Energy and Power En gi neering, 2010, 2, 248-253
doi:10.4236/epe.2010.24036 Published Online November 2010 (http://www.SciRP.org/journal/epe)
Numerical Simulation of Water Droplets Deposition on the
Last-Stage Stationary Blade of Steam Turbine
Danmei Xie, Xinggang Yu, Wangfan Li, Youmin Hou, Yang Shi, Sun Cai
School of Power and Mechanical Engineering, Wuhan University, Wuhan, China
Received June 29, 2010; revised August 10, 2010; accepted September 17, 2010
Abstract
Based on the method of discrete phase, the law of droplets’ deposition in the last stage stationary blade of a
supercritical 600 MW Steam Turbine is simulated in the first place of this paper by using the Wet-steam
model in commercial software FLUENT, where the influence of inlet angle of water droplets of the station-
ary blades is also considered. Through the calculation, the relationship between the deposition and the di-
ameter of water droplets is revealed. Then, the amount of droplets deposition in the suction and pressure sur-
face is derived. The result is compared with experimental data and it proves that the numerical simulation
result obtained in this paper is reasonable. Finally, a formula of the relationship between the diameter of wa-
ter droplets and the inlet angle is fit, which could be used for approximate calculation in the engineering ap-
plications.
Keywords: Stea m Turbine, Stationary Blade, Wet Steam, Water Droplets Deposition, Discrete Phase,
Numerical Simulation
1. Introduction
In the energy industry, high efficiency steam turbine is
widely used as a driving machine and its dominant posi-
tion in the power industry will stay for a long time. For
the last stage of steam turbine in the th ermal power plant
and nuclear station, steam will spontaneously condensate
and generate a large number of tiny water droplets,
called the primary water droplets with the diameter of
about 0.01-2.0 μm, when it expands to wet steam zone in
the cascade channel. The amount of primary water drop-
lets accounts for nearly 90% of the liquid mass in wet
steam [1]. The remainder part will form water film or
stream of steam, due to inertia and turbulent diffusion
deposition on the blade surface, wh ich will be brought to
the trailing edge by the steam and split out to form sec-
ondary water droplets under action of pulling force of
steam. Compared with the primary water droplets, the
number of the secondary water droplets is smaller and
the diameter is larger (less than 200 μm), but the mass of
a single droplet is larger. It is difficult to speed up the
flow speed. Severe erosion will b e caused on the moving
blade when it impacts the surface of moving blade with a
high relative velocity. For the steam turbine running in
the wet steam turbine area, such erosion is obvious and it
threatens the safe operation [2]. Moreover, according to
Baumann moisture loss formula, 1% of the moisture
produces 1% of loss [3], it can be seen that this erosion
will reduce the efficiency of units. It is necessary to
study the law of droplets deposition on the blades of the
final stage of the steam turbine.
At present, the numerical simulation result of the
droplets deposition on the channel of stationary blades of
steam turbine differ greatly from that obtained with the
experimental data [4-6]. That is because most of the cal-
culations are based on such assumptions as the direction
of flow is parallel to the blade setting angle, and the in-
fluence of inlet angle for water droplets to the d eposition
amount and location of deposition is ignored, when the
discrete phase model is used to calculate the droplets
deposition. By considering the influence of inlet angle of
water droplets of the stationary blade, this paper studies
the relationship between the deposition and the diameter
of water droplets and the amount of droplets deposition
in the suction and pressure surface.
2. Steam Control State Equations
Generally, the three physics laws of conservation, namely:
mass conservation law, momentum conservation law and

D. M. XIE ET AL. 249
energy conservation law, must be followed in fluid cal-
culations. According to the characteristics and nature of
the flow, other constraint equations should be included,
such as: viscosity transport equation, fraction conserva-
tion equation, etc.
2.1. Mass Conservation Equation (Continuity
Equation)

im
i
uS
tx
 

 (1)
The equation is suitable to either compressible flows
or incompressible flows. Source Sm is the mass trans-
ferred from the dispersed secondary phase to continuous
phase (for example, mass caused by the liquid evapora-
tion), the source term can also be the source defined by
customer.
2.2. Momentum Conservation Equation
The momentum conservation equation in the direction of
i of the inertial (non-accelerating) coordinate system is:
 
ij
iij ii
jij
p
uuu gF
tx xx

 

 

 (2)
where ρ is the static pressure, τij is the stress tensor, ρgi
and Fi are the gravitational body force and external body
forces of the i direction, such as the lift force due to in-
teraction of the discrete phase. Fi includes other relevant
source terms, such as porous medium and customer de-
fined source term.
2.3. Energy Conservation Equation
() () () ()
()()()
T
ppp
TuTvTwT
txy z
kTkTkT S
xc xyc yzc z

  

 
  


(3)
where ρ is fluid density, t is time. u, v, w is the velocity
component in x, y, z axis of fluid infinitesimal vector u. T
is temperature, k is the heat transfer coefficient for the
fluid, cp is specific heat capacity of fluid, ST is the inner
heat source for the fluid.
2.4. State Equation
Virial state equation [7] is used as the steam state equa-
tion use.


