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Measurement of viscosity of highly viscous non-Newtonian ﬂuids by means of ultrasonic guided waves Rymantas Kazys, Liudas Mazeika, Reimondas Sliteris, Renaldas Raisutis ⇑ Ultrasound Institute of Kaunas University of Technology, Studentu St. 50, Kaunas LT-51368, Lithuania

a r t i c l e

i n f o

Article history: Received 7 June 2013 Received in revised form 24 December 2013 Accepted 8 January 2014 Available online 21 January 2014 Keywords: Ultrasonic measurements Ultrasonic guided waves Dynamic viscosity

a b s t r a c t In order to perform monitoring of the polymerisation process, it is necessary to measure viscosity. However, in the case of non-Newtonian highly viscous ﬂuids, viscosity starts to be dependent on the vibration or rotation frequency of the sensing element. Also, the sensing element must possess a sufﬁcient mechanical strength. Some of these problems may be solved applying ultrasonic measurement methods, however until now most of the known investigations were devoted to measurements of relatively low viscosities (up to a few Pa s) of Newtonian liquids. The objective of the presented work is to develop ultrasonic method for measurement of viscosity of high viscous substances during manufacturing process in extreme conditions. For this purpose the method based on application of guided Lamb waves possessing the predominant component of in-plane displacements (the S0 and the SH0 modes) and propagating in an aluminium planar waveguide immersed in a viscous liquid has been investigated. The simulations indicated that in the selected modes mainly inplane displacements are dominating, therefore the attenuation of those modes propagating in a planar waveguide immersed in a viscous liquid is mainly caused by viscosity of the liquid. The simulation results were conﬁrmed by experiments. All measurements were performed in the viscosity standard Cannon N2700000. Measurements with the S0 wave mode were performed at the frequency of 500 kHz. The SH0 wave mode was exited and used for measurements at the frequency of 580 kHz. It was demonstrated that by selecting the particular mode of guided waves (S0 or SH0), the operation frequency and dimensions of the aluminium waveguide it is possible to get the necessary viscosity measurement range and sensitivity. The experiments also revealed that the measured dynamic viscosity is strongly frequency dependent and as a characteristic feature of non-Newtonian liquids is much lower than indicated by the standards. Therefore, in order to get the absolute values of viscosity in this case an additional calibration procedure is required. Feasibility to measure variations of high dynamic viscosities in the range of (20–25,000) Pa s was theoretically and experimentally proved. The proposed solution differently from the known methods in principle is more mechanically robust and better ﬁtted for measurements in extreme conditions. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Modern chemical industry requires development of completely new sensing techniques, which are suitable for non-invasive express analysis of dynamical behaviour and quality control of such processes as polymerisation, melting, etc. One of the main parameters to be measured is viscosity. Usually, for viscosity measurements in laboratory conditions the capillary and rotational viscometers are used, however they are slow, do not have a sufﬁcient mechanical strength and are not suitable for in-line measurements. Other serious problem is

⇑ Corresponding author. Tel.: +370 37 351162; fax: +370 37 451489. E-mail addresses: [email protected] (R. Kazys), [email protected] (R. Raisutis). 0041-624X/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2014.01.007

that in many cases the liquids may be very viscous (dynamic viscosity up to 25,000 Pa s or more) and behave as non-Newtonian ﬂuids. In this case the viscosity starts to be dependent on the vibration or rotation frequency of the sensing element. Some of those problems may be solved applying ultrasonic measurement methods. Historically the ﬁrst attempt to measure viscosity of liquids using ultrasonic waves was based on measurement of the acoustic shear impedance of a viscous liquid [1]. For that reﬂection of the shear wave at the interface solid–viscous liquid was analysed. The viscosity was obtained from amplitude or phase measurements of the reﬂected signals [1–4]. It was noticed that at lower shear impedances and low frequencies (10–300 Hz) liquids may be considered as Newtonian liquids, however at ultrasonic frequencies they should be treated as non-Newtonian liquids [3].

