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1 The Mechanical and Nuclear Engineering Department NUCE 470 - FALL 2013 POWER PLANT SIMULATION with TRACE FINAL PROJECT Prepared by: ANDREW DUNNING AYSENUR TOPTAN RICKY VIVANCO Delivery date: 12 / 16 / 2013 Due date: 12 / 16 / 2013 Dunning, Toptan, Vivanco NucE 470 Final Project 2013 2 ABSTRACT The main objective of this project was to create a model that simulated a pressurized water reactor (PWR) at steady state. In order to simulate a PWR, we used the Symbolic Nuclear Analysis Package (SNAP) and TRACE coding software to create models of each component of a PWR. A steam generator, reactor core and core vessel, reactor coolant pump and pressurizer models were created and simulations were run until the steady state results coincided with the given parameter. After we were satisfied with each component, we created four copies of the steam generator and reactor coolant models and connected them to the reactor core vessel to create a four loop PWR model. The components were connected in the order of typical PWR loops and the pressurizer was connected by surge line to the hot leg of the first loop. We then ran simulations until the compiled model approached the parameters given in the assignment. The observed a turbine output of about 850 MW which is 27% of the 3,125 MW thermal power output simulated by our core model. After completing a 5% increase in thermal power to 3281.25 MW, we observed an accurate temperature, pressure and mass flow increase. With Richardson Extrapolation of the steam generator and C++ coding, we were able to achieve a .0065% error in mass flow rate at the exit nozzle. Dunning, Toptan, Vivanco NucE 470 Final Project 2013 3 TABLE OF CONTENTS Abstract ............................................................................................................................................................ 1 Table of Contents ........................................................................................................................................ 1 List of Figures ................................................................................................................................................ 1 List of Tables .................................................................................................................................................. 1 Nomenclature ............................................................................................................................................... 1 I. Introduction ............................................................................................................................................... 1 II. Model Development ............................................................................................................................. 1 1. Steam Generator Model ............................................................................................................... 2 2. Reactor Vessel/ Core and Pressurizer Model .......................................................................... 2 3. Pump Model ..................................................................................................................................... 2 4. Turbine Model ................................................................................................................................. 2 III. Subsystem Results ................................................................................................................................ 1 5. Steam Generator Results .............................................................................................................. 2 6. Reactor Core/ Vessel Results ...................................................................................................... 2 7. Plant Base Steady State Results ................................................................................................... 2 8. Plant Transient Results .................................................................................................................. 2 IV. Steam Generator Subsystem Richardson Error Analysis ................................................. 4 V. Conclusion ................................................................................................................................................. 4 VI. References ................................................................................................................................................ 4 Appendix A: List of Enclosures ............................................................................................................ 4 Appendix B: Richardson Error Analysis ........................................................................................... 4 Dunning, Toptan, Vivanco NucE 470 Final Project 2013 4 FIGURES Figure 1.1: Steam Generator Geometry................................................................................................... 8 Figure 1.2: Nodalization used for UTube Steam Generator in TRACE ............................................ 9 Figure 1.3: Feedwater Mass Flow Controller ....................................................................................... 14 Figure 1.4: Downcomer/Boiler Region Mass Flow Controller ......................................................... 14 Figure 2.1: Cut-out of a Reactor Pressure Vessel ............................................................................... 15 Figure 2.2: Simplified Reactor Pressure Vessel Geometry ................................................................ 