effects of relative position between a stator and a rotor

1

International Journal of Fluid Machinery and Systems DOI: http://dx.doi.org/10.5293/IJFMS.2018.11.1.001

Vol. 11, No. 1, January-March 2018 ISSN (Online): 1882-9554

Original Paper

Effects of Relative Position between a Stator and a Rotor

on Steam Condensing Flow in Rotating Machinery

Changhyun Kim1, JaeHyeon Park2 and Jehyun Baek1

1Department of Mechanical Engineering, POSTECH

77 Cheongam-Ro, Nam-Gu, Pohang, 37673, Republic of Korea,

[email protected], [email protected] 2Thermal & Fluid Research Team, Doosan Heavy Industries & Construction

112 Suji-Ro, Suji-Gu, Yongin, 16858, Republic of Korea, [email protected]

Abstract

Flow of steam is different from other gas flows and involves droplet generation in the flow field. This phase-

transition affects not only flow aspects, but also machine performance in a negative way. These days, CFD is widely

used in machine design and optimization processes, so phase-transition phenomena in steam flows should be considered

in CFD to predict the internal flows precisely. In the past, non-equilibrium wet-steam model was implemented on in

house code T-Flow and it was applied to steady calculations of a steam turbine model with changing stator/rotor

interface. The results showed that mixing plane method is not appropriate to simulate steam condensing flow and frozen

rotor method can be affected by relative position between a stator and a rotor. Therefore, steady wet-steam flows with

non-equilibrium phase-transition were simulated for a steam turbine model in this study with 4 different stator-rotor

positions and their effects on the wet-steam flow fields were investigated in detail.

Keywords: Non-equilibrium wet-steam model, Rotating machinery, Steam condensing flow.

1. Introduction

Flow of steam is different from other gas flows and involves droplet generation in the flow field. This phase-transition affects not

only flow aspects, but also machine performance in a negative way. In addition, it is totally harmful for machine structures as blades

and casing. For an instance, phase-transition in the last stages of LP steam turbine [1] changes the internal flow field and lowers the

turbine performance. Therefore, prevention/preparation of droplet generation in steam flow is quite important issue in fluid machinery.

These days, computational fluid dynamics (CFD) which is a powerful tool to predict internal flow field is widely used in machine

design and optimization processes. Therefore, calculations using CFD should consider the droplet generation phenomena to simulate

internal flows precisely. However, all of the phase-transition phenomena including cavitation and condensation are still the challenging

issues in numerical analysis due to complexity, and various models are still developing to model them accurately.

Many studies analyzed steam condensing flows in Laval nozzles and blade cascades, and simulations agreed well with experiments

in some cases. Simpson et al. [2] studied steam condensing flow in a Moore nozzle C using numerical calculation. Dykas et al. [3]

simulated wet-steam flows in arc Laval nozzles using SFM and TFM, and compared each other. Halama [4] studied inviscid, laminar

flows in the blade cascade by considering non-equilibrium phase-transition. Zhu et al. [5] simulated steam condensing flows in a

Barschdorff nozzle and a Bakhtar cascade with QMOM. Nevertheless, wet-steam flows which include non-equilibrium phase-

transition phenomena are still difficult to predict, especially in rotating cases as steam turbines due to 3D complex flow structures.

Therefore, several studies were conducted to simulate steam condensing flows in steam turbines. Bohn et al. [6] focused on nucleation

phenomena occurred in LP steam turbine. Dong et al. [7] simulated steam condensing flow with inhomogeneous model in LP steam

turbine. Chandler et al. [8] studied unsteady low-pressure turbine flows with non-equilibrium wet-steam calculations. Halama et al. [9]

simulated transonic flow of wet-steam in nozzles and turbines. Grubel et al. [10] and Schatz et al. [11] conducted both of experiment

and simulation in LP turbines, and compared them in detail. Starzmann et al. [12] tried to simulate unsteady wet-steam flow in LP

steam turbine, which requires high computational costs.

