Investigation on thrust and moment coefficients of a centrifugal turbomachine

In radial pumps and turbines, the centrifugal through-flow is quite common, which has strong impacts on the core swirl ratio, pressure distribution, axial thrust and frictional torque. The impact of centrifugal through-flow on above parameters are still not sufficiently investigated with different circumferential Reynolds numbers and dimensionless axial gap widths. A test rig is designed at the University of Duisburg-Essen and descirbed in this paper. Based on the experimental results, correlations are determined to predict the impact of the centrifugal through-flow on the core swirl ratio, the thrust coefficient and the moment coefficient with good accuracy. Part of the 3D Daily&Nece diagram from a former study of the authors is extended with centrifugal through-flow. The results will provide a data base for calculation of axial thrust and moment coefficient in order to design radial pumps and turbines with smooth impellers.


INTRODUCTION
Rotor-stator cavities are common devices in radial pumps and turbines. The typical geometry of a rotor-stator cavity is shown in Figure 1.

Figure 1. Geometry of a rotor-stator cavity
The through-flow in such cavities can be either radial inward or radial outward and it impacts the radial pressure distribution acting on the turbomachine rotor in a certain manner. The study of the flow in a rotor-stator cavity has significant relevance to many problems encountered in turbomachinery. The thrust coefficient and the moment coefficient are two major concerns in radial pumps and turbines. The investigation of the flow in rotor-stator cavities can provide more confidence for calculating the axial thrust (direction see Figure 1) and the frictional torque M in radial pumps and turbines.
Since evaluating and is quite important for the design of turbomachinery, a lot of researches are accomplished on these topics. Von Kármán [1] and Cochran [2] gave a solution of the ordinary differential equation for the steady, axisymmetric, incompressible flow. Daily and Nece [3] examined the flow of an enclosed rotating disk both analytically and experimentally. Kurokawa et al. [4~6] studied and in a rotor-stator cavity with both centrifugal and centripetal through-flow. Poncet et al. [7] studied the centrifugal through-flow in a rotor-stator cavity and obtained two equations of the core swirl ratio K for both the Batchelor type flow and the Stewartson type flow based on the local flow rate coefficient (positive for centrifugal through-flow). Schlichting and Gersten [8] organized an implicit relation based on the results of Goldstein [9] for under turbulent flow conditions. Debuchy et al. [10] determined an explicit equation of K for the Batchelor type flow which is valid over a wide range of the local flow rate coefficient: 0.5 (negative for centripetal through-flow). Launder et al. [11] provided a review of the current understanding of instability pattern that are created in rotor-stator cavities leading to transition and eventually turbulence. Will et al. [12~14] investigated the flow in the side chamber of a radial pump. Recent experimental investigations for large global Reynolds number with or without through-flow have been conducted by Coren et al. [15], Long et al. [16] and Barabas et al. [17]. Based on the experimental results, Bo Hu et al. [18] determined a b correlation to calculate the values of in a rotor-stator cavity with centripetal through-flow. They also extended part of the 2D Daily&Nece diagram into 3D by distinguishing the tangential velocity profiles with a third axis of through-flow coefficient based on the simulation results. Based on the experimental results, two equations were determined to describe the impact of , Re and the dimensionless axial gap width G on for regime III (merged disk boundary layer and wall boundary layer, namely Couette type flow) and regime IV (separated disk boundary layer and wall boundary layer, namely Batchelor type flow).
This study is focused on the impact of centrifugal through-flow on and , so that the influence of both the centripetal (Bo Hu et al. [18]) and the centrifugal through-flow can be better understood. The definitions of the significant dimensionless parameters in this study are given in Eq. (1.1~1.10).

