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Computational modeling of non-equilibrium condensing steam flows in low-pressure steam turbines
Journal Pre-proof Computational Modelling of Non-Equilibrium Condensing Steam Flows In LowPressure Steam Turbines Ahmed M. Nagib Elmekawy, Mohey Eldeen H.H. Ali PII:
S2590-1230(19)30065-9
DOI:
https://doi.org/10.1016/j.rineng.2019.100065
Reference:
RINENG 100065
To appear in:
Results in Engineering
Received Date: 28 August 2019 Revised Date:
19 November 2019
Accepted Date: 20 November 2019
Please cite this article as: A.M. Nagib Elmekawy, M.E.H.H. Ali, Computational Modelling of NonEquilibrium Condensing Steam Flows In Low-Pressure Steam Turbines Results in Engineering, https:// doi.org/10.1016/j.rineng.2019.100065. 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 The Author(s). Published by Elsevier B.V.
Credit Author Statement Elmekawy, Ahmed: Conceptualization, Reviewing and Editing, Supervision.
Methodology,
Software,
Writing-
Ali, Mohey Eldeen: Software, Data curation, Writing- Original draft, Visualization, Investigation, Validation preparation..
COMPUTATIONAL MODELLING OF NON-EQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES Ahmed M. Nagib Elmekawy*, Mohey Eldeen H. H. Ali. Mechanical Engineering Department, Alexandria University, Alexandria, Egypt
Article info
Abstract
Key words: Computational Fluid Dynamics Condensation Low pressure steam turbine Corresponding author [email protected]
Condensation occurring in low pressure stages of steam turbines contributes to many losses in efficiency and damage to the stator blades such as corrosion and pitting. This problem has been observed profusely in power generation industries. The purpose of this study is to properly simulate the condensation and shock wave phenomenon at the last stage of the steam turbine and accordingly understand this phenomenon. Numerical simulation was conducted by using ANSYS Fluent and applying k-omega SST turbulence model. The results agreed with the experimental values.
Introduction Steam Turbines are one of the most essential components in power generation using mechanical techniques such as driving compressors or electric generators. Large steam turbines have 3 stages; high pressure, intermediate, and low pressure stage. In the low pressure stage steam flowing in the turbine condenses forming small droplets of water which causes corrosion and pitting on the blades of the turbine which in turn causes deterioration of the turbine’s efficiency. White et al. [1] organized stationary turbine cascade experiments of condensing steam flow with test conditions and provided measurement data of pressure distribution, wetness level, droplet radii at certain locations, Schlieren profiles and normalized entropy. Yousif, et al. [2] performed experiments for non-equilibrium condensation in transonic flow of steam for LP steam turbine, studying the influence of changing of outlet pressure on phase change and flow condition between saturated steam and fine droplets of water. Dykas, et al. [3] conducted experiments in linear blade cascade
for condensing steam flow. Theoretical and Numerical studies began by Laplace [4], who is the founder of the classical theory of nucleation. Thomson [5] used the results of Laplace for first theoretical expression and recognized the existence of supersaturation in steam of curved surface. Helmholtz [6] and Gibbs [7] obtained vital equations which contain the thermodynamic equilibrium systems. Based on their work, the famous Kelvin-Helmholtz or Gibbs-Thomson equation was developed. Volmer and Webber [8] made improvement in the nucleation theory. Farkas [9] described the kinetic mechanism of supersaturated vapors and attained a form of the steady state nucleation rate and after the contribution of many other scientists to the progress of the nucleation theory it became known as ’the classical nucleation theory’. Gerber and Kermani [10] adopted the 'EulerianEulerian' approach for condensing flows. In the works of White and Hounslow [11], White [12], and Gerber and Mousavi [13], the technique of moments and quadrature method of moments were applied for condensing steam flow modeling. In present days, due to the improvement of computers computational
power, the modeling of 3D multistage LP turbine flows is attempted by several researchers such as Yamamoto et al. [14], Starzmann et al. [15], Chandler et al. [16], Starzmann et al. [17] , and Grübel, [18] . The Experiment setup [3] used in this paper is of a steam passed over linearly cascaded blades where these blades are identical to those of the last stage of the low-pressure steam turbine being studied. The properties of the steam entering the test section are also identical to the upstream conditions of the steam entering the last stage of the low-pressure steam turbine. Moreover, the exit conditions are identical to that of the condenser of the steam turbine. The objective of this research is to properly simulate the condensation and shockwave phenomena occurring in the last stage of the steam turbine in order to thoroughly understand them and ultimately make the steam exit the turbine with the lowest pressure possible while eliminating or reducing the effect both the condensation and the shockwave.
Physical Modeling This section covers the discussion of the thermodynamics and fluid mechanics aspects for the flow behavior.
Nucleation and Condensation The definition of the condensation process is the transformation of the steam from the gas state to the liquid state [19]. But physically the transformation process is not that simple as it passes through multiple stages, at first, nucleation process starts with very small droplets of water in the fluid flows. Nucleation occur under two types homogeneous or heterogeneous. In this paper, homogenous nucleation is assumed which means no foreign soluble salts or suspended particles are exists in the flow. Condensation process occurs due to the drop in pressure and temperature of the steam which causes the formation of the water droplets that have a very small size. When the small droplets form, two types of formation occur. Droplets
continue growing in size to form larger water droplets and this is the condensation process according to the classical nucleation theory, or the evaporation of these water droplets into steam again, this mainly depends on the partial pressure of the water droplets and it’s temperature whether it is above the boiling temperature specified at the partial pressure or not according to droplet growth theory [20]. For turbine blade stages, condensation process only occurs at the last stages of the turbine where a very low pressure exists and that depends on the condenser conditions that fixes the required exit pressure from the turbine to obtain the maximum power possible to make sure that condensation process occurs. Beside the inlet and outlet conditions of the turbine, condensation process depends on the arrangement of the station such as is there a reheat or not, and other aerodynamic phenomena such as shock wave which will be discussed later. Condensation process depends on other parameters such as the blade shape and the trailing edge shape. There are three used shapes of the blade trailing edge: sharp edge, circular edge and square edge [21]. Each shape affects directly the condensation process. The square edge causes large zone of separation behind the blade leading to very large drop in the pressure causing condensation process to occur rapidly. For the circular edge, used in this paper, also causes large zone of separation but it is better than the square edge in the zone of separation as shown in simulation results in this paper. The best shape of the trailing edge is the sharp edge which practically does not exist practically as there must be slight curvature at the edge. This paper is performed for the condensation process occurs at the last stages of the turbine to reduce the losses due to condensation. There are three losses types in steam turbine blades:
1. Mechanical losses (Erosion) that caused by the contact between the moving water layer and the blade material and turbine casing. 2. Thermodynamic losses due to the cooling effect due to the existence of the liquid. 3. Aerodynamic losses due to the collision between the liquid phase and the material of the blade whether it’s stationary or moving, and due to the shock waves that occurs at the flow domain. Two types of shock waves that occur at the blade section, the condensation shock takes place in the divergent part at the blade curve (convex) after the throat between two stator blades in LP part last stages and an oblique shock wave occurs between the pressure side of the down blade and the end of the suction side of the upper blade. The condensation shock that occurs at the end of the trailing edge of all blades is not a shock wave, as their properties are very different from the properties of the aerodynamic shock wave, it occurs due to the released latent heat from the water droplets heat addition that holds the flow back. This heat addition leads to increases in pressure and temperature as a result of the condensation shock wave, pushing the main shock wave downstream. This explains why the main shock forward shift appears up till the condensation shock wave appears. as illustrated in Ref [22], [23], [20]. As for oblique shock waves which is not normal on the flow direction and causes slight reduction in the Mach No. of the flow but keep it supersonic, it also causes a slight increasing in the flow static pressure and temperature and slight reduction in the flow total pressure while the total temperature remains constant as shown in Fig. 1.
