COUPLED THERMAL HYDRAULICS/NEUTRONICS ANALYSIS OF NUCLEAR REACTORS

NÜKLEER REAKTÖRLERİN TERMAL HİDROLİK VE NÖTRONİK EŞZAMANLI ANALİZİ

SEFA KEMAL UZUN

Submitted to HACETTEPE UNIVERSITY

THE INSTITUTE FOR GRADUATE STUDIES IN SCIENCE AND ENGINEERING

in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE

in NUCLEAR ENGINEERING

2008

To the Directory of the Institute for Graduate Studies in Science and Engineering,

This study has been accepted as a thesis for the degree of MASTER OF SCIENCE

in NUCLEAR ENGINEERING by our Examining Committee.

Head: _______________________ Assoc. Prof. Dr. Cemal Niyazi SÖKMEN Member : _______________________ Prof. Dr.Mehmet TOMBAKOĞLU Member : _______________________ Assist. Prof Dr. İlker TARI This is to certify that the Board of Directors of the Institute for Graduate Studies in

Science and Engineering has approved this thesis on …/…/……

Prof.Dr................................... Director

The Institute for Graduate Studies in Science and Engineering

i

COUPLED THERMAL HYDRAULICS/NEUTRONICS ANALYSIS OF NUCLEAR REACTORS

Sefa Kemal UZUN

ABSTRACT Nuclear reactor calculations include two main branches. These are thermal-

hydraulics and neutronics. Coupling of these calculations is important for nuclear

reactor dynamics and safety. In this study, coupled analysis of Pebble Bed Modular

Reactor is considered by using the DONJON code. OECD-PBMR400 cross section

database is used to realize coupling. The database contains temperature dependent

cross section set for defined region wise geometry. Temperatures of regions are

calculated by the thermal module which is embedded in DONJON. With an

Interpolation module cross sections are generated from temperatures. Generated

cross sections are used for neutronics calculations. Neutronics calculations results

gives the power generation for each region in the core and those power generations

are input for thermal module. So totally closed data cycle achieved between the

neutronics and thermal hydraulics modules. Temperature increase in core region

causes decrease of multiplication factor so that the PBMR is a stable nuclear reactor

design for the steady state conditions with the OECD-PBMR400 cross section

database.

Keywords: coupling, neutronics, thermal-hydraulics, PBMR, DONJON

Advisor: Prof. Dr. Cemal Niyazi SÖKMEN, Hacettepe University, Department of

Nuclear Engineering, Nuclear Engineering Section

ii

NÜKLEER REAKTÖRLERİN TERMAL HİDROLİK VE NÖTRONİK EŞZAMANLI ANALİZI

Sefa Kemal UZUN

ÖZ Nükleer reaktör hesaplamaları iki ana daldan oluşmaktadır, bunlar termalhidrolik ve

nötronik hesaplamalardır. Nötronik ve termalhidrolik hesaplamaların eşzamanlı

yapılması nükleer reaktör dinamiği ve güvenliği bakımından önemlidir. Bu çalışmada

Çakıl Yataklı Modüler Reaktörün DONJON kodu kullanılarak eşzamanlı analizi

yapılmıştır. Eşzamanlı hesabı gerçekleştirmek için OECD-PBMR400 tesir kesiti veri

tabanı kullanılmıştır. Veritabanı, bölgelere ayrılmış geometri için sıcaklığa bağlı tesir

kesitlerini içermektedir. Bölgelerdeki sıcaklık DONJON kodunun içine gömülen termal

modül tarafından hesaplanmaktadır. Interpolasyon modülü kullanılarak bu

sıcaklıklardan tesir kesitleri üretilmektedir. Uretilen tesir kesitleri nötronik

hesaplamalarda kullanılmaktadır. Nötronik hesaplamaların sonuçları reaktör

kalbindeki her bir bölge için güç üretimlerini verir ve bu güç üretimleri de termal

modülün girdi değerlerini oluşturur. Böylece nötronik ve termalhidrolik modüller

arasında tamamen kapalı bir veri akış döngüsü sağlanır. Reaktör kalbindeki sıcaklık

artışı çarpım faktöründe düşüşe sebep olur. Eşzamanlı analiz sonuçları da

göstermektedir ki Çakıl Yataklı Modüler Reaktör, düzenli hal durumunda ve OECD-

PBMR400 tesir kesitlerine göre durağan bir reaktör tasarımıdır.

Anahtar Sözcükler: eşzamanlı,nötronik,termalhidrolik,PBMR,DONJON

Danışman: Prof.Dr. Cemal Niyazi SÖKMEN , Hacettepe Üniversitesi, Nükleer Enerji Mühendisliği Bölümü, Nükleer Enerji Mühendisliği Anabilim Dalı

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ACKNOWLEDGEMENTS I would like to thank my advisor, Assoc. Prof. Dr. C. Niyazi SÖKMEN for his helpful

guidance and endurance. I would also like to thank the committee members, Prof. Dr.

Mehmet TOMBAKOĞLU, and Assist. Prof. Dr. İlker TARI for their valuable

comments. I would also like to thank my managers from General Directorate of

Security, IT Department, and Software Development Section and all members of

Hacettepe University Nuclear Engineering Department.

iv

CONTENTS Page

ABSTRACT........................................................................................................................... i ÖZ......................................................................................................................................... ii ACKNOWLEDGEMENTS.................................................................................................. iii CONTENTS......................................................................................................................... iv LIST OF FIGURES................................................................................................................v LIST OF TABLES ............................................................................................................... vi 1. INTRODUCTION..............................................................................................................1

1.1 Objective of Thesis .......................................................................................................1 1.2 Overview of PBMR ......................................................................................................2

1.2.1 Core Specifications of PBMR ................................................................................2 1.2.2 The Specifications of Fuel Sphere (Pebble) ............................................................3

2. THEORETICAL BACKGROUND of COUPLING............................................................4 2.1 Theory of Neutronics ....................................................................................................4

2.1.1 Multiplication Factor and Criticality.......................................................................4 2.1.2 Main Features of Neutronics ..................................................................................4 2.1.3 Two Group Diffusion Method ................................................................................5 2.1.4 Xenon Poisoning....................................................................................................6

2.2 Theory of Thermal Hydraulics ......................................................................................6 2.3 Feedback Effects...........................................................................................................7

