Thermodynamic wetness loss calculation in a steam turbine rotor tip section: nucleating steam flow

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1 Journal o Physics: Conerence Series PAPER OPEN ACCESS Thermodynamic wetness loss calculation in a steam turbine rotor tip section: nucleatin steam low To cite this article: Joby Joseph et al 016 J. Phys.: Con. Ser Related content - Identiication o paddy ield usin Landsat imae in Karawan Reency, West Java Bamban Riadi and Ahmad Budiman Suriadi - Discussion on "The vacuum system o the Birminham proton synchrotron" - Development o corrosion risk map or Peninsular Malaysia usin climatic and air pollution data Fathoni U, Zakaria C M and Rohayu C O View the article online or updates and enhancements. This content was downloaded rom IP address on 05/0/018 at 13:41

2 Thermodynamic wetness loss calculation in a steam turbine rotor tip section: nucleatin steam low Joby Joseph 1, S Sathyanarayanan 1,, Viney K 3,B V S SS Prasad 1*,D Biswas 4 and T Jimbo 4 1 Department o Mechanical Enineerin, Indian Institute o Technoloy Madras, India Lloyd s Reister Consultin, Mumbai, India. 3 Delt University o Technoloy, The Netherlands. 4 Toshiba Corporate R&D Centre, Kawasaki, Japan. Abstract: Rapid expansion o steam in the last staes o a steam turbine causes condensation. The ormation o liquid droplets due to condensation results in wetness losses, which include aerodynamic losses (due to riction between liquid droplets and the vapour), thermodynamic losses (due to irreversible heat addition), and brakin losses (due to the impact o liquid droplets on the blade). The thermodynamic loss contributes up to 80% to the wetness losses when the diameter o the droplets ormed is less than 1 μm. In this study, the thermodynamic loss in a two-dimensional steam turbine rotor tip section is numerically investiated or various operatin and o-desin conditions. A pressure based, Eulerian-Eulerian approach is used to model the non-equilibrium condensation process. The entropy chane due to condensation is used to compute the thermodynamic losses. 1. Introduction Durin power eneration in a steam turbine, steam underoes rapid expansion across each stae o the turbine. In the last staes, the hih-speed steam crosses the vapor saturation line and reaches a meta-stable state beore condensin. At this state, ine liquid droplets o less than one micron are ormed, resultin in wetness losses in the turbine. The total wetness loss in a steam turbine due to condensation has two major contributors, the aerodynamic and the thermodynamic losses. For droplets o size less than 1 μm in diameter, the aerodynamic loss is less than 1 % o the total work output [1]. The present study investiates the thermodynamic loss in the last stae o a steam turbine, numerically, based on evaluatin the speciic entropy rise in terms o the interphase mass and heat lux, an approach suested oriinally by Youn []. Further, the variation o this loss with pressure ratio, inlet subcoolin and inlet anle is also predicted. Althouh early modelin methods o non-equilibrium condensation were proposed in onedimensional nozzles, e.. [1], Eulerian-Laranian approach was suested or complex twodimensional analysis in turbine cascades in the 1990s [3-4]. As the Laranian approach is computationally expensive, the Eulerian-Eulerian approach is ollowed or studyin the condensin wet steam lows in a one-dimensional nozzle [5] by assumin the liquid droplets to be in continuum. Ishazaki et al. [6] ollowed the same approach or transonic non-equilibrium condensation lows throuh a steam turbine cascade. Gerber and Kermani [7] used a pressure based * Correspondin author. address: Content rom this work may be used under the terms o the Creative Commons Attribution 3.0 licence. Any urther distribution o this work must maintain attribution to the author(s) and the title o the work, journal citation and DOI. Published under licence by Ltd 1

