CFD Analysis of the Aspirator Region in a B&W Enhanced Once-Through Steam Generator

Transcription

1 CFD Analysis of the Aspirator Region in a B&W Enhanced Once-Through Steam Generator Adam M. Spontarelli Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirement for the degree of Masters of Science in Mechanical Engineering May 7, 2013 Blacksburg, VA Keywords: Computational Fluid Dynamics, Once-Through Steam Generator, OpenFOAM, Condensation

2 Abstract This analysis calculates the velocity profile and recirculation ratio in the aspirator region of an enhanced once-through steam generator of the Babcock & Wilcox design. This information is important to the development of accurate RELAP5 models, steam generator level calculations, steam generator downcomer models, and flow induced vibration analyses. The OpenFOAM CFD software package was used to develop the three-dimensional model of the EOTSG aspirator region, perform the calculations, and post-process the results. Through a series of cases, each improving upon the modeling accuracy of the previous, insight is gained into the importance of various modeling considerations, as well as the thermal-hydraulic behavior in the steam generator downcomer. Modeling the tube support plates and tube nest is important for the accurate prediction of flow rates above and below the aspirator port, but has little affect on the aspirator region itself. Modeling the MFW nozzle has minimal influence on the incoming steam velocity, but does create a slight azimuthal asymmetry and alter the flow pattern in the downcomer, creating recirculation patterns important to inter-phase heat transfer. Through the development of a two-phase solution that couples the aspirated steam and liquid feedwater, it was found that the ratio of droplet surface area to volume plays the most important role in determining the rate of aspiration. Calculations of the velocity profile and recirculation ratio are compared against those of historical calculations, demonstrating the possibility that these parameters were previously underpredicted. Such a conclusion can only be confidently made once experimental data is made available to validate the results of this analysis.

3 For Elise who encouraged me and Isaac who inspired me. iii

4 Acknowledgements I would like to thank my advisor, Dr. Danesh Tafti, for allowing me to come to Virginia Tech and work under his direction. I would like to thank him for his patience in dealing with this long distance endeavor and all of the inconveniences that it entailed. His guidance in this research has been invaluable. I would like to thank Anca Hatman for taking the time to share her wisdom and extensive experience in performing CFD analyses of the B&W OTSG. I would like to thank John Klingenfus for his continual guidance and mentorship. From the inner workings of the B&W nuclear plants, to the details of LOCA analysis and the RELAP5 code, and even his musings on the void fraction of a rainstorm, he has shared with me more knowledge than I could ever hope to retain. Most of all, I would like to thank my loving wife, Elise, for her support and understanding. iv

5 Contents 1 Introduction Background Information History Previous Analyses Flow Induced Vibration Methodology OpenFOAM Geometry Overview Outline Assumptions Key Inputs Power Level Steam Pressure Main Feedwater Tube Plugging End of Cycle Average Temperature Reduction Orifice Plate Setting Geometry Flow Rates Summary of Key Inputs Phase I - Simple Geometry Mesh Generation v

6 CONTENTS y Plus Error Estimation Iteration Error Estimation Discretization Error Estimation Inputs Boundary and Initial Conditions Fluid Properties k-epsilon Turbulence Model Solver Description Results Phase II - Inclusion of Tube Bundles and Support Plates Mesh Generation Porous Regions th Span th TSP th Span th TSP Pressure Drop Verification and Validation Inputs Solver Description Results Phase III - Inclusion of MFW Nozzle Geometry Mesh Generation Inputs Solver Description Results Phase IV - Solution Improvement Case 4 - Void Fraction Model Results Case 5 - Lagrangian Spray Mesh Generation vi

7 CONTENTS Inputs Boundary and Initial Conditions Main Feedwater Spray Solver Description Results Case 6 - Coalescence Inputs Solver Description Results Case 7 - Breakup Inputs Solver Description Results Case 8 - Condensation Inputs Solver Description Results Summary and Conclusions References 137 Appendices 141 A blockmeshdict 142 B Hydraulic Diameter Script 159 C GCI Script 163 D Plotting Scripts 166 D.1 Error Plots D.2 Mass Flow Rate Plots E spraycondensationfoam Solver Source Code 168 vii

8 CONTENTS viii

9 List of Figures 1.1 B&W Lowered-Loop Nuclear Steam Supply System [1] Simplified Schematic of Primary and Secondary Loops [2] B&W OTSG Cross-Sectional View [3] Region Between the 9th and 10th TSPs (a) Cross-Section of the Aspirator Region (b) Labeled Schematic [1] B&W Enhanced Once-Through Steam Generator [1] EOTSG Model Geometry EOTSG Labeled Drawing EOTSG Model Dimensions (inches) y Plus Values Grid 1 Mesh (1,124,000 Cells) (a) Top View: y-axis (b) Side View: z-axis Grid 2 Mesh (570,000 Cells) (a) Top View: y-axis (b) Side View: z-axis Grid 3 Mesh (242,400 Cells) (a) Top View: y-axis (b) Side View: z-axis Axial Velocity Profiles Through Aspirator Case1 Mesh ix

10 LIST OF FIGURES 4.7 Case1 Mesh (Aspirator Region) Case1 Residuals Case1 Continuity Errors Case1 Pressure Distribution Case1 Streamlines Case1 Azimuthal Velocity Profile Case1 Radial Velocity (x-dir) OTSG vs. EOTSG Aspirator Elevation Case1 Crossflow Velocity Comparison Case1 Crossflow Velocity Comparison Normalized Broached TSP Hole [4] Broached TSP Hole (accurate scaling) Axial Pressure Distribution in the Tube Bundle Region ONS Pressure Taps Case2 Residuals Case2 Continuity Errors Case2 Pressure Distribution Case2 Velocity Magnitude Case2 Crossflow Velocity Comparison Case3 Geometry Case3 Residuals Case3 Continuity Errors Crossflow Velocity (Case3 - Case2 ) Velocity Streamline Comparison Case3 (left) vs. Case2 (right) Case3 Crossflow Velocity Comparison Case4 Residuals Case4 Continuity Errors Case4 Crossflow Velocity Comparison Case5 Aspirator Mass Flow Rate Comparison Case5a Iteration Errors (a) Residuals x

11 LIST OF FIGURES (b) Errors Case5b Iteration Errors (a) Residuals (b) Errors Case5c Iteration Errors (a) Residuals (b) Errors Case5b Feedwater Temperature Distribution Case5b Feedwater Temperature Contours Case5 Crossflow Velocity Comparison Case6 Aspirator Mass Flow Rate Comparison Case6 Iteration Errors (a) Residuals (b) Errors Case6 Feedwater Temperature Comparison Downcomer Steam Temperature Comparison Case6 Feedwater Temperature Contours Case6 Downcomer Temperature Streamlines Case6 Crossflow Velocity Comparison Case7 Aspirator Mass Flow Rate Comparison Case7 Iteration Errors (a) Residuals (b) Errors Downcomer Steam Temperature Comparison Downcomer Feedwater Temperature Comparison Case7 Feedwater Temperature Contours Case7 Crossflow Velocity Comparison Case8 Aspirator Mass Flow Rate Comparison Case8 Parcel Distribution Case8 Parcel Size xi

12 LIST OF FIGURES xii

13 List of Tables 3.1 Key Inputs for EOTSG at 80% FP (2054MWt) Boundary Mass Flow Rates (Phase I through Phase III) Case1 y plus Values at Convergence Grid Sizes and Representative Cell Sizes Case1 Boundary Conditions EOTSG TSP and Tube Nest Porosity Inputs OTSG Tube Nest Pressure Data [5] Case4 (Void Fraction Model) Boundary Conditions OTSG Tube Nest and Downcomer Pressure Data [5] Case6 Droplet Size Comparison Case7 Droplet Size Comparison Recirculation Ratio Comparison Case8 Droplet Size Comparison xiii

14 LIST OF TABLES xiv

15 GCI grid convergence index. LOFW loss of feedwater. MFW main feedwater. Glossary NRC Nuclear Regulatory Commision. NSSS nuclear steam supply system. ONS Oconee Nuclear Station. OTSG once-through steam generator. ANO Arkansas Nuclear One. B&W Babcock & Wilcox. CFD computational fluid dynamics. DC downcomer. EOC end of cycle. EOTSG enhanced once-through steam generator. FA fuel assembly. FIV flow induced vibration. FP full power. PISO pressure implicit with splitting of operators. PWR pressurized water reactor. RCS reactor coolant system. SG steam generator. SGTP steam generator tube plugging. SIMPLE semi implicit method for pressure linked equations. TSP tube support plate. USCS United States customary system. xv

16 Glossary xvi

17 Chapter 1 Introduction Unexpected tube wear in the once-through steam generators (OTSGs) and enhanced once-through steam generators (EOTSGs) of the Babcock & Wilcox (B&W) design has spawned projects intended to determine potential damage mechanisms, one of which is flow induced vibration (FIV) [6]. In terms of FIV, the areas of major interest have been the inlet, outlet, and aspirator regions where tube bundle crossflow is most prominent. Detailed calculations of the operating fluid conditions over the lower and upper spans of the OTSGs have been performed [7]; however, conditions in the aspirator port (sometimes referred to as the bleed port) have been found using more rudimentary calculations [8][9]. In addition to the concerns of FIV, the flow field in the aspirator region, particularly the effects of main feedwater (MFW) spray condensation, plays an important role in accurate RELAP5-MOD2 B&W [10] modeling and steam generator (SG) level calculations [11]. The purpose of this research is to develop an accurate computational fluid dynamics (CFD) model able to calculate the velocity and density profiles through the aspirator port in a B&W EOTSG. By doing so, a further understanding of fluid behavior in the aspirator region of the OTSG and EOTSG can be attained, improving FIV analysis as well as RELAP5 based thermal hydraulic calculations which depend on SG heat transfer. 1

18 1. INTRODUCTION 1.1 Background Information This research deals with the Babcock & Wilcox pressurized water reactor (PWR) power plant design, specifically with the secondary side of the OTSGs and EOTSGs. The major components of the PWR are shown and labeled in Figure 1.1, which depicts what is referred to as the nuclear steam supply system (NSSS). The NSSS encompasses both the primary and secondary loops, as shown in Figure 1.2. The primary side consists of the reactor vessel which houses the nuclear fuel, reactor coolant pumps which force pressurized water through the core and steam generators, and the pressurizer which maintains a constant operating pressure. The secondary side removes heat from the primary by way of the steam generators. These components are essentially counterflow heat exchangers in which the hot, primary fluid enters the top and flows down the inside of the tubes; while a pool of cold fluid on the secondary side is boiled and the resulting steam is used to drive a turbine generator. Figure 1.1: B&W Lowered-Loop Nuclear Steam Supply System [1] Figure courtesy of AREVA NP Inc. 2

19 1.1 Background Information The B&W OTSGs are designed to remove the heat generated in a nuclear reactor vessel and utilize the once-through concept to produce dry, superheated steam at the outlets of the steam generator. Reactor coolant enters the top of the generator under hot conditions; the flow is then diverted through several thousand tubes which extend the full length of the generator, and exits through two outlet nozzles at the lower plenum. Meanwhile, on the secondary side, feedwater enters the annulus area via the feedwater nozzle connections just above the midpoint of the shell, and is sprayed downward. The cold feedwater draws steam from the tube bundle region through the aspirator port, which is the gap between the upper and lower shrouds. The aspirated steam is intended to bring the feedwater to saturation temperature in order to prevent thermal shock to the OTSG shell. The saturated feedwater is then boiled by removing heat from the lower tube bundle, and then superheated along the remaining tube bundle length. The superheated steam is directed downward through the steam annulus and leaves the OTSG through two steam nozzles, which are just above the feedwater nozzle connections. Figure 1.2: Simplified Schematic of Primary and Secondary Loops [2] 3

20 1. INTRODUCTION Figure 1.3 shows a cross-sectional view of the OTSG. The MFW nozzles penetrate near the midpoint of the steam generator. Figure 1.3: B&W OTSG Cross-Sectional View [3] Figure courtesy of AREVA NP Inc. 4

21 1.1 Background Information A 3D cross-sectional view of the aspirator port is shown in Figure 1.4a, which provides the relative locations of various features such as the circumferential spacing of the spray nozzles, the tube bundle density, and the tube support plate spacing. This annular port spans 360 of the SG, directing nearly saturated steam into the SG downcomer in order to raise the MFW temperature to saturation conditions. Figure 1.4: Region Between the 9th and 10th TSPs Figure courtesy of AREVA NP Inc. (a) Cross-Section of the Aspirator Region (b) Labeled Schematic [1] During normal operation of the 177 fuel assembly (FA) B&W plants, subcooled feedwater enters the downcomer and as the SG elevation increases, the secondary side fluid traverses the boiling curve and progresses from saturated liquid to subcooled nucleate boiling, transition boiling, film boiling, and finally radiative heat transfer to saturated vapor. The points at which these transitions occur are indistinct; however, a reliable approximation of the transition to saturated vapor is found by temperature measurements of the primary and secondary sides. With counter-current flow as exists in the OTSGs, the primary side fluid at the highest elevation can be expected to cool down at a relatively slow rate, since the outside of the tubes is surrounded with vapor. As 5

22 1. INTRODUCTION the primary fluid continues down through the tubes, there will be a point at which the secondary side will contain liquid droplets, a condition which produces very high heat transfer rates, and it is at this point that the primary side temperature will decrease dramatically. This elevation can also be determined by examining secondary side temperatures. Up until the transition from subcooled boiling to saturated steam, the fluid temperature will remain at saturation. However, once steam is generated, the temperature can begin to increase. The elevation at which the transition to saturated vapor takes place is referred to as the boiling length. The boiling length is dependent on reactor thermal power. As power increases, the heat removal rate of the secondary side must increase as well. If the temperature of the main feedwater and the turbine inlet conditions are to remain the same, the rate of heat transfer must increase by increasing the main feedwater mass flow rate, subsequently increasing the mixture level. According to experimental measurements taken from loss of feedwater (LOFW) testing [10], under full power conditions (2568 MWt), the boiling length is at approximately 35 feet from the upper face of the lower tubesheet. This means that two phase conditions exist near the aspirator port (31.9 ft). Solving for the flow field under such conditions is complex; it involves droplet evaporation due to heat transfer from the tube surface, droplet growth, droplet breakup from impact with solid surfaces and collisions with other droplets, and droplet collisions which can result in breakup or coalescence. Not only would the model necessary to solve such a problem require incredible complexity, but the computational requirements would be prohibitive, given the available resources. Instead, this analysis will focus on a reduced power level which results in nearly saturated steam in the aspirator region. Such a model will be valuable in the event that a plant must reduce power due to a limiting condition of operation, such as pump maintenance, and it will offer insight into what may be happening at full power operation as well. 6

23 1.2 History 1.2 History According to the SG design requirements outlined by the Nuclear Regulatory Commision (NRC) [12], the SG and all its parts shall be designed to withstand vibrations, including flow induced vibrations during normal operation or environmental vibrations. In addition, applicants should perform detailed analysis of potential adverse flow effects (flow induced vibration) that can severely impact reactor pressure vessel internal components, as well as other main steam system components such as the steam generator internals [12]. FIV analyses performed up until January 1, 2004 for the B&W 177-FA plants with OTSGs are summarized in the AREVA thermal hydraulic inputs for OTSG structural analyses [7]. Calculation of crossflow velocities in the aspirator region use hand calculations based upon the assumption that steam flow drawn through the aspirator is precisely the amount necessary in order to raise the MFW flow to saturated liquid conditions. This assumption is made in the calculations performed at core thermal powers of up to 2772 MWt [8], as well as 3014 MWt [13], which simply applies a scaling factor to the results of 2772 MWt. Similar FIV analyses have been performed for plants equipped with the EOTSG [14]. The EOTSG (Figure 1.5) was designed by Framatome ANP in 2002 [15] as a replacement to the OTSG design, with many design enhancements that improve its FIV performance over the original OTSG. The design changes incorporated into the EOTSG and their effect upon the FIV response of the tube bundle adjacent to the aspirator are elaborated below. The OTSG design had an open tube lane to allow inspection within the tube bundle. This open tube lane created flow velocities in the lane region that exceeded the general non-lane tube bundle flow velocities. For this reason, the OTSG tubes adjacent to the open tube lane experienced the greatest flow induced vibrations. The EOTSG design eliminated the open lane, thereby reducing the maximum flow velocities in the EOTSG tube bundle. The OTSG and the EOTSG tube support plate (TSP) thicknesses are 1.5 and 1.18 inches, respectively. The thinner TSP thickness will provide lower flow losses, 7

24 1. INTRODUCTION however the amount of available wear area will be lower for the EOTSG from an analytic perspective. If all aspects related to tube wear between the OTSG and EOTSG were identical, the EOTSG tube would therefore experience a higher tube wear rate. The shroud opening area in the aspirator region has been increased from 4 to 4.5 inches. Figure 1.5: B&W Enhanced Once-Through Steam Generator [1] Figure courtesy of AREVA NP Inc. Various B&W plants, including Arkansas Nuclear One (ANO) unit 1, upgraded from OTSGs to EOTSGs in FIV analyses were performed for the lower span, aspirator port, and upper span [14], in accordance with the methodology established for the FIV analysis of tube bundles as outlined in the ASME B&PVT Code [16]. The FIV analyses 8

