Development and implementation of a Nuclear Power Plant steam turbine model in the system code ATHLET

Master Thesis von cand. Ing. JORDI BASSAS Matrikelnummer: 03283589 Development and implementation of a Nuclear Power Plant steam turbine model in the system code ATHLET Betreuer TUM: Betreuer GRS: Ausgegeben: Abgegeben: Prof. Dr. Rafael Macián-Juan Dipl.-Phys. Philipp Schöffel 15.01.2011 22.07.2011 ii iii Erklärung Hiermit versichere ich, die vorliegende Arbeit selbstständig und ohne Hilfe Dritter angefertigt zu haben. Gedanken und Zitate, die ich aus fremden Quellen direkt oder indirekt übernommen habe, sind als solche kenntlich gemacht. Diese Arbeit hat in gleicher oder ähnlicher Form noch keiner Prüfungsbehörde vorgelegen und wurde bisher nicht veröffentlicht. Ich erkläre mich damit einverstanden, dass die Arbeit durch den Lehrstuhl für Nukleartechnik der Öffentlichkeit zugänglich gemacht werden kann. München, den 22. Juli 2011 JORDI BASSAS iv v Abstract In order to improve the simulation of the whole secondary loop with the system code ATHLET a steam turbine model has to be implemented. This paper deals with the development of a thermo-hydraulic model of a Nuclear Power Plant steam turbine and its implementation in the system code ATHLET. The model is based on Stodola’s cone law and simulates the pressure drop and the enthalpy drop along the different turbine stages as well as the steam and water extractions. The influence of the steam and water extractions on the turbine behaviour as well as the importance of an accurate model for the steam and water extractions are carefully explained. Heat and mass balances of the Nuclear Power Plant Philippsburg 2 are used for reference purposes as well as for validation purposes of the implemented model. The comparison between steady state simulations and the real plant data indicate a satisfactory accuracy of the model and of the thermodynamic approach used. vi vii List of Contents Erklärung ................................................................................................................. iii Abstract .................................................................................................................. v List of Contents ........................................................................................................ vii List of Figures ............................................................................................................ xi List of Tables............................................................................................................. xv Acknowledgments .................................................................................................. xvii List of Acronyms ..................................................................................................... xix 1 Introduction .............................................................................................. 1 2 Steam turbines ......................................................................................... 3 2.1 The Rankine Cycle .................................................................................... 3 2.2 Types and construction of turbines ............................................................ 5 2.3 Particularities of steam turbines in Nuclear Power Plants; the saturated steam process ........................................................................................... 8 2.3.1 Steam production in a Nuclear Power Plant ............................................... 8 2.3.2 The saturated steam process..................................................................... 9 3 Physical models ..................................................................................... 11 3.1 Stodola’s cone law ................................................................................... 11 3.2 Steam properties across the turbine ........................................................ 14 3.3 Extractions ............................................................................................... 18 viii 4 Data base................................................................................................ 19 4.1 Reference plant ....................................................................................... 19 4.2 Available data on the reference plant ....................................................... 20 5 ATHLET .................................................................................................. 28 5.1 Description of ATHLET ............................................................................ 28 5.1.1 Modules of ATHLET ................................................................................ 28 5.1.2 The Thermo-Fluid dynamic Module ......................................................... 29 6 Implementation of the turbine model ................................................... 36 6.1 Alternative implementation strategies ...................................................... 36 6.2 Chosen modelling strategy ...................................................................... 38 6.2.1 Pressure drop model ............................................................................... 38 6.2.2 Power extraction model ........................................................................... 39 6.3 Implementation in ATHLET ...................................................................... 42 6.3.1 The pump model as a basis ..................................................................... 43 6.3.2 Modelling of the water and steam extractions .......................................... 47 6.3.3 Momentum Flux term ............................................................................... 50 6.3.4 Basic thermo-fluid dynamic models used ................................................. 52 6.3.5 Turbine data required by user input ......................................................... 52 7 Results ................................................................................................... 54 7.1 Simulation of all the stages separately ..................................................... 57 7.1.1 Pressure at the inlet of every stage .......................................................... 57 7.1.2 Enthalpy at the outlet of every stage ........................................................ 60 7.2 Simulation of the whole LP-turbine with steam and water extractions, and constant diameter of the “turbine pipe” .............................................. 61 7.2.1 Pressure evolution along the LP-turbine .................................................. 62 7.2.2 Enthalpy evolution along the LP-turbine ................................................... 64 ix 7.2.3 Evaluation of results ................................................................................ 65 7.3 Simulation of the whole LP-turbine with steam and water extractions and conic geometry ................................................................................. 67 7.3.1 Pressure evolution along the LP-turbine .................................................. 68 7.3.2 Enthalpy evolution along the LP-turbine ................................................... 69 7.3.3 Evaluation of results ................................................................................ 70 7.4 Simulation of the whole LP-turbine with steam and water extractions and conic geometry with qualitative water extractions .............................. 70 7.4.1 Pressure evolution along the LP-turbine .................................................. 71 7.4.2 Enthalpy evolution along the LP-turbine ................................................... 72 7.4.3 Water extractions ..................................................................................... 73 7.4.4 Evaluation of results ................................................................................ 74 8 Assessment ........................................................................................... 76 8.1 Geometry data ......................................................................................... 76 8.2 Application of the cone law ...................................................................... 76 8.3 Enthalpy calculation ................................................................................. 76 8.4 Models to be developed ........................................................................... 77 8.5 Extension ................................................................................................. 77 9 Summary and Outlook ........................................................................... 