Jiří Kozák

Pavel Rudolf

Martin Hudec

David Štěfan

Matěj Forman

Numerical and experimental investigation of the cavitating flow within Venturi tube

Keywords:

Hydrodynamic cavitation represents complex physical phenomenon undesirably affecting operation as well as lifespan of many hydraulic machines from small valves to the large hydro power plants. On the other hand, the same phenomenon and its concomitants such as pressure pulsations can be exploited in many positive ways. One of them which seems to be very promising and perspective is the cavitation utilization for reduction of the microorganisms such as cyanobacteria within large bulks of water. Mutual effect of the swirl induced by the upstream mounted generator and flow constriction in convergingdiverging nozzle has been experimentally investigated. The main scope of this paper is numerical investigation complementing the experimental results. The multiphase simulations were carried out using the OpenFoam 1606+ and its interPhaseChangeFoam solver. The present study focuses on CFD results of the cavitating velocity field within the nozzle and analysis of the loss coefficient within the nozzle.

INTRODUCTION

The investigation of the cavitation is one of the key research directions of hydraulic engineering and multiphase fluid mechanics in general. In case of the pressure drop below the value of saturated water vapor, cavities filled mainly by the vapor occurred within the liquid. One of the most important type of cavitation in the field of hydraulic engineering is represented by so-called hydrodynamic cavitation, where the pressure drop is induced by the local increase of the flow velocity. [START_REF] Ch | Cavitation and Bubble Dynamics[END_REF] Vapor cavities within the flow induce pressure pulsations, vibrations and noise. Above that, the surfaces of hydraulic machines and pipelines exposed to the collapsing vapor cavities are threatened by so-called cavitation erosion due to the enormous local variations of pressure and temperature. These reasons led to exhaustive research of this phenomenon from the beginning of the last century to the present days. Since the abovementioned concomitants of cavitation are usually undesirable, the research of this phenomenon is focused mainly on the cavitation suppression.

Typical example could be investigation of the off-design operation of the hydraulic turbines. [START_REF] Dörfler | Flow-induced pulsation and vibration in hydroelectric machinery: engineer's guidebook for planning, design and troubleshooting[END_REF] This issue has gained importance mainly with the expansion of renewable sources, where hydro power plants play important role in the electrical grid regulation. On the other hand, the off-design operation of certain hydraulic turbines types leads to the inception of the vortex structures downstream the turbine runner due to the presence of the significant residual swirl within the flow. This problem is particularly severe in case of Francis turbines, due to the fixed angle of the runner blades and frequent utilization of this type of hydraulic turbine.

Cavitating vortex rope is mostly present due to the pressure drop in the core of the swirl. This reasons evoked extensive investigation of this problematics. Since the investigation on the full-size turbines represents considerable finance burden and even the utilization of scaled down turbine models can be unbearably expensive, the exploitation of so-called swirl generators is common practice. [START_REF] Štefan | Numerical and experimental investigation of swirling flow in a conical diffuser[END_REF], [START_REF] Hirt | Volume of fluid (VOF) Method for the dynamics of free boundaries[END_REF] The swirl generator designed for this particular application, specifically for the investigation of the swirl decay and dissipation within the diffuser, has been exploited for the investigation of the cavitation induced by the swirl within the Venturi tube. A Venturi tube is a converging-diverging nozzle with short straight throat in between both sections. Investigation is motivated mainly by the idea to apply hydrodynamic cavitation for cyanobacteria elimination [START_REF] Jančura | Selective method for cyanobacterial bloom removal: hydraulic jet cavitation experience[END_REF] or generally for disinfection of large bulks of water. This application of the cavitation seems to be effective and above that, thanks to its physical nature (e.g. absence of chemical agents), environmentally-friendly. This is particularly important in case of water reservoirs which are used for the recreation or as a drinking water sources.

While the principle of the cavitation reduction of the cyanobacteria within the water seems to be an effective approach, the complete mechanism of the disruption of this organism has to be analyzed. For this purpose, cavitation and its dynamics within the Venturi tube have been investigated and results were discussed in several contributions. [START_REF] Rudolf | Characterization of the cavitating flow in convergingdiverging nozzle based on experimental investigation[END_REF][6] [START_REF] Kozák | Transition of cavitating flow to supercavitation within Venturi nozzle -hysteresis investigation[END_REF] Introduction of swirl using the swirl generator represents the further step of this investigation. The cavitation patterns downstream the throat of the nozzle have been highly modified by the presence of the swirl. The additional pressure drop causes step up of the cavitation amount. Above that the redistribution of the flow across the cross section of the Venturi tube and shift of the vapor clouds away from the walls can be suitable from the cavitation erosion point of view.

While the experimental data acquired during the investigation provide rich source of information, the additional CFD analysis was convenient due to the uninvestigated velocity field downstream the swirl generator information. Moreover, the numerical analysis of this phenomenon brings better insight into the dynamics of the cavitating structures complementary to the high-speed video records and other experimental data.

The following chapters consist of the experimental measurement description and following numerical investigation. The main scope of the paper will be focused on the combination of the experimental and numerical results to provide complex description of the cavitation induced by the presence of the swirl within the Venturi tube. The results are discussed in the direction from the supercavitation to the initial stage of cavitation.

Experimental measurement

Experimental measurement was performed exploiting closed cavitation circuit equipped with the pressure tank, where the pressure was set using the pressurized air and vacuum pump (fig. 1). Discharge within the circuit was set using the pump controlled by frequency convertor. This setup allowed to investigate different cavitation regimes for the constant discharge. Discharges between 4 and 7 l/s have been investigated with and without upstream mounted swirl generator.

Figure 1 Scheme of cavitation circuit equipped with the transducers

The cavitation has been studied within the axisymmetric Venturi tube manufactured from plexiglass, thus it was possible to capture the investigated phenomenon using the high-speed (HS) camera. The main dimensions of the Venturi tube are depicted in the following figure 2.

Figure 2 Main dimensions of the exploited Venturi Tube

Geometry of the swirl generator (SG) is depicted in the figure 3. Swirl generator consists of 9 fixed blades with the linear change of the trailing edge deviation angle from axial direction (30° to 50°, from hub to shroud respectively).

Figure 3 Model of the swirl generator

The various properties of the cavitating flow have been measured during the investigation as it is depicted in the Figure 1. While the data of acoustic emission and noise represent rich source of information, for the scope of this paper the pressure and vibration records as well as the high speed (HS) video records are important. These records, in conjunction with the record of the flow discharge Q and water temperature T enable to describe the static and dynamic properties of the cavitating flow. List of the important transducers is shown on the following The sampling frequency of Kistler and Aura transducer was 51.2 kHz, whereas the sampling frequency of the HS records was set to 20 000 fps. Grayscale H-S video records with the resolution of 1024 by 296 px have been captured using the Photron FastcamSA-X2 1000K camera. Venturi tube has was illuminated using two led matrix light sources using the dark background.

The amount of air within water has not been measured during the measurement. Test rig has been operated under the supercavitation regime prior the measurement for about 30 minutes. This method led to the final concentration between 6 to 8 mg/l of the oxygen based on our previous experience.

Hydraulic loss coefficient evaluation

Static characteristics of the flow have been evaluated using hydraulic loss coefficients. The additional properties such as water density ρ and saturated vapor pressure pva were calculated according to IAPWS based on the measured water temperature. It should be noted that the range of the investigated cavitation regimes was restricted by the test rig characteristics. The most problematic part of the rig represented pump shaft sealing, where the suction of air has been observed under certain circumstances.

To characterize the flow, the cavitation number σvel was evaluated (eq. 1). While the calculation of this value using bulk velocity within the throat of the nozzle vth is reasonably accurate in case of the axial flow, in case of the SG the swirl significantly distorts velocity field. This issue will be in detail discussed in the part devoted to numerical simulations.

σ vel = p 3 -p vap 0.5ρv th 2 (1)
Hydraulic loss coefficient ξ has been evaluated using the formula [START_REF] Dörfler | Flow-induced pulsation and vibration in hydroelectric machinery: engineer's guidebook for planning, design and troubleshooting[END_REF].

ξ = p 1 -p 3 0.5 ρv th 2 (2)
Using these results, it was possible to capture hydraulic loss coefficient evolution with respect to cavitation number. As it can be seen in the figure 4, the hydraulic loss coefficient is significantly higher in case of the SG presence. On the other hand, the significant amount of kinetic energy contained within the swirl is neglected. This discrepancy will be discussed in the chapter devoted to the numerical analysis. It should be also noted that the value of ξ is not affected by the discharge considering individual investigated configurations.

Figure 4 Hydraulic loss coefficient versus cavitation number

(measurement uncertainty for selected operating points)

Vibration induced by the cavitating flow

The discussion of the cavitation dynamics will be focused mainly on the analysis of the test rig vibrations using the accelerometer and the analysis of H-S records. It should be noted that the analysis will be done for the flowrates of 4 l/s and 6 l/s. In general, it can be stated that the noise as well as vibrations of the rig were significantly higher in case of pure axial inflow (AX). While the influence of additional source of dampening presented by the SG cannot be fully neglected, the main reason for this behavior is significantly modified flow structure downstream the generator. These observations are supported by the analysis of the mean value of the pipe wall acceleration magnitude depicted in the figure 5. Acceleration is imprint of the vibrations induced by collapsing vapor clouds. The vibrations induced during the operation of the test rig under the supercavitation regime varied from 1.4 to 1.5 g while the maximum mean value of the vibrations reached 24.4 g, corresponding to the axial inflow and 6 l/s. It should be noted that the static value of measured acceleration on the test rig has was 0.14 g, when the pump was turned off.