25 1.3 Previous Analyses require thermal hydraulic inputs of velocities and densities in order to assess the region of interest. For ANO-1, these flow field calculations were performed in an EOTSG FIV inputs analysis [9], and the subsequent FIV analysis [14]. Unlike the flow field calculations for the aspirator region of the OTSG, these were performed using the PORTHOS CFD code [17],[18]. 1.3 Previous Analyses This document focuses primarily on two historical analyses described in the previous section: that which supports the OTSG [8], and that which supports the EOTSG [9] flow field calculation. As stated previously, the OTSG flow field calculation is performed by assuming that exactly enough saturated steam is drawn through the aspirator such that the incoming main feedwater would be heated to saturated liquid conditions. For the input conditions specified (2568 MWt, lbm/hr FW at 465F ), the steam mass flow rate is calculated to be 746, 000 lbm/hr, resulting in an average velocity of 3.29 ft/sec for both 2568 MWt and 2772 MWt plants. The velocity profile depicted in Figure 6 of the original OTSG analysis [8] appears to be highly symmetric about the aspirator port. This profile is calculated by what is referred to as a graphical analysis, in which the available flow area from each location in the span between the 9th and 10th TSPs is represented as a semicircle, and the velocity through each portion of that circle is inversely proportional to the available flow area. This method fails to account for the azimuthal asymmetry caused by the existence of the MFW nozzle, and also the momentum of the upward flowing steam, which would be expected to skew the velocity profile axially. The EOTSG analysis of the aspirator region [9] is performed using the PORTHOS CFD code. PORTHOS is a computer code for calculating three-dimensional steadystate or time dependent two-phase flow in porous or non-porous media. It calculates secondary side velocity, pressure, void fraction, and enthalpy distribution. The code employs a finite difference technique to solve the complete set of two-fluid equations. It was originally developed for use in recirculating steam generators of pressurized water reactor (PWR) nuclear power plants; however, B&W Nuclear Technologies adapted the code for use in B&W designed OTSGs and EOTSGs [18]. The modifications did 9

26 1. INTRODUCTION not allow for transient analysis, as the new model lacked a two-way coupling between the aspirated steam and feedwater flows [18]. The resulting B&W PORTHOS model was benchmarked against test measurements taken at the Alliance Research Center for 19-tube [19] and 30-tube [20] tests and the full scale start-up test results for the Oconee Unit 1 B steam generator [5]. Good agreement was found with the 19-tube tests; however, there was poor agreement with the 30-tube and full size SG. PORTHOS is well suited for the single phase conditions of the lower and upper spans; however, its ability to model the conditions in the aspirator region are unreliable. Additionally, it is important to note that a major assumption of this code is that the feedwater is heated to saturation with the resulting mixed stream of feedwater and bleed flow entering the tube bundle at a thermodynamic quality of zero. [18] This is the same assumption made by the OTSG analyses, and is necessary without a two-way coupling between the MFW and aspirated steam. This means that while PORTHOS is able to provide a more refined solution with model improvements such as porous tube bundle treatment, it is ultimately limited to the same over-constrained problem as that defined for the OTSG: the velocity through the aspirator port is predetermined. At the time, PORTHOS was considered to be the best engineering tool available to B&W for evaluation of fluid velocity and moisture fraction distribution in the OTSG. However, given the recent advancements in computational fluid dynamics and available computing power, a re-evaluation of the OTSG using modern tools is warranted. 1.4 Flow Induced Vibration While this work does not perform an FIV analysis, it does calculate the inputs for such an analysis, and it is therefore important to note the significant forms of flow induced vibration in the steam generator. Compared to crossflow, axial flow is a much less severe excitation mechanism. According to FIV analyses performed by AREVA [14], none of the conditions necessary for axial flow turbulence-induced vibration exist in the EOTSG. It is for this reason that crossflow is the focus of this analysis. In the SG tube bundle region, vortex induced vibration and fluid-elastic instability are the most likely mechanisms of FIV. 10

27 1.4 Flow Induced Vibration Vortex induced vibration affects isolated structures subject to crossflow. In the SG, the outermost tubes fit this criteria. As vortices are shed from the tube, an alternate force is exerted onto the tube. When the shedding frequency is close to the structural frequency, the two systems can become strongly coupled, resulting in a non-linearly vibrating fluid-structure system. This phenomena is known as lock-in, and it almost always results in failure. Tubes located within the bundle are more likely to experience fluid-elastic instability, in which the motion of one tube can affect surrounding tubes, causing high stresses and leading to failure. This mechanism is discussed at length in ASME Section III, Appendix N-1300 [16] as well as by Au-Yang [21]. In order to provide inputs necessary to assess these two excitation mechanisms of interest, this analysis will provide crossflow velocity profiles throughout the 10th span at a spectrum of radial locations. 11

28 1. INTRODUCTION 12

29 Chapter 2 Methodology In solving for the flow field in the aspirator region, this research intends to build on the basic understanding of fluid conditions in this region and provide boundary conditions necessary for FIV analysis of the B&W EOTSG. It is for this reason that a power level which produces a boiling length below the aspirator is sought after first, from which insights into full power operation will be attained. According to experimental data [10], at 80% full power (2054 MWt), the boiling length exists at approximately 27 ft, 4 ft below the aspirator port. Start-up data from Oconee Nuclear Station [5] indicates a boiling length of approximately 30 ft at 80% full power. The difference can be attributed to difference in MFW temperature, saturation pressure, or SG tube plugging. Regardless of these small differences in operating conditions, it is clear that the boiling length exists below the aspirator port. This analysis will calculate the velocity field of the aspirator region in an EOTSG using the methodology described below. Results will be compared against already existing data calculated for generic B&W plants [7] as well as the Arkansas Nuclear One (ANO) specific calculations made using PORTHOS [9], and conclusions will be drawn based upon the findings. 13

30 2. METHODOLOGY 2.1 OpenFOAM The OpenFOAM (Open Field Operation and Manipulation) CFD Toolbox [22] is an open source CFD software package with an extensive range of features to solve anything from complex fluid flows involving chemical reactions, turbulence and heat transfer, to solid dynamics and electromagnetics. It includes tools for meshing, and for pre- and post-processing. Almost everything (including meshing, and pre- and post-processing) runs in parallel as standard. This analysis takes advantage of various OpenFOAM solvers and utilities which are described throughout the following sections. In addition, a new solver is developed by modification of the existing C++ classes, in order to include a Lagrangian condensation model (Section 7.5). 2.2 Geometry Overview The area of interest is between the 9th and 10th tube support plates (10th Span) at the periphery of the SG, near the aspirator port, as shown in Figure 1.4a. The base model represents this region with a 1/32 nd azimuthal slice from the top of the 8th TSP to just above the 10th TSP, including the downcomer. An isometric view of the geometry is provided in Figure 2.1, while a two-dimensional, labeled view of the model is presented in Figure 2.2. The geometries created for this analysis are done so in the Cartesian coordinate system. Starting from the right side of Figure 2.1, the outer curved face represents the shell wall. The coinciding volume is the downcomer region, the bottom of which contains an outflowing two-phase mixture of feedwater and steam. Most of the downcomer is separated from the tube bundle by the shroud wall; however, they are connected by a 4.5 inch opening referred to as the aspirator port. The leftmost volume is the tube bundle region which is truncated by the left curved face in order to reduce geometry size. A depth of penetration into the tube bundle was chosen such that the flow from the innermost face of the model does not affect flows in the aspirator region. This depth of one foot was chosen based upon results of previous analyses [9]. The bottom face of the tube bundle is the steam inlet, while the top face is the steam outlet. The main 14

31 2.2 Geometry Overview Figure 2.1: EOTSG Model Geometry feedwater nozzle is not shown here, but penetrates the shell wall in the upper portion of the downcomer, just below the bottom of the aspirator port. 15

32 2. METHODOLOGY Figure 2.2: EOTSG Labeled Drawing 16

33 2.3 Outline 2.3 Outline OpenFOAM version [23][24] is used in all stages of this analysis: mesh generation, solving, and post-processing. The native blockmesh application is used for the mesh creation and is detailed in Section 4.1. A grid sensitivity study is performed against the mesh in Section Various solvers are used throughout this analysis, depending on the needs of the simulation. Each case section discusses the solver used and the inputs to that solver. The results of each case are presented at the end of their respective sections, with comparisons made against the results calculated in prior analyses [7][9]. The results tabulate the calculated velocity through the aspirator port, as well as other parameters of interest such as pressure drop along the axial length, recirculation ratio, spray velocity, temperature, and droplet size distribution when relevant. This problem has been split into four phases, each being an iteration of the previous and each increasing in model accuracy as well as complexity. Below is a list of the cases analyzed in this document along with a summary of the conditions and solvers used. 1. Simple Geometry (Section 4): Base model which consists of a 1/32nd slice of the EOTSG from the top of the 8th TSP to just above the 10th TSP. This model contains the porous TSP and tube bundle regions but does not apply any resistance to them. The velocity boundary conditions reflect an aspirated steam flow rate which is exactly enough to raise the MFW to saturation conditions. The simplefoam solver is used. 2. Inclusion of TSPs and Support Plates (Section 5): Adds resistance to the porous regions. Uses the same boundary conditions as Case1. The poroussimplefoam solver is used. 3. Inclusion of MFW Nozzle (Section 6): The MFW nozzle geometry is incorporated into the model. Uses the same boundary conditions as Case2. The poroussimplefoam solver is used. 4. Inclusion of MFW Spray Effects (Section 7): This phase contains multiple cases. First, Case3 is repeated, but the boundary conditions are adjusted to 17

34 2. METHODOLOGY reflect a best-estimate aspirator flow rate based upon experimental measurements. The poroussimplefoam solver is used. Next, a separate model is created which includes explicit modeling of the MFW spray in a Lagrangian framework. The aspirated flow rate is no longer specified, but is instead driven by the drag on the spray droplets. The sprayfoam solver is used. This Lagrangian model is divided into multiple cases such that the separate effects of droplet coalescence and breakup can be evaluated. Finally, a condensation model is incorporated into the Lagrangian model. A custom solver named spraycondensationfoam is developed and used. 2.4 Assumptions The following key assumptions are made by this analysis. 1. The head losses in the porous regions are calculated using velocities based on the assumption that the aspirated flow rate is exactly enough to bring the MFW to saturation conditions. If an iterative process were used to calculate more accurate losses, it is expected that the difference in values would be negligible. 2. The MFW droplet diameter is assumed to be approximately equal to the nozzle hole diameter. 3. The MFW spray is assumed to breakup immediately upon exiting the nozzle. In reality, the high velocity spray would likely exit the nozzle as a stream and breakup further down the downcomer. 4. The O Rourke collision model [25] is used in Phase IV, which assumes that the frequency of collision is much less than the time step, and also that the Weber number is less than 100. For Weber numbers greater than 100, droplet shattering is more likely to occur than coalescence. An approximation made in indicates that the Weber number is less than The Pilch-Erdman droplet breakup model assumes rigid, spherical droplets. 18

35 2.4 Assumptions 6. The condensation model is homogeneous, it assumes rigid, spherical droplets, and the heat capacity of the droplet is neglected because of the substantially greater contribution made by the enthalpy of vaporization. 19

36 2. METHODOLOGY 20

37 Chapter 3 Key Inputs This chapter addresses the treatment of each parameter of interest relative to the aspirator flow rate. A summary of key inputs can be found at the end of this section in Table 3.1. The plant conditions used by this analysis reflect normal operating conditions for EOTSGs in 177 fuel assembly lowered loop B&W plants with unplugged tubes. The Oconee Nuclear Station (ONS) operating conditions are used as inputs for this analysis. ONS is used because as stated previously, this analysis focuses on plant conditions at 80% of full power (FP). Conditions at this power level must either be provided explicitly, or else conditions at a lower power level must be supplied such that the 80% FP conditions can be interpolated. Operating conditions at 100% FP and 50% FP are well known for ONS, allowing for interpolation of conditions at 80% FP. Additionally, there exists start-up data for ONS at 75% and 85% FP. While the input parameters are derived from the ONS plant, the results of this analysis will also be compared against that of ANO. This is done because previous FIV analyses of the aspirator port, including more recent and more sophisticated, were performed for ANO. The differences between these two plants will be highlighted and their data will be normalized such that the results can be appropriately compared. Section 4.4 will make these distinctions. 21

38 3. KEY INPUTS 3.1 Power Level The power level analyzed is 2054 MWt. This is 80% of full power operation for most B&W plants ( = 2054MW t). The 80% power level is chosen because it results in a mixture level below the bottom of the aspirator port. Based upon experimental data [10], at 80% FP, the boiling length is approximately 27 ft, between the 8th and 9th tube support plates in the EOTSG (Figure 3.1). The bottom of the aspirator port in the EOTSG is 383 inches (31.92 ft) above the upper face of the lower tubesheet, according to the EOTSG drawings [26]. 3.2 Steam Pressure A turbine header pressure of 900 psia is targeted for all power levels. The steam pressure in the aspirator region of the SGs depends on the head loss in the main steamlines and is therefore different for every plant. The ONS start-up data provides measured pressures at the SG exit but not in the aspirator region. The data indicates that exit pressure is relatively unaffected by power level, changing by approximately 3 psia between 20% and 100% FP. At 100% FP, the pressure in the aspirator region is approximately 944 psia for ONS, while at 50% FP it is approximately 940 psia. This analysis assumes a pressure of 944 psia at the top of the 8th TSP at 80% FP. This is a reasonable assumption because the absolute pressure has a relatively small affect on the results of this analysis; the saturation enthalpy changes by less than one percent over a pressure range of 900 psia to 1000 psia. What is more important to this analysis is the pressure drop in the tube bundle and tube support plate regions, which is calculated in Section ANO uses a SG exit pressure of 915 psia [9], plus a head loss of approximately 15 psia [4] between the SG exit nozzle and the 10th span, giving a pressure of 940 psia in the aspirator region. 22

39 3.3 Main Feedwater 3.3 Main Feedwater The main feedwater sprays are able to control the rate of heat transfer from the primary to secondary system by adjustment of the temperature and flow rate. The primary system is also referred to as the reactor coolant system (RCS); it is the loop of the PWR that is responsible for transferring heat from the nuclear fuel to the SG. The secondary side is physically separated by the SG tubes; it encompasses the loop from the MFW nozzles, along the outside of the SG tubes, through the main steam lines and turbine, through the condenser, and finally back to the MFW nozzles. For ONS, the MFW temperature is set to 460F for 100% rated power and 385F for 50% rated power analyses [27]. Linear interpolation is used to calculate the MFW temperature at 80% power. T MF W = 385F + (80% 50%) 460F 385F 100% 50% = 430F At 100% FP, the MFW flow rate is targeted at 1500 lbm/s [27], however, this can change depending on the SG conditions (SG tube plugging, orifice plate setting) and ultimately depends on a heat balance between the primary and secondary side. A heat balance at 80% FP between the primary and secondary side is calculated as follows, Q primary = Q secondary Q primary = ṁ MF W (h g h in ) where Q primary is equal to the core power plus the power produced by the reactor coolant pumps. At 80% FP, this is approximately 2054 MWt plus 16 MWt of reactor coolant pump heat [28]. The vapor enthalpy, h g, is that of superheated steam, and the liquid enthalpy, h in, is determined by the MFW temperature. At 80% FP, the OTSG steam exit temperature is approximately 594 F [5]. The pressure at the SG exit is approximately 915 psia (Figure 7 of the ONS start-up report [5]); resulting in a vapor enthalpy of h g = 1254 Btu/lbm. ṁ MF W = Btu/s MW t (2568MW t MW t) = 2320lbm/s 1254Btu/lbm 408Btu/lbm The MFW flow rate for each SG is therefore 1160 lbm/s. Given that this analysis models a 1/32 nd slice of the SG, the MFW flow rate used will be 36.2 lbm/s. 23

40 3. KEY INPUTS For ANO at 100% FP, the MFW flow rate is equal to 1527 lbm/s-sg while the MFW temperature is 464 F, as indicated by Section 4.3 of the ANO FIV inputs [9]. These values are within one percent of the values used for ONS at 100% FP. 3.4 Tube Plugging As SG tubes wear over time due to friction between the tubes and the tube support plates, they are plugged to prevent any flow through the primary side. This necessarily reduces the rate of heat transfer from the primary to secondary side and requires an adjustment in MFW flow or temperature in order to compensate. This analysis assumes zero tube plugging, as does the ANO analysis. In order to apply the results to SGs with tube plugging, it is recommended that the scaling factors calculated for ANO be used [9]. 3.5 End of Cycle Average Temperature Reduction Similar to tube plugging, reducing the reactor coolant system average temperature, Tavg, reduces the heat transferred to the secondary side. This generally occurs at the end of a fuel cycle when burnup has depleted the fuel to the point that the reactor would initiate a power coastdown. As a means of extending the life of the fuel and sustaining full power operation for several days, RCS boron concentration is first reduced as much as possible, followed by a reduction in the average RCS temperature in order to add positive reactivity to the core. This process is known as end of cycle (EOC) Tavg reduction. This analysis assumes a nominal RCS average temperature. It is recommended that scaling factors be used to modify the results of this analysis for insight into the effects of Tavg reduction, as discussed in Section 4.3 of AREVA s thermal-hydraulic inputs for structural analysis [7]. 24

