The Small-scale Large efficiency Inherent safe Modular Reactor A thermal hydraulic feasibility study in terms of safety

Transcription

1 The Small-scale Large efficiency Inherent safe Modular Reactor A thermal hydraulic feasibility study in terms of safety D. E. Veling BSc Technische Universiteit Delft

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3 THE SMALL-SCALE LARGE EFFICIENCY INHERENT SAFE MODULAR REACTOR A THERMAL HYDRAULIC FEASIBILITY STUDY IN TERMS OF SAFETY by D. E. Veling BSc in partial fulfillment of the requirements for the degree of Master of Science in Applied Physics at the Delft University of Technology, to be defended publicly on Wednesday November, 04 at 0:30 AM. faculty department section Applied Sciences Radiation Science and Technology Nuclear Energy and Radiation Applications supervisors Dr.ir. M. Rohde Dr.ir. J. L. Kloosterman thesis committee Dr.ir. M. Rohde TU Delft, Applied Sciences, RST, NERA Dr.ir. J. L. Kloosterman TU Delft, Applied Sciences, RST, NERA Prof.dr. D. J. E. M. Roekaerts TU Delft, Applies Sciences, PE, FM

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5 ABSTRACT As renewable energy technologies i.e. wind and solar) will not be able to meet the world demand of energy in the near future, it is expected that nuclear power will remain an important player in a more sustainable energy-mix. Besides, the Fukushima accident demonstrated that the safety features of current nuclear reactors can still be improved. In response, this thesis investigates a new reactor design that offers an innovative combination of sustainability and safety, named SLIMR Small-scale, Large efficiency, Inherently safe, Modular Reactor). This small modular reactor operates at a power of 50 MWth, is cooled by supercritical water for a higher thermal efficiency, runs on a natural circulation flow to eliminate the need for pumps and external power for decay heat removal, and is cooled by natural circulation by submerging the vessel in a large pool of water. Hence, the decay heat can be transfered completely passively via the vessel wall to the environment. The SLIMR design is composed of the integral design of the MASLWR Multi-Application Small Light Water Reactor), in which the secondary containment vessel is removed. The thickness of the pressure vessel has been changed to 0.37 [m] in order to withstand a pressure of 5 MPa, and the core is replaced by fuel assemblies as defined for the European HPLWR High Performance Light Water Reactor). This thesis has assessed for which dimensions the SLIMR design is inherently safe under both normal and accidental situations. It is performed by a transient thermal hydraulic system-code developed as part of this project. The one-dimensional code simulates the flow inside the pressure vessel, and the heat transfer from the downcomer via the pressure vessel to the pool, where the pool is modelled by a single heat balance i.e. the flow around the vessel is not modelled). The thermal hydraulic system of the SLIMR is represented by this numerical model. It is concluded that the SLIMR is feasible in terms of thermal hydraulic safety. Firstly, it was found that the SLIMR is safe during nominal operation, having a stable flow, and no deterioration of heat transfer occurs for a core height of 4.0 [m]. Secondly, in an accident scenario of a SCRAM accompanied with a station blackout, the SLIMR can transfer its decay heat completely passively to the environment via a stable natural circulation flow i.e. no oscillations), a maximum coolant temperature that does not exceed 385 [ C], and without occurrence of heat transfer deterioration. It is also determined that in this design the minimum vessel width must be 3.6 [m] and the minimum vessel height 3. [m]. In addition, the heat loss during nominal operation is significant and a pool free surface of 600 [m ] is necessary to passively transport the heat from the SLIMR to the environment. Furthermore, the evaporation of the pool water requires a continuous supply of feed water. It is recommended to find a solution to decrease the size of the pool. One option is to decrease the heat loss by the use of a secondary containment, as can be found in the NuScale reactor. The removal of decay heat, however, should be accomplished by other means of passive decay heat removal systems inside the secondary containment. Moreover, future work should deal with e.g. reactor physics related issues such as criticality and reactivity feedback, as this work merely covers the thermal-hydraulics of SLIMR. iii

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7 CONTENTS Abstract iii Introduction. Small Modular Reactors mpower NuScale Comparison of the Reactors Natural Circulation in Nuclear Power Plants Instabilities in Natural Circulation Flows Advantages and Disadvantages An Innovative Element: Supercritical Water Advantages and Challenges The SLIMR Preliminary Design Thesis Objective Phase A Phase B Phase C Phase D Thesis Outline Theory 3. Thermal Hydraulics Conservation of Mass Conservation of Momentum Conservation of Energy Heat Transfer Conduction Convection Radiation Heat Transfer Deterioration Boiling Regimes Natural Convection Boiling Nucleate Boiling Pool Evaporation of Pool Water Decay Heat Numerical Model 9 3. Previous work Numerical Methods Finite Volume Method Staggered and Unstaggered Grid Spatial Discretization: First Order Upwind Scheme v

8 vi CONTENTS 3..4 Temporal Discretization: Semi-Implicit Euler Method Courant Number Fourier Number The SLIMR Model Flow Model Heat Transfer Model Pool Heat Balance Solver Tri-diagonal Systems Cyclic Tri-diagonal Systems Results 5 4. Phase A - Natural Circulation A - Simulation Procedure and Initial Setup A -Variation of the Riser Length A - Variation of the Core height A3 - Variation of the Riser Diameter A4 - Variation of the Downcomer Diameter A5 - Variation of the Core Inlet Friction A6 - Variation of the Core Inlet Temperature A7 - Variation of the Core Power A - Stability A - Summary Phase B - Heat Transfer B - Simulation Procedure and Initial Setup B - Variation of the Downcomer Diameter B - Variation of the Riser Length B3 - Variation of the Pool Temperature B4 - Variation of the Core Inlet Temperature B5 - Variation of the Thickness of vessel B - Summary Phase C - Pool C - Simulation Procedure and Initial Setup C - Pool Width C - Environmental Temperature C3 - Evaporation C - Summary Phase D - Accident Scenario D - Simulation Procedure and Initial Setup D - Variation of the Downcomer Diameter D - Variation of the Riser Length D - Summary Conclusions and Recommendations Conclusions Recommendations

9 CONTENTS vii Bibliography 89 List of Abbreviations 93 List of Symbols 95 A Additional Figures of the Results 97 B Discretization of the One-Dimensional Flow Equations 09 B. Mass Balance B. Enthalpy Balance B.3 Momentum Balance B.4 Pressure Correction C Discretization of the Radial Heat Transfer Equation 9 D Deriving Analytical Model 5

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11 INTRODUCTION The world energy consumption has increased sharply over the last century and this growth is expected to continue in time. Moreover, the increase will not only be due to the rapid growth of the world population, but also because of the increasing energy consumption per capita. At the same time, electric power becomes more and more economically accessible to the majority of the world population. In addition, the world s total electricity coverage increases. Besides the increasing growth of the energy demand, it is expected that the worldwide emission of CO will increase disturbingly, and with it the concern for climate change. Furthermore, fossil fuels i.e. oil and gas) are expected to become rare within decades. This creates the urge for an alternative energy source that has abundant resources and a low carbon emission. However, renewable energy technologies wind, biomass, geothermal, hydro and solar) are not yet sufficient to meet the world demand of energy in the near future. It is therefore expected that nuclear power remains an important contributor to a more sustainable energy mix. However, as it is widely known that nuclear energy has some disadvantages. For example, the production of nuclear power goes hand in hand with the production of radioactive waste, which must be stored in deep geological repositories. Furthermore, the fuel in nuclear power plants distinguishes itself from all other sorts of fuel. In this process the nuclear reactor core produces decay heat, also when the fission chain reaction has fully stopped. The removal of the decay heat in large-scale reactors is therefore dependent on active systems that operate on external power sources. The possible consequences of this dependency is demonstrated by the Fukushima Daiichi accident. Here the facility was cut off from the power grid due to the earthquake, after which the diesel generators were destroyed by a tsunami, and the batteries flooded; due to the complete absence of external power the reactors were not able to cool the core from its decay heat, this led to a melt down of the fuel rods in reactor to 3. Before this accident the support of nuclear energy had grown in the world 3. By the press this growth was even termed as the nuclear renaissance, but unfortunately, the extraordinary events in Fukushima led to a severe accident. It is to this end that there is now a large negative perception It is expected that the world population will grow to 8. billion people in 030, and the average energy consumption will double.[] The electricity coverage increases from 8% to 87% in 030. The distribution of the main regions without electricity coverage: Africa 58%, developing Asia 9%, Latin America 7%, Middle East %, developing countries 5%.[] 3 In 009, by an investigation over 0,000 people in more than 0 countries, it was found that more than 66% of the respondents believed that their country should begin or increase the use of nuclear energy.[]