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Another group of methods is based on exploitation of guided waves propagating in solid waveguides immersed into the liquid viscosity of which is measured. The attenuation and velocity of guided waves depends on the viscosity of the liquid, but usually attenuation is measured. The most popular measurement method is based on application of ultrasonic wave propagating in the cylindrical rods or thin wires, which are immersed into a viscous liquid. In this case only in-plane displacements on the surface of the waveguide exist, attenuation of which depends on viscosity of the liquid. Therefore, usually attenuation of the torsional wave in a waveguide is measured [5–8]. It was noted that exploitation of a torsional wave propagating in hollow cylinders may improve sensitivity of measurements signiﬁcantly [6]. The detailed analysis of immersed into a viscous liquid solid cylinder in which propagate guided lowest order torsional and longitudinal waves was presented in [8]. In the case of the longitudinal mode the total attenuation depends not only on the viscosity induced attenuation but also on a radiation into a liquid caused by off-plane displacements. Therefore, measurements of the shear viscosity using the fundamental longitudinal mode L(0, 1) must be performed at relatively low frequencies at which off-plane displacements are quite low [9,10]. Viscosity of a polymer melts at high temperatures was successfully measured by means of the guided shear and ﬂexural F(1, 1) mode waves [11,12]. In most cases measurements were performed in the Newtonian viscoelastic liquids with a not very high dynamic viscosity, e.g. not exceeding a few Pa s. Another drawback of those methods is a relatively low mechanical robustness which is not acceptable for the in-line measurement application in industry. A higher mechanical robustness possess the strip-shape waveguides with rectangular cross-sections which were also used for viscosity measurements [13–15]. For this purpose longitudinal, shear-horizontal (SH) and quasi-Scholte modes of a guided waves propagating in a planar waveguide were used. In the case of the SH mode sensitivity of the method is proportional to the ratio l/d, where l is the length of the waveguide immersed in a viscous liquid and d is the thickness of the waveguide. The shear-horizontal SH mode was excited and picked-up using the shear mode transducers attached to the end of the strip. Usually the width of the shear mode piezoelectric transducer cannot be made less than 3–4 mm, therefore the ratio l/d < (4–5) and sensitivity of the method is limited [15]. In order to enhance the sensitivity bent waveguides were used [15]. In the case of the quasi-Scholte mode it was demonstrated that the attenuation of this mode depends on viscosity of the loading liquid, however measurements were carried out only in a Newtonians liquids [14]. In practice, the measurement methods are validated using the reference media. However, a few authors reported an anomalous measurement results obtained using the viscosity standards such as the Cannon N2700000. The viscosity values obtained at ultrasonic frequencies were a few times lower than obtained by a conventional rotational viscometer used by the manufacturer for calibration [7,8]. This correlates with the results obtained by the shear impedance methods [2,3]. The reason for it is that these standards are the non-Newtonian viscoelastic liquids made of polymers [7]. Due to that the viscosity starts to be dependent on a strain rate of the liquid [15,16]. On the other hand some authors came to conclusion that the strain rate during ultrasonic measurements is quite low and reduction of the measured viscosity is due to the viscoelastic relaxation at ultrasonic frequencies [8]. The objective of this research was to develop and investigate ultrasonic method for measurement of viscosity of the highly viscous ﬂuids suitable for applications in extreme conditions, for example, during the manufacturing process of the polymer materials.

