Linear analysis of a natural circulation driven supercritical water loop

Transcription

1 TU Delft Bachelor Thesis Linear analysis of a natural circulation riven supercritical water loop D J van er Ham supervise by Dr. Ir. M. Rohe July 3, 216

2 Nomenclature Symbol Units Description A m 2 Length A w m 2 Wall-cross-sectional area C i m 3 Precursor concentration m Hyraulic iameter E f J Energy per fission event H J kg 1 Specific enthalpy K Pressure loss coefficient L m Length without subscript: total core length M g mol 1 Molar mass N m 3 Number ensity N A mol 1 Avogaro constant P in m Perimeter Q J s 1 Core heating power Qw J s 1 Fuel-to-wall heat flow R Reactivity T K Temperature V m 3 Volume V f m 3 Fuel volume W kg s 1 Mass flow rate c p J kg 1 K 1 Flui specific heat c p,w J kg 1 K 1 Wall material specific heat f Darcy friction factor g m s 2 Gravitational acceleration h J kg 1 Specific enthalpy n m 3 Neutron concentration p N m 2 Pressure t s Time v n m s 1 Neutron velocity Λ s Mean generation time Σ f m 1 Macroscopic neutronic cross-section for thermal fission α r m 3 kg 1 Density reactivity feeback coefficient β Delaye neutron fraction ɛ Enrichment θ K Temprature pertubation λ Eigenvalue λ f J s 1 m 1 Flui thermal conuctivity K 1 λ i s 1 Precursor ecay constant i

3 ii λ w J s 1 m 1 K 1 Wall thermal conuctivity µ N s m 2 Dynamic viscosity ρ kg m 3 Density ρ w kg m /3 Wall material ensity σ f m 3 Microscopic neutronic cross-section for thermal fission τ s Fuel heat transfer time constant Dimensionless numbers N fr N sub N h N u Froue number Subcooling number Pseuo phase change number Nusselt number Subscripts Value at: Bottom core noe 1 Top core noe B Buffer vessel noe D Downcomer noe F Fuel noe R Riser noe pc Pseuo-critical point w Wall noe low heating moel w, Top wall noe high heating moel w, 1 Top wall noe high heating moel Other X X ˇx Nu W.66 m.66 K.66 Commom abbreviations BWR DR DWO GIF HPLWR SCWR Steay state variable Dimensionless variable Pertubation on variable Ajuste Nusselt number Boiling Water Reactor Decay Ratio Density Wave Oscillations Feneration IV International Forum High Performance Light Water Reactor Supercritical Water-coole Reactor

4 Abstract The High Performance Light Water Reactor HPLWR is a European reactor base on the Generation IV Supercritical Water-coole Reactor SCWR concept. This reactor is esigne to be more environmentally frienly an more efficient. Furthermore, it can be esigne to rely on natural circulation as its riving force, eliminating active pumps thus making it more safe. This oes cause the reactor to be unstable uner certain operating conitions. This thesis will use an exten an existing moel to investigate an better unerstan these instabilities by looking at them uner ifferent parameters as well as by analyzing the frequency an time epenent behavior of the perturbations on the systems variables. In this moel the reactor is reuce to several noes. Conservation balances are set up an linearize for all noes. The remaining equations are written as a matrix equation, reucing the problem to an eigenvalue problem. The steay state solutions are foun through an iterative process an the stability is etermine for all operating conitions. The results can then be plotte on an instability map Through a parametric stuy in which the hyraulic iameter an length of the riser were varie simultaneously, it was foun that for a small hyraulic iameter increasing the length makes the system less stable. For a high hyraulic iameter, occurrence of instabilities associate with gravity increase, but those associate with friction ecrease. The resonance frequency was foun to be almost constant on a line that passes through the origin of an instability plot. Using this result, a linear relationship was foun between the resonance frequency an the imensionless ensity an frequency of water the part of the core where its subcritical. A time epenent solution for the problem was foun. Using this solution friction prove to be a bigger factor in the perturbation of the solution than gravity for most operating conitions, particularly in the riser. The frequency of all perturbations for constant parameters was foun to be the same for ifferent perturbations. iii

5 Contents 1 Introuction Backgroun of Research Supercritical Fluis High Performance Light Water Reactor Instabilities in the reactor Previous research Thesis outline Physical moel HPLWR in noes Assumptions an simplifications Balance equations Metho of stability investigation Matrix equation Time epenent solution Operating conitions Computational Implementation Algorithm Numerical experiments Parametric stuy Frequency analysis Gravity an friction analysis Results Depenency on riser length an hyraulic iameter Frequency plot Perturbe variables in time Gravity an friction in time Conclusions an iscussion Overall conclusions Discussion an recommenations Bibliography 32 Appenix A Dimensionless variables 34 iv

6 CONTENTS v Appenix B Balance equations 36 B.1 Transport balances low heating moel B.2 Transport balances high heating moel B.3 Dimensionless balances low heating moel B.4 Dimensionless balances high heating moel B.5 Linearise balances low heating moel B.6 Linearise balances high heating moel Appenix C Matrices 43 C.1 Low heating moel C.2 High heating moel Appenix D Reference case parameters 49