12 . INTRODUCTION of nuclear power plants NPP). In 000 the generation IV International Forum was founded, which resulted in six innovative reactor designs to enhance the economics and safety of NPPs. Now, after the Fukushima accident, it has become of even greater importance that major improvements in the safety of NPPs are enforced. Due to this there are developments towards so-called Small Modular Reactors SMRs). This reactor type is very promising, as they have a respectively low power density in relation to the size of the reactor. This creates the possibility to remove decay heat completely passively 4 i.e. without the use of external power sources) in the event of a station blackout, increasing the safety of nuclear reactors... SMALL MODULAR REACTORS Although the concept of smaller size reactors has been known for a long time, the prejudice of large scale reactors, and hence economies of scale have prevailed for a long time. However, in the course of recent years, the validity of this paradigm has been questioned, as the complexity of reactors increased extensively. This goes hand in hand with rapidly increasing costs, delay in licensing, delay in construction and operation, which will eventually lead to a decrease in profitability. In addition, these reactors are commonly adjusted to the interests of the individual buyer, making all of these large reactors mostly one-of-a-kind [4]. In contrast, SMRs are designed to be Nth-of-a-kind 5 and are generally based on conventional reactor designs [5]. Moreover, the modular construction of SMRs makes factory production possible, and this will very likely result in lower costs, but also higher quality parts increase the safety of the reactor. To this end the design, safety and deployment advantages can overcome the economies of scale of large reactors [4]. As most SMRs are based on conventional reactors, there are multiple types of SMRs. This work focuses on pressurized light water reactors with an integral structure, based on uranium fuel. This is decided, as these reactors have the largest possibilities to proceed to the construction phase in the near future [4]. In this category the focus is on designs of the mpower and NuScale reactors that are both designed and developed in the USA; these reactors are in the most advanced stage and are expected to be built in the early 0s.... MPOWER The mpower reactor, shown in Figure.a, is designed by Babcock and Wilcox Nuclear Energy Inc. and the Bechtel Power Corporation together Generation mpower LLC). As part of the Obama Administration, in order to continue America s clean energy innovation, the US Department of Energy DoE) selected mpower to be supported in the design and licensing, in order to commercialize SMRs in the United States [6]. Therefore, the mpower design was awarded with a governmental investment of $50 million, which had to be matched by at least 50% of private investments [7]. It was expected by Generation mpower LLC that the overall capital costs of this design are comparable to large-scale power plants. Besides, a capital cost reduction is incorporated, which is based on the phased construction scheme, in which the first reactor units are introduced 4 Passive safety systems are according to the IAEA defined as: Either a system which is composed entirely of passive components and structures or a system which uses active components in a very limited way to initiate subsequent passive operation ; whereby a passive component is indicated a component which does not need any external input to operate.[3] 5 The learning curve for such a reactor is overcome; for the new generation it is estimated that no major design costs need to be incorporated.

13 .. SMALL MODULAR REACTORS 3 a) Cross-sectional view of the mpower reactor. b) Cutaway view of the complete NuScale reactor. Figure.: A representation of the mpower and NuScale reactor. and start production before all units are finished. However, Babcock and Wilcox failed to find customers or investors. This resulted in an announcement, in April 04, declaring that it was cutting its expenditure on mpower from $50 million a year to $5 million a year [8]. While initially it was expected that the mpower would be in commercial operation by 0, the future of the mpower is now unclear [7]. CHARACTERISTICS AND SPECIFICATIONS This Light Water Reactor LWR) is an integrated version of a conventional pressurized water reactor PWR). With an integral design is meant that the Reactor Pressure Vessel RPV) consist of a selfcontained assembly with reactor core, control rod drive mechanisms, reactor coolant pumps, steam generator 6 and pressurizer. Furthermore, the mpower design is based on a forced circulation flow. The mpower reactor produces a thermal power of 45 MWth, of which an electric output of 5 MWe can be generated. This can be expanded by building multiple units. The reactor uses standard LWR fuel based on uranium with less than 5% enrichment. The core consists of 69 assemblies, which have a 7 x 7 structure, and has a fuel cycle of 4 years. At the end of the fuel lifetime, the entire core will be replaced in one batch. SAFETY Due to the integral design the safety increases, as the risks of primary loop penetrations are reduced. In addition, there are no large external pipes connected to the primary loop, leading to reduced risk of Loss Of Coolant Accidents LOCA). A system based on gravity is proposed to remove decay heat from the reactor, where there is no reliance on emergency AC power for at least 7 [h] following an accident [4]. For further safety the facility is built completely underground. 6 In conventional PWRs the primary coolant flows inside tubes and the secondary fluid flows around these tubes. In case of an integrated PWR this is inverted.[4]

14 4. INTRODUCTION... NUSCALE The NuScale reactor, shown in Figure.b, is designed and developed by NuScale Power LLC, which was founded to commercialize the Multi-Application Small Light Water Reactor MASLWR). This preliminary NuScale design is developed in a cooperation of the Oregon State University, the Idaho National Engineering Laboratory and Nexant-Bechtel. As in the case of mpower it is expected by NuScale Power Inc. that the overall capital costs can be decreased by a phased construction scheme. In this case a multi-modular configuration of reactors can be created, of which each NuScale reactor is self contained and independent of the others. It is also estimated by NuScale Power LLC, that for a multi-modular configuration facility the plants will operate at full power for about 95% of the time. This can be explained by the fact that in the event of maintenance only one reactor has to be shut down, which makes it a very reliable system i.e. high utilization rate). Building on Obama s Climate Action Plan, the US DoE selected NuScale Power LLC to support this project to design, certify and commercialize innovative SMRs in the USA [9]. In May 04, the DoE and NuScale Power LLC signed a contract for funding and thus NuScale receives up to $7 million in matching funds over five years [8]. NuScale Power, in contrast to mpower, found investors to back at least 50% of the DoE investments. NuScale believes that its first planned reactor in Idaho could begin commercial operation by 03. CHARACTERISTICS AND SPECIFICATIONS Similar to the mpower design the NuScale reactor is an integrated version of a conventional PWR pressure in primary circuit.7 [MPa]). By this means, the heat generated in the core is transferred within the primary circuit into the secondary circuit by the helical steam generator heat exchanger) that is integrated in the downcomer section of the RPV. However, in this design no pumps are needed as the flow of the primary coolant is based on natural circulation. The NuScale reactor produces a thermal output of 60 MWth, from which an electric output of 45 MWe can be generated. The core is based on a conventional LWR, in which the assemblies consist of a 7 x 7 configuration, and the fuel has an enrichment of 4.95%. SAFETY The NuScale design incorporates 7 barriers between the fuel and the environment in order to prevent contamination of the environment. Their 7 layers include: the fuel cladding, the reactor pressure vessel, the containment vessel; the reactor pool heat sink during accident), the concrete pool walls and floor with a stainless steel liner improving heat transfer by conduction) below ground level; the biological shield on top of the free surface of the pool), and an HVAC heating, ventilating, and air-conditioning) reactor building Seismic Category ) [0]. In addition, just like the mpower design, the integral system decreases the risks of penetrations of the primary loop. Moreover, without large external pipes the risk for a LOCA decrease. Furthermore, by the introduction of the natural circulation flow, the need for primary coolant pumps, pipes and valves is absent, and hence failures and maintenance of these parts are prevented. Moreover, and even more important, no external power is needed to circulate the coolant; by this passive system the reactor can cool for extended periods of time without the need for power. In order to let the decay heat reach the pool the Decay Heat Removal System DHRS) is designed. The DHRS uses two independent channels for feed-water to the steam generator tube bundles the helical steam generator). In this event water is drawn from the containment cooling pool, and by

15 .. NATURAL CIRCULATION IN NUCLEAR POWER PLANTS 5 HOT LEG COLD LEG a b Figure.: Representation of a simple natural circulation loop. means of spargers the steam enters the pool where it condenses. The initiation of the natural circulation flow is provided by feed-water accumulators. During this event both flows in the primary and the secondary loop are based on natural circulation. The amount of pool water is large enough to provide a 7 [h] day cooling supply, obtaining a final bulk temperature of 93 [ C]. In order to prevent large heat losses during nominal operating power, a containment vessel is built around the reactor pressure vessel, between which a vacuum is maintained. This reduces heat loss during normal operation significantly, and thereby no insulation is required around the reactor vessel...3. COMPARISON OF THE REACTORS It is found that the SMR designs have a couple of general similarities: they are both based on a conventional pressurized LWR design; the reactors have an integral design; and both are placed underground. In addition, both reactors are designed to have a strategy against extreme events, and both can deposit decay heat passively no need for external power) for over 7 [m]. However, there are also some differences. So the NuScale design, contrary to the mpower, is based on a natural circulation of the coolant. Besides, the NuScale reactor is smaller, but the thermal power of the mpower is.7 times larger. In general, it is found that the mpower design has a resemblance with conventional large pressurized light water reactors, but in compact and integral size. In contrast, the NuScale reactor has implemented safety features that are not widely used in pressurized water reactors, and most importantly, the coolant circulation is based on natural circulation... NATURAL CIRCULATION IN NUCLEAR POWER PLANTS In a natural circulation loop the coolant is transported from the heat source to the heat sink without the presence of pumps. In this case the pipes between the source and the sink need to be connected in such a way that it forms a continuous path. In the event that the flow path is completely filled with a fluid, a natural circulation flow could be initiated by addition of heat and in the presence of a body force like gravity. Under the condition that the source and sink conditions are maintained constant, it is expected that a steady-state circulation can be achieved. Moreover, in the event that the completeness of the closed loop is maintained the steady-state flow can continue indefinitely. The driving force behind the natural circulation is buoyancy; this is the result of the differences in