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2. Measurement method From the presented above analysis it is possible to conclude that the viscosity of the highly viscous liquids can be estimated by the measurement of attenuation of the guided waves propagating in the waveguide immersed into the liquid. However the waveguide and the selected guided wave mode should meet following requirements: 1. The guided wave mode and the operation frequency should be selected in such a way that only an in-plane displacement of the waveguide surface is dominant. 2. The surface area of the waveguide contacting with a viscous liquid should be large enough in order to get a required sensitivity and a measurement range, especially in the case of highly viscous liquids. 3. The material of the waveguide should be sufﬁciently strong in order to withstand in situ conditions, for example a high pressure, temperature, etc. Taking into account those requirements the method for viscosity measurement based on an application of the symmetrical or shear-horizontal (SH) guided wave modes propagating in an aluminium planar waveguide immersed into a viscous liquid has been proposed (Fig. 1). In order to get the necessary mechanical robustness the thickness of the waveguide was selected 0.9 mm. The properties of the Lamb wave modes propagating in a plate depend on elastic properties of the plate material, plate thickness and the properties of the liquid into which the plate is submerged. The properties of the guided waves are described by the dispersion curves. For calculation of the dispersion curves the commercially available software ‘‘DISPERSE’’, based on the global matrix model [17] was applied. The calculated phase and group velocity dispersion curves of the Lamb waves are presented in Fig. 2. At the frequencies lower than 1.5 MHz mainly the two primary fundamental modes A0 and S0 may propagate. The S0 mode possesses the essentially higher group velocity than the A0 mode, so the signals can be easily separated in the time domain. This fact starts to be very important in the case of interference of a several modes. When the plate is submerged into a viscous liquid, a part of propagating wave energy is transferred to this liquid. If only inplane displacements are dominating (for example in the case of the S0 mode), only shear stresses are coupled to the surrounding liquid. Therefore, in this case the attenuation of the guided wave is mainly caused by a shear leakage [3]. The normal (off-plane) and in-plane displacements of the S0 mode at 500 kHz (the region of not signiﬁcant dispersion) in the aluminium plate having thickness of 0.9 mm are shown in Fig. 3. It can be seen that the predominant component of the longitudinal S0 mode is caused mainly by an inplane displacement. Therefore, it is possible to assume that the attenuation of the S0 mode propagating along the submerged waveguide should be mainly affected by the viscosity of the surrounding liquid, so this mode may be used for the viscosity measurement.

Fig. 1. Propagation of guided waves along the aluminium plate submerged into the viscous liquid.

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mechanical movement such as it is used in a rotational viscometer will be different from the viscosities values measured using the high frequency ultrasonic waves. This is not a systematic error of the measurement, but it means that the viscosity is dependent on a frequency x. As a consequence the attenuation of the guided wave propagating in the waveguide will be dependent on a frequency not only due to the properties of the guided waves and material of the waveguide, but also due to the fact that the viscosity values are frequency dependent. So, in order to estimate the viscosity from the measured attenuation of the guided wave a relation between the attenuation and viscosity of the liquid should be known. In order to investigate performance of the proposed technique the Cannon N2700000 viscosity reference standard has been used. It was considered to be a non-Newtonian liquid. The relation between the attenuation of a guided wave propagating in a waveguide submersed in a viscous liquid and the viscosity of the liquid is known only in the case of Newtonian liquids. For example, at the low frequencies the asymptotic approximation (LFAA) relates the attenuation a(x) of the S0 mode to the viscosity g and the density q of the liquid and is given by [10]:

1 aðx0 Þ ¼ d

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x0 gðx0 Þ q ; 2 E qs

ð1Þ

where d is the thickness of the plate, x is the angular frequency, E is the Young modulus of the aluminium plate (E = 70.758 GPa), q is the density of the viscous ﬂuid (q = 911 kg/m3), qs is the density of the aluminium plate (qs = 2700 kg/m3), g(x0) is the frequency dependent dynamic viscosity at the frequency x0. From Eq. (1) follows that the dynamic viscosity g(x0) at the frequency x0 may be found from the measured attenuation of the S0 mode: 2

gðx0 Þ ¼

2 d aðx0 Þ2 E qs ; q x0

ð2Þ

However, it is necessary to determine in what viscosity ranges Eq. (2) is applicable. Fig. 2. Phase (a) and group (b) velocity dispersion curves of the Lamb wave modes propagating in the aluminium plate having thickness of 0.9 mm.

