investigation on thrust and moment coefficients of a...

Investigation on thrust and moment coefficients of a centrifugal turbomachine

Bo Hu1*, Dieter Brillert1 , Hans Josef Dohmen1 , Friedrich-Karl Benra1

Abstract

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.

Keywords

Rotor-stator cavity — Centrifugal through-flow — Core swirl ratio — Pressure —Thrust coefficient

—Moment coefficient

1 Department of Mechanical Engineering, University of Duisburg-Essen, Duisburg, Germany

*Corresponding author: [email protected]

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

𝑎

𝑠

𝛺

𝑠𝑏

𝑧

b

𝑟

𝑡

Investigation on thrust and moment coefficients of a centrifugal turbomachine — 2

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).

(1.1)

(1.2)

(1.3)

(1.4)

, (1.5)

(1.6)

(1.7)

| |

(1.8)

( ) ( ) ，

(1.9)

∫ ( )

(1.10)

1. 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. Based on the experimental results from Poncet et al. (2005), the transition zone of the two flow types is

(a) Batchelor type flow ( )

(b) Stewartson type flow ( )

Figure 2. Velocity profiles for both Batchelor type flow and Stewartson type flow

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) and Debuchy et al. (2008) respectively determined Eq. (2.1) and Eq. (2.2) to predict the core swirl ratio K for Batchelor type flow. Poncet et al. (2005) derived a correlation of K for Stewartson type flow. The results from the three equations are depicted in 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. (2.1) to predict the values of K for Batchelor type flow.

Batchelor type flow:

Poncet et al. (2005):

( )

(2.1) Where Debuchy et al. (2008):

( )

(2.2)

Where Stewartson type flow:

(2.3) Where

K

(a) Comparison of the results from different equations

0

1

2

3

-0,6 -0,4 -0,2 0 0,2 0,4 0,6

𝑉𝑟

𝑉𝜑 x

𝑉𝑟

𝑉𝜑

x

𝐶𝑞𝑟

𝜁

𝜁


K

(b) Transition zone from Batchelor type flow to Stewartson type flow Eq. (2.1) Eq. (2.2) Eq. (2.3)

Figure 3. Main K- curves

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.

(3. 1)

(3. 2)

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.

( ) ∫

(

)

(4)

Where:

∫

∑

(

)

(m) ，

( )

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).

(5) (6)

( ) ( ) (7)

( )

(8)

2. 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.

(a) Cross section of the cavity model

(b) Simulation domain

Figure 4. Domain for numerical simulation at G=0.072

The mesh on the cross section at the position “I” and position “II” (see Figure 4) are depicted in Figure 5.

0

0,1

0,2

0,3

0,4

0,5

0 0,01 0,02 0,03 0,04 0,05

𝐶𝑞𝑟

Tran

siti

on

zo

ne

Eq. (

2.2

) an

d E

q. (

2.3

)

Disk

Simulation domain

Wall

Shaft

Pressure

inlet

Mass flow outlet

Tran

siti

on

zo

ne

Eq. (

2.1

) an

d E

q. (

2.3

)

I

II

Simulation domain

Q

Q


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.

(a) Position “I” (b) Position “II”

Figure 5. Mesh on the cutting plane

3. 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.

Figure 6. View of the test rig

Table 1. Parameters of the experiments b (mm) n (/min) Q (m3/s) s (mm) sb (mm) a (mm) t (mm)

110 0~2500 ~5.56 2~8 8 23 10

(V)

(I)

(VI) (II) (VII) (III) (VIII) (IV)

(IX) (X)

(I). 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

Figure 7. Cross section of the test rig

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. For computing the torque, the values when the shaft without the disk is rotating at different speeds of rotation are subtracted from the measured values. The relative error, , of the pressure transducers is 1% (FS). The

value of for the torque transducer is 0.1% (FS). The

value of for the axial thrust transducers is 0.5% (FS). All the experimental results are the ensemble average of 1000 samples. The uncertainties of the

Connected to an

electric motor

Q 𝐹𝑎

Disk Disk

Outlet

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.