16 6. INTRODUCTION density between the path from the source to the sink and vice versa. The working principle can best be explained by introducing the simple loop in Figure.. In the source the fluid gains heat, increases in temperature and decreases in density; the fluid becomes lighter and rises. In the sink the fluid loses energy, decreases in temperature and increases in density; the fluid becomes heavier and sinks. If the conditions of the source and sink are kept constant - the fluid absorbs an equal amount of energy in the source as is rejected in the sink - a steady-state natural circulation is established. In this process the density of the fluid in the hot leg is assigned by ρ hot, and in the cold leg by ρ cold. To this end the hydrostatic pressure can be calculated for the extrema in the bottom of the loop; the pressure in a) is p a = ρ hot g H, and in b) it is p b = ρ cold g H. Where H is the height of the loop, and g the gravitational acceleration; it is clearly that for ρ hot < ρ cold, there exist a pressure difference p a < p b. Equivalently, the driving pressure is enhanced by the loop height and the density difference. Here the driving pressure will be balanced out by the frictional pressure, p dr i ving = p f r iction, and a steady-state natural circulation flow will set in.... INSTABILITIES IN NATURAL CIRCULATION FLOWS Natural circulation NC) systems for both one-phase and two-phase flows) are more vulnerable for instabilities than systems with a forced flow, which is due to the regenerative feedback that is inherent to NC systems. For example, if the driving force is disturbed during a steady-state, the NC flow will be affected, which in turn will affect the driving force. This will consequently lead to an oscillatory behavior, which can be stable or unstable []. Here it is defined that a system is stable if it returns to the original steady-state after a perturbation. On the other hand, if the system oscillates with increasing amplitude the system is unstable. In a nuclear reactor unstable flows are undesirable as they could deteriorate the control and performance of the reactor. In addition, and even more severely, flow oscillations could deteriorate the heat transfer dry-out), which in a worst-case scenario could lead to damage to the core. It is therefore of great importance in the design of a reactor to have knowledge about the stability of the operation conditions. Over the years multiple types of flow instabilities have been witnessed in NC systems. The types that can occur in channel flows can be classified as static or dynamic instabilities []. A typical static instability is the Ledinegg instability, which refers to a system with multiple steady-state solutions. This type can be predicted by a steady-state analysis of the operation point. Of the dynamic instability the Density Wave Oscillation DWO) is most common. To explain the mechanisms of DWOs one must consider a system that operates at steady-state, see Figure.. By a perturbation of the driving force this instability can be triggered, and consequently, fluid packages with fluctuating mass flow rate are heated in the core. This will result into fluid packages with a low mass flow rate leaving the core with a lower density than the relatively high mass flow rate packages. The result is that a density wave oscillation will be traveling through the system, which can affect other variables i.e. reactivity feedback) in the reactor. If there is a positive feedback in the system, e.g. an increase in the mass flow rate at the entrance of the core, it results in a decrease of the pressure drop in the riser, which in turn stimulates the increases of mass flow rate of density wave. The DWOs will grow in amplitude and the system is unstable, the decay ratio >. For decaying DWOs, the decay ratio <, and the system is stable.... ADVANTAGES AND DISADVANTAGES The major advantage of an NC system is its simplicity; the elimination of pumps in the system greatly simplifies the construction, operation and maintenance. Moreover, the elimination of pumps

17 .3. AN INNOVATIVE ELEMENT: SUPERCRITICAL WATER 7 eliminates the scenarios with loss of flow, or related accident scenarios with pump failure. Furthermore, the absence of pumps reduces the amount of pipes, which in turn changes the scenario of a LOCA favorably. Another major advantage that focuses on the safety of the reactor in an accident scenario, is that an NC system enables the possibility that decay heat can be removed passively during a station blackout. The primary disadvantage of a PWR NC system is the low buoyancy force due to the small density difference in this system), resulting in a low mass flow rate. To increase the mass flow rate for a fixed power, either the height of the loop must be increased, or the resistance in the loop must be decreased. Moreover, it is due to the low mass flow rate that the maximum channel power is lower, and thereby, core volume is larger compared to forced circulation systems. Besides, large core volumes could lead to stability problems, where NC systems are already inherently less stable than forced circulation systems. In addition, a low mass flux affects the Critical Heat Flux CHF; exceeding could lead to burnout, dry-out) and to this end several measures must be taken in order to increase the heat flux for NC systems. Finally, NC systems must be able to start up from a stagnant fluid, low pressure and low temperature. In order to avoid premature occurrence of CHF, unstable zones must be avoided during the pressure and power raising process. Therefore, it is essential that there is knowledge of a start-up procedure, which avoids these instabilities. It is evident here that the start-up of an NC system is more complicated than for a forced circulation system..3. AN INNOVATIVE ELEMENT: SUPERCRITICAL WATER A large step further towards sustainability can be made with an SMR that is based on supercritical water. WHAT IS SUPERCRITICAL WATER? Water can be distinguished in three physical states: solid, liquid and vapor, see Figure.3a. By increasing the temperature and pressure along the vapor-liquid equilibrium 7 both phases could coexist, although in this process the vapor becomes denser and the liquid less dense. At one point on this line only one phase can exist in the system, which is given as the critical point p c =. [MPa], and T c = 374 [ C]). By continuation the saturation line becomes the pseudo critical line. For the region that is initiated by the critical point no boiling will occur. The pseudo critical point that is used in the operation of the HPLWR is p ps = 5 [MPa], and T ps = 385 [ C]..3.. ADVANTAGES AND CHALLENGES The primary advantage of a system based on supercritical water is that higher core exit temperatures can be achieved than in conventional PWRs. The result of this is that a high thermal efficiency in the conversion of heat to electricity can be achieved, i.e. 44% compared to 34% for conventional PWRs. The usage of supercritical water is not new and already being applied in fossil fuel power plants around the world [3]. Moreover, the SCWR is one of the innovative reactor designs by the Generation IV International Forum, making it one of the promising features in the future of nuclear reactors. Another advantage is that the boiling of the coolant is eliminated when the reactor operates above critical pressure. In this event the transition from liquid to vapor is a continuous process, so the coolant remains in a single-phase throughout the whole system. The result of this is that some simplifications could be made in the design, by eliminating steam separators and dryers. 7 Also known as the saturation line.

18 8. INTRODUCTION a) Pressure versus temperature diagram for water, whit pseudo critical line. b) Several properties of water versus the temperature, near the pseudo-critical point at a pressure of 5 [MPa]. Figure.3: Representations of the different physical phases of water, and the properties of water near the pseudo-critical point. The third advantage is that supercritical water has a large drop in the density around the pseudocritical point. The result of this is that a larger buoyancy force can be achieved, which in turn could result in higher mass flow rates. Hence, the disadvantage of a low buoyancy force in NC systems is eliminated, and thus lower vessel heights can be expected. Finally, Figure.3b shows that there is a peak in the specific heat capacity at the pseudo- critical point. Due to this peak large quantities of energy can be absorbed and transported to the heat exchanger. In addition, in case of an accident this peak can serve as a buffer, in which large quantities of energy can be stored to avoid excessive temperatures. The primary challenge that must be overcome is the corrosion of the materials that are in contact with the supercritical water. During nominal operation these materials are affected by combined thermal, radiative and thermochemical stresses, and must withstand these over the lifetime of the reactor. To this end, the feasibility of this concept depends on whether these materials can be found..4. THE SLIMR By introducing the SLIMR a Small-scale, Large efficiency, Inherently safe, Modular Reactor) it has been tried to combine all the elements mentioned above into one innovative small modular reactor. The SLIMR is, as the NuScale, an integral reactor based on a naturally circulated flow. However, the SLIMR operates at 50 MWth at a pseudo-critical pressure of 5 MPa, which increases the efficiency of the SMR. Moreover, the SLIMR has been designed as a semi supercritical water reactor, to this end the temperature interval is [ C] compared to [ C] for the large scale HPLWR. The result of this is that the materials in the SLIMR will suffer less corrosion. The minimum temperature is kept at 80 [ C], since it is indicated by Dobashi et al. [4] that this temperature has an optimal thermal efficiency. Additionally, the supercritical water leads to higher driving forces, which leads to smaller reactor designs with larger surface-to-volume ratio. This accompanied with a low thermal core power, enables the possibility to deposit decay heat through the skin the RPV ) of the reactor. Moreover, this

