatomic, or polyatomic molecules. Based on the fixed heat capacity ratio and average molar mass, traditional methods analyze the compositions of binary gas mixtures by measuring sound speed [9][10][11]. However, this analysis cannot be applied to multi-component gas mixtures or gas mixtures with the same sound speeds because of the ill-posedness of the mathematical problem.

Inflection points of acoustic velocity dispersions for gases
According to acoustic relaxation theory, C eff V (ω) reflects the footprint of the molecular relaxation of gases [24]. For a gas mixture with W kinds of molecules, including N types of vibrational modes, C eff V (ω) changes with acoustic frequency and is expressed as [25] Figure 1. Frequency dependences of acoustic velocity and absorption spectrum for CO 2 at T = 297 K, which were calculated from Zhang's model [19].
where C ∞ V and C ∞ l are the translational and rotational specific heats of the gas mixture and molecule l, respectively; a l is the mole fraction of gas molecule l; C int i and τ i are the vibrational specific heat and relaxation time of the ith single-relaxation process.
On the one hand, in equation (1) the imaginary part of k eff (ω) determines the relaxation absorption α r (ω), which is conventionally represented by the dimensionless absorption coefficient α r (ω) λ [26], where is the acoustic wavelength. Thus α r (ω) λ is given by Re (k eff (ω)) .
(4) Figure 1 shows the variation of α r (ω) λ versus frequency for gaseous CO 2 , which results from the molecular vibrational-vibrational and vibrational-translational relaxation processes [21]. The bell-shape absorption spectrum is characterized by the peak amplitude reflecting the relaxation strength. The corresponding frequency is the effective relaxation frequency, which is the reciprocal of the relaxation time. As a result, the peak contains the relaxation characteristics of gas mixtures. On this basis, Hu et al [15] employed the location of the spectral peak to identify the gas compositions of CO 2 -N 2 and CH 4 -N 2 mixtures.
On the other hand, based on equations (1) and (2), the real part of k eff (ω) determines the acoustic velocity V (ω). V (ω) can be expressed as a function of the effective isochoric molar heat capacity C eff V (ω): ) . Figure 1 also plots the variation of V (ω) against frequency, which also results from the same molecular relaxation processes as α r (ω) λ [21]. Clearly, V (ω) monotonically increases with increasing frequency f on the S-shape acoustic velocity dispersion curve. The velocity V m of the inflection point on the curve can be defined as [27] where V 2 (0) and V 2 (∞) are the minimum and maximum values of the squared acoustic velocities in the lowest and highest limits of acoustic frequencies, respectively. V 2 (0) and V 2 (∞) are obtained from ) , where C eff V (0) denotes the sum of the external and internal isochoric molar heat capacities, and C eff V (∞) represents the external isochoric molar heat capacity of the gas mixtures.
In figure 1, the frequency f m of the inflection point has the same value as the absorption spectrum peak because both represent the effective relaxation frequency of the acoustic relaxation process. The inflection point is a unique point on the acoustic velocity dispersion curve and carries the information of the relaxation frequencies. Similar to Hu et al's method, which employed the location of the spectral peak to detect gas compositions [15], we propose using the location of the inflection point to identify gas compositions.

Locations of the inflection points of acoustic velocity dispersion for gas sensing
To determine the composition of a gas mixture, the measurement of the inflection point must be compared with the prediction of a theoretical model. For most gases, molecular vibrational relaxation dominates the relaxation processes, and the rotational relaxation is regarded as constant at ambient temperature [23,24,28,29]. However, for hydrogen, molecular rotational relaxation plays a key role in the relaxation processes, while vibrational relaxation is negligible at ambient temperature [30]. Recently, combining Zhang's vibrational model for gas mixtures such as CH 4 , N 2 , CO 2 [25], we proposed a mixed rotational and vibrational relaxation model to calculate the acoustic characteristic of hydrogen gas mixtures [19]. Based on the mixed relaxation model [19], theoretical acoustic velocity dispersion in this paper is obtained and a theoretical inflection point is acquired for gas mixtures. The theoretical detection areas of these inflection points are constructed at different temperatures, as shown in figure 2. The green curves refer to the influence of temperature on the locations of the inflection points at a constant composition of the gas mixtures. The blue and red curves indicate how the locations of the inflection points vary with the compositions of gas mixtures at fixed temperatures of 273 K and 323 K, respectively.
Gas mixtures with different compositions have different inflection point locations. The effective detection area of CO 2 -N 2 is located at the bottom and left of figure 2, while that of CH 4 -N 2 is located at the top and left. Meanwhile, the effective detection area of CO 2 -H 2 is located at the bottom and right, and that of CH 4 -N 2 is located at the top and right. Clearly, gas mixtures of CH 4 -N 2 , CO 2 -N 2 , CH 4 -H 2 , and CO 2 -H 2 can be qualitatively identified by their respective effective detection areas. By comparison, it can be seen from figure 2 that the acoustic velocity of gas mixtures CO 2 -N 2 ranges from approximately 262 m s −1 to 370 m s −1 while that of gas mixtures CO 2 -H 2 is, similarly, approximately 267 m s −1 to 350 m s −1 . The corresponding relaxation frequency of CO 2 -N 2 varies from approximately 11 kHz-52 kHz, while that of CO 2 -H 2 ranges from 460 kHz-6670 kHz. Due to their overlapping ranges of acoustic velocity, traditional methods fail to identify the gas mixtures, considering only sound speed, but the different compositions of CO 2 -N 2 and CO 2 -H 2 give them independent effective inflection point detection areas, which can be used to distinguish them.