7 1 Introuction 1.1 Backgroun of Research By the year 25, human population is expecte to have grown to ten billion iniviuals [7]. Combine with the ever increasing global average wealth means that the eman for energy will continue to increase for many years to come [11]. Since restraining that energy consumption leas to a lower stanar of living an halts economic growth [11] an fossil fuels are consiere a main contributor to global warming [7], the nee for new energy sources continues to be high. To meet these emans, the U.S. Department of Nuclear Energy, Science an Technology foune the Generation IV International Forum GIF. The goal of this international organization is to carry out research an evelopment neee to evaluate the next generation, or generation IV, nuclear energy systems [6]. To this en, GIF has selecte six systems for further evelopment. They are looking to eploy the first commercial generation IV reactors in about two ecaes. One such system is the Supercritical Water-coole Reactor SCWR. This is the reactor that The Nuclear Energy an Raiation Applications section of Delft University of Technology NERA performs research in [12]. The SCWR utilizes a similar principle as the secon an thir generation reactors Boiling water reactor BWR an Pressurize water reactor PWR, which right now are the two most common reactors [17]. All these reactors rely on water to both moerate the neutrons an act as a coolant. The ifference is that the SCWR operates uner much higher temperature an pressure, which bring the water above its supercritical point. 1.2 Supercritical Fluis The equilibrium state of water is epenent on both the temperature an the pressure, as seen in the phase iagram in figure 1.1. There are various ifferent states water can be in as a result of these two. The three colore lines are calle change lines, as they inicate for which conitions water transition from one of the three most common states to another. These lines meet at the triplet point, where water can be in any of the three states. The other interesting point marke in the iagram is the pseuo-critical point ubbe critical point in figure 1.1. At Table 1.1: Properties of water at the the pseuo-critical point, the ensities of vapor create by the pseuo-critical point increase in in temperature an the liqui water are ientical. Water in this state is calle supercritical. Thus any combination Property Value of temperature an pressure where both are equal or T pc C higher then their values at the pseuo-critical point makes the p pc 22.6 MPa water supercritical. It can not be sai that there is an actual h ph J/ kg phase change, but its properties o change quite rastically. ρ pc kg/m 3 The properties of water at the pseuo-critical point can be seen v ph m 3 / kg in table 1.1. Note that a SCWR generally maintains a constant pressure µ ph Pa s of 25 MPa [7], well above the critical pressure p c = 22.6 MPa. The average temperature of the water 1

8 CHAPTER 1. INTRODUCTION 2 Figure 1.1: Phase iagram supercritical fluis [7] in a SCWR is assume to go from about 28 C to 5 C. Since the critical temperature of water in the reactort pc = C, in the reactor it always transitions from a liqui phase to a supercritical flui see figure 1.1. This is mostly esirable because of the large variation of the specific heat capacity aroun the pseuo-critical point. Specific heat capacity, along with several other properties such as ensity an viscosity, shows highly non-linear temperature behaviour aroun the pseuo-critical point. As can be seen in figure 1.2, the specific heat capacity has a sharp an fairly high peak at T = T pc. This allows significantly more heat to be store in the water with only small temperature changes. 1.3 High Performance Light Water Reactor The basic workings of a SCWR can be seen in figure 1.3. Liqui water enters the core, where nuclear reactions heat it to be supercritical. The supercritical water then flows through a turbine, where the heat is converte to electricity through a generator. Then the water enters a conenser, where it is coole back to the inlet temperature, after which it can enter the core again. SCWRs have a multitue of avantages over current reactor [5]. These inclue a higher thermal efficiency of 45% vs 34-36% for current reactors. Some SCWR esigns also make the inclusion of a pump obsolete. Another property of supercritical water is that its ensity rops very rapily with temperature aroun the pseuo-critical point, meaning there is a large ensity ifference in the reactor. This is sufficient to provie a riving force for the loop. Due to the heat capacity peak, less water is neee to cool the core, making smaller reactors more viable. One of the reactors that utilizes Supercritical water is the High Performance Light Water Reactor HPLWR, a European version of the SCWR. The HPLWR is what this report will analyze. HPLWR reactors mostly set themselves apart from other SCWRs through their core esign, which can be seen in figure 1.4. The core of a HPLWR is a three-pass core esign, meaning it consists of three heating sections with mixing chambers between them. The water is first brought to a uniform 31 C

9 CHAPTER 1. INTRODUCTION 3 Figure 1.2: Specific heat capacity vs temperature for all temperatures foun in SCWR at 25 MPa. The specific heat capacity peaks at the pseuo-critical point at T pc =384.9 C [7] Figure 1.3: Simplifie overview of a SCWR. [7]

10 CHAPTER 1. INTRODUCTION 4 Figure 1.4: Concept of the core of a HPLWR [3] an passes through the evaporator, which heats it to an average temperature of 39 C. In the upper mixing chamber the water temperature is mae uniform. It then flows through the first superheater, another mixer an another heater, bringing the final temperature to the 5 C mentione earlier. This leas to the ensity of the water to ecrease from approximately 78 kg/m 3 to a mere 9 kg/m 3, proviing a significant riving force for the water to flow [7]. This report will analyze a reactor where the force provie by the ensity ifference throughout the loop is the only riving force, meaning it exclusively relies on natural convection. To enhance the flow, a riser is place on top of the core, increasing gravitational pressure rop. Much to the same effect, a owncomer is ae after the water is coole own. In pump riven reactors, technical failures or natural isasters may cause a power outage, preventing the pumps from working an thereby eliminating the moeration. The elimination of pumps thus ecreases the risk of such events influencing the reactor, making it more safe in emergency situations. 1.4 Instabilities in the reactor In orer to be a viable working reactor, a system nees to be stable. This is the case for systems where a small perturbation from its steay state operating variables trigger a negative feeback loop, causing the system to return to sai steay state. In talking about stability it is convenient to consier the ecay ratio, which is efine as the ratio between two consecutive peaks. If a system respons to a perturbation with an oscillation with a ecay ratio larger then one, every peak has a larger amplitue then the previous one. A ecay ratio of larger then one thus means an unstable system, while a ecay ratio smaller then one makes a system stable. This is visualize in figure 1.5. The system of the left graph has a small ecay ratio, therefore returning to its steay state value. The right system has a larger ecay ratio, making it oscillate with an ever increasing amplitue. Since the HPLWR relies on water as a moerator, instabilities in its properties greatly ecrease the reactors safety an shoul thus be avoie. Two types of instabilities are present in natural circulation riven BWRs [16]. Both of them are cause by local ensity ifferences an are known as Density Wave Oscillations DWO. Specifically, type-i an type-ii DWO s are present in BWRs. Type-I an type-ii iffer in how they are cause.