16 Figure 2.3: General view of the Reactor Vessel and Core Model ................................................... 16 Figure 2.4: Input data section for geometry and connections of Reactor Vessel ........................ 17 Figure 2.5: Input Data Section For Volumetric and Edge Data of Reactor Vessel....................... 18 Figure 2.6: Control System for the Temperature Difference Calculation Between Hot Leg and Cold Leg ............................................................................................................................................ 18 Figure 4.1: Control Block Schematic for Turbine Output Power ................................................... 23 Figure 5.1: Primary side Hot and Cold Leg Temperatures .............................................................. 24 Figure 5.2: Boiler liquid level ................................................................................................................... 25 Figure 5.3: Downcomer liquid level ...................................................................................................... 25 Figure 5.4: Steam mass flow rate ........................................................................................................... 26 Figure 6.1: Reactor mass flow rate ...................................................................................................... 27 Figure 6.2: Temperature difference between hot leg and cold leg ................................................ 28 Figure 6.3: Hot leg mass flow rate ....................................................................................................... 29 Figure 6.4: Temperature difference between hot and cold legs ..................................................... 29 Figure 7.1: Pressurized Water Reactor Plant ........................................................................................... 30 Figure 7.2: Primary Mass Flow Rate ......................................................................................................... 31 Figure 7.3: Hot Leg Temperature ............................................................................................................. 31 Figure 7.4: Primary Side Pressure ............................................................................................................. 32 Figure 7.5: Pump pressure ......................................................................................................................... 32 Figure 7.6: Reactor Power ......................................................................................................................... 33 Figure 7.7: Steam Mass Flow Rate ............................................................................................................. 33 Figure 7.8: Steam Temperature ................................................................................................................. 34 Figure 7.9: Boiler Water Level .................................................................................................................. 34 Figure 7.10: Control Blocks Schematic ..................................................................................................... 35 Figure 7.11: Turbine Output Power during the steady-state ................................................................. 36 Figure 8.1: Steam Generator Feedwater and Exit Steam Mass Flow Rate (General Trend) ................. 37 Figure 8.2: Steam Generator Feedwater and Exit Steam Mass Flow Rate (closer view to transient) .. 37 Figure 8.3: Transient Primary Mass Flow Rate ....................................................................................... 38 Figure 8.4: Transient Reactor Power ........................................................................................................ 38 Figure 8.5: Cold leg temperature during the transient ............................................................................ 39 Figure 8.6: Hot leg temperature during the transient ............................................................................. 39 Dunning, Toptan, Vivanco NucE 470 Final Project 2013 5 Figure 8.7: Steam Generator boiler water level during the transient (general view) ........................... 40 Figure 8.8: Steam Generator boiler water level during the transient (closer view to transient) ........ 40 Figure 8.9: Downcomer water level during the transient ....................................................................... 41 Figure 8.10: Downcomer water level during the transient (closer view to transient) ......................... 41 Figure 8.11: Primary side pressure during the transient ........................................................................ 42 Figure 8.12: Pump pressure during the transient ................................................................................... 42 Figure 8.13 Turbine outlet power during the transient .......................................................................... 43 Figure 8.14: Turbine inlet temperature during the transient .................................................................. 44 Figure 8.15: Turbine inlet pressure during the transient ........................................................................ 44 Figure 9.1: Steam exit mass flow rate for Richardson Error Analysis ................................................. 45 Tables Table 1.1: Steam Generator Primary Side Parameters ..................................................................... 10 Table 1.2: Steam Generator Secondary Side Parameters ................................................................. 