Two oblique shock waves occurred called fishtail shock waves as shown in Fig. 2, the main shock begins to take place downstream of the trailing edge of the blade and the shock reflected from the suction side. The major reason of occurring the shock wave is the high back pressure after the blades with the expansion between blades that causes an increase in the pressure difference between back pressure and pressure between blades which forms the shock wave. The shock wave formed "fishtail shock" has two legs, first leg "left one" between the trailing edge and the suction surface of the blade affected by the condensation shock because the pressure difference across the shock wave will decrease in result of decreasing the shock wave strength or pushing it downstream, therefore, the disturbances to the boundary layer become smaller and separation may disappear due to pressure rise across the condensation shock.
Figure 2: Fish tale shock wave
Figure 1. Oblique shock wave and its effect on flow properties.
Table 1 “The effect of oblique shock wave”. Property P Po V Ma T To Ρ S
Change Increases Decreases Decreases Decreases Increases remains constant Increases Increases
Table 1 shows the effect of oblique shock wave on the flow properties. Another type of shock wave occurs is the condensation shock. The blade geometry was obtained from the experimental data published by Dykas [3].
cascade and the equations for each section of pressure side of blade cascade. Same method were used for the suction side of the blade.
Figure 3. Blade geometry [3].
Geometry, meshing and mathematical modeling The simulation was done using ANSYS FLUENT.
1- Geometry The geometry specified in the experimental study conducted by Dykas [3] was used [as shown in Fig. 3. To find the blade curve between points an interpolation was conducted using Microsoft Excel and Matlab. Then, we divided the blade geometry profile of the cascade into three sections for each of the pressure side and suction side of the blade cascade and two arcs, one for the leading edge and the other is for trailing edge. In the pressure side of blade cascade, the first section starts from X= 14.1 mm and ends at X= 105.63 mm, the second section starts from X= 105.63 mm and ends at X= 165.55 mm and the third section starts from X= 165.55 mm and ends at X= 172.9 mm. For each section of the pressure side of blade cascade, we obtained an equation which is a relation between X(mm) and Y(mm) using Microsoft Office Excel. Figures 4, 5, 6 and 7 show the geometry profile of pressure side of blade cascade for each section of pressure side of blade
Figure 4. Co-ordinates of blade geometry in Microsoft Office Excel
Figure 5. Geometry profile of first section of pressure side of blade cascade in Microsoft Office Excel.
that, it's important to use inflation layer meshing to precisely realize the boundary layer region for any wall around turbulent flows. As shown in Fig. 8 the difference between a favorable pressure gradient and adverse pressure gradient with the separation of flow is illustrated. As it’s shown in Fig. 8., the adverse pressure gradient (APG) increases the turbulence of the flow at the boundary layer and the flow becomes very unstable It's obvious, that the flow behavior in the near the region of the wall is very complicated and needs to be captured in a proper way to have any assurance in our CFD results.
Figure 6. Geometry profile of second section of pressure side of blade cascade in Microsoft Office Excel.
Figure 8 Near-wall velocity profiles
Figure 7. Geometry profile of third section of pressure side of blade cascade in Microsoft Office Excel. 2- Meshing As free stream flow towards to the wall in fluid domain, non- linear reduction of velocity can be recognized above the point where the flow will be zero velocity at wall that is called (no slip condition). For the plot the velocity profile in the nearwall region, a big variation of velocity perpendicular to the wall can be realized, so it's very significant in CFD simulation that capture this gradient in correct way. For do
Proper inflation mesh selecting for the shape is related to the option of the Turbulence Model, and the flow field we are concerned to be captured. For resolving the complete profile of the boundary layer can be done by using empirical wall functions to minimize the cell. As shown in Fig. 9, we recognize that the boundary layer profile with a minimized cell, that is typical of a wall function option and the boundary layer profile is resolved all way to the wall; this will give a more precise resolution of the boundary layer. For specified simulations as flows with strong wall-bounded effects, this resolution is very important for our case.
Figure 9. Representation of wall function vs fully resolving the boundary condition From this point, we start to recognize the location of first node in near wall mesh is very important. Non-dimensional distance (based on local cell fluid velocity) will be used from the wall to first node, that can be termed Y+ as shown in Fig 10. To use a wall function approach for specific turbulence model with confidence that need to make sure that y+ values are less than 3.
Figure 9. Y+ definition. Tetrahedral mesh was used to obtain suitable skewness and orthogonal quality for that complex geometry. The flow domain is divided into numerous zones to have more control on the number of elements and the mesh size in each zone such as blade suction side and outlet zone where more refined mesh than other zones. Inflation layers were done to obtain more accurate results near the wall surface as shown in Fig. 10 and 11. Sixty-Eight hundred thousand elements is used with an average skewness and orthogonal quality of 0.0076 and 0.9545; respectively.
Figure 10. Mesh for different zones
Figure 11. Inflation layer over the blade geometry. 3Mathematical modeling Numerical setup is performed for the twodimensional flow in FLUENT by using, Transient flow, density based solver, wet steam model and k- SST turbulence model as suggested by other researchers [3] and by comparing this turbulence model with the K-ϵ turbulence model it was found that the earlier gives more realistic results than the later. For condensation process several assumptions were used: the velocity slip between the two phases and the interaction between droplets are assumed negligible. The mass fraction of water droplets and volume of condensed water are small. Droplets are spherical. Boundary conditions used are
1. Inlet total pressure =89000 Pa 2. Outlet static pressure =39000 Pa measured at 80 mm from the blade trailing edge. 3. Inlet temperature= 100 C. Assumptions • Turbulent • Unsteady “Transient’ • Compressible • 2-D Flow “X&Y’ • The velocity slip between the two phases is negligible ‘means that the surface interface of both faces moves with the same speed no drop. • interaction between droplets is negligible • The mass fraction of water droplets is small < 0.2 • Volume of condensed water is negligible as its size is very small
Mathematical equations + ∇. =
Fluent solves Naiver-Stokes equations [20]
It represents the general form of the continuity equation The term represents the amount of mass added from one phase to another for example “the amount of mass that turned into liquid from vapor state”. In words, represents the mass flow rate that leaves the domain.
Represents the change in the density that occurs as time passes. While ∇. represents the mass flow rate at the main coordinates “x & y” that enters the domain. + ∇. = −∇ + ∇. ̿ + ̅ +
The term represents the rate of change
in momentum with time at the main coordinates. The term ∇. represents the change in momentum within the main coordinates at the main coordinates. −∇ Represents the static pressure applied from the
main coordinates. ∇. ̿ Represents the shear stress tensor and will be described later. ̅ + Represents the gravitational force and
the body forces.
