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Int. J. Nuclear Energy Science and Technology, Vol. 11, No. 2, 2017
Study on the temperature distributions in fuel
assemblies of lead-cooled fast reactors
Gilberto Espinosa-Paredes
Área de Ingeniería en Recursos Energéticos,
Universidad Autónoma Metropolitana-Iztapalapa,
Av. San Rafael Atlixco 186, Col. Vicentina,
México DF 09340, México
and
Facultad de Ingeniería,
Universidad Nacional Autónoma de México,
Programa de Estancias Sabáticas del CONACyT,
Morelos, México
Email: [email protected]
Juan-Luis François*
Departamento de Sistemas Energéticos,
Facultad de Ingeniería,
Universidad Nacional Autónoma de México,
Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México
Email: [email protected]
*Corresponding author
Heriberto Sánchez-Mora
Área de Ingeniería en Recursos Energéticos,
Universidad Autónoma Metropolitana-Iztapalapa,
Av. San Rafael Atlixco 186, Col. Vicentina,
México DF 09340, México
and
Departamento de Sistemas Energéticos,
Facultad de Ingeniería,
Universidad Nacional Autónoma de México,
Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México
Email: [email protected]
Alejandría D. Pérez-Valseca
Área de Ingeniería en Recursos Energéticos,
Universidad Autónoma Metropolitana-Iztapalapa,
Av. San Rafael Atlixco 186, Col. Vicentina,
México DF 09340, México
Email: [email protected]
Copyright © 2017 Inderscience Enterprises Ltd.
183
184
G. Espinosa-Paredes et al.
Cecilia Martín-del-Campo
Departamento de Sistemas Energéticos,
Facultad de Ingeniería,
Universidad Nacional Autónoma de México,
Paseo Cuauhnáhuac 8532, 62550 Jiutepec, Morelos, México
Email: [email protected]
Abstract: The aim of this paper is to make a comparative study of two
concepts of Lead-Cooled Fast Reactor (LFR) fuel assemblies, from a point of
view of the thermofluids performance. The sub-channel analysis approach was
applied to determine the temperature distribution in the fuel, in the cladding
and in the lead-coolant. The mathematical model is fully transient and takes
into account the heat transfer in an annular fuel pellet design. The thermofluid
is modelled with a mass, energy and momentum balance with thermal
expansion effects. The neutronic processes are modelled with point kinetic
equations for power generation with feedback fuel temperature and expansion
effects. The numerical experiments consider steady-state and transient
behaviours. The numerical comparison shows that a hexagonal assembly is an
option to compact the size of the LFR core design. This option leads to higher
temperature in the fuel and the cladding than in the case of a rectangular
assembly design. Results show the LFR with square array is more sensitive to
power changes than the hexagonal array at the same nominal power and with
the same transient conditions.
Keywords: lead-cooled reactor; fast reactor; ELSY; annular fuel; fuel
assembly; sub-channel analysis.
Reference to this paper should be made as follows: Espinosa-Paredes, G.,
François, J-L., Sánchez-Mora, H., Pérez-Valseca, A.D. and Martín-del-Campo,
C. (2017) ‘Study on the temperature distributions in fuel assemblies of
lead-cooled fast reactors’, Int. J. Nuclear Energy Science and Technology,
Vol. 11, No. 2, pp.183–203.
Biographical notes: Gilberto Espinosa-Paredes received his bachelor in
Energy Engineering in 1984, his master in Mechanical Engineering in 1992
and his PhD in Chemical Engineering (two-phase flow and heat transfer in
nuclear reactors) in 1998. Currently he is Professor at the Division of Basic
Sciences and Engineering of the Metropolitan Autonomous University,
Campus Iztapalapa.
Juan-Luis François received his bachelor in Energy Engineering in 1979, his
master in Nuclear Engineering in 1984 and his PhD in Physical Sciences
(Reactor Physics) in 1987. Currently he is Professor at the College of
Engineering of the National Autonomous University of Mexico.
Heriberto Sánchez-Mora received his bachelor in Energy Engineering in 2015.
Currently he is doing his master studies in Energy Engineering at the National
Autonomous University of Mexico.
Alejandría D. Pérez-Valseca received her bachelor in Energy Engineering in
2010 and her Master of Science (Energy and Environment) in February of
2017. Currently she is a student of PhD in Sciences (Energy and Environment)
at the Metropolitan Autonomous University.
Study on the temperature distributions
185
Cecilia Martín-del-Campo is Professor at the Faculty of Engineering of the
National Autonomous University of Mexico since 1999. She received her
bachelor in Energy Engineering in 1979, her master in Nuclear Engineering in
1984 and her PhD in Physical Sciences (Reactor Physics) in 1987.
1
Introduction
The Generation IV reactor concepts must meet the following criteria: (1) economic
competitiveness, (2) inherent safety, (3) minimisation of waste, (4) non-proliferation
enhancement, and (5) social acceptance. The Lead-Cooled Fast Reactor (LFR) has been
proposed as a possible answer to the compelling need for safer and sustainable
nuclear energy production (Generation IV International Forum, 2002; Generation IV
International Forum, 2015). The LFR has been included in GIF-IV because of its fastneutron spectrum, which allows for a closed fuel cycle for efficient conversion of fertile
uranium and the management of actinides. The LFR design increased safety margins,
even under the most severe accidental conditions and has the propitious characteristics of
an inert coolant such as lead and bismuth.
Liquid metal cooled reactors are expected to play an important role in the future of
nuclear energy production because of their possible efficient use of uranium, and have
the ability to reduce the radioactive lifetime and volume of nuclear waste. Thermofluids
is recognised as a key scientific topic in the development of such reactors. Two important
challenges for the design of liquid metal fast reactors are the fuel assembly and the
analysis of thermofluids (Roelofs et al., 2013).
The high boiling point and relatively low melting point make lead attractive as a
coolant in fast reactor designs. Lead is chemically inert in water, which may be regarded
as an advantage. However, the main drawbacks are the corrosion and erosion issues on
the reactor internal components (Wolniewicz et al., 2015). The LFR has been the subject
of several studies, and numerous investigations have been carried out to assess their
effectiveness in the light of the requirements of the GIF-IV (Loewen and Tokuhiro, 2003;
