Reactor behavior comparisons for two liquid metal-cooled fast reactors during an event of loss of coolant

Case Studies in Thermal Engineering 16 (2019) 100556

Contents lists available at ScienceDirect

Case Studies in Thermal Engineering journal homepage: http://www.elsevier.com/locate/csite

Reactor behavior comparisons for two liquid metal-cooled fast reactors during an event of loss of coolant Alejandría-D. P�erez-Valseca a, b, Sergio Quezada-García c, *, �mez-Torres d, Alejandro Va �zquez-Rodríguez a, Armando-M. Go a Gilberto Espinosa-Paredes a

Universidad Aut� onoma Metropolitana-Iztapalapa, Divisi� on de Ciencias B� asica e Ingeniería, Av. San Rafael Atlixco 186, Vicentina, Ciudad de M�exico, 09340, Mexico b Universidad Aut� onoma Metropolitana-Iztapalapa, Divisi� on de Ciencias Biol� ogicas y de la Salud, Av. San Rafael Atlixco 186, Vicentina, Ciudad de M�exico, 09340, Mexico c Universidad Nacional Aut� onoma de M�exico, Facultad de Ingeniería, Av. Universidad 3000, Ciudad Universitaria, Coyoac� an, 04510, Mexico d Instituto Nacional de Investigaciones Nucleares, Carretera M�exico – Toluca, La Marquesa s/n, Ocoyoacac, Estado de M�exico, 52750, Mexico

H I G H L I G H T S

� The decrease of the coolant flow generates a major impact on the neutron density of lead-cooled fast reactor. � After the loss of flow, sodium-cooled fast reactors have higher temperature differences regarding its steady state. � Lead as a coolant, has a greater capacity for heat removal in a nuclear reactor during a ULOF. A R T I C L E I N F O

A B S T R A C T

Keywords: Liquid metal-cooled fast reactors Lead-cooled fast reactor Sodium-cooled fast reactor Unprotected loss-of-flow Subchannel analysis

The aim of this paper is to present a fast-numerical tool based on the average channel approach to analyze the response of fast reactors under unprotected loss of flow events. The comparison of the transient behavior of a Lead-cooled Fast Reactor and a Sodium-cooled Fast Reactor during a loss of flow event is presented. The coolant flow to the core inlet was reduced to 90%, 70% and 50% of the nominal value. The parameters compared were power, fuel, cladding, and coolant tempera­ tures, as well as heat removal. In order to compare the results between both reactors, the values of neutronic density and removed heat obtained during the transient were normalized with respect to the values in the steady state, as a result, the percentages of increase or decrease of the pa­ rameters selected for the model were analysed. In the case of analysis of fuel, gap, clad and coolant, the increments of temperature are presented. With the obtained results, the capacity of the coolants for the removal of the heat generated during the transient can be identified. Therefore, lead has a better capacity to remove heat in fast nuclear reactors.

1. Introduction Since the beginning of the development of nuclear reactors, the Liquid Metal-cooled Fast Reactors (LMFR) were developed, being the first metal used mercury, but then, the sodium and lead were considered due to several advantages. In the last years, the Generation * Corresponding author. E-mail address: [email protected] (S. Quezada-García). https://doi.org/10.1016/j.csite.2019.100556 Received 30 September 2019; Received in revised form 19 October 2019; Accepted 20 October 2019 Available online 22 October 2019 2214-157X/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Nomenclature Af Ci Cp drod Dh g G h k KD lp L m n n0 Nu P Pm P0 Pr q q’’’ r r0 Re t T V v z

