A metastable wet steam turbine stage model

Transcription

1 Nuclear Engineering and Design 16 (00) A metastable wet steam turbine stage Wageeh Sidrak Bassel *, Arivaldo Vicente Gomes Instituto de Pesquisas Energeticas e Nucleares, Traessa R 400-Cidade Uniersitaria, CEP São Paulo, SP, Brazil Received 17 April 001; received in revised form 6 November 001; accepted 10 January 00 Abstract A for the prediction of the efficiency of axial flow steam turbine stage is described, where the flow through turbine cascade is considered non-homogeneous and metastable. At the exit an oblique shock brings it to equilibrium. The losses in the cascade are expressed according to Dunham and Came (Trans. ASME (1970)) and Kacker and Okapuu (J. Eng. Power (198)) which is imprivemente of Ainley and Mathieson (1951) method. Two phase flow frictional multiplier is used as a correction factor for pressure coefficient. The is compared with data of performance evaluation of large steam turbines of PWR power plants and results shows a good agreement. 00 Elsevier Science B.V. All rights reserved. 1. Introduction * Corresponding author. addresses: wsbassel@net.ipen.br (W.S. Bassel), avgomes@net.ipen.br (A.V. Gomes). The performance of axial flow turbine was determined by means of loss first deduced by Ainely and Metheison (1951). The work has been revised by Dunham and Came (1970) and later by Kacker and Okapuu (198). The loss is widely used in gas turbine and the efficiency can be determined within accuracy of 1.5%. The application of this method in steam turbine results of higher errors. Craig and Cox (1971) proposed 1% less of efficiency for each 1% of mean stage wetness. The solution of three dimensions Navier Stokes equation by numerical method for compressible single phase was developed for turbine cascade flow. According to Cofer (1996), the application of similar method for two phase flow of wet steam turbine not yet established up to now. The present work is simple and can be used for wet steam turbine stage with adequate accuracy. The is based upon system of non-linear algebraic conservation equations of mass, momentum in axial and tangential directions, energy and loss. In these equations it is considered slip between vapor and liquid phases and the loss is corrected by two phase flow frictional multiplier. Since the residence time of steam when passing through blade row in the order of 0.1 ms, it is assumed that there is no heat transfer between the two phases and there is no sufficient time for formation of new liquid drops so both the vapor and liquid phases are in metastable state. The metastable state is transformed in stable equilibrium state at the entrance of new subsequent row by means of oblique equilibrium shock /0/$ - see front matter 00 Elsevier Science B.V. All rights reserved. PII: S (0)0008-6

2 114 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) mu g x + 1 x S V (C L cos m C D sin m )ch m g g Fig. 1.. Governing physical equations Fig. 1 shows steam turbine blade terminology used in this work. The governing physical equations which describes the flow for fixed and moving blades are the conservation equations of mass, momentum and energy, pressure loss coefficient and equation of state. Equation of conservation of mass: g sh g + (1 ) l S = g1 sh 1 1 g1 + (1 1) l1. (1) S 1 Equation of conservation of momentum in axial direction: p sh +m g x + 1 x S V +(C L sin m +C D cos m )ch m g g, () =p 1 sh 1 +m g1 x1 + 1 x 1 S 1 where h m =(h 1 +h )/. Equation of conservation of momentum in tangential direction:. (3) =mu g1 x1 + 1 x 1 S 1 Equation of conservation of energy: V g x 1 x +H = V g1 x1 1 x 1 +H1, S S 1 (4) H =x H g +(1 x )H l, (5) V g = g +u g, (6) mean flow angle m : tan m = 1 g1 + u u g1 g. (7) The exit flow angle : tan = u g. g The inlet flow angle 1 : (7a) tan 1 = u g1. (7b) g1 The pressure loss coefficient Y bifasic is defined by, Y two phase = p 1+(1/) g1 V 1 (p +(1/) g V ), (1/) g V (8) where Y two phase =Y, (8a) where is the two phase frictional multiplier, and Y=Y p +Y sec +Y tet+ Y tc. (8-b) Y p is the profile loss coefficient, Y sec secondary loss coefficient, Y tet trailing edge coefficient and Y tc tip clearance coefficient. These coefficients are determined according to Dunham and Came (1970), Kacker and Okapuu (198). The two phase frictional multiplier is the ratio of the two phase pressure drop estimated by Friedel correlation to vapor phase pressure drop estimated at blade inlet.

