Genevieve Palardy

Huajie Shi

Arthur Lévy

Steven Le Corre

Irene Fernandez Villegas

Arthur Levy

Steven Lecorre

A study on amplitude transmission in ultrasonic welding of thermoplastic composites

Keywords: A/ Thermoplastic resin, B/ Vibration, C/ Process Modelling, E/ Joints/Joining

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Introduction

Composite materials, in particular carbon fibre reinforced polymers (CFRP), have been increasingly used for applications in aircraft structures due to their great potential for weight reduction [START_REF] Kaw | Mechanics of Composite Materials[END_REF]. As opposed to thermoset composites, and as a result of the linear and entangled molecular structure of thermoplastics, thermoplastic composites (TPCs) can be re-melted and re-shaped easily. It is possible to manufacture and join TPC components using low-cost manufacturing technologies, such as press forming or welding. Since only local heating of the structure is required, welding processes are usually faster, more energy efficient and less expensive, which makes thermoplastic composites attractive for use in the aerospace industry. Ultrasonic welding (USW) is a very efficient technique for TPCs with fast welding times (< 5 s), great potential for mass production and does not need introduction of foreign material at the weldline [START_REF] Ageorges | Advances in fusion bonding techniques for joining thermoplastic matrix composites: a review[END_REF][START_REF] Yousefpour | Fusion Bonding/Welding of Thermoplastic Composites[END_REF]. The principle of USW is shown in Figure 1. An energy director (ED) made of the same thermoplastic as the adherends' matrix is placed between the lower and upper adherends. In this work, instead of the classical triangular shapes, flat energy directors, which consist of thin films, are used [START_REF] Villegas | A comparative evaluation between flat and traditional energy directors for ultrasonic welding of CF/PPS thermoplastic composites[END_REF]. A sonotrode applies pressure on the parts to be welded, as well as high frequency (20-3 40 kHz) and low amplitude vibrations (2.5-250 μm peak-to-peak amplitude) [START_REF] Benatar | Ultrasonic welding of PEEK graphite APC-2 composites[END_REF]. Heat generation is localized at the interface due to the lower compressive stiffness of the energy director. The welding energy is generated from a combination of surface friction and viscoelastic heating [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF]. A consolidation phase follows the vibration phase to create a high quality bond.

There is considerable research literature showing experimental investigations of the USW process for TPCs at the coupon level [START_REF] Villegas | A comparative evaluation between flat and traditional energy directors for ultrasonic welding of CF/PPS thermoplastic composites[END_REF][START_REF] Benatar | Ultrasonic welding of PEEK graphite APC-2 composites[END_REF][START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF][START_REF] Villegas | In situ monitoring of ultrasonic welding of thermoplastic composites through power and displacement data[END_REF][START_REF] Villegas | Ultrasonic welding of advanced thermoplastic composites: An investigation on energy-directing surfaces[END_REF][START_REF] Villegas | Optimizing ultrasonic welding of carbon-fibre reinforced polyetherimide[END_REF][START_REF] Palardy | On the effect of flat energy directors thickness on heat generation during ultrasonic welding of thermoplastic composites[END_REF][START_REF] Villegas | Ultrasonic welding of CF/PPS composites with integrated triangular energy directors: melting, flow and weld strength development[END_REF][START_REF] Harras | Optimization of the ultrasonic welding of PEEKcarbon composites[END_REF][START_REF] Liu | Factors affecting the joint strength of ultrasonically welded polypropylene composites[END_REF][START_REF] Liu | Optimizing the Weld Strength of Ultrasonically Welded Nylon Composites[END_REF]. These studies involve understanding the effect of various parameters such as welding force, vibration amplitude, solidification force and time and shape of the energy director on the welding process and on the quality of the welded joints. While experimental research provides solid insight into the inner workings of ultrasonic welding of thermoplastic composites, the complex heating mechanisms behind the process remain insufficiently understood and hence not accurately modelled. As USW is moving toward up-scaling and industrialization, the development of a reliable finite element model has become essential in increasing the robustness and reliability of sequential and even continuous welding of larger and more complex parts [START_REF] Lu | Sequential Ultrasonic Welding of PEEK/Graphite Composites Plates[END_REF][START_REF] Palardy | Up-Scaling of the Ultrasonic Welding Process for Joining Thermoplastic Composites[END_REF][START_REF] Senders | Zero-flow: a novel approach to continuous ultrasonic welding of CF/PPS thermoplastic composite plates[END_REF].

To date finite element analyses of the USW process for TPCs are mostly limited to prediction of the temperature field at the weld interface and in the adherends based on the shape of the energy director [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF][START_REF] Levy | A level set based approach for the finite element simulation of a forming process involving multiphysics coupling: Ultrasonic welding of thermoplastic composites[END_REF][START_REF] Levy | Ultrasonic welding of thermoplastic composites: modeling of the process using time homogenization[END_REF][START_REF] Wang | FEM Investigation of the Temperature Field of Energy Director During Ultrasonic Welding of PEEK Composites[END_REF][START_REF] Yan | The Effects of Energy Director Shape on Temperature Field During Ultrasonic Welding of Thermoplastic Composites[END_REF]. However, obtaining accurate measurements of temperature distribution at the welding interface and within the adherends is difficult and hence experimental validation of such models remains a challenge. More recently, Levy et al. developed a numerical multiphysical model to predict power dissipation during the initial transient heating phase of the USW process, which was also applied to flat energy directors [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF][START_REF] Levy | Ultrasonic welding of thermoplastic composites: a numerical analysis at the mesoscopic scale relating processing parameters, flow of polymer and quality of adhesion[END_REF]. This approach is particularly attractive since the power dissipated during the USW process is affected by the physical changes occurring at the welding interface [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF] and is provided as a power-time curve by microprocessor-controlled ultrasonic welders. Consequently, dissipated power throughout the welding process constitutes a meaningful and relatively straightforward data source for experimental validation of USW numerical models. Coupling the physics of adhesion, heat transfer and solid mechanics showed promising results, but several empirical parameters had to be estimated to obtain good agreement with the experimental data [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF]. One of those parameters was the hammering coefficient, . Hammering is the loss of contact between sonotrode and upper adherend during the vibration phase of the USW process [START_REF] Nonhof | Estimates for process conditions during the ultrasonic welding of thermoplastics[END_REF]. As a result, it leads to reduced friction and viscoelastic dissipation and is therefore critical for heat generation at the weld interface.

As previously mentioned, heat is generated through frictional and viscoelastic heating during USW. It is thought that frictional dissipation ( ) at the interface between the energy director and the adherends mostly occurs before the glass transition temperature is reached. It can be calculated from the average dissipated mechanical work associated with friction during an ultrasonic cycle, given by Eq. ( 1) [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF][START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF]:

(

) 1
where e is called the hammering efficiency, i.e. the ratio between actual heat generation (with hammering) and ideal heat generation (without hammering), is the pulsation of vibration, is the vertical stress on the horizontal interface and is the horizontal displacement discontinuity across the interface. Viscoelastic heating, which is maximized around the glass transition temperature of the thermoplastic polymer, can be quantified, based on the work by Tolunay et al. ( [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF]), with Eq. ( 2):

(

where is the loss modulus of the material and is the amplitude strain tensor. In previous work [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF], the hammering efficiency, , i.e. the ratio between the actual heat generation (with hammering) and the ideal heat generation (without hammering), was incorrectly written as , where was defined as the hammering coefficient. The hammering coefficient accounted for the amplitude transmitted between the sonotrode and the upper adherend according to the following equation, where is the amplitude transmitted to the substrate and is the amplitude of the sonotrode:

(

It should be noted that for the sake of clarity is renamed in the present paper as amplitude transfer ratio. In appendix A of the present paper, it is shown that in fact, . Despite its importance on frictional and viscoelastic heat generation during the USW welding process, neither nor have ever been experimentally measured.

