A comprehensive investigation of finding the best location for hot steam injection into the wet steam turbine blade cascade

Journal Pre-proof A comprehensive investigation of finding the best location for hot steam injection into the wet steam turbine blade cascade Mohammad Ali Faghih Aliabadi, Esmail Lakzian, Iman Khazaei, Ali Jahangiri PII:

S0360-5442(19)32092-4

DOI:

https://doi.org/10.1016/j.energy.2019.116397

Reference:

EGY 116397

To appear in:

Energy

Received Date: 6 May 2019 Revised Date:

23 August 2019

Accepted Date: 18 October 2019

Please cite this article as: Aliabadi MAF, Lakzian E, Khazaei I, Jahangiri A, A comprehensive investigation of finding the best location for hot steam injection into the wet steam turbine blade cascade, Energy (2019), doi: https://doi.org/10.1016/j.energy.2019.116397. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

A comprehensive investigation of finding the best location for hot steam injection into the wet steam turbine blade cascade

Mohammad Ali Faghih Aliabadi1, Esmail Lakzian2*, Iman Khazaei1, Ali Jahangiri1 1

Department of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran

2*

Center of Computational Energy, Department of Mechanical Engineering, Hakim Sabzevari University, Sabzevar, Iran Corresponding author: [email protected] Tel: +98-5144012818; Mob: +98-915-352-7142 Abstract In the power plant industry, a major part of energy losses in the thermodynamic cycle is related to the steam turbines, which are not made to work with wet steam flow, and the presence of the liquid phase causes efficiency reduction and mechanical damages such as erosion, which leads to high expenses due to the high cost of steam turbines. The present study used the hot steam injection (HSI) to decrease condensation loss and erosion rate. Location for HSI effects on the pressure distribution, Mach number, and liquid mass fraction of flow at the suction and pressure surfaces. For this purpose, several locations on suction and pressure sides were considered to select an appropriate location for injection. In this study, EEHIC (entropy generated, erosion rate, HSI ratio, inlet mass flow rate to the blade and condensation loss) method is used to select the most suitable HSI location. Results indicated that injecting on the suction side is more effective on mentioned parameters than the pressure side. HSI on the suction side (the lowest temperature in the center line of blade) decreased the condensation loss and erosion rate ratio by 81% and 99 %, respectively, compared to the case of no injection. Keywords: hot steam injection; wet steam; condensation loss; entropy generation; liquid mass fraction.

1

Nomenclature 

 a

     ℎ ℎ   !" !# %&

Area, (
 )

Second Virial coefficient Third Virial coefficient Specific heat capacity at constant pressure, (J/kg K) Droplet diameter, (m) Total energy, (J) Erosion Safety factor Bulk Gibbs free energy change, (J/kg) Enthalpy, (J/kg) Hardness Nucleation rate, (1/m3s) Kinetic energy Boltzmann's constant Erosion coafficient

'

+,

Order of accuracy

*

 0 S 3 4

5,  7

8

9



Volume fraction

Condensation loss

)

Greek symbols  



effective diffusion coefficients Heat capacity ratio

∗

Critical condition

(

Liquid

Hot steam injection ratio-HSI ratio

Kronecker delta function Relative error  Non-isothermal correction coefficient  Mass generation rate, (kg/m3 s)  Thermal conductivity, (W.m.K)  Dynamic viscosity, (Pa s)  Density, (kg/m3)  Liquid surface tension, (N/m)  Viscous stress tensor, (Pa)  Loss rate ω hardness coefficient ∅ Superscript

Number of liquid droplets per unit mass, (1/kg) Pressure, (Pa)

Subscript 

Vapor

liquid-vapor ( Saturation Injection .'/ Stagnation 0 Abbreviation HSI Hot Steam injection EEHIC Entropy generated, Erosion rate, HSI ratio, Inlet mass flow rate to the blade and Condensation loss GCI Grid Convergence Index

Condensation coefficient Droplet radius, (m) Gas constant, (J/kg K) Entropy, (J/kg K) Time, (s) Temperature, (K) Velocity components, (m/s) Cartesian direction, (m) Liquid phase mass fraction Sound speed

2

1. Introduction Today, the condensation phenomenon occurs in many of industrial equipment [1-3] such as supersonic nozzles [4, 5], thermo-compressors [6], steam turbine blades [7], ejectors [8], and condensers [9]. During the expansion process in the last stages of low-pressure steam turbines, superheat steam passes the saturation line and enters the two-phase flow region and a large number of fine droplets of liquids are formed. The presence of the liquid phase within the turbine blade causes thermodynamic losses, aerodynamic losses, blade erosion, erosion damage, and thermal efficiency reduction [10]. On the other hand, an increase in the wetness decreases the efficiency of the wet steam turbine [10]. Recently, a large body of research has been conducted to provide information for better understanding the steam condensation flow, and experimental data to validate numerical solution methods. Bakhtar et al. [11] and Dykas et al. [12] measured pressure distribution in the turbine blade on the pressure and suction sides. In another study, Bakhtar et al. [13] evaluated the effect of Courtney and Kantrowitz corrections on the nucleation equation. For simulating the wet steam flow, Lagrangian-Eulerian [14] and Eulerian-Eulerian [15] approaches are utilized. Thus, various numerical models have been used to solve the wet steam flow. Dykas et al. [16] compared the single-fluid model with the two-fluid model in Eulerian-Eulerian approach. In addition, Dykas and Wroblewski [17] applied the moment method for calculating droplet radius in a two-fluid model. Sharifi et al. [18] and Mazzelli et al. [19] investigated the flow behavior in thermocompressor and ejector by considering the effects of non- equilibrium condensation. Yamamoto et al. [20] studied the effects of inlet wetness on wet steam flow in blades of the turbine and compressor. Researchers are interested in reducing the irreversibility in connection with heat transfer and viscous shear stresses in engineering systems towards minimum states, to reach better performance. Therefore, entropy generation needs to be considered a criterion for measuring the irreversibilities in designing the engineering systems. Evidently, there are approaches to reduce thermodynamic losses and entropy generation in a large number of recent studies [21-28]. Gerber and Kermani [29] studied thermodynamic and aerodynamic losses in wet steam flow in the nozzle. Lakzian and Shaabani [30] examined the effect of the coalescence process on entropy generation in condensing wet steam flow in a convergent-divergent nozzle. Based on the results, 3