19 .4. THE SLIMR 9 is in comparison to the NuScale design even more passive 8. Furthermore, since the SLIMR design runs on natural circulation, the possibility to remove decay heat during a station blackout is created. In addition, the supercritical water has a simulating effect on the heat transfer as it increases the temperature gradient over the RPV from the downcomer to the pool), and thus increasing the feasibility of the proposed passive decay heat removal system. Finally, the idea is to submerge the SLIMR completely in a pool, which is on its own not an innovative idea. However, the combination of an SMR with supercritical water as coolant that is driven on natural convection is. This pool must be sufficiently large so that it can passively transport the heat lost by the SLIMR during nominal power, and it should be capable of removing the decay heat entirely in a passive way. The primary advantage of the pool is that pipes for the supply of cooling water can be eliminated, together with other active decay heat removal systems that might fail i.e. valves). In addition, the pool with its concrete walls is an extra barrier between the fuel and the surroundings, and thus decreasing the chances of contamination of the environment. By combining an integral small modular reactor that is cooled by supercritical water and running on natural circulation it is the goal to deliver a safe and sustainable reactor that meets today s needs and wishes. See Figure.4 for a representation of the SLIMR. Figure.4: Schematic representation of the SLIMR submerged in a pool, the geometry of the SLIMR is not to scale. In order to study the feasibility of the SLIMR, a preliminary design must be developed. This is done by combining the best elements of the NuScale, MASLWR and HPLWR in the design of the SLIMR..4.. PRELIMINARY DESIGN The first step that had to be made was to make a preliminary design for integral design of the SLIMR. For this purpose multiple small modular reactors were assessed of which the designs of the NuScale 9 8 In the NuScale design, the decay heat is removed by a natural circulation flow in the secondary loop, which sucks up cool water from the pool and returns steam back into the pool. However, this system needs to be mechanically valves) initiated. 9 The NuScale reactor is the commercialized version of the MASLWR, but similar in design.

20 0. INTRODUCTION Figure.5: Blueprint of the MASLWR design, the initial measurements of the SLIMR geometry. reactor and the MASLWR matched the characteristics of the SLIMR best. Moreover, the secondary containment is left out in order to make the heat transport possible via the vessel wall, see Figure.5 for a representation of only the primary containment of the MASLWR reactor. CORE Due to the fact that supercritical water is the coolant in the SLIMR, the reactor core in this design is not based on the MASLWR, which in turn is based on a conventional LWR core. This type of core is not applicable, due to the respectively large density drop that is expected over the core section, and results in the need of extra moderation at the end of the core. This has been overcome in most designs by the use of an extra flow area in the middle of the assembly, which provides extra moderation in this region. In order to increase the feasibility of this reactor the aim is to work with designs that are already in development. To this end the fuel assemblies as defined for the European HPLWR are utilized in the design of the SLIMR. The core is designed for an average linear heat rate of 9.75 [K W /m], with a maximum of 5.0 [K W /m] [5]. These core assemblies are designed to have an active length of 4.0 [m], and in order to make the least possible changes to existing techniques it is highly recommended that this length is maintained. REACTOR PRESSURE VESSEL For the large scale HPLWR it is found that the thickness RPV is 45 [cm] [6], although it is expected that this will be less for the SLIMR because of the smaller vessel. Besides, it is found that the MASLWR has a RPV thickness of.7 [cm], and operates at a pressure of 8.6 [MPa]. For a vessel of similar design and geometry the thickness of the vessel can be determined by a linear interpolation. To this end it can be determined that the vessel thickness of 37 [cm] will be sufficient in the preliminary design of the SLIMR.

21 .5. THESIS OBJECTIVE.5. THESIS OBJECTIVE The main objective of this thesis is to perform the first calculations on the SLIMR in terms of thermal hydraulic safety. Specifically, the goal is to answer the following question: Is it feasible to design a SLIMR that is inherently safe under both normal and accidental situations? To answer this question the work is divided into four phases: Phase A - Is it possible to obtain a geometry in which the SLIMR has a safe nominal operation point, which is stable and in which no heat transfer deterioration occurs? Phase B - What is the steady-state heat loss of such a SLIMR design during normal operation? Phase C - With what pool dimensions is it possible to transfer this heat passively to the environment and maintain safe working conditions i.e. a pool water temperature around 40 C)? Phase D - Is it possible to obtain a geometry of the SLIMR design that is within the boundaries of a safe nominal operation ànd allows safe deposition of decay heat to the environment under accidental situations in a fully passive way, without damaging the core?.5.. PHASE A The first step is to determine if it is possible for the SLIMR design to safely operate at nominal conditions. Therefore we are primarily interested in whether it is possible to achieve natural circulation in steady-state at the desired operation point. Secondly, it is important to know if the mass flow provided by natural circulation is sufficient to cool the fuel and cladding. To assess the sufficiency of natural circulation one has to investigate whether heat transfer deterioration - i.e. the rapid decrease of heat transfer capabilities at temperatures close to the pseudo-critical point of the supercritical coolant - occurs or not. In this thesis the criteria for this assessment are based on the work of Pioro et al. [7]. The last step is that it must be determined if the steady-state operation points are stable or unstable. This must be investigated, as it is known that natural circulation systems are sensitive to dynamic instabilities, because of the large density differences in the flow channels. For this last step the steady-state operation points are perturbed in order to find out if they return to their original steady-state operation point or not. All the issues mentioned above will be investigated as a function of the most basic parameters of the SLIMR. The parameters that will be varied are as follows: the riser diameter, the outer annulus diameter the outer diameter of the downcomer channel), the height of the vessel the riser length), the core length, the core inlet friction, the core inlet temperature and the core power. By assessing the influence of the individual parameters the domain of safe nominal operation is determined i.e. possible restrictions on the parameter values). To summarize, the sub-questions that must be answered during Phase A are: Is it possible to obtain natural circulation in a steady state under nominal conditions? Are the obtained steady-state nominal operation points stable? Is deterioration of the heat transfer avoided during normal operation? What is the sensitivity of the above sub-questions to the investigated design parameters?

22 . INTRODUCTION.5.. PHASE B In the assumption that appropriate steady-state operation points are found in Phase A, the second step is to calculate the heat transfer from the SLIMR to the pool during normal operation. For this the total heat loss from the SLIMR, the exterior surface temperature of the reactor vessel and the dominating heat transfer phenomena between the vessel and the surrounding water in the pool will be determined. In this step the following basic parameters are varied: the width of the vessel the outer annulus diameter), the height of the vessel the riser length), the vessel wall thickness, the temperature of the pool and the temperature of the coolant in the downcomer core inlet temperature). To emphasize, the most important questions to answer during Phase B are: What is the total heat loss of the SLIMR during normal operation? How do the basic parameters affect the results?.5.3. PHASE C In Phase C it is determined whether it is possible to transfer the heat from the SLIMR via the pool to the environment during nominal conditions in a completely passive way. To answer this question the results of the previous phases will be used. The goal is to design a pool with an average temperature around 40 C in steady state. Furthermore, the evaporation rate 0 of such a pool will be investigated. Summarizing Phase C, we search for the answers to: What is the dimension of the pool to achieve a desirable steady-state temperature of 40 C? What is the expected water evaporation rate in this pool at this desirable condition?.5.4. PHASE D In the last phase it will be determined whether SLIMR is able to safely deposit its decay heat during a station blackout i.e. without external power). For the design to be passively safe it must be verified that the coolant temperature does not exceed the maximum design temperature of 600 [ C] and that no deterioration of heat transfer occurs during the cooling period. During this step it must be investigated whether the SLIMR returns to a stable natural circulation flow after the first stages of the transient. Furthermore it is determined which heat transfer phenomenon at the external surface of the vessel) is dominant. To do so the effects of the following basic parameters are studied: the width of the vessel the outer annulus diameter) and the height of the vessel the riser length). In conclusion, Phase D investigates the following questions: Is it possible to find a SLIMR geometry within the safe parameter domain obtained in Phase A that is able to deposit its decay heat completely passively, without causing any damage to the internal structures of the reactor? How do the investigated parameters affect the transient results? Is the dimension of the pool as determined in Phase C sufficient for the investigated blackout accident situation as well?.6. THESIS OUTLINE The outline of the thesis is as follows. After this introductory chapter, the theory is introduced in Chapter. The solution algorithm is elaborated in Chapter 3. The results of the transient calculations are presented in Chapter 4, followed by conclusions and recommendations in Chapter 5. 0 It must be noted that during steady-state the pool is continuously filled in order to make up for the evaporation.