Flow area [m2] Concentration of the i-th delayed neutron precursor [-] Specific heat capacity [J kg 1 K 1] Rod diameter [m] Equivalent diameter [m] Gravitational acceleration [m s 2] Mass flux [kg m 2 s 1] Heat transfer coefficient [J m 2 s 1 K 1] Thermal conductivity [W m 1 K 1] Doppler constant [-] Rod pitch [m] Subchannel length [m] Number of delayed neutron groups [-] Normalized neutron density [-] Neutronic density in the stationary state [-] Nusselt number [-] Thermal power in the subchannel [W] Wet perimeter [m] Nominal thermal power per fuel rod [W] Prandtl number [-] Energy due to fluid flux [W] Heat source [W m 3] Radius [m] Annulus radius [m] Reynolds number [-] Time [s] Temperature [K] Volume [m3] Velocity [m s 1] Active length of fuel [m]

Geek symbols Thermal expansion coefficient [K 1] β Total fraction of the delayed neutron [-] Total pressure drop [Pa] ΔPT Δr Spatial step in r [m] Δt Time step [s] Δz Spatial step en z [m] λi Decay constant of delayed neutron precursor i [s Λ Mean neutron generation time [s] Friction coefficient [-] ξfr ρ Density of material [kg m 3] ρ0 Reactivity in stationary state [pcm] ρt Total reactivity [pcm] ψ Axial power distribution [-]

α

1

]

List of subscripts c Coolant cl Clad f Fuel g Gap in Entrance 0 Reference state Mathematical operators 〈�〉 Average

2

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

IV International Forum (GIF) established the design of reactors who are developed in the IV generation of fission reactors, and it was decided to select the Sodium-cooled Fast reactor and Lead-cooled fast reactor among others for this generation. The investigations on LMFR is in course in a lot of countries, mainly in Europe and Asia. These two technologies arose as competitors in order to demonstrate which one is the best technology. In our case, we wanted to present a new fast running tool with good capabilities for safety evaluations of fast reactors no matter which technology they use, i.e., to demonstrate that can be applied to both technologies and verify quali­ tatively the tool by means of safety related transients. The Lead-cooled Fast Reactor (LFR) and the Sodium-cooled Fast Reactor (SFR) are part of the proposed Generation IV reactors (GenIV). This generation was established for the Generation IV International Forum (GIF), is the agency responsible for monitoring the research and development activities necessary to determine the viability and performance capacity of this generation of nuclear power systems [1]. GIF establishes four priority areas for the development of nuclear energy for generation IV reactors: sustainability, safety and reliability, economic competitiveness and physical protection [1]. The LFR and SFR reactors are cooled with liquid metals, the LFR can be cooled with lead or a eutectic mixture of lead-bismuth, and the SFR is cooled with sodium or a sodium-potassium mixture. Although these reactors are classified as fourth generation reactors, the development of this technology has its origin after the Second World War as well as thermal reactors [2]. The development of SFR technology has a long history, with approximately 350 reactor-years of experience [3]. Its overall per­ formance has been remarkable, with important achievements such as the demonstration of the reproduction and viability of the fast reactor fuel cycle, thermal efficiencies that reach values of 43–45%, the highest in nuclear practice, and the accumulation of indis­ pensable experience in the dismantling of several reactors of this type. In the case of LFR, there is less experience, although the lead (Pb) or the eutectic mixture of lead-bismuth (LBE) was proposed and investigated as a coolant for fast reactors since the 1950 [4]. The first LFR system was a 70 MWth prototype reactor, which reached criticality and started its operation at full power in Russia, in 1959. In total, seven nuclear submarines of Project 705/705K cooled with LBE were built and operated. Currently, a significant number of countries are expending large research and development programs to the LFR, considered a valid and promising alternative to sodium as a fast reactor coolant. Despite their history, fast reactors require great international research and development efforts due, in large part, to the nature of fission by fast neutrons and to the technologies proposed or adopted for such systems. As a result, new design criteria, often stricter, are imposed for the development of fast neutron reactors. The research and development of rapid reactor technologies include many areas, including the measurement and evaluation of nuclear data, reactor physics and thermo-fluid calculations (thermal and fluid-dynamic refrigerant), fuel, materials, refrigerants, and refrigeration technologies, as well as the design of instrumentation and control systems. Within the development work of the SFR and LFR technologies, security analyses are carried out. These analyses include the simulation of transient events; in general, transients could be protected or unprotected. Protected transients correspond to accidents when the reactivity control system works as designed and the initiating event of the transient is followed by SCRAM. During the unprotected transients, however, there is no SCRAM. The classical unprotected transients under consideration are [5–7]: � Unprotected Loss-of-Flow (ULOF) � Unprotected Transient-Over-Power (UTOP) � Unprotected Loss-of-Heat-Sink (ULOH) A ULOF (Unprotected Loss-of-Flow) transient is initiated when the power of the primary pump is lost, or a shaft break occurs. Due to the lack of SCRAM, the only mechanism for decreasing the reactor power is the sum of reactivity feedbacks [7]. A UTOP (Unprotected Transient-Over-Power) transient is initiated when a control rod begins to move out of the core; the control system does not induce SCRAM and the pumps maintain the nominal coolant flow through the core [7]. The thermal-hydraulic core analysis of a nuclear reactor can be studied by an average channel model [8], with this analysis it is possible to have a representative model of the core from a model of one or more subchannels. In a previous work [9] a comparative study of two concepts of Lead-Cooled Fast Reactor (LFR) fuel assemblies was done, from a point of view of the thermal-fluids performance. The subchannel analysis approach was applied to determine the temperature dis­ tribution in the fuel, in the cladding, and in the coolant. The subchannel analysis is the most suitable approach to the design of fuel assemblies [10–12]. The model used in these comparisons is the base to develop the present work, considering the neutronic point kinetic equations to describe the neutron processes and a heat transfer model to obtain temperature profiles in each section of fuel rod and coolant. The subchannel model developed in this work is fully transient. In the present work, a ULOF event is analysed for a 300 MWth LFR reactor and for an SFR of 3600 MWth, the coolant inlet flow is reduced to 90%, 70%, and 50%. To do this, a comparison was made of the behaviour of power, fuel, cladding and coolant temper­ atures, and heat removed. 2. Mathematical model The model used to describe the process in the nuclear reactor includes three main parts: (1) neutronic process, modeled by the neutron point kinetics equation (2) heat transfer process in the fuel rod, and (3) thermal fluid process of coolant. In this work, the model used to simulate the LFR and SFR is based in the subchannel analysis, the model for both reactors is the same, but for each one, the parameters are changed. This means, for point kinetic equations the values of delayed neutron, neutron 3