80 10 Bibliography ........................................................................................... 81 xvi Table 7.10 Pressure at the LP-turbine stages inlets, plant working at 80%, 60% and 40% of the nominal power ....................................................................... 68 Table 7.11 Enthalpy before and after every stage of the LP-turbine, plant working at 80%, 60% and 40% of the nominal power................................................ 69 Table 7.12 Compared error in the enthalpy (red) and enthalpy (red) results for the different operation points. ........................................................................ 70 Table 7.13 Pressure at the LP-turbine stages inlets, plant working at 80%, 60% and 40% of the nominal power. ...................................................................... 71 Table 7.14 Enthalpy before and after every stage of the LP-turbine, plant working at 80%, 60% and 40% of the nominal power:............................................... 72 Table 7.15 Water extracted for the plant working at 80%, 60% and 40% of the nominal power ....................................................................................................... 73 Table 7.16 Compared error in the enthalpy (red) and pressure (blue) results for the different operation points ......................................................................... 75 Table 7.17 Error in the water mass flow extracted in the third fourth and fifth extraction of the LP-turbine (extractions A3, A5 and A6) .......................................... 75 xvii Acknowledgments This paper was developed during my practicum at GRS. I want to thank my supervisors in GRS, Dipl.-Phys. Philipp Schöffel and Dr. Ing. Fabian Weyermann for their constant support and invaluable help during the development of this paper as well as to Dipl. Ing. (FH) Georg Lerchl for his help with ATHLET. My gratitude also to my tutor Professor Rafael Macián-Juan from the Technische Universität München for his support during my progress and for making the development of my Master Thesis at GRS possible. Thanks also to all the workers and friends at GRS for their readiness to help me at any time and for their support. I also would like to express to Ms Imogen Helen Sexton Kakuschky my sincere gratitude for her correction of this Master Thesis and for her support. Finally I want to thank my friends and family for being there, and my parents for making it all possible. xviii xix List of Acronyms Area m2 Boiling Water Reactor - Absolute velocity m/s Control Volume - Frequency Hz Mass flow through a junction in ATHLET kg/s Enthalpy J/kg Total enthalpy J/kg High Pressure - Low Pressure - Light Water Reactor - 󰇗 Mass flow kg/s Mass (in ATHLET equations) kg Torque Nm xx Polytropic exponent kg/s Rotational Speed 1/min Nuclear Power Plant - Pressure bar Power W Pressurized Water Reactor - Heat J Entropy J/(kg·K) Single Junction Pipe - Evaporation Enthalpy - Graphite‐moderated boiling water reactor (Russian type) - Temperature K Time Dependent Volume - Thermo-Fluid dynamic - Thermo-Fluid dynamic Object - Internal energy m3/kg xxi Specific volume m3/kg Fluid velocity m/s Work J Steam quality J Correction factor - Steam void fraction - Efficiency - Moment of inertia kg/m2 Isentropic exponent - Density kg/m3 Time constant s Interphase mass exchange per unit volume kg/s/m3 Angular velocity rad/s xxii 1 1 Introduction All the thermal power plants need an element to transform the heat power into electrical power. In a Nuclear Power Plant the heat produced by the nuclear fission is used to produce high pressure steam. This steam expands through a turbine in which the heat stored in the steam is transformed into mechanical energy used to drive a generator thus producing electricity. In the field of nuclear safety, so called system codes (e.g. RELAP, TRACE, CATHARE or ATHLET) have been developed to simulate the behaviour of the plant. The aim of this paper is to develop a model for the steam turbine of a Nuclear Power Plant in the computer code ATHLET (acronym for Analysis of Thermal-hydraulics of Leaks and Transients) developed by the company Gesellschaft für Anlagen- und Reaktorsicherheit (GRS). ATHLET is a 1-D best estimate code and therefore the whole cooling system including the steam turbine should be simulated with the maximum accuracy. In order to do that, and to be able to simulate the behaviour of the plant as a whole in situations such as full and partial load, and abnormal situations such as load rejections, and the operation of the plant supplying energy only for the plant itself isolated from the net, the development of the steam turbine model is necessary. Also the users of ATHLET have being asking for a turbine model in the past. As the aim of this paper is to model the steam turbine of a Nuclear Power Plant in operation, a short and simple description of the basic NPP features will be given. The chosen thermodynamic approach, with the model delivering a pressure drop and a power extraction makes the presentation of the basic thermodynamic background necessary. The concepts of the Rankine cycle and its particularities for the case of a Nuclear Power Plant as well as the principles behind the operation of steam turbines will also be explained. The system code ATHLET will be presented and described in order to improve the understanding of the chosen approach. 2 Turbine manufacturers do not publish any relevant data about steam turbines, this makes the development of a model quite complicated. The goal of this paper is to develop a model which requires only data accessible by the final user. The reference turbine used (the Low Pressure turbine of the Nuclear Power Plant Philippsburg 2) will be described and analysed and the assumptions and hypotheses made will be developed and justified. The models developed will be explained and justified before alternative approaches are commented. Finally the implementation in ATHLET will be presented as well as the results of the simulations. The application range of the model as well as the possible extensions will be explained in the last part of this paper. 3 2 Steam turbines 2.1 The Rankine Cycle The cycle described by the steam in a NPP is known as the Rankine cycle. In the Rankine cycle a working fluid is alternatively condensed at low pressure and evaporated at high pressure, water being the most common working fluid. Water steam is produced in a high pressure boiler and then expanded through a turbine (where the conversion into mechanical work is produced). The low pressure steam is condensed in a condenser. The condensate is then pumped into the boiler thus closing the cycle. Figure 2.1 T-s diagram of the Rankine cycle (Ainsworth, 2007). The Rankine cycle consists of four processes; the red numbers in Figure 2.1 indicate the different states: Process 1-2: The pump compresses the working fluid from the low condenser pressure to the high boiler pressure. 10 pands and its thermal energy is transformed into kinetic energy and then into mechanical energy in the turbine as detailed before. This loss of heat of the steam results in condensation of the steam thus decreasing the steam quality. The humidity in form of water drops causes energy losses and, should it be allowed to increase, could result in a rapid erosion of the turbine blades. In order to avoid this, the water is partly extracted after every stage. This water extraction results in an increase of the specific enthalpy of the remaining steam as the water extracted has a much lower enthalpy than the steam. The fluid enthalpy, that is, the specific enthalpy multiplied by the steam mass has decreased in the amount of the absolute enthalpy of the extracted water. Before entering the LP-turbine the steam gets through a moisture separator and a reheater, entering into the turbine as superheated steam. This increases the efficiency of the cycle; however this improvement is minimal (Strauß, 2006) as part of the high quality steam from the steam generator has to be used for this re-heating instead of expanding through the turbine. The main objective of the reheating is to minimize the appearance of moisture in the LP-turbine thus reducing the erosion of the blades. A much more effective measure to improve the efficiency of the plant is the feed water preheating. This is done by extracting steam from the turbine and using it to preheat the feed water. The steam extracted from the low pressure stages from the turbine has all its condensation heat but only a fraction of its original capacity to perform work at the turbine. 