 3
g
2
ggg 1
 DCBRTP (4)
where ρg is the density, T is temperature, P is pressure. B,
C and D are the first, the second and the third order virial
coefficients respectively.
Above Equations (1)-(4) co nstitute closed equ ations of
solution to the problem. If two-phase flow is investig ated,
the impact of particles on the continuous phase formula
should be added to the above equations.
3. Calculation Model and Boundary
Condition
3.1. Calculation Model and Structure
The physical dimensions in the cross section of 73%
height of a 600 MW steam turbine stationary blade are
used referr ed in physical modeling in this paper. Becau se
that the internal flow of steam turbine is within limited
space in irregular regions and its solution is within that
area both for contraction and expansion conditions, non-
uniform mesh are used in meshing and the mesh spacing
are adjusted according to speed and pressure gradient
and liquid phase deposition in this paper, to ensure that
each part of the node spacing is relatively stable. The
wall of blade is the place to where droplets are attached.
The blade trailing edge is the zone where the secondary
droplet is formed when water film is torn due to steam
separation. Intensive energy exchange occurs in the
blade trailing edge with severe turbulence and pressure
fluctuation. So refining meshing is applied in these two
regions. Figure 1 shows the final computing mesh in this
paper.
3.2. Boundary Condition
Inlet and outlet boundary conditions are set as pressure
inlet and pressure outlet, the blade wall is treated as no
slip wall, and the upper and lower boundary is set as pe-
riodic boundary conditions. The design parameters of a
Figure 1. Two-dimensional grid of stationary blade.

D. M. XIE ET AL. 250
600 MW steam turbine are listed as followings: mean
total pressure of inlet is 18793.76 Pa; mean total tem-
perature of outlet is 330.96 K; mean back pressure of
outlet is 11924.83 Pa; outlet pressure of inner hollow
stationary blade is 9200Pa; inlet humidity of wet steam is
0.0794, outlet humidity is 0.083.
In this paper, wet-steam module in multi-phase is used,
and some user-defined function code is compiled. The
Realizable k-ε turbulence model is selected from turbu-
lence equations, SIMPLE algorithm is set for the pres-
sure and velocity coupling, and second order upwind
scheme are used for discrete equations.
3.3. The Relationship between the Inlet Angle of
Droplets and the Diameter of Droplets
The literature [8] showed th at the depositio n water on the
surface of last stage stationary blade follows such law as:
most water on the surface of stationary blade is from
inert deposition of second ary water droplets, and the dis-
tribution of secondary water droplets with different size
is related to the circumferential speed of the secondary
last moving blade. As the speed and direction of secon-
dary water droplets into the flow of stationary blade are
greatly related with the circumferential speed of secon-
dary last moving bucket, the axial clearance and the
steam flow. In general, the larger the circumference
speed or the smaller the axial clearance, the larger the
inlet angle of droplets when sprayed into the stationary
blade. The direction of secondary water droplets with
about 10 µm diameter into the statio nary b lad e chan n el is
basically close to the inlet angle of steam. The water
droplets of the level of average diameter of about 20-25
μm will enter the stationary blade channel with angle
slightly greater than 90 °. As for water droplets of 40 μm,
due to a great inertia force, will flow into the channel
with large negative incidence angle. Droplets of larger
diameter will be directly thrown to outer edge or dia-
phragm owing to small proportion of the total droplets’
mass or great inertia force. Therefore, the impact of lar-
ger diameter of the droplets in the stationary blade can be
basically negligible.
Based on the above analysis, the diameter of partial
droplets and their corresponding proportion of mass as
well as the entry angle needed for the numerical simula-
tion are calculated and collected. Here α is taken as the
supplementary angle of inlet angle. Some data is shown
in Table 1.
4. Parameter Setting of Discre te Phase
In this paper, the discrete phase model (DPM) is used to
investigate the movement of water droplets in the last
stage of stationary blade and the impact of water droplets
to the vapor phase flow field. In FLUENT, the discrete
phase model is applied by defining the initial position,
the speed, the size and the temperature of particles. Ini-
tial condition of particles defined by the physical prop-
erty of particle can be used to initialize the trajectory of
particles and heat and mass transfer calculations.
In this paper, the injection type is group injection, and
the particle type is inert. 1000 particle streams are de-
fined in this calculation [9].
In this paper, a distribution method called Rosin-
Rammler is employed to define the distribution of parti-
cle size. In this way, the overall range of particle size is
divided into discrete size groups, while each size group
is represented by a single particle stream of group injec-
tion. For Rosin-Rammler distribution, it’s assumed that
there is an exponential relationship between the particle
diameter d and the mass fraction Yd of particles whose
diameter is larger than d, as:
(/)
n
dd
d
Ye

