Levent Kavurmacioglu
email: kavurmacio@itu.edu.tr

Hidir Maral

Cem Berk Senel

A Parametric Approach to Turbine Tip Leakage Aerodynamic Investigation for Axial Flow Turbine

Keywords:

This current paper deals with the development of a computational system dealing with parametric assessment of aerodynamic losses near the tip region of a high pressure turbine blade. Performance of an axial flow turbine is strongly related to the flow structure in the tip gap between blade tip and casing. Flow structure in the tip gap is a significant source of inefficiency. Special emphasis is paid to developing a sufficiently accurate 3D RANS based loss estimation system for the optimization of tip section geometry. An HP turbine tip section will be optimized for minimum aerodynamic losses and also for minimized heat loss. The present study deals with a preliminary study of effective parametric grid generation and turbulent flow model implementation and assessment under realistically simulated turbine flow conditions. Initial development of this computational model is performed in a linear turbine cascade arrangement.

Numerical experiments with parametrically generated multizone structured grid topologies and unstructured grids pave the way for the 3D optimization of the HP turbine blade tip region. The future efforts will include modified squealer tips, tip trenches and tip carving investigations in an effort to obtain optimal 3D tip shapes as far as aerodynamic losses and heat losses are concerned.

INTRODUCTION

In turbomachinery field to allow relative motion of blades and to protect the blade surface which is exposed to hot gas stream a gap is required. Performance of turbomachine is strongly related to the flow structure within this gap and flow in this gap is 3-D and highly complex. A pressure driven flow between pressure side and suction side of the blade throughout the gap results in approximately one-third of the aerodynamic loss in rotor of axial gas turbine [START_REF] Mischo | Flow physics and profiling of recessed blade tips: Impact on performance and heat load[END_REF]. The leakage flow generally rolls into vortical structure near the suction side of the blade. Leakage flow, tip vortex and interaction of these flow structures with secondary flow, endwall boundary layer and main stream flow reduce the aerodynamic performance of the turbomachine. Besides, a reduction in work output of the turbine is observed since the flow passes over the blade tip is not turned as the main flow [START_REF] Mischo | Flow physics and profiling of recessed blade tips: Impact on performance and heat load[END_REF][START_REF] Heyes | The effect of blade tip geometry on the tip leakage flow in axial turbine cascades[END_REF][START_REF] Krishnababu | Aerothermal investigations of tip leakage flow in axial flow turbines-Part I: effect of tip geometry and tip clearance gap[END_REF]. Leakage flow is also a significant source of higher thermal loads at blade tip [START_REF] Key | Comparison of turbine tip leakage flow for flat tip and squealer tip geometries at highspeed conditions[END_REF]. Many studies have been carried out in order to clarify structure of leakage flow and reduce the effects of it. Unshrouded turbines have been studied intensively compared to shrouded turbines since shrouded turbines bring difficulties about both mechanical and thermal loadings. Moore and Tilton investigated the leakage flow analytically and experimentally by assuming the leakage flow as a flow through an orifice [START_REF] Moore | Tip leakage flow in a linear turbine cascade[END_REF]. Morphis and Bindon investigated the effects of blade tip geometry and radius of pressure side corner on aerodynamic loss [START_REF] Bindon | Development of axial turbine leakage loss for two profiled tip geometries using linear cascade data[END_REF]. Heyes et all obtained that squealer tip geometries could reduce the leakage flow as tip gap is blocked by separation bubble [START_REF] Heyes | The effect of blade tip geometry on the tip leakage flow in axial turbine cascades[END_REF]. Experiments by Yaras and Slojander showed that leakage flow rate was decreased by the effect of moving casing that modeled the relative motion between blades and casing [START_REF] Yaras | Effects of simulated rotation on tip leakage in a planar cascade of turbine blades: Part I-tip gap flow[END_REF]. Secondary flow near the endwall of planar cascade was visualized using multiple smoke wires by Wang et al. [START_REF] Wang | Flow visualization in a linear cascade of high performance turbine blades[END_REF].

Passive control methods are the most common approach in order to reduce the loss related leakage flow. Squealer, partial squealer and winglet blade tip geometries are widely used. Mischo et al. showed that performance of an axial flow turbine could be improved by confining the leakage flow within the cavity of a squealer tip geometry [START_REF] Mischo | Flow physics and profiling of recessed blade tips: Impact on performance and heat load[END_REF]. Heyes et al. observed that application of squealer geometry could reduce the tip leakage losses [START_REF] Heyes | The effect of blade tip geometry on the tip leakage flow in axial turbine cascades[END_REF]. In a numerical study on the effect of squealer tip geometry on heat transfer and efficiency, Ameri et al. obtained that leakage flow rate was reduced whereas heat transfer to the blade tip was increased [START_REF] Ameri | Effect of squealer tip on rotor heat transfer and efficiency[END_REF]. Camci and Dey found out that using tip platform extensions could affect the aerodynamic performance by weakening the tip vortex [START_REF] Camci | Aerodynamic tip desensitization of an axial turbine rotor using tip platform extensions[END_REF]. Measurements by Azad et al. indicated that implementation of suction side squealer had better results with respect to pressure side squealer [START_REF] Azad | Effect of squealer geometry arrangement on a gas turbine blade tip heat transfer[END_REF]. An experimental study on heat transfer was carried out by Kwak et al. to determine the effect of the position of squealer [START_REF] Kwak | Heat transfer coefficients on the squealer tip and near-tip regions of a gas turbine blade with single or double squealer[END_REF]. Considerable reduction in heat transfer was achieved in a recent study that modeled the tip gap as convergent-divergent channel by Zhang and Le [START_REF] Zhang | Tip shaping for HP turbine blade aero-thermal performance management[END_REF].

There have been less study about tip carving in the In this paper, flow structure of leakage flow is tried to be investigated in a comprehensive way. A parametric study on the effect of tip gap height is going to be carried out numerically as seen in Figure 1. Aerodynamic loss related to the leakage flow will be determined for the tip clearance values of 0.7 %, 1.0 %, and 1.5 %. This parametric approach will pave the way for passive control methods and optimization process for tip shaping. Axial turbine blade tip profile used in calculations belongs to Pennsylvania State University Axial Flow Turbine Research Facility (AFTRF) [START_REF] Zaccaria | An experimental investigation into the steady and unsteady flow field in an axial flow turbine[END_REF].

METHODS

A numerical study has been carried out in this study. In the field of turbomachinery experimental measurements may become difficult, expensive and time consuming. In such circumstances Computational Fluid Dynamics (CFD) method takes an important role to understand the flow structure.

Parametric assessment of aerodynamic loss corresponds to the most important part of this study. Grid generation process has been carried out via ANSYS Workbench. Computational domain which comprised multiple blocks was obtained using SpaceClaim module in ANSYS. As seen from Figure 2, computational domain was modeled as a linear cascade arrangement for a single blade. Dimensions to obtain the flow field were defined parametrically on the interface of ANSYS Workbench. Computational domain was divided into multi blocks in order to provide a parametric definition and achieve a fully hexagonal grid in a simple way. Creating multi blocked flow domain enables to use multizone method in ANSYS Meshing module. Each of inlet and outlet blocks involve one block whereas rotor domain is divided into three blocks in this study. Rotor consists of basically two blocks; rotorblade domain and gap domain as seen from Figure 2. Then gap domain is separated into 2 blocks to be able to control the mesh sizes to achieve the required y + values. Once the computational domain is formed, geometry is transferred to meshing module on ANSYS Workbench. ANSYS Meshing module was used to generate a structured mesh in a parametric assessment using multizone method. Basic variables to for mesh generation can also be introduced as parameters. Similar to procedure in solid modelling, grid sizing functions were defined parametrically. Number of division in the tip gap was introduced as a parameter different from all other definitions in meshing process. Application of parametric approach in grid generation enabled an efficient working environment in CFD calculations.

Afterwards, the mesh file was transferred to CFX-Pre for modeling and then CFX Solver for the simulation. Mass flow rate and static pressure boundary conditions have been imposed at inlet and exit respectively. Turbulence intensity and length scale at inlet section were specified as 0.5% and 0.123 m. Boundary conditions represent the realistic conditions of AFTRF. A comparison to experimental results are not available since AFTRF is a rotational test set-up. Also, blade tip geometry has been used, not solid model of the blade itself.

Considering the number of cases for CFD methods, solid modelling and grid generation becomes significant productivity issues. Parametric study removes the necessity of modeling of the computational steps at each time. Grid generation corresponds to the most problematic step in numerical calculations. It may take a long time for a single model. Within this study we experienced that parametric approach reduced time at each step for numeric calculation. Parametric assessment reduced the required time for a complete numerical solution to one-sixth excluding simulation time.

Parametric definition is illustrated in Figure 4. Basic parameters for current numerical study are tip clearance ratio in terms of blade height and number of division in tip gap. The rest of the parameters given under geometry part have been defined depending on the clearance ratio. Basic difference among the cases investigated within the scope of this effort is tip gap height. Tip clearance is defined in term of blade height; ratio of tip gap height to blade height. Three different tip clearance cases 0.7% (FLAT0007) 1.0% (FLAT0010), and 1.5% (FLAT0015), are going to be investigated.

