View - The Applied Modelling and Computation Group (AMCG)

Space-dependent kinetics simulation of a gas-cooled fluidized 

bed nuclear reactor 

C.C. Pain a , J.L.M.A. Gomes a , M.D. Eaton a , C.R.E. de Oliveira a, , 

A.P. Umpleby a , A.J.H. Goddard a ,H. van Dam b , T.H.J.J. van der Hagen b , 

D. Lathouwers b 

a Computational Physics and Geophysics, T.H. Huxley School of the Environment, Earth Sciences and Engineering, Imperial College of 

Science, Technology and Medicine, Prince Consort Road, London SW7 2BP, UK 

b Interfaculty Reactor Institute (IRI), Delft University of Technology, Mekelweg 15, NL 2629 JB Delft, The Netherlands 

Abstract 

Received 12 September 2001; received in revised form 16 April 2002; accepted 28 May 2002 

In this paper we present numerical simulations of a conceptual helium-cooled fluidized bed thermal nuclear reactor. 

The simulations are performed using the coupled neutronics/multi-phase computational fluid dynamics code finite 

element transient criticality which is capable of modelling all the relevant non-linear feedback mechanisms. The 

conceptual reactor consists of an axi-symmetric bed surrounded by graphite moderator inside which 0.1 cm diameter 

TRISO-coated nuclear fuel particles are fluidized. Detailed spatial/temporal neutron flux and temperature profiles have 

been obtained providing valuable insight into the power distribution and fluid dynamics of this complex system. The 

numerical simulations show that the unique mixing ability of the fluidized bed gives rise, as expected, to uniform 

temperature and particle distribution. This uniformity enhances the heat transfer and therefore the power produced by 

the reactor. 

# 2002 Elsevier Science B.V. All rights reserved. 

1. Introduction 

Nuclear reactor concepts based on gas fluidization 

of fine uranium fuel pellets have attracted 

considerable attention over the years. Reasons 

behind this interest lies in their excellent heat 

transfer capabilities (Molerus and Wirth, 1997) 

and the mixing ability of the fluidized bed. The 

Corresponding author. Tel.: /44-20-7594-9319; fax: /44- 

20-7594-9341 

E-mail address: c.oliveira@ic.ac.uk (C.R.E. de Oliveira). 

Nuclear Engineering and Design 219 (2002) 225 /245 

0029-5493/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. 

PII: S 0 0 2 9 - 5 4 9 3 ( 0 2 ) 0 0 2 1 5 - 7 

www.elsevier.com/locate/ned 

latter unifies the temperature of the bed, and 

increases the active surface area from which heat 

transfer occurs. In addition, the constant mixing of 

the bed potentially leads to a uniform burnup of 

the uranium particles. A self-controlling feature is 

also present in that as the bed is fluidized and the 

gas flow increases the power achieves a maximum 

at a particular bed height. At this height, the 

power will be that at which heat production is 

balanced by heat losses. 

A possible disadvantage of such a reactor is the 

chaotic particle flow characteristics of the fluidized

226 

C.C. Pain et al. / Nuclear Engineering and Design 219 (2002) 225 /245 

bed in which large bubbles and slugs propagate 

through it (Davidson et al., 1985), changing the 

geometry and nuclear criticality. This will impact 

on the fission rate which will also be highly 

variable*/although it is possible that the power 

output obtained from the heated gases may not be 

as variable. This variability and chaotic unpredictability 

requires thorough investigation in order 

that the concept can be assessed. 

Power variability in such a system has been 

studied by van Dam et al., 1998 who investigated 

the sensitivity of the reactor to voidage fluctuations. 

They concluded that, due to the slow 

neutron kinetics of the reactor (a consequence of 

the long neutron lifetime*/large scattering crosssections, 

Hetrick, 1993) the amplitude of the 

fission-power fluctuations would be small. The 

present paper aims to check this conclusion with 

fully coupled transient fluidized bed simulations. 

The stability of the reactor is provided by the 

instantaneous negative reactivity temperature 

feedback in the coated fuel particles. 

Fluidized bed nuclear reactor concepts adopt 

some aspects of the pebble bed reactor Pebble bed 

reactor (Gerwin and Scherer, 1987) and the fuel 

particles are of a prefabricated design (Gulden and 

Nickel, 1977). Optimization of this fuel particle is 

described in (Golovko et al., 1999). However, the 

concept investigated here is not the only fluidized 

concept; for example Sefidvash (1996) suggests 

fluidizing 0.2 cm diameter fuel particles with 

supercritical steam in a reactor designed to be 

non-fluctuating. 

The modelling approach we have developed 

applies detailed spatial/temporal modelling so 

that the reactor dynamics evolve naturally. This 

is in contrast to point kinetics models (Hetrick, 

1993) which, although often having adequate 

accuracy, require correlation with existing data 

when the material evolves within the transient, 

such as in fissile liquid transients, (Mather et al., 

1994; Mather, 1991; Mather and Barbry, 1991) 

and nuclear fluidized beds. 

Others have used space-dependent kinetics models 

mostly to model transients in fissile liquids, 

such as the multi-region model of Kimpland and 

Korneich (1996), the finite difference model of 

Yamamoto (1995) and the nodal model of Rifat et 

al. (1993). However, there are a limited number 

point kinetics models available for powders, (see 

for example Rozain, 1991; Basoglu et al., 1994). 

Golovko et al. (2000a) investigated the nuclear 

fluidized bed (similar to the one studied here) 

using point kinetics models linked to expressions 

for heat loss and bed expansion, looking at start 

up transients of the reactor see and various 

accident scenarios such as loss of heat sink (coolant 

gas is not cooled adequately) and change of 

gas inlet temperature (Golovko et al., 2000c). The 

model used in these studies is described in Golovko 

et al., (2000b). 

Without a doubt, the most satisfactory approach 

is an integrated neutrons/fluids/heat transfer 

method, such as that contained in the finite 

element transient criticality (FETCH) code (Pain 

et al., 2001b). The neutronics model solves the 

neutron Boltzmann transport equation in full 

phase-space using an second-order variational 

principle, (de Oliveira et al., 1998). The fluids 

algorithm is a multi-phase compressible/incompressible 

flow model which solves the conservation 

equations for both gas and solid particle phases. 

This unique fundamentally based combined methodology 

is potentially capable of modelling the 

complex non-linear reactivity feedback mechanisms 

which occur in nuclear reactor designs such as 

the one studied in this paper. 

Although no means of directly validating the 

overall FETCH model against experimental data 

is available, we have made every effort to validate 

the transient criticality (Pain et al., in press Pain et 

al., 2001b,d, 1998a) and the fluidized bed modelling 

(Pain et al., 2001a) individually, with careful 

comparison with experimental results for transient 

criticality in fissile solutions (Barbry, 1987; Ogawa 

et al., 1999). These studies have provided a strong 

foundation from which to investigate a fluidized 

bed nuclear reactor. 