While the approach proposed by Levy et al. [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF] is an important first step toward successful implementation of the physics behind USW of TPCs, it can benefit from empirical evaluation and quantification of the hammering effect. To further improve the proposed model and achieve closer agreement with the experimental data, it is necessary to properly determine the actual amplitude transmitted to the parts being welded, as it can greatly influence heat generation and hence power dissipated during the process. Therefore, in this study, we proposed and assessed a strategy to measure and estimate amplitude transmission to the upper adherend during ultrasonic welding of thermoplastic coupons focusing on the vibration phase of the USW process.

The experimental measurements were used as input into a COMSOL Multiphysics model to simulate dissipated power during the vibration phase of USW for a specific welding force and estimated transmitted amplitude. The predicted dissipated power was then compared to the experimental power curve provided by the ultrasonic welder.

Methods

In this paper, a combination of experimental and finite element analysis approaches were employed. The experiments consisted of composite adherends manufacturing and welding, as well as laser sensor measurements of amplitude transmission on the surface of the upper adherend. A numerical multiphysical model, based on the framework developed by Levy et al. [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF], was used to predict the power curves obtained from the ultrasonic welding process by taking into consideration the amplitude transmission measurements.

Experimental

Laminates were manufactured from six layers of carbon fibre/polyetherimide (CF/PEI) 5 harness satin T300 prepreg, provided by TenCate, The Netherlands. They were stacked with a [0/90]3S sequence and consolidated in a hot-platen press at 320°C and 2 MPa for 30 min. After consolidation, the laminates were water-jet cut into 101.6 mm by 25.4 mm adherends for the welding process. A rectangular neat PEI film, with a thickness of 0.25 mm, was used as the energy director. Before welding, the composite adherends and energy directors were dried in an oven at 135°C for 6 hours and 1 hour, respectively.

A 20 kHz Rinco Dynamic microprocessor controlled ultrasonic welder with a maximum power output of 3000 W was used to weld specimens in a single lap shear configuration with an overlap area of 12.7 mm x 25.4 mm. A rectangular sonotrode with a surface area of 14.9 mm x 30.1 mm and a gain (i.e. amplification of the ultrasonic wave in the sonotrode) of 2.75 was used. A custom-made welding jig was used to clamp the lower and upper adherends during welding (Figure 2). It prevented in-plane movement but allowed the upper adherend to remain perfectly parallel to the bottom one during the whole process. A welding force of 1500 N and a vibration amplitude amounting to 36.3 μm (half peak-to-peak value), were used for measurements during the welding process. The duration of the vibration phase of the USW process was indirectly controlled by the displacement of the sonotrode, called "travel". The optimum travel was selected as 60% of the energy director thickness (i.e. 0.15 mm for a 0.25 mm-thick energy director), following the empirical procedure detailed in [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF]. After welding, the joints underwent a solidification phase with a consolidation force of 1000 N applied for 4 seconds. It should be noted that the parameters of the welding process as well as the thickness of the adherends, corresponding to typical values for ultrasonic welding of thermoplastic composites,

were not modified in this study, since its main focus was the amplitude transmission during welding. Detailed results on the effect of the welding parameters and the thickness of the adherends on the welding process and its outcome can be found in [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF]. It should also be noted that even though the nominal vibration frequency of the ultrasonic welder used in this study was 20 kHz, slight variations (within approximately a 1% range) in the actual frequency (provided by the welder after the welding process) are typically observed from weld to weld. The exact impact of those frequency variations in the welding process is unknown although it is assumed to be small owing to the consistent welding output.

As previously described in Section 1, when hammering occurs during the application of ultrasonic vibration, there is a loss of contact between the sonotrode and the upper adherend. To determine the actual amplitude transmitted from the sonotrode to the weld assembly, the displacement of the upper adherend was measured with a laser displacement sensor. A CMOS sensor by KEYENCE®, LK-H052 sensor (range ±10mm, accuracy 0.025µm) with LK-G5000P controller, was used in this study. The laser sensor was set to work at a sampling rate of 200 kHz for a maximum recording time of 6 seconds. More specifically, 10 points could be measured in each vibration cycle since the welder worked at a frequency of 20 kHz. The accuracy of the sensor was assessed by measuring the displacement at the base the sonotrode for a certain nominal vibration amplitude. For these tests, referred to hereafter as sensor calibration tests, the sonotrode was vibrated freely in the air and the laser sensor was placed perpendicular to the base of the sonotrode.

For maximum accuracy in the measurement of the vibration transmitted to the upper adherend during the ultrasonic welding process, displacement measurements on the adherend should be taken directly underneath the sonotrode, but since this was physically not possible, displacement measurements were taken at different distances d from the sonotrode to evaluate the impact of d on the sensor measurements and find the most appropriate d value (Figure 3). As depicted in the welding schematic and displacement measuring setup used in this work and shown in Figure 3, the laser beam had to be placed at an angle  from the overlap in order to allow small d values. Since in this setup the vertical vibrational movement of the composite adherend was not parallel to the direction of the laser beam, the measured displacement Dmeas was converted to the real displacement D by using Eq. ( 4):

By comparing the measurement results for a constant movement from a vertically installed sensor and a tilted sensor at , it was found that the error introduced when was less than 1.6% and increased when . Therefore, in order to measure the displacement as close as possible to the overlap, a 45° was selected as the sensor tilt angle.

Modelling

In this paper, a 2D COMSOL Multiphysics 5.2® heat transfer model, previously developed in [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF], was improved and subsequently used to predict the welding power dissipation, i.e. power curves, during the USW process. Figure 4 shows that only a quarter of the actual geometry was considered in the model (one adherend and half the thickness of the energy director) in order to reduce computing time. It should be noted that such symmetric system is a simplification and does not accurately reflect the geometry of the welding set-up and specimen. However it is based on the assumption that the welding overlap (which does have two axes of symmetry) is the main source of power dissipation during the welding process. The assumption also implies that the friction behavior of the two adherend/energy director interfaces is the same, thus disregarding any potential instability of the friction behavior. The symmetry assumption could hence be a source of errors in the numerical prediction and should be addressed in future work. The present study focused on the heating of both the energy director and the adherend, but did not account for the flow that occurs at the interface upon melting of the energy director and adherend. The model takes into consideration viscoelastic dissipation ((2), as well as friction, intimate contact evolution and adhesion evolution at the interface between the energy director and the adherend.

In order to improve on the previous modelling framework presented in [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF], three main changes were applied: 1) the amplitude transmission from the sonotrode to the upper adherend was experimentally determined (in this paper) and was used instead of the previously used hammering coefficient, 2) viscoelastic heating from both the energy director and the adherends was taken into account, and 3) the simulation was stopped when the minimum temperature in the ED exceeded Tg + 100 o C (315 o C) in order to capture the effect of viscoelastic heating over a representative duration of the vibration phase of the welding process (~0.7s, [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF]). The model parameters and material properties are detailed in [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF].

Results and discussion

Sensor calibration measurements

As mentioned before, calibration of the laser displacement sensor was performed by applying a known vibration amplitude hereafter referred to as nominal amplitude, taking into account the accuracy of the laser sensor (0.025 m), the latter was assumed to be a suitable system to measure transmitted amplitude during welding.