Figure 5 Analysis of test rig vibrations induced by cavitation

Increase of the acceleration magnitude related to the fully developed cavitation is steep in both of the investigated configurations (region 3).

The maximum magnitude of the acceleration has been significantly lower in case of the SG presence. The maximum vibrations have been observed between the fully developed and initial stage of cavitation. Above that these maximum values have been observed closed to the σvel value of 0.44.

Seeing the decrease of the acceleration magnitude from the fully developed cavitation to the initial stage (region 4), it can be stated that the decrease is steeper in case of axial flow. Interesting observation is the fact, that the vibrations magnitude of the cavitating flow influenced by the additional swirl is higher compared to the axial flow down to a certain value of cavitation number. Unlike the value of cavitation number corresponding to the maximum acceleration, this threshold value is significantly less constant. In case of higher discharge (6 l/s), the vibrations of the axial flow became more significant close to the cavitation number of 0.83, while in case of the lower discharge (4 l/s) this transition has been observed close to the cavitation number value 0.69.

High speed video records analysis

Using the captured high-speed video records, it was possible to investigate typical cavitation patterns (Figure 6) for both test rig configurations. Beside the overall description of cavitation behavior, dominant frequencies of the cavitating structures were studied using the proper orthogonal decomposition (POD).

Cavitation regimes -influence of the induced swirl

Region 1 in the Figure 5: Supercavitation regime is typical for its low vibrations, pressure pulsations and noise induced by the cavitating flow regardless of the SG presence. Nevertheless, the cavitation patterns are completely different. In case of axial flow the liquid jet surrounded by the vapor was observed downstream the nozzle.

On the other hand, the coherent cavitating vortex surrounded by liquid water was observed in case of SG presence. Separation of the cavitating boundary layer occurred within the diffuser. Beside the analysis of the experimentally obtained data, usually velocity field, it is possible to exploit this method for analysis of the video records. Since number of analyzed spatial points was rather high, so-called method of snapshots has been applied [START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF]. POD was utilized by Danlos et al. for study of passive control of cavitation using the surface grooves. Decomposition has been exploited for the detection of the cavitation regimes [START_REF] Danlos | Cavitation regime detection through Proper Orthogonal Decomposition:Dynamics analysis of the sheet cavity on a grooved convergentdivergent nozzle[END_REF]. Another application of the POD can be found in investigation of aerated cavitation dynamics within the Venturi nozzle with rectangular cross-section [START_REF] Tomov | POD study of aerated cavitation in a venturi nozzle[END_REF]. In-depth investigation of the swirl structures downstream the runner of swirl generator designed by team at UPT led by Resiga was done by Štefan et al. [START_REF] Štefan | Proper Orthogonal Decomposition of Self-Induced Instabilities in Decelerated Swirling Flows and Their Mitigation Through Axial Water Injection[END_REF]. Investigation of the swirling flow dynamics downstream the runner of Francis turbine using the POD was done by Rudolf et al. [START_REF] Rudolf | Decomposition of the swirling flow field downstream of Francis turbine runner[END_REF].

The results presented in this contribution are successive to the published analyses of the transitions between fully developed and supercavitation regime of cavitation Venturi tube [START_REF] Kozák | Transition of cavitating flow to supercavitation within Venturi nozzle -hysteresis investigation[END_REF]. POD analysis has been carried out considering the constrained region of interest containing the upper half of the captured images. The main reason for the image was to minimize influence of uneven scene illumination. For the purpose of the analysis, each 10 th image of record was considered. Most significant spatial coefficients of the pixels intensities fluctuations with and without the upstream SG corresponding to the initial, fully developed and supercavitation regimes are depicted in the figure 8. Additional swirl significantly influences the spatial coefficients structure. This is especially noticeable in case of supercavitation, where the most important fluctuations are small instabilities close to interface layer and in case of initial phase of cavitation where the small but coherent axisymmetric vortex was observed in case of SG presence. Spatial coefficient and time coefficient define the variation of the POD mode significance in time. Thus, it is possible to study dynamics of the POD modes using the spectral analysis of the temporal coefficients. Dominant frequencies of the most significant POD modes versus σvel are depicted in the figure 9. The frequencies are the higher, the higher is the discharge. Presence of the swirl induced by the SG increases the frequencies slightly. This increase of frequency is more significant in case of 6 l/s. Comparing the results of the frequency analysis with the chart of the pipe wall acceleration magnitude, several observations can be described. In the region of fully developed and initial stage of cavitation, the growth of the frequency value is nearly linear. in case of transitions regimes different results can be found.

Figure 9 Dominant frequencies obtained by analysis of the temporal coefficients corresponding to the most important dynamic POD modes

Main source of instability represents highly stochastic structures of the interphase region in case of supercavitation.

In the fully developed stage of cavitating flow, the increase of dominant frequencies is more rapid, compared to the initial stage of cavitation. Cavitation number range of this rapid frequency increase is in good agreement with the steep increase of the acceleration magnitude. Faster increase of the frequency magnitude, compared to the initial stage of cavitation can be attributed to the position of the frequency downstream the nozzle, or to the faster increase of the vapor phase amount within the flow and corresponding faster increase of the medium compliance. Both of these assumptions led to the conclusion that the decrease of the vapor phase volume is more rapid in case of the fully developed cavitation compared to the further decrease of the vapor amount during the initial stage of cavitation.

Increasing value of the frequencies is related to the decreasing mean magnitude of vibrations in the initial regime of cavitation. In other words, slower pulsation of massive cavitating structure produces stronger vibrations compared to the faster pulsations in this region. On the other hand, the lower is the cavitation number in case of fully developed cavitation, the lower is the observed magnitude of the acceleration.

With the increasing cavitation number, the abrupt drop in dominant frequency can be seen. Regardless of the frequency magnitude drop occurred, acceleration of vibrations remains still higher by order compared to the vibrations induced by supercavitation. It should be noted that these frequency drops occurred earlier in case of axial inflow.

Relative significance of the first dynamic modes computed as a fraction of the POD mode eigenvalue to sum of all eigenvalues is depicted in the figure 10. Comparing with the chart of the acceleration magnitude, remarkable match of the results can be found. Cavitation number corresponding to the maximum relative significance is in good agreement with maximum of the mean acceleration magnitude. Above that, while in the wide range of operating regimes, the relative significance of the most important POD mode is higher in case of the axial flow. Decrease below the value obtained in case of SG presence can be found. On the other hand, it should be noted that σvel of this relative significance drop is higher, than in case of pipe wall acceleration analysis for 4 l/s discharge. Numerical investigation has been carried out using OpenFoam 1606+ and its multiphase interPhaseChange solver based on VOF method. The main reason for the numerical investigation was to complement the experimental data, especially the velocity field downstream the generator. Nevertheless, the CFD results can provide much more. It should be emphasized that the numerical investigation will be focused on the configuration including SG. The numerical analysis of the cavitating flow within the investigated Venturi tube considering only axial inflow can be found in [START_REF] Kozák | Analysis of Pressure Pulsations Of Cavitating Flow In Converging-Diverging Nozzle[END_REF].

Computational grids

The computational grids were created using the cfMesh and its predominantly hexahedral algorithm cartesianMesh. The cfMesh provides wide range of the mesh refinement tools as well as boundary layers tools, while the process of the mesh properties setting is not complicated and mesh generation is relatively fast.

In the first step of the numerical analysis, the whole experimental section including the SG has been considered (figure 13). Two different computational grids were utilized for the numerical analysis: coarse (CG1) and refined (CG2). The comparison of the coarse and refined meshes is depicted in the figure 11.

Numerical settings

The interPhaseChangeFoam solver was exploited. This solver is using so-called VOF method to capture interface of two incompressible, isothermal, immiscible fluids where the phase-change is considered. VOF method was proposed by Hirth. [START_REF] Hirt | Volume of fluid (VOF) Method for the dynamics of free boundaries[END_REF] Beside the single continuity and momentum equations of the mixture, the transport equation of the volume fraction α had to be solved (eq. 3).

∂(αρ l ) ∂t + ∇ • (αρ l 𝐔) + ∇ • [α𝐔 C (1 -α)] = R e -R c (3)
Where Re and Rc represent the source terms of evaporation and condensation respectively. These terms are calculated using cavitation models, usually based on Rayleigh-Plesset equation.

Model proposed by Kunz et al was considered for the purpose of this investigation [START_REF] Kunz | A Preconditioned Implicit Method for Two-Phase Flows with Application to Cavitation Prediction[END_REF]. The choice was based on the reasonably accurate results of the previous numerical investigation considering axial inflow into the Venturi tube [START_REF] Kozák | Transition of cavitating flow to supercavitation within Venturi nozzle -hysteresis investigation[END_REF].

In case of Kunz cavitation model, the source terms Re and Rc are computed using the formulas ( 4) and [START_REF] Rudolf | Characterization of the cavitating flow in convergingdiverging nozzle based on experimental investigation[END_REF].