11 3 Physical models In order to simulate the turbine a series of key physical models have to be developed. The development of these models has to be directed towards the proper representation of those variables relevant for the purpose of the modelling. In this case, the model is expected to represent the turbine behaviour in the thermo-hydraulics simulation code ATHLET. In order to integrate this model into ATHLET, it has to provide a series of variables as pressure drop across the turbine, enthalpy drop across the turbine, power output, pressure at the extraction lines, etc. Given the data available (see Chapter 4) and that only the above stated variables are necessary a detailed fluid dynamics model of the behaviour of the fluid through the moving and fixed blades is not necessary. Instead of that, a simpler thermodynamic approach is used. 3.1 Stodola’s cone law Stodola’s Cone Law (Stodola, 1922), and its different versions (Traupel, 2001) display, given the design parameters, the relationship between the inlet- and outlet pressure at the turbine and the mass flow through the turbine. The cone law equation is (Traupel, 2001): 󰇗 󰇗 √ √ ( ) ( ) (3.1) ‘n’ being the polytrophic exponent, ‘p’ the pressure and ‘v’ the specific volume. The sub index “a” stands for the inlet value, “b” for the outlet value and “0” for the design values. For wet steam the calculation of the polytrophic exponent is (Traupel, 2001): 12 ( ) ( ) (3.2) “κ” is the isentropic exponent, “ ” and “ ” the specific volume of steam and water respectively, “r” the evaporation enthalpy and “ηp” the overall efficiency of the turbine. The term ( ) depends only on the pressure (Traupel, 2001), taking a mean value of p for the range of pressures of the reference turbine: ( ) For wet steam and considering that the steam quality is never under 0.8, κ=1.135 (Grote, 2009) (Traupel, 2001). The expansion polytrophic exponent for dry steam is (Ray, 1980), (Traupel, 2001): ( ) (3.3) For dry steam and for the range steam parameters in a NPP-turbine, κ=1.3. If the approximation is done (Grote, 2009), (Ray, 1980), (Stodola, 1922), (Traupel, 2001), (Zimmer, 2008) and considering the steam an ideal gas, the equation (3.1) can be simplified to: 󰇗 󰇗 √ √ ( ) ( ) (3.4) 13 Figure 3.1 Graphic representation of the cone law (Stodola, 1922) If the temperature varies moderately from the design temperature, the influence of the temperature is rather limited (about 5%). The same analogy can be done without considering the ideal gas simplification for the product of pressure and specific volume. However, in order to maintain the accuracy it has been chosen not to neglect the influence of temperature variations. In order to describe the changes in the operating conditions, the mass flow has to be constant throughout the whole group of stages, making the application of equation (3.4) only possible in those sections of the turbine with the same mass flow i.e. stages between two consecutive extractions 1 . Steam turbines in NPP have several extraction lines which extract steam and/or condensate for feed water preheating and also to limit the quantity of condensate in the turbine (see Figure 3.2). This means that in order to describe faithfully the behaviour of the whole turbine, several interconnected sections will be necessary. 1 For practical reasons, every group of stages will be referred to as a stage (see section 4.2). 14 Figure 3.2 Turbine with several extraction lines, on the right the subdivision in sections can be seen. 3.2 Steam properties across the turbine The thermodynamic state of water can be defined by two thermodynamic properties. For overheated steam or undercooled pressure and temperature give a definite state of the steam, for humid vapour, pressure and temperature are dependent on each other thus making the use of a third variable necessary, e.g. steam quality or specific volume. The steam expansion through a turbine is a polytrophic process; therefore the isentropic enthalpy drop Δhs has to be multiplied by an internal efficiency factor ηi, called isentropic efficiency (see equation (2.6)). (3.5) Or, what is the same: (3.6) An isentropic process occurs at constant entropy, whereas in the real process the entropy increases. This implies that the final enthalpy is higher than the isentropic enthalpy (see Figure 3.3). At the end of the real process this can be seen as a higher tem- 15 perature in the case of overheated steam or a higher steam quality in the case of wet steam. Figure 3.3 Comparison between the real process and the ideal process in the hs-Diagram The isentropic efficiency can be calculated for every given turbine, once the pressure and the enthalpy at every reference point are known (e.g. see Figure 4. and Figure 4.3). The process would be analogue to the one described above. Given the steam properties at the inlet of the turbine and knowing the isentropic efficiency of every stage, only the pressure at every point of the turbine is necessary to know all the thermodynamic properties at that point. With the entropy and the pressure drop, the isentropic enthalpy difference can be calculated. Multiplying the isentropic enthalpy difference by the isentropic efficiency, the real enthalpy drop can be calculated; and knowing the pressure and the enthalpy at a certain point, all the properties are known (see Figure 3.4). So, for a given turbine stage and given ηi, pa, ha, Ta, xa and pb, the algorism above described would be: 16 Figure 3.4 Calculation of the enthalpy at the exhaust of a turbine when inlet properties (sub index a), the exhaust pressure (sub index b) and the isentropic efficiency of the turbine are known. As stated above, the internal efficiency can be calculated, provided that the rest of the parameters are given. However, when simulating off design operation, these parameters are not known. The internal efficiency is influenced by many design factors including blade construction and operation point and it reaches its maximum at nominal load. Equation (3.7) is a semi-empirical formula that describes the variations of the internal efficiency as a function of the angular velocity; the design efficiency and the isentropic enthalpy drop (Ray, 1980). [ √ √ ] (3.7) 17 Where α is a positive constant. For the purpose of this paper it can be considered that α=2 and ηi,0=0.87. In (Grote, 2009) equation (3.8) is used; however he quotes (Ray, 1980). [ √ ] (3.8) Although both equations behave similarly in the surroundings of the design point beyond a certain point, they give very different results. The simplest hypothesis is that there was a spelling mistake. Figure 3.5 Evolution of the isentropic efficiency depending of the isentropic enthalpy difference for constant angular velocity. 18 In Figure 3.5 both equations plotted. The red line corresponds to equation (3.7) and the blue one to equation (3.8). Notice that if instead of plotting after (Δhs /Δhs,0) the plot is done after (Δhs,0 / Δhs), the plot resulting then is identical but corresponding the red line to equation (3.8) and the blue one to equation (3.7). For the purpose of this paper, equation (3.7) will be used and the above stated hypothesis will be accepted. 3.3 Extractions Extractions are of great importance to this paper (see section 4.2). It will be considered that there are two different kinds of extractions, the steam extraction and the water extraction. The extractions increase the efficiency of the cycle by preheating the feed water and extracting the condensed water of the turbine keeping the quality of the steam in it inside the margins thus avoiding erosion problems in the blades and minimizing the efficiency losses due to condensation. The extraction of water increases the quality of the remaining steam in the turbine, which increases the enthalpy. This steam with higher quality is partly extracted by the steam extraction and the rest of it enters the following stage. 19 4 Data base An unexpected difficulty was the unavailability of reliable and abundant turbine data. There were a few heat balances of NPP available, but mostly only for the full power output configuration, so reference data was partly available, but no possibility to compare the results with real data. Manufacturers are very reserved with their data. Considering the fact that the potential user of ATHLET is expected to have a very limited access to relevant data; it has been decided to develop a model which relies as much as possible on data obtainable by the user. This lack of detailed data had a great influence in the development of this work making the first fluid dynamic approaches developed unpractical. All the geometry based models had to be abandoned as the geometry was completely unknown. Even if access to detailed geometry would have been granted, ATHLET is a 1-D code and a blade geometry based solution would have required a 3-D approach. 