Chandra and Roelofs, 2010; Colombo et al., 2010; Alemberti et al., 2011; Bandini et al.,
2011; Aufiero et al., 2013; Bortot et al., 2013; Wang et al., 2013; Chen et al., 2015; Guo
et al., 2015; Sartori et al., 2016; Yang et al., 2016).
LFRs offer great potential for plant simplifications and higher operating efficiencies
compared to other coolants. However, the safety characteristics and design-basis
represent challenges, which need to be analysed and studied, in depth, using numerical
analysis and computational tools. The thermofluids and point kinetics model for the
ELSY lead fast reactor, with an open square core option, was used to verify the core
temperatures at beginning- and end-of-cycle conditions for a nominal power of
1500 MWth (Bandini et al., 2011).
In this work we are comparing the thermofluids processes in the core, the fuel heat
transfer, and the thermal power, in two fuel assemblies in an LFR: rectangular and
hexagonal. In LFR core design, the prediction of the maximum fuel temperature and
coolant are crucial to security requirements. In previous work, an ELSY reactor model,
which considers thermofluids and reactor point kinetics, was developed with RELAP5
code and has been used to verify the core temperatures at a nominal power of
1500 MWth (Bandini et al., 2011).
186
G. Espinosa-Paredes et al.
In this study the sub-channel analysis approach was applied to determine the
temperature distribution in the fuel, the cladding and lead. The sub-channel analysis is
the most suitable approach in the design of fuel assemblies (Wang et al., 2013; Chen
et al., 2015). The sub-channel model developed in this work is fully transient. The
mathematical models govern the different physical phenomena occurring in a subchannel of the LFR. The fuel pin is annular with a distribution of temperatures in a onedimensional radial direction and is transient. The thermofluids in the sub-channel is onedimensional in the axial direction and is also transient, taking into account the thermal
expansion effects of lead. The neutronics power is modelled with point kinetics equations
for power generation with fuel temperature feedback, and the power distribution in the
axial direction is non-uniform. The coupling of the neutron processes, heat transfer in the
fuel, and thermofluids, is discussed.
2
Lead-cooled fast reactor (LFR) description
The ELSY reactor with a thermal power of 1500 MWth, which is a pool type reactor
cooled with pure lead, was applied in this study, in order to perform the comparison
between the two fuel assemblies: rectangular and hexagonal. The European Lead Fast
Reactor has been developed in the framework of the European lead system project
funded by the Sixth Framework Programme of EURATOM. The project points out the
possibility of designing a competitive and safe fast critical reactor using simple
engineered technical features, while fully complying with the Generation IV goals
(Alemberti et al., 2011). Table 1 shows the specifications of the ELSY reactor.
Table 1
Main design data of ELSY
Feature
Thermal power
Value
1500 MWth
Efficiency
40%
Coolant
Lead
Length
1.2 m
Coolant inlet temperature
673.15 K
Coolant outlet temperature
750.15 K
Speed of lead
2 m/s
In this work we analyse and compare the temperature distribution between the hexagonal
and square assemblies in the ELSY core reactor. Figure 1 shows a schematic illustration
of the core and fuel assembly for both arrangements, as well as each zone of the core,
control rods and reflectors.
The characteristics of the fuel assemblies studied in this work are presented in
Table 2. In this table the hexagonal array has more fuel assemblies than the square array;
however, the square assembly has more fuel rods per assembly. The difference between
the number of rod fuels of the square and the hexagonal arrays implies that the masses of
fuel and coolant are different for each configuration; the comparison is based on the same
thermal power for both reactors.
Study on the temperature distributions
Figure 1
187
Scheme of LFR: (a) core in hexagonal array, (b) core in square array, (c) hexagonal fuel
assembly, (d) square fuel assembly
Inner zone
Middle zone
Control rods
Outer zone
Reflectors
(a)
(b)
(c)
(d)
In each of these arrangements, the core distribution assemblies depend on the Pu
enrichment. In the case of the square array, enrichments are: 14.45% Pu in the inner
zone, 17.53% Pu in the middle zone and 20.50% Pu in the outer zone (Artioli et al.,
2009). While those for the hexagonal array are: 14.9% Pu in the inner zone%, 15.5% Pu
in the middle zone and 17.4% Pu in the outer zone (Sobolev at al., 2009).
Table 2
Features of the core in the hexagonal and square arrays
Feature
Hexagonal array
Square array
Total assemblies in the core
325
170
Rods in the assembly
165
428
27.31
20.62
Thermal power/rod (kW/rod)
188
3
G. Espinosa-Paredes et al.
Sub-channel analysis
The LFR uses two liquid metal coolant materials. The first is Pb which has a melting
temperature of 327.45°C, and an atmospheric boiling temperature of 1743°C. The second
is lead-bismuth eutectic (LBE) which is composed of 45.0 at% Pb and 55.0 at% Bi with a
melting temperature of 124.5°C and an atmospheric boiling temperature of 1670°C. The
Pb and LBE densities, at 480ºC, are 10,470 kg/m3 and 10,100 kg/m3, respectively. Pb has
a high boiling temperature that is well above the temperatures at which the steel cladding
and structures lose their strength and melt.
3.1 Neutronic power
The thermal power in the sub-channel is given by:
P  t , z   P0 n  t   z 
(1)
where P0 is the nominal thermal power per fuel rod, n(t) is the neutron density, and (z)
is the axial power distribution. According to Table 2 P0 is 27.31 kWth/sub-channel for
the hexagonal array and 20.62 kWth/sub-channel for the square array. The axial power
distribution (z) used in this work is shown in Figure 2. The axial distribution of power
in the hottest and coldest pins of the warmest assembly in the inner zone is similar
(Sobolev at al., 2009). The axial neutron flux profiles in LFR within the active height
were obtained with the SERPENT code and compared with the COMSOL code results
(Aufiero et al., 2013), which are very close to trend shown in Figure 2. The axial
distribution is the same for all the fuel rods and is estimated for each node in de rod.
Figure 2
Axial power distribution in the sub-channel (see online version for colours)
The neutron density is calculated with neutron point kinetics equations with six
precursors of delayed neutrons:
dn  t 
dt