 (5)
where d is the mean diameter (average diameter), n is the
spread pa r ameter.
Figure 2 shows the droplets size distribution and
Table 1. Data of diameter of droplets, proportion of mass
and inlet angle.
Diameter of Droplets
D/μm Proportion of Mass Q Entry Angle α/º
2 0.00066 88
7 0.0102 82
12 0.02878 78
17 0.04638 73
21 0.05101 70
25 0.04417 67
30 0.02601 62
35 0.010058 57
40 0.002437 53
Figure 2. The diameter of droplets and the curve of mass
distribution [9].

D. M. XIE ET AL. 251
droplets cumulative curve gained by the experiment re-
sults of literature [9]. The averag e diameter of droplets is
23.21 μm, of which about 95% of the water droplets di-
ameters are less than 100 μm. Therefore, the diameter
range of secondary water droplets, in the simulation of
this paper, is 2 μm-40 μm.
5. The Results of Simulation
5.1. Movement Law of Different Water Droplets
Diameters
In numerical simulation of water droplet movement tra-
jectory, different water droplet diameters are selected,
they are 2 μm, 7 μm, 12 μm, 17 μm, 21 μm, 25 μm, 30
μm, 35 μm, and 40 μm. Simulated trajectories of 2 μm,
21 μm and 40 μm diameter are shown in Figure 3.
5.2. Calculation of Water Droplets Deposition
According to the results of discrete-phase simulation, the
number of trap droplets in the suction surface and pres-
sure surface is calculated. According to the research of
literature [10], the ratio of discrete phase droplets in the
suction surface and pressure surface corresponds to the
ratio of water droplets deposition in the suction surface
and pressure surface of experiment and actual operation.
Calculations are summarized in Table 2.
From Ta ble 2 it can be seen that the trap total amount
increases first and then tends to decrease with the in-
creasing of diameter, if the inlet angle of water droplet
considered. By using the distribution of mass fraction of
water droplet shown in Figure 2 and the trap total
amount relative to diameter of water droplet, the deposi-
tion rate either in the suction surface or in the pressure
surface can be calculated, shown in Figure 4. And it will
be fitted into the Formula (6).
2
2( )
02
c
xx
w
A
yy e
w 


 
 (6)
where values are as follows:
The total deposition of secondary water droplets:
y0 = −6 .87455 × 10-4; xc = 23.35519; w = 14.19257; A
= 0.66905.
The Deposition Rate in the Suction Surface:
y0 = −4.32448 × 1 0-4; xc = 25.44163; w = 14.62533; A
= 0.09274.
The Deposition Rate in the Pressure Surface:
y0 = −5.98832 × 1 0-4; xc = 23.03415; w = 13.98533; A
= 0.57449.
Using Formula (6), for a given water droplet diameter,
the deposition rate either in the suction surface or in the
pressure surface can be calculated.
By integrating the calculated formula of the percent-
age of deposition, it can be drawn that: the deposition of
blade surface accounts for 63.58% of the total amount of
inlet secondary water droplets; in which the deposition
on suction surface accounts for 8.89% and the deposition
on pressure surface accounts for 54.66% of total deposi-
tion.
Figure 5 shows the percentage of deposition of sec-
ondary water droplets got by the numerical simulation in
this paper and the experimental data by J. Valha [11].
From the figure, it can be seen that th e trend of two is the
same. It can prove the reliability of above study for inlet
angle of droplets. In this paper, a formula of the diameter
of droplets and the inlet angle is fit.
0.9044 88.9931yx