2.3.1 Fuel Temperature Coefficient.................................................................................8 2.3.2 Moderator Temperature Coefficient .......................................................................8

2.3.3 Feedback Effects for High Temperature Gas Cooled Reactors....................................8 3. PROBLEM DEFINITION ..................................................................................................9

3.1 Problem Setup ..............................................................................................................9 3.1.1 Neutronics Definition.............................................................................................9 3.1.2 Thermal-hydraulics Definition .............................................................................10 3.1.3 Cross Section Tables Definition ...........................................................................12

4. COUPLING WITH DONJON CODE...............................................................................13 4.1 General Aspects of DONJON .....................................................................................13 4.2 Coupling Mechanism..................................................................................................13

5. RESULTS .......................................................................................................................15 5.1 Steady State Neutronics Results ..................................................................................15 5.2 Coupled Results ..........................................................................................................22

5.2.1 Results with Different Initial Guesses...................................................................30 REFERENCES.....................................................................................................................36 APPENDIX-A......................................................................................................................37

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LIST OF FIGURES FIGURE 1.1 ............................................................................................................................................. 3 FIGURE 3.1 ............................................................................................................................................. 9 FIGURE 3.2 ........................................................................................................................................... 10 FIGURE 4.1 DATA FLOW SCHEMA OF COUPLING........................................................................... 14 FIGURE 4.2 MODULAR DATA FLOW SCHEMA OF COUPLING ...................................................... 14 FIGURE 5.1 AXIAL POWER DENSITY DISTRIBUTION...................................................................... 16 FIGURE 5.2 PARTICIPANT’S AXIAL POWER DENSITY DISTRIBUTION.......................................... 16 FIGURE 5.3 RADIAL POWER DENSITY DISTRIBUTION ................................................................... 17 FIGURE 5.4 PARTICIPANT’S RADIAL POWER DENSITY DISTRIBUTION....................................... 17 FIGURE 5.5 RADIAL FAST FLUX DISTRIBUTION.............................................................................. 18 FIGURE 5.6 PARTICIPANT’S RADIAL FAST FLUX DISTRIBUTION ................................................. 18 FIGURE 5.7 AXIAL FAST FLUX DISTRIBUTION................................................................................. 19 FIGURE 5.8 PARTICIPANT’S AXIAL FAST FLUX DISTRIBUTION..................................................... 19 FIGURE 5.9 RADIAL THERMAL FLUX DISTRIBUTION...................................................................... 20 FIGURE 5.10 PARTICIPANT’S RADIAL THERMAL FLUX DISTRIBUTION ....................................... 20 FIGURE 5.11 AXIAL THERMAL FLUX DISTRIBUTION....................................................................... 21 FIGURE 5.12 PARTICIPANT’S AXIAL THERMAL FLUX DISTRIBUTION........................................... 21 FIGURE 5.13COUPLED RADIAL FAST FLUX DISTRIBUTION .......................................................... 22 FIGURE 5.14 PARTICIPANT’S COUPLED RADIAL FAST FLUX DISTRIBUTION ............................. 22 FIGURE 5.15 COUPLED AXIAL FAST FLUX DISTRIBUTION ............................................................ 23 FIGURE 5.16 PARTICIPANT’S COUPLED AXIAL FAST FLUX DISTRIBUTION ............................... 23 FIGURE 5.17 COUPLED RADIAL THERMAL FLUX DISTRIBUTION ................................................. 24 FIGURE 5.18 PARTICIPANT’S COUPLED RADIAL THERMAL FLUX DISTRIBUTION ..................... 24 FIGURE 5.19 COUPLED AXIAL THERMAL FLUX DISTRIBUTION .................................................... 25 FIGURE 5.20 PARTICIPANT’S COUPLED AXIAL THERMAL FLUX DISTRIBUTION........................ 25 FIGURE 5.21 COUPLED AXIAL TEMPERATURE DISTRIBUTION .................................................... 26 FIGURE 5.22 PARTICIPANT’S AXIAL TEMPERATURE DISTRIBUTION........................................... 26 FIGURE 5.23 COUPLED RADIAL TEMPERATURE DISTRIBUTION.................................................. 27 FIGURE 5.24 PARTICIPANT’S COUPLED RADIAL TEMPERATURE DISTRIBUTION .................... 27 FIGURE 5.25 COUPLED AXIAL POWER DENSITY DISTRIBUTION ................................................. 28 FIGURE 5.26 PARTICIPANT’S COUPLED AXIAL POWER DENSITY DISTRIBUTION .................... 28 FIGURE 5.27 COUPLED RADIAL POWER DENSITY DISTRIBUTION............................................... 29 FIGURE 5.28 PARTICIPANT’S COUPLED RADIAL POWER DENSITY DISTRIBUTION .................. 29 FIGURE 5.29 MULTIPLICATION FACTOR VS. ITERATION NUMBER OF INITIAL GUESS 1 .......... 30 FIGURE 5.30 MULTIPLICATION FACTOR VS. ITERATION NUMBER OF INITIAL GUESS 2 .......... 30 FIGURE 5.31 MULTIPLICATION FACTOR VS. ITERATION NUMBER OF INITIAL GUESS 3 .......... 31 FIGURE 5.32 MULTIPLICATION FACTOR VS. ITERATION NUMBER OF INITIAL GUESS 4 .......... 31 FIGURE 5.33 CONTOUR PLOT OF FAST FLUX OF PBMR ............................................................... 32 FIGURE 5.34 CONTOUR PLOT OF THERMAL FLUX OF PBMR ....................................................... 32 FIGURE 5.35 CONTOUR PLOT OF TEMPERATURE DISTRIBUTION OF PBMR............................. 33 FIGURE 5.36 CONTOUR PLOT OF XE DISTRIBUTION OF PBMR CORE........................................ 33 FIGURE 5.37 CONTOUR PLOT OF ZERO ITERATION POWER DENSITY DISTRIBUTION............ 34 FIGURE 5.38 CONTOUR PLOTOF SEVEN ITERATION POWER DENSITY DISTRIBUTION........... 34

vi

LIST OF TABLES TABLE 1: MAJOR DESIGN AND OPERATING CHARACTERISTICS OF THE PBMR......................... 2 TABLE 2 RISER PROPERTIES OF PBMR .......................................................................................... 12 TABLE 3 TABLE OF INITIAL GUESS................................................................................................... 30