 t   

6
n  t    i Ci  t 
i 1
(2)
Study on the temperature distributions
dCi  t 
dt
i


189
n  t   i Ci  t  , for 1, 2,3, , 6
 T fuel
(3)

   f  T fuel   c  Tclad   l  Tlead

 T fuel  
  
in 

Clad expansion
Lead expansion

 Fuel expansion
  0  1.1K D ln 
out
(4)
Doppler
where  is the reactivity,  is the total fraction of delayed neutrons,  is the mean neutron
generation time,  is the decay constant of delayed neutron precursor, Ci is the
concentration of the i-th delayed neutron precursor. The first and second terms on the
right side of equation (3) represents the rate of formation of the precursors and
radioactive decay of the i-th group, respectively.
The nuclear parameters used in this study are for the dominant fissile nuclide Pu239
and are shown in Table 3.
Nuclear parameters for Pu239 (Waltar et al., 2012)
Table 3
Group
i  s 1 
i
1
0.0129
0.0000817
2
0.0311
0.000602
3
0.134
0.0004644
4
0.331
0.0007052
5
1.26
0.00022145
6
3.21
0.00007525
6
The total fraction of delayed neutrons is given by     i  0.00215 , and the mean
i 1
neutron generation time is   6.116  107 s (Ponciroli et al., 2014).
Equation (4) computes the total reactivity that assumes various contributions: initial
reactivity margin (0), Doppler effect, fuel expansion, cladding expansion, and the effect
of lead density, whose parameters are: Doppler constant KD = –555 pcm, reactivity
coefficients f = –0.232 pcm/K, c = 0.045 pcm/K, and l = –0.247 pcm/K (Ponciroli
et al., 2014; Waltar et al., 2012). It is important to note that the c includes the effects of
the axial and radial expansion of the clad.
The changes in the average temperature in equation (4) are defined as:
 Tfuel  Tfuel  Tfuel ,  Tclad  Tclad  Tclad 0 , and  Tlead  Tlead  Tlead 0 , where
0
the subscript 0 represents the reference temperature. The average temperatures to
compute equation (4) are defined as follows:
T fuel
in
T fuel
out