3. Application of the symmetric S0 mode The objective of the experimental investigations was to verify a feasibility of viscosity measurements of the highly viscous nonNewtonian liquids using the presented above approach. For this purpose measurements of the S0 mode attenuation in an aluminium waveguide versus viscosity were carried out. The experimental set-up used for the through-transmission measurements is presented in Fig. 4. In order to carry out the measurements for the cases of the different viscosities and at the same time to avoid an inﬂuence of the density variations, all measurements were performed using the same viscosity standard N2700000 only at the different temperatures. The dependence of the dynamic viscosity of the Cannon N2700000 type standard liquid on temperature g0(T) is known and is presented in Fig. 5. This dependency can be expressed by:

g0 ðTÞ ¼ a eðbðTþcÞÞþdÞ ; Fig. 3. Normalized displacements of the S0 mode propagating in the aluminium plate having thickness of 0.9 mm (at 500 kHz): 1 – normal (off-plane) displacements and 2 – in-plane displacements (along the axis of the waveguide).

However, differently from measurement of other physical quantities such as for example density, the viscosity depends on the applied measurement technique and its parameters like a shear rate or frequency used in the measurement procedure. It means that the viscosity values obtained using a low frequency shear

ð3Þ

where T is the temperature in degrees of Celsius, a = 73049, b = 0.0875, c = 5.6084 and d = 0.0155 are the approximation coefﬁcients. For generation of the S0 mode waves the rectangular ultrasonic transducers operating in a transverse extension mode at the frequency 500 kHz were bonded to the planar surface of the Al waveguide (Fig. 4. The transducers were oriented in such a way that the in-plane displacements would be directed along the waveguide,

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The fastest S0 mode signals were selected from the recorded waveforms of the received signals using the Hanning window and the frequency spectra were calculated using the fast Fourier transform (FFT). The attenuation of the signals caused by the viscosity was obtained by comparing the frequency spectra of the signals measured in the cases of the aluminium plate loaded by the liquid viscosity standard and the free aluminium plate contacting with surrounding air only:

af ¼500kHz ¼

Fig. 4. The experimental set-up used for the through-transmission type measurements of the dynamic viscosity of the standard N2700000 versus temperature.

e.g., in the direction of propagation of the S0 mode. Dimensions of the waveguide were 0.9 mm 15 mm 370 mm. The width of the ultrasonic transducer was selected to be equal to the width of the waveguide, e.g. 15 mm, therefore the ultrasonic beam occupied the whole width of the waveguide. That has been conﬁrmed by measurements using the contact point-type probe. The aluminium plate was submerged into the viscosity reference standard N2700000. The length of the aluminium plate l contacting with the viscosity standard inside the measurement vessel was 40 mm. The necessary temperature of the measurement vessel with the viscous liquid and the sensor inside was obtained by means of a thermostat. Generation and reception of ultrasonic signals were performed using the Ultralab ultrasonic measurement system, which was been developed at the Ultrasound Institute of Kaunas University of Technology. The transmitter was excited using 3 periods burst with the frequency 500 kHz. When the guided S0 mode waves enters and exits the immersed part of the waveguide some part of the wave is reﬂected due to the change of the acoustic impedance, but how it was shown in [18,19] those reﬂections are small enough and may be neglected.

