An acoustic transmission sensor for the longitudinal viscosity of fluids.

Physical fluid parameters like viscosity, mass density and sound velocity can be determined utilizing ultrasonic sensors. We introduce the concept of a recently devised transmission based sensor utilizing pressure waves to determine the longitudinal viscosity, bulk viscosity, and second coefficient of viscosity of a sample fluid in a test chamber. A model is presented which allows determining these parameters from measurement values by means of a fit. The setup is particularly suited for liquids featuring higher viscosities for which measurement data are scarcely available to date. The setup can also be used to estimate the sound velocity in a simple manner from the phase of the transfer function.

Introduction

As the knowledge of (physical) liquid properties plays an important role in modern process control, there is an increasing demand for robust and reasonably priced sensors, especially for sensors sensing physical liquid properties, e.g., mass density, sound velocity and viscosity. In particular, sensors with online measurement capabilities are preferred to standard laboratory equipment, which is often bulky, service intensive and mostly less or not suited for online measurements. As discussed, e.g., in [1] and [2], recently a lot of effort has been spent on the investigation of sensors for (shear-) viscosity usually in combination with other parameters like the mass density or speed of sound. Acoustic viscosity sensor principles utilizing shear waves suffer from the drawback that due to the small penetration depth (in the range of a few microns depending on the excitation frequency) of the excited shear waves into the liquid only a thin fluid layer is being sensed [1,2]. Therefore these approaches are prone to surface contamination and less suited for sensing complex liquids such as emulsions or suspensions featuring particle sizes in the range or greater than the penetration depth of the utilized acoustic shear wave [3].

In this contribution, we report on a recently introduced [4] sensor setup using a piezoelectric PZT (lead zirconate titanate) transmitter and receiver for exciting and detecting pressure waves (rather than shear waves) in a small test chamber containing the liquid to be investigated. Utilizing pressure waves senses the so called longitudinal viscosity coefficient rather than the shear viscosity coefficient, which can also be used for condition monitoring applications. Furthermore the bulk of the probe is being sensed which therefore overcomes the drawbacks associated with the small penetration depths which is characteristic for shear wave approaches.

Theory and modeling

The basic concept of our sensor setup is based on the well-known theory of the viscous attenuation of pressure waves [5]. Fig. 1 depicts the elementary setup.

Fig. 1

Elementary sensor setup.

The setup consists of two planar, rigid boundaries separated by a distance h. These two boundaries form the sample chamber containing the liquid to be investigated. In one boundary, a PZT transmitting transducer (diameter d, thickness l) is flush mounted while in the opposite boundary another flush-mounted PZT transducer (diameter d, thickness l) serves as receiving device. The transducers are standard PZT disks, which are further specified below. When operated in the thickness-extensional mode (also called the “33-mode”), the transmitter excites pressure waves into the liquid, which are detected by the receiver upon impingement, thus a transfer function can be obtained.

Fig. 2 sketches the model of the whole sensor setup. The transducers represent a connection between the electric and acoustic domain. In the equivalent circuit, the normal forces (i.e. the pressure times the active transducer area and the velocities at the acoustic ports are represented by voltages and currents, respectively. The setup can be operated in a transient and a continuous mode; in this contribution, we consider the latter (for an example of a burst-based approach, see, e.g. [6]). In contrast to [6] our approach operates using a fixed distance between the transducers, which simplifies the setup. Furthermore our setup intentionally operates in the transducer's near field and utilizes a continuous wave excitation of the transmitting transducer. The transmitter is driven by an AC source (internal impedance Z) providing an open circuit voltage V exciting pressure waves which are detected by the receiver generating an electric output signal V at the electric load impedance Z. Particularly for the case of low attenuation of the waves in the fluid in the sample chamber, i.e. low viscosities, the reflections occurring at the transducers occur lead to characteristic interference patterns in the frequency response of the voltage transfer function V/V. Fig. 2 shows an equivalent circuit of the setup where the PZT transducers are represented by commonly used three-port systems featuring one electrical port and two acoustic ports representing the two faces of the transducer disks [7,8]. The acoustic backing of the PZT elements is represented by the acoustic impedances Zac1 for the transmitter and Zac1 for the receiver. In our case this backing is given by the surrounding air which, due to its comparatively low characteristic acoustic impedance, can be approximated as an acoustic short circuit.

Fig. 2

Model of the whole sensor system with AC signal source (left), transmitter, acoustic transmission line, receiver, and load impedance (right).