1 zL
T fuel
L z  0
1 zL
T fuel
L z  0
r  ra
dz
r  rf
dz
(5)
(6)
190
G. Espinosa-Paredes et al.
T fuel 
Tclad 
Tlead 
1
V fuel
1
Vclad

T fuel dV
(7)

Tclad dV
(8)
V fuel
Vclad
1 zL
Tead dz
L z  0
(9)
where ra and rf are radiuses of the inner annular region and the fuel region of the pin,
respectively, as is illustrated in Figure 3. In these equations L represents the active length
of the core, and V is the volume.
The numerical solution of the neutronics power considers two methods. The first is
the Runge–Kutta 4th order method, and is applied for the numerical solution of neutrons
density, given by equation (2), and the second is the Euler method, and is applied for the
numerical solution of concentration of the i-th neutron delayed precursor, given by
equations (2 and 3). The step length used in this work was of 0.0001 s.
3.2 Fuel heat transfer
The fuel mathematical model calculates the heat transfer in annular fuel pellets of
hexagonal and square arrays, as is illustrated in Figure 3. The fuel pellets have
differences in dimensions for each fuel-assembly design, as presented in Table 4.
Figure 3
Annular fuel pellets (see online version for colours)
In Figure 3, there is a gap void between the fuel and the cladding. Although the thickness
of the gap is quite thin, the low thermal conductivity of gases causes a large temperature
drop across the gap. In the real case the gap spacing is not uniform and the heat
conduction transfer process is very complex, but in this work, in order to simplify the
Study on the temperature distributions
191
calculations, it is considered uniform. The cladding temperature is determined in terms of
coolant temperature, and, eventually, in terms of the coolant inlet conditions and the
reactor power level.
Table 4
Fuel pellets dimensions (Figure 3)
Hexagonal fuel assembly
(Sobolev et al., 2009)
Radius (mm)
Square fuel assembly
(Bandini et al., 2011)
ra
1
1
rf
4.5
4.44
rg
4.65
4.55
rcl
5.25
5.25
The fuel heat transfer formulation is based on the following fundamental assumptions:
(1) axis-symmetric radial heat transfer, (2) the heat conduction in the axial direction is
negligible with respect to the heat conduction in the radial direction, (3) the volumetric
heat rate generation in the fuel is uniform in the radial direction, (4) the gap spacing is
uniform, and (5) the annular region is in thermodynamic equilibrium with the temperature
of the inner surface of the fuel. Under these assumptions, the transient temperature
distribution in the annular fuel pin, initial and boundary conditions is given by:
T fuel
  Cp  fuel
  Cp  gap
T
Tgap
  Cp clad
T

k fuel   T fuel
r
r r  r

k gap   Tgap 
r
 , Gap rf  r  rg
r r  r 
Tclad kclad   Tclad

r
T
r r  r

  q   t , z  , Fuel ra  r  rf


 , Clad rg  r  rcl

(10)
(11)
(12)
The initial condition is given by T(r, 0) = f(r), and the boundary conditions are:
dT fuel
dr
k gap
 0, at r  ra
dTgap
dr
 hgap T fuel  Tgap  , at r  rf
(13)
(14)
kclad
dTclad
 hgap Tgap  Tclad  , at r  rg
dr
(15)
kclad
dTclad
 hlead Tclad  Tlead  , at r  rcl
dr
(16)
In these equations  is the density, Cp is the specific heat, k is the thermal conductivity,
hlead is the lead heat transfer coefficient, hgap is the gap conductance, and q   t , z  is the
heat source given by:
q   t , z  
P t, z 
Vf
(17)
192
G. Espinosa-Paredes et al.
where P(t, z) is the sub-channel power given by equation (1), and Vf is the fuel volume.
Table 4 presents the fuel pellet dimensions for each core array. The physical properties of
fuel consisting of 0.15 molar fraction of plutonium oxide, and 0.85 of U-238 as a
function of temperature are presented in Table 5. Table 6 shows the physical properties
of the gap (helium) and cladding (T91).
Table 5
Fuel properties (Carbajo et al., 2001)