ð4Þ

where ua(t) is the ultrasonic signal picked-up by the receiver when the waveguide is contacting with air, u1(t) is the ultrasonic signal picked-up by the receiver when the waveguide is loaded by the viscosity standard N2700000 at different temperatures, l is the length of the waveguide immersed into a viscous liquid. In this way the signal losses at the regions where the waveguide contacts with the pipe walls are eliminated. The measurements were carried out in the temperature range 30–60 °C. At these temperatures the zero frequency viscosity, which is usually measured by rotational viscometer, varies in the range of 200–3500 Pa s. Please note that the length l in this case is equal to the internal diameter of the measurement vessel. The experimentally obtained attenuation values of the S0 mode versus the low frequency viscosity are presented in Fig. 6. The dynamic viscosity values given here were measured by a rotational viscometer. The experimental results with a very good accuracy may be approximated using equation

a¼

Fig. 5. Dynamic viscosity of the standard N2700000 versus temperature: solid line – approximation of the a priori measured values using the rotational viscometer, dots – reference data provided by the Cannon Instrument Company (at T = 25 °C and T = 60 °C).

20 FFT½u1 ðtÞ lg ; l FFT½ua ðtÞ

K 1 K 5 g0 K 5 g0 þ K 2

K 3

þ K4;

ð5Þ

where K1 = 2.5282, K2 = 1.4876, K3 = 0.9540, K4 = 0.5479 and K5 = 0.8689 103 are the appropriate coefﬁcients. If to use the obtained attenuations values and the low frequency approximation according to Eq. (2) it is possible to get a relation between the attenuation of the guided wave and the expected viscosity values gk at the measurement frequency 500 kHz. The obtained results are presented in Fig. 7. As it can be seen, the obtained viscosity values are essentially different comparing to the viscosities measured using the rotational viscometer. We would like to point out that by several authors already has been noticed that the shear viscosity of the Cannon standard N2700000 obtained at the ultrasonic frequencies is essentially lower than the measured by a conventional rotational viscometer

Fig. 6. Attenuation of the S0 mode at 500 kHz in the Al waveguide (dimensions 0.9 mm 15 mm 370 mm) versus the dynamic viscosity of the standard N2700000.

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In order to obtain a relation between the measured attenuation of the guided wave and the shear dynamic viscosity at ultrasonic frequencies the Maxwell model may be applied. This model enables to relate high frequency measurement results to the values declared by the manufacturer. As it was shown in [2,7] the Maxwell model gives a rather good relation between the viscosity g(x) of the standard Cannon N2700000 at an arbitrary angular frequency x and the viscosity g0 at the zero frequency:

gðxÞ ¼

Fig. 7. Relation between the dynamic viscosity and attenuation of the S0 mode at the frequency 500 kHz.

[7,8]. There may be two reasons for that. One reason is that the shear viscosity of the non-Newtonian liquids depends on a strain rate and as a rule is reducing with increasing strain rate [16,18]. Another reason may be that the reduction of the shear viscosity is due to the viscoelastic relaxation at ultrasonic frequencies [8]. Let us estimate the expected range of the strain rate in our case, e.g., when displacements in the standard N2700000 are created by the guided wave in the metallic waveguide. The S0 mode propagating in a planar waveguide produces inplane displacements which are of opposite direction each halfwavelength k/2, where k is the wavelength of the S0 mode (Fig. 8). Let us assume that this maximal displacement between two peaks of the opposite polarity is 2Dx. On the boundary waveguide-liquid those displacements drag the boundary layer of the viscous liquid to the opposite directions along the same distance Dx. The strain in this case is given by

e ¼ Dx=ðk=4Þ:

ð6Þ

This strain is produced during half-period of the oscillations. Consequently, the strain rate is given by

c ¼ e f;

ð7Þ

where f is the central frequency of the propagating S0 mode. In our case f = 500 kHz, the displacement Dx obtained from ﬁnite element modelling Dx = 0.2 109 m, k = 6 mm and the strain rate c = 0.26 s1, e.g. is rather low. It means that in our case the actual viscosity at ultrasonic frequencies is lower not due to the strain rate, but very likely due to a viscoelastic relaxation [8].

g0 1 þ x2 s 2

;

ð8Þ

where s is the relaxation time of the liquid. It is necessary to state that in general the dynamic viscosity g0 and the relaxation time s depend on temperature. The relaxation time s at constant temperature may be found from Eq. (8) if at least two viscosity values at different frequencies are known; one value may be taken at zero frequency. In this case we obtain