We have chosen to use the two-fluid granular 

temperature method of modelling which has a gas 

and a solid fluid phase. Within the solid phase, 

particle modelling is based on an analogy between 

the kinetic theory of gases and binary particle / 

particle collisions (Savage, 1983; Shahinpoor and 

Ahmadi, 1983; Lun et al., 1984; Johnson and 

Jackson, 1987; Jenkins and Savage, 1983; Chap-

man and Cowling, 1970). These models are proving 

to be accurate for a wide range of gas /solid 

fluidization scenarios, (Cao and Ahmadi, 1995; 

Samuelsberg and Hjertager, 1996; Ding and Gidaspow, 

1990) 

The remainder of this paper is structured as 

follows: in the next section the FETCH coupled 

fluid dynamics/neutronics code is described. This 

is followed by a description of the reactor in 

Section 3 which also presents static modelling. 

Section 4 describes the transient modelling. Conclusions 

are drawn in the final section. 

2. The FETCH code 

The FETCH code is used here to simulate the 

dynamics of a nuclear fluidized bed. It is comprised 

of three modules: two transient 3D finite 

element modules*/the neutron transport code 

EVENT (de Oliveira, 1986) and the computational 

fluid dynamics (CFD)/multi-phase code FLUID- 

ITY (Mansoorzadeh et al., 1998), and an interface 

module which provides the coupling between 

neutronics and fluids. 

2.1. Neutronics 

C.C. Pain et al. / Nuclear Engineering and Design 219 (2002) 225 /245 227 

The Boltzmann neutron transport equation is 

solved using finite elements in space, spherical 

harmonics (PN) in angle, multi-group in energy 

and implicit two level time discretization methods. 

These methods have been applied using the 

second-order even-parity variational principle in 

the EVENT computer code. Its lowest mode of 

angular resolution is equivalent to diffusion theory. 

Further details of the numerical formulation 

implemented in EVENT can be found in de 

Oliveira et al. (1998). 

At each time-step the interface module organizes 

the feedback from FLUIDITY of spatial temperature, 

density and delayed neutron precursor distributions 

into the EVENT neutronics module and 

also, in the light of these fields, updates the spatial 

distribution of multi-group neutron cross-sections. 

For a given element of the finite element (FE) 

mesh, a cross-section set is obtained by interpolating 

in temperature and gas content a cross-section 

data-base. This database has been group-condensed 

taking into account resonance self shielding 

and thermal temperature effects, into six groups 

using the WIMS8A code (WIMS8A, 1999) and a 

representative geometry. The neutronics module 

generates for FLUIDITY spatial distributions of 

fission-power and delayed neutron generation 

rates. 

Material cross-sections are generated as follows 

using the lattice cell code WIMS8A. First the 

cross-sections were self-shielded using the equivalence 

theory method in WHEAD (part of WIMS) 

which relates the heterogeneous problem to an 

equivalent homogeneous model. A subgroup resonance 

calculation was then performed using the 

WPROC (part of WIMS) collision probability 

routine which calculates collision probabilities 

using a synthetic approximation for a system of 

spherical grains packed in annular geometry. 

Group cross-sections were then obtained for 

temperatures ranging from 550 to 2000 K by 

condensing to six groups the standard WIMS 69 

group library. 

2.2. Multi-phase fluids modelling 

Conservation equations for the particles and the 

helium gas are expressed in Eulerian form using a 

two phase continuum description. The momentum 

equations are discretized with an implicit nonlinear 

Petrov /Galerkin method, (Hughes and 

Mallet, 1986), and the other conservation equations 

are solved using an implicit high resolution 

method which is globally second-order accurate in 

space and time, (Leonard, 1991). The second-order 

fluxes for the high resolution method are obtained 

from a finite element interpolation of the solution 

variables. These methods are embodied in the 

CFD code FLUIDITY, (Pain et al., 2001c). The 

delayed neutrons are solved for and transported in 

FLUIDITY and are passed to EVENT through 

the interface code. 

The governing equations which include delayed 

neutron precursor concentrations are listed in 

Table 1 and Table 2 and interfacial momentum 

and energy exchanges between phases are listed in 

Table 3. The convective and conductive heat 

transfer correlations used here are based on the

228 

Table 1 

Conservation equations used in the simulations 

Continuity equation @ 

@t (okrk ) 

@ 

(okrk vki ) 

@xi 0 

Momentum equation @ 

@t (okrkn kt) 

@ 

(okrkv kivkj) @xj @rg ok @xi okrkg i b(vki vki) @ 

(tkij) @xi Gk Thermal energy 

equations 

DTg Cpg rgo g 

Dt 

pg @ 

ogvgi @xi @ 

osvsi @xi @ 

@xi @Tg ogkg @xi a(Ts Tg) DTs Gwg; cps rsos Dt 

Granular energy 

equation 

3 

2 

@(oxr xU) @t 

@ 

(osrs vsjU) @xj @vsi tsij @xj @qj @xj g 3bU 

Equation for d th 

delayed neutron 

group precursor 

oncentration 

@Cd (r; t) 

@t 

@vsj Cd (r; t) 

@xj ldCd (r; t) bdg0 v X 

f(r; E; t)dE 

f 

Neutron transport 

equation 

1 @c(r; V; E; t) 

v @t 

V×9c(r; V; E; t) Hc(r; V; E; t) S(r; V; E; t) 

work of Schmidt and Renz (1999), Molerus et al. 

(1995a,b), Natarajan and Hunt (1998), Hunt 

(1997), Hsiau (2000). Delayed neutron precursors 

are assumed to exist only in the solid phase and are 

in six delayed neutron precursor group form 

Duderstadt and Hamilton (1976). Thermal radiation 

heat transfer is neglected in the present study 

since its role would be to unify the temperature 

distribution in the reactor (Molerus et al., 1995a) 

and the calculated temperature distributions, see 

Section 4, havebeen found to be fairly homogeneous 

even without it. However, thermal radiative 

heat transfer may play a significant role in the 

course of the transient in accident scenarios in 

which large temperature differences could occur 

across the reactor. 

The fluids equations are solved only in the fluids 

occupied domain, shown in Fig. 1, which extends 

to a height of 500 cm. Because the particulate fuel 

does not expand into the remainder of the 600 cm 

gas/fuel filled cavity, it is excluded from the fluids 

calculation domain. The boundary conditions at 

the inlet are, for the gas, a superficial velocity 

1 

normal to the inlet boundary of 120 cm s */ 

Umf /25.0 cm s 


1 at 6 MPa pressure and 

230 8C according to the Ergun equation Table 2. 

