The Small-scale Large efficiency Inherent safe Modular Reactor · 2017-09-14 · THE SMALL-SCALE...

The Small-scale Large efficiencyInherent safe Modular ReactorA thermal hydraulic feasibility study in terms ofsafety

D. E. Veling BSc

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THE SMALL-SCALE LARGE EFFICIENCYINHERENT SAFE MODULAR REACTOR

A THERMAL HYDRAULIC FEASIBILITY STUDY IN TERMS OFSAFETY

by

D. E. Veling BSc

in partial fulfillment of the requirements for the degree of

Master of Sciencein Applied Physics

at the Delft University of Technology,to be defended publicly on Wednesday November 12, 2014 at 10:30 AM.

faculty Applied Sciencesdepartment Radiation Science and Technology

section Nuclear Energy and Radiation Applications

supervisors Dr.ir. M. RohdeDr.ir. J. L. Kloosterman

thesis committee Dr.ir. M. Rohde TU Delft, Applied Sciences, RST, NERADr.ir. J. L. Kloosterman TU Delft, Applied Sciences, RST, NERAProf.dr. D. J. E. M. Roekaerts TU Delft, Applies Sciences, PE, FM

ABSTRACT

As renewable energy technologies (i.e. wind and solar) will not be able to meet the world demand ofenergy in the near future, it is expected that nuclear power will remain an important player in a moresustainable energy-mix. Besides, the Fukushima accident demonstrated that the safety features ofcurrent nuclear reactors can still be improved. In response, this thesis investigates a new reactor de-sign 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 powerof 150 MWth, is cooled by supercritical water for a higher thermal efficiency, runs on a natural cir-culation flow to eliminate the need for pumps and external power for decay heat removal, and iscooled by natural circulation by submerging the vessel in a large pool of water. Hence, the decayheat can be transfered completely passively via the vessel wall to the environment. The SLIMRdesign is composed of the integral design of the MASLWR (Multi-Application Small Light Water Re-actor), in which the secondary containment vessel is removed. The thickness of the pressure vesselhas been changed to 0.37 [m] in order to withstand a pressure of 25 MPa, and the core is replaced byfuel 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 nor-mal and accidental situations. It is performed by a transient thermal hydraulic system-code de-veloped as part of this project. The one-dimensional code simulates the flow inside the pressurevessel, and the heat transfer from the downcomer via the pressure vessel to the pool, where the poolis modelled by a single heat balance (i.e. the flow around the vessel is not modelled). The thermalhydraulic 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 foundthat the SLIMR is safe during nominal operation, having a stable flow, and no deterioration of heattransfer occurs for a core height of 4.20 [m]. Secondly, in an accident scenario of a SCRAM accom-panied with a station blackout, the SLIMR can transfer its decay heat completely passively to theenvironment via a stable natural circulation flow (i.e. no oscillations), a maximum coolant temper-ature that does not exceed 385 [°C], and without occurrence of heat transfer deterioration. It is alsodetermined that in this design the minimum vessel width must be 3.6 [m] and the minimum vesselheight 13.1 [m]. In addition, the heat loss during nominal operation is significant and a pool freesurface of 1600 [m2] 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 decreasethe heat loss by the use of a secondary containment, as can be found in the NuScale reactor. Theremoval of decay heat, however, should be accomplished by other means of passive decay heatremoval 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 coversthe thermal-hydraulics of SLIMR.

iii

CONTENTS

Abstract iii

1 Introduction 11.1 Small Modular Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 mPower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 NuScale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Comparison of the Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Natural Circulation in Nuclear Power Plants . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Instabilities in Natural Circulation Flows . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Advantages and Disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 An Innovative Element: Supercritical Water . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Advantages and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 The SLIMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Preliminary Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.1 Phase A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.2 Phase B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.3 Phase C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.4 Phase D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Theory 132.1 Thermal Hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Conservation of Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Heat Transfer Deterioration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Boiling Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Natural Convection Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Nucleate Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5.1 Evaporation of Pool Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Decay Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Numerical Model 293.1 Previous work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.2 Staggered and Unstaggered Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3 Spatial Discretization: First Order Upwind Scheme . . . . . . . . . . . . . . . . 31

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vi CONTENTS

3.2.4 Temporal Discretization: Semi-Implicit Euler Method . . . . . . . . . . . . . . 31

3.2.5 Courant Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.6 Fourier Number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 The SLIMR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Heat Transfer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 Pool Heat Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.1 Tri-diagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.2 Cyclic Tri-diagonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Results 51

4.1 Phase A - Natural Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 A - Simulation Procedure and Initial Setup . . . . . . . . . . . . . . . . . . . . . 51

4.1.2 A1 -Variation of the Riser Length . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.3 A2 - Variation of the Core height. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.4 A3 - Variation of the Riser Diameter . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.5 A4 - Variation of the Downcomer Diameter . . . . . . . . . . . . . . . . . . . . 58

4.1.6 A5 - Variation of the Core Inlet Friction . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.7 A6 - Variation of the Core Inlet Temperature . . . . . . . . . . . . . . . . . . . . 60

4.1.8 A7 - Variation of the Core Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.9 A - Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.10 A - Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 Phase B - Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1 B - Simulation Procedure and Initial Setup . . . . . . . . . . . . . . . . . . . . . 64

4.2.2 B1 - Variation of the Downcomer Diameter . . . . . . . . . . . . . . . . . . . . 66

4.2.3 B2 - Variation of the Riser Length . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.4 B3 - Variation of the Pool Temperature . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.5 B4 - Variation of the Core Inlet Temperature . . . . . . . . . . . . . . . . . . . . 69

4.2.6 B5 - Variation of the Thickness of vessel . . . . . . . . . . . . . . . . . . . . . . 69

4.2.7 B - Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Phase C - Pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 C - Simulation Procedure and Initial Setup . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 C1 - Pool Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.3 C2 - Environmental Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.4 C3 - Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.5 C - Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Phase D - Accident Scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.1 D - Simulation Procedure and Initial Setup. . . . . . . . . . . . . . . . . . . . . 74

4.4.2 D1 - Variation of the Downcomer Diameter . . . . . . . . . . . . . . . . . . . . 75

4.4.3 D2 - Variation of the Riser Length . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.4 D - Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Conclusions and Recommendations 85

5.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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 109B.1 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.2 Enthalpy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.3 Momentum Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113B.4 Pressure Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

C Discretization of the Radial Heat Transfer Equation 119

D Deriving Analytical Model 125

1INTRODUCTION

The world energy consumption has increased sharply over the last century and this growth is ex-pected to continue in time. Moreover, the increase will not only be due to the rapid growth of theworld population, but also because of the increasing energy consumption per capita1. At the sametime, electric power becomes more and more economically accessible to the majority of the worldpopulation. In addition, the world’s total electricity coverage increases2.

Besides the increasing growth of the energy demand, it is expected that the worldwide emission ofCO2 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 alternativeenergy source that has abundant resources and a low carbon emission. However, renewable energytechnologies (wind, biomass, geothermal, hydro and solar) are not yet sufficient to meet the worlddemand of energy in the near future.

It is therefore expected that nuclear power remains an important contributor to a more sustain-able energy mix. However, as it is widely known that nuclear energy has some disadvantages. Forexample, the production of nuclear power goes hand in hand with the production of radioactivewaste, which must be stored in deep geological repositories. Furthermore, the fuel in nuclear powerplants distinguishes itself from all other sorts of fuel. In this process the nuclear reactor core pro-duces decay heat, also when the fission chain reaction has fully stopped. The removal of the decayheat in large-scale reactors is therefore dependent on active systems that operate on external powersources. The possible consequences of this dependency is demonstrated by the Fukushima Daiichiaccident. Here the facility was cut off from the power grid due to the earthquake, after which thediesel generators were destroyed by a tsunami, and the batteries flooded; due to the complete ab-sence of external power the reactors were not able to cool the core from its decay heat, this led to amelt down of the fuel rods in reactor 1 to 3.

Before this accident the support of nuclear energy had grown in the world3. By the press thisgrowth was even termed as the ‘nuclear renaissance’, but unfortunately, the extraordinary eventsin Fukushima led to a severe accident. It is to this end that there is now a large negative perception

1It is expected that the world population will grow to 8.2 billion people in 2030, and the average energy consumption willdouble.[1]

2The electricity coverage increases from 81% to 87% in 2030. The distribution of the main regions without electricitycoverage: Africa 58%, developing Asia 19%, Latin America 7%, Middle East 11%, developing countries 25%.[1]

3In 2009, by an investigation over 10,000 people in more than 20 countries, it was found that more than 66% of therespondents believed that their country should begin or increase the use of nuclear energy.[2]

1

2 1. INTRODUCTION

of nuclear power plants (NPP).

In 2000 the generation IV International Forum was founded, which resulted in six innovative reactordesigns to enhance the economics and safety of NPPs. Now, after the Fukushima accident, it hasbecome 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 reactortype is very promising, as they have a respectively low power density in relation to the size of thereactor. This creates the possibility to remove decay heat completely passively4 (i.e. without theuse of external power sources) in the event of a station blackout, increasing the safety of nuclearreactors.

1.1. SMALL MODULAR REACTORSAlthough the concept of ‘smaller size’ reactors has been known for a long time, the prejudice of ‘largescale’ reactors, and hence economies of scale have prevailed for a long time. However, in the courseof recent years, the validity of this paradigm has been questioned, as the complexity of reactors in-creased extensively. This goes hand in hand with rapidly increasing costs, delay in licensing, delayin 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 theselarge 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 re-actor 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 thereactor. To this end the design, safety and deployment advantages can overcome the economies ofscale of ‘large reactors’ [4].

As most SMRs are based on conventional reactors, there are multiple types of SMRs. This workfocuses on pressurized light water reactors with an integral structure, based on uranium fuel. Thisis decided, as these reactors have the largest possibilities to proceed to the construction phase inthe near future [4]. In this category the focus is on designs of the mPower and NuScale reactors thatare both designed and developed in the USA; these reactors are in the most advanced stage and areexpected to be built in the early 20s.

1.1.1. MPOWER

The mPower reactor, shown in Figure 1.1a, 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, theUS Department of Energy (DoE) selected mPower to be supported in the design and licensing, inorder to commercialize SMRs in the United States [6]. Therefore, the mPower design was awardedwith a governmental investment of $150 million, which had to be matched by at least 50% of privateinvestments [7]. It was expected by Generation mPower LLC that the overall capital costs of this de-sign 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

4Passive safety systems are according to the IAEA defined as: ‘Either a system which is composed entirely of passivecomponents and structures or a system which uses active components in a very limited way to initiate subsequentpassive operation’; whereby a passive component is indicated ‘a component which does not need any external input tooperate’.[3]

5The learning curve for such a reactor is overcome; for the new generation it is estimated that no major design costs needto be incorporated.

1.1. SMALL MODULAR REACTORS 3

(a) Cross-sectional viewof the mPower reactor.

(b) Cutaway view of thecomplete NuScale reactor.

Figure 1.1: A representation of the mPower and NuScale reactor.

and start production before all units are finished. However, Babcock and Wilcox failed to find cus-tomers or investors. This resulted in an announcement, in April 2014, declaring that it was cuttingits expenditure on mPower from $150 million a year to $15 million a year [8]. While initially it wasexpected that the mPower would be in commercial operation by 2022, the future of the mPower isnow unclear [7].

CHARACTERISTICS AND SPECIFICATIONS

This Light Water Reactor (LWR) is an integrated version of a conventional pressurized water reac-tor (PWR). With an integral design is meant that the Reactor Pressure Vessel (RPV) consist of a self-contained assembly with reactor core, control rod drive mechanisms, reactor coolant pumps, steamgenerator6 and pressurizer. Furthermore, the mPower design is based on a forced circulation flow.

The mPower reactor produces a thermal power of 425 MWth, of which an electric output of 125 MWecan be generated. This can be expanded by building multiple units. The reactor uses standard LWRfuel based on uranium with less than 5% enrichment. The core consists of 69 assemblies, whichhave a 17 x 17 structure, and has a fuel cycle of 4 years. At the end of the fuel lifetime, the entire corewill 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 riskof Loss Of Coolant Accidents (LOCA). A system based on gravity is proposed to remove decay heatfrom the reactor, where there is no reliance on emergency AC power for at least 72 [h] following anaccident [4]. For further safety the facility is built completely underground.

6In conventional PWRs the primary coolant flows inside tubes and the secondary fluid flows around these tubes. In caseof an integrated PWR this is inverted.[4]

4 1. INTRODUCTION

1.1.2. NUSCALE

The NuScale reactor, shown in Figure 1.1b, is designed and developed by NuScale Power LLC, whichwas founded to commercialize the Multi-Application Small Light Water Reactor (MASLWR). Thispreliminary NuScale design is developed in a cooperation of the Oregon State University, the IdahoNational Engineering Laboratory and Nexant-Bechtel.

As in the case of mPower it is expected by NuScale Power Inc. that the overall capital costs can bedecreased by a phased construction scheme. In this case a multi-modular configuration of reactorscan be created, of which each NuScale reactor is self contained and independent of the others. It isalso estimated by NuScale Power LLC, that for a multi-modular configuration facility the plants willoperate at full power for about 95% of the time. This can be explained by the fact that in the event ofmaintenance only one reactor has to be shut down, which makes it a very reliable system (i.e. highutilization rate).

Building on Obama’s Climate Action Plan, the US DoE selected NuScale Power LLC to support thisproject to design, certify and commercialize innovative SMRs in the USA [9]. In May 2014, the DoEand NuScale Power LLC signed a contract for funding and thus NuScale receives up to $217 millionin matching funds over five years [8]. NuScale Power, in contrast to mPower, found investors to backat least 50% of the DoE investments. NuScale believes that its first planned reactor in Idaho couldbegin commercial operation by 2023.

CHARACTERISTICS AND SPECIFICATIONS

Similar to the mPower design the NuScale reactor is an integrated version of a conventional PWR(pressure in primary circuit 12.7 [MPa]). By this means, the heat generated in the core is transferredwithin 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 neededas the flow of the primary coolant is based on natural circulation.

The NuScale reactor produces a thermal output of 160 MWth, from which an electric output of 45MWe can be generated. The core is based on a conventional LWR, in which the assemblies consistof a 17 x 17 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 pre-vent contamination of the environment. Their 7 layers include: the fuel cladding, the reactor pres-sure vessel, the containment vessel; the reactor pool (heat sink during accident), the concrete poolwalls and floor with a stainless steel liner (improving heat transfer by conduction) below groundlevel; the biological shield (on top of the free surface of the pool), and an HVAC (heating, venti-lating, and air-conditioning) reactor building (Seismic Category 1) [10]. In addition, just like themPower design, the integral system decreases the risks of penetrations of the primary loop. More-over, 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. More-over, and even more important, no external power is needed to circulate the coolant; by this passivesystem 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 (thehelical steam generator). In this event water is drawn from the containment cooling pool, and by

1.2. NATURAL CIRCULATION IN NUCLEAR POWER PLANTS 5

HOT LEG

COLD LEG

a b

Figure 1.2: Representation of a simple natural circulation loop.

means of spargers the steam enters the pool where it condenses. The initiation of the natural circu-lation flow is provided by feed-water accumulators. During this event both flows in the primary andthe secondary loop are based on natural circulation. The amount of pool water is large enough toprovide a 72 [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 builtaround the reactor pressure vessel, between which a vacuum is maintained. This reduces heat lossduring normal operation significantly, and thereby no insulation is required around the reactor ves-sel.

1.1.3. COMPARISON OF THE REACTORS

It is found that the SMR designs have a couple of general similarities: they are both based on aconventional pressurized LWR design; the reactors have an integral design; and both are placedunderground. In addition, both reactors are designed to have a strategy against extreme events, andboth can deposit decay heat passively (no need for external power) for over 72 [m]. However, thereare also some differences. So the NuScale design, contrary to the mPower, is based on a naturalcirculation of the coolant. Besides, the NuScale reactor is smaller, but the thermal power of themPower is 2.7 times larger. In general, it is found that the mPower design has a resemblance withconventional 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 waterreactors, and most importantly, the coolant circulation is based on natural circulation.

1.2. NATURAL CIRCULATION IN NUCLEAR POWER PLANTSIn a natural circulation loop the coolant is transported from the heat source to the heat sink withoutthe presence of pumps. In this case the pipes between the source and the sink need to be connectedin such a way that it forms a continuous path. In the event that the flow path is completely filledwith a fluid, a natural circulation flow could be initiated by addition of heat and in the presence ofa body force like gravity. Under the condition that the source and sink conditions are maintainedconstant, it is expected that a steady-state circulation can be achieved. Moreover, in the event thatthe 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

6 1. INTRODUCTION

density between the path from the source to the sink and vice versa. The working principle canbest be explained by introducing the simple loop in Figure 1.2. In the source the fluid gains heat,increases in temperature and decreases in density; the fluid becomes lighter and rises. In the sinkthe fluid loses energy, decreases in temperature and increases in density; the fluid becomes heavierand sinks. If the conditions of the source and sink are kept constant - the fluid absorbs an equalamount of energy in the source as is rejected in the sink - a steady-state natural circulation is es-tablished. In this process the density of the fluid in the ‚hot leg’ is assigned by ρhot , and in the‘cold leg’ by ρcold . To this end the hydrostatic pressure can be calculated for the extrema in the bot-tom of the loop; the pressure in (a) is pa = ρhot g H , and in (b) it is pb = ρcold g H . Where H is theheight of the loop, and g the gravitational acceleration; it is clearly that for ρhot < ρcold , there ex-ist a pressure difference pa < pb . Equivalently, the driving pressure is enhanced by the loop heightand the density difference. Here the driving pressure will be balanced out by the frictional pressure,∆pdr i vi ng =∆p f r i ct i on , and a steady-state natural circulation flow will set in.

1.2.1. INSTABILITIES IN NATURAL CIRCULATION FLOWS

Natural circulation (NC) systems (for both one-phase and two-phase flows) are more vulnerable forinstabilities than systems with a forced flow, which is due to the regenerative feedback that is in-herent to NC systems. For example, if the driving force is disturbed during a steady-state, the NCflow will be affected, which in turn will affect the driving force. This will consequently lead to anoscillatory behavior, which can be stable or unstable [11]. Here it is defined that a system is stable ifit returns to the original steady-state after a perturbation. On the other hand, if the system oscillateswith increasing amplitude the system is unstable.

In a nuclear reactor unstable flows are undesirable as they could deteriorate the control and per-formance of the reactor. In addition, and even more severely, flow oscillations could deterioratethe heat transfer (dry-out), which in a worst-case scenario could lead to damage to the core. It istherefore of great importance in the design of a reactor to have knowledge about the stability of theoperation conditions.

Over the years multiple types of flow instabilities have been witnessed in NC systems. The types thatcan occur in channel flows can be classified as static or dynamic instabilities [12]. A typical staticinstability 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 instabil-ity the Density Wave Oscillation (DWO) is most common. To explain the mechanisms of DWOs onemust consider a system that operates at steady-state, see Figure 1.2. By a perturbation of the drivingforce this instability can be triggered, and consequently, fluid packages with fluctuating mass flowrate are heated in the core. This will result into fluid packages with a low mass flow rate leaving thecore with a lower density than the relatively high mass flow rate packages. The result is that a densitywave oscillation will be traveling through the system, which can affect other variables (i.e. reactivityfeedback) in the reactor.

If there is a positive feedback in the system, e.g. an increase in the mass flow rate at the entranceof the core, it results in a decrease of the pressure drop in the riser, which in turn stimulates theincreases of mass flow rate of density wave. The DWOs will grow in amplitude and the system isunstable, the decay ratio > 1. For decaying DWOs, the decay ratio < 1, and the system is stable.

1.2.2. ADVANTAGES AND DISADVANTAGES

The major advantage of an NC system is its simplicity; the elimination of pumps in the systemgreatly simplifies the construction, operation and maintenance. Moreover, the elimination of pumps

1.3. AN INNOVATIVE ELEMENT: SUPERCRITICAL WATER 7

eliminates the scenarios with loss of flow, or related accident scenarios with pump failure. Further-more, the absence of pumps reduces the amount of pipes, which in turn changes the scenario of aLOCA favorably. Another major advantage that focuses on the safety of the reactor in an accidentscenario, is that an NC system enables the possibility that decay heat can be removed passively dur-ing a station blackout.

The primary disadvantage of a PWR NC system is the low buoyancy force (due to the small densitydifference in this system), resulting in a low mass flow rate. To increase the mass flow rate for afixed power, either the height of the loop must be increased, or the resistance in the loop mustbe decreased. Moreover, it is due to the low mass flow rate that the maximum channel power islower, and thereby, core volume is larger compared to forced circulation systems. Besides, largecore volumes could lead to stability problems, where NC systems are already inherently less stablethan 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 orderto increase the heat flux for NC systems. Finally, NC systems must be able to start up from a stagnantfluid, low pressure and low temperature. In order to avoid premature occurrence of CHF, unstablezones must be avoided during the pressure and power raising process. Therefore, it is essential thatthere is knowledge of a start-up procedure, which avoids these instabilities. It is evident here thatthe start-up of an NC system is more complicated than for a forced circulation system.

1.3. AN INNOVATIVE ELEMENT: SUPERCRITICAL WATERA large step further towards sustainability can be made with an SMR that is based on supercriticalwater.

WHAT IS SUPERCRITICAL WATER?

Water can be distinguished in three physical states: solid, liquid and vapor, see Figure 1.3a. Byincreasing the temperature and pressure along the vapor-liquid equilibrium7 both phases couldcoexist, although in this process the vapor becomes denser and the liquid less dense. At one pointon this line only one phase can exist in the system, which is given as the critical point (pc = 22.1[MPa], and Tc = 374 [°C]). By continuation the saturation line becomes the pseudo critical line. Forthe region that is initiated by the critical point no boiling will occur. The pseudo critical point thatis used in the operation of the HPLWR is pps = 25 [MPa], and Tps = 385 [°C].

1.3.1. ADVANTAGES AND CHALLENGES

The primary advantage of a system based on supercritical water is that higher core exit tempera-tures can be achieved than in conventional PWRs. The result of this is that a high thermal efficiencyin the conversion of heat to electricity can be achieved, i.e. 44% compared to 34% for conventionalPWRs. The usage of supercritical water is not new and already being applied in fossil fuel powerplants around the world [13]. Moreover, the SCWR is one of the innovative reactor designs by theGeneration IV International Forum, making it one of the promising features in the future of nuclearreactors.

Another advantage is that the boiling of the coolant is eliminated when the reactor operates abovecritical pressure. In this event the transition from liquid to vapor is a continuous process, so thecoolant remains in a single-phase throughout the whole system. The result of this is that some sim-plifications could be made in the design, by eliminating steam separators and dryers.

7Also known as the saturation line.

8 1. INTRODUCTION

(a) Pressure versus temperature dia-gram for water, whit pseudo criticalline.

(b) Several properties of water versus the temperature,near the pseudo-critical point at a pressure of 25 [MPa].

Figure 1.3: Representations of the different physical phases of water, and the properties of water near the pseudo-criticalpoint.

The third advantage is that supercritical water has a large drop in the density around the pseudo-critical point. The result of this is that a larger buoyancy force can be achieved, which in turn couldresult in higher mass flow rates. Hence, the disadvantage of a low buoyancy force in NC systems iseliminated, and thus lower vessel heights can be expected.

Finally, Figure 1.3b shows that there is a peak in the specific heat capacity at the pseudo- criticalpoint. Due to this peak large quantities of energy can be absorbed and transported to the heat ex-changer. In addition, in case of an accident this peak can serve as a buffer, in which large quantitiesof energy can be stored to avoid excessive temperatures.

The primary challenge that must be overcome is the corrosion of the materials that are in contactwith the supercritical water. During nominal operation these materials are affected by combinedthermal, radiative and thermochemical stresses, and must withstand these over the lifetime of thereactor. To this end, the feasibility of this concept depends on whether these materials can be found.

1.4. THE SLIMR

By introducing the SLIMR (a Small-scale, Large efficiency, Inherently safe, Modular Reactor) it hasbeen 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, theSLIMR operates at 150 MWth at a pseudo-critical pressure of 25 MPa, which increases the efficiencyof the SMR. Moreover, the SLIMR has been designed as a semi supercritical water reactor, to thisend the temperature interval is 280-400 [°C] compared to 280-500 [°C] for the large scale HPLWR.The result of this is that the materials in the SLIMR will suffer less corrosion. The minimum tem-perature is kept at 280 [°C], since it is indicated by Dobashi et al. [14] that this temperature has anoptimal thermal efficiency.

Additionally, the supercritical water leads to higher driving forces, which leads to smaller reactordesigns with larger surface-to-volume ratio. This accompanied with a low thermal core power, en-ables the possibility to deposit decay heat through the ‘skin’ (the RPV ) of the reactor. Moreover, this

1.4. THE SLIMR 9

is in comparison to the NuScale design even more passive8. Furthermore, since the SLIMR designruns on natural circulation, the possibility to remove decay heat during a station blackout is cre-ated. In addition, the supercritical water has a simulating effect on the heat transfer as it increasesthe temperature gradient over the RPV (from the downcomer to the pool), and thus increasing thefeasibility 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 inno-vative idea. However, the combination of an SMR with supercritical water as coolant that is drivenon natural convection is. This pool must be sufficiently large so that it can passively transport theheat lost by the SLIMR during nominal power, and it should be capable of removing the decay heatentirely in a passive way. The primary advantage of the pool is that pipes for the supply of coolingwater 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 thesurroundings, and thus decreasing the chances of contamination of the environment.

By combining an integral small modular reactor that is cooled by supercritical water and runningon natural circulation it is the goal to deliver a safe and sustainable reactor that meets today’s needsand wishes. See Figure 1.4 for a representation of the SLIMR.

Figure 1.4: 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 doneby combining the best elements of the NuScale, MASLWR and HPLWR in the design of the SLIMR.

1.4.1. 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 NuScale9

8In the NuScale design, the decay heat is removed by a natural circulation flow in the secondary loop, which sucks upcool water from the pool and returns steam back into the pool. However, this system needs to be mechanically (valves)initiated.

9The NuScale reactor is the commercialized version of the MASLWR, but similar in design.

10 1. INTRODUCTION

Figure 1.5: 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 secondarycontainment is left out in order to make the heat transport possible via the vessel wall, see Figure1.5 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 designis not based on the MASLWR, which in turn is based on a conventional LWR core. This type of coreis 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 mostdesigns by the use of an extra flow area in the middle of the assembly, which provides extra moder-ation in this region. In order to increase the feasibility of this reactor the aim is to work with designsthat are already in development. To this end the fuel assemblies as defined for the European HPLWRare 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 25.0 [K W /m] [15]. These core assemblies are designed to have anactive length of 4.20 [m], and in order to make the least possible changes to existing techniques it ishighly recommended that this length is maintained.

REACTOR PRESSURE VESSEL

For the large scale HPLWR it is found that the thickness RPV is 45 [cm] [16], although it is ex-pected that this will be less for the SLIMR because of the smaller vessel. Besides, it is found thatthe MASLWR has a RPV thickness of 12.7 [cm], and operates at a pressure of 8.6 [MPa]. For a vesselof similar design and geometry the thickness of the vessel can be determined by a linear interpola-tion. To this end it can be determined that the vessel thickness of 37 [cm] will be sufficient in thepreliminary design of the SLIMR.

1.5. THESIS OBJECTIVE 11

1.5. THESIS OBJECTIVEThe main objective of this thesis is to perform the first calculations on the SLIMR in terms of ther-mal 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 situa-tions?

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 operationpoint, 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 en-vironment 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 bound-aries of a safe nominal operation ànd allows safe deposition of decay heat to the environmentunder accidental situations in a fully passive way, without damaging the core?

1.5.1. PHASE AThe first step is to determine if it is possible for the SLIMR design to safely operate at nominal con-ditions. Therefore we are primarily interested in whether it is possible to achieve natural circulationin steady-state at the desired operation point. Secondly, it is important to know if the mass flowprovided by natural circulation is sufficient to cool the fuel and cladding. To assess the sufficiencyof natural circulation one has to investigate whether heat transfer deterioration - i.e. the rapid de-crease of heat transfer capabilities at temperatures close to the pseudo-critical point of the super-critical coolant - occurs or not. In this thesis the criteria for this assessment are based on the workof Pioro et al. [17]. The last step is that it must be determined if the steady-state operation pointsare stable or unstable. This must be investigated, as it is known that natural circulation systems aresensitive to dynamic instabilities, because of the large density differences in the flow channels. Forthis last step the steady-state operation points are perturbed in order to find out if they return totheir original steady-state operation point or not.

All the issues mentioned above will be investigated as a function of the most basic parameters ofthe SLIMR. The parameters that will be varied are as follows: the riser diameter, the outer annulusdiameter (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 assessingthe 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 beanswered 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?

12 1. INTRODUCTION

1.5.2. PHASE BIn the assumption that appropriate steady-state operation points are found in Phase A, the secondstep is to calculate the heat transfer from the SLIMR to the pool during normal operation. For thisthe total heat loss from the SLIMR, the exterior surface temperature of the reactor vessel and thedominating heat transfer phenomena between the vessel and the surrounding water in the poolwill be determined.

In this step the following basic parameters are varied: the width of the vessel (the outer annulusdiameter), the height of the vessel (the riser length), the vessel wall thickness, the temperature of thepool 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?

1.5.3. PHASE CIn Phase C it is determined whether it is possible to transfer the heat from the SLIMR via the pool tothe environment during nominal conditions in a completely passive way. To answer this questionthe results of the previous phases will be used. The goal is to design a pool with an average tem-perature around 40 °C in steady state. Furthermore, the evaporation rate10 of such a pool will beinvestigated. 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?

1.5.4. PHASE DIn the last phase it will be determined whether SLIMR is able to safely deposit its decay heat during astation blackout (i.e. without external power). For the design to be passively safe it must be verifiedthat the coolant temperature does not exceed the maximum design temperature of 600 [°C] andthat no deterioration of heat transfer occurs during the cooling period. During this step it mustbe investigated whether the SLIMR returns to a stable natural circulation flow after the first stagesof the transient. Furthermore it is determined which heat transfer phenomenon (at the externalsurface 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 Athat is able to deposit its decay heat completely passively, without causing any damage to theinternal 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 blackoutaccident situation as well?

1.6. THESIS OUTLINEThe outline of the thesis is as follows. After this introductory chapter, the theory is introduced inChapter 2. The solution algorithm is elaborated in Chapter 3. The results of the transient calcula-tions are presented in Chapter 4, followed by conclusions and recommendations in Chapter 5.

10It must be noted that during steady-state the pool is continuously filled in order to make up for the evaporation.

2THEORY

This chapter reports on several basic topics relevant to this study, and are included in the transientmodel that describes the behaviour of the SLIMR. We approach this work from the inside out, de-scribing 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.

2.1. THERMAL HYDRAULICSThe fluid flow in the reactor is described by three conservation equations [18]; the continuity equa-tion 2.1 converses the mass in the system, the momentum equation 2.2 keeps track of the conserva-tion of momentum, and the conservation of energy is preserved by equation 2.3 in terms of internalenergy u:

∂ρ

∂t+∇· (ρ~v) = 0 (2.1)

∂

∂t(ρ~v)+∇·ρ~v~v =−∇p +∇ ¯τ+ρ~g (2.2)

∂

∂t(ρu)+∇·ρu~v =−∇·~q" +q

′′′ −∇p ·~v + ( ¯τ : ∇~v). (2.3)

In the following subsections the conservation equations are further described, and the energy equa-tion will be converted to a equation in terms of enthalpy. Further all balance equations are rewrittento a one-dimensional form allowing them to be implemented in the system code.

2.1.1. CONSERVATION OF MASS

The one-dimensional continuity equation is derived by integrating equation 2.1 over a control vol-ume dV to obtain equation 2.4. By utilizing Gauss’s divergence theorem [19] the convective termof the first part of equation 2.4 can be rewritten, where the integration is now carried out over thecross-sectional area A, in the direction perpendicular to the flow, this is given as:Ñ

V

∂ρ

∂tdV +

ÑV

∇· (ρ~v)dV =V∂ρ

∂t+

ÏA

(ρ~v)d A. (2.4)

Where ρ is the density and ~v the velocity of the fluid, both variables are constant over the cross-sectional 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 acomponent in the x-direction. By defining the control volume with a fixed cross-sectional area, the

13

14 2. THEORY

width can be set to an infinitesimal limit ∆x; this term ∆x is divided out of equation 2.4, into:

A∂ρ

∂t+ lim

∆x→0

[Aρvx ]x+∆xx

∆x= A

∂ρ

∂t+ ∂M

∂x= 0, (2.5)

resulting in the one-dimensional continuity equation, where M is the mass flow rate.

2.1.2. CONSERVATION OF MOMENTUM

Analogously, an expression for the one-dimensional momentum equation 2.2 can be found, result-ing in:

A∂(ρvx )

∂t+ ∂

∂x(Aρv2

x ) =−A∂p

∂x−

∮Pw

τw dPw +ρ~g A. (2.6)

In this expression the first term on the left side is the inertia term, and the second one is the con-vective term. The right side starts with the pressure term, followed by the friction term, and themost right term is the gravity term. In this expression the friction term is still notated as an integralover the perimeter of the wall, the wetted perimeter Pw . This perimeter doesn’t only contain theouter perimeter of the tube, but it also includes possible obstacles in the flow path, which have theirlength in the flow direction. Therefore the integral over τw runs along the complete cross-sectionalgeometry of the flow path.