3 Eulerian-Eulerian model to predict non-equilibrium condensation in transonic low. The paper estimates the pressure distribution on a typical cascade o a turbine rotor tip section and validated the results aainst the experimental data o Bakhtar et al. [8]. A similar approach is used in this paper to model non-equilibrium condensation and rom the predicted results; thermodynamic wetness loss is calculated usin post processin techniques. Nomenclature Cp isobaric Speciic Heat (J/k K) ζ enthalpy loss coeicient E total Enery per mass (J/k) λ heat transer coeicient (W/m K) h speciic enthalpy (J/k) ρ density (k/m 3 ) I nucleation rate (No. o droplets/m 3 s) σ liquid surace tension (N/m) N number o droplets per volume (No. o droplets/m 3 ) τ shear stress tensor (N/m ) P pressure (Pa) q heat lux (W/m ) Subscripts Q rate o heat dissipation due to irreversibility (W) 0 total properties dis R characteristic as constant o steam (461.5 J/k K) 1 state at inlet r Kelvin-Helmholtz critical radius (m) state at outlet * averae radius (m) liquid T temperature (K) as u velocity component (m/s) i x-direction component V resultant velocity vector (m/s) j y-direction component x spatial variable (m) p droplet sc sub-cooled Greek alphabets sat saturation α liquid mass raction sh superheat γ ratio o speciic heats S chane in speciic entropy (J/kK) d volume o a control volume (m 3 ). Computational Model In order to model the non-equilibrium condensation process, two additional conservation equations (or liquid mass raction and droplet density) are solved alon with the compressible Navier-Stokes equations or the sinle low mixture; an approach similar to [7] is used or solvin the overnin set o equations..1. Scalar transport equation or liquid mass conservation This scalar equation is solved or conservin the liquid mass raction, deined as the mass o liquid to the mass o vapor, in each control volume. u t x j The source term o this equation S - the liquid mass enerated, is the sum o mass increase due to nucleation (the ormation o critically sized droplets)and the rowth/demise o these droplets, based on the classical nucleation theory [9], is iven by 4 3 r S Ir* 4r N () 3 t here, ' I ' is the droplet nucleation rate, discussed in the subsequent section. The term r * ives the critical radius deined as the threshold value o radius, only above which the droplet would j S (1)

4 sustain in the liquid phase. It is derived based on the Kelvin-Helmholtz equation and is simpliied as iven in Youn []. r * (3) G RT ln S Lare clusters o very ine droplets are ormed when the ree-enery barrier or the ormation o droplets is overcome. This occurs at hih derees o subcoolin, in the rane o K. ' S ' is the supersaturation ratio deined as the ratio o vapor pressure to the saturation pressure at the correspondin vapor temperature. It takes a value reater than unity when the vapor is in the subcooled reion. Supersaturation ives the deree o subcoolin attained by the vapor beore nucleation and is a measure o the deviation o vapor rom its thermodynamic equilibrium. The second term o S in equation () is the increase/decrease in liquid mass raction due to the rowth o the droplet, iven as the product o the interacial surace area o all the droplets and the droplet rowth rate ( r ). To compute the surace area o the droplet, we use the averae radius t o a droplet r, which is derived based on the assumption o a spherical droplet and is iven by 3 m r (4) p N 4 N The liquid droplets ormed row when the vapor surroundin it condenses on its surace. The rate o droplet rowth is obtained by perormin an enery balance over the spherical droplet. The heat released by a condensin droplet, i) increases the droplet temperature and ii) heats up the surroundin sub-cooled vapor. The ormer component is nelected since the size o the droplet is very small and a uniorm droplet temperature is assumed. Thereore, the remainin terms ive 1/ 3 r h h 4r T T 4 r p p (5) t The heat transer coeicient accounts or the Knudsen number eects. The rowth rate is a stron unction o the heat convected away rom the droplet and based on this, a simpliied equation available in Youn [10] is used. r P 1 (6) CpT p T t h RT The term T p is the droplet temperature, calculated based on the capillarity eect described in [11]. For a droplet havin a size r < 1μm, its temperature is calculated rom r* Tp Tsat PTsc (7) r.. Scalar transport equation or droplet density conservation The second transport equation to obtain the number o droplets per unit mass o vapour (N) is N u j N S N (8) t x The variable in the source term classical nucleation [1] iven by N qc I 3 1 m j S = I, is the droplet nucleation rate per volume, based on the 1 4r exp 3KT (9) 3