Wet steam model Overview on the wet steam model: During the expansion of steam, condensation occurs when the pressure of steam is reduced. The expansion of steam causes the formation of water droplets due to the nucleation of steam. Wet steam model in ANSYS uses the EulerianEulerian approach for modeling wet steam flow. The flow mixture is modeled using the compressible Navier-Stokes equations, two transport equations is used for the liquid-phase mass-fraction, and the number of liquiddroplets per unit volume. The phase change model which is the main reason for the droplet formation, is based on the nucleation theory. In Eulerian-Eulerian approach, both phases are treated separately, means that each phase has its own time and space that occupy, so at this approach each phase has its own conservation equations. In another word, pure vapor is solved separately from the liquid droplets, but the formation of the water droplets due to vapor condensation is modeled using the classical nucleation theory which was discussed at the condensation chapter. Limitation on wet steam model 1- Pressure inlet, mass-flow inlet, and pressure outlet are the only inflow and outflow boundary conditions available. 2- Density based solver only Wet steam model equations 1) liquid phase equations Wetness mass fraction equation
+ ∇. = Γ
While Γ is the mass generation rate due to condensation and evaporation. Note that the mass fraction of liquid means the percentage of the liquid mass that exists in the mixture mass. As Γ increases this means that the liquid mass increases that leads to witness mass fraction increasing.
+ ∇. = Ι
Number of water droplets
While Ι is the nucleation rate ‘number of new droplets per unit volume’ from the mixture density equation and from the average droplet volume equation the number of the water droplets per unit volume can be found
=
4 π#̅ $ 3
# is the droplet radius
=
%
#̅ 1 8+1 = :; 7< − 7 ℎ.3 452-#73 28 It’s noticed that the rate of growth of droplet is a function of the liquid and vapor properties such as liquid enthalpy, vapor pressure, liquid density and specific heat ratio. Also, it is a function of 7< which is the temperature of the droplet.
Nucleation rate
>? 30 2D 'GHIJ∗ L P A B C $ F $MNO == 1 + @ . E K
While >? is the evaporation coefficient, QR is Boltzmann constant and E is the mass of one molecule. Anon-isothermal correction factor @ is given by
* &'% () + *,
@=
28 − 1 ℎ.3 ℎ.3 G P G − 0.5P 8 + 1 S7 S7
1 = 3 S71 + V3 + :30
2) Phase change equations Mass generation rate #̅ 4 Γ = -. Ι# ∗ $ + 4-. #̅ 0 3
3)
The first term represents the droplet formation and the second term represents the growth of the droplet with time. Note that the mass generation rate is calculated at the mean radius. # ∗ Is Kelvin-Helmholtz critical radius. There is a critical radius at which the droplet size will turn from demise to growth and it depends on the mixture properties. If the radius of the droplet is smaller than the critical means the heating effect is larger than the cooling effect that leads into demise, and the opposite is true.
The Mach number, Liquid Mass Fraction, and Pressure contours are illustrated in Fig. 12,13 and 14 respectively with Mach number and liquid mass fraction contours compared to reference [3] where (a) represents the contours of this paper while (b) represents that of the cited paper. From the Pressure contour shown in Fig. 14, some of the steam entering the test section inlet region collides with the blades at the leading edge leading to creation of stagnation points with very high static pressure and zero velocity. As the steam passes between the blades its velocity increases while the pressure and temperature decreases. From the liquid
While growth rate
Equation of state
Results and Discussion
mass fraction contour shown in Fig. 13, at the trailing edge of the blade and the suction side of the following blade steam starts to condensate as its pressure drops below the saturation pressure entering the wet region. In the outlet region, the high rates of expansion causes great decrease in the pressure which in turn causes the increase in the liquid mass fraction thus increasing the condensation of the steam. This condensation further decreases the pressure due to the decreasing in volume of the condensed steam, but because of the high back pressure of the condenser an Aerodynamic shock wave occurs and the pressure rises again where the steam’s condition reverts away from the saturation pressure. The direction of the Streamlines in Fig. 15 shows that the aerodynamic shockwave is an oblique shockwave where the direction of the shockwave is not normal to the flow.
Figure 13. Liquid Mass Fraction Contours.
Figure 14. Pressure Contour from numerical solution.
Figure 12. Mach number Contours.
The reason for the decreased Mach number, increased liquid mass fraction and formation of eddies in the region downstream the trailing edge as shown in Fig. 15 and 16, as mentioned in Ref [21], is due to the circular shape of the blade trailing edge which forms large zones of separation downstream the trailing edge.
Figure 15. Velocity Streamlines. Figure 17. Pressure Distribution [the dots represent the experimental data and the lines represent the simulation].
In comparison with the numerical results of the cited paper shown in Fig , this numerical paper could capture the physical phenomena, such as the separation downstream the blade trailing edge, oblique shock wave and condenation shocks, in a way more accurate than the cited paper and that is because the large number of mesh elements used for the shock zone. Figure 16. Eddies at the trailing edge of the blade. The pressure distribution on the blade surface in Fig. 17 is superimposed on the figure of the cited paper [3]. The percentage error between experimental and simulation results are listed in Table 2 showing the average and maximum error in both this paper and the cited paper.
The experimental values of pressure distribution was conducted by Dykas [3]. Comparing this paper simulation results with the experimental data showed an improvement in the average and maximum errors between the numerical results and the experimental data as shown in Table 2. The average and maximum errors for this paper were less than those that were calculated in the cited paper.
Mesh independence results Results showed in Fig. 18 illustrate mesh independence tests. Solving for number of elements of 200000, 400000, 600000 and 800000, It was found that as the number of elements increases the solution accuracy
increases but it requires more computational power so the decision of dividing the flow domain and limit the number of elements to 680000 elements was taken to obtain more
accurate results with computational power.
optimizing
the
Figure 17. Mesh independency test Results. Pressure distribution on the blade for different meshes. Tab le 2. Error Calculation. Point 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Point 9 10 11 12 13 14 15 16 17 18 19 20
Blade Suction Side Experimental (Pa) Numerical (Pa) Percentage Error Cited Paper (Pa) Percentage Error 71100.2 71580.7065 0.675815961 71712 0.860475779 66296.7 66862.3638 0.853230704 66986 1.039719926 60392 61189.795 1.32102762 61336 1.563120943 57260.1 58211.1644 1.660954836 57997 1.286934532 60608.2 61060.1934 0.745762785 60822 0.352757548 58726 58966.9289 0.41025934 58973 0.42059735 53188.3 52638.3003 1.034061438 52192 1.873156314 47504.6 46881.1488 1.312401746 45411 4.407152149 42063.2 46564.4859 10.70124456 37603 10.60356796 48070 47397.1499 1.399729769 40000 16.78801747 45328.2 41374.5585 8.722255682 34562 23.75166011 42593 36035.4827 15.39576292 40274 5.444556617 38804.2 34287.8658 11.63877673 37500 3.360976389 36061.8 33518.1262 7.053651787 34521 4.272665258 34044.1 32720.7989 3.887020365 33288 2.220942836 Blade Pressure Side Experimental (Pa) Numerical (Pa) Percentage Error Cited Paper (Pa) Percentage Error 79131.3 79015.9435 0.145778598 79470 0.428022792 76437.4 76235.5026 0.264134311 76541 0.135535746 72767.3 73246.2267 0.658161977 73459 0.950564333 70497.3 70352.2974 0.205685324 70377 0.170644833 68161.1 69091.1911 1.364548254 69041 1.290912265 67068.2 68042.2612 1.45234433 67911 1.25663131 66396.5 67236.8325 1.265627706 67295 1.353233981 66573.9 66676.9599 0.154805262 66575 0.001652299 66396.5 66198.2133 0.29864029 66267 0.195040401 65508.5 65628.5888 0.18331789 65856 0.530465512 64718.5 64763.6736 0.069800134 64521 0.305167765 63417.8 63476.5536 0.092645283 63600 0.287301042 Average Numerical Error Average cited paper Error 2.702497985 3.153758277 Maximum Numerical Error Maximum cited paper Error
15.39576292
23.75166011
Conclusion This paper presents the numerical study that was carried out by ANSYS Fluent on the blades of a low-pressure steam turbine to clarify the condensation process and the effect of the blade geometry on it. The results of this paper proved that ANSYS Fluent can be used to solve such a complicated process for this geometry with an acceptable result. The turbulence model that can be used to solve such a problem with acceptable results is the K-ω SST model. The separation mentioned in Ref. [20] occurs due to the shape of the trailing edge can be clearly identified from the numerical results. Aerodynamic oblique shock wave was noticed from the solution and it
occurs due to the high back pressure of the test section. Condensation shock was clearly noticed in the region that exists downstream the trailing edge of the blade. The simulation results of this paper matched the experimental data. Further study would be carried to optimize the shape of the geometry in order to reduce the losses.