Numerical Method

All simulations in present study were carried out using commercial software ANSYS as pointed out previously. SpaceClaim module was used for modeling of computational domain, meshing module for grid generation and CFX for the numerical calculations. Computations were performed for a single blade with periodic boundary conditions in pitchwise direction.

Blade tip geometry was used to create solid model of the axial turbine. Profile at blade tip was extruded to obtain a cylindrical blade. Specifications of the blade geometry was given in table 1. Length of inlet domain is equal to 3 axial chord length whereas it is equal to 6 axial chord for outlet domain in order to reduce gradients in streamwise direction. Reynolds number based on inlet velocity and axial chord length is calculated as 143500. Multizone method was used for grid generation. Multizone, which is a type of blocking approach similar to ICEM CFD, uses automated topology decomposition and generates structured hexa mesh where blocking topology is available [START_REF]ANSYS Meshi-Intro_16.0_L04_Meshing Methods[END_REF]. Fully hexagonal elements have been used in calculations in order to reduce solution time and increase the accuracy. In order to resolve the boundary layer effects inflation (boundary layer mesh) was applied. Grid around rotor domain is seen in Figure 1.

The 3D incompressible RANS equations were solved using a finite volume discretization with the assumption of steady state flow at 25°C.

A two equation turbulence model was used in the calculations. Shear Stress Transport (SST) turbulence model was performed. SST model is a combination of standard (k-ε) and (k-ω) models to overcome the shortcomings of each model by a blending function depending on the distance away from the wall [START_REF]ANSYS CFX Intro_16_L10 Turbulence[END_REF]. For using SST model y+ value should be less than 2, in some resources 1. In present study for each tip clearance heights this condition was satisfied; y+<2. Number of elements in the cases are approximately 14 million. This high numbers are due to number of elements in tip gap region to resolve the flow structures accurately and to reduce aspect ratio to an acceptable level. Initial height of the first cell away from the wall should be as small as to keep y+ values around 1, which results in higher aspect ratios.

RESULTS AND DISCUSSION 2.1 Results and Discussion

In this section numerical results are going to be given in details with flow visualizations in different sections of the computational domain.

In numerical calculations grid dependency study is an important issue to deal with. More importantly, y + is a crucial measure for boundary layer. Firstly, a grid dependency study has been performed including coarse, medium and fine cases. Number of elements at each case and number of division in tip gap are given in Table 2. Tip gap height has been divided into 30, 35, and 40 respectively for COARSE, MEDIUM and FINE cases.

Table 2: Mesh specifications for grid dependency

Distribution of static pressure coefficient has been examined for grid dependency study. Figure 5 shows static pressure coefficient, CP, distribution near blade tip. CP is defined as Pref is the area averaged pressure at inlet plane, and Um reference velocity. Reference velocity corresponds to exact value of test rig. Figure 5 indicates that there are small differences between three cases whereas MEDIUM and FINE CP curves are closer to each other compared to COARSE case. FINE and MEDIUM cases differ slightly from each other. Considering the highly complex flow structure in tip gap, blade tip becomes a critical region for mesh dependency study. Thus, near blade tip region has been used instead of blade tip itself.

Figure 5: Mesh dependency study for FLAT0010

In numerical studies y + is considered as an important issue. Height of first cell from wall in non-dimensional form is defined as y + . In order to resolve the flow within the boundary layer, it should be kept as small as. Flow through the tip gap is highly complex and turbulent, thus order of y + becomes very important to investigate the flow field. Also, for turbulence modeling, order of y + is a necessary condition. In present study order of y + is plotted at two different spanwise terminals including midpspan and near blade tip. Figure 6 and Figure 7 indicate that y + condition is satisfied. Maximum y + value is 0.27 and average 0.15 for all cases at midspan. Besides, y + values around the blade profile is much smaller than 2. Blade tip region is a critical region due to high shear effects to ensure recommended y + level. As seen from Figure 6, maximum value is around 1.3 which means y + is kept at reasonable level. Average value for y + is around 0.5 for all cases. Less than 1.3 is achieved at lower spanwise stations. For flow visualization streamlines are widely used to provide a better understanding. Streamlines released upstream of the blade leading edge in the vicinity of tip clearance visualize the formation of tip leakage vortex. Figure 9 depicts how leakage flow rolls into tip vortex near suction side of the blade. Pressure driven flow passes over the blade tip tends to evolve into a vortical structure. Formation of tip vortex verifies that flow becomes highly complex near suction side. Streamlines in Figure 9 indicates intensity of shear region near suction side increases. Increase in tip gap height results in larger tip vortex. Thus total pressure loss due to highly shear region is expected to be higher for larger tip clearances. Effect of tip gap height on tip vortex is visualized near suction side of the blade using streamlines. Figure 10 provides a better understanding to see the influence of tip clearance. Tip leakage flow evolves into a larger tip vortex for the case FLAT0015. Obviously seen that tip vortex is dominant to passage vortex. A weak passage formation is valid for all three tip clearances. Total pressure distribution at exit plane confirms this phenomena. It can be concluded that tip vortex suppresses passage vortex. For larger tip gap heights, tip vortex spreads a wider region as streamlines released upstream of leading edge apparently. Influence area of tip vortex is extended to lower spanwise locations.

Leakage flow rate is an important criteria to determine the aerodynamic loss. Reducing leakage flow rate is one of the main goals to improve the aerodynamic performance. Thus, tip gap height is tried to be smaller as possible as. However, there is a physical limit to avoid mechanical problems. Flow rate along streamwise direction is shown in Figure 11. Leakage flow distribution exhibits similar trends for all cases. At same axial location, flow rate takes the smallest value for FLAT0007 and largest value for FLAT0015. Leakage flow rate reaches its highest value around 80% of axial chord for all cases. Total pressure distribution is another important variable for comparing aerodynamic performance. Total pressure distribution at exit plane is going to be used to calculate aerodynamic loss. Exit plane is placed 1.05 axial chord distance from the leading edge to be perpendicular to the camber line of the blade.

Total pressure contour (non-dimensional) downstream of the trailing edge is a comprehensive tool for predicting the flow structures from hub to tip. Figure 12 indicates tip vortex, passage vortex just beneath the tip vortex, endwall effects and hub vortex. In all three cases tip vortex is dominant to the passage vortex. Vortical structure near hub region is similar to each other. In addition, it can be concluded that end wall losses are at the same order of magnitude. However, tip vortex is different for all cases. Tip vortex is weaker for case FLAT0007 compared to other two cases. From figure it is obvious that increasing tip clearance, strengthens the tip vortex. Tip vortex becomes larger for the highest tip gap and expands to a wide region in pitchwise direction. It covers almost the whole region up to periodic side for case FLAT0015.

Total pressure loss coefficient, ∆Cp0, was defined as the difference between mass flow averaged total pressure coefficients at exit and inlet planes. Total pressure coefficient, Cp0, was defined similar to static pressure coefficient, Cp. ∆Cp0, was calculated for each case as seen in Figure 12. Then, the results were recalculated with reference to 1 % case. Calculations indicate that reducing tip gap height results in decrease in total pressure loss which means improvement in aerodynamic performance. Total pressure loss is determined (-4.28)% for tight gap (FLAT0007) with respect to base (FLAT0010) and 11.71% for larger gap.

Conclusion

A numerical prediction for flow in tip gap was presented. A detailed investigation has been carried out to make the flow structure in tip gap clear.

The remarkable point in present study is to form computational domain and generate mesh by introducing a parametric approach. Parametric approach has reduced the time considerably for numerical calculations. Thus, it paves an efficient method in CFD.

Results from numerical calculations consistent with previous studies in literature. Increase in tip clearance height leads to an increase in total pressure loss since tip vortex becomes effective almost the whole region up to the periodic surface. Leakage flow rate shows tendency to increase for higher values of tip gap height. Effect of passage vortex is very small with respect to tip vortex. Total pressure contour and leakage flow distribution are useful tools to offer new designs. In future studies to reduce the aerodynamic loss these results will be used.

Figure 1 .

 1 Figure 1. Grid generation obtained parametrically

Figure 2 .

 2 Figure 2. Computational domain obtained in a parametric way Figure 3 shows the definition of the tip gap height in SpaceClaim. By setting up the dimensions in a parametric way, time required to form a new solid model has been reduced considerably. Tip clearance ratio has been introduced as a design parameter in terms of blade height in present study.

Figure 3 .

 3 Figure 3. Definition of tip gap height.

Figure 4 .

 4 Figure 4. Parametric definitions on Workbench interface.

Figure 6 .

 6 Figure 6. y + distribution at blade tip.

Figure 7 .

 7 Figure 7. y + distribution at midspan.Blade tip region is a critical region due to high shear effects to ensure recommended y + level. As seen from Figure6, maximum value is around 1.3 which means y + is kept at reasonable level. Average value for y + is around 0.5 for all cases. Less than 1.3 is achieved at lower spanwise stations.

Figure 8 .

 8 Figure 8. CP distribution at blade tip

Figure 9 .

 9 Figure 9. Velocity streamlines in the tip gap.

Figure 10 :

 10 Figure 10: Tip vortex visualization near suction side.

Figure 11 .