11 CHAPTER 1. INTRODUCTION 5 Figure 1.5: Example of reaction on permutation by a stable left an an unstable right system. [15] Type-I DWO instabilities are associate with gravitational pressure rops an oscillates with low frequencies. Type-II on the other han is cause by frictional pressure rops an oscillates at slightly higher frequencies aroun Hz [8]. In BWRs the water unergoes a phase change from liqui to vapor, while SCWRs are consiere single phase. However, the ensity behavior of water aroun the pseuo-critical point is similar to its behavior aroun the boiling point, so the same types of instabilities are assume to be present in the HPLWR. Another type of instability that is often consiere in two-phase flows is the Leinegg instability. However, this report will focus on the two DWO instabilities iscusse above as they are far more prevalent. 1.5 Previous research Stability in natural circulation riven loops have been analyze extensively by NERA [1][16]. An experimental setup name DeLight - short for Delft Light Water Reactor Facility - was built at TU Delft for this exact purpose [1]. Spoelstra an Schenerling performe numerical research on the setup, but their results varie from those of the experiment [14]. Note that Schenerling improve upon Spoelstra s moel by taking into account the thermal inertia, causing his results to agree with the experiment more then Spoelstra s see figure 1.6. The stability of the HPLWR core was investigate by Ortega Gómez [7]. Here it was foun that uner force circulation there were no Leinegg instabilities an no pressure rop oscillations at steay state in the HPLWR. Also, DWO s were foun to be the main type of instability present in the reactor. Another numerical analysis was performe by Krijger [8]. A significant ifference between Krijger s metho an Spoelstra s is that Krijger ivie the reactor into four or five noes, epening on if the water reaches its pseuo-critical point. Insie these noes, the variables are assume to increase linearly. Krijger erive mass-, energy- an momentum balances for all these noes an linearize these equations. The result was an eigenvalue problem he solve using a Matlab coe. Krijger foun that increasing the riser length ha a estabilizing effect. These results o not agree with previous research one in BWRs by van Bragt [16], where increasing the riser length increase type I instabilities but ecrease type II instabilities. Krijger also foun that increasing the buffer volume improve the stability of the system. Lippens [9] an Zonnevel [19] later improve on Krijger s moel by aing an enhancing equations to inclue the thermal inertia an neutronic-thermal-hyraulic coupling respectively. Thermal inertia was foun to improve the system s stability, while neutronic-thermal-hyraulic coupling ecrease stability. De Vries [2] further expane on this moel by eveloping a metho for fining the time-epenent behavior of system variables. The variables showe to have a ecay ratio of larger for the expecte conitions, but his results ha some very peculiar an likely incorrect values, making

12 CHAPTER 1. INTRODUCTION 6 Figure 1.6: The experimental values foun in DeLight with numerical results of Spoelstra s an Schenerling s numerical moel. [14] interpreting them meaningless. 1.6 Thesis outline This theses will continue improving an expaning on Krijger s moel. An improve parametric stuy will be performe, changing several parameters for ifferent values of the riser length to investigate its estabilizing effect. The frequency of the unstable regions will also be investigate. Finally, the time-epenency of the perturbe variables will be investigate. The next chapter will cover the moel use an the equations that are erive from it. This inclues the structure of the loop an linearizing the equations that follow from it. Chapter 3 escribes how the equations chapter 2 ens on are converte to a matrix equation. It will then erive how the time-epenency of the perturbe variables can be etermine. Chapter 4 will iscuss how the conclusions of chapter 3 are implemente in a Matlab coe in orer to fin all the eigenvalues an eigenvectors an all the steay state values of the system variables are foun. Chapter 5 inclues the parametric stuies performe an introuces the analysis of the frequency in unstable regions of the system. Chapter 6 presents the results obtaine from the analysis presente in chapters 3 through 5. A brief interpretation of all the results is also inclue. Finally, Chapter 7 gives the overall conclusions rawn from the analyses an iscusses them, ening with recommenations for further research.

13 2 Physical moel This chapter will introuce the moel the rest of the report uses to analyse the reactor. The equations that were erive from the assumptions mae are also iscusse. 2.1 HPLWR in noes In 213, Krijger [8] aapte a moel evelope by Rohe, who again aapte a moel use on boiling water reactors by Guio et al [4]. Purpose of sai moel was to etermine the stability regions of a reactor. Lippens an Zonnevel took this moel an expene on it. The visualization of the final moel use by Zonnevel can be seen in figure 2.1. Figure 2.1: The HPLWR broken own into noes. Water flows in the clockwise irection. Noe labels are as follows: F : Fuel, w : Wall noe, w1 : Wall noe 1, : Core noe, 1 : Core noe 1, R: Riser. B: Buffer, D: Downcomer. [19] Subcritical water enters the core of the reactor at the bottom of noe. Here it absorbs heat, causing its temperature to rise. Depening on the operating conitions of the reactor, the temperature can increase enough to make the water supercritical, but this is not necessarily the case. If the water oes reach its pseuo-critical point, the rest of the core in which the water is supercritical is ubbe noe 1. This is seen in the right loop of figure 2.1. If the water is not heate enough the entire core will be noe, as seen in the left loop in figure 2.1. These situations are calle the high heating moel an the low heating moel respectively. Note that in the high heating moel, the length of both noe 7