4.1 Reference plant The reference data for this paper has mainly been the heat and mass balances of the Nuclear Power Plant Phillipsburg 2 with the old turbine. For this NPP we have data about more operation points that for any other, namely for 100%, 80%, 60% and 40% power output. The plant consists of one 2-flow HP turbine and 3 identical 2-flow LP turbines. For comparison purposes the reference turbine in (Grote, 2009) has also been used. It is an industrial extraction turbine in a steel mill in Salzgitter used for the production of electricity and process steam, it was installed in 2006 by MAN Turbo and its generator has an electrical power output of 45-55 MW. 26 Taking this into account, the specific enthalpy after every stage has to be recalculated. At every extraction a known quantities of steam and water are extracted. The pressure, the enthalpy, and the mass flow of both steam and water are known. The mass flow through stage i (the stage before the extractions) is also known. So to calculate the specific enthalpy of the steam coming out of stage T1 (in the HP turbine): 󰇗 󰇗 󰇗 󰇗 (4.10) Being: the enthalpy at the outlet of stage T1. 󰇗 the mass flow through T1 (stage before the extraction). 󰇗 the mass flow of water extracted at A1 the enthalpy of the water extracted at A1 the enthalpy of the steam extracted at A1 After recalculating the enthalpy after every stage, the power output is slightly above the generator power. The resulting efficiency of the turbine and the generator together is of 98%. This efficiency is very high but not much higher than the expected efficiency (95 to 97%). Although it is considered acceptable, there are some factors which could explain this high efficiency. The main reason is the assumption that the quality of the steam extracted is identical to the quality of the steam remaining in the turbine, so that only the water extraction itself is responsible for the steam quality improvement. This assumption does not take into consideration the centrifugal forces in the turbine. Due to these forces, it is expected that the steam extracted has a lower quality than the steam remaining in the turbine. This fact means that the steam quality improvement at every stage is greater than calculated. 27 The internal efficiency of the turbine stages calculated after these assumptions varies greatly depending on the stage. For stage T5 it almost reaches a value of 1, which is impossible. This is because the enthalpy after stage T5 is actually higher than the enthalpy calculated supposing an identical enthalpy of the steam extracted and the steam in the turbine. At the last stage, the internal efficiency is 0,4, the reason for this is that the steam coming out of the last stage (as well as the steam along the turbine) has a high velocity and therefore a high kinetic energy and as it enters the condenser it loses its velocity which then results in an enthalpy increase (see equation (2.7)). Despite of this, it has been decided to maintain the steam enthalpy at the last stage equal to the enthalpy of the steam in the condenser. The fact that the results are within the margins, and that there is no way to determine the real quality improvement, are considered sound arguments in favour of maintaining all the aforenamed assumptions and considerations in this paper. An interesting fact is that the total power output of the turbine remains constant no matter what steam quality value is used in the extraction A5, as a higher steam quality brings an increase of the power output at the stage T7 but this increase is compensated by the decrease of the power output in the stage T6 and vice versa. This seems to be true for all the extractions, as what are important for the power calculation are the total mass extraction and the quality improvement resulting of the extraction. In (Grote, 2009) a large number of geometrical data of the reference turbine was available, such as Volumes of the spaces between stages, exact enthalpies and pressure at the inlet and at the outlet of every stage and isentropic efficiencies of every stage at one of the design points, however, the information for other design points was limited (Grote, 2009) and it could not be used as reference data to be compared with the results provided by the implemented models. 28 5 ATHLET The turbine model is to be developed in ATHLET, so a general description of ATHLET is necessary in order to justify the solutions chosen. For those areas necessary to understand the development of the model, a detailed description is provided. For further detail see (GRS, 2009). 5.1 Description of ATHLET The thermal-hydraulic computer code ATHLET (Analysis of Thermal-Hydraulics of LEaks and Transients), developed by the Gesellschaft für Anlagen- und Reaktorsicherheit (GRS), aims to cover the whole spectrum of design basis and beyond design basis accidents (without core degradation) for light water reactors such as PWR, BWR, VVER and RBMK. 5.1.1 Modules of ATHLET ATHLET is composed of several basic modules which simulate the phenomena involved in the operation of LWR. These basic modules are: Thermo-fluid dynamics: This module is based upon a 5-equation model with a mixture momentum equation, and separate conservation equations for vapour and liquid, or a 2-fluid model with 6-equations which has a momentum equation for vapour and another for liquid. Heat Transfer and Heat Conduction: This module allows the simulation of the heat conduction in all those components needed. Neutron Kinetics: Models the nuclear heat generation. General Control Simulation Module: Is a block-oriented simulation language for the description of control, protection and auxiliary systems. GCSM allows the representa- 29 tion of fluid dynamic systems in a very simplified way requiring very little computation time to do so. So far, the turbine has been modelled by a large number of GCSM signals. The solution of the differential equation system is performed implicitly by the ODE- solver FEBE. The coupling of other independent modules can easily be performed in the general interface. Although major plant components can be modelled by connecting TFOs and HCO via input data, some of them are available as special objects. These special objects are simplified and compact models. Additional models for the simulation of valves, pumps, accumulators, steam separators, single ended breaks, double ended breaks, fills, leaks and boundary conditions for pressure and enthalpy, are provided. The purpose of this paper is to add a turbine to this list. 5.1.2 The Thermo-Fluid dynamic Module The leading module in ATHLET is the thermo-fluid-dynamic (TFD) module. Given that the turbine model is to be a thermodynamic model, the TFD module has to be explained comprehensibly in order to fully understand the proposed solution. The basic equations describing the thermal-hydraulic behaviour of the system are based on the conservation laws of mass, energy and momentum. They are time and space dependant partial differential equations which have to be solved numerically, as it is not possible to solve them analytically. The system configuration to be simulated is modelled connecting basic thermo-fluid dynamic objects (TFO) and heat conduction objects (HCO) via input data. There are different TFOs categories; however only the pipe objects are relevant for this paper. Pipe objects apply for a one-dimensional TFD-Model with partial differential equations describing the transport of fluid. In the input data the nodalization is defined. Beyond that point a pipe object is treated as consecutive Control Volumes united to each other 30 by junctions. The control domain of every junction is defined by the CVs centres at its right and left (see Figure 5.1). The momentum differential equations provide the mass flow rates at the boundaries of each CV. A single junction pipe consists of a single junction without any control volumes. The mass and energy based partial differential equations are integrated using the CVs as integration domain, the pressure, the vapour and the liquid temperatures, and the steam quality being the solution variables. For the momentum based partial differential equations the integration domain is the junction (staggered grid), the mass flow rate being the solution in the 5-equation model and the phase mass velocities in the 6- equation model. This is known as a staggered grid (see Figure 5.1). The quantities resulting from these integrations represent the local average physical state and are only time dependant. Figure 5.1 Staggered grid with CV and junctions (GRS, 2009) For the 5-equation model the mass and energy balances for vapour and liquid in the CV are solved separately. The solution variables in the CV are: The mass quality, the liquid temperature, the vapour temperature and the pressure. 