 11 Figure 11. Leakage flow rate along streamwise direction Determination of location where leakage flow becomes larger can be used in passive control methods to reduce the flow rate.Total pressure distribution is another important variable for comparing aerodynamic performance. Total pressure distribution at exit plane is going to be used to calculate aerodynamic loss. Exit plane is placed 1.05 axial chord distance from the leading edge to be perpendicular to the camber line of the blade.Total pressure contour (non-dimensional) downstream of the trailing edge is a comprehensive tool for predicting the flow structures from hub to tip. Figure12indicates tip vortex, passage vortex just beneath the tip vortex, endwall effects and hub vortex. In all three cases tip vortex is dominant to the passage vortex. Vortical structure near hub region is similar

Figure 12 .

 12 Figure 12. Total pressure distribution at exit plane and aerodynamic loss.

Figure 13 :

 13 Figure 13: Total pressure contour along streamwise direction.

Figure 13

 13 Figure13shows total pressure contours along chordwise direction. It can be deduced from figure that formation of tip vortex is observed after 20 % of axial chord. Besides, evolution of tip vortex throughout the streamwise direction at different axial terminals is demonstrated. Total pressure contour enables to make further suggestions about passive control methods to reduce the loss related to the leakage flow.

Table 1 :

 1 Blade specifications

	Blade height [mm]	123
	Axial chord [mm]	85.04
	Flow angle (°) (inlet)	71.3
	Pitch [mm]	99.274

ACKNOWLEDGMENTS

The authors grateful to Prof. Cengiz Camci for his great support and contriubution. Prof. Camci allowed to use AFTRF blade tip geometry in present study.


Daichi Nakai
email: nakai@cfs.ifs.tohoku.ac.jp

Teppei Furusawa

Donghyuk Kang

Yuka Iga

Experimental Study of Thermodynamic Effect of Tip Leakage Cavitation in Hot Water

Keywords: Tip Leakage Cavitation, Hot Water, NACA0009, Temperature Measurement, Thermistor

Thermodynamic effect on cavitation appears in cryogenic fluids. The thermodynamic effect is considered to suppress the development of cavitation. The cavitation performance of inducer of liquid rocket turbopump with liquid hydrogen and oxygen is improved by the thermodynamic effect. The authors have conducted experimental studies about thermodynamic effect on cavitation around single hydrofoils in hot water in previous study, where sheet cavity and sheet/cloud cavity occurs. However, cavitation in inducer is not sheet cavity which is separation type but tip leakage cavity which is vortex type. Therefore, in the present study, experiment is conducted about tip leakage cavitation in order to clarify the intensity thermodynamic effect on vortex type cavitation. In the present study, cavitation experiments is conducted with high temperature and high pressure cavitation tunnel, NACA 0009 hydrofoil is chosen for a test body. In the experimental results, development of the pattern of the tip leakage cavitation did not change by the change of freestream temperature from room temperature up to 90 o C. The result of sheet cavitation is in our previous study. The effect of freestream temperature on a temperature reduction inside the cavity is investigated by the direct measurement technique of temperature inside the tip leakage cavity.

INTRODUCTION

Cavitation occurs when the pressure of fluid is decreased roughly below the saturated vapor pressure. The occurrence of cavitation deteriorates the performance of inducer of liquid propellant rocket turbopump with liquid hydrogen and oxygen. So, influence of cavitation cannot be neglected in the design of inducer of liquid propellant rocket turbopump and to understand the behavior of cavitation is necessary. A number of cavitation studies have been conducted for cavitation in room temperature. However, when working fluid is replaced to cryogenic fluid, the behavior of cavitation is different from that of cavitation in room temperature water. Latent heat to be supplied from liquid around cavity is needed for vaporization, and temperature of liquid around cavity is decreased. As a result, local saturated vapor pressure is decreased and the growth of cavity is suppressed. Thus, the performance of inducer of liquid propellant rocket turbopump is improved by the thermodynamic effect. This suppression of cavity growth and improvement of performance of inducer are called thermodynamic effect. Thermodynamic effect on cavitation in water of room temperature is negligibly small, on the other hand, that in cryogenic fluids or hot water or refrigerant becomes significant and cannot be neglected, which is usually utilized in a condition near the critical temperature. The extent of thermodynamic effect is known to increasing temperature of working fluid, however, Cervone et al.

reported diverse behavior of cavitation in hot water of 70 o C, the occurrence of cloud cavitation and super cavitation shifts toward higher cavitation number compared to water of 25 o C [START_REF] Cervone | Thermal Cavitation Experiments on a NACA0015 Hydrofoil[END_REF].

The result indicates that the intensity of thermodynamic effect is still unclear, and that is considered to depend not only on thermal property of the fluid but also on cavitation pattern or the unsteadiness. The authors conducted experimental studies about thermodynamic effect on cavitation around single hydrofoils in hot water [START_REF] Yamaguchi | Thermodynamic Effect on Cavitation in High Temperature Water[END_REF], where sheet cavity and sheet/cloud cavity occurs. However, cavitation in inducer is not sheet cavity which is separation type but tip leakage cavity which is vortex type. Also, an author conducted experiments using room temperature water and numerical calculation in order to clarify the mechanism of cavitation instabilities in real inducer [START_REF] Kang | Cause of Cavitation Instabilities in Three Dimensional Inducerl[END_REF]. According to the study, there was close interaction between tip leakage cavitation and cavitation instabilities. Therefore, in the present study, the cavity types in leakage flow and temperature depression inside the tip leakage cavity are experimentally observed in order to clarify the intensity of thermodynamic effect on vortex type cavitation. The experiment was conducted with NACA0009 hydrofoil with tip clearance, and temperature of cavity was measured by using thermistor probe.

THERMODYNAMIC EFFECT

The thermodynamic parameter Σ proposed by Brennen [START_REF] Brennen | Cavitation and Bubble Dynamics[END_REF] is defined as,

2 / 1 ∞ 2 2 2 = Σ l pl l v α T c ρ ρ L (1)
where L is latent heat of liquid, ρv is density of vapor, ρl is density of liquid, cpl is heat capacity of liquid, T∞ is temperature of free stream and αl is thermal diffusivity of liquid. This parameter is considerd to describe the cause of thermodynamic effect, but the result of cavity volume reduction on each hydraulic machinery is still unclear as mentioned before. To investigate the incidence of thermodynamic effect, many experimental research had been conducted. Franc et al. investigated a cavitating inducer in R114 and estimate the temperature depression in cavity by comparing the experimental results of cavitating inducer in room temperature water [START_REF] Franc | An Experimental Investigation of Thermal Effects in a Cavitating Inducer[END_REF]. Niiyama et al. visualized cavitation hydrofoil in liquid Nitrogen and measure temperature depression in the cavity [START_REF] Niiyama | Experimental Investigation of Thermodynamic Effect on Cavitation in Liquid Nitrogen[END_REF]. Many experimental research of thermodynamic effect were conducted using cryogenic fluids or refrigerant as working fluid. However, the experimental results conducted in these fluids cannot be directly compared with the results obtained in room temperature water because it may includes the influence of the difference of fluid. Therefore, in order to conduct cavitation experiments with and without thermodynamic effect in same fluid, high temperature and high pressure cavitation tunnel had been used in present study. The working fluids of this tunnel is water and the free stream temperature can be varied from room temperature to 140 o C, in which the thermodynamic parameter Σ can be varied for wide range.

In present study, water is used as working fluid and by varying temperature of water the thermodynamic effect on tip leakage cavitation is investigated. Figure 1 shows thermodynamic parameter Σ of some cryogenic fluids, refrigerant and water. Thermodynamic parameter Σ of room temperature is in the order of 10 0 and the extent of thermodynamic effect on cavitation is too small to observe at that temperature. By increasing temperature of water, thermodynamic parameter increases. At the temperature about 140 o C the value of Σ reaches the order of 10 4 , this value is corresponding to Σ of Liquids Nitrogen. By increasing temperature of water, water can simulate cryogenic fluids such as Liquid Nitrogen and Liquid Oxygen in the aspect of thermodynamic parameter. Also water show wide range of Σ, this enables to study cavitation with and without thermodynamic effect by using same fluid. In this study, a cavitation tunnel which can vary temperature of water from room temperature to 140 o C had been constructed. Temperature of water was increased and controlled by using electric heater with accuracy of 0.1 o C. The pressure of free stream was measured at upstream and downstream of test body with pressure transducer. The flow velocity and its distribution were measured with LDA (FlowLite 2D, Dantec Dynamics). The vertical distribution of streamwise velocity is shown in Fig. 3. The velocity distribution of vertical direction was measured without test body. The thickness of boundary layer was about few millimeters. Uniform flow velocity distribution was achieved around the center of the test section. NACA0009 hydrofoil with 30 mm chord length, 2.7 mm maximum thickness and 19.73 mm span width was used as a test body. M-Dreyer et al. investigated the influence of the gap width on both trajectory and intensity of the tip vortex cavitation with dimensionless gap width τ/h from 0 to 2 [START_REF] Dreyer | Mind the gap -tip vortex in axial turbines[END_REF]. Their study showed a maximum influence at the dimensionless gap width 0.3. In the present study, for first stage of the experiment, gap width τ was set as constant value in 0.27 mm, dimensionless gap width 0.1.