After integration, the average wall shear stress τw is substituted by a relation for the wall friction.This relation proposed by Darcy-Weisbach, is described in work of Todreas and Kazimi [18], giving:∮

Pw

τw dPw = τw Pw = fM 2

2Dh Aρ. (2.7)

Two new parameters are introduced here: the Darcy-Weisbach friction factor f which will be men-tioned later in this section, and the hydraulic diameter Dh which is given as:

Dh = 4A

Pw(2.8)

In this equation the hydraulic diameter, Dh , is a characteristic length that can be used when han-dling flows in non-circular geometries.

By incorporating equation 2.7 into equation 2.6 closer to the final one-dimensional momentumequation. However, the design of the SLIMR does not merely consist of straight tubes. Elementsthat could obstruct the flow should be taken into account. One should think of bends, contrac-tions, expansions, the core inlet friction and others. These obstructions will be incorporated intothe momentum equation as a local pressure loss term, ∆pi . The pressure term now becomes:

A∂p

∂x= A

∂

∂x

(p +∑

i∆pi Hstep (xi )

)

= A∂p

∂x+ A

∑i∆piδ(x −xi ). (2.9)

Here the pressure loss term is described as:

∆pi = KiM 2

2A2ρ. (2.10)

2.1. THERMAL HYDRAULICS 15

Here Ki , an empirically found friction coefficient for diverse kinds of flow obstructions, is given inwork of Janssen and Warmoeskerken [20].

The final one-dimensional momentum equation can now be obtained by substitution of equations2.7 and 2.9 into equation 2.6. The momentum balance now becomes:

∂M

∂T+ ∂

∂x

(M 2

Aρ

)=−A

∂p

∂x−∑

iKi

M 2

2ρAδ(x −xi )− f

Pw M 2

8A2ρ+ Aρg (2.11)

In its final form, the last parameter that needs to be defined is f , the Darcy-Weisbach friction fac-tor1. Usually, this factor is presented as a function of the Reynolds number. For this reason theReynolds 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 thefluid. Osborn Reynolds discovered that the flow regime mainly depends on the ratio of inertia forcesto 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|vx |Dh

µ= |M |Dh

Aµ. (2.12)

Here the Reynolds number is given for a one-dimensional system with a cross-sectional area aver-aged velocity vx , or mass flow rate M . The characteristic length is given by the hydraulic diameterDh , which makes equation 2.12 also applicable for non-pipe flows, and µ is the dynamic viscosityof the fluid.

The flow regimes can be categorized - based on the Reynolds number - in three classes2; laminarwhen Re < 2300, transient for 2300 < Re < 4000, and turbulent for 4000 < Re. As in this work thesystem is modeled in one-dimension, the physical effects, due to the flow regimes, are approximatedby 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) forfully developed flows, where each is dedicated to an interval of Reynolds. In this work the mostreferenced correlations are used; combined they cover the whole range of Reynolds that is of ourinterest. These correlations are given by:

Poisseuille Re < 2 ·103 f = 64 ·Re−1 (2.13)

Blasius Re < 3 ·104 f = 0.316 ·Re−0.25 (2.14)

McAdams 3 ·104 <Re < 106 f = 0.184 ·Re−0.20 (2.15)

Haaland 4 ·103 <Re < 108 f =(−1.8 · log

[6.9

Re+

(ε

3.7Dh

)10/9])−2

(2.16)

Ishigai Re < 108 f = 0.2187 ·Re−0.224 (2.17)

These correlations are developed for circular tubes but can be used analogously for other flow ge-ometries, using the hydraulic diameter as equivalent of the diameter.

1The Darcy-Weisbach friction factor should not be confused with the Fanning friction factor, which is four times smaller.2The values that mark the boundaries of the flow regimes in this text are characteristic for pipe flows.

16 2. THEORY

However, for the laminar regime - where the shear of the velocity gradient is significant throughoutthe entire flow cross-section (see Figure 2.1a) - the correlation is specific to the flow geometry3. Incase of a turbulent flow regime - in which the velocity gradient is mainly near the wall (see Figure2.1b) - the geometry of the cross-sectional flow area is is of less importance. Therefore the conceptusing of the hydraulic diameter is more accurate in predicting the friction factor in the turbulentregime.

(a) Laminar regime. (b) Turbulent regime.

Figure 2.1: The temperature (T is shown relative to Ts ) and velocity gradients at the wall for the laminar and turbulentregime. It can be seen that gradient for the turbulent regime is steeper and moreover, mainly near the wall. As a result thewall shear stress and heat transfer rate are larger in the turbulent regime than they are in the laminar regime. [21]

In this work the correlations are linked in a way that the transitions between the correlations aresmooth, this is to exclude instabilities that can be induced by step wise transitions. In the simula-tions the flow is predominantly turbulent, mostly utilizing the Haaland correlation for high Reynoldsnumbers with the exception of the warming up stage and the SCRAM (emergency shutdown of a nu-clear reactor, and an acronym for Safety Control Rod Axe Man) of the system.

The last relation 2.17 (stretching over the complete range of Reynolds numbers) is known as theIshigai correlation, which correlation is infrequently used. However, it is of importance in thiswork, since it is used to benchmark the one-dimensional model derived in this work with the one-dimensional model developed by Chatoorgoon et al.[22] This correlation is also developed by Cha-toorgoon et al. using the data that are obtained by Ishigai et al.[23], who studied the friction andheat transfer for water flow in tubes at supercritical pressures.

2.1.3. CONSERVATION OF ENERGY

Finally, the conservation of energy will be derived in one-dimension. Prior to this, the energy bal-ance - given in the form of the internal energy equation 2.3 - will be rewritten in terms of enthalpyh. Therefore the definition of enthalpy is introduced by:

h ≡ u + p

ρ. (2.18)

Applying this definition into the equation for the internal energy (equation 2.3), we obtain, aftersome rearranging, the energy equation in terms of enthalpy:

∂

∂t(ρh)+∇·ρh~v =−∇·~q" +q

′′′ + Dp

Dt+ ( ¯τ : ∇~v). (2.19)

This equation can now be simplified, focusing on the system that is modeled in this work. There-fore, since generation does not exist in this fluid, the production term can be eliminated. Also the

3The Poissuille correlation is developed for circular tubes.

2.2. HEAT TRANSFER 17

shear term can be neglected, because for a system with moderate velocity gradients in combinationwith a low viscous fluid like water, heat generation by wall friction will be insignificant. The workdone by the pressure can also be neglected since its contribution is insignificant in respect to thethermal power of the core.

In the next step, equation 2.19 is reduced to a one-dimensional energy equation. Employing thesame procedure as for the conservation of mass and momentum gives:

A∂ρh

∂t+ ∂Mh

∂x= q ′. (2.20)

The conservation of energy by this enthalpy equation is a reduced form of the energy equation 2.3.This clarifies why this form has been chosen, with only a few variables to account for, making thenumerical implementation much easier.

The last variable that is defined is the heat rate q ′, averaged over the wall perimeter, also indicated asthe linear heat rate. In this work the assumption is made that the thermal core power is transferredinstantaneously from the core (the heat source) to the coolant. Analogously, an equal amount ofheat is transfered instantaneously from the coolant to the heat exchanger (the heat sink). The linearheat rate is obtained by dividing the thermal core power by the effective core height, or the lengthof the heat exchanger section. For instance, the core heat rate is defined as,

q ′Cor e =

PCor e

LCor e. (2.21)

And vice versa for the heat exchanger, the linear heat rate is defined as:

q ′H X = −PCor e

LH X. (2.22)

The heat transfer between the SLIMR and the environment will be discussed in the next section.

2.2. HEAT TRANSFERHeat flows from hot objects to cold ones; e.g. the heat flow from the respectively hot RPV to the coolpool it is submerged in. The basic three physical mechanisms that are responsible for this transportof energy can be listed as: conduction, convection, and radiation. These mechanisms can workindependently or together, as for this problem. In this section the mechanisms will be explainedand specified on the heat transfer problem [21].

2.2.1. CONDUCTION

In the situations where a temperature gradient exists over a solid material, like the stainless steelcontainment of the RPV, or stagnant fluid, the mechanism of heat transfer is conduction. Heat trans-fer 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-stationarydiffusion equation without convection and internal heat production, given as:

ρcp∂T

∂t=λ∇2T. (2.23)

18 2. THEORY

In an analogous manner as applied for the mass, momentum and energy equation in the previoussection 2.1, an one-dimensional expression for equation 2.23 can be found. The one-dimensionalradial conduction equation can now be written as:

ρcp∂T

∂t= 1

r

∂

∂r

(rλ

∂T

∂r

). (2.24)

Here cp is the specific heat, ρ the density, T the temperature and λ the thermal conductivity. Thethermal conductivity is a property that can be considered as a heat resistance coefficient of themedium. The lower this coefficient the worse the conductive heat transfer. For a metal the thermal

Table 2.1: The density, specific heat, and thermal conductivity for some materials represented in this work.

ρ [kg /m3] cp [J/kg · °C] λ [W /m · °C]

stainless steel 8.00 ·103 0.49 ·103 19 (at 225°C)Water 1.00 ·103 4.21 ·103 0.58 (at 20°C)Sandy Soil 1.60 ·103 0.80 ·103 0.30

conductivity stainless steel is relatively low. However, in contrast is the thermal conductivity of fullystagnant water or sandy soil is very low. Here the thermal conductivity of sandy soil is representativefor other ground types. Table 2.1 presents the ρ, cp and λ for a selection of media.

In the cooling problem of the SLIMR, the stainless steel vessel is trapped between two fluid layers. Tosolve this conduction problem two boundary conditions need to be applied [24]. The first boundarycondition is applied to the inner surface of the vessel where a convection boundary condition isutilized, and the heat flux is specified by Newton’s law of cooling, given as:

λ∂T

∂r= h (Tw −Tb) . (2.25)

In which h is the heat transfer coefficient, Tw is the wall temperature and Tb the bulk temperatureof the water in the downcomer. Here the values of h and Tb are given, and the wall temperature Tw

is yet unknown. The second boundary condition is applied to the outer surface of the vessel, and isspecified as a heat flux, given as:

λ∂T

∂r= q ′′ = q

2πr H. (2.26)

Here q ′′ is the heat flux, q the heat rate and H the height of the cylinder. This boundary condition ischosen because the heat transfer to the pool is given by multiple correlations, that are different foreach boiling regime.

2.2.2. CONVECTION

Solid materials adjacent to stagnant fluids will transfer heat by conduction at all times. However, inthe presence of a bulk fluid motion, the physical mechanism of convection prevails. During con-vection regions with hot fluid are replaced with cooler fluid. It is a combination of conduction andrefreshing of the adjacent fluid, which effectively enhances the heat transfer. Here is the refresh rateof the fluid is a strong function of the fluid motion; the higher the velocity of the fluid, the higherthe convective heat transfer.

The initiated motion of the fluid can be natural or forced and is used to classify the convective heattransfer. In natural (or free) convection, the fluid motion is caused by differences in density. By thisnatural means, the warmer fluid rises by buoyancy and the cooler fluid falls. In the event of forcedconvection the fluid motion is initiated by an external force. In this way the surface is provided with


a continuous fresh feed of fluid.

As for conduction, the driving force in convection is the difference in temperature. The governingequation of heat convection is given by Newton’s law of cooling, written as:

qconv = hs(Ts −T∞). (2.27)

In which Ts is the surface temperature of the solid medium, T∞ the temperature of the bulk fluidsufficiently far from the surface, and hs is the convectional heat transfer coefficient. The convec-tional heat transfer coefficient hs is analogous to the thermal conductivity λ, but is not a propertyof the fluid. It is a parameter that is experimentally determined and depends on the nature of thefluid motion, the fluid velocity, several properties of the fluid, the surface geometry and the boilingregime. The typical orders of magnitude for convectional heat transfer coefficient can be seen in Ta-ble 2.2. The heat transfer coefficient can be calculated with the use of the Nusselt number Nu, the

Table 2.2: Typical order of magnitude of the convectional heat transfer coefficient.[21]

Type of convection h [W /m · °C]

Free convection of gases 2-25Free convection of liquids 10-1000Forced convection of gasses 25-250Forced convection of liquids 50-2500

characteristic length Lc and the thermal conductivity of the fluid λ. Rearranged the Nusselt numberis given as:

Nu = convective heat transfer

conductive heat transfer= hsLc

λ, (2.28)

Here the Nusselt number is the ratio of convective to conductive heat transfer at the boundary be-tween solid and fluid. It represents the improvement of the heat transfer through the fluid layer iffluid motion is present. The larger the Nusselt number, the more efficient the heat transfer due toconvection.

For many engineering problems a correlation for the Nusselt number can be found. This empiricalrelation is in most cases dependent on the Prandtl number. The Prandtl number - which is also adimensionless parameter - describes the relative thickness of the thermal and velocity boundarylayer, and is defined as:

Pr = molecular diffusivity of momentum

molecular diffusivity of heat= µcp

λ. (2.29)

The Nu and Pr number are used to describe both forced and natural convection, which will furtherbe 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 isforced by natural circulation. This natural circulation flow is approximated by the one-dimensionalequations derived in the previous section 2.1. Due to the fact that it is a one-dimensional flow, allproperties are area averaged over the cross-section perpendicular to the flow direction. The heattransfer correlations that are used in this work must therefore be evaluated with only the bulk prop-erties of the fluid.

20 2. THEORY

To begin with the laminar flow regime, where analogous to the Darcy-Weisbach friction factor theheat transfer is dependent on the geometry (see subsection 2.1.2). The Nu correlation for an annularduct 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 byequation 2.30. However, the effect that this approximation causes will be of a conservative nature, inwhich the heat transfer is estimated to be lower. Furthermore, the flow is predominantly turbulentand therefore the turbulent regime is preferably modeled with more care.

Laminar flow Re < 2 ·103 − Nu = 3.66 (2.30)

Dittus–Boelter 1 ·104 <Re 0.7 ÉPr É 160 Nu = 0.023 ·Re0.8Prn (2.31)

The heat transfer in the turbulent regime is less geometry dependent (see subsection 2.1.2). There-fore, the annular duct can be treated as a non-circular duct, in which the hydraulic diameter of theannulus Dh = Do −Di . The empirical correlation utilized in this work is the conventional Dittus-Boelter correlation, equation 2.31, 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 (2 · 103<Re<1 · 104),the Dittus-Boelter correlation is used as a rough estimate, whereby the closer to Re=1 ·104, the moredesirable the approximation is.

The Dittus-Boelter correlation is the most widely used heat transfer correlations at subcritical pres-sures. However, in this work the convective heat transfer must be estimated at supercritical pres-sures. It was found that the Dittus-Boelter correlation might produce some unrealistic results [17].These errors concentrate in particular around the critical and pseudo-critical points, where theDittus-Boelter equation is very sensitive to property variations. Due to difficulties around the crit-ical and pseudo-critical points, especially for turbulent flows and at high heat fluxes, multiple cor-relations for supercritical fluids are derived by experimental data. These derived correlations, ofwhich many are based on the conventional Dittus-Boelter equation, showed only to be more or lessaccurate within a particular dataset.

In this work the heat loss is evaluated from the bulk fluid in the downcomer to the stainless steelRPV. Here the fluid properties are predominantly below the pseudo-critical point, and the heat fluxis respectively low. Therefore it is reasonable to utilize the basic Dittus-Boelter equation calculatingthe convective heat transfer from the bulk fluid in the downcomer to the stainless steel RPV. In thissubcritical regime (liquid), the correlation predicts the heat transfer with an overall-weighted aver-aged error of 10.4%[17]. On the other hand, if the fluid in the downcomer comes in the superheatedregime (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 section2.4), heat will be transferred by natural convection. And by the same phenomenon, the pool waterexchanges 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 orientationof the surface. In this work, the geometries of the RPV and pool consist of simple shapes as well asstandard orientations (vertical or horizontal). Therefore the Nu number can be given by empiricalcorrelation, that are most well-known and frequently used in engineering. In this thesis the RPV isgiven by a correlation for a vertical cylinder.


The correlations for natural convection are mostly functions of the Grashof number, which is a di-mensionless number that is given by:

GrL = buoyancy force

viscous force= gβ(Ts −T∞)L3

c

ν2 (2.32)

Here subscript L designates the characteristic length of the geometry it is based on, g is the grav-itational acceleration, β the volumetric expansion coefficient, Ts the surface temperature, T∞ thetemperature of the fluid sufficiently far from the surface, and ν is the kinematic viscosity. Thereforeall fluid properties need to be evaluated at the film temperature, given as T f = (Ts +T∞)/2 [21]. TheGrashof number is analogous to the Reynolds number in forced convection. The fluid flow becomesturbulent as the Grashof number exceeds a certain critical value4. Together with the Pr number theyform the Rayleigh number, given as:

RaL = GrLPr. (2.33)

In which the Rayleigh number is used in most empirical correlations, as a simple function of RaL .

When modeling the RPV, the surface of a vertical cylinder can be treated as a vertical plate, and theNu number is given by equation 2.35. Applying this approximation the following criteria must besatisfied:

D ≥ 35L

Gr 1/4L

. (2.34)

When this condition is met, the diameter of the cylinder is sufficiently large so that the effects by thecurvature can be neglected. In this correlation the characteristic length is given by the length of thecylinder.

Vertical wall Entire range of Ra Nu ={

0.825+ 0.387Ra1/6L[

1+ (0.492/Pr)9/16]8/27

}2

(2.35)

The convective heat transfer at the free surface of the pool is modeled using of a correlation that isspecific for a pool with a still water surface. It determines the convectional heat transfer coefficientof the free surface of the pool with and without a wind blowing over the free surface. This correlationby Hahne and Kubler (1994)[25] can be written as:

h[W /m · °C] = 3.1+4.1 · v[m/s] (2.36)

In which v is the air velocity in [m/s]. Due to the fact that the pool is roofed it is hereby assumedthat there is no wind blowing over the pool, and v = 0.

2.2.3. RADIATION

Heat transfer by radiation is fundamentally different from conduction and convection. Heat trans-fer by radiation is the only heat transfer mechanism that can overcome a vacuum. Every object andmedium is continuously emitting heat by radiation. In this process heat is transferred by electro-magnetic radiation (or photons). The radiative heat rate of an object can be evaluated by combiningthe Stefan- Boltzmann’s law with the emissivity of the objects surface, given as:

qr ad = εσSB T 4s (2.37)

Here Ts is the surface temperature of the object in [K], σSB = 5.670 · 10−8 [J/sm2K4] is the Stefan-Boltzmann’s constant, and ε is the emissivity of the object surface material; a value between 1 and

4For a vertical plate the flow regime becomes turbulent for Grashof numbers bigger than 109.

22 2. THEORY

0.

When modelling the heat transfer from the SLIMR to the pool, the radiative heat transfer is ne-glected, which is due to the low emissivity of the RPV (polished stainless steel has an emissivityεss = 0.17 [21]), and due to the respectively low temperature difference between the RPV and thesurrounding surfaces. Moreover, the effect of radiation is minimal compared to the other mecha-nisms of heat transfer. As a result the evaluated heat transfer from the RPV to the pool is lower thanthe actual heat transfer, which makes the approximation conservative.

When modelling the heat loss from the pool, the radiative heat transfer is often significant comparedwith conduction or natural convection (i.e. for gasses). To this end, heat loss by radiative heattransfer is included in the model. Furthermore, water has a high emissivity εw = 0.96 [26]. In theevaluation of the heat loss, the net radiative heat transfer is of interest. With this it is assumed thatthe pool is completely enclosed by a much larger surface (the ceiling of the facility), that is separatedby a medium (air) which will not intervene with the radiation. In this special case the surface areaof the surrounding surface has no effect on the net radiative heat transfer.

2.3. HEAT TRANSFER DETERIORATIONThe Heat Transfer Deterioration (HTD) is characterized by lower values of the wall heat transfercoefficient compared to normal heat transfer. Here the normal heat transfer refers to the forcedconvective heat transfer at subcritical pressures far from critical or pseudo-critical regions, and eval-uated with the conventional Dittus-Boelter equation 2.31. The phenomenon of HTD is extremelyundesirable for the core section of the SLIMR. In an event where deterioration of heat transfer oc-curs, the energy that is released in the core can accumulate. This will lead to an increase of the innertemperature of the core, and may lead to temperatures above the maximum allowable temperatureof 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. (1971, 1967)[27] [28], and is brought back into the spotlight by Pioro et al. (2005) [17]. Vikhrev et al. conductedexperiments on supercritical water flowing through vertical pipes. These experiments were per-formed 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 param-eters can be found in table 2.3. In general, the deterioration of heat transfer occurs at high heat

Table 2.3: Range of investigated parameters for experiments with water flowing in vertical circular tubes at supercriticalpressures.

Reference p [MPa] T [°C] [h in kJ/kg] q [MWth/m2] G [kg/m2s] Flow geometry

Vikhrev et al. (1967) 24.5, 26.5 hb = 230–2750 0.23–1.25 485–1900 St. st. tube (D = 7.85, 20.4 mm; L = 1.515, 6 m)

fluxes. It was found by Vikhrev et al. that for a mass flux of 495 [kg/m2s] two types of deterioratedheat transfer exist:

1 The first type of HTD is due to the flow structures at the entrance region of the tube. Accordingto Vikhrev et al. (1971) this type of HTD can be avoided, to this end the following conditionmust be satisfied:

L

D> 40−60. (2.38)

In which L is the heated length and D the tube diameter. Moreover, this type of HTD onlyoccurs at low mass fluxes and high heat fluxes, whereas at high mass fluxes this type of HTDwill disappear.

2.4. BOILING REGIMES 23

2 The second type of HTD appears at any section of the tube (within a certain enthalpy range,see Table 2.3), and occurs when the wall temperature exceeds the pseudo-critical tempera-ture. This type of HTD can, according to Vikhrev et al. (1967), be prevented when the followingcondition is satisfied:

q ′′

G= Pcor e /Api n

M/Acor e< 0.4 [kJ/kg]. (2.39)

In which q ′′ is the heat flux in [kW/m2], and G is the mass flux in [kg/m2s], Pcor e is the totalcore power in [kW], Api ns is the total surface area of the all fuel pins, M is the mass flow rate,and Acor e the cross-sectional flow area of the core.

However, these conditions are not enough for a clear identification of HTD; but they can be usedas criteria in engineering. In this work both conditions are applied, and are met in the design ofthe SLIMR. Therefore, HTD is excluded by design and the wall temperature of the fuel pins will notexceed the pseudo-critical temperature.

The first condition is met by integrating the core design of the HPLWR into the SLIMR design. Tosatisfy the second condition is more complex, this is so since equation 2.39 is a function of the heatflux and mass flux. In this work the mass flux - due to the natural circulation - is a function of thecore power and geometry of the SLIMR. The solution to equation 2.39 which will be referred to bythe ’HTD rate’.

2.4. BOILING REGIMESThe RPV transfers its heat by natural convection during operational power. In an event where thepool water temperature increases to its saturation temperature Tsat (100°C at 1 atm), saturated boil-ing occurs. In addition, boiling can be qualified as sub-cooled when the bulk temperature of thefluid is below the saturation temperature of the liquid.

Depending on the excess temperature5 and the bulk temperature of the pool water itself, the boilingregime 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 inFigure 2.2. In this graph the heat flux is presented in relation to the excess temperature. The shapeof this curve depends on the fluid, the material of the surface and the pressure, but is nearly inde-pendent 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 burnoutpoint is given by point C in the boiling curve. In this phenomenon the decreasing heat transfer tothe fluid, and constant supply of energy by the mechanism of conduction makes the surface tem-perature jump (e.g. the Texcess in Figure 2.2 rises from ∼ 30°C at point C to ∼ 1200°C at point E). Theconsequences of the burnout effect are critical in the design of the SLIMR. As the temperature of theouter surface of the RPV increases significantly, it will have a deteriorating effect on the heat trans-fer. In this process the temperature in the reactor will increase, with, as an ultimate consequence, apossible meltdown. Therefore the SLIMR must be designed in order for the excess temperature tobe lower than 30°C.

2.4.1. NATURAL CONVECTION BOILING

The first stadium in the boiling curve is subjected to the natural convective boiling, this region endin point A, see Figure 2.2. In this event bubbles will not emerge in the fluid. The fluid that is super-

5The excess temperature is evaluated by the difference between the surface and the saturation temperature of the fluid,given by Texcess = Ts −Tsat .

24 2. THEORY

Figure 2.2: Pool boiling in water on a horizontal wire at atmospheric pressure.[21]

heated6 near the surface, causes natural convective currents. As a result the heat transfer from thesurface to the fluid can be modeled as natural convective heat transfer, as discussed in sub-section2.2.2.

2.4.2. NUCLEATE BOILING

In region from point A to point C the excess temperature is large enough to form bubbles at nucle-ation sites on the heated surface, see Figure 2.2. This stage in the boiling curve is called nucleateboiling. In this regime the heated surface is essentially independent of the orientation and geome-try of the heated surface area. Therefore, the surface of the vertically oriented SLIMR is treated as apan filled with water on a stove.

This region can be separated into two sub-regions. Starting with region A − B , the bubbles areformed at an increasing rate for increasing excess temperature. As the vapor bubbles rise by buoy-ancy, the respectively cooler water fills up the vacated space. This effectively increases the heattransfer compared to convection boiling. The bubbles in this sub-regime dissipate - they condenseand collapse - before they can reach the free surface of the water, making them effective energytransporters 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 pointC , 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 areformed at such a rate that they form a continuous vapor strip to the water surface where the vaporis released. In this form it is assumed that the heat is not transferred to the bulk fluid of the pool,

6A superheated fluid is heated to a few degrees above the saturation temperature of the fluid.

2.5. POOL 25

but directly to the environment.

The most widely used empirical correlation for the rate of heat transfer in the nucleate boilingregime is given by the Rohsenow equation [21]:

qnucl eate =µl h f g

[g (ρl −ρv )

σ

]1/2[

cp (Ts −Tsat )

cs f h f g Prnl

]3

. (2.40)

Here µl is the dynamic viscosity of the liquid, h f g the enthalpy of vaporization, g the gravitationalacceleration, ρl the density of the liquid, ρv the density of the vapor, σ the surface tension of liq-uid–vapor interface in N /m, cp the specific heat of the liquid, cs f the experimental constant thatdepends on the surface–fluid combination, Prl the Prandtl number of the liquid and n an experi-mental constant that depends on the fluid. The fluid properties are to be evaluated at the saturationtemperature Tsat .

This heat transfer relation for pool boiling only applies to smooth surfaces, which applies to thepolished stainless steel of the RPV. Further, it must be noted that the results obtained with theRohsenow equation need to be handled with care. This is because errors of ±100% are possiblein the calculation of the heat transfer rate with a given excess temperature.

2.5. POOLOne of the safety features is that the SLIMR is submerged in a pool. This allows the SLIMR to loseits decay heat in case of an emergency. However, during nominal operation heat will also transferfrom the SLIMR to the pool. By this means the pool is continuously heated. From the pool heat istransported 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 [29].

In this work the pool is modelled as a single node, assuming that the water is ideally mixed. Furtherit is assumed that the water in the pool is incompressible, and that its density ρ and specific heat cp

are constant. The energy and mass balance of the pool can now be written as:

cpρVpdTp

d t=QSLI MR −Qcond −Qconv −Qr ad −Qeva −Qmakeup (2.41)

ρdVp

d t= mmakeup +meva . (2.42)

In which Vp is the volume of the pool water. During nominal operation the makeup mass ratemmakeup equals the evaporation mass rate meva , resulting in dVp /d t ≈ 0.

Of the heat transfer mechanisms that are modelled in this work, the heat loss by conduction to theground is considered small enough to be neglected, which in most circumstances accounts less forthan 1% of the total energy loss from the pool [26]. However, the heat losses from the free pool sur-face by convection and radiation are significant, these are modelled as given in sub-section 2.2.2 and2.2.3. Furthermore, evaporation causes the largest losses (at least 60% of the total heat losses[25]),and therefore its prediction is most important. To this end, the latest evaporation correlation isimplemented in our model, which will be discussed in the following sub-section.

2.5.1. EVAPORATION OF POOL WATER

The free surface of the pool will transfer energy and mass. Analogous to heat, which is transferredalong a temperature gradient, mass is transported along a concentration gradient. However, in this

26 2. THEORY

case, for a region with a relatively high concentration of moister in the air, it will be transferred tothe relatively low concentration zone. By this means, the saturated vapor layer on the water surfaceof the pool will transfer heat to the region with less moist air, the rest of the facility and the environ-ment.

This mass transfer can be described as follows. The air that is in contact with the water surface be-comes saturated with moisture and, thus, becomes lighter [30]. Analogous to natural convectionheat transfer, the air with water vapor (lighter) will rise by buoyancy, and the dryer (heavier) air fallsdescends to replace the volume that has risen. In addition to this, the convective currents will beenhanced when the dry-bulb temperature in the facility is lower than the temperature of the poolwater. In this event the air layer that is in contact with the free surface of the pool gains heat, andthe heated air enhances the buoyancy. When the dry-bulb temperature in the facility is higher thanthe pool temperature, the effect is reversed.

The rate of evaporation from an undisturbed pool into quiet air (air without forced flow) can be de-scribed by a equation published by Shah et al. (2003) [31]. This correlation uses the analogy betweenheat and mass transfer from to a horizontal plate with the heated face upward. The evaporation rateis given by the following correlation:

Evap = K ·ρw (ρ f −ρw )1/3 · (Ww −W f ). (2.43)

In which Evap is the evaporation flux per hour [kg /m2h], ρ is the density of air in [kg/m3], W thespecific humidity of air [kg of moisture/kg of dry air], subscript w denotes ’saturated at water surfacetemperature’, subscript f denotes ’at facility temperature and humidity’, and K is a constant that isgiven by:

K ={

35, for (ρ f −ρw ) > 0.0240, for (ρ f −ρw ) < 0.02

(2.44)

This correlation is verified with numerous data sets, and is applicable in the ranges are presented inTable 2.4, with a mean deviation of the data of only 14.5%.

Table 2.4: Verified range of Shah et al. formulas for evaporation from pools. [32]

Range of data

Pool area [m2] 0.073−425Water temperature [°C] 7−94Air temperature [°C] 6−35Air relative humidity [%] 28−98pw −p f [Pa] 210−80, 156ρ f −ρw [kg/m3] −0.004 to +1.002

The values for ρ and W are obtained by the psychometric equations and splines obtained fromASHREA (2009) Handbook - Fundamentals [33]. To begin with the humidity saturated at the watersurface temperature Ww = Ws , a condition at which the moist air is in equilibrium with the poolwater at given pressure and temperature; the saturated humidity can be given as:

Ws = 0.62198pw s

p −pw s. (2.45)

Where p = 101325[Pa] is the total pressure, and pw s is the saturation pressure (of water vapor inabsence of air), which is a function of temperature.

2.6. DECAY HEAT 27

Figure 2.3: Decay heat power ratio for U235 fuel as a function of time after shutdown. [34]

The density of the moist air, saturated at water surface temperature, is evaluated at the temperatureof the pool water. The function for density ρw is based on the ideal gas law and is given as:

ρw = p (1+Ww )

(Tp +273.15)(Ra +Ww Rw )(2.46)

Here Ra = 286.9[J/kg·K] is the gas constant for air, Rw = 461.5[J/kg·K] is the gas constant for watervapor, and Tw is the pool water temperature in [K].

2.6. DECAY HEATIn an event that a reactor needs to be shut down, due to maintenance or in the event of an acci-dent, control rods are inserted in the core (this is also known as a SCRAM). From this point on theheat generation continues by decay heat, even though the power that is generated by fission hascompletely ceased. The major source of shutdown power is the decay of fission products, in whichunstable 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 heatcould 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 heatproduction is dependent on the amount of fission products, in turn, is dependent on the operationtime of the reactor. To limit the computational effort, using costly isotope generation and depletioncodes, the decay heat is estimated by a semi-empirical Way-Wigner type of function [34], which iswritten as:

P = 6.48 ·10−3 · [t−0.2 − (t +T0)−0.2] ·P0. (2.47)

Here P0 is the constant power production before shutdown, t the time after shutdown in [days], andT0 the operation time of the reactor in [days]. This equation is based on the empirical correlationsthat evaluate the energy release by β and γ emission due to decaying fission products. The numberof fission products are evaluated using 193 MeV per fission [35].

28 2. THEORY

The estimated decay heat can be seen in Figure 2.3, where the decay heat is plotted as a function oftime for various reactor operation times. In this graph shows that the initial decay power is relativelyindependent of the reactor operation time. However, for the determination of the long term decayheat the reactor operation time is relatively important. To this end, the transient analysis of the ofSLIMR after a SCRAM is performed for an infinite reactor operation time7. The reactor operationtime is considered to be infinite for a period of 1013 seconds.

7This conservative approach is also performed in the analysis of the NuScale reactor.