The acoustic transmission between the transducers can be modeled by an acoustic transmission line connecting the corresponding acoustic ports of the transducers. As discussed in [9-11], especially for low viscous fluids diffraction effects have to be taken into account. Diffraction effects lead to an additional attenuation and phase shift of the pressure waves [10,11]. According to [10] the diffraction loss factor D (assuming d = d) for emission from a circular source can be calculated as,with s defined as

The relation given in (1) holds if one of the following conditions is fulfilled [10]:these conditions have to be fulfilled in order for the approximation given in (1) to be valid. For our setup (d = d = 10 mm, λ in the order of 0.35 mm @ 4 MHz) the first of these conditions is fulfilled.

In (1) and (2) J0 and J1 are Bessel functions of the first kind, λ is the wavelength of the pressure waves in the fluid (not to be confused with the second coefficient of viscosity λ introduced below), and j is the imaginary unit. The complex-valued factor D provides information about the additional damping and phase shift due to diffraction. We now can define additional attenuation constant and phase constant Δα and Δβ, respectively, accounting for the effects of diffraction by setting

Note that this assignment holds only for a particular distance h, i.e. the so defined Δα and Δβ depend on h. The 1D-transmission line model does not account for diffraction effects but can be extended to include the additional attenuation and phase shift due to diffraction as discussed below.

The electric transfer function between the electric ports can be obtained by using the ABCD-matrix (chain matrix) approach known from the theory of electrical networks relating the input voltage V and current I to the output voltage V (=V) and current I (=−I) [7]. Fig. 3 (top) shows an elementary two-port element. The ABCD-matrix is defined as

Fig. 3

Elementary two-port (top) and chain of two-ports (bottom).

For a chain of two-ports as shown in Fig. 3, the overall ABCD-matrix is determined by multiplying the ABCD-matrices of the individual two-ports, i.e.

From the components of the voltage transfer function G of the entire system can be calculated as [7];

The ABCD-matrix for the PZT elements relating the electric port and the acoustic port connected to the fluid can be obtained from KLM-equivalent circuit [7,8]. The other acoustic port in the KLM-model (see also Fig. 2) is connected to constant impedances Zac1 and Zac1 representing the air backing.

The ABCD-matrix of the transmission line representing the fluid is derived in analogy to an electric transmission line with a length h, a characteristic impedance Z and a propagation constant γ (associated with a plane pressure wave) as [12]

Here the approximation holds for the common case that . As it can be seen in Eq. (7), in this 1D-model the transfer function of the system is affected by the following fluid parameters: sound velocity c, mass density ρ, and the longitudinal viscosity term (2μ + λ), where μ denotes the shear viscosity and λ denotes the second coefficient of viscosity. We note that there are different conventions as to how to define the second coefficient of viscosity, we adopt that also used by White [12,13], see also below. α0 and β0 represent the attenuation constant and the phase constant. In the equivalent 1D-transmission line model, the propagation constant is related to the transmission line parameters by (the distributed resistance of the transmission line R′ is assumed as R′ = 0 [14]),

In (8) the equivalent distributed inductance, conductance and capacitance are represented by L′, G′ and C′ respectively. The approximation in (8) holds for the case of small losses, i.e. G′/(ωC′) ≪ 1. For the sake of completeness, we mention that in the approximations given in (7) and (8) the following relationships have been utilized (first order taylor series approximation),

These approximations are justified for x ≪ 1 which corresponds to the assumptions given above which are particularly fulfilled for the liquids investigated. By equating the approximations derived in (7) and (8) the approximate equivalent transmission line parameters can be identified as

The diffraction effects can now be formally included in the model by modifying the propagation constant to obtain a new constant γ and accordingly, modify the values for the distributed conductance and capacitance, modeling the additional attenuation constant Δα and phase constant Δβ. The modified parameters for the fluidic transmission line (subscript “D”) considering diffraction can be obtained as (parameters with subscript “0” refer to the case neglecting diffraction)

So the ABCD matrix for the fluid-line in the diffraction free case is given as

For the case of including diffraction, is obtained by replacing Z and γ by Z and γ in (11).

Simulation and experimental results

To obtain experimental data, the prototype device shown in Fig. 4 has been built. The sample chamber consists of 1.5 mm FR4 copper coated PCB material. In each chamber wall, a PI-ceramic PIC255 PZT disk (d = d = 10 mm, l = l = 0.5 mm, fres ≈ 4 MHz), connected to an SMA connector providing the electric connection is flush mounted. Since due to the mounting one side of each PZT transducer is facing air, it is approximated by an acoustic short circuit in the modeling as described above. The distance h according to Fig. 1 is 34 mm for the setup investigated in this work. This distance is below the Rayleigh distance N, given by [15] such that we are operating in the near field of the transducer.

Fig. 4

Prototype device with the PZT transducers embedded in the chamber walls. SMA connectors provide the electrical connection.