5 here, the term qc is the condensation coeicient deined as the raction o vapour molecules incident on the surace o the droplet that condenses on it. Its value is taken to be unity [11] in this model. 'm' is the mass o one molecule o water and 'K' is the Boltzmann constant. A correction actor or non-isothermal eects ( ) is used or non-equilibrium nucleation, as proposed by Kantrowitz [13]. 1 h h 1 (10) 1 RT RT where ( h ) is the equilibrium latent heat. and are the liquid surace tension and liquid density obtained as a unction o temperature. The other terms used in this section are iven in the nomenclature. The eects o liquid ormation and its rowth in the low physics are iven as additional source terms in the low equations. These source terms are discussed by Gerber [7]. The equations (1) to (10) alon with the Navier-Stokes equation are used to predict the condensation o expandin steam. For rapidly expandin steam lows, the vapor properties have to be evaluated at the sub-cooled reion since it crosses the saturation line beore condensin. The equation-o-state (EOS) o steam, the virial coeicients and its thermodynamic properties are evaluated based on the equations by Youn [10, 14]. 3. Validation The developed numerical model and the property unctions are interated to the commercial code ANSYS Fluent usin User-Deined Scalars (UDS) and User-Deined Functions (UDF) respectively. The inite volume discretization method is used to solve the conservation equations. A pressure based solver, with non-sereated coupled approach, is used or pressure-velocity couplin. A complex case o condensation in a steam turbine rotor-tip cascade [8] is used or validation. The same blade proile and computational domain is used in this study to compute the thermodynamic losses. The blade proile is that o a low-pressure (penultimate) stae o a steam turbine where the condensation is maximum. The eometric details o the cascade are iven in [8]. 1.0 present study Experiment (Bakhtar) 0.9 Pressure side (P/P 0 ) 0.6 Suction side condensation shock Fiure 1: Validation o the numerical model A sinle blade passae (iure 1) with supersonic exit low condition is used. A case with a pressure ratio o.34 and an inlet subcoolin o 1K as operatin conditions is studied. The choice o turbulence model used is not o much siniicance in predictin condensation, thereore, the hih Reynolds number K-ω SST model appropriate or lows with separation is adopted in this study. 0. X/C 4