3NUMERICAL MODEL

The numerical model in this work is based on the theory in Chapter 2, and is developed in orderto perform transient calculations on the SLIMR. Hereby a one-dimensional loop code is developedto study the transient effects of the thermal-hydraulic stability of the natural circulation flow of theSLIMR. This loop model is coupled to a one-dimensional model that is developed in order to studythe heat transfer from the downcomer through the RPV to the pool the SLIMR is submerged in. Thisheat transfer model is again linked to a single heat balance that represents the pool.

This chapter begins with a retrospect on the previous work on the in-house written one-dimensionalloop codes. Hereafter, in section 3.2 the numerical methods used in this work are explained. Thenthe complete transient algorithm is described in section 3.3. Finally, the solver algorithm is declaredin section 3.4.

3.1. PREVIOUS WORKThe STEALTH-code is the starting point of the in-house written one-dimensional thermal-hydrauliccode that was the beginning of the SLIMR-code. This code by Koopman (2008)[36] was developedto analyze the thermal-hydraulic stability in natural circulation reactors. For the discretization ofthe governing fluid flow equations of this one-phase flow (representing a two-phase flow by usingthe homogeneous equilibrium model), a first order implicit upwind scheme was chosen. Althougha one-dimensional flow can flow forward and backward, the spatial discretization in the STEALTHcode is configured in such away that only a forward flow can be simulated. This code only dealswith the thermal-hydraulic mechanisms; void reactivity feedback was not accounted for.

The solution algorithm of the STEALTH-code was an example in the development of the DeLight-code, which was developed by Kam (2011) [37]. The DeLight-code was developed to study thethermal-hydraulic stability in the DeLight facility. For the discretization of the governing equationof this one-phase flow (supercritical Freon), a first order implicit upwind scheme was chosen aswell. As for the STEALTH-code, the DeLight-code is configured so that only a forward flow can besimulated. The physical properties (temperature, density and dynamic viscosity) of the fluid aretaken from thermo-dynamic properties package NIST. This code deals not only with the thermal-hydraulics, but also accounts for void reactivity feedback.

Later the DeLight-code was extended by Spoelstra (2012) [38]. In this version the discretization intime was improved by a semi-implicit Euler method. To gain computational speed splines wereintroduced containing the state equations of temperature, density, and the dynamic viscosity. Fur-thermore, the energy equation was simplified. Here, the pressure dependency is omitted to assure

29

30 3. NUMERICAL MODEL

proper convergence at smaller time steps. The pressure term is found to be small and does not sig-nificantly affect the solution. This is also neglected by several other authors, e.g. Ambrosini [39],and Jain [40].

The latest DeLight-code is the starting point in this work and has been completely overhauled. Thishas been done, since it was found that the DeLight-code is hardcoded to the geometry of the DeLightfacility, and the DeLight-code can only simulate forward flows, whereby in case of the SLIMR theflow direction (forward or backward) after a SCRAM is unpredictable. The governing equations thatdescribe the flow are now discretized in such a way that both forward and backward flows can besimulated. Further, since the code is to be re-written it is chosen to program the code in Matlab (theprevious codes are programmed in Fortran 90), which makes the code better accessible for studentsin further work.

3.2. NUMERICAL METHODSThe numerical methods that are used in the SLIMR-code will be explained in this section. Firstit will be explained what method is used for the spatial discretization. After this, the type of gridconfigurations that are used and how the variables are interpolated will be accounted for. Then thetime discretization scheme will be explained. Finally, the Courant number and the Fourier numberwill be introduced. These dimensionless numbers are used in order to avoid numerical instabilities.

3.2.1. FINITE VOLUME METHOD

The discretization method that is used in this work is the Finite Volume Method (FVM). Here, thedomain of interest is divided into a finite number of non-overlapping cells or control volumes overwhich the conservation is enforced in a discrete manner. The FVM is a common technique in Com-putational Fluid Dynamics (CFD) since it provides a natural way of preserving conservative prop-erties, making the physical interpretation of the conserving equations simpler [41]. To this end, itis also a convenient way to discretize the governing equation for conduction [24]. The technique ofFVM can be summarized in two steps: firstly, the governing equations are integrated over a controlvolume;, secondly, the integrated equations are discretized into a set of algebraic equations. To de-rive this set of algebraic equations the arrangement of the control volumes must be determined, aswill be explained next.

3.2.2. STAGGERED AND UNSTAGGERED GRID

The arrangement of control volumes that is used in the discretization of the governing flow equa-tions is known as the staggered grid. In a staggered configuration the control volumes are not equalfor each conserving equation. A representation of the staggered mesh is shown in Figure 3.1. Hereone can see a distinction between the main control volume (A) and the staggered control volume(B). In the centroid of the main control volume the cell-averaged value of pressure and fluid prop-erties are stored; these scalar quantities are denoted with S. The velocity components - or the massflow rate, M = ρv , as in this work - are stored on the faces of the main control volume, and are thecentroids of the staggered control volume. The velocity components are denoted with ~V .

To this end, the main control volume (A) is used in the conservation of mass and energy and thestaggered control volume (B) is used for the conservation of momentum.

In this process the pressure and velocity (mass flow rate) are decoupled. This results in some majoradvantages [41]:

• No need for odd-even decoupling between the pressure and velocity. [42]

3.2. NUMERICAL METHODS 31

• Discretizing the mass and energy balance, no further interpolation is needed since the massflow rates are located where required.

• The pressure produces a surface force, and therefore the natural place to define the pressureis at the cell boundaries.

• Discretizing the momentum balance, the pressure gradient term can be written directly interms of the pressures on the faces of the staggered control volume without interpolation.

However, it is not without any disadvantage. The arrangement of the staggered grid configurationincreases the geometrical complexity and thereby increases the application difficulty. This is due tothe increased difficulty handling in the different variables for the different control volumes and tokeep track of their indices.

For the discretization of the governing equation that describes the conduction through the vesselof the SLIMR there is no need for a staggered grid configuration, and an unstaggered grid satisfies.This grid contains only one type of control volume, the main control volume (A). In the centroid ofthis control volume the temperature and other properties of the stainless steal are stored. A rep-resentation of the unstaggered grid can also be visualized in Figure 3.1(note: in this case one mustignore the staggered control volume (B)).

3.2.3. SPATIAL DISCRETIZATION: FIRST ORDER UPWIND SCHEME

By the use of a staggered grid in the discretization of the governing flow equations not all variablesare located where they are required. These variables need to be interpolated. In this work the inter-polation is accomplished by a first order upwind scheme, and is relatively easy to implement. In thisscheme the fluid property or mass flow rate required at the face of a control volume is interpolatedby the value of the fluid property or mass flow rate at the grid point upstream side of this face. Inthis process the interpolation is dependent on the flow direction, i.e. for a forward1 flow the valuerequired at location j−1 (left face of control volume A) is interpolated by a value of the variable Si−1,and in case of a backward flow the value is interpolated form the variable Si .

The downside of the first order upwind scheme is that it is diffusive and hereby enhances the mixingof the fluid after a SCRAM. In addition, numerical diffusion has an artificial stabilizing effect onmass flow rate oscillations. It is therefore important that the discretization step is sufficiently small,checked by a grid independency test.

3.2.4. TEMPORAL DISCRETIZATION: SEMI-IMPLICIT EULER METHOD

In time, the model is discretized using the semi-implicit Euler method. The Euler method, a firstorder procedure, is the most basic method in numerical integration of differential equations. Thereare two variations to the Euler method; the backward or implicit Euler method, and the explicit Eu-ler method.

The advantage of the implicit scheme is that it is unconditionally stable, even for large time steps.However, it is computationally more expensive. This results from the increase of iterations neededfor a system of coupled equations (i.e. the governing flow equations) to calculate the new time step.

On the other hand, the explicit method is more accurate and also computationally cheaper. How-ever, to ensure numerical stability, the criteria by Courant and Fourier must be met; these will beelaborated later in this section. The disadvantage is that this model requires sufficiently small time

1A forward flow in Figure 3.1 flows from the left of the page to right.


𝑆𝑖𝑉𝑗−1 𝑉𝑗

𝑥𝑖−12

𝑥𝑖−1 𝑥𝑖 𝑥𝑖+12

𝑥𝑖+1

(𝐴)

𝑆𝑖 𝑆𝑖+1𝑉𝑗−1 𝑉𝑗

(𝐵)

𝑡𝑛

𝑡𝑛+1

∆𝑡

∆x

𝑉𝑗+1

𝑆𝑖+1𝑆𝑖−1→ →

→ → →

Figure 3.1: One-dimensional staggered discretization grid. The control volume for mass and energy is given by (A), andthe control volume for the momentum balance is given by (B). The horizontal lines are the grid lines, and the vertical linesrepresent the time lapse.

3.3. THE SLIMR MODEL 33

steps to assure numerical stability. Decreasing the time-steps has, however, major consequences inthe computational costs.

Combining the explicit and the implicit methods results in the semi-implicit Euler method. In thisscheme the factor θ is used to indicate what numerical time scheme is used. For θ = 1 the numericalscheme will be fully implicit, for θ = 1

2 a Crank-Nicolson scheme and for θ = 0 the scheme is fullyexplicit.

3.2.5. COURANT NUMBER

The Courant number is a dimensionless number that is used in the numerical analysis of fluid flows,which is given as:

Co = flow transport rate

storage rate= v

∆t

∆x. (3.1)

In which v is the velocity of the fluid flow, ∆x the length of the control volume, and ∆t the time step.A high Courant number could lead to too fast propagations of the flow information, which couldlead to instabilities.

Since the fully implicit scheme (θ = 1) is unconditionally stable a Courant number larger than 1 ispossible. In case of an explicit scheme (θ = 0) a Courant number smaller than 1 is required. In caseof 0 < θ < 1, the criteria of explicit scheme prevails.

3.2.6. FOURIER NUMBER

A dimensionless number that is often used in the numerical analysis of heat conduction is theFourier number, defined as:

Fo = diffusive transport rate

storage rate= α∆t

(∆r )2 (3.2)

Where α is the thermal diffusivity in [m2/s], and can be given as α= λ/ρcp . The Fourier number isused in conduction problems, just like the Courant number is used in flow problems.

As already noted, the implicit scheme (θ = 1) is unconditionally stable, modeling the phenomenonof conduction. Therefore there is no Fourier criterion that applies or must be met with in thisscheme. On the other hand, the explicit scheme (θ = 0) requires a condition by Fourier to ensurenumerical stability, but then the Fourier number must not exceed 1/2.

3.3. THE SLIMR MODELThis section covers all transient algorithms that are used in this work in which the heart of the code isthe one-dimensional flow model. The flow model is coupled to the heat transfer model, and the heattransfer model is again coupled to the heat balance of the pool. The models can work coupled orseparately. A flow chart of the complete SLIMR-code can be seen in Figure 3.2, and will be explainedfurther in this section model-by-model.

3.3.1. FLOW MODEL

The flow model loop is the main part of this work, and is a variation of the Semi-Implicit Methodfor Pressure- Linked Equations (SIMPLE) algorithm. After the introduction of the SIMPLE algorithmthe procedure of the flow model is explained step by step.

THE SIMPLE ALGORITM

The best-known solution algorithm that has been developed and used to solve the incompressiblefluids flow equations is the SIMPLE algorithm. The SIMPLE algorithm was devised by Patankar and


Figure 3.2: The flow chart of the transient SLIMR Model, the box indicates the time loop. This model contains the FlowModel (shown in Figure 3.3), the Heat Transfer Model (shown in Figure 3.5), and the Pool Heat Balance (shown in Figure3.6).


Spalding in 1972 [43].It is this algorithm that is typically in the core of most commercial CFD andsystem codes. The method is very robust, even on coarse grids, but its downside is that it has a lowexponential convergence rate. That is why it is an effective tool for rough solutions to even verycomplicated problems, but its efficiency can quickly be lost by refining the mesh in order to get ahigher spatial accuracy. As a time-stepping procedure it is therefore not designed to solve transientproblems for very large and refined meshes on three-dimensional structures, therefore it is too inef-ficient. Nevertheless, it is a perfect algorithm for transient calculations on a one-dimensional grid.[41]

Since its publication numerous modifications have been made to the basic SIMPLE algorithm, mainlyto improve the exponential convergence rate. These schemes, such as SIMPLER, SIMPLEC, SIM-PLEM utilize a more elaborated procedure in obtaining equation B.37. The method used in thisthesis is the method of Bijl [44]. Here the convection term is neglected2 in the pressure correctionscheme obtaining the mass flow correction equation, making it a technique that is simpler to imple-ment but is less sophisticated than the SIMPLE algorithm. However, once the solution is converged,it is not influenced; the solution of this method is equal to the SIMPLE method. The disadvantage ofthis method is that it decreases the exponential convergence rate, since it approaches the solutionin a less sophisticated way. However, in this work the one-dimensional code converges typically in1 iteration step. This makes it irrelevant to make use of more sophisticated algorithms, in order toacquire a higher rate of convergence.

BENCHMARK OF THE FLOW MODEL

The SLIMR flow model is benchmarked by the transient models of Chatoorgoon [22][45], Jain [40],and by a self-developed steady-state code (Appendix D). This is performed for a set benchmark ge-ometry for several heating rates. By comparing the results a very good agreement was found. Espe-cially for the case in which the same Darcy-Weishbach friction factor correlation was used, e.g. theIsgigaih correlation. The maximum deviation between the SLIMR and the model of Chatoorgoonwas in this case less than 1%, and the maximum deviation with the self-developed steady-state codewas only 2%.

THE ITERATION SCHEME OF THE FLOW MODEL

Next is the iteration procedure that is utilized in this work is shown in great detail, an illustration ofthis scheme is shown in Figure 3.3.

Before the iteration procedure is started at the beginning of each time step the core power is up-dated. This results in a linear heat rate at the core and the heat exchanger. In the event that theheat transfer model is included, the heat rate in the downcomer is estimated by the heat rate thatis obtained in the previous time step by the heat transfer model. Here, the heat rate for each cell inthe downcomer is calculated separately. The heat transfer model is elaborated in next sub-section3.3.2. Both the linear heat rate and the heat rate in the heat transfer model are constant throughoutone time step.

ITERATION STEP 1 IN FIGURE 3.3 - ENTHALPY BALANCE

The iteration starts by solving the energy balance. To this end, the enthalpy hn+1,k+1 of this itera-tion step is determined. The notation of the superscripts can be declared as follows; n denotes theprevious time step, n +1 the current time step, k the previous iteration step, and k +1 the currentiteration step. The subscripts cover the spatial indices of the staggered grid, i and j as defined in

2NOTE: The convection term is not neglected in the governing equations of the fluid flow, but only in the SIMPLE algo-rithm correcting the pressure.


Figure 3.3: The flow chart of the numerical model of the Flow Model. The box indicates the iteration loop of the pressurecorrection scheme.


Figure 3.4: Specific heat capacity vs. temperature around the pseudo-classical temperature for several system pressures.[46]

Figure 3.1. In case of backward (or negative) flows the index of the upwinded terms must be altered;the indices corresponding to backward flows are indicated within round brackets. This being said,the full numerical energy equation reads:

Aiρni

[hn+1,k+1

i −hni

]∆xi+[

θ(M n+1,k

j hn+1,k+1i (i+1) −M n+1,k

j−1 hn+1,k+1i−1(i )

)+ (1−θ)

(M n

j hni (i+1) −M n

j−1hni−1(i )

)]∆tn+[

θ(M n+1,k

j hn+1,k+1i −M n+1,k

j−1 hn+1,k+1i

)+ (1−θ)

(M n

j hn+1,k+1i −M n

j−1hn+1,k+1i

)]∆tn = (3.3)[

θq ,n+1i + (1−θ)q ,n

i

]∆xi∆tn + [

θqn+1i + (1−θ)qn

i

]∆tn .

Solving this equation estimates the enthalpy for this iteration step. Here the mass flow rate of theprevious iteration step is used to approximate the mass flow rate current iteration step:

hn+1,k+1(M n+1,k+1) ≈ hn+1,k+1(M n+1,k ). (3.4)

In the event that it is the first iteration step for this time step, the converged mass flow rate M n ofthe previous time step will be used as initial guess in the current time step.

ITERATION STEP 2 IN FIGURE 3.3 - EQUATIONS OF STATE

The next step in the iteration scheme is to acquire physical properties of the fluid needed in thesteps that follow. However, these properties are dependent on temperature and pressure. In thiswork it can be assumed that the physical properties are only temperature dependent. With this ap-proximation a constant pressure of 25 MPa is assumed. This is a reasonable assumption, since itis found that the pressure deviation - that is mainly hydrostatic - divert the absolute pressure by amaximum of 0.5%. The minor influences of the pressure diversions to the fluid parameters can beillustrated by highlighting the most sensitive parameter; the heat capacity cp , see Figure 3.4.

In principle the equations of state can be noted in terms of temperature or in terms of enthalpy [46].In this numerical scheme the energy balance is expressed in terms of enthalpy and therefore alsothe fluid properties are indicated by the enthalpy. From this point onwards the temperature, density,dynamic viscosity, thermal conductivity and Prandtl number can be derived. The state equations


are given by:

T = T (h)p ρ = ρ(h)p µ=µ(h)p λ=λ(h)p Pr = Pr (h)p . (3.5)

The fluid properties versus the enthalpy range used in the numerical model are illustrated in Ap-pendix 3.3.2. In order to calculate the fluid properties we make use of splines. This is a method forfast and accurate calculation of fluid properties at constant pressure.

A brief description of how a spline-based calculation of fluid properties will be illustrated here. If aproperty, let this be ρ, then a grid of values of this property can be created. This can be constructedby dividing the enthalpy range in sub domains, bins. The bin width is uniform and chosen in sucha way that the large gradients are captured around the pseudo-critical point and are captured prop-erly, which is the first requirement to the spline. The second requirement is that it needs to be acontinuous property function [47]. These requirements can be achieved with a third order polyno-mial fit, which can be written as:

T (i ) =CT (N ,1) · [h(i )−h(N )]3 +CT (N ,2) · [h(i )−h(N )]2 +CT (N ,3) · [h(i )−h(N )]+CT (N ,4). (3.6)

In this fit, ′N ′ is the bin number that is linked to the N th bin enthalpy that has the closest fit to theestimated enthalpy of the i th grid cell. grid cell. The coefficients are linked to the bin number of theenthalpy that has the best fit, and are obtained from a prepossessed data file. The data used for thepreprocessing is extracted from the NIST REFPROP 7.0 database. The fluid properties derived withthe spline coefficients give maximum relative error with the NIST data of 0.02%.

ITERATION STEP 3 IN FIGURE 3.3 - REYNOLD NUMBER AND FRICTION FACTOR

In this step the Reynolds number must be evaluated in order to determine which empirical frictioncorrelation must be used, after which the Darcy-Weisbach friction factor can be calculated. TheReynolds number is thus approximated, using the mass flow rate of the previous iteration step, thiscan be given as:

Ren+1,k+1 ≈ M n+1,k ·Dh

A ·µn+1,k+1, (3.7)

Where M n+1,k = vn+1,kρn+1,k A. With the evaluated Reynolds number the Darcy-Weisbach frictionfactor can be calculated. However, due to the fact that the Reynolds number is an approximation,the friction factor is an estimation as well:

f n+1,k+1(M n+1,k+1) ≈ f n+1,k+1(M n+1,k ). (3.8)

ITERATION STEP 4 IN FIGURE 3.3 - MOMENTUM BALANCE

In this step of the iteration scheme the mass flow M n+1,∗ will be estimated, i.e. the momentumequation is solved in this step. The fully discretized momentum balance used in this numerical


scheme reads:[M n+1,∗

j −M nj

]∆x j+θ

M n+1,kj ( j+1)M n+1,∗

j ( j+1)

ρn+1,k+1i+1 Ai+1

−M n+1,k

j−1( j )M n+1,∗j−1( j )

ρn+1,k+1i Ai

+ (1−θ)

(M n

j ( j+1)M nj ( j+1)

ρni+1 Ai+1

−M n

j−1( j )M nj−1( j )

ρni Ai

)∆tn =− (3.9)

[θAi

(pn+1,k

i+1 −pn+1,ki

)+ (1−θ)Ai

(pn

i+1 −pni

)]∆tn−θK j

M n+1,kj M n+1,∗

j

2ρn+1,k+1i (i+1) Ai (i+1)

+ (1−θ)K j

M nj M n

j

2ρni (i+1) Ai (i+1)

∆tn−θ f n+1,k+1i (i+1) Pi (i+1)

M n+1,kj M n+1,∗

j

8ρn+1,k+1i (i+1)

(Ai (i+1)

)2 + (1−θ) fi (i+1)Pi (i+1)

M nj M n

j

8ρni (i+1)

(Ai (i+1)

)2

∆x j∆tn+[θAi (i+1)ρ

n+1,k+1i (i+1) gi (i+1) + (1−θ)Ai (i+1)ρ

ni (i+1)gi (i+1)

]∆x j∆tn .

This balance utilizes the pressure of previous iteration pn+1,k , and the approximated friction factordetermined in iteration step 3. Here the link between the pressure and the mass flow rate is the mainreason why solving the momentum balance results in an estimated mass flow rate. To this end, themass flow rate is written as:

M n+1,k+1(pn+1,k+1) ≈ M n+1,k+1(pn+1,k ) ≡ M n+1,∗. (3.10)

What can also be noticed is the quadratic term M n+1,∗M n+1,∗,which is linearized by replacing oneof the unknown terms by the known mass flow rate M n+1,k from the previous iteration, which is arequirement of our solver that can not solve quadratic terms. This approximation can be given as:

M n+1,∗M n+1,∗ ≈ M n+1,k M n+1,∗. (3.11)

ITERATION STEP 5 IN FIGURE 3.3 - PRESSURE CORRECTION

In this step the pressure pn+1,k+1 of the current iteration step is calculated. Here the pressure devi-ation between the current and the previous iteration step is defined as the pressure correction, andcan be given as:

pn+1,′ = pn+1,k+1 −pn+1,k , (3.12)

This pressure deviation is induced by the error of estimating the mass flow rate as determined initeration step 4.

To calculate this deviation pn+1,′ we have to find a relation with the estimated mass flow rate M n+1,∗.The full derivation of this equation can be found in the appendix section B.4. Briefly explained, therelation is obtained by eliminating M n+1,k+1 out of the mass balance B.12 with the use of the masscorrection equation B.38. To this end, solving the pressure correction equation yields the pressuredeviation and likewise the mass conservation is preserved, this equation yields,[

Ai

(ρn+1,k+1

i −ρn+1,ki

)]∆xi+

[θ

(M n+1,∗

j −M n+1,∗j−1

)+ (1−θ)

(M n

j −M nj−1

)]∆tn = (3.13)

θ

[θ∆tn

∆x j−1Ai−1

(pn+1,′

i−1 −pn+1,′i

)− θ∆tn

∆x jAi

(pn+1,′

i −pn+1,′i+1

)]∆tn .

Now solving the pressure correction equation we obtain pn+1,′ , and to this end the pressure pn+1,k+1

of the current iteration step is acquired by:

pn+1,k+1 = pn+1,k +pn+1,′ . (3.14)


ITERATION STEP 6 IN FIGURE 3.3 - MASS FLOW CORRECTION

In Step 5 an error has been made while estimating the mass flow rate with M n+1,k+1 ≈ M n+1,∗. Inthis step the mass flow rate will be corrected for this error. The mass flow rate correction reads:

M n+1,′ = M n+1,k+1 −M n+1,∗. (3.15)

Combining equations 3.14, 3.17 and a stripped version of the momentum equation - which is typicalto the method of Bijl - are used to obtain a relation between the pressure deviation pn+1,′ and themass flow rate correction M n+1,′ . This equation is known as the mass flow rate correction equationand can be written as:

M n+1,′j = θ

∆tn

∆x j

[Ai

(pn+1,′

i −pn+1,′i+1

).]

(3.16)

Now the corrected mass flow rate is given by:

M n+1,k+1 = M n+1,′ +M n+1,∗. (3.17)

ITERATION STEP 7 IN FIGURE 3.3 - CONVERGENCE

This step comprises a check for convergence. Step 1 to 6 are repeated until the converging criteriaare reached. There are two criteria applied that, separately from one another, can stop the iterationroutine. The first criterion is given as,

M ax

(∣∣∣∣∣ pn+1,k −pn+1,k+1

pn+1,k+1

∣∣∣∣∣)<C , (3.18)

in which C is a maximum allowable relative difference of pressure deviation in respect to the cor-rected pressure. The second criterion is that only a finite number of iteration steps can be taken,and assures that it does not end in an infinite loop when the pressure is not converging. In caseof convergence the pressure deviation will decrease to zero, pn+1,′ → 0. The same applies for themass flow rate correction M n+1,′ → 0, meaning that the mass flow rate equals the corrected massflow rate, M n+1,∗ → M n+1,k+1. Ultimately, equation B.39 results in the mass balance as derived inthe appendix section B.1, given as,

Ai

[ρn+1,k+1

i −ρni

]∆xi +

[θ

(M n+1,k+1

j −M n+1,k+1j−1

)+ (1−θ)

(M n

j−1 −M nj−1

)]∆tn = 0. (3.19)

3.3.2. HEAT TRANSFER MODEL

After the SLIMR flow model has converged, the SLIMR heat transfer model can be started. Thismodel evaluates the heat loss from the downcomer to the pool. This is done by a one-dimensionalmodel - representing the stainless steel vessel of the SLIMR - that is adjacent to each cell in thedowncomer section of the one-dimensional flow model. The heat transfer at the lower and upperplenum are not incorporated in this model. This makes the calculations conservative, i.e. a lowerestimation of the decay heat removal.

The heat transfer model incorporates correlations for the forced convective heat transfer from thesupercritical water to the Inner Surface (IS) of the reactor vessel. With the use of a transient con-duction model the heat will transfer to the Outer Surface (OS) of the reactor vessel. From the outersurface of the reactor vessel heat transfers to the pool, incorporating correlations for natural con-vection and nucleated boiling.

The method that is used to simulate the unsteady heat transfer is derived from the work of Cam-pos et al.[24]. Their work focuses on the heat transfer problem for a hollow cylinder heated from


the central axis in an unsteady state, applying the finite volume method with Half Control Volume(HCV). Their model showed good accuracy, compared to other models. In addition, this method isrelatively easy to implement, and the same solver could be used as for the SLIMR flow model.

The algorithm of this heat transfer model is illustrated in Figure 3.5, and will be explained further inthis sub-section step by step.

HEAT TRANSFER STEP 1 IN FIGURE 3.5 - B.C. WITH DEFINED CONV. HEAT TRANSFER COEFF.The heat transfer scheme starts with the calculation of the heat transfer coefficient. This coefficientis needed at the boundary condition of the inner surface of the conduction problem of a hollowcylinder (reactor vessel). The fluid properties and variables Re and Pr that are needed in this step,are obtained from the control volume - one-dimensional fluid flow - that is adjacent to first controlvolume of the one-dimensional RPV slab.

The calculation of the heat transfer coefficient by forced convection starts by the Nusselt number:

Nun+1f c =

{3.66 for Ren+1 < 2 ·103

0.023 ·Ren+1(Pr n+1

)melse

(3.20)

In which m in the Dittus-Boelter correlation is given by:

m ={

0.3 for T ni s < T n+1

b0.4 for T n

i s > T n+1b

(3.21)

In which T ni s is the temperature of the inner surface of the RPV of the previous time step, and T n+1

bthe bulk fluid temperature of the flow in the downcomer at the current time step. To this end, theconvective heat transfer coefficient can be evaluated:

hn+1i s =

Nun+1f c λn+1

Lc(3.22)

In which λn+1 is the thermal conductivity of the water at super critical pressure, and Lc = Do−Di thecharacteristic diameter of the annulus, where Di is the inner diameter, and Do the outer diameterof the annulus.

HEAT TRANSFER STEP 2 IN FIGURE 3.5 - BOUNDARY CONDITION WITH DEFINED HEAT RATE.In this step the heat rate from the RPV slab to the pool is calculated. This heat rate is needed for theboundary condition at the outer surface of the conduction problem of the hollow cylinder.

To evaluate the heat rate, the phenomenon that is responsible for the heat transfer must be deter-mined first. To this end the excess temperature is evaluated, given as:

T n+1excess = T n+1

os −Tsat . (3.23)

In which T n+1os is the temperature of the outer surface of the RPV in [°C], and Tsat = 100 [°C] for water

at a constant pressure of 101325[Pa].

If T n+1excess < 5, the heat transfer mechanism is Natural Convection .

In the continuation of this step the fluid properties must be determined at the film tempera-ture [21], given as:

T n+1f =

T np +T n

os

2(3.24)


Figure 3.5: The flow chart of the Heat Transfer Model. The box indicates a loop, where the scheme is repeated until the HeatTransfer Loop is executed for all one-dimensional RPV slabs that are adjacent to a control volume in the downcomer of theFlow Model.


Here the film temperature of the current time step is estimated by the pool temperature T np

and the outer surface temperature of the RPV T nos of the previous time step.

The fluid properties at the film temperature are derived with equations of state. Here it is as-sumed that the pool is at constant atmospheric pressure. The equations of state are noted interms of the pool temperature. From this point onwards the thermal expansion coefficient,the kinematic viscosity, the thermal conductivity, and the Pr number of the fluid can be ob-tained:

β=β(T f )p ν= ν(T f )p λ=λ(T f )p Pr = Pr(T f )p (3.25)

Then the Grashof number can be calculated:

Grn+1 =gβn+1

(T n

os −T np

)L3

c(νn+1

)2 . (3.26)

In which Lc is the characteristic length of the cylinder, in this work it is the height of the RPV.The temperature gradient is estimated by the temperatures of the previous time step. Afterthis the Rayleigh number can be given:

Ran+1 = Prn+1Grn+1. (3.27)

After which the Nu number can be evaluated:

Nun+1nc =

(0.825+ 0.387 · (Ran+1)1/6

(1+ (0.492/Prn+1)9/16)8/27

)2

; (3.28)

Now the convective heat transfer coefficient can be calculated:

hn+1os = Nun+1

nc λn+1

Lc. (3.29)

Finally, the heat rate can be calculated with Newton’s Law:

qn+1os = hn+1

os Aos

(T n

os −T np

). (3.30)

In which Aos is the outer surface area of the RPV, and the temperature gradient is estimatedby the temperatures of the previous time step.

If 5 < T n+1excess < 30, the heat transfer mechanism is Nucleate Boiling .

In this step the fluid properties are evaluated at saturation temperature Tsat . Multiplying theRohsenow correlation by the outer surface area Aos of the RPV slab, now the nucleate boilingheat rate is determined:

qn+1os =µl h f g

[g (ρl −ρv )

σ

]1/2[

Cp T n+1excess

Cs f h f g Prml

]3

Aos . (3.31)

In which the coefficient Cs f = 0.0130, and m = 1 for the fluid-surface combination of wa-ter–stainless steel that is mechanically polished.


HEAT TRANSFER STEP 3 IN FIGURE 3.5 - HEAT CONDUCTION.In this step the unsteady one-dimensional conduction equation is solved. The variables needed atthe boundary conditions at the end of the one-dimensional RPV slab are derived in the previoussteps. Therefore the convective heat transfer coefficient hn+1

i s is evaluated in step 1. This is neededfor the is boundary condition at the inner surface of the RPV adjacent to the flow in the downcomer,control volume i = 1. In step 2 the heat rate qn+1

nc is given, needed for the outer surface of the RPVadjacent nc to the pool water, control volume i = I . The discretized unsteady one-dimensionalconduction equation in radial coordinates is given as:

ρcp(T n+1

i −T ni

)(r 2j+1 − r 2

j

2

)= (3.32)

(θ r2λ

T n+12 −T n+1

1∆r2

+ (1−θ) r2λT n

2 −T n1

∆r2+θ r1hn+1

i s

[T n+1

1 −T n+1b

]+ (1−θ) r1hni s

[T n

1 −T nb

])∆t , i = 1

(θ r j+1λ

T n+1i+1 −T n+1

i∆r j+1

+ (1−θ) r j+1λT n

i+1−T ni

∆r j+1−θ r jλ

T n+1i −T n+1

i−1∆r j

− (1−θ) r jλT n

i −T ni−1

∆r j

)∆t ,1 < i < I

(θ

qn+1os

2πH + (1−θ)qn

os2πH −θ r J−1λ

T n+1I −T n+1

I−1∆r J−1

− (1−θ) r J−1λT n

I −T nI−1

∆r J−1

)∆t , i = I

Here H is the height of the a part of the reactor vessel, which is equal to the length of a cell in theloop. Solving the equation above results in the temperature profile T n+1

sl ab of the RPV for the currenttime step.

For this transient one-dimensional conduction model it is verified in time, if a steady-state temper-ature distribution is achieved after an unsteady initial state. In addition, the time scale of the heatpenetration is verified by the „Penetratie Theorie” [48]. For both cases a good agreement has beenfound: a steady state has been achieved, and the time scale of the heat penetration is similar to thetime scale that has been determined in the „penetration theory”.

HEAT TRANSFER STEP 4 IN FIGURE 3.5 - HEAT RATE

After determining the temperature profile of the RPV slab, the inner T n+1i s = T n+1

sl ab(i ) and outerT n+1

os = T n+1sl ab(I ) surface temperatures of the current time step are given. Where in step 1 the B.C.1

is utilized to determine T n+1i s in step 3, now the heat rate from the downcomer to the RPV can be

calculated with Newton’s law multiplied by the inner surface area of the RPV slab:

qn+1i s = hn+1

i s Ai s(T n+1

i s −T n+1b

)(3.33)

Here qn+1i s is the heat rate that will be subtracted from the adjacent cell in the thermal-hydraulic

loop is for the following time step, Ai s , the surface area of the RPV is adjacent to the fluid flow.