14 CHAPTER 2. PHYSICAL MODEL 8 an 1 are variables that epen on the operating conitions, while in the low heating moel the length of noe is constant. The riser an buffer noes are the relatively straightforwar parts of the reactor, as they are just tubes that the water flows through. The buffer noe represents the turbine an heat exchanger together. It is where the water is coole an the resulting heat is converte to electricity epicte by the q in figure 2.1. The fuel an wall noes o not interact with the water irectly. The wall noe is ae to moel the thermal inertia, while the fuel noe accounts for heat transfer from the fuel to the wall. 2.2 Assumptions an simplifications Several simplifications were mae for this moel. An important one involves the equation of state. The pressure of water is assume to be a constant 25 MPa throughout the entire reactor, espite in reality there being local frictional an gravitational pressure rops. From this assumption we can say that the ensity of water is only epenent on its enthalpy. If water is subcritical, its ensity is assume to ecrease linearly with enthalpy. Once the enthalpy is large enough to make the water supercritical, the specific volume, which is the inverse of ensity, linearly increases with enthalpy. This leas to the formula seen in equation 2.1. To fin the two constants C 1 an C 2 the equations is fitte through ata from National Institute of Stanars an Technology [13] see figure 2.2. { 1 ρ v i = pc+c 1H i h pc H i < h pc 2.1 v pc + C 2 H i h pc H i h pc Figure 2.2: Equation 2.1 fitte through the ata from NIST. [8] The temperature is also approximate, using a quaratic function in both the sub- an supercritical region. This approximation an its slope nee to be continuous aroun the pseuo-critical point. With the constants α = KJ 2 an α 1 = KJ 2, the following equation can be foun: T i = α i H i h pc 2 + H i h pc cp, pc + T pc 2.2

15 CHAPTER 2. PHYSICAL MODEL 9 Figure 2.3: Equation 2.2 fitte through the ata from NIST. [9] Finally, the thermal conuctivity is approximate with a linear function in the subcritical region an an exponential function in the high enthalpy region. Again, the approximation an slope are continuous at the pseuo-critical point. { β H + λ f,b, H i < h pc λ f = 2.3 λ z e β1h1 + λ f,b,1 H i h pc Figure 2.4: Equation 2.3 fitte through the ata from NIST. [9] The fit seen in figure 2.4 yiels the following values [9]:

16 CHAPTER 2. PHYSICAL MODEL 1 β = β 1 = kgj 1 λ f,b, = WM 1 K 1 λ f,b,1 = WM 1 K 1 λ z = WM 1 K Balance equations With the efinitions in the previous section an the moel introuce in section 2.1, transport equations can be erive for the mass, energy an momentum in every noe. These equations can then be mae imensionless an linearize, leaving equations with only constants an perturbations. These balances have all been erive by Krijger, Lippens an Zonnevel, so for this report most of them are merely inclue in appenix B. To illustrate how they are brought from their initial form to the linearize equations in the en, the high heating momentum balance will be use as an example. Note that oppose to the mass an energy balances, the momentum balance is integrate over the entire loop rather than over a single noe. The initial high heating momentum balance is as follows: A t W L + A t W 1L 1 + AL R t W R + t V BW + AL D t W = f L, + K W 2 2ρ f 1L 1 + K 1 W 1 2 f RL R + K r W R 2 f DL D + K D W 2...,1 2ρ 1,r 2ρ R,D 2ρ in A 2 gρ L A 2 gρ 1 L 1 A 2 gρ R L R A 2 gρ i nl D Note that L an L 1 are both variables, as the length it takes water to reach its pseuo-critical point may vary. Yet when combine they must always total the length of the core, meaning L can be substitute for L L 1, with L being the total core length. For the steay state solution for equation 2.4, the steay state mass flow nees to be calculate. In steay state, gravity an friction terms nee to cancel each other perfectly, as there is no net force on the water. This means the following equation for the steay state mass flow is foun. W HHM 2 = ga 2 ρ in L D ˆρ L ˆρ R L R fl, + K 1 2ρ + f1l1,1 + K 1 1 2ρ 1 + f RL R,r + K r 1 2ρ R + f DL D 2.5,D + K D 1 2ρ in The variables in in equation 2.4 are then substitute for the imensionless variables foun in appenix A. W t L L 1 t W + t W 1L 1 + L R t W R + V B t W + L D t W =... W R W R... W f L + K W 2 f 1L 1 + K 1 W 1 2 f RL R ρ in, 2ρ,1 2ρ 1... f DL D + K D W 2 ρ L ρ 1L 1,D 2ρ in N fr N fr,r ρ RL R N fr + K r W 2 R... 2ρ R ρ inl D N fr 2.6 With N fr being the imensionless Froue number, which is efine as the ratio between the flow inertia of the water an the gravitational acting force on it. N fr = W 2 v 2 pc gla 2 2.7

17 CHAPTER 2. PHYSICAL MODEL 11 The equations will be solve using linear methos, so they must first be linearize. All time epenent variables are replace by their steay state solution an a perturbation, enote by a capital an a lower case letter respectively. X = X + ˇx 2.8 As long as the perturbations are small, the prouct of two perturbations an a perturbation multiplie by the time erivative of a perturbation can be neglecte. XY = X + ˇxȲ + ˇy XȲ + X ˇY + Ȳ ˇx + ˇxˇy 2.9 Applying this to all variables in equation 2.6 will yiel the final momentum equation solely base on the perturbations. 1 + L 1 + L D + V B t w + L 1 t w 1 + L R t w 1 f υ R = f 1υ l1 2 N F r υ N F r υ 1 f L fd L D + K υ + + K D υ in υ in w f1 L 1 + K 1 υ 1 w f1 L 1 + K 1 + L 1 C 1 N h h pc h 2 N F r υ 2 1 R υpc fr L R + K R + υ in υ R w R + 1 fr L R + K R + L R C 1 N h h pc h υ R 2 N F r υ 2 R R υpc