the amount of entropy generation increased by considering the coalescence process. Lakzian and Masjedi [31] investigated the effect of slip between vapor and liquid phases in the wet steam flow in a one-dimensional supersonic nozzle. Given that no empirical proof has been provided regarding the anticipated effects due to volumetric heating and HSI, But researchers have taken this issue into account. Mahpeykar et al. [32-35] studied the effects of volumetric cooling and heating on the parameters of two-phase flow, condensation shock location and entropy generation, by using the one-dimensional analytical solution in convergent-divergent nozzles. Ahmadpour et al. [36] investigated the effects of inlet superheating and volumetric cooling as two practical choices to control condensation in wet steam flow. They demonstrated that both techniques are effective to reduce the wetness losses and enhance the performance of thermo-compressors and steam turbines. Lakzian et al. [37] reduced the amount of produced wetness by volumetric heating in the turbine blades. They indicated that using a certain amount of volumetric heating could reduce the generated entropy. Han et al [38] developed and installed an endwall fence in a stator cascade channel. They explored the effect of different heating intensities as well as different placement positions on the performance of the cascade. Boroomand and Mirhoseini [39-40] injected hot steam into a wet steam flow in the convergent-divergent nozzle and studied its effect on the twophase flow parameters. By using the genetic algorithm, they proposed the optimal amount of HSI in the convergent-divergent nozzle and indicated that wetness and droplet radius can be reduced by HSI. Controlling the condensation phenomenon is a crucial issue regarding the performance of the steam turbine blade. One of the methods to control the condensation phenomenon is the HSI into the passage of blades. HSI at the trailing edge of the blades was suggested to be done by Xu et al [41] to decrease the outlet liquid mass fraction. Moreover, the injection effects on the liquid droplet were explored by Gribin et al. [42-43]. However, the above papers have not studied at the optimum location of steam injection in detail. Given that the pressure distribution, Mach number, and liquid mass fraction near the suction side are different from the pressure side, the present study considered different locations on the suction and pressure sides to select the best location for HSI. Thus, EEHIC method was used for this purpose. EEHIC method was used for each location of injection and no injection case, also by comparing the result of this method, the best

4

location of injection can be selected. Furthermore, the Eulerian-Eulerian and ::4  − < turbulence models were used to simulate the viscous flow in turbine blade. 2. Numerical method 2.1. Governing equations In this study, the conservation equations for mass, momentum and energy along with droplet number and wetness equations are solved using the Eulerian-Eulerian approach and the singlefluid model [44-45]. In this model, the liquid and vapor phase are calculated simultaneously and the governing equations are solved for combining the droplet and vapor fluids. The slip velocity between the two phases is neglected due to the small liquid droplet radius or small drag force. Eqs. (1-3) relate the liquid and vapor phases. ) = )> = )?

(1)

ℎ = B1 − 8)ℎ> + 8ℎ?

(3)

1 1−8 8 = +  > ?

(2)

where ) represents the pressure,  indicates the density, and ℎ shows enthalpy. The subscripts 

and ( indicate vapor and liquid phases, respectively. The equations of continuity, momentum, energy, liquid mass fraction, and droplet number are as follows [16]:

C CB5 ) + =0 C3 C7

(4)

CB5 ) CB5 5 + )  ) C  + − =0 C3 C7 C7

(5)

CB) CB5 D) CB+ − 5  ) + + =0 C3 C7 C7

(6)

CB8) CB5 8) + = EF, + GHIJ C3 C7

(7)

CB') CB5 ') + = C3 C7

(8)

5

Abadi et al. [46] represented that ::4  − < turbulence model has a better adaptation with the

experimental points in the turbine blade. The ::4  − < model is used for simulating the its specific loss rate < for vapor and liquid phases are written as follows:

turbulent flow between the steam turbine blades. The equations for turbulent kinetic energy k and CB) CB5 ) C C + = BK ) + LK − MK + NK C3 C7 C7 C7

CB<) CB<5 ) C C< OP Q + LP − MP + RP + NP + = C3 C7 C7 C7

(9) (10)

where K and P are the effective diffusion coefficients of k and ω, respectively, LK represents

the turbulent kinetic energy associated with medium velocity gradients, and LP indicates the specific loss rate. Further, MK and MP denote the k and ω losses associated with turbulence,

respectively, NK and NP are the source terms for turbulent kinetic energy equations and its specific loss rate, respectively, and RP is the cross-diffusion [37]. 2.2. Nucleation and droplet growth equations The formation of droplets in non-equilibrium conditions without particles and ions is called

homogeneous nucleation. To this aim, molecular clusters must overcome free critical energy barriers in order to form a droplet with a critical radius. The Gibbs free energy required to form a spherical droplet is obtained from Eq. (11). ∆ = 4U  H −
04> (' B

) ) ) B4> )

(11)

The surface tension σW in this study is considered equal to the surface tension of the flat plate. [46]. The radius corresponding to this point is called the critical radius and is represented by  ∗

For each supercooled single phase steam, Gibbs free energy variations contain a maximum point

[46].