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ g0 gðx0 Þ s¼ ; gðx0 Þ x20

ð9Þ

where x = 2pf0 is the central frequency of the S0 mode. In order to estimate the relaxation time the date presented in [7] were used. In this case the viscosity value g(x0) = 391 Pa s at the frequency f0 = 320 kHz was obtained. At the zero frequency the dynamic viscosity of the standard Cannon N2700000 is g0 = 5593 Pa s [7]. Substituting those values into Eq. (9) we obtain the relaxation time s = 1.814 ls. Then Eq. (8) enables to predict the dynamic viscosity g(x0) of the standard Cannon N2700000 at the frequency f0 = 500 kHz used in our investigation. The zero frequency viscosities of the N2700000 at temperatures used in the experiment have been varying in the range of 200–3500 Pa s. Application of the Maxvell relation (Eq. (8)) enables to estimate the dynamic viscosities at the zero frequency and corresponding viscosities at the frequency 500 kHz. This dependency is presented in Fig. 9 by a solid line. In general it demonstrates the linear dependency between the zero frequency viscosities and the viscosities measured at the frequency 500 kHz. However if to take the viscosities gk estimated from the measured attenuation using the low frequency approximation, then a non-linear dependency is obtained (Fig. 9, dots). One of the explanations of this difference may be that the relaxation time s of the N2700000 standard liquid depends on a temperature T. In our case the different viscosity values were obtained changing the set temperature of the standard liquid. The temperature dependence of the relaxation time s(T) may be obtained from the viscosity values gk estimated from the measured attenuation a of the guided wave. Let us assume that the dependence of the relaxation time s versus

Fig. 8. Estimation of a strain rate caused by a propagating guided wave. The arrows indicate displacements of the waveguide and a viscous liquid during one half-period of the propagating guided wave. Please note that the arrows are not to scale.

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relaxation time into Eq. (8) the non-linear theoretical Maxwell dependency of the viscosities measured at the frequency of 500 kHz versus zero frequency viscosities is obtained (Fig. 9) which corresponds very well to the experimental results. From the results presented follows that if to assume that the relaxation time depends on temperature then exploiting the Maxwell model of the liquid it is possible to relate the measured attenuation of the guided wave to the frequency-dependent viscosity of the liquid. 4. Application of the SH0 mode

Fig. 9. The dynamic viscosity at the frequency f = 500 kHz versus the low frequency viscosity g0 at different temperatures: (1) calculated according to the Maxwell model with the constant relaxation time; (2) dots – measurement results; and (3) calculated according to the Maxwell model with an assumption that the relaxation time depends on temperature.

Fig. 10. Relaxation time s versus viscosity g0 at different temperatures: 1 – constant relaxation time and 2 – according to the Maxwell model with an assumption that the relaxation time depends on temperature.

temperature dependent zero frequency viscosity g0(T) may be approximated by a similar function as given by Eq. (5):

sðg0 Þ ¼

K s1 K s5 g0 K s5 g 0 þ K s2

K s3 þ K s4 ;