 (7)
Table 2. Water droplets deposition.
Diameter of
Droplets/μm Trap Total Amount
Trap Amount
in the Suction
Surface
Trap Amount in
the Pressure
Surface
The Total Deposition
Rate
The Deposition Rate
in the Suction
Surface
The Deposition
Rate in the Pressure
Surface
2 15 0 15 0.00000990 0 0.000009900
7 112 7 105 0.00114240 0.000173400 0.000969000
12 314 29 285 0.00903692 0.000836420 0.008200500
17 542 55 487 0.02514000 0.002550900 0.022589100
21 693 83 610 0.03535000 0.004233830 0.031116170
25 797 112 685 0.03520349 0.004947040 0.030256450
30 906 159 747 0.02357000 0.004135590 0.019434410
35 942 210 732 0.00947000 0.002112180 0.007357820
40 933 277 659 0.00227000 0.000667848 0.001602152

D. M. XIE ET AL. 252
(a)
(b)
(c)
Figure 3. Water droplets trajectory of each diameter. (a) water droplets trajectory of diameter of 2 μm; (b)
water droplets trajectory of diameter of 21 μm; (c) water droplets trajectory of diameter of 40 μm.

D. M. XIE ET AL. 253
Figure 4. The percentage of water droplets deposition.
Figure 5. The ratio of secondary water droplets of experi-
ment data [11].
where x is the diameter of droplet.
Using Formula (7), for a given water droplet diameter,
the deposition rate of second water droplet on each sur-
face can be calculated. It can be easily used in approxi-
mate engineering calculations of inlet angle in the last
stage of stationary blade of steam turbine.
6. Conclusions
Taking a last stage stationary blade of a supercritical 600
MW steam turbine as an object, droplets with different
diameter are selected to simulate the movement trajec-
tory of particles, and the influence of the inlet angle of
water droplets changing with the droplets diameter are
taken into account.
By simulating calculation, a formula of the diameter of
droplets and the inlet angle is fit and a curve of water
droplets deposition is drawn according to the formula.
compared the curve with the experimental data, it can be
found out that the tendency of results by the two methods
are same. Furthermore, the relationship between the di-
ameter of water droplets and inlet angle is fit as a for-
mula in the paper, which can be used for approximate
calculation in the engineering application.
7. References
[1] Z. P. Feng, L. Li and G. J. Li, “The Status and Progress
of Numerical Study on Two-Phase Condensate Flows of
Steam Turbine,” Shanghai, 2002, pp. 1-10.
[2] Y. M. Hou, “Optimization Design of New Type Hollow
Stationary Dehumidity Blade in the Last Stage of SC
Steam Turbine,” Wuhan University, Wuhan, 2010.
[3] M. J. Moore and C. H. Sieverding, “Low Pressure Tur-
bine and the Condenser Aerothermodynamics,” Z. M.
Weng, M. Z. Yu, D. J. Cheng and B. Liu Translate, Xi’an
Jiaotong University Press, Xi’an, Vol. 66, 1992.
[4] S. A. Slater, A. D. Lee Ming and J. B. Young, “Particle
Deposition from Two Dimensional Turbulent Gas
Flows,” International Journal of Multiphase Flow, Vol.
29, No. 5, 2003, pp. 721-750.
[5] C. G. Li, X. J. Wang, et al., “Numerical Investigation of
Movement and Deposition of Water Droplets on Steam
Turbine Stationary Cascade,” Power Engineering, Vol.
42, No. 9, 2008, pp. 22-26.
[6] Y. F. Zhao, Y. Z. Wang, Y. F. Liu and B. Sun, “Removal
Method of Depositional Droplets and Energy Analysis in
Hollow Cascade,” Power Engineering, Vol. 24, No. 3,
2004, pp. 515-519.
[7] L. Li, Z. P. Feng and G. J. Li, “Numerical Simulation of
Wet Steam Flow with Spontaneous Condensation in Tur-
bine Cascade,” Journal of Engineering Thermophysics,
Vol. 23, No. 3, 2002, pp. 309-311.
[8] M. Z. Yu and Y. Huang, “Dewetting Method and Deposi-
tion Law of Steam Turbine at the Last Stage,” Turbine
Technology, Vol. 5, 1988, pp. 44-50.
[9] X. J. Wang, C. Lu, J. C. Liu, et al. “Slot Location of the
Hollow Stationary Blade to Water Performance Test,”
Engineering for Thermal Energy and Power, Vol. 22, No.
5, 2007, pp. 12-15.
[10] S. Cai, D. M. Xie, et al. “Numerical Simulation of Suc-
tion Slot Structure on the LP Last Stage Stationary Blades
of Steam Turbine,” Asia-Pacific Power and Energy En-
gineering Conference, Chengdu, 2010.
[11] J Valha, “Liquid Phase Movement in the Last Stages of
Large Condensing Steam Turbines,” Proceedings of the
Institution of Mechanical Engineers, Conference Pro-
ceedings 1964-1970, Vol. 178-184.