In POSTECH, non-equilibrium wet-steam model was implemented on in house code T-Flow and steam condensing flows in various

Laval nozzles and blade cascades were numerically studied [13,14]. In addition, it was applied to steady calculations of a steam turbine

model with changing stator/rotor interface [15]. Resultingly, it was found that using mixing plane method can distort and overestimate

This paper was presented at 7th International Symposium on Fluid Machinery and Fluids Engineering(ISFMFE), October 18-22, 2016, Jeju, Korea

Received February 13 2017; accepted for publication August 30 2017: Review conducted by Hyung Hee Cho. (Paper number O17005S)

Corresponding author: Jehyun Baek, [email protected]

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wetness distribution in the rotor passage during mixing process, so use of mixing plane method is not recommended when simulating

steady wet-steam flow. On the other hand, results using frozen rotor method can be affected by relative positions of stator and rotor

blades, so further study on the effects of relative position between a stator and a rotor for the steam condensing flow in rotating

machinery was demanded.

Therefore, steady wet-steam flows with non-equilibrium condensation were simulated for a steam turbine model in this study with 4

different stator-rotor positions and their effects on the wet-steam flow fields were investigated in detail.

2. Theoretical Backgrounds

2.1 T-Flow

In-house code T-Flow used in this study was started to develop from 1990s to calculate 3D flow fields. It is used to predict the

internal flows in rotating machinery and was applied to many studies of tip leakage flow, clocking effect, stator/rotor interaction

in centrifugal, axial compressors [16-18] and axial turbines [19,20]. T-Flow solves compressible RANS (Reynolds-Averaged

Navier-Stokes) equations given in eq. (1).

( ) ( ) ( )0v v vE E F F G GQ

t x x x

(1)

2

2

2

[ , , , , ]

[ , , , , ( )]

[ , , , , ( )]

[ , , , , ( )]

[0, , , , ]

[0, , , , ]

[0, , , ,

T

T

T

T

T

v xx xy xz xx xy xz x

T

v yx yy yz yx yy yz y

v zx zy zz zx

Q u v w e

E u u p uv uw u e p

F v uv v p vw v e p

G w uw vw w p w e p

E u v w q

F u v w q

G u v

]T

zy zz zw q

Here, Q is the vector which consists of conservative variables, E~G are inviscid flux vectors and Ev~Gv are viscous flux

vectors. x~z are 3 directions in Cartesian coordinate system and u~w are velocity components in each direction. T is used for

transpose matrices, t, e, q and τ are time, internal total energy, heat flux and Reynolds stress, respectively. The governing

equations are discretized by finite volume method in space and Euler implicit time marching scheme is used. Flux difference

splitting method suggested by Roe (or flux vector splitting method by Van-Leer), upwind scheme and MUSCL techniques are

used for inviscid flux terms. Central difference scheme is used for viscous flux terms. Local time stepping method based on CFL

number and MPI technique are used for rapid calculation.

2.2 Wet-Steam Model

Condensation, a phase-transition phenomenon in steam flow, frequently takes place in non-equilibrium states during rapid

expansion. In this case, the boundary of vapor-to-liquid transition is not a saturation line, but a Wilson line where the supercooled

vapor starts to condense. To consider this kind of unusual phase-transition in calculation, non-equilibrium wet-steam model is

needed and two additional equations related with wet-steam parameters are implemented on T-Flow [21].

( ) ( ) ( ) ( )u v w

t x y z

(2)

( ) ( ) ( ) ( )u v wI

t x y z

(3)

The main parameters in eq. (2) and eq. (3) are the liquid mass fraction (wetness, β) and the number of droplets per unit volume

(droplet number, η). The source term of eq. (2) is eq. (4), which is related to newly created droplets and growth of extant droplets.

Radius of newly created droplets from nucleation is represented as r* and its size is determined by the pressure and surface tension

of water as in eq. (5). The growth of droplets, eq. (6), is defined by temperature difference between the phases. The source term of

eq. (3) is eq. (7) which is the number of newly created droplets per unit time and unit volume (nucleation rate). The θ term in

nucleation rate is a non-isothermal correction factor (eq. (8)). Finally, the droplet size can be determined from calculated wetness

and droplet number values as in eq. (9).