23 THEORY This chapter reports on several basic topics relevant to this study, and are included in the transient model that describes the behaviour of the SLIMR. We approach this work from the inside out, describing the fluid flow in the reactor, the heat transfer from the flow in the downcomer to the pool, the boiling regimes, the heat transfer phenomena to environment and ultimately the decay heat... THERMAL HYDRAULICS The fluid flow in the reactor is described by three conservation equations [8]; the continuity equation. converses the mass in the system, the momentum equation. keeps track of the conservation of momentum, and the conservation of energy is preserved by equation.3 in terms of internal energy u: ρ + ρ v) = 0 t.) t ρ v) + ρ v v = p + τ + ρ g.) t ρu) + ρu v = q" + q p v + τ : v)..3) In the following subsections the conservation equations are further described, and the energy equation will be converted to a equation in terms of enthalpy. Further all balance equations are rewritten to a one-dimensional form allowing them to be implemented in the system code.... CONSERVATION OF MASS The one-dimensional continuity equation is derived by integrating equation. over a control volume dv to obtain equation.4. By utilizing Gauss s divergence theorem [9] the convective term of the first part of equation.4 can be rewritten, where the integration is now carried out over the cross-sectional area A, in the direction perpendicular to the flow, this is given as: V ρ t dv + V ρ v)dv = V ρ t + A ρ v)d A..4) Where ρ is the density and v the velocity of the fluid, both variables are constant over the crosssectional area in this one-dimensional system. Therefore the density loses its averaging parameter, and the direction subscript on the velocity vector can be omitted since the velocity will only have a component in the x-direction. By defining the control volume with a fixed cross-sectional area, the 3

24 4. THEORY width can be set to an infinitesimal limit x; this term x is divided out of equation.4, into: A ρ t + lim x 0 [Aρv x ] x+ x x x = A ρ t + M = 0,.5) x resulting in the one-dimensional continuity equation, where M is the mass flow rate.... CONSERVATION OF MOMENTUM Analogously, an expression for the one-dimensional momentum equation. can be found, resulting in: A ρv x) t + p x Aρv x ) = A x P w τ w dp w + ρ g A..6) In this expression the first term on the left side is the inertia term, and the second one is the convective term. The right side starts with the pressure term, followed by the friction term, and the most right term is the gravity term. In this expression the friction term is still notated as an integral over the perimeter of the wall, the wetted perimeter P w. This perimeter doesn t only contain the outer perimeter of the tube, but it also includes possible obstacles in the flow path, which have their length in the flow direction. Therefore the integral over τ w runs along the complete cross-sectional geometry of the flow path. After integration, the average wall shear stress τ w is substituted by a relation for the wall friction. This relation proposed by Darcy-Weisbach, is described in work of Todreas and Kazimi [8], giving: P w τ w dp w = τ w P w = f M D h Aρ..7) Two new parameters are introduced here: the Darcy-Weisbach friction factor f which will be mentioned later in this section, and the hydraulic diameter D h which is given as: D h = 4A P w.8) In this equation the hydraulic diameter, D h, is a characteristic length that can be used when handling flows in non-circular geometries. By incorporating equation.7 into equation.6 closer to the final one-dimensional momentum equation. However, the design of the SLIMR does not merely consist of straight tubes. Elements that could obstruct the flow should be taken into account. One should think of bends, contractions, expansions, the core inlet friction and others. These obstructions will be incorporated into the momentum equation as a local pressure loss term, p i. The pressure term now becomes: Here the pressure loss term is described as: A p x = A p + ) p i H step x i ) x i = A p x + A p i δx x i )..9) i p i = K i M A ρ..0)

25 .. THERMAL HYDRAULICS 5 Here K i, an empirically found friction coefficient for diverse kinds of flow obstructions, is given in work of Janssen and Warmoeskerken [0]. The final one-dimensional momentum equation can now be obtained by substitution of equations.7 and.9 into equation.6. The momentum balance now becomes: M T + x M Aρ ) = A p x i M K i ρa δx x i ) f P w M 8A + Aρg.) ρ In its final form, the last parameter that needs to be defined is f, the Darcy-Weisbach friction factor. Usually, this factor is presented as a function of the Reynolds number. For this reason the Reynolds number will be introduced first, followed by the Darcy-Weisbach friction factor. REYNOLDS NUMBER The wall friction - but also forced convective heat transfer - are influenced by the flow regime of the fluid. Osborn Reynolds discovered that the flow regime mainly depends on the ratio of inertia forces to viscous forces in a fluid. He defined this ratio as the Reynolds number, a dimensionless quantity, which can be expressed as: Re = inertia forces viscous forces = ρ v x D h = M D h µ Aµ..) Here the Reynolds number is given for a one-dimensional system with a cross-sectional area averaged velocity v x, or mass flow rate M. The characteristic length is given by the hydraulic diameter D h, which makes equation. also applicable for non-pipe flows, and µ is the dynamic viscosity of the fluid. The flow regimes can be categorized - based on the Reynolds number - in three classes ; laminar when Re < 300, transient for 300 < Re < 4000, and turbulent for 4000 < Re. As in this work the system is modeled in one-dimension, the physical effects, due to the flow regimes, are approximated by empirical relations. DARCY-WEISBACH FRICTION FACTOR The effect of the flow regime on the shear stress is imposed in the Darcy-Weisbach friction factor. This factor is given by several empirical correlations i.e. functions of the Reynolds Number) for fully developed flows, where each is dedicated to an interval of Reynolds. In this work the most referenced correlations are used; combined they cover the whole range of Reynolds that is of our interest. These correlations are given by: Poisseuille Re < 0 3 f = 64 Re.3) Blasius Re < f = 0.36 Re 0.5.4) McAdams <Re < 0 6 f = 0.84 Re 0.0.5) [ ) ]) Haaland <Re < f =.8 log Re + ɛ 0/9.6) 3.7D h Ishigai Re < 0 8 f = 0.87 Re 0.4.7) These correlations are developed for circular tubes but can be used analogously for other flow geometries, using the hydraulic diameter as equivalent of the diameter. The Darcy-Weisbach friction factor should not be confused with the Fanning friction factor, which is four times smaller. The values that mark the boundaries of the flow regimes in this text are characteristic for pipe flows.

26 6. THEORY However, for the laminar regime - where the shear of the velocity gradient is significant throughout the entire flow cross-section see Figure.a) - the correlation is specific to the flow geometry 3. In case of a turbulent flow regime - in which the velocity gradient is mainly near the wall see Figure.b) - the geometry of the cross-sectional flow area is is of less importance. Therefore the concept using of the hydraulic diameter is more accurate in predicting the friction factor in the turbulent regime. a) Laminar regime. b) Turbulent regime. Figure.: The temperature T is shown relative to T s ) and velocity gradients at the wall for the laminar and turbulent regime. It can be seen that gradient for the turbulent regime is steeper and moreover, mainly near the wall. As a result the wall shear stress and heat transfer rate are larger in the turbulent regime than they are in the laminar regime. [] In this work the correlations are linked in a way that the transitions between the correlations are smooth, this is to exclude instabilities that can be induced by step wise transitions. In the simulations the flow is predominantly turbulent, mostly utilizing the Haaland correlation for high Reynolds numbers with the exception of the warming up stage and the SCRAM emergency shutdown of a nuclear reactor, and an acronym for Safety Control Rod Axe Man) of the system. The last relation.7 stretching over the complete range of Reynolds numbers) is known as the Ishigai correlation, which correlation is infrequently used. However, it is of importance in this work, since it is used to benchmark the one-dimensional model derived in this work with the onedimensional model developed by Chatoorgoon et al.[] This correlation is also developed by Chatoorgoon et al. using the data that are obtained by Ishigai et al.[3], who studied the friction and heat transfer for water flow in tubes at supercritical pressures...3. CONSERVATION OF ENERGY Finally, the conservation of energy will be derived in one-dimension. Prior to this, the energy balance - given in the form of the internal energy equation.3 - will be rewritten in terms of enthalpy h. Therefore the definition of enthalpy is introduced by: h u + p ρ..8) Applying this definition into the equation for the internal energy equation.3), we obtain, after some rearranging, the energy equation in terms of enthalpy: t ρh) + ρh v = q" + q Dp + Dt + τ : v)..9) This equation can now be simplified, focusing on the system that is modeled in this work. Therefore, since generation does not exist in this fluid, the production term can be eliminated. Also the 3 The Poissuille correlation is developed for circular tubes.