EXPERIMENTAL SET UP

Table 1 shows experimental condition, corresponding thermodynamic parameter and Reynolds number. For reference, in case of a real inducer in liquid propellant rocket engine in Japan, working fluid is liquid oxygen, Reynolds number Re = UD / ν (D is diameter of inducer, U is peripheral speed of inducer tip) is about 1.2 × 10 6 [START_REF] Kamijo | Developmental History of Liquid Oxygen Turbopumps for the LF-7 Engine[END_REF].

To measure temperature of cavity, thermistor probe (Nikkiso-Thermo Co., Ltd.) was inserted into the test section with stainless steel pipe of diameter 2 mm and thickness 0.5 mm. The configuration of test body and inserted thermistor probe is shown in Fig. 4. The angle of attack was 12 degree and temperature of cavity was measured at 14 mm downstream from hydrofoil leading edge by temperature probe Tc shown in Fig. 4. Additionally, temperature probe is inserted from backside wall in order to place the thermosensitive part inside the tip leakage cavity which is in upper part of tip clearance in frontside wall. The reference temperature was measured at 10 mm below and 20 mm ahead the center of flow channel by the probe T∞. The thermosensible part of thermistor was placed inside a tip leakage cavitation which is upper part of tip clearance of the hydrofoil. To reduce the influence of temperature variation of free stream, the temperature of the cavity and free stream were measured simultaneously. All thermistor were calibrated before conducting experiment with quartz thermometer. Thermistor shows large change of electric resistance depending on temperature, so measurement of temperature with high accuracy is enabled. The resistance of thermistor was measured with digital multi meter, and the uncertainly of temperature measurement was less than 0.02 K. The influence of heat conduction from tunnel wall exists and steady-state heat conduction analysis was conducted. The estimated difference between temperature of thermistor and temperature of cavity was less than 4 %. Estimated time constant of the temperature probe was less than 5.42 s. The temperature of cavity was obtained by averaging 30 s of measurement results.

RESULTS AND DISCUSSION

Aspect of tip leakage cavitation

First, the cavity type on a hydrofoil with the leakage flow was observed. Higashi et al. classified the leakage vortex cavity in four regions from the configurations of the cavity [START_REF] Higashi | Tip Leakage Vortex Cavitation from the Tip Clearance of a Single Hydrofoil[END_REF]. In our experimental result, four kinds of tip leakage cavity were also observed, although the pattern was somewhat different from that in Higashi et al. In experimentally investigated the characteriscs of oscillating cavitation on a flat plate hydrofoil in a water tunnel [START_REF] Sato | Observations of Oscillating Cavitation on a Flat Plate Hydrofoil[END_REF]. It is thought that this tendency agrees with the case of the tip leakage cavity. Cavitation disappears and appears unsteadily during Region B and Region C, so this area is set of transition region. After that, in Region D, as the cavitation number decreases further, the cavity becomes to a leakage supercavitation.

Fig. 6 shows comparison of the occurence map of cavitation pattern in each angle of attack and cavitation number between low temperature water 30 o C and high temperature water 90 o C. For both figures, experimental condition is U∞ = 10 m/s. As shown in Fig. 6, there is no significant difference of configurations of the cavity as a whole between 30 o C and 90 o C. In our previous experimental studies about sheet cavity and sheet/cloud cavity of separation type in water of 30 o C and 80 o C [START_REF] Yamaguchi | Thermodynamic Effect on Cavitation in High Temperature Water[END_REF], there was also no significant change of cavitation pattern between 30 o C and 80 o C. So, it is shown that cavitation which is vortex type shows the similar tendency of which is separation type up to 90 o C. However, incipient cavitation number at 90 o C is higher than 30 o C in case of that angle of attack is 12 deg. It is thought that this is because scale effect in high Reynolds number condition is easy to appear at high temperature, and tip vortex is strong with high angle of attack [START_REF] Boulon | An experimental insight into the effect of confinement on tip vortex cavitation of an elliptical hydrofoil[END_REF], so tip vortex cavitation remains undefeated in this case.

Measurement of temperature

In this study, the temperature of free stream was varied from room temperature to 90 o C and temperature inside a tip leakage cavity is measured by temperature probe. The aspect of a flow field with the temperature probe is shown in Fig. 7. This figure is snapshot of supercavitation at σ = 0.76, U∞ = 10 m/s and angle of attack is 12 deg. In this figure, it is confirmed that cavitation is not caused by thermistor probe T∞.

Fig. 8 shows the result of influence of cavtitation number on temperature depression in tip leakage cavity on the condition of T∞ = 90 o C, U∞ = 10 m/s and AoA = 12 deg. As cavitation number decreases, the temperature depression in tip leakage cavity becomes large. This tendency is in agreement with the experimental result of temperature depression in separation type cavity with hot water [START_REF] Yamaguchi | Thermodynamic Effect on Cavitation in High Temperature Water[END_REF], and the experimental and numerical analysis of cavitating orifice flow with liquid nitrogen [START_REF] Niiyama | Thermodynamic Effects on Cryogenic Flow in an Orifice[END_REF]. In the region of 0.9 < σ < 1.4, Region C, cavity length cyclically changes from small partial cavitation and tip leakage cavitation to super cavitation, so the thermistor probe was covered with both cavity and water alternately. In the region of σ > 1.4, Region B, shedding of cavity was observed. So the thermistor was also coverd with both cavity and water cyclically. Therefore, the measured teperature was some sort of time averaged temperature. In Region D, maximum temperature depression reaches over 0.4 o C in tip leakage cavitation in super cavitation condition, in which the accuracy of temperature measurement is relatively higher than that in Region B and C. Fig. 9 shows the result of influence of free stream temperature on temperature depression in tip leakage cavity

CONCLUSION

Figure 1 .Figure 2 .

 12 Figure 1. Thermodynamic parameter Σ of some cryogenic fluids, R114 and water

Figure 4 .

 4 Figure 4. Configulation of temperature probe and NACA0009 hydrofoil (chord length 30 mm, angle of attack 12 deg)

Fig. 5 .

 5 the aspect of cavitation in NACA0009 with chord length C = 30 mm and tip clearance τ = 0.27 mm in the condition of flow temperature T∞ = 30 o C, U∞ = 10 m/s, AoA = 12 deg are shown. These photographs were taken without inserting temperature probe. In Region A, at higher cavitation number, tip vortex cavitation was observed intermittently near the leading edge at the tip. As the cavitation number decreases, in Region B, tip vortex cavitation, leakage vortex cavitation, and partial cavitation appear. In this region, small partial cavitation and leakage vortex cavitation oscillate with smaller amplitude and realtively high frequency. The oscillation of small partial cavity is well known through some experiments by several researchers. Arndt et al. conducted numerical and experimental investigation about sheet/cloud cavitation, and experiments were conducted at two different water tunnel with 2D NACA0015 hydrofoil [10]. Kawanami et al. experimentally investigated genaration mechanism of cloud cavitation on a hydrofoil by controlling cloud cavitation with an obstacle [11]. As the cavitation number decreases more, in Region C, the cavity cyclically changes from small partial cavitation and tip leakage cavitation to super cavitation with low frequency oscllation. Some experimental studies about the kind of oscillating cavitation, specifically in case of sheet/cloud cavity, are conducted. Wade et al. brought to attention about oscillating cavitation [12]. Sato et al.

Figure 5 .

 5 Figure 5. Appearance of cavity in water (T∞ = 30 o C, U∞ = 10 m/s, AoA = 12 deg).

Figure 6 .Figure 7 .

 67 Figure 6. Mapping of cavitation pattern (U∞ = 10 m/s)

Figure 8 .Figure 9 .

 89 Figure 8. Influence of cavitation number on the temperature depression inside tip leakage cavity (T∞ = 90 o C, U∞ = 10 m/s, AoA = 12 deg)

  Thermodynamic effect on tip leakage cavitation on a hydrofoil with tip clearance in high temperature water was studied with high temperature and high pressure water cavitation tunnel. ・ Cavitation pattern of the tip leakage cavitation in NACA0009 hydrofoil was classified into four types. ・ It was shown that there was no significant change in occurrence region of the each type of tip leakage cavitation in a temperature range from 30 o C to 90 o C. This tendency is similar to cavitation which is separation type on a hydrofoil without tip clearance. ・ Temperature depression in tip leakage cavity was measured with thermistor probe. Observed maximum temperature depression was about 0.4 o C on the condition of super cavitation in water of 90 o C. Further study will be conducted for following topics. ・ Experiments with tip clearance variation in order to investigate the relationship of gap width and thermodynamic effect. ・ Estimation of vorticity by numerical simulation in order to consider influence of vorticity on thermodynamic effect. NOMENCLATURE cpl heat capacity of liquid [J] p∞ free stream pressure [MPa] pv pressure of vapor [MPa] L latent heat of vaporization [J/kg] T∞ temperature of free stream [ o C] Tc temperature in cavity [ o C] ΔT temperature difference between free stream and cavity [ o C] U∞ free stream velocity [m/s] αl thermal diffusivity of liquid [m 2 /s] σ cavitation number = (p∞ -pv) / (ρU∞ 2 /2) [dimensionless] ρl density of liquid [kg/m 3 ] ρv vapor density [kg/m 3 ] Σ thermodynamic parameter [m/s 3/2 ] τ gap width [mm] h foil maximum thickness [mm] Re Reynolds number U∞C / ν [dimensionless] ν kinematic viscosity [m 2 /s] C chord length [mm]


Sebastian Schuster
email: sebastian.schuster@stud.uni-due.de

Friedrich-Karl Benra

Dieter Brillert

Droplet deposition in radial turbines

Keywords: Condensation, radial turbine, droplet deposition

In this paper droplet deposition in radial turbines used for waste heat recovery in the chemical industry is investigated. During expansion, the water vapour in the working fluid undergoes sub-cooling, at a certain point a cloud of fine droplets begins to form and a fraction of these droplets is deposited on the blade surfaces. In this paper, droplet deposition is calculated by incorporating turbulent fluctuations and Brownian motion into a Lagrange particle tracking algorithm. After introducing the calculation approach, a validation of the enhanced CFD code by means of deposition experiments in a straight tube is presented. For a radial turbine, the functional relation between the location of maximum sub-cooling and the onset of droplet deposition is pointed out. Even though the validation of the numerical code is quite satisfactory, some uncertainties arise from the nucleation modelling. This uncertainty will be addressed in the publication as well. The paper concludes with suggestions on how to increase the reliability of the calculations.