Displacement measurements for upper adherend

As explained in Section 2.1, the amplitude transmission between the sonotrode and the upper adherend was estimated through measurement of the vertical displacement of the latter during the USW process. As shown in Figure 7 to its initial position at the onset of the vibration phase) for each case (dashed, red line). Generally, there was good agreement in the trend shown by both curves consisting of an initial plateau (corresponding to the heating of the energy director [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF]) followed by a steady downward displacement of the sonotrode (corresponding to the squeeze flow of the energy director under the welding pressure [START_REF] Villegas | Strength development versus process data in ultrasonic welding of thermoplastic composites with flat energy directors and its application to the definition of optimum processing parameters[END_REF]). It should be noted that whereas the displacement measured by the laser sensor was able to capture the high-frequency oscillation, the displacement measured by the welder did not capture this oscillation. The average amplitude over the entire duration of the vibration phase for different d values was calculated and compared to the nominal amplitude of the welding machine (45.4 m) as well as to numerical calculations, as

shown in Figure 9. The numerical calculations were obtained from simplified dynamic numerical simulations, performed with COMSOL Multiphysics 5.2®, for the boundary value problem described in Figure 4. In these simulations, the mechanical properties of the adherends and energy director were taken from [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF] (values at room temperature). To simulate the hammering effect in a simplified manner, an intermittent sinusoidal force at =20 kHz, F (t), was applied on the upper adherend just under the sonotrode, as described by Eq. ( 5):

(

This analytical form only considers the negative part of the applied force, i.e. force applied on the adherend, which combines a constant contribution (holding force F0 amounting to 1500 N) and a fast oscillating contribution, F1 sin(t). The parameter F1, equal to 66 kN, was estimated from the normal stress necessary to impose the nominal amplitude of the vibration onto the welding stack, i.e. adherends and energy director [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF]. It should be noted that owing to the relatively small thickness of the welding stack the oscillating contribution to the force was significantly higher than the static contribution as discussed in [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF].

Several observations can be made from the data plotted in Figure 9. First and foremost, the distance from the sonotrode had a strong effect on the amplitude transmission measurements. The experimental and the numerical amplitude values showed a similar trend characterized by a non-uniform distribution of the vibration amplitude along the upper adherend. This indicates that, for the specific set of boundary conditions considered in this study, the superposition of the different vibration modes excited by the sonotrode in the upper adherend played a major role on the effect of the distance from the sonotrode on the measured amplitude.

Deformation of the springs beneath the sliding platform onto which the upper adherend was clamped (not considered in the numerical calculations) was also believed to contribute to the observed effect. Secondly, the numerical results indicated that the average amplitude at d=2 mm could be considered relatively close to the average amplitude directly underneath the sonotrode. Consequently, d=2 mm was selected as the most appropriate location for the transmitted amplitude measurements.

The average amplitude measured at d=2 mm was 42.0 ± 8 m which was also relatively close to the nominal amplitude of the sonotrode, 45.4 m. The scatter in these measurements was high, with a coefficient of variation amounting to almost 20%, but still lower than the overall scatter in measurements obtained at d>2 mm.

Since, as shown in Figure 9, the amplitude measured at d=2 mm could be expected to be close to but not the same as right at the welding overlap, simulation could eventually enable to define a quantitative correction factor to estimate the transmitted amplitude at the welding overlap, i.e. right underneath the sonotrode, from the measurements taken at d= 2 mm. This would however require fine tuning of all simulation parameters and is beyond the scope of this paper. Moreover the FFT of the displacement measurements at d=2 mm (see Figure 10) showed, in addition to a main peak at 20 kHz, secondary peaks attributed to the different vibration models in the upper adherend, at approximately 40, 60 and 80 kHz. These secondary frequencies were, however, not considered when using the displacement measurements at d=2 mm to estimate the transmitted amplitude at the welding overlap.

Strategy to determine amplitude transmission

As briefly explained in Section 1, the coefficient of hammering, , can be expressed as a function of the duration of the actual contact between the sonotrode and the upper adherend. However, experimentally determining this duration of contact from laser sensor data alone can be particularly complex. Therefore, the alternative and more straightforward approach followed in this paper entailed considering the hammering coefficient as the ratio between the actual amplitude transmitted to the upper adherend, i.e. the compressive displacement induced in the adherend by the sonotrode, and the nominal amplitude in the sonotrode.

To simplify the laser sensor data (shown in Figure 8) and facilitate its input into the 2D model described in Section 2.2, a strategy to estimate the amplitude transmitted to the upper adherend was developed. This corresponds to intervals of about 80 ultrasonic periods and ensures that the true peak is captured. To circumvent the high variability of the data points and to apply it to the numerical model, the data was fitted using a linear function (Figure 12).

Considering that the ultrasonic welder used in this study, establishes full vibration in the sonotrode in approximately 0.05 s, the piecewise affine function shown in Figure 12 was defined for the amplitude transmission. According to this function, the amplitude transmission increased sharply from 0 to 19.7 m within the first 0.05 s, and then increased gradually until it reached 33.0 m at the end of the vibration phase of the welding process.

It is interesting to note that the amplitude transmission seemed to improve over time, meaning that hammering decreased. It is believed that after the initial establishment of the vibrations, the decrease in stiffness, caused by heating and melting of the energy director at the interface, improved the compliance between the sonotrode and upper adherend, hence promoting a better transmission of amplitude.

Implementation into numerical multiphysical model

In the 2D COMSOL model presented in Section 2.2, the hammering coefficient was implemented as the ratio between the bilinear function shown in Figure 12 and the nominal amplitude of the sonotrode, i.e. 36.3 μm, over the duration of the vibration phase of the welding process (0.7s). The reader should note that the hammering efficiency e in equations ( 1) and ( 2) was taken as equal to as justified in appendix A. Figure 13 shows representative experimental and simulation results

for the case shown in Section 3.3: 1500 N welding force and 36.3 m nominal amplitude. The "Exp power" curve was obtained from the welding machine. The simulation results included four different power curves: frictional heating between the energy director and adherends ("Friction heating"), viscoelastic heating from the energy director ("ED Viscoelastic heating"), viscoelastic heating from the adherends ("Ad Viscoelastic heating") and the sum of all heating sources resulting in the "Total heating" curve.

When comparing total simulated power to experimental power, it generally followed a similar behaviour: two main power peaks (circled areas for both simulated and experimental curves) with a maximum power value of between 1300 and 1600 W.

As shown experimentally, the first peak coincides with the moment at which melting first occurs within the energy director [START_REF] Villegas | In situ monitoring of ultrasonic welding of thermoplastic composites through power and displacement data[END_REF]. The second power peak was associated with melting occurring within the composite adherends [START_REF] Villegas | In situ monitoring of ultrasonic welding of thermoplastic composites through power and displacement data[END_REF]. Similarly, the first power peak in the simulated power curve was, as seen in Figure 13, mostly attributed to the viscoelastic heating in the energy director, with secondary contributions from viscoelastic heating of the adherends and frictional heating. Likewise, the second power peak in the simulated curve was clearly a result of the viscoelastic heating of the adherends. The delay observed between the occurrence of simulated and experimental peaks may have been caused by assuming in the model that energy dissipation was purely concentrated at the interface. In reality, part of the welding energy is dissipated in the welding setup, tooling and anvil, as well as in the adherends outside the overlap, which means it would take more time for the weld to be created.

It should be noted that the shape of the first peak in the simulated power curve is likely influenced by the loss modulus, , in (2, which is dominant around Tg (~215 o C) [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF]. A closer examination of the predicted temperature profile across the thickness of the energy director and upper adherend at 0.12 s (Figure 14 Improvement over the previous version of the model presented in [START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF] can be highlighted in two ways. Firstly, the total simulated power values were at a similar level as the experimental ones without the use of a global efficiency factor and both the simulated and experimental curves showed two main power peaks. This was achieved by a combination of better empirical determination of the amplitude transmitted to the upper adherend (no hammering coefficient required) and consideration of the contribution of viscoelastic heating from the adherends.

Secondly, the temperature evolution was generally captured accurately as both power peaks coincided with the moment when Tg was first reached in the ED and secondly, in the adherend.

Nevertheless, the simulated results still show discrepancies with the experimental results, one of the most significant ones being the inability of the model to accurately predict the time at which the main power peaks occurred causing the simulated and experimental power curves to not overlap. Discrepancies between simulated and experimental results are believed to be a consequence of assumptions and limitations of the model. For instance, the model was limited by the fact that flow, which in the experiment shown in Figure 13 was observed to commence at around 0.5 s, was not taken into consideration. Flow should be considered in future iterations of this model in order to fully capture heat generation and heat transfer in the ultrasonic welding process. Likewise, the low values of friction heating compared to viscoelastic heating, even at the onset of the welding process seem to contradict experimental findings [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF],

which suggests that a revision of the approach used in the friction heating model might be needed.