1

1. INTRODUCTION A nuclear reactor design must be reliable, efficient and safe to be an acceptable

design. Design should prove that it includes all of the criteria. Safety assurance of a

reactor is the most important design criteria because historical experiences show that

unsafe reactors cause biting results for all people. To assure the safety of a reactor,

its behaviors should be predictable for expected and unexpected cases. This

requires coupling of all calculations about the reactor. Neutronics and thermal

hydraulics are the main calculations of a reactor design, therefore they should be

coupled. The stability of a reactor depends on the feedback effects of thermal

hydraulics to the neutronics events. Main concern, when sudden temperature

increase occurs in the core of the reactor, how it will effect the power generation of

the reactor. Coupled analysis gives the answer of the question. Furthermore, in the

process of developing a new type reactor like Pebble Bed Modular Reactor, results of

coupled analysis are very important to understand the reactor dynamics. With the

sponsorship of Nuclear Energy Agency of OECD (Organization for Economic Co-

operation and Development) “PBMR COUPLED NEUTRONICS/THERMAL

HYDRAULICS TRANSIENT BENCHMARK THE PBMR-400 CORE DESIGN” is

prepared to improve coupling methods and to simulate special transient cases of

PBMR. Benchmark definition includes two phases; first phase is about steady state

coupled case of thermal hydraulics/neutronics, and second phase is about some well

defined transient cases of PBMR. Results of the first phase are initial conditions for

transient cases. This study is focused on only steady state coupling case (phase 1)

of the benchmark problem.

1.1 Objective of Thesis The specific goal of this study is making a coupled steady state analysis of PBMR.

Coupled analysis should give an idea about effects of neutronics to the thermal

hydraulics calculations and vice versa. Another objective of this study is improving a

useful coupling method for next transient cases.

2

1.2 Overview of PBMR The PBMR is a pebble-type High Temperature Gas Cooled Reactor (HTGR). This

PBMR power plant incorporates a closed cycle primary coolant utilizing helium to

transport heat energy directly from the modular pebble bed reactor to a recuperative

Brayton cycle power conversion unit with a single shaft

turbine/compressor/generator. This replacement of the steam cycle that is common

in present nuclear power plants (NPP) with a direct gas cycle provides the benefits of

simplification and a substantial increase in overall system efficiency and inherent

safety with attendant lowering of capital and operational costs.

Table 1: Major Design and Operating Characteristics of the PBMR

PBMR Characteristics Value Installed thermal capacity 400 MW(t) Installed electric capacity 165 MW(e) Load following capacity 100-40-100% Availability ≥ 95 % Core Configuration Vertical with fixed centre graphite reflector Fuel TRISO ceramic coated U-235 in graphite

spheres Primary coolant Helium Primary coolant pressure 9 MPa Moderator Graphite Core outlet temperature 900 º C. Core inlet temperature 500 º C. Cycle type Direct Number of circuits 1 Cycle efficiency ≥ 41 % Emergency planning zone 400 meters

1.2.1 Core Specifications of PBMR

• An annular core outer diameter 3.7 m and a fixed central reflector with an

outer diameter 2.0 m

• An effective cylindrical core height of 11.0 m

3

• A graphite side reflector of ~ 90 cm.

• Reactivity control system consist 24 partial control rods (12 upper and 12

lower in the side reflector with 6.5 m effective length).

• The core contains ~ 452000 fuel spheres or “pebbles” with a packing fraction

of 0.61.

1.2.2 The Specifications of Fuel Sphere (Pebble)

• The uranium loading is 9 g per fuel-sphere with U235 enrichment at 9.6 wt%.

• The inner 5 cm of the fuel sphere contains about 15 000 UO2 TRISO-coated

micro spheres within a graphite matrix and surrounded by an outer graphite

fuel free zone.

Figure 1.1

4

2. THEORETICAL BACKGROUND of COUPLING

2.1 Theory of Neutronics

2.1.1 Multiplication Factor and Criticality Nuclear fission is the basic phenomenon of the nuclear reactors. As a nature of

fission reaction, fuel material nucleus absorbs a neutron before fission, and emits

neutrons after fission. Average number of neutrons emitted from fission varies to fuel

material type. This number is at the interval of two and three. Neutrons emitted from

fission slow down (for thermal reactors) and causes other fissions. This is a chain

reaction and multiplication factor is the main indicator of this chain reaction.

Multiplication factor (k) is the ratio of the number of neutrons in one generation to the

number of neutrons in the preceding generation. If k is smaller than 1.0, it means that

number of neutrons is decreasing, so reactor will shut down itself automatically a

period later. If k is greater than 1.0 it means number of neutrons is increasing, so

power will increase exponentially. Consequently, unless multiplication factor is equal

to 1.0, reactor will not work at steady state conditions. This is the “criticality condition”

which is the basic the design parameter of a nuclear reactor.

2.1.2 Main Features of Neutronics All of the neutronics equations about nuclear reactors derived from viewpoint of

neutron balance. This means that, number of neutrons generated in a control volume

plus comes in from out of volume should be equal to number of neutrons absorbed in

the volume plus leaks out. Source term for the balance equation comes from the

fission reaction of fuel material. Furthermore, fission cross sections of fissile materials

strongly depend on energy level of neutrons. Another important aspect of neutronics

equations is the scattering of neutrons which causes slowing down of neutrons.

When a neutron collide a nucleus, nucleus can absorb the neutron or scatter an

arbitrary angle. Angular neutron flux may be also important; therefore energy, time,

space and angle dependent neutron balance equation is called “neutron transport

equation”. Analytical solution of transport equation is very difficult without considering

any simplification. There are various approximations to simplify neutron transport

5

equation. One of these, grouping of neutrons due to energy level is a common and

reasonable approximation to reduce the energy dependence of the equation. Most

useful method to eliminate the angular dependency of the “neutron transport

equation” is the diffusion approximation. As a result of these approximations “neutron

diffusion equation” is obtained which is the adequate equation for most of the

neutronics calculations of nuclear reactors. Time dependent one speed diffusion

equation for semi infinite cylindrical geometry is shown at equation 2.1. Φ shows

neutron flux (neutron/cm2/s) as a scalar quantity, D is the diffusion coefficient derived

from neutron transport equation. V is the mean velocity of neutrons, ν is the number

neutrons released per fission.