11 Table 2.1: Reactor Core and Reactor Core Vessel Parameters .................................................. 19 Table 3.1: Reactor Coolant Pump Parameters ................................................................................... 21 Table 3.2: Reactor Coolant System Parameters ................................................................................. 21 Table 8.1: Transient Power Table ........................................................................................................... 37 Table 8.2: Transient Secondary Side Steam Fill Table ........................................................................ 37 Table 9.1: Prescribed steam generator feedwater flow rate transient .......................................... 45 Dunning, Toptan, Vivanco NucE 470 Final Project 2013 6 NOMENCLATURE LCSP LOWER CORE SUPPORT PLATE PWR PRESSURIZED WATER REACTOR SG STEAM GENERATOR UCSP UPPER CORE SUPPORT PLATE RCS REACTOR COOLANT SYSTEM SS STEADY STATE Dunning, Toptan, Vivanco NucE 470 Final Project 2013 7 I. INTRODUCTION When studying the safety and reliability nuclear power plant operations, thermal-hydraulic coding is an essential tool. When designing a nuclear reactor, these codes are helpful in determining precise estimations of reactor systems parameters even while design feature are still being modified. This leads to more efficient power production while minimizing. In addition to precision calculations, the analysis codes can also predict accident scenarios when simulating a reactor system model. This proves helpful since accident scenarios typically cannot be performed on any real life platform due to cost, feasibility and safety concerns. The pressurized water reactor (PWR) design is the most popular in the nuclear power industry. The PWR design uses light water (H2O) under high temperatures and pressure to generate electricity. The system is comprised of a primary and secondary loop. The primary loop, which includes the reactor pressure vessel and core, steam generator, pressurizer and reactor coolant pump, runs heated and cooled liquid water in closed recirculation. The PWR usually consists of 4 separate primary loops connected to one reactor core and pressure vessel. The secondary loop recirculates of water that is heated by the primary loop via the steam generator and then cooled when run through the connected turbine that generates electricity. Each component of the primary and secondary loop of the PWR design was modeled for this project. For our models, the Symbolic Nuclear Analysis Package (SNAP) software and TRACE coding software were used to simulate the function of PWR model. SNAP provides visuals when generating the components of the PWR system and allows modifications of the boundary conditions for each component. Using data given, various parameters were estimated to generate each component of the PWR system. To being, we modeled the steam generator and reactor core and vessel separately to minimize error in comparison to modeling them together. After our models were able to reach steady state, they were connected and simulated together. The system parameters were then slightly adjusted in many areas to ensure the compiled system could reach steady state as well. Finally, we studied the effects of change in power on the other reactor systems through transient analysis of our model. The complete development of our system is described in this report. Key Plant Data Parameter Value Parameter Value Steam Pressure 6825809 Pa Core Thermal Power 3125 MW Steam Flow Rate per Loop 480 ks/s Net Electrical Power 1000 MW Pressurizer Volume 75.0 m^3 Efficiency 32% Number of Fuel Assemblies 193 Hot Leg Temperature 598 K Fuel Lattice 17 x 17 Cold Leg Temperature 565 K Active Fuel Length 3.70 m RCS Mass Flow per Loop 4400 kg/s Rods Per Assembly 264 Primary System Pressure 1551323 Pa Number of Control Rods 53 Dunning, Toptan, Vivanco NucE 470 Final Project 2013 8 II. MODEL DEVELOPMENT 1. STEAM GENERATOR MODEL In this section, we evaluate and analyze the steam generator model created for our PWR. The TRACE file from a previous steam generator model was used and modified to meet our specifications given in Tables 1.1 and 1.2. The steam generator is modeled to be a U-Tube type steam generator, whose basic geometries can be seen in figure 1.1. The steam generator is comprised of two loops, a primary side and secondary side. The primary side of the steam generator comes from the Hot leg of the reactor coolant system that runs the highly pressurized heated water from the reactor pressure vessel through the steam generator. The primary coolant enters the steam generator through the bottom plenum which separates into thousands of small U-tubes. The water then travels up and down the steam generator into the exit plenum. The water runs up and down only once, hence the pipes being named U-tubes. While the water runs through the U-tubes, the secondary side provides water at a lower temperature and pressure through the downcomer of the steam generator, whose outlet is at the bottom of the structure. The water fills the steam generator to steady water level and is heated by the U-tubes it is surrounding. The heated water in the U-tubes heat the secondary water until a steady flow of steam is forced to the top of the steam generator and through a nozzle that connects to the turbine generator. In summary, the hot primary side liquid transfers heat to the secondary side to produce steam that moves the turbine. Figure 1.1. Steam Generator Geometry Dunning, Toptan, Vivanco NucE 470 Final Project 2013 9 Figure 1.2 Nodalization used for UTube Steam Generator in TRACE A simplified