3 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) The equations of state, enthalpy and entropy in metastable state are estimated by curve fitness from data of Keenen and Keys steam table as follows.equation of state: p =1+ g RT g B(T)+ g C(T), (9) B(T) = /T /T, (9a) C(T)= /T, (9b) where p is the pressure in Pa;, density in kg m 3 ; T, temperature in K; R, gas constant for steam, equal J kg 1 K. The enthalpy in kj kg 1 is: H g =H 0 + RT g (B(T) T 1000 ) db(t) dt + g (C(T) 1 T ) dc(t), (10) dt H 0 = T T T 3. (10a) The entropy in kj kg 1 K is: i g =i o + R p ln db(t) T g dt + gb(t) dc(t) g C(T)+ g T, (11) dt i o = T T T 3, (11a) i =x i g +(1 x )i l, (1) the slip ratio S is given as S =( l / g ) 0.33, (13) and void fraction is given as: 1 = 1+((1 x )/x )( g / l S ). (14) In the above system of equations the inlet flow velocity and all the thermodynamic properties are known. Considering that there is no heat transfer between as the two phases and no formation of new liquid drops it can be assumed: x =x 1, H l =H l1, l = l1, i l i l1. The solution of non-linear system of algebraic equations from Eq. (1) to Eq. (14) gives the numerical values of the following variables: ( g, u g, V g, p, T, g, C L, C D, H g, H, i g, i, S, ). The flow, after exit plane of the blade, suffers from obstacle represented by the subsequent blade row, due to this it is assumed that an oblique shock is established in plane parallel to exit blade plane transforming the flow to stable equilibrium conditions. The governing physical equations for the equilibrium shock are: the tangential velocities, relative to the shock plane, before and after the shock are equal. u gs =u g, (15) conservation of mass: m gs +m ls =m, (16) m ls = gs (1 s )sh ls /S s, (17) m gs = gs s sh gs. (18) Equation of conservation of momentum: p s sh +m gs gs +m ls gs /S s =p sh +m g g +m l g /S. (19) Equation of conservation of energy m gs Hgs + gs +mls Hls + gs S s =m g Hg + g +ml Hl + g S. (0) The equations from Eq. (16) to Eq. (0) joint with the seven equations of pressure as a function of temperature, and density, enthalpy and entropy of liquid and vapor phase as a function of temperature too, in thermodynamic equilibrium state,

4 116 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) Table 1 Characteristics of turbine stage illustrative example Description Fixed blade Moving blade Spacing (s, mm) Cord (c, mm) Inlet blade height (h 1, mm) Exit blade height (h, mm) Main radius (mm) Exit flow angle ( ) Number of blades Velocity at mean radius (m s 1 ) Type 64.8 Unshrouded forms a system of non-linear algebraic equations, where the solution gives (m gs, m ls, s, gs, p s, T s, H gs, H ls, gs, ls, i gs, i ls ). The work done is given by the equation: W=U F, (1) where U is mean radius speed, F is the change of the tangencial momentum of the moving blade, based on absolute velocity, that is given by the equation: Table Inlet flow conditions Mass flow rate (m, kgs 1 ) Static pressure (p 1, bar).393 Steam quality (x 1 ) Vapor phase enthalpy (H g1,kgkg 1 ) Liquid phase enthalpy (H l1,kjkg 1 ) Mixture enthalpy (kj kg 1 ) Vapor phase entropy (i g1,kjkg 1 K) Liquid phase entropy (i l1,kjkg 1 K) Mixture entropy (i 1,kJkg 1 K) Vapor phase density (kg m 3 ) Liquid phase density (kg m 3 ) Mixture density (m 3 kg 1 ) Inlet axial velocity ( 1,ms 1 ) Inlet tangential velocity (u 1,ms 1 ) 9 Slip ratio (S 1 ) 8.69 Void fraction ( 1 ) Table 3 Exit flow conditions of fixed blades Exit conditions before shock plane Frictional two phase flow multiplication 1.4 factor () Two phase flow pressure drop coefficient 0.18 (Y two phase ) Exit axial velocity (,ms 1 ) 80.9 Exit tangential velocity (u,ms 1 ) 61.9 Exit velocity (V,ms 1 ) Exit static pressure (p, bar) Static temperature (T, C) Vapor phase density ( g,kgm 3 ) Lift coefficient (C L ) Drag coefficient (C D ) Vapor phase enthalpy (H g,kjkg 1 ) Mixture enthalpy (H,kJkg 1 ) Vapor phase entropy (i g,kjkg 1 C) Mixture entropy (i,kjkg 1 C) Slip ratio (S ) Void fraction ( ) Conditions after shock plane Vapor phase mass rate of flow (m gs, kg s 1 ) Liquid phase mass rate of flow (m ls, kg s 1 ) Steam quality (x s ) Void fraction ( s ) Slip ratio (S s ) 9.7 Vapor phase axial velocity ( gs,ms 1 ) 8.09 Static pressure (p s, bar) Static temperature (T s, C) Vapor phase enthalpy (H gs,kjkg 1 ) Liquid phase enthalpy (H ls,kjkg 1 ) Vapor phase density ( gs,kgm 3 ) Liquid phase density ( ls,kgm 3 ) Vapor phase entropy (i gs,kjkg 1 C) Liquid phase entropy (i ls,kjkg 1 C) Mixture entropy (i s,kjkg 1 C) F= x (u g U)+(1 x ) u g S U nmoving + x s u gs +(1 x s ) u gs. () S s fixed The dynamic enthalpy drop is given by: H dynamic =H static +KE inlet KE exit, (3) where H dynamic is the dynamic enthalpy drop, H static is the static enthalpy drop, KE inlet is the n