R e = C v ρ v α t ∞ (0.5ρ l U ∞ 2 ) min[0, p -p v ] (4)
R c = C c ρ v α 2 (1 -α) t ∞ (5)
Where 𝑡 ∞ is mean flow time scale and 𝑈 ∞ represents mean stream velocity [START_REF] Kunz | A Preconditioned Implicit Method for Two-Phase Flows with Application to Cavitation Prediction[END_REF]. Values of the empirical constants Cv (900) and Cc (12000) were set-up based on [START_REF] Kozák | Analysis of Pressure Pulsations Of Cavitating Flow In Converging-Diverging Nozzle[END_REF]. The realizable k-ε model has been chosen for the numerical analysis. The model of turbulence was derived from the standard k-ε model by Shih et al. [START_REF] Shih | A New K-epsilon Eddy Viscosity Model for High Reynolds Number Turbulent Flows: Model Development and Validation[END_REF]. The proposed modifications of turbulent viscosity formulation and transport equation for the dissipation rate led to the significantly lower turbulence dampening. This model was successfully applied by Štefan et al for the swirl dissipation investigation within the diffuser downstream the SG [START_REF] Štefan | Numerical and experimental investigation of swirling flow in a conical diffuser[END_REF]. While this research involved only the single-phase computations, the good results were obtained by main author in case of the cavitating flow in Venturi tube (considering axial inflow) [START_REF] Kozák | Analysis of Pressure Pulsations Of Cavitating Flow In Converging-Diverging Nozzle[END_REF]. Numerical simulations of the multiphase flow within the Venturi tube for the purpose of discharge measurement using Star-ccm+ and realizable k-ε was done by Brinkhorst [START_REF] Brinkhorst | Experimental and Numerical Investigation of the Cavitation-induced Choked Flow in a Herschel Venturi-Tube[END_REF]. Inlet velocity has been set according to the experimentally measured discharges. The inlet turbulent properties were prescribed using the turbulentMixingLengthDisssipationRateInlet in case of ε and turbulentIntensityKineticEnergyInlet with 5% of intensity in case of k. The outlet pressure has been set according to the static pressure experimentally measured by transducer p3 placed 700 mm downstream the nozzle.

Evaluation of the hydraulic losses

As it has been mentioned, the bulk velocity within the throat of the nozzle, which was considered as a reference velocity for the computation of σvel as well as for ξ, neglects the tangential and radial components of the velocity. These components are significantly higher in case of the swirl induced by the SG. Loss coefficient analysis based on the bulk velocity is depicted in the figure 14.

Figure 14 Hydraulic loss coefficientinfluence of the computational grid

It is clearly depicted that the results of the CFD analysis are significantly closer to the experimental data in case of refined grid CG2. While the decrease of the hydraulic loss coefficient is linear considering both of the utilized computational grids, the inclination is closer to the experimental data in case of the CG2. For these reasons, the CG2 has been picked as reference for the further investigation.

The energy correction coefficient αcorr given by equation ( 6) has been evaluated in the inlet of the nozzle throat. This region was chosen as a reference due to the stability of the flow.

α corr = ∬ v ax •v mag 2 dS v bulk 3 S (6)
Where vax represents axial velocity, vmag is velocity magnitude and vbulk is bulk velocity.

The evaluated mean value of αcorr is 1.173 based on the analysis of three operating points utilizing the CG2. The values of αcorr were nearly constant and the difference between the lowest and highest value was less than 0.5 %. Only slight increase of αcorr was observed for increasing σ. The αcorr evaluated in case of the axial inflow is 1.01.

The analysis of the hydraulic loss coefficients as well as σvel have been corrected using the αcorr in case of 6 l/s and the SG presence to respect real kinetic energy magnitude. The results are depicted in the following figure 15.

Figure 15 Correction of the hydraulic loss coefficient using the energy correction factor

It is clearly depicted that the values of the corrected hydraulic loss coefficients are approximately lower by 0.22 compared to the uncorrected results. Due to the fact that this decrease is devoted to the inclusion of the swirl within the throat, it is plausible to assume that the remaining difference between the hydraulic loss curves of the corrected results and the results of the axial inflow analysis is caused mainly by the presence of the swirl generator.

If the corrected values of the cavitation number are applied for the pipe wall acceleration magnitude analysis, the discrepancy between the peaks positions of maximum vibrations for the axial flow and flow affected by the induced swirl can be discerned (figure 16).

Figure 16 Pipe wall acceleration magnitude versus cavitation number corrected using αcorr

As it can be seen in the figure 16, the cavitation number corresponding to the maximum acceleration magnitude is lower by 33.5% in case of SG presence compared to the axial inflow.

To discuss this result, another definition of the cavitation number based on the pressure difference was exploited (eq. 7). This definition of cavitation number is also widely used in case of cavitation analysis within valves. The results are depicted in the figure 17. Several conclusions can be drawn comparing the figures 16 and 17. Similar to the correction of σvel, the cavitation number σpres corresponding to the maximum magnitude of acceleration is lower in case the SG presence. On the other hand, the observed difference is 48.2% between the investigated experimental configurations in case of σpres, which is certainly larger difference compared to the correction by αcorr. Regardless this difference, the results obtained using the corrected σvel are in good qualitative agreement with the analysis of the acceleration exploiting σpres.

𝜎 𝑝𝑟𝑒𝑠 = 𝑝 3 -𝑝 𝑣𝑎𝑝 𝑝 1 -𝑝 3 (7) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1

Discussion of the numerical results

The value of the swirl number Sn evaluated according to the eq. 8 at the inlet of the throat downstream the SG is 0.234.

Sn = ∫ v ax • v tan r dS R ∫ v ax 2 dS (8)
Where vtan represents tangential velocity, r radial coordinate and R is a reference radius (i.e. radius of the throat).

Although the results of the numerical analysis were in reasonably good agreement with the experimental observations from the hydraulic loss point of view, two major differences of the flow fields should be described. Thin cavitating vortex filament was observed downstream the spike of SG during the experiments (Figure 18). Using the abovementioned settings and computational grids, the vortex filament has not been captured numerically. On the other hand, the significant pressure drop in the region of the nozzle axis can be seen in the figure 19 and above that the contour of the averaged vapour void fraction (0.25) extends into the diffuser upstream the throat of the nozzle. The second qualitative discrepancy between the numerical and experimental flow downstream the swirl generator is represented by the periodical pulsations of the cavitating boundary layer within the diffuser downstream the throat of the nozzle observed in case of the CFD analysis (Figure 19). The cavitation in this region was relatively seldom in case of the experiments. The significant cavitation of boundary layer was observed only during the supercavitation regime in experimental observation. The absence of the cavitating vortex filament in case of CFD could be partially devoted to the rough surface of the real swirl generator, where the small imperfections of the spike play an important role in the process of the cavitation nucleation. On the other hand, the overestimated cavitation of the boundary layer points to slightly different redistribution of the flow downstream the nozzle throat in case of CFD analysis.

Regardless the abovementioned flaws of numerical computations, the obtained results are in good agreement with the experimental observations. Downstream the coherent conical cavitating structure, the relatively short, but significant, helical vortex can be found in case of H-S image corresponding to the cavitation number of 0.98 (fig. 20). It was possible to capture this phenomenon using the refined computational grid CG2 which is depicted in the figure 11 using the vapour void fraction of 0.25.

Figure 20 Cavitating vortex captured using the CFD

The helical vortex breakdown in the throat of the nozzle observed during the experimental measurement was also captured using the CFD (figure 21). Again, the contour of 0.25 of vapour void fraction was exploited for the cavitation visualization.

Figure 21 Helical vortex breakdown captured using the CFD

The comparison of the axial velocity and vapour void fraction fields with the images captured using the HS camera is provided in the following set of images (figure 22). The images correspond to the maximum length of the cavitation structures, prior the beginning of its collapse. First of all, it should be emphasized that while the HS images represent 3D instantaneous spatial distribution of the cavitation within the flow, the images of 2D longitudinal crosssection was chosen for the numerical results visualization. As it is clearly depicted, the amount of the gaseous phase is the higher, the lower is cavitation number. Above that, the predicted length of the cavitating structures prior the beginning of their collapse is in good agreement with the experimental observation. Several conclusions can be stated, comparing the water volumetric fraction and field of axial velocity. The region of the significant backflow can be found within the core of the cavitating vortex, which has been also observed on the HS records.

As it is clearly depicted in the set of figures 22, the significant amount of liquid water can be found within the core of cavitating vortex in case of the cavitation numbers of 0.71 and 0.48. While the H-S records do not provide information of the phase distribution across the cross-section of the nozzle, it was possible to observe that the liquid water was sucked in to the cavitating vortex in the cavitation closure region (Figure 23).

CONCLUSIONS

The cavitation within the Venturi tube has been investigated using the experimental and numerical methods with particular regard to effect of the induced swirl on the cavitating flow properties.

The magnitude of the average pipe wall acceleration has been significantly higher in case of the axial inflow. For both of the investigated configurations it was possible to observe steep increase of the pipe wall acceleration magnitude during the fully developed cavitation. In the certain instant, the acceleration magnitude reached the maximum value and then started to decrease for both investigated configurations. The hydraulic loss coefficients have been also analyzed. It was clearly depicted that while the inclination of the hydraulic loss coefficients curve versus cavitation number was nearly the same for both of the configurations, the values were significantly higher in case of the swirl generator presence. These results were subsequently corrected using the numerical methods.

The dynamics of the cavitating structures was investigated exploiting the POD. The dominant frequencies of the most significant dynamic modes were found. As it was clearly depicted, the increase of the dominant frequencies was steeper during the fully developed cavitation compared to the initial stage. Thus, it is possible to assume that the decrease of the cavitation volumetric, as an additional source of compliance within the flow, is faster in this stage.

Relative significance of the most important dynamic modes has been studied using the analysis of the corresponding eigenvalues. Results were in surprisingly good agreement with the results of the acceleration magnitude analysis.

Consequently, the considerable mutual interdependence of the coherent cavitation fluctuations with the magnitude of pipe wall acceleration (i.e. vibrations) can be assumed.