3 2

*

44

3d d

rIr r

t

(4)

*

2

lnd c

rRT S

,

( )sat c

PS

P T (5)

3

1

( )22

p d c

lv d c

r PC T T

t h RT

(6)

2 2

*

3

42exp

(1 ) 3

c c

d m b c

q rI

M K T

(7)

2( 1)

0.5( 1)

lv lv

c c

h h

RT RT

(8)

33

4 d

r

(9)

From the calculation of wet-steam parameters using eqs. (2) and (3), the source terms of each phase can be defined and two

sets of governing equations are required for vapor and liquid phases in principle (inhomogeneous approach) [22]. However, since

the main solver T-Flow is a density based solver which is used to calculate single-phase flow, two-phase flow can be modelled by

assuming that the vapor and liquid are well mixed in a single mixture (homogeneous approach) [13-15]. The net amounts (vapor +

liquid) of inter-phase transfer rates of mass and momentum are assumed to be zero owing to no external gain or loss. On the other

hand, the latent heat released from condensation works as an energy source, so that the net amount of inter-phase transfer rate of

energy is not zero and can be represented as follows (eq. (10)).

3 / ( ) ( )c d lvr q q mh (10)

3α/r means the surface density of droplet and qc, qd are the transferred heat flux to vapor and liquid phases from released latent

heat, respectively. The product of these terms is same as the product of inter-phase mass transfer rate and latent heat. The amount

of heat flux transferred to each phase can be determined by droplet size as in Table 1 [22].

Table 1 Heat flux transferred to vapor and liquid phases [22].

Droplet diameter Vapor phase (qc) Liquid phase (qd)

Less than 0.1nm - -

Less than 1μm ( )2

cc sat c

kNu T T

r where

2

1 3.18cNu

Kn

-

Greater than 1μm ( )2

cc sat c

kNu T T

r where 1/2 1/32 0.6Re Prc d dNu ( )

2

dd sat d

kNu T T

r where 6dNu

The properties used in the above equations (e.g., latent heat, thermal conductivity of vapor, liquid) are calculated from user

defined function made by fitting the original values of IAPWS-97. In addition, temperature values of the phases (Tc, Td) are

required to get qc, qd values precisely. The calculation with mixture assumption just can give the temperature of a mixture. Though,

the temperature of liquid phase can be estimated from the temperature of vapor phase and droplet information as in eq. (11)

[22,23]. Moreover, it is assumed that the temperature of vapor phase (Tc) is not significantly different from the temperature of a

mixture (T) in the case of small wetness.

*( )d sat sat c

rT T T T

r (11)

After calculating conservative variables by solving eq. (1) in matrix form, density, velocity and internal total energy come out.

Because T-Flow uses ideal gas law as equation of state, the pressure is usually derived from the internal total energy. However,

the pressure in multi-phase flow can be changed due to the water droplets, the approximation for pressure in wet-steam flows (eq.

(12)) is used based on the study of Mei et al. [24] The accuracy of flow variables can be modified by changing the equation of

state as Virial equation or IAPWS-97, so further studies are needed to check the effects of equation of state used.

2( 1)(1 ) 1

1 ( 1) 2lvp e V h

(12)

In addition, the inner-iteration method is adopted to reduce calculation time. When RANS equations (eq. (1)) and wet-steam

equations (eqs. (2) and (3)) are solved simultaneously in a single matrix (7 variables), the accuracy in each step might be slightly

increased. However, it will be a time consuming job, because small time-step is required for wet-steam calculation and then this

time-step is used for both of RANS & wet-steam calculations. On the other hand, if RANS & wet-steam equations are solved

individually, like turbulence models of tracking method, computational time can be considerably reduced by using different time

step in each equation. Because time-step used in wet-steam calculation is relatively small, inner-iterations are necessary to match

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up with the time marching of RANS calculation.

3. Numerical Analysis

3.1 Single stage steam turbine

Using implemented wet-steam model, T-Flow can predict steam condensing flows in various cases. At first, it was used to

simulate the flow in a Moore nozzle C for validating the implemented code [13] and then steam condensing flows in a Moore

nozzle B, a Moses and Stein nozzle and a White cascade were numerically studied [14]. In those studies, non-equilibrium phase-

transitions with pressure, temperature increase (condensation shock) were observed and predicted pressure, droplet radius agreed

well with the experimental data as in Figs. 1 and 2.