Conclusions

In this paper, an experimental determination of the amplitude transmitted to the upper adherend during ultrasonic welding of thermoplastic composites was carried out. The experimental results were used to improve upon a previously developed COMSOL Multiphysics model by providing more accurate input parameters regarding the hammering effect. Based on both experimental and modelling findings, the following observations can be made:

 When using the laser sensor on the surface of the upper adherend, measurement data varied significantly when the distance between the sonotrode and the sensor increased. In order to obtain measurements similar to what the upper adherend experienced underneath the sonotrode, it was recommended to place the sensor as close as practically possible to the sonotrode (i.e. 2 mm, in our case).

 The amplitude transmitted to the upper adherend was calculated and averaged at selected points over the duration of the process. It was described by a bilinear function, with positive slopes, indicating less hammering toward the end of the process cycle. This behaviour suggested that after onset of the vibrations, the decrease in stiffness of the energy director, induced by heating and melting at the interface, led to an improved compliance between the sonotrode and upper adherend, hence promoting better transmission of amplitude.

 The experimental results were implemented into a numerical multiphysical model as the imposed sonotrode displacement for prediction of dissipated power based on frictional and viscoelastic heating. Improved agreement between simulated power and experiments was achieved. Taking into consideration viscoelastic heating from both the energy director and the adherends revealed the presence of two power peaks corresponding to the main stages of the ultrasonic welding process in thermoplastic composites. To further improve this model, flow of the energy director at the interface should be taken into consideration, so that physical behaviour of the polymer past Tg can be represented more realistically. Moreover the friction heating models should be revised.

which is different from what was given in previous works that assumed . Selfheating efficiency can therefore be read directly from amplitude measurements as proposed in this work. where the black dashed line represents the estimated mean displacement curve.

Figure Captions

Figure 12: Evolution of the estimated amplitude transmitted to the upper adherend (1500 N welding force and 36.3 m nominal amplitude). Values were calculated at 0.004 s time intervals over the duration of the process, an affine function was fitted through the data points and a linear increase of the amplitude was proposed during the period needed for full amplitude to be established in the sonotrode.

Figure 13: Comparison between experimental (Exp power) and simulated (Total heating) power curves for a representative welding case (1500 N welding force and 36.3 m nominal amplitude) using average amplitude from Figure 12. . Viscoelastic heating from adherends: Ad Viscoelastic heating, viscoelastic heating from energy director: ED Viscoelastic heating, frictional heating: Friction heating. The circled areas represent two characteristic power peaks for both simulated and experimental curves.

Figure 14: Predicted temperature profile across the thickness of energy director and upper adherend over a vertical distance of 1 mm at three locations: overlap centreline (--), overlap quarter (-) and overlap edge (-). (a) t = 0.12 s (left circled peak in Figure 13), and (b) t = 0.50 s (right circled peak in Figure 13).

Figure A1: Displacement of the sonotrode and upper adherend versus time. The sonotrode displacement, , is a sine (with a period T and an amplitude ). However, due to the hammering effect, the displacement of the upper adherend,

, is truncated with a loss of contact time tc.

Figure A2: Viscoelastic efficiency e vs amplitude transfer ratio . The viscoelastic efficiency quantifies how much the viscoelastic heating, crucial to the welding process, is reduced because of the hammering effect. The amplitude transfer ratio is the ratio between the amplitude of the vibration transmitted to the upper substrate and the sonotrode amplitude . The contact time ratio is the ratio of contact duration over the period. The process efficiency (in terms of energy dissipation) appears as roughly proportional to the amplitude transmitted.

A study on amplitude transmission in ultrasonic welding

of thermoplastic composites, Figures , is a sine (with a period T and an amplitude ). However, due to the hammering effect, the displacement of the upper adherend, , is truncated with a loss of contact time tc.

44

Figure A2: Viscoelastic efficiency e vs amplitude transfer ratio . The viscoelastic efficiency quantifies how much the viscoelastic heating, crucial to the welding process, is reduced because of the hammering effect. The amplitude transfer ratio is the ratio between the amplitude of the vibration transmitted to the upper substrate and the sonotrode amplitude . The contact time ratio is the ratio of contact duration over the period. The process efficiency (in terms of energy dissipation) appears as roughly proportional to the amplitude transmitted.

  to the sonotrode and comparing it to the vertical displacement measured on the bottom surface of the sonotrode by the laser sensor. In these calibration tests, the sonotrode was vibrating freely in the air and the laser sensor was placed perpendicular to the bottom surface of the sonotrode. It should be noted that no additional information about the actual vibration amplitude was available in this study, hence the measurements of the laser sensor could only be compared to the nominal amplitude entered in the ultrasonic welder. Sensor displacement measurements were taken at five different positions (three measurements per position), as illustrated in Figure5. Sonotrode displacement measurements at position 1 for a nominal amplitude of 45.4 m and a duration of 0.5 ms are shown in Figure6(a) as a representative example of the results of the calibrations tests. The measured average amplitude over all positions (1 to 5) was 44.9 ± 2.3 m, thus displaying close agreement (within 1.2%) with the nominal value, 45.4 m. Furthermore, Figure6(b),corresponding to measurements at a different position, demonstrates that the measured oscillation frequency obtained through Fast Fourier Transform (FFT) was similar to the nominal value given by the supplier of the ultrasonic welder, 20 kHz.Note that the differences in amplitude between Figure6(a) and Figure 6(b) correspond to the variability observed in measurements taken at different positions in the bottom surface of the sonotrode (44.9±2.3 µm). These results proved the suitability of the chosen laser sensor to provide accurate measurements of the ultrasonic vibrations during the USW process despite the relatively low number of points (10 points) captured during each vibration cycle. It should be noted that owing to the fact that the calibration tests were conducted independently from the research on amplitude transmission during welding, the nominal amplitude value used for the calibration tests (45.4 m) was not the same as the nominal amplitude used in the welding tests (36.3 m). Nevertheless based on the results of the calibration tests and

  , such measurements were carried out at different distances d from the sonotrode to identify the position leading to the highest accuracy. The d values were 2 mm, 5 mm, 15 mm and 25 mm. Measurements were repeated three times for each distance d. Representative displacement curves are presented in Figure 8 for (a) 2 mm, (b) 5 mm, (c) 15 mm, and (d) 25 mm. The blue curves, corresponding to the data obtained from the laser sensor measurements, were compared to the displacement curves provided by the ultrasonic welder (i.e. vertical displacement of the sonotrode relative

Figure 11 (

 11 a) shows representative laser sensor data (d=2mm) for the vibration phase in a welding process with a force of 1500 N and a nominal amplitude of 36.3 μm. First, as indicated by the dashed line in the close-up in Figure 11 (b), a curve was fitted through the data points to establish the average sonotrode displacement. Then, the transmitted amplitude was calculated by taking half its peak-to-peak displacement value as shown by the arrow in Figure 11 (b). Due to the large amount of measured data points, the amplitude values were computed for time intervals of 0.004 s, as shown in Figure 12.

  (a)), i.e. the time at which a distinct and very sharp power peak occurs, revealed that at that moment, the temperature in the ED was between 200 o C and 250 o C, which indicated the predominance of viscoelastic heating over frictional heating. Regarding the second peak in the simulated power curve, occurring at around 0.50 s, the predicted temperature profile clearly showed that temperature values around Tg were expected within the upper adherend, contributing to viscoelastic heat generation within the adherend itself (Figure14 (b)).

Figure 1 :Figure 2 :

 12 Figure 1: Principle of ultrasonic welding of thermoplastic composites using a flat energy director

Figure 3 :

 3 Figure 3: Determination of amplitude transmission via displacement measurement of the upper adherend during ultrasonic welding using a tilted sensor. Measured displacement: Dmeas, real displacement: D, sensor tilt angle: , and distance between sonotrode and laser: d. Dimensions are not to scale.

Figure 4 :

 4 Figure 4: Boundary conditions of 2D model to predict the power dissipated during the ultrasonic welding process. A quarter model was used based on the symmetry (Sym.) of the system. Dimensions are not to scale.