41 3.6 Orifice Plate Setting 3.6 Orifice Plate Setting The orifice plate exists at the bottom of the SG downcomer and serves as an adjustable means of flow restriction. Depending on the operating conditions of the SG (tube plugging, MFW settings, etc.) the orifice plate will be adjusted to either increase or decrease pressure drop such that the liquid level remains close to the nominal location. By assuming a liquid level below the aspirator port, this analysis assumes that the orifice plate is adjusted to ensure this is true. 3.7 Geometry This section calculates the areas and hydraulic diameters of important SG regions used throughout this analysis. The dimensions necessary for this calculation are provided by Table 3.1. A dimensioned drawing of the model is provided in Figure 3.1. The open area in the bundle region is calculated by subtracting the area of the tubes and support rods from the empty area inside the shroud. A = A shroud A tubes A rods ( ) in 2 A shroud = π = 76.43ft in/ft ( ) 0.625in 2 A tubes = π 15, 597tubes = 33.23ft in/ft There are two in OD support rods and fifty in OD support rods between spans 2 through 15. ( ) 0.625in 2 ( ) 0.787in 2 A rods = 2rods π + 50rods π = 0.17ft in/ft 2 12in/ft A = 76.43ft ft ft 2 = 43.03ft 2 The wetted perimeter in the tube bundle region is the summation of all wetted surfaces. P = P shroud + P tubes + P rods ( ) in P shroud = π = 30.99ft 12in/f t 25

42 3. KEY INPUTS ( 0.625in P tubes = π 12in/f t ( 0.625in 12in/f t P rods = 2rods π ) 15, 597tubes = ft ) + 50rods π ( ) 0.787in = 10.63ft 12in/f t P = 30.99ft ft ft = ft The hydraulic diameter for the open flow path is calculated as follows: D H Bundle = 4 A P = ft f t = 0.066ft The hydraulic diameter of the downcomer is more easily calculated as it is an annular region. The diameters used are found in Figure 3.1. D H DC = π(D2 o D 2 i ) π(d o + D i ) = D o D i D H DC = 70in 60.69in = 9.31in = 0.78ft 26

43 3.7 Geometry Figure 3.1: EOTSG Model Dimensions (inches) 27

44 3. KEY INPUTS 3.8 Flow Rates This section calculates the steam mass flow rates and velocities at the inlet, outlet, and aspirator port by assuming the exact aspirator flow necessary to bring main feedwater to saturation conditions. This is the same technique used by the existing FIV input analyses [8][9]. These values will be used for Phase I through Phase III of this analysis. ṁ MF W h MF W + ṁ Asp h Asp = ṁ D h D The unknown downcomer flow rate is found by performing a mass and energy balance, ṁ D = ṁ MF W + ṁ Asp ṁ MF W h MF W + ṁ Asp h Asp = (ṁ MF W + ṁ Asp ) h D The unknown downcomer enthalpy is assumed at saturated liquid conditions. Solving for ṁ Asp, ṁ Asp = ṁ MF W h MF W h D h D h Asp ( )Btu/lbm = 1160lbm/s ( )Btu/lbm = 220.1lbm/s The downcomer mass flow rate is therefore, ṁ D = ṁ MF W + ṁ Asp = 1160lbm/s lbm/s = lbm/s Knowing that the SG liquid level remains constant during normal operation, the SG outlet mass flow rate must equal the MFW mass flow rate. ṁ MF W = ṁ out = 1160 lbm/s The accompanying steam velocities are calculated as follows: v in = v out = v asp = ṁin ρ A = ṁout ρ A = ṁasp ρ A D = lbm/s 2.1lbm/ft ft 2 = 15.3ft/s lbm/s 2.1lbm/ft ft 2 = 12.8ft/s 220.1lbm/s 2.1lbm/ft ft 2 = 4.0ft/s 28

45 3.8 Flow Rates The Reynolds numbers at the boundaries can now be calculated. Where Re = v D H ν ν = µ ρ = lbm/sec ft 2.1lbm/ft 3 = ft 2 /sec Using a hydraulic diameter, D H, of ft as calculated in Section 3.7, Re in = 15.3ft/s 0.066ft ft 2 /sec 168, 000 Re out = 12.8ft/s 0.066ft ft 2 /sec 141, 000 The Reynolds number at the downcomer outlet is calculated similarly, using a hydraulic diameter of 0.78 ft. Re asp = 4.0ft/s 0.78ft ft 2 520, 000 /sec 29

46 3. KEY INPUTS 3.9 Summary of Key Inputs Table 3.1: Key Inputs for EOTSG at 80% FP (2054MWt) Parameter Analysis Value Reference SG Pressure (top of 8th TSP) 944 psia Section 3.2 Steam Density 2.1 lbm/ft 3 Calculated MFW Flow Rate 1160 lbm/s-sg Section 3.3 MFW Temperature 430 F Section 3.3 Mixture Level (boiling length) 27 ft Section 3.1 Aspirator Elevation (bottom) ft [26] Aspirator Height 4.5 in [26] MFW Nozzle Elevation (top) ft [26] SG Tube OD in [29] SG Tube OD Average Surface Roughness, 57 microinch [4] R a SG Shell Average Surface Roughness, R a 125 microinch [4] Number of SG Tubes 15,597 [30] Shroud ID in [31] TSP OD in [32] Table 3.2: Boundary Mass Flow Rates (Phase I through Phase III) Location Mass Flow Rate (lbm/s) Velocity (ft/s) Inlet Outlet Aspirator (Downcomer exit) The steam density is calculated using a constant pressure at saturated conditions throughout the model. This is appropriate because the degree to which pressure changes between the 8th and 10th TSPs has little affect on the steam density. ρ g (944psia) = 2.1lbm/ft 3 30

47 Chapter 4 Phase I - Simple Geometry Phase I consists of a single case, Case1, which develops a working three-dimensional model of the region of interest in the EOTSG and solves for an aspirator velocity profile given a set of boundary conditions corresponding to 80% of full power operation. Case1 analyzes the empty SG and aspirator region, meaning that the SG internals such as the tube support plates, support rods, and tubes are not modeled. The following subsections detail the mesh generation process (4.1), the inputs (4.2), the solver procedure (4.3), and the results (4.4). 4.1 Mesh Generation The OpenFOAM mesh generation utility, blockmesh, is used to develop the meshes used in all four phases of this analysis. The mesh is generated from a dictionary file named blockmeshdict located in the constant/polymesh directory of each case. blockmesh reads this dictionary, generates the mesh and writes out the mesh data to points and faces, cells and boundary files in the same directory. The dictionary file used to create the final mesh including TSPs, tube bundles, and the MFW nozzle can be found in Appendix A. The dictionary begins by defining the vertices to later be connected in three dimensional hexahedral blocks. The block definitions follow next and are defined by a set of eight vertices, one at each corner of the hexahedron. For those blocks which represent a porous region, a name is provided to the block; for example, Span9. The 31

48 4. PHASE I - SIMPLE GEOMETRY second entry in the block definition is the number of cells in the x, y, and z directions. Finally, the cell expansion ratios (also known as grading) are defined in the third entry. A value of 0.1 indicates that beginning cell in the direction specified will be ten times larger than the end cell. The rate at which the cells expand is then made uniform. A value of 10 would produce the opposite effect, making the beginning cell one tenth the size of the end cell. There are also some blocks which use negative values for cell grading, which produce a cell expansion or reduction in both directions on the axis specified, referred to as double grading. This feature is not available by default in OpenFOAM version but was produced separately by Pei [33]. Double grading was used in regions surrounded by walls, such as the x-direction of the downcomer region which is bounded by the shroud at the inner wall and the shell at the outer wall. The feature was also used to grade axial regions of the tube bundle such that the mesh coarsens at the midpoint between TSPs where flow is relatively quiescent, and is gradually refined closer to the TSPs where flows begin to accelerate. Next in blockmeshdict are the edge definitions which allow a reshaping of the blocks such that straight edges can be made into arcs. Because the model is a 1/32 nd azimuthal slice of the steam generator, the walls must be curved accordingly. This is done first by specifying the edge to be modified by the two vertices that define it, then by specifying a point at which the circular arc passes through. The boundaries are defined next. Each boundary is given a name and patch type (e.g. wall, patch, cyclic, etc.), and is defined by a set of faces which are composed of four vertices. For example, the inlet boundary consists of two adjacent faces, one associated with the inner tube bundle region, and another for the empty gap between the tube bundle and shroud wall. The inlet boundary condition type is patch, which is the basic patch type for a condition that contains no geometric or topological information about the mesh. Finally, patches are merged in the mergepatchpairs section in order to join blocks which do not have the same number of cells at the shared faces. The faces are essentially stitched together by dividing the cells of the neighboring faces such that they match one another. In generating the three dimensional mesh used in this analysis, a 2D mesh was first created. The blockmeshdict file associated with the 2D mesh can also be found in Appendix A. The original EOTSG drawings [34][26][35] were used as the source of dimensions to generate this file, and the most important dimensions have been repeated 32

49 4.1 Mesh Generation in Figure 3.1. A 2D mesh was created in order to run scoping cases testing various boundary conditions. Once the mesh structure was finalized and the boundary conditions established, the 2D mesh was converted to a 3D mesh using the Python script named 2Dto3D.py presented in Appendix A. The 2D mesh contains a set of 43 vertices for the front face of the geometry and 43 vertices for the back face. The conversion script simply translates these vertices radially outward so that the flat 2D geometry becomes a 3D azimuthal slice representing degrees of the full EOTSG. The resulting 3D geometry is shown in Figure y Plus The y plus value marks the transition of a turbulent flow from the inner wall region of the boundary layer where viscous forces dominate, to the outer region where turbulence overwhelms the viscous forces. When employing a Reynolds Average Stress turbulence model, as is done in this analysis, it s necessary to define the location of this transition. This analysis makes use of the standard k-epsilon turbulence model because of its wide range of applicability and because of its second order accuracy [36]. The k-epsilon turbulence model is further discussed in Section 4.2.3, however, it is mentioned here in order to justify the choice of a target y plus value between 30 and 500. The scale of turbulent eddies near a wall can be very small and the rate at which properties change is very high and would require extremely fine cell sizes near the wall that would result in impractical computational demand. Rather than explicitly modeling such fine cells, standard wall functions are used to resolve the turbulent behavior near the wall. The wall functions calculate the laws-of-the-wall for the mean velocity, temperature, k, and epsilon. The general guidance when using wall functions is to place the centroid of the cell closest to the wall within the fully turbulent loglaw zone. This region is typically between y plus values of 30 and 300, but the upper bound increases with increasing Reynolds number. Figure 5.7 of Computational Fluid Mechanics and Heat Transfer [37] provides a visual understanding of the various zones that comprise a turbulent boundary layer for incompressible flow over a flat plate. The mesh for Case1 was generated with a target y plus value of 150. The location of 33

50 4. PHASE I - SIMPLE GEOMETRY the first cell centroid, y, can therefore be calculated from, y + = yu ν Where u is the frictional velocity in the wall layer. u = ( ) 1/2 τw ρ Where τ w is the shear stress at the wall. τ w = f Dρu 2 8 Where f D is the Darcy friction factor, which is found from the Moody chart [38]. The shell is a machined surface while the shroud is made from a rolled plate surface, both with an average roughness of approximately 125 micro-inches (Table 3.1). This analysis treats the shroud and shell as a smooth pipe given their low roughness, large diameter, and high flow rates. Using the Moody chart with a Re of 168,000 and a smooth pipe assumption, a Darcy friction factor of is found. The wall shear stress is then calculated, τ w = f Dρu 2 8 τ w = lbm/ft3 (15.3ft/s) 2 8 τ w = 0.983lbf/ft 2 The frictional velocity is then calculated as, u = ( ) 1/2 τw ρ ( 0.983lbf/ft u 2 = 2.1lbm/ft 3 u = 0.68ft/s ) 1/2 Finally, the y-value can be calculated, y = y+ ν u 34

51 4.1 Mesh Generation y = ft 2 /s 0.68f t/s y = ft The cell size of the cells adjacent to the wall are twice the length of the y-value, since the y-value is the position of the cell centroid. Using the calculated y-value, the cell coarseness can be optimized as follows: An expansion ratio α = 0.1 is used in order to allow for a reduction in the number of cells necessary near the walls while still providing an accurate solution. The width of the region to be discretized is necessary. The gap between the tube support plate and the shroud wall is approximately 0.45 inches or ft. The number of cells necessary to fill this region with an expansion ratio of 0.1 and a target minimum cell size of ft is therefore: ft α f t = n = 6.25cells Using the yplusras utility provided with OpenFOAM, the actual y plus data is calculated for the converged solution of the coarse grid (570,000 cells). The value calculated by yplusras is sometimes referred to as y in CFD codes such as FLUENT. The incompressible y calculation can be found in RASModel.C and is referred to as yp in the code. y = C 1 4 µ y k 1 2 ν The results of the yplusras utility are tabulated in Table 4.1. Table 4.1: Case1 y plus Values at Convergence Patch Name Minimum y plus Maximum y plus Average y plus Shroud Shell A contour plot of the y plus values is presented in Figure 4.1, which shows that the high y plus values exist only in local regions which do not play an important role in calculating the aspirator velocity profile. Conversely, the regions of importance, such as the shroud and the lower wall of the aspirator port, have values in the range of 100 to 200. Therefore, from a y plus perspective, this mesh is acceptable for further use. 35

52 4. PHASE I - SIMPLE GEOMETRY Figure 4.1: y Plus Values Error Estimation In order to adequately quantify the errors associated with the numerical solutions presented in this analysis, the iteration errors and discretization errors are addressed in the following subsections Iteration Error Estimation OpenFOAM computes the norm of the residuals, while the convergence criteria are specified in the system/fvsolutions file. For Case1, the convergence criteria relative to the initial residual are set to 1e-2 for the pressure, and 1e-3 for the velocity, k, and epsilon. A plot of the residuals versus solution step for the final version of Case1 is presented in Figure 4.8. Also, a plot of the continuity errors can be found in Figure 4.9. At the time of convergence, the total continuity error is approximately , a negligible value. 36

53 4.1 Mesh Generation Discretization Error Estimation Because this mesh serves as the basis for the later cases, a mesh sensitivity study is performed using the grid convergence index (GCI) method proposed by Celik [39]. This method is based on the Richardson extrapolation technique, in which multiple solutions to the numerical calculation are found by adjusting a parameter (grid size) and are used to extrapolate a more accurate solution. In the case of the GCI method, the various numerical solutions are compared against the extrapolated solution in order to calculate a relative error. This method is recommended by the Journal of Fluids Engineering and has been evaluated over several hundred CFD cases [39]. follows; 1. Calculate a representative or average cell size, h. 1 N (δv i ) 1/3 N i=1 The procedure is as Where δv i is the volume of the i th cell and N is the total number of cells. This calculation can be performed directly by the OpenFOAM solver by implementing the following code: scalar h=0; forall(mesh.c(),celli) { } h += (1 / (celli+1)) * mesh.v()[celli]; double H = std::pow(h,(1/3.0)); Info << h = << H << endl; The forall loop iterates through every cell in the mesh and stores the cell index as the variable celli. The average cell volume is stored in h and summed over each iteration. The average cell size is then calculated as the cube root of h. Finally, the average cell size is printed to the standard output. 2. Select three significantly different sets of grids such that the grid refinement factor, r = hcoarse h fine is greater than 1.3. Run the same simulation on each grid and determine the values of key variables important to the study. For example, this analysis examines the values of velocity. 37

54 4. PHASE I - SIMPLE GEOMETRY 3. Let h 1 < h 2 < h 3 and r 21 = h 2 h 1, r 32 = h 3 h 2 and calculate the apparent order, p, of the method using, p = 1 log r 21 log ɛ 32 + q(p) ɛ 21 q(p) = log rp 21 s r p 32 s s = 1 sign ɛ 32 ɛ 21 where ɛ 32 = φ 3 φ 2, and φ denotes the transport variable of interest. A Python script has been written in order to calculate the GCI and can be found in Appendix C. The initial mesh generated consists of 1,124,000 hexahedral cells, which is most easily found by using the checkmesh utility. In addition to cell count, checkmesh provides other cell-related information, including max aspect ratio, orthogonality, volume, and max skewness. A more coarse mesh is generated consisting of 570,000 cells, and the coarsest mesh consists of 242,400 cells. Each of these three meshes passed the checkmesh tests but issued a warning about finding approximately 500 non-orthogonal cells. Upon inspection, the non-orthogonal cells were found to be at the 10th TSP where the stitching results in highly skewed connections to the exiting tube bundle. Because these cells are in a region far from the aspirator port, the warning is ignored. Figures 4.2a through 4.4b display each of the meshes. Table 4.2 summarizes the important mesh information for this sensitivity. 38