Umf is the minimum fluidization gas velocity of the 

fuel particles. The gas was assumed to enter at 

230 8C and at a density dictated by a 6 MPa 

pressure. No heat loss conditions are applied at the 

vertical graphite walls of the reactor. Zero stress 

conditions are applied to the gas at the outlet 

boundary and on the walls slip and no normal 

flow conditions were applied. At the outlet (top 

plane of fluid domain) gas can enter depending on 

the evolving gas dynamics near the outlet, it is 

assumed that this gas is at 230 8C and 6 MPa 

pressure. To ensure the top boundary does not 

provide a external heat source the temperature and 

pressure of any incomming gas at the top plane 

boundary must be set to equal to the inlet 

conditions. For the solid phase a specified shear 

stress condition was applied as described in Pain et 

al. (2001c) and no normal flow conditions were 

enforced. The granular temperature boundary 

conditions are described in Pain et al. (2001c) 

and we have assumed particle /particle, wall / 

particle and friction coefficients of 0.97, 0.9 and 

0.1, respectively. 

All simulations are impulsively so that after the 

first time-step the gas inlet velocity is at 120.0 

cm s 

@ 

@x i 

@Ts osks @xi a(T g T s) G wg S f 

1 . This is a stern test of the robustness of the 

nuclear fluidized bed because this initialization 

results in rapidly expansion of the bed and a 

corresponding rapid change in the nuclear criticality 

of the bed. That is the ramp reactivity

Table 2 

The two-fluid granular temperature constitutive equations used in the simulations 

Gas phase Newotnian viscous stress 

tgij 2ogmg 1 

2 

@vgi @xj @xi 1 @vgk 3 @xk Solid phase stress tensor 

tsij ps @vsk oszs @xk dij 2osms 1 

2 

Solids pressure ps osrs[1 2(1 e)osg0]U Solids shear viscosity 

zs 4 

3 osrsd sg0(1 e) U 

p 

0:5 

Radial distribution function 

1 

os 1 

3 

1 

Collisional energy dissipation 

Flux of fluctation energy 

insertion is potentially very large. In practice the 

reactor would be initiated with a gradual increase 

in fluidization velocity up to a maximum and one 

would therefore expect a smaller ramp reactivity 

insertion and so a smaller in magnitude initial 

fission response and therefore lower temperature. 

The finite element discretization and solution of 

the multi-phase flow equations are described in 

Pain et al. (2001c). In summary, this involves the 

use of a mixed finite element formulation with 

rectangular elements. Both velocity components 

are centered on the four nodes of the rectangle and 

thus result in a bi-linear variation of velocity 

through out each element. Pressure, temperatures 

of both gas and solid phases, volume fractions, 

densities and delayed neutron precursor concentrations 

all have piece-wise constant variations, 

with a constant variation though out each element. 

An adaptive time-stepping method is used here 

and allows transient behavior of all fields to be 

resolved, (Pain et al., 2001d). 

3. The reactor 


The reactor is drawn to scale in 3D in Fig. 1a 

with part of the reactor removed to reveal the 

g 0 

o i 

g 3(1 e 2 )o 2 r sg 0U 4 

q j 

2r so 2 

s g 0d s 

U 

p 

@v gj 

U 

p 

ds 0:5 

@U 

@xj 0:5 

@v s 

@x k 

@v si 

@x j 

@v sj 

@x i 

internal cavity. A schematic of the reactor is 

shown in Fig. 1b. 

The particles are formed in layers as detailed in 

Table 4. They have a 300:1 moderator (in the form 

of carbon compounds) to uranium oxide fuel ratio. 

However, they are still under-moderated and thus 

the fuel responds with positive reactivity feedback 

to the additional moderation provided by the 

surrounding graphite walls. The reactors leakage 

and moderating properties must be such that as 

the bed height increases (due to fluidization) from 

maximum packing (under moderated) the criticality 

increases to a maximum and decreases on 

further expansion of the bed (over moderated). 

This provides a method of controlling the fission 

rate (power) with fluidizing flow rate and provides 

a safety mechanism for decreasing the criticality of 

the system in a rapid transient with rapid expansion 

of the gas along with bed height, (Golovko et 

al., 1999). In this demonstration we have chosen to 

impulsively start the gas flow to a fixed rate. This 

is likely to impulsively start the fission rate with a 

form that will depend in part on the level of the 

‘fixed’ neutron source. The power level might then 

expect to decrease as the bed temperature rises. No 

account is taken of fission product poisons influencing 

reactivity in this study. 

1 

3 

@v si 

@x k

230 

Table 3 

Correlations used in the simulations conducted 

Gas /solid friction coefficient 

Drag coefficient 

Convective heat transfer 

coefficient 

Conductive heat transfer 

coefficient 

The particles are chosen to be 0.1 cm in diameter 

which is small enough that the neutron flux 

distribution across each particle is fairly uniform 

resulting in uniform burning of the fuel, (Shmakov 

and Lyutov, 2000), and also allowing one to 

assume that the particles form a continuum for 

spatial homogenization and group collapsing purposes. 

This would be invalid for pebble bed 

reactors (Gerwin and Scherer, 1987) as the pebbles 

are typically of the order of 5.0 cm in diameter. 

The larger the particles in the bed, the larger the 

Table 4 

Material composition of TRISO fuel particle 

Material Density 

(g cm 3 ) 

UO2 kernel 10.88 0.26 

Porus carbon buffer 

layer 

1.1 0.77 

PyC coating 1.9 0.85 

SiC coating 3.2 0.92 

PyC coating 1.9 1.00 

b 

C D 

o 

150 

2 s ms (1 os )d2 7 osrgjv g vsj s 4 ds 3 

4 C (1 os )osrg jvg vsj D 

(1 os ) 

ds 2:65 

o 

20(0:225 os) 150 

2 s mg (1 os )d2 8 

>< 

7 osrgjv g v 

>: 

sij 

15(os s 4 ds 24 

Rep (1 os ) f1 0:15[(1 os )Rep ]0:687 8 

>< 

g if RepB1000 >: 0:44 if RepE1000 a 6o g 

Outer diameter 

(mm) 

The uranium is enriched to 16.76 wt.% with an overall 

particle density of 1.92 g cm 3 . 

o 

0:175)C 

srgjv g 

D 

hgs hgs dg kg [(7 10og 5o 

ds 2 

g )(I 0:7Re1=5 p Pr1=3 ) (1:33 2:4og 1:2o 2 

g )Re7=10 P Pr1=3 qffiffiffiffiffiffiffiffiffiffiffi 

] 

ogkg (1 1 og)kgas o s k s 

g? 0 


ffiffiffiffiffiffiffiffiffi 

p 

osrs Cxds 3 p 

U 

16 7o s 

16(1 o s ) 2 

32g? 0 

flow rate required to fluidized them which can 

enhance heat transfer with the particles. However, 

larger particles have a smaller heat transfer rate 

because of the relatively small surface area per unit 

mass compared to small particles. Thus, some 

compromise is required. In addition, the particles 

must not be so large that the bed dynamics are 

those of very large slugs which will make the 

fission rate difficult to control. In fact the chosen 

particle are D particles in the Geldart classification 

and thus prone to producing large voids/slugs 

when fluidized with gas, (Geldart, 1986). 