Obtaining the inflection point by simple measurement method
To apply our proposed method to practical industry, simple measurement is necessary. Conventional methods [31][32][33] measure the acoustic velocities over a wide frequency range to obtain the acoustic velocity dispersion. However, it is impractical to operate this by changing the frequency in real time due to a longer measurement procedure and numerous transducers. To overcome this problem, we recently developed a simple measurement approach to reconstruct the whole acoustic velocity dispersion of gas mixtures by solely measuring acoustic velocities at 3-5 frequencies [22]. The frequency-dependent acoustic velocity dispersion of the multi-relaxation processes where τ i and ε i are the relaxation time and relaxation strength of the ith single relaxation process of gas mixtures, respectively. In equation (8), for a single relaxation process, V(ω) can be expressed as a function of V(∞), ε, and τ . Because the vast majority of multi-relaxation processes of gases represent one primary relaxation process, V(∞), ε and τ can be obtained to reconstruct V(ω) in practice by measuring the acoustic velocities at three frequencies [22]. Therefore, one can determine the location of the inflection point from reconstructed acoustic velocity dispersion using equations (5) and (8).

Experimental apparatus
The prototype sensor shown in figure 3(a) is used to measure the acoustic velocities of gases. The sensor has six pairs of transmitterreceiver transducers matching frequencies of 25 kHz, 40 kHz, 75 kHz, 100 kHz, 300 kHz, and 400 kHz. The stepper motor drives the translation stage to move the emitter, while the receiver was kept stationary. The experimental apparatus shown in figure 3(b) is composed of the sensor, the chamber, control panel, gas bottle, vacuum pump (on the back) and measurement instruments. The sensor in figure 3(a) is sealed in the chamber with a diameter of 400 mm and a length of 620 mm. The chamber is made of cylindrical stainless steel, and it can sustain a pressure from −0.001 MPa to 2.5 MPa. The control panel turns the valves of the pipeline on or off to control gases in or out of the chamber. The test gas mixtures 86.9% CO 2 -13.1% N 2 , supplied by Newruide Special Gas Company in China, were mixed by CO 2 and N 2 with a claimed purity of 99.999%. The vacuum pump is used to evacuate gases in the chamber, where the degree of vacuum is on the order of −0.001 MPa. Two digital pressure measurements are used to sense pressure, one with a range from 0100 kPa works for negative pressure and the other with a range from 0 MPa3 MPa is for positive pressure. One digital thermometer with a probe is placed near the receiver and the other is near the door of the chamber, which were calibrated on ITS-90. The type B uncertainty contributions of the instruments are estimated and shown in table 1. Three representative points on the acoustic velocity dispersion curve are selected and the contribution of the pressure to the combined uncertainty of the acoustic velocity is estimated in table 2.  Trusler et al [34][35][36][37] accurately measured the speed of sound in several gases at variable pressures along different isotherms by means of an interferometer or resonator. In contrast, we measured acoustic velocity directly by moving the distance of the emitter and relative delay time between the maximum wave peaks of the receiver over a wide frequency/pressure range, where the accuracy is lower than that of Trusler's method due to experimental limits. A similar measurement procedure was carried out by Ejakov et al [23], the only difference being that they measured acoustic attenuation but we measured acoustic velocity. According to the relaxation theory of gases, lowering the pressure is equal to raising the frequency [23]. One usually changes the gas pressure to broaden the measured frequency range to obtain adequate experiment  figure 4