31 Integrating the mass conservation equations over a CV Vi, liquid mass balance equation (5.1) and vapour mass balance equation (5.2) are obtained. ∑ ∑ (5.1) ∑ ∑ (5.2) With: (5.3) (5.4) With: (5.5) From the phase mass balances above the differential equation for the mass quality is derived: (5.6) With (5.7) (5.8) 32 Integrating the energy balance equations over the CV Vi and after making some simplifications, the ordinary differential equations for the phase temperatures (5.9) and (5.10) are obtained. ( | ) (5.9) ( | ) (5.10) Where ∑ ( ) ∑ ( ) ( ) (5.11) ∑ ( ) ∑ ( ) ( ) (5.12) QEI is the interfacial heat exchange due to condensation or evaporation, QI the heat source to the control volume and wi the average fluid velocity in the CV. The differential equation for the pressure is: (5.13) With: 33 | | And [ | | ( | )] [ | | ( | )] In the junction a mixture momentum balance is solved. (5.14) is the differential equation for the mixture flow rate over a junction j connecting CVs i1 and i2. ∫ [ ] (5.14) ∫ (5.15) ΔpI is the pressure difference between the CVs at both sides of the junction ΔpMF is the momentum flux term ΔpWR is the relative velocity term Δpgrav is the elevation term Δpfric is the friction and loss pressure drop Δpρ is the density derivative term ΔpI is the external source term, e.g. pump differential pressure term 34 From all these terms and in the context of this paper, only the external source term, the friction and loss pressure drop term, and the momentum flux term need a further analysis. The external source term and its influence and importance in the solution chosen will be explained in chapter 6.2. The friction and loss pressure drop term is: ∫ (5.16) The momentum flux is calculated as follows: ∫ 󰇍 󰇍 󰇍 󰇍 (5.17) Where: And | | ( ) ( ) [ ] With 35 [ ] Two models relevant for this paper are FILLs and Pressure‐Enthalpy Boundary Component. Fills are junction related models used for the simulation of mass sources and sinks. If it is to be a mass source, the mass flow to be injected in the system has to be defined in the GCSM module as well as the total specific enthalpy. If a sink is to be simulated, the mass flow has to have a negative sign and the enthalpy does not need to be defined as it is calculated as in normal junctions, i.e. they are calculated from the upstream conditions. Pressure‐Enthalpy Boundary Component - also referred to as 'time dependent volume (TDV)’ – is a CV related model which permits to establish, via GCSM signals, a pressure enthalpy boundary at the edge of the system. In this way and depending upon the conditions in the system, mass will flow into the TDV or from the TDV into the system. 42 This approach relies on the use of an average density. While this can be accepted in the case of incompressible fluids where the density variation can be neglected, the density variation throughout a turbine stage is considerable 2 . This density variation makes the assumption implied in equation (6.17) quite bold. However, the accuracy of the achieved results endorses the applicability of the proposed approach. 6.3 Implementation in ATHLET Once the equations have been developed, the next step has been the implementation in ATHLET of the proposed model. Before starting with the implementation in ATHLET, all the equations have been tried in MATLAB in order to observe the response of the model. Although the results are not free of errors, the decision has been taken to carry on the implementation in ATHLET and to make any further modifications there. The TFD system chosen is a fill junction connected to a pipe which is connected to a time dependent volume (see Figure 6.2). As it is a first approach a pipe with a constant diameter is user in order to minimize any influence beside than the one of the turbine junction. 2 For example in the case of the first stage in the LP turbine 43 Figure 6.2 TFD system. Fill on the left, pipe in the middle and TDV on the right. The turbine junction is in the middle of the pipe. In order to observe the behaviour of the turbine junction and of the whole TFD, the first simulation has been made by setting in the turbine junction, the pressure and enthalpy drop. After observing the adequacy of the configuration, further modifications have been done in order to implement the model. 6.3.1 The pump model as a basis The turbine model has been developed taking the pump model as a basis. What the pump junction does is to introduce a pressure difference as part of the momentum equation of the junction (see equation (5.14)) and adding a pump power to the fluid in both adjacent CVs (see equations (5.11) and (5.12)). The development of the turbine model taking the pump model as a basis is much simpler and takes much less modifications in the code than choosing the approach suggested in 6.1. That approach, besides of the initial instability problems explained, would have taken major modifications in the code including the development of entirely new subroutines. 44 All what is required is to modify the power extraction to the fluid and the way the pressure drop is inserted by introducing the equations (6.5) and (6.18) in the corresponding subroutines. 6.3.1.1 The modifications in the pump model In the pump junction the power of the pump is added at both adjacent CV. In the turbine junction, instead of adding power, it is subtracted thus only a sign modification is necessary to represent the work performed by the steam in the shaft. In a steam turbine the power transfer from the steam to the shaft takes place at that stage; however the present model does not model that stage internally. Instead it models a new junction type which is then accommodated in a pipe object. The pressure drop is added as part of the momentum equation of the junction; however the power extraction from the fluid cannot take place in the junction as the energy balance equation is not solved in the junction (see chapter 5). The power must therefore be extracted from the fluid in the CV after the turbine. The difference with the energy added in the pump model is that in the latter the pump energy is added to the CV before the pump and to the CV after the pump; in the turbine model all the energy is extracted from one single CV after the turbine junction. In the turbine the work is done only by the steam; water in the turbine has a lower velocity than steam and actually receives energy from the turbine. Therefore, instead of extracting power from the liquid and steam phase, the energy extraction takes place only in the steam phase. The modifications done in order to achieve this can be seen in Figure 6.3. The energy extraction done by the pump in the CV left of the junction (index JILJ) as well as the energy extraction to the liquid (QLI) in both CV have been turned off. Instead all the power extraction takes place in the CV right of the turbine junction and only to the steam. 45 Figure 6.3 Part of the subroutine dkturb.f. Comments are in green. The calculation of the pressure drop and the power extraction is calculated in subroutine ktutr.f (see Figure 6.4). In order to perform the power drop calculation after the 46 equation (6.5) the pressure and temperature in the CV right of the turbine junction have to be known. This data is available in a variables array. Figure 6.4 Subroutine ktutr.f calculates the pressure drop and the power extraction. For the internal efficiency, given the considerations stated in section 4.2, a constant value has been chosen for all the stages. A typical value of 0.87 has been chosen (Ray, 1980) for the simulations. Nominal values of the turbine junction Δpi calculation after equation (6.5) Power extraction after equation (6.18) 47 6.3.2 Modelling of the water and steam extractions In order to couple several stages, steam and water extractions have to be included in the simulation. This has been done adding two fill junctions after every stage; one for the steam and the other for the water. Given the fact that a fill must be always be directed toward the TFD system, the only way to add a fill between two stages (i.e. not at the leftmost junction of a pipe) is via a Single Junction Pipe (for a complete description of the SJP see (GRS, 2009)). Between every stage, the leftmost extraction is the water extraction and the rightmost extraction is the steam extraction (see Figure 4.4). In order to simplify the explanation, the concept stage will hereinafter comprehend the set of a turbine junction and the extractions after that junction (at the right of that junction). The mass flow to be extracted by every fill is set by the user in the input data set (see Figure 6.5). Notice that there are two values for every extraction. The purpose of this is to avoid that extraction takes place before a semi-stationary main mass flow is achieved as it could result in instability and, in the case of water extraction, in the paradox of extracting more water than the moisture present in a given stage resulting in an abrupt end of the simulation. 