3.3.3. POOL HEAT BALANCE

This algorithm evaluates heat and mass balance of the pool. Here the pool is modeled as singlenode.The algorithm of the transient pool model is illustrated in Figure 3.6, and will be explainedfurther in this sub-section step by step.

POOL HEAT BALANCEL STEP 1 IN FIGURE 3.6 - HEAT GAIN BY THE SLIMR.In this step the total heat rate from the SLIMR to the pool is determined. This is accomplished bysummation over all heat rates of the RPV slabs to the pool, determined in Heat Transfer Step 2, andcan be given as:

QSLI MR =I∑

i=1qn+1

os(i ). (3.34)


Figure 3.6: The flow chart of the Pool Heat Balance.


POOL HEAT BALANCE STEP 2 IN FIGURE 3.6 - HEAT LOSS BY NATURAL CONVECTION.In this step the heat loss by natural convection from the free surface of the pool is determined. Herethe convective heat transfer coefficient is hconv = 3.1, given by the correlation of Hahne and Kubler(1994). Now the heat rate can be calculated with Newton’s law multiplied by the free surface area ofthe pool:

Qconv = hconv Apool (T np −T f aci l i t y ). (3.35)

Here the pool temperature is estimated by the previous time step, and T f aci l i t y is the temperatureof the facility, which is kept constant.

POOL HEAT BALANCE STEP 3 IN FIGURE 3.6 - HEAT LOSS BY MAKE-UP WATER.To make up for the evaporated water the pool is continuously supplemented by make-up waterwhere it is uniformly mixed. The rate of energy that is required to bring the make-up water to thetemperature of the pool water is given by:

Qmakeup = M nvap cp

(T n

p −T f aci l i t y

). (3.36)

In which Mvap is the evaporated mass that is calculated in the previous time step, and the pool tem-perature Tp is estimated by the previous time step. The temperature of the make-up water equalsthe temperature of the facility T f aci l i t y , and is kept constant.

POOL HEAT BALANCE STEP 4 IN FIGURE 3.6 - HEAT LOSS BY RADIATION.In this step the heat loss by radiation from the free surface of the pool is determined. The net heattransfer can be given as:

Qr ad = εσSB

((T n

p +273.15)4 − (Tr oom +273.15)4)· Apool . (3.37)

In which the pool temperature is estimated by the previous time step, the emissivity εw = 0.96 [26],and the temperature of the surrounding surface is at the temperature of the facility T f aci l i t y , whichis kept constant.

POOL HEAT BALANCE STEP 5 IN FIGURE 3.6 - HEAT LOSS BY FREE SURFACE

In this step the mass and heat transfer by evaporation at the free surface of the pool is modeled. Forthis, the saturation pressure must be evaluated first, which is a function of temperature. To this enda spline is developed describing the equation of state (it is assumed that the facility is at a constantpressure of 101325[Pa]) of the saturation pressure in terms of temperature. The psychometric datathat are used is obtained from ASHREA (2009) Handbook - Fundamentals. The equation of state canbe given as:

pw s = pw s(Tpool )p . (3.38)

In which Tpool is the pool temperature. Next, the saturated humidity can be calculated, given as:

W n+1w = 0.062198

pn+1w s

p −pn+1w s

(3.39)

Here p is the pressure at 101325 [Pa]. The density of the moist air, saturated at water surface tem-perature, is evaluated at the temperature of the pool water, and is given as:

ρn+1w = p

(1+W n+1

w

)(T n

p +273.15)(Ra +W n+1

w Rw) (3.40)

Now the constant K can be determined for the current time step:

K ={

35, for (ρ f −ρw ) > 0.0240, for (ρ f −ρw ) < 0.02

(3.41)

3.4. SOLVER 47

The properties in the facility are kept constant at the temperature of the environment and atmo-spheric pressure. The humidity in the facility is, with a relative humidity of Wr = 0.8, given by:W f =Wr ·Ws . The density of the moist air in the facility is given by equation 3.40. Now the evapora-tion can be calculated with the correlation of Shah et al., given as:

E n+1vap = K n+1 ·ρw (ρr −ρn+1

w )1/3 · (W n+1w −Wr ). (3.42)

In which E n+1vap is the evaporation in [kg /m2h], the evaporation rate over the total free water surface

is given as:M n+1

vap = E n+1vap Aps f /3600. (3.43)

The heat rate by evaporation can be evaluated, resulting in:

Qvap = M n+1vap ·

(hvap + cp

(Tsat −T n

p

)). (3.44)

Where hvap = 2257 ·103 is the evaporation enthalpy in [J/kg ].

The calculation of the evaporation rate in this model is verified by the work of Asdrubali [49]. Here,the model of Shah [31] is compared to an experiment, where several parameters such as water tem-perature, air temperature and relative humidity were varied. It was found that the model used inthis thesis matches the predictions of Shah, and the work of Asdrubali showed that the model ofShah was in good agreement with the experimental results.

POOL HEAT BALANCE STEP 6 IN FIGURE 3.6 - POOL TEMPERATURE

In this step the new pool temperature is calculated, given by:

T n+1p = T n

p + QSLI MR −Qconv −Qr ad −Qmakeup −Qvap

ρcp(3.45)

3.4. SOLVERNow that the physical system is modeled we want to compute its solutions. To this end the finite vol-ume discretized balance equations are re-written to form linear systems of equations. This methodis also utilized in the work of Koopman [36], Kam [37] and Spoelstra [38]. For each linearized bal-ance equation we end up with N equations and N unknowns, where N is the number of grid nodesin the system. This can be written in matrix form as:

A ·φ= s. (3.46)

In this expression Ais the matrix carrying all terms that are dependent on φ. All unknowns in thissystem are collected in vector φ, the unknowns can be either h, M∗, p, or Tsl ab . The vector s is thesolution vector, here all terms are brought together that are not dependent on the unknowns φ.

3.4.1. TRI-DIAGONAL SYSTEMS

Due to the first order upwinding scheme is matrix A tri-diagonal. A tri-diagonal system is a specialset of linear equations, where only the elements on the diagonal - and the plus and minus onecolumn of the diagonal - are nonzero:

b1 c1 0 · · ·a2 b2 c2 · · ·

· · ·· · · aN−1 bN−1 cN−1

· · · 0 aN bN

·

φ1

φ2

· · ·φN−1

φN

=

s1

s2

· · ·sN−1

sN

. (3.47)


This shows that the super-diagonal (diagonal plus one column) formed by the ci and the sub-diagonal (diagonal minus one column) formed from the bi . The tri-diagonal structure can be usedas an advantage to reduce the computational effort needed to solve the problem. The recipe offinding φ is given by:

SET :

w1 = b1

φ1 = s1

w1

FOR i = 2,3, ...,N−1,N

zi = ci−1

wi−1

wi = bi −ai zi (3.48)

yi = si −ai yi−1

wi

END

FOR j = N−1,N−2, ...,2,1

φ j = y j − z j+1φ j+1

END

This algorithm [50] is used solving the pressure correction step in the pressure correction loop, butalso to solve the one-dimensional conduction in the heat transfer loop.

3.4.2. CYCLIC TRI-DIAGONAL SYSTEMS

Another type of system present in this work is the cyclic tri-diagonal system. Where A is cyclic dueto the nature of the closed loop geometry, in which the first node is dependent on the last node andvice-versa. This system is given by:

b1 c1 0 · · · β

a2 b2 c2 · · ·· · ·· · · aN−1 bN−1 cN−1

α · · · 0 aN bN

·

φ1

φ2

· · ·φN−1

φN

=

s1

s2

· · ·sN−1

sN

. (3.49)

As can be seen A is tri-diagonal except for the elements α and β in the corners. o solve this system,the Sherman-Morrison method is utilized to describe the cyclic tri-diagonal matrix A as an ordinary

non-cyclic tri-diagonal matrix A′ with a correction of u ⊗ v . The system of equations can now begiven as:

Aφ= (A′+u ⊗ v)φ= s. (3.50)

In this notation vectors u and v can be defined as:

u =

γ

0· · ·0α

, v =

10· · ·0

β/γ

. (3.51)

Here is γ is an independent variable that can be set to any value in order to avoid loss of precisionby subtraction, shown in equation 3.52. Then matrix A′, which is the tri-diagonal part of the matrix

3.4. SOLVER 49

A′, can be modified by to terms:

b′1 = b1 −γ, b′

N = bN −αβ/γ. (3.52)

If for the vectors y and z the following applies:

A′y = s, A′z = u. (3.53)

These systems of equations can be solved by tri-diagonal algorithm 3.48, and the solution to system3.49 is given by:

φ= y −[ v · y

1+ v · z

]z. (3.54)

Here the Sherman-Morrison method reduces the problem of the cyclic system to solving the non-cyclic systems of equations 3.53.

4RESULTS

In this chapter all results will be discussed, which are divided into four parts. These parts correspondto the phases A, B, C, and D as introduced in the thesis objective. The section ’Phase A - Natural Cir-culation’ discusses the stability of the steady-state operation points and their physical quantities;the mass flow rate, the core enthalpy step1, the core exit temperature, and the heat transfer deteri-oration rate, for several parameters; the riser diameter, the outer diameter of the downcomer, theriser length, the core height, the core inlet friction, the core inlet temperature and the core power.The section ’Phase B - Heat Transfer’ discusses the steady-state heat transfer during normal opera-tion by the total heat loss and the surface temperature of the RPV, for several parameters; the outerdiameter of the dowcomer, the riser length, the core inlet temperature, the pool temperature, andthe thickness of the RPV. The section ’Phase C - Pool’ discusses the steady-state of the pool, e.g. thetemperature and evaporation, for several steady-state heat losses of the SLIMR, and pool dimen-sions. The section ’Phase D - Accident Scenario’ discusses the transient behavior of the SLIMR aftera SCRAM by the overall final coolant temperature after 72 [h], the maximum coolant temperatureafter 72 [h], and the overall maximum heat transfer deterioration rate, for several riser lengths anddowncomer diameters.

4.1. PHASE A - NATURAL CIRCULATIONIn this phase of the thesis objective, shown in section 1.5, the central question is:

Is it possible to obtain a geometry in which the SLIMR has a safe nominal operating point, whichis stable and in which no heat transfer deterioration occurs?

This will be assessed for the basic geometry of the SLIMR, for which it will be investigated how themost basic parameters in this design will affect the results.

4.1.1. A - SIMULATION PROCEDURE AND INITIAL SETUP

In order to obtain the results the algorithm described in chapter 3 is utilized and in the frameworkof this Phase we focus only on the internal flow of the coolant in the SLIMR. The flow model devel-oped in this thesis can simulate the natural circulation flow in the SLIMR, and works independentlyfrom the other models. To this end, only the flow model within the time stepping loop in Figure 3.2is utilized. Moreover, only the thermal-hydraulics in this model are dealt with, and the reactivityfeedback and heat loss via the downcomer to the environment are not accounted for. Energy can

1The core enthalpy step is the difference of coolant enthalpy between the exit and the entrance of the core.

51

52 4. RESULTS

only enter the system by the core, and is extracted by the heat exchanger.

In these simulations the appropriate time step and grid size are selected by a time and grid size in-dependency test. Here it is found that a smaller temporal step is required for the stability check,this was also found by Spoelstra [38]. In general, it is found that by increasing the refinement ofthe time step and the grid size, the accuracy of the solution increases. However, it is at the cost of anon-linear increase of the computation time. The time steps and grid size in this phase are basedon a compromise between the discretization error, the computation time and the Courant number.

The simulation procedure is split up in 4 steps that are connected in which the next step is based onthe previous step. Step A1 - Reaching Nominal Operating Power, Step A2 - Damping of Mass FlowRate Oscillations, Step A3 - Power Increment, Step A4 - Determine the Stability. These steps simulatedifferent time intervals, see table 4.1. The interval of the simulation times are chosen in such a waythat it is ensured that the objective of each step is achieved. The procedure and objectives for eachstep can be explained as follows:

Table 4.1: Selected simulation times and their corresponding time step, and spatial grid size for each of the procedures inPhase A. Calculating Steady-State the fully implicit scheme is used, and for the stability check the semi-implicit parameteris set to Θ= 0.6.

Time

Steady-State Step A1 - Reaching Nominal Operating Power 1000 sStep A2 - Damping of Mass Flow Rate Oscillations 1000 sTime step used in Step A1 and A2 0.05 s

Stability Step A3 - Power Increment 10 sStep A4 - Determine the stability of Mass Flow Rate Oscillations 90 sTime step used in Step A3 and A4 0.01 s

Space

Grid size 0.25 m

STEP A1 - REACHING NOMINAL OPERATING POWER.This step simulates 1000 seconds of operation time. The objective of this step is to initiate a steady-state natural circulation flow. In the initial state the coolant is completely stagnant, and the temper-ature is homogeneously distributed and equals the predefined core inlet temperature, and the coreproduces no power.

By starting the simulation the core power will increase from zero, via a S-shaped heating curve, tonominal operating power, see Figure A.11a. This smooth heating curve is enforced to minimizeabrupt power gradients, which can cause mass flow rate oscillations, with which mass flow rateoscillations could make the simulation unstable, or increase the simulation time needed in findinga steady-state solution.

STEP A2 - DAMPING OF MASS FLOW RATE OSCILLATIONS.This second step simulates another 1000 seconds. The objective of this step is to ensure that pos-sible mass flow rate oscillations are completely dampened. At the end of this interval, the physicalquantities of the steady-state working point will be determined; the mass flow rate, the core exittemperature, the core enthalpy step and heat transfer deterioration rate are fully converged.

STEP A3 - POWER INCREMENT.The third step simulates 10 seconds. The aim of the step is to provoke mass flow rate oscillations.Here the simulation started with the steady-state solution is determined in step 2. Over the interval

4.1. PHASE A - NATURAL CIRCULATION 53

of this step the core power is subjected to an abrupt power increment of 1% starting from t = 0, thiscan be seen in Figure A.12a. This figure shows the power versus time for Step A3 and A4.Validation and verification of the code made clear that the decrease of the time step in the tran-sition from Step 2 to Step 3, e.g. from 0.05 to 0.01 seconds, could already cause flow oscillations.Nevertheless, a power increment is induced to ensure that the steady-state situation is disturbed.

STEP A4 - DETERMINE THE STABILITY OF MASS FLOW RATE OSCILLATIONS.The final step simulates 90 seconds of operation time. The goal of this step is to determine stabilityof the provoked mass flow rate oscillations. At this point the core power is abruptly brought backfrom the incremented power to nominal operation power, which can be seen in Figure A.12a. Val-idation and verification of the code made clear that this interval is long enough to determine thestability of the oscillations. A steady-state working point is defined as stable when the decay ratio ofthe mass flow rate oscillations is smaller than 1.

In order to present the results of the stability analysis in a more general way we utilize the stabilitymap. For this purpose a set of non-dimensional parameters NPC H and NSU B are used as x- and y-axis, respectively. The sub-cooling number NSU B scales the inlet sub-cooling at the core inlet to theenthalpy at the pseudo-critical point, hpc = 2.16 ·106 [J/kg], and the phase-change number NPC H

represents the core power to the mass flow rate. Both can now be defined as:

NSU B = hpc −hi n

hpc, NPC H = Pcor e

M ·hpc. (4.1)

BASIC DIMENSIONS OF THE SLIMR GEOMETRY

In this study only one parameter is varied for each simulation, the parameters that are not variedare kept constant to the values that can be found in Table 4.2.

Table 4.2: Initial parameters of the SLIMR design, for CASE A.

Reactor Dimensions

Core Height 4.25 mEquiv. Diameter 1.37 mInlet Friction, K 0 -

Riser Length 9.00 mDiameter 0.94 m

Heat Exchanger Length 2.50 mDowncomer Outer Annulus Diameter 2.44 mRPV Vessel Thickness 0.37 m

Operation Point

System Pressure 25 MPaCore Thermal Power 150 MWth

Inlet Temperature 300 °C

4.1.2. A1 -VARIATION OF THE RISER LENGTH

The riser length of the MASLWR is 6 [m]. It is expected that this length can be reduced because ofthe increased buoyancy due the supercritical water. However, it is possible that a large riser lengthis needed in order to deposit the decay heat to the environment. On that account, we chose to varythe riser length from 3.5 [m] to 10 [m] in steps of 0.25 [m] for several constant core heights rangingbetween 2 [m] and 6 [m]. Within this range it is found that all steady-state working points are stable.The sub-cooling number over this range is constant, NSU B = 0.382, and the phase-change numberranges between NPC H = 0.119−0.156.

54 4. RESULTS

A1 - EFFECT ON THE MASS FLOW RATE

In Figure 4.1a the dependence of the mass flow rate to the riser length can be seen for various coreheights. The mass flow rate increases over the interval of the increasing riser length within in a rangeof 31.7% for a core height of 2 [m] to 24.8% for a core height of 6 [m].

Increasing the riser length results in an increasing mass flow rate. This is because the density dif-ference (between the hot leg and the cold leg) of the fluid is affected by gravity by the increasingdifference in elevation between the reactor core and the heat exchanger, but also the height of thevessel. This produces an increasing buoyancy force that drives the fluid. However, by increasingthe mass flow rate, the residence time of the fluid in the core decreases and thereby declining theenthalpy step in the core, which in turn decreases the density difference. Therefore the growth ofthe mass flow rate declines for increasing riser lengths. Moreover, by increasing the mass flow ratethe friction in the loop also increases, which in turn also has a declining effect on the growth of themass flow rate.

A1 - EFFECT ON THE CORE ENTHALPY STEP AND CORE EXIT TEMPERATURE

The enthalpy step in the core versus the riser length for several core heights can be seen in Figure4.2a. Over the interval of the increasing core height, the core enthalpy step decreases within in arange of 24.0% for a riser length of 3.5 [m] to 19.9% for a riser length of 10 [m].

As it is defined in these simulations, unless otherwise specified, the core power and the core inlettemperature (core inlet enthalpy) are kept constant. Thus the enthalpy step is only a function of themass flow rate, in which an increasing mass flow rate decreases the residence time of the fluid in thecore, which leads to a decreasing enthalpy step.

Here the core exit temperature is only a function of the core enthalpy step. Decreasing the enthalpystep will result in a decreasing core exit temperature. The core exit temperature decreases over theinterval of the increasing riser length within in a range of 3.2% for a core height of 2 [m] to 2.7% fora core height of 6 [m]. The maximum core exit temperature found is 368.4 [°C].

A1 - CHECK FOR HEAT TRANSFER DETERIORATION

The heat transfer deterioration rate versus the riser length for several core heights can be seen inFigure 4.3a, here it shows that the heat transfer deterioration rate stays below the set criteria. Overthe interval of the increasing riser length, the heat transfer deterioration rate decreases within in arange of 24.0% for a core height of 2 m to 19.9% for a core height of 6 m.

For a constant core geometry the heat transfer deterioration is only a function of the mass flow rate.By increasing the riser length, the mass flow rate increases, and results in a decreasing heat transferdeterioration rate.

4.1.3. A2 - VARIATION OF THE CORE HEIGHT

The core height of the MASLWR is 2 [m] and is based on a conventional LWR core. In case of theSLIMR the core is based on the HPLWR, which has a core height of 4.20 [m]. It is expected that due tothe lower power density of the SLIMR - in respect to the HPLWR - the core height could be reduced.However, it is also possible that a larger core height is needed in order to avoid deterioration of theheat transfer, because it is expected that the mass flow rate for SLIMR is lower than for the HPLWR.To this end, the core height is varied from 2 [m] to 6 [m] in steps of 0.25 [m] for several constantriser lengths ranging between 3.5 [m] and 10 [m]. Within this range it is found that all steady-stateworking points are stable. Here the sub-cooling number over this range is constant, NSU B = 0.382,and the phase-change number ranges between NPC H = 0.156−0.205.



The results of the steady-state mass flow rate where the core height is varied for several riser lengthsare shown in Figure 4.1b. The mass flow rate decreases over the interval of the increasing core heightwithin in a range of 23.7% for a riser length of 3.5 [m] to 27.7% for a riser length of 10 [m].

That the mass flow rate decreases for increasing core height could not be predicted in advance. Thisis because the core height increases buoyancy force in the loop, but it also has a wetted perimeterthat is approximately a 100 times larger than for the riser section. Here it is found that frictioninflicted by the core section dominates. As a consequence, by increasing the friction, the resistancein the loop increases, which in turn causes a decreasing mass flow rate.Further, it shows that the decrease of the mass flow rate declines for an increasing core height. Thiscan be explained by the increasing enthalpy step and with it the increasing density difference, whichin turn increases the buoyancy in the loop.

A2 - EFFECT ON THE CORE ENTHALPY STEP AND EXIT TEMPERATURE

The enthalpy step in the core versus the core height for several riser lengths can be seen in Figure4.2b. Over the interval of the increasing core height, the enthalpy step in the core increases withina range of 30.9% for a riser length of 3.5 [m] to 38.2% for a riser length of 10 [m]. As in case A1, theenthalpy step in the core is only correlated to the mass flow rate. An increasing mass flow rate, willlead to a decreasing enthalpy step.

Likewise for case A1, the core exit temperature is a function of the core enthalpy step. Increasing theenthalpy step will result in a increasing core exit temperature. The core exit temperature increasesover the interval of the increasing core height within a range of 3.4% for a riser length of 3.5 [m] to3.9% for a riser length of 10 [m]. The declining growth of the exit temperature also shows in thedeclining growth of the enthalpy step. This decline is amplified as the exit temperature approachesthe pseudo-critical temperature, with which an increase of the temperature requires more energy asthe specific heat of the coolant increases. The maximum core exit temperature found is 368.4 [°C].


Increasing the core height, means a decrease in the heat transfer deterioration rate, see Figure 4.3b.Here it shows that all results fall within safe criteria. Over the interval of the increased core height,the heat transfer deterioration rate decreased within in a range of 56.3% for a riser length of 3.5 [m]to 53.9% for a riser length of 10 [m].

Due to the decreasing mass flow rate one would expect an increasing heat transfer deteriorationrate. However, in this case the core dimensions are varied by increasing the core height, this resultsin a decreasing linear heat rate. The decreasing heat rate is dominant and consequently the heattransfer deterioration rate decreases.

4.1.4. A3 - VARIATION OF THE RISER DIAMETER

The riser diameter of the MASLWR is 0.91 [m]. This parameter in the design of the SLIMR is varied toinvestigate its effect on the natural circulation flow. Here it should be taken into account that thereis enough room in the riser section for the control rod mechanisms, and that a riser diameter thatis too large could cause suppression of the flow in the heat exchanger section. Having taken all thisinto account, the riser diameter is varied from 0.49 [m] to 1.17 [m] in steps of 0.02 [m] for severalconstant downcomer diameters ranging between 2.43 [m] and 4.50 [m]. Also for this case it is foundthat all steady-state working points are stable. The sub-cooling number over this range is constant,NSU B = 0.382, and the phase-change number NPC H = 0.160 remains practically unchanged as well.

56 4. RESULTS

4 6 8 10300

350

400

450

500

550

600

Riser Length [m]

Mas

s F

low

Rat

e [k

g/s]

2.00 m3.00 m4.25 m6.00 m

(a) Varied riser length for several constant coreheights.

2 3 4 5 6300

350

400

450

500

550

600

Core Length [m]

Mas

s F

low

Rat

e [k

g/s]

4.00 m6.00 m8.00 m10.00 m

(b) Varied core height for several constantriser lengths.

Figure 4.1: Steady-state mass flow rate at nominal operating power. The other parameters, that are kept constant in thesesimulations, can be found in Table 4.2.

4 6 8 10250

275

300

325

350

375

400

425

450

Riser Length [m]

Cor

e ∆

h

[kJ/

kg]

2.00 m3.00 m4.25 m6.00 m

(a) Varied riser length for several constant coreheights.

2 3 4 5 6250

275

300

325

350

375

400

425

450

Core Length [m]

Cor

e ∆

h

[kJ/

kg]

4.00 m6.00 m8.00 m10.00 m

(b) Varied core heights for several constantriser lengths.

Figure 4.2: Steady-state core enthalpy step, difference between core inlet and outlet, at nominal operating power. Theother parameters, that are kept constant in these simulations, can be found in Table 4.2.

From here on not all figures are shown as their correlations are repetitive. Besides, one can findrepresentative figures of their correlations in cases A1 and A2. The original figures can be found inthe Appendix A.


In Figure 4.4a the dependence of the mass flow rate versus the riser diameter can be seen for var-ious downcomer diameters. Over the interval of the increased riser diameter, the mass flow rateincreases within a range of 0.9% for a downcomer diameter of 2.43 [m] to 1.3% for a downcomerdiameter of 4.25 [m].

By increasing the riser diameter, the mass flow rate generally increases. This can be explained bythe increased cross-sectional flow area and with it the velocity of the coolant in the riser sectiondecreases, which in turn decreases the friction. In addition, the hydraulic diameter increases, andless friction is inflicted to the coolant (i.e. the coolant ‘feels’ less wall). By decreasing the friction inthe riser section, the resistance in the loop decreases, which in turn increases the mass flow rate. In


4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Riser Length [m]

HT

D r

ate

q/G

[kJ

/kg]

2.00 m3.00 m4.25 m6.00 m

(a) Varied riser lengths for several constantcore heights.

2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Core Length [m]

HT

D r

ate

q/G

[kJ

/kg]

4.00 m6.00 m8.00 m10.00 m

(b) Varied core heights for several constantriser lengths.

Figure 4.3: Steady-state heat transfer deterioration rate in the core at nominal operating power. The other parameters,that are kept constant in these simulations, can be found in Table 4.2.

general it is found that the riser diameter has only a small effect on the natural circulation flow, dueto the small contribution to the friction.

However, increasing the riser diameter> 0.9 [m] (keeping the diameter of the downcomer constant),could have a negative effect on the hydraulic radius of the heat exchanger section2. By increasingthe riser diameter the cross sectional flow area of the heat exchanger section decreases, and with italso the hydraulic radius in the heat exchanger section. On this account, the resistance in the loopincreases and the mass flow rate decreases, this can be seen clearly for a constant downcomer di-ameter of 2.49 [m].

Furthermore, decreasing the riser diameter < 0.7, the mass flow rate decreases rapidly, which indi-cates a non-linear increase of the friction. The increase of the friction can be explained by the steepincrease of the coolant velocity in the riser section, which is due to the non-linear decrease of thecross-sectional flow area of the riser.


The enthalpy step in the core is in these simulations only a function of the mass flow rate, just likethe correlations in previous cases. The results range between an overall minimum and maximumof 340−345 [kJ/kg]. Over the interval of the increased riser diameter, the enthalpy step decreaseswithin a range of 0.8% for a downcomer diameter of 2.43 [m] to 1.3% for a downcomer diameter of4.25 [m].

The found overall minimum and maximum core exit temperature range between 356.5−357.2 [°C],and just like the previous cases the core exit temperature is only correlated to the enthalpy step.Hereby the core exit temperature decreases over the interval of the increasing riser diameter withina range of 0.5% for a downcomer diameter of 2.43 [m] to 1.1% for a downcomer diameter of 4.25 [m].

2Given that the heat exchanger section contains 1000 cooling rods, and is hereby responsible for a wetted perimeter thatis approximately 160 larger than for the riser section.

58 4. RESULTS


Similar to case A1, the heat transfer deterioration is in this case3 only a function of the mass flow rate,ranging with very minor variations around 0.15 [kJ/kg]. By increasing the riser diameter, the massflow rate increases and results in a decreasing heat transfer deterioration rate. Over the interval ofthe increased riser diameter, the heat transfer deterioration rate decreases within a range of 0.8%for a downcomer diameter of 2.43 [m] to 1.3% for a downcomer diameter of 4.25 [m].

0.6 0.8 1434

435

436

437

438

439

440

441

442

Riser Diameter [m]

Mas

s F

low

Rat

e [k

g/s]

2.49 m3.01 m3.49 m3.99 m4.25 m

(a) Varied riser diameter for several constantdowncomer (outer annulus) diameters.

2.5 3 3.5 4434

435

436

437

438

439

440

441

442

Annulus Outer Diameter [m] M

ass

Flo

w R

ate

[kg/

s]

0.49 m0.69 m0.89 m1.09 m1.17 m

(b) Varied downcomer (outer annulus) diam-eter for several constant riser diameters.

Figure 4.4: Steady-state mass flow rate at nominal operating power.The other parameters, that are kept constant in thesesimulations, can be found in Table 4.2.

4.1.5. A4 - VARIATION OF THE DOWNCOMER DIAMETER

The downcomer diameter of the MASLWR is 2.74 [m]. It is expected that this parameter has a largeeffect on the heat transfer from the SLIMR to the pool. By varying this parameter it should be takeninto account that by decreasing the downcomer diameter there is enough room in the heat ex-changer section for the cooling rods, and that by increasing the downcomer diameter the SLIMRcan still be transported by truck or train [51]. Having taken all this into account, the downcomerdiameter is varied from 2.43 [m] to 4.50 [m] in steps of 0.02 [m] for several constant riser diame-ters ranging between 0.49 [m] to 1.17 [m]. Within this range it is found that all steady-state workingpoints are stable. The sub-cooling number over this range is constant, NSU B = 0.382, and the phase-change number ranges between NPC H = 0.158−0.160.


In Figure 4.4b the dependence of the mass flow rate versus the downcomer diameter can be seen forvarious riser diameters. Here it is shown that the overall minimum and maximum mass flow ratesrange between 434−441 [kg/s]. Over the interval of the increased downcomer diameter, the massflow rate increases within a range of 0.2% for a riser length of 0.49 [m] to 0.6% for a riser length of1.17 [m].

Likewise, as for case A3, the effect will be explained with the use of the hydraulic diameter. The in-crease of the downcomer diameter increases the hydraulic diameter in the heat exchanger section.Consequently, the friction in the heat exchanger section decreases, which in turn decreases the re-sistance in the loop and increases the mass flow rate. As for case A3, it is found that the effect on thenatural circulation flow is small due to the small contribution of the friction.3In this case, A3, the core geometry and power is unchanged, consequently the HTD is only affected by the mass flow

rate.



The enthalpy step in the core is, like previous simulations, only a function of the mass flow rate.Hereby the minimum and maximum correspond to case A3. Over the interval of the increaseddowncomer diameter, the core enthalpy step decreases within a range of 0.2% for a riser lengthof 0.49 [m] to 0.6% for a riser length of 1.17 [m].

The core exit temperature is only correlated to the enthalpy step in the core, and the overall mini-mum and maximum corresponds to case A3. Over the interval of the increased downcomer diame-ter, the core exit temperature decreases within a range of 1.1% for a riser length of 0.49 [m] to 1.7%for a riser length of 1.17 [m].


Just like in cases A1 and A3, the heat transfer deterioration is in this case4 only a function of themass flow rate, and yet again the overall minimum and maximum of the results correspond to caseA3. Over the interval of the increased downcomer diameter, the heat transfer deterioration ratedecreases within a range of 0.2% for a riser length of 0.49 [m] to 0.6% for a riser length of 1.17 [m].

4.1.6. A5 - VARIATION OF THE CORE INLET FRICTION

It is expected that due to the supercritical water the buoyancy force is so strong, that the desired coreexit temperature cannot be obtained (i.e. due to too high mass flow rate, the core exit temperatureis too low). To acquire the desired core exit temperature the resistance in the loop can be regulatedby the core inlet friction5 in natural circulation reactors. For the natural circulation BWR it is foundthat the typical inlet friction is Kcor ei n = 30 [52], and it is assumed that this typical value can also beapplied to the SLIMR. Therefore, the core inlet friction Kcor ei n is varied from 0 [-] to 30 [-] in stepsof 1 [-] for several constant riser lengths ranging between 3.50 [m] to 10.0 [m]. Within this range itis found that all steady-state working points are stable. The sub-cooling number over this range isconstant, NSU B = 0.382, and the phase-change number ranges between NPC H = 0.193−0.262.


The overall minimum and maximum mass flow rates found with these simulations range between275−460 [kg/s]. Over the interval of the increased core inlet friction, the mass flow rate decreaseswithin a range of 26.4% for a riser length of 3.50 [m] to 27.7% for a riser length of 10.0 [m].

As the inlet friction factor rises, the resistance in the loop increases, and consequently the mass flowrate decreases. Therefore, as already discussed in case A2, the enthalpy step in the core increasesand thereby the density difference also. As a result the decrease of the mass flow rate declines forincreasing core inlet friction.


Just like the previous cases, the core enthalpy step is a function of the mass flow rate. The core en-thalpy step increases over the interval of the increasing core inlet friction within a range of 35.9% fora riser length of 3.50 [m] to 38.3% for a riser length of 10.0 [m]. The overall minimum and maximumvalues that are found, range between 325−550 [kJ/kg].

The core exit temperature correlates to the core enthalpy step. Over the interval of the increasedcore inlet friction, the core exit temperature increases within a range of 4.9% for a riser length of3.50 [m] to 5.3% for a riser length of 10.0 [m]. The minimum and maximum core exit temperaturefound ranges between 340−370 [°C].