∗ =

2H

* ? 04> (' B ) * B4> )

(12)

6

Eq. (12), which is known as the Kelvin–Helmholtz equation. The classical nucleation equation expresses the number of supercritical droplets produced per unit mass of vapor per unit time [46]. ,?Y

2H [\/ > −4U ∗  H = +, Z
^7* B ) U ? 3!" 4> +,

(13) one,

!" indicates the Boltzmann constant,
demonstrates the mass of a molecule. Later, various where

represents

the

condensation

coefficient,

which

is

equal

to

corrections including Kantrowitz corrections were applied to classic nucleation equation, which is used in Eq. (14). KY =

1  1 + ∅ ,?Y

(14)

where ∅ represents temperature correction coefficient and is obtained from Eq. (15). ∅=2

B − 1) ℎ?> ℎ?> 1 ` − a B + 1) 04> 04> 2

in which ℎ?> indicates the latent heat. The phase change is defined by two mass sources [46]. EF, =

4 U  ∗ \  3 ?

GHIJ = 4U? ' 

 3

(15)

(16) (17)

EF, displays the source mass rate generated by droplets produced in the nucleation process and GHIJ denotes the mass rate of droplets condensed in the droplet growth process. where  and

bH bc

are droplet radius and droplet growth, respectively [37].

7

 ) +1 = B ) e B4? − 4> ) 3 ℎ?> ? √2U04 2

(18)

Droplet temperature is calculated by Gyarmathy approximation, which has highly been considered by researchers [37]. 4? = 4 B)) − B4 B)) − 4> ) 2.3. State equation

∗ 

(19)

The Virial equation is used to calculate steam properties in low-pressure flow [47]. ) = > 04> B1 + > + > )

(20)

where B and C are regarded as the second and third Virial coefficients which are calculated by the following equation [47]:

i   = fg B1 + )[g + f ^ h B1 − ^ [h ) f\  

where B is given in m3/kg,  = 0.000942, and f\ = -0.0004882. = fB − j )^ [lh + m

(21)

with T given in Kelvin, =10000, fg = 0.0015, f =

gijj k

(22)

where C is given in m6/kg2,  = nop.rn with T given in Kelvin, j = 0.89780,  = 11.16, f = 1.722, and b =1.5E -0.06.

k

The sound speed is calculated from an equation which is widely used in mixture models. 9=

1

f 1−f t B + ) > 9> ? 9?



(23)

8

where f is the vapor volume fraction, > and ? are respectively vapor and liquid density, 9> and 9? are respectively vapor and liquid sound speed. ) 9> = Z >

9? = −4.168w o + 12.64w \ − 58.37w  − 84.99w + 1529 w=

4> − 386.3 66.12

(24)

(25) (26)

2.4. Boundary conditions The total pressure and temperature, and the number of droplets and liquid mass fraction are specified for both the blade subsonic inlet boundaries and injection slot. All thermo-physical conditions at blade outlet are extrapolated from the adjacent computational domain due to the nature of supersonic flow. Moreover, adiabatic and no slip boundary conditions are applied to wall boundaries. Fig. 1 illustrates boundary conditions on the blade. For the slot, the stagnation temperature and pressure are specified and the amount of wetness and the droplet number are zero.

Fig. 1. Boundary Conditions on the Blade

9

2.5. Erosion rate The erosion caused by liquid droplet is regarded as one of the biggest concerns in designing steam turbine since it causes mechanical damages. Lee et al. [48] proposed a model for calculating the erosion rate, which is based on the collision rate of droplet flow, collision speed of the droplet, droplet size, and material hardness:  = # B


{? }? i.g  ~ ∅> )B ) B ) 10
{H#| }H#| H#|

(27)

where # represents the erosion coefficient, ∅ indicates the hardness coefficient, D shows the hardness, which relies on the blade material,
{? displays the droplet flow rate, } demonstrates

the droplet velocity, and  is droplet diameter. According to the blade material, € is between

2 − 4.5, which is considered as 4.5 in this paper. The parameters of
{H#| ،}H#| , and H#| are

0.01743

‚

ƒƒ„ …

, 568 , and 30.6 μm, respectively. ƒ †


{? = 8
{cIc
{cIc =
{E?#c ~8
E?#c } i.g  o.i

(28) (29) (30)

2.6 Condensation Loss When the water droplets are nucleated and grown, the thermodynamic and dynamic properties of the steam are changed and the steam ability to work is decreased. These variations are called the wetness loss and its calculations are important in order to design the steam turbines [49]. Condensation losses are the losses which are created because of the condensation of the steam molecules on the droplet surfaces [50]. When the nucleation phenomenon occurs, steam molecules on the droplet surface condensate and droplets grow. The condensation process is an irreversible process and the condensation losses are produced. %& = ℎ?> B
{?,IFc −
{?,E )+,

(31)

10

Where +, is the condensation coefficient, %& is the condensation losses,
{?,IFc is the outlet liquid

mass flow rate, and
{?,E is the inlet liquid mass flow rate. 2.7 Numerical simulation

The governing equations were solved using the finite volume method approach on a densitybased technique. Additionally, the Roe [51] method was implemented to compute the convective fluxes. Finally, the upwind method with second-order accuracy was used for space discretization. The experimental data of Bakhtar et al. [11] was used to validate the numerical solution of wet steam flow in turbine blade. Table 1 and table 2 present the geometrical specifications and the of the blade are 172 kPa and 380.66 K, respectively. Flow at the blade inlet is subsonic and

boundary conditions of this blade. The stagnation pressure and stagnation temperature at the inlet

supersonic at the blade outlet. Three sets of quadrilateral computational grids: fine (30840 cells), medium (14440 cells) and coarse (7432 cells) are calculated and analyzed by Grid Convergence parameter, the specific calculation results of the GCI analysis are shown in Table 3. Where * is Index (GCI) [52] to obtain an optimal computational mesh. Taking the mass flow rate as

the order of accuracy, Ž is the safety factor,  is the relative error between two grids and 1 is a coarse mesh, 2 is a medium mesh, and 3 is a fine mesh, respectively. It can be seen from the calculation results of GCI in Table 2, GCI values of grid 2-3 are small. The medium mesh is selected to save computation time and cost. Table 1 Geometrical specifications for the turbine blade cascade [11]. Length Chord Pitch 76 mm