3.1. Static modelling 

d s 

vsj (1 os) 1:65 

if o s 

0:225 

if o s 50:175 

if 0:175Bo s50:225 

The first step in obtaining an understanding of 

the reactor is to perform a series of Keff eigenvalue 

(criticality) calculations. The critical eigenvalue 

Keff provides an indication of the initial quantity 

of fuel required in the reactor and the feedback 

mechanism resulting from changes in temperature 

and fuel geometry. The mass of fuel particles used 

here is 1.619 /10 6 g which corresponds to a 

collapsed core height of 136.0 cm with a maximum 

packing factor of 0.62.


Fig. 1. The fluidized bed nuclear reactor 3D domain and schematic: (a) 3D domain showing internal cavity; (b) 2D schematic of 

FLUBER reactor. 

Using the assumption that, as the fluidized bed 

expands, the fuel particles distribution remains 

uniform, Keff versus fluidized bed height is calculated 

and plotted in Fig. 2a. This graph confirms 

that changing flow rates, which change the bed 

expansion, may provide a means of controlling the 

power output of the reactor in addition to the 

inherent stabilization. The maximum temperature 

achievable would be that associated with the 

height at which Keff is at a maximum and the 

maximum power output would be at a height 

larger than this*/due to the power output being a 

function of the quantity of gas heated, that is the 

fluidization flow rate and therefore the height of 

the bed. 

The effect of changing temperature of the 

particles for a bed of height 172 cm is shown in 

Fig. 2b. The graph shows the strong negative 

reactivity feedback effect with increasing temperature 

which provides the main passive control of 

criticality. The temperature reactivity coefficient, 

which equals the gradient of the graph Fig. 2b at 

230 8C is /4.7 /10 5 K 1 . For neutronics 

purposes the temperature of the graphite moderator 

surrounding the inner core, is assumed to be 

230 8C in all static and transient calculations.

232 


Fig. 2. Critical eigenvalue results with a fuel particle mass of 1.619 /10 6 g the height versus Keff (a) is for this constant mass (b) show 

the strong negative temperature coefficient: (a) K eff versus expanded core height; (b) K eff versus temperature. 

As well as providing information on reactivity 

feedback effects, the static calculations also provide 

an indication of the power distributions in the 

reactor. Since most of the fissions occur in the 

thermal groups, the scalar flux distribution of 

thermal group 6, for the eigenvalue calculation, 

shows that much of the fission energy is deposited 

next to the graphite walls from which thermalized 

Fig. 3. Thermal (a) and fast (b) neutron scalar flux contours for a fuel particle mass of 1.619 /10 6 g and a collapsed bed height of 

136.0 cm. (c) finite element mesh used in both transient and eigenvalue calculations. The whole computational domain consists of 2000 

quadrilateral elements and 2121 nodes, the fluids domain contains 750 quadrilateral elements and 656 nodes. The central axis of the 

axi-symmetric model is on the left hand side of the diagrams.


Fig. 4. Fission rate and cumulative fissions for the three simulations with differing fissile mass content in the reactor: (a) low power; (b) 

intermediate power; (c) high power. 

neutrons emanate, see Fig. 3a. As one might 

expect this flux distribution is very similar to the 

particle importance map in the central cavity, 

shown in van der Hagen et al. (1997), which 

estimates the importance to criticality of a particle 

at a given position in the reactor. The fast group, 

group 1, flux distribution is shown in Fig. 3b. 

4. Transient modelling 

The aim of this section is to report the reactor 

dynamics when the power is allowed to evolve. To 

this end, we present three transient simulation 

results with differing particle mass (fuel mass) 

content in the reactor of 1.809 /10 6 , 1.735 /10 6 

Fig. 5. Maximum temperature and temperature at three sensors at the bottom of the reactor. High power simulation: (a) maximum 

and central temperature; (b) temperature at the two sensors.

234 


Fig. 6. Selected fields at 6 s (just after fission spike) into the simulation with high power. The central axis of the axi-symmetric model is 

on the left hand side of the diagrams: (a) solids fraction; (b) gas temperature (8C); (c) third longest-lived delayed concentration (cm 3 ); 

(d) shortest-lived delayed concentration (cm 3 ). 

and 1.661 /10 6 g which corresponds to a collapsed 

core height, assuming a maximum packing 

factor of 0.62, 152.0, 145.6 and 139.2 cm, respectively. 

These three simulations will be referred to 

Fig. 7. Maximum pressure deviation from 6 MPa overpressure and velocity of both phases for the high power simulation: (a) 

maximum pressure deviation; (b) maximum velocity.


Fig. 8. Maximum temperature and temperature at three sensors at the bottom of the reactor. Intermediate power simulation: (a) 

maximum and central temperature; (b) temperature at the two sensors. 

as high power, intermediate power and low power. 

All three transients are initiated with zero neutron 

fluxes and have a fixed source of 0.3 neutrons 

cm 3 1 

s in each of the six neutron energy groups 

and in the lower 172.0 cm of the inner cavity. Each 

simulation took approximately 2 weeks on a 500 

MHz Compaq AXP1000 workstation in single 

precision. 

Fig. 9. Maximum temperature and temperature at three sensors at the bottom of the reactor. Low power simulation: (a) maximum and 

central temperature; (b) temperature at the two sensors.

236 

4.1. Fission-power and temperature 


Fig. 10. Particle volume fraction at the three sensors versus time into the simulation for (a) low power; (b) intermediate power; and (c) 

high power reactor fuel loading. 