48 Figure 6.5 Detail of input data set. Mass flows in every extraction. WGSTART/ GSTART and WGENDE/GENDE are the mass flows of water/steam extractions at the beginning and after a given time of the simulation. The fill junctions in every extraction can be divided in water and steam extractions. The fills of the steam extractions are not different from any fill. Given the fact that it is an extraction, the mass flow has to be set negative. This mass flow extracted will have exactly the same properties as the steam in the turbine. The water extractions however cannot be represented as ordinary fills, because, as explained in chapter 5, the properties of the fluid extracted will be the ones of the fluid in the adjacent CV (i.e. the turbine). In order to extract only water and to achieve the steam quality improvement effect described by Figure 4.4, some modifications had to be made to the fill subroutine. In Figure 6.6 a detail of subroutine dfk1ha.f can be seen. For a regular discharge fill junction, the steam quality through that junction (XXM) is given by the steam quality in the CV upstream of that junction. By multiplying the steam quality and the extraction mass flow, the quantity of water and steam to extract is obtained. 49 Figure 6.6 Detail of subroutine dfk1ha.f where the quality of the steam to be extracted can be set. The modification consists in adding a new input camp in the declaration of a fill junction in the input data set. This input value set by the user is to be one in the case of a water extraction and zero for all other cases. In the case of a water extraction, the steam quality is automatically set to zero, thus extracting only water, being the mass flow of water extracted determined by GCSM signals, i.e. by the user. The real water extractions occur as a consequence of the internal conditions in the turbine and the operation mode, therefore a more realistic option has been developed as an alternative (see Figure 6.7 and Figure 6.8) where the user only has to fill in the input data of the percentage of water in a stage to be extracted (i.e. the steam quality improvement). The way this model operates is quite simple; the steam quality in the CV at the outlet of the closest (upstream) turbine junction together with the mass flow through that same turbine junction is read by the subroutine thus calculating the water mass flow. The mass flow to be extracted results from the product of the water mass flow and the ABGRAD variable set by the user (proportion of water) A model for these percentages is still to be developed so that the user does not have to set these values for every operation point. 50 Figure 6.7 Detail of the input data set for a water extraction. ISANZ equal to one implies that it is a water extraction and ABGRAD is the percentage of water in the stage to be extracted. Figure 6.8 Detail of subroutine dfk1ha.f where the water mas flow to be extracted is set by a percentage of the water flow through the stage. 6.3.3 Momentum Flux term Several stages including their corresponding steam and water extractions have been coupled. At the end the whole LP-turbine has been coupled. The steam and water extractions have been set only in the stages needed. So for instance, the steam after the first stage of the LP-turbine is overheated steam thus making a water extraction unnecessary. The geometry of the ‘turbine pipe’ has been set so that critical flow is not been achieved. In order to do that, the first approach has been to use a pipe with a constant diameter which is big enough to avoid critical flow at any point. This geometry however is not realistic. In a real turbine the diameter increases after every stage. In order to achieve this, the diameters increase along the turbine in order to have similar Mach 51 numbers at every stage. The geometry resulting is similar to a cone as it can be seen in Figure 6.9. Figure 6.9 Final configuration of the ‘turbine pipe’. The vertical black lines with the brown dots represent the turbine junctions and the arrows represent the steam and water extractions. The basis of the turbine junction and the turbine model is that the only terms different to zero in the momentum balance of equation (5.14) are Δps and the source term ΔpI. The former being the pressure difference between CV i and CV i+1 and the latter the turbine differential pressure calculated in the turbine junction according to equation (6.5). In order to achieve this the momentum flux term and the friction term have been set to zero (the friction term has been set close to a zero value in order to avoid instabilities). The conic geometry proposed above implies a flow cross section increase. This cross section increase causes a pressure recovery when the momentum flux is calculated, in order to avoid this, in the initial calculation of the simulation; the friction term is adjusted to compensate this pressure recovery. After the initial calculation however and given that the JDPA term has been set to zero (which means that the momen- 58 Table 7.2 Pressure at the inlet of the LP-turbine stages, plant working at 60% of the nominal power. Stage Simulation inlet pressure Real inlet pressure Per cent error T3 0,58711156 0,58990826 -0,474% T4 0,2379122 0,23302752 2,096% T5 0,10404303 0,10550459 -1,385% T6 0,03512752 0,03614679 -2,820% T7 0,0184393 0,02018349 -8,642% T8 0,00791484 0,00880734 -10,134% Table 7.3 Pressure at the inlet of the LP-turbine stages, plant working at 40% of the nominal power. Stage Simulation inlet pressure Real inlet pressure Per cent error T5 0,06948744 0,0706422 -1,635% T6 0,02341212 0,02431193 -3,701% T7 0,01208187 0,01357798 -11,019% T8 0,00533608 0,00605505 -11,874% Notice that the pressure at stages T3 and T4 has not been simulated; the reason for this is that there is no real data available about these stages at 40% power output and comparison with the simulation is therefore not possible. As it can be seen the error is only significant for the last stages for the 60% and 40 % operation points. As a first approximation, however, it is considered acceptable. 59 Figure 7.2 Calculated pressure behaviour along stages T3, T4 and T5 of the LP turbine Figure 7.3 Calculated enthalpy along stages T3, T4 and T5 of the LP turbine 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Pressure as p/p100 Length LP 1st stage LP 2nd stage LP 3rd stage 0,84 0,86 0,88 0,9 0,92 0,94 0,96 0,98 1 1,02 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 Enthalpy as h/h100 Length LP 1st stage LP 2nd stage LP 3rd stage 60 7.1.2 Enthalpy at the outlet of every stage The results for the enthalpy after every stage can be seen in Table 7.4, Table 7.5 and Table 7.6, for 80%, 60% and 40%, respectively. Table 7.4 Enthalpy at the outlet of the LP-turbine stages for the plant working at 80% of the nominal power. Stage Simulation outlet enthalpy Real outlet enthalpy Per cent error T3 0,95214298 0,95340749 -0,133% T4 0,91037398 0,91053411 -0,018% T5 0,8619126 0,85296453 1,049% T6 0,83476118 0,82358954 1,356% T7 0,79694549 0,8055026 -1,062% T8 0,75333379 0,79496539 -5,237% Table 7.5 Enthalpy at the outlet of the LP-turbine stages for the plant working at 60% of the nominal power. Stage Simulation outlet enthalpy Real outlet enthalpy Per cent error T3 0,959080547 0,96029478 -0,126% T4 0,91702779 0,91690485 0,013% T5 0,868961741 0,86247058 0,753% T6 0,772795206 0,77378909 -0,128% T7 0,805454733 0,80837214 -0,361% T8 0,778101519 0,79606736 -2,257% 61 Table 7.6 Enthalpy at the outlet of the LP-turbine stages for the plant working at 40% of the nominal power. Stage Simulation outlet enthalpy Real outlet enthalpy Per cent error T5 0,87594752 0,873446055 0,286% T6 0,84850339 0,759540481 11,713% T7 0,81307931 0,811627977 0,179% T8 0,7861073 0,799269947 -1,647% The error for the enthalpy is also acceptable, except after stage T6 operating at a 40% of nominal power. The real value of the enthalpy after stage T6 is not known as in extraction A5 only water is extracted (see section 4.2). Therefore it can be only said that there is a deviation from the interpolated value, not necessarily an error. Although the errors in the simulation of every separate stage cannot be neglected, it can be accepted that the proposed turbine junction model is accurate enough to proceed to the coupling of several stages with their corresponding water and steam extractions. 