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Table 1 Parameters of LFR and SFR [13,14]. Parameter

LFR

SFR

Thermal Power (MWth) Maximum cladding temperature (stationary) (� C) Maximum cladding temperature (transient) (� C) Inlet coolant temperature (� C) Outlet coolant temperature (� C)

300 550 750 400 480

3600 550 750 395 545

Table 2 Nuclear parameters. Parameter

LFR

SFR

Parameter

LFR

SFR

λ1 (s 1) λ2 (s 1) λ3 (s-1) λ4 (s 1) λ5 (s 1) λ6 (s 1) λ7 (s 1) λ8 (s 1) Λ (s)

0.0125 0.0283 0.0425 0.133 0.292 0.666 1.63 3.55 7.66 � 10

0.0127023 0.0301099 0.112331 0.327449 1.22596 8.14883 4.48274 � 10

β1 β2 β3 β4 β5 β6 β7 β8 β

6.47 � 10 5 5.90 � 10 4 2.12 � 10-4 4.72 � 10 3 1.09 � 10 3 4.00 � 10 4 3.76 � 10 4 1.56 � 10 4 0.007609

8.78 � 10 5 8.16105 � 10 5 6.51854 � 10-4 1.7707 � 10 3 7.88203 � 10 4 2.05715 � 10 4 0.003586