INTRODUCTION

At the end of many procedures in the process industry, a gas mixture at high pressure containing water and steam is present. The exergy stored in such mixtures can be converted into mechanical work by using a radial turbine. Condensation may occur during expansion, but before noticeable condensation emerges, sub-cooling of around 20 °C is necessary. Once a cloud of fine droplets has formed, the droplets rapidly grow to a few micrometers. Directly after the fog forms, a part of the fine droplets is deposited on the surface of the blade, mainly by Brownian motion. With increasing droplet diameter, the dominant deposition mechanism changes from Brownian motion to turbulent migration, and finally to inertia-moderated motion.

Condensation in turbo machines has been investigated since the beginning of the 20th century, starting with the work of Baumann [START_REF] Baumann | Recent developments in steam turbine practice[END_REF]. In the mid-20th century, Gyarmathy [START_REF] Gyarmathy | Grundlagen einer Theorie der Naßdampfturbine[END_REF] developed the first theory of the condensation process in axial steam turbines. A fundamental element of this theory is the size of the droplets. This is calculated from the initial size at the birth of the droplets and with a growth law. Bakhtar et al. [START_REF] Bakhtar | Classical nucleation theory and its application to condensing steam flow calculations[END_REF] provide an overview of nucleation models. Different numerical methods can be used to calculate the gas-liquid flow: These are the fully Euler methods; the mixed Euler-Lagrange methods; and the quadrature methods of moments (QMOM). A comprehensive comparison of these methods can be found in White [START_REF] White | A comparison of modelling methods for polydispersed wet steam flow[END_REF].

Yau and Young [START_REF] Yau | The deposition of fog droplets on steam turbine blades by turbulent diffusion[END_REF], as well as Young and Yau [START_REF] Young | The inertial deposition of fog droplets on steam turbine blades[END_REF], investigate droplet deposition in axial turbo machines. Crane [START_REF] Crane | Droplet deposition in steam turbines[END_REF], Starzmann et al. [START_REF] Starzmann | On kinematic relaxation and deposition of water droplets in the last stage of low-pressure steam turbines[END_REF], and Starzmann, Casey and Mayer [START_REF] Starzmann | Water droplet flow paths and droplet deposition in low-pressure steam turbines[END_REF] show investigations of droplet deposition in axial steam turbines. These investigations focus on the amount of water deposited and the comparison between two calculation approaches. In one calculation approach the deposition rate is calculated only taking into account the inertia force on a curved stream line. In the other calculation approach the turbulent fluctuation is incorporated as well.

Guha [START_REF] Guha | Transport and deposition of particles in turbulent and laminar flow[END_REF] offers a review on deposition calculation and the equations presented are used in this paper. An important conclusion of the work of Young, Yau, and Walters [START_REF] Young | Fog droplet deposition and coarse water formation in low-pressure steam turbines: A combined experimental and theoretical analysis[END_REF] is that the theory of droplet deposition and experiments are consistent. Discrepancies in older publications are explained by an averaged diameter being used for calculation instead of a diameter spectrum. Therefore, in this publication, the Euler-Lagrange method is used for the calculation of the gas-liquid flow, as it can easily handle a diameter spectrum.

Once a droplet hits a wall an interaction with the wall has to be accounted for. This interaction is investigated by Gomaa and Weigand [START_REF] Gomaa | Modelling and investigation of the interaction between drops and blades in compressor cascades with a droplet-laden inflow[END_REF] as well as by Schmehl et al. [START_REF] Schmehl | CFD analysis of spray propagation and evaporation including wall film formation and spray/film interactions[END_REF]. Their correlation allows the calculation of the behaviour of a droplet when it hits a surface. From these investigations, it can be deduced that very small droplets -barely larger than a micrometer -form a liquid film on the surface of the blade. In the presented case, where the temperature is below saturation temperature and the Weber number is quite small, all droplets dissolve in a liquid film at the surface.

The liquid film motion is mainly influenced by the wallshear stress and the centrifugal force. Even at an assumed film thickness of a micrometer, the direction of motion is mainly radially outward oriented (Schuster et al. [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF]). Such thin films have never been measured in applications related to turbomachines and it is likely that the film is thicker as it breaks up into rivulets. While the film gets thicker, the centrifugal force becomes even more dominant. Droplets originating from the water films might cause damage to the turbine and that is the reason why investigating droplet deposition in radial turbines is necessary.

In this paper the functional relationship between the location of maximum sub-cooling and deposition onset is determined. Whereas, in axial steam turbines, the amount of water is important to calculate the erosion rate, in radial turbines, the location of deposition onset is the key factor to determine liquid film motion.

Because the focus of investigations has not been before on the onset of deposition, the nucleation model used is investigated for parameters having an influence on the results.

METHOD

Condensation

The calculation of the gas phase is based on the Reynolds-averaged Navier-Stokes equations in integral form. With the k-ω SST turbulence model purposed by Menter [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF], the system of equations is closed. This results in seven partial differential equations which are transformed into an algebraic system of equations by applying the finite volume method on a block-structured mesh. Due to nucleation droplets begin to form, and their motion is calculated in consideration of Newton's second law. Interaction between the gas phase and the droplets is calculated by the particle-source-in-cell method [START_REF] Crowe | Particlesource-in-cell (PSICELL) model for gas-droplet flows[END_REF]. In each finite volume, the number of droplets formed by nucleation is calculated. Once the calculated number n dr is above a predetermined nucleation rate threshold value of NRT = 1.0 x 10 +15 1/(m³•s), a particle path consisting of n dr droplets is initialised in the finite volume. All particle paths exchange mass, momentum, and energy with the surrounding gas phase. Therefore, the calculation of the two sets of equations is conducted one after another until a converged solution for both systems is obtained. This approach is often called two-way coupling. The number of droplets formed is calculated with classical nucleation theory and the droplet growth using heat and mass transfer models. A large number of publications are available on condensation models [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF], [START_REF] Gerber | A pressure-based Eulerian-Eulerian multi-phase model for non-equilibrium condensation in transonic steam flow[END_REF][START_REF] Simpson | Viscous and unsteady flow calculations of condensing steam in nozzles[END_REF][START_REF] Yang | Numerical simulation on nonequilibrium spontaneous condensation in supersonic steam flow[END_REF][START_REF] Wroblewski | Steam condensing flow modeling in turbine channels[END_REF][START_REF] Patel | The effect of turbulence and real gas models on the two-phase spontaneously condensing flow in nozzle[END_REF][START_REF] Moraga | Numerical sensitivity study and calibration of non-equilibrium wet steam model[END_REF][START_REF] Schuster | Influence of different gas models on the numerical results of high-velocity condensation[END_REF][START_REF] Schuster | Investigation about the sensitivity of condensation calculations on model coefficients[END_REF]. The one used here is described in Schuster et al. [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF]. The empirical factors in the models are set to: q c = 1.0, ν = 0.0, NBTF = 0.975. Schuster et al. [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF] also perform a validation of the code. Therefore, the flow in a Laval nozzle is calculated and static pressure along the nozzle axis as well as droplet diameter at one point on the nozzle axis are compared to the experimental results. A general good agreement between measured and calculated values is obtained in terms of static pressure. A larger deviation exists for the droplet diameter but is still in the same magnitude. A discussion and conclusions related to the discrepancy and empirical factors are given by Schuster et al. [START_REF] Schuster | Sensitivity analysis of condensation model constant on calculated liquid film motion in radial turbines[END_REF] [START_REF] Schuster | Investigation about the sensitivity of condensation calculations on model coefficients[END_REF].