55 4.1 Mesh Generation Figure 4.2: Grid 1 Mesh (1,124,000 Cells) (a) Top View: y-axis (b) Side View: z-axis Figure 4.3: Grid 2 Mesh (570,000 Cells) (a) Top View: y-axis (b) Side View: z-axis Figure 4.4: Grid 3 Mesh (242,400 Cells) (a) Top View: y-axis (b) Side View: z-axis 39

56 4. PHASE I - SIMPLE GEOMETRY Table 4.2: Grid Sizes and Representative Cell Sizes Grid Name Cell Count Representative Grid Size, h (ft) Grid Grid Grid Figure 4.5 presents an axial velocity profile through the aspirator port at x=4.789 ft and z=0 ft. The local order of accuracy ranges from to 10.70, with a global average, p ave, of 5.2. This global order of accuracy is limited to a value of 2.0 so as not to exceed the formal order of accuracy of the discretization scheme. Second order accurate Gaussian quadrature is used for this analysis as a means of integration as defined in the fvschemes file. The GCI values are then calculated and plotted in the form of error bars as shown in Figure 4.5. The maximum discretization uncertainty is 54%. This high value is relative to a low velocity and corresponds to a maximum uncertainty of about ±0.03f t/s. 40

57 4.1 Mesh Generation Figure 4.5: Axial Velocity Profiles Through Aspirator Based on the GCI calculation, the coarse grid of 570,000 cells is appropriate for use throughout the remainder of this analysis. For Case1, the mesh contains regions to represent the tube bundles and support plates, but they do not have any porosity applied to them. The base model utilizes a structured, non-uniform, hexahedral mesh, as shown in Figure 4.6. A closer view of the aspirator port is shown in Figure

58 4. PHASE I - SIMPLE GEOMETRY Figure 4.6: Case1 Mesh 42

59 4.1 Mesh Generation Figure 4.7: Case1 Mesh (Aspirator Region) 43

60 4. PHASE I - SIMPLE GEOMETRY 4.2 Inputs Case1 utilizes SG conditions under the assumption that exactly enough steam is aspirated in order to bring the MFW to saturation conditions. This section describes the structure of the OpenFOAM cases analyzed herein as well as the specific inputs utilized. The information associated with each case is located in a separate directory and comprises multiple sub-directories. A visual representation of the basic case file structure is shown below. Root Directory/ 0/ epsilon k nut nutilda p U constant/ polymesh/ blockmeshdict RASProperties transportproperties system/ controldict fvschemes fvsolution sampledict In the root directory, there exists a 0 directory, where boundary and initial conditions are defined; a constant directory, where transport properties and mesh information is stored; and a system directory, which specifies the solver and solution behavior. Each of these directories will be examined in further detail in the subsections below. 44

61 4.2 Inputs Boundary and Initial Conditions Case1 contains a total of eight boundary patches: shroud, shell, inlet, outlet1, outlet2, newback, newfront, and insidewall. The assignment of these boundaries is shown in Figure 2.1. The shroud and shell boundaries are walls, the newback and newfront patches are cyclic or periodic boundaries, the insidewall is a slip boundary that prevents interference with the modeled flow field, and finally inlet, outlet1, and outlet2 are considered the flow boundaries. Each of these eight boundaries must specify a boundary condition for five distinct parameters when using the k-epsilon turbulence model: fluid velocity (U), pressure (p), kinematic turbulent viscosity (nut), turbulent kinetic energy (k), turbulent dissipation rate (epsilon). The conditions applied to each of these parameters can be found in the respective files of the 0 directory. The boundary conditions applied at the flow boundaries are summarized in Table 4.3. The turbulence-related boundaries are addressed in Section Table 4.3: Case1 Boundary Conditions Boundary Velocity (ft/s) Pressure (ft 2 /s 2 ) Inlet v = Outlet 1 (SG outlet) 12.8 ft/s P = 0 Outlet 2 (Downcomer outlet) -4.0 ft/s P = 0 Although the velocities were hand calculated at each flow boundary, a pressure is applied at the inlet in order to improve the rate of convergence. Because this is an incompressible calculation, the pressure is supplied in units of ft 2 /s 2. The 944 psia pressure is therefore, 944lbf/in 2 144in2 1ft lbm ft/s2 lbf 1ft3 2.1lbm = ft2 /s 2 The resulting velocity at the inlet should, however, equal that provided in Table Fluid Properties The fluid properties are specified in the transportproperties file in the constant directory. Since Case1 is an incompressible analysis, only the kinematic viscosity must 45

62 4. PHASE I - SIMPLE GEOMETRY be specified. This is done through the following line: nu nu [ ] 6.0e-6 The values in the brackets determine the dimensions of the value being specified. The meaning of each digit is well explained by the OpenFOAM user s manual [23]. In this case, the kinematic viscosity is set equal to ft 2 /sec. A dynamic viscosity of lbm/ft sec is found from the IAPWS steam tables for saturated steam at 944 psia. ν = µ ρ = lbm/ft sec 2.1lbm/ft 3 = ft 2 /s k-epsilon Turbulence Model The turbulence model is selected in the constants/rasproperties file. Throughout this analysis, the standard k-epsilon turbulence model is used by setting RASModel equal to kepsilon [36]. Among the two-equation turbulence models, k-epsilon has the benefit of being among the most widely validated turbulence models as well as maintaining accuracy across a wide range of applications. OpenFOAM has the capability of using the RNG and realizable k-epsilon models; however, these models are not necessary for this analysis. The RNG model improves accuracy for rapidly strained and swirling flows, but neither condition is expected to occur in the highly symmetric EOTSG geometry. Similarly, the realizable k-epsilon model provides greater accuracy for flows involving rotation and strong pressure gradients, making it unnecessary for this analysis. By implementing the k-epsilon model in the simplefoam solver, boundary conditions must be specified for k and epsilon at each boundary. As with the other boundary conditions, this is done in the files of the 0 directory. For epsilon, the turbulent dissipation rate, a wall function called epsilonwallfunction is used at the shell and shroud boundaries. Similarly, the turbulent kinetic energy, k, is calculated using the kqrwallfuction. Wall functions are semi-empirical formulas necessary in order to assist the k-epsilon model in resolving near-wall transport properties. They are applied 46

63 4.3 Solver Description in the viscous sub-layer region and in the law-of-the-wall region discussed in Section The turbulent dissipation rate is computed from the turbulent length scale as follows, ɛ = C µ k 3 2 l Where C µ is a constant of the k-epsilon model equal to The initial value of k is set using the fluctuating component of velocity. k = 1 2 U U In the Cartesian coordinate system, this becomes k = 1 2 (U 2 x + U 2 y + U 2 z ) In order to calculate initial values for these parameters, estimations must be made as to the degree of the fluctuating components of velocity. The initial turbulence is assumed to exist only in the axial direction (y-dir) and is set equal to 5% of the inlet velocity, while l is equal to 20% of the hydraulic diameter of the SG tube region. The initial values of k and epsilon are therefore k = 1 2 ( ft/s)2 = 0.29ft 2 /s 2 ɛ = C0.75 µ k 1.5 l = (0.29ft 2 /s 2 ) ft ɛ = 1.94ft 2 /s 3 These values are applied at the walls and at the inlet. Other boundary patches specify a zerogradient initial value. 4.3 Solver Description The solver used for Phase I is referred to as simplefoam. Its formulation and implementation into OpenFOAM is detailed by Jasak [40]. It is a steady-state solver for 47

64 4. PHASE I - SIMPLE GEOMETRY turbulent, incompressible flow, and is based upon the SIMPLE algorithm developed by Patankar [41]. The main loop of the simplefoam solver is implemented as follows: while (simple.loop()) { Info<< "Time = " << runtime.timename() << nl << endl; // --- Pressure-velocity SIMPLE corrector { #include "UEqn.H" #include "peqn.h" } turbulence->correct(); runtime.write(); Info<< "ExecutionTime = " << runtime.elapsedcputime() << " s" << " ClockTime = " << runtime.elapsedclocktime() << " s" << nl << endl; } 1. All field values are set to an initial guess. For the initial iteration, these values are provided in the 0 directory, while subsequent iterations use the solution to the previous iteration. 2. The under-relaxed momentum predictor equation is assembled and solved for in the call to UEqn.H. Line 60 also makes the first guess at the solution of the eddy viscosity based upon the values of k and epsilon. 3. The pressure corrector is implemented in peqn.h as well as calculation of the conservative fluxes. The pressure field is updated with an appropriate underrelaxation. The explicit velocity correction is performed using, U p = H(U) a p 1 a p p 48

65 4.4 Results 4. The other equations in the system are solved using the available fluxes, pressure and velocity fields. In order to improve convergence, the other equations are under-relaxed in an implicit manner using, a P α φn P + a N φ n N = R P + 1 α α a P φ o P 5. The convergence criteria are checked for all equations. If the system is not converged, a new iteration is started on step 2. An outer iteration of the turbulent kinetic energy and dissipation rate is then made by line Results The simplefoam solver converged after 299 iterations using the residual criteria discussed in Section The residual and continuity data are plotted in Figure 4.8 and Figure 4.9. Both the global and cumulative continuity errors are presented, where the global represents the error summed over all control volumes in the system, while the cumulative is the summation of the global error since the first iteration. These figures were produced using the gnuplot scripts located in Appendix D. The pressure and velocity fields through the central x-y plane are shown below. The pressure distribution shown in Figure 4.10 is gradual, as expected. This is because the tube support plates and tube bundles are not modeled in Case1. 49

66 4. PHASE I - SIMPLE GEOMETRY Figure 4.8: Case1 Residuals Figure 4.9: Case1 Continuity Errors 50

67 4.4 Results Figure 4.10: Case1 Pressure Distribution 51

68 4. PHASE I - SIMPLE GEOMETRY Figure 4.11: Case1 Streamlines 52

69 4.4 Results Figure 4.12: Case1 Azimuthal Velocity Profile 53

70 4. PHASE I - SIMPLE GEOMETRY Figure 4.11 presents the velocity streamlines in the 10th span and downcomer. This figure indicates that within a short radial distance into the tube nest, the fluid velocity is essentially unaffected by the aspirator flow. The streamlines also provide a visualization of the flow pattern through the aspirator and in the downcomer. The recirculating pattern in the downcomer will in reality be prevented by the spray from the feedwater nozzles. The existence of this pattern may contribute to the skew of the velocity profile by counteracting flow in the lower part of the aspirator. The two-phase analysis of PhaseIV will provide further insight into this issue. Figure 4.12 shows a cross section through the y-z plane at a radial distance of ft from the center of the steam generator. One can see that the velocity is entirely symmetric about the z-axis. This result is expected to change once the MFW nozzle is added to the model in Phase III. Figure 4.13 presents the the x-component of the velocity field at various radial positions in the 10th span. A positive value indicates flow from the SG interior (tube bundle) to the downcomer. This plot provides quantitative values of the crossflow velocity profile through the aspirator port, which is essential to FIV analysis. One can see that the velocity increases sharply at the opening of the aspirator port ( ft), and similarly decreases at the exit ( ft). The shape of the profile has a slight skew toward the exit, which is shown by the fact that the peak of the profile is above ft, which is the location of the center of the 4.5 inch aspirator port. This is expected, given that the momentum of the upward flowing steam is likely to delay the change in direction. Progressing deeper into the tube nest, the peak of the velocity profile decreases, while at the same time broadening. The broadening is due to the fact that the aspirator port has a wider view of the tube nest at increasing radial distances. The results of Case1 are compared against those of historical calculations in Figure The step-wise curve is the hand calculation performed as a generic calculation for all 177 FA B&W plants with OTSGs at 2568 MWt [8]. The ANO-1 curve is produced by the PORTHOS CFD code and represents the ANO-1 plant at 2568 MWt with 25% steam generator tube plugging (SGTP) and OTSGs [9]. The second ANO-1 curve represents the same ANO-1 plant, but without SGTP and with EOTSGs. Finally, the Case1 curve represents the results from this analysis. The velocities for the generic 177 FA plants are calculated at a radial position of ft (location of the outermost tube 54

71 4.4 Results Figure 4.13: Case1 Radial Velocity (x-dir) in the OTSG), while all other curves present the velocities extracted from a radius of ft, which is close to the position of the TSP edge in the EOTSG. The shape of the historical curves is somewhat surprising. At a radial position of ft ( inches from the OTSG shroud wall) and ft (1.722 inches from the EOTSG shroud wall), one would expect the crossflow velocity to be near zero at elevations below and above the aspirator port. However, as seen in Figure 4.15, this is not the case. The reason that the generic 177 FA analysis calculated non-zero velocities above and below the aspirator port is because of the method used by the original calculation [8]. The velocities calculated are a function of the available flow area and do not take into account the line of sight from the aspirator port. Essentially, the calculation assumes that the shroud wall is not present. As for the PORTHOS calculations, it is not clear why such high crossflow velocities would be calculated near the wall. It may be an effect of coarse meshing vertically and radially. There are only two mesh points that span the 4.5 inches of the aspirator port and three additional mesh points spanning 1.5 feet below the aspirator. These non-zero crossflow velocities at the wall are not 55

72 4. PHASE I - SIMPLE GEOMETRY addressed in the historical calculations and without additional information and insight into the results, it is difficult to determine the true cause. The historical calculations produced velocity profiles for B&W plants at 2568 MWt. Since the results of Case1 are only applicable at 2054 MWt (80% FP), it is difficult to draw conclusions from direct comparisons of the velocities. Instead, comparisons are made against the shape of the velocity profiles. This is done by normalizing each of the curves between a value of zero and one. The normalized results are presented in Figure When comparing against the generic results of the 177 FA calculation [8], one must remember that OTSGs were used in that analysis, and that the distance between the TSPs, the position of the aspirator port, and the size of the aspirator port differs from that of the EOTSG. Figure 4.14 gives a perspective of the relative elevations with respect to the upper face of the lower tubesheets for each SG. Figure 4.14: OTSG vs. EOTSG Aspirator Elevation 56

73 4.4 Results Figure 4.15: Case1 Crossflow Velocity Comparison Figure 4.16: Case1 Crossflow Velocity Comparison Normalized 57

74 4. PHASE I - SIMPLE GEOMETRY As previously stated, the historical calculations overpredict the crossflow velocity above and below the aspirator port opening. Given that the OTSG aspirator port is slightly higher than the EOTSG aspirator port (Figure 4.14), the hand calculated generic 177 FA profile seems to make a reasonable prediction of the profile. Though it does not show the same vertical skewing as produced by Case1, the slope of the curve matches very closely (Figure 4.16). The PORTHOS curves lie on top of one another because the EOTSG PORTHOS results were found by applying a constant scaling factor to the results of the OTSG PORTHOS analysis. These curves show a more gradual change in velocity, which, as previously stated, is likely due to the coarse mesh that was used. The PORTHOS analysis was also the only calculation that found negative crossflow velocities at the top of the aspirator (Figure 4.15). The results of Case1 indicate that the model was properly prepared. The magnitude of the velocity profile is lower than that found by historical calculation which is expected given that the power level has been reduced by 20%. The shape of the profile matches that of the generic 177 FA calculation. This is expected given that the generic calculation and Case1 do not account for the effects of the tube bundles and tube support plates. The following phases of this analysis will incorporate the effects of the tube bundles and TSPs to better compare against the results found by PORTHOS. In addition, features not modeled in the historical calculations, such as the MFW nozzle and MFW droplet dynamics, will be incorporated in Phases III an IV. 58

75 Chapter 5 Phase II - Inclusion of Tube Bundles and Support Plates Case2 modifies the first case by adding porous regions which represent the tube bundles and tube support plates. A porous treatment is used because of the computational demand necessary to model these regions explicitly. While this model only consists of a 1/32 nd slice of the EOTSG penetrating one foot into the tube nest, it would still contain approximately 200 tubes, each nearly 7 feet long, accompanied by their broached holes in the TSPs. By modeling these regions as porous media, the total number of grid cells is significantly reduced while still maintaining the impact that these regions have on the resulting flow field. 5.1 Mesh Generation Porous Regions The loss coefficients and resulting pressure drops across each porous region of the model- 9th span, 9th TSP, 10th span, and 10th TSP- are calculated below following the methodology provided by Smotrel [4]. For each region, a pressure drop is calculated across that region by multiplying the velocity head by the loss coefficient. The loss coefficients are sometimes derived from empirical correlations and other times by use of the Darcy-Weisbach equation for head loss. Once the pressure drop is known, a loss 59