Density  kg m3 ; Specific heat Cp  J kgK  ; Thermal conductivity k  W mK 
 fuel 
9.9672  10
1
11043.5
 1.179  105 T  2.429  10 9 T 2  1.219  1012 T 3 
 90.998  106 A
1.620  1012 
Cp fuel  0.85  
 1.6926  102 T 

2
T2
  A  1 T

111.275  106 B

0.15  
 2.9358  102 T 
2
  B  1 T


1
6400  e 16.35  
k fuel  1.158  

;
4 2
5 2
 0.1205  2.6455  10 T

A  e548.68 T ; B  e18541.7 T ;  T 1000
hgap  6000 W m 2  K (Glasstone and Sesonske, 1981).
The heat transfer coefficient of equation (16) is given by hlead  klead Nu Dh , where
klead is the thermal conductivity (given in Table 7), Gh is the equivalent diameter
(see Section 3.3), and the Nusselt number for each fuel-assembly arrangement, is given
by Todreas and Kazimi (1990):
4  0.025  Pr Re 0.8 , for square array
Nu  
0.8
7  0.025  Pr Re  , for hexagonal array
(18)
where Pr and Re are the Prandtl and Reynolds numbers, respectively.
Table 6
Gap and clad properties (Glasstone and Sesonske, 1981)
Property
Density   kg m
3

Specific heat Cp  J kgK 
Thermal conductivity k  W mK 
Gap
Cladding
2.425
7700.0
5191.0
622.0
15.8  10–4T0.7
26.0
The annular fuel pellet temperature distribution is obtained considering 19 radial nodes at
each of the 24 axial nodes in the core. Ten nodes were considered in the fuel, four nodes
in the gap, and five in the cladding. The differential equations described previously are
transformed into discrete equations using the control volume formulation technique in an
Study on the temperature distributions
193
implicit form. Application of the control volume formulation enables the equations for
each region (fuel, gap and clad) to be written as a single set of algebraic equations for the
sweep in the radial direction:
t
a j T jt  t  b j T jt1t  c j T jt
1  d j for j  1, 2,3,  , M
(19)
where aj, bj, cj and dj are coefficients that are computed at the time t. The procedure
solution applied in this work is the Thomas algorithm (Patankar, 1980).
3.3 Thermofluids analysis
The thermofluids are modelled with mass, energy and momentum balance that takes into
account thermal expansion effects.
 th lead
dTlead G

 0, Mass balance
z
dt
(20)
Tlead Pm hlead Tclad  Tlead 
G dTlead


, Energy balance
t
Af lead Cplead
lead dz
(21)
 fr  G 2    G 2 
G


 
  lead g , Momentum balance
t
2  lead L  z  lead 
(22)
In these equations lead  lead T  , th is the thermal expansion coefficient, G is the
mass flux, Pm is the wetted perimeter (given by d rod   4   4l p ; see Figure 4), and Af is
the flow area (cross-sectional area). In the momentum balance given by equation (16),
the friction coefficient is calculated with the following relation:
 fr 
0.32
 
0.210 L   l p

1
1



 
Re0.25 Dh   d rod
 

(23)
where the rod pitch for the hexagonal array is lp = 15.5 mm (Sobolev et al., 2009), and
for the square array is lp = 13.9 mm (Bandini et al., 2011), and drod is the rod diameter
(see Table 4) as is illustrated in Figure 4, as well as the flow area (Af) indicated by the red
line. The hydraulic diameter for each array is given by:
 4