volumetric nodel scheme was used to model the structures of the steam generator. The way the structures were divided can be seen in the figure above. The figure also shows the cells of each structure. The cells were used to make sure that the correct values of flow area and flow rate are running when the simulation is tested. Some structures, such as the U-tubes, are not modeled exactly. So, calculations had to be made to assign parameters that averaged the bundle of U-tubes. Other calculations were also performed to model each structure and assign the parameters needed for TRACE to simulate the model as accurately as possible. These parameters were calculated from given parameters for the primary and secondary side. Tables 1.1 and 1.2 give a summary of the overall parameters of the primary and secondary side of the steam generator. Dunning, Toptan, Vivanco NucE 470 Final Project 2013 10 Table 1.1. Steam Generator Primary Side Parameters Parameter SI Units British Units Tube outer diameter 0.0222 m 0.874in Tube wall thickness 0.00105 m 0.0413in Height of tube bundle 9 m 30ft Hot leg inner diameter 0.7 m 3ft Cold leg inner diameter 0.7 m 3ft Number of SG tubes 5000 5000 Hot leg plenum inlet flow area * 0.3848451 m2 596.511sqin Plenum exit flow area * 1.58654356 m2 17.1 ft2 Volume average flow area of the plenum * 4.75963068 m2 4.14 ft2 Total plenum volume a 4.75963068 m3 4.14 ft2 Primary side inlet temperature 598 K 617 deg F Primary side outlet temperature 565 K 617 deg F Primary side pressure 1551323 Pa 225psi SG primary side flow rate 4400 kg / s 301.5 slug/s U-tube inner diameter * 0.0201 m 0.791 in Average tube length * 17.4 m 57.08 Hydraulic diameter of primary side * 0.0201 m 0.791 in Wetted perimeter of boiler primary side * 315.7300617 m 1035 ft SG tube inner flow area * 1.58654356 m2 17.1 ft2 SG tube inner surface area * 5493.703073 m2 59133 ft2 Total tube volume * 33.67557988 m3 362.5285 ft2 * Calculated parameters a Since the precise shape of the plenum is unknown, the volume has been calculated by multiplying the area by a height of one meter Dunning, Toptan, Vivanco NucE 470 Final Project 2013 11 Table 1.2. Steam Generator Secondary Side Parameters Parameter SI Units British Units SG overall height 20 m 65.6ft Feedwater inler diameter 0.364 m 14.3in Downcomer height 10.0177 m 32.8ft Lower shell outer diameter 3.5 m 11.5ft Lower shell thickness 0.0668 m 2.19ft Upper shell outer diameter 4.5 m 14.8 ft Upper shell thickness 0.0889 m 3.5 in Feedwater temperature 503.15 K 445.73 deg F Secondary side pressure 6825809 Pa 990 psi SG secondary side flow rate 480 kg/s 32.9 slug/s Lower shell inner diameter * 3.3664 m 11.04ft Upper shell inner diameter * 4.3222 m 14.18ft Boiler outer diameter * 1.214207708 m 3.98ft Boiler inner diameter * 1.154207708 m 3.79ft Hydraulic diameter of boiler region * 0.034024861 m 1.34in Hydraulic diameter of downcomer * 2.152182292 m 7.06ft Wetted perimeter of secondary side * 701.0596196 m2 7546.0 Hydraulic diameter of feedwater * 0.4554 m 1.49ft Wetted perimeter of downcomer area * 14.39040352 m 47.21ft SG tube outer surface area * 6067.672051 m2 65304.6 ft2 SG tube total flow area * 1.58654356 m2 17.07 ft2 Flow area of boiler region * 5.963364 m2 64.12 ft2 Area of boiler region * 1.157912793 m2 12.46 ft2 Downcomer flow area * 7.742728887 m2 83.3 ft2 Lower shell area * 8.90064168 m2 95.8 ft2 Boiler wall thickness * 0.03 m 1.18in Boiler flow area - 5.963364 m2 64.18 ft2 Boiler flow area + 1.046303846 m2 11.26 ft2 Feedwater flow area * 0.104062115 m2 1.12 ft2 U-tube pitch a 0.02442 m 0.96in * Calculated parameters - excluding tubes + including tubes a chosen parameter for the pitch of a U-tube unit cell The following are the equations used to determine the unknown par1ameters of the steam generator. Tables 1.1 and 1.2 show which parameters were given and which needed to be calculated based of the given parameters. For the following calculations, the square lattice is assumed and calculations are done being based on that assumption. Dunning, Toptan, Vivanco NucE 470 Final Project 2013 12 Primary Side Calculations: Inner U-tube diameter for the Steam Generator is determined by subtracting the twice the shell thickness from the outer shell diameter. ( ) Flow area of the inlet plenum: Total flow area of inner Utubes: Flow area of the exit plenum: Volume averaged flow area of the plenum: Inlet plenum volume: ( ) Wetted perimeter of primary side: Hydraulic diameter of Primary side: Steam Generator Tuber Inner surface area: Steam Generator Tube area: Dunning, Toptan, Vivanco NucE 470 Final Project 2013 13 Secondary Side Calculations: Lower inner shell diameter: ( ) Upper shell diameter: ( ) Boiler inner diameter: ( ) Boiler outer diameter: ( ) Total surface area of outer Utubes: Area of the boiler region: Downcomer flow area: ( ) Lower shell Area: ( ) Flow area of boiler region: ( ) Area of inner boiler region: Feedwater flow area: Secondary side wetted perimeter: Downcomer wetted perimeter: ( ) Hydraulic diameter of boiler region: Hydraulic diameter of downcomer: Dunning, Toptan, Vivanco NucE 470 Final Project 2013 14 Description of the feedwater flow controller: Figure 1.3 Feedwater Mass Flow Controller Mix mass flow from the steam generator feedwater is subtracted from the steam generator boiler. Change in the mass flow rate will be determined according to the difference between the two. Figure 1.4 Downcomer/Boiler Mass Flow Controller Mass flow from the steam generator downcomer is divided by the absolute value of mass flow from the steam generator boiler. This ratio will give us how mass flow changes in the regions of downcomer and boiler. 