76 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES factor, F, used as input to the OpenFOAM code can be calculated. A summary of these values can be found at the end of this section in Table 5.1. A schematic of the porous regions is displayed in Figure th Span Crossflow (x- and z-direction) The 9th span is the tube bundle region between the 8th TSP and the 9th TSP. For Case2, this is the first porous region adjacent to the inlet boundary. The 9th span is surrounded by the shroud wall on all sides, so unlike the 10th span, this region expects to see very little crossflow. The porous parameters of this region will therefore be calculated using an assumed crossflow velocity of 0.5 ft/s. As stated previously, this section will calculate the pressure drop and loss factor of this region. The pressure drop, P, is the product of the loss coefficient, K, and the velocity head, V H. P = K V H The crossflow loss coefficient through the SG tube bundle is a function of the tube pattern, the direction of flow, and the flow angle of attack relative to the vertical tube. The loss coefficient will be calculated using the empirical correlation developed by AREVA for flow through staggered tubes [4]. The effect of the tie rods is not considered because the outermost tie rods are approximately 8.5 inches from the shroud wall, a distance at which aspirated steam does not impact flow. The Reynolds number at this location can be calculated with an estimated crossflow velocity of 0.5 ft/s over inches, the full length of the 9th Span (Figure 3.1). ( ) 0.5ft/s 37.82in 12in/ft Re av = ft 2 = 263, 000 /sec The loss coefficient is therefore, K = 9.04 The pressure drop from the center to the outside of the tube bundle can now be calculated. P = K ρv 2 2g 60

77 5.1 Mesh Generation P = lbm/ft3 (0.5ft/s) ft/s 2 = 0.07lbf/ft 2 = lbf/in 2 The pressure drop in the radial direction is effectively zero. This is expected given that the 9th span is surrounded by walls. OpenFOAM makes use of the Darcy-Forchheimer equation when implementing porous media [42]. S i = (µd + 12 ) ρ u kk F ij u i Where S i is a sink term which creates a pressure drop across the medium. Because the porous regions of this analysis are non-permeable, the permeability term can be eliminated, leaving only the inertial loss term. S i = 1 2 ρu i u kk F ij OpenFOAM accepts the scalar components of the inertial resistance factor, F ijk, as an input to the porosity calculations. In order to achieve the target pressure drop over the porous region, this factor must be input as a loss coefficient per unit length. The pressure drop in the x- and z-direction is therefore calculated as, p x = f i x 1 2 ρu i u The i- and k-component of the inertial resistance factor is therefore, 32.2ft lbm s 2 lbf f i = 2 p x xρu 2 = lbf/ft in/ft 2.1lbm/ft3 (0.5ft/s) = 2 0.9ft 1 Parallel Flow (y-direction) Flow along the tube bundle is calculated by use of the Darcy-Weisbach equation. ( ) L P = f D ρv 2 D H 2g The Darcy friction factor is found from the Moody Diagram using a Reynolds number of 166,000 (as calculated in Section 3.8) and a relative roughness of [4]. f D =

78 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES The length, L, is equal to the distance between tube support plates. For the 9th span, this value is equal to inches, as shown in Figure 3.1. The hydraulic diameter, D H, was calculated for the tube bundle to be ft in Section 3.7. The pressure drop for flow parallel to the tube bundle in the 9th span is therefore, ( ) 37.82in P = lbm/ft3 (15.3ft/s) 2 12in/ft 0.066ft ft/s 2 P = 6.2lbf/ft 2 = 0.04lbf/in 2 As stated previously, OpenFOAM makes use of the Darcy-Forchheimer equation when implementing porous media. The pressure drop in the y-direction is therefore calculated as, p y = f j y 1 2 ρu j u The j-component of the inertial resistance factor is therefore, 32.2ft lbm s 2 lbf f j = 2 p y yρu 2 = 2 6.2lbf/ft ft 2.1lbm/ft3 (15.3ft/s) = ft th TSP The tube support plates provide intermediate support for the SG tubes between the upper and lower tubesheets. The TSPs in the region of interest contain broached holes which allow for flow along the tubes (Figure 5.1). The TSPs are supported by tie rods and are not attached to the SG shroud. The small gap between the outside of the TSP and the shroud can be seen in Figure 1.4a. Crossflow is not possible in this region, so an arbitrarily high inertial resistance factor of is applied. In order to calculate the axial pressure drop across the TSPs, the total available broached hole area must be calculated first. Rather than discretizing the complex geometry of the broached hole into shapes which can be used to calculate the area, a method is developed which takes advantage of the scaling and digital nature of the figure. By knowing the distance between two points in an accurately scaled drawing (The original.png source of Figure 5.2), a constant pixel per length can be derived. In this case, the length of 0.32 inches is used as the base measurement. This length is traversed by 173 pixels, resulting in a pixels inch 62

79 5.1 Mesh Generation Figure 5.1: Broached TSP Hole [4] Figure courtesy of AREVA NP Inc. Figure 5.2: Broached TSP Hole (accurate scaling) ratio. This ratio is applicable in all directions, meaning that there are pixels inch in the x- and y-directions. The number of pixels inside the broached hole is then counted and converted into an area. This process is performed in a script named getpixel.py, the details of which can be found in Appendix B. The total broached hole area calculated by getpixel.py is, A Broached = 0.484in 2 The area occupied by the tube must then be subtracted in order to find the available 63

80 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES flow area. A BroachedF low = A Broached A T ube ( ) 0.625in 2 A BroachedF low = 0.484in 2 π = 0.177in 2 2 The available flow area is then multiplied by the total number of broached holes. A BroachedF lowt otal = 15, 597tubes A BroachedF low A BroachedF lowt otal = in 2 = in 2 = 19.17ft 2 The wetted perimeter of the broached flow area is also needed to calculate the hydraulic diameter of the flow path. Similar to the wetted area, this value consists of the broached hold perimeter plus the tube wetted perimeter. The broached hole perimeter is also calculated in getpixel.py by counting the number of pixels which outline the broached hole and applying the pixel-to-length conversion factor. Again, the details of this process are discussed in Appendix B. P BroachedF low = P Broached + P T ube P Broached = 2.40in P T ube = π(0.625in) = 1.963in P BroachedF low = 2.40in in = 4.363in The hydraulic diameter of the broached hole is therefore, DH BroachedF low = 4 A BroachedF low P BroachedF low = in in = 0.162in Using an empirical formula developed during a test campaign in a Framatome laboratory, the loss coefficient for flow through broached TSP holes can be calculated as a function of the change in available flow area [43]. The upstream and downstream flow area = A Span = 43.03ft 2 The area ratio, σ = A T SP A Span = = 0.45 K T SP =

81 5.1 Mesh Generation The velocity head through the broached hole is calculated next. In order to calculate the actual steam velocity through the TSP region, the area of the gap between the TSP and the shroud is also needed. From Table 3.1, the outside diameter of the TSP is inches, while the inside diameter of the shroud is inches. A gap = π 4 (118.38in in 2 ) = 168.6in 2 = 1.17ft 2 The total available flow area across the 1.18 inch length of the TSP region is therefore, A T otal = in in 2 = in 2 = 20.34ft 2 The velocity through this region can now be calculated by scaling the span velocity. The velocity head is therefore, u T SP = u Span A Span A T otal u T SP = 15.3ft/s 43.03ft ft 2 = 32.4ft/s V H T SP = 2.1lbm/ft3 (32.4ft/s) ft lbm s 2 lbf = 34.23lbf/ft 2 = 0.24lbf/in 2 The predicted pressure drop across the tube support plate can now be calculated. P = K V H P = lbf/in 2 = 0.15lbf/in 2 = 22.2lbf/ft 2 While the pressure drop was calculated using the actual velocity expected through the broached holes, the resistance factor supplied to OpenFOAM must be calculated using a superficial velocity. This is because the porous region will not actually restrict the area, and so the code calculated velocity in this region will reflect the velocity of the fully open span, 43.03ft 2. Using the Darcy-Forchheimer equation, the j-component of the inertial resistance factor is therefore, 32.2ft lbm s 2 lbf f j = 2 p y yρu 2 = lbf/ft in 12in/ft 2.1lbm/ft3 (15.3ft/s) = ft 1 65

82 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES th Span The aspirator port exists in the 10th span (Figure 2.2). This means that the lower portion of the 10th span experiences a higher mass flow rate than the upper portion which loses some flow through the aspirator port. This difference is expected to have a minor effect on the calculated pressure drop and is therefore simplified by modeling the 10th span as a single region with constant resistance factors reflecting the mass flow rate of the lower portion. Crossflow (x- and z-direction) The crossflow loss coefficient through the 10th span is calculated using the same method as used for the 9th span (Section ). The difference is that the 10th span expects a much higher crossflow velocity since it connects to the downcomer by way of the aspirator port. In order to calculate the pressure drop and inertial resistance in this region, the loss coefficient must first be calculated. The Reynolds number at this location can be estimated by the aspirator velocity through the 4.5 inch aspirator port. to, The total area of the aspirator port is equal A asp = π in 4.5in = in 2 = 11.62ft 2 The average velocity through the aspirator port is therefore, u avg = ṁ ρa = Re av = The loss coefficient is therefore, 220.1lbm/s 2.1lbm/ft ft 2 = 9.02ft/s ( ) 9.02ft/s 4.5in 12in/ft ft 2 = 564, 000 /sec K = 1.30 The pressure drop from the center of the tube bundle to the aspirator port can now be calculated. P = K ρv 2 P = lbm/ft3 (9.02ft/s) ft/s 2 = 3.45lbf/ft 2 = 0.02lbf/in 2 2g 66

83 5.1 Mesh Generation Using the Darcy-Forchheimer equation, the i- and k-component of the inertial resistance factor is calculated as, Parallel Flow (y-direction) 32.2ft lbm s 2 lbf f i = 2 p x xρu 2 = lbf/ft in/ft 2.1lbm/ft3 (9.02ft/s) = ft 1 The pressure drop and loss factor for the vertical, parallel flow in the 10th span is equal to that of the 9th span. This is because of the assumption that the entire 10th span is subject to the mass flow rate upstream of the aspirator port. Therefore, the steam velocities in this region will be the same as that of the 9th span th TSP The 10th tube support plate is geometrically the same as the 9th. The only difference is that the wedges that block some of the gap between the TSP and the shroud are slightly different in size. The wedges are not included in this analysis because of their small size and negligible impact on the total available area. Therefore, this analysis will treat the 10th TSP as if it were identical to the 9th by applying the same loss factor of 29.6 ft 1. The expected pressure drop across the 10th TSP can then be calculated using the outlet superficial velocity of 12.8 ft/s. p y = f j 1 2 yρu2 p y = 29.6ft in 12in/ft 2.1lbm/ft3 (12.8ft/s) ft lbm s 2 lbf p y = 15.6lbf/ft 2 = 0.11lbf/in 2 Table 5.1: EOTSG TSP and Tube Nest Porosity Inputs Region Loss Factor, F ijk (ft 1 ) Pressure Drop, P (lbf/in 2 ) 9th Span (0.9, 0.258, 0.9) ( , 0.04, ) 9th TSP (1E3, 29.6, 1E3) (N/A, 0.15, N/A) 10th Span (0.272, 0.258, 0.272) (0.02, 0.04, 0.02) 10th TSP (1E3, 29.6, 1E3) (N/A, 0.11, N/A) 67

84 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES Pressure Drop Verification and Validation Proper implementation of the loss factors presented in Table 5.1 can be verified by examination of the pressure distribution in the results of this case. The pressure drops can be validated against the experimental values found by the ONS start-up tests [5]. The pressure data was extracted from Case2 along a vertical line at the interior of the tube bundle. Figure 5.7 shows the pressure distribution. These results can be used to verify the intended pressure drops in Table 5.1. The 9th span was intended to produce a pressure drop of 0.04 psia. The results of Case2 show an inlet pressure of 944 psia and an exit of , a difference of 0.04 psia. The location of the 9th TSP is made clear by the sharp pressure drop in Figure 5.7. The code-calculated pressure drop is approximately 0.16 psia, within the uncertainty of the intended value of 0.15 psia. The pressure at the inlet of the 10th span is approximately psia while the exit is at psia, giving a difference of psia. The target pressure drop across this region was 0.04 psia. Given that the target value was calculated using a constant velocity in a region which undergoes a velocity change once steam is drawn through the aspirator, the small difference between values is expected. Because the difference is small and because the loss factor for the 10th span can be expected to be the same as the 9th span, the value of 0.258ft 1 is acceptable for use. The 10th TSP induces a pressure drop of approximately 0.1 psia, a value close to the target of 0.11 psia. The resulting pressure drops calculated by poroussimplefoam verify that the porosity model was implemented correctly. Pressure measurements from the ONS-1 OTSG [5] are used to validate the pressure drops calculated in Table 5.1. Figure 5.4 presents the pressure tap labels for the ONS OTSG, while the elevations are found in the source drawings [44][45]. All of the SG pressure taps used at ONS were differential pressure transmitters designed to monitor local static pressure. The uncertainty associated with these transmitters is not provided. While the ONS start-up tests were performed using OTSGs, the data can still be used by this analysis to ensure that reasonable pressure drops are being calculated. The pressure drop from the upper P 7 pressure tap to the upper P 8 tap is used for comparison. These taps are both in the tube nest region and are located near 68

85 5.1 Mesh Generation Figure 5.3: Axial Pressure Distribution in the Tube Bundle Region the inlet boundary of this analysis and just below the 10th TSP. The pressures were measured at 75% and 85% of full power, so the value at 80% must be interpolated (Table 5.2). The pressure drop measured for ONS at 80% FP from the top of the 8th TSP to the bottom of the 10th TSP is approximately 0.4 psid. For this same region, Case2 calculates a pressure drop of = 0.23psid. While this is almost half of the experimental value, it provides a reasonable level of validation. The EOTSG can be expected to produce a lower pressure drop due to the thinner TSPs, and the uncertainty of the data collected by ONS is unknown. Table 5.2: OTSG Tube Nest Pressure Data [5] 80% FP (interpolated) P 7 (inh2o) P 8 (inh2o) Difference (inh2o) 11.1 Difference (psid)

86 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES Figure 5.4: ONS Pressure Taps Figure courtesy of AREVA NP Inc. 70

87 5.2 Inputs 5.2 Inputs The boundary and initial conditions, fluid properties, and turbulence model used for Case2 are the same as those used by Case1, with the exception of the porosity inputs described in the previous section and summarized in Table Solver Description poroussimplefoam is used to solve Case2. This solver uses the same methodology as the simplefoam solver, with the addition of porous terms to the momentum equation. For the steady-state solver, this is done through the addition of a sink term to the momentum equations [42]. t (ρu i) + u j (ρu i ) = p + µ τ ij + S i x j x i x j where S i is the Darcy-Forchheimer equation, S i = (µd ij + 12 ) ρ u kk F ij u i where the first term is the viscous loss term and the second is the inertial loss term. The combination of these two losses result in a pressure drop proportional to the velocity and velocity squared; however, as stated previously, only the inertial loss term is used in this analysis. The inputs to these loss terms are provided in the /constant/porouszones file. 5.4 Results The purpose of Case2 is to examine the influence of the tube nest and tube support plates on the steam flow field in the aspirator region. Comparisons between the results of Case2 and Case1 will therefore be made, as well as comparisons against the PORTHOS calculated results [9]. Case2 converged after 345 iterations (Figure 5.5). The total continuity error was less than (Figure 5.6). 71

88 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES The influence of the tube support plates and tube bundles can be seen in the pressure and velocity contours of Figure 5.7 and Figure 5.8, respectively. There is a gradual pressure drop across the 9th span, followed by a sharp drop across the 9th TSP. The same behavior occurs over the 10th span and 10th TSP. The velocity distribution in the tube nest region is significantly different from Case1 (Figure 5.8). There is now an increased velocity in the open region between the tubes and shroud. The steam can also be seen to jet through the small gap between the TSP and shroud and gradually increase before being drawn into the aspirator port. The crossflow velocity profile through the aspirator port is compared against that of Case1 in Figure 5.9. Beginning at the inlet of the 10th span, Case2 shows a positive crossflow where Case1 did not. The source of this low elevation crossflow is the steam being drawn from the dense tube nest into the open gap between the outermost tube and shroud wall. The positive velocity indicates that the steam is always being drawn into the gap. Looking again at Figure 5.8, this would indicate that the continual increase of the velocity with increasing elevation in the 10th span is caused by the increasing availability of steam, rather than the flow of steam from the gap into the tube nest. The peak of the velocity profiles is similar, with Case2 being slightly less than Case1 but at the same elevation. Case2 shows a greater negative flow out of the upper portion of the aspirator. This upper location is a dead-end notch that seems to develop a recirculating pattern of flow. The fact that the velocity at this location is negative does not indicate that there is steam being drawn from the downcomer into the tube nest. It is simply caused by the outflow of the steam that was drawn into the dead-end notch. Finally, the greater negative flow rate out of the notch causes a higher positive flow rate back into the gap near the top of the 10th span. This is consistent with the other results that show a tendency for steam to be drawn into the gap and away from the tube nest. The results of Case2 provide some insight into the unexpected results calculated by PORTHOS and discussed previously. Because Case2 models the tube bundles and tube support plates, it serves as a better comparison against the PORTHOS results which also model these regions through the use of distributed resistance. The non-zero crossflow at low elevations in the 10th span can be attributed by the tendency for steam to draw into the open gap between the tube bundle and the shroud wall. However, the difference between the Case2 and PORTHOS crossflows in this region are substantial. 72