27 .. HEAT TRANSFER 7 shear term can be neglected, because for a system with moderate velocity gradients in combination with a low viscous fluid like water, heat generation by wall friction will be insignificant. The work done by the pressure can also be neglected since its contribution is insignificant in respect to the thermal power of the core. In the next step, equation.9 is reduced to a one-dimensional energy equation. Employing the same procedure as for the conservation of mass and momentum gives: A ρh t + Mh x = q..0) The conservation of energy by this enthalpy equation is a reduced form of the energy equation.3. This clarifies why this form has been chosen, with only a few variables to account for, making the numerical implementation much easier. The last variable that is defined is the heat rate q, averaged over the wall perimeter, also indicated as the linear heat rate. In this work the assumption is made that the thermal core power is transferred instantaneously from the core the heat source) to the coolant. Analogously, an equal amount of heat is transfered instantaneously from the coolant to the heat exchanger the heat sink). The linear heat rate is obtained by dividing the thermal core power by the effective core height, or the length of the heat exchanger section. For instance, the core heat rate is defined as, And vice versa for the heat exchanger, the linear heat rate is defined as: q Core = P Core L Core..) q H X = P Core L H X..) The heat transfer between the SLIMR and the environment will be discussed in the next section... HEAT TRANSFER Heat flows from hot objects to cold ones; e.g. the heat flow from the respectively hot RPV to the cool pool it is submerged in. The basic three physical mechanisms that are responsible for this transport of energy can be listed as: conduction, convection, and radiation. These mechanisms can work independently or together, as for this problem. In this section the mechanisms will be explained and specified on the heat transfer problem [].... CONDUCTION In the situations where a temperature gradient exists over a solid material, like the stainless steel containment of the RPV, or stagnant fluid, the mechanism of heat transfer is conduction. Heat transfer will continue until a thermal equilibrium sets in, reaching an isothermal state of the medium. Hereby, the driving force of this heat transfer mechanism is the temperature difference in the medium. The larger the temperature difference, the larger the heat flux. The cooling or heating of a body through conduction in time, is described by the non-stationary diffusion equation without convection and internal heat production, given as: ρc p T t = λ T..3)

28 8. THEORY In an analogous manner as applied for the mass, momentum and energy equation in the previous section., an one-dimensional expression for equation.3 can be found. The one-dimensional radial conduction equation can now be written as: ρc p T t = r r r λ T )..4) r Here c p is the specific heat, ρ the density, T the temperature and λ the thermal conductivity. The thermal conductivity is a property that can be considered as a heat resistance coefficient of the medium. The lower this coefficient the worse the conductive heat transfer. For a metal the thermal Table.: The density, specific heat, and thermal conductivity for some materials represented in this work. ρ [kg /m 3 ] c p [J/kg C] λ [W /m C] stainless steel at 5 C) Water at 0 C) Sandy Soil conductivity stainless steel is relatively low. However, in contrast is the thermal conductivity of fully stagnant water or sandy soil is very low. Here the thermal conductivity of sandy soil is representative for other ground types. Table. presents the ρ, c p and λ for a selection of media. In the cooling problem of the SLIMR, the stainless steel vessel is trapped between two fluid layers. To solve this conduction problem two boundary conditions need to be applied [4]. The first boundary condition is applied to the inner surface of the vessel where a convection boundary condition is utilized, and the heat flux is specified by Newton s law of cooling, given as: λ T r = h T w T b )..5) In which h is the heat transfer coefficient, T w is the wall temperature and T b the bulk temperature of the water in the downcomer. Here the values of h and T b are given, and the wall temperature T w is yet unknown. The second boundary condition is applied to the outer surface of the vessel, and is specified as a heat flux, given as: λ T r = q = q πr H..6) Here q is the heat flux, q the heat rate and H the height of the cylinder. This boundary condition is chosen because the heat transfer to the pool is given by multiple correlations, that are different for each boiling regime.... CONVECTION Solid materials adjacent to stagnant fluids will transfer heat by conduction at all times. However, in the presence of a bulk fluid motion, the physical mechanism of convection prevails. During convection regions with hot fluid are replaced with cooler fluid. It is a combination of conduction and refreshing of the adjacent fluid, which effectively enhances the heat transfer. Here is the refresh rate of the fluid is a strong function of the fluid motion; the higher the velocity of the fluid, the higher the convective heat transfer. The initiated motion of the fluid can be natural or forced and is used to classify the convective heat transfer. In natural or free) convection, the fluid motion is caused by differences in density. By this natural means, the warmer fluid rises by buoyancy and the cooler fluid falls. In the event of forced convection the fluid motion is initiated by an external force. In this way the surface is provided with

29 .. HEAT TRANSFER 9 a continuous fresh feed of fluid. As for conduction, the driving force in convection is the difference in temperature. The governing equation of heat convection is given by Newton s law of cooling, written as: q conv = h s T s T )..7) In which T s is the surface temperature of the solid medium, T the temperature of the bulk fluid sufficiently far from the surface, and h s is the convectional heat transfer coefficient. The convectional heat transfer coefficient h s is analogous to the thermal conductivity λ, but is not a property of the fluid. It is a parameter that is experimentally determined and depends on the nature of the fluid motion, the fluid velocity, several properties of the fluid, the surface geometry and the boiling regime. The typical orders of magnitude for convectional heat transfer coefficient can be seen in Table.. The heat transfer coefficient can be calculated with the use of the Nusselt number Nu, the Table.: Typical order of magnitude of the convectional heat transfer coefficient.[] Type of convection h [W /m C] Free convection of gases -5 Free convection of liquids Forced convection of gasses 5-50 Forced convection of liquids characteristic length L c and the thermal conductivity of the fluid λ. Rearranged the Nusselt number is given as: convective heat transfer Nu = conductive heat transfer = h sl c λ,.8) Here the Nusselt number is the ratio of convective to conductive heat transfer at the boundary between solid and fluid. It represents the improvement of the heat transfer through the fluid layer if fluid motion is present. The larger the Nusselt number, the more efficient the heat transfer due to convection. For many engineering problems a correlation for the Nusselt number can be found. This empirical relation is in most cases dependent on the Prandtl number. The Prandtl number - which is also a dimensionless parameter - describes the relative thickness of the thermal and velocity boundary layer, and is defined as: molecular diffusivity of momentum Pr = molecular diffusivity of heat = µc p λ..9) The Nu and Pr number are used to describe both forced and natural convection, which will further be discussed below. FORCED CONVECTION The internal wall of the vessel exchanges heat with the cooling fluid during nominal operation and, more importantly, from its decay heat after a SCRAM. The fluid flowing through the downcomer is forced by natural circulation. This natural circulation flow is approximated by the one-dimensional equations derived in the previous section.. Due to the fact that it is a one-dimensional flow, all properties are area averaged over the cross-section perpendicular to the flow direction. The heat transfer correlations that are used in this work must therefore be evaluated with only the bulk properties of the fluid.

30 0. THEORY To begin with the laminar flow regime, where analogous to the Darcy-Weisbach friction factor the heat transfer is dependent on the geometry see subsection..). The Nu correlation for an annular duct with fully developed laminar flow, in which one surface is isothermal and the other adiabatic, can be approximated by a fully developed flow in a circular tube with isothermal surface, given by equation.30. However, the effect that this approximation causes will be of a conservative nature, in which the heat transfer is estimated to be lower. Furthermore, the flow is predominantly turbulent and therefore the turbulent regime is preferably modeled with more care. Laminar flow Re < 0 3 Nu = ) Dittus Boelter 0 4 <Re 0.7 Pr 60 Nu = 0.03 Re 0.8 Pr n.3) The heat transfer in the turbulent regime is less geometry dependent see subsection..). Therefore, the annular duct can be treated as a non-circular duct, in which the hydraulic diameter of the annulus D h = D o D i. The empirical correlation utilized in this work is the conventional Dittus- Boelter correlation, equation.3, which makes use of the bulk properties of the fluid only. Here, n = 0.4 for heating the fluid, and n = 0.3 for cooling. In the transition regime 0 3 <Re< 0 4 ), the Dittus-Boelter correlation is used as a rough estimate, whereby the closer to Re= 0 4, the more desirable the approximation is. The Dittus-Boelter correlation is the most widely used heat transfer correlations at subcritical pressures. However, in this work the convective heat transfer must be estimated at supercritical pressures. It was found that the Dittus-Boelter correlation might produce some unrealistic results [7]. These errors concentrate in particular around the critical and pseudo-critical points, where the Dittus-Boelter equation is very sensitive to property variations. Due to difficulties around the critical and pseudo-critical points, especially for turbulent flows and at high heat fluxes, multiple correlations for supercritical fluids are derived by experimental data. These derived correlations, of which many are based on the conventional Dittus-Boelter equation, showed only to be more or less accurate within a particular dataset. In this work the heat loss is evaluated from the bulk fluid in the downcomer to the stainless steel RPV. Here the fluid properties are predominantly below the pseudo-critical point, and the heat flux is respectively low. Therefore it is reasonable to utilize the basic Dittus-Boelter equation calculating the convective heat transfer from the bulk fluid in the downcomer to the stainless steel RPV. In this subcritical regime liquid), the correlation predicts the heat transfer with an overall-weighted averaged error of 0.4%[7]. On the other hand, if the fluid in the downcomer comes in the superheated regime steam) the errors are significantly higher; overall-weighted average error of 75.3%. NATURAL CONVECTION The outer wall of the RPV exchanges heat with the adjacent pool water. If the pool water is calm, and the temperature of the RPV surface is below the saturation temperature of the fluid see section.4), heat will be transferred by natural convection. And by the same phenomenon, the pool water exchanges its heat with air adjacent to the free surface of the pool. In this event the heat transfer is dependent on the geometry of the surface as well as the orientation of the surface. In this work, the geometries of the RPV and pool consist of simple shapes as well as standard orientations vertical or horizontal). Therefore the Nu number can be given by empirical correlation, that are most well-known and frequently used in engineering. In this thesis the RPV is given by a correlation for a vertical cylinder.