Conflict of interest No conflict of interest
Acknowledgments We would like to thank E-JUST University for assistance with the simulation and our colleagues and teammates for their large contribution in this work.
References
[1] A. White, J. young and P. walters, "Experimental validation of condensing flow theory for a stationary cascade of steam turbine blade," 1996. [2] A. Yousif, A. Al-Dabagh and R. Al-Zuhairy, "Non-equilibrium spontaneous condensation in transonic steam flow," Int. J. Therm. Sci., 2013. [3] S. Dykas, M. Majkut, M. Strozik and K. Smołka, "Experimental Study of Condensing Steam Flow in Nozzles and Linear Blade Cascade," nternational Journal of Heat and Mass Transfer, 2015. [4] L. P, Traite de Mechanique Celeste, 1806. [5] T. W, "The Equilibrium of Vapour at a Curved Surface of Liquid.". [6] v. H. R, "Untersuchung über Dämpfe und Nebel, besonders über solche von Lösungen," 1886. [7] J. Thomson, "Applications of dynamics to physics and chemistry," 1888.
[8] V. M and W. A, "Nucleation of supersaturated structures," 1926. [9] F. L, "nucleation rate in supersaturated vapors," 1927. [10] A. Gerber and M. Kermani, "A Pressure Based Eulerian-Eulerian Multi-Phase Model for NonEquilibrium Condensation in Transonic Steam Flow," Heat Mass Transfer, 2004. [11] A. White and M. Hounslow, "Modelling droplet size distributions in polydispersed wetsteam flows," Int. J. Heat Mass Transfer, 2000. [12] W. A. J, "A comparison of modelling methods for polydispersed wet-steam flow," international Journal for Numerical Methods in Engineering, 2003. [13] A. GERBER and A. MOUSAVI, "Application of quadrature method of moments to the polydispersed droplet spectrum in transonic steam flows with primary and secondary nucleation," Applied Mathematical Modelling, 2007. [14] S. Yamamoto, Y. Sasao, H. Kato, H. Satsuki, H. Ooyama and K. Ishizaka, "Numerical and experimental investigations of unsteady 3-D wet-steam flows through two-stage stator-rotor cascade channels," ASME Turbo Expo, 2010. [15] J. Starzmann, M. Schatz, M. V. Casey, J. F. Mayer and F. Sieverding, "Modelling and Validation of Wet Steam Flow in a Low Pressure Steam Turbine," ASME Turbo Expo, 2011. [16] K. Chandler, A. White and J. Young, "Non-equilibrium wet-steam calculations of unsteady lowpressure turbine flows," 2013. [17] J. Starzmann, M. Casey, M. J.F and F. Sieverding, "Wetness loss prediction for a low pressure steam turbine using computational fluid dynamics," Inst. Mech. Eng. Part J. Power Energy, 2013. [18] M. Grübel, J. Starzmann, M. Schatz, T. Eberle, D. M. Vogt and F. Sieverding, "Two-Phase Flow Modeling and Measurements in Low-Pressure Turbines," ASME Turbo Expo, 2014. [19] R. Sidin, Droplet size distribution in condensing flow, Enschede: University of Twente, 2009. [20] h. Mohey Eldeen, M. Yousef, A. A, A. Ahmed, S. Alaa, E. Dalia, R. Mohamed, H. Mamdouh, M. Mustafa, A. Nada, M. Nada and A. Yasmine, COMPUTATIONAL MODELLING OF NONEQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES, Alexandria,Egypt: Faculty of Engineering , Alexandria University, 2017. [21] N. B. Vargaftik, B. N. Volkov and L. D. Voljak, International Tables of the Surface Tension of Water, 1983.
[22] Y. Cengel, Fluid mechanics, Fundamentals and application. [23] J. Anderson, Modern Compressible Flow with Historical Perspective, 2003.
WX WY ^ \. X] _`
W X] ^ WY ^ ] ^ \. X] \e
\. gh l Xk ^n
WXo WY ^ o \. X] q
WXs WY ^ s \. X] t Xv Z∗
Appendix Density variation with time Mass flow rate for main coordinates (X&Y) Amount of mass transformed from one phase to another Rate of change in momentum with time The change in the momentum of the flow at the main directions (X&Y) Static pressure applied on the flow from the main directions (X&Y) Shear stress tensor Gravitational force Body force Rate of change in witness fraction Variation in witness fraction with the main coordinates (X&Y) Mass generation rate Rate of change in the number of droplets Variations (X&Y) in the number of droplets with the main coordinates Nucleation rate Liquid density Kelvin-Helmholtz critical radius
Z[
WZ[ WY ab
Droplet radius Growth rate of the liquid droplet Temperature of water droplet
T
Mean temperature
cd
Evaporation coefficient
f
ij m` B ,C W Xi WY W Xp WY ri rp
qi ,qp
ui ,up _i ,_p
non-isothermal correction factor Boltzmann constant Mass of one molecule empirical functions Rate of change in turbulence kinetic energy The rate of change in the specific rate of dissipation The generation of turbulence kinetic energy due to mean velocity gradients The generation of the specific rate of dissipation due to mean velocity gradients The effective diffusivity of i and p
The dissipation of i and p due to turbulence User-defined source terms defined in the software
COMPUTATIONAL MODELLING OF NON-EQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES Highlights • • • •
Computational Fluid Dynamics Condensation Low pressure steam turbine Simulation
Conflict of Interest No conflict of interest
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Please change the graph number of the second figure to 18 as shown below:
Best Regards From: Nagappan, Seenuvasan (ELS-CON) Sent: Tuesday, December 3, 2019 2:33 PM To: Ahmed Nagib Subject: RE: Publication of your article [RINENG_100065] in Results in Engineering is on hold due to file problems Dear Dr. Elmekawy, Thank you for your response. As per the below email we have also raised query for 17th figure . But we have received response for 9th figure only. Kindly check and advise also on the 17th figure. Thanks & Regards Seenuvasan Nagappan Journal Administrator ELSEVIER | Global Journals Production [email protected]
From: Ahmed Nagib Sent: 03 December 2019 03:37 To: Nagappan, Seenuvasan (ELS-CON)
Subject: Re: Publication of your article [RINENG_100065] in Results in Engineering is on hold due to file problems *** External email: use caution ***
Dear Sir, Please consider the first figure only.