18 3 Metho of stability investigation The linearize equations that escribe the system can be solve to fin values for all perturbe variables. Once this is one, the stability of the system as well as the properties like its frequency can be etermine. This chapter will cover how the solutions are foun. 3.1 Matrix equation The final equations from section 2.3 are all linear, first orer ifferential equations. Moreover, there is an equal amount of variables an equations for both moels 13 for the low heating moel an 16 for the high heating moel. They can thus be be written in a matrix equation. A x = B x 3.1 t Matrices A an B contain the operating values an constants an can be seen in appenix C, while x contains all the perturbe variables. The matrix equation has solutions of the form x = ve λt. In this solution, v contains the amplitue an phase of the initial perturbation. λ can be complex an can thus be ecompose as λ = a + bi. Since λ is in the exponent for the solution, a contains the ecay ratio an b contains the frequency of the perturbation. If a is negative, the perturbation will ecrease with time, making it stable. If a is positive, the perturbation will be unstable. Substituting the propose solution for x in equation 3.1 an iviing by e λ t gives the following: Aλ v = B v 3.2 This is a generalize eigenvalue problem, which can be solve by fining solutions to etb At =. Thus the values for λ, an by exten the system s stability an frequency, are foun. The solution is etermine using a Matlab script. Chapter 4 will expan on how this is one. 3.2 Time epenent solution When investigating the behavior of the perturbation in time, only having λ no longer suffices. Instea, the full solution for x must be etermine. As state in the previous section, the solutions of equation 3.1 are of the form ve λt. The general solution is a linear combination of all these solutions [18]. xt = n c i v i e λit 3.3 i=1 All values for λ i an v i can be foun from equation 3.2. The values of c i can be etermine using an initial value vector x. Because at t = the exponent of the solution equals one, the constants are 12

19 CHAPTER 3. METHOD OF STABILITY INVESTIGATION 13 calculate as follows: c 1 1 c 2. = v 1 v 2... v n x 3.4 c n The matrix V contains all the eigenvectors v i in its columns, the first column being v 1, secon column being v 2 an so on. To fin x, all that nees to be one is filling in equation 3.3 in matrix form. e λ1t e λ2t c 1 xt = v 1 v 2... v n e λ3t c c e λn n 3.3 Operating conitions The stability maps that are to be calculate with this moel utilizes two operating conitions to escribe the systems parameters. One is epenent on the inlet temperature of the core, the other on the heating power. Krijger, Lippens, Zonnevel an e Vries all use the same imensionless numbers as conitions. The first is the phase change number, a measure for the ifference in enthalpy that water unergoes as it flows through the core. N h = Q W h pc 3.6 The other imensionless variable is the subcooling number inicates the inlet enthalpy of the coolant. Since it is assume that the enthalpy always lies between an h pc, N sub varies from to 1. N sub = 1 H in h pc 3.7 Note that N h = N sub lies exactly on the borer of the low- an the high heating moel. For these conitions Q W = h pc h in, meaning the water leaves the core exactly at its pseuo-critical enthalpy. For N h N sub water has an enthalpy ifference larger than neee to become supercritical. Therefore, the high heating moel hols for these conitions, while for N h N sub the low heating moel oes.

20 4 Computational Implementation In chapters 2 an 3, the theoretical an mathematical interpretation of the reactor are explaine. The results of sai interpretations are acquire using a Matlab script. This chapter explains how the script solves all the equation mentione an generates the results seen in chapter 6 that come from them. 4.1 Algorithm The coe use followe the same general structure as Krijger s, Lippens, Zonnevel s an e Vries coe. This layout is isplaye in figure 4.1. The only ifferences between figure 4.1 an the figure in Zonnevel s report, which is where it was aapte from, are the introuction of a thir FOR-loop an the creation of the frequency map. These aitions are inicate by the re boxes an arrows in the figure. The extra FOR-loop allows for a multitue of instability maps to be generate with a single script, all for ifferent parameters. This mostly offers convenience an is particularly useful when changing multiple parameters in one run of the coe. To give a brief overview of the coe, it starts by clearing the workspace. Then all the constants an input variables which are to remain constant through the entire script are specifie, incluing the resolution of the instability map an the steay state calculations. N h an N sub are assigne a linear spacing of values that cover their entire range, an the variable parameters are ascribe all their values in a vector. Then, for the first value of the variable parameter, the coe calculates the eigenvalues for every combination of N sub an N h using two FOR-loops. The istinction of the low- an high heating moel is mae using an IF loop, with the conition N h N sub meaning the low heating moel is use. In the IF-loop the steay state values are calculate iteratively using a WHILE-loop. Once etermine, these are use to calculate all flow-epenent variables. This gives enough information to fill in the matrices seen in section 3.1. If any of eigenvalues foun from solving the equation has a positive real part, a is assigne to that combination of N h an N sub to inicate it is unstable. If none of the eigenvalues are negative a 1 is assigne for that combination being stable. The imaginary part of the eigenvalue is use to calculate the frequency. This is one for all combinations of N h an N sub, which then allows for the instability map an frequency map to be rawn. The whole process is then repeate for all values of the variable parameter, resulting in several instability maps an frequency maps. 14

21 CHAPTER 4. COMPUTATIONAL IMPLEMENTATION 15 Figure 4.1: General esign of the algorithm. This figure is an aaptation from Zonnevel [19]

22 5 Numerical experiments With the moel an computational implementation thereof explaine in the previous chapters, this chapter etails how everything covere so far is use to investigate the workings of the reactor. This inclues three ifferent stuies. The first is a parametric stuy where both the length an hyraulic iameter of the riser are varie. The secon is a frequency analysis on specific parts of the instability map. The thir is the analysis of the time-epenent solutions of the perturbations an their impact on gravitation an friction relate behavior of the system. 5.1 Parametric stuy In section 1.5, it was mentione that Krijger foun that increasing the riser length ecrease the stability of the system, as both regions of instability increase in size. This contraicte with earlier research one by van Bragt [16]. Van Bragt s research concerne a boiling water reactor. However, the pressure rop in a BWR is quite similar to that of the HPLWR escribe by the moel use by Krijger. Moreover, the instabilities in both reactors are mostly type I an II DWO in both reactors. The reactors were thus expecte to show similar behavior. Van Bragt conclues in his analysis that increasing the riser length increases the size of the unstable area on the stability map associate with type I DWO, but ecreases the unstable region relate to type II DWO. This woul imply that increasing the riser length increases the gravitational ensity fluctuations relative to those cause by friction. In Krijger s analysis ensity fluctuations from both causes increase. The purpose of this parametric stuy is to explain the ifference between Krijger s an van Bragt s finings. As the ifference between the stuies lies in the area where instabilities are mostly cause by friction, to fin the cause of the ifference the friction instabilities will be analyze. The pressure rop ue to friction can be calculate as such: P friction = fl + K W 2 2ρ In his research, Krijger allowe every one of these parameters to vary per noe except one: the hyraulic iameter was consiere constant throughout the system. Worth noting however is that influences the steay state massflow W, as seen in equation 2.5, which in turn influences the friction factor. 5.1 f =.316Re Re = W Aµ Changing the hyraulic iameter coul thus influence the instabilities cause by friction quite significantly, although it is not irectly apparent how. Keeping it constant throughout the entire loop coul thus prove to be too big of a simplification