4In this case, A4, the core geometry and power is unchanged, consequently the HTD is only affected by the mass flowrate.

5By increasing the resistance in the loop, the mass flow rate decreases, which in turn increases the core exit temperature

60 4. RESULTS


Similar to the previous cases the heat transfer deterioration is a function of the mass flow rate. Theheat transfer deterioration rate increases over the interval of the increasing core inlet friction withina range of 35.9% for a riser length of 3.50 [m] to 38.3% for a riser length of 10.0 [m]. The minimumand maximum HTD rate found ranges between 0.14−0.24 [kJ/kg].

4.1.7. A6 - VARIATION OF THE CORE INLET TEMPERATURE

In the design of the HPLWR the core inlet temperature is 280 [°C]. This same temperature is main-tained in the design of the SLIMR. It is expected that the core inlet temperature affects the heatloss of the SLIMR to the pool during nominal operation. Besides, in this sub-section it will be in-vestigated how the core inlet temperature affects the nominal operation of the SLIMR. Taking intoaccount that the optimal thermal efficiency is achieved at 280 [°C] [14], the core exit temperatureis varied from 275 [°C] to 310 [°C] in steps of 1 [°C] for several constant core powers ranging be-tween 100 [MWth] and 250 [MWth]. It is found within this range that all steady-state working pointsare stable. Here the sub-cooling number over this range is constant, NSU B = 0.361−0.442, and thephase-change number ranges between NPC H = 0.118−0.121.


In Figure 4.5a the dependence of the mass flow rate versus the core inlet temperature can be seenfor various core powers. Over the interval of the increased core inlet temperature, the mass flow rateincreases with 3.1% for the whole range of the varied core powers.

As the core inlet temperature increases, the core exit temperature comes closer to the pseudo-critical temperature, with which the decline of the density becomes steeper when approaching thepseudo-critical temperature. Due to this, the average density difference between the ‘hot leg’ (coreand riser) and the ‘cold leg’ (heat exchanger and downcomer) increases for the increasing core inlettemperature. This increases the buoyancy in the loop, which in turn increases the mass flow rate.


The overall minimum and maximum steady-state core enthalpy steps that are found range between250− 465 [kJ/kg]. Over the interval of the increased core power, the core enthalpy step decreaseswith 3.0% for the whole range of the varied core powers. The increasing mass flow rate decreasesthe enthalpy step in the core, as explained for previous cases.

In Figure 4.6a a linear correlation can be seen between the core inlet temperature and the core exittemperature for a several core powers. It also shows that the growth of exit temperatures declinewhen they are approaching the pseudo-critical temperature, which is a result of the increasing heatcapacity. Over the interval of the increased core inlet temperature, the core exit temperature in-creases within a range of 8.3% for a core inlet temperature of 100 [MWth] to 4.9% for a core inlettemperature of 250 [MWth]. The maximum core exit temperature found is 373.9 [°C].


The heat transfer deterioration rate is a function of the mass flow rate. Over the interval of theincreased core inlet temperature, the heat transfer deterioration rate decreases with 3.0% for thewhole range of the varied core powers. The minimum and maximum HTD rate found ranges be-tween 0.11−0.20 [kJ/kg], and thus meeting the criteria to prevent for deteriorated heat transfer tothe core.

4.1.8. A7 - VARIATION OF THE CORE POWER

The nominal operation power of the MASLWR is 150 [MWth]. It is expected, by using supercriticalwater as coolant in an natural circulation system, that higher operating powers can be achieved.


Besides, a lower core power may be needed in order to make it possible that the decay heat can bedeposited passively to the environment. On that account, we chose to vary the core power from100 [MWth] to 250 [MWth] in steps of 5 [MWth] for several constant core inlet temperatures rangingbetween 275 [°C] to 310 [°C]. Within this range it is found that all steady-state working point arestable. The sub-cooling number over this range is constant, NSU B = 0.361−0.442, and the phase-change number ranges between NPC H = 0.215−0.208.


The dependency of the mass flow rate versus the core power for several inlet temperatures can beseen in Figure 4.5b. Over the interval of the increased core power, the mass flow rate increases with40.8% for the whole range of the varied core inlet temperatures.

The increasing core power increases the energy deposit in the core and results in a larger enthalpystep, which in turn raises the average density difference between the ‘hot leg’ and the ‘cold leg’. Thisdifference increases the buoyancy force, which in turn increases the mass flow rate. The declining

280 290 300 310380

400

420

440

460

480

500

520

540

560

Core Inlet Temperature [°C]

Mas

s F

low

Rat

e [k

g/s]

100 MW125 MW150 MW175 MW200 MW225 MW250 MW

(a) Varied core inlet temperature for severalconstant thermal core powers.

100 125 150 175 200 225 250380

400

420

440

460

480

500

520

540

560

Core Power [MW]

Mas

s F

low

Rat

e [k

g/s]

280 °C290 °C300 °C310 °C

(b) Varied thermal core power for several con-stant core inlet temperatures.

Figure 4.5: Steady-state mass flow rate at nominal operating power. The other parameters, that are kept constant in thesesimulations, can be found in Table 4.2.

280 290 300 310320

330

340

350

360

370

380


Cor

e E

xit T

empe

ratu

re [°

C]

100 MW125 MW150 MW175 MW200 MW225 MW250 MW

(a) Varied core inlet temperature for severalconstant thermal core powers.

100 125 150 175 200 225 250320

330

340

350

360

370

380

Core Power [MW]

Cor

e E

xit T

empe

ratu

re [°

C]

280 °C290 °C300 °C310 °C

(b) Varied thermal core power for several con-stant core inlet temperatures.

Figure 4.6: Steady-state core exit temperature at nominal operating power. The other parameters, that are kept constantin these simulations, can be found in Table 4.2.

62 4. RESULTS

growth of the mass flow rate6 can be explained by the increasing wall friction for correspondinghigher velocities, which, in turn, increases the resistance in the loop.


The core enthalpy step increases over the interval of the increasing core power within a range of77.6% for a core inlet temperature of 275 [°C] to 77.5% for a core inlet temperature of 310 [m]. Theresults range between an overall minimum and maximum of the steady-state corresponding to caseA6. In previous cases a correlation could be found where the core enthalpy step decreased for in-creasing mass flow rate. However, in this case the increasing energy deposit overrules this correla-tion completely.

In Figure 4.6b the core exit temperature versus the core power are shown for several core inlet tem-peratures. By increasing the core enthalpy step the core exit temperature increases. Over the intervalof the increased core power, the core exit temperature increases within a range of 9.6% for a core in-let temperature of 275 [°C] to 6.2% for a core inlet temperature of 310 [m]. The maximum core exittemperature found is 373.9 [°C].


The increase of the mass flow rate decreases the heat transfer deterioration rate. However the in-creasing core power outweighs the increase of the mass flow rate, and as a result the heat transferdeterioration rate increases. Over the interval of the increased core power, the heat transfer dete-rioration rate decreases with 77.6% for the whole range of the varied core inlet temperatures. Theminimum and maximum HTD rate found correspond to case A6, and thus meeting the criteria toprevent heat transfer deterioration.

4.1.9. A - STABILITY

During these simulations, no thermo-hydraulic instabilities were found in the considered range ofthe parameters. It must be noted here that reactivity feedback is not included in this model, whichcan still have a major impact on the stability. In Figure 4.7 one can find the stability map with thestable operation region that is obtained from the parameter study. The system operating condi-tions situated above the dashed dark blue line indicates that the core exit temperature is below thepseudo-critical temperature; the liquid is sub-cooled at the core exit.

Here it is expected that the stability map is similar to that of the HPLWR, which is again similar tothat of natural circulation BWR [53]. It is found that the obtained region on the stability map is alsostable for the HPLWR and natural circulation BWR. This stable region could therefore be consideredas a safe zone for operation. Further, it is found for the HPLWR and natural circulation BWR thatthe areas to the left and above the studied region in this work are stable, and this is also expected forthe SLIMR. These areas can be exploited during the start-up of the reactor; moving from zero powerand a low inlet temperature towards high power and a high inlet temperature, as the reactor vesselis pressurized beforehand.

For the HPLWR it is found that there is also a stable area underneath the dashed dark blue line. It isexpected that this area also exists for the SLIMR, but it is not found in this work. This area could beutilized to obtain a higher core exit temperature, which favors the thermal efficiency.

4.1.10. A - SUMMARY

To summarize, it is found in this phase that for all variations on the basic parameters of the SLIMRdesign, a steady-state natural circulation flow is established. In addition, it is found for all of these

6The mass flow rate can eventually decrease for high enough core powers.


0.1 0.2 0.3 0.4 0.5 0.60.1

0.2

0.3

0.4

0.5

0.6

NPCH

[−]

NS

UB [−

]

Core Outlet Enthalpy hout

=hpc

Stable Working Points

Figure 4.7: Stability map of the SLIMR, obtained from the data in the parameter study of this section. It is found that allworking points were stable.

normal operation points that they are stable, i.e. no occurrence of density wave oscillations. Further,a good agreement is found by comparing the stable region with the stability maps of the HPLWR andthe natural circulation BWR, from which it can be concluded that the obtained operation pointswere all stable. In addition, it can be considered that there is a trajectory that can be utilized to startup the reactor from a stationary state.

Secondly, it is found that the heat transfer deterioration rate ranges between 0.11−0.32 [kJ/kg] overall variations on the basic parameters. In this case the heat transfer deterioration rate does notexceed the condition of HTD<0.4 [kJ/kg] and excessive temperatures in the fuel rod are avoided.However, the heat transfer deterioration rate as evaluated in this work is an average. This is due tothat fact that the linear heat rate in the core - as modeled in this work - is constant, which is notthe case in a real reactor. For the HPLWR, the maximum linear heat is approximately 2.5 times theaverage linear heat rate. To comply to this for a core power of 150 MWth, it is found that the coreheight must be larger than 4 [m]. To this end, the HPLWR designed core height of 4.20 [m]7 will bemaintained in the design of the SLIMR.

Of all parameters - over their varied interval - it is found that the largest effect on the normal op-eration of the SLIMR are given by: the riser length (A1), the core height (A2), the inlet friction (A5),and the core power (A7), which is shown in Table 4.3. Moreover, it is confirmed that by increasingthe buoyancy in the loop - i.e. by increasing the height of the vessel (by increasing the riser length),by increasing the density over the core (by the core inlet temperature or the core power) - the massflow rate increases. In addition, it is also confirmed that by inflicting more resistance in the loop -i.e. by increasing the sections with a large wetted perimeter (like the core height), by decreasing thehydraulic diameter of the heat exchanger and riser sections and by increasing the core inlet friction- the mass flow rate decreases.

Furthermore it is found (for cases A1; A2; A3; A4; A5) that the core enthalpy step, core exit tempera-ture and the heat transfer deterioration rate are directly proportional but reversed to mass flow rate.Besides, for increasing mass flow rate, the deposit of energy in the core decreases which results ina declining core enthalpy step, which in turn results in a decrease of the core exit temperature. Theheat transfer deterioration rate decreases as the mass flux in the core increases, for a constant coregeometry, which is the case when the mass flow rate increases. The other cases refer to subsections4.1.7 and 4.1.8.

7Because of the grid size in our model, 0.25 [m], the core height used in the simulations is 4.25 [m].

64 4. RESULTS

Table 4.3: Effect of the parameters on the design of the SLIMR. Here M is the mass flow rate, ∆h the core enthalpy step, Tthe core exit temperature and HTD the heat transfer deterioration rate.

Parameter Interval M ∆h T HTD Stable

(A1) Riser length 3.5 - 10.0 [m] ↑ 24.8 - 31.7 % ↓30.9 - 38.2% ↓ 2.7 - 3.2 % ↓19.9 - 24.0% Yes

(A2) Core height 2.0 - 6.0 [m] ↓ 23.7 - 27.7 % ↑30.9 - 38.2% ↑ 3.4 - 3.9 % ↓53.9 - 56.3% Yes

(A3) Riser Dia. 0.49 - 1.17 [m] ↑ 0.9 - 1.3 % ↓0.8 - 1.3% ↓ 0.5 - 1.1 % ↓0.8 - 1.3% Yes

(A4) Ann. Dia. 2.43 - 4.25 [m] ↑ 0.2 - 0.6 % ↓0.2 - 0.6% ↓ 1.1 - 1.7 % ↓0.2 - 0.6% Yes

(A5) Inlet Friction 0 - 30 [-] ↓ 26.4 - 27.7 % ↑35.9 - 38.3% ↑ 4.9 - 5.3 % ↑35.9 - 38.3% Yes

(A6) Inlet Temp. 275 - 310 [°C] ↑ 3.1 % ↓3.0% ↑ 4.9 - 8.3 % ↓3.0% Yes

(A7) Power 100 - 250 [MWth] ↑ 40.8 % ↑77.5 - 77.6% ↑ 6.2 - 9.6 % ↑77.6% Yes

The overall minimum and maximum values that are found in this parameter study are listed as fol-lows: the mass flow rate ranges between 275− 560 [kg/s]; the core enthalpy step ranges between250 − 550 [kJ/kg]; the core exit temperature ranges between 326 − 374 [°C]. Here it can be notedthat the obtained temperatures are below the desired core exit temperature of 400 [°C], and conse-quently, a lower thermal efficiency is obtained than desired.

4.2. PHASE B - HEAT TRANSFERIn the design of the SLIMR it is of great importance that in the event of an emergency shutdownthe reactor cools down by its passive design. However, by the same passive design heat will alsobe transported from the coolant to the pool during nominal operation. In this phase of the thesisobjective, shown in section 1.5, the central question is:

What is the steady-state heat loss of such a SLIMR design during normal operation?

This will be studied by varying the dimensions of the most basic geometrical parameters of theSLIMR: the downcomer diameter, the height of the RPV (varying the riser length), and the stainlesssteel slab thickness of the RPV. Besides the geometrical parameters, it is also investigated what theeffect of the core inlet temperature, and pool temperature is on the heat loss to the pool.

4.2.1. B - SIMULATION PROCEDURE AND INITIAL SETUP

In order to obtain the results in this section the algorithm that is described in chapter 3 is utilized.The framework of this section focuses on the heat transfer from the SLIMR to the pool. Because ofthis, the pressure correction loop and the heat transfer loop are coupled in the time stepping loop,see Figure 3.2, in which the pool temperature is kept constant in time. Moreover, in this model onlythe thermal-hydraulics are dealt with, in which reactivity feedback is not accounted for.

It is expected that, if a neutronics model is incorporated, a lower inlet temperature - which increasesthe density and as a result increases the moderation - will lead to a higher core power. To this end,the heat loss from the downcomer to the pool will be compensated by a higher core power. How-ever, this is not the case in this model, where the core power equals the heat that is extracted in theheat exchanger. In case there is also a heat loss in the downcomer, this will result in a net heat loss.

4.2. PHASE B - HEAT TRANSFER 65

The consequence of this net heat loss is that no steady-state situation can be established, and theoverall temperature in the SLIMR would decrease in time.

In order to prevent this, the heat that is lost to the pool is not subtracted from the internal flow in thedowncomer of the SLIMR. As a result, the temperature of the fluid in the downcomer, between theheat exchanger inlet and the core inlet, is homogeneously distributed in axial direction and equal tothe exit temperature of the heat exchanger.

However, by this approach an error is introduced, because now there is no temperature gradient inthe downcomer section, where it is expected. The error due to this is however insignificantly small,since the temperature gradient in the complete downcomer would only be between 1 - 4 [°C]. Hereit should be noted once again that in this section the heat loss to the pool is of interest.

The simulation algorithm can be split up in 3 steps. These steps run for several time intervals, seetable 4.4, which lengths are chosen to ensure that the objective of each step is achieved. The timesteps and grid size in this phase are based on a compromise between the discretization error, thecomputation time, the Courant number and the Fourier number. The procedure and objectives foreach step can be explained as follows:

Table 4.4: Selected simulation times and their corresponding time step, and spatial grid size for each of the procedures inCASE B. During all steps the fully implicit scheme is used.

Time

Steady-State Reaching Nominal Operating Power 500 sReaching Steady-State Mass Flow Rate 500 sReaching Steady-State Heat Transfer 36000 sTime step 0.1 s

Space

Grid size 0.25 m

STEP B1 - REACHING NOMINAL OPERATING POWER

This step is equal to step A1 and runs for 500 seconds. The objective of this step is to initiate thenatural circulation flow. In the initial state the temperature of the coolant is distributed homoge-neously and equals the set core inlet temperature, and additionally starts the fluid at rest.

The temperature profile over the RPV slab is distributed by a temperature gradient, where the tem-perature of the inner surface of the RPV slab equals the set core inlet temperature, and the outersurface of the RPV equals the pool temperature.

STEP B2 - REACHING STEADY-STATE MASS FLOW RATE

The second step is equal to step A2 and runs for 500 seconds. The objective of this step is to ensurethat possible mass flow rate oscillations are completely dampened.

STEP B3 - REACHING STEADY-STATE HEAT TRANSFER

The third and last step runs for 10 hours. The objective of this step is to ensure that a steady-stateheat transfer from the SLIMR to the pool is established; the spatial distribution of the temperature inthe RPV does not change any further, i.e. the derivative of the temperature distribution with respectto time is zero.

66 4. RESULTS


In this study for each simulation only one parameter is varied, the parameters that are not variedare kept constant to the values that can be found in Table 4.5.

Table 4.5: Initial parameters of the SLIMR design, for CASE B.

Reactor Dimensions


Riser Length 9.00 mDiameter 1.00 m

Heat Exchanger Length 2.50 mDowncomer Outer Annulus Diameter 2.44 mRPV Vessel Thickness 0.37 m

Operation Point



Pool

Water Temperature 40 °C

4.2.2. B1 - VARIATION OF THE DOWNCOMER DIAMETER

It is expected that the width of the vessel has a large effect on the heat loss of the SLIMR as the heatexchanging surface area increases. This effect is investigated by varying the downcomer diameter- as in case A4 - from 2.25 [m] to 4.25 [m] in steps of 0.125 [m] for several constant riser lengthsranging between 3.5 [m] and 10 [m].

B1 - EFFECT ON THE TOTAL HEAT LOSS

In Figure 4.8a the dependence of the total heat loss of the SLIMR versus the downcomer diametercan be seen for various riser lengths. The total heat loss increases over the interval of the increasingdowncomer diameter within in a range of 29.4% for a riser length of 3.5 [m] to 42.7% for a riser lengthof 10 [m]. By increasing the downcomer diameter, the heat exchanging surface areas - internal andexternal of the vessel - increases, and consequently the total heat loss from the SLIMR to the poolincreases.

A part of the downcomer is the heat exchanger section. This is a constant factor in the SLIMR de-sign; 2.5 [m] in length, consisting of 1000 helical shaped heat exchanging rods, which are equallyspread over the cross sectional flow area of the annulus (downcomer). It is due to these rods thatthe hydraulic diameter approximately 160 times smaller than for the rest of the downcomer. Thisresults in a less turbulent flow in comparison to the rest of the downcomer. As a consequence it ismodelled that the fluid is mixed less, and has a respectively large thermal boundary layer8 at theinner surface of the vessel.

On the other hand, it is expected that due to the rods the mixing of the fluid is stimulated. It is there-fore possible that the heat transfer model in this work is inadequate for this part of the downcomer9

. However, it is found that the heat flux for this part of the downcomer is approximately 2-3 times

8The thermal boundary layer is defined as the distance from the surface at which the temperature difference T −Ts equals0.99(T∞−Ts ).

9A possible mistake is made by considering the geometry of the heat exchanger section as an annular duct, just like forthe rest of the downcomer, by Dh = Di −Do . This comment only applies to the heat transfer model.


smaller than for the rest of the downcomer, which in turn make the results in this work more con-servative in terms of safety.

For the heat exchanger in this sub-section it is found that the heat loss is approximately 0.16 [MW]for a downcomer diameter of 2.25 [m], and for 4.25 [m] its contribution decreases by approximately60% to only 0.06 [MW]. This decrease can be explained by the increasing cross sectional flow areaof the downcomer, which results in a decrease of the fluid velocity, which in turn decreases theturbulence of the flow. Therefore, the heat transfer decreases from the internal flow to the wall ofthe RPV for the increasing downcomer diameter.

B1 - EFFECT ON THE EXTERIOR TEMPERATURE OF THE VESSEL

The exterior temperature of the RPV versus the downcomer diameter can be seen in Figure 4.9a.Over the interval of the increasing downcomer diameter, the temperature decreases within in arange of 1.5% for a riser length of 3.5 m to 1.1% for a riser length of 10 m. During the increase ofthe downcomer it is found that the heat flux at the exterior surface of the vessel decreases. Thisexplains the decrease of the temperature at this surface.

2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25

0.8

1

1.2

1.4

1.6

1.8

2

Annulus Outer Diameter [m]

Tot

al H

eat L

oss

[MW

]

3.50 m4.50 m5.50 m6.50 m7.50 m8.50 m9.50 m

(a) Varied downcomer (outer annulus) diam-eter for several constant riser lengths.

4 5 6 7 8 9 10

0.8

1

1.2

1.4

1.6

1.8

2

Riser Length [m]

Tot

al H

eat L

oss

[MW

]

2.250 m2.625 m3.000 m3.375 m3.750 m4.125 m

(b) Varied riser length for several constantdowncomer (outer annulus) diameters.

Figure 4.8: Steady-state total heat loss during nominal operating power. The other parameters, that are kept constant inthese simulations, can be found in Table 4.5.

2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.2552.8

53

53.2

53.4

53.6

53.8

54

54.2


Ext

erio

r T

empe

ratu

re R

PV

[°C

]

3.50 m4.50 m5.50 m6.50 m7.50 m8.50 m9.50 m

(a) Varied downcomer (outer annulus) diam-eter for several riser lengths.

4 5 6 7 8 9 1052.8

53

53.2

53.4

53.6

53.8

54

54.2

Riser Length [m]

Ext

erio

r T

empe

ratu

re R

PV

[°C

]

2.250 m2.625 m3.000 m3.375 m3.750 m4.125 m

(b) Varied riser length for several downcomer(outer annulus) diameter.

Figure 4.9: Steady-state external surface temperature of the SLIMR during nominal operating power. The other parame-ters, that are kept constant in these simulations, can be found in Table 4.5.

68 4. RESULTS

4.2.3. B2 - VARIATION OF THE RISER LENGTH

It is expected that the vessel height has a large effect on the heat loss of the SLIMR, since it affectsthe heat exchanging surface area of the SLIMR and the natural circulation of the flow. This effect isinvestigated - as in case A1 - varying the riser length from 3.5 [m] to 10 [m] in steps of 0.25 [m] forseveral constant downcomer diameters ranging between 2.25 [m] and 4.25 [m].


Figure 4.8b shows the dependence of the total heat loss from the RPV to the pool versus riser lengthfor several downcomer diameters. The heat loss increases over the interval of the increasing riserlength within in a range of 83.4% for a downcomer diameter of 2.25 [m] to 102.2% for a downcomerdiameter of 4.25 [m]. The increase of the total heat loss can be explained by the increase of the heatexchanging surface area of the RPV due to the increasing riser length. Additionally, by increasingthe riser length the natural flow in the SLIMR is affected (as found in case A1). Here the velocity ofthe coolant increases, which in turn stimulates the forced convective heat transfer from the coolantto the RPV wall.

The heat loss that is specific to the part of the RPV next to the heat exchanger, increases over theinterval of the increasing downcomer diameter within a range of 5.3% for a downcomer diameter of2.25 [m] to 6.7% for a downcomer diameter of 4.25 [m]. The explanation for this increase is similarto the increase of the total heat loss.


The exterior temperature of the RPV versus the downcomer diameter is shown in Figure 4.9b. Overthe interval of the increasing riser length, the surface temperature increases within in a range of0.1% for a downcomer diameter of 2.25 m to 0.5% for a downcomer diameter of 4.25 m. The surfacetemperature that is specific to the part of the RPV next to the heat exchanger increases over theinterval of the increased riser length within a range of 0.8% for a downcomer diameter of 2.25 [m]to 0.4% for a downcomer diameter of 4.25 [m]. Since it is determined in this sub-section that theheat flux increases, the minor increase of the surface temperature can be explained by Newton’s lawof cooling, see equation 2.27. By increasing the riser length, the heat flux increases, which in turnincreases the surface temperature and free convective heat transfer coefficient.

4.2.4. B3 - VARIATION OF THE POOL TEMPERATURE

The MASLWR is submerged in a pool with a temperature of 40 [°C]. Higher pool temperatures couldbe more efficient as the heat loss to the environment is smaller. Besides, lower pool temperaturesare more desirable for working activities around the pool and it decreases the water losses by evap-oration. On that account, we chose to vary the pool temperature from 25 [°C] to 75 [°C] in steps of2.5 [°C] for several constant core inlet temperatures ranging between 275 [°C] and 310 [°C].


The overall minimum and maximum values of total heat loss found by these simulations range be-tween 1.09−1.55 [MW]. Here, the total heat loss decreases over the interval of the increasing pooltemperature within in a range of 18.6% for a core inlet temperature of 275 [°C] to 16.2% for a coreinlet temperature of 310 [°C]. A similar correlation is found for the heat loss to the part of the RPVnext to the heat exchanger. However, the overall minimum and maximum contribution of the heatloss in this part ranges between 0.11−0.16 [MW]. The decrease of the heat loss can be explained bythe decline of the temperature gradient over the RPV slab for increasing pool temperature.


The overall minimum and maximum RPV surface temperature found for these simulations are 43−86 [°C], respectively. By increasing the pool temperature, the RPV surface temperature rises within


a range of 106.6% for a core inlet temperature of 275 [°C] to 102.0% for a core inlet temperatureof 310 [°C]. A similar correlation is found for RPV surface temperature specific to the part of theRPV next to the heat exchanger. However, for this part the overall minimum and maximum surfacetemperature are found to be 36−83 [°C], respectively. This can be explained by the fact that the outersurface temperature of the vessel must follow the increasing pool temperature in order to transferthe excess of heat from the SLIMR to the pool.

4.2.5. B4 - VARIATION OF THE CORE INLET TEMPERATURE

The core inlet temperature equals the temperature of the coolant in the downcomer. Hence, it af-fects the heat transfer to the pool. The core inlet temperature is varied - as in case A6 - from 275 [°C]to 310 [°C] in steps of 2.5 [°C] for several constant pool temperatures ranging between 25 [°C] and 75[°C]. The overall minimum and maximum values of the results in this case correspond to case B3.


In this case the total heat loss increases over the interval of the increasing core inlet temperaturewithin in a range of 14.6% for a pool temperature of 25 [°C] to 18.0% for a pool temperature of 75[°C]. For the part of the RPV adjacent to the heat exchanger a similar correlation is found for the totalheat loss. The increase in heat loss can be explained by the increased temperature gradient, which isthe driving force for heat transfer. By increasing the temperature of the coolant in the downcomer,the temperature gradient of the downcomer to the pool increases. Thus the heat loss increases.


The exterior surface temperature of the RPV increases over the interval of the increasing core inlettemperature within a range of 3.9% for a pool temperature of 25 [°C] to 1.5% for a pool temperatureof 75 [°C]. For the outer surface temperature of the part of the RPV that is adjacent to the heat ex-changer a similar correlation is found.

Due to the increasing average heat flux, which in turn is caused by the increasing the core inlettemperature, the exterior surface temperature of the SLIMR increases.

4.2.6. B5 - VARIATION OF THE THICKNESS OF VESSEL

It is expected that the thickness of the vessel has a large effect on the heat transfer to the pool,which will be investigated in this sub-section. The vessel thickness of an HPLWR is 0.45 [m], of theMASLWR the vessel thickness is 0.127 [m], and the vessel thickness of the SLIMR is 0.37 [m]. On thisaccount the thickness of the RPV is varied from 0.10 [m] to 0.50 [m] in steps of 0.02 [m].


The overall minimum and maximum values of total heat loss, found by these simulations, rangebetween 1.1− 3.4 [MW]. Here it decreases over the interval of the increasing thickness of the RPVwith 69.2%. The contribution to the total heat loss from the part of the RPV that is next to the heatexchanger ranges between 0.13−0.19 [MW]. For this part a similar correlation is found; in this casethe heat loss decreases by 34.4% for increasing thickness of the RPV wall.

This can be explained by the increasing thickness of the RPV, which increases the resistance of heattransfer from the internal flow to the pool, and consequently, the heat transfer from the SLIMR tothe pool decreases. For the part of the RPV that is next to the heat exchanger the effect is smallerdue to lower heat transfer by forced convection, as discussed in case ‘B1 - Total Heat Loss’.

The explanation for the decrease of the average heat loss is equal to that of the total heat loss.

70 4. RESULTS


The overall minimum and maximum RPV surface temperature found in these simulations are 51−72[°C], respectively. Here, over the interval of the increasing core inlet temperature, the exterior sur-face temperature of the RPV decreases with 28.9%. A decrease of the exterior surface temperatureis also found for the part of the RPV that is adjacent to the heat exchanger, which is 9.7%. For thissection there is a minimum and maximum RPV surface temperature of 48−53 [°C], respectively.

The decrease of the surface temperature is due to the decrease in the average heat flux, as is dis-cussed.

4.2.7. B - SUMMARY

To summarize, it is found that the SLIMR has a significant heat loss - in the order of 1 [MW] - duringnominal operating power. This high heat loss is inherent to the passive design of this reactor.

Further, it is found that of the studied parameters in this section the largest effects over their variedinterval on the heat loss are given by: the riser length (B2), the thickness of the vessel (B5) and thedowncomer diameter (B1), respectively. The effect of all parameters over their varied intervals isshown in Table 4.6. Moreover, it is confirmed that by increasing the heat exchanging surface area -i.e. by increasing the height or the width of the SLIMR - the heat loss increases. Also decreasing theresistance - i.e. by decreasing the thickness of the RPV - results in an increasing heat loss. In addi-tion, by increasing the temperature gradient over the RPV - i.e. by decreasing the pool temperatureor increasing the core inlet temperature - the heat loss increases.

Table 4.6: Effects of the parameters on the steady-state heat transfer of the SLIMR during nominal operating power.

Parameter Interval Total Heat Loss Average Heat Flux RPV Ext. Temp.

(B1) Ann. Dia. 2.25 - 4.25 [m] ↑ 29.4 - 42.7 % ↓ 8.2 - 6.1 % ↓ 1.5 - 1.1 %

(B2) Riser length 3.5 - 10.0 [m] ↑ 83.4 - 102.2 % ↑ 0.0 - 2.3 % ↑ 0.1 - 0.5 %

(B3) Pool Temperature 25 - 75 [°C] ↓ 16.2 - 18.6 % ↓ 16.3 - 18.7 % ↑ 102.2 - 106.6 %

(B4) Core Inlet Tem. 275 - 310 [°C] ↑ 14.6 - 18.0 % ↑ 14.6 -18.1 % ↑ 1.5 - 3.9 %

(B4) Thickness RPV 0.10 - 0.50 [m] ↓ 69.2 % ↓ 78.0 % ↓ 28.9 %

The overall minimum and maximum values that are found in this parameter study are listed as fol-lows: the total heat loss from the SLIMR, found for cases B1 - B4, ranges between 0.72−1.88 [MW],for an RPV thickness of 0.37 [m]. Moreover, by decreasing the RPV wall thickness to 0.10 [m], evenhigher heat losses are found to 3.4 [MW]. This vessel thickness corresponds to the thickness of theMASLWR, and the NuScale reactor, and explains the secondary containment in their design, whichis there to decrease the heat loss to the environment during normal operation.

Besides, it is found that the minimum and maximum exterior surface temperature ranges between43.8−72.6 [°C] (by excluding case B3, the increase of the pool temperature), which are all below thesaturation temperature. Hence, the heat transfers from the outer surface of the SLIMR to the pool bythe phenomenon of natural convection, and nucleate boiling is excluded during nominal operatingpower.

4.3. PHASE C - POOL 71

4.3. PHASE C - POOLAs it is found in section ‘B - Heat Transfer’, the heat transfer to the pool during nominal operationis significantly large. Therefore it is important in the design of the facility that the pool can transferthis heat passively to the environment. In this phase of the thesis objective, shown in section 1.5,the central question is:

With what pool dimensions is it possible to transfer this heat passively to the environment andmaintain safe working conditions (i.e. a pool water temperature around 40 °C)?

4.3.1. C - SIMULATION PROCEDURE AND INITIAL SETUP

In order to obtain the results for this section the algorithm that is described in chapter 3 is utilized,in which the framework of this section only focuses on the pool in which the SLIMR is submerged,and it is limitted only to section 3.3.3. Furthermore, the pool in this section is not coupled to theSLIMR, and for that reason the heat loss of the SLIMR is approximated. The heat loss in this case iskept constant at 1 [MW]. This value is obtained from phase B, and represents the order of magnitudeof the heat loss. Consequently, the steady-states could be obtained in an iterative way.