(a)Distance from throat (m)

Pre

ss

ure

rati

o,p

/p0

()

Dro

ple

tra

diu

s,r

(m)

-0.1 0 0.1 0.2 0.3 0.4 0.50.2

0.3

0.4

0.5

0.6

0.7

0

1E-08

2E-08

3E-08

4E-08

5E-08

6E-08

7E-08Pressure (Exp)

Pressure (T-Flow)

Radius (Exp)

Radius (T-Flow)

(b)Distance from throat (m)

Pre

ss

ure

rati

o,p

/p0

()

Dro

ple

tra

diu

s,r

(m)

0 0.02 0.04 0.060.2

0.3

0.4

0.5

0.6

0

1E-08

2E-08

3E-08Pressure (Exp)

Pressure (T-Flow)

Radius (Exp)

Radius (T-Flow)

Fig. 1 Pressure and droplet radius distributions along the centerline of

(a) Moore nozzle - B and (b) Moses and Stein nozzle - case 252 [14].

(a)Normalized surface distance ()

Pre

ss

ure

rati

o,p

/p0

()

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2Pressure (Exp)

Pressure (T-Flow)

(b)Normalized surface distance ()

Pre

ss

ure

rati

o,p

/p0

()

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2Pressure (Exp)

Pressure (T-Flow)

(c)Normalized surface distance ()

Pre

ss

ure

rati

o,p

/p0

()

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2Pressure (Exp)

Pressure (T-Flow)

Fig. 2 Pressure distributions along the blade surfaces of

White cascade (a) H1, (b) L1, and (c) W1 [14].

However, calculation of steam condensing flow is required for not only nozzles and cascades, but also rotating machinery as

steam turbines. Therefore, wet-steam flow in a single stage steam turbine (Fig. 3) was simulated in the previous study [15]. As a

result, droplet generation with condensation shock was also found in the steam turbine model. In addition, entropy rise was

promoted by generated droplets, so that entropy rise was maximized near the rotor trailing edge where the high wetness zone was

located. More important thing was that the selection of stator/rotor interface in steady calculation affected not only the flow field,

but also distributions of wet-steam parameters. In the case of using mixing plane method, pressure increase by the condensation

was higher and high wetness zone seemed to be overestimated in size and shape. It is because mixing plane method can change or

spoil the droplet distributions (e.g., droplet radius, wetness) during mixing process, so the study contended that use of mixing

plane is not recommended in steady multi-phase calculation. On the other hand, results using frozen rotor method can be affected

by circumferential positions of a stator and a rotor, so further study on the effects of relative position between a stator and a rotor

on steam condensing flow was demanded.

Fig. 3 Single stage steam turbine.

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Therefore, prior to the unsteady calculation, steady multi-phase calculations were conducted on a steam turbine model in this

study with 4 different stator-rotor positions (Fig. 4). The turbine model is a single stage steam turbine from ANSYS CFX tutorial

with some modifications in geometry and is the used one in the previous study [15]. Since the stator domain possessed 6.0 degrees

in circumferential direction, the rotor domain moved in counter-clockwise direction (=rotating direction) with 1.5 degrees in each

case. The original (0.0 degrees) rotor position is the position which was tested with changing stator/rotor interface [15] and the

leading edge (LE) of rotor blade is close to the trailing edge (TE) of stator blade at that position. In other words, rotor LE

approaches to stator TE in the cases of -3.0 degrees to -1.5, 0.0 degrees, and rotor LE recedes from stator TE in the cases of 0.0

degrees to +1.5, -3.0 degrees. These explanations on relative blade positions correspond to the casing region and can be changed

by span location due to the positively leaned rotor blade. However, we attach more importance to the casing region where high

wetness and entropy rise appear.

-3.0degrees -1.5degrees

0.0 (original) +1.5degrees

Fig. 4 Test cases with 4 different stator-rotor positions.