23 CHAPTER 5. NUMERICAL EXPERIMENTS 17 Contrary to Krijger, van Bragt assigne a separate value to the hyraulic iameter of every noe in his moel [16]. Expecte is that the ifference can be assigne to the ifferent hyraulic iameters they choose. In reality the hyraulic iameter of the riser is significantly bigger than in the rest of the system. A large hyraulic iameter increases the volume : surface area ratio, which in turn increases the gravity : friction ratio. Although it is not possible to give the influence precisely, it oes feel intuitive to say that a large hyraulic iameter makes gravitational cause instabilities more important relative to friction cause instabilities. This woul mean that increasing the length of a riser with a high hyraulic iameter woul shift the instabilities towar the type I area rather than making the system less stable overall, as seen in van Bragt s thesis. To confirm the expectation that the ifferent approaches to the hyraulic iameter is the cause of their contraicting results, the hyraulic iameter of the riser,r will be given a separate value from the rest of the loop. The riser length L R will be varie in the same manner as Krijger i for multiple values of,r. The results of this stuy will be compare to both the results foun by Krijger an van Bragt. 5.2 Frequency analysis The frequencies of the system can be investigate by looking at the imaginary part of the eigenvalues. To fin the frequency of a perturbation, the imaginary part of the eigenvalue with the largest real part is taken, as that eigenvalue will ominate the behavior of the perturbations. This imaginary part is then consiere the frequency of the system, also known as the resonance frequency. Because time has been mae imensionless in the equations from section 2.3, the resonance frequency taken from the eigenvalue is imensionless as well. The frequency can be calculate using the table in appenix A. f = W υ pc AL f 5.4 The value of the resonance frequency can then be assigne to the operating conitions for which the calculations were performe. Once this is one for all combinations of N h an N sub, a frequency map can be mae. The result of oing this for the reference case can be seen in figure 5.1. Figure 5.1: Frequency uner all ifferent operating conitions for reference case. First thing to note is that there appears to be a higher frequency for high N h an for low N sub. This conclusion is the same as what was foun by Krijger, Lippens an Zonnevel [8][9][19]. In all their theses, this is where their analysis ene. This section will go more in epth an set up a ifferent investigation. In an attempt to see how the frequency of the system is etermine, a closer look is taken at figure 5.1. It appears that on a straight line that goes through the origin on the frequency map, the

24 CHAPTER 5. NUMERICAL EXPERIMENTS 18 frequency is constant. In orer to analyze this further, a imensionless constant K is introuce for which the following hols: K = N sub N h 5.5 The hypothesis is that for each K, the frequency will be the same for any N sub an N h. Substituting equations 3.6 an 3.7 gives a better insight of what K actually means: K = h pc h in = 1 L 1 h out h in L So K is equal to the imensionless length of the subcritical part of the core noe. Note that L is the total core length, which is a constant. In orer to relate K to the resonance frequency of the system, the frequency of the noes is analyze. The frequency of a given noe is efine as the inverse of the time water is insie it. The average time water resies in a noe is epenent on its length L i an the water s spee, which will be calle u i. The spee of the water insie the noe can be calculate as follows: u i = W v i 5.7 A The time water remains in that noe τ i is then calculate as the length of the noe over the spee of the water in it. Since equation 5.6 shows that K is effectively L, the frequency is erive in this manner for noe. The length can then be isolate an the expression is mae imensionless. K = τ W ρ AL The equation can be greatly simplifie by making both τ an ρ imensionless. By substituting accoring to table A.1 most parameters fall out of the expression K = τ ρ 5.9 These are both values that can be preicte when setting the operating conitions of a reactor. If a relation between the resonance frequency an the slope K is foun, substituting equation 5.9 into sai relation coul enable for preictions of the resonance frequency when setting the operating conitions. In the next chapter, the relation between K an the resonance frequency will be foun using a fit an this erivation will be one. 5.3 Gravity an friction analysis In section 3.2 it was explaine how the time-epenent perturbations of the system coul be foun. Among other things, these results are useful for analyzing how behavior of the ifferent perturbations is relate. One particular application comes from the linearize momentum balance. In the momentum balance, there are both terms clearly relate to friction an ones relate to gravity. Terms that inclue the Darcy friction factor f i or friction coefficient K i are friction relate, while terms with the Froue Number N F r are gravitational. By taking out the friction an gravity relate terms separately an summing them, two new equations are foun, or four if both the low heating moel an the high heating moel are consiere. By filling in the perturbe variables foun as escribe in section 3.2 the temporal behavior of the system can be better unerstoo. In particular it is useful to see which operating conitions cause the system to be mostly ictate by friction, an for which gravity is more important. The friction an gravity formula of the high heating moel are isplaye below.