C - ASSUMPTIONS AND LIMITATIONS

The relative humidity of the air in the facility is constant at 80% (yearly averaged for de Bilt, KNMIthe Netherlands). Here it is assumed that the air in the facility is continuously refreshed. Further itis assumed that the air directly above the pool is stagnant; there is no forced airflow in the facility.The evaporated pool water is replaced by make-up water, and its temperature equals the set envi-ronmental temperature. Besides, it is simulated that the water level remains constant and by themixing of the make-up water the overall pool temperature decreases. In order to calculate the netradiative heat loss, it is assumed that the roof of the facility is kept at a constant temperature, andequals the set environmental temperature. The pool is modeled as a single node, which cannot sim-ulate the free convection of the water in the pool. The consequence of this is, that the temperatureof the pool in this situation is therefore homogeneously distributed. Hence, it is assumed that thefluid is perfectly mixed by natural circulation and by the addition of make-up water.

4.3.2. C1 - POOL WIDTH

In this case the width of the square shaped pool is varied from 10.0 [m] to 40.0 [m] in steps of 2.5 [m]for several constant environmental temperatures ranging between 10 [°C] and 35 [°C]. The depen-dence of the average steady-state pool temperature is shown in Figure 4.10a. A maximum averagepool temperature of 84 [°C] is found for a pool width of 10 [m] (in this case the free surface of thepool is 10 [m] x 10 [m] = 100 [m2]). By increasing the width of the pool, the free surface area thatcan exchange heat with the environment increases, and also the volume of the pool. Due to theincreased free surface area the temperature of the pool decreases over the interval of the increasedpool width with 57.6% for an environmental temperature of 10 [°C] to 42.9% for an environmentaltemperature of 35 [°C]. Therefore the lowest average pool temperatures are found for a pool widthof 40 [m], where the free pool surface is 1600 [m2]; for an environmental temperature of 10−35 [°C]steady-state pool temperatures are determined that are 35−47 [°C], respectively.

4.3.3. C2 - ENVIRONMENTAL TEMPERATURE

Here the temperature of the environment is varied from 10 [°C] to 35 [°C] in steps of 1 [°C] for severalconstant pool widths ranging between 10.0 [m] and 40.0 [m]. For this case, the dependence of theaverage pool temperature is shown in Figure 4.10b. It shows that the steady-state average pool tem-perature increases over the interval of the increasing environmental temperature within in a rangeof 3.0% for a pool width of 10.0 [m] to 38.7% for a pool width of 40.0 [m]. The minimum and maxi-

72 4. RESULTS

10 20 30 4030

40

50

60

70

80

90

Pool Width [m]

Poo

l Wat

er T

empe

ratu

re [°

C]

10.0 °C17.5 °C27.5 °C35.0 °C

(a) Varied pool width for several constant en-vironmental temperatures.

10 15 20 25 30 3530

40

50

60

70

80

90

Environment Temperature [°C]

Poo

l Wat

er T

empe

ratu

re [°

C]

5.0 m17.5 m27.5 m40.0 m

(b) Varied environmental temperature forseveral constant pool widths.

Figure 4.10: Steady-state average pool temperature for an average heat loss from the SLIMR of 1 [MW] during nominaloperating power.

40 50 60 70 800

50

100

150

200

250

300

350

400

450

Pool Water Temperature [°C]

Eva

pora

tion

[kg/

m2 *d

ay]

10.0 °C17.5 °C27.5 °C35.0 °C

(a) Varied average pool temperatures for sev-eral constant environmental temperatures.

10 15 20 25 30 350

50

100

150

200

250

300

350

400

450

Environment Temperature [°C]

Eva

pora

tion

[kg/

m2 *d

ay]

40.0 °C55.0 °C72.5 °C85.0 °C

(b) Varied environmental temperature forseveral constant average pool temperatures.

Figure 4.11: Steady-state pool evaporation during nominal operating power.

mum temperatures correspond to case C1.

For pools with a large free surface area it is found that the steady-state average pool temperatureis affected more dominantly by the environmental temperature. This can be explained by loweraverage pool temperatures, that are found for large free surface areas, with which the relative tem-perature difference between the pool and the environment is much larger for relatively low averagepool temperatures than for high average pool temperatures.

4.3.4. C3 - EVAPORATION

This subsection is focused on the pool water loss by evaporation. In this case the temperature ofthe pool water is varied from 40 [°C] to 85 [°C] in steps of 1 [°C] for several constant environmentaltemperatures ranging between 10 [°C] and 35 [°C]. The dependence of the evaporation rate (i.e. theevaporated water mass per square meter per day) for this case is shown in Figure 4.11a. It showsthat the lowest evaporation rates can be found for lowest pool water temperatures. By increasingthe pool water temperature over the defined interval, the evaporation rate increases within a range

4.3. PHASE C - POOL 73

of 1.90 ·103% for an environmental temperature of 10 [°C] to 6.44 ·103% for an environmental tem-perature of 35 [°C]. The minimum and maximum evaporation rates found range between 5.6−414.9[kg/m2day], respectively. These results are obtained by the empirical correlation of Shah[32]. Thisincrease can be explained by the average velocity of the water molecules, which is correlated to theenergy of the water molecule and in turn the temperature of the water. Furthermore, the velocitiesof the water molecules have a Gaussian distribution. Here the molecules that exceed a thresholdvelocity, are needed to escape the free water surface, break out and become vapor. Therefore, byincreasing the temperature of the water, the average velocity of the molecules increases, resulting inmore molecules that could escape the water surface which increases the evaporation rate.

In the second case the environmental temperature is varied from 10 [°C] to 35 [°C] in steps of 1 [°C]for several constant pool water temperatures ranging between 40 [°C] and 85 [°C]. The results can beseen in Figure 4.11b. Here it shows that the lowest environmental temperatures cause the highestevaporation rates, which causes the evaporation rate to decrease over the interval of the increasingenvironmental temperature within a range of 73.0% for a pool temperature of 40 [°C] to 11.6% fora pool temperature of 85 [°C]. The minimum and maximum evaporation rates correspond to thevalues that were mentioned previously. The decrease of the evaporation rate can be explained by thedecreasing temperature gradient; from the saturated vapor layer adjacent to the free water surfaceto the ambient temperature far from the water surface. Because of this the convective currents -which are incorporated in the correlation of Shah - decrease, which in turn has a declining effect onthe evaporation rate.

4.3.5. C - SUMMARY

To summarize, for all cases a steady-state pool temperature is found that is below the saturationtemperature of 100 [°C], i.e. no boiling occurs during normal operation. Here it is found that thesteady-state pool temperature decreases for the increasing pool surface, and that the environmen-tal temperature affects the pool temperature more dominantly for low pool temperatures.

The maximum temperature found for a free pool surface of 100 [m2] in combination with an envi-ronmental temperature of 35 [°C] is 84 [°C], for a decreasing environmental temperature a steady-state pool temperature of 81 [°C] is found. For a free pool surface of 1600 [m2] a steady-state pooltemperature of 47 [°C] and 35 [°C] are found, respectively.

As elaborated in the theory, Chapter 2, the greater part of the heat transfer is due to evaporation ofthe pool water at the free surface. Here it is found, for a steady-state pool temperature of 40 [°C] anda room temperature of 20 [°C], that the evaporation rate is 16 [kg/m2day]. By increasing the pooltemperature to 85 [°C], the evaporation rate increases to 398 [kg/m2day].

In order to obtain a desirable pool temperature of 40 [°C] at an environmental temperature of 20 [°C](room temperature for good working conditions around the pool), the surface of the pool must beat least 1600 [m2] (this is 1.3 times the size of an Olympic swimming pool). The evaporation rate fora pool of this size under the conditions mentioned before is 25.6 ·103 [kg/day]. In the event that thepool can not be refilled during one day, this result in a decrease of the pool water level by 1.6 [cm].This is a small decrease, which makes it unnecessary to refill the pool continuously. However, thedesign is dependent on a fresh feed of water, as the evaporated water must be replaced. This makesthe design therefore less suitable for deployment at a remote location without access to water.

74 4. RESULTS

4.4. PHASE D - ACCIDENT SCENARIO

The previous sections of this chapter merely focus on the steady-state operation of the SLIMR. Hereit is found that the reactor operates safely (stable) at nominal operating power for all the simulatedcases. Furthermore, it is found that the reactor has a relative high heat loss, and consequently, theSLIMR needs to be submerged in a large pool with a free pool surface of at least 1600 [m2] (1.3 timesan Olympic swimming pool) in order to passively carry through the heat of the SLIMR. In this phaseof the thesis objective, section 1.5, the central question is:

Is it possible to obtain a geometry of the SLIMR design that is within the boundaries of a safenominal operation ànd allows safe deposition of decay heat to the environment under acciden-tal situations in a fully passive way, without damaging the core?

This phase is the ultimate goal in this thesis, in which it will be verified whether it is possible to coolthe SLIMR completely passively from its decay heat in the event of a SCRAM and the active heatexchangers in the downcomer are not working. Moreover, it is expected that the reactor keeps pres-surized at 25 [MW]. Here it is expected that the produced decay heat is transported via the coolantfrom the core to the downcomer by means of a natural circulation flow. In the downcomer thecoolant can exchange heat with the pool water, which are separated from each other by the stain-less steel RPV wall.

In this phase the geometrical parameters that most dominantly affect the heat transfer of the SLIMRare varied. These parameters are found to be the downcomer diameter (heat transfer case B1), andthe riser length (natural circulation flow case A1; heat transfer case B2). By studying these parame-ters, the feasibility of the SLIMR during an accident will be investigated in terms of safety.

4.4.1. D - SIMULATION PROCEDURE AND INITIAL SETUP

In order to obtain the results in this section the algorithm that is described in chapter 3 is utilized, inwhich the framework of this section focuses on the internal flow of the SLIMR and the heat transferfrom the SLIMR to the pool. For this reason the flow model (see Figure 3.3) and the heat trans-fer model (see Figure 3.5) are coupled in the time stepping loop, see Figure 3.2. Further, as it isdetermined in phase C, it is necessary for the pool to have a significant free surface area, and, con-sequently, a large volume. Here it is found that fluctuations of the heat loss from the SLIMR to thepool are insignificant due to this volume. To this end, the model for the transient calculation of thepool temperature (see Figure 3.6) is not included, and the pool temperature is kept constant at 40[°C].

The simulation algorithm can be split up in 4 steps. These steps run for several time intervals, seetable 4.7, and their lengths are chosen to ensure that the objective of each step is achieved. The timesteps and grid size in this phase are based on a compromise between the discretization error, thecomputation time, the Courant number and the Fourier number. The procedure and objectives foreach step can be explained as follows:

STEP D1 - REACHING NOMINAL OPERATING POWER

This step is similar to step A1, and runs for 500 seconds. The objective of this step is to increase thecore power, the nominal operating power and initiate the natural circulation flow. Furthermore, thisstep only deals with thermal-hydraulics, with which heat enters the system via the core and can byextracted by the heat exchanger.

4.4. PHASE D - ACCIDENT SCENARIO 75

Table 4.7: Selected simulation times and their corresponding time steps, for CASE D.

Time

Steady-State Flow Reaching Nominal Operating Power 500 sReaching Steady-State Operation 500 sTime step 0.1 s

Steady-State Heat Temperature Distribution RPV 6 hTime step 1 s

SCRAM Simulating Decay Heat Removal 72 hTime step 1 s

Space

Grid size 0.25 m

STEP D2 - REACHING STEADY-STATE OPERATION

This step is similar to step A2, and runs for 500 seconds. The objective of this step is to ensure thatthe mass flow rate oscillations dampen. Just like in step D1, energy can only enter the system via thecore and be subtracted by the heat exchanger.

STEP D3 - TEMPERATURE DISTRIBUTION RPVThe third step is equal to step B3 and runs for 6 hours. The objective of this step is to ensure that asteady-state temperature profile over the RPV slab is achieved, i.e. the derivative of the temperaturedistribution with respect to time is zero. Just like in section B, the energy that is gained by the RPVis not extracted from the internal flow.

STEP D4 - SIMULATING DECAY HEAT REMOVAL

This step is initiated with a SCRAM of the system, and from this moment decay heat is produced inthe SLIMR. Here the decay heat is estimated by an analytical function (equation 2.47), for a core thathas been in operation for an infinite time. This step lasts for 72 [h]; for this period it will be assessedwhether the overall temperature in the SLIMR decreases or escalates.

To make a judgment on the feasibility of the SLIMR in terms of safety the following results will bereviewed: the maximum temperature of the coolant in the core within the 72 [h] (here the maximumcore temperature is determined over the interval starting 10 [s] after the SCRAM), the temperatureof the coolant in the core at the end of the 72 [h], and the maximum heat transfer deterioration ratewithin the 72 [h] (focusing on the first minutes after the SCRAM).


In this study for each simulation only one parameter is varied, the parameters that are not variedare kept constant to the values that can be found in Table 4.8.

4.4.2. D1 - VARIATION OF THE DOWNCOMER DIAMETER

The downcomer diameter is varied from 2.25 [m] to 4.25 [m] in steps of 0.125 [m] for several con-stant riser lengths ranging between 3.5 [m] and 10 [m]. First the behavior of the system - after aSCRAM accompanied with a station blackout - will be described in general, after which the resultson the variation of the downcomer will be given.

After the SCRAM the respectively hot coolant in the ‘hot leg’ rises to the top of the SLIMR, and therespectively cooler fluid in the ‘cold leg’ falls to the bottom of the SLIMR. This causes an abruptdrop in the mass flow rate; the decrease of the mass flow rate at normal operation to this mini-mum covers approximately 30 seconds. Due to the production of decay heat the temperature of the

76 4. RESULTS

(a) T = 250 s (b) T = 500 s (c) T = 750 s (d) T = 1000 s

(e) T = 1250 s (f) T = 1500 s (g) T = 1750 s (h) T = 2000 s

(i) T = 2250 s (j) T = 2500 s (k) T = 2750 s

Tem

pera

ture

[°C

]

270

280

290

300

310

320

330

340

350

(l) Colorbar

Figure 4.12: Temperature distribution of coolant, shown as a cross-sectional view of the SLIMR. In these figures it is shownhow the coolant in the SLIMR reacts on a SCRAM accompanied by a station blackout. Here the focus is on the intervalfrom 250 - 2750 [s] after the SCRAM is initiated. Each sub-figure, indicated by (a) to (k), corresponds to a moment in timein Figure 4.13.


500 1000 1500 2000 2500 3000 3500

10−1

100

101

102

Time [s]

Pow

er [M

W]

(a)(b) (c) (d) (e) (f) (g) (h) (i) (j) (k) Core Power

Heat Loss

Peak in Heat Loss due to Peak in MFR

(a) Thermal core power (in Cyan) and the total heat loss (in Dark Blue) versus time.

500 1000 1500 2000 2500 3000 35000

15

30

45

60

75

90

105

120

135

Mas

s F

low

Rat

e [k

g/s]

Time [s]

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

500 1000 1500 2000 2500 3000 35000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Hea

t Tra

nsfe

r D

eter

iora

tion

Rat

e [k

J/kg

]

Max Peak in MFR During SCRAM

(b) Mass flow rate of the coolant (in Cyan) and the heat transfer deterioration rate (in Dark Blue) versus time.

500 1000 1500 2000 2500 3000 3500265

275

285

295

305

315

325

335

345

355

Cor

e E

xit T

empe

ratu

re [°

C]

Time [s]

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

500 1000 1500 2000 2500 3000 35000

50

100

150

200

250

300

350

400

450

Cor

e E

ntha

lpy

Ste

p [k

J/kg

]

(c) Core exit temperature (in Cyan) and the core enthalpy step (in Dark Blue) versus time.

Figure 4.13: These figures represent the interval 250 - 3600 [s] after the SCRAM is initiated. Each intersection, indicatedby (a) to (k), corresponds to a cross section of the temperature distribution in the SLIMR, see Figure 4.12. It shows that themass flow rates oscillate, with which the respectively hot coolant slowly spreads throughout the downcomer and moves inthe direction of the lower plenum. Once this prop of coolant leaves the downcomer the average density in the downcomerincreases. At that point the temperature of the coolant that enters the core increases, and thus the average density in theriser decreases. This increases the buoyancy in the loop and causes the peak in the mass flow rate, which in turn increasesthe heat loss to the pool.

78 4. RESULTS

Table 4.8: Initial parameters of the SLIMR design, for CASE D.

Reactor Dimensions


Riser Diameter 1.0 mHeat Exchanger Length 2.50 mRPV External Wall 0.37 m

Operation Point



Pool

Water Temperature 40 °C

coolant in the ‘hot leg’ rises, which increases the buoyancy. Slowly the slug flow of the hot coolant- which has been gathered in the upper plenum - moves via the downcomer (and mixes with thecooler coolant) by an oscillating mass flow rate to the lower plenum, which covers approximately45 minutes. Now the hot slug flow of coolant moves out of the downcomer into the lower plenumand enters the core, the mass flow rate increases rapidly until a respectively uniform temperaturedistribution is obtained. From this moment the bulk temperature of the coolant in the SLIMR rises,after which time it shall reveal whether the SLIMR has a net heat loss. The whole process is visu-alized; the interval 250-3600 [s] is shown in Figure 4.12 and 4.13, and the rest is shown in Appendix A.

Generally, it is found that coolant temperature stays within the ranges of the splines, which rangebetween 1 − 1000 [°C]. However, for a riser length of 3.5 [m] and smaller, in combination with adowncomer diameters that is smaller than 2.875 [m] this did not apply. For these dimensions it wasfound that by exceeding the pseudo-critical temperature the coolant temperature rapidly rises tothe maximum spline temperature of 1000 [°C], after which the simulation had to be stopped. How-ever, this is not a problem, as a bulk temperature of 1000 [°C] already exceeds the design limits ofthe reactor.

In the event that the bulk coolant temperature exceeds the pseudo-critical temperature, it is foundthat the heat loss decreases severely. Furthermore, it is known that from this point the heat capacitydecreases steeply, resulting in a more rapid increase of the bulk coolant temperature for an equalamount of decay heat. As a result, the coolant density decreases steeply, and therefore the coolantmass decreases. A consequence of the decrease of the coolant mass is that the bulk temperatureincreases steeper for an equal amount of decay heat. It is due to the foregoing reasons that the bulktemperature region above the pseudo-critical temperature is a slippery slope; once the bulk coolanttemperature exceeds this threshold it is found that the temperature only increases.

For the cases where the coolant temperature did not exceed the pseudo-critical temperature, theeffect of the downcomer diameter on the maximum temperature, the last temperature and the heattransfer deterioration rate is elaborated below.

D1 - EFFECT ON THE MAXIMUM BULK TEMPERATURE IN 72H

Figure 4.14a shows the dependence of the maximum temperature within 72 hours on the down-comer diameter and riser length. The maximum coolant temperature over the interval of the in-creasing downcomer diameter decreases within a range of 9.0% for a riser length of 4.5 [m] to 9.2%


for a riser length of 10 [m].

The decrease of the maximum temperature is due to the increase of the volume of the downcomer,and thus the amount of water present in the reactor vessel. As the amount of decay heat remains thesame for all cases, the more water the vessel contains, the lower the change in enthalpy and thus,the lower the maximum bulk temperature. Note that in all cases the maximum bulk temperaturewas found after a relatively long period, in which the temperature of the water showed an almostentirely homogenized temperature distribution throughout the entire vessel, see e.g. Figure A.16.

Secondly, the heat exchanging surface area of the SLIMR increases for an increasing downcomerdiameter. As a result, the rate of heat loss to the pool increases, as we have seen in sub-section 4.2.2,where we have found that the rate of heat loss is a strong function of the exchanging surface area.Hence the rate at which the vessel loses energy becomes larger than the rate of production of decayheat for an increasing downcomer diameter. That is why the tipping point occurs earlier in time,see e.g. Figure A.15. After this point the bulk coolant temperature in the SLIMR will only decrease.Therefore, the maximum coolant temperature decreases for increasing downcomer diameter.

Returning to the heat loss, it is found for almost the entire time interval that the heat loss is lowerthan during nominal operating power. In exception of one moment in time, which can be foundwithin the first hour after the SCRAM. For this moment, a respectively large peak in the heat loss(from the internal flow to the RPV) is caused by a peak in the mass flow rate, with which the overallcoolant temperature in the downcomer is higher than during nominal operating power. Due to thethickness of the RPV the effects of this peak are not noticeable at the exterior surface of the SLIMR.To this end, it is found for all simulations that the exterior temperature of the SLIMR is below thesaturation temperature of 100 [°C]. Consequently, it is found that the phenomenon of heat transfer- at the exterior surface of the SLIMR - is given by natural convection for all events.

D1 - EFFECT ON THE BULK TEMPERATURE AFTER 72H

The temperature of the coolant in the core after 72 hours versus the downcomer diameter for severalriser lengths are shown in Figure 4.15a. Over the interval of the increasing downcomer diameter, thefinal temperature of the coolant increases within a range of 6.5% for a riser length of 4.5 [m] to 11.4%for a riser length of 10 [m].

As explained previously, the water volume in the SLIMR increases for an increasing downcomer di-ameter, and with it the energy that is stored in SLIMR. Here the surface-to-volume ratio decreasesfor an increasing downcomer diameter. Translated to this case, the ratio of the net heat loss to theenergy stored in the SLIMR decreases for an increasing downcomer diameter. Consequently, it takesmore time to cool the SLIMR when this ratio decreases, and thus higher coolant temperatures arefound 72 [h] after the scram.

Further, it shows that the effect on the final temperature is smaller for the SLIMR designs that havemaximum coolant temperatures (see ‘D1 - Effect on the Maximum Temperature in 72h’) that ap-proach the pseudo-critical temperature. This can be explained since more energy is stored in thecoolant for these temperatures, which is due to the peak in the heat capacity. Further, it must benoted that all bulk temperatures after 72 h are lower than the maximum bulk temperature in 72 h,this can be seen in Figure 4.15a.

D1 - EFFECT ON THE MAXIMUM HTD RATE IN 72H

The maximum heat transfer deterioration rate in 72 hours versus the downcomer diameter can beseen in Figure 4.16a. Over the interval of the increasing downcomer diameter, the heat transfer de-

80 4. RESULTS

terioration rate decreases within in a range of 79.9% for a riser length of 4.5 m to 82.1% for a riserlength of 10 m.

The maximum in the heat transfer deterioration rate is found within the first minute, and corre-sponds to the drop of the mass flow rate, which we elaborated at the beginning of this sub-section.At this moment the production of the decay heat is relatively large, and decreases exponentially intime.

As explained earlier, the HTD rate is a function of the linear heat rate in the core and the mass flowrate. In the simulations performed in this section, the linear heat rate is not varied. However, theproduction of decay heat decreases in time. Consequently, the HTD rate decreases as the drop inthe mass flow rate occurs later in time. In addition, the HTD rate can also decrease when the mini-mum after the drop of the mass flow rate becomes smaller, see Figure A.14.

Here it is found, that for increasing an downcomer diameter, the minimum value of the mass flowrate increases, and its occurrence shifts to a later point in time. The time shift can be explained,because the mass flow rate during nominal operation power is larger for an increasing downcomerdiameter (see case A4), and here less resistance is implied on the fluid. Due to this, it takes moretime to slow the fluid down. In addition, the minimum value of the mass flow rate will also risedue to the decrease of the friction. As a result the maximum HTD rate decreases for an increasingdowncomer diameter.

4.4.3. D2 - VARIATION OF THE RISER LENGTH

The riser length is varied from 3.5 [m] to 10 [m] in steps of 0.25 [m] for several constant downcomerdiameters ranging between 2.25 [m] and 4.25 [m].

For the cases where the bulk coolant temperature did not exceed the pseudo-critical temperaturethe effect of the riser length will be elaborated below.

D2 - EFFECT ON THE MAXIMUM BULK TEMPERATURE IN 72H

Figure 4.14b shows the dependence of the overall maximum coolant temperature in 72 hours versusthe riser length for various downcomer diameters. It is found that the maximum coolant tempera-ture decreases over the interval of the increasing riser length within in a range of 16.8% for a down-comer diameter of 2.875 [m] to 17.0% for a downcomer diameter of 4.25 [m].

This can be explained by the increase of the heat loss for increasing riser length, see section ‘B2 -Riser Length’. Here it was elaborated that the heat loss also increased due to the rise of the buoyancyfor increasing riser length. As a result, the tipping point - the moment where the system has a netheat loss - occurs earlier in time, and therefore lower maximum temperatures are obtained.

Furthermore, it can be seen that for the largest riser lengths the maximum temperature flattens; aclear example is seen for a downcomer diameter of 2.25 [m] with a riser length larger than 8 [m].This is due to the way the data are processed, in which only the core exit temperatures are stored10 [s] after the initiation of the SCRAM until the end of the simulation. Of these temperatures onlythe maximum temperature is selected. It is expected that the trend of the decreasing maximumtemperature decreases for increasing riser length.

D2 - EFFECT ON THE BULK TEMPERATURE AFTER 72H

The temperature of the coolant flowing through the core after 72 hours versus the riser length forseveral downcomer diameters is shown in Figure 4.15b. Over the interval of the increasing riser


2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25310

320

330

340

350

360

370

380

390


Max

Tem

pera

ture

in 7

2h [°

C]

3.50 m4.50 m5.50 m6.50 m7.50 m8.50 m9.50 m

(a) Varied annulus outer diameter for severalconstant riser lengths.

4 5 6 7 8 9 10310

320

330

340

350

360

370

380

390

Riser Length [m]

Max

Tem

pera

ture

in 7

2h [°

C]

2.250 m2.625 m3.000 m3.375 m3.750 m4.125 m

(b) Varied riser lengt for several constant an-nulus outer diameters.

Figure 4.14: Maximum bulk coolant temperature in the core after a SCRAM of the SLIMR.

2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25220

240

260

280

300

320

340

360

380


Tem

pera

ture

afte

r 72

h [°

C]

3.50 m4.50 m5.50 m6.50 m7.50 m8.50 m9.50 m


4 5 6 7 8 9 10220

240

260

280

300

320

340

360

380

Riser Length [m]

Tem

pera

ture

afte

r 72

h [°

C]

2.250 m2.625 m3.000 m3.375 m3.750 m4.125 m

(b) Varied riser length for several constant an-nulus outer diameters.

Figure 4.15: Final bulk coolant temperature in the core, 72 [h] after a SCRAM of the SLIMR.

2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.250

0.1

0.2

0.3

0.4

0.5

0.6


Max

HT

D r

ate

in 7

2h [k

J/kg

]

3.50 m4.50 m5.50 m6.50 m7.50 m8.50 m9.50 m


4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

Riser Length [m]

Max

HT

D r

ate

in 7

2h [k

J/kg

]

2.250 m2.625 m3.000 m3.375 m3.750 m4.125 m

(b) Varied riser length for several constant an-nulus outer diameters.

Figure 4.16: Maximum heat transfer deterioration rate in the core after a SCRAM of the SLIMR.

82 4. RESULTS

length, the final coolant temperature decreases within a range of 36.6% for a downcomer diameterof 2.875 [m] to 31.4% for a downcomer diameter of 4.25 [m]. This is due to the increase of the heatloss to the pool (‘B2 - Riser Length’) without changing the surface-to-volume ratio, in which theincrease of the buoyancy simulates the heat transfer. As a result, the final temperature of the coolantin the SLIMR decreases effectively for an increasing riser length.

D2 - EFFECT ON THE MAXIMUM HTD RATE IN 72H

The maximum heat transfer deterioration rate in 72 hours versus the downcomer diameter is shownin Figure 4.16b. Over the interval of the increasing riser length, the maximum heat transfer deteri-oration rate decreases within in a range of 28.2% for a downcomer diameter of 2.875 m to 24.3% fora downcomer diameter of 4.25 m. By increasing the riser length it takes more time to slow the fluiddown. This can be explained by higher steady-state mass flow rates during nominal operation forlarger riser lengths, as found case A1. In addition, the path length, that the heated fluid in the coremust travel until it reaches the upper plenum, rises for increasing riser length. Here the drop in themass flow rate shifts to a later moment in time. The effect of this is also elaborated on in case D1,‘Effect on the Maximum HTD rate in 72h’. Besides, it is known that the production of decay heatdecreases in time, and due to this the heat transfer deterioration rate decreases.

Furthermore, an odd correlation can be found for a SLIMR design with a downcomer diameter of2.25 [m]. This behavior is hard to interpret, and increases for a decreasing downcomer diameter.

4.4.4. D - SUMMARY

To summarize, it is found for the major part of the simulated geometries that the SLIMR can coolitself passively in the event of a SCRAM in which the active heat exchangers in the downcomer arenot working. Here, it is found that for all these cases the maximum coolant temperature does notexceed the pseudo-critical temperature of 384 [°C]. However, in the events that the overall coolanttemperature exceeds the pseudo-critical temperature it is found that there is no way back. Herethe heat loss keeps decreasing and in addition the amount of coolant in the SLIMR is declining; theoverall coolant temperature in the SLIMR rises rapidly, and thus the boundaries of the splines over-flow.

In order to present the feasibility of the SLIMR design the contour map in Figure 4.17 is introduced.Here the simulated cases in which the coolant exceeds the pseudo-critical temperature are given inred; it is found for this design that these dimensions are highly unsuitable. The area next to it, givenin orange, ranges the simulations for which the overall maximum coolant temperatures is between384−375 [°C]. For this area it is found that SLIMR cools down safely, although the maximum overallcoolant temperatures are close to the pseudo-critical temperature and makes this region suitablebut questionable. The green area is a very suitable area. For these dimensions it is found that themaximum coolant temperature is below 375 [°C] and thus at a safe distance from the pseudo-criticaltemperature. Further it can be seen in Figure 4.17 that the boundaries are very angular, which is dueto the resolution of the data; where simulations are performed for every 0.25 [m] riser length, and0.125 [m] downcomer diameter.

The decrease of the maximum temperature for an increasing downcomer diameter can be explainedby the increasing coolant volume in the SLIMR. Here the coolant acts like a buffer where more andmore energy can be stored for an increasing volume. In addition, the heat exchanging surface areaincreases for an increasing downcomer diameter, and therefore the moment where the heat loss ishigher than the decay heat production occurs earlier in time. By increasing the riser length the heatexchanging surface area increases as well. In addition, the riser length affects the natural circulationof the coolant positively, and consequently the heat loss. Thus the heat loss overcomes the produc-


Riser Length [m]

Ann

ulus

Out

er D

iam

eter

[m]

4 5 6 7 8 9 10

2.5

3

3.5

4

Figure 4.17: The feasibility map of the SLIMR for varied riser lengths and downcomer diameters. The red area is a veryunsuitable area where the coolant temperature exceeds 1000 [°C]; for the orange area it is found that the SLIMR can coolitself passively, but the maximum temperature approaches the pseudo-critical temperature at short range; for the greenarea the SLIMR is completely safe, the maximum temperature is below 375 [°C].

tion of decay heat earlier in time.

Furthermore, it is found that by increasing the riser length the overall coolant temperature decreasesfaster in time. While by the increase of the downcomer diameter it can be found that overall coolanttemperatures after 72 [h] decrease less, which can be explained by the decreasing surface-to-volumeratio.

Finally, it is found for all simulations that the heat transfer deterioration rate is below the conditionof HTD<0.4 [kJ/kg], and for that reason it is ensured that the surface temperature of the fuel pinswill not reach the pseudo-critical temperature.

5CONCLUSIONS AND RECOMMENDATIONS

The NERA research group proposes a nuclear reactor type that has an innovative combination ofsafety and sustainability; this reactor is called the SLIMR (Small-scale, Large efficiency, Inherentlysafe, Modular Reactor). This small modular reactor is cooled by supercritical water, which enhancesthe thermal efficiency of the reactor; and runs by natural circulating flow, which eliminates the needfor pumps during operation and external power in case of an accident. In addition, the SLIMR issubmerged in a pool, which enables the reactor to passively deposit its decay heat through the wallof the RPV during a station blackout. In this work a feasibility study in terms of safety is performedin order to determine whether this new reactor type can be inherently safe under both normal andaccidental situations.

5.1. CONCLUSIONSIt can be concluded that it is feasible to design a SLIMR that is inherently safe under both normaland accidental situations.

This work shows that the SLIMR can operate safely at nominal conditions, being stable and withoutoccurrence of deteriorated heat transfer. Hereby it is determined that the recommended core heightof 4.20 [m] is necessary to avoid onset of deteriorated heat transfer over the whole core length. Fur-ther, no restrictions are imposed on the geometry of the SLIMR for normal operation.

The SLIMR is also safe under accidental situations. In the event of a SCRAM accompanied with astation blackout it is found that the SLIMR can deposit its decay heat completely passively to the en-vironment via a stable natural circulation flow, a maximum overall coolant temperature that doesnot exceed 385 [°C], and without occurrence of deteriorated heat transfer. Here the dimensions ofthe SLIMR should be at least 3.6 [m] for the vessel width and 13.1 [m] for the vessel height.

Besides, it has been found that if the overall coolant temperature exceeds the pseudo-critical tem-perature after a SCRAM, the coolant temperature can only increase, whereby a severe accident willbe inevitable. This is a major disadvantage in the design of this reactor.

Further, it has been found that the heat loss during normal operation is significantly large and at alltimes larger than the heat loss after a SCRAM. On that account, a large pool is necessary to passivelytransport the heat from the SLIMR to the environment during normal operation. Moreover, it isdetermined that in order to obtain a desirable pool temperature of 40 [°C] this pool should be atleast 1600 [m2] (1.3 times an Olympic swimming pool), and has a significant loss 25.6 ·103 [kg/day]

85

86 5. CONCLUSIONS AND RECOMMENDATIONS

of pool water a day (by evaporation). This large water loss makes the SLIMR dependent on a supplyof feedwater and therefore the design is less suitable to be located in remote places.