As AC-source, an Agilent 81150A function generator with an internal impedance of 50 Ω has been used. The measurement of V and V was performed with a LeCroy Waverunner 44Xi oscilloscope resulting in a load impedance Z of 1 MΩ in parallel with a capacitance of 16 pF.

In principle, the model can be fitted to measurement results yielding the unknown parameter (i.e. the longitudinal viscosity 2μ + λ). However, in order to so, the other model parameters have to be firmly established. Especially the PZT material data provided by the manufacturer are prone to large tolerances [16,17]. To obtain a correction for the PZT parameters tolerances a reference measurement for the electric impedance of the PZT elements in air has been performed. By fitting the measured values of the PZT's electrical impedance in air to the theoretical values obtained from a standard three-port PZT model [7], corrected parameters can be determined.

Fig. 5 shows the measured values for the magnitude and phase of the electric impedance of the PZT transmitter and receiver in air (no fluid in the sample chamber) together with the simulation values (KLM model [7,8]) using the PZT data provided by the manufacturer [18] as well as with the fitted PZT parameters. The influence of the tolerances of the PZT parameters can clearly be seen. Furthermore in the measurement results, spurious influences of higher order radial modes which are not covered by the 1D-model can be seen. We note that these higher order radial modes are not due to the mounting of PZT transducers in the walls of the sample chamber, but due to the geometry of the disk transducer. A higher diameter to thickness ratio would decrease the influence of these spurious radial modes. In order to obtain a setup being as small as possible, we nonetheless chose a disk featuring a relatively small diameter. In the literature [19-21], 2D-models covering this coupling between the thickness and the planar modes are discussed, however, due to their complexity and the number of additional parameters, these models are hardly suited for a parameter fit based on an impedance measurement as used in our approach. For all simulations given below, the so fitted PZT parameters have been used. We performed measurements using an S200 viscosity standard oil at three temperatures (20 °C, 25 °C and 40 °C) where the nominal shear viscosity μ of the oil was used in the simulation runs. Furthermore, the mass density and the sound velocity of the S200 standard were measured using an Anton-Paar DSA5000 density and sound velocity meter. Table 1 shows the S200 fluid parameters used for the simulation runs (as the value of λ is not known yet it was set to zero, so the longitudinal viscosity 2μ + λ for the simulation runs is de facto given by 2μ):

Fig. 5

Magnitude and phase of the PZT transducers electric impedance in air (simulated and measured values). The impedance characteristics for the receiving and transmitting transducer were virtually identical.

Table 1

Material parameters for test fluid (S200 viscosity standard oil) at different temperatures (the second coefficient of viscosity is not provided in the literature).

Temperature [°C]	ρfl [kgm−3]	cfl [ms−1]	μ [Ns m−2]
20	886.8	1497.96	602.2 × 10−3
25	883.8	1478.97	418.1 × 10−3
40	874.6	1425.09	160.5 × 10−3

Fig. 6 shows the transfer function G obtained from measurement and simulation. A mismatch is expected due to the assumption λ = 0 in the simulation. This parameter can now be varied to obtain a fit between these characteristics (Fig. 7). The resulting values for (2μ + λ) and the often used parameter η (bulk viscosity) are depicted in Fig. 8 (using the model with and without diffraction). As discussed in [12], η is another common parameter used to characterize the second coefficient of viscosity and is related to the parameters μ and λ by η = (2/3μ + λ). η describes the viscosity associated with volume changes due to isotropic (compressive) stresses [22]. The determined η virtually does not change with temperature while (2μ + λ) decreases with increasing temperature. Note that the sample shows a bulk viscosity that is considerably higher than that of other fluids which have thus far been investigated in literature using acoustic spectroscopy setups, e.g., [23]. The present more compact setup featuring smaller propagation paths and thus smaller attenuation is specifically targeting at the determination of the longitudinal viscosity of highly viscous liquids (taking spurious losses associated with near field distortion into account, see also below). In turn, the measurement of low viscous liquids with our setup will lead to increased errors as spurious attenuation and interference effects (which are not accurately covered by the model) will be more pronounced in comparison to the targeted viscous attenuation effect.

Fig. 6

Measurement and simulation results (with and without diffraction included) for G for different temperatures (20 °C, 25 °C and 40 °C). Note that for the simulation λ was set to zero.

Fig. 7

Measured and fitted (with a fit of 2μ + λ and diffraction effects included) values for G for different temperatures (20 °C, 25 °C and 40 °C).

Fig. 8

Nominal datasheet values for the shear viscosity μ and the 2μ + λ and η values obtained from the fit with and without diffraction effects included.