6 The pressure ratio (P/P o ) on both the suction and pressure sides o the blade are compared (iure). It is observed that values on the pressure side o the blade are in areement with that o the experiments while there is a sliht deviation in the computational result in the suction side. The location o the condensation shock is almost same while its strenth is slihtly under-predicted. This said, the deviation on the suction side is very minimal which is acceptable. With this validation, the above model is used or predictin the thermodynamic losses or nucleatin lows. 4. Computation o Thermodynamic Loss The major source o thermodynamic loss in a steam turbine stae is due to the non-equilibrium between the vapour and the liquid phases. The dierence in the rate o latent heat released (rom the liquid droplet) and the nucleation rate is responsible or the thermal non-equilibrium. This latent heat transer leads to an increase in entropy in the system which leads to the thermodynamic loss. Consider a droplet at temperature T that releases latent heat durin condensation to the p surroundin vapor which is at temperaturet. Due to this irreversible heat transer, there is an increase in entropy in the low ield. This increase in entropy is a direct measure o the thermodynamic loss, which is calculated usin the second law o thermodynamics under the assumption that each phase is in thermal equilibrium experiencin an internally reversible heat transer. The entropy production rate is iven by S Q T Q T p 1 1 S h d, (11) T T The rate o heat release Q (in W) is iven as the product o liquid mass enerated S and the latent heat ( h ) released durin eneration. The heat dissipation rate due to irreversibility in a control volume can be expressed as 1 1 Qdis S. Tmean S h Tmeand (1) T Tp For droplets o size less than 1μm, the mean temperature can be taken to be equal to either the sub-cooled vapour temperature ( T ) or liquid droplet temperature ( T p ), with a small loss in accuracy [15]. The heat dissipated in a control volume, interated over the entire domain ives the total enery dissipated. Durin transonic lows in the blade passae, shockwaves are observed downstream, near the trailin ede. Across the shock, temperature and pressure o the vapour increases which leads to the evaporation o droplets at these reions. This results in droplets ainin latent heat rom the vapour which aain increases the entropy. This phenomenon is accounted in equation (1), (by takin the absolute value o the inverse o temperature dierence). To study the losses only due to condensation (and evaporation), it is appropriate to compare the condensin steam low conditions with the exit case as reerence. In this study, the thermodynamic loss is quantitatively expressed in terms o enthalpy loss coeicient (ζ), deined as the ratio o chane in static enthalpy due to condensation ( h, wet h, ) to the total to static enthalpy chane at the exit or steam ( h 01, h, ). h, wet h, Qdis h01, h, V (13), m p 5

7 where m, and V, are the mass low rate o steam and the averae velocity at the exit plane respectively or a steam condition. Usin the equation (13), the enthalpy loss coeicient is obtained or various operatin conditions Eect o pressure ratio Durin operation, the pressure ratio varies in these turbine staes with varyin load. A prediction o chane in thermodynamic loss with chane in pressure ratio is carried out in this section. The exit pressure is varied by maintainin a constant inlet sub-coolin o 1K. The low is maintained transonic/supersonic as the exit Mach number is varied rom 1.04 to The numerical results show that the nucleation rate is lower in the low pressure ratio cases, which implies that equilibrium conditions may not have reached at the exit o the cascade. Due to the hiher rate o expansion in the hiher pressure ratio conditions, the nucleation rate is hih at upstream locations (iure.1 and.), where the pressure and temperature are hih. The values o latent heat at these upstream locations are hiher, since latent heat is a stron unction o temperature. The hih value o latent heat results in the increased thermodynamic loss (equation (1)). The liquid mass enerated is also hih or hiher pressure ratio, contributin urther to the loss (Table 1). Downstream o the cascade, neative values o liquid mass eneration are noticed, these are due to the presence o oblique shock, which result in evaporation (iure.3 and.4). Pressure ratio=1.81 Pressure ratio=.8 Fiure.1 Liquid mass raction. Fiure. Liquid mass raction. Fiure.3 Liquid mass eneration rate. Fiure. Liquid mass eneration rate. 6

8 Across the shock, the temperature and pressure o steam increases, which leads to local zones where the droplets are surrounded by hiher temperature steam. Liquid droplets receive heat rom the surroundin steam, which is at a hiher temperature to vapourize. This irreversible process also leads to a rise in entropy. 4.. Eect o inlet sub-coolin In multi-stae steam turbines operatin at hih speeds, the state o inlet steam to the last staes is subcooled in most cases. To examine the eect o inlet temperature on condensation and thermodynamic loss, dierent inlet subcoolin (6K, 1K and 15K) conditions are analyzed with the same rate o expansion i.e. at constant pressure ratio. 6 K sub-coolin 15 K Sub-coolin Fiure 3.1 Liquid mass raction. Fiure 3. Liquid mass raction Liquid mass raction K subcoolin 15K subcoolin Position alon the mid passae between blades (m) Fiure 3.3 Liquid mass raction comparison The exit liquid mass raction or 6K inlet subcoolin scenario is about 5 % whereas or 15K subcoolin case its value increased to 6 % (iure 3.1 and 3.). Also or the lower inlet subcoolin (6K) case, the condensation commences later in the bulk steam compared to the hiher inlet subcoolin condition (iure 3.3). Thereore, the non-equilibrium and heat dissipation is more with hiher inlet sub-coolin. 7