Axial velocity (ms -1 )

The numerical analysis carried out using the OpenFoam 1606+ and its multiphase interPhaseChangeFoam solver provided rich source of additional information. The influence of the computational grid has been discussed using the analysis of the hydraulic loss coefficient, where good agreement with the experimental data was obtained using the finer mesh. Therefore, the finer mesh has been chosen for the further analysis. Using the evaluation of the kinetic energy correction factor at the inlet part of the throat of the nozzle, the experimental results were corrected in case of the swirl generator presence. This correction has been done due to the fact, that the bulk velocity (and consequently kinetic energy) exploited for the ξ and σvel computation is significantly lower compared to the actual velocity magnitude in case of the flow affected by the presence of the swirl.

Using the corrected data, the hydraulic losses as well as acceleration magnitude were analyzed. In case of pipe wall acceleration magnitude, the difference between σvel corresponding to the maximum vibrations based on the test rig configuration was found. This result was confirmed using alternative definition of the cavitation number, where similar discrepancy of the acceleration peaks has been observed. Thus, it is possible to assume, that this approach is correct.

From the qualitative point of view, two major flaws of the CFD simulations were described, absence of the cavitating filament downstream the spike of swirl generator and overestimated cavitation of the boundary layer downstream the throat. Other properties such as length of the cavitating structures and presence of the specific structures were captured correctly. The redistribution of the flow downstream the throat has been described using the CFD.

Overall, it can be stated that the cavitating flow within the Venturi tube is significantly modified when the swirl is introduced. The cavitating patterns are completely different compared to the axial inflow. Dominant frequencies of the axial pulsations induced by cavitation are slightly higher in case of SG presence, while the vibrations are significantly more severe in case of axial inflow. Cavitation of the boundary layer was considerably less significant and the volumetric of the cavitation was surrounded by the liquid water in case of SG presence. Thus, it can be assumed, that the surfaces of nozzle are far less threatened by the occurrence if the cavitation erosion compared to the axial inflow.

Figure 6

 6 Figure 6 Set of images capturing cavitating structures and influence of the induced swirl over the wide range of σ.Region 2 in the Figure5: Transition of the supercavitation to the fully developed regime of cavitation shows the alternation of the coherent liquid jet surrounded by the vapor phase and massive cavitating clouds in case of axial flow. Additionally, cavitating vortex induced by the SG is disturbed from time to time, while the backflow within the core of the structure can be observed prior this collapse. The observed magnitude of the mean pipe-wall acceleration remain low. Region 3 in the Figure5: Fully developed cavitation is typical by periodical pulsations of the cavitating structures, which can reach significant length. The massive cavitating clouds were observed in case of the axial flow. The clouds are transported by the flow to the region of higher pressure, where clouds decay. Cavitating instabilities near the axis of the nozzle can be observed in case of the axial flow. Conical cavitating vortex was observed in case of SG presence. While its length remains relatively constant, the significant presence of small vortices can be seen in the cavitation closure region. Similar to the region 2, the backflow can be observed within the core of the vortex. When the backflow reaches throat of the nozzle the helical type of vortex breakdown occurs. Rapid decrease of the gaseous phase amount led to the decrease of fluid compliance and consequent rapid increase of pipe-wall acceleration up to the maximum value close to the region 4. Region 4 in the Figure5: Cavitation is characterized by periodical creation of the vortex rings in case of the axial flow. The lower is cavitation number, the more massive rings are. In case of the SG presence, the conical cavitating structure can be observed. The length of the structure is the shorter, the higher is the cavitation number. Stable nature of the cavitating structure affected by the swirl led to the more gradual decrease of the pipe-wall acceleration compared to the axial inflow. Region 5 in the Figure5:In case of the axial flow, pulsations of the cavitating boundary layer in the throat of the nozzle have been observed as well as strongly stochastic instabilities of the closure region. With the additional swirl (SG), the straight cavitating vortex filament (Fig.7) has been observed downstream the generator. This filament became conically shaped with the decreasing cavitation number. Periodical helical type of vortex breakdown in throat of the nozzle was observed. Contrary to the previous regions, the pipe-wall acceleration is more significant in case of SG presence. In case of the axial inflow, the cavitating structures became relatively stable, especially in case of the highest investigated values of σvel.

Figure 7

 7 Figure 7 Detail of the straight cavitating vortex within the throat of the nozzle 1.1.1. Proper Orthogonal decomposition Proper Orthogonal decomposition (POD) represents mathematical method of the dynamical phenomena investigation. Data are decomposed to the set of POD modes consisting of spatial coefficient φ and temporal coefficient a.Beside the analysis of the experimentally obtained data, usually velocity field, it is possible to exploit this method for analysis of the video records. Since number of analyzed spatial points was rather high, so-called method of snapshots has been applied[START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF]. POD was utilized by Danlos et al. for study of passive control of cavitation using the surface grooves. Decomposition has been exploited for the detection of the cavitation regimes[START_REF] Danlos | Cavitation regime detection through Proper Orthogonal Decomposition:Dynamics analysis of the sheet cavity on a grooved convergentdivergent nozzle[END_REF]. Another application of the POD can be found in investigation of aerated cavitation dynamics within the Venturi nozzle with rectangular cross-section[START_REF] Tomov | POD study of aerated cavitation in a venturi nozzle[END_REF]. In-depth investigation of the swirl structures downstream the runner of swirl generator designed by team at UPT led by Resiga was done by Štefan et al.[START_REF] Štefan | Proper Orthogonal Decomposition of Self-Induced Instabilities in Decelerated Swirling Flows and Their Mitigation Through Axial Water Injection[END_REF]. Investigation of the swirling flow dynamics downstream the runner of Francis turbine using the POD was done by Rudolf et al.[START_REF] Rudolf | Decomposition of the swirling flow field downstream of Francis turbine runner[END_REF].The results presented in this contribution are successive to the published analyses of the transitions between fully developed and supercavitation regime of cavitation Venturi tube[START_REF] Kozák | Transition of cavitating flow to supercavitation within Venturi nozzle -hysteresis investigation[END_REF]. POD analysis has been carried out considering the constrained region of interest containing the upper half of the

Figure 8

 8 Figure 8 Spatial coefficients of the most significant POD modes red max values, bluemin values

Figure 10

 10 Figure[START_REF] Tomov | POD study of aerated cavitation in a venturi nozzle[END_REF] Relative significance of the most important dynamic POD modes contributions

Figure 11

 11 Figure 11 Comparison of the coarse (CG1) with the refined (CG2) computational gridthroat of the nozzle Comparison of the computational grids size, as well as number of the hexahedral cells within the grids is depicted in the following figure 12.

Figure 12

 12 Figure 12 Computational grids

Figure 13

 13 Figure[START_REF] Hirt | Volume of fluid (VOF) Method for the dynamics of free boundaries[END_REF] Computational domain

Figure 17

 17 Figure 17 Pipe wall acceleration magnitude versus σpres

Figure 18 Figure 19

 1819 Figure18Cavitating vortex filament downstream the SG

Figure 22

 22 Figure 22The images captured using HS camera in comparison with the numerical results.

Figure 23

 23 Figure 23 Detail of the cavitation closure region with velocity vectors colored by the axial velocity magnitude (ms -1 ).

table .

 .

	Sgn. px p2k Q T acc	Transducer pressure pressure fluct. discharge water. temp. vibrations	Type BD Sensors DMP 331 Kistler -211B4 ELA -MQI 99 -SN Rawet -PTP 55 Aura SV129

Table

l/s: SG -uncor. 6 l/s -AX 6l/s: SG -cor.

ACKNOWLEDGMENTS

Czech Science Foundation is gratefully acknowledged for support of this research under project No 16-18316S "Principles and mechanisms causing microorganism elimination by hydrodynamic cavitation ".


Sebastian Schuster

Dieter Brillert

Uwe Martens

Viktor Hermes

Friedrich-Karl Benra
email: friedrich.benra@uni-due.de

Investigation of the evaporation process of liquefied hydrocarbons in front of a compressor

Keywords: Liquid injection, evaporation, droplet size, liquid films, required compressor power

Inlet fogging and wet compression are well known methods to lower the inlet temperature of air compressors of gas turbines and in this way lead to a more efficient compression process. Injection of water into the gas stream is another established method to reduce the compressor stage inlet temperature during the compression of crack gas. The lower inlet temperature of the compressor stage reduces its power demand and additionally, the discharge temperature remains in a suitable range. Applying the method of liquid injection, all injected droplets should be evaporated before entering the compressor impeller to avoid damage at the blades. A good knowledge about the evaporation process of injected liquid in a free stream is necessary to determine the required distance for complete evaporation of all droplets.

The aim of this paper is to provide clear statements on technical possibilities and requirements for an optimized and reliable compression process under liquid injection. The evaporation process of injected hydrocarbon droplets into a gas stream is investigated under realistic boundary conditions. Using conservation laws of mass and energy, a 1D numerical model is derived which allows the calculation of the liquid hydrocarbon evaporation in a free stream. The required fluid properties are taken from the National Institute of Standards and Technology (NIST). A comparison between the 1D model and a 3D Navier-Stokes solution for injected CH 4 droplets into a CH 4 gas stream shows good agreement regarding the decrease of the droplet diameter during the evaporation with respect to the amount of injected mass of liquid fluid and gives a good indication of the requested distance for complete evaporation. In addition, the 1D model predicts the temperature decrease of the gas during evaporation in good accordance to a 3D Navier-Stokes solver. Applying the simple 1D model, it is possible to evaluate the thermodynamic inlet parameters of a compressor for given boundary conditions quickly. In addition, the results allow a statement regarding the complete evaporation of all sizes of injected droplets at the compressor inlet and the formation of liquid films at the walls.