ð10Þ

In the case of the pure fundamental SH0 mode propagating in a planar waveguide there should be no off-plane displacements, e.g., all displacements should take place in the plane of the waveguide only, therefore from this point of view, this mode is similar to the torsional mode in a cylindrical rods and therefore is attractive for a shear viscosity measurements. The experimental set-up used for the measurements is shown in Fig. 11. The Al waveguide was the cylindrical rod the middle part of which was machined to the ﬂat plate. The length of the ﬂat part was 38 mm and the thickness 1.15 mm. The SH0 mode in the Al waveguide was excited and picked-up by the 580 kHz SH mode piezoelectric transducers attached to the tips of the waveguide. Only the ﬂat part of the waveguide was immersed into the Cannon N2700000 viscosity standard. The calculated phase and group velocity dispersion curves in the ﬂat part of the waveguide are presented in Fig. 12. The calculations were performed using the computational package ‘‘DISPERSE’’ [17]. At the frequencies lower than 1.1 MHz mainly only one fundamental SH0 mode may propagate. Therefore, this mode with the frequency of 580 kHz was selected for measurements. The measurement and calculation results versus the low frequency viscosity g0 of the viscosity standard N2700000 obtained at different temperatures (at the frequency f = 580 kHz) are presented in Fig. 13. If to compare the measurement results obtained for the S0 and the SH0 modes it can be seen that the estimated variations of the relaxation times slightly differ for those two cases (Fig. 14). The relaxation time is a physical property of the material, therefore it should not depend on a guided wave mode and the frequency used in measurements. It is likely that this difference may be caused by the fact that the equations used in the low frequency approximation (Eqs. (1) and (2)) were derived for the case of 2D approach, which assumes the uniform distribution of the wave parameters including the amplitude of displacements across the wave

where Ks1 = 0.4960 103, Ks2 = 2.316, Ks3 = 1.758, Ks4 = 0.1442 105 and Ks5 = 0.0018 are the approximation coefﬁcients. The coefﬁcients Ks were found by minimizing the mean-squared error:

F¼

min

K s1 ;K s2 ;K s3 ;K s4 ;K s5

5 X fg½g0 ; sðg0 ;K s1 ; K s2 ; K s3 ; K s4 ;K s5 Þ gk g2 ;

ð11Þ

k¼1

where the dynamic viscosity g according to the Maxwell model is given by:

g½g0 ; sðg0 ; K s1 ; K s2 ; K s3 ; K s4 ; K s5 Þ ¼

g0 1 þ x2 sðg0 ; K s1 ; K s2 ; K s3 ; K s4 ; K s5 Þ2

:

ð12Þ The obtained dependence of the relaxation time s(g0) versus viscosity g0(T) at different temperatures is presented in Fig. 10. We should like to point out that the absolute values of this relaxation time vary around the constant value obtained by other authors (Fig. 10, the straight line) [7]. By substituting this

Fig. 11. Experimental set-up for the viscosity measurements using the SH0 mode.

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by means of additional correction coefﬁcients, which evaluate a non-uniform distribution of the tangential component of the displacement or the particle velocity:

acorr ¼ a K Dx ;

ð13Þ

where acorr is the corrected attenuation, a is the measured attenuation, KDx is the correction coefﬁcient given by

K Dx ¼

Fig. 12. Phase velocities of the SH modes propagating in an aluminium plate (thickness 1.15 mm) immersed into the viscosity standard N2700000.

propagation direction. In the case of the planar waveguide with ﬁnite dimensions the distribution of the wave parameters such as the amplitude of tangential component of the particle velocity across the plate is not uniform and is different for the S0 and the SH0 modes. Therefore, attenuation of a guided waves propagating in a waveguide with a ﬁnite lateral dimensions will be different comparing to theoretically predicted using the low frequency approximation (Eq. (2)). This difference may be taken into account

1 d

Z

þd=2

d=2

uns ðyÞ dy;

ð14Þ

where y is the axis oriented across the waveguide, d is the width of the waveguide, uns ðyÞ is the spatial distribution of the particle velocity across the waveguide normalized with respect to the maximal value of it. The spatial distributions uns ðyÞ were determined using the ﬁnite element modelling results presented in [20]. From those results follows that in the case of the S0 mode the coefﬁcient KDx = 0.67 and in the case of the SH0 mode – KDx = 0.52. In the case of a uniform distribution of particle velocity, what happens when the S0 or the SH0 modes propagate in a plate with inﬁnite lateral dimensions the coefﬁcient is KDx = 1, what means that no correction is needed. The corrected attenuation acorr has been used for determination of the dynamic viscosity at the measurement frequency (Eq. (2)). The obtained new values of the dynamic viscosity gcorr versus low frequency viscosities g0 are presented in Fig. 15 by dots. The viscosity obtained using the S0 mode is slightly higher and it can