31 .. HEAT TRANSFER The correlations for natural convection are mostly functions of the Grashof number, which is a dimensionless number that is given by: buoyancy force Gr L = viscous force = g βt s T )L 3 c ν.3) Here subscript L designates the characteristic length of the geometry it is based on, g is the gravitational acceleration, β the volumetric expansion coefficient, T s the surface temperature, T the temperature of the fluid sufficiently far from the surface, and ν is the kinematic viscosity. Therefore all fluid properties need to be evaluated at the film temperature, given as T f = T s +T )/ []. The Grashof number is analogous to the Reynolds number in forced convection. The fluid flow becomes turbulent as the Grashof number exceeds a certain critical value 4. Together with the Pr number they form the Rayleigh number, given as: Ra L = Gr L Pr..33) In which the Rayleigh number is used in most empirical correlations, as a simple function of Ra L. When modeling the RPV, the surface of a vertical cylinder can be treated as a vertical plate, and the Nu number is given by equation.35. Applying this approximation the following criteria must be satisfied: D 35L Gr /4 L..34) When this condition is met, the diameter of the cylinder is sufficiently large so that the effects by the curvature can be neglected. In this correlation the characteristic length is given by the length of the cylinder. Vertical wall Entire range of Ra Nu = { Ra /6 L [ /Pr) 9/6 ] 8/7 }.35) The convective heat transfer at the free surface of the pool is modeled using of a correlation that is specific for a pool with a still water surface. It determines the convectional heat transfer coefficient of the free surface of the pool with and without a wind blowing over the free surface. This correlation by Hahne and Kubler 994)[5] can be written as: h[w /m C] = v[m/s].36) In which v is the air velocity in [m/s]. Due to the fact that the pool is roofed it is hereby assumed that there is no wind blowing over the pool, and v = RADIATION Heat transfer by radiation is fundamentally different from conduction and convection. Heat transfer by radiation is the only heat transfer mechanism that can overcome a vacuum. Every object and medium is continuously emitting heat by radiation. In this process heat is transferred by electromagnetic radiation or photons). The radiative heat rate of an object can be evaluated by combining the Stefan- Boltzmann s law with the emissivity of the objects surface, given as: q r ad = εσ SB T 4 s.37) Here T s is the surface temperature of the object in [K], σ SB = [J/sm K 4 ] is the Stefan- Boltzmann s constant, and ε is the emissivity of the object surface material; a value between and 4 For a vertical plate the flow regime becomes turbulent for Grashof numbers bigger than 0 9.

32 . THEORY 0. When modelling the heat transfer from the SLIMR to the pool, the radiative heat transfer is neglected, which is due to the low emissivity of the RPV polished stainless steel has an emissivity ε ss = 0.7 []), and due to the respectively low temperature difference between the RPV and the surrounding surfaces. Moreover, the effect of radiation is minimal compared to the other mechanisms of heat transfer. As a result the evaluated heat transfer from the RPV to the pool is lower than the actual heat transfer, which makes the approximation conservative. When modelling the heat loss from the pool, the radiative heat transfer is often significant compared with conduction or natural convection i.e. for gasses). To this end, heat loss by radiative heat transfer is included in the model. Furthermore, water has a high emissivity ε w = 0.96 [6]. In the evaluation of the heat loss, the net radiative heat transfer is of interest. With this it is assumed that the pool is completely enclosed by a much larger surface the ceiling of the facility), that is separated by a medium air) which will not intervene with the radiation. In this special case the surface area of the surrounding surface has no effect on the net radiative heat transfer..3. HEAT TRANSFER DETERIORATION The Heat Transfer Deterioration HTD) is characterized by lower values of the wall heat transfer coefficient compared to normal heat transfer. Here the normal heat transfer refers to the forced convective heat transfer at subcritical pressures far from critical or pseudo-critical regions, and evaluated with the conventional Dittus-Boelter equation.3. The phenomenon of HTD is extremely undesirable for the core section of the SLIMR. In an event where deterioration of heat transfer occurs, the energy that is released in the core can accumulate. This will lead to an increase of the inner temperature of the core, and may lead to temperatures above the maximum allowable temperature of the core. This must be avoided at all times. Research on this topic - relevant to the work in this thesis - is conducted by Vikhrev et al. 97, 967) [7] [8], and is brought back into the spotlight by Pioro et al. 005) [7]. Vikhrev et al. conducted experiments on supercritical water flowing through vertical pipes. These experiments were performed for equal flow geometries, the same mass flux and heat flux, but at various inlet enthalpies, in order to cover a wide range of bulk fluid enthalpies. The range of values of investigated parameters can be found in table.3. In general, the deterioration of heat transfer occurs at high heat Table.3: Range of investigated parameters for experiments with water flowing in vertical circular tubes at supercritical pressures. Reference p [MPa] T [ C] [h in kj/kg] q [MWth/m ] G [kg/m s] Flow geometry Vikhrev et al. 967) 4.5, 6.5 h b = St. st. tube D = 7.85, 0.4 mm; L =.55, 6 m) fluxes. It was found by Vikhrev et al. that for a mass flux of 495 [kg/m s] two types of deteriorated heat transfer exist: The first type of HTD is due to the flow structures at the entrance region of the tube. According to Vikhrev et al. 97) this type of HTD can be avoided, to this end the following condition must be satisfied: L > ) D In which L is the heated length and D the tube diameter. Moreover, this type of HTD only occurs at low mass fluxes and high heat fluxes, whereas at high mass fluxes this type of HTD will disappear.

33 .4. BOILING REGIMES 3 The second type of HTD appears at any section of the tube within a certain enthalpy range, see Table.3), and occurs when the wall temperature exceeds the pseudo-critical temperature. This type of HTD can, according to Vikhrev et al. 967), be prevented when the following condition is satisfied: q G = P core/a pin < 0.4 [kj/kg]..39) M/A core In which q is the heat flux in [kw/m ], and G is the mass flux in [kg/m s], P core is the total core power in [kw], A pins is the total surface area of the all fuel pins, M is the mass flow rate, and A core the cross-sectional flow area of the core. However, these conditions are not enough for a clear identification of HTD; but they can be used as criteria in engineering. In this work both conditions are applied, and are met in the design of the SLIMR. Therefore, HTD is excluded by design and the wall temperature of the fuel pins will not exceed the pseudo-critical temperature. The first condition is met by integrating the core design of the HPLWR into the SLIMR design. To satisfy the second condition is more complex, this is so since equation.39 is a function of the heat flux and mass flux. In this work the mass flux - due to the natural circulation - is a function of the core power and geometry of the SLIMR. The solution to equation.39 which will be referred to by the HTD rate..4. BOILING REGIMES The RPV transfers its heat by natural convection during operational power. In an event where the pool water temperature increases to its saturation temperature T sat 00 C at atm), saturated boiling occurs. In addition, boiling can be qualified as sub-cooled when the bulk temperature of the fluid is below the saturation temperature of the liquid. Depending on the excess temperature 5 and the bulk temperature of the pool water itself, the boiling regime changes. The boiling regimes are divided into natural convection boiling, nucleate boiling, transition boiling and film boiling. These regimes are illustrated by the boiling curve for water in Figure.. In this graph the heat flux is presented in relation to the excess temperature. The shape of this curve depends on the fluid, the material of the surface and the pressure, but is nearly independent of the geometry and orientation of the heated surface. In the design of SLIMR it is extremely important to avoid the danger of a burnout. The burnout point is given by point C in the boiling curve. In this phenomenon the decreasing heat transfer to the fluid, and constant supply of energy by the mechanism of conduction makes the surface temperature jump e.g. the T excess in Figure. rises from 30 C at point C to 00 C at point E). The consequences of the burnout effect are critical in the design of the SLIMR. As the temperature of the outer surface of the RPV increases significantly, it will have a deteriorating effect on the heat transfer. In this process the temperature in the reactor will increase, with, as an ultimate consequence, a possible meltdown. Therefore the SLIMR must be designed in order for the excess temperature to be lower than 30 C..4.. NATURAL CONVECTION BOILING The first stadium in the boiling curve is subjected to the natural convective boiling, this region end in point A, see Figure.. In this event bubbles will not emerge in the fluid. The fluid that is super- 5 The excess temperature is evaluated by the difference between the surface and the saturation temperature of the fluid, given by T excess = T s T sat.