15 2. REACTOR VESSEL / CORE AND PRESSURIZER MODEL In this section, we evaluate and analyze the reactor core and reactor core vessel of our PWR system. The TRACE file we started with was a given model which we modified to meet our specifications given in Tables 2.1. The model and the changes we made to it, were designed to simulate a general reactor core and reactor vessel used in a typical, four loop PWR. To being the path through the vessel, pressurized water at high temperature in single phase is driven into the reactor vessel through four inlet nozzles. The flow path is set toward the bottom of the vessel through the downcomer region. The flow is then driven upward through the lower core support plate, core, and upper core support plate. This core region is where the water is heated by the thermal power of the fuel rods. Then the flow exits the reactor vessel through four exit nozzles that are each connected a separate steam generator. The primary goal of the reactor pressure vessel and reactor core is to heat the primary loop water while keeping it in liquid state through pressurization. Figure 2.1: Cut-Out of a Reactor Pressure Vessel 16 Figure Error! No text of specified style in document.2: Simplified Reactor Pressure Vessel Geometry A simplified volumetric nodel scheme was used to model the structures of the reactor core and pressure vessel. The way the structures were divided can be seen in figure 2.3-2.5. The figures also show the cells and different view of the structure. The cells were used to make sure that the correct values of flow area and flow rate are simulated when the simulation is running. Some structures, such the core region, are not modeled exactly. So, calculations had to be made to assign parameters that averaged the flow areas of different regions. Other calculations were also performed to model each region of the core and vessel and assign the parameters needed for TRACE to simulate the model as accurately as possible. These parameters were calculated from given parameters for the primary and secondary side. Table 2.1 gives a summary of the overall parameters of reactor core and reactor core vessel. Figure 2.3 : General view of the Reactor Vessel and Core Model 17 Figure 2.4. Input data section for geometry and connections of Reactor Vessel 18 Figure 2.5. Input Data Section For Volumetric and Edge Data of Reactor Vessel Figure 2.6. Control System for the Temperature Difference Calculation Between Hot Leg and Cold Leg 19 Table 2.1. Reactor Core and Reactor Core Vessel Parameters Parameter SI Units British Units Vessel outer diameter 4.75 m 15.5ft Vessel wall thickness 0.256 m 10.07in Downcomer width 0.3 m 11.8in Core barrel thickness 0.1524 m 6in Reflector thickness 0.321 m 1.05ft Fuel rod diameter 0.009499 m 0.373in Fuel rod cladding thickness 0.000559 m 0.022in Fuel rod gas gap thickness 0.000191 m 0.00751 Control rod diameter 0.009677 m 0.381in Holes in LCSP 80 80 LCSP hole diameter 0.3048 m 12in Holes in UCSP 80 80 UCSP hole diameter 0.3048 m 12in Active fuel length 3.70 m 12.13 Inlet temperature 565 K 557.33deg F Outlet temperature 598 K 616.73 deg F Primary side pressure 1551320 Pa 225psi Flow rate per hot leg 4400 kg / s 30.15 slug/s Fuel rods per assembly 264 264 Number of fuel Assemblies 193 193 Surface area, control rods * 1.61132978 m2 17.3 ft2 Fuel pellet diameter * 0.008001 m 0.315in Core area diameter * 3.20926974 m 10.5ft Area of core region * 8.09 m2 87.1 ft2 Vessel inner diameter * 4.238 m 13.9ft Diameter of core barrel * 3.3332 m 10.9ft Area of Core Barrel * 8.72594814 m2 93.9 ft2 Flow area of Core Barrel * 8.72594814 m2 93.9 ft2 Barrel wetted perimeter * 10.4715566 m 34.3ft Hydraulic diameter of core barrel * 3.3332 m 10.9ft Hydraulic diameter of core * 0.01175642 m 0.463ft Hydraulic diameter of downcomer * 0.31142712 m 12.2 Downcomer wetted perimeter * 24.7431837 m 81.0 ft. Core wetted perimeter * 1522.21637 m 4994.09ft Flow area of LCSP * 5.83727015 m2 62.8 ft2 Flow area of Core * 4.47396 m2 48.1 ft2 Flow area of Downcomer * 1.92642461 m2 20.7 ft2 Number of Control Rods 53 53 Pitch a 0.01260 m 0.04134ft * Calculated parameters a typical PWR fuel rod pitch 20 Calculations: Fuel Pellet Diameter: ( ) Core Area Diameter: ( ) Area of Core Region: Vessel Inner Diameter: Core Barrel Diameter: Core Barrel Diameter: Core Barrel Area: Core Barrel Flow Area: Core Barrel Wetted Perimeter: Core Barrel Hydraulic Diameter: Core Hydraulic Diameter: Downcomer Hydraulic Diameter: Downcomer Wetted Perimeter: ( ) Core Wetted Perimeter: Support Plate Flow Area : Core Flow Area: Downcomer Flow Area: ( ( ) ) Upper & Lower Core Support Plate Loss Coefficient: ( ) 21 3. PUMP MODEL For the pump models, we decided to use the given parameters for pump number 2 from the built in model within SNAP. The model used built in Westinghouse pump curves. We connected the four pump models to the cold legs of our loops and adjusted the flow rate until we were able to reach steady state. The final pump parameters can be seen in table 3.1 and 3.2. Table 3.1. Reactor Coolant Pump Parameters Parameter Pump 2 Parameter Reactor Coolant Pump MOI 3455.0 kg-m2 Reactor Coolant Pump Hr 911.0 m2/s2 Reactor Coolant Pump Tr 35933.0 N-m Reactor Coolant Pump Q''' 6.0 m3/s Reactor Coolant Pump r 754.0 kg/m3 Reactor Coolant Pump r 124.5 rad/s Reactor Coolant Pump Flow Area 0.45673 m2 Reactor Coolant Pump Dh 0.762 m In The pressurizer model we assumed a 60% water volume ratio and were given that the total volume of the pressurizer was 75 m3. After choosing an inner diameter of 3 m, we calculated the rest of the dimensions and parameters. The final parameters of the