Best Regards From: Nagappan, Seenuvasan (ELS-CON) Sent: Monday, December 2, 2019 3:45 PM To: Ahmed Nagib Subject: RE: Publication of your article [RINENG_100065] in Results in Engineering is on hold due to file problems Dear Dr. Elmekawy, Thank you for your e-mail. We have received 9th and 17th figure as two times with different legend in manuscript. kindly check and advise which one should be follow.(refer attached screenshot). Thanks & Regards Seenuvasan Nagappan Journal Administrator ELSEVIER | Global Journals Production [email protected]
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From: Ahmed Nagib Sent: 01 December 2019 02:14 To: Nagappan, Seenuvasan (ELS-CON) Subject: Re: Publication of your article [RINENG_100065] in Results in Engineering is on hold due to file problems *** External email: use caution ***
Dear Sir, I am not able to understand what you mean from you comment “We have received two set of figure(9,17), kindly check and advise which one should be follow”. Can you please clarify it to me? Ahmed Nagib Elmekawy, PhD, P.E. > On Nov 30, 2019, at 6:43 AM, "[email protected]" wrote: > > * We have received two set of figure(9,17), kindly check and advise which one should be follow.
S2590-1230(19)30065-9
DOI:
https://doi.org/10.1016/j.rineng.2019.100065
Reference:
RINENG 100065
To appear in:
Results in Engineering
Received Date: 28 August 2019 Revised Date:
19 November 2019
Accepted Date: 20 November 2019
Please cite this article as: A.M. Nagib Elmekawy, M.E.H.H. Ali, Computational Modelling of NonEquilibrium Condensing Steam Flows In Low-Pressure Steam Turbines Results in Engineering, https:// doi.org/10.1016/j.rineng.2019.100065. 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 The Author(s). Published by Elsevier B.V.
Credit Author Statement Elmekawy, Ahmed: Conceptualization, Reviewing and Editing, Supervision.
Methodology,
Software,
Writing-
Ali, Mohey Eldeen: Software, Data curation, Writing- Original draft, Visualization, Investigation, Validation preparation..
COMPUTATIONAL MODELLING OF NON-EQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES Ahmed M. Nagib Elmekawy*, Mohey Eldeen H. H. Ali. Mechanical Engineering Department, Alexandria University, Alexandria, Egypt
Article info
Abstract
Key words: Computational Fluid Dynamics Condensation Low pressure steam turbine Corresponding author [email protected]
Condensation occurring in low pressure stages of steam turbines contributes to many losses in efficiency and damage to the stator blades such as corrosion and pitting. This problem has been observed profusely in power generation industries. The purpose of this study is to properly simulate the condensation and shock wave phenomenon at the last stage of the steam turbine and accordingly understand this phenomenon. Numerical simulation was conducted by using ANSYS Fluent and applying k-omega SST turbulence model. The results agreed with the experimental values.
Introduction Steam Turbines are one of the most essential components in power generation using mechanical techniques such as driving compressors or electric generators. Large steam turbines have 3 stages; high pressure, intermediate, and low pressure stage. In the low pressure stage steam flowing in the turbine condenses forming small droplets of water which causes corrosion and pitting on the blades of the turbine which in turn causes deterioration of the turbine’s efficiency. White et al. [1] organized stationary turbine cascade experiments of condensing steam flow with test conditions and provided measurement data of pressure distribution, wetness level, droplet radii at certain locations, Schlieren profiles and normalized entropy. Yousif, et al. [2] performed experiments for non-equilibrium condensation in transonic flow of steam for LP steam turbine, studying the influence of changing of outlet pressure on phase change and flow condition between saturated steam and fine droplets of water. Dykas, et al. [3] conducted experiments in linear blade cascade
for condensing steam flow. Theoretical and Numerical studies began by Laplace [4], who is the founder of the classical theory of nucleation. Thomson [5] used the results of Laplace for first theoretical expression and recognized the existence of supersaturation in steam of curved surface. Helmholtz [6] and Gibbs [7] obtained vital equations which contain the thermodynamic equilibrium systems. Based on their work, the famous Kelvin-Helmholtz or Gibbs-Thomson equation was developed. Volmer and Webber [8] made improvement in the nucleation theory. Farkas [9] described the kinetic mechanism of supersaturated vapors and attained a form of the steady state nucleation rate and after the contribution of many other scientists to the progress of the nucleation theory it became known as ’the classical nucleation theory’. Gerber and Kermani [10] adopted the 'EulerianEulerian' approach for condensing flows. In the works of White and Hounslow [11], White [12], and Gerber and Mousavi [13], the technique of moments and quadrature method of moments were applied for condensing steam flow modeling. In present days, due to the improvement of computers computational
power, the modeling of 3D multistage LP turbine flows is attempted by several researchers such as Yamamoto et al. [14], Starzmann et al. [15], Chandler et al. [16], Starzmann et al. [17] , and Grübel, [18] . The Experiment setup [3] used in this paper is of a steam passed over linearly cascaded blades where these blades are identical to those of the last stage of the low-pressure steam turbine being studied. The properties of the steam entering the test section are also identical to the upstream conditions of the steam entering the last stage of the low-pressure steam turbine. Moreover, the exit conditions are identical to that of the condenser of the steam turbine. The objective of this research is to properly simulate the condensation and shockwave phenomena occurring in the last stage of the steam turbine in order to thoroughly understand them and ultimately make the steam exit the turbine with the lowest pressure possible while eliminating or reducing the effect both the condensation and the shockwave.
Physical Modeling This section covers the discussion of the thermodynamics and fluid mechanics aspects for the flow behavior.
Nucleation and Condensation The definition of the condensation process is the transformation of the steam from the gas state to the liquid state [19]. But physically the transformation process is not that simple as it passes through multiple stages, at first, nucleation process starts with very small droplets of water in the fluid flows. Nucleation occur under two types homogeneous or heterogeneous. In this paper, homogenous nucleation is assumed which means no foreign soluble salts or suspended particles are exists in the flow. Condensation process occurs due to the drop in pressure and temperature of the steam which causes the formation of the water droplets that have a very small size. When the small droplets form, two types of formation occur. Droplets
continue growing in size to form larger water droplets and this is the condensation process according to the classical nucleation theory, or the evaporation of these water droplets into steam again, this mainly depends on the partial pressure of the water droplets and it’s temperature whether it is above the boiling temperature specified at the partial pressure or not according to droplet growth theory [20]. For turbine blade stages, condensation process only occurs at the last stages of the turbine where a very low pressure exists and that depends on the condenser conditions that fixes the required exit pressure from the turbine to obtain the maximum power possible to make sure that condensation process occurs. Beside the inlet and outlet conditions of the turbine, condensation process depends on the arrangement of the station such as is there a reheat or not, and other aerodynamic phenomena such as shock wave which will be discussed later. Condensation process depends on other parameters such as the blade shape and the trailing edge shape. There are three used shapes of the blade trailing edge: sharp edge, circular edge and square edge [21]. Each shape affects directly the condensation process. The square edge causes large zone of separation behind the blade leading to very large drop in the pressure causing condensation process to occur rapidly. For the circular edge, used in this paper, also causes large zone of separation but it is better than the square edge in the zone of separation as shown in simulation results in this paper. The best shape of the trailing edge is the sharp edge which practically does not exist practically as there must be slight curvature at the edge. This paper is performed for the condensation process occurs at the last stages of the turbine to reduce the losses due to condensation. There are three losses types in steam turbine blades:
1. Mechanical losses (Erosion) that caused by the contact between the moving water layer and the blade material and turbine casing. 2. Thermodynamic losses due to the cooling effect due to the existence of the liquid. 3. Aerodynamic losses due to the collision between the liquid phase and the material of the blade whether it’s stationary or moving, and due to the shock waves that occurs at the flow domain. Two types of shock waves that occur at the blade section, the condensation shock takes place in the divergent part at the blade curve (convex) after the throat between two stator blades in LP part last stages and an oblique shock wave occurs between the pressure side of the down blade and the end of the suction side of the upper blade. The condensation shock that occurs at the end of the trailing edge of all blades is not a shock wave, as their properties are very different from the properties of the aerodynamic shock wave, it occurs due to the released latent heat from the water droplets heat addition that holds the flow back. This heat addition leads to increases in pressure and temperature as a result of the condensation shock wave, pushing the main shock wave downstream. This explains why the main shock forward shift appears up till the condensation shock wave appears. as illustrated in Ref [22], [23], [20]. As for oblique shock waves which is not normal on the flow direction and causes slight reduction in the Mach No. of the flow but keep it supersonic, it also causes a slight increasing in the flow static pressure and temperature and slight reduction in the flow total pressure while the total temperature remains constant as shown in Fig. 1.