Droplet transport and droplet deposition

The droplet trajectory results from both the droplet origin and the forces acting on the droplet. Droplets are tracked in consideration of Newton´s second law. Because the droplets originate from the gas phase it is assumed that the droplet velocity at the origin point is equal to the gas phase velocity. If a droplet trajectory crosses a wall the conclusion is drawn that the droplet is deposited on that wall. The droplet is then removed from the calculation. As mentioned above, the origin of a droplet is calculated using classical nucleation theory. As the droplets originate from the gas flow, it is assumed that the droplet velocity at the formation point is equal to the gas velocity at that point. Once the droplet is exposed to an altered gas velocity, a force acts on the droplet. This force results from the flow around the droplet and is called aerodynamic drag force F D . Consequently, the droplet velocity U d converges to the gas phase velocity U ̅ g . On the other hand, the same force with a negative sign acts on the gas phase. The aerodynamic drag force F D can be calculated with the non-dimensional drag coefficient C D . In contrast to the drag calculation on a fixed sphere, the slip velocity between gas phase and droplet must be incorporated into the equations. Due to the small droplet diameter d d being no larger than a few micrometers, the particle Reynolds number Re d is well below 1,000. Thus, the functional relationship between the Reynolds number and the drag coefficient can be calculated using the Schiller-Naumann correlation [START_REF] Schiller | Über die grundlegende Berechnung bei der Schwerkraftaufbereitung[END_REF]. In the case of condensation, the droplets are so small, that the flow around the droplets cannot be described using continuum mechanics. Hence, a correction of the drag coefficient is necessary in conjunction with the condensation calculation, as stated by Gyarmathy [2]. This correction was originally suggested by Cunningham [START_REF] Cunningham | On the velocity of steady fall of spherical particles through fluid medium[END_REF]. All further drag force calculations are based on Eq. 1 with the corrected drag coefficient [START_REF] Gyarmathy | Grundlagen einer Theorie der Naßdampfturbine[END_REF] given by Eq. 2 and the original Schiller-Naumann correlation [START_REF] Schiller | Über die grundlegende Berechnung bei der Schwerkraftaufbereitung[END_REF] given by Eq. 3. In addition to the variables mentioned in Eqs. 1-2, ρ g is the gas density and Kn is the Knudsen number.

  2 D d g D , C u n g d g d 8 d C          F U UU U (1) D,Cun D 1 1 2.53 C C   Kn ( 2 )   0.687 D d d 24 1 0.15 C   Re Re ( 3 )
As long as the gas phase velocity U ̅ g is directly taken from the solution of the Reynolds averaged Navier-Stokes equations, turbulent fluctuations are not incorporated in the droplet path calculation. This approach is used by Gerber [START_REF] Gerber | Inhomogeneous multi-fluid model for prediction of non-equilibrium phase transition and droplet dynamics[END_REF], Dykas, and Wroblewski [START_REF] Dykas | Single-and two-fluid models for steam condensing flow modeling[END_REF] as well as by Sasao et al. [START_REF] Sasao | Eulerian-Lagrangian numerical simulation of wet steam flow through multi-stage steam turbine[END_REF]. It can be considered state of the art against the background of condensation calculations. In particular, for cases where droplet deposition is not in the focus of investigation but on the change in the thermodynamic state, researchers trust in this procedure. However, if investigations focus on droplet deposition, results are not accurate enough if the influence of turbulence fluctuations is not incorporated in droplet calculations. The need for an improved model in the context of condensation calculations in turbines is obviously clear from the conclusion of Starzmann et al. [START_REF] Starzmann | On kinematic relaxation and deposition of water droplets in the last stage of low-pressure steam turbines[END_REF]: '…further work on the problematic of droplet deposition is needed and a next step should be to implement more sophisticated treatments in 3D-CFD codes to verify the accuracy of the presented results'.

A general insight on deposition mechanism is given by experimental investigations in straight tubes flooded with a particle-laden gas. The obtained values are converted into non-dimensional values. Thereby, results obtained at different Reynolds numbers can be compared. In this context, the Reynolds number is calculated with the gas velocity and the tube diameter. The non-dimensional values are the deposition rate G dep+ (Eq. 4), defined for example by Wu and Young [START_REF] Wu | The deposition of small particles from a turbulent air flow in a curved duct[END_REF], and the non-dimensional particle relaxation time τ + (Eq. 5), used by Liu and Agarwal [START_REF] Liu | Experimental observation of aerosol deposition in turbulent flow[END_REF] as well as by Wu and Young [START_REF] Wu | The deposition of small particles from a turbulent air flow in a curved duct[END_REF]. In Eq. 4, the deposition rate G dep is the mass flow of particles that are deposited on a surface divided by the area of the surface. The particle density ρ is the mass of particles divided by the gas volume. The friction velocity u τ is calculated from the square root of the wall-shear stress divided by the gas density. The particle relaxation time τ d is deduced from the force balance between inertia force and aerodynamic drag force on the assumption that Stokes' drag law is valid. A detailed derivation is given by Guha [START_REF] Guha | Transport and deposition of particles in turbulent and laminar flow[END_REF]. In Eq. 5, ν g is the kinematic viscosity, ρ d is the liquid density, and ρ g is the gas density.

τ dep dep+ G G u    ( 4 ) τ τ 2 2 d d d d+ g g g 1 18 u du                  ( 5 )
The results from droplet deposition experiments in straight tubes are shown in Fig. 1. Experimental results from Liu and Agarwal [START_REF] Liu | Experimental observation of aerosol deposition in turbulent flow[END_REF] as well as Sehmel [START_REF] Sehmel | Aerosol deposition from turbulent airstream in vertical conduits[END_REF] are plotted. CFD results are also shown to which chapter 1.3 refers to. It must be mentioned that the results from Sehmel shown in the figure are not directly noted in the original publication. Instead, the non-dimensional particle relaxation time and non-dimensional deposition rate are calculated from the values given in the publication. The non-dimensional deposition rate as a function of the nondimensional relaxation time can be divided into three regimes. Thereby, in Fig. 1, the borders shown are taken from Wu and Young [START_REF] Wu | The deposition of small particles from a turbulent air flow in a curved duct[END_REF]. In regime 1, droplet deposition is mainly caused by diffusion, while in regime 3 the dominant effect is the large-scale turbulence structure. Finally, in regime 2, all sizes of turbulent fluctuations have an important role in the deposition process.

If the droplet path is calculated using Eq. 1, the turbulent fluctuation is not incorporated, nor is the diffusion process. In order to incorporate the turbulent fluctuation, the gas phase velocity U̅ g resulting from Reynolds averaging must be replaced by the mean velocity U ̅ g plus the fluctuation velocity U`. One way to handle this is by using direct numerical simulation (DNS). But summarizing this approach cannot be applied to the calculation of the flow in radial turbines due to the high computational effort. Large eddy simulation (LES) would reduce computational requirements but only give information about the large-scale turbulence. But LES give no benefit as in area 2, all scales have an impact on the deposition rate. Hence, the only practical way to handle deposition calculation in radial turbines is to deduce the fluctuation velocity from the turbulent kinetic energy k and the turbulent dissipation rate ε. The following approach is based on the work of Gosman and Ioannides [START_REF] Gosman | Aspects of Computer Simulation of Liquid-Fueled Combustors[END_REF]. A review is given in the publication of Guha [START_REF] Guha | Transport and deposition of particles in turbulent and laminar flow[END_REF]. A droplet enters a turbulence structure at a time t 0 and it is assumed that the droplet does not leave the structure before the structure disappears at time t 1 . This means the force resulting from the fluctuation velocity of the turbulence structure acts on the droplet as long as the structure exists.

The turbulent fluctuation velocity is calculated using Eq. 6 from the turbulent kinetic energy k and a random number ξ. In contrast to the original publication a own random number is calculated for each coordinate. This fluctuation velocity is added to the mean velocity so that the turbulent fluctuation is incorporated in Eq. 1. Besides the fluctuation velocity, the time interval in which the turbulence structure exists is needed. This interval is calculated using Eq. 7. In this equation, l e is the length of the structure calculated using Eq. 8, where c μ is a turbulence model constant. The exponent applied to the turbulence model constant is different to the one used in the publication of Gosman and Ioannides [START_REF] Gosman | Aspects of Computer Simulation of Liquid-Fueled Combustors[END_REF] and is in accordance with the calculation of turbulence length scale in the used CFD code [35]. The random number ξ is calculated separately for each coordinate with the normal distribution (Eq. 9). ς is generated by a random number generator for equally distributed numbers between -0.5 and 0.5. If the standard deviation σ is set to 0.1, a good agreement is achieved between calculation and experiment, as will be shown in the validation. For different turbulence models different values might be necessary. In the future it might be desirable to use an advanced Reynolds Stress model in order to have a better prediction of the fluctuation velocity in wall normal direction.

     ( 8 ) 2 2 1 1 exp if 0 2 2 1 1 exp if < 0 2 2                                                    ( 9 )
If the particles are very small the Brownian motion has a major impact on the deposition rate. Ounis, Ahmadi and McLaughlin [START_REF] Ounis | Dispersion and Deposition of Brownian Particles from Point Sources in a Simulated Turbulent Channel Flow[END_REF] state that the Brownian motion becomes important for submicrometer-size particles: "The Brownian motion significantly affects the dispersion of submicrometersize particles near a wall.". Wu and Young [START_REF] Wu | The deposition of small particles from a turbulent air flow in a curved duct[END_REF] mention that the diffusion process becomes important below a particle relaxation time of 0.3.