Figure 5 :

 5 Figure 5: Displacement sensor positions on the bottom surface of the sonotrode for calibration measurements.

Figure 6 :

 6 Figure 6: (a) Representative example of the measured sonotrode displacement using a laser sensor at position 1 (as shown in Figure 5) with a nominal amplitude of 45.4 μm. (b) Corresponding Fast Fourier Transformation (FFT).

Figure 7 :

 7 Figure 7: Distances d from the sonotrode at which displacement was measured by the laser sensor, as in Figure 3.

Figure 8 :

 8 Figure 8: (a) Displacement curve measured by the sensor on the upper adherend at d = 2 mm, the dashed, red line represents the displacement of the sonotrode provided by the USW machine. Similar curves are shown for other d values: (b) 5 mm, (c) 15 mm and (d) 25 mm. In all cases, the welding force and nominal amplitude were 1500 N and 45.4 m, respectively.

Figure 9 :

 9 Figure 9: Average amplitude of the displacement curves over the entire duration of the vibration phase for all d values shown in Figure 8. Experimental data points are compared with numerical dynamic simulations for a nominal amplitude of the sonotrode of 45.4 m.

Figure 10 :

 10 Figure 10: Fast Fourier Transform (FFT) of representative example of sensor measurements for displacement of the upper adherend at a distance d = 2 mm from the sonotrode (as shown in Figure 3).

Figure 11 :

 11 Figure 11: (a) Representative displacement curve measured using the laser sensor at d = 2 mm on the upper adherend during welding (1500 N welding force and 36.3 µm nominal amplitude), and (b) Close-up of the rectangular area in (a)

Figure 7 :

 7 Figure 7: Principle of ultrasonic welding of thermoplastic composites using a flat energy director.

Figure 8 : 32 Figure 9 :

 8329 Figure 8: Ultrasonic welding setup used in this work. 1: Sonotrode, 2: clamp for lower adherend, 3: clamp for upper adherend, 4: moving platform, and 5: springs supporting the moving platform.

Figure 10 :Figure 11 :Figure 12 :

 101112 Figure 10: Boundary conditions of 2D model to predict the power dissipated during the ultrasonic welding process. A quarter model was used based on the symmetry (Sym.) of the system. Dimensions are not to scale.

Figure 7 :

 7 Figure 7: Distances d from the sonotrode at which displacement was measured by the laser sensor, as shown in Figure 3.

Figure 8 :

 8 Figure 8: (a) Displacement curve measured by the sensor on the upper adherend at d = 2 mm, the dashed, red line represents the displacement of the sonotrode provided by the USW machine. Similar curves are shown for other d values: (b) 5 mm, (c) 15 mm and (d) 25 mm. In all cases, the welding force and nominal amplitude were 1500 N and 45.4 µm, respectively.

Figure 9 :

 9 Figure 9: Average amplitude of the displacement curves over the entire duration of the vibration phase for all d values shown in Figure 8. Experimental data points are compared with numerical dynamic simulations for a nominal sonotrode amplitude of 45.4 m.

Figure 10 : 3 Figure 11 :Figure 12 :

 1031112 Figure 10: Fast Fourier Transform (FFT) of representative example of sensor measurements for displacement of the upper adherend at a distance d = 2 mm from the sonotrode (as shown in Figure 3).

Figure 13 :

 13 Figure 13: Comparison between experimental (Exp power) and simulated (Total heating) power curves for a representative welding case (1500 N welding force and 36.3 m nominal amplitude) using average amplitude from Figure 12. Viscoelastic heating from adherends: Ad Viscoelastic heating, viscoelastic heating from energy director: ED Viscoelastic heating, frictional heating: Friction heating. The circled areas represent two characteristic power peaks for both simulated and experimental curves.

Figure A1 :

 A1 Figure A1: Displacement of the sonotrode and upper adherend versus time. The sonotrode displacement,, is a sine (with a period T and an amplitude ). However, due to the hammering

Genevieve Palardy 3 , Huajie Shi 1 , Arthur Levy 2 , Steven LeCorre 2 and Irene Fernandez Villegas 1

Appendix A

In previous works on modelling and simulation of the process by the same group [START_REF] Levy | A level set based approach for the finite element simulation of a forming process involving multiphysics coupling: Ultrasonic welding of thermoplastic composites[END_REF][START_REF] Levy | Ultrasonic welding of thermoplastic composites: modeling of the process using time homogenization[END_REF][START_REF] Levy | Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors[END_REF][START_REF] Levy | Ultrasonic welding of thermoplastic composites: a numerical analysis at the mesoscopic scale relating processing parameters, flow of polymer and quality of adhesion[END_REF], the dissipated power (dissipated by viscoelasticity or by friction) used as a source term in the heat transfer problem, was obtained from the elastic problem.

The load was considered sinusoidal, and the stresses and strains integrated over an ultrasonic period. In order to account for the hammering effect, an approximate correction coefficient was used in the expression of the source terms. In this appendix, this correction factor is theoretically derived from the amplitude effectively transmitted to the upper adherend.

The sonotrode tip has a displacement that is sinusoidal as shown in (A1):

where is the amplitude, the pulsation, and the frequency usually in the order of 20 kHz. Because of the hammering effect and the loss of contact between the sonotrode and the composite, the imposed displacement on the top surface of the top adherend is not equal to . Rather, it is a truncated sine as illustrated in Figure A1. The contact time ratio can be defined as:

where is the loss of contact time during an ultrasonic period . ranges between 0 and 1 and reaches 1 for a perfect contact with no hammering.

The amplitude of the imposed displacement fulfils:

where is the amplitude transfer ratio which is the ratio between the amplitude transmitted to the substrate and the sonotrode amplitude .

For the sake of clarity, let us consider the average viscoelastic dissipation per period. When there is no contact, the dissipated power is zero. Thus:

where is the Cauchy stress tensor and the strain rate tensor. Following the methodology developed in previous works [START_REF] Benatar | Ultrasonic welding of PEEK graphite APC-2 composites[END_REF], [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF], under linear viscoelasticity assumptions, this becomes:

where is the amplitude of the strain tensor obtained when solving the elastic problem with the sonotrode displacement . Integrating the squared sinus and noting that leads to:

(A6) which corresponds, when using the contact time ratio , to:

. (A7)

In this final equation, denotes the reference viscoelastic dissipation obtained theoretically if no hammering occurs (i.e. the dissipation obtained in previous works). The viscoelastic efficiency e of the process is thus:

which can be expressed as a function of the amplitude transfer ratio using Eq. (A2).

Figure A2 shows the dependency of the efficiency versus the amplitude transfer ratio . In first approximation, one may choose: (-) and overlap edge (-). (a) t = 0.12 s (left circled peak in Figure 13), and (b) t = 0.50 s (right circled peak in Figure 13).


Simon Andres
email: andres@ikdg.rwth-aachen.de

Joachim Funcke

Manfred Wirsum

Thomas Polklas

S Y Mpo

O Tatin

Suitability of light sca ering technique for measurements of water droplets in turbine wet steam flows

Keywords: Mie theory, light sca ering, wet steam, droplet measurements

Suitability of light scattering technique for measurements of water droplets in turbine wet steam flows

INTRODUCTION

e measurement of droplets in condensing flow in low pressure steam turbines has been a challenging task ever since. Optical techniques are capable of determining droplet sizes as well as overall wetness. However, by classification into primary and secondary droplets (fog and coarse droplets respectively) only certain of these technologies satisfy the required measuring needs. e distinction between primary and secondary droplets is made by means of their formation. Primary droplets are condensing spontaneously during expansion at the Wilson point. Secondary droplets break off from liquid films forming due to primary droplets we ing downstream blades.

ese coarse secondary droplets are responsible for blade erosion so that prevention of their generation or at least knowledge of their behaviour is desired (see for example [ ]). In his extensive work on wet steam flows Gyarmathy [ ] states that in every turbine stage approximately -% of primary droplets strike flow leading structures which potentially causes secondary droplets that have been breaking off blade trailing edges. He estimates that -% of wetness is provided by secondary droplets at the turbine exit. Contrary Williams and Lord [ ] find the secondary droplet fraction to be maximally % of total wetness where Cai et al. [ ] measure values somewhere in between. However, for determination of the wetness fraction and the explicit thermodynamic state of the two-phase flow respectively, both primary and secondary droplets have to be measured. e importance for measurements of primary droplets is thereby increased by moving upstream from the turbine exit whereas secondary droplets mostly appear and have to be measured towards the turbine outlet.