pressurizer can be seen in table 3.2 Table 3.2. Reactor Coolant System Parameters Parameter SI Units British Units Cold Leg ID 0.762 m 2.5ft Hot Leg ID 0.762 m 2.5ft Crossover Leg ID 0.762 m 2.5ft Length of Cold Leg 7.620 m 2.5ft Length of Hot Leg 7.620 m 2.5ft Length of Crossover Leg 15.240 m 50ft Pressurizer ID 3.000 m 9.842ft Pressurizer Heater Power 1.860 MW ft-lbf/s Pw Pressurizer 9.425 m 1341.0 hp Pressurizer Flow Area 2.356 m2 25.4 ft2 Dh Pressurizer 1.000 m 3.28ft Surge Line Length 10.000 m 32.8 Surge Line ID 0.356 m 1.17ft Pressurizer Volume 75.000 m3 2648.6 ft3 Pressurizer Height 10.610 m 34.8 Reactor Coolant Pump FA 0.457 m2 4.91 ft2 Reactor Coolant Pump Dh 0.762 m 2.5ft 22 Calculations for the RCS parameters Lengths of hot leg and cold are considered to be 10 times the inner diameter of the hot and cold lengths respectively. For the length of the crossover leg, the length is calculated based on the 20 times the inner diameter of the crossover leg. Wetted perimeter of the pressurizer is determined by the following relation To calculate the pressurizer flow area, relation of has been used. For the calculation of the hydraulic diameter of pressurizer, 4 times flow area of the pressurizer is divided by the wetted perimeter of pressurizer which have been calculated soon. In order to calculate the pressurizer height, volume of the pressurizer is divided by pressurizer flow area. Flow area of the reactor coolant pump is obtained via . Finally, the hydraulic diameter of the reactor coolant pump is assumed to be same with cold leg inner diameter to be consistent. 23 4. Turbine Model In the project, the turbine was not required as a different model in the med file. Therefore, the turbine output work was evaluated via the control blocks. The control block scheme for the turbine output power is illustrated in Figure **. The control blocks are constructed to perform an energy balance across the turbine to estimate produced turbine work. To calculate the output work, the following relation was used Figure 4.1 Control Blocks Schematic for the Turbine Output Power Estimation with the assumption of 0.90 isentropic efficiency and the ideal plant parameters. 24 This relation bases on the total energy produced by turbine from the inlet and outlet streams through the turbine. Four streams were considered entering the turbine since the reactor was a four-sensor Pressurized Water Reactor. Signal blocks were used to obtain the steam exit slow rates and steam enthalpies from the exits of steam generators to calculate the total inlet energy for the turbine. To calculate the exit energy, exit enthalpy was assumed as a constant and exit flow rate which was same with the inlet mass flow rate with corresponding to conservation of mass. The exit enthalpy was calculated with the assumption of 0.90 isentropic efficiency at atmospheric pressure conditions as ( ) The values used during calculations were obtained from the reference [****]. 2.33x J/kg was accepted as a constant during the calculation of the produced turbine work. III. SUBSYSTEM RESULTS 5. STEAM GENERATOR RESULTS Figure 5.1 Primary side hot and cold leg temperatures 25 Figure 5.1 above shows the Primary side hot and cold leg temperatures of the standalone Steam Generator results of the pre-finalized steam generator model. This figure shows the temperature difference across the primary side before the boiler fills. The boiler can be seen to fill steadily in figure 5.2 below. As stated this was the pre-finalized model and was corrected in the base steady state model by adjusting the steam mass flow rate of the secondary side. Figure 5.2: Boiler liquid level Figure 5.3: Downcomer liquid level 26 Figure 5.2 and Figure 5.3 simply show that the water levels and were the primary source in determining the accuracy of the model. As stated above, the model continued to fill causing a fatal error until the secondary steam mass flow rate was adjusted prior to the model implementation into the Base Steady State model. Figure 5.4: Steam mass flow rate In Figure 5.4, it can be seen that the Steam mass flow rate of the secondary side reaches a steady state of 320 kg/s. This value is lower than the final value that can be seen from the Base Steady state model in the sections to follow. 27 6. REACTOR CORE / VESSEL RESULTS Figure 6.1: Reactor mass flow rate In Figure 6.1 shows the Reactor Core mass flow to be 4400 kg/s. Figure 6.2 shows the temperature difference across the core before the time step was adjusted. 28 Figure 6.2: Temperature difference between hot leg and cold leg Time-step is changed to 1.0e-05 for minimum and 0.1 for maximum. This change enhances the convergence of the steady-state solution. 29 Figure 6.3: Hot leg mass flow rate Figure 6.4: Temperature difference between hot and cold legs 30 Figure 6.3 and Figure 6.4 show the core mass flow rate and temperature difference of the pre-finalized core model. The heat structures to model the vessel and boiler wall are to be inserted, the results of which can be seen in the base steady state model section to follow. 7. PLANT BASE STEADY STATE RESULTS After developing and correcting the major components of each separate Subsystem, that is the Steam Generator and the Reactor Core Models, were combined to form a four loop compound PWR model that can be seen in figure 7.1 below. This model included the addition of the hot leg, cold leg, and cross-over piping, as well as a pressurizer system on loop one. Figure 7.1: Pressurized Water Reactor Plant 31 This primary loop model was run to steady state in approximately one thousands seconds. The Steady state primary mass flow rate, seen in figure 7.2, can be seen to be slightly smaller than the 4400 kg/s that was expected. The hot and cold leg temperature difference can be seen only around 30 K in figure 7.3. Figure 7.2: Primary Mass Flow Rate Figure 7.3: Hot Leg Temperature 32 Figure 7.4: Primary Side Pressure Figure 7.5: Pump pressure The pressure of the system is seen as expected in figure 7.4, but the pump pressure run with some discrepancy in figure 7.5. 