In each calculation, flow in single passages of a stator and a rotor was modelled using periodic condition. Computational grids

of stator/rotor passages were generated using ICEM CFD 14.0 in structured format, and they are composed of 745,000 and

646,000 nodes respectively. As boundary conditions, total pressure and temperature were given at stator (nozzle) inlet in normal

direction and static pressure was given at rotor outlet. Only frozen rotor method was used for stator/rotor interface. No slip and

adiabatic conditions were used at the blade surfaces and the end walls. No droplet incomed through the stator inlet. The details

including number of blades and numerical settings are listed in Table 2. Multi-phase calculations were started after the

convergences of single-phase calculations for numerical stability.

Table 2 Boundary conditions of a single stage steam turbine.

Parameters Values

Number of blades (Stator/Rotor) 60/113 [EA]

Rotating speed 5000 [rpm]

Inlet total pressure (Po) 26.5 [kPa]

Inlet total temperature (To) 328.5 [K]

Outlet static pressure 6.62 [kPa]

Inlet wetness 0

Inlet droplet number 0

3.2 Numerical results

These are the numerical results of steady steam condensing flows in a single stage steam turbine with 4 different stator-rotor

positions. At first, pressure distributions on 10%, 50%, 90% span locations are shown in Fig. 5. In all cases, pressure gradually

decreased as flow went downstream and pressure increased as span increased. Especially, in the rotor passage, pressure was the

highest in the vicinity of rotor LE near casing. However, pressure distribution in constant span was altered as the rotor position

changed. When the rotor blade approached to the stator blade (-3.0 degrees to -1.5, 0.0 degrees), pressure in the overall passage

was lowered. On the other hand, pressure was recovered when the rotor blade receded from the stator blade (0.0 degrees to +1.5, -

3.0 degrees). Therefore, it can be said that the relative position between a stator and a rotor definitely affects the main flow field

including pressure and its effects on other flow/wet-steam parameters are going to be represented.

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-3.0 degrees -1.5 degrees 0.0 (original) +1.5 degrees

90%

span

50%

span

10%

span

Fig. 5 Pressure distributions with different circumferential positions of a rotor.

Nucleation is a significant phenomenon of generating droplets in steam flow and it has a close relationship with supercooling

level. Supercooling level is defined by the difference between saturation temperature and vapor temperature (Ts-Tc) and reaches to

almost 30~40 Kelvin in most cases of non-equilibrium steam condensing flow [13,14]. Since similar distributions of supercooling

level were observed throughout the span locations at constant rotor position, just the distributions near casing region were

depicted in Fig. 6. In a single stage steam turbine, high supercooling level was observed at the nozzle throat and it reached to

almost 35 Kelvin. However, unlike pressure distribution in Fig. 5, rotor position hardly influenced supercooling level distribution.


90%

span

Fig. 6 Supercooling level distributions with different circumferential positions of a rotor.


90%

span

Fig. 7 Nucleation rate distributions with different circumferential positions of a rotor.

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Nucleation also showed similar characteristics along the span locations, so just the nucleation distributions near casing region

were shown in Fig. 7. Because the nucleation is initiated from the supercooled vapor, high intensity of nucleation was found at the

nozzle throat and the contours in Figs. 6 and 7 look alike. Like supercooling level distribution, rotor position did not affect the

nucleation rate distribution. It is because the nozzle shape was not changed throughout the span locations and the nucleation

usually occurred in the nozzle passage which was extrinsic to the position of rotor passage. As a result, distributions of

supercooling level and nucleation remained unaffected by circumferential position of a rotor.

Although rotor position hardly changed the trend of nucleation, the distributions of wet-steam parameters were extremely

changed by altering the rotor position. In Fig. 8, droplet radius distributions are shown and those in nozzle passages were quite

similar in 4 cases of different rotor positions; droplet consistently grew up after the nucleation and it was maximized at the nozzle

outlet. (Since droplets were generated by the condensation at the nozzle throat, uncommonly large droplets in the upstream of the

nozzle passage are meaningless.) However, droplet sizes in the rotor passage were considerably affected by rotor’s circumferential

position, which might not be observed in the case of using mixing plane method. Near the casing region where we’re interested

due to high wetness and entropy rise, many large droplets were distributed in the case of -3.0 degrees. On the other hand,

relatively small droplets were distributed in the case of 0.0 degrees. It is because the trend of droplet radius in the rotor passage is

determined by incoming droplets through the rotor inlet. In the nozzle passage, large droplets were located at the middle of the

passage, while small droplets were located at the wake region. Therefore, if rotor LE is far from stator TE (-3.0 degrees), large

droplets are flow into the middle of the rotor passage without any interference. On the other hand, if rotor LE is close to stator TE

(0.0 degrees), large droplets are hindered by rotor blade. For that reason, droplet sizes in rotor passage are considerably affected

by the relative position between a stator and a rotor.