Fig. 4a /c show the fission-power in fissions per 

second (3.2 /10 11 J /1 fission) together with the 

cumulative fissions for the three cases. For the 

high power case, there is a large fission peak which 

occurs, 4 s after, initiation of gas flow. The large 

magnitude of the fission peak heats the fuel 

particles along with fluidizing gases to a maximum 

temperature of 1200 8C, see Fig. 5a. This results 

Fig. 11. A comparison of fission rate versus time for the high fuel loading case with a large and a small neutron source. The 

corresponding maximum gas temperatures for the two simulations is also shown: (a) comparison of fission rates; (b) comparison of 

maximum temperatures.


in strong negative temperature reactivity feedback 

(via spectral shift and Doppler broadening, (Duderstadt 

and Hamilton, 1976) which reduces the 

fission rate and thus the temperature gradually 

decreases, see Fig. 5a, as the cooling gas extracts 

heat from the particles. This rapid and large 

deposition of heat energy expands the cooling 

gas and results in a rapid expansion of the bed, see 

Fig. 6a, which also has a negative feedback effect, 

see Fig. 2a. Fig. 6 show the (a) solids fraction, (b) 

temperature, (c) third longest-lived delayed neutron 

concentration and (d) shortest-lived delayed 

neutron concentration at 6 s into the transient, at 

the bed’s most expanded state. The maximum 

velocity of the particles and gas appears to increase 

after the fission spike, see Fig. 7b, along with the 

maximum pressure deviation from the 6 MPa over 

pressure, Fig. 7a. The small pressure deviations 

suggest that gas density differences are mostly 

attributed to temperature changes. 

The intermediate power simulation shows a 

much smaller fission peak and maximum temperature, 

see Fig. 8a. This temperature quickly decreases 

as the reactor approaches a quasi steadystate. 

The frequency spectrum of the fission rate, 

for this intermediate power case, shows a dominant 

frequency of 1 Hz. Although, as with all the 

simulations, the fission rate (power) oscillates 

vigorously and by about an order of magnitude, 

the temperature of the bed varies smoothly, see 

maximum temperature versus time graphs Figs. 5 

and 8 and Fig. 9, which is a gauge of the steadiness 

of the energy output of the reactor. This suggests 

that the power extracted from the gas would be 

steady also. To generate the temperature versus 

time graphs, Figs. 5 and 8 and Fig. 9, and the 

particle volume fraction versus time graphs, Fig. 

10, three sensors were placed in the bottom of the 

reactor cavity. These are labelled; bottom center 

which refers to the sensor situated along the 

Fig. 12. Selected fields at 40 s into the simulation with low power. The central axis of the axi-symmetric model is on the left hand side 

of the diagrams: (a) solids fraction; (b) gas temperature (8C); (c) third longest-lived delayed concentration (cm 3 ).

238 


Fig. 13. Selected fields at 70 s into the simulation with intermediate power. The central axis of the axi-symmetric model is on the left 

hand side of the diagrams: (a) solids fraction; (b) Gas temperature (8C); (c) third longest-lived delayed concentration (cm 3 ). 

central axis, bottom corner which refers to the 

sensor in the corner of the domain where the 

vertical walls meet the floor of the cavity, and 

bottom mid-center sensor which is positioned half 

way between the bottom center and bottom corner 

sensors. 

The lowest power simulation produces enough 

fission energy to heat up the reactor at about 20 s 

into the simulation. This is due to the smallness of 

criticality and suggests (as would usually be good 

practice) that a larger neutron source is required at 

reactor start up. The temperature has no large 

peak, see Fig. 4, and seems to quickly reach a quasi 

steady-state*/in a time averaged sense. It is 

recognized that, by analogy with a continuous 

filling fissile solution criticality the initial form of 

the power rise will depend strongly on the fixed 

source density, (Pain et al., 1998b). We investigate 

its effect here on the coarse of the high power 

transient by repeating the high powered case with 

the source strength increased to 3 /10 3 neutrons 

cm 3 1 

s . The resulting fission rate and maximum 

temperature versus time graphs are compared 

in Fig. 11a and b, respectively. 

Notice that the fission peak is much smaller for 

the case with the larger source. This is because the 

reactor undergoes a ramp reactivity insertion due 

to the movement of the particles in the bed. The 

larger the source the quicker the neutron population 

builds up, during this ramp, to which 

increases the bed temperature. The negative temperature 

reactivity feedback effects then stabilize 

the temperature. Thus, the excess reactivity at the 

point at which the initial fission spike occurs 

governs the initial power of the system. As one 

would expect the temperature of the bed for the 

simulation with the source is much smaller, see 

Fig. 11b. 

The unsteadiness of the reactor is seen in the 

particle volume fractions at three sensors placed at


Fig. 14. Selected fields at 40 s into the simulation with high power. The central axis of the axi-symmetric model is on the left hand side 

of the diagrams: (a) solids fraction; (b) gas temperature (8C); (c) third longest-lived delayed concentration (cm 3 ). 

Fig. 15. Relationship between outlet temperature and power output from the three simulations: (a) power versus outlet temperature; 

(b) reactor power output

240 

the bottom of the reactor, see Fig. 10. The 

temperature for all three simulations is plotted at 

the sensors at the bottom of the bed, in Figs. 5 and 

8 and Fig. 9. The quickest variability in temperature 

is observed at the bottom of the bed, due to 

the cooling influence of the incoming helium gas. 

The initial temperature rise is fairly rapid for all 

three cases and since much of the heat is deposited 

in the bottom outer edge of the reactor (as 

indicated by the thermal flux distribution, Fig. 

3a) this is the point at which the temperature is 

largest as the temperature initially rises, Fig. 5b, 

Fig. 8b and Fig. 9b. However, on larger time scales 

advection takes place and thus this is no longer the 

case*/as seen in these figures. The solid phase 

temperature (not shown here) is very similar to the 

gas phase temperature, indicating rapid gas /solid 

heat transfer rates. 

Increasing the power output of the reactor 

increases the height to which the particles fluidize, 

due to the expanding fluidizing gases with temperature, 

compare Fig. 12a, Fig. 13a and Fig. 14a. 

4.2. Gas power output 

The power (fission rate), although highly variable 

by an order of magnitude, deposits much of 

its energy into the particles as they move into the 

vicinity of the bottom outer edge of the reactor. In 

this way the maximum temperature for all cases is 

fairly steady and is another reason why the 

temperature distribution is fairly uniform. A 

consequence of this uniformity is that the heat 

transfer coefficient for the reactor as a whole, 

f Gout 


r g C g DT g u z dG; which is a measure of the power 

output of the reactor, is fairly steady, see Fig. 15b, 

for all three reactors. In which Gout is the top 

outlet boundary of the fluids domain, DTg is the 

deviation of the gas outlet temperature from the 

inlet temperature and uz is the normal velocity 

component to the outlet boundary. Occasionally, 

the heat flux from the gas as shown in Fig. 15b 

oscillates because relatively cool gas (at 230 8C) is 

dragged into the domain along G out, increasing the 

hot gas flow rate out of the system and thus 

resulting in a peak in heat flux output. This peak is 

followed by a dip in the heat flux as these cool 

gases are expelled. Thus this oscillation is due to 

the restricted domain size and boundary conditions. 