5 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) kinetic energy of inlet flow based upon absolute velocity, KE exit is the kinetic energy of exit flow based upon absolute velocity. =W/H dynamic, (4) where is the stage efficiency. 3. Illustrative example The above was applied in the analysis of steam turbine stage of a low pressure turbine of a Table 4 Exit flow conditions of moving blades Exit conditions before shock plane Frictional two phase flow multiplication 1.47 factor () Two phase flow pressure drop coefficient (Y two phase ) Exit axial velocity (,ms 1 ) 8.3 Exit tangential velocity (u,ms 1 ) Exit velocity (V,ms 1 ) Exit static pressure (p, bar) Static temperature (T, C) Vapor phase density ( g,kgm 3 ) Lift coefficient (C L ) Drag coefficient (C D ) Vapor phase enthalpy (H g,kjkg 1 ) Mixture enthalpy (H,kJkg 1 ) 59.9 Vapor phase entropy (i g,kjkg 1 C) Mixture entropy (i,kjkg 1 C) Slip ratio (S ) Void fraction ( ) Conditions after shock plane Vapor phase mass rate of flow (m gs,kgs 1 ) Liquid phase mass rate of flow (m ls,kgs 1 ) Steam quality (x s ) Void fraction ( s ) Slip ratio (S s ) 10.0 Vapor phase axial velocity ( gs,ms 1 ) Static pressure (p s, bar) Static temperature (T s, C) Vapor phase enthalpy (H gs,kjkg 1 ) Liquid phase enthalpy (H ls,kjkg 1 ) Vapor phase density ( gs,kgm 3 ) Liquid phase density ( ls,kgm 3 ) Vapor phase entropy (i gs,kjkg 1 C) 7.0 Liquid phase entropy (i ls,kjkg 1 C) Mixture entropy (i s,kjkg 1 C) Table 5 Comparison between metastable and homogeneous equilibrium s (case 1) Parameter Metastable Homogeneous equilibrium Inlet pressure (bar) Inlet steam quality (%) Two phase frictional multiplier for stator 1.47 Not applicable Pressure loss stator Two phase frictional multiplier for rotor Pressure loss Not applicable rotor Exit stage pressure (bar) Exit stage steam quality (%) Stage efficiency Stage efficiency after correction Not applicable 83.8 Estimated stage efficiency from heat balance 85 nuclear power plant. Table 1 shows the stage specification, Table shows inlet flow conditions, Table 3 shows the result of calculation of the fixed blades. Table 4 shows the result of calculations of the moving blades. The calculated results are as follow: W=68. kj/kg =88.55% (specific work done), (efficiency). The estimated stage efficiency, obtained during performance evaluation of steam turbine, by measuring exit power and exit enthalpy indirectly, was 86.3%; the difference is due to some other types of losses, like leakage in the stator blades and losses due to expansion of steam from exit fixed blade to inlet moving blades, that was not considered in this calculation.