90%

span

50%

span

10%

span

Fig. 8 Droplet radius distributions with different circumferential positions of a rotor.

Furthermore, the droplet number distributions are depicted in Fig. 9. In the nozzle passage, the droplet number was low at the

middle of the passage and it was high at the wake region. Accordingly, the trends of droplet number near the casing region of the

rotor passage were opposite to those of droplet size; small amount of droplets was found in the case of -3.0 degrees, while large

amount of droplets was placed in the case of 0.0 degrees. Although the number of droplets was changed by the span location and

the rotor position, droplet number was maximized near rotor TE in all cases. Resultingly, the effect of rotor position on wetness

which is the combination of droplet radius, droplet number can be found in Fig. 10.

8


90%

span

50%

span

10%

span

Fig. 9 Droplet number distributions with different circumferential positions of a rotor.


90%

span

50%

span

10%

span

Fig. 10 Wetness distributions with different circumferential positions of a rotor.

Wetness was also maximized near rotor TE like droplet number in each span and rotor position. However, the effect of rotor

position on wetness distribution was similar to that on droplet radius distribution in Fig. 8. Since wetness is the product of droplet

number and the cube of droplet radius (eq. (9)), droplet radius might have more influence on wetness rather than droplet number.

In the nozzle passage, wetness was high at the middle of the nozzle and low at the wake region, like droplet radius. Therefore,

high wetness was achieved in the cases of -3.0, +1.5 degrees where rotor LE was far from stator TE. On the other hand, relatively

9

low wetness was achieved in the case of -1.5, 0.0 degrees where rotor LE was close to stator TE. In addition, low wetness was

observed on the blade walls in every case and it might be caused by neglecting the sticking droplets, which are favorable to solve

in Lagrangian frame.

In the end, entropy rise distributions are plotted in Fig. 11. Entropy increased throughout the passage can be calculated using

eq. (13). As shown in the previous study [15], the trends of entropy rise were analogous to those of wetness in all cases; entropy

rise was maximized near rotor TE where high wetness zone was created. However, different from wetness distribution, entropy

rise near rotor TE was high in the broader region in the case of 0.0 degrees compared to the case of -3.0 degrees. Nevertheless,

entropy rise near rotor TE decreased as rotor LE approached to stator TE (-1.5 degrees) and it increased as rotor LE receded from

stator TE (+1.5 degrees). Therefore, it can be said that entropy rise is strongly affected by wetness, and the effects of rotor’s

circumferential position on wetness and entropy rise distributions are quite obvious.

ln( / ) ln( / )o p o oS S C T T R P P (13)


90%

span

50%

span

10%

span

Fig. 11 Entropy rise distributions with different circumferential positions of a rotor.

In addition, the quantitative trends of flow and wet-steam parameters at rotor outlet are shown in Figs. 12 and 13. In general,

the maximum values were considerably influenced by rotor position, while the averaged values were relatively changed in little.

In Fig. 12, the maximum and averaged values of droplet size had no common trend on circumferential position of a rotor.

However, the maximum droplet size was the biggest when rotor LE was far from stator TE (-3.0 degrees). Furthermore, the

maximum and averaged values of droplet number were the smallest when rotor LE was close to stator TE (0.0 degrees).

On the other hand, wetness which represents the overall amount of water-liquid showed quite obvious trend on rotor position

in Fig. 13. Both of the maximum and averaged values of wetness were the highest in the case of -3.0 degrees and the lowest in the

case of 0.0 degrees. Therefore, it showed the same trend with the undulation of wetness in the rotor passage shown in Fig. 10.

Entropy rise also showed analogous tendency on the rotor position; maximum and averaged values were the highest in the case of

-3.0 degrees and the lowest in the case of 0.0 degrees.

As mentioned in the above, the averaged value undulated in a narrow range by changing circumferential position of a rotor.