1

( , ) ( , )a fD r t r tV t

φφ φ ν φ

∂− ∇ ∇ + Σ = Σ

∂ (2.1)

Σf is the macroscopic fission cross section of fuel material. Σa is total macroscopic

absorbtion cross section defined as like following ;

moderator structure coolant fuela a a a a

fuel fuel fuela fγ

=

=

∑ ∑ +∑ +∑ +∑

∑ ∑ +∑ (2.2)

fuelγ∑ represents the cross section of radiative capture is a interaction of neutron with

fuel material nuclei. That is nuclei absorbs neutron and emit gamma radiation.

2.1.3 Two Group Diffusion Method Neutrons traveling in a reactor have energies in a scale of 10 MeV to less than 0.01

eV. To get more realistic results energy dependence must be considered also. By

choosing discrete energy levels (E0, E1…..EN), neutron’s groups will be determined

between these energy levels. So N number of diffusion equation will be solved for N

group of neutron.

Separating neutrons into two groups (fast and thermal) due to a cut off energy

around 3 eV (for gas cooled reactors) is a reasonable method. So diffusion equation

6

will be solved for fast and thermal groups. Steady state fast and thermal diffusion

equations shown at equations 2.3 and 2.4.

1 1 21 1 1 1 1 2 2

1( )φ φ ν φ ν φ− ∇ ∇ + ∑ = ∑ + ∑r f fD

k (2.3)

2 1 22 2 2 1a sD φ φ φ

→− ∇ ∇ + ∑ = ∑ (2.4)

Some assumptions considered at the above equations. Thus, there is no up

scattering from thermal to fast group, fission neutrons born in fast group.

1 11 2→∑ = ∑ + ∑r s a (2.5)

2.1.4 Xenon Poisoning Xenon is the most important fission product which affects the neutron balance of the

reactor because it is a strong neutron absorber. So Xenon concentration must be

considered in the neutronics calculations. Xenon occurs by two mechanisms. One of

these direct from fission reaction xenon yields. The other source term of xenon

comes from decay of iodine. Steady state xenon equilibrium is shown at equation

2.6.

1 2

1 2

1 2

1 2

( )( )

( )f f x I

x a a

X eφ φ γ γ

λ σ φ σ φ

∑ + ∑ +=

+ + (2.6)

1φ , 2φ indicates fast and thermal fluxes,1f

∑ and 2f

∑ denotes the fast and thermal

fission cross sections. xγ ,

Iγ denotes effective fraction of fission products which are

Xe135 and I135. 1aσ ,

2aσ denotes the fast and thermal microscopic absorption cross

sections of xenon. xλ indicates the beta decay constant of xenon.

2.2 Theory of Thermal Hydraulics Universal principals of “conservation of energy” and “conservation of momentum” are

also base principals for nuclear reactor thermal hydraulics calculations. Therefore, to

7

get the temperature distribution of the nuclear reactor, solution of heat equation is an

obligation. In this study Pebble Bed Modular Reactor is modeled with solution

coordinate two dimensional cylindrical (r, z) coordinates. This heat equation for the

core region is shown at equation 2.7.

, ,( )1

0r

p f out f inmc T TT Tkr k q

r r z z V

−∂ ∂ ∂ ∂ + + − =

∂ ∂ ∂ ∂

&& (2.7)

, ,( )p f out f in s lm

mc T T hA T− = ∆& (2.8)

, ,

,

,

( ) ( )

ln

f in f o u t

lm

f in

f o u t

T T T TT

T T

T T

− − −∆ =

− −

(2.9)

Thus, q& represents the volumetric heat generation rate, ,f outT , ,f inT shows the fluid

(helium) outlet and inlet temperatures. T is the surface temperature the pebbles. m& is

the mass flow rate of the helium, pc is the specific heat capacity of helium. Equation

2.8 shows that the energy transported with helium is equal to convective heat

transfer between the fuel pebbles and helium. lmT∆ Is the log-mean temperature

difference which is defined at equation 2.9.

2.3 Feedback Effects The stabilization of nuclear reactors strongly depends on the feedback effects of

thermal-hydraulics calculations. Major feedback effects come from the fuel

temperature coefficient, moderator temperature coefficient and thermal expansion

terms. Unless, total of reactivity feedback terms is not negative, reactor do not

behave stable. The sign of the total reactivity feedback terms may alter according to

the operating conditions.

8

2.3.1 Fuel Temperature Coefficient Increase of fuel temperature causes decrease of resonance escape probability due

to Doppler broadening. This means that less neutron reach thermal region and cause

fission. So as shown at four factor formula (equation 2.10), decrease of p (resonance

escape probability), reduce the multiplication factor.

k p fε η∞ = (2.10)

2.3.2 Moderator Temperature Coefficient The moderator temperature variation, affects the multiplication factor with various

mechanisms. Increase of temperature, causes decrease of the moderator density

and also increase the average energy of moderator atoms. These factors hardens

neutron energy spectrum. Another parameter affected by moderator temperature

changes is fuel utilization factor ( f ) but it is not easy to predict the sign and

magnitude of this effect. Moreover, the term predominates depend on the choice of

moderator and of fuel configuration.

2.3.3 Feedback Effects for High Temperature Gas Cooled Reactors Density changes of the solid moderator with temperature are very small, so

moderator temperature coefficients of solid moderated reactors are less than liquid

moderated reactors. Although moderator temperature coefficient is relatively small in

magnitude, it is positive in sign. It is positive because the fuel loading contains

substantial amounts of uranium-233, a nuclide for which temperature coefficients of η

is positive. But this factor may not be detrimental for reactor safety, if the fuel

temperature coefficient is sufficiently negative.

PBMR fuel loading doesn’t contain U233 therefore this assures that there is no

prompt positive feedback effect in the core region. Although, because of structure of

9

PBMR, it contains large solid volume so temperature response of the reactor is slow.

After a transient case prompt effects preserves reactor stability but late cooling of the

moderator may bring a positive feedback effect.

3. PROBLEM DEFINITION

3.1 Problem Setup

3.1.1 Neutronics Definition According to the benchmark definition guide neutronics domain for coupled cases is

given like figure 3.1. The x direction of the figure represents the radial direction, and

the y direction of the figure represents the axial direction. Number 1 regions shows

the centre and side reflectors, number 2 regions shows the control rod, and number 3

region shows the core barrel. Void regions represented with number 0. Core region is

shown 101 to 522. For uncoupled case cross section sets are given according to the

figure 3.2.