Table 2. Uncertainties of the measured results

p (bar) (N) M (Nm) Re

4.04 2.43 3.00 9.01 4.1

4. RESULTS AND DISCUSSION

4.1 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. The dimensionless radial velocities

are not exactly zero in the central cores ( ),

shown in Figure 8 (a~c). From the distribution of

tangential velocity , there are central cores at all the

investigated radial positions where the values of are

almost constant along , shown in Figure 8 (d~f). The

values of the tangential velocity are smaller at

when 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].

x=0.955 x=0.79 x=0.57

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

=1262

=3787 =5050

Figure 8. Velocity profiles for Re= and G=0.072

The velocity profiles at the three radial coordinates for Re=1.9× and G=0.018 (small gap) are shown in

-0,1

-0,05

0

0,05

0,1

0 0,5 1

-0,1

-0,05

0

0,05

0,1

0 0,5 1

-0,1

-0,05

0

0,05

0,1

0 0,5 1

0

0,2

0,4

0,6

0,8

1

0 0,5 1

0

0,2

0,4

0,6

0,8

1

0 0,5 1

0

0,2

0,4

0,6

0,8

1

0 0,5 1

-0,01

-0,005

0

0,005

0,01

0 0,5 1-0,01

-0,005

0

0,005

0,01

0 0,5 1

-0,01

-0,005

0

0,005

0,01

0 0,5 1

𝜁 𝜁 𝜁

Disk Wall

𝜁 𝜁 𝜁

𝜁 𝜁 𝜁


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].

x=0.955 x=0.79 x=0.57

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

=1262

=3787 =5050

Figure 9. Velocity profiles for Re= and G=0.018

4.2 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

(x=0.455) to r=0.105 m (x=0.954). Since the radial

distances between the adjacent pressure tubes are small,

the application of the average values of K between the

tubes may not result in large errors. The values of for

the experimental results are calculated with

.

The values of K ( ) therefore can be verified

based on the pressure measurement with Eq. (10).

( ) ( )

(

)

(

)

(9)

-0,1

-0,05

0

0,05

0,1

0,15

0 0,5 1-0,05

0

0,05

0,1

0 0,5 1

-0,1

-0,05

0

0,05

0,1

0 0,5 1

0

0,2

0,4

0,6

0,8

1

0 0,5 10

0,2

0,4

0,6

0,8

1

0 0,5 1

0

0,2

0,4

0,6

0,8

1

0 0,5 1

-0,01

-0,005

0

0,005

0,01

0 0,5 1

-0,01

-0,005

0

0,005

0,01

0 0,5 1

-0,01

-0,005

0

0,005

0,01

0 0,5 1

𝜁 𝜁 𝜁

Disk Wall

𝜁 𝜁 𝜁

𝜁 𝜁 𝜁


√ ( ) ( )

(

)

(

)

(10)

Based on the results from both numerical

simulations and pressure measurements, Eq. (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.

Batchelor/Couette type flow ( ):

( )

( )

Stewartson type flow ( ):

( ) (

) (11)

K

G=0.018, Eq. (11) G=0.036, Eq. (11) G=0.054, Eq. (11) G=0.072, Eq. (11)

G=0.018, Sim G=0.018, Exp G=0.036, Sim G=0.036, Exp

G=0.054, Sim G=0.054, Exp G=0.072, Sim G=0.072, Exp

Figure 10. Mean - curves

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.

(11). The correctness of Eq. (12) will be further verified

based on the LDV measurements in the future.