Identifying the compositions of the gas mixtures with the same molecular weights
Next, we begin to identify the above gas mixtures qualitatively and quantitatively using the locations of the inflection points. First, we identify gas mixtures qualitatively by locating the inflection points of gas mixtures in the effective detection areas of figure 2. If the location of the inflection point in figure 4(a) is in the effective detection area of CO 2 -N 2 , the unknown gas mixture is identified as CO 2 -N 2 ; meanwhile, if the location of the inflection point in figure 4(b) is in the effective detection area of CO 2 -H 2 , it is identified as CO 2 -H 2 . Secondly, the composition of the gas mixtures is determined quantitatively as follows.  The synthesized inflection points from acoustic velocity dispersions for 95% CO 2 -5% H 2 and 95% CO 2 5% H 2 from Behnen et al [38]. because they have the same molecular weight, and traditional methods struggle to differentiate them. However, our proposed method can distinguish them because different molecules and compositions of gas mixtures result in different relaxation frequencies.
H 2 and pH 2 are spin isomers with exactly the same molecular weight. At ambient temperature, the difference between the ideal-gas heat capacity ratio of pH 2 and H 2 is small (about 2%) [39]. Both the same molecular weight and a small difference in heat capacity ratio lead to about 0.1 m s −1 difference of acoustic velocities between 95% CO 2 -5% pH 2 and 95% CO 2 -5% H 2 . The above coincidence cannot be distinguished by traditional gas sensing methods using sound speed only. However, our method can differentiate 95% CO 2 -5% pH 2 and 95% CO 2 -5% H 2 by their different relaxation frequencies.
The frequency of the inflection point for the former mixture is 388 kHz which falls in the effective detection area of CO 2 -pH 2 (black lines in figure 5(b)), while that of the latter is 699 kHz falling in the area of CO 2 -H 2 (red lines in figure 5(b)). Clearly, it is still difficult to identify the gas mixtures because some part of the effective detection area of CO 2 -pH 2 overlaps with that of CO 2 -H 2 . However, we can eliminate the incorrect result. In the effective detection area of CO 2 -H 2 , the location of the inflection point of 95% CO 2 -5% H 2 (red '□' in figure   5(b)) is also identified as 90.5% CO 2 -9.5% pH 2 at T = 291 K. By comparing T = 291 K with the ambient temperature, i.e. 303.15 K (measured by a precise thermometer), the final result is gas mixture 95% CO 2 -5% H 2 and not 90.5% CO 2 -9.5% pH 2 . Therefore, using the locations of the inflection points, one can analyze the compositions of gas mixtures with similar or even the same acoustic velocities.

Analysis of the measurement errors
In this section, we evaluate the influence of one possible error in the determination of the acoustic velocity on the location of the inflection point. According to the reconstruction method of acoustic velocity dispersion, the inflection point on velocity dispersion for most gases can be obtained by measuring acoustic velocities at three frequencies [22]. We first suppose that there is one measurement error at the three frequencies, and then assume that there are two measurement errors at the three frequencies. If there are three measurement errors at respective frequencies, for example, all the velocities are too large or too small. In this situation, the obtained velocity dispersion is likely to overlap with the other velocity dispersion and cannot be corrected. Thus, three measurement errors at three frequencies are not considered here. The error correction and analysis in this section all are based on the gas mixtures 86.9% CO 2 -13.1% N 2 at the temperature 303.15 K.
Suppose that the measurement error of acoustic velocity comes from one of three frequencies f 1 , f 2 and f 3 , while others have no errors. Figure 6(a) considers the case when V 1 at f 1 = 4.667 kHz may be in error by about 1%, which is the worst case of measurement error, as shown in figure 4(a), while figure 6(b) considers the case when V 2 at f 2 = 40 kHz is in error by the same relative amount. It can be observed from figure 6(a) that the position of the obtained inflection point increases from left to right with an increase in V 1 . Therefore, a larger V 1 leads to a larger V m . Similarly, the effect of a measurement error affecting V 3 would produce the same effect with the same trend and is not plotted in figure 6 to avoid repetition. In contrast, as V 2 increases and decreases in figure 6(b), the velocity value V m of the inflection point remains unchanged, while the corresponding relaxation frequency f m slightly decreases and increases, respectively. This is because the velocity value of the inflection point depends only on the sum of V 2 (∞) and V 2 (0).
According to the identification procedure, the consequence of velocity errors may lead to errors in the determination of the composition of the mixture and the temperature which are calculated from the dispersion model. Figure 7 shows how the relative detection errors of the gas concentration ξ (86.9% CO 2 in gas mixtures 86.9% CO 2 -13.1% N 2 ) and those of the calculated temperature T by the dispersion model to the real temperature vary with relative measurement errors of the acoustic velocities V 1 and V 2 , respectively. The relative error of V 1 is proportional to the errors of ξ and T in figure 7(a), while that of V 2 is inversely proportional to those of ξ and T in figure 7(b). The relative measurement error of V 3 has the same tendency as that of V 1 . Furthermore, it can be observed from figure 7 that the relationship between the relative error of acoustic velocity and that of temperature is approximately linear. According to computer analysis, the relationship between the relative error of velocity ∆V and that of temperature ∆T is In the following section, based on equation (9), we use the difference between the temperature determined by the dispersion model and the temperature measured by a thermometer to correct the measurement error of the acoustic velocities of the gas mixtures.