In contrast to the shear viscosity, which is much better investigated for most standard fluids, data for the longitudinal viscosity are rare, which is confirmed by recent papers on this subject (see, e.g., [6] where the longitudinal viscosity is termed acoustic viscosity). This makes it difficult to verify our results by comparing to reference data for liquids. In [24] the ratio η/μ for some liquids is given. This ratio is related to the ratio (2μ + λ)/μ by (2μ + λ)/μ  = 4/3 + η/μ. As an indicator, it is stated that η/μ is rarely greater than 20 or less than 0.1 with a common value around about one [24]. It is further stated that the bulk viscosity shows a similar temperature dependence as the shear viscosity, where in particular, the ratio η/μ and thus also (2μ + λ)/μ (which is equal to 4/3 + η/μ) only shows a slight temperature dependence. For our data, it can be observed that (i) the ratio (2μ + λ)/μ lies in the proper order of magnitude and (ii) the temperature dependence of (2μ + λ) is similar to that of the shear viscosity μ. However, the ratio (2μ + λ)/μ shows a significant temperature dependence, which is also documented by the fact that the values for η do not show a similar temperature dependence as the shear viscosity μ (see Fig. 8).

This behavior could be explained by considering that, apart from the diffraction, there are other loss mechanisms not covered by our model. Examples for such mechanism are additional viscoelastic damping in the transducer which is not covered by our fit of the transducer's impedance in air and signal losses due the non-uniform shape of the wavefront in the near field, which leads to spurious signal reduction when the wave amplitudes are averaged over the planar area of the receiving transducer. As these signal losses are not included in our model, they would lead to an overestimate in the longitudinal viscosity (2μ + λ), which, according to our model, is directly proportional to the damping coefficient of the acoustic wave in the liquid given by see also (7). Thus, the correct longitudinal viscosity would be that obtained from the fit using our model (see Fig. 8) reduced by a correction term which is related to the spurious additional damping yielding a corrected longitudinal viscosity (2μ + λ)corr. Assuming for instance that the subtractive term does not show a significant temperature dependence, one can find that the temperature dependence of the ratio (2μ + λ)corr/μ can be minimized conforming to the behavior described in [24]. For our experimental data, the corresponding subtractive term would be in the order of 190 mPas. Due to the involved assumptions (e.g., temperature-independent subtractive term), this simple consideration should not be over-interpreted and in the present form particularly certainly cannot serve as a basis for a numerical correction of the measurement data. But it provides an indication that we are still missing relevant loss mechanisms in our model as discussed above. Other works using similar approaches [6] also introduced correction terms accounting for various loss mechanisms, which were then determined in a calibration step.

Regardless of the errors introduced by the spurious additional damping, the longitudinal viscosity derived from the measurements shows the qualitatively correct behavior and correct order of magnitude such that it can serve as monitoring parameter in condition monitoring applications where changes in the observed fluid need to be sensed.

We finally note that the presented setup can also be used to estimate the sound velocity of the liquid in the sample chamber. Neglecting the phase shifts of the transducers, the wavenumber k of the pressure waves is related to the phase shift φ of the transfer function by (for a given distance of the transducers h as depicted in Fig. 1),

The group velocity v of the waves is defined as,

Thus we have,which, for negligible dispersion, corresponds to the sound velocity in the liquid. Fig. 9 depicts the values for the sound velocity obtained by a measurement with an Anton-Paar DS5000 density and sound velocity meter, compared to the values obtained from the simulation and measurement results. The values show the same tendency although there is a mismatch in the absolute values. The difference can be related to additional phase shifts from the PZT transducers not considered in the above formula. As can be seen in Figs. 8 and 9 both parameters, the longitudinal viscosity term (2μ + λ) and the sound velocity decrease with increasing temperature. This behavior is well known for a lot of liquids. For the sound velocity we confirmed this behavior by the measurements with the commercial Anton-Paar DSA5000 laboratory instrument. For the longitudinal viscosity this behavior is also well-known (see the discussion above).

Fig. 9

Sound velocity, measured with an Anton-Paar DSA5000 and estimated from simulated transfer function with and without diffraction effects, and estimated from the measured transfer function. Note that the simulation and measurement results have not been corrected for the additional phase shifts introduced by the PZT transducers which explains the deviations.

Conclusions

We presented an approach for a transmission sensor setup utilizing pressure waves. Based on the KLM-model for a PZT transducer, the acoustic wave transmission theory and the ABCD-matrix approach, a simple 1D-model for the sensor was introduced. By fitting the simulated transfer functions to measurement results, an estimate for the longitudinal viscosity, and in turn, the bulk viscosity of the sample liquid can be determined. The setup is particularly suited for liquids featuring bulk viscosities above ∼150 mPas for which measurement results have been scarcely reported so far. As an example we determined the of a S200 viscosity standard oil. Beside the suitability of evaluating the second coefficient of viscosity the setup can be used to estimate the sound velocity from the measured phase shift in a straightforward manner.