34 4. THEORY Figure.: Pool boiling in water on a horizontal wire at atmospheric pressure.[] heated 6 near the surface, causes natural convective currents. As a result the heat transfer from the surface to the fluid can be modeled as natural convective heat transfer, as discussed in sub-section NUCLEATE BOILING In region from point A to point C the excess temperature is large enough to form bubbles at nucleation sites on the heated surface, see Figure.. This stage in the boiling curve is called nucleate boiling. In this regime the heated surface is essentially independent of the orientation and geometry of the heated surface area. Therefore, the surface of the vertically oriented SLIMR is treated as a pan filled with water on a stove. This region can be separated into two sub-regions. Starting with region A B, the bubbles are formed at an increasing rate for increasing excess temperature. As the vapor bubbles rise by buoyancy, the respectively cooler water fills up the vacated space. This effectively increases the heat transfer compared to convection boiling. The bubbles in this sub-regime dissipate - they condense and collapse - before they can reach the free surface of the water, making them effective energy transporters from the heated surface to the bulk fluid. In region B C, the heat transfer is enhanced as it is region A B. However, getting closer to point C, the bubble production is so large that the fresh liquid has difficulties to reach the heated surface. To this end, the heat flux reaches a maximum in point C. In this whole sub-regime the bubbles are formed at such a rate that they form a continuous vapor strip to the water surface where the vapor is released. In this form it is assumed that the heat is not transferred to the bulk fluid of the pool, 6 A superheated fluid is heated to a few degrees above the saturation temperature of the fluid.

35 .5. POOL 5 but directly to the environment. The most widely used empirical correlation for the rate of heat transfer in the nucleate boiling regime is given by the Rohsenow equation []: q nucleate = µ l h f g [ g ρl ρ v ) σ ] [ ] / 3 cp T s T sat )..40) c s f h f g Pr n l Here µ l is the dynamic viscosity of the liquid, h f g the enthalpy of vaporization, g the gravitational acceleration, ρ l the density of the liquid, ρ v the density of the vapor, σ the surface tension of liquid vapor interface in N/m, c p the specific heat of the liquid, c s f the experimental constant that depends on the surface fluid combination, Pr l the Prandtl number of the liquid and n an experimental constant that depends on the fluid. The fluid properties are to be evaluated at the saturation temperature T sat. This heat transfer relation for pool boiling only applies to smooth surfaces, which applies to the polished stainless steel of the RPV. Further, it must be noted that the results obtained with the Rohsenow equation need to be handled with care. This is because errors of ±00% are possible in the calculation of the heat transfer rate with a given excess temperature..5. POOL One of the safety features is that the SLIMR is submerged in a pool. This allows the SLIMR to lose its decay heat in case of an emergency. However, during nominal operation heat will also transfer from the SLIMR to the pool. By this means the pool is continuously heated. From the pool heat is transported to the environment. In this process the pool loses heat due to conduction, convection, evaporation, radiation, but also by the addition of make-up water [9]. In this work the pool is modelled as a single node, assuming that the water is ideally mixed. Further it is assumed that the water in the pool is incompressible, and that its density ρ and specific heat c p are constant. The energy and mass balance of the pool can now be written as: c p ρv p dt p dt ρ dv p dt = Q SLI MR Q cond Q conv Q r ad Q eva Q makeup.4) = ṁ makeup + ṁ eva..4) In which V p is the volume of the pool water. During nominal operation the makeup mass rate ṁ makeup equals the evaporation mass rate ṁ eva, resulting in dv p /dt 0. Of the heat transfer mechanisms that are modelled in this work, the heat loss by conduction to the ground is considered small enough to be neglected, which in most circumstances accounts less for than % of the total energy loss from the pool [6]. However, the heat losses from the free pool surface by convection and radiation are significant, these are modelled as given in sub-section.. and..3. Furthermore, evaporation causes the largest losses at least 60% of the total heat losses[5]), and therefore its prediction is most important. To this end, the latest evaporation correlation is implemented in our model, which will be discussed in the following sub-section..5.. EVAPORATION OF POOL WATER The free surface of the pool will transfer energy and mass. Analogous to heat, which is transferred along a temperature gradient, mass is transported along a concentration gradient. However, in this

36 6. THEORY case, for a region with a relatively high concentration of moister in the air, it will be transferred to the relatively low concentration zone. By this means, the saturated vapor layer on the water surface of the pool will transfer heat to the region with less moist air, the rest of the facility and the environment. This mass transfer can be described as follows. The air that is in contact with the water surface becomes saturated with moisture and, thus, becomes lighter [30]. Analogous to natural convection heat transfer, the air with water vapor lighter) will rise by buoyancy, and the dryer heavier) air falls descends to replace the volume that has risen. In addition to this, the convective currents will be enhanced when the dry-bulb temperature in the facility is lower than the temperature of the pool water. In this event the air layer that is in contact with the free surface of the pool gains heat, and the heated air enhances the buoyancy. When the dry-bulb temperature in the facility is higher than the pool temperature, the effect is reversed. The rate of evaporation from an undisturbed pool into quiet air air without forced flow) can be described by a equation published by Shah et al. 003) [3]. This correlation uses the analogy between heat and mass transfer from to a horizontal plate with the heated face upward. The evaporation rate is given by the following correlation: E vap = K ρ w ρ f ρ w ) /3 W w W f )..43) In which E vap is the evaporation flux per hour [kg /m h], ρ is the density of air in [kg/m 3 ], W the specific humidity of air [kg of moisture/kg of dry air], subscript w denotes saturated at water surface temperature, subscript f denotes at facility temperature and humidity, and K is a constant that is given by: { 35, for ρ f ρ w ) > 0.0 K =.44) 40, for ρ f ρ w ) < 0.0 This correlation is verified with numerous data sets, and is applicable in the ranges are presented in Table.4, with a mean deviation of the data of only 4.5%. Table.4: Verified range of Shah et al. formulas for evaporation from pools. [3] Range of data Pool area [m ] Water temperature [ C] 7 94 Air temperature [ C] 6 35 Air relative humidity [%] 8 98 p w p f [Pa] 0 80, 56 ρ f ρ w [kg/m 3 ] to +.00 The values for ρ and W are obtained by the psychometric equations and splines obtained from ASHREA 009) Handbook - Fundamentals [33]. To begin with the humidity saturated at the water surface temperature W w = W s, a condition at which the moist air is in equilibrium with the pool water at given pressure and temperature; the saturated humidity can be given as: p ws W s = ) p p ws Where p = 035[Pa] is the total pressure, and p ws is the saturation pressure of water vapor in absence of air), which is a function of temperature.

37 .6. DECAY HEAT 7 Figure.3: Decay heat power ratio for U 35 fuel as a function of time after shutdown. [34] The density of the moist air, saturated at water surface temperature, is evaluated at the temperature of the pool water. The function for density ρ w is based on the ideal gas law and is given as: p +W w ) ρ w = T p )R a +W w R w ).46) Here R a = 86.9[J/kg K] is the gas constant for air, R w = 46.5[J/kg K] is the gas constant for water vapor, and T w is the pool water temperature in [K]..6. DECAY HEAT In an event that a reactor needs to be shut down, due to maintenance or in the event of an accident, control rods are inserted in the core this is also known as a SCRAM). From this point on the heat generation continues by decay heat, even though the power that is generated by fission has completely ceased. The major source of shutdown power is the decay of fission products, in which unstable fission products decay via β and γ emission to stable isotopes. It is of great importance that this heat is extracted from the system. Accumulation of decay heat could otherwise lead to fuel damage, and melting or even evaporation of the core. To this end, the transient prediction and analysis of the SLIMR after a SCRAM is of vital interest, in which the decay heat production must be determined accurately. The magnitude of this heat production is dependent on the amount of fission products, in turn, is dependent on the operation time of the reactor. To limit the computational effort, using costly isotope generation and depletion codes, the decay heat is estimated by a semi-empirical Way-Wigner type of function [34], which is written as: P = [t 0. t + T 0 ) 0.] P 0..47) Here P 0 is the constant power production before shutdown, t the time after shutdown in [days], and T 0 the operation time of the reactor in [days]. This equation is based on the empirical correlations that evaluate the energy release by β and γ emission due to decaying fission products. The number of fission products are evaluated using 93 MeV per fission [35].