Sensing the characteristic acoustic impedance of a fluid utilizing acoustic pressure waves.

Ultrasonic sensors can be used to determine physical fluid parameters like viscosity, density, and speed of sound. In this contribution, we present the concept for an integrated sensor utilizing pressure waves to sense the characteristic acoustic impedance of a fluid. We note that the basic setup generally allows to determine the longitudinal viscosity and the speed of sound if it is operated in a resonant mode as will be discussed elsewhere. In this contribution, we particularly focus on a modified setup where interferences are suppressed by introducing a wedge reflector. This enables sensing of the liquid's characteristic acoustic impedance, which can serve as parameter in condition monitoring applications. We present a device model, experimental results and their evaluation.

Introduction

Liquid property sensors gain more and more importance in the monitoring of many technical processes (e.g. drain intervals of lubrication oil can be optimized by online-measurement of selected physical parameters). A lot of recent work has focused on viscosity sensing often combined with the determination of other fluid parameters like mass density or speed of sound in the liquid to be monitored. Standard laboratory equipment used for viscosity measurements mostly involves motors and/or rotating objects and is consequently bulky and maintenance-intensive. Moreover, they require manual sample withdrawal and they are therefore often less suited for online monitoring. Previously investigated miniaturized sensors utilizing shear vibrations suffer from the drawback that due to the strong attenuation of shear waves in the liquid only a thin fluid layer is being sensed [1]. We recently devised a novel sensor concept primarily based on the viscous attenuation of pressure waves, which enables sensing the bulk of the liquid sample. From the theory of the propagation of acoustic waves in gases the attenuation of pressure waves in fluids is well known [2]. In [3], we presented a setup utilizing this principle where the longitudinal viscosity coefficient rather than the shear viscosity coefficient is sensed. Recently also acoustic spectroscopy was utilized to determine the longitudinal viscosity [4,5]. Compared to these approaches, we focused on an integrated sensor system rather than using laboratory equipment. The sensor setup basically utilizes interferences appearing in a resonant chamber containing the liquid sample and can also be used for the measurement of sound velocity [6].

In the present contribution, we briefly discuss the devised model and associated numerical simulation results for the resonating setup. Then we focus on another sensing application using this setup, i.e. the determination of the characteristic acoustic impedance of the sample liquid, which can be achieved by a slight modification. We present an associated theoretical model, experimental results and their evaluation.

Theory and modeling

The sensing concept is based on a previously proposed sensor setup [6] utilizing the attenuation of pressure waves in a liquid sample. The basic sensor setup is sketched in Fig. 1. The arrangement consists of two rigid boundaries separated by a distance h, forming the chamber for the liquid sample.

Fig. 1

Basic sensor setup.

In one of the boundaries, a PZT transducer with a diameter d and a thickness l is flush-mounted. This transducer generates (resonating) pressure waves in the liquid. Owing to the mounting the backing of the transducer is air, resulting in a defined acoustic impedance at the transducer's back side. Even though an air backing reduces the wave amplitude that can be generated in the liquid, it yields the advantage that no complicated models with possibly unknown material parameters of a backing structure have to be included in the model presented below.

Due to the mismatch of the acoustic impedances of the liquid and the rigid boundary, the pressure waves in the liquid are reflected at the opposite chamber wall, leading to a comb-like interference pattern in the spectrum of the transducer's electrical impedance. The spacing of the comb-like resonances is governed by the speed of sound in the liquid under test [6]. The whole setup can be modeled by combining a standard 3-port PZT transducer model with an acoustic transmission line representing the pressure wave propagation in the liquid and two acoustic load impedances accounting for the PZT backing (Z) and the second boundary (Z) as shown in Fig. 2.

Fig. 2

One-dimensional model of the sensor system with the transducer as 3-port model combined with an acoustic transmission line representing the fluid.