Kleitz and Dorey [ ] as well as Moore and Sieverding [ ] give a good overview on instrumentation of wet steam in general and also cover measurement of the immanent droplets. It is emphasized that optical measurement technologies have been proven superior to other approaches. Secondary droplets have been under investigation by means of several measurement techniques. e most recent developments have been made by Bosdas et al. [ ] who use an optical backsca er probe with a sha diameter of as small as 5 mm, Cai et al. [ ] who designed an integrated probe for combined measurement of primary and secondary droplets using extinction and forward sca ering that has a diameter of 20 mm and Bartos et al. [ ], who also use a photogrammetric approach. Other optical probes have been introduced in earlier years. Kercel et al. [ ] developed a probe that utilizes two interfering beams to form the measuring volume; Fan et al. [ ] designed a video-probe that takes pictures of very small exposure time to determine droplet sizes. Further approaches like shadowgraphy, holography and diffractometry can also be found and are described in Kleitz' and Dorey's work [ ]. Opposite to the variety of measurement approaches for coarse droplets the measurement of primary droplets almost exclusively is done with light extinction probes. Early a empts by Walters [ ], Tatsuno et al. [ ] . Fundamentals e quantity of interest for the measurements is the intensity of the electromagnetic field sca ered by a spherical particle which is specified by the absolute value of the Poynting vector

I = |S| = |E × H| ( )
with the electrical field E and the magnetic field H and its complex conjugate H respectively. For S follows the vector product

S = E × H = e 1 e 2 e 3 E 1 E 2 E 3 H 1 H 2 H 3 ( )
and for conversion into spherical coordinates

S = - e r e φ e θ E r E φ E θ H r H φ H θ . ( )
Note that only the sca ered far field needs consideration in the discussed application so that it is assumed r ≫ λ. Because the a enuation of the radial components is proportional to r 2 and of the tangential components only to r,

E r ∧ H r ∼ λ 2 r 2 , E φ ∧ H φ ∼ λ r , E θ ∧ H θ ∼ λ r , ( )
the vector product above simplifies to

S = - e r e φ e θ 0 E φ E θ 0 H φ H θ = E θ H φ -E φ H θ e r .
( ) e energy of the diffracted wave therefore is only radiated into the e r direction as two intensity components perpendicular to each other

I = I 0 λ 2πr 2 cos 2 φ S 2 ∧ I ⊥ = I 0 λ 2πr 2 sin 2 φ S 1 . ( ) e amplitude functions S 1 = ∞ n=1 2n + 1 n (n + 1) (a n π n + b n τ n ) ( )
and

S 2 = ∞ n=1 2n + 1 n (n + 1) (a n τ n + b n π n ) ( )
are composed of the sca ering coefficients

a n = mψ n (mx) ψ ′ n (x) -ψ n (x) ψ ′ n (mx) mψ n (mx) ξ ′ n (x) -ξ n (x) ψ ′ n (mx) ( ) and b n = ψ n (mx) ψ ′ n (x) -mψ n (x) ψ ′ n (mx) ψ n (mx) ξ ′ n (x) -mξ n (x) ψ ′ n (mx) ( )
and the angular functions

π n = 1 sin θ P 1 n (cos θ) ( )
and

τ n = -sin θ P 1 n (cos θ) d (cos θ) , ( )
where ψ n and ξ n are a function of the spherical Bessel and Hankel functions of the first kind and P 1 n is the Legendre polynomial.

e sca ering coefficients include the Mie-coefficient x which is given by the product of wavenumber and particle diameter

x = kd 2 = πd λ ( )
and which is one of the most important parameters since its value allows conclusions about the type of sca ering -for both small and large Mie coefficient values simplifications to the theory can be made. m denotes the ratio of complex refractive indices of particle and medium:

m = m Particle m Medium . ( )
e relative intensity of the sca ered light is finally given as

I s I 0 = λ 2πr 2 sin 2 φ|S 1 (θ)| 2 + cos 2 φ|S 2 (θ)| 2 .
( )

It can be seen that the relative light sca ering intensity depends on the incident light wavelength(s), the refractive indices of particle and surrounding media, the distance to the sca erer, the observation and the sca ering angle, the light's polarization direction and of course the droplet size. Hereina er the influence of these parameters on the relation between sca ered light intensity and droplet size is discussed.

. Theory utilization e upper discussed theory can be implemented into computational tools with some li le effort. A MATLAB tool developed by the author is capable of numerically calculating the relative sca ered light intensity, accounting for all the aforementioned parameters according to the set and given boundary conditions such as the geometry of the probe, the light source and the exposed media.

However, simplifications are made by assuming unpolarized incident light and a constant refractive index. Unpolarized implies that the polarization of the light, which is in fact arbitrary in its orientation but distinct at every point in time and statistically evenly spread, changes too fast to be measured or -and this is the important assumption -to have an influence on the physical result of a measurement [ ].

For unpolarized light, or to be more accurate, for the mean value of all occurring polarization states, integration over the whole polarization angle from 0 to 2π gives the relative sca ered light intensity

I s I 0 = 1 2 λ 2πr 2 |S 1 (θ)| 2 + |S 2 (θ)| 2 ( ) since 1 2π 2π ∫ 0 cos 2 φ dφ = 1 2π 2π ∫ 0 sin 2 φ dφ = 1 2 .
( ) e refractive index was set to a real value of m = . . Dependence of the refractive index subject to wavelength and fluid temperature has been modelled but neglected in the following calculations.

Due to the complexity of the underlying equations there is no possibility for optimization of the relevant parameters regarding the sca ered light intensity to droplet diameter relation. e shown correlations therefore only show trends but cannot be considered as an optimum. An extensive parameter study would be needed to be confident of the optimum parameter values.

All given figures only show relative sca ering intensities. It should be emphasized that a relative intensity value is sufficient for theoretical understanding since the measurements require a calibration anyway where absolute intensity values are assigned to known droplet sizes.

. Dependence on wavelength

To counterbalance this circumstance the incident light might consist of multiple wavelengths. A change of wavelength for a certain particle size results in a change of the Mie coefficient, see eq. , and hence the relation between intensity and particle diameter differs. In fig. the change of sca ered light intensity with particle diameter is shown for different wavelengths at a constant sca ering angle of °. e curve progressions seem to be arbitrary without any regularity, especially for droplets greater µm in diameter. However it can be assumed that proper assembling of multiple wavelengths leads to smoothing of the curve progression. Since the water droplets of interest are in a size range from to µm plo ing of sca ered light intensity for larger droplets will be omi ed -see sec. . small mean sca ering angles are favourable for measurements. e choice of a fixed observation angle in combination with an average sca ering angle will determine the curve progression of sca ered light intensity regarding the particle diameter. As seen in fig. small sca ering angles relate to forward sca ering where also the absolute sca ered light intensity increases which benefits the signal strength and hence its processing.

As a conclusion of this section fig. contrasts the scattered light intensity progression for a basic single wavelength and sca ering angle calculation to an optimized calculation, where the above described enhancing factors have been applied. For the optimized configuration the relation of scattered light intensity and particle diameter is explicit and therefore the measurement technique is theoretically suitable for primary droplet measurements.

Figure .

Relative sca ered light intensity as a function of particle size for an optimized and a basic calculation

. MEASUREMENT ERRORS

In this section a short treatise about measurement errors occurring during light sca ering measurements shall be conducted. For each measurement error an estimation on it's significance shows the influence on the measurement results.