Figure 3.1.

10

For uncoupled case simplified cross section database is given according to the figure

3.2

Figure 3.2.

For both uncoupled and coupled cases, linear polynomial elements are used. In

radial direction, 18 regions from center split 4 pieces, 19.th, 20.th regions split 8 and

32 pieces. In axial direction, each region split 8 pieces, except top and bottom

regions split 32 pieces.

3.1.1.1 Boundary Conditions For radial outer, top and bottom boundaries black condition is given. Black condition

means that, neutrons leaks out from the reactor but there is no neutron flow towards

to the reactor.

3.1.2 Thermal-hydraulics Definition To get the temperature distribution necessary for this study, T-BED code is used as a

sub module of DONJON code. T-BED code solves the equation 2.7 (heat equation

for 2D cylindrical coordinates) with finite difference method. Convection through the

11

bottom reflector is modeled with average core outlet temperature as temperature of

coolant. Convection through the riser is modeled with average temperature of helium

in the riser. Void model of the core as defined like equation 3.1.

11 2

1 2

( ) ( )(1 exp( )) (1 exp( ))

r r R rC C

N Nε ε ∞

− −= + − × + − (3.1)

Where; 1 100r cm= , 185R cm= , 0.39ε∞ = , 1 2 1.36C C= = , 1 25

pd

N N= = 6p

d = cm

The core is divided into 3 regions according to model the void distribution.

0.432 1 1.1417

0.390 1.1417 1.7083

0.436 1.7083 1.85

m r m

m r m

m r m

ε

< <

= < < < <

Heat transfer correlation for the heat transfer between helium and pebbles;

(1 / 3 ) 0 .5

0 .3 6 0 .8 6

1 .1 8 1 .0 7

P r P r1 .2 7 R e 0 .0 3 3 R eN u

ε ε= + (3.2)

k

h N ud

= (3.3)

Effective thermal conductivity of the fuel element and reflector graphite;

2115 / / 6.09 10pebblek W m K= Φ = × 2123 / / 3.09 10reflectork W m K= Φ = ×

Pressure Drop Correlation for Pebble bed core:

2

3

0 . 1

1

2

3 2 0 6R e R e

( )1 1

hp u

d

ε ρψ

ε

ψ

ε ε

−∆ =

= +

− −

(3.4)

12

The core region has 10 nodes in radial direction and 20 nodes in axial direction. On

this way, there are 10 flow paths within core. There is no advection between

channels. So, mass, momentum and energy interchange of helium between channels

are totally neglected. An initial guess for mass flux distribution and temperature of the

core is assigned. Due to the void and temperature distribution within core, mass flux

of the channels is different. Mass flux distribution is calculated by taking the axial

pressure drop of each channel equal to each other for each step of the iteration.

1 2 1 0

1 2 1 0

. . .

. . .t o t a l

P P P

m m m m

∆ = ∆ = = ∆

= + + +& & & & (3.5)

3.1.2.1 Flow Parameters and Boundary Conditions

• Helium temperature at the riser inlet 488.1 C

• Inlet mass flow rate 185.31 kg/s

• At the top, bottom and center boundaries adiabatic boundary condition is

applied

• At r = 2.75 m convection boundary condition applied with the parameters

h=200W/m2K and T= 430 C.

Table 2 Riser Properties of PBMR

Number of Gas Riser in Side Reflector

36

Radius of Risers 0.085 m

Mean Radius of the Location of the Risers in the Side Reflector

2.526 m

3.1.3 Cross Section Tables Definition The benchmark definition includes two cross section databases. One of these

simplified cross section database which is used for only neutronics calculations

(without thermal-hydraulics). The other one contains temperature, xenon and

13

buckling dependent cross section sets. According to the benchmark, sets are

generated from MICROX using the equilibrium core number densities with two

energy groups with a thermal cut-off of 2.38 eV. Cross section sets are given as

region wise as shown in figure 3.1. There are 34 tables for each region. The cross

sections are depending on 5 state parameters. The parameters are fuel temperature,

moderator temperature, fast buckling, thermal buckling and xenon concentration.

Buckling table given at Appendix A, in this study buckling terms assumed constant

for reflector and fuel regions. Fast and thermal buckling is assumed 1E-5, 1E-6. To

generate the cross sections from the tables, interpolation is required. FORTRAN

source code LINT5D.f is given to make linear interpolation on the cross section sets.

4. COUPLING WITH DONJON CODE

4.1 General Aspects of DONJON DONJON is a neutronics code which is collection of FORTRAN77 modules. It solves

the multigroup neutron diffusion equation by using the finite element method. The

execution of DONJON is controlled by the generalized GAN driver. It is appropriate

for coupling because of modular structure.

4.2 Coupling Mechanism To achieve the coupling, data transfer between the neutronics and thermal-hydraulics

modules must be acquired. DONJON code has a modular structure and it transfers

the data between modules by using LCM memory objects. LCM memory object can

save output of a module, and also can be input of another module. And also data on

a LCM memory object can be copied another memory object. This structure is very

suitable for coupling. Anyway DONJON includes neutronics modules but thermal-

hydraulics and interpolation modules put in the sources and recompiled. The CLE-

2000 control language allows loops and conditional statements to manage calling of

the modules. Figure 4.1 shows the general coupled data flow and the figure 4.2

shows the detailed data flow between the modules.

14

Figure 4.1. Data Flow Schema of Coupling

Figure 4.2. Modular Data Flow Schema of Coupling

15

THERMA module generates temperature distribution and this data used for cross

section interpolation by DENEME module. Beside temperature distribution for cross

section generation xenon concentration is required. Xenon concentration calculated

from equation 2.6 in the CONSXE module. Interpolated cross sections are used for

solving 2 group neutron diffusion equations in the FLUD module. By using the output

of neutronics calculations power distribution is obtained by OUT module. The power

distribution is also input for the THERMA module. As a result, completely closed data

cycle is designed. To initialize the cycle in the first iteration temperatures read from

an external file.

5. RESULTS The results are compared with the benchmark participant’s results to make

validation. And also some extra figures are added to the results understand

coupling effect.