7.2 Simulation of the whole LP-turbine with steam and water extractions, and constant diameter of the “turbine pipe” A new TFD system has been developed including all the stages and its corresponding extractions. As it can be seen in Figure 7.4, the turbine pipe configuration remains unchanged except by the addition of the turbine junctions and the extractions. 62 Figure 7.4 TFD system. The points are the turbine junctions and the arrows the steam and water extractions. Figure 7.5 Detail of Figure 7.4. Turbine junctions and extractions can be seen clearly. Between some stages, there is only one extraction as only steam (between stages T3 and T4) or only water (between stages T6 and T7) is extracted. 7.2.1 Pressure evolution along the LP-turbine Because the pressure drop occurs only at the turbine junctions (i.e. at the stages), the pressure after one stage is exactly the same as the pressure before the next stage. 63 The pressure after stage T8 (i.e. in the condenser) is not displayed as it is a boundary condition and is not calculated by the TFD system, but set by the user. Table 7.7 Pressure at the LP-turbine stages inlets, plant working at 80%, 60% and 40% of the nominal power 80% 60% 40% Sta ge Simulation pressure Real pressure Per cent error Simulation pressure Real pressure Per cent error Simulation pressure Real pressure Per cent error T3 0,7937 0,7917 0,251% 0,5897 0,5899 -0,038% 0,3843 0,3844 -0,037% T4 0,3217 0,3165 1,647% 0,2397 0,2330 2,870% 0,1609 N/A N/A T5 0,1488 0,1422 4,653% 0,1096 0,1055 3,882% 0,0732 0,0706 3,684% T6 0,0480 0,0483 -0,532% 0,0357 0,0361 -1,276% 0,0238 0,0243 -2,073% T7 0,0270 0,0269 0,399% 0,0201 0,0202 -0,356% 0,0135 0,0136 -0,568% T8 0,0119 0,0117 1,723% 0,0089 0,0088 0,596% 0,0061 0,0061 0,217% Figure 7.6 Calculated pressure behaviour along the LP turbine, plant working at 80%, 60% and 40% of the nominal power. As can be seen in Table 7.7 the results are significantly better for the simulation of the whole turbine than for the simulation of the stages separately, the error being below 5% in all the three simulations. Figure 7.6 displays clearly the pressure drop after every stage. Notice that the main pressure drop takes place at the first stage. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 Pressure as p/p100 Length 80% of Nominal power 60% of Nominal power 40% of Nominal power 64 7.2.2 Enthalpy evolution along the LP-turbine The enthalpy variations in the turbine take place along the stages and during the water extractions. Therefore the enthalpy has to be displayed before and after every stage, as if a water extraction takes place, it is no longer true as it is for the pressure, that the enthalpy after a stage is equal to the enthalpy before the following stage. In Table 7.8 the enthalpy at the inlet and at the outlet of every stage is given. Hence between two given stages two enthalpies are displayed, the one at the outlet of the first stage and the other at the inlet of the following stage or, accordingly, one enthalpy value before the water extraction and another after the water extraction. Table 7.8 Enthalpy before and after every stage of the LP-turbine, plant working at 80%, 60% and 40% of the nominal power 80% 60% 40% St ag e Simulation enthalpies Real enthalpies Per cent error Simulation enthalpies Real enthalpies Per cent error Simulation enthalpies Real enthalpies Per cent error T3 1,0078 1,0078 0,000% 1,0166 1,0166 0,000% 1,0252 1,0252 0,000% 0,9529 0,9534 -0,057% 0,9604 0,9603 0,006% 0,9692 N/A N/A T4 0,9529 0,9534 -0,057% 0,9604 0,9603 0,006% 0,9692 N/A N/A 0,9122 0,9105 0,188% 0,9185 0,9169 0,175% 0,9260 0,9242 0,190% T5 0,9122 0,9105 0,188% 0,9185 0,9169 0,175% 0,9260 0,9242 0,190% 0,8610 0,8530 0,945% 0,8678 0,8625 0,614% 0,8744 0,8734 0,111% T6 0,8681 0,8600 0,938% 0,8716 0,8662 0,616% 0,8744 0,8734 0,111% 0,8420 0,8236 2,237% 0,8455 0,7738 9,263% 0,8485 0,7595 11,706% T7 0,8515 0,8331 2,215% 0,8536 0,8384 1,807% 0,8542 0,8444 1,161% 0,8146 0,8055 1,130% 0,8165 0,8084 1,006% 0,8178 0,8116 0,758% T8 0,8281 0,8189 1,124% 0,8286 0,8203 1,011% 0,8281 0,8218 0,768% 0,7597 0,7950 -4,440% 0,7635 0,7961 -4,094% 0,7732 0,7993 -3,260% 65 Figure 7.7 Calculated enthalpy behaviour along the LP-turbine, plant working at 80%, 60% and 40% of the nominal power. In Figure 7.7 the effect of the water extractions can be clearly seen after stages T5, T6 and T7, and how the importance of these extractions in the enthalpy increase after every extraction cannot be neglected. 7.2.3 Evaluation of results Table 7.9 shows the error made at every stage. Compared with the results obtained from the simulation of the stages separately (section 7.1), there is a substantial improvement. 0,75 0,8 0,85 0,9 0,95 1 1,05 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 Enthalpy as h/h100 Length 80% of Nominal power 60% of Nominal power 40% of Nominal power 66 Table 7.9 Compared error in the enthalpy (red) and enthalpy (red) results for the different operation points. The error in the pressure calculation stays below 5% in all the simulations and is only above 3% between stages T4 and T5. The magnitude of the error at that point for all the simulations (compared with the other errors) could be explained by some inaccuracy in the measurements at that point or by the assumptions made. The real cause, however, remains unclear. The error in the enthalpies is quite small for all the points but two. At the condenser (after stage T8) it is due to the deceleration of the steam which transforms its kinetic energy into heat thus increasing the enthalpy, the steam enthalpy at the condenser in the heat balance is considered after this deceleration thus being higher than the actual enthalpy at the outlet of the turbine itself where the steam still has a considerable kinetic energy. At the outlet of stage T6 the largest error is made by far which might be explained by the fact that the real enthalpy at that point is not known and the value used as reference has been interpolated (see section 4.2). These considerations being made, the performance of the model is considered quite satisfactory. Enthalpies Pressure Enthalpies Pressure Enthalpies Pressure 0,000% 0,251% 0,000% -0,038% 0,000% -0,037% -0,057% 0,006% 0,000% -0,057% 1,647% 0,006% 2,870% 0,000% 0,188% 0,175% 0,190% 0,188% 4,653% 0,175% 3,882% 0,190% 3,684% 0,945% 0,614% 0,111% 0,938% -0,532% 0,616% -1,276% 0,111% -2,073% 2,237% 9,263% 11,706% 2,215% 0,399% 1,807% -0,356% 1,161% -0,568% 1,130% 1,006% 0,758% 1,124% 1,723% 1,011% 0,596% 0,768% 0,217% -4,440% -4,094% -3,260% T6 Stage Per cent error at 60% Per cent error at 40% T3 T4 T5 Per cent error at 80% T7 T8 67 7.3 Simulation of the whole LP-turbine with steam and water extractions and conic geometry The next step has been to adjust the geometry of the turbine pipe to a cone adjusting it to the changing conditions of the steam along it. As explained in subsection 6.3.3 the diameters at the inlet and at the outlet of the “turbine pipe” have been set in order to avoid a critical flow at any point. The reference power plant has a 6 flow LP-turbine, so that there are 6 LP-turbines. This has also been reflected in the model, the dimensions of it being the corresponding to only one of these 6 LP-turbines. The simulation has been set so that the steam mass flow is divided between 6 identical turbines. Figure 7.8 Image of the geometry used. The fill and the TDV can be seen at the left and at the right end respectively. 74 Figure 7.13 Qualitative steam quality evolution across the LP-turbine. Notice the increase after every extraction. 7.4.4 Evaluation of results The error in the water extractions is of importance as can be seen in Table 7.15, the reason for this is that a small change in the enthalpy of the steam has significant effect on the steam quality; this, together with the large amount of steam flowing across the turbine, causes large variations on the water quantity at the extraction point. As the water extracted depends on the water present at the CV before the extraction point, this water quantity variation is responsible for the errors. 