Two oblique shock waves occurred called fishtail shock waves as shown in Fig. 2, the main shock begins to take place downstream of the trailing edge of the blade and the shock reflected from the suction side. The major reason of occurring the shock wave is the high back pressure after the blades with the expansion between blades that causes an increase in the pressure difference between back pressure and pressure between blades which forms the shock wave. The shock wave formed "fishtail shock" has two legs, first leg "left one" between the trailing edge and the suction surface of the blade affected by the condensation shock because the pressure difference across the shock wave will decrease in result of decreasing the shock wave strength or pushing it downstream, therefore, the disturbances to the boundary layer become smaller and separation may disappear due to pressure rise across the condensation shock.
Figure 2: Fish tale shock wave
Figure 1. Oblique shock wave and its effect on flow properties.
Table 1 “The effect of oblique shock wave”. Property P Po V Ma T To Ρ S
Change Increases Decreases Decreases Decreases Increases remains constant Increases Increases
Table 1 shows the effect of oblique shock wave on the flow properties. Another type of shock wave occurs is the condensation shock. The blade geometry was obtained from the experimental data published by Dykas [3].
cascade and the equations for each section of pressure side of blade cascade. Same method were used for the suction side of the blade.
Figure 3. Blade geometry [3].
Geometry, meshing and mathematical modeling The simulation was done using ANSYS FLUENT.
1- Geometry The geometry specified in the experimental study conducted by Dykas [3] was used [as shown in Fig. 3. To find the blade curve between points an interpolation was conducted using Microsoft Excel and Matlab. Then, we divided the blade geometry profile of the cascade into three sections for each of the pressure side and suction side of the blade cascade and two arcs, one for the leading edge and the other is for trailing edge. In the pressure side of blade cascade, the first section starts from X= 14.1 mm and ends at X= 105.63 mm, the second section starts from X= 105.63 mm and ends at X= 165.55 mm and the third section starts from X= 165.55 mm and ends at X= 172.9 mm. For each section of the pressure side of blade cascade, we obtained an equation which is a relation between X(mm) and Y(mm) using Microsoft Office Excel. Figures 4, 5, 6 and 7 show the geometry profile of pressure side of blade cascade for each section of pressure side of blade
Figure 4. Co-ordinates of blade geometry in Microsoft Office Excel
Figure 5. Geometry profile of first section of pressure side of blade cascade in Microsoft Office Excel.
that, it's important to use inflation layer meshing to precisely realize the boundary layer region for any wall around turbulent flows. As shown in Fig. 8 the difference between a favorable pressure gradient and adverse pressure gradient with the separation of flow is illustrated. As it’s shown in Fig. 8., the adverse pressure gradient (APG) increases the turbulence of the flow at the boundary layer and the flow becomes very unstable It's obvious, that the flow behavior in the near the region of the wall is very complicated and needs to be captured in a proper way to have any assurance in our CFD results.
Figure 6. Geometry profile of second section of pressure side of blade cascade in Microsoft Office Excel.
Figure 8 Near-wall velocity profiles
Figure 7. Geometry profile of third section of pressure side of blade cascade in Microsoft Office Excel. 2- Meshing As free stream flow towards to the wall in fluid domain, non- linear reduction of velocity can be recognized above the point where the flow will be zero velocity at wall that is called (no slip condition). For the plot the velocity profile in the nearwall region, a big variation of velocity perpendicular to the wall can be realized, so it's very significant in CFD simulation that capture this gradient in correct way. For do
Proper inflation mesh selecting for the shape is related to the option of the Turbulence Model, and the flow field we are concerned to be captured. For resolving the complete profile of the boundary layer can be done by using empirical wall functions to minimize the cell. As shown in Fig. 9, we recognize that the boundary layer profile with a minimized cell, that is typical of a wall function option and the boundary layer profile is resolved all way to the wall; this will give a more precise resolution of the boundary layer. For specified simulations as flows with strong wall-bounded effects, this resolution is very important for our case.
Figure 9. Representation of wall function vs fully resolving the boundary condition From this point, we start to recognize the location of first node in near wall mesh is very important. Non-dimensional distance (based on local cell fluid velocity) will be used from the wall to first node, that can be termed Y+ as shown in Fig 10. To use a wall function approach for specific turbulence model with confidence that need to make sure that y+ values are less than 3.
Figure 9. Y+ definition. Tetrahedral mesh was used to obtain suitable skewness and orthogonal quality for that complex geometry. The flow domain is divided into numerous zones to have more control on the number of elements and the mesh size in each zone such as blade suction side and outlet zone where more refined mesh than other zones. Inflation layers were done to obtain more accurate results near the wall surface as shown in Fig. 10 and 11. Sixty-Eight hundred thousand elements is used with an average skewness and orthogonal quality of 0.0076 and 0.9545; respectively.
Figure 10. Mesh for different zones
Figure 11. Inflation layer over the blade geometry. 3Mathematical modeling Numerical setup is performed for the twodimensional flow in FLUENT by using, Transient flow, density based solver, wet steam model and k- SST turbulence model as suggested by other researchers [3] and by comparing this turbulence model with the K-ϵ turbulence model it was found that the earlier gives more realistic results than the later. For condensation process several assumptions were used: the velocity slip between the two phases and the interaction between droplets are assumed negligible. The mass fraction of water droplets and volume of condensed water are small. Droplets are spherical. Boundary conditions used are
1. Inlet total pressure =89000 Pa 2. Outlet static pressure =39000 Pa measured at 80 mm from the blade trailing edge. 3. Inlet temperature= 100 C. Assumptions • Turbulent • Unsteady “Transient’ • Compressible • 2-D Flow “X&Y’ • The velocity slip between the two phases is negligible ‘means that the surface interface of both faces moves with the same speed no drop. • interaction between droplets is negligible • The mass fraction of water droplets is small < 0.2 • Volume of condensed water is negligible as its size is very small
Mathematical equations + ∇. =
Fluent solves Naiver-Stokes equations [20]
It represents the general form of the continuity equation The term represents the amount of mass added from one phase to another for example “the amount of mass that turned into liquid from vapor state”. In words, represents the mass flow rate that leaves the domain.
Represents the change in the density that occurs as time passes. While ∇. represents the mass flow rate at the main coordinates “x & y” that enters the domain. + ∇. = −∇ + ∇. ̿ + ̅ +
The term represents the rate of change
in momentum with time at the main coordinates. The term ∇. represents the change in momentum within the main coordinates at the main coordinates. −∇ Represents the static pressure applied from the
main coordinates. ∇. ̿ Represents the shear stress tensor and will be described later. ̅ + Represents the gravitational force and
the body forces.