33 Figure 7.6: Reactor Power The Reactor power seen above in Figure 7.6 is constant as expected for the steady state model, this graph simply serves as a steady state basis. Figure 7.7 below shows the Steady State secondary side Steam Mass flow rate of the steam generator. The flow rate is below the goal of 480kg/s but must be maintained at this level in order not to fill the boiler. Figure 7.7: Steam Mass Flow Rate 34 Figure 7.8: Steam Temperature The steam temperature of the secondary side is shown to reach steady state, and is in an acceptable range as seen in Figure 7.8 above. It can be seen to approach a lower temperature prior to the time at which the boiler water level achieves a steady state value which is seen below in figure 7.9. Figure 7.9: Boiler Water Level 35 The control block Schematic in figure 7.10 above simply illustrates that the boiler water level is a signal variable parameter. Figure 7.10: Control Blocks Schematic for the Boiler Water Level Estimation 36 Figure 7.11: Turbine Output Power during the steady-state Figure 7.11 shows the calculated turbine output power at steady-state conditions. The steady-state value of the turbine power is about 850 MW which corresponds to an approximate value for the plant efficiency 27%. Under all the assumptions, the resulting plant efficiency is quite close to the given plant efficiency 32 %. The relative error is calculated as15%. 37 8. PLANT TRANSIENT RESULTS Upon developing a compound four loop base steady model, and seeing it runs to a reasonable Steady state solution, the model was run using a five percent power transient modeled with the table lookup Power option under the power component in the restart case. The secondary side flow rate was also set to a transient state to compensate for the increase in temperature from the resulting power step. The power transient was a five percent increase and can be shown below in table 8.1 and the Secondary side flow rate transient can be seen in table 8.2 next to it. This can also be seen graphically in figure 8.1 and 8.2 below which show the Steam Generator feed water and exit steam mass flow rate on both a macro and microscopic level. Figure 8.1 Steam Generator Feedwater and Figure 8.2 Steam Generator Feedwater and Exit Steam Mass Flow Rate (General Trend) Exit Steam Mass Flow Rate (closer view to transient) Table 8.1 Transient Power Table Table 8.1 Transient Secondary Side Fill Table 38 Figure 8.3: Transient Primary Mass Flow Rate The Figure 8.3 above, shows the Primary Mas Flow rate evaluated at the hot leg which is for an unknown reason affected by the power transient but stays mainly consistent with the steady state model. And Figure 8.4 below shows the transient jump in reactor power. Figure 8.4: Transient Reactor Power 39 Figure 8.5: Cold leg temperature during the transient Above in Figure 8.5 is the microscopic and microscopic cold leg temperature near the transient which suffer only just over a one degree change during the transient. While Figure 8.6 below shows the micro and macroscopic Hot leg temperature just after the transient. Figure 8.6: Hot leg temperature during the transient 40 Figure 8.7: Steam Generator boiler water level during the transient (general view) Next, we will look at the Steam generator boiler water level. We can see the macro in figure 8.7 above and the micro in figure 8.8 below. There is a small drop in water level of about three centimeters, which is intuitive because as the temperature increases with power, more water boils off. Even though this is counteracted by the increase in secondary side steam mass flow, a differential of three centimeters was deemed acceptable. Figure 8.8: Steam Generator boiler water level during the transient (closer view to transient) 41 Figure 8.9: Downcomer water level during the transient We also recorded the Down comer water level on a macro, figure 8.9 above, and micro, figure 8.10 below. The downcomer water level increase is caused by the increase in Steam mass flow. Figure 8.10: Downcomer water level during the transient (closer view to transient) 42 Figure 8.11: Primary side pressure during the transient Finally we model the system pump pressure which encounters a minor jump after the transient seen in figure 8.11 above. And the system pump pressure in figure 8.12 below which has the same differential between pumps that we saw in the base steady state calculation. Figure 8.12: Pump pressure during the transient 43 Figure 8.13 Turbine outlet power during the transient As shown in the Figure 8.13, the turbine outlet power reaches the value of 870 MW which is same with the value at steady-state conditions as one can expect to obtain. It is already known that the applied transient is before 500 seconds and the system compensates that value. At the beginning of the solution, the decrement in the secondary mass flow rate leads to decrease in the produced power through the turbine. 44 Figure 8.14: Turbine inlet temperature during the transient Namely, Figure 8.14 shows the decrement in the secondary mass flow rate is causing the turbine inlet temperature decrease. Figure 8.15. Turbine inlet pressure during the transient It is clear to observe that there is no obvious change in the turbine inlet pressure as one can expect. 