According to [7], the electric impedance Z3 of a PZT transducer loaded with acoustic impedances Z and Z at the acoustic ports can be calculated by (assuming complex notation and a time dependence exp(jωt)), where j is the imaginary unitwhere Z and Z represent acoustic load impedances defined byHere C0 denotes the capacitance of the mechanically clamped transducer, Z represents the acoustic impedance of the transducer with the area A, ω is the angular frequency, and is the effective wavenumber of longitudinal waves within the PZT transducer. F1,2 terms the normal forces at the transducer faces, T the associated normal stresses, v1,2 the velocities (the indices 1 and 2 refer to the acoustic ports, 1 = transducer backside, 2 = side facing the sample chamber), and k is the electromechanical coupling factor of the piezoelectric material (PZT). For a more detailed general discussion about the transducer modeling see [6,7]. Due to the air backing of the transducer, Z can be approximately set to zero representing an acoustic shortcut because of the low acoustic impedance of air compared to the typical acoustic impedances of PZT and of liquids.

Using a one-dimensional approximation, the pressure waves in the liquid can be modeled by the following equations (we assume that the wave propagates in y-direction)Here T, u, ζ, ρ, and c are the normal stress, the displacement amplitude in the liquid, the adiabatic compressibility, the mass density, and the sound velocity of the fluid, respectively. Using the analogy V (electric voltage) ↔ T and I (electric current) ↔ v, the characteristic acoustic impedance Z and the propagation constant γ of the fluid can be found as (analogous to electric transmission line theory) [8]Owing to the viscous losses, Z and γ are complex-valued. Furthermore, for realistic material data (e.g. water or glycerol) and neglecting the viscous losses, Z and γ can be approximated as given in Eq. (5). The attenuation of the pressure waves is determined by the longitudinal viscosity, which is given by λ + 2μ, where μ and λ are the first and second coefficients of viscosity, respectively, and thus affects the Q-factor of the resonances. The definition for the viscosity coefficients is unfortunately not unique [2,5]. In this paper, we adopt the notation of White [9] where the two viscosity coefficients are labeled as μ and λ (not to be confused with the Lame constants in the theory of linear elasticity, which are often labeled using the same symbols). Using this notation, the linearized Navier–Stokes equation appears in the formHere, denotes the displacement vector and the dot above the symbol its derivative with respect to time. μ is the coefficient representing the shear viscosity (often only termed viscosity) while λ describes the dilatational viscosity (associated with compressional stress components). Using another common notation (see, e.g., Landau–Lifshitz [2]), the terms in Eq. (6) are replaced by . Obviously, the two definitions (White and Landau–Lifshitz notation) can be converted into each other using the formulas η = μ, η = (2/3)μ + λ. The longitudinal viscosity is then given by (2μ + λ) = [(4/3)η + η].

As mentioned before, the liquid in the sample chamber is modeled by a terminated acoustic transmission line. According to common transmission line theory (see, e.g., [10]), the input impedance of the acoustic transmission line terminated with an impedance Z can be calculated asAn ideal rigid boundary can be represented by an acoustic open circuit which means that the acoustic impedance Z is infinite. As described above, the interference pattern in the impedance can be used to determine the liquid's properties, in particular the velocity of compressional waves and the longitudinal viscosity of the liquid. Alternatively, if interferences are suppressed, the characteristic acoustic impedance of the liquid can be determined, which is essentially given by the density and the compressional wave velocity of the liquid while the influence of the longitudinal viscosity can be most often neglected (see Eq. (5)). To suppress the interferences, an infinitely long acoustic transmission line or a matched load impedance Z would have to be used. Therefore, we placed a wedge reflector in the transmission path which deviates the propagating wave such that virtually no interference occurs, which corresponds to an infinitely long transmission line in the model. In that way, the characteristic acoustic impedance of the liquid affects the transducer's electrical impedance and the comb-like resonances disappear, i.e., the acoustic impedance Z is given by Z rather than Eq. (7)

Eq. (1) can be rewritten (assuming Z = 0 for the air backing) asEq. (9) states that the electrical impedance of the air backed transducer, apart from the frequency, primarily depends on the fluid's characteristic acoustic impedance Z, all other parameters in Eq. (9) are determined by the PZT transducer.

Simulation results

The 1D-model described above was implemented in MATLAB for simulation purposes. For all simulations the value of the second coefficient of viscosity λ was arbitrarily set to zero as values for most materials are not securely established yet and since the values do not affect the results significantly. This means that the assumed longitudinal viscosity used in the simulation is given by 2 μ. For the simulations of the full 1D-model the following parameters have been used:

PZT disk: diameter d = 10 mm, thickness l = 0.5 mm, material PI-ceramic PIC-255, simulation data see [11].