Wet steam model Overview on the wet steam model: During the expansion of steam, condensation occurs when the pressure of steam is reduced. The expansion of steam causes the formation of water droplets due to the nucleation of steam. Wet steam model in ANSYS uses the EulerianEulerian approach for modeling wet steam flow. The flow mixture is modeled using the compressible Navier-Stokes equations, two transport equations is used for the liquid-phase mass-fraction, and the number of liquiddroplets per unit volume. The phase change model which is the main reason for the droplet formation, is based on the nucleation theory. In Eulerian-Eulerian approach, both phases are treated separately, means that each phase has its own time and space that occupy, so at this approach each phase has its own conservation equations. In another word, pure vapor is solved separately from the liquid droplets, but the formation of the water droplets due to vapor condensation is modeled using the classical nucleation theory which was discussed at the condensation chapter. Limitation on wet steam model 1- Pressure inlet, mass-flow inlet, and pressure outlet are the only inflow and outflow boundary conditions available. 2- Density based solver only Wet steam model equations 1) liquid phase equations Wetness mass fraction equation
+ ∇. = Γ
While Γ is the mass generation rate due to condensation and evaporation. Note that the mass fraction of liquid means the percentage of the liquid mass that exists in the mixture mass. As Γ increases this means that the liquid mass increases that leads to witness mass fraction increasing.
+ ∇. = Ι
Number of water droplets
While Ι is the nucleation rate ‘number of new droplets per unit volume’ from the mixture density equation and from the average droplet volume equation the number of the water droplets per unit volume can be found
=
4 π#̅ $ 3
# is the droplet radius
=
%
#̅ 1 8+1 = :; 7< − 7 ℎ.3 452-#73 28 It’s noticed that the rate of growth of droplet is a function of the liquid and vapor properties such as liquid enthalpy, vapor pressure, liquid density and specific heat ratio. Also, it is a function of 7< which is the temperature of the droplet.
Nucleation rate
>? 30 2D 'GHIJ∗ L P A B C $ F $MNO == 1 + @ . E K
While >? is the evaporation coefficient, QR is Boltzmann constant and E is the mass of one molecule. Anon-isothermal correction factor @ is given by
* &'% () + *,
@=
28 − 1 ℎ.3 ℎ.3 G P G − 0.5P 8 + 1 S7 S7
1 = 3 S71 + V3 + :30
2) Phase change equations Mass generation rate #̅ 4 Γ = -. Ι# ∗ $ + 4-. #̅ 0 3
3)
The first term represents the droplet formation and the second term represents the growth of the droplet with time. Note that the mass generation rate is calculated at the mean radius. # ∗ Is Kelvin-Helmholtz critical radius. There is a critical radius at which the droplet size will turn from demise to growth and it depends on the mixture properties. If the radius of the droplet is smaller than the critical means the heating effect is larger than the cooling effect that leads into demise, and the opposite is true.
The Mach number, Liquid Mass Fraction, and Pressure contours are illustrated in Fig. 12,13 and 14 respectively with Mach number and liquid mass fraction contours compared to reference [3] where (a) represents the contours of this paper while (b) represents that of the cited paper. From the Pressure contour shown in Fig. 14, some of the steam entering the test section inlet region collides with the blades at the leading edge leading to creation of stagnation points with very high static pressure and zero velocity. As the steam passes between the blades its velocity increases while the pressure and temperature decreases. From the liquid
While growth rate
Equation of state
Results and Discussion
mass fraction contour shown in Fig. 13, at the trailing edge of the blade and the suction side of the following blade steam starts to condensate as its pressure drops below the saturation pressure entering the wet region. In the outlet region, the high rates of expansion causes great decrease in the pressure which in turn causes the increase in the liquid mass fraction thus increasing the condensation of the steam. This condensation further decreases the pressure due to the decreasing in volume of the condensed steam, but because of the high back pressure of the condenser an Aerodynamic shock wave occurs and the pressure rises again where the steam’s condition reverts away from the saturation pressure. The direction of the Streamlines in Fig. 15 shows that the aerodynamic shockwave is an oblique shockwave where the direction of the shockwave is not normal to the flow.
Figure 13. Liquid Mass Fraction Contours.
Figure 14. Pressure Contour from numerical solution.
Figure 12. Mach number Contours.
The reason for the decreased Mach number, increased liquid mass fraction and formation of eddies in the region downstream the trailing edge as shown in Fig. 15 and 16, as mentioned in Ref [21], is due to the circular shape of the blade trailing edge which forms large zones of separation downstream the trailing edge.
Figure 15. Velocity Streamlines. Figure 17. Pressure Distribution [the dots represent the experimental data and the lines represent the simulation].
In comparison with the numerical results of the cited paper shown in Fig , this numerical paper could capture the physical phenomena, such as the separation downstream the blade trailing edge, oblique shock wave and condenation shocks, in a way more accurate than the cited paper and that is because the large number of mesh elements used for the shock zone. Figure 16. Eddies at the trailing edge of the blade. The pressure distribution on the blade surface in Fig. 17 is superimposed on the figure of the cited paper [3]. The percentage error between experimental and simulation results are listed in Table 2 showing the average and maximum error in both this paper and the cited paper.
The experimental values of pressure distribution was conducted by Dykas [3]. Comparing this paper simulation results with the experimental data showed an improvement in the average and maximum errors between the numerical results and the experimental data as shown in Table 2. The average and maximum errors for this paper were less than those that were calculated in the cited paper.
Mesh independence results Results showed in Fig. 18 illustrate mesh independence tests. Solving for number of elements of 200000, 400000, 600000 and 800000, It was found that as the number of elements increases the solution accuracy
increases but it requires more computational power so the decision of dividing the flow domain and limit the number of elements to 680000 elements was taken to obtain more
accurate results with computational power.
optimizing
the
Figure 17. Mesh independency test Results. Pressure distribution on the blade for different meshes. Tab le 2. Error Calculation. Point 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Point 9 10 11 12 13 14 15 16 17 18 19 20
Blade Suction Side Experimental (Pa) Numerical (Pa) Percentage Error Cited Paper (Pa) Percentage Error 71100.2 71580.7065 0.675815961 71712 0.860475779 66296.7 66862.3638 0.853230704 66986 1.039719926 60392 61189.795 1.32102762 61336 1.563120943 57260.1 58211.1644 1.660954836 57997 1.286934532 60608.2 61060.1934 0.745762785 60822 0.352757548 58726 58966.9289 0.41025934 58973 0.42059735 53188.3 52638.3003 1.034061438 52192 1.873156314 47504.6 46881.1488 1.312401746 45411 4.407152149 42063.2 46564.4859 10.70124456 37603 10.60356796 48070 47397.1499 1.399729769 40000 16.78801747 45328.2 41374.5585 8.722255682 34562 23.75166011 42593 36035.4827 15.39576292 40274 5.444556617 38804.2 34287.8658 11.63877673 37500 3.360976389 36061.8 33518.1262 7.053651787 34521 4.272665258 34044.1 32720.7989 3.887020365 33288 2.220942836 Blade Pressure Side Experimental (Pa) Numerical (Pa) Percentage Error Cited Paper (Pa) Percentage Error 79131.3 79015.9435 0.145778598 79470 0.428022792 76437.4 76235.5026 0.264134311 76541 0.135535746 72767.3 73246.2267 0.658161977 73459 0.950564333 70497.3 70352.2974 0.205685324 70377 0.170644833 68161.1 69091.1911 1.364548254 69041 1.290912265 67068.2 68042.2612 1.45234433 67911 1.25663131 66396.5 67236.8325 1.265627706 67295 1.353233981 66573.9 66676.9599 0.154805262 66575 0.001652299 66396.5 66198.2133 0.29864029 66267 0.195040401 65508.5 65628.5888 0.18331789 65856 0.530465512 64718.5 64763.6736 0.069800134 64521 0.305167765 63417.8 63476.5536 0.092645283 63600 0.287301042 Average Numerical Error Average cited paper Error 2.702497985 3.153758277 Maximum Numerical Error Maximum cited paper Error
15.39576292
23.75166011
Conclusion This paper presents the numerical study that was carried out by ANSYS Fluent on the blades of a low-pressure steam turbine to clarify the condensation process and the effect of the blade geometry on it. The results of this paper proved that ANSYS Fluent can be used to solve such a complicated process for this geometry with an acceptable result. The turbulence model that can be used to solve such a problem with acceptable results is the K-ω SST model. The separation mentioned in Ref. [20] occurs due to the shape of the trailing edge can be clearly identified from the numerical results. Aerodynamic oblique shock wave was noticed from the solution and it
occurs due to the high back pressure of the test section. Condensation shock was clearly noticed in the region that exists downstream the trailing edge of the blade. The simulation results of this paper matched the experimental data. Further study would be carried to optimize the shape of the geometry in order to reduce the losses.