5.2. RECOMMENDATIONS

The recommendations are for clarity separated in ‘Further Work’, ‘SLIMR Design’ and the ‘Code’.

FURTHER WORK

In this work reactivity feedback has not been accounted for. However, reactivity feedback could af-fect the stability of the SLIMR. Thus, it would be a valuable addition to investigate the effect of atransient neutronic model on the stability of the SLIMR.

Further, in this work the deposition of the core power has been given by a constant linear heat rate.For a real core this is not the case and could affect the stability. It is therefore recommended to in-vestigate the effect of a more realistic linear heat rate to the stability of the SLIMR.

Furthermore, in order to avoid heat transfer deterioration, this work made use of the criteria of Pi-oro et al.. By meeting these criteria we assumed that the surface temperature of the fuel rods wouldnot exceed the pseudo-critical temperature, and thereby avoid temperatures in the center of thefuel rod that could exceed its design limit. However, it would be desirable that the temperature dis-tribution over the fuel rod could be calculated in time, and thereby eliminating the Pioro criteria.It is therefore recommended to utilize a transient one-dimensional conduction model to representa fuel rod. The one-dimensional model that has been derived in this work would be very suitable,whereby only a few minor adjustments need to be made.

Finally, in this work the upper plenum and lower plenum have been modeled as one node. Thevolumes of these nodes are significantly larger than the rest of the loop, and hereby stimulating themixing of the fluid, and the damping of mass flow rate osculations. It is therefore recommended toanalyze the effect of these nodes on stability during normal operation, and their effect during thefirst hour after a SCRAM. This could firstly be approached by spitting this cell in multiple parts, afterwhich these plenums could be modeled in two or three-dimensions.

SLIMR DESIGN

As it found in Phase B is the heat loss during normal operation significant. To deposit this heat pas-sively to the environment a pool with a very large free area is needed. This results again in a verylarge water loss per day. A solution to this could be to install a passive cooling system in the pool,that allows the pool to be smaller.

A second option is to design the SLIMR with an extra containment, just as in the designs of themPower, NuScale and MASLWR. In this design during nominal operation a vacuum is maintainedbetween the primary and secondary containment. Hereby the heat loss to the pool has become in-significant.

Continuing on this last design one must think of a new method to passively deposit the heat fromthe SLIMR to the pool in the event of an accident. This could be similar to the design of the NuScale.Here a secondary natural circulation system is used, in which pool water flows through the heatexchangers, and enters the pool as steam. However, this system is dependent on valves (i.e. thesecondary loop to the generator must be closed), and therefore not completely passive.

5.2. RECOMMENDATIONS 87

CODE

In this work the balance equations are discretized in such a way that both forward flows and back-ward flows are possible. This method works perfectly in the event that the mass flow rate flips frompositive to negative instantaneously. However, if the mass flow rate balances around 0 the modelwill not converge and crash.

This can be explained by the linear solver that is used in this work. Here the mass flow rate of theprevious iteration step has been used to solve the mass flow rate of the next iteration step. In theevent that the new mass flow rate is expected to be negative and is estimated by a positive mass flowrate, the tridiagonal solver crashes (the solver demands that both the estimated and the new massflow rate to be or positive or negative). To overcome this problem a quadratic solver can be used.These types of solvers are very costly though, but it is possible to use this quadratic solver only inthe event that the mass flow rate approaches 0.

In this work the one-dimensional flow code has been benchmarked, and the one-dimensional con-duction model has been validated. However, this has been done independently of one another. Tothis end it would be recommended to benchmark the two combined versus a code such as Relap5(U.S. system code) or CATHARE (European system code).

BIBLIOGRAPHY

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http://www.energy.gov/articles/energy-department-announces-new-investment-us-small-modular-reactor-design-and

http://www.energy.gov/articles/energy-department-announces-new-investment-us-small-modular-reactor-design-and

http://www.energy.gov/articles/expanding-options-nuclear-power

http://www.world-nuclear-news.org/NN-Federal-funding-agreed-for-NuScale-2905144.html

http://www.energy.gov/articles/energy-department-announces-new-investment-innovative-small-modular-reactor

http://www.nuscalepower.com/reactorbuildingsandbarriers.aspx

http://www.alstom.com/

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http://www.cfd-online.com/Wiki/Staggered_grid

LIST OF ABBREVIATIONS

AC Alternating CurrentASHREA American Society of Heating, Refrigerating and Air conditioning EngineersBWR Boiling Water ReactorCFD Computational Fluid DynamicsCHF Critical Heat FluxDeLight Delft Light water reactorDHRS Decay Heat Removal SystemDWO Density Wave OscillationEOS Equation Of StateFVM Finite Volume MethodHPLWR High Performance Light Water ReactorHTD Heat Transfer DeteriorationLLC Limited Liability CompanyLOCA Loss Of Coolant AccidentsLWR Light Water ReactorMASLWR Multi-Application Small Light Water ReactorNC Natural CirculationNPP Nuclear Power PlantsPWR Pressurized Water ReactorRPV Reactor Pressure VesselSCWR SuperCritical Water cooled ReactorSIMPLE Semi-Implicit Method for Pressure-Linked EquationsSLIMR Small-scale, Large efficiency, Inherently safe, Modular ReactorSMR Small Modular Reactor

93

LIST OF SYMBOLS

LATIN SYMBOLS

A Cross sectional area [m2]cp Specific heat [J/kg °C]Dh Hydraulic Diameter [m]Evap Evaporation flux per hour [kg /m2 h]f Darcy-Weisbach friction factor [−]g Gravitational acceleration [m/s2]G Mass flux [kg /m2 s]h Enthalpy [J/kg ]H Height [m]h f g Enthalpy of vaporization [J/kg ]hs convectional heat transfer coefficient [W /m °C]K Local friction factor [−]L Length [m]m Mass rate [kg /s]M Mass flow rate [kg /s]p Pressure [Pa]P Perimeter [m]q ′ Linear heating rate [J/m s]q ′′ Surface heating rate [J/m2 s]q ′′′ Volumetric heating rate [J/m3 s]q heat rate [J/s]r Radius [m]R Gas constant [J/kg K ]t Time [s]T Temperature [°C]u Internal energy [J ]v Velocity [m/s]V Volume [m3]W Specific humidity of air [−]

GREEK SYMBOLS

α Thermal diffusivity [m2/s]β Thermal expansion coefficient [1/K ]ε Emissivity [−]λ Thermal conductivity [W /m °C]µ Dynamic viscosity [Pa s]ν Kinematic viscosity [m2/s]ρ Density [kg /m3]σ Surface tension [N /m]τ Shear stress [kg /m s2]

95

SUBSCRIPTS

w wallb bulkf films surfacep poolss stainless steelv vaporl liquids saturatedr relativew saturated at water surface temperaturef at facility temperature and humidityj Spatial discretization indexi Spatial discretization index staggered grid(i ) Spatial discretization index staggered grid for negative flows surfacei innero outeri s inner vessel surfaceos outer vessel surfacenc natural convectionp f s pool free surface

SUPERSCRIPTS

n previous time stepn +1 current time stepk previous iteration stepk +1 current iteration step∗ estimate′ correction

DIMENSIONLESS NUMBERS

Co Courant numberFo Fourier numberGr Grashof numberNSU B Sub-cooling numberNPC H Phase-change numberNu Nusselt numberPr Prandtl numberRa Rayleigh numberRe Reynolds number

96

AADDITIONAL FIGURES OF THE RESULTS

0 5 10 15 20 25 30250

300

350

400

450

500

Core Inlet Friction − Coefficient K [−]

Mas

s F

low

Rat

e [k

g/s]

4.00 m6.00 m8.00 m10.00 m

(a) Mass flow rate.

0 5 10 15 20 25 30300

350

400

450

500

550

600


Cor

e ∆

h

[kJ/

kg]

4.00 m6.00 m8.00 m10.00 m

(b) Enthalpy step core.

0 5 10 15 20 25 30335

340

345

350

355

360

365

370

375


Cor

e E

xit T

empe

ratu

re [°

C]

4.00 m6.00 m8.00 m10.00 m

(c) Core exit temperature

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4


HT

D r

ate

q/G

[kJ

/kg]

4.00 m6.00 m8.00 m10.00 m

(d) Heat transfer deterioration rate.

Figure A.1: Varied core inlet friction for several constant riser lengths. The results are obtained at steady-state duringnominal operating power. These figures contain the results of CASE A5.

97

98 A. ADDITIONAL FIGURES OF THE RESULTS

0.6 0.8 1340

341

342

343

344

345

Riser Diameter [m]

Cor

e ∆

h

[kJ/

kg]

2.49 m3.01 m3.49 m3.99 m4.25 m

(a) Varied riser diameter for several constantouter annulus diameters. This figure belongsto CASE A3.

2.5 3 3.5 4340

341

342

343

344

345

346


Cor

e ∆

h

[kJ/

kg]

0.49 m0.69 m0.89 m1.09 m1.17 m

(b) Varied outer annulus diameter for severalconstant riser diameters. This figure belongsto CASE A4.

Figure A.2: Steady-state enthalpy step at nominal operating power. These figures contain the results of CASES A3 and A4.

0.6 0.8 1356.5

356.6

356.7

356.8

356.9

357

357.1

357.2

Riser Diameter [m]

Cor

e E

xit T

empe

ratu

re

[°C

]

2.49 m3.01 m3.49 m3.99 m4.25 m


2.5 3 3.5 4356.5

356.6

356.7

356.8

356.9

357

357.1

357.2


Cor

e E

xit T

empe

ratu

re

[°C

]

0.49 m0.69 m0.89 m1.09 m1.17 m


Figure A.3: Steady-state core exit temperature at nominal operating power. These figures contain the results of CASES A3and A4.

0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Riser Diameter [m]

HT

D r

ate

q/G

[kJ

/kg]

2.49 m3.01 m3.49 m3.99 m4.25 m


2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


HT

D r

ate

q/G

[kJ

/kg]

0.49 m0.69 m0.89 m1.09 m1.17 m


Figure A.4: Steady-state heat transfer deterioration rate in the core at nominal operating power. These figures contain theresults of CASES A3 and A4.

99

280 290 300 310250

300

350

400

450

500


Cor

e ∆

h

[kJ/

kg]

100 MW125 MW150 MW175 MW200 MW225 MW250 MW

(a) Varied core inlet temperature for severalconstant thermal core powers. This figure be-longs to CASE A6.

100 150 200 250250

300

350

400

450

500

Core Power [MW] C

ore

∆ h

[k

J/kg

]

280 °C290 °C300 °C310 °C

(b) Varied thermal core power for several con-stant core inlet temperatures. This figure be-longs to CASE A7.

Figure A.5: Steady-state enthalpy step at nominal operating power. These figures contain the results of CASES A6 and A7.

280 290 300 3100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


HT

D r

ate

q/G

[kJ

/kg]

100 MW125 MW150 MW175 MW200 MW225 MW250 MW

(a) Varied core inlet temperature for severalconstant thermal core powers. This figure be-longs to CASE A6.

100 125 150 175 200 225 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Core Power [MW]

HT

D r

ate

q/G

[kJ

/kg]

280 °C290 °C300 °C310 °C

(b) Varied thermal core power for several con-stant core inlet temperatures. This figure be-longs to CASE A7.

Figure A.6: Steady-state heat transfer deterioration rate in the core at nominal operating power. These figures contain theresults of CASES A6 and A7.


275 280 285 290 295 300 305 3101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8


Tot

al H

eat L

oss

[MW

]

25 °C35 °C45 °C55 °C65 °C75 °C

(a) Varied core inlet temperature for severalconstant pool temperatures. This figure be-longs to CASE B4.

25 30 35 40 45 50 55 60 65 70 751

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Pool Temperature [°C]

Tot

al H

eat L

oss

[MW

]

275 °C280 °C285 °C290 °C295 °C300 °C305 °C310 °C

(b) Varied pool temperature for several con-stant core inlet temperatures. This figure be-longs to CASE B3.

Figure A.7: Steady-state total heat loss during nominal operating power. These figures belong to CASES B3 and B4.

275 280 285 290 295 300 305 31030

40

50

60

70

80

90


Ext

erio

r T

empe

ratu

re R

PV

[°C

]

25 °C35 °C45 °C55 °C65 °C75 °C

(a) Varied core inlet temperature for severalconstant pool temperatures. This figure be-longs to CASE B4.

25 30 35 40 45 50 55 60 65 70 7530

40

50

60

70

80

90

Pool Temperature [°C]

Ext

erio

r T

empe

ratu

re R

PV

[°C

]

275 °C280 °C285 °C290 °C295 °C300 °C305 °C310 °C

(b) Varied pool temperature for several con-stant core inlet temperatures. This figure be-longs to CASE B3.

Figure A.8: Steady-state external surface temperature of the SLIMR during nominal operating power. Theses figure belongto CASES B3 and B4.

101

0.1 0.2 0.3 0.4 0.51

1.5

2

2.5

3

3.5

RPV Wall Thickness [m]

Tot

al H

eat L

oss

[MW

]

(a) Total heat loss versus the RPV wall thick-ness.

0.1 0.2 0.3 0.4 0.550

55

60

65

70

75

RPV Wall Thickness [m]

Ext

erio

r T

empe

ratu

re R

PV

[°C

]

(b) Surface temperature of the SLIMR versusthe RPV wall thickness.

Figure A.9: Steady-state surface temperature at the outer side of the SLIMR during nominal operating power. Thesefigures belong to CASES B5.


(a) T = 0 s (b) T = 100 s (c) T = 200 s (d) T = 300 s

(e) T = 400 s (f) T = 500 s (g) T = 600 s (h) T = 700 s

(i) T = 800 s (j) T = 900 s (k) T = 1000 s

Tem

pera

ture

[°C

]

260

270

280

290

300

310

320

330

340

350

(l) Colorbar

Figure A.10: Temperature distribution of coolant, shown as a cross-sectional view of the SLIMR. In these figures the SLIMRreaches nominal operating power within 1000 [s]. Each sub-figure, indicated by (a) to (k), corresponds to a moment intime in Figure A.11.

103

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

140

160

Time [s]

Pow

er [M

W]

( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) ( i ) ( j ) ( k )

(a) Thermal core power versus time.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

350

400

450

Time [s]

Mas

s F

low

Rat

e [k

g/s]

( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) ( i ) ( j ) ( k )

(b) Mass flow rate versus time.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

50

100

150

200

250

300

350

400

450

Time [s]

Cor

e E

ntha

lpy

Ste

p [k

J/kg

] ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) ( i ) ( j ) ( k )

(c) The core enthalpy step versus time.

Figure A.11: Obtaining a steady-state operation point for nominal thermal core power. It shows that for increasing corepower the mass flow rate increases: due to the increasing core power the core enthalpy increases as well. After 1000 [s] ofsimulation, another 1000 [s] is simulated to dampen the mass flow rate oscillation (too small to visualize in this figure).Each intersection, indicated by (a) to (k), corresponds to a cross section of the temperature distribution in the SLIMR, seeFigure A.10.

0 10 20 30 40 50 60 70 80 90 100145

150

155

160

165

170

Time [s]

Pow

er [M

W]

(a) Thermal core power versus time.

0 10 20 30 40 50 60 70 80 90 100390

395

400

405

410

415

420

Time [s]

Mas

s F

low

Rat

e [k

g/s]

(b) Mass flow rate versus time.

0 10 20 30 40 50 60 70 80 90 100360

370

380

390

400

410

420

Time [s]

Cor

e E

ntha

lpy

Ste

p [k

J/kg

]

(c) The core enthalpy step versus time.

Figure A.12: After obtaining a steady-state operation point for nominal thermal core power, this points stability is checked.In these figures it can be seen that the thermal core power is abruptly increased by 10% after which it is brought back tonominal operating power. On that account, the buoyancy increases abruptly and the mass flow rate begins to oscillate.In this case it shows that the flow falls back quickly into its equilibrium point.


(a) T = 0 s (b) T = 5 s (c) T = 35 s (d) T = 65 s

(e) T = 95 s (f) T = 125 s (g) T = 150 s (h) T = 175 s

(i) T = 200 s (j) T = 225 s (k) T = 250 s

Tem

pera

ture

[°C

]

270

280

290

300

310

320

330

340

350

(l) Colorbar

Figure A.13: Temperature distribution of coolant, shown as a cross-sectional view of the SLIMR. In these figures it is shownhow the coolant in the SLIMR reacts on a SCRAMS. Here the focus is on the first 250 [s] after the SCRAM is initiated. Eachsub-figure, indicated by (a) to (k), corresponds to a moment in time in Figure A.14.

105

0 50 100 150 200 250

10−1

100

101

102

Time [s]

Pow

er [M

W]

(a)(b) (c) (d) (e) (f) (g) (h) (i) (j)

(k)

Core PowerHeat Loss


0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Mas

s F

low

Rat

e [k

g/s]

Time [s]

(a)(b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Hea

t Tra

nsfe

r D

eter

iora

tion

Rat

e [k

J/kg

]

Max Peak in HTD During SCRAM

(b) Mass flow rate of the coolant and the heat transfer deterioration rate versus time.

0 50 100 150 200 250265

275

285

295

305

315

325

335

345

355

Cor

e E

xit T

empe

ratu

re [°

C]

Time [s]

(a)(b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Cor

e E

ntha

lpy

Ste

p [k

J/kg

]

(c) Core exit temperature and the core enthalpy step versus time.

Figure A.14: These figures represent the 275 [s] after that the SCRAM is initiated. Each intersection, indicated by (a) to (k),corresponds to a cross section of the temperature distribution in the SLIMR, see Figure A.13. Here it shows that the corepower drops, which in turn causes a drop in the mass flow rate. Moreover, due to the drop of the mass flow rate and dueto the fact that the production of decay heat is respectively high, the heat transfer deterioration rate increases. Here thedecay heat increases the temperature of the coolant in the core; slowly the mass flow rate increases. Consequently thehot fluid in the upper plenum enters the downcomer.


(a) T = 1 h (b) T = 3 h (c) T = 5 h (d) T = 7 h

(e) T = 11 h (f) T = 20 h (g) T = 30 h (h) T = 40 h

(i) T = 50 h (j) T = 60 h (k) T = 70 h

Tem

pera

ture

[°C

]

240

260

280

300

320

340

(l) Colorbar

Figure A.15: Temperature distribution of coolant, shown as a cross-sectional view of the SLIMR. In these figures it is shownhow the coolant in the SLIMR reacts on a SCRAM. Here the focus is on the interval from 1 to 72 [h] after the SCRAM isinitiated. Each sub-figure, indicated by (a) to (k), corresponds to a moment in time in Figure A.16.

107

10 20 30 40 50 60 70

10−1

100

101

102

Time [h]

Pow

er [M

W]

(a)(b) (c) (d) (e) (f) (g) (h) (i) (j)

(k)

Core PowerHeat Loss

The Heat Loss by the RPV Overcomes the Decay Heat of the Core Ripple caused by the discrete transition between the Nusselt correlations


10 20 30 40 50 60 7020

25

30

35

40

45

50

55

60

65

Mas

s F

low

Rat

e [k

g/s]

Time [s]

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

10 20 30 40 50 60 700

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Hea

t Tra

nsfe

r D

eter

iora

tion

Rat

e [k

J/kg

]

Ripple in the MFR is because of the discrete transition between the Nusselt correlations

(b) Mass flow rate of the coolant (in Cyan) and the heat transfer deterioration rate (in Dark Blue) versus time.

10 20 30 40 50 60 70245

255

265

275

285

295

305

315

325

335

Cor

e E

xit T

empe

ratu

re [°

C]

Time [s]

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k)

10 20 30 40 50 60 705

10

15

20

25

30

35

40

45

50

Cor

e E

ntha

lpy

Ste

p [k

J/kg

]

Max Temperature Inside RPV

(c) Core exit temperature (in Cyan) and the core enthalpy step (in Dark Blue) versus time.

Figure A.16: These figures represent the interval 1 - 72 [h] after the SCRAM is initiated. Each intersection, indicated by (a)to (k), corresponds to a cross section of the temperature distribution in the SLIMR, see Figure A.15. Here it shows that theoverall coolant temperature in the SLIMR keeps increasing until the tipping point is reached. At this point the total heatloss is larger than the production of the decay heat and the overall coolant temperature will decrease. NOTE: Time is inHOURS.

BDISCRETIZATION OF THE

ONE-DIMENSIONAL FLOW EQUATIONS

The discretization of mass, energy and momentum is described in several steps. The method of discretizationis the finite volume method (FVM), a technique that is widely used to solve partial differential equations(PDE’s). It is especially suitable for physical conservation laws and is commonly applied for discretizing ofthe governing flow equations, such as in this work.

B.1. MASS BALANCEThe PDE of the mass balance is given as:

A∂ρ

∂t+ ∂M

∂x= 0. (B.1)

When discretizing equation B.1 in space and time whit the FVM; control volumes have to be defined. In spacethis control volume is given by the domain [xi− 1

2, xi+ 1

2] and in time the interval is [tn , tn+1]. Integration over

this control volume yields:

∫ tn+1

tn

∂

∂t

∫ xi+ 1

2

xi− 1

2

Aρd x

d t +∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

∂M

∂xd x

d t = 0. (B.2)

We start with the integration of the inertia term, the left term of equation (B.1). The mass balance controlvolume, can be shown in figure B.1. Here the fluid properties and geometrical characteristics are defined inthe middle of the grid cell. The mass flow rates are defined at the cell faces. Discretizing in space is completedby solving the inner integral. As can be seen in figure B.1 the density is not defined at the faces of the cell,which are the boundaries of the interval of integration. An approximation has to be made to integrate thisterm. This can be done by assuming that when the volume is small enough the values for A and ρ are constant

∆𝑥𝑖

ρ𝑖𝐴𝑖

Figure B.1: Discretization grid with a representation of a row of spatial control volumes for the mass balance. Specific forthe FVM integration of the inertia term; all parameters of interest, and associated grid nodes are visualized.

109

110 B. DISCRETIZATION OF THE ONE-DIMENSIONAL FLOW EQUATIONS

𝑀𝑖+12𝑀

𝑖−12

Figure B.2: Discretization grid with a representation of a row of spatial control volumes for the mass balance. Specific forthe FVM integration of the convection term; all parameters of interest, and associated grid nodes, are visualized. The bluearrows represent the in and out flow of mass in this grid cell.

in the cell, thus the average of the A and ρ over the grid cell can be taken, this reads:

∫ tn+1

tn

∂

∂t

∫ xi+ 1

2

xi− 1

2

Aρd x

d t ≈∫ tn+1

tn

∂

∂t

(A(xi )ρ(xi , t )∆xi

)d t . (B.3)

Integrating the time derivative gives:∫ tn+1

tn

∂

∂t

(A (xi )ρ (xi , t )∆xi

)d t = [

A (xi )ρ (xi , t )∆xi]tn+1

tn= A (xi )

[ρ(xi , tn+1)−ρ(xi , tn)

]∆xi . (B.4)

To obtain a numerical scheme, the discretized representation is introduced. An example, of how to writedown the indices that indicate a parameter in space and time; the subscript is used to indicate the grid celland the superscript denotes the time step, which is ρ (xi , tn+1) := ρn+1

i . When this is applied to equation (B.4)the full discretized inertia term is yielded:

A (xi )[ρ(xi , tn+1)−ρ(xi , tn)

]∆xi = Ai

[ρn+1

i −ρni

]∆xi . (B.5)

Now we can continue with the convection term, which is the second term of equation (B.1). Again start withthe inner integral. For the control volume of the mass balance the mass flow rates are defined at the faces ofthe grid cell, the boundaries of our interval of integration. A visualisation of the nodes of interest for the in-tegration of the convection term can be seen in figure B.2. Integrating the derivative needs no approximationand gives: ∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

∂M

∂xd x

d t =∫ tn+1

tn

([M (x, t )]

xi+ 1

2x

i− 12

)d t . (B.6)

In the following step a time integration is needed. Since in the convection term there is no derivative, anapproximation is needed. Here the assumption is that when the time interval is small enough the mass flowsdo not differ significantly between the time steps:∫ tn+1

tn

([M (x, t )]

xi+ 1

2x

i− 12

)d t ≈

[[M (x, tn)]

xi+ 1

2x

i− 12

]∆tn =

[M

(xi+ 1

2, tn

)−M

(xi− 1

2, tn

)]∆tn . (B.7)

Since the mass flow was specified at the boundaries, it does not need an interpolation. In order to make thetime-stepping scheme more sophisticated an intermediate time between tn and tn+1 = tn+∆tn is introduced.Letting tn +θ∆tn be some in-between time, a step size which can be customized by the parameter 0 É θ É 1,and this can be represented as:

tn +θ∆tn := tn+θ . (B.8)

Furthermore, it is assumed that the time-increment ∆t may be so brief that in this case the mass flow, inand out at the faces of the control volume are approximately constant and arbitrarily close to each other atany intermediate time for the interval ∆t . With this approximation the equation (B.7) can be rewritten thatincludes the intermediate time:[

M(xi+ 1

2, tn

)−M

(xi− 1

2, tn

)]∆tn ≈

[M

(xi+ 1

2, tn+θ

)−M

(xi− 1

2, tn+θ

)]∆tn . (B.9)

B.2. ENTHALPY BALANCE 111

∆𝑥𝑖

ρ𝑖ℎ𝑖𝐴𝑖

Figure B.3: Discretization grid with a representation of a row of spatial control volumes for the energy balance. Specific forthe FVM integration of the inertia term; all parameters of interest, and their associated grid nodes, are visualized.

To deduct the intermediate time notation out of the superscript the following definition can be used:

M n+θ := θM n+1 + (1−θ) M n . (B.10)

Using this definition and the discrete representations the full numerical scheme for the convection term canbe derived (B.6), given as:[

M n+θi+ 1

2−M n+θ

i− 12

]∆tn =

[θ

(M n+1

i+ 12−M n+1

i− 12

)+ (1−θ)

(M n

i+ 12−M n

i− 12

)]∆tn . (B.11)

Combining (B.5) and (B.11) results in a fully discretized notation of mass balance. Now the discretization gridcan be re-written to a computational grid, this is done for coding purposes where only integers can denotea grid point. Here we introduce an approach that removes the problem of half indices without requiringstorage of every point of the staggered grid. Another notation for the grid nodes containing the mass flow isintroduced, as xi− 1

2= x j−1 and xi+ 1

2= x j . Now all cell values of dependent variables are taken into account,

and are arranged by i and j over the total set of grid cells. Moreover, applying this the same amount of datastorage is used as for an unstaggered grid 3.1. The mass balance can be written in the non-fractional indexnotation:

Ai[ρn+1

i −ρni

]∆xi +

[θ

(M n+1

j −M n+1j−1

)+ (1−θ)

(M n

j −M nj−1

)]∆tn = 0. (B.12)

B.2. ENTHALPY BALANCEThe PDE of the energy equation is given as:

A∂ρh

∂t+ ∂Mh

∂x= q (B.13)

The control volume to be used by the FVM for the energy balance is the same as for the mass balance. Inspace this control volume is given by the domain [xi− 1

2, xi+ 1

2] and in time the interval is [tn , tn+1]. Integration

over this control volume yields:

∫ tn+1

tn

∂

∂t

∫ xi+ 1

2

xi− 1

2

Aρhd x

d t +∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

∂Mh

∂xd x

d t =∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

qd x

d t (B.14)

Again we start with the integration of the inertia term, which is the left term of equation (B.13). For this termand the terms that will follow during the discretization of the energy balance, the same tools are used as in thederivation of the numerical scheme of the mass balance. A representation of the integration of this term ( theenergy control volume) can be shown in Figure B.1, where the fluid properties and geometrical characteristicsare defined in the middle of the grid cell, and mass flow rates are defined at the cell faces. The inertia termreads:∫ tn+1

tn

∂

∂t

∫ xi+ 1

2

xi− 1

2

Aρhd x

d t ≈∫ tn+1

tn

∂

∂t

(A(xi )ρ(xi , t )h(xi , t )∆xi

)d t = A(xi )

[ρ(xi , t )h(xi , t )

]tn+1tn

∆xi

= Ai[ρn+1

i hn+1i −ρn

i hni

]∆xi (B.15)


𝑀𝑖+12ℎ𝑖𝑀

𝑖−12ℎ𝑖−1

𝑀𝑖−12ℎ𝑖

𝑀𝑖+12ℎ𝑖+1

ℎ𝑖−1 ℎ𝑖 ℎ𝑖+1

Figure B.4: Discretization grid with a representation of a row of spatial control volumes for the energy balance. Specific forthe FVM integration of the convection term; all parameters of interest, and their associated grid nodes, are visualized. Alsothe interpolation of the enthalpy at the cell faces can be seen, the light blue represents a mass flow in positive direction anddark blue a flow in negative direction.

∆𝑥𝑖

∙𝑞𝑖

Figure B.5: Discretization grid with a representation of a row of spatial control volumes for the energy balance. Specific forthe FVM integration of the production term; all parameters of interest, and their associated grid nodes, are visualized inthis figure.

The discretization of the convection term gives a new problem, since fluid properties are not defined at thecell faces and it is necessary to introduce another approximation. Therefore, the upwind value of the fluidproperties are used at the faces of the grid cell, a representation is shown in Figure B.4. An example is shownto elaborate on the previous figure: if the mass flow at face xi− 1

2is in positive direction the enthalpy will be

hi− 12≈ hi−1. The same applies for a negative mass flow, then the upwind value is hi− 1

2≈ hi . Upwinding is the

final step in the discretization of the convection term (the changed variables are highlighted for clarification):

∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

∂Mh

∂xd x

d t =∫ tn+1

tn

[M(x, t )h(x, t )]x

i+ 12

xi− 1

2

d t ≈ [M(x, tn+θ)h(x, tn+θ)]x

i+ 12

xi− 1

2

∆tn

≈[θ

(M n+1

i+ 12

hn+1i+ 1

2−M n+1

i− 12

hn+1i− 1

2

)+ (1−θ)

(M n

i+ 12

hni+ 1

2−M n

i− 12

hni− 1

2

)]∆tn (B.16)

≈[θ

(M n+1

i+ 12

hn+1i (i+1) −M n+1

i− 12

hn+1i−1(i )

)+ (1−θ)

(M n

i+ 12

hni (i+1) −M n

i− 12

hni−1(i )

)]∆tn .

The outcome of (B.16) is a result for both positive and negative mass flows at the cell faces, at each face theenthalpy is given by its upwind value for its corresponding mass flow. The explanation for the subscript isthat the first index is an interpolation for a positive flow and the index corresponding to a negative flow isindicated within brackets. The last term that needs to be discretized is the production term of (B.13), whereq[ J

s·m ] is the linear heat rate (must be q ′, mistake in this appendix). As for the inertia term this discretizationis straightforward. The spatial representation can be seen in Figure B.5. The discretized production termreads:

∫ tn+1

tn

∫ xi+ 1

2

xi− 1

2

qd x

d t ≈ q(xi , tn+θ)∆xi∆tn = [θqn+1

i + (1−θ)qni

]∆xi∆tn . (B.17)

B.3. MOMENTUM BALANCE 113

Combining the equations (B.15), (B.16) and (B.17) results in the fully discretized form of the energy balance:

Ai[ρn+1

i hn+1i −ρn

i hni

]∆xi+ (B.18)[

θ

(M n+1

i+ 12

hn+1i (i+1) −M n+1

i− 12

hn+1i−1(i )

)+ (1−θ)

(M n

i+ 12

hni (i+1) −M n

i− 12

hni−1(i )

)]∆tn = (


i

)∆xi∆tn .

In the numerical model the enthalpy balance is the first equation that needs to be solved at each time step. Tosolve equation (B.18) the density of the new time step needs to be known, (ρn+1

i ), which is not known yet. Thedensity could be estimated, but elimination of this term is preferred. This is done by subtracting the massbalance equation (B.12) which is then multiplied by hn+1

i , and appears as:

Ai[ρn+1

i −ρni

]∆xi hn+1

i +[θ

(M n+1

j −M n+1j−1

)+ (1−θ)

(M n

j −M nj−1

)]∆tnhn+1

i = 0. (B.19)

Now subtracting equation (B.19) from equation (B.18) results in the final form of the enthalpy balance. In thisstep the numerical scheme from the discretization grid is also rewritten to a computational grid. The energybalance can now be written in the non-fractional index notation:

Aiρni

[hn+1

i −hni

]∆xi+[

θ(M n+1

j hn+1i (i+1) −M n+1

j−1 hn+1i−1(i )

)+ (1−θ)

(M n

j hni (i+1) −M n

j−1hni−1(i )

)]∆tn− (B.20)[

θ(M n+1

j hn+1i −M n+1

j−1 hn+1i

)+ (1−θ)

(M n

j hn+1i −M n

j−1hn+1i

)]∆tn = [


i

]∆xi∆tn .