Geometry: h = 29 mm

Fluids (reference data taken from [12,13], for a temperature of 20 °C), as the value of λ is not known for most liquids yet, it was arbitrarily set to zero such that the longitudinal viscosity is given by 2 μm). The following table gives the used values for the mixtures:

For air, the following data according to [14] have been used:

Fig. 3 shows the simulation results for the electrical impedance of the PZT disk when the sample chamber is filled with air, distilled water, and a 70% glycerol–water mixture without using the reflecting wedge. In this case, interferences are not suppressed and consequently characteristic comb-like resonances can be observed. The interference patterns are superposed to the transducer's own characteristic impedance spectrum featuring a series resonance followed by a parallel resonance. As mentioned before, the spacing of the superposed resonances is determined by the sound velocity of the fluid in the test chamber. Furthermore the lower Q-factor and higher damping of the resonating pressure waves in the 70% glycerol–water mixture due to the higher viscosity of glycerol compared to distilled water can be observed.

Fig. 3

Simulated electric impedance of the PZT transducer for air, distilled water and a 70% glycerol–water mixture without using the metallic wedge reflector so that the comb-like resonances can be seen.

The simulation results for different water–glycerol mixtures using the metallic wedge to supress the interference effects are shown in Fig. 4. The comb-like resonances [3] disappear and the dependence on the fluid's characteristic acoustic impedance Z can be seen. With increasing Z, the electrical impedance of the PZT transducer decreases/increases at the intrinsic parallel/series resonance of the transducer.

Fig. 4

Simulated electric impedance of the PZT transducer for air and different glycerol–water mixtures using the metallic wedge reflector.

Experimental results

A prototype setup shown in Fig. 5 was built. The sample chamber is made of 1.5 mm FR4 PCB material. One of the wall features an embedded PIC255 PZT disk (PI Ceramic, diameter d = 10 mm, thickness l = 0.5 mm) which is connected to an SMA connector. The disk faces free space (air) on one side and the fluid sample on the other. For this device the distance h according to Fig. 1 is 50 mm.

Fig. 5

Prototype device with the PZT transducer (diameter 10 mm) connected to the SMA connector embedded in the left wall. The metallic wedge can be seen in the fluid chamber.

An Agilent 4294 impedance analyzer was used to measure the electric impedance of the PZT transducer. The metallic wedge is needed for sensing the liquid's characteristic acoustic impedance as described above. The reflection face of the wedge encloses an angle of 60° with its base. The wedge is furthermore rotated (see Fig. 5) to avoid that the reflected wave is in the same plane as the incident wave. The exact wedge position was adjusted such that virtually no interference phenomena could be observed in the impedance of the PZT disk. This justifies the model using an infinitely long acoustic transmission line.

Fig. 6 shows the measured magnitude and phase of the transducer's electrical impedance when the sample chamber is filled with different media and using the metallic wedge to suppress the interference effects. The qualitative behavior predicted by our model can be observed. The electrical impedance decreases/increases with increasing Z at parallel/series resonance of the PZT transducer, respectively. However, it can be seen that the simulation values differ from the experimental values. Reasons for the differences are, e.g., losses in the PZT disk itself, acoustic radiation into air, and additional damping due to the mounting of the transducer. Furthermore, particularly for the measurement in air, additional resonances can be seen at 4.12 MHz and 4.25 MHz. These resonances are supposed to be higher order modes of the PZT disk's radial mode. The simple 1D-model used in our modeling approach does not account for these modes. Further improvements in the model and the prototype device are currently investigated (e.g., including transducer losses, PZT parameter fitting, alternative modeling which account for the radial modes). As shown in [15,16], the PZT material data provided by the manufacturer has very high tolerances. This is one of the reasons for the shift in the resonance frequencies between the simulation and the measured data. To get a feeling for the behavior of our setup, we performed a parameter fit for the PZT data (k, ɛ33, c33 = c33RE + jc33IM, where the imaginary part of c33 accounts for damping effects) assuming a reference value for the acoustic impedance Z which was obtained from [12,13]. For the sake of comparison, this fitting procedure for the PZT parameters was performed three times for three mixing ratios, 0%, 50% and 100% weight of glycerol (yielding three slightly different sets of PZT parameters). We then took each set of the so obtained PZT parameters and calculated the impedance spectra for a series of mixing ratios, which were fitted to the respective measurement data by fitting only Z. The so-obtained fitted Z-value is then considered as the measured value. A hill climbing fit algorithm with a variable step size [17] implemented in MATLAB was used to perform the fit procedure. The flowchart of the whole fitting is shown in Fig. 7.