5.1 Steady State Neutronics Results This part includes results about uncoupled pure neutronics calculations. Simplified

OECD-PBMR400 cross section set is used for this calculation. By taking the

convergence criteria 10-6, convergence occurred at 45. iteration.

K-effective = 0.996726 (DONJON)

K-effective = 1.000317 (Benchmark Participants Average).

16

012

34567

89

10

0 200 400 600 800 1000 1200

AXIAL POSITION(CM)

PO

WE

R D

EN

SIT

Y (

W/C

M3

)

Figure 5.1. Axial Power Density Distribution

0

1

2

3

4

5

6

7

8

9

10

0 200 400 600 800 1000 1200

AXIAL POSITION(CM)

PO

WE

R D

EN

SIT

Y(W

/CM

3)

Figure 5.2. Participant’s Axial Power Density Distribution This axial power density is calculated from average of radial power densities for each

row of core region. Axial power density result of the study fits the benchmark

participant’s results. Thus, maximum axial power occurs at a location around 350 cm

far from top of top reflector. And it has value approximately 9.5 W/cm3.

17

3

3.5

4

4.5

5

5.5

6

100 110 120 130 140 150 160 170 180 190 200

RADIAL POSITION (CM)

PO

WE

R D

EN

SIT

Y (

W/C

M3

)

Figure 5.3. Radial Power Distribution

3

3.5

4

4.5

5

5.5

6

100 110 120 130 140 150 160 170 180 190 200

RADIAL POSITION(CM)

PO

WE

R D

EN

SIT

Y(W

/CM

3)

Figure 5.4. Participant’s Radial Power Distribution The radial power density distribution of PBMR core is calculated from average of

axial power densities for each column in PBMR core. Study and benchmark results

looks like same. And also maximum radial power density occurs at r=117 cm and its

value around 5.6 W/cm3.

18

Figure 5.5. Radial Fast Flux Distribution

0

1E+13

2E+13

3E+13

4E+13

5E+13

6E+13

7E+13

8E+13

9E+13

0 50 100 150 200 250 300

RADIAL DISTANCE

FL

UX

Figure 5.6. Participant’s Radial Fast Flux Distribution Radial flux is calculated from axial average of each column of defined geometry at

figure 3.1 or figure 3.2. Fast flux results are reasonable because source of fast flux is

the fission in the core region. Anyway, fast flux increases in the core region. It has a

maximum value at the radial center of the core.

19

0.00E+00

1.00E+13

2.00E+13

3.00E+13

4.00E+13

5.00E+13

6.00E+13

7.00E+13

-200 0 200 400 600 800 1000 1200

Axial Position(cm)

Flu

x(n

/cm

2/s

)

Figure 5.7. Axial Fast Flux Distribution

0.0E+00

1.0E+13

2.0E+13

3.0E+13

4.0E+13

5.0E+13

6.0E+13

7.0E+13

-200 0 200 400 600 800 1000 1200

AXIAL DISTANCE

FL

UX

Figure 5.8. Participant’s Axial Fast Flux Distribution Axial fast flux distribution derived from radial average of fast fluxes for each row in

the figure 3.1.

20

Figure 5.9. Radial Thermal Flux Distribution

0

2E+13

4E+13

6E+13

8E+13

1E+14

1.2E+14

1.4E+14

1.6E+14

0 50 100 150 200 250 300

RADIAL DISTANCE

FL

UX

Figure 5.10. Participant’s Radial Thermal Flux Distribution

Radial thermal flux distribution derived from axial average of power densities for each

column in the figure 3.1. Thermal flux decreases in the core region significantly as

expected because fuel material absorbs thermal neutrons via fission reaction. And

also decreases towards to radial outer boundary due to black condition.

21

0.00E+00

2.00E+13

4.00E+13

6.00E+13

8.00E+13

1.00E+14

1.20E+14

1.40E+14

1.60E+14

1.80E+14

2.00E+14

-200 0 200 400 600 800 1000 1200

Axial position(cm)

Flu

x(n

/cm

2/s)

Figure 5.11. Axial Thermal Flux Distribution

0.00E+00

2.00E+13

4.00E+13

6.00E+13

8.00E+13

1.00E+14

1.20E+14

1.40E+14

1.60E+14

1.80E+14

2.00E+14

-200 0 200 400 600 800 1000 1200AXIAL DISTANCE

FL

UX

Figure 5.12. Participant’s Axial Thermal Flux Distribution

Axial thermal flux distribution derived from average of radial thermal fluxes for each

row in the figure 3.1.

22

5.2 Coupled Results Following results are obtained with 7 iterations between thermal hydraulics and

neutronics modules. Multiplication factor is converged to 1.009317. Converged

Average Fuel and Moderator Temperature of the core is 888 K. Participant’s average

multiplication factor is 1.00099. Participant’s converged average fuel temperature is

1080 K and average moderator temperature is 1065 K.

01E+132E+133E+134E+135E+136E+137E+138E+139E+131E+14

0 50 100 150 200 250 300

Radial Position(cm)

Flu

x(n

/cm

2/s)

Figure 5.13. Coupled Radial Fast Flux Distribution

01E+132E+13

3E+134E+135E+136E+137E+13

8E+139E+131E+14

0 50 100 150 200 250 300RADIAL DISTANCE (CM)

FL

UX

(n/c

m2 /s

)

Figure 5.14. Participant’s Coupled Radial Fast Flux Distribution

23

0.00E+001.00E+132.00E+133.00E+134.00E+135.00E+136.00E+137.00E+138.00E+139.00E+13

-200 0 200 400 600 800 1000 1200

Axial Position(cm)

Flu

x(n

/cm

2/s)

Figure 5.15. Coupled Axial Fast Flux Distribution

Figure 5.16. Participant’s Coupled Axial Fast Flux Distribution Study’s fast flux distribution is a bit higher than participant’s around 300 cm below

from top of the reactor. 600 cm below from top of the reactor fast flux of the study

smaller than participant’s result.