INTRODUCTION

Injection liquid in front of a compressor is a method used to decrease the inlet temperature of the compressor by evaporation and thereby reduce the necessary power demand for the compression process. A large field of various applications is known and executed in several implementations. The most established method is water injection into the suction chamber of a heavy-duty gas turbine (HDGT) to increase power output on hot days, for example; but there are other fields of compression processes which can benefit from the advantages of liquid injection. For example, cracked gas compressors are the most important ones to be listed. All the installed machines have a huge power demand and are worth being investigated with regard to power saving due to liquid injection. In addition, the lower discharge temperature prevents the polymerization of cracked gas. In liquefied natural gas (LNG) supply chains, the costs can be considerably reduced by injecting liquid hydrocarbons into the gas stream.

Injecting water droplets in front of a gas turbine compressor has been the subject of research for several decades. So called wet compression has been applied in several power stations to augment the power output on hot days due to the decreased inlet temperature and an increased mass flow [1 -4]. The investigation of single water droplet evaporation in an environment of ambient air at rest for low pressures and low temperatures is performed by Matz et al. [5]. Experimental results of droplet evaporation at elevated pressure and temperature levels in a moving air flow are published by Schnitzler [6] and Kefalas [7]. Based on their measurements a droplet evaporation model introduced by Abramzon and Sirigano [8] is extended for polydisperse sprays and droplet breakup. In addition, the gas moves with a velocity close to real compressor fluid velocities and the slip between the droplet and the air flow is taken into consideration [9].

Usually, wet compression supersaturates the inlet air and water droplets remain in the flow entering the impeller of the compressor. Also, interstage injection provides a multi-phase flow of humid air and water droplets which must be processed by the compressor. This method has the advantage of continuously cooling the fluid during the compression process by evaporation of the water droplets inside the machine. Thermodynamic analyses of wet compression processes are given by Zheng et al. [10,[START_REF] Zheng | Thermodynamic Performance of Wet Compression and Regenerative (WCR) Gas Turbine[END_REF] and Bianchi et al. [3]. In Deschwanden et al. [START_REF] Deschwanden | Droplet evaporation in the context of interstage injection[END_REF] the interstage injection process is described for a multi-stage axial compressor. It is shown that under application of multiple water injections a drastic decrease of the fluid temperature compared to a dry compression process can be obtained.

The risks of multiple or inter-stage injections include the formation of water films at compressor walls and compressor blade damage due to droplets which hit the blades. The impact of droplets on the blades will likely produce erosion and degrade the blades. Water films can accumulate at certain positions in the machine and can subsequently produce large secondary droplets which have a high potential of destruction. In order to avoid these risks, that a droplet should be evaporated completely before it has the chance to hit a wall of a compressor component. Thus, knowledge about the evaporation process of injected liquid into a flow stream is necessary to determine the required distance for complete evaporation under the given boundary conditions.

Generally, injecting liquid in front or into a compressor gives new opportunities for high efficient compressors. Isothermal compression has multiple advantages; not only reduced required compressor power and decreased fluid temperature, but also a reduced number of compressor stages.. Injecting water into an air stream is regarded as a multi-component system because of the different compositions of liquid and gas. To investigate the evaporation process, a multi-component model must be applied. In case of injecting the same compound as a liquid into the gas stream (for example CH 4 into CH 4 ) a one-component evaporation model can describe the process.

The purpose of the present contribution is to provide a method which gives clear statements on the requirements for a reliable compression process under liquid injection without sucking liquid droplets into the machine. In contrast to wet compression, injected liquid hydrocarbon droplets into a gas stream are the focus of this paper. Since the liquid is the same substance as the gas, a one-component evaporation model should be applied.

In this paper, a simple 1D model which computes the evaporation of liquid CH 4 (Methane) droplets in gaseous CH 4 is presented. The evaporation of a droplet in a straight pipe is investigated under reliable boundary conditions. In addition, an exemplary compression process of a radial compressor impeller for CH 4 is discussed under injection of CH 4 droplets.

EVAPORATION MODEL

The continuum mechanical methods for heat and mass transfer are applied to a single liquid droplet in a gas environment. At the boundary surface between the droplet and the environment the heat and mass flow can be balanced using the established conservation equations. In case of a high gas temperature and a lower liquid temperature, the gas phase transfers heat to the droplet surface. From there, the heat is conveyed into the droplet. In Fig. 1 a droplet in a gas environment is depicted together with the heat and the mass transfer in case of evaporation. Assuming that both heat flows are of the same amount, at first the droplet will heat up continuously. As such, the difference of temperature gradient from the droplet surface to the droplet center becomes smaller and smaller. Hence, the heat transfer from the surface into the droplet decreases. Because of the arising imbalance of energy, a mass flow emerges from the droplets to the gas flow which levels out the energy balance. Determining the balance of the heat and mass flows the temperature at the surface of the droplet has to be known. In a one-component evaporation process the temperature of the gas environment typically is higher than the saturation temperature of the fluid and the droplet temperature is lower than it. At the droplet surface the temperature must be equal to the saturation temperature of the fluid. q g q l ṁ = (q g + q l )/L < 0 Evaporation rate:

ṁ • L Figure 1: Evaporating droplet in a gas environment
The heat transfer can be determined with the following Nusselt-correlations:

( ) = ⋅ ⋅⋅ - g dr g g s g q d Nu T T πλ (1) ( ) =⋅ ⋅⋅ - l dr l l s l q d Nu T T πλ (2)
In case of evaporation, the heat transported away from the droplet surface into the droplet is smaller than the heat transported from the gas to the surface. The difference is compensated by mass transfer using Eq. (3):

lg dr qq dm m dt L + = =  (3)
Empirical data are required for the Nusselt-Numbers in Eqs. (1) and (2). According to Ranz and Marshall [START_REF] Ranz | Evaporation from drops Part 1[END_REF][START_REF] Ranz | Evaporation from drops Part 2[END_REF], the Nusselt-Number for small velocity differences between gas and droplets can be represented as Nu g = 2. For the heat transfer within the droplet, a Nusselt-Number of Nu l = 6 was proposed by Gerber [START_REF] Gerber | Inhomogeneous Multifluid Model for Prediction of Nonequilibrium Phase Transition and Droplet Dynamics[END_REF].

MASS AND ENERGY CONSERVATION

In order to calculate the evaporation rate information must be available on the flow field -in particular on the gas and liquid temperatures. A 1D approach and a 3D approach are used for the investigations. The 3D approach is able to consider friction and temperature gradients orthogonal to the main flow direction.

One dimensional calculation

The conservation of mass along a streamline for an evaporation process can be expressed by Eq. (4):

g dr 0 +=   dr dm dm N dt dt (4)
This expression considers the changing gas mass flow ṁ g and the changing droplet mass m dr multiplied by the number of droplets Ṅ dr .

The temperature of a droplet along a streamline can be calculated using Eq. ( 5):

l = + , dr dr p l l dm dT mc q L dt dt (5)
In an evaporation process the mixing enthalpy is constant along a streamline. At the beginning of the process the mixing enthalpy can be determined from the starting / inlet conditions (Index 0):

0 m dh dx = (6) ( ) 0 0 00 0 0 00 0 0 1 g g ll m g l g l m h mh h w h wh m m + = = - + +     , , , , , , , , (7)
And at any downstream point the enthalpy of the gas phase can be computed as follows:

( ) ( ) ( ) 1 m l ref p l l ref g h wh c T T h w -⋅ + - = - ,, (8)
With an appropriate equation of state and isobaric conditions, the gas temperature can be determined.

The number of droplets is kept constant during the complete evaporation process:

=   l dr dr m N m (9)

3D flow calculation

The 3D flow calculations are done with an extension of ANSYS CFX. The Eulerian-Lagrangian framework is chosen to account for the conservation of mass, momentum and energy of the gas, and the liquid phase.

In terms of evaporation, the main difference is that the gas total enthalpy is conserved and not the mixture enthalpy. For this approach, a source term is required in the energy equation to account for the energy transport with the evaporated mass. The convective heat transfer and the evaporated mass are calculated with the same equations presented in paragraph 1; namely Eqs. (1), (2) and (3).

FLUID PROPERTIES

The employed fluid properties for CH 4 are taken from NIST database [START_REF] Lemmon | Thermophysical Properties of Fluid Systems[END_REF]. In the case of evaporation in a pipe, the pressure is nearly constant along the pipe and fluid properties can be calculated with sufficient accuracy as functions of the temperature only. In order to accelerate the computation speed in the required range of temperature all fluid data are represented by polynomial descriptions as function of the temperature. Compared to the NIST data the polynomials give nearly no deviation to the reported data.

RESULTS

Pipe

In order to test the results for plausibility the computed results for the evaporation process of liquid CH 4 injected into a pipe are presented in this section and compared to each other and to the temperature after complete evaporation in an isobaric and isenthalpic process. The conveyed gas in the pipe is also CH 4 . The velocity of the injected CH 4 droplets is equal to the gas velocity all the time.

Comparison of calculation methods

All calculations in this section are conducted under the assumption that the velocity of the droplets keeps constant along the complete evaporation distance. In addition, it is assumed that the gas volume also stays constant during the evaporation process and thus the gas velocity is constant along the distance.