THEORETICAL ANALYSIS
In a rotor-stator cavity with centrifugal through-flow, Batchelor type flow and Stewartson type flow are quite common. Their main profiles of the dimensionless tangential velocity and the dimensionless radial velocity along are shown in Figure 2. To predict the axial thrust, the pressure distribution along the radius of the disk should be estimated. The pressure distribution can be calculated with the core swirl ratio K. With the increase of , the flow type may change from Batchelor type flow to Stewartson type flow. Using a two-component LDV system, Poncet et al. (2005) Figure 3 (a). Figure 3 (b) depicts the transition zone from the Batchelor type flow to the Stewartson type flow. Since the transition zone is very small, Eq. (2.2), which is valid for a wider range, is selected for modification in this paper instead of Eq.  A plenty of researches, such as those by Kurokawa et al. [6] and Poncet et al. [7], show that the pressure distribution along the radius of the disk can be estimated with the core swirl ratio K with Eq. (3.1) both with and without through-flow. Will et al. [12~14] determined Eq. (3.2) to evaluate the pressure distribution along the radius of the disk for the incompressible, steady flow. It is obtained directly from the radial momentum equation when the turbulent shear stress is neglected. In a rotor-stator cavity, the cross sectional area changes in the radial direction. Consequently, the pressure must also change since the mean velocity changes in the radial direction according to the continuity equation. Based on Eq. (3.2), the pressure along the radius can be calculated with Eq. (4) based on the values of K. K is a variable along the radius of the disk. A simplification is made as follows: K is a fixed value every 1 mm in the radial direction. Then, the approximate pressure distribution along the disk can be calculated with Eq. (4).
represents the pressure at x=1. Due to the construction of the geometry, there is no pressure tube at x=1. The closest pressure tube in the front cavity is at x=0.955. The value of is calculated combining the measured pressure at x=0.955 with Eq. (4) based on the core swirl ratio along the radius.
The difference of the force on both sides of the disk is the main source for the axial thrust , calculated with Eq. (5).
(calculated with Eq. (6)) and respectively represent the force and the thrust coefficient on the front surface of the disk (in the front chamber, shown in Figure 1), while (calculated with Eq. (7)) and are those on the back surface of the disk (in the back chamber). represents the radius of the hub (see Figure 1). The back chamber (G=0.072), shown in Figure 1, is viewed as an enclosed cavity. The values of are obtained when =0 and the axial gaps of the both cavities have the same size for different Re (under that condition = ). After obtaining those values, the values of with different values of can be calculated with Eq. (8).

NUMERICAL SIMULATION
To predict the cavity flow, numerical simulations are carried out using the ANSYS CFX 14.0 code. Considering the axial symmetry of the problem, a segment (15 degree) of the whole domain is modeled and a rotational periodic boundary condition is applied. Structured meshes are generated with ICEM 14.0. The domain for numerical simulation when G=0.072 is depicted with yellow color in Figure 4.  The mesh on the cross section at the position "I" and position "II" (see Figure 4) are depicted in Figure 5. The simulation type is set as steady state. Barabas et al. (2015) found that the simulation results from the SST turbulence model in combination with the scalable wall functions are in good agreement with the measured pressure in a rotor-stator cavity with air. The deviations of the pressure measurements are less than 1%. Hence, in this study, the same turbulence model and wall functions are used. The turbulent numeric is set as second order upwind. The non-slip wall condition is set for all the walls. The boundary conditions at the inlet and the outlet are pressure inlet and mass flow outlet, respectively. The values of the pressure at inlet are set according to the pressure sensor at the pump outlet. The convergence criteria are set as in maximum type. The maximum value of in all the simulation model is 13.4.

TEST RIG DESIGN AND EXPERIMENTAL SET-UP
The test rig is supplied with water by a pump system, shown in Figure 6. The shaft is driven by an electric motor. A frequency converter is used to adjust the speed of rotation (0~2500/min) with the absolute uncertainty of 7.5/min. In this study, only the axial gap of the front chamber is changed by installing six sleeves with different length. Other parameters of the experiments in this study are given in Table 1.The cross section of the test rig is shown in Figure 7.
Sleeves (to change the axial gap), (II). Guide vane (24 channels), (III). Front chamber, (IV). Disk, (V). Back cover, (VI). Linear bearing, (VII). Tension compression sensor, (VIII). Thrust plate, (IX). Nut, (X). Shaft The transducers in the test rig include two pressure transducers (36 pressure tubes, 12 in the front chamber, 24 in the back chamber), a torque transducer and three tension compression transducers. A thrust plate is fixed by a ball bearing and a nut from both sides to convey the axial thrust to the tension compression transducers. A linear bearing is used to minimize the frictional resistance during the axial thrust measurements. The measured of the disk is 1 μm. The values of on all the other surfaces of the test rig are below 1.6 μm.
During the measurements of axial thrust, the calibration of the axial thrust transducers is performed when changing the axial gap width of the front chamber.  Wall measured results are estimated with the root sum squared method. The measured range of the torque meter is 0~10 The measured range of the pressure transducer is 0~2.5 bar (absolute pressure). The measured range of the thrust transducers is -100~100 N. The input voltage signals are the following ranges: 0~10 V for the pressure transducers and the torque transducer, -10 V~10 V for the axial thrust transducers. The absolute accuracy of the data acquisition system (with NI USB-6008) is 4.28 mV in this study. The random noise and zero order uncertainty are neglected because they are very small. The uncertainties of the measured results, noted as , are calculated in a former study of the authors (Bo Hu et al. [18]), given in Table 2.