35.76 mm

18.26 mm

Axial chord

Inlet flow angle

25.27 mm

0

Table 2 Boundary condition of the turbine blade cascade [11]. Pj 172 kPa Tj = T’ BPj ) − 8 380.66 K P“”• = 0.48Pj 82.56 kPa Inlet Flow Angle 0j

11

Table 3. Grid Convergence Index (GCI) method is used to determine the appropriate grid resolution. Grid 1-2 Grid 2-3 1- coarse,2-medium 2-medium,3-fine g,  ¡g, ,\  ¡,\ Ž * B%) B%) B%) B%) Inlet mass flow rate 3 1.83 0.27 1.4 0.11 0.6

Fig. 2 indicates the distribution of static pressure at the suction and pressure sides and the droplet 2 depicts two pressure increase on the suction side. The increase in pressure at 17.68 mm on the radius in the center line of the blade, which are compared with the experimental results [11]. Fig.

pressure at 20.46 mm on the suction side is due to the aerodynamic shock. The average radius of

suction side is due to the condensation shock occurred after the throat. However, the increase in

the droplets has a difference with the experimental data. This kind of difference is also shown in Bakhtar’s study [11]. 180

10-7

160

140

-8

r (m)

P (kPa)

10

120

10-9

100

80

Pressure calculated Experimental pressure data Droplet radius calculated Experimental radius data

60 0

5

10

15

20

10-10 25

x (mm) Fig. 2. Static pressure distribution on suction side and pressure side and droplet radius distribution in the blade

central line compared to the experimental data [11]

For more validation, Moore's nozzles [53] were used to validate the numerical solution of the specifications with respect to wet steam condensing flow. From the Moore et al. [53] work, the geometries of nozzles A and B were used, with the specified inlet boundary conditions: stagnation pressure of 25 kPa and stagnation temperature of 354.6 K for nozzle A, and 357.6 K 12

for nozzle B (Table 4). To obtain an optimal computational mesh GCI method is used to determine the appropriate grid resolution. A quadrilateral mesh with 9200 cells is used for nozzle A. Also for nozzle B quadrilateral grids with 15000 cells are used. In Fig. 3, comparisons are made, and a good agreement was achieved between the static pressure distribution along the nozzle centerline, obtained from the simulation and the experimental Moore's nozzle A and B data [54]. But the average radius of the droplets has a slight difference compared to the experimental data, which can be seen in some references [29,54]. Table 4 Operating conditions for nozzle

Pj (kPa) 25 25

Moore A Moore B

25

10

-7

10

-8

25

10-7

20

-9

10

Pressure calculated Experimental pressure data Droplet radius calculated Experimental radius data

5

-0.2

-0.1

0

0.1

0.2

0.3

10

0.4

10

-9

10

0 -0.2

0.5

-8

15

Pressure calculated Experimental pressure data Droplet radius calculated Experimental radius data

5

-10

10

r (m)

10

P (kPa)

15

r (m)

P (kPa)

20

0

Tj (K) 354.6 357.6

-0.1

0

0.1

0.2

0.3

Axial coordinate (m)

Axial coordinate (m)

(Nozzle A)

(Nozzle B)

10-10

0.4

0.5

Fig. 3. Comparisons between simulation results and experimental data

3. Results and discussion 3.1. Hot steam injection (HSI) The main purpose of the present study is to inject hot steam into an appropriate location in order to eliminate the effects of wetness and droplet radius and to reduce generated entropy. Since bakhtar’s turbine blade has an experimental radius size, this blade is used to study HSI. The HSI 13

is performed in different locations on the suction and pressure sides. Fig. 5 demonstrates the computational grid. For each injection location, the effect of the number of meshes on the distribution of pressure on the central line was done. Then, by selecting the optimal mesh, the impact of the injection site is examined.

Fig. 4. Hot steam injection from suction side and pressure side

Fig. 5. Computational mesh in blade in hot steam injection a) on suction side 7 = 20.21 mm and b) on pressure side 7 = 19.45 mm

(a)

(b)

3.2. Evaluating the effect of HSI location When the wet steam flows on the turbine blades, the cross-section, pressure, and temperature are reduced at the blade convergent part while the Mach number rises to allow the flow to be sonic in the throat. The blade structure is shown in Fig. 6. After that, the pressure and temperature continue to decrease despite the increase in the Mach number and the cross-section of the flow. 14

In addition, the degree of super-cooling, which is the criterion of being non-equilibrium of the steam, increases and consequently, the liquid phase is formed to return to the equilibrium. The occurrence of condensation shock and release of latent heat, first, increase the droplet temperature and then, the steam phase temperature. According to Rayleigh flow characteristics, giving heat to the supersonic flow increases the pressure and reduces the Mach number. Among the effects of the liquid phase is the reduction of the efficiency of each turbine stage by about 1% in the presence of 1% wetness [33]. In this study, due to the damage caused by erosion and the efficient reduction caused by the presence of wetness, the use of HSI to reduce wetness is proposed. Considering the limitations in pressure, temperature, and HSI rate, selecting appropriate HSI location, which greatly affects the EEHIC, plays an important role in this regard.

Fig. 6. Blade structure (convergent part, throat and divergent part)

pressure sides with the injection angle of 90°, the injection slot width of 0.09 E?#c , injection In this section, the effect of HSI location is evaluated in different locations on suction and

steam temperature of 500 K, and injection pressure of 170 kPa for pressure side and 160 kPa for suction side. On the suction side, five injection locations were considered to examine the effects of injection location. In addition, the injection locations of No. 1 and 2 were at the blade convergent part (in subsonic part), No. 3 and 4 were at the blade divergent part (in supersonic part), and No. 5 was after the occurrence of the first aerodynamic shock on the suction side and near the blade outlet (Fig. 7). On the pressure side, the injection locations of 6 to 10 were at the blade convergent part (in subsonic) (Fig. 7). The heat and mass were added to flow at the HSI 15

location. Initially, the basic concepts of the effect of adding heat and mass to the steam flow should be investigated in this regard.