Conflict of interest No conflict of interest
Acknowledgments We would like to thank E-JUST University for assistance with the simulation and our colleagues and teammates for their large contribution in this work.
References
[1] A. White, J. young and P. walters, "Experimental validation of condensing flow theory for a stationary cascade of steam turbine blade," 1996. [2] A. Yousif, A. Al-Dabagh and R. Al-Zuhairy, "Non-equilibrium spontaneous condensation in transonic steam flow," Int. J. Therm. Sci., 2013. [3] S. Dykas, M. Majkut, M. Strozik and K. Smołka, "Experimental Study of Condensing Steam Flow in Nozzles and Linear Blade Cascade," nternational Journal of Heat and Mass Transfer, 2015. [4] L. P, Traite de Mechanique Celeste, 1806. [5] T. W, "The Equilibrium of Vapour at a Curved Surface of Liquid.". [6] v. H. R, "Untersuchung über Dämpfe und Nebel, besonders über solche von Lösungen," 1886. [7] J. Thomson, "Applications of dynamics to physics and chemistry," 1888.
[8] V. M and W. A, "Nucleation of supersaturated structures," 1926. [9] F. L, "nucleation rate in supersaturated vapors," 1927. [10] A. Gerber and M. Kermani, "A Pressure Based Eulerian-Eulerian Multi-Phase Model for NonEquilibrium Condensation in Transonic Steam Flow," Heat Mass Transfer, 2004. [11] A. White and M. Hounslow, "Modelling droplet size distributions in polydispersed wetsteam flows," Int. J. Heat Mass Transfer, 2000. [12] W. A. J, "A comparison of modelling methods for polydispersed wet-steam flow," international Journal for Numerical Methods in Engineering, 2003. [13] A. GERBER and A. MOUSAVI, "Application of quadrature method of moments to the polydispersed droplet spectrum in transonic steam flows with primary and secondary nucleation," Applied Mathematical Modelling, 2007. [14] S. Yamamoto, Y. Sasao, H. Kato, H. Satsuki, H. Ooyama and K. Ishizaka, "Numerical and experimental investigations of unsteady 3-D wet-steam flows through two-stage stator-rotor cascade channels," ASME Turbo Expo, 2010. [15] J. Starzmann, M. Schatz, M. V. Casey, J. F. Mayer and F. Sieverding, "Modelling and Validation of Wet Steam Flow in a Low Pressure Steam Turbine," ASME Turbo Expo, 2011. [16] K. Chandler, A. White and J. Young, "Non-equilibrium wet-steam calculations of unsteady lowpressure turbine flows," 2013. [17] J. Starzmann, M. Casey, M. J.F and F. Sieverding, "Wetness loss prediction for a low pressure steam turbine using computational fluid dynamics," Inst. Mech. Eng. Part J. Power Energy, 2013. [18] M. Grübel, J. Starzmann, M. Schatz, T. Eberle, D. M. Vogt and F. Sieverding, "Two-Phase Flow Modeling and Measurements in Low-Pressure Turbines," ASME Turbo Expo, 2014. [19] R. Sidin, Droplet size distribution in condensing flow, Enschede: University of Twente, 2009. [20] h. Mohey Eldeen, M. Yousef, A. A, A. Ahmed, S. Alaa, E. Dalia, R. Mohamed, H. Mamdouh, M. Mustafa, A. Nada, M. Nada and A. Yasmine, COMPUTATIONAL MODELLING OF NONEQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES, Alexandria,Egypt: Faculty of Engineering , Alexandria University, 2017. [21] N. B. Vargaftik, B. N. Volkov and L. D. Voljak, International Tables of the Surface Tension of Water, 1983.
[22] Y. Cengel, Fluid mechanics, Fundamentals and application. [23] J. Anderson, Modern Compressible Flow with Historical Perspective, 2003.
WX WY ^ \. X] _`
W X] ^ WY ^ ] ^ \. X] \e
\. gh l Xk ^n
WXo WY ^ o \. X] q
WXs WY ^ s \. X] t Xv Z∗
Appendix Density variation with time Mass flow rate for main coordinates (X&Y) Amount of mass transformed from one phase to another Rate of change in momentum with time The change in the momentum of the flow at the main directions (X&Y) Static pressure applied on the flow from the main directions (X&Y) Shear stress tensor Gravitational force Body force Rate of change in witness fraction Variation in witness fraction with the main coordinates (X&Y) Mass generation rate Rate of change in the number of droplets Variations (X&Y) in the number of droplets with the main coordinates Nucleation rate Liquid density Kelvin-Helmholtz critical radius
Z[
WZ[ WY ab
Droplet radius Growth rate of the liquid droplet Temperature of water droplet
T
Mean temperature
cd
Evaporation coefficient
f
ij m` B ,C W Xi WY W Xp WY ri rp
qi ,qp
ui ,up _i ,_p
non-isothermal correction factor Boltzmann constant Mass of one molecule empirical functions Rate of change in turbulence kinetic energy The rate of change in the specific rate of dissipation The generation of turbulence kinetic energy due to mean velocity gradients The generation of the specific rate of dissipation due to mean velocity gradients The effective diffusivity of i and p
The dissipation of i and p due to turbulence User-defined source terms defined in the software
COMPUTATIONAL MODELLING OF NON-EQUILIBRIUM CONDENSING STEAM FLOWS IN LOW-PRESSURE STEAM TURBINES Highlights • • • •
Computational Fluid Dynamics Condensation Low pressure steam turbine Simulation
Conflict of Interest No conflict of interest
*** External email: use caution ***
Please change the graph number of the second figure to 18 as shown below:
Best Regards From: Nagappan, Seenuvasan (ELS-CON)
From: Ahmed Nagib
Subject: Re: Publication of your article [RINENG_100065] in Results in Engineering is on hold due to file problems *** External email: use caution ***
Dear Sir, Please consider the first figure only.
Best Regards From: Nagappan, Seenuvasan (ELS-CON)
-----Original Message-----
From: Ahmed Nagib
Dear Sir, I am not able to understand what you mean from you comment “We have received two set of figure(9,17), kindly check and advise which one should be follow”. Can you please clarify it to me? Ahmed Nagib Elmekawy, PhD, P.E. > On Nov 30, 2019, at 6:43 AM, "[email protected]"