Therefore, the force F B , as a result of the Brownian motion, must be incorporated in the droplet path calculation. This force is calculated using Eq. 10, which is given by Guha [START_REF] Guha | Transport and deposition of particles in turbulent and laminar flow[END_REF] and is based on the publication of Ounis, Ahmadi and McLaughlin [START_REF] Ounis | Brownian Diffusion of Submicrometer Particles in the Viscous Sublayer[END_REF]. In this equation, k B is the Boltzmann constant, T g the temperature of the gas phase, D B the diffusion coefficient, and Δt the integration time step. The diffusion coefficient D B is calculated using Eq. 11. In addition to the previously mentioned variables, this equation includes the dynamic viscosity μ g , the droplet diameter d d , and the Cunningham correction factor C C (Eq. 12). To be consistent with the publication of Guha [START_REF] Guha | Transport and deposition of particles in turbulent and laminar flow[END_REF], in the calculation, the Cunningham correction factor is taken from this publication and is slightly different to the one from Gyarmathy [START_REF] Gyarmathy | Grundlagen einer Theorie der Naßdampfturbine[END_REF] used in Eq. 2.

2 2 B g B 1 3 B 2 k T D t       ... F ( 1 0 ) B g B C g d 3 k T D C d       ( 1 1 ) C 1 2.7 C    Kn ( 1 2 )
There are other forces acting on the droplets that are incorporated in the calculation but have only a minor effect on the onset of deposition. Hence, only the literature where the equations to calculate this force can be found are given. Caused by a pressure gradient, the so-called pressure gradient force acts on a droplet (Lantermann [START_REF] Lantermann | Simulation der Transport-and Depositionsvorgänge von Nanopartikeln in der Gasphase mittels Partikel-Monte-Carlo-and Lattice-Boltzmann-Methoden[END_REF]). A droplet in an inhomogeneous flow field is exposed to a lift force calculated based on the equation given by Sommerfeld [START_REF] Sommerfeld | Theoretical and Experimental Modelling of Particulate Flows[END_REF]. A temperature gradient in a flow field results in the so-called thermophoretic force and is calculated using the equation published by Talbot et al. [START_REF] Talbot | Thermophoresis of particles in a heated boundary layer[END_REF]. It must be mentioned that the influence of the thermophoretic force can get extraordinarily high if high temperature gradients exist in the flow. For example, Ryley and Davis [START_REF] Ryley | Effect of thermophoresis on fog droplet deposition on low-pressure steam turbine guide blades[END_REF] summarize that droplet deposition on steam turbine blades can be reduced by 30% to 90% if the blades are heated.

Validation tube

The experimental results of droplet deposition in straight tubes shown in Fig. 1 can be used for code validation. As the deposition rate is independent from the Reynolds number in the non-dimensional form of the diagram, the CFD boundary conditions must not fit the experimental ones. The Reynolds number in the CFD calculations is 6,500. The density of the gas is calculated using the state equations for ideal gases. The properties of the gas are: dynamic viscosity μ g = 1.831 x 10 -05 Pa•s; molar mass M = 28.96 g/mol. The outlet pressure is set to p = 1 bar. The temperature is constant at T g = 20 °C. The diameter of the droplets d d with a density of ρ d = 997.0 kg/m³ is varied between 2 μm (τ d+ = 0.1) and 220 μm (τ d+ = 1000), and is constant from the inlet to the outlet. The deposition rate is determined in the second half part of the tube where the deposition rate is constant. The CFD results are shown additionally to the experimental results in Fig. 1. To ensure that the results are independent from the number of particle tracks calculated, in each calculation the number of tracks is doubled starting from an initial point. For the same reason, the integration time step is reduced by one magnitude from a starting point until an integration time stepindependent solution is obtained. If τ d+ is greater than nine, an independent solution is obtained at 1,000 tracks and an integration time step of 1.0 x 10 -6 s. For smaller relaxation times (τ d+ equal 1 and 0.1), 8,000 particle tracks and an integration time step of 1.0 x 10 -7 s is necessary for an independent solution. For τ d+ = 1, the results of different time steps are shown in Fig. 1 to illustrate the influence of the integration time step. The CFD results are in satisfactory agreement with the experimental results. For the regime of turbulent migration and the regime inertia moderated this is achieved by setting the standard deviation σ to 0.1. For different turbulence models different values might be necessary.

RESULTS AND DISCUSSION

Radial turbine under investigation

A radial turbine is chosen for the investigations that is used for waste heat energy recovery in the process industry. The rotor blades are radially oriented at the leading edge. Figure 2 shows a meridional plot of the radial turbine under investigation with the reference planes and coordinates. The plane before the stator is marked with 0, the one before the impeller with 1, and the one after the impeller with 2. The axis of rotation is the z-axis. By convention, z is zero at the intersection point of hub and leading edge. All values of z are with regard to a value z 0 . The radial position is marked with r/r 0 . A part of the results will be shown as a function of S at different relative channel heights. By definition, the channel height is 0 % at the hub and 100 % at the casing. Results that are averaged in the circumferential direction and in the direction from hub to casing at constant S are shown as a function of S M . For a clear distinction between pressure side and suction side, in diagrams that are showing a meridional plot, the results obtained on the suction side are shown in the first quadrant of the coordinate system. Hence, the z- coordinate starts from zero to positive values. The through flow direction is from top left to bottom right. The results on the pressure side are shown in the second quadrant. The z- coordinate ranges from zero to negative values. The through flow is from top right to bottom left. Thus, Fig. 2 is showing the suction side. Besides, the meridional plot results are shown on S 1 -surfaces, which are defined as surfaces at a constant relative channel height. Total temperature and total pressure are specified as inlet boundary conditions before the stator in the CFD calculations. At the turbine outlet, the static pressure is specified. At all walls, the fluid has no slip and the walls are adiabatic. The fluid is a mixture of air and water steam defined as a two-component mixture. The absolute pressure is determined from the partial pressure of the two components. In each node of the computational mesh, the air temperature is equal to the water steam temperature. The air underlying the equation of the state of perfect gases has a molar mass of M = 28.96 g/mol with a constant isobaric heat capacity c p = 1004.4 J/(kg•K), a constant dynamic viscosity μ g = 1.831 x 10 -05 Pa•s, and constant heat conductivity λ = 0.0261 W/(m•K). The water steam properties are calculated using the International Association for the Properties of Water and Steam (IAPWS) equations. The release IAPWS 97 is used. All equations and coefficients can be found in the available literature [START_REF] Wagner | International Steam Tables[END_REF]. At the inlet, the mixture composition is specified and remains constant throughout the domain.

Figure 3 shows the radial turbine impeller. The hub and blade are shown in grey. A S 1 -surface is plotted and coloured with the sub-cooling between 15 °C and 25 °C. Sub-cooling T sc is defined as the saturation temperature T sat minus the gas temperature T g . A positive value means that the gas temperature is below the saturation temperature and, hence, that potential for the formation of droplets exists. The blue points give a qualitative impression of the locations of droplet deposition. The direction of the view is towards the rotor blade suction side. At the impeller inlet, sub-cooling is below 15 °C. In flow direction, the sub-cooling increases up to a value of approximately 25 °C. At this value, the condensation is high enough for a significant release of latent heat accompanied with a gas temperature increase and, hence, a reduction of sub-cooling. Once droplets have been formed, they might be deposited on the blade surfaces. The blue points show the locations where droplets are deposited. For the analysis of droplet deposition at a constant relative channel height, the location in the stream-wise direction at which deposition occurs first is marked. This is done at different channel heights and the connection of the determined points results in the red line shown in Fig. 3. To compare the location of maximum sub-cooling and the onset of deposition, the point of maximum sub-cooling is needed. This point is determined from surface streamlines on the S 1surfaces. On each streamline, the location at which the subcooling has a local maximum is picked. The location of the streamline closest to the surface is used for further comparison in the meridional view. On the suction side, an uncertainty arises in the evaluation due to a secondary flow phenomenon. This can be seen in Fig. 3. Close to the suction surface, the sub-cooling first increases up to a value of 20 °C before dropping slightly (highlighted with a red oval). After that drop, the sub-cooling increases again before dropping down rapidly at a value of 25 °C (highlighted with a black oval). The first maximum is not caused by latent heat release. It is caused by transport of hotter gas from the hub to the casing. 4a the sub-cooling T sc-sr , averaged in circumferential and in channel height direction with regard to a reference value, is plotted as a function of the stream-wise coordinate S M for different mesh resolutions. The solution on the medium mesh can be called mesh-independent. The same is true for the droplet deposition on the pressure side, as In Fig. 5, droplet deposition is shown incorporating all mentioned forces on the one hand and only the aerodynamic drag force calculated with the mean velocity on the other. The grey full lines enclose the part between stator-impeller gap, hub, and casing. The dashed lines illustrate the rotor blade contour. The pressure side is shown on the left and the suction side on the right. Results based on all mentioned forces are plotted using squares filled in red. The others are marked with red crosses. On the pressure side, a shift in deposition begin when incorporating all forces is clearly visible above 50 % relative channel height. But the difference between the two approaches is smaller at the hub. The reason for this is the higher blade curvature at the hub in comparison to the blade curvature at the casing. Hence, droplet deposition is dominated by inertia forces on a curved streamline at the hub, whereas it is dominated by turbulent migration and diffusion at the casing. On the suction side, no deposition occurs due to inertia on a curved stream line; hence, no data marked with crosses can be found on the suction side. However, deposition occurs due to diffusion and turbulent migration. The results clearly show the need to incorporate Brownian motion and turbulent fluctuation in the calculations. 3 provides an initial indication that a relationship between the locations of maximum sub-cooling and deposition start is present. Fig. 6, therefore, offers more detailed insight on the relation between both locations on the rotor blade pressure side. Three different calculations are performed using different inlet temperatures, where T o-r valued at zero is the design point temperature. With increasing inlet temperature, the location of maximum sub-cooling -shown using the black lines -is shifted in direction to the trailing edge. Up to a channel height of 90 %, the deposition beginshown with the red symbols -is also shifted to the trailing edge with increasing inlet temperature. At above 90%, the deposition starts near the blade's leading edge. Droplets that are deposited there originate from the tip clearance flow.