Fig. 6

Measured electric impedance of the PZT transducer for air and different glycerol–water mixtures using the metallic wedge reflector.

Fig. 7

Flowchart for the fit procedure. For the sake of comparison, the entire fit operation has been performed three times by using different glycerol ratios (0%, 50% and 100%) for the PZT-parameter fit (step 2).

The results and the resulting errors (which are, of course, approximately zero for the particular mixture where the PZT parameters were fitted) are shown in Table 1 and Fig. 8. It can be seen that the maximum relative error compared to the reference values [12,13] stays below 10% within the whole measurement range so this parameter fit procedure can be seen as a kind of calibration procedure.

Table 1

Reference values, fit values for Z ≈ ρc, and the resulting relative errors.

Glycerol ratio	ρflcfl (ref. values)	ρflcfl (fit 0%)	ρflcfl (fit 50%)	ρflcfl (fit 100%)	Rel. error 0%	Rel. error 50%	Rel. error 100%
[%]	[kg/(m2 s)]	[kg/(m2 s)]	[kg/(m2 s)]	[kg/(m2 s)]	[%]	[%]	[%]
0	1.4946E+06	1.4964E+06	1.6267E+06	1.5615E+06	–	8.8	4.5
20	1.6928E+06	1.6201E+06	1.7493E+06	1.6812E+06	−4.3	3.3	−0.7
40	1.9035E+06	1.7748E+06	1.9063E+06	1.8403E+06	−6.8	0.1	−3.3
50	2.0042E+06	1.8741E+06	2.0081E+06	1.9452E+06	−6.5	–	−2.9
60	2.0979E+06	1.9621E+06	2.0953E+06	2.0303E+06	−6.5	−0.1	−3.2
80	2.2565E+06	2.1578E+06	2.2905E+06	2.2224E+06	−4.4	1.5	−1.5
100	2.3721E+06	2.3108E+06	2.4451E+06	2.3782E+06	−2.6	3.1	–

Fig. 8

Reference values and fitted values for Z ≈ ρc.

Conclusion and outlook

An approach for using acoustic pressure waves to sense fluid parameters was presented. Based on a standard 3-port model for a PZT disc and the transmission theory for acoustic pressure waves a simple 1D-model for the sensor setup was devised. This 1D-model enables qualitative predictions of the sensor behavior. A prototype device to obtain first experimental data to validate the simulation results was introduced. The experimental data obtained with the prototype device show a clear dependence of the electrical impedance on fluid parameters (e.g., sound of speed – frequency spacing, viscosity – damping) and validates the predicted sensor behavior from the simulation data. In the present contribution, the influence of the liquid's characteristic acoustic impedance was considered in particular. It was found that in the investigated range of acoustic impedances a simple calibration procedure (providing estimates for the material parameters of the involved materials) yields measurement errors below 10%. Still, the experimental data show also that the modeling needs some further refinements, e.g., accounting for acoustic losses in the setup, additional damping due to the PZT mounting, and consideration of higher order radial modes of the PZT disk. Further research will cover a more detailed model for the sensor setup and improvements of the prototypes.

	ρfl [kg m−3]	cfl [m s−1]	μ [Ns m−2]	λ [Ns m−2]
Distilled water	998.0	1499.3	1.0049 × 10−3	0
70% Glycerol	1184.1	1847.2	23.0936 × 10−3	0

	ρfl [kg m−3]	cfl [m s−1]	μ [Ns m−2]	λ [Ns m−2]
Air	991.161	1343	18.600 × 10−6	0