25 CHAPTER 5. NUMERICAL EXPERIMENTS 19 F HHM = 1 f υ f 1υ l1 1 f L fd L D + K υ + + K D υ in w 2 f1 L 1 + K 1 υ 1 w f1 L 1 C 1 N h h pc + K 1 h 1 2 υpc fr L R + K R υ R w R + 1 fr L R C 1 N h h pc + K R h R υpc G HHM = 1 1 N F r υ N F r υ 1 l1 + L 1 N F r υ 2 R C 1 N h h pc h 1 + υpc L R N F r υ 2 R C 1 N h h pc h R 5.11 υpc Since all perturbe variables are expecte to oscillate, both equations above will as well. It is thus possible to etermine their amplitue an frequency. From the amplitues a ratio of friction an gravity be etermine. R fg = A friction A gravity 5.12 R fr makes it very simple to see if the system is mostly governe by gravity or friction, a high R fg meaning the system is mostly rule by friction an a low one implying its mostly rule by gravity. Because the information is store in one number it is also very easy to compare ifferent operating conitions. This, among other things, gives insight in the type of instability when choosing operating conitions that make the system unstable. The frequency of the time epenent friction an gravity perturbations is not etermine using the eigenvalues as in section 5.2. Rather, since there is a calculate time signal it is now possible to etermine the perio T of the signal. The frequency is then foun by taking the inverse of sai perio. f = 1 T 5.13 Note that since time has been mae imensionless, the frequency calculate this way will also be. To fin the frequency in Hertz, equation 5.4 is use. The frequencies combine with the frictiongravity ratio allow for thorough examination of the instabilities. The two expecte instabilities type I an type II are cause by gravity an friction respectively an have a ifferent frequency. The first makes R fg useful for analyzing the instabilities, the later makes the frequency effective.

26 6 Results Chapter 5 explaine the ifferent stuies that were performe. This chapter will iscuss why specific choices were mae in terms of chosen conitions, as well as giving the results of sai stuies. 6.1 Depenency on riser length an hyraulic iameter The parametric stuy etaile in 5.1 has been execute with the values seen in table 6.1. The first value of the hyraulic iameter of the riser,r was chosen to be the same as in Krijger s stuy. This is one to ensure that the results are similar espite the fact that a ifferent coe is use an the moifications that are mae to the moel by Lippens an Zonnevel. For this value the hyraulic iameter of the riser is equal to that of the core an the rest of the system. In reality the riser hyraulic iameter tens to be much higher, as the core is smaller an the water flows through multiple ifferent pipes, increasing the surface:volume ratio. Therefore the other values for,r are chosen to be bigger than the core hyraulic iameter. The riser length can really be just about anything, as there are no restrictions to it. The values of 4.2 m is chosen because this is the value of L R in Krijger s reference case. The other values are fairly ranom values within a reasonable interval. Table 6.1: Values chosen for the length an hyraulic iameter of the riser Values D R Values L R m 2 m m 4.2 m m 8 m m 12 m The instability borer was calculate for all combinations of the riser length an hyraulic iameter of the riser from the table. To make the results easily comparable to Krijger an Bragt, the calculations with the same hyraulic iameter were all plotte in the same graph. This lea to four ifferent graphs for every value of,r with four ifferent instability borers for every riser length in each of them. The graphs can be seen in figure 6.1. In the top two graphs in figure 6.1 are the ones where the hyraulic iameter of the riser is the smallest. In these graphs, a higher riser length lowers the instability borer. Note that the purple line, which is the highest riser length, is the lowest line. This means a high riser lenght makes the system less stable overall, much like the case with Krijger s analysis. The top left graph, where,r is the smallest, is more extreme in this regar. The lines on the right sie of the graph lie quite far apart, where in the case of the slightly higher hyraulic iameter in the top right graph the ifferent plots are quite close to each other. In the top left graph, a thir unstable area emerges in the top left graph. The origins of this region are unknown.. The bottom two graphs show very ifferent behavior. Here, the high riser length still has the lower borer on the left half of the graph, with the purple line once again being the lowest. On the right sie however, the orer of the lines has been flippe, meaning a high riser length makes the system more stable for these conitions. The increase of the riser length when its hyraulic iameter is high thus leas to an increase of type I DWO instabilities, but a ecrease of type II DWO. This aligns quite well with Bragt s finings. A frequency map was also mae for all the ifferent operating conitions. 2

27 CHAPTER 6. RESULTS 21 Figure 6.1: Instability lines for ifferent riser lengths plotte in same graph, with ifferent hyraulic iameter of the riser in every graph. Since it is impossible to plot multiple frequency maps in a single graph, there were 16 ifferent ones, too many to show all here. 6.2 Frequency plot To confirm the hypothesis that the resonance frequency of the system is constant for any given K, in which case equation 5.9 hols, the resonance frequency has been calculate for several ifferent values of K an plotte against N sub. N h increases appropriately as N sub oes to stay on the lines with constant K. The plots can be seen in figure 6.2. Note that one of the plots shows an empty graph. This is because the system is empty for that value of K, meaning the coe oes not have any points with a resonance frequency assigne to it. As is clear from the figure, the resonance frequency is not perfectly constant over the chosen lines. That sai, the scale on the y-axis is very small, meaning only very small changes occur especially for the lower values of K. Similar plots for the mass flow an ensity show similar results, with the mass flow barely changing. Note that the irection of the graph changes for every value of chosen K. At low K, the frequency ecreases with N sub an at high K it increases. Following these finings, for the remaining part of this section the resonance frequency will be consiere constant for constant K. The next logical step is then to see how the frequency is epenent on K. To answer this, a ranom point on a multitue of lines associate with ifferent K-values was chosen. The frequency on these points was calculate an that frequency was then consiere the frequency for all points on that line on the instability plot. Because of the way the coe is structure, the lines that o not cross into unstable regions are not assigne any frequency. These values of K can thus be recognize as the points for which frequency is zero. The foun frequencies were plotte against K, which resulte in the graph seen in figure 6.3. First thing to note is that there is a very clear ifference between type I an type II Density wave oscillations in this graph. At low K, which leas to lines going through the right unstable area on the instability map, the frequencies are high. Although the frequency ecreases as K increases, the ecrease is not enough to explain the significant ip seen between K =.5 an K =.7. The two ifferent types of instability o provie an explanation for this. Something else that is noteworthy is the increase of the frequency once K > 1. What is also