24

0.0E+00

2.0E+13

4.0E+13

6.0E+13

8.0E+13

1.0E+14

1.2E+14

1.4E+14

1.6E+14

0 50 100 150 200 250 300

Radial Position(cm)

Flu

x(n

/cm

2s)

Figure 5.17. Coupled Radial Thermal Flux Distribution

0

2E+13

4E+13

6E+13

8E+13

1E+14

1.2E+14

1.4E+14

1.6E+14

0 50 100 150 200 250 300

RADIAL DISTANCE (CM)

FL

UX

(n

/cm

2 /s)

Figure 5.18. Participant’s Coupled Radial Thermal Flux Distribution Radial thermal flux doesn’t vary for uncoupled and coupled cases and also similar

results obtained with participants despite serious difference between thermal

hydraulics results.

25

0.00E+00

3.00E+13

6.00E+13

9.00E+13

1.20E+14

1.50E+14

1.80E+14

2.10E+14

2.40E+14

-200 0 200 400 600 800 1000 1200

Axial Position(cm)

Flu

x(n

/cm

2/s

)

Figure 5.19. Coupled Axial Thermal Flux Distribution

Figure 5.20. Participant’s Coupled Axial Thermal Flux Distribution

Around 300 cm below from top of the reactor thermal flux result of the study higher

than participants results. 800 cm below from top of the reactor lower results obtained

than participant’s.

26

700

750

800

850

900

950

1000

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

Axial Position

Tem

per

atu

re(K

)

Figure 5.21. Coupled Axial Temperature Distribution

500.00550.00600.00650.00700.00750.00800.00850.00900.00950.00

0 100 200 300 400 500 600 700 800 900 1000 1100 1200

AXIAL DISTANCE FROM TOP (CM)

FU

EL

TE

MP

. (°C

)

Figure 5.22. Participant’s Coupled Axial Fuel Temperature Distribution

A serious difference occurred between temperature distribution result of the study

and participant’s result. This shows that there is a fault in the thermal hydraulics part

of coupling. The difference between the thermal hydraulics definition of the

benchmark and solution definition of T-BED may be cause of this fault. And also,

some modifications made on the T-BED code to use in the DONJON code. While

changing the code, there may be an error on the code.

27

860

865

870

875

880

885

890

895

900

100 110 120 130 140 150 160 170 180 190

Radial Position

Te

mp

era

ture

(K)

Figure 5.23. Coupled Radial Temperature

760

780

800

820

840

860

880

900

100 110 120 130 140 150 160 170 180 190

RADIAL DISTANCE (CM)

FU

EL

TE

MP

. (°

C)

Figure 5.24. Participant’s Coupled Radial Fuel Temperature Distribution As a result of the fault in the thermal hydraulics module radial temperature

distribution results are completely different.

28

0

2

46

8

10

12

0 200 400 600 800 1000 1200

axial position(cm)

Po

wer

Den

sity

(W/c

m3)

Zero iteration7 iterationZero iteration With Benchmark Temp.

Figure 5.25. Coupled Axial Power Density Distribution

Figure 5.26. Participant's Coupled Axial Power Density Distribution Figure 5.25 shows that zero iteration (thermal-neutronics) axial power density and

seven iteration (thermal-neutronics) axial power density. At seven iteration power

density a little shifts up at the axial location from 0 to 375 cm. After 375 cm power

density is lower than one iteration power density. And also participant power density

is comparable with power density of the study. With the benchmark temperature

distribution input, power density results are close to the other results.

29

4.34.54.74.95.15.35.55.7

100 120 140 160 180 200

radial position(cm)

Po

wer

Den

sity

(W/c

m3)

Zero iteration7 iterationZero iteration with Benchmark Temp.

Figure 5.27. Coupled Radial Power Density Distribution

4.3

4.5

4.7

4.9

5.1

5.3

5.5

5.7

100 110 120 130 140 150 160 170 180 190 200

RADIAL POSITION(CM)

PO

WE

R D

EN

SIT

Y(W

/CM

3)

Figure 5.28. Participant’s Coupled Radial Power Density Distribution

Radial power density doesn’t change drastically for uncoupled and coupled cases.

And also zero iteration and seven iteration cases results are close to each other.

Participant’s radial power density is similar with study results except r=185 cm there

is a small difference between results.

30

5.2.1 Results with Different Initial Guesses Table 3 Table of Initial Guess INITIAL GUESS NUMBER

AVERAGE FUEL TEMP.(K)

AVERAGE MODERATOR TEMP.(K)

REFLECTOR and CONTROL ROD REGION TEMP.(K)

CORE OUTER BARREL TEMP.(K)

Xe CONCENTRATION (#/barn-cm)

1 1080 1050 800 600 0 2 680 650 500 400 0 3 1080 1050 800 600 1E-10 4 680 650 500 400 1E-10

1.00161.00241.00321.004

1.00481.00561.00641.00721.008

1.00881.00961.0104

0 1 2 3 4 5 6 7 8

ITERATION NUMBER

K-E

FF

EC

TIV

E

figure 5.29. Multiplication Factor vs. Iteration Number of Initial Guess 1

1.0081.009

1.011.0111.0121.0131.0141.0151.0161.0171.0181.019

0 1 2 3 4 5 6 7 8

ITERATION NUMBER

K-E

FF

EC

TIV

E

figure 5.30. Multiplication Factor vs. Iteration Number of Initial Guess 2

31

0.9980.999

11.0011.0021.0031.0041.0051.0061.0071.0081.009

1.01

0 1 2 3 4 5 6 7 8

ITERATION NUMBER

K-E

FF

EC

TIV

E

figure 5.31. Multiplication Factor vs. Iteration Number of Initial Guess 3

1.00881.00961.01041.0112

1.0121.0128

1.01361.0144

0 1 2 3 4 5 6 7 8

ITERATION NUMBER

K-E

FF

EC

TIV

E

figure 5.32. Multiplication Factor vs. Iteration Number of Initial Guess 4

Starting solution with different initial guesses doesn’t affect convergence point of the

multiplication factor. Decreasing of the fuel, moderator and reflector temperature

increases the multiplication factor. The dominant factor increasing the multiplication

factor is the decrease of fuel temperature. Furthermore increase of fuel temperature

decreases the multiplication factor. Xe concentration also is another factor

decreasing the multiplication factor.

32

figure 5.33. Contour Plot of Fast Flux of PBMR

figure 5.34. Contour Plot of Thermal Flux of PBMR

33

Contour plots shows that, thermal and fast flux distributions are as like as expected.