The heat transfer rate out of a reactor system is 

perhaps more accurately estimated from the maximum 

temperature versus time graphs, shown in 

Fig. 5a, Fig. 8a and Fig. 9a. Using these temperatures 

combined with the graph of the power output 

of the reactor versus its temperature (Fig. 15), 

gives the steady heat flux out of the system. In a 

time averaged sense this will equal the heat flux 

out of the system given by Fig. 15b. At the end of 

the three simulations the power output is 23.0 

MWt (34.5 KW kgU 1 ), 11.0 MWt (17.3 

KW kgU 1 ) and 6.0 MWt (9.8 KW kgU 1 ) for 

the high power, intermediate power and low power 

simulations, respectively. Typical power outputs of 

commercial reactors are: 3600.0 MWt (37.9 

KW kgU 1 ) for pressurized water reactors; 

3579.0 MWt (25.9 KW kgU 1 ) for boiling water 

reactors and 3000.0 MWt (77.0 KW kgU 1 ) for 

high-temperature gas reactors (Duderstadt and 

Hamilton, 1976). Due to the large variablility in 

the designs of these reactors these power outputs 

are meant only as a rough guide. 

4.3. Fission heat source 

The shortest-lived neutron precursor concentration 

distributions at a quasi steady-state, for all 

three simulations Fig. 12d, Fig. 13d and Fig. 14d 

provide an indication of the instantaneous power 

distribution which is at a maximum near the 

bottom outer edge and vertical wall of the reactor, 

again as indicated by the thermal flux distribution 

in static criticality, see Fig. 3a. The delayed 

neutron precursor concentrations, with half lives 

of 0.18, 0.50, 2.2, 6.0, 22 and 55 s (Duderstadt and 

Hamilton, 1976), provide an indication of time 

averaged (averaged over time scale of half-life) 

heat source. Delayed neutron precursors are unstable 

fission products, which are advected with 

the particles and on decay result in a neutron 

emission. For modelling purposes it is convenient 

to lump the precursors into a small number of 

delayed groups each with a characteristic half-life. 

A small fraction, 0.7%, of fissions are delayed 

which provides a means of controlling the power 

variation of this and all other nuclear reactors.


Fig. 16. Selected time-averaged fields-averaged over 50 /70 s of the intermediate power simulation. The central axis of the axisymmetric 

model is on the left hand side of the diagrams: (a) solids fraction; (b) vertical gas velocity (cm.s 

1 

); (c) vertical particle 

velocity (cm s 

1 

); (d) shortest-lived delayed concentration (cm 

3 

). 

Since the delayed precursor generation rate is 

approximately proportional to the fission rate 

(power), the delayed neutron concentration can 

be viewed as time averaged heat sources, over time 

scales associated with the decay rate. 

The third longest-lived delayed neutron concentration 

distributions (half-life of 6 s), Fig. 12c, Fig. 

13c and Fig. 14c provide an indication of the 

history of the particles and also evidence to suggest 

that in the three simulations all particles have been 

subject to approximately the same heat source 

from fissions over a time scale of 6 s. In the three 

simulations the second and longest-lived delayed 

precursor concentration distributions are also very 

similar to the particle concentrations. Thus the 

longest-lived delayed neutron concentrations will 

reflect the particle concentrations, as seen in Fig. 

12c, Fig. 13c and Fig. 14c when the particles are 

subject to the same heat source. This similarity 

between time averaged heat source and particle 

concentration can only come about from the 

movement of the particles around the bed and 

through areas of large heat source. The uniformity 

of the gas phase temperature distribution throughout 

the bed, see Fig. 12b, Fig. 13b and Fig. 14b, is 

a result of the uniformity of this heating (over a 6 s 

time scale) and also the rapid gas movement 

through the bed. 

4.4. Time averaged results 

Particles, in a time averaged sense, move down 

the center of the reactor and up the sides, see Fig. 

16c. These time averaged results were obtained 

from the intermediate power calculation and 

averaged over the final 20 s of the simulation. 

This particle recirculation provides the global 

mixing mechanism. However, this flow is contrary 

to that typically observed in fluidized beds, and so 

is the large accumulation of particles near the

242 

center of the reactor, see Fig. 16a. These are a 

result of imposing axi-symmetry on the flow. The 

time averaged shortest-lived delayed neutron precursor, 

Fig. 16d, reflects the time averaged power 

distribution of the reactor and shows that as well 

as being peaked near the walls, the power is also 

peaked along the central axis, due to the large 

density of particles near the center. The time 

average vertical gas velocity distribution is shown 

in Fig. 16b, which shows that a gas circulation is 

set up in the bed which provides an additional 

method of transporting particle heat and unifying 

the temperature of the bed. 

5. Conclusions 


In this work we have demonstrated how a 

nuclear fluidized bed reactor can be modelled 

using space-dependent kinetics. It was shown 

here how increasing the fuel content in the reactor 

increases the temperature and power output of the 

reactor and as a consequence expands the gas and 

results in an increase in freeboard height. The 

superb mixing abilities of the fluidized bed were 

demonstrated with the uniformity in the calculated 

temperature distributions. This uniformity enhances 

the heat transfer out of the bed and 

therefore the power output of the reactor. 

The modelled fission-power varied by about an 

order of magnitude however, the gas temperature 

was fairly steady after the initial transient. Thus 

the power output extracted from the simulated 

beds would be relatively steady. The fission-power 

fluctuations are large and further work is required 

to eliminate the possibility that they might lead to 

an uncontrolled criticality excursion. The simulations 

were conducted over a relatively short period 

of time, namely tens of seconds, and to use more 

reasonable time variation of coolant gas inflow 

rate. There is thus a need to conduct similar 

investigations over larger time scales*/minutes. 

In addition, it was observed that all particles were, 

remarkably, exposed to approximately the same 

heat source quantity, over a short time scale of 6 s. 

Although, the axis-symmetric model used in this 

investigation results in an unrealistic accumulation 

of particles along the central axis we believe the 

model does provide an insight into the complex 

dynamics of this reactor. Future work will involve 

using a 3D model as well as a range of other 

models, including this one, to further investigate 

the dynamics of the reactor. In the future these 

models should be useful in further optimizing the 

design of this and other reactors. 

Acknowledgements 

Mr Gomes is supported by CAPES/Brazil and 

by UERJ(PROCASE)/Brazil. 