28 CHAPTER 6. RESULTS 22 Figure 6.2: Resonance frequency against N sub for along ifferent lines in the instability plot Figure 6.3: The frequency for all values of K. When f=, the system is fully stable for that value of K

29 CHAPTER 6. RESULTS 23 Figure 6.4: The frequency for all values of 1/K with a linear fit through the type II DWO point. striking is that for these K, the frequency is perfectly constant. A closer look at the frequency plots shows that in this area, the frequency is inee constant. The conition that K > 1 translates to the operating conitions not making the water supercritical, which means using the low heating moel is applie. In section 5.2 it was shown that K influences the system by etermining the length of the parts of the core where water is subcritical an supercritical. In the low heating moel, the water is always subcritical, meaning L = 1 an L 1 =. Since L cannot increase beyon one, the relation between L an K no longer hols in the low heating moel. Changing K thus appears stops affecting the frequency of system altogether. In section 5.2 it was erive that with a relation between K an the resonance frequency, the resonance frequency coul be preicte given some values are known. To fin this relation, the assumption is mae from figure 6.3 that f response is linearly epenent on 1 K. To test this assumption as well as to fin the relation, the frequency is plotte against 1 K an using matlab s curve fit tool, a linear fit is mae. The resulting plot can be seen in figure 6.4 The parameters of the fit result in the following relation: f response = K Taking a look back at equation 5.9, once equation 6.1 is rewritten to isolate K, there are now two expressions for K. By equating these, the final expression for the resonance frequency is foun as a function of the imensionless frequency of the subcritical part of the core an the imensionless ensity of water in this core: f response =.2491 f ρ In the high heating moel, the ensity of the water leaving noe is constant as water always leaves that noe when it becomes supercritical. The ensity of water entering the core is epenent on the operating conitions only. Since the ensity of water is assume to always increase linearly, ρ is constant for any combination of operating conitions an can be calculate using equation 2.1. That makes the resonance frequency linearly epenent on the frequency of the subcritical part of the core. 6.3 Perturbe variables in time The eigenvalue problem was solve as escribe in section 3.2, which lea to values for all the perturbe variables in time. The initial values use to calculate the constants from the linear combination to fin xt were chosen to be.1 for the perturbe length of noe 1, the part of the core where water is

30 CHAPTER 6. RESULTS 24 supercritical, with others all being zero. These conitions were chosen because it is effectively seeing what happens if the power of the reactor fluctuates a bit. An increase in power will lea to the water being heate faster, making it supercritical earlier in the core, which means ľ1 will increase. Other starting conitions were also use, but since there were no major ifferences in the outcomes the initial conitions will be kept constant throughout this entire section as well as the next one. To see how the isturbance in the length of noe 1 influences the system, the perturbe mass flow is plotte in both core noes an the riser. This is one in for two ifferent operating conitions, one leaing to a stable flow an the other making the system unstable. The plots can be seen in figure 6.5 an 6.6. As expecte, uner stable conitions the amplitue of the perturbation in all noes ecreases albeit being ifficult to see in figure 6.5, while for unstable conitions they all increase. Since one perturbation being unstable will impact all other perturbations, these are the only two possibilities: either all the perturbations converge or they all iverge. The plots agree with this for all the operation conitions that were trie. Figure 6.5: Perturbe mass flow in the two core noes an riser for stable operating conitions N h = 1.9 an N sub =.9.

31 CHAPTER 6. RESULTS 25 Figure 6.6: Perturbe mass flow in the two core noes an riser for stable operating conitions N h = 1.95 an N sub = Gravity an friction in time As escribe in section 5.3 an analysis was performe to see the impact that both friction an gravity ha on the system. For this, four ifferent operating conitions were chosen. Two of the chosen points were in the area where type I DWO instabilities are expecte, which means that gravity is likely to be more prevalent then friction. Two points were chosen in the type II DWO area, where the opposite applies. In both the type I an type II one point was chosen for which the system was stable, an one for which it was unstable. The chosen operating points can be seen in the instability plot in figure 6.7. Note that all operating conitions fall uner the high heating moel, as they are to the right of the black line inicating N h = N sub. Once the perturbe variables are calculate, they are plugge into equations 5.1 an Then all the terms that affect the same noe are taken out of sai equations an summe to get the behavior of the friction terms an gravity terms in a single noe. This is one for both the supercritical part of the core noe 1 an the riser. F HHM,1 = 1 f υ f 1υ l1 1 f1 L 1 + K 1 υ 1 w f1 L 1 + K G HHM,1 = 1 N F r υ N F r υ 1 l1 + L 1 N F r υ 2 R C 1 N h h pc h 1 υpc 6.3 C 1 N h h pc h 1 υpc 6.4

32 CHAPTER 6. RESULTS 26 Figure 6.7: The chosen operating conitions in section 6.4, inicate by the black squares. fr L R F HHM,R = + K R υ R w R + 1 fr L R + K R 2 L R G HHM,R = N F r υ 2 R C 1 N h h pc h R 6.5 υpc C 1 N h h pc h R 6.6 υpc The plots that were mae with the aforementione operating conitions can be seen in figures 6.8 through As escribe in section 5.3, the frequency of both the friction an gravity signal are calculate by taking the inverse of the perio formula The ratio of friction an gravity R fg was also calculate as escribe in formula The ratio in noe 1 is calle R fg,1 an the ratio in the riser is ubbe R fg,r. The results of these calculations can be seen in table 6.2. Table 6.2: Calculate values of gravity an friction plots N h N sub fhz R fg,1 R fg,r Note that only one frequency is isplaye for every combination of N h an N sub. This is because for constant operating conitions, the frequency of all the perturbations is the same. This results in all the formula s 6.3 through 6.6 also have the same frequency. The frequency in the table thus applies to all four signals shown in the graph belonging to those operating conitions.