Velocity distribution
All the velocities are made dimensionless by dividing .The velocity profiles at three radial positions for Re=1. 9 and G=0.072 (wide gap) are shown in Figure 8 increases, depicted in Figure 8 (d, e). The trend of are in good agreement with the measured in the literature (such as from Poncet et al. [7] and Debuchy et al. [10]). The values of | | become smaller towards the shaft. The velocities for are in the reference [18]. Disk Wall Figure 9. The dimensionless radial velocities vary along , shown in Figure 9 (a~c). The values of increase with the increase of in general. At and , all the values of are positive (all the boundary layer are centrifugal). The flow type is therefore Stewartson type flow. The tangential velocity decreases constantly from the disk to the wall, which is the characteristic of the regime III, shown in Figure 9 (d~f). The values of | | are very small, compared with those in Figure 9. This indicates that the axial circulation of the fluid is weaker for small axial gap width. The velocities for are in the reference [18].

Main K curves
To evaluate the pressure distribution, the values of K should be estimated. Although some correlations are determined to predict the values of K with centrifugal through-flow, such as Eq. (2.1), Eq. (2.2) and Eq. (2.3), there is still an uncertainty on the impact of G on K. The geometry of the cavity，especially at the inlet and the outlet, will also have large influence on K. Based on Eq. (4), the pressure difference between the two pressure tubes number e and number e+1 can be calculated with Eq. (9). ̅ represents the average value of K between the two adjacent pressure tubes. There are 12 pressure tubes in the front chamber from r=0.05 m  11) is determined to describe the impact of G on K. The experimental results based on pressure measurements are compared with those from simulation and those calculated by Eq. (11) in Figure 10.
The results from Eq. (11) are in good agreement with those from numerical simulations and experiments.
Relatively large errors only occur when 0.01, which can be attributed to the application of the average values of K in and around the transition zone of the two flow types, where K decreases dramatically. In the future, more pressure taps will be manufactured at the low radius to eliminate the error.  The influence of G on K is weak based on the results in Figure 10. Poncet et al [7] and Debuchy et al [10] ignored the impact of G on K based on the results from LDV measurements. In most of the radial pumps and turbines, G is a variable along the radius. A simplified correlation is required with good accuracy over the whole range of G. Based on the measurements, Eq. (12) is correlated to predict the values of K when G ranges from 0.018 to 0.072. The results are compared in Figure 11. The results from Eq. (12) are in good accordance with both the simulation results and the experimental results. In this paper, Eq. (12) is applied during the calculation of the pressure instead of Eq.   Figure 11. Mean -curves There are some results of K, which, however, do not fit the resuts from Eq. (12), especially at x=0.955 for wider gaps. Some of the results are shown in Figure 12 (a). Near the outlet, there is a area change from the front cavity to the channel in the guide vane for G=0.036, 0.054 and 0.072. (see Figure 1 and Figure 4).
The measured pressure at x=0.955 is strongly influenced by the geometry at the outlet of the testrig (see Figure 12

Pressure coefficient
A reference pressure is taken at the dimensionless radial coordinate x=1. Due to the restriction of the geometry of the test rig, there is no pressure tube at x=1. The closest tube is at x=0.955. The pressure values at x=1 are from Eq. (4) based on the values of K from Eq. (12). The values of pressure coefficient are positive because the pressure drops towards the shaft. In Figure 13, the values of are plotted versus the non-dimensional radial coordinate x. The through-flow coefficient is used as a parameter. The experimental results show that decreases with the increasing , Re and G in general. The experimental results are in good agreement with those from equations. When Re=2. 79 , the uncertainty of the is 1. 3 , which is very small compared with the measured results. Hence, the error bars are neglected in Figure 13 (d~f).   Figure 13. Distribution of along the radius