Appendix A: Nomenclature 

v velociy, m s 

t time, s 

r position vector 

x coordinate 

g gravitational constant, m s 

2 

p pressure, Pa (N m 2 ) 

Cp specific heat capacity, J kg 1 K 1 

T temperature, K 

q flux of fluctuation energy, 

kg m 1 s 

3 

CD drag coefficient 

g0 

radial distribution function 

g 0? 

radial distribution function for effective 

conductivity 

e particle /particle restitution coefficient 

d diameter, m 

Re Reynolds number 

Pr Prandtl number 

h fluid-particle heat transfer coefficient, 

W m 2 K 1 

Cd(r, t) dth delayed group precursor concentration 

ld 

decay constant (b decay) of dth 

precursor group, s 

1 

bd fraction of all fission neutrons (both 

prompt and delayed) emitted per 

fission that appear from the dth 

precursor group 

f(r, E, t) neutron scalar flux, cm 2 s 

1 

eV 

1 

1

f(r, V, E, t) neutron angular flux, 

cm 2 s 

1 

eV 

1 

sr 

1 

E neutron energy (eV) 

Sf(r, t) fission heat source, cm 3 s 

1 

S(r, V, E, t) neutron source, 

cm 3 s 

1 

eV 

1 

sr 

1 

Sf(r, t) macroscopic fission cross-section, 

cm 1 

/H/ scattering-removal operator 

MWt megawatt thermal 

KW kgU 1 kilowatt per kilogram of Uranium 

minimum fludization velocity 

Umf 

Greek symbols 

o volume fraction 

r density, kg m 3 

b interphase drag constant, 

kg m 3 s 

t viscous stress tensor, N m 2 

G frictional force exerted on the wall 

by the phase, N m 4 s 

U granular temperature, m 2 s 

2 

g collisional energy dissipation, 

kg m 1 s 

m viscosity, N s m 2 

z bulk viscosity, N s m 2 

k thermal conductivity, W m 1 K 1 

V direction of neutron travel 

Subscripts 

k phase (g, gas; s, solid) 

i, j x, y-directions 

p particle 

w wall 

gas pure gas 

References 


3 

Barbry, F., 1987. Fissile solution criticality accidents-review of 

pressure wave measurements experiments in the SILENE 

reactor. Institut de Protection et de Surete Nucleaire, 

Technical note SRSC No. 87.96. 

Basoglu, B., Brewer, R.W., Haught, C.F., Hollenbach, Wilkenson, 

A.D., Dodds, H.L., Pasqua, P.F., 1994. Simulation 

of hypothetical criticality accidents involving homogeneous 

damped low-enriched UO2 powder systems. Nuclear Technology 

105, 14 /30. 

1 

Chapman, S., Cowling, T.G., 1970. The Mathematical Theory 

of Non-uniform Gases. Cambrige University Press, Cambridge, 

UK. 

Cao, J., Ahmadi, G., 1995. Gas-particle two-phase turbulent 

flow in a vertical duct. International Journal of Multiphase 

Flow 21, 1203 /1228. 

Davidson, J.F., Clift, R., Harrison, D., 1985. Fluidization. 

Academic Press, London. 

de Oliveira, C.R.E., 1986. An arbitrary geometry finite element 

method for multigroup neutron transport with anisotropic. 

Progress in Nuclear Energy 18, 227 /236. 

de Oliveira, C.R.E., Pain, C.C., Goddard, A.J.H., 1998. The 

finite element method for time-dependent radiation transport 

applications. Proceedings of the 1998 Radiation 

Protection and Shielding Topical Conference, Nashville, 

USA, 343. 

Ding, J., Gidaspow, D., 1990. A bubbling fluidization model 

using kinetic theory of granular flow. A.I.Ch.E. 36, 523 / 

538. 

Duderstadt, J.J., Hamilton, L.J., 1976. Nuclear Reactor 

Analysis. Wiley, New York. 

Geldart, D., 1986. Gas Fluidization Technology. Wiley, Chichester, 

UK. 

Gerwin, H., Scherer, W., 1987. Treatment of the upper cavity in 

a pebble-bed high temperature gas-cooled reactor by diffusion 

theory. Nuclear Science and Engineering 97, 9 /19. 

Golovko, V.V., Kloosterman, J.L., van Dam, H., van der 

Hagen, T.H.J.J., 1999. Fuel particle design for a fluidized 

bed reactor. Proceedings of Jahrestagung Kerntechnik ’99, 

Annual Meeting on Nuclear Technology ’99, Karlsruhe, 

Germany, 625 /628. 


Hagen, T.H.J.J., 2000a. Investigation of a hypothetical 

start-up transient of a fluidized bed nuclear reactor. 

Proceedings of Jahrestagung Kerntechnik 2000, Annual 

meeting on Nuclear Technology 2000, Bonn, Germany. 


Hagen, T.H.J.J., 2000b. Dynamic core stability analysis of a 

fluidized bed nuclear reactor, PHYSOR 2000, Pittsburg, 

Pennsylvania, USA. 


Hagen, T.H.J.J., 2000c. Analysis of transients in a fluidized 

bed nuclear reactor, PHYSOR 2000, Pittsburg, Pennsylvania, 

USA. 

Gulden, T.D., Nickel, H., 1977. Preface coated particle fuels. 

Nuclear Technology 35, 206 /213. 

Hetrick, D.L., 1993. Dynamics of nuclear reactors. 555 N, 

Kensington Avenue, La Grange Park, Illinois 60525 USA: 

American Nuclear Society. 

Hughes, T.J.R., Mallet, M., 1986. A new finite element 

formulation for computational fluid dynamics: IV. A 

discontinuity-capturing operator for multi-dimensional advection-diffusion 

systems. Computing Methods in Applied 

Mechanics and Engineering 58, 329 /336. 

Hunt, M.L., 1997. Discrete element simulations for granular 

material flows: effective thermal conductivity and self-

244 


diffusity. International Journal of Heat and Mass Transfer 

40, 3059 /3068. 

Hsiau, S.S., 2000. Effective thermal conductivities of a single 

species and a binary mixture of granular materials. International 

Journal of Multiphase Flow 26, 83 /97. 

Jenkins, J.T., Savage, S.B., 1983. A theory for the rapid flow of 

identical, smooth, nearly elastic spherical particles. Journal 

of Fluid Mechanics 130, 187 /202. 

Johnson, Jackson, P.C., 1987. Frictional-collisional constitutive 

relations for granular materials with application to plane 

shearing. Journal of Fluid Mechanics 176, 67 /93. 

Kimpland, R.H., Korneich, D.E., 1996. A two-dimensional 

multi-region computer model for predicting nuclear excursions 

in aqueous homogeneous assemblies. Nuclear Science 

and Engineering 122, 204 /211. 

Leonard, B.P., 1991. The ULTIMATE conservative difference 

scheme applied to unsteady one-dimensional advection. 

Computing Methods in Apllied Mechanics and Engineering 

88, 17 /74. 

Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N., 1984. 

Kinetic theories for granular flow: inelastic particles in 

couette flow and slightly inelastic particles in a general flow 

field. International Journal of Multiphase Flow 140, 223 / 

256. 