From the energy balance (B.20) the enthalpy in each node for each time step is gained. However, the un-knowns in this conservation balance, enthalpy hn+1 and mass flow M n+1 are coupled. To obtain a solutionthis requires an iterative method. The energy balance is therefore rewritten and complemented by a super-script for an iteration step index. The notation for the enthalpy for the current time step and current iterationstep is now given as hn+1,k+1. The notation for the enthalpy of the current time step and previous iterationstep is given as hn+1,k . In this conservation equation, the mass flow is used as being the value calculated inthe previous iteration step M n+1,k . In case it is the first iteration step of the current time step the convergedmass flow from the previous time step is used:

Aiρni

[hn+1,k+1

i −hni

]∆xi+[

θ(M n+1,k

j hn+1,k+1i (i+1) −M n+1,k

j−1 hn+1,k+1i−1(i )

)+ (1−θ)

(M n

j hni (i+1) −M n

j−1hni−1(i )

)]∆tn− (B.21)[

θ(M n+1,k

j hn+1,k+1i −M n+1,k

j−1 hn+1,k+1i

)+ (1−θ)

(M n

j hn+1,k+1i −M n

j−1hn+1,k+1i

)]∆tn = [


i

]∆xi∆tn .

B.3. MOMENTUM BALANCENow the mass and energy balances are discretized a more complicated derivation of the FVM for the momen-tum balance follows. The starting point for the discretization of the momentum equation is the followingPDE:

∂M

∂t+ ∂

∂x

(M 2

ρA

)=−A

∂p

∂x−∑

iKi

M 2

2ρAδ(x −xi+ 1

2)− f

Pw M 2

8ρA2 + Aρg . (B.22)

First a control volume needs to be specified, which in this case is different than for the mass and energyconservations. The momentum control volume for a staggered grid is in space given by the domain [xi , xi+1]and in time the interval is [tn , tn+1]. Integration over this control volume yields:∫ tn+1

tn

∂

∂t

(∫ xi+1

xi

Md x

)d t +

∫ tn+1

tn

(∫ xi+1

xi

∂

∂x

M 2

ρAd x

)d t =−

∫ tn+1

tn

(∫ xi+1

xi

A∂p

∂xd x

)d t− (B.23)∫ tn+1

tn

(∫ xi+1

xi

KM 2

2ρAδ

(x −xi+ 1

2

)d x

)d t −

∫ tn+1

tn

(∫ xi+1

xi

fPw M 2

8ρA2 d x

)d t +

∫ tn+1

tn

(∫ xi+1

xi

Aρg d x

)d t .

We start with the first integral on the left side of the momentum balance (B.13), named the inertia term.A spatial representation of the control volume and all parameters of interest can be seen in Figure B.6. Inthis Figure the momentum control volume is different from that in the mass and energy balance. For thediscretization of this term and the terms that will follow obtaining; the same tools and notations are used as


Inertia term

𝑀𝑖+12

(∆𝑥𝑖+∆𝑥𝑖+1)/2

Figure B.6: Discretization grid with a representation of a row of spatial control volumes for the momentum balance. Specificfor the FVM integration of the inertia term; all parameters of interest, and their associated grid nodes, are visualized.

ρ𝑖 𝐴𝑖𝑀𝑖+12

𝑀𝑖+12

𝑀𝑖+32

𝑀𝑖−12

ρ𝑖 𝐴𝑖𝑀𝑖−12

ρ𝑖+1 𝐴𝑖+1𝑀𝑖+12

ρ𝑖+1 𝐴𝑖+1𝑀𝑖+32

𝐴𝑖 𝐴𝑖+1

Figure B.7: Discretization grid with a representation of a row of spatial control volumes for the momentum balance. Specificfor the FVM integration of the convection term; all parameters of interest, and their associated grid nodes, are visualized.Also the interpolation of the mass flow at the cell faces can be seen, the light blue represents a mass flow in positive directionand dark blue a flow in negative direction.

in the derivation of the numeric scheme for the mass and energy balance. A representation for the integrationof the inertia term, can be shown in Figure B.6. Now the the mass flow is defined in the middle of the gridcell, and the fluid properties and geometrical characteristics are defined at the cell faces. The discretizationyields: ∫ tn+1

tn

∂

∂t

(∫ xi+1

xi

Md x

)d t ≈

[M(xi+ 1

2, t )

]tn+1

tn∆x j =

[M n+1

i+ 12−M n

i+ 12

]∆x j . (B.24)

Discretizing the convection term, the second term on the left-hand side, is somewhat more difficult. A spatialrepresentation can be found in Figure B.7. It can be seen that the mass flow is not defined at the cell walls.Thus, upwinding is used. Discretization of this term yields:∫ tn+1

tn

(∫ xi+1

xi

∂

∂x

M 2

ρAd x

)d t =

∫ tn+1

tn

([M(x, t )2

ρ(x, t )A(x)

]xi+1

xi

)d t ≈

[M(x, tn+θ)2

ρ(x, tn+θ)A(x)

]xi+1

xi

∆tn

≈[θ

( (M n+1

i+1

)2

ρn+1i+1 Ai+1

−(M n+1

i

)2

ρn+1i Ai

)+ (1−θ)

( (M n

i+1

)2

ρni+1 Ai+1

−(M n

i

)2

ρni Ai

)]∆tn (B.25)

≈

θ

(M n+1

i+ 12 (i+ 3

2 )

)2

ρn+1i+1 Ai+1

−

(M n+1

i− 12 (i+ 1

2 )

)2

ρn+1i Ai

+ (1−θ)

(

M ni+ 1

2 (i+ 32 )

)2

ρni+1 Ai+1

−

(M n

i− 12 (i+ 1

2 )

)2

ρni Ai

∆tn .

Next is the discretization of the pressure term, this is the first term on the right side of equation B.22. Thisterm represents the pressure driving force acting on the control volume. In Figure B.8 a representation of allparameters and a visualization of the pressure acting on the face between grid cell xi and xi+1 are visualized.By definition the representative cross-sectional area at node i+ 1

2 is given by Ai , this is a result of the staggeredgrid and is unrelated to previous approximations. The discretization yields:∫ tn+1

tn

(∫ xi+1

xi

A∂p

∂xd x

)d t =

∫ tn+1

tn

[A(x)p(x, t )

]xi+1xi

d t ≈ [A(xi )

(p(xi+1, tn+θ)−p(xi , tn+θ)

)]∆tn

≈ Ai[θ

(pn+1

i+1 −pn+1i

)+ (1−θ)(pn

i+1 −pni

)]∆tn . (B.26)


Pressure term

𝐴𝑖

𝑝𝑖

𝑝𝑖+1

Figure B.8: Discretization grid with a representation of a row of spatial control volumes for the momentum balance. Specificfor the FVM integration of the pressure term all parameters of interest, and their associated grid nodes, are visualized. Theblue arrows represent the pressure which face each other at the node where the mass flow is defined.

𝑀𝑖+12

ρ𝑖 𝐴𝑖

ρ𝑖+1 𝐴𝑖+1

𝐾𝑖+12

𝐴𝑖+1𝐴𝑖

Local friction term

Figure B.9: Discretization grid with a representation of a row of spatial control volumes for the momentum balance. Specificfor the FVM integration of the local friction term all parameters of interest, and their associated grid nodes, are visualized.

For the second integral at the right side of the momentum equation, the local friction term is defined bythe Dirac delta distribution. A spatial representation of the parameters of interest and visualization of theinterpolation can be found in Figure B.9. The discretization yields:

∫ tn+1

tn

(∫ xi+1

xi

KM 2

2ρAδ

(x −xi+ 1

2

)d x

)d t ≈

∫ tn+1

tn

K (xi+ 12

)

(M(xi+ 1

2, t )

)2

2ρ(xi+ 12

, t )A(xi+ 12

)

d t

≈

K (xi+ 12

)

(M(xi+ 1

2, tn+θ)

)2

2ρ(xi+ 12

, tn+θ)A(xi+ 12

)

∆tn

≈

θKi+ 12

(M n+1

i+ 12

)2

2ρn+1i+ 1

2

Ai+ 12

+ (1−θ)Ki+ 12

(M n

i+ 12

)2

2ρni+ 1

2

Ai+ 12

∆tn

≈

θKi+ 12

(M n+1

i+ 12

)2

2ρn+1i (i+1) Ai (i+1)

+ (1−θ)Ki+ 12

(M n

i+ 12

)2

2ρni (i+1) Ai (i+1)

∆tn . (B.27)

The third integral from the right side of the equation (B.22) is the wall friction term. A representation of thisterm can be seen in Figure B.10, the blue bars represent the difference in geometrical properties using upwind


𝑀𝑖+12

ρ𝑖 𝐴𝑖 𝐹𝑖 𝑃𝑖

𝐴𝑖+1𝐴𝑖

(∆𝑥𝑖+∆𝑥𝑖+1)/2

ρ𝑖+1 𝐴𝑖+1 𝐹𝑖+1 𝑃𝑖+1

Friction term

Figure B.10: Discretization grid with a representation of a row of spatial control volumes for the momentum balance.Specific for the FVM integration of the wall friction term all parameters of interest, and their associated grid nodes, arevisualized. The blue bars represent wall friction by the interpolation of the geometrical properties for positive and negativeflow.

𝑀𝑖+12

ρ𝑖 𝐴𝑖 𝑔𝑖

𝐴𝑖+1𝐴𝑖

ρ𝑖+1 𝐴𝑖+1 𝑔𝑖+1

(∆𝑥𝑖+∆𝑥𝑖+1)/2

Gravity term

Figure B.11: Discretization grid with a representation of a row of spatial control volumes for the momentum balance.Specific for the FVM integration of the gravitational term all parameters of interest, and their associated grid nodes, arevisualized.

interpolation for positive and negative flows. Discretization will result in:

∫ tn+1

tn

(∫ xi+1

xi

f PM 2

8ρA2 d x

)d t ≈

∫ tn+1

tn

f (xi+ 12

, t )P (xi+ 12

)

(M(xi+ 1

2, t )

)2

8ρ(xi+ 12

, t )(

A(xi+ 12

))2 ∆x j

d t

≈

f (xi+ 12

, tn+θ)P (xi+ 12

)

(M(xi+ 1

2, tn+θ)

)2

8ρ(xi+ 12

, tn+θ)(

A(xi+ 12

))2

∆x j∆tn

≈

θ f n+1i+ 1

2Pi+ 1

2

(M n+1

i+ 12

)2

8ρn+1i+ 1

2

(Ai+ 1

2

)2 + (1−θ) f ni+ 1

2Pi+ 1

2

(M n

i+ 12

)2

8ρni+ 1

2

(Ai+ 1

2

)2

∆x j∆tn (B.28)

≈

θ f n+1i (i+1)Pi (i+1)

(M n+1

i+ 12

)2

8ρn+1i (i+1)

(Ai (i+1)

)2 + (1−θ) f ni (i+1)Pi (i+1)

(M n

i+ 12

)2

8ρni (i+1)

(Ai (i+1)

)2

∆x j∆tn .

The last term that needs to be discretized is the gravitational term, a representation of all parameters of in-terest can be found in Figure B.11. The straightforward discretization yields:∫ tn+1

tn

(∫ xi+1

xi

Aρg d x

)d t ≈

∫ tn+1

tn

(A(xi+ 1

2)ρ(xi+ 1

2, t )g (xi+ 1

2)∆x j

)d t

≈[

A(xi+ 12

)ρ(xi+ 12

, tn+θ)g (xi+ 12

)]∆x j∆tn

≈[θAi+ 1

2ρn+1

i+ 12

gi+ 12+ (1−θ)Ai+ 1

2ρn

i+ 12

gi+ 12

]∆x j∆tn

≈[θAi (i+1)ρ

n+1i (i+1)gi (i+1) + (1−θ)Ai (i+1)ρ

ni (i+1)gi (i+1)

]∆x j∆tn . (B.29)


Combining the discretized equations B.22, B.24, B.25, B.26, B.27, B.28 and B.29 results in the complete dis-cretized momentum equation. Here the computational grid is also implemented. The new annotation forthe cell faces are now xi− 1

2= x j−1, xi+ 1

2= x j and xi+ 3

2= x j+1. Re-writing yields:

[M n+1

j −M nj

]∆x j+[

θ

(M n+1

j ( j+1)M n+1j ( j+1)

ρn+1i+1 Ai+1

−M n+1

j−1( j )M n+1j−1( j )

ρn+1i Ai

)+ (1−θ)

(M n

j ( j+1)M nj ( j+1)

ρni+1 Ai+1

−M n

j−1( j )M nj−1( j )

ρni Ai

)]∆tn =− (B.30)[

θAi(pn+1

i+1 −pn+1i

)+ (1−θ)Ai(pn

i+1 −pni

)]∆tn−[

θK j

M n+1j M n+1

j

2ρn+1i (i+1) Ai (i+1)

+ (1−θ)K j

M nj M n

j

2ρni (i+1) Ai (i+1)

]∆tn−θ f n+1

i (i+1)Pi (i+1)

M n+1j M n+1

j

8ρn+1i (i+1)

(Ai (i+1)

)2 + (1−θ) f ni (i+1)Pi (i+1)

M nj M n

j

8ρni (i+1)

(Ai (i+1)

)2


n+1i (i+1)gi (i+1) + (1−θ)Ai (i+1)ρ

ni (i+1)gi (i+1)

]∆x j∆tn .

Due to the coupled parameters, that were mentioned earlier when discretizing the energy balance, the su-perscript consists of the time notation and an iteration step index. To obtain the new mass flow M n+1,k+1 bysolving the momentum balance, the density ρn+1,k+1 is used that has been calculated earlier in this iterationstep. Furthermore, the pressure pn+1,k of the previous iteration step is used, and since the friction factor iscoupled to the mass flow it is approximated by f n+1,k ≈ f (ρn+1,k+1, M n+1,k ). Thus, re-writing equation (B.30)yields:

[M n+1,k+1

j −M nj

]∆x j+θ

M n+1,k+1j ( j+1) M n+1,k+1

j ( j+1)

ρn+1,k+1i+1 Ai+1

−M n+1,k+1

j−1( j ) M n+1,k+1j−1( j )

ρn+1,k+1i Ai

+ (1−θ)

(M n

j ( j+1)M nj ( j+1)

ρni+1 Ai+1

−M n

j−1( j )M nj−1( j )

ρni Ai

)∆tn =− (B.31)

[θAi

(pn+1,k

i+1 −pn+1,ki

)+ (1−θ)Ai

(pn

i+1 −pni

)]∆tn−θK j

M n+1,k+1j M n+1,k+1

j

2ρn+1,k+1i (i+1) Ai (i+1)

+ (1−θ)K j

M nj M n

j

2ρni (i+1) Ai (i+1)

∆tn−θ f n+1,k

i (i+1) Pi (i+1)

M n+1,k+1j M n+1,k+1

j

8ρn+1,k+1i (i+1)

(Ai (i+1)

)2 + (1−θ) fi (i+1)Pi (i+1)

M nj M n

j

8ρni (i+1)

(Ai (i+1)

)2


n+1,k+1i (i+1) gi (i+1) + (1−θ)Ai (i+1)ρ

ni (i+1)gi (i+1)

]∆x j∆tn

Since, the linear solver can not handle quadratic terms the quadratic mass flow term M n+1,k+1M n+1,k+1 needsto be linearized. This is fulfilled with the following approximation M n+1,k+1M n+1,k+1 ≈ M n+1,k M n+1,∗, usingthe mass flow of the previous iteration assuming it is close to the mass flow to be calculated. Now equation


(B.31) could be re-written as:[M n+1,∗

j −M nj

]∆x j+θ

M n+1,kj ( j+1)M n+1,∗

j ( j+1)

ρn+1,k+1i+1 Ai+1

−M n+1,k

j−1( j )M n+1,∗j−1( j )

ρn+1,k+1i Ai

+ (1−θ)

(M n

j ( j+1)M nj ( j+1)

ρni+1 Ai+1

−M n

j−1( j )M nj−1( j )

ρni Ai

)∆tn =− (B.32)

[θAi

(pn+1,k

i+1 −pn+1,ki

)+ (1−θ)Ai

(pn

i+1 −pni

)]∆tn−θK j

M n+1,kj M n+1,∗

j

2ρn+1,k+1i (i+1) Ai (i+1)

+ (1−θ)K j

M nj M n

j

2ρni (i+1) Ai (i+1)

∆tn−θ f n+1,k

i (i+1) Pi (i+1)

M n+1,kj M n+1,∗

j

8ρn+1,k+1i (i+1)

(Ai (i+1)

)2 + (1−θ) fi (i+1)Pi (i+1)

M nj M n

j

8ρni (i+1)

(Ai (i+1)

)2


n+1,k+1i (i+1) gi (i+1) + (1−θ)Ai (i+1)ρ

ni (i+1)gi (i+1)

]∆x j∆tn .

The error that is made with these replacements for the mass flow rate is defined as the mass flow rate correc-tion:

M n+1,′ = M n+1,k+1 −M n+1,∗. (B.33)

The correction of this error is treated in the following section of this appendix.

B.4. PRESSURE CORRECTION

The error that is made in the momentum balance, by the estimation of the mass flow rate to be M n+1,k+1 ≈M n+1,∗, is defined at the mass flow rate correction equation (B.33). In the same manner a pressure correctioncan be defined, which is the deviation of the pressure from the previous iteration step and current pressure,which is given as:

pn+1,′ = pn+1,k+1 −pn+1,k . (B.34)

The relation between both the mass flow correlation (B.33) and the pressure correlation (B.34) is known asthe mass flow rate correction equation. This is found by subtracting equation (B.32) from equation (B.31), inwhich the terms of convection, friction and body forces are neglected. The mass flow rate correction equationis given as:[

M n+1,k+1j −M n

j

]∆x j −

[M n+1,∗

j −M nj

]∆x j =−

[θAi

(pn+1,k+1

i+1 −pn+1,k+1i

)+ (1−θ)Ai

(pn

i+1 −pni

)]∆tn

(B.35)

+[θAi

(pn+1,k

i+1 −pn+1,ki

)+ (1−θ)Ai

(pn

i+1 −pni

)]∆tn . (B.36)

Eliminating the terms of the previous time step we can re-write equation (B.35) into:

M n+1,k+1j ∆x j −M n+1,∗

j ∆x j =−θ[

Ai

(pn+1,k+1

i+1 −pn+1,k+1i

)− Ai

(pn+1,k

i+1 −pn+1,ki

)]∆tn . (B.37)

With the correlations in equation B.33 and B.34 the equation B.37 can be re-written as:

M n+1,′j = θ

∆tn

∆x j

[Ai

(pn+1,′

i −pn+1,′i+1

)]. (B.38)

Equation B.38 is the mass flow rate correction equation in its final form. Now, the mass flow correction equa-tion has been derived out of the mass flow correlation, and the pressure correction equation is based on thepressure correlation. The pressure correction equation describes the relation between between pn+1,′ andM n+1,∗. This relation is obtain by eliminating M n+1,k+1 from the mass balance B.12, by usage of the earlierderived mass flow correction equation B.38. The pressure correction equation yield:[

Ai

(ρn+1,k+1

i −ρn+1,ki

)]∆xi+

[θ

(M n+1,∗

j −M n+1,∗j−1

)+ (1−θ)

(M n

j −M nj−1

)]∆tn = (B.39)

θ

[θ∆tn

∆x j−1Ai−1

(pn+1,′

i−1 −pn+1,′i

)− θ∆tn

∆x jAi

(pn+1,′

i −pn+1,′i+1

)]∆tn .

To conclude, the derived the pressure correction equation which is implemented in the numerical scheme iscompleted, it corrects the pressure while mass conservation is enforced.

CDISCRETIZATION OF THE RADIAL HEAT

TRANSFER EQUATION

This appendix presents the transient heat transfer problem in the cylinder heated from the central axis inan unsteady state, using the Finite Volume Method with a Half Control Volume. The time dependent, onedimensional heat conduction in cylindrical coordinates can be described by the following governing differ-ential equation:

ρcp∂T

∂t= 1

r

∂

∂r

(r k

∂T

∂r

). (C.1)

The discretization can be derived by integration over the volume and time interval. The elementary controlvolume in cylindrical coordinated is given as:

V = r∆r∆θ∆z. (C.2)

Integration yields:

ρcp

∫ tn+1

tn

∫ 2π

0

∫ zi+ 1

2

zi− 1

2

∫ ri+ 1

2

ri− 1

2

∂T

∂tr dr d z dθ

d t =∫ tn+1

tn

∫ 2π

0

∫ zi+ 1

2

zi− 1

2

∫ ri+ 1

2

ri− 1

2

1

r

∂

∂r

(rλ

∂T

∂r

)r dr d z dθ

d t . (C.3)

Solving integrals for the angular coordinate and height, gives us:

ρcp

∫ tn+1

tn

2π∆z∫ r

i+ 12

ri− 1

2

∂T

∂tr dr

d t =∫ tn+1

tn

2π∆z∫ r

i+ 12

ri− 1

2

1

r

∂

∂r

(rλ

∂T

∂r

)r dr

d t . (C.4)

There are no variations considered in∆z and∆θ, to this end 2π∆z cancel out in both terms. A one-dimensionalgrid of control volumes can be shown in Figure C.1. In this figure, a control volume is represented as a gridnode Ti boxed by two faces, respectively r j is the left and r j+1 the right face. Here j is the index for the faces

𝑇1

𝑟1 𝑟2

𝑇2 𝑇𝑖−1 𝑇𝑖 𝑇𝑖+1 𝑇𝐼𝑇𝐼−1

𝑟𝑗 𝑟𝑗+1 𝑟𝐽𝑟𝐽−1

𝐵.𝐶. 1 𝐵.𝐶. 2

Figure C.1: Representation of a one-dimensional grid of control volumes in radial direction. Also the boundary conditionsfor the outer faces of the slab are visualized.

119

120 C. DISCRETIZATION OF THE RADIAL HEAT TRANSFER EQUATION

𝑇1

𝑟1 𝑟2



𝐵.𝐶. 1 𝐵.𝐶. 2

𝑆𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛 − 𝐼

Figure C.2: Representation of a one-dimensional grid of control volumes in radial direction. The boundary condition forthe internal radius of the hollow cylinder is by Situation I.

and i is used for the grid nodes. The terms in in equation C.4 will be evaluated separately. We start with theterm on the left:

ρcp

∫ tn+1

tn

∫ ri+ 1

2

ri− 1

2

∂T

∂tr dr

d t ≈ ρcp

∫ tn+1

tn

∂

∂t

[T (ri , t )

r 2

2

]ri+ 1

2

ri− 1

2

d t (C.5)

≈ ρcp

∫ tn+1

tn

∂

∂t

T (ri , t )r 2

i+ 12− r 2

i− 12

2

d t

≈ ρcp(T n+1

i −T ni

)( r 2j+1 − r 2

j

2

).

Here there is first integrated in the r -domain. In this step we approximate the temperature with a constantover the volume interval. Assuming that the volume is small enough the faces of the cell are arbitrary close toeach other, the temperature T at the grid node is average for the grid cell. Integration over time is straightfor-ward. The second term, the right term of equation C.4, will for now be integrated in the r -domain. Integrationover time is straightforward and yields:

∫ tn+1

tn

∫ ri+ 1

2

ri− 1

2

1

r

∂

∂r

(rλ

∂T

∂r

)r dr

d t ≈∫ tn+1

tn

([rλ

∂T

∂r

]r j+1

−[

rλ∂T

∂r

]r j

)d t . (C.6)

Combining the final terms of equation C.5 and C.6 gives us:

ρcp(T n+1

i −T ni

)( r 2j+1 − r 2

j

2

)=

∫ tn+1

tn

([rλ

∂T

∂r

]r j+1

−[

rλ∂T

∂r

]r j

)d t . (C.7)

To complete the scheme in equation C.7 for the whole RPV, the problem must be split up in subproblems. Asin Figure C.1, the domain can divide in a problem for grid nodes 1, i , and I . Where for nodes 1 and I , adjacentto the external face of the RPV, are imposed to boundary conditions.

SITUATION I

This boundary condition is enforced on the internal radius of the hollow cylinder. This is the wall that is incontact with the water that it flowing through the downcomer. For this situation we make use of the followingcondition:

∂T

∂r

∣∣∣∣r1

=−h

λ[T (r1, t )−T (z, t )] . (C.8)

121

𝑇1

𝑟1 𝑟2



𝐵.𝐶. 1 𝐵.𝐶. 2

𝑆𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛 − 𝐼𝐼

Figure C.3: Representation of a one-dimensional grid of control volumes in radial direction. Situation II gives the mainheat transfer section for the hollow cylinder.

Here T (z, t ) is water temperature in the downcomer, and z is the height of our slab. Now the right term ofequation C.7 can be re-written as:∫ tn+1

tn

([rλ

∂T

∂r

]r2

−[

rλ∂T

∂r

]r1

)d t =

∫ tn+1

tn

([rλ

∂T

∂r

]r2

+ r1h(z, t )) [T (r1, t )−T (z, t )]

)d t (C.9)

=∫ tn+1

tn

(r2λ

T (r2, t )−T (r1, t )

∆r2+ r1h(z, t )) [T (r1, t )−T (z, t )]

)d t

≈(r2λ

T (r2, tn+θ)−T (r1, tn+θ)

∆r2+ r1h(z, t )) [T (r1, tn+θ)−T (z, tn+θ)]

)∆t

≈(

r2λT n+θ

2 −T n+θ1

∆r2+ r1hn+θ

z

[T n+θ

1 −T n+θz

])∆t .

Re-writing the last term of equation C.9, yields:(θ r2λ

T n+12 −T n+1

1

∆r2+ (1−θ) r2λ

T n2 −T n

1

∆r2+θ r1hn+1

z

[T n+1

1 −T n+1z

]+ (1−θ) r1hnz

[T n

1 −T nz

])∆t . (C.10)

Finally the fully discretized equation that describes the heat transfer through the internal wall of the hollowcylinder is:

ρcp(T n+1

1 −T n1

)( r 22 − r 2

1

2

)=(

θ r2λT n+1

2 −T n+11

∆r2+ (1−θ) r2λ

T n2 −T n

1

∆r2+θ r1hn+1

z

[T n+1

1 −T n+1z

]+ (1−θ) r1hnz

[T n

1 −T nz

])∆t (C.11)

SITUATION IIThis is the main part of the RPV slab. Its discretization is straightforward and yields:∫ tn+1

tn

([rλ

∂T

∂r

]r j+1

−[

rλ∂T

∂r

]r j

)d t =

∫ tn+1

tn

(r j+1λ

T (ri+1, t )−T (ri , t )

∆r j+1− r jλ

T (ri , t )−T (ri−1, t )

∆r j

)d t (C.12)

=(r j+1λ

T (ri+1, tn+θ)−T (ri , tn+θ)

∆r j+1− r jλ

T (ri , tn+θ)−T (ri−1, tn+θ)

∆r j

)∆t

=(

r j+1λT n+θ

i+1 −T n+θi

∆r j+1− r jλ

T n+θi −T n+θ

i−1

∆r j

)∆t ..

Re-writing the last term of equation C.12. Finally the fully discretized equation that describes the heat transferthrough the internal wall of the hollow cylinder is:

ρcp(T n+1

i −T ni

)( r 2j+1 − r 2

j

2

)=(

θ r j+1λT n+1

i+1 −T n+1i

∆r j+1+ (1−θ) r j+1λ

T ni+1 −T n

i

∆r j+1−θ r jλ

T n+1i −T n+1

i−1

∆r j− (1−θ) r jλ

T ni −T n

i−1

∆r j

)∆t . (C.13)

122 C. DISCRETIZATION OF THE RADIAL HEAT TRANSFER EQUATION

𝑇1

𝑟1 𝑟2



𝐵.𝐶. 1 𝐵.𝐶. 2

𝑆𝑖𝑡𝑢𝑎𝑡𝑖𝑜𝑛 − 𝐼𝐼𝐼

Figure C.4: Representation of a one-dimensional grid of control volumes in radial direction. The boundary condition forthe external radius of the hollow cylinder is by Situation III.

SITUATION III

This boundary condition is inflicted to the external radius of the hollow cylinder. This is the wall that is incontact with the pool water. Since different correlation correlation are imposed to simulate the heat flow thethe pool water which depend on the yet unknown temperature. Another boundary condition is imposed,giving the following condition:

∂T

∂r

∣∣∣∣rI

=− q(z, t ))

2π∆z rλ. (C.14)

Here q(z, t ) is heat rate [J/s] from the RPV to the pool. This heat rate is from the RPV to the pool is hereforcoupled explicitly. Now the right term of equation C.7 can be re-written as:

∫ tn+1

tn

([rλ

∂T

∂r

]r J

−[

rλ∂T

∂r

]r J−1

)d t =

∫ tn+1

tn

([−rλ

q(z, t )

2π∆z rλ

]r J

−[

rλ∂T

∂r

]r J−1

)d t (C.15)

=∫ tn+1

tn

(−q(z, t )

2π∆z− r J−1λ

T (r I , t )−T (r I−1, t )

∆r J−1

)d t

≈(−q(z, tn+θ)


T (r I ,n+θ )−T (r I−1,n+θ )

∆r J−1

)∆t

≈(

qn+θz


T n+θI −T n+θ

I−1

∆r J−1

)∆t .

Re-writing the last term of equation C.15, yields:

(θ

qn+1z

2π∆z+ (1−θ)

qnz

2π∆z−θ r J−1λ

T n+1I −T n+1

I−1

∆r J−1− (1−θ) r J−1λ

T nI −T n

I−1

∆r J−1

)∆t . (C.16)

Finally the fully discretized equation that describes the heat transfer through the internal wall of the hollowcylinder is:

ρcp(T n+1

I −T nI

)( r 2J − r 2

J−1

2

)=(

θqn+1

z

2π∆z+ (1−θ)

qnz

2π∆z−θ r J−1λ

T n+1I −T n+1

I−1

∆r J−1− (1−θ) r J−1λ

T nI −T n

I−1

∆r J−1

)∆t . (C.17)

123

Combining equation C.11, C.13 and C.17 results in the fully FVM discretized, semi-implicit Euler, one-dimensionalheat transfer equation on an unstaggered grid:

ρcp(T n+1

i −T ni

)( r 2j+1 − r 2

j

2

)= (C.18)

(θ r2λ

T n+12 −T n+1

1∆r2

+ (1−θ) r2λT n

2 −T n1

∆r2+θ r1hn+1

z

[T n+1

1 −T n+1z

]+ (1−θ) r1hnz

[T n

1 −T nz

])∆t , i = 1

(θ r j+1λ

T n+1i+1 −T n+1

i∆r j+1

+ (1−θ) r j+1λT n

i+1−T ni

∆r j+1−θ r jλ

T n+1i −T n+1

i−1∆r j

− (1−θ) r jλT n

i −T ni−1

∆r j

)∆t ,1 < i < I

(θ

qn+1z

2π∆z + (1−θ)qn

z2π∆z −θ r J−1λ

T n+1I −T n+1

I−1∆r J−1

− (1−θ) r J−1λT n

I −T nI−1

∆r J−1

)∆t , i = I

DDERIVING ANALYTICAL MODEL

The goal of this model is to find the steady-state mass flow rate for the benchmark case. The starting point forthis model is the one-dimensional momentum balance:

∂M

∂t+ ∂

∂x

(M 2

ρA

)=−A

∂p

∂x−∑

jK j

M 2

2ρAδ(x −x j )− f

P M 2

8ρA2 + Aρ−→g . (D.1)

In a steady state all forces that are acting on the loop are in equilibrium, for this situation the time dependanceis not longer relevant and cancels out:

∂

∂x

(M 2

ρA

)=−A

∂p

∂x−∑

jK j

M 2

2ρAδ(x −x j )− f

P M 2

8ρA2 + Aρg cos(θ). (D.2)

To obtain equation D.2 in terms of pressure, it is divided by the flow area A. This since a difference in the flowarea for the hot leg and the cold leg would result in a net force, when the other variables are held constant. Toget the terms of pressure in equilibrium a line integral over the whole loop will be carried out. This reads:∮

1

A

∂

∂x

(M 2

ρA

)=−

∮∂p

∂x−

∮ ∑j

K jM 2

2ρA2 δ(x −x j )−∮

fP M 2

8ρA3 +∮

ρg cos(θ). (D.3)

When the pressure gradient is integrated over a closed loop this term becomes zero. Another term that be-comes zero is the convection term, this since the mass flow rate is constant in steady state and as consequencethe derivative of the mass flow rate is zero. Now equation D.1 is given as:∮ ∑

jK j

M 2

2ρA2 δ(x −xi )+∮

fP M 2

8ρA3 =∮

ρg cos(θ). (D.4)

Solving the line integrals results in:∑j

K jM 2

2ρ j A2j

+∑i

fiPi M 2

8ρi A3i

li = g∑

iρi li cos(θi ). (D.5)

The complete loop is here represented as a summation of length elements, with indices i . These length ele-ments have been obtain splitting up the loop in regions with shared properties. Rewriting previous equationwill give a steady state solution for the mass flow rate:

M 2 = g∑

i ρi li cos(θi )∑j

K j

2ρ j A j

2 +∑i fi

Pi

8ρi A3i

li

. (D.6)

The benchmark geometry is split up in 8 regions and 2 local frictions, equation D.5 becomes:

M 2 = g∑8

i=1ρi li cos(θi )∑2j=1

K j

2ρ j A2j+∑8

i=1 fiPi

8ρi A3i

li

. (D.7)

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The Small-scale Large efficiency Inherent safe Modular Reactor · 2017-09-14 · THE SMALL-SCALE...

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