Axial thrust
Based on the measurements, Bo Hu et al. [18] determined an empirical equation for the thrust coefficient in a rotor-stator cavity with centripetal through-flow. It is organized based on the experimental results for centripetal through-flow. When compared with the experimental results in this paper, it is modified for centrifugal through-flow. In this study, is positive for centrifugal through-flow. It is written as: The comparison of the results of for different G and are shown in Figure 14. Bp represents the calculated thrust coefficient based on the pressure calculation along the radius of the disk, which are calculated combining the measured pressure with Eq. (4) based on the values of K (calculated based on every 1 mm along the radius for Batchelor type flow or Stewartson type flow) from Eq. (12). In the transition zone, the equation of K for Batchelor type flow is used. The values of are smaller for wider axial gaps in general. The values of decrease with increasing . In a rotor-stator cavity with centripetal through-flow ( is negative) studied by Bo Hu et al. [18], however, the values of increase with increasing | |. The experimental results of are in good agreement with those based on the pressure calculation and Eq. (13).

Part of 3D Daily&Nece diagram
The moment coefficients can be predicted according to the flow regimes. The typical tangential velocity profiles for regime III and regime IV are shown in Figure 15. In this study, the 2D Daily&Nece diagram is extended with centrifugal through-flow by classifying the tangential velocity profiles at x=0.945, x=0.79 and x=0.57 based on the results of numerical simulation. Currently, four distinguishing lines are found, depicted in Figure 16 (a). Below and above the distinguishing lines are regime III (small axial gap, turbulent flow, merged boundary layers) and regime IV (large axial gap, turbulent flow, separated boundary layers), respectively. The distinguishing surface is drawn through the distinguishing lines, shown in Figure 16 (b). Near the distinguishing surface, there is a mixing zone, where regime III and regime IV coexist in the cavity (ignored in this study). The distinguishing surface for centripetal through-flow (Bo Hu et al. [18]) is also plotted to make it a complete diagram.

Moment coefficient
According to the experimental results from Han et al. [19], the moment coefficient on the cylinder surface of the disk, noted as , can be estimated with Eq. (14) for smooth disks.
Comparing the torque measurements by the authors with the results from Daily and Nece [3] and Dorfman [20] [21], two correlations are determined to predict the moment coefficient (for a single surface of the disk), given in Eq. (15) and Eq. (16).
Regime IV ( ) The experimental results of are compared with those from Eq. (15) and Eq. (16), depicted in Figure 17.   On the distinguishing lines (see Figure 16), the results from Eq. (15) should be equal to those from Eq. (16). The results of / at the distinguishing lines are presented in Figure 18. The differences, attributed to the existence of the mixing zone, are very small in general and cover an amount less than 5%. Based on the results from Figure 18, Eq. (15) and Eq. (16) can be used to predict the transition from regime III to regime IV with good accuracy. All the results show that the moment coefficient can be predicted with Eq. (15) and Eq. (16) based on the 3D Daily&Nece diagram. ). The applications of the equations will become wider by introducing the impact of surface roughness of the disks in the next step. All the results of K will be verified based on the velocity measurements with a two-component LDV system. The distinguishing lines will be modified based on the measured velocity components in both tangential and radial directions in the future. The outlet geometry has a relatively large influence on the results of K, which deserves further study. The impacts of boundary condition (at both inlet and outlet) and internal flow structures on and should also be investigated in the future.

CONCLUSIONS
The influence of centrifugal through-flow on the velocity, radial pressure distribution, axial thrust and frictional torque in a rotor-stator cavity with different G is strong.
Based on the pressure measurements, an empirical correlation is determined to predict the impact of Re, on K when G ranges from 0.018 to 0.072. A correlation is determined, which enables to predict the influence of G, Re and on the thrust coefficient for a smooth disk ( ). Part of the 3D Daily&Nece diagram is obtained by distinguishing the tangential velocity profiles for centrifugal through-flow. Four distinguishing lines and the approximate distinguishing surface are presented.
Two correlations are determined to predict the influence of centrifugal through-flow on for the two zones with good accuracy for the smooth disk ( ). At the distinguishing lines, the results from the two equations are very close. The values of for centrifugal through-flow exceed those for centripetal through-flow at the same values of | |.
Using the equations for the axial thrust coefficient and the moment coefficient, the influence of the centrifugal through-flow can be better predicted when designing radial pumps and turbines with smooth impellers. This makes the correlations of a huge worth for the designers.
Some more attention will be drawn in the future to the impact of the disk roughness. The 3D Daily&Nece diagram and Eq. (12) will also be modified based on the velocity measurements with a two-component LDV system.

ACKNOWLEDGMENTS
This study is funded by CSC (China Scholarship Council) and the chair of turbomachinery at University of Duisburg-Essen.