The computational fluid dynamics (CFD) simulations in this section are conducted under the assumption of no friction at the walls of the pipe in order to allow a comparison to the 1D calculations. Comparing the calculated temperature of an evaporation process of CH 4 in a free gas stream also consisting of CH 4 results of both methods are in an excellent agreement for a wide range of liquid mass flow rates (see Fig. 2). The calculated decrease in temperature due to evaporation shows an excellent agreement for both methods (see Fig. The decrease in temperature as a function of the residence time t in the pipe is shown in Fig. 4 for a certain injected liquid mass and for droplets having a diameter of 20 µm. As can be seen, the temperature decreases continuously from the injection point and reaches a stable temperature after 30 ms. All droplets are evaporated at that point.

Evaporation considering gas volume increase and slip between droplets and gas

Evaporation of the liquid generates a volume of additional gas which tends to increase the gas velocity in a pipe having a constant diameter. Furthermore, the temperature of the gas decreases as another result of the evaporation. Both phenomena have an opposing effect on the gas velocity and are considered in this section. Additionally, in this chapter the CFD simulations are conducted considering friction at the pipe walls and also the slip between the droplets and the gas flow. The resistance force acting on the droplets and vice versa on the gas is determined using the Schiller-Naumann correlation [START_REF] Schiller | Über die grundlegende Berechnung bei der Schwerkraftaufbereitung[END_REF]. Fig. 5 shows the required time1 for complete evaporation as a function of the droplet diameter and the injected liquid mass. It is obvious that the droplet diameter should be as small as possible to achieve a short evaporation distance. In addition, the larger the injected liquid mass the longer the evaporation distance. This result proposes a clear target: small droplet diameters and a precisely adjusted liquid mass flow rate are necessary to achieve short evaporation distances. The results of the 1D model and the CFD simulation are again in good accordance which means that the friction at the pipe walls and the slip between droplets and gas velocity plays a minor role. Increasing the gas velocity at the inlet of the pipe leads to a linear extension of the evaporation distance.

Injection of a high amount of liquid

This section of the paper investigates which consequences will arise when injecting a large amount of liquid into the gas stream. The amount (50% of mass) is much bigger than the amount which can be evaporated under the given boundary conditions. The CFD simulation shows a strong temperature decrease starting at the injection point (see Fig. 6). The temperature is much smaller than the achieved temperature under injection of 12.7% mass only (Fig. 4), but not all droplets evaporate. The possible deposition of droplets is investigated using another 1D model. Based on the existing inlet boundary conditions, no liquid evaporation is considered for this case. This conservative assumption is the worst case and leads to the highest possible deposition rate.

The liquid is injected parallel to the gas flow which prevents droplet deposition by laminar inertia effects. Thus, only effects of diffusion and turbulent fluctuations are investigated. The deposition rate G dep+ can be described using the non-dimensional particle relaxation time τ dr+ and the particle flow density r flow .

⋅ ⋅∆ = = = ⋅⋅ ⋅⋅ ⋅⋅    ( 12
)
The wall shear forces are computed according to Schlichting [START_REF] Schlichting | Grenzschichttheorie[END_REF] as:

To determine the deposition rate experiments about particle deposition in pipes from Sehmel [START_REF] Sehmel | Aerosol deposition from turbulent airstream in vertical conduits[END_REF] as well as Liu and Agarwal [START_REF] Liu | Experimental observation of aerosol deposition in turbulent flow[END_REF] are used. These results are depicted in Fig. 7 together with the linear approximation for the three distinguished areas. [START_REF] Sehmel | Aerosol deposition from turbulent airstream in vertical conduits[END_REF][START_REF] Liu | Experimental observation of aerosol deposition in turbulent flow[END_REF] The thickness of the developed liquid film is determined using an adapted approach from Schuster et al. [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF]. In an iterative procedure, the film thickness is calculated considering a shear force between liquid and gas phase impressed from the gas phase and the velocity of the liquid film. The wall shear force of the liquid is computed under the assumption of a laminar velocity profile.

The results are depicted in Fig. 8 for very small droplets (1 µm) and for droplets of a standard size (20 µm). The figure clearly shows that the thickness of the wall film increases rapidly within the first distance after injection of the big droplets. For the very small droplets the wall film thickness increases to a thickness of only 2 µm. It is likely that the film breaks-up into rivulets having a larger thickness.

Compressor

Injecting liquid in front of a compressor impeller reduces the temperature increase during the change of state. The investigation of droplet evaporation in a compressor impeller can be treated almost the same as in a pipe. A major difference is that rothalpy keeps constant:

2 2 rot m 11 22 H h u w const. = -+= (17)
To determine the mixing enthalpy from Eq. ( 17) information on the circumferential velocity u and on the relative velocity w are required. The computation of these values from velocity triangles is straightforward. The definition of velocity triangles instead of using arbitrary geometry is a comprehensible way to investigate the benefits of evaporation in the impeller. The velocity triangles are defined at the inlet and the outlet, together with a suitable change in relative flow angle between inlet and outlet. In a later step, the impeller geometry can be computed from the velocity triangles and from the gas properties; however, this is beyond the scope of this paper. For the following investigations, common assumptions are used: The flow is swirl-free at the inlet; meridional velocity is constant; flow is strictly radially-oriented at the outlet; the flow angle (β) changes linear between inlet and outlet. Therefore, all requested information about the flow are given and result in the velocity triangles depicted in Fig. 9, together with the meridional view. The shroud contour is only for illustration purposes and must be computed after determining the density along the mean line. As the fluid properties are a function of the temperature and the pressure, the static pressure should be known at the inlet of the impeller. With the given information, the pressure increase along a streamline can be described in an approximate manner by the following equation:

m p vdp y h h r ∆ = = ⋅∆ ≈ ∫ ( 18
)
For a given polytropic efficiency, which is assumed to be constant from impeller inlet to impeller outlet, the pressure can be calculated from Eq. ( 18) for small steps along the radius of the streamline through the impeller. In Figs. 10a and10b, the pressure increase and temperature increase in a radial compressor impeller are depicted for a complete dry (no injection) process and for a process with injection of a small amount of liquid mass (y = 5 %) and small droplets (5 µm).

It´s obvious that during the compression process the outlet pressure slightly increases when injecting liquid due to the higher density of the fluid, but a remarkable outlet temperature decrease occurs in comparison to the complete dry process. This situation provides incentive to inject liquid in front of a compressor which should evaporate during the compression process. The evaporation of the droplets in the compressor impeller allows operators to reach nearly the same pressure ratio but with a much smaller outlet temperature for the same power input. The advantage of liquid injection becomes clearer when the geometry of the compressor impeller is adjusted in order to reach the same outlet pressure compared to the dry compression. Reducing the outlet diameter of the impeller and keeping the impeller outlet flow angle at the same magnitude (β 2 = 90°) allows to calculate the requested compressor power from the Euler equation multiplied by the overall mass flow (gas + evaporated liquid + liquid).

In Fig. 11, a comparison of the power requirement of a radial compressor for injection of different amounts of liquid is plotted versus the droplet diameter. Under the assumption of complete evaporation at the impeller outlet, the biggest power savings can be realized with the highest amount of injected liquid (y = 15 %) and with the smallest droplet diameter (5 µm). This result is not surprising but clearly shows the trend: the smaller the droplets are, the greater the power savings and the more liquid can be injected.

The reduction of required compressor power approaches zero for injected liquid droplets larger than 50 μm independent of the injected amount, because as shown before in Fig. 5 the droplets will not evaporate within the available distance.

CONCLUSIONS

This paper provides an insight into the injection of liquid hydrocarbons into a gas stream of the same composition. To evaluate the evaporation of the injected droplets, two different calculation methods are applied. Under the given assumptions the results of the 1D method are comparable to the results of the CFD simulations. Using the easy-to-handle 1D method which gives results in a short response time it is possible to quickly estimate the required distance for complete evaporation of the droplets. In addition, for a defined evaporation time the parameters of the applied injection nozzles can be specified in order to get droplets as small as required.

If it is indispensable to evaporate all injected droplets before they reach the impeller of the compressor, the temperature decrease of the gas and the mass flow gained are the benefits of the liquid injection. But if it´s permissble for the compressor to suck in certain size droplets, the evaporation during the compression process will be another benefit which reduces the outlet temperature of the gas and the power requirement of the compressor. Droplet diameters below 10 μm are necessary to gain a remarkable benefit of evaporation in the impeller.

The results presented in this paper show the feasibility of the technology and justify further investigation. The next steps of research should be the investigation of the impact of droplet/droplet interaction and the interaction of droplets with the surrounding walls on the evaporation process. In this context the behavior of liquid films and the generation of secondary droplets should be a focus. Another point of investigation is the definition of a droplet diameter spectrum because no available nozzles can produce a mono dispersed spray.

NOMENCLATURE

Figure 2 :Figure 3 :

 23 Figure 2: Decrease in temperature after complete evaporation of CH 4 droplets in a CH 4 gas stream In Fig. 3 the necessary evaporation time is depicted versus the injected liquid mass (3a) and versus the droplet diameter (3b).

Figure 4 :

 4 Figure 4: Temperature decrease along the pipe

Figure 5 :

 5 Figure 5: Evaporation time as a function of droplet diameter and injected liquid mass

Figure 6 :

 6 Figure 6: Trend of decrease in gas temperature at a high amount of injected liquid. Computed with CFD

Figure 7 :

 7 Figure 7: Measured deposition rates in pipes[START_REF] Sehmel | Aerosol deposition from turbulent airstream in vertical conduits[END_REF][START_REF] Liu | Experimental observation of aerosol deposition in turbulent flow[END_REF]

Figure 8 :

 8 Figure 8: Thickness of liquid film at pipe wall as a function of the through-flow time

Figure 9 :

 9 Figure 9: Meridional view with the mean streamline (red) and velocity triangles of a radial impeller as an example

Figure 10 :

 10 Figure10: Course of pressure ratio and temperature increase along the radial coordinate in a radial impeller under liquid evaporation

Figure: 11

 11 Figure: 11 Possible reduction of required compressor power due to liquid injection

The time is calculated from the evaporation distance and the velocity at the inlet

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial support and the permission for publication of the results for this project from Siemens AG, Dresser-Rand business, Duisburg, Germany.