Fig. 7. HSI locations at the blade

Although simplistic assumptions were used in appendixes A and B for analytical study of the flow, a two-dimensional numerical solution is used to evaluate the effect of HSI because of the complexity of two-phase flow conditions and turbine blade geometry. Fig. 8 and Fig. 9 indicate the effect of injection location on suction and pressure sides on pressure and temperature ratio in the center line of the blade and compare with the case of no injection. The HSI on suction and pressure sides in the blade convergent part (injection of 1 and 2 on the suction side and the injection of 6 to 10 on the pressure side) increased pressure and temperature in the center line of the blade at the injection location. Further, the HSI in the supersonic part (injections No. 3, 4, and 5) caused a sudden rise in pressure and temperature at the injection location, as depicted in Fig. 8 and Fig. 9.

16

180

160

160

140

140

No Injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

120

100

80

60

P (kPa)

P (kPa)

180

0

5

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

120

100

80

10

15

20

60

25

0

5

10

x (mm)

15

20

25

x (mm)

(Suction side)

(Pressure side)

385

385

380

380

375

375

370

370

365

365

360

No injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

355 350 345 340 335 330 325

T (K)

T (K)

Fig. 8. Variations of pressure to different locations of injection, on suction side and pressure side compared to no injection state as well as compared to each other, in the central line of the blade.

0

5

10

360

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

355 350 345 340 335 330

15

20

325

25

0

5

10

15

x (mm)

x (mm)

(Suction side)

(Pressure side)

20

25

Fig. 9. Variations of temperature to different locations of injection, on suction side and pressure side compared to no injection state as well as compared to each other, in the central line of the blade.

Fig. 10 illustrates the Mach number for the center line of the blade in different injection locations. The HSI on the convergent part of the blade decreases the Mach number while it leads to a sudden change in Mach number in the divergent part of the blade. As shown in Fig. 10, the flow fails to become sonic in a throat in the injections No.3 and 4, and the Mach number increases after throat and in divergent part of the blade. The reason is that the HSI creates a 17

virtual throat for the flow, resulting in increasing Mach number in the center line in injections No.3 and 4. In injection 5, which is almost at the end of the blade, it is shown that the flow has a similar behavior to the case of no injection in the convergent part of the blade, leading to an increase in the pressure and temperature and decrease in Mach number in the divergent part and near the injection location. Fig. 11, Fig. 12 and Fig. 13 demonstrate the effect of HSI in different locations on the nucleation graph, liquid mass fraction, and average droplet radius at the center line of the blade. The HSI slightly delays the start of nucleation at the centerline of the blade. In the nucleation diagram on the suction side, the maximum nucleation value of injections 1, 2, and 3 is greater than the case of no injection while the maximum nucleation value in injection No.4 is much smaller than that of the case of no injection. In injection 5, there is no significant variation compared to the case of no injection. On the pressure side of the nucleation diagram, the HSI slightly increases the maximum amount of nucleation. However, the HSI on the suction side decreases the amount of the liquid mass fraction at the center line of the blade and it is observed that the liquid mass fraction is zero in injection No.4. The HSI on the pressure side would not significantly alter the liquid mass fraction at the center line of the blade. In the droplet average radius diagram, the HSI on pressure and suction sides decreases droplet average radius. The droplet radius is approximately zero in injection No.4. 1.2

1.2

No injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

1

Mach number

0.9 0.8 0.7 0.6

1 0.9

0.5

0.8 0.7 0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0

5

10

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

1.1

Mach number

1.1

15

20

0.1

25

x (mm)

0

5

10

15

20

25

x (mm)

(Suction side)

(Pressure side)

Fig. 10 Comparisons of variations of Mach number to different locations of injection and no injection, on suction side and pressure side, in the central line of the blade.

18

Log10 (Droplet nucleation rate) (1/m .s)

No injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

20

16

12

8

4

0

0

5

24

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

3

Log10 (Droplet nucleation rate) (1/m3.s)

24

10

15

20

25

20

16

12

8

4

0

x (mm)

0

5

10

15

20

25

x (mm)

(Suction side)

(Pressure side)

Fig. 11. Comparisons of variations of nucleation rate to different locations of injection and no injection, on suction side and pressure side, in the central line of the blade.

0.05

0.05

No injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

0.03

0.04

Liquid mass fraction

Liquid mass fraction

0.04

0.02

0.01

0

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

0.03

0.02

0.01

0

5

10

15

20

0

25

0

5

10

15

x (mm)

x (mm)

(Suction side)

(Pressure side)

20

25

Fig. 12. Comparisons of variations of liquid mass fraction to different locations of injection and no injection, on

suction side and pressure side, in the central line of the blade.

19

0.04

No injection Injection 1 Injection 2 Injection 3 Injection 4 Injection 5 Location 1 Location 2 Location 3 Location 4 Location 5

0.03

0.02

Droplet radius (micron)

Droplet radius (micron)

0.04

0.01

0

0

5

10

15

20

0.03

0.02

0.01

0

25

No injection Injection 6 Injection 7 Injection 8 Injection 9 Injection 10 Location 6 Location 7 Location 8 Location 9 Location 10

0

5

x (mm)

10

15

20

25

x (mm)

(Suction side)

(Pressure side)

Fig. 13. Comparisons of variations of droplet average radius to different locations of injection and no injection, on suction side and pressure side, in the central line of the blade.

Table 5 provides data used to study the effect of injection location on suction and pressure sides, as follows: •

trailing edge of the blade such that the average liquid mass fraction B8¢IFc ) decreased by Compared to no injection condition, the HSI decreases the liquid mass fraction at the

HSI decreases the average droplet radius at the trailing edge of the blade B̅IFc ), so that it

81% in injection No.4. •

blade B}¢IFc ) is decreased as the injection location approaches to the end of the blade.

is reduced by 47% in injection No.4. The average flow velocity at the trailing edge of the •

The erosion rate depends on the radius, droplet velocity, inlet mass flow rate, and liquid mass fraction. The HSI reduces the erosion rate by 99% compared to the no injection condition.