6 118 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) Table 6 Comparison between metastable and homogeneous equilibrium s (case ) Table 7 Comparison between metastable and homogeneous equilibrium s (case 3) Parameter Metastable Homogeneous equilibrium Parameter Metastable Homogeneous equilibrium Inlet pressure (bar) Inlet steam quality (%) Two phase frictional 1.4 Not applicable multiplier for stator Pressure loss stator Two phase frictional 1.47 Not applicable multiplier for rotor Pressure loss rotor Exit stage pressure (bar) Exit stage steam quality (%) Stage efficiency (%) 88.5% Near 100% Stage efficiency after correction Estimated stage Not applicable efficiency from heat balance Inlet pressure (bar) Inlet steam quality (%) Two phase frictional 1.4 Not applicable multiplier for stator Pressure loss stator Two phase frictional 1.4 Not applicable multiplier for rotor Pressure loss rotor Exit stage pressure Note (bar) Exit stage steam quality (%) 88.5 Note Stage efficiency (%) note 1 Note Stage efficiency after Not applicable Note correction Estimated stage 0.71 efficiency from heat balance The above calculations was repeated using concept of thermodynamic equilibrium between phases, vapor and liquid, during flow through the blade rows. Correlations of enthalpy, entropy, specific volume of saturated vapor and saturated liquid, also the pressure temperature relationship of saturation state was developed. These correlations substituted the Eqs. (9) (11) of metastable state. The pressure loss coefficient Y is used without multiplication of two phase frictional multiplier. Thus represents the treatment of the problem by the same method used in gas turbine, yielding homogeneous equilibrium solution of wet steam turbine stage. Carig and Cox suggested reduction of 1% of efficiency for each 1% mean humidity. Three real cases were analyzed with both methods, the metastable and homogeneous equilibrium for comparison, the results are shown in Tables 5 7. Note 1 Case 3 represents the final stage of low pressure steam turbine, the exit kinetic energy is completely lost, so the efficiency in this case is expressed by the work done divided by the isentropic static enthalpy drop. The rotor blade is highly tapered by reducing the blade cord and thickness from hub to tip, in order to improve the mechanical vibration aspect. Due to blade taper the estimated efficiency at midline estimated by the present work is 9% greater than the real mean blade efficiency. Note The calculated exit conditions of the stator of equilibrium homogeneous had no physical meaning even the mathematical solution converged thus there were negative change in en-

7 W.S. Bassel, A.V. Gomes / Nuclear Engineering and Design 16 (00) tropy, negative absolute pressure and temperature. The expected exact solution is flow with stationary or moving shock within the blade row. The treatment of such phenomena is out of scope of this work. Appendix A. Nomenclature c cord (m) C D drag coefficient C L lift coefficient h blade height (m) H enthalpy (kj kg 1 ) m mass flow rate through one blade spacing (kg s 1 ) p static pressure (N m ) s spacing (m) S slip ratio x steam quality V relative flow velocity (m s 1 ) component of V in axial direction (m s 1 ) u component of V in tangential di- rection (m s 1 ) void fraction flow angle (with axial direction) density (kg m 3 ) Subscript 1 blade exit s g1 l1 g l gs ls References blade inlet before shock plane blade exit, after shock plane vapor phase at blade inlet liquid phase at blade inlet vapor phase at blade exit liquid phase at blade exit vapor phase after shock plane liquid phase after shock plane Ainley, D.G., & Mathieson, G.C.R. A method of performance estimation for axial flow turbine. British ARC, R&M 974, Cofer, J.I., Advances in steam path technology. Journal of Engineering for Gas Turbines and Power 118. Craig, H.R.M., Cox, H.J.A., A performance estimation of axial flow turbines. Proc. Inst. Mech. Engnr. Vol 185, pp Dunham, J., Came, P.M., Improvment to the Ainley- Mathieson method of turbine performance prediction method. ASME Trans. Series A J. Engineering for Power, Vol. 9 no. 3, July 1970 pp Kacker, S.C., Okapuu, U., 198. A mean line prediction method for axial flow turbine efficiency. ASME Trans. Series A J. Engineering for Power, Vol. 104 no. 1, Jan. 198 pp