Abbreviations


Toru Shigemitsu

Katsuhiko Hirosawa

Keisuke Miyazaki

Hiroaki Fukuda

Performance and Internal Flow of Contra-Rotating Small-Sized Cooling Fan

Keywords: Cooling fan, Small-sized axial fan, Contra-rotating rotors, Performance, Internal flow

High pressure and large flow rate small-sized cooling fans are used for servers in data centers and there is a strong demand to increase its performance because of increase of quantity of heat from servers. Therefore, high rotational speed design is conducted, and rotational speed over 10,000min -1 is employed for cooling fans of servers. Contra-rotating rotors have been adopted for some of high pressure and large flow rate cooling fans to meet the demand. The company's research and development period for the contra-rotating small-sized cooling fan is short and its internal flow condition is not clarified well. Therefore, the internal flow condition was investigated by the numerical analysis.

In the present paper, fan static pressure curves of the contra-rotating small-sized cooling fan with a 40mm square casing are shown by the experimental. Furthermore, the influences of the geometrical shape and design specification of the contra-rotating small-sized cooling fan on the internal flow condition are clarified by the numerical analysis.

INTRODUCTION

Data center has been built because of spread of cloud computing, establishment of ubiquitous networking society and increase of electric parts in machines. Then, power consumption in data centers, IT devices and machines has been increasing significantly [START_REF] Hata | Energy Saving Service for Data Centers[END_REF] . Electrical power used for cooling of IT devices for data centers is huge the same as that used for IT devices itself in data centers and the electric power consumption of it is growing rapidly. In the point of view of global warming and energy savings, there is a strong demand for reduction of power consumption in above facilities and equipments [START_REF] Miyahara | Fan Cooling Technology for Small Electronic Device[END_REF] . Axial fans are used as cooling fans of servers and desk top computers. Then, researches of axial fans on fan noise and effect of inlet geometry were conducted [START_REF] Ito | Experimental Research for Performance and Noise of Small Axial Fan[END_REF], [START_REF] Liu | Effect of Inlet Geometry on Fan Performance and Inlet Flow Fields in a Semi-opened Axial Fan[END_REF] . High pressure and large flow rate small-sized cooling fans are used for servers in data centers and there is a strong demand to increase its pressure and flow rate because of increase of quantity of heat from servers. The increase of the pressure and flow rate by increase of a fan diameter is restricted because of limitation of space. Therefore, high rotational speed design is conducted, and rotational speed over 10,000min -1 is employed for cooling fans of servers. Contra-rotating rotors have been adopted for some of high pressure and large flow rate cooling fans to meet its demand. On the other hand, low rotational speed design [START_REF] Furukawa | Performance Test and Flow Measurement of Contra-Rotating Axial Flow Pump[END_REF] and advantages on the performance of contrarotating fans and pumps were verified by experimental results [START_REF] Furukawa | Experimental Study of Pump Characteristics of Contra-Rotating Axial Flow Pump[END_REF]- [START_REF] Nouri | Design and Experimental Validation of a Ducted Counter-Rotating Axial-Flow Fans System[END_REF] . The company's research and development period for the contra-rotating small-sized cooling fan is short and its internal flow condition is not clarified well. Therefore, the performance of each front and rear rotor and internal flow condition were investigated by the numerical analysis with the simple numerical model without the spokes [START_REF] Shigemitsu | Performance Characteristics and Flow Conditions of High Pressure and Large Flow Rate Cooling Fan[END_REF], [START_REF] Shigemitsu | Internal Flow Condition of High Power Contra-Rotating Small-Sized Axial Fan[END_REF] . In the case of the contra-rotating rotors, it is necessary to design the rear rotor considering the unsteady circumferential velocity distributions at the outlet of the front rotor [START_REF] Shigemitsu | Experimental Study on Rear Rotor Design in Contra-Rotating Axial Flow Pump[END_REF] . Further the fan noise becomes larger than the conventional rotor stator type fan because of the interaction between the front and rear rotors. Then, the passive noise reductions with the peforated blade was propsed for the contra-rotating fan [START_REF] Wang | Passive Noise Reduction for a Contrarotating Fan[END_REF] . It is important to clarify the influence of the wake from the front rotor to the rear rotor and the potential interaction between the front and rear rotors to increase the performance and to reduce the fan noise [START_REF] Sanders | Multi-Blade Row Interactions in a Transonic Axial Compressor: Part I -Stator Particle Image Velocimetry (PIV) Investigation[END_REF] . The blade row distance between front and rear rotors is a key parameter to consider the wake and potential interaction for the contra-rotating fan. The influence of the blade row distance between the front and rear rotors and pressure fluctuation on the casing wall were investigated for the counter rotating fan with fan diameter D=375mm [START_REF] Nouri | Design and Experimental Validation of a Ducted Counter-Rotating Axial-Flow Fans System[END_REF], [START_REF] Nouri | Experimental Study of the Instationary Flow Between Two Ducted Counter-Rotating Rotors[END_REF] . On the other hand, the conventional design method and the theory for the turbo machinery should be modified for smallsized axial fans because small-sized axial fans applied to electrical devices belong to extremely small size field in the turbo machinery [START_REF] Shigemitsu | Performance and Flow Condition of Small-Sized Axial Fan and Adoption of Contra-Rotating Rotors[END_REF] . Therefore, there is the strong demand to establish the design method for small-sized axial fans based on the internal flow between the front and rear rotors. Furthermore, the performance characteristics of each front and rear rotor and internal flow condition of the contra-rotating small-sized cooling fan having the spokes are not clarified well. Therefore, a design method of the contra-rotating small-sized cooling fan is not established.

In the present paper, the performance characteristics of each front and rear rotor of contra-rotating small-sized axial fan with 40mm square casing are shown by the experimental and numerical results. Then, the internal flow condition of the contra-rotating small-sized cooling fan having spokes are discussed based on the numerical analysis results.

EXPERIMENTAL APPARATUS AND METHOD

A picture and primary dimensions of the high pressure and large flow rate contra-rotating small-sized cooling fan(R40W-A) are shown in Fig. 1 and Table 1 respectively. The rotors of R40W-A are set in a 40mm square casing, and the hub tip ratio of the front and rear rotors are Dhf/Dtf=25mm/37.2mm=0.67 and Dhr/Dtr=26mm/37.2mm =0.70 respectively. Operating flow rate is Qo=0.55m 3 /min and a tip clearance is c=0.6mm. Each rotational speed of the front and rear rotors of R40W-A is extremely high as Nf=15000min -1 and Nr=14000min -1 . Figure 2 shows a schematic diagram of an experimental apparatus. The experimental apparatus was designed based on the Japanese Industrial Standard and air blown in a test section passes the rotors, a chamber, a measurement duct and a booster fan and blow out in the ambient atmosphere. Each rotor is driven by a brushless motor set inside of the hub and the motor is supported by spokes connected to the casing. The rotational speed of each rotor was kept constant (Nf=15000min -1 , Nr=14000min -1 ) by PWM control using a function generator when a performance test was conducted. The fan static pressure (∆P) is measured by a pressure difference between static holes downstream of the rotor installed at the chamber and ambient air. Further, the rotational speed was evaluated using a pulse of the motor measured by an oscilloscope. Flow rates were measured by an orifice meter set at the measurement duct and the pressure curve from a cutoff flow rate to a large flow rate was investigated in the experiment. There are three spokes downstream of the front and the rear rotors to support the motors and the casing has curved corners at the inlet and outlet of it and circular diffuser and nozzle between the front and rear rotor as can be confirmed by a sectional view of R40W-A in Fig. 3. Therefore, there are some geometrical restrictions for the high pressure and large flow rate smallsized cooling fan.

NUMERICAL ANALYSIS CONDITIONS

Commercial software ANSYS-CFX16.2 was used to investigate the flow condition which couldn't be measured by the experiment. Three dimensional unsteady numerical analysis was conducted because the complicated structure existed in this cooling fan. Figure 4 shows the numerical model. In the numerical model, a simplification of the geometry was conducted by extension of the length of the front rotor hub to the rear rotor inlet as shown in Fig. 4 in order to reduce the computational cost. Numerical grids used for the numerical analysis are shown in Fig. 5. The numerical domains comprise inlet, front rotor, spoke, rear rotor, chamber and outlet duct regions. The numerical grid points are 617,090 for the inlet region, 1,479,336 for the chamber region and 237,628 for the outlet duct region respectively. The numerical grid points are 3,924,468, 362,848 and 3,107,155 for the front rotor, spoke and rear rotor regions respectively. The tip clearance was kept 0.6mm as the same with the experimental apparatus in the numerical analysis. At an inlet boundary, constant flow rate was given and constant pressure was given as an outlet boundary condition. The coupling between the front and rear rotors was accomplished by the transient rotor stator. The scalable wall function was used as a near wall treatment and the LRR Reynolds Stress model, which provides high accuracy for some complex flows, was used as a turbulence model. The unsteady numerical flow analysis was conducted at the operating flow rate Qo=0.55m 3 /min and 4 other flow rate points 0.4Qo, 0.7Qo, 1.3Qo and 1.6Qo. A time step number per one rotor rotation was 140 and the time step was t=2.857×10 -5 s. The data of one rotor rotation were obtained after 10 rotor rotations in the unsteady numerical analysis and the computational time for 10 rotor rotations at each flow rate was about 1 week. The fan static pressure for the numerical analysis was calculated by the static pressure difference between the static holes positions installed at the chamber and the inlet boundary of the numerical domain, which was almost the same with the experiment. The air was assumed incompressive fluid, so the density of the air was constant in this research. The static pressure efficiency was evaluated by a ratio of a multiplication of the static pressure and the flow rate to the shaft power obtained by the numerical analysis.