  d rod
Dh  
 4
 d
 rod
2
 3 2  d rod
lp 

4
 2
2
 2  d rod

 lp 
,
4 


 , for hexagonal array

(24)
for square array
The physical properties of the lead used in this work are given in Table 7.
The numerical solution applied to balance equations of mass, energy and momentum
was the Euler method.
194
4
G. Espinosa-Paredes et al.
Multiphysics coupling
Numerical simulation of the nuclear reactors is a technological tool that allows for the
safety analysis and reliability of future reactor designs. Nuclear reactors are multiphysics
and multiscale systems that involve the nuclear process, fuel heat transfer, and
thermofluids analysis. The multiphysics coupling between different mathematical models
(presented in the previous section) allows the simulation of the behaviour of the reactor
core of an LFR. The coupling of the physical processes involves a complex dynamic
interaction of variables among nuclear processes, of fuel heat transfer, and thermofluids.
The simulation of nuclear processes with the neutron point kinetics approach is coupled
with fuel heat transfer, through average temperatures of the fuel and cladding, and at the
same time is coupled with thermofluids through average temperature of the lead. The fuel
heat transfer requires a nuclear heat source and the thermal properties of lead. And the
thermofluids in the core requires the cladding wall temperature.
Figure 4
Schematic sub-channel: (a) hexagonal fuel-assembly arrangement, (b) hexagonal
fuel-assembly arrangement (see online version for colours)
(a)
Table 7
(b)
Lead physical properties (Sobolev et al., 2008)
Property
Relation
Density  kg m3 
lead  11441  1.2795T
Heat capacity  J kgK 
Cplead  175.1  4.961  102 T  1.985  105 T 2  2.099  109 T 3
Thermal conductivity  W mK 
klead  9.2  0.011T
1.524  106 T 2
Thermal expansion coefficient (K–1)  th 
Viscosity (Pa s)
1
8942  T
 1069 
lead  4.55  104 exp 