¥“”• . Eqs (27-29) indicates how to calculate 8¢IFc , ̅IFc , and V

¦ 8IFc IFc ∑ ',#??,IFc ̅IFc = ∑ ',#??,IFc ∑ },#??,IFc }¢IFc = ∑ ,#??,IFc 8¢IFc =

(32) (33) (34)

20

Table 5 Analyzing the effect of HSI on suction side and pressure side HSI 7E 8¢IFc ̅IFc }¢IFc number m [10[r m] [mm] [ ] s

Suction side

Pressure side

1 2 3 4 5 6 7 8 9 10 No injection

13.90 16.67 19.45 20.21 22.23 13.90 16.67 19.45 20.21 22.23 -

451 426 375 320 309 471 471 472 472 469 467

0.025 0.020 0.013 0.007 0.015 0.035 0.035 0.034 0.035 0.035 0.037

2.38 1.72 1.43 1.25 1.47 2.49 2.46 2.44 2.15 1.52 2.50

3.3. Selecting a suitable location for HSI The present study aimed to determine a suitable location for HSI. The EEHIC method plays an important role in selecting the best location for HSI. This section compares the mentioned parameters in EEHIC method in injection locations on the suction and pressure sides with each other (Fig. 14). One of important parameters is the entropy generation, which demonstrates the losses. entropy generation «:G#E ¬ is increased by HSI, compared to the case of no injection and injection

Minimizing the generated entropy reduces the thermodynamic losses. It is observed that the

on the pressure side has less entropy generation than that of the suction side and the injection locations 8 to 10 are appropriate for HSI. It is worth noting that the difference in generating entropy in injection locations is insignificant. Given the high cost of the turbine and its components, reducing the erosion rate is of great importance to increase the turbine lifetime. The parameter

­H

­H ®¯ °±²³´µ

shows the ratio of erosion

rate in HSI to that no injection condition, which is calculated at the end cross-section of the blade. HSI on the suction side decreases the erosion rate and it is observed that the erosion rate ratio drops significantly by more than 99% in injection No.4. Additionally, the HSI on the 21

pressure side reduces the erosion rate, although its reduction is considerably less than that of the suction side. Considering the significant reduction in erosion rate in the HSI, the suction side is the proper side and injection No.4 is a suitable location for injection (Fig. 14). Steam generating for HSI is a costly process and the high amount of HSI ratio may not be economical. α represents the HSI ratio obtained from the ratio of the HSI flow rate to the inlet mass flow rate. Regarding the variation in the inlet mass flow rate, the inlet flow rate of the case of no injection is used to calculate the HSI ratio. Fig. 14 shows the HSI ratio in each injection location. In the present study, due to the variability of the inlet flow rate at each injection location, the inlet mass flow rate of the case of no injection was used to calculate the HSI ratio. The HSI ratio in each injection location is considered less than 10%. The inlet mass flow rate is effective in work produced by the blade. The HSI reduces the inlet mass flow rate; in other words, as the injection location gets closer to the throat, the inlet mass flow rate decreases and the HSI rate increases. As mentioned before, the flow velocity is reduced by HSI, and considering that the velocity reduction is greater than the density increase, it is expected that the amount of flow rate is decreased (5 = − ). On the other hand, the amount F ¶

of pressure is reduced as getting closer to the throat, and thus, more flow rate is injected into the flow by considering injection parameters as constant. In addition, HSI reduces flow rate due to the Rayleigh line (heat greater than critical heat). As mentioned, the inlet steam rate is decreased as the injection location gets closer to the throat. As demonstrated in Fig. 14, the injections No.4 and 5 have the highest inlet steam rate. By comparing the inlet steam rate for each injection location, the suction side is more suitable for HSI and injections No.4 and 5 are suitable locations on the suction side. Since the condensation of the steam on droplets at a different temperature is an irreversible process, a condensation loss is produced. The liquid mass flow rate indicates the liquid generated as the nucleation process and liquid droplet growth. As stated above, the efficiency of each turbine stage is reduced by about 1% with a 1% wetness. Fig. 14 displays the condensation loss for each injection location. The injections 1 to 5 and injections 6 to 10 are on the suction side and pressure side, respectively. The HSI on pressure side does not significantly affect the condensation loss. On the suction side, the HSI reduces the condensation loss. The suction side is 22

appropriate to reduce the condensation loss and injection locations No.3 and 4 are suitable locations for decreasing the condensation loss. Considering the specified parameters to select the optimal location for HSI, the importance of condensation loss and erosion rate, and the low range of entropy variations at different injection locations, the injection location No.4 is a suitable location for HIS, due to the lowest erosion rate and condensation loss and the acceptable value of other determinant parameters.

Fig. 14. EEHIC method for each injection

23

3.4. The effect of HSI on entropy Thermodynamic losses are one of the most important losses in steam turbines and occur when the flow deviates from its equilibrium state and the phenomenon of nucleation and becoming two-phase happen. The two-phase flow is an irreversible process and reduces the performance of steam turbines. Eq. (35) mentions that the entropy of the flow depends on the entropy of the steam and the liquid.

:·I,Y? = B1 − 8):> + 8:?

(35)

where :> indicates the entropy generated by the steam phase, :? represents the entropy generated

by the liquid phase, and 8 shows as the liquid mass fraction. This section studies the entropy generation by the HSI in the wet steam flow.