3.

EXPERIMENTAL AND NUMERICAL ANALYSIS RESULTS

Performance curve of R40W-A

Figure 6 shows the fan static pressure curves of the test fan at the design rotational speed Nf/Nr=15000min - 1 /14000min -1 obtained by the experiment. Numerical analysis results are also shown in Fig. 6. A horizontal axis is the flow rate Q and a vertical axis is the fan static pressure ∆Ps. The fan static pressure of each front and rear rotor obtained by the numerical analysis, in which the static pressure difference in the spoke region is excluded, is also shown in Fig. 6. The test fan showed the negative slope of the pressure curve from the operating flow rate Qo=0.55 m 3 /min to the maximum flow rate Q=1.08 m 3 /min and the maximum fan static pressure ∆Ps=510Pa was obtained at the operating flow rate Qo=0.55 m 3 /min in the experiment. The maximum fan static pressure of the numerical analysis was also obtained at the operating flow rate Qo=0.55 m 3 /min and its value was ∆Ps=500Pa, which was within the 2% accuracy of the experiment. A positive slope of the pressure curve appeared from the partial flow rate Q=0.50 m 3 /min to the operating flow rate Qo=0.55 m 3 /min. After that, the pressure curve became flat in partial flow rates region Q=0.35-0.50 m 3 /min. Then, the positive slope of the pressure curve appeared again in partial flow rates region Q=0.25-0.35 m 3 /min. It was confirmed that the operating conditions of the test fan were in severe condition as the fan static pressure was set extremely high and the design flow rate existed near the flow rate range with the positive slope of the fan static pressure curve. The whole of the numerical data accorded with the experimental data and the trend of the fan static pressure curve of the experiment could be captured well by the numerical analysis quantitatively. Then, the performance of each front and rear rotor, which could not be obtained by the experimental method, was investigated by the numerical analysis. The fan static pressure of the front rotor was extremely lower than that of the rear rotor and the fan static pressure decreased according the increase of the flow rate. On the other hand, the fan static pressure of the rear rotor was high and about 77% of the total fan static pressure was obtained by the rear rotor at the operating flow rate Qo=0.55m 3 /min -1 . The fan static pressure of the rear , where the maximum total fan static pressure obtained. The static pressure efficiency of each front rotor, rear rotor and its total including static pressure difference in spoke region are shown in Fig. 7. The static pressure efficiency was obtained by the unsteady numerical analysis. A horizontal axis is the flow rate Q and a vertical axis is the static pressure efficiency ηs. The maximum total fan static pressure obtained at the operating flow rate Qo=0.55m 3 /min -1 was low as ηs =46.7% because of the design specification of high pressure and large flow rate cooling fan with the 40mm square casing. The fan static pressure of the rear rotor was higher than that of the front rotor in all flow rates, where numerical analysis conducted and the difference of the static pressure efficiency between the front and rear rotors became large with the increase of the flow rate. It was found from the performance curve of the test fan that the static pressure increased significantly in the rear rotor and its tendency became large with the increase of the flow rate. Therefore, the internal flow condition of each front and rear rotor was investigated using the numerical analysis results to clarify the difference of each front and rear rotor performance.

Internal flow condition of R40W-A at operating flow rate Qo

The internal flow conditions at the operating flow rate Qo=0.55m 3 /min were investigated by the numerical analysis results. Blade-to-blade relative velocity vectors and static pressure distribution at each radial positions r/rc=0.74, r/rc=0.85 and r/rc=0.96 are shown in Fig. 8 and Fig. 9. The flow rate is the operating flow rate Qo. r and rc means radius, where the data obtained, and inner radius of the casing. It was observed from the internal flow near the hub in Fig. 8(a) and Fig. 9(a) that the stagnation point at the leading edge of the front and rear rotors existed near the suction surface of the blade at the operating flow rate Qo=0.55m 3 /min and the small separation occurred on the suction surface of the rear rotor. On the other hand, the relative velocity of the suction surface of the rear rotor was high because of the circumferential velocity at the outlet of the front rotor and decreased suddenly at the outlet of the rear rotor. Therefore, the static pressure increased significantly in the rear rotor, although the static pressure increase of the front rotor was small. These flow condition was similar to those at other radial positions in Fig. 8(b),(c) and Fig. 9(b),(c) and corresponded to the result that the fan static pressure of the rear rotor was large compared to the front rotor in Fig. 6. The leakage flow from the pressure surface to the suction surface near the mid of the blade chord were confirmed for both front and rear rotors at shroud region in Fig. 8(c). The meridional velocity vectors and static pressure on the vertical plane at the operating flow rate Q0=0.55m 3 /min are shown in Fig. 10 and Fig. 11. The rotational direction of the front rotor is front side of the paper and that of the rear rotor is back side of the paper. The vortex occurred at the inlet corner curve and this vortex could be caused by the separation of the main flow around the inlet corner. In general, the corner separation occurred at the inlet of the casing and this separation vortex influenced on the inlet flow condition. In this test fan, this vortex was basically got stuck in the inlet corner, so the flow condition near the inlet shroud wasn't influenced by this vortex. The leakage flow from the blade tip was observed for both front and rear rotors in Fig. 10 and these low velocity regions related to the leakage flow was observed in wide region near the shroud on the meridional plane. There is the circular diffuser and nozzle region on the shroud between the front and rear rotors to improve the performance and noise.

The back flow occurred in the diffuser region and the static pressure was high in this region. The low velocity region spread radially inner section between the front and rear rotors by the influence of the back flow region. Then, the main flow inclined to radial mid and hub region between the front and rear rotors. The back flow region disappeared in the circular nozzle region with the increase of the velocity and uniform flow was achieved at the inlet of the rear rotor. The static pressure increased significantly in the rear rotor and its distribution was uniform in radial direction as could be seen in Fig. 11.

CONCLUDING REMARKS

The performance characteristics of the high pressure and large flow rate small-sized contra-rotating axial flow fan were investigated by the experiment and the numerical analysis. Then, the internal flow conditions at the operating flow rate were clarified by the numerical analysis results. As a result, following concluding remarks were obtained.

1. The fan static pressure of the front rotor was extremely lower than that of the rear rotor and the fan static pressure of the front rotor decreased according the increase of the flow rate. On the other hand, the fan static pressure of the rear rotor was high and about 77% of the total fan static pressure was obtained by the rear rotor at the operating flow rate Qo=0.55m 3 /min -1 .

2. The maximum total fan static pressure obtained at the operating flow rate Qo=0.55m 3 /min -1 was low as ηs =46.7% because of the design specification of high pressure and large flow rate cooling fan with the 40mm square casing. The fan static pressure of the rear rotor was higher than that of the front rotor in all flow rates.

3. The stagnation point at the leading edge of the front and rear rotors existed near the suction surface of the blade at the operating flow rate Qo=0.55m 3 /min. On the other hand, the relative velocity of the suction surface of the rear rotor was high because of the circumferential velocity at the outlet of the front rotor and decreased suddenly at the outlet of the rear rotor. Therefore, the static pressure increased significantly in the rear rotor, although the static pressure increase of the front rotor was small. 4. In general, the corner separation occurred at the inlet of the casing and this separation vortex influenced on the inlet flow condition. In this test fan, this vortex was basically got stuck in the inlet corner, so the flow condition near the inlet shroud wasn't influenced by this vortex. 5. There is the circular diffuser and nozzle region on the shroud between the front and rear rotors. The back flow occurred in the diffuser region and the static pressure was high in this region. The low velocity region spread radially inner section between the front and rear rotors by the influence of the back flow region. Then, the main flow inclined to radial mid and hub region between the front and rear rotors. The back flow region disappeared in the circular nozzle region with the increase of the velocity and uniform flow was achieved at the inlet of the rear rotor. The static pressure increased significantly in the rear rotor and its distribution was uniform in radial direction.

Figure. 1 Figure. 3

 13 Figure.1 Picture of R40W-A

Fig. 2

 2 Fig.2 Schematic diagram of experimental apparatus

Figure. 4 Figure. 6

 46 Figure.4 Sectional view of numerical model

FigureFigure. 8

 8 Figure.7 Static pressure efficiency

  Figure.9 Static pressure distribution at Qo (c) r/rc=0.96

Figure. 10

 10 Figure.10 Meridional velocity vectors at Qo

Table . 1

 . Primary dimensions of rotors

			Hub	Mid Tip
		Diameter[mm]	25	31.1 37.2
		Blade Number		7
	Front Rotor	Solidity	1.61	1.4 1.22
		Stagger Angle	18.2° 31.2° 37.2°
		Blade Number		5
	Rear Rotor	Solidity	1.03 1.17 1.1
		Stagger Angle	40.6°	40° 45.5°

ACKNOWLEDGMENTS

The authors wish to show our special thanks to the supports by Nidec Corporation, the project research aid from Tokushima University, Japan Science and Technology Agency and Komiya research aid.