However, it was found that altering rotor position causes the fluctuations of flow and wet-steam parameters, and the maximum

values undulated in quite high amplitudes (Δβ≒3.44%, ΔS≒30.84J/kgK). Therefore, further study with unsteady multi-phase

calculation on rotating machinery is still demanded and will be conducted near future.

10

Circumferential position of a rotor (o)

Dro

ple

tra

diu

s(m

)

-4.5 -3 -1.5 0 1.5 36E-09

8E-09

1E-08

1.2E-08

1.4E-08

1.6E-08Max value @ outlet

Avr value @ outlet

LE of rotor blade is locatednear TE of stator blade


Dro

ple

tn

um

be

r(/

m3)

-4.5 -3 -1.5 0 1.5 30

1E+19

2E+19

3E+19

4E+19

5E+19

6E+19Max value @ outlet

Avr value @ outlet


Fig. 12 Maximum and averaged values of droplet radius and droplet number at rotor outlet.


We

tne

ss

()

-4.5 -3 -1.5 0 1.5 3

0.06

0.08

0.1

0.12

0.14

Max value @ outlet

Avr value @ outlet



En

tro

py

ris

e(J

/kg

K)

-4.5 -3 -1.5 0 1.5 3500

550

600

650

700

750

800

850

900Max value @ outlet

Avr value @ outlet


Fig. 13 Maximum and averaged values of wetness and entropy rise at rotor outlet.

4. Conclusion In this study, steady wet-steam flows with non-equilibrium condensation were simulated for a steam turbine model with 4

different stator-rotor positions and their effects on the wet-steam flow fields were found as follows.

1. Main flow field including pressure distribution was affected by circumferential position of a rotor, but distributions of

supercooling level and nucleation rate were hardly changed by altering rotor position. It is because the nucleation usually

occurred in the nozzle passage which is extrinsic to the position of rotor passage.

2. Droplet size, droplet number, wetness and entropy rise distributions in the rotor passage were extremely changed by

altering the rotor position. When rotor LE approached to stator TE, generated droplets by the condensation were hindered

by rotor blade. Therefore, wetness and entropy rise in the rotor passage were decreased. On the other hand, when rotor LE

receded from stator TE, generated droplets flew into the middle of the rotor passage and then wetness and entropy rise in

the rotor passage were increased.

3. The rotor position also caused the fluctuations of wet-steam parameters at rotor outlet. The maximum, averaged values of

wetness and entropy rise were minimized when rotor LE was close to stator TE. On the contrary, they were maximized

when rotor LE was far from stator TE. In addition, the maximum values undulated in quite high amplitudes unlike the

average values.

As a result, it was found that flow and wet-steam parameters are responsive in regard to rotor’s circumferential position in

steady multi-phase calculations of rotating machinery. Therefore, flow field might be extremely changed in the real cases where

rotor moves consistently and further study with unsteady multi-phase calculation is demanded.

11

Nomenclature

α

β

γ

Γ

η

θ

π

ρ

σ

C

hlv

I

Kb

Volume fraction []

Water mass fraction, wetness []

Specific heat ratio (Cp/Cv) []

Mass generation rate [kg/m3s]

Number density of droplet [/m3]

Non-isothermal correction factor []

Circular constant (pi) []

Density (w/o subscript, mixture) [kg/m3]

Surface tension [N/m]

Speed of sound [m/s]

Enthalpy of vaporization [J/kg]

Nucleation rate [/m3s]

Boltzmann constant [J/K]

Kn

k

Mm

Nu

p

Pr

R

r

Re

S

T

V

Knudsen number []

Conductivity [W/m∙K]

Water molecular mass [kg]

Nusselt number []

Pressure (w/o subscript, static) [N/m2]

Prandtl number []

Gas constant [J/kg∙K]

Droplet radius [m]

Reynolds number []

Entropy [J/kg∙K]

Temperature (w/o subscript, mixture) [K]

Fluid velocity [m/s]

Subscript

c

d

p

sat

Continuous phase, steam

Dispersed phase, water

Isobaric

Saturated

o

v

*

Inlet, Total

Isochoric

Critical

References

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effects of relative position between a stator and a rotor

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