45 IV. STEAM GENERATOR SUBSYTEM RICHARDSON ERROR ANALYSIS Richardson extrapolation is a sequence acceleration method which is used to improve the rate of convergence of a sequence in numerical analysis. The Richardson extrapolation error analysis was performed using the isolated steam generator model in which feedwater flow transient is generated as in the following Table *. The feedwater flow transient is applied three cases which have different steam generator meshes. Steam exit mass flow rate change with time is shown at Figure *. Table 9.1: Prescribed steam generator feedwater flow rate transient t [s] Feedwater Flow Rate [kg/s] 0 350 200 350 201 330 300 330 301 350 400 350 Figure 9.1: Steam exit mass flow rate versus time with the initiation of the feedwater flow rate transient. 46 The mass flow rates for the Richardson Error Analysis were evaluated at the 25 seconds for each cases. The point was chosen in order to estimate the error changes since it was required to obtain the steam mass flow rate error at a point which it undergone the largest change. A code which had been written in C++ language was included at Appendix II to evaluate the error analysis by utilizing the Richardson extrapolation error analysis. The calculated error was assigned to finest mesh. Under the desired considerations, a time of 205 seconds was chosen to obtain the steam exit mass flow rates for each case. The obtained steam exit mass flow rates were 352.4356 kg/s, 354.3546 kg/s and 354.6854 kg/s respectively for the cases base, x2 and x4 meshes. The mesh sensivity studies were performed in accordance with the number of cells being doubled with the refinement ratio progressively. The spatial order of accuracy is calculated via the written code as ( ) ( )=-2.536324 The spatial error being associated with the finest mesh was evaluated as ( )=2.318701 kg/s 47 V. CONCLUSIONS The major components of a PWR were developed separately to model a typical PWR design. The parameters of each system component was calculated and adjusted based on simulation results until steady state could be achieved at a desired value. We were able to achieve steady state when the components were put together. Our outputs also came relatively close to our desired plant data as shown in our results section. After achieving steady state, a transient was initiated by imposing a 5% increase in reactor power. The changes in the reactor systems, such as temperature, pressure and mass flow rate, were as expected. As a result of the core power increase, the output power of the turbine increase proportionally as well. The model perfectly capable of modeling the transient and the marginal power increase did not fatally perturb the reactor system or components. Achieving symmetry between each primary loop and determining the proper mass flow rate within them took the most effort. Overall, our model and simulation were created successfully and supported us in gaining further insight into the capabilities of SNAP and TRACE. 48 VI. REFERENCES [1] NUCE470 final report Group F. Sample report for the final project. [2] Numerical Methods for Ordinary Differential Equations, 2nd edition by J.C. Butcher [3] Duderstadt, J. J., & Hamilton, L. J. (1942). Nuclear Reactor Analysis. Ann Arbor, Michigan: John Wiley & Sons. [4] Lamarsh J. R., & Baratta, A. J. (2001). Introduction to Nuclear Engineering. Upper Saddle River NJ: Prentice Hall. [5] Buongiorno, J. (Fall 2010). PWR Description. 22.06: Engineering of Nuclear Systems. MIT, Cambridge, MA. Retrieved (Fall 2013) from http://ocw.mit.edu/courses/nuclear-engineering/22-06-engineering-of- nuclear-systems-fall-2010/lectures-and-readings/MIT22_06F10_lec06a.pdf 49 APPENDIX A: LIST OF ENCLOSURES File Name Description FinalProject.med Annotated SNAP Model Editor File for the plant RichardsonExtrap_Model.med SNAP Model Editor File Richardson Extrapolation FinalSteadyState_Input.inp Plant steady state run input file FinalSteadyState_output.out Plant steady state run output file Restart_Input.inp Plant transient run input file Restart_Output.out Plant transient run output file RichardsonExtrap_C++Code.cpp C++ code for the Richardson Error Analysis 50 APPENDIX B: Richardson Error Analysis Richardson extrapolation consists of calculating a result in a manner that depends on a small parameter, and for which the error in the calculation varies systematically as the parameter varies. By using a sequence of values of the parameter, much of the effect of the errors can be eliminated so that improved accuracy results [2]. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )51 Written C++ code for the Richardson Error Analysis // RICHARDSON EXTRAPOLATION #include #include #include #include #define _CRT_SECURE_NO_DEPRECATE #define PI 3.142857 using namespace std; FILE *pout; errno_t err = fopen_s(&pout, "richardson.txt", "w+"); int main(void) { // Open for write if (err != NULL) { printf("\nThe files were not opened...\n", err); } else { printf("\nThe files were opened...\n"); long double r_value = 0.0; long double func[3] = { 0, 0, 0 }; long double error[3] = {0, 0, 0}; long double p[3] = {0, 0, 0}; printf("enter the values of the functions respectevily. f1, f2 .. fn\n"); for (int i = 0; i < 3; i++) { printf("f(%d) : \t",i); cin >> func[i]; } printf("enter the value of r \n"); cin >> r_value; for (int i = 2; i < 3; i++) { p[i] = (func[i] - func[i - 1]) / (func[i-1] - func[i - 2]); p[i] = log(p[i]) / log(r_value); error[i] = func[i - 1] - func[i - 2]; error[i] /= pow(r_value, p[i]) - 1; fprintf(pout, "%llf \t %llf \n", p[i], error[i]); } } fclose(pout); return 0; }