89 5.4 Results Case2 calculates a maximum velocity of approximately 0.25 ft/s, while PORTHOS calculates a velocity between 3.5 and 4.5 ft/s. The results of this analysis indicate that while some low elevation crossflow can be expected, such high velocity crossflow at a location 1.7 inches from the shroud wall is unrealistic. PORTHOS did not model the notch in the shell wall just above the aspirator port. Given that this was the source of negative crossflow velocities calculated by Case2, it is odd that PORTHOS would also calculate negative velocities at this elevation and above. The PORTHOS curves seem to indicate that steam is being drawn out of the open gap and into the tube nest at these upper elevations, a result that contradicts previous findings. While the inclusion of the tube nest and tube support plates altered the flow path in the tube region, it had little impact on the velocity profile through the aspirator port. 73

90 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES Figure 5.5: Case2 Residuals Figure 5.6: Case2 Continuity Errors 74

91 5.4 Results Figure 5.7: Case2 Pressure Distribution Figure 5.8: Case2 Velocity Magnitude 75

92 5. PHASE II - INCLUSION OF TUBE BUNDLES AND SUPPORT PLATES Figure 5.9: Case2 Crossflow Velocity Comparison 76

93 Chapter 6 Phase III - Inclusion of MFW Nozzle Geometry Case3 uses the same model used in Case2, but incorporates the MFW nozzle geometry. This is done to assess any effects that the nozzle might have on the upstream aspirator velocity profile. It is expected that the nozzle will block the centerline flow path (z = 0) and will result in higher aspirator velocities occurring further from the centerline. 6.1 Mesh Generation The MFW nozzle geometry is modeled by removing cells from the mesh produced by Case2. The MFW nozzles penetrate the SG shell wall such that the top of nozzle is in line with the bottom of the aspirator port. The EOTSG MFW nozzle dimensions are found in the design drawing [46]. The modeled MFW nozzle is simplified in that it is modeled as a rectangular block without any rounding of the edges. A region to represent the MFW nozzle is first created by using the OpenFOAM tool toposet. This tool is able to select all cells within a boundary and group them into a set named mfwnozzle. The cells that comprise mfwnozzle are then subtracted from the Case2 geometry by using the OpenFOAM tool subsetmesh. 77

94 6. PHASE III - INCLUSION OF MFW NOZZLE GEOMETRY subsetmesh -overwrite mfwnozzle -patch shell The cells are subtracted from the mesh and the new boundary is attached to the shell patch. The resulting mesh contains 564,260 hexahedral cells. The geometry is shown in Figure 6.1. Figure 6.1: Case3 Geometry 6.2 Inputs Case3 uses the same inputs (boundary conditions, fluid properties, and turbulence model) as Case Solver Description As with Case2, Case3 uses the poroussimplefoam solver. 78

95 6.4 Results 6.4 Results Case3 converged after 372 iterations. The total continuity error was approximately 0.06 lbm (Figure 6.3). Figure 6.2: Case3 Residuals Figure 6.3: Case3 Continuity Errors 79

96 6. PHASE III - INCLUSION OF MFW NOZZLE GEOMETRY Figure 6.4 shows a slice through the aspirator port normal to the x-axis. The slice is at a radial position of 4.94 ft, which is directly at the aspirator inlet. The plotted values represent the difference between the x-direction velocities of Case3 and Case2. Positive values indicate that Case3 has a higher velocity into the downcomer, while negative values indicate the opposite. It was expected that the addition of the MFW nozzle would cause increased flow in z-locations less than -0.2 ft and greater than 0.2 ft. The results seem to indicate this is true, particularly at the top of the port. Additionally, flow is reduced in the lower, center region of the port where the nozzle closely resides. Figure 6.4: Crossflow Velocity (Case3 - Case2 ) Figure 6.5 presents a comparison of the streamlines in the aspirator port at z = 0 between Case3 and Case2. It can be seen that the MFW nozzle in Case3 is deflecting steam in a recirculating pattern, back toward the aspirator port, effectively lifting the incoming flow. This appears to be the primary cause of the reduced flow rate at the lower, central location of the aspirator port, as well as the increased flow rate in the upper half of the port, as shown in Figure 6.4. Similar to the previous cases, comparisons of the crossflow velocity at the centerline (z = 0) are made in Figure 6.6. The comparison plot shows negligible difference above and below the aspirator port, and a slightly lower crossflow predicted by Case3 in the aspirator port. Given the results shown by Figure 6.4, the lower velocity at the 80

97 6.4 Results Figure 6.5: Velocity Streamline Comparison Case3 (left) vs. Case2 (right) centerline is expected, while there exists a higher velocity ( 0.6f t/s) further from the MFW nozzle. 81

98 6. PHASE III - INCLUSION OF MFW NOZZLE GEOMETRY Figure 6.6: Case3 Crossflow Velocity Comparison Although the MFW nozzle has little affect on crossflow velocity through the aspirator port, the streamlines make clear that the flow pattern changes significantly. With the implementation of an explicit spray and condensation model, this could play an important role. The increased vorticity caused by the MFW nozzle would likely alter the mixing between vapor and liquid, altering the rate of condensation, and ultimately changing the aspirator crossflow. 82

99 Chapter 7 Phase IV - Solution Improvement Unlike the previous phases, in which each phase consists of a single case, Phase IV contains five cases. This section aims to calculate a more accurate solution by incorporating experimental data and insights into the models of the previous phases. In order to better understand the fluid flow behavior in the aspirator port, a better characterization of the MFW spray and its effects on the aspirated flow is required. First, Phase IV develops a model based upon the void fraction data of the ONS start-up testing [5][47]. A two-phase Euler-Lagrangian model is then developed in this section in order to explicitly simulate the interactions between the MFW spray and the surrounding steam. This two-phase model is developed in portions. The base model is created in Case5, a droplet collision model is then included in Case6, followed by the addition of a breakup model in Case7, and finally a condensation model is developed and applied in Case Case 4 - Void Fraction Model This section makes use of prior research performed on the OTSG downcomer properties including pressure, density, and void fraction distributions [47]. If there were only enough aspirated steam to bring the MFW to saturation conditions, the void fraction, 83

100 7. PHASE IV - SOLUTION IMPROVEMENT α, at the downcomer exit would be equal to zero. This is the assumption applied to the previous cases of this analysis. Keep in mind that because MFW spray is not explicitly modeled, the downcomer (DC) exit in the CFD model should reflect the thermal-hydraulic conditions of the actual DC exit. The model assumes that all of the mixing is done by the time the DC exit is reached. According to Clark [47], at 80% full power, the steam quality, χ, is equal to at inches above the upper face of the lower tubesheet. Assuming there has been sufficient mixing throughout the downcomer so that the liquid and steam are in thermal equilibrium, this would indicate that excess steam is aspirated, bringing the feedwater to saturation and drawing an additional 0.03 lbm of steam for every pound of total downcomer fluid flow. χ = m g m g + m f 0.03 This means that the enthalpy of the downcomer fluid no longer reflects that of saturated liquid. Instead, the enthalpy is calculated as follows. h D = 0.97h L h V A downcomer exit pressure of 944 psia is used to calculate the downcomer enthalpy. Although this pressure will be higher at lower elevations due to the increased hydrostatic head, the increase in pressure will increase the growth of h L and h V at approximately the same rate over the range of pressures that could exist at the downcomer elevation. h D = Btu/lbm Btu/lbm = 553.6Btu/lbm The aspirated flow necessary to generate a downcomer enthalpy of Btu/lbm can now be calculated. ṁ MF W h MF W + ṁ Asp h Asp = (ṁ MF W + ṁ Asp ) h Downcomer ṁ Asp = ṁ MF W h MF W h D h D h Asp ( )Btu/lbm ṁ Asp = 1160lbm/s ( )Btu/lbm = 262.8lbm/s 84

101 7.1 Case 4 - Void Fraction Model The resulting aspirator steam mass flow rate is 19% ( 262.8lbm/s 220.1lbm/s the flow rate required to saturate the MFW. v asp = ṁasp ρ A D = 262.8lbm/s 2.1lbm/ft ft 2 = 4.74ft/s = 1.19) greater than With a total MFW flow of 1160 lbm/s, the recirculation ratio can be calculated by, R = ṁ Asp ṁ Asp + ṁ MF W = 262.8lbm/s 262.8lbm/s lbm/s = 18% The boundary conditions used by Case4 are summarized in Table 7.1. Table 7.1: Case4 (Void Fraction Model) Boundary Conditions Boundary Velocity (ft/s) Pressure (ft 2 /s 2 ) Inlet v = Outlet 1 (SG outlet) 12.8 ft/s P = 0 Outlet 2 (Downcomer outlet) ft/s P = Results The results of Case4 ultimately rely on the accuracy of the pressure measurements taken during the ONS start-up testing, as these are the source of the vapor quality used to calculate the boundary conditions. Case4 converged after 361 iterations. The total continuity error was approximately 0.1 lbm (Figure 7.2). 85

102 7. PHASE IV - SOLUTION IMPROVEMENT Figure 7.1: Case4 Residuals Figure 7.2: Case4 Continuity Errors 86

103 7.1 Case 4 - Void Fraction Model Similar to the previous cases, comparisons of the crossflow velocity at the centerline (z = 0) are made in Figure 7.3. Above and below the aspirator port, the crossflow velocities of Case3 and Case4 are nearly identical, as expected. At the aspirator port itself, Case4 shows a greater velocity, peaking at approximately 6.2 ft/s, compared to the 5.5 ft/s of Case3. In the dead-end notch just above the aspirator port, Case4 shows a less negative velocity than Case3. Figure 7.3: Case4 Crossflow Velocity Comparison Case4 is developed from the experimental results of ONS start-up testing. The higher crossflow velocities of this case indicate that the assumptions implicit in the prior cases lack a driving force that significantly alters the results. Those earlier cases apply a Dirichlet boundary condition at all three boundaries out of necessity. Because the MFW is not explicitly modeled, the driving forces caused by the interaction of liquid and vapor cannot be captured, and must instead be accounted for through the choice in downcomer outlet boundary condition. Case4 still makes an assumption about the interaction between liquid and vapor, but the assumption is more reliable in that it is 87

104 7. PHASE IV - SOLUTION IMPROVEMENT derived from experiment. In order to confirm and better support the results of Case4, the subsequent cases will model MFW explicitly. This way, the forces most important to the problem (drag and gravity) can be modeled, and the results can be compared to that of Case4. 88

105 7.2 Case 5 - Lagrangian Spray 7.2 Case 5 - Lagrangian Spray Case5 and the subsequent cases use a Lagrangian particle tracking method of modeling interaction between the main feedwater and the aspirated steam. The feedwater is treated as a Lagrangian field and the steam as an Eulerian field. The OpenFOAM software package includes a solver known as sprayfoam, which serves as a transient, PIMPLE solver for compressible, turbulent flow with reacting Lagrangian parcels. The solver also includes classes to easily model a parcel injection source. It is capable of calculating evaporation of the fluid parcels; however, it is not capable of modeling droplet nucleation and growth by way of condensation. Case5 makes use of the sprayfoam solver, without the effects of condensation. This case serves to demonstrate the momentum transfer from the liquid droplets to the aspirated steam by way of drag only. Selecting appropriate boundary conditions for this problem is a challenge. The steam velocity at the SG exit is well known, and the operating pressure can be reasonably ascertained at various elevations. The problem arises in defining a boundary condition at the downcomer exit. Without a Dirichlet type downcomer exit boundary condition, a unique solution to the problem does not exist. The solution can be satisfied with an inlet mass flow rate of 100 lbm/s or 1,000 lbm/s, it only requires that the aspirated flow increase, which is caused by a decrease in the downcomer exit pressure. A steam velocity cannot be specified without making assumptions about the amount of mass and heat transferred between the aspirated steam and MFW. In Phases I through III, it was assumed that the aspirated steam was exactly enough to raise the MFW temperature to saturation conditions. In Case 4, a similar assumption was made, but instead of assuming saturation conditions at the downcomer exit, a void fraction was used to derive a downcomer enthalpy, which could then be used to calculate a velocity. A pressure boundary can be specified at the downcomer exit, but results are highly sensitive when pressure is specified at both an inlet and exit. To use this approach, experimental pressure measurements would be needed to serve as a basis for the boundary condition, and while pressure measurements for the OTSG were made during ONS 89

106 7. PHASE IV - SOLUTION IMPROVEMENT start-up testing [5], the measurements are not accurate enough to produce reliable results. By taking a different route to the same elevation, a different pressure measurement can be calculated, indicating that the various taps were of different quality and accuracy. For example, beginning at the upper face of the lower tube sheet, a pressure difference to the downcomer at the elevation of the 7th TSP can be found by, 6in + P 9 + P 3 Alternatively, it can be found by, 12in + P 8 P 10 However, depending on the power level, the resulting dp of these two paths can differ by as much as 15 inches of water ( 3500P a). Defining a constant temperature at the downcomer exit is possible, but also has its challenges. The ONS start-up tests recorded temperature at a single location in the downcomer which was approximately 6 ft below the MFW nozzle. The report makes clear that in all tests, the thermocouple measured saturation temperature at the local pressure. While this information is certainly helpful, it is insufficient. It is possible that saturation temperature was reached prior to contacting the thermocouple. If this were the case, it would mean that applying a saturation temperature at the 6 ft elevation would underpredict the aspirator flow rate, as more steam would be needed to heat the liquid drops to saturation in a shorter period of time. Additionally, the depth of the thermocouple penetration is not clear. If the measuring junction was located close to the shell wall, then it may only indicate the temperature of the unmixed steam rather than the liquid spray, which would clearly be at saturation temperature at all power levels. Given the condition of the existing experimental data, this analysis will present a spectrum of possible solutions, able to be validated once more detailed downcomer data is made available. A spectrum of solutions will highlight the potential steam velocities through the aspirator port and clarify whether or not the region is likely to experience flow induced vibration. The spectrum will vary pressure as the outlet boundary condition because reasonable approximations can be made with regard to this property. Section describes the chosen boundary conditions in more detail. 90

107 7.2 Case 5 - Lagrangian Spray Mesh Generation The mesh used by Case5 and the subsequent Lagrangian cases is the same as that used by Case4 ; however, all inputs, including geometry coordinates, have been converted to SI units. This was done for two reasons: the parcel density reported by OpenFOAM when using USCS units appeared to be in SI units, which brought in to question whether or not the correct unit conversions were being made during code calculation. Also, the use of SI units allowed for a more seamless integration of the freesteam water property code in Case8 [48]. Case5 does not include porous regions to represent the tube bundle and tube support plates. The results of Case2 indicate that these regions do not significantly affect the aspirator flow rate or velocity profile. However, the pressure distribution throughout the domain will be altered. This is dealt with by incorporating the approximate pressure drop that would be caused by these regions into the boundary condition calculation. This calculation is performed in Section Inputs Many of the inputs to Case5 are the same as used in previous cases. However, with the addition of the energy equation and a second phase, additional properties must be defined. The file structure is as follows: 91

108 7. PHASE IV - SOLUTION IMPROVEMENT Root Directory/ 0.org/ alphat epsilon H2O k mut N2 O2 p T U Ydefault chemkin/ chem.inp constant/ chemistryproperties combustionproperties g polymesh/ blockmeshdict radiationproperties RASProperties spraycloudproperties thermophysicalproperties turbulenceproperties makefaceset.setset system/ controldict createpatchdict_cyclics createpatchdict_mfwinlet decomposepardict fvschemes fvsolution toposetdict 92

109 7.2 Case 5 - Lagrangian Spray The following subsections will address the important parameters associated with each sub-directory of Case Boundary and Initial Conditions The boundary and initial conditions are defined in the files under the 0.org directory. The U, k, and epsilon files are the same as those used in the earlier phases of this analysis. The mut is the same as the nut file, but used by the compressible k-epsilon turbulence model. The pressure definition file, p, has been modified to assign a Dirichlet pressure condition at the downcomer exit. The other files in 0.org are new. Because the sprayfoam solver includes the energy equation, the temperature boundaries must be defined in the T file. This case assigns a constant temperature of K at the inlet and SG exit, while the downcomer exit is set to a zero gradient condition. In reality, the steam temperature would be greater at the SG exit than it is at the inlet, but since the heat transfer from the reactor coolant system isn t modeled, this is not the case. The H2O, N2, and O2 files define the mass fraction of each respective molecule. This analysis uses 1.0 for the H2O and 0.0 for the N2 and O2. Alphat defines the turbulent thermal diffusivity of the continuous phase. The initial Prandtl number (P r t ) is specified in alphat as 0.92 based on experimental data for turbulent water flows at high temperature [49]. The thermal diffusivity is then code calculated by the k-epsilon model using, α t = µ t P r t Ydefault defines the mass fraction of those species declared in chemkin.in that do not have an associated input file in the 0.org directory. Since this case is only modeling water, the Ydefault values are set to zero. The DC exit pressure could be provided by experimental data, but as mentioned in Section 7.2, the only available data appears inadequate. To demonstrate the problem, the pressure boundaries will be calculated based upon the ONS start-up data. The pressure tap associated with P 7 can be used as the inlet pressure of this analysis since it is located just above the 8th TSP (Figure 5.4). The pressure at the downcomer exit can be approximated by calculating the difference between the tap at P 7 and P 4. In order to do this, an unbroken chain of measurements must be used to get 93