Batchelor/Couette type flow ( ):

( )

Stewartson type flow ( ):

(

) (12)

0,01

0,11

0,21

0,31

0,41

0,001 0,01 0,1

Tra

nsitio

n z

on

e o

f

the tw

o f

low

typ

es

Batchelor/ Couette type flow

Stewartson type flow

𝐶𝑞𝑟


K

G=0.018, Sim G=0.018, Exp G=0.036, Sim G=0.036, Exp G=0.054, Sim

G=0.054, Exp G=0.072, Sim G=0.072, Exp Eq. (12)

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 (b)). Based on the simulation results,

there are small vortices near the outer radius of the

disk. Hence, part of the measured values at x=0.955

are not used during the calculation of K.

K

(a) Part of the results do not fit Eq. (12) (b) Surface streamlines near the outlet (from simulation)

Figure 12. Large differences of K attributed to the geometry near the outlet

4.2 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).

0,01

0,11

0,21

0,31

0,41

0,001 0,01 0,1

0

0,1

0,2

0,3

0,4

0,5

0,001 0,01 0,1

𝐶𝑞𝑟

𝑥

Batchelor/ Couette type flow Stewartson type flow

𝑥

𝐶𝑞𝑟

Tra

nsitio

n z

on

e o

f

the tw

o f

low

typ

es


x x x

(a) Re=0.76 , G=0.072 (b) Re=0.76 , G=0.036 (c) Re=0.76 , G=0.018

x x x

(d) Re=2.79 , G=0.072 (e) Re=2.79 , G=0.036 (f) Re=2.79 , G=0.018

=0, Exp

=0, Eq. (4) =1262, Exp

=1262, Eq. (4)

=3787, Exp

=3787, Eq. (4) =5050, Exp

=5050, Eq. (4)

Figure 13. Distribution of along the radius

4.3 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:

( ) ( )

( ) (13)

Where 0.018 G 0.072, 5050, Re .

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).

(a) G=0.018 (b) G=0.036

0,4

0,6

0,8

1

0 0,05 0,1 0,150,4

0,6

0,8

1

0 0,05 0,1 0,150,4

0,6

0,8

1

0 0,05 0,1 0,15

0,4

0,6

0,8

1

0 0,05 0,1 0,150,4

0,6

0,8

1

0 0,05 0,1 0,150,4

0,6

0,8

1

0 0,05 0,1 0,15

0

0,01

0,02

0,03

0,04

0,3 1,3 2,3 3,3

0

0,01

0,02

0,03

0,04

0,3 1,3 2,3 3,3

𝑅𝑒 ( )

𝐶𝐷

𝑅𝑒 ( )

𝐶𝐷

𝐶𝑝 𝐶𝑝

𝐶𝑝

𝐶𝐷 𝐶𝐷

𝐶𝑝

𝐶𝑝

𝐶𝐷

𝐶𝑝

𝐶𝐷

𝐶𝐷 𝐶𝐷


(c) G=0.054 (d) G=0.072

=0 Exp Bp Eq. (13)

= Exp Bp Eq. (13)

= Exp Bp Eq. (13)

= Exp Bp Eq. (13)

= Exp Bp Eq. (13)

Figure 14. Mean - curves 4.4 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.

(a) Regime III (b) Regime IV

Figure 15. Typical tangential velocity profiles for regime III and regime IV

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.

G G

(a) Distinguishing lines (b) Distinguishing surface

Figure 16. Part of the 3D DailyNece diagram

0

0,01

0,02

0,03

0,04

0,3 1,3 2,3 3,3

0

0,01

0,02

0,03

0,04

0,3 1,3 2,3 3,3

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,1 1 10

𝐶𝐷 [18]

𝐶𝐷

𝐶𝐷

𝐶𝐷

𝜁

𝐶𝐷

𝑅𝑒 ( )

Centripetal

through-flow

Centrifugal

through-flow

𝐶𝐷 𝐶𝐷

𝑅𝑒 ( ) 𝑅𝑒 ( )

𝑅𝑒 ( )

𝐶𝐷

𝐺

𝐺

𝐺

𝐺

𝜁


4.5 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.

| |

𝑡

(𝑙𝑔

𝜐)

(14)

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 III

(

) (15)

Regime IV

(

) (16)

The experimental results of are compared with

those from Eq. (15) and Eq. (16), depicted in Figure 17.