75 Table 7.16 Compared error in the enthalpy (red) and pressure (blue) results for the different operation points Table 7.17 Error in the water mass flow extracted in the third fourth and fifth extraction of the LP-turbine (extractions A3, A5 and A6) Despite the errors in the water mass flow extractions, there is no qualitative and little quantitative difference from the errors committed in the pressures and the enthalpies in the fixed extraction simulations (see Table 7.12), the main error being in the enthalpy after stage T6. Enthalpy Pressure Enthalpy Pressure Enthalpy Pressure 0,000% 0,198% 0,000% -0,094% 0,000% -0,094% -0,052% 1,658% 0,012% 2,886% 0,000% -0,052% 1,658% 0,012% 2,886% 0,000% 0,206% 4,675% 0,192% 3,922% 0,206% 3,727% 0,206% 4,675% 0,192% 3,922% 0,206% 3,727% 0,933% -0,411% 0,599% -1,082% 0,129% -1,923% 0,857% -0,411% 0,587% -1,082% 0,129% -1,923% 2,153% 0,464% 9,236% 0,072% 11,736% -1,923% 2,010% 0,464% 1,693% 0,072% 1,155% -0,171% 1,009% 0,157% 1,008% 0,153% 0,897% 0,324% 0,929% 0,157% 0,935% 0,153% 0,850% 0,324% -2,790% 0,000% -2,333% 0,000% -1,748% 0,000% Percent error at 60% Percent error at 40% T8 Stage Percent error at 80% T3 T4 T5 T6 T7 Extraction Mass flow error at 80% Mass flow error at 60% Mass flow error at 40% A3 -12,700% -11,715% 0,000% A5 -11,469% -12,018% -9,205% A6 -6,194% -7,429% -11,241% 76 8 Assessment 8.1 Geometry data The two conic models developed have their geometrical data set in order to avoid a critical flow of the steam and are not based on real data. The knowledge of the geometry and the size of the turbine are necessary for the transient calculation, as the speed with which changes in the inlet steam conditions reflect on the conditions across the turbine depends on the quantity of steam stored in the turbine at a given point, i.e. the turbine internal volume. The lack of geometry data and the unavailability of transient data made it impossible to assess the adequacy of the model for transient calculations. 8.2 Application of the cone law The representation of the pressure behaviour has been seen in chapter 7, the error always being below 5%. Considering the absolute deviations, the accuracy of the model seems to improve as the largest deviation takes place between the second and the third stages (T4 and T5) and it is below 0,075 bar which could be in the area of the measurement tolerances (Grote, 2009). It can therefore be considered that the cone law in the form it has been used in the present model (see equation (6.5)) as well as the model as a whole depict the steady state pressure behaviour through the turbine with great accuracy. 8.3 Enthalpy calculation The representation of the enthalpy behavior presents the main problems and errors. Despite the simplified model used for its calculation (see equation (6.18)), the problems seem to arise due to an incomplete knowledge of some relevant data (see Chapter 4) and due to some of the assumptions made. However, should accurate data be availa- 77 ble, it is to be expected that the model would depict the enthalpies steady state behavior with an even more satisfactory accuracy. 8.4 Models to be developed A model for the extractions still needs to be developed. Some empirical formula should be able to describe the amount of fluid extracted and the improvement of the steam quality in the turbine resulting from this extraction. The modifications made in the water extractions (via the fill junction, see Figure 6.8) allow the implementation of this relationship once it is developed. In the present simulation, given the lack of reliable data (see subchapter 6.3.1.1), a constant efficiency has been used and the variation of this efficiency depending on the power output of the turbine has been neglected. The real internal enthalpy drop at every stage at nominal power is not known, and therefore, given the available data, it has not been possible to adjust the internal efficiency of every stage according to equation (3.7). Should the extraction model be developed and the nominal stage efficiencies be known, it would be possible to introduce in the subroutine ktutr.f (see Figure 6.4) a calculation for the real internal efficiency of a given stage after equation (3.7). In order to display the total power extracted from the fluid or, accordingly, the total power output of the turbine, a variable has to be developed which adds the energy extracted from the fluid at every stage (equation (6.18)) and makes it possible to directly see the power output of the turbine (see equation (8.1) ). ∑ 󰇗 (8.1) 8.5 Extension Although not part of the turbine itself, some models need to be developed in order to successfully couple the turbine model with a plant simulation. 78 In order to model the whole turbine, i.e. the HP and the LP-turbine in a same simulation, a water separator as well as a re heater have to be included in the simulation. The re heater can be modelled as a heat exchanger and for the moisture separator the moisture separator of the steam generator can be modified. An alternative option is to implement it as a modified water extraction junction. Given that the steam quality of the steam at the outlet of the moisture is known and constant for all the operation points and that both the inlet mass flow as well as the inlet steam quality are known ( , 󰇗 and respectively in Figure 8.1), the modelling of the moisture separator is quite simple. Figure 8.1 Moisture separator (detail of Figure 4.) Making a mass balance: 󰇗 󰇗 󰇗 (8.2) And knowing that: 79 󰇗 󰇗 (8.3) The water mass flow to be extracted is: 󰇗 󰇗 ( ) (8.4) All the variables on the right side of equation (8.5) are known. So that the moisture separator can be easily modelled using the modified fill junctions used for the water extractions in the turbine (see subsection 6.3.2) For the transient calculations that involve a variation of the angular speed of the turbine, such as load rejection and rump up, a momentum balance determines the variation of angular speed. The power in a rotational system, i.e. a turbine is defined by the product of the torque M and the angular velocity ω. (8.5) The momentum balance equation of the whole system equation (8.6) results from a breaking torque, the moment of inertia Θ of the whole rotating machine and the torque provided by the turbine. (8.6) ( ) (8.7) Equation (8.7) describes the angular velocity variation. The breaking torque term Mbr includes all the components of the equation different from the turbine torque and have to be modelled. 80 9 Summary and Outlook The object of this paper was to develop and to implement a turbine model in ATHLET. Although some aspects of the turbine have not been considered yet, the basic equations and the general layout of the model have been successfully implemented. The thermo-hydraulic model developed needs only the thermodynamic properties of the steam at some points of the plant (basically extractions, turbine inlet as well as turbine outlet) operating at nominal power. The performed steady-state simulations predict the behaviour of the steam pressure as well as the enthalpy with very good precision, the error being below 5% at all the simulations. The fact that almost all the necessary input data is to be found at the plant heat and mass balances easies the use of the model as it is data to which the potential user is expected to have access. So far steady state calculations for a KONVOI LP-turbine have been successfully carried out. In principle the model should describe the transient conditions with the similar accuracy as it describes the steady state conditions. However, the fact that the water extraction model needs to be improved is expected to diminish the accuracy of the results. In order to perform such calculations, variables such as the geometry of the turbine (internal volume), the real isentropic efficiency of every stage, the moment of inertia of the turbine-generator complex as well as the nominal pressure at the HP-turbine inlet are necessary. In order to couple the turbine model with the secondary loop, the models for the water extractions still need to be improved. The final result however is more than satisfactory as the main aspects of the turbine modelling have been successfully addressed thus laying the basis for the implementation of the whole turbine once the necessary reference and validation data is available. The Steam turbine being one of the most complex components of the secondary loop, the simulation capabilities of ATHLET regarding the secondary loop have been significantly extended by the provision of a turbine model. 81 10 Bibliography Ainsworth, A. (2007, August 21). English Wikipedia. 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