Fig. 15 represents the flow entropy variations for injection No.4 and no injection condition, in which the entropy generation is increased by making wet steam flow. Moreover, this procedure is repeated in the case of HSI case of no injection, although the effect of HSI on entropy generation is greater than that of making wet steam flow. The HSI significantly increases the amount of entropy generation in the injection location and the downstream because hot steam contains high entropy due to its high temperature. Fig. 16 demonstrates the effect of HSI into the turbine blades on the Mach number. The Mach number drops sharply at the downstream of the injection location. Regarding the equation 4j = 4B1 +

¸[g 

¹f )º»¼ , a decrease in the Mach º

number increases the temperature due to the increase of stagnation temperature and the HSI. The injection is perpendicular to the mainstream, so the mixture of the injection and mainstream is very serious and the local entropy generation increases.

24

Fig. 15. Effect of HSI on the local entropy generation

Fig.16. The effect of HSI on the Mach number

4. Conclusion After validating the numerical solution in the turbine blade, the present study studied the effect of HSI on two-phase flow parameters for selecting the best injection location. Several injection locations on the suction and pressure sides were selected due to the different behaviors of the flow near the suction and pressure sides. Further, EEHIC method was considered to select the best injection location. Then, the effect of HSI on five parameters was studied for each injection location. The authors of this paper will examine the impacts of pressure, temperature, slot angle and slot width, on the injection in their future studies. 25

The results indicated that the HSI from the suction side has a greater impact than that of the pressure side. Further, the location of HSI played a significant role in the parameters of condensate flow. The HSI reduced the inlet mass flow rate in the blade and increased the pressure, temperature, and the Mach number in the convergent part, in addition to a sudden rise in the pressure, temperature, and the Mach number in the divergent part. It is suggested that hot steam should be injected in a location before the shock and with the lowest temperature in the flow. In the present study, the injection location No.4 reduced condensation loss and the erosion rate ratio by 81% and more than 99%, compared to the case of no injection, respectively.

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[29] Kermani MJ, Gerber AG. A general formula for the evaluation of thermodynamic and aerodynamic losses in nucleating steam flow. International journal of heat and mass transfer. 2003 Aug 1;46(17):3265-78. [30] Lakzian E, Shaabani S. Analytical investigation of coalescence effects on the exergy loss in a spontaneously condensing wet-steam flow. International Journal of Exergy. 2015;16(4):383403. [31] Lakzian E, Masjedi A. Slip effects on the exergy loss due to irreversible heat transfer in a condensing flow. International Journal of Exergy. 2014;14(1):22-37. [32] Mahpeykar MR, Rad EA, Teymourtash AR. Analytical investigation of simultaneous effects of friction and heating to a supersonic nucleating Laval nozzle. Scientia Iranica. 2014 Oct 12;21.

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[33] Mahpeykar MR, Teymourtash AR, Rad EA. Theoretical investigation of effects of local cooling of a nozzle divergent section for controlling condensation shock in a supersonic twophase flow of steam. Meccanica. 2013 May 1;48(4):815-27. [34] Mahpeykar MR, Teymourtash AR, Amiri Rad E. Reducing entropy generation by volumetric heat transfer in a supersonic two-phase steam flow in a Laval nozzle. International Journal of Exergy. 2011 Jan 1;9(1):21-39. [35] Somesaraee MT, Rad EA, Mahpeykar MR. Analytical investigation of simultaneous effects of convergent section heating of Laval nozzle, steam inlet condition, and nozzle geometry on condensation shock. Journal of Thermal Analysis and Calorimetry. 2018 Aug 1;133(2):1023-39. [36] Ahmadpour A, Abadi SN, Meyer JP. On the performance enhancement of thermocompressor and steam turbine blade cascade in the presence of spontaneous nucleation. Energy. 2017 Jan 15;119:675-93. [37] Vatanmakan M, Lakzian E, Mahpeykar MR. Investigating the entropy generation in condensing steam flow in turbine blades with volumetric heating. Energy. 2018 Mar 15;147:70114. [38] Han X, Zeng W, Han Z. Investigation of the comprehensive performance of turbine stator cascades with heating endwall fences. Energy. 2019 Mar 13. [39] Mirhoseini MS, Boroomand M. Control of wetness fraction and liquid droplet size in wet steam two phase flows with hot steam injection. Meccanica. 2018 Jan 1;53(1-2):193-207. [40] Mirhoseini MS, Boroomand M. Multi-objective optimization of hot steam injection variables to control wetness parameters of steam flow within nozzles. Energy. 2017 Dec 15;141:1027-37. [41] Xu L, Yan P, Huang H, Han W. Effects of hot steam injection from the slot at the trailing edge on turbine nozzle vane flow field. Journal of Thermal Science. 2008 Dec 1;17(4):298-304. [42] Gribin V, Tishchenko A, Tishchenko V, Gavrilov I, Khomiakov S, Popov V, Lisyanskiy A, Nekrasov A, Nazarov V, Usachev K. An experimental study of influence of the steam injection on the profile surface on the turbine nozzle cascade performance. InASME Turbo Expo 2014: Turbine Technical Conference and Exposition 2014 Jun 16 (pp. V01BT27A050V01BT27A050). American Society of Mechanical Engineers. [43] Filippov GA, Gribin VG, Tishchenko AA, Gavrilov IY, Tishchenko VA, Khomiakov SV, Popov VV, Sorokin IY. Steam injection impact on the performances of nozzle grid in wet-vapor stream. Thermal Engineering. 2016 Apr 1;63(4):233-8.

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30

APPENDIX A By using the gas dynamics equations, Appendix A presented that adding heat to subsonic flow in a constant cross-sectional state (Rayleigh flow) and causes the fluid flow to reach a state with the maximum entropy in a given mass flow rate. Adding more heat (critical heat) can only reduce the mass flow rate of the flow in the nozzle, or in another hand, create a jump to another Rayleigh line with a lower mass flow rate. Therefore, there is a maximum amount to add heat for each mass flow rate in the nozzle and adding further heat can reduce the flow rate and change the Rayleigh line. By increasing heat to the maximum value, the static pressure decreases and then, adding more heat increases static pressure. Under constant inlet stagnation temperature and pressure conditions and regarding the reduction of inlet mass flow rate and Mach, as well as the equations of 4j = 4B1 +

¸[g 

¹f )º»¼ and )j = )B1 + º

¸[g 

¹f )º»¼ , giving heat more than the º

maximum heat increases the static pressure and temperature at the inlet.