. Extinction

A laser beam that propagates straightly from the probe window to the measuring volume forms a cylinder. A laser beam that di-or converges from the probe window to the measuring volume forms a truncated cone. In these volumes enclosed droplets cause an a enuation of the laser beam, generally known as extinction. e sca ered light emi ed from the droplet in the direction of the receiver suffers from extinction as well. It should be self-evident that -for a constant droplet size distribution -the extinction rises the more droplets lie within the upper mentioned cylinders or truncated cones. e main influences on the magnitude of the extinction error are therefore the droplet density in the measuring environment and the size and shape of the laser beam imaging the measuring volume including the distance between the probe window and the measuring volume.

Total avoidance of the extinction error cannot be accomplished as long as droplets pass the beam that propagates towards the measuring volume and from the MV towards the receiver. Nevertheless, extinction can be minimized by proper selection of the optical path if the given particle density in the investigated flow is constant. In case of assurance that droplets only pass the measuring volume -for example through local seeding -extinction can be neglected.

Estimation of this error can be carried out in respect to assumed droplet size distributions. e effect of a monodispersed droplet size distribution is contrasted to a polydispersed size distribution. Firstly, the droplet density per volume needs to be considered. As seen in sec. droplet densities in LP steam turbines are said to be to per mm . e general relation for a enuation of a beam due to extinction is

ln I 0 I 0 -I ext = 4πk z λ . ( )
For multiple droplets of single size inside a medium the equation expands to

ln I 0 I 0 -I ext = NQ ext πd 2 4 z ( )
and for multiple droplets that have a certain distribution in size to

ln I 0 I 0 -I ext = i N(d i )Q ext πd 2 i 4 z. ( )
Q ext denotes the so called extinction efficiency . Beam expansion e beam expansion error comes hand in hand with extinction since extinction equals the absorption of the particle medium plus its sca ering.

Q ext = 2λ dπ 2 Re{S(0 • )} ( )
In consequence, if we assume absorption in water to be neglectible -the imaginary part of the refractive index of water which denotes the absorption is in the range of 10 -to 3.35 × 10 -for visible light [ ] -all a enuation of the laser beam is due to sca ering at the extinction causing droplets. It could therefore be a possibility that the amount of sca ered light into the forward direction -towards the measuring volume -affects the intensity distribution in a non-neglectible manner so that the striven sharpness of the measuring volume is reduced. e significance of this error is assessable when falling back on the calculated extinction error. At worst all light that is lost due to extinction can make a change in intensity distribution. As for the upper calculated case with the droplet density distribution from Petr and Kolovratnik [ ] this is about . % of the incident light. Hence the beam expansion error seems to be neglectable, at least for this case. Reconsideration may be important for other circumstances.

. Coincidence

To achieve reliable results it has to be ensured that only one droplet at a time is situated inside the measuring volume. e shape and the size of the measuring volume are a key to this. However there always is a certain possibility of two or more droplets being inside the measuring volume at the same time.

e misinterpretation of multiple droplets as one is called coincidence. A special type of coincidence occurs when one particle leaves while another enters the measuring volume instantaneously. Depending on the measurement analysis system and setup it can also be a source of error which is called dynamic coincidence.

As Kleitz and Dorey [ ] mention the coincidence as "one of the most serious limitations" of light sca ering technique there shall be a closer look on how to deal with this error.

A way to reduce coincidence is by adapting the measuring volume size to the maximal droplet density. e catch is that the droplet density has to be known previous to the measurements and adaptions to the optical system of the probe are required. e measuring volume then can be generated as small as necessary that only a single droplet is lit simultaneously.

Detection of coincidence can be done at high measuring card sampling rates.

e slope of the sca ered pulse edge should not change while a droplet passes the measuring volume. In case there is a change of slope this could be an indicator for the entry of a second droplet into the measuring volume. ese pulses should be excluded from the evaluation. Depending on their percentage share in regard to the overall measurement and assuming that those droplets follow the overall size distribution the droplet number can be corrected subsequently to obtain the actual quantity. Another practical way to identify coincidence requires a uniform intensity profile across the measuring volume and the beam axis. An elaborate design of the probe's optics can assure a reasonably even distribution. Droplets entering the MV at different times are distinguishable by means of the overall pulse height.

Dynamic coincidence -however small the possibility may be -can neither be exposed nor quantified.

. Illumination error is error is caused by a non-uniform intensity distribution inside the measuring volume, mostly one of Gaussian shape. In particular, a smaller droplet flying through the middle of the measuring volume produces the same sca ering intensity as a bigger droplet flying through an outlying area. Statistical correction of this error can be applied for assuming a uniform droplet distribution over time and space. Since the same amount of droplets with the same sizes cross each part of the measuring volume correction factors can be applied to the pulse distribution that increase the amplitude of a certain share of all pulses just by the ratio of actual and maximum beam intensity. Physical correction by applying different optics resulting in a constant intensity distribution inside the MV is also a good possibility.

. Side error

Besides multiple droplets staying inside the measuring volume at the same time it is also possible that a droplet only scratches the edge of the illuminated volume. In this case just a part of the sca ered light is received by the sensor and the droplet will be underestimated in size. is is called side error. Besides the small pulses that are generated by side errored droplets these pulses are normally very short in time so that detection is rather difficult. is however helps avoiding side errors if for example a pulse width limit can be applied to the measurement values.

. Multiple sca ering

Another possible error arises from multiple sca ering. If a droplet A lies within the measuring volume and is additionally illuminated by the sca ered light of a droplet B (that for example lies behind the measuring volume but is also illuminated by the laser beam) the sca ered light of droplet A will be a superposition from the sca ered light with the laser's incident intensity and droplet B's sca ered intensity. Since forward sca ered light intensity is normally much bigger compared to intensities in all other angles -this effect diminishes for smaller droplets -the additional light sca ering intensity introduced by droplet B will be negligible over the sca ered light intensity originating from the laser beam. In addition, lit droplets inside the measuring volume sca er light in not only the direction of the sensor but everywhere. Droplets outside the measuring volume are subject to illumination by this sca ered light and sca er light themselves into all directions -also towards the sensor.

is share seems neglectable as well when compared to the main sca ered light intensity originated from the laser beam. Jones [ ] says that the distance between particles has to be three times their diameter to neglect multiple sca ering.

. APPLICATION TO DROPLET MEASURE-MENTS IN LP STEAM TURBINES

Mie theory can only be applied if certain requirements are met. Droplets need to be spherical and homogeneous without surface charge as well as isolated and surrounded by a homogeneous and loss-free medium [ ]. e incident light has to be a planar harmonic wave of infinite extent with known intensity -which also implies that phase and amplitude of the wave should be constant across the particle. For particle clouds Jones [ ] additionally expresses that the particles need random arrangement yielding incoherent sca ering, have no interaction of their electromagnetical fields among each other and are only lit by unsca ered light -that multiple sca ering is neglectable, respectively. e satisfaction of these requirements will be part of this section.

Droplets of special interest in this paper are the so called primary droplets that nucleate because of expansion of the steam. Due to its surface tension a droplet levitating in space will always try to a ain a spherical shape [ ]. It is due to other on the droplet acting forces that it will deform and eventually break up into smaller droplets. e ratio of these forces and the droplet surface tension, expressed by the Weber number

W e = ρ g w 2 rel d σ , ( )
is an indicator for deformation and break up of droplets. Since the relative velocity w rel between gaseous and liquid phase for droplets forming within the flow during condensation can be assumed to be zero there is no droplet deformation. Slip between droplets and steam caused by flow deflection is negligible for droplets smaller µm [ ]. On the other hand secondary droplets e.g. from blade trailing edges are much bigger, have a high difference in velocity compared to the steam and are therefore very sensitive to deformation and break up. A good overview on breakup modes depending on Weber numbers is given by Kékesi et al

. [ ].
Homogeneity of the primary droplets is given material and temperature wise due to their small size. In water dissolved salts and other foreign objects could induce inhomogeneity but since the power plant feed water is usually completely demineralized this is disregarded.