Figure 7 shows the results on the suction side in the same manner as is shown in Fig. 6 for the pressure side. From the hub in direction to the casing, deposition starts at nearly constant stream-wise position first. Thereafter, droplet deposition is shifted to the leading edge. Qualitatively, the same is true for the location of maximum sub-cooling. The discrepancy between both locations on the suction side is caused by the secondary-flow phenomenon visible in Fig. 3. The relation between both locations shows a dependency on the coordinate S. While the location of maximum sub-cooling is near the leading edge, deposition starts before maximum sub-cooling is reached. On the other hand, if the location of maximum sub-cooling is near the trailing edge, deposition starts after maximum sub-cooling is reached. The reason is the increasing boundary layer thickness along the blade. As the thickness of the boundary layer increases, droplets have to travel a longer way from the nucleation point to the surface.

From the results it can be concluded that: If the interest is in the onset of deposition in stream-wise direction, it is sufficient to calculate the line of maximum sub-cooling in steam-wise direction as this is in good accordance with the line of deposition onset in stream-wise direction.

Influence of NRT on deposition calculation

As soon as steam undergoes sub-cooling, the potential for droplet formation exists. If the grade of sub-cooling is not sufficient, these droplets collapse in a little while. The macroscopic nucleation model used does not allow calculating the process of droplet collapse. Consequently, the formation and growth of droplets in even low sub-cooled steam will be calculated. However, the mass fraction of water droplets calculated in low sub-cooled steam is too small to have a noticeable influence on the temperature of the gas phase. Effectively, the macroscopic nucleation model gives the right results even though details are neglected. Nevertheless, the calculation of droplets in low sub-cooled vapour has an influence on the computing time. To speed up the CFD calculation, a threshold value NRT is specified, below which no droplets are initialized. This approach is also used by Gerber [START_REF] Gerber | Two-phase Eulerian/Lagrangian model for nucleating steam flow[END_REF] and Fakhari [START_REF] Fakhari | On the effects of unsteadiness on the condensation process in low-pressure steam turbines[END_REF]. The basic value is NRT = 1.0 x 10 +15 1/(m³•s) and is equal to that used by the previously mentioned authors. Lower threshold values should not have an influence on the sub-cooling as the additional droplets have no noticeable influence on the gas phase. Moreover, the subcooling course is the same for the basic value and a threshold value of 1.0 x 10 +10 1/(m³•s), as shown in Fig. 8. This figure shows the sub-cooling minus a reference value T sc-sr as a function of S M in an averaged manner. The maximum subcooling value and its location are identical for both calculations; the same is true for the increase and drop of the graph. Obviously, the amount of droplets generated below a NRT of 1.0 x 10 +15 1/(m³•s) is to small to reduce the subcooling. The heat released from the droplets to the gas phase is not big enough for heating-up the gas phase.

Even though the threshold value has no influence on the sub-cooling, it might have an influence on the deposition. This influence is shown in Fig. 9. The figure shows the blade pressure side in meridional view. The surface is coloured with the deposition rate in a logarithmic manner. No deposition takes place in areas that are coloured in black. One can see that the deposition onset is shifted upstream if the threshold value is reduced from 1.0 x 10 +15 1/(m³•s) to 1.0 x 10 +10 1/(m³•s). In this additional area of deposition, the rate is six to 10 magnitudes lower than in the rest of the area. A further decrease of the threshold value leads, once again, to a shift in the upstream direction. The deposition rate in the new area is again several magnitudes lower. The results show that the influence of NRT is low for the deposition rate integrated over the surface but is high for the onset of deposition in stream wise direction.

In conclusion it can be said that the nucleation model is accurate enough to calculate the integrated deposition rate but must be improved to accurate and reliable calculate the deposition onset in stream-wise direction.

Conclusions

Droplets in a broad diameter spectrum are deposited on the impeller blades of radial turbines when expanding a fluid below the saturation temperature. It is not sufficient to incorporate deposition due to inertia on a curved streamline alone. The influence of turbulent fluctuation and Brownian motion must be considered to predict the point of first deposition in a stream-wise direction to the right. For this calculation, a two-equation turbulence model can be used and the deposition calculation can be calibrated with the standard deviation.

With the used nucleation model and the widely accepted NRT value, it can be stated that deposition starts at the location of maximum sub-cooling. Consequently, the blade area that undergoes deposition is the area downstream of the line of maximum sub-cooling. In conclusion, it is not necessary to perform deposition calculations to determine the point of first droplet deposition in a stream-wise direction. It is sufficient to perform condensation calculations and determine the point of maximum sub-cooling.

Furthermore, the influence of NRT on the deposition is discussed. In conclusion, the position of deposition onset is shifted upstream when NRT gets smaller. Thereby, the deposition rate is reduced by several magnitudes. In a next step, two questions have to be answered. The first is a validation of the calculation. As mentioned, details of the nucleation process are not incorporated in the model. To estimate the correctness of the results, at first, condensation experiments in a Laval nozzle should be performed with simultaneous observation of the deposition. When the results are in line with the calculations of small NRT, then the next question must be answered: How do the small water portions behave?

Figure 1 .

 1 Figure 1. Deposition rate (Eq. 4) in a straight tube as a function of the non-dimensional relaxation time (Eq. 5)

Figure 2 .

 2 Figure 2. Sketch of the radial turbine

Figure 3 .

 3 Figure 3. Sub-cooling in the impeller flow channel and droplet deposition at the impeller flow channel surfaces of the investigated radial turbine Figure 4a/b show the results of the mesh independence study. In Fig.4athe sub-cooling T sc-sr , averaged in circumferential and in channel height direction with regard to a reference value, is plotted as a function of the stream-wise coordinate S M for different mesh resolutions. The solution on the medium mesh can be called mesh-independent. The same is true for the droplet deposition on the pressure side, as

T

  sc / °C on the S 1 -surface r zshown in Fig.4b. The points plotted are the location at which first deposition takes place in the stream-wise direction at constant relative channel height. The grey full lines show the hub and casing. The dashed lines illustrate the rotor blade contour.

Figure 4a .

 4a Figure 4a. Mesh independence study -sub-cooling in an extract of the impeller flow channel

Figure 5 .

 5 Figure 5. Influence of different droplet deposition mechanisms on the deposition onset on the impeller blade

Figure

  Figure3provides an initial indication that a relationship between the locations of maximum sub-cooling and deposition start is present. Fig.6, therefore, offers more detailed insight on the relation between both locations on the rotor blade pressure side. Three different calculations are performed using different inlet temperatures, where T o-r valued at zero is the design point temperature. With increasing inlet temperature, the location of maximum sub-cooling -shown using the black lines -is shifted in direction to the trailing edge. Up to a channel height of 90 %, the deposition beginshown with the red symbols -is also shifted to the trailing edge with increasing inlet temperature. At above 90%, the deposition starts near the blade's leading edge. Droplets that are deposited there originate from the tip clearance flow.Figure7shows the results on the suction side in the same manner as is shown in Fig.6for the pressure side. From the hub in direction to the casing, deposition starts at nearly constant stream-wise position first. Thereafter, droplet deposition is shifted to the leading edge. Qualitatively, the same is true for the location of maximum sub-cooling. The discrepancy between both locations on the suction side is caused by the secondary-flow phenomenon visible in Fig.3. The relation between both locations shows a dependency on the coordinate S. While the location of maximum sub-cooling is near the leading edge, deposition starts before maximum sub-cooling is reached. On the other hand, if the location of maximum sub-cooling is near the trailing edge, deposition starts after maximum sub-cooling is reached. The reason is the increasing boundary layer thickness along the blade. As the thickness of the boundary layer increases, droplets have to travel a longer way from the nucleation point to the surface.From the results it can be concluded that: If the interest is

z / z 0

 0 Pressure side, Inertia force only Pressure side, all forces Suction side, all forces

Figure 6 .

 6 Figure 6. Droplet deposition and sub-cooling on the blade pressure side

Figure 7 .

 7 Figure 7. Droplet deposition and sub-cooling on the blade suction side

Figure 8 .Figure 9 .

 89 Figure 8. Influence of NRT on the sub-cooling in the impeller flow channel