For G=0.018 and G=0.036, most of the flow regimes are

regime III, shown in Figure 17 (a) and Figure 17 (b). The

experimental results of are in good accordance with

those from Eq. (15) in general. When G increase to

0.036, the flow regimes change from regime III to

regime IV with the increase of Re for =0, 1262 and

2525, which can also be found based on the

experimental results of . For G=0.054 and G=0.072,

most of the flow regimes are regime IV. The results of

from experiments are in good agreement with those

from Eq. (16) in general. The regime III may occur at

small Re and large . At the same values of

, the

intersection points of the curves from Eq. (15) and those

from Eq. (16) are close to those in Figure 16 (a)

between the curves for G and . The difference can

be attributed to the existence of the mixing zone. The

amounts of increase with the increase of . The

values of drop faster with the increase of Re for

smaller values of G. Compared with those for

centripetal through-flow (Bo Hu et al. [18]), the

centrifugal through-flow will result in larger values of

at the same values of | |, which is in accordance with

the conclusion of Dibelius et al. [22].

(a) G=0.018 (b) G=0.036

(c) G=0.054 (d) G=0.072

=0 Exp Eq. (15) Eq. (16)

Exp Eq. (15) Eq. (16)

= Exp Eq. (15) Eq. (16)

Exp Eq. (15) Eq. (16)

= Exp Eq. (15) Eq. (16)

Figure 17. Main - curves in dependence of G

0,0005

0,001

0,0015

0,30 1,30 2,30 3,30

Millionen

0,0005

0,001

0,0015

0,30 1,30 2,30 3,30

Millionen

0,0005

0,001

0,0015

0,3 1,3 2,3 3,3

0,0005

0,001

0,0015

0,30 1,30 2,30 3,30

Millionen

𝑅𝑒 ( )

𝑅𝑒 ( )

𝑅𝑒 ( )

𝑅𝑒 ( )


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.

/

1262

2525

3787

5050

Figure 18. Results of / at the distinguishing lines

There are still some limitations of this work. All the

experimental results are obtained with the smooth disk

( ). 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.

5. 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.

NOMENCLATURE

a Hub radius

Bp Based on the pressure calculation

b Outer radius of the disk

Through-flow coefficient

Axial thrust coefficient

on the front surface

on the back surface

Moment coefficient

Moment coefficient on the cylinder surface of

the disk

for regime III

for regime IV

Pressure coefficient

Local flow rate coefficient

c Constant

Relative error of the transducer

e Pressure tube number

Fa Axial thrust

Faf Force on the front surface of the disk

Fab Force on the back surface of the disk

FS Full scale

G Dimensionless axial gap

K Core swirl ratio at =0.5

Average value of K between the two adjacent

0,950,960,970,980,99

11,011,021,031,041,05

0,3 1,3 2,3 3,3

𝑅𝑒 ( )

5%

1%


pressure tubes

LDV Laser Doppler Velocimetry

Frictional torque

Frictional resistance on the cylinder surface of

the disk

Mass flow rate

n Speed of rotation

Uncertainty of the measured results

p Pressure

Pressure at r=b

Dimensionless pressure

Q Volumetric through-flow rate

Re Global circumferential Reynolds number

Reφ Local circumferential Reynolds number

r Radial coordinate

SST Shear stress transport

s Axial gap of the front chamber

sb Axial gap of the back chamber

t Thickness of the disk

Dimensionless radial velocity

Dimensionless axial velocity

Dimensionless tangential velocity

x Dimensionless radial coordinate

Axial coordinate

Dimensionless axial coordinate

Dynamic viscosity of water

Kinematic viscosity of water

Density of water

Angular velocity of the disk

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