Bohren and Huffman [ ] give some information on charged spherical particles. A quantitative evaluation is not made but it is assumed that surface charge only has a slight impact on sca ering and absorption properties.

e droplets' surrounding medium -steam -is isolated from the droplets due to its gaseous state and assumed homogeneous since any inhomogeneities would lead to further sca ering at these impurities. Steam is considered loss-free -meaning it has negligible absorption -as are the water droplets.

Illumination of the particle depends on the optical system and whether the beams is collimated or converging. A collimated beam can be seen as a planar wave since its focal point lies at infinity. A convergent beam can rather be approximated as a spherical wave. Weyl [ ] however shows that a spherical wave can be expressed as a superposition of many plane waves of miscellaneous direction.

Of the three requirements expressed by Jones [ ] for particle clusters the first and third need no consideration since the formation of droplets and hence also their arrangement are arbitrary and multiple sca ering has already been treated during error estimation in sec. . . Interaction of the droplets' electromagnetic fields is dependent on their quantity per volume. Jones therefore gives maximum particle concentrations that are 10 19 per m for . µm particles, 10 16 per m for . µm particles and so forth. Droplet densities in wet steam turbines have been found to be to per m [ ][ ]. Gyarmathy [ ] and Seibold [ ] predict primary droplet sizes of up to 0.6 µm. In other literature also sizes of up to µm reported but directly neglected due to their marginal occurrence. Most authors believe primary droplets ranging in sizes from . to µm. From this point of view interaction among the droplets in LP wet steam flow should have no importance for Mie theory application.

. DESIGN OF A MEASUREMENT SYSTEM

A probe for measurements of primary droplets inside LP steam turbines needs to meet certain conditions. As Bosdas et al. [ ] point out access holes in steam turbines usually are in a diameter range of to mm. e probe's diameter should therefore also lie in this range to ensure maximum accessibility. Other conditions also arise from the above cited sections.

e light source should be strong in power to compensate for the relatively small sca ered light intensity and polychromatic to improve the correlation of sca ered light intensity and droplet diameter. A multi-line laser therefore seems appropriate. Insertion of the laser beam into the probe should be done through optical fibers. In order to allow for decoherization of the incident polychromatic light by smoothing the speckle contrast a multimode fiber of an appropriate length can be used for transportation of the laser beam into the probe head [ ]. Another advantage of a laser is its focusability. eoretically for monochromatic light a focal point with almost no extent is feasible. Even though the use of polychromatic light should be of choice it can generate an adequate spot size if some effort is put into the design of the imaging system.

To cope with coincidence the measuring volume should be as small as possible and the beam intensity inside the measuring volume should be of different than a Gaussian shape, preferably uniform. Based on a droplet concentration of -droplets per m the minimum edge length of the measuring volume, implying a cubic volume, calculates to .µm. However, regarding the droplet size distribution calculated by Petr and Kolovratnik [ ] that state droplet sizes from as small as . µm a lower resolution limit has to be set in order to distinguish measured droplet pulses from measurement noise. Assumption of a lower resolution limit of . µm leads to exclusion of about % of droplets which however are due to their small size only responsible for % of the wetness fraction. e permissible edge length then rises to . -. µm. Hence it gives an advantage to set a higher minimum resolution limit allowing the enlargement of the measuring volume without coincidence of two or more droplets. On the other hand the measurement error rises because of the increased minimum resolution limit just by the wetness fraction of the excluded droplets. For a uniformly lit MV it can be enlarged and coincident droplets can be detected by the superposition of their sca ered light intensities assuming that they are not entering the measurement volume at the same time. To prevent droplet deposition on the optical components of the probe a purging air flow on these surfaces should be realized.

Accornero and Mare o [ ] found axial Mach numbers between . and . at the exit of a MW steam turbine which imply velocities of up to m / s . For measurement of primary droplets that follow the steam flow without slip a high sampling measured value acquisition is needed. For the upper case a droplet passes a measurement volume of µm in diameter in ns. Hence a system with a sampling rate of approximately GHz should be used.

Due to the primary droplets' small size only a few measurement techniques are feasible for their determination. Most approaches include the light extinction method that admi edly has its disadvantages. Only light sca ering technique is able to measure single droplets and hence gives an exact droplet size distribution. Extinction suffers from matching distribution functions to the actual measurement results and can therefore only estimate the real size distribution. An accurate understanding of the nucleation of droplets in wet steam flow can therefore only be achieved by application of the light sca ering technique.

. CONCLUSION AND OUTLOOK is paper theoretically proves the suitability of light sca ering techniques for droplet size measurements in low pressure steam turbines.

rough the treatment of the underlying physical theory it is shown that the choice of the incident light source as well as the geometrical setup of the probe and the position of the measuring volume respectively have a major impact on the technique's applicability. Furthermore conditions in the low pressure turbine as a measurement environment determine the setup and design of the probe and the measurement system.

On one hand, due to the measurement of single droplets, this technique helps further understanding of the phenomenology of nucleating wet steam flows. On the other hand it helps to reduce the still existing mismatch between calculated condensing flows in CFD simulations by providing the opportunity for adjustment of the underlying models.

Further

Figure .

 . Figure . Logarithmic representation of the sca ered light intensity around a spherical water droplet for a Mie coefficient of x = 7

Figure .

 . Figure . Logarithmic angular sca ered light amplitude for particles with varying diameter and Mie coefficient respectively

Figure .

 . Figure .Relative sca ered light intensity as a function of particle size for various wavelengths at a constant sca ering angle of °

Figure .

 . Figure . Relative sca ered light intensity as a function of particle size for various observation angles at a constant sca ering angle of °for a single wavelength

Figure .

 . Figure . Relative sca ered light intensity as a function of particle size for various mean sca ering angles at a constant observation angle of °that

Figure .

 . Figure . Extinction coefficient for polychromatic light and resulting beam intensity for different particle sizes assuming a monodispersed droplet size distribution

  and Young et al. [ ] provided a basis for all following probe designs. Recent studies on primary droplets have been performed by Schatz and Eberle [ ] who use a wedge probe that is 10 mm in di-ameter and that is also capable of measuring total and static pressures. Bosdas et al. [ ] introduce a slightly smaller probe with heating to prevent contamination of the optical components and any other undesired beam disturbances. Only Bohn et al. [ ] -to the authors best knowledgeconsider the light sca ering technique for the measurement of primary droplets. e present work reinforces this judgement and points out principles, requirements, limitations and uniqueness of this technique.

	NOMENCLATURE
	d	Droplet or particle diameter
	E	Electrical field
	H	Magnetic field
	I	Electromagnetic wave intensity
	m	Complex refractive index
	Q ext	Extinction efficiency
	r	Radial distance
	S	Poynting vector
	W e	Weber number
	x	Mie coefficient
	Symbols
	θ	Sca ering angle
	λ	Wavelength
	ρ	Density
	σ	Surface tension
	φ	Polarization angle
	Subscripts
	0	Incident
	r	Radial
	rel	Relative
	s	Sca ered
	Abbreviations
	CFD	Computational fluid dynamics
	LP	Low pressure
	MV	Measuring volume
	. LIGHT SCATTERING THEORY
	At the very beginning shall be an introduction into the un-
	derlying physics. Discussion covers the intensity of light
	sca ered by a spherical particle. Interested readers may be
	referred to Mie's [ ] and Debye's [ ] original work, as well
	as to Bohren and Huffmann [ ], van de Hulst [ ] or Jones
	[ ] who, among others, give a good insight into sca ering
	theory.

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ACKNOWLEDGEMENTS

e investigations were conducted as part of the joint research program COOREFLEX-turbo in AG Turbo. e work was supported by the Bundesministerium für Wirtscha und Technologie (BMWi) under file number ET S on the basis of a resolution of the German Parliament. e authors gratefully acknowledge AG Turbo and MAN Diesel & Turbo SE for their support and permission to publish this paper.

e responsibility for the content of this publication lies with the authors.