In core region fast flux reaches the maximum value, in the centre reflector region

thermal flux reaches the maximum value. Rectangle in the plot denotes the

boundaries of the PBMR core region.

figure 5.35. Contour Plot of Temperature Distribution of PBMR

figure 5.36. Contour Plot of Xenon Distribution of PBMR Core

34

figure 5.37. Zero iteration Power Density Distribution of PBMR Core

figure 5.38. 7 iteration Power Density Distribution of PBMR Core

35

6. CONCLUSION The results of the study are commonly concordant with the benchmark participant’s

results except the thermal results. Temperature values in the core region are

approximately 200 K lower than the participant’s average. And also multiplication

factor of the study is higher than participant’s average. Therefore there is a reverse

balance between multiplication factor and temperatures in the core. Results get with

different initial guesses shows that Xe effect on the multiplication factor. There is no

drastic difference between the study and benchmark power densities and fluxes

despite the temperature difference. So temperature dependencies of the cross

sections are small. This means that temperature feedbacks will more effective at fast

transients than slow transients. Consequently, at the steady state conditions

feedback effects are not so important. To make a coupled transient analysis

DONJON may be a good tool for neutronics part of coupled case and also it can

manage the data flow between the thermal hydraulics and neutronics modules.

36

REFERENCES OECD/NEA/NSC, PBMR Coupled Neutronics/ Thermal Hydraulics Transient Benchmark The PBMR-400 Core Design, 20 June 2007 Lewis. E. E, Nuclear Power Reactor Safety, a Wiley-Interscience Publication John Wiley & Sons (1977), 157-186. Duderstadt J. James, Hamilton J. Louis, Nuclear Reactor Analysis, John Wiley & Sons, 569-571. Reitsma F, Strydom G, Haas J B M De, Ivanov K, Tyobeka B, Mphahlele, Downar T J, Seker V, Gougar, Da Cruz D. F., Sikik U. E, The PBMR steady-state and coupled kinetics core thermalhydraulics benchmark test problems Incropera P. F, DeWITT P. D, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, 569-571. 382 Roy Robert, Hebert Alain, THE GAN Generalized Driver, Instiute De genie nucleaire Ecole Polytechnique de Montreal, March 2000. Roy Robert, THE CLE-2000 TOOLBOX, , Instiute De genie nucleaire Ecole Polytechnique de Montreal, December 1999. Varin E., Hebert A., Roy R, Koclas J, A User Guide for DONJON Version 3.01,2005/08/15. Varin E., Hebert A., Data Structures of DONJON VERSION 3.00 rev C – 2003/11/28.

37

APPENDIX-A DETAIL ON OECD-PBMR 400 CROSS SECTION LIBRARY

The ranges chosen for each parameter were selected based on the reactor conditions for normal operation as well as for accident conditions. The following values for the 5 state parameters were selected: Fuel temperature (Doppler temperature): 300K, 800K, 1400K, 2400K Moderator temperature: 300K, 600K, 800K, 1100K, 1400K, 1800K, 2400K Xenon concentrations expressed as homogenised concentrations: 0.0 (or very small 1.0E-15), 2.0E-11, 8.0E-10 [#/barn.cm] Buckling terms: Separate values for fast and thermal buckling and also for fissionable ana non-fissionable material were defined; 3 values each FUEL REFLECTOR Fast Thermal Fast Thermal -1.0E-04 -2.5E-03 -6.5E-01 - 1.1E-03 1.0E-04 -1.0E-05 -1.0E-04 5.0E-05 4.0E-03 5.0E-03 1.0E-02 1.0E-02 * ******************************** * OECD-PBMR400 * ******************************** * *

* Each material set has: * * - ID number and description in quotes * - Number of parameters * - Number of data points for each parameter * *************************************************************************** * The first records of the data in each table is the points (or values) of * all the state parameters specified for the material. The order is * thus very important. The first four values are fuel temperatures, * the next seven values are moderator temperatures, the following * three are the fast buckling, then three thermal bucklings and the * last three are the Xenon number densities. * * * THE TABLES REFER TO,IN ORDER OF APPEARANCE: * * - Diffusion Coefficient Radial, fast group [cm] * - Diffusion Coefficient Axial, fast group [cm] * - Macroscopic Transport Corrected Total Cross-section, fast group [cm-1] * - Macroscopic Absorption Cross-section, fast group [cm-1] * - Macroscopic Nu-fission Cross-section, fast group [cm-1]

38

* - Macroscopic Fission Cross-section, fast group [cm-1] * - Macroscopic Scattering Cross-Section from fast(1) to thermal(2) group [cm-1] * - Inverse velocity, fast group [s.cm-1] * - Fraction Beta(1) of delayed neutrons, fast group [dimensionless] * - Fraction Beta(2) of delayed neutrons, fast group [dimensionless] * - Fraction Beta(3) of delayed neutrons, fast group [dimensionless] * - Fraction Beta(4) of delayed neutrons, fast group [dimensionless] * - Fraction Beta(5) of delayed neutrons, fast group [dimensionless] * - Fraction Beta(6) of delayed neutrons, fast group [dimensionless] * - Kappa, Energy release per fission, fast group [MeV] * - Microscopic Absorption Xenon Cross-section, fast group [cm2] * * * - Diffusion Coefficient Radial, thermal group [cm] * - Diffusion Coefficient Axial, thermal group [cm] * - Macroscopic Transport Corrected Total Cross-section, thermal group [cm-1] * - Macroscopic Absorption Cross-section, thermal group [cm-1] * - Macroscopic Nufission Cross-section, thermal group [cm-1] * - Macroscopic Fission Cross-section, thermal group [cm-1] * - Macroscopic Scattering Cross-Section from thermal(2) to fast(1) group [cm-1] * - Inverse velocity, thermal group [s.cm-1] * - Fraction Beta(1) of delayed neutrons, thermal group [dimensionless] * - Fraction Beta(2) of delayed neutrons, thermal group [dimensionless] * - Fraction Beta(3) of delayed neutrons, thermal group [dimensionless] * - Fraction Beta(4) of delayed neutrons, thermal group [dimensionless] * - Fraction Beta(5) of delayed neutrons, thermal group [dimensionless] * - Fraction Beta(6) of delayed neutrons, thermal group [dimensionless] * - Kappa, Energy release per fission, thermal group [MeV] * - Microscopic Absorption Xenon Cross-section, thermal group [cm2]

39