 T 
Study on the temperature distributions
5
195
Results and discussion
The numerical analysis shows the comparison of hexagonal fuel assemblies versus square
fuel assemblies in steady-state analysis and transient behaviour.
5.1 Steady-state analysis
Figure 5 shows the temperature field (axial and radial) in the fuel rod at different powers:
100%, 75%, 50%, 25% of rated power, in which can be clearly seen the zones of high,
medium and low temperatures.
Table 8 shows the cladding and fuel pin peak temperatures at 25%, 50%, 75% and
100% of rated power that are representative results of Figure 5. According with this table
the cladding temperatures for both arrays are close with a maximum difference of 1.52 K.
However, the difference maxima of the peak temperatures between square and hexagonal
arrays are 304 K, which represents a crucial difference that must be considered in the
LFR core design.
Results of the temperature distributions in the axial direction are presented in
Figure 6, where the average definitions applied are given in equations (6)–(9). It is
important to note that there are significant differences in temperature in the fuel and in
the gap, but in the cladding they were very close in both arrangements. This is important
from the point of view of safety.
When comparing the steady-state results between arrays, the square array design is
attractive from the point of view of temperatures, but it would be difficult to make a
competitive compact design in comparison to the hexagonal array owing to the triangular
features of the hexagonal array design (see Table 2 and Figure 3).
Table 8
Temperature comparison in the fuel rod
Rated power *Clad temperature *Clad temperature
(%)
square (K)
hexagonal (K)
Peak temperature
square (K)
Peak temperature
hexagonal (K)
25
675.95
676.09
821.80
868.18
50
679.06
679.39
993.16
1102.31
75
682.35
682.94
1191.87
1385.28
100
685.83
686.75
1423.99
1727.51
Note:
*Obtained with equation (8).
5.2 Transient analysis
Transient responses of the two LFR fuel assemblies (square and hexagonal arrangements)
were investigated and compared. Three transients were simulated: (1) transient due to
velocity changes in the inlet lead-coolant, (2) transient due to temperature changes in the
inlet lead-coolant, and (3) transient due to reactivity changes. These transients were
performed instantaneously based on the steady-state condition. The first transient
considers effects in the reduction and in the increase of the mass flux in the LFR core due
to the main coolant pump action; the second transient considers effects due to possible
faults in the steam generator; and the third transient corresponds to the extraction and
insertion of a control rod. All the transients started at nominal power steady-state
conditions (Table 1).
196
Figure 5
G. Espinosa-Paredes et al.
Temperature field at 100%, 75%, 50%, and 25% of rated power
Study on the temperature distributions
Axial temperatures distribution
Figure 6
Fuel temperature
1.2
1
0.8
z [m]
z [m]
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
700
Cladding temperature
1.2
Hex 100%
Sq 100%
Hex 25%
Sq 25%
1
800
900
1000
1100
T [K]
1200
1300
1400
680
700
720
T [K]
740
760
780
Lead temperature
1.2
Hex 100%
Sq 100%
Hex 25%
Sq 25%
1
Hex 100%
Sq 100%
Hex 25%
Sq 25%
0
660
1500
Gap temperature
1.2
1
0.8
z [m]
0.8
z [m]
197
0.6
0.6
0.4
0.4
0.2
0.2
0
650
700
750
800
850
900
950
0
670
Hex 100%
Sq 100%
Hex 25%
Sq 25%
680
690
700
710
T [K]
720
730
740
750
760
T [K]
The numerical results of these transients are shown in Figure 7, where Sq and Hex were
used to indicate square and hexagonal, respectively. The steady-state condition
(at time initial, which was used as a reference) was indicated using a continuous black
line, and was the reference case at 100% of rated power. In this figure the behaviour of
normalised power, and total reactivity, for both fuel-assembly arrangements is presented.
1
Transient due to the velocity variation in the inlet lead-coolant
The speed of the inlet lead-coolant is the boundary condition to obtain the flux mass in
the LFR core, which is 2 m/s at steady-state condition (Figure 7 (a)). The velocity
variation transient consists of the reduction and increases around 2 m/s. This figure
shows that the hexagonal array has lower power changes (increase or decrease) relative
to the square array. This is because the Doppler feedback contribution to reactivity is
higher in the hexagonal array, mainly due to the greater fuel rods pitch of the hexagonal
versus the square lattice, and the slightly higher volume of fuel in the rods of the
hexagonal assembly. With respect to the transient due to temperature behaviour, a
reduction of lead speed (mass flux) from 2 m/s to 1 m/s is due to a partial loss of cooling,
and therefore higher temperatures in fuel and lead, where the power decreases by fuel
temperature rise. An increase of lead speed from 2 m/s to 3 m/s increases cooling
capacity, decreases the temperature and increases the power.
198
2
G. Espinosa-Paredes et al.
Transient due to the temperature changes in the inlet lead-coolant
This transient consists of changes in the inlet temperature of lead-coolant, maintaining
the speed inlet (mass flux) invariable. This transient compares both fuel-assembly
arrangements when the inlet temperature of lead-coolant suffers changes. The coolant
inlet temperature in steady-state condition is 673.15 K, which corresponds to nominal
power, and the transient applies 10 K at around this temperature in one time step. The
transient behaviour of this scenario is shown in Figure 7 (b).
A reduction of coolant temperature of 673.15 K to 663.15 K produces an insertion of
positive reactivity due to the Doppler effect (being the main positive contribution in the
reactivity followed by the lead temperature decrease), and therefore the power increases.
When the coolant temperature increases from 673.15 K to 683.15 K the behaviour of
power and reactivity is exactly opposite in regards to the reduction of coolant
temperature, i.e. the power decreases due to negative reactivity.
Regarding the comparison between the hexagonal and square arrangements, the
numerical results show that the hexagonal array presents lower power changes than the
square array, although these changes are marginal. This transient is important for the
LFR, because it is an inherent alternative power control linked to the coolant temperature
(lead) in the inlet of the core.
3
Transient due to reactivity changes
This transient corresponds to the extraction and the insertion of a control rod equivalent