Correction for measurement error
The correction procedure where the measurement error arises from one of three frequencies is discussed in figure 8 . By locating this inflection point in the detection area of CO 2 -N 2 , gas mixture 92.0% CO 2 -8.0% N 2 at T = 310.6 K is obtained. Clearly, the identification result is incorrect by comparing the temperature of 310.6 K with the real one of 303.15 K. First, suppose the measurement error comes from V 1 at f 1 since we do not know at which frequency point the measurement error occurs. The relative error ∆T between the obtained 310.6 K and real 303.15 K is 2.46%. The corresponding correction factor ∆V = 0.016% for V 1 is calculated by equation (9). After V 1 decreases by the correction factor of ∆V, the new inflection point is synthesized. By repeating the correction procedure iteratively, V 1 can be continuously corrected until ∆T converges to zero. The symbols '⃝' in figure 8(b) reflect the traces of the corresponding inflection points and the arrow refers to the correction direction. The final detection result is gas mixture 86.9% CO 2 -13.1% N 2 at temperature 303.6 K. Secondly, suppose that the measurement errors from V 2 at f 2 , and V 3 at f 3 are also considered, respectively. The traces of the corresponding inflection points during the temperature correction processes are shown as '+' and '□' in figure 8(b), respectively, and the detection results are listed in table 5. Therefore, for the situation of one measurement error, even if one does not know which particular measurement point is affected by an error, our correction method can restrict the maximum absolute error to 0.2% based on the volume concentration of CO 2 .
To further demonstrate the correction approach, the measurement errors of acoustic velocities from two of three frequencies are considered.  figure 9(a), which also come from our raw experimental data for gas mixtures 86.9% CO 2 --13.1% N 2 at T = 303.15 K. The initial detection result is 83.6% CO 2 -16.4% N 2 at T = 294.4 K. Because we do not know which two of the three are errors, suppose firstly the measurement errors come from V 1 and V 3 . Both V 1 and V 3 are increased by the same factor of ∆V by comparing 294.4 K with the real T = 303.15 K, and a new inflection point is synthesized. Iterating the correction steps, the traces of the corresponding inflection points are shown as ' * ' in figure 9(b). The final location of the inflection point determines that the gas mixture is 86.8% CO 2 -13.2% N 2 as shown in table 6. The measurement errors from two situations of V 2 and V 3 , V 1 and V 2 are also considered. Their traces of the corresponding inflection points are shown as symbols '△' and '+' in figure 9(b) and the results are listed in table 6.
In conclusion, whether there is one or two measurement errors of acoustic velocities at three frequencies and whether or not one knows where the measurement errors occur, our correction method based on temperature can improve the detection results. A similar temperature correction was conducted to determine gas mixture 40% CO 2 -60% N 2 at T = 293.7 K [15]. Compared with Hu et al's [15] detection error of 4.75% for a CO 2 composition, our maximum abso-   lute error of 0.2% is more accurate. More importantly, Hu et al [15] only discussed the cases where the measured data must be lower than the theoretical value at low frequency and higher than the theoretical value at high frequency. However, our correction method does not require such severe correction conditions.

Conclusion
In this paper, we demonstrated that the inflection point of acoustic velocity dispersion is related to molecular relaxation, and proposed a method to detect gas mixtures with similar, or even the same, sound speeds based on the locations of their inflection points. The inflection point can be synthesized from acoustic velocity dispersion by only measuring the acoustic velocity. To distinguish the gas composition, detection areas are constructed by a mixed relaxation model, and the detection error is effectively eliminated by the temperature correction function. Traditional methods identify gas mixtures by sound speeds, which are invalid to gas mixtures with the same molar mass, but our proposed method can overcome the traditional problem and synthetically obtain both molar mass and relaxation information to distinguish them. Furthermore, the classical absorption becomes very large in the MHz frequency range, and the corresponding relaxation absorption of gases is so small that it is indistinguishable from noise. Thus, locating the inflection point for gas sensing is not only a simple method, but also the only approach applicable above the MHz frequency range for acoustic relaxation technology. Consequently, the proposed method has the potential to monitor gas mixtures in industry.