110 7. PHASE IV - SOLUTION IMPROVEMENT from one tap to the other. Beginning with the pressure tap at P 4, the P 9 pressure is added, then 6 inches of water are subtracted in order to reach the bottom tap associated with P 7, and finally the measurement of P 7 is subtracted. The interpolated pressure measurements at 80%FP are provided in Table 7.2. Table 7.2: OTSG Tube Nest and Downcomer Pressure Data [5] Pressure Difference in 80%FP (interpolated) inches of std water P P P P The pressure difference from the upper face of the lower tubesheet to the DC pressure tap is = 97.6inH2O. The additional six inches of water accounts for the distance between the lowest pressure tap and the top face of the lower tubesheet. Similarly, for the tube bundle location, the pressure difference is = inH2O, where the twelve inches represents the difference between the tap and the upper face of the lower tubesheet. The pressure drop in the downcomer region must be adjusted for the elevation difference between P 4 and P 7. The difference in elevation between these two taps is 30 inches (350in - 320in) [45]. The pressure at the downcomer exit boundary can be approximated by calculating the difference between P 4 and P 3 and weighting the difference such that the value represents 30 inches below P 4. P 4 and P 3 are located inches apart. 30 P 4 P = = A pressure difference between the inlet boundary and the downcomer exit boundary can now be calculated = 16.3 The negative value indicates that there is a greater pressure drop in the tube region, thus, the downcomer region is at a higher pressure. Converting from inches of standard water to psia, ρgh = = 84.76lbf/ft2 = 0.58lbf/in 2 94

111 7.2 Case 5 - Lagrangian Spray According to the experimental data, the downcomer exit should therefore be 0.58 lbf/in 2 (4000 Pa) greater than the inlet boundary, which would certaintly result in reverse flow from the downcomer into the tube nest. As an alternative to using the experimental data, the pressure results of Case3 could be used, however, this would be somewhat arbitrary. The result of doing so would likely increase the aspirated flow rate, which would tempt the conclusion that prior analyses underpredicted the recirculation ratio and the resulting aspirator velocity profile. However, it could be reasoned that the single phase cases do not accurately predict pressure drop because they lack the two-phase interaction in the downcomer. Instead of attempting to pinpoint an exact pressure difference between the inlet and outlet boundaries, and in effect giving false confidence in the results, a spectrum of cases will be run which can be validated by experiment once more reliable data is made available. As is discussed in Section 7.2, because the porous media equation is not solved in this case, the pressure drop created by the tube nest and TSPs must be accounted for through the calculation of the pressure boundaries. For the first of the three subcases, Case5a, the downcomer exit pressure pressure is set equal to 1000 Pa greater than the inlet pressure. Because the resistance produced by the tube nest and TSPs does not exist, what this really represents is a pressure difference of 1000 Pa plus the pressure loss from these regions. Based on the calculation below, this first case therefore represents an actual pressure difference of -590 Pa. In other words, the DC exit is at a pressure 590 Pa lower than the inlet boundary. P = P DC (P in + P porous ) From Table 5.1, the pressure drop from the model inlet to the aspirator port is approximately 1590 Pa (0.23 psia). P a = P a ( P a P a) = 590P a Case5b implements an exit pressure equal to the inlet. P b = P a ( P a P a) = 1590P a 95

112 7. PHASE IV - SOLUTION IMPROVEMENT Case5c implements an exit pressure 1000 Pa less than the inlet. P c = P a ( P a P a) = 2590P a To reiterate, the boundary condition pressure differences of +1000, 0, and Pa are used to simulate what would in reality be measured as -590, -1590, and Pa, respectively. For context, these values can be compared against the results of Case3 and the measurements at Oconee. Case3 calculated a pressure difference of Pa between these two boundaries, placing it between Case5b and Case5c. The pressure measurements at ONS would suggest the use of a pressure difference of Pa, well outside the range chosen here. The results of these three cases will be compared in Section Main Feedwater Spray The MFW spray is modeled using a cone nozzle injection. The properties associated with this model are defined in the spraycouldproperties file inside of the conenozzleinjectioncoeffs variable. At 80% FP, MFW injects at 1160 lbm/s (526.2 kg/s) per SG. This value is then divided by 32 to find the flow rate from a single MFW nozzle, giving 16.5 kg/s. The injection model is set up to inject a specified total mass over a specified total time. The total time chosen is 10 seconds in order to bound the length of the transient, therefore, the total mass input is kg. The disc injection method is chosen, meaning that a circular surface is used as the injection point. An inner and outer diameter must therefore be specified to define the surface. An inside diameter of 0 is used, and the outer diameter is calculated to reflect the same spray area of the MFW nozzle holes. Drawings of the MFW nozzle in the EOTSG [46] show each of the nozzle holes to have a diameter of m. There are 91 holes in each nozzle, resulting in a total flow area of m 2. Since the model groups these holes into one large circle, the diameter of that injecting circle, D inj, must be calculated. D inj = m 2 π = m 96

113 7.2 Case 5 - Lagrangian Spray The outside diameter of the injecting disc is therefore set to m. The position of the injection point is at the bottom face of the MFW nozzle that was created in Case3, which is at x,y,z = (1.67, 1.521, 0). The injection direction is specified as negative in the y-direction. The number of parcels per second is set to 10,000. This chosen value is somewhat arbitrary, but is chosen as a value that strikes a balance between computational cost and numerical accuracy. Clearly, as the number of parcels approaches the number of particles, the model accuracy increases, but so does computational cost. Because the parcel flow in a single, consistent direction in this simulation, it is believed that a coarse representation of the particles is sufficient. The total surface area of the particles is conserved regardless of how many particles there are per parcel. What s lost is the ability to travel in different directions. Since the liquid spray is directed downward at a high velocity, this should not affect the results. The spray discharge coefficient is set equal to 1.0, and the spray angle is set equal to 0 degrees due to the high exit velocity. The droplet diameter is determined by a uniform distribution between a minimum value of 4 mm and a maximum of 6 mm, a range that bounds the nozzle hole diameter of 5 mm [46]. The Ranz-Marshall correlation, Nu = Re 1 2 P r 1 3, is used to model heat transfer between the continuous phase steam and the spherical droplets [50] Solver Description sprayfoam is a transient PIMPLE solver for compressible, turbulent flow with reacting Lagrangian parcels. The PIMPLE algorithm is a combination of the semi implicit method for pressure linked equations (SIMPLE) and pressure implicit with splitting of operators (PISO) algorithms. The general outline of the solver is as follows, 1. The parcels are first generated at the specified injection point. This is done in the call to the evolve function of the SprayCloud class (SprayCloud.C). 2. The parcel cloud is prepared in the solve function of the KinematicCloud class (KinematicCloud.C) followed by a call to the evolve function. The evolve function calls the inject function of the InjectionModel class (InjectionModel.C). The inject function finds the cell associated with the position of the injection point. A parcel 97

114 7. PHASE IV - SOLUTION IMPROVEMENT is then randomly generated within that cell and within the property bounds specified by the user in the constant/sprayproperties file. The parcel is moved and its mass is accounted for. 3. The PIMPLE corrector loop is entered for a set number of iterations, depending on user input in the system/fvsolution file. In this case, only a single iteration is made (nnonorthogonalcorrectors 0). (a) The discretized momentum equation is assembled and solved in UEqn.H in order to find an intermediate velocity field. (b) YEqn.H performs the combustion calculations and resulting mass fractions; however, this step is ignored by this case since combustion is disabled. (c) The discretized energy equation is assembled and solved in hseqn.h. (d) The pressure corrector loop is entered and the pressure equation is solved. The number of iterations made by the pressure corrector loop is also defined in system/fvsolutions. This case makes two iterations (ncorrectors 2). (e) After the pressure corrector loops are complete, the turbulence equations are then solved. This case uses the standard compressible k-epsilon equations. 4. The PIMPLE corrector loop is then completed and the mass fluxes are corrected using the new pressure field. 5. The time step is then incremented and the process repeated. The subsequent Lagrangian cases are run until a steady-state solution is reached. This occurs once the downcomer has filled with liquid droplets and the aspirated flow rate does not change significantly between time steps Results A total of three variations of Case5 were run, varying the downcomer outlet pressure. The resulting figures and curves will be marked based upon this property. 98

115 7.2 Case 5 - Lagrangian Spray Figure 7.4 shows a plot of aspirator mass flow rate versus time for each of the three sub-cases. The transients were terminated at two seconds, a time at which all three cases had reached a reasonably steady state. Figure 7.4: Case5 Aspirator Mass Flow Rate Comparison Case5a resulted in a negative mass flow rate, indicating that the DC outlet pressure being slightly greater than the aspirator port pressure is enough to overcome the drag force of the liquid droplets and produce reverse flow from the downcomer into the tube nest. Had the ONS data been used to apply an exit pressure 4000 Pa (minus the TSP and tube nest losses) greater than the inlet, the reversed flow would be substantially greater. The residuals and continuity errors for each case are presented in the figures below. The values plotted reflect those calculated at the beginning of the PIMPLE loop. The residuals become more oscillatory as the downcomer exit pressure is decreased. This resulted in the need for more iterations in the pressure-corrector loop which is the reason for the increase in total number of iterations. The Case5c continuity error returns to 99

116 7. PHASE IV - SOLUTION IMPROVEMENT zero near the end of the transient. This is because of a code restart that was performed, resetting the cumulative error. 100

117 7.2 Case 5 - Lagrangian Spray Figure 7.5: Case5a Iteration Errors (a) Residuals (b) Continuity Errors Figure 7.6: Case5b Iteration Errors (a) Residuals (b) Continuity Errors Figure 7.7: Case5c Iteration Errors (a) Residuals (b) Continuity Errors 101

118 7. PHASE IV - SOLUTION IMPROVEMENT At two seconds in all cases, the total liquid mass introduced to the system is equal to 32.6 kg, confirming that the 16.3 kg/s MFW flow rate was modeled correctly. The feedwater temperature distribution in the downcomer of Case5b is shown in Figure 7.8. Figure 7.9 calibrates the temperature range with respect to the minimum and maximum feedwater temperatures. This figures also shows the spray flow pattern and droplet diameter. The parcel diameter shown is four times larger than what was modeled, this is done simply to help with visualization. Also, remember that each parcel represents 10,000 droplets. The spray shows little to no spreading throughout all elevations of the downcomer, once again highlighting the importance of the depth of the thermocouple penetration. Figure 7.8: Case5b Feedwater Temperature Distribution 102

119 7.2 Case 5 - Lagrangian Spray Figure 7.9: Case5b Feedwater Temperature Contours The aspirator velocity profile is presented in Figure 7.10 in United States customary system (USCS) units for comparison against previous cases. Since the tube bundles and support plates were not modeled, the velocity profile should be most like that of Case1, in which the elevations above and below the aspirator port show crossflow velocities close to zero. The range of results between the three subcases is significant. With a difference of 2000 Pa between Case5a and Case5c, the calculated crossflow can range from negative values to nearly 14 ft/s, significantly exceeding even the values calcu- 103

120 7. PHASE IV - SOLUTION IMPROVEMENT lated by Case4. This once again makes clear the importance of accurate experimental measurements in validating these results. Figure 7.10: Case5 Crossflow Velocity Comparison Moving forward, only a single case will be analyzed at each stage of model development, as opposed the spectrum of boundary conditions that was applied in Case5. The point has already been made that the results require experimental validation. Rather than reiterating this point, the following sections will serve to show the differences introduced by each of the model changes. Since Case5b produces results most consistent with the previous cases, its boundary conditions will be used for the remainder of this analysis. 104

121 7.3 Case 6 - Coalescence 7.3 Case 6 - Coalescence Case6 begins with the model used by Case5, but introduces the effects of droplet coalescence. This is done using the O Rourke collision model [25] as described in Section Inputs With the exception of the collision model, Case6 uses the same input parameters as Case5. Unlike Case5, Case6 consists of a single transient that applies a downcomer exit pressure equal to the inlet pressure. Results will therefore be compared against Case5b. The O Rourke collision model is enabled in constants/sprayfoamproperties. The coalescence variable is Boolean; either on or off. If the coalescence is turned off, then a bouncing type of collision is calculated, otherwise, the parcels coalesce upon collision. stochasticcollisionmodel ORourkeCoeffs { coalescence on; } ORourke; Solver Description Every droplet has a potential to collide with any other droplet in the system. Therefore, for every N droplets, there are N-1 possible collisions. As the value of N becomes large, the probability of interaction becomes 1 2 N(N 1) = 1 2 N 2. Since droplet i colliding with droplet j is the same as j colliding with i, the probability is cut in half. As stated previously, OpenFOAM works with the concept of parcels, in which droplets are grouped together. This technique can significantly reduce the number of particles modeled, but can still result in a demanding collision probability calculation. This analysis injects 10,000 parcels per second, which would require one-hundred million 105

122 7. PHASE IV - SOLUTION IMPROVEMENT collision calculations every time step. used to significantly reduce this computational cost. Fortunately, the O Rourke algorithm can be The first major improvement in efficiency stems from the fact that the O Rourke collision model requires that two parcels must share the same Eulerian cell in order to be considered as potential collision partners. This feature is implemented in the update member function of the ORourkeCollision class as follows, if ((celli!= cellj) (m1 < VSMALL) (m2 < VSMALL)) { } return false; If the two droplets are not in the same cell or either of the parcels is below the minimum size threshold (user-defined in spraycloudproperties ), the collision calculation is skipped, otherwise a probability of collision is calculated by the ratio of the two parcel volumes as follows, ν = π(d 1 + d 2 ) 2 v 2 v 1 dt 4V The probability distribution of the number of collision is governed by a Poisson distribution, which OpenFOAM implements as, collp rob = e ν If the collision probability is greater than a random, sampled value between 0 and 1, then a collision is assumed to occur, and the code proceeds by calling the collidesorted function. This function ensures the conservation of mass, momentum, and energy upon coalescing two droplets Results The Case6 transient was terminated once the flow rate through the aspirator port had reached a steady state. Figure 7.11 shows a plot of aspirator mass flow rate versus time for Case6 and Case5b. 106

123 7.3 Case 6 - Coalescence Figure 7.11: Case6 Aspirator Mass Flow Rate Comparison Case6 was terminated at two seconds. The residuals are presented in Figure 7.12a. Similar to Case5c, this case was restarted, which reset the cumulative mass error back to zero. The rate of convergence and number of iterations necessary to calculate two seconds of simulation time are similar to that of Case5b, indicating that the addition of the O Rourke collision model added little computational overhead. 107

124 7. PHASE IV - SOLUTION IMPROVEMENT Figure 7.12: Case6 Iteration Errors (a) Residuals (b) Continuity Errors The droplet properties of each of Case5 and Case6 cases is presented in the following table. These values are extracted directly from the output files of each case. The table compares the arithmetic mean diameter, D10, the Sauter mean diameter, D32, and the maximum particle diameter, Dmax. The variations of Case5 produce small, but consistent differences. As the steam mass flow rate through the downcomer increases, the average droplet diameter increases. This small increase is due to the volume expansion resulting from a difference in average parcel temperature. In the case with the lowest relative velocity between droplet and steam, the heat transfer is smallest, the parcel temperature is lowest, and the density is highest. As the relative velocity between phases is increased, the heat transfer increases, which increases the droplet temperature and therefore increases the average droplet diameter. The difference is small, however, especially when compared to the effects of coalescence. The Sauter mean diameter of the Case6 droplets is approximately 1.5 times greater than Case5. This difference indicates that given the same volume of liquid droplets, the Case5 droplets will have a 1.5 times greater surface area. 108

125 7.3 Case 6 - Coalescence Table 7.3: Case6 Droplet Size Comparison Case D10 (µm) D32 (µm) Dmax (µm) Case5a Case5b Case5c Case Using the same method as was used to create Figure 7.8, the feedwater droplet temperature distribution was extracted from the results of Case6. A linear fit was then calculated to the data and compared against that of Case5b (Figure 7.13). Figure 7.13: Case6 Feedwater Temperature Comparison Figure 7.13 shows that the droplets of Case6 are at a lower temperature than Case5 in the higher elevations, and a higher temperature at the lower elevations. The lower temperature at high elevations is because of the lower heat transfer rate of Case6. However, as the droplets of Case6 coalesce with decreasing elevation, there are fewer droplets to absorb the heat, and even though the heat transfer rate is less than Case5b, the droplet temperature is higher. This could be confirmed by examining the heat transfer rate, 109