Mansoorzadeh, S., Pain, C.C., de Oliveira, C.R.E., Goddard, 

A.J.H., 1998. Finite element simulations of incompressible 

flow past a heated/cooled sphere. International Journal for 

Numerical Methods in Fluids 28, 903. 

Mather, D., Barbry, F., 1991. Examination of some fissile 

solution scenarios using CRITEX, Proceedings of the 

Fourth International Conference on Nuclear Criticality 

Safety, Oxford, UK. 

Mather, D., 1991. Validation of the CRITEX code. Proceedings 

of the Fourth International Conference on Nuclear Criticality 

Safety, Oxford, UK. 

Mather, D., Buckley, A., Prescott, A., 1994. CRITEX-a code to 

calculate the fission release arising from transient criticality. 

AEA Report CS/R1007/R. 

Molerus, O., Burschka, A., Dietz, S., 1995aa. Particle migration 

at solid surfaces and heat transfer in bubbling fluidized 

beds-I: particle migration measurement systems. Chemical 

Engineering Science 50, 871 /877. 

Molerus, O., Burschka, A., Dietz, S., 1995bb. Particle migration 

at solid surfaces and heat transfer in bubbling fluidized 

beds-II: prediction of heat transfer in bubbling fluidized 

beds. Chemical Engineering Science 50, 879 /885. 

Molerus, O., Wirth, K.E., 1997. Heat Transfer in Fluidized 

Beds. Chapman & Hall, London. 

Natarajan, V.V.R., Hunt, M.L., 1998. Kinetic theory analysis 

of heat transfer in granular flows. International Journal of 

Heat and Mass Transfer 41, 1929 /1944. 

Ogawa, K., Nakajima, K., Yanagisawa, H., Sono, H., Aizawa, 

E., Morita, T., Sugawara, S., Sakuraba, K., Ohno, A., 1999. 

Measurment of power profile during nuclear excursions 

initiated by various reactivity additions using tracy. Proceedings 

of the Sixth International Conference on Nuclear 

Criticality Safety, Versailles, France. 

Pain, C.C, Mansoorzadeh, S., de Oliveira, C.R.E., 2001aa. A 

study of bubbling and slugging fluidized beds using the twofluid 

granular temperature model. International Journal of 

Multiphase Flow 27, 527 /551. 

Pain, C.C., de Oliveira, C.R.E., Goddard, A.J.H., Umpleby, 

A.P., Criticality behaviour of dilute plutonium solutions. 

Nuclear Science and Technology, in press. 

Pain, C.C., de Oliveira, C.R.E., Goddard, A.J.H., Umpleby, 

A.P., 2001bb. Transient criticality in fissile solutions*/ 

compressibility effects. Nuclear Science and Engineering 

138, 78 /95. 

Pain, C.C., Mansoorzadeh, S., de Oliveira, C.R.E., Goddard, 

A.J.H., 2001cc. Numerical modelling of gas-solid fluidized 

beds using the two-fluid approach. International Journal of 

Numerical Methods in Fluids 36, 91 /124. 

Pain, C.C., de Oliveira, C.R.E., Goddard, A.J.H., 2001dd. 

Non-linear space-dependent kinetics for criticality assessment 

of fissile solutions. Progress in Nuclear Energy 39, 53 / 

114. 

Pain, C.C., Goddard, A.J.H., de Oliveira, C.R.E., 1998a. The 

finite element transient criticality code FETCH-verification 

and validation. Proceedings of the Second NUCEF International 

Symposium on Nuclear Fuel Cycle, Hitachinaka, 

Ibaraki, Japan, 139. 

Pain, C.C., de Oliveira, C.R.E., Goddard, A.J.H., 1998b. 

Modelling the criticality consequences of free surface 

motion in fissile liquids. Proceedings of the Second NUCEF 

International Symposium on Nuclear Fuel Cycle, Hitachinaka, 

Ibaraki, Japan. 

Rifat, M., Al-Chalabi, R.M., Turinsky, P.J., Faure, F.X., 

Sarsour, H.N., Engrand, P.R., 1993. NESTLE: a nodal 

kinetics code. Transactions of the American Nuclear Society 

68, 432 /433. 

Rozain, J., 1991. Criticality excursions in wetted powder. 

Proceedings of the Fourth International Conference on 

Nuclear Criticality Safety, Oxford, UK. 

Samuelsberg, A., Hjertager, B.H., 1996. An experimental and 

numerical study of flow patterns in a circulating fluidized 

bed reactor. International Journal of Multiphase Flow 22, 

575 /591. 

Savage, S.B., 1983. Granular Flows at High Shear Rates. 

Academic Press, London. 

Schmidt, A., Renz, U., 1999. Eulerian computation of heat 

transfer in fluidized beds. Chemical Engineering Science 54, 

5515 /5522. 

Sefidvash, F., 1996. Status of the small modular fluidized bed 

light water nuclear reactor. Nuclear Engineering Design 

167, 203 /214. 

Shahinpoor, M., Ahmadi, G., 1983. A kinetic theory for the 

rapid flow of rough identical spherical particles and the 

evolution of fluctuation. In: Shahinpoor, M. (Ed.), Advances 

in Mechanics and the Flow of Granular Materials, 

II. Trans. Tech. Pub, Andermannsdorf, Switzerland, pp. 

641 /667. 

Shmakov, V.M., Lyutov, V.D., 2000. Effective cross sections 

for calculations of criticality of dispersed media. PHYSOR 

2000, Pittsburg, Pennsylvania, USA.


van der Hagen, T.H.J.J., van Dam, H., Harteveld, W., 

Hoogenboom, J.E., Khotylev, V., Mudde, R.F., 1997. 

Studies on the inhomogeneous core density of a fluidized 

bed nuclear reactor. Proceedings of Global 97-International 

Conference on Future Nuclear Systems, Pacifico Yokohama, 

Yokohama, Japan, 1050 /1055. 

van Dam, H., van der Hagen, T.H.J.J., Hoogenboom, J.E., 

Khotylev, V.A., Mudde, R.F., 1998. Statics and dynamics 

of a fluidized bed fission reactor. Proceedings of the 

International Conference of Emerging Nuclear Energy 

Systems, ICENES ’98 Tel-Aviv, Israel, 609 /616. 

WIMS8A: user guide for version 8, 1999. AEA Technology 

Report ANSWERS/WIMS (99)9. 

Yamamoto, Y., 1995. Space-dependent kinetics analysis of a 

hypothetical array criticality accident involving units of 

aqueous uranyl fluoride. Proceedings of the Fifth International 

Conference on Nuclear Criticality Safety, Albuquerque, 

New Mexico, 10 /19.

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