Fig. 13. The measurement and calculation results versus the low frequency viscosity g0 of the viscosity standard N2700000 obtained at different temperatures (at the frequency f = 580 kHz): a – attenuation of the SH0 mode in the Al waveguide; b – dynamic viscosity at the frequency f = 580 kHz: (1) calculated according to the Maxwell model with the constant relaxation time; (2) dots – measurement results; and (3) – calculated according to the Maxwell model with an assumption that the relaxation time depends on temperature; c – relaxation time at different temperatures: (1) constant relaxation time and (2) according to the Maxwell model with an assumption that the relaxation time depends on temperature.

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1111

Fig. 16. The relaxation time s of the standard N2700000 versus the low frequency viscosity g0 estimated using the corrected attenuation acorr of the S0 and the SH0 modes: 1 – calculated according to the Maxwell model with the constant relaxation time; 2 – calculated according to the Maxwell model with an assumption that the relaxation time depends on temperature in the case of the SH0 mode; 3 – calculated according to the Maxwell model with an assumption that the relaxation time depends on temperature in the case of the S0 mode.

Fig. 14. Comparison of the viscosity measurement results obtained using the S0 and the SH0 modes: (a) dots – viscosities measured using the S0 mode (approximation denoted by the dashed line (1)); dots – viscosities measured using the SH0 mode (approximation denoted by the solid line (2)); (b) relaxation time in the case of the S0 (2) and the SH0 (3) mode (the case of constant relaxation time is presented by the solid line (1)).

Fig. 15. Dynamic viscosities gcorr of the standard N2700000 versus the low frequency viscosity g0 estimated using the corrected attenuation acorr of the S0 and the SH0 modes: 1 – S0 mode (dots – experiment, the dashed line 1 – approximation); 2 – SH0 mode (dots – experiment, the solid line 2 – approximation).

The obtained results correspond very well to the Maxwell theory with assumption that the relaxation time is temperature dependent. The relaxation times got for the S0 and the SH0 modes versus low frequency viscosities at different temperatures calculated using Eqs. (10)–(12) are presented in Fig. 16. As can be seen the relaxation times of the standard N2700000 in this case completely coincide for the cases of both waveguides. In general it can be stated that in this case the results obtained using a different waveguides and a different guided waves modes coincide almost completely. The estimated variations of the relaxation time obtained by both techniques are very close also. So, it means that the developed techniques enable to estimate the dynamic viscosity in a wide viscosity range. On the other hand the proposed waveguides are mechanically robust and may be used for in-line measurements of viscous liquids under in situ conditions. 5. Conclusions Feasibility to measure variations of the high dynamic viscosities in the range of (20–25,000) Pa s was theoretically and experimentally proved. The proposed solution in comparison to the known methods in principle is more mechanically robust and better ﬁtted for measurements in extreme conditions. By selecting a particular mode of guided waves (S0, SH0), the operation frequency and dimensions of the aluminium planar waveguide it is possible to get the necessary viscosity measurement range and sensitivity. The experiments also revealed that the measured dynamic viscosity of the viscosity standard N2700000 is strongly frequency dependent and as a rule is much lower than indicated by the manufacturer. That is characteristic feature of the non-Newtonian liquids. Therefore, in order to get the absolute viscosity values in other non-Newtonian liquids an additional calibration procedure in each case is required. The proposed viscosity estimation techniques based on application of a planar metallic waveguide of a rectangular cross-section is mechanically robust and may be used for in-line measurements of viscous liquids during manufacturing process. Acknowledgements

be explained by the fact that the measurements were carried out at the slightly different frequencies: 500 kHz for the S0 mode and 580 kHz for the SH0 mode.

The part of this work was sponsored by the European Union under the Framework-7 project Polysense ‘‘Development of a low

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