126 7. PHASE IV - SOLUTION IMPROVEMENT however, this data is not available. Instead, the steam temperature in the downcomer will be presented. Keep in mind that the steam temperature data is artificial. Without a condensation model, the steam is able to decrease to any arbitrary temperature, based upon the heat transfer rate. In reality, the steam temperature would remain saturated, and heat transfer would instead result in a phase change. Nevertheless, the steam temperature data can give insight into the rate of heat transfer. The temperature distribution of the steam phase through the center of the downcomer from the lower face of the MFW nozzle to the downcomer exit is shown in Figure The temperature drops quickly from K (539.7 F) to about 532 K (500 F), then begins to increase slightly as steam exits the central path and enters a re-circulation pattern that extends from the top to the middle of the modeled downcomer. This re-circulation path present in Case6 is shown by a streamline plot in Figure This figure shows three distinct re-circulation zones. The first is above the MFW nozzle, as was seen in Case3 (Figure 6.5), after the introduction of the MFW nozzle geometry. The next two are in the spray region, and while these patterns seem to encourage mixing, they do not appear significant enough to affect the spray flow, as the droplets continue to fall along the same path with little to no spreading. 110

127 7.3 Case 6 - Coalescence Figure 7.14: Downcomer Steam Temperature Comparison The comparison between downcomer temperature distributions in Figure 7.14 confirms that there is a greater heat transfer rate in Case5b. This is expected, given the smaller droplet size (Table 7.3). 111

128 7. PHASE IV - SOLUTION IMPROVEMENT Figure 7.15: Case6 Feedwater Temperature Contours 112

129 7.3 Case 6 - Coalescence Figure 7.16: Case6 Downcomer Temperature Streamlines 113

130 7. PHASE IV - SOLUTION IMPROVEMENT The aspirator velocity profile is presented in Figure 7.17 in USCS units for comparison against previous cases. The shape of the profile for Case6 is quite similar to Case5, it is only the maximum velocity that differs between the cases. Figure 7.17: Case6 Crossflow Velocity Comparison 114

131 7.4 Case 7 - Breakup 7.4 Case 7 - Breakup Case7 introduces the effects of droplet breakup to the Case6 model. This is done using the Pilch-Erdman acceleration-induced liquid droplet breakup model [51] Inputs With the exception of the breakup model, Case7 uses the same input parameters as Case6. The Pilch-Erdman breakup model is enabled in constants/sprayfoamproperties. By default, an option exists to enable or disable the oscillation equation; however, this equation is intended for use with the Taylor Analogy Breakup model, not the Pilch-Erdman model. It was disabled by setting the option to false. The B 1 and B 2 coefficients were set to the values described in Section breakupmodel PilchErdman; PilchErdmanCoeffs { solveoscillationeq false; B ; B ; } Solver Description The Pilch-Erdman breakup model was derived for liquid drops in a higher velocity flow field which is less dense than the droplets [51]. This makes it a suitable choice for use in the OTSG downcomer region. The Weber number, a ratio of the droplet s inertial force to its surface tension, is used to determine which of the five breakup mechanisms is active. W e = ρv 2 D σ Where ρ is the density of the flow field, V is the relative velocity between the flow field and the drop, D is the droplet diameter, and σ is the droplet surface tension. The 115

132 7. PHASE IV - SOLUTION IMPROVEMENT Weber number in the downcomer region can be approximated by estimating a relative velocity. Given a total MFW nozzle injection area of m 2 (Section ) and a liquid mass flow rate of 16.4 kg/s, the droplet velocity is calculated. v d = v d = ṁ ρa 16.4kg/s 842kg/m m 2 12m/s Based on the previous cases of this analysis, the steam velocity entering the downcomer is approximately 1 m/s, resulting in a relative velocity of 11 m/s. The expected Weber number can then be calculated. W e = 33.6kg/m3 (11m/s) m N/m = 96 Referring to the qualitative description of breakup mechanisms provided by Pilch and Erdman, this places the breakup model in range of the bag-and-stamen mechanism, in which a thin bag of liquid is blown downstream while a column of liquid (stamen) is formed along the drop axis parallel to the approaching flow [51]. As the MFW droplets drag against the aspirated steam and begin to equilibrate with one another, the Weber number will decrease, potentially undergoing bag breakup or even vibrational breakup once the velocities approach one another. The relative velocity of 11 m/s also indicates that the incompressible correlation should be used, as it is well below the 0.5 Mach number ( 245m/s) necessary for compressible calculations. Once a Weber number is calculated by the code, the total breakup time is calculated by the appropriate correlation, where the total time is when the drop and its fragments no longer undergo further breakup. For the bag-and-stamen mechanism, total breakup time, τ, is calculated by, τ = 14.1 (2W e 12).25 This time is then used to calculate the droplet velocity, V d. V d = U rel ρc ρ d ( B 1 τ + B 2 τ 2) 116

133 7.4 Case 7 - Breakup Where ρ c is the continuous phase density, ρ d is the droplet density, and the B 1 and B 2 coefficients are defined as, B 1 = 3 4 C d B 2 = 3β Where C d is the droplet drag coefficient, equal to 0.5 for incompressible flow and β is equal to The droplet velocity is used to calculate the distance traveled. 2W e c σ D s = ( ) V d U rel ρc Urel 2 Where W e c is the critical Weber number and σ is the droplet surface tension. Together, the total breakup time and the droplet distance are used to implicitly calculate a new droplet diameter by weighting the droplet distance by the time step. d new = d old + dt τ b D s 1 + dt τ b Finally, the number of particles is corrected to conserve mass, np article = np article d3 old d 3 new Results The Case7 transient was terminated at two seconds, a time after reaching steady state. Figure 7.18 shows a plot of aspirator mass flow rate versus time for Case7, Case6, and Case5b. While the addition of the coalescence model decreased the aspirated steam flow rate, the addition of the breakup model returned the mass flow to that of Case5b, approximately 145 kg/s. With a total MFW flow of 526 kg/s, this equates to a recirculation ratio of 22%. R = ṁ Asp ṁ Asp + ṁ MF W = 145kg/s 145kg/s + 526kg/s = 22% The residuals are presented in Figure 7.19a. Similar to Case6 and Case5c, this case was restarted, which reset the cumulative mass error back to zero. The rate of convergence and number of iterations necessary to calculate two seconds of simulation time are similar to those cases as well. This indicates that the addition of the Pilch-Erdman breakup model added little computational overhead. 117

134 7. PHASE IV - SOLUTION IMPROVEMENT Figure 7.18: Case7 Aspirator Mass Flow Rate Comparison Figure 7.19: Case7 Iteration Errors (a) Residuals (b) Continuity Errors The droplet size comparison table found in Section is repeated here with the addition of the results from Case7. There are two primary observations to be made from these results. The first is that the addition of the breakup model still allows for the 118

135 7.4 Case 7 - Breakup development of large droplets, as the maximum diameter is much closer to that of Case6 than Case5. This is possible because the large droplets develop near the bottom of the spray where the relative velocity begins to decrease. The second observation is of the importance of the Sauter mean diameter. Notice that the arithmetic average diameter for Case7 is significantly less than calculated by any of the Case5 transients. However, the Sauter mean diameter, D32, is much closer to the results of Case5. This indicates that while the arithmetic average droplet diameter may be equal, the distribution of droplet sizes is not uniform. In this case, there must be a greater number of small droplets, which increase the surface area to volume ratio. The droplet size results indicate that the combination of the coalescence and breakup models produces the greatest droplet surface area. 119

136 7. PHASE IV - SOLUTION IMPROVEMENT Table 7.4: Case7 Droplet Size Comparison Case D10 (µm) D32 (µm) Dmax (µm) Case5a Case5b Case5c Case Case This greater droplet surface area in Case7 is apparent in its effect on downcomer steam temperature distribution. A comparison against the previous two cases is shown in Figure At the centerline of the downcomer exit boundary, the steam temperature is approximately 10 F less than that of Case5b. Again, these temperatures are artificial and are only presented to provide a sense of the heat transfer rate. Figure 7.20: Downcomer Steam Temperature Comparison The feedwater droplet temperatures are also compared against the previous cases for use in experimental validation. Figure 7.21 shows the importance of accurate modeling 120

137 7.4 Case 7 - Breakup of the droplet size distribution. The smaller droplets of Case7 undergo significantly greater heatup than the previous cases. Figure 7.21: Downcomer Feedwater Temperature Comparison The feedwater droplet temperature contours are presented in Figure This figure shows the spray flow pattern and droplet diameter, which is scaled four times larger than the modeled diameter. These contours can be compared against the same plot generated for Case6 (Figure 7.15). 121

138 7. PHASE IV - SOLUTION IMPROVEMENT Figure 7.22: Case7 Feedwater Temperature Contours 122

139 7.4 Case 7 - Breakup The aspirator velocity profile of Case3, Case4, Case5b, Case6, and Case7 is presented in Figure The Case7 curve overlaps that of Case5b. As was stated previously, the shape of the curve for the Lagrangian cases differs above and below the aspirator port because the TSPs and tube nest are not modeled. The inclusion of the droplet coalescence and breakup models produces the greatest aspirator velocity, with a maximum crossflow of 6.7 ft/s at 32.1 ft above the upper face of the lower tubesheet. Figure 7.23: Case7 Crossflow Velocity Comparison 123

140 7. PHASE IV - SOLUTION IMPROVEMENT The recirculation ratios of the various cases throughout this analysis as well as those of the historical calculations are summarized in Table 7.5. Table 7.5: Recirculation Ratio Comparison Case MFW Flow Rate (lbm/s) Aspirated Flow Rate (lbm/s) Recirculation Ratio ANO MWt, EOTSG [9] Case Case Case5a Case5b Case5c Case Case

141 7.5 Case 8 - Condensation 7.5 Case 8 - Condensation While the previous cases examined the effects of spray droplet behavior, they did not consider the potential for condensation. It is expected that condensation plays an important role in accurately modeling the downcomer region and subsequently calculating the aspirator flow rate. In this case, the cold feedwater has the potential to remove heat from the incoming steam, dropping the temperature below saturation, and causing a phase change from vapor to liquid. By transforming steam into liquid, the reduction in volume will result in a pressure drop that may, depending on the magnitude, result in additional flow through the aspirator. The condensed liquid droplets would also increase the droplet surface area to volume ratio, increasing the heat transfer between phases, as was seen in Case7. OpenFOAM is not equipped by default to model condensation; therefore, a model is developed in this section and applied to the Lagrangian framework. This new model begins with the sprayfoam solver and makes changes to include the effects of dropwise condensation. This new solver is named spraycondensationfoam Inputs Case8 uses the same inputs (boundary conditions, fluid properties, and turbulence model) as Case7. The maximum Courant limit has been increased from 0.5 to 1.0 to reduce run time. For the same reason, the mesh has been significantly coarsened to a total of 85,700 cells. The minimum condensed droplet diameter has been set to m in order to reduce the number of calls to the condensation function Solver Description The condensation model is implemented primarily in spraycondensationfoam.c and InjectionModel.C. The source code for these files is provided in Appendix E. 125

142 7. PHASE IV - SOLUTION IMPROVEMENT The condensation logic proceeds as follows: at the beginning of each time step, the local temperature of every cell is checked against the saturation temperature of that cell. Since steam saturation properties are not readily available in OpenFOAM, modifications were made to interface OpenFOAM with freesteam [48], a steam table application written in C. If a cell is found to have a temperature below the saturation temperature, then a routine is entered which calculates the Kelvin-Helmholtz critical droplet radius, r, that would be required in order for condensation to occur. Where S = p cell p sat(t cell ) r = 2σ rho f RT cell ln S is the supersaturation ratio. This critical radius indicates the size above which a droplet will grow and below which a droplet will disperse because the local pressure is unable to overcome the energy required to form and maintain the droplet s surface tension. Using this critical radius, the rate of nucleation per unit volume per second, J, is calculated using the classic nucleation equation [52] amended by the nonisothermal correction factor, φ [53]. J = q c 1 + φ ( ρ 2 g ρ f ) ( ) 4πr 2σ 2 σ m 3 π e 3kT cell Where q c is the condensation coefficient, m is the mass of a single molecule, and k is Boltzmann s constant. The nonisothermal correction factor φ is given by, ( ) ( ) 2(γ 1) hfg hfg φ = 0.5 (γ + 1) RT cell RT cell Where γ 1.32 for low-pressure steam in Young s original implementation [53]. The number of nucleations per unit volume per second are then multiplied by the volume of the cell being analyzed and the time step size. The resulting number of droplets is then stored in a single parcel. This is done to reduce computational cost. This number of droplets per parcel is then passed to the condensation function located in InjectionModel.C. This function randomly generates a point inside the cell where the parcel will be created and assigns the parcel a velocity equal to that of the carrier phase. By assuming the droplet velocity is equal to the carrier phase velocity, there 126

143 7.5 Case 8 - Condensation is no exchange of momentum at the time of nucleation. The parcel is then added to the existing spraycloud object and the properties including diameter, velocity, and thermodynamic conditions are set based on a single droplet. The parcel temperature is set to the saturation temperature based on the cell pressure. The parcel mass is calculated by multiplying the droplet mass by the number of particles per parcel, and the result is used to calculate the mass transferred from the vapor to the liquid phase, as well as the enthalpy transfer. if (td.cloud().solution().coupled()) { scalar dm = -(np0 * mass0); td.cloud().rhotrans(gid)[celli] += dm; // Mass td.cloud().hstrans()[celli] += dm*hs; // Energy td.cloud().phasechange().addtophasechangemass(dm); } The mass and energy transfer is stored in the spray cloud object named td which is incorporated into the conservation equations through the source term. The implementation of this condensation code can be considered preliminary. There are many aspects which could be greatly improved, most noticeably is the issue of performance. By forcing the code to loop through every cell in order to check for condensation conditions, a large CPU overhead is created. A better method may be to predefine a region of cells which can support condensation, similar to the way that porous zones are defined. Additionally, the spraycondensationfoam solver does not support parallel execution. The loop over every cell accesses the td.cloud object and will quickly cause errors when different nodes attempt to read or write to that object simultaneously. Additionally, the number of parcels created by the condensation routine can be quite large, even with the arbitrary limit of 1 parcel per cell per time step. The additional parcels significantly increase computation time when compared against the sprayfoam solver. It is for these reasons that the model used by Case8 is significantly less detailed than the previous cases. The mesh has been coarsened from a cell count of 564,260 in Case7 to 85,700 in Case8. Clearly, the accuracy with respect to turbulence modeling is lost and the results of the GCI analysis performed in Section are no longer applicable. Case8 does however still provide insight and serve as a foundation 127

144 7. PHASE IV - SOLUTION IMPROVEMENT for future work and further development of a Lagrangian condensation model for the OpenFOAM CFD code Results In its current state, the spraycondensationfoam solver is impractical. Even with the measures taken to increase solution speed, a week of wall-clock time only produces 0.2 seconds of simulated time. Between the cost of performing the calculation in series, the number of calls that are made to the condensation function, and high number of parcels that must be tracked, the time necessary to reach a steady state for this problem cannot be achieved in any reasonable amount of time. This section will therefore present the results as they relate to the implementation of a condensation model, rather than making comparisons to previous cases. The results presented below reflect the solution at 0.8 seconds. Figure 7.24: Case8 Aspirator Mass Flow Rate Comparison 128

145 7.5 Case 8 - Condensation In the earlier time of the transient, the mass flow rate through the aspirator port is negative (Figure 7.24). This causes the liquid spray to remove heat from the tube bundle region and reduce the local temperature below saturation. With the condensation model enabled, this results in a significant number of parcels being created through the tube nest. This causes the calculation to slow down tremendously for parcel interactions that have no significance in the steady state solution. Attempts were made to avoid this problem by mapping the final solution from Case7 to the initial state of Case8, but this resulted in tens of thousands of condensed parcels being created with each time step, causing a discontinuity that soon crashed the solver. At the time the transient was terminated, the mass flow rate had not yet reached a steady-state. Figure 7.25: Case8 Parcel Distribution The parcel distribution at 0.8 seconds is shown in Figure The condensed parcels appear to encourage more spreading of the liquid droplets, which is beneficial for mixing and heat transfer. The figure also confirms that the condensed parcels are at saturation temperature, while the sprayed parcels are at 494 K (430 F). The condensed parcels are significantly smaller than the sprayed parcels, primarily on the order of 10 8 to 10 6 m, as shown by Figure It was expected that these small parcels would increase the average droplet surface area, resulting in increased aspiration. Table 7.6 compares the droplet sizes between the previous Lagrangian cases. The results show a smaller average droplet size, but the Sauter mean diameter is slightly increased. This is likely due to the fact that the solution has not reached a steady condition, but it does indicate that the surface area to volume ratio is similar 129

146 7. PHASE IV - SOLUTION IMPROVEMENT to the previous cases. This may be due to the small condensed fluid mass relative to the sprayed liquid mass. Figure 7.26: Case8 Parcel Size 130