to 20 pcm step reactivity (Figure 7 (c)). After the instantaneous step insertion of
positive reactivity (+20 pcm) the power suddenly increases, after that, there is a small
decrease to finally achieve the steady-state condition, as can be seen in Figure 7 (c). After
the insertion of negative reactivity (–20 pcm) the behaviour of the power and reactivity is
exactly the opposite. As in the previous cases (Figure 7 (a) and (b)) the power changes
are greater for the square arrays. The main feedback is due to fuel temperature and lead
temperature.
The transient of insertion of positive reactivity produces a power peak of about
109.5% for the square array, and about 108% for the hexagonal array. The final power of
steady state (107% and 105% for the square and hexagonal arrays, respectively) is
greater than those obtained in transients of increase of lead speed (Figure 7 (a)), and a
reduction of lead-coolant temperature. However, the lowest final power of steady state is
obtained with a reduction of lead speeds from 2 m/s to 1 m/s (Figure 7 (a)), about 92%
for the square array and about 94% for the hexagonal array. In general, according to
results, the LFR with a square array is more sensitive to power changes than the
hexagonal array at the same nominal power and transient conditions.
Figure 8 presents the transient field temperature in the LFR (fuel, gap, clad and lead)
for both fuel arrangements during insertion of a positive reactivity (+20 pcm), for elapsed
times of simulation of 1 s, 5 s and 200 s. In this figure there is a temperature increase
(the intensity of the colours is higher) as the time elapses. The clad and peak temperatures
for 1 s, 2 s, and 200 s of elapsed time of simulation are presented in Table 9, for square
and hexagonal arrangements.
Study on the temperature distributions
199
Transients due to changes in: (a) inlet velocity of lead-coolant, (b) coolant inlet
temperature, (c) reactivity
Figure 7
(a) Variation of lead inlet velocity
1.04
Variation of lead inlet velocity
2
Power [normalized]
1.02
0
Reactivity [pcm]
1
Sq 3 m/s
Sq 1 m/s
Hex 3 m/s
Hex 1 m/s
0.98
0.96
0.94
-2
-4
-6
Sq 3 m/s
Sq 1 m/s
Hex 3 m/s
Hex 1 m/s
0.92
-8
0.9
0
20
40
60
80
100
120
140
160
180
0
200
20
40
60
80
100
120
140
160
180
200
t [s]
t [s]
(b)
Variation of lead inlet temperature
1.02
Variation of lead inlet temperature
1.5
1.015
1
Reactivity [pcm]
Power [normalized]
1.01
1.005
1
Sq 663K
Sq 683K
Hex 663K
Hex 683K
0.995
0.99
0.5
0
-0.5
Sq 663K
Sq 683K
Hex 663K
Hex 683K
-1
0.985
0.98
0
20
40
60
80
100
120
140
160
180
200
-1.5
0
20
40
60
80
t [s]
100
120
140
160
180
200
t [s]
(c) Variation of reactivity
1.1
20
15
1.04
10
Reactivity [pcm]
Power [normalized]
1.06
1.02
1
Sq 20pcm+
Sq 20pcmHex 20pcm+
Hex 20pcm-
0.98
0.96
5
0
-5
-10
0.94
Sq 20pcm+
Sq 20pcmHex 20pcm+
Hex 20pcm-
-15
0.92
0.9
Variation of reactivity
25
1.08
-20
0
20
40
60
80
100
t [s]
120
140
160
180
200
-25
0
20
40
60
80
100
t [s]
120
140
160
180
200
200
Figure 8
G. Espinosa-Paredes et al.
Transient temperature fields during insertion of a positive reactivity
As shown in this table the clad temperatures of both arrays are very close with a
difference of less than 1 K at any simulation time. However, the maximum differences of
the peak temperatures between the square and hexagonal arrays are 310.08 K at 1 s,
315.15 K at 5 s, and 310.73 K at 200 s.
Table 9
Temperature comparison: transient insertion of positive reactivity
Time (s)
*Clad temperature
square (K)
*Clad temperature
hexagonal (K)
Peak temperature
square (K)
Peak temperature
hexagonal (K)
1
686.08
687.03
1440.92
1751.50
5
686.61
687.46
1476.92
1792.07
200
686.81
687.56
1494.41
1805.14
Note:
*Obtained with equation (8).
Results of the clad and lead temperature distributions in the axial direction at different
elapsed times of simulation are presented in Figure 9. In this figure, the temperature
difference (both clad and lead) between 5 s and 200 s are very similar for the hexagonal
array, but for the square array it is significantly higher. The physical explanation of these
temperature differences, between clad and lead, at 5 s and 200 s, is that the thermal
resistance of the LFR with the square array is greater than the hexagonal array, which is
related to the thermal conductivity of the fuel, clad, lead and heat transfer coefficient
(given by equation 18).
Study on the temperature distributions
Figure 9
Axial temperatures distribution: transient insertion of positive reactivity
Square array
1
1
0.8
0.8
0.6
T lead,1s
T clad,1s
T lead,5s
T clad,5s
T lead,200s
T clad,200s
0.4
0.2
0
660
680
700
720
T [K]
Hexagonal array
1.2
z [m]
z [m]
1.2
6
201
740
760
780
0.6
T lead,1s
T clad,1s
T lead,5s
T clad,5s
T lead,200s
T clad,200s
0.4
0.2
0
660
680
700
720
740
760
780
T [K]
Conclusions
With the development of relatively simple mathematical models based on the subchannel analysis approach, interesting results were obtained for two concepts of LFR fuel
assemblies, from a point of view of the thermofluids performance.
Regarding the steady-state condition, comparisons show that the hexagonal assembly
leads to higher temperature in the fuel than in the case of the rectangular assembly
design, with a peak temperature difference of 304 K between both arrays at 100% rated
power. This is important from the point of view of safety.
Concerning the kinetic behaviour, for the transient due to the velocity variation in the
inlet coolant, it was found that the hexagonal array has lower power changes relative to
the square array, due to the higher Doppler contribution to reactivity in the hexagonal
array, as it was explained in Section 5.2 (1). Regarding the other two analysed transients:
the temperature changes in the inlet lead-coolant and the reactivity variation, similar
behaviours were observed as far as the power changes are concerned. In conclusion:
results show that the LFR with square array is more sensitive to power changes than the
hexagonal array at the same nominal power and with the same transient conditions. Also,
from the transient analysis it was found that the thermal resistance of the LFR with the
square array is greater than the hexagonal array.
Future work is focused on the coupling of an improved neutronics, where a detailed
full core model of the ELSY reactor has been set up with the Serpent Monte Carlo code
(Juárez-Martínez, 2017).
Acknowledgements
J.L. François acknowledges the support of the National Autonomous University of
Mexico through the project PAPIIT-IN115517.
202
G. Espinosa-Paredes et al.
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