9 4.3. Eect o incidence anle The incidence anle o steam, (measured rom the tanent to the camber line at the leadin ede o the blade) varies with chane in mass low rate o steam, blade speed and durin o-desin conditions. This variation in incidence aects the location o onset o condensation and thereby losses. To account or these variations, a rane o incidence anle between -5 to +5 deree is investiated. It is observed that, as the low incidence anle increases rom -5 to +5 deree, the onset o condensation moves upstream on the suction side (iure 4). For the 0 deree incidence, the nucleation commences at about 40 % o the chord, near the throat o the blade passae (iure ), whereas or cases with hiher positive incidence, the low turnin near the leadin ede is more which results in hiher expansion rate. This leads to hiher derees o subcoolin and, thereore, early onset o nucleation and hiher values o heat dissipation. However, this chane in location o onset does not siniicantly aect the value o maximum outlet liquid mass raction (α) in the domain. When the incidence becomes neative, the velocity component in the horizontal direction is lesser and the low is directed more towards the suction surace o the blade. As a result, the expansion rate o the vapor is lower, resultin in a low value o heat dissipation and, thereore, thermodynamic loss. This arument is exclusively valid only or loss due to condensation as the other components o losses (aerodynamic) may vary dierently. -5 deree +5 deree Fiure 4.1 Liquid mass raction. Fiure 4. Liquid mass raction. Case No. Pressure ratio Table 1: Thermodynamic losses or various operatin conditions. Subcoolin (K) Incidence anle (deree) Exit mass low(k/s) ( case) Averae exit Velocity(m/s) ( case) Thermodynamic Loss Coeicient

10 5. Conclusion This paper utilizes an Eulerian-Eulerian approach to predict the non-equilibrium condensation based on classical nucleation theory. A numerical approach is used to compute the thermodynamic loss, associated with non-equilibrium condensation, based on entropy eneration. The thermodynamic loss component is a siniicant actor and must be accounted in the estimation o losses in a wet steam low. This method computes losses associated with condensation as well as evaporation. The various operatin conditions that alter the location o onset o condensation and thereby heat dissipation are investiated. From this study, it can be inerred that: a) When subcooled steam enters the LP stae, the thermodynamic losses are hiher or hih pressure ratio (due to early lare expansion). b) Hiher the deree o subcoolin, hiher the loss. c) Durin hiher positive incidence, the low turnin near the leadin ede is more leadin to reater expansion rates and thereore losses. 6. Reerences [1] Moore M J, Walters P T, Crane R I, and Davidson B J 1973 IMechE Conerence [] Youn J B 199 J. Turbomachinery [3] White A J and Youn J B 1993 J. Propulsion. Power. 9 (4): [4] Bakhtar F, Mahpeykar M R and Abbas K K 1995 ASME J. Fluids En [5] McCallum M and Hunt R 1999 Int. J. Numerical Methods En [6] Ishazaki K, Ikohai T and Daiuji H 1995 Proc 6th ISCFD [7] Gerber A G and Kermani M J 004 Int. J. Heat and mass Transer [8] Bakhtar F, Ebrahimi M and Webb R A 1995 Proc.IMechE. Part C [9] K Ishazaki, T Ikohai and H DaiujiIn Proceedins o the 6th International Symposium on Computational Fluid Dynamics [10] Youn J B 198 PCH. 3() [11] Gyarmathy G 1976 Hemisphere, London [1] McDonald J E 1963 Am. J. Phys [13] Kantrowitz 1951 J. Chem. Phys. 19 (9) [14] Youn J B 1988 J. En or Gas Turbines and Power [15] Gerber A G and Kermani M J 003 Int. J. Heat and Mass Transer