This section examines the effect of heating on a one-dimensional control volume with a constant cross-section (Rayleigh flow). By applying the heat δ+, the energy equation is as follows:

[ℎj ] = + e 4j = + 5  ½ℎ + ¾ = + 2 ℎ + 55 = +

(a-1) (a-2) (a-3) (a-4)

Equation (a-2) indicates that if δq = 0, then flow is adiabatic and the value of Tj is constant. The

T-S diagram, which called the Rayleigh line, is used to determine the effect of heat on the Mach

number in a mass flow rate. The entropy variations for a perfect gas are written as (derived from the John Gas Dynamics Book) [35]:

 )  4 - − -g 4  − 1 B1 + ¹g ) ÂB1 + ¹g − 4¹g Ã4g Ä = (' − (' B ± ) e 4g  2 2

(a-5)

Fig. A.1 indicates the effect of heating and cooling on supersonic and subsonic on the Rayleigh

31

line. The flow is supersonic at the point Bf) By heating the flow - =

Å& k

, the amount of entropy

generation increases and the flow moves to point B. The point B is the highest point in the Rayleigh curve. By adding heat, the flow moves from point B to point A and the Mach number reaches one. The point A is the maximum entropy in a mass flow rate and adding more heat reduces the mass flow rate. In other words, the flow jumps from a Rayleigh line to another Rayleigh line with a lower mass flow rate. By adding heat to the subsonic flow, the Mach number rises and reaches one. By using the momentum equation, the velocity increases and the static pressure decrease. The static temperature increases to B and then drops in this line [39].

Fig. A.1. The effect of heating and cooling on the behavior of subsonic and supersonic flow on the Riley line [55]

32

APPENDIX B Appendix B addressed the mass injection in a one-dimensional control volume with a constant cross-section. Then, the equations of continuity, momentum, energy, and state were written by injection location (ℎj,E = ℎj,g ) and a relationship was obtained for velocity and pressure

equaling the stagnation enthalpy of injection with the stagnation enthalpy of the flow before the

variations. The equations indicated that the velocity and pressure variations have opposite sign,

that is, an increase in the velocity in the control volume decreases the pressure. In addition, the equations demonstrated that the variations of density, temperature, and pressure have the same sign. The velocity variations extracted from equations depend on injection parameters (injection pressure, injection speed, and injection cross-section), flow parameters before the injection location (pressure, velocity, Mach number) and HSI ratio. By substituting the values of flow and injection parameters in the obtained equation for the velocity variation in appendix B (Eq. 29-b), it is expected that velocity variation has a decreasing trend for all considered locations on suction and pressure sides while the pressure variation has an increasing trend in all injection locations. On the other hand, the HSI has a higher temperature than that of flow, resulting in heating the flow and increasing the pressure, according to appendix A. Regarding the mass injection and heating effect, the pressure and temperature are expected to increase at all injection locations and the velocity reduces. The mass injection equations and their effects on the flow were investigated in this section. First, we study the energy equation.


{E
{
{ ℎj, −
{g ℎj,g =
{E ℎj,E ℎj, − B1 − €)ℎj,g = €ℎj,E ℎj, − ℎj,g = €Bℎj,E − ℎj,g ) €=

(b-1) (b-2) (b-3) (b-4)

By assuming ℎj,E = ℎj,g , the equations are as follows: ℎj,E = ℎj,g

(b-5)

33

ℎj, = ℎj,g ℎj = 0

(b-6)

5 Bℎ + ) = 0 2

(b-7) (b-8)

ℎ + 55 = 0 e 4 = 55 5 4 = 5 e

(b-9) (b-10) (b-11)

The mass conservation equation in the axis of a one-dimensional control volume with a constant cross section is written as follows:

Fig. B.2. The control volume is intended to investigate the effect of mass injections

[
{] =
{E [5] =
{E
{E =
{  5 + =  5

(b-12) (b-13) (b-14) (b-15)

By neglecting the frictional force, the momentum conservation equation is as follows: 34

∑ Æ = ) + )E E − B) + )) = «
{ +
{E ¬B5 + 5 ) −
{5 −
{E 5E )E E
{E ) + B1 + )55 = + B5E − 5)   ) + B1 + )55 = !

(b-16)

(b-17)

By using the perfect gas relationship: ) = 04 )  4 = + )  4

(b-18) (b-19)

By using the Mach number relationship: 5  = ¹ B04) 255 = 2¹04¹ + ¹ 04 5 ¹ 1 4 = + 2 4 5 ¹

(b-20) (b-21) (b-22)

By substituting the equations (11-b) and (15-b) in (19-b), we have:

) 5 5 5 1 =− + 5 =  + ` − a 5 ) 5 e 4 e 4 5 5 1 ) = ) + ` − a )5 e 4 5

(b-23) (b-24)

By substituting ) from the equation (17-b):

! − ) (b-25) 5 1 B1 + )5 + Ã e 4 − 5 Ä ) Equation (17-b) implies that the ) variation is in the opposite direction of 5 variation, 5 =

order to accurately examine the 5, the equation (25-b) is written in terms of the Mach number.

representing that an increase in the pressure in the control volume decreases the velocity. In

5 =

! − ) 504 04 B1 + )5 + Ã − Ä e 4 5

(b-26)

35

)E E
{E + B5E − 5) − )   5 = 0 1 5[B1 + ) + − ] e ¹

(b-27)

36

Highlights

•

Location for hot steam injecting (HSI) in turbine blade is studied.

•

Five parameters are used to select the most suitable HSI location.

•

HSI from the suction side has a greater impact than that of the pressure side.

•

HSI reduced the condensation loss and erosion rate ratio by 81% and 99%.