7

7

generation time and decay constants of delayed neutron precursor were different. For the heat transfer analysis, the fuel pin is annular with a distribution of temperatures in a one-dimensional radial direction, the fuel is MOX and the properties of fuel, gap and clad are considered equal in both reactors for the sake of comparison, the material of cladding is 15-15Ti, as it was reported in Grasso et al. [13] to LFR and for SFR in the work of Matuzas et al. [14]. For the thermofluid, the analysis is one-dimensional in the axial direction, considering the thermal expansion effects of the liquid metal; the properties of coolants are considered constants. Table 1 shows the design parameters of LFR and for SFR. In the next sections, the model and the parameters used in the work are described. 2.1. Neutronic model The neutron density is calculated with neutron point kinetics equations with six precursors of delayed neutrons for SFR and eight groups for LFR, the model is given by [15]: m X dnðtÞ ρt ðtÞ β nðtÞ þ ¼ λi Ci ðtÞ Λ dt i¼1

(1)

dCi ðtÞ β ¼ nðtÞ dt Λ

(2)

λi Ci ðtÞ;

for ​ ​ ​ ​ i ¼ 1; 2; 3; :::; m

where n(t) is normalized neutron density, t is the time, ρt is the total reactivity, β is the total fraction of the delayed neutron, Λ is the mean neutron generation time, λi is the decay constant of delayed neutron precursor, Ci is the concentration of the i-th delayed neutron precursor and m is the number of delayed neutron groups. The total reactivity considers the external reactivity, Doppler effects and expansion effects in fuel, clad and coolant, the reactivity coefficients for SFR were obtained using the stochastic code Serpent version 2.1.28 [16]. � � 〈Tf 〉out ρt ¼ ρ0 þ KD ln þ αf Δ〈Tf 〉 þ αcl Δ〈Tcl 〉 þ αc Δ〈Tc 〉 (3) |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} 〈Tf 〉 |fflfflfflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflinfflfflfflfflffl } Fuel ​ expansion Clad ​ expansion Coolant ​ expansion Doppler

where ρ0 is the reactivity in stationary state, KD is Doppler constant, α is thermal expansion coefficient and the subscripts f, cl and c, correspond to fuel, clad and coolant, respectively. The nuclear parameters used in this study are presented in Table 2. In the case of LFR are considered eight groups of neutrons precursors, which are presented in the work of Grasso et al. [17] and for SFR are considered six groups of precursors, the nuclear parameters were obtained using the stochastic code SERPENT version 2.1.28 [16]. The total reactivity (Eq. (3)) is calculated considering the Doppler effect, fuel expansion, clad expansion, and coolant expansion. In Table 3 the coefficients for each reactor are presented. The changes in the average temperature in Eq. (3) are defined as [18]: Δ〈Tf 〉 ¼ 〈Tf 〉 〈Tf 〉0 , Δ〈Tcl 〉 ¼ 〈Tcl 〉 〈Tcl 〉0 and Δ〈Tc 〉 〈Tc 〉0 , where the subscript 0 represents the reference temperature, and 〈 � 〉 represents the average temperature on volume in the fuel core, clad and coolant. The thermal power in the subchannel is given by: 4

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Table 3 Reactivity coefficients [17,18]. Coefficient

LFR

SFR

KD (pcm) αf (pcm/K) αcl (pcm/K) αc (pcm/K)

555 0.232 0.045 0.271

834 0.303 0.0405 0.4505

Fig. 1. Axial power distribution for LFR and SFR, respectively.

Fig. 2. Annular fuel pellet.

(4)

Pðt; zÞ ¼ P0 nðtÞψ ðzÞ

where P0 is nominal thermal power per fuel rod and ψ (z) is the axial power distribution. For LFR, the P0 is 13.81 kWth/subchannel, considering 21717 rods distributed in 127 fuel assemblies with 171 fuel rods each one. In the case of SFR, P0 is 29.32 kWth/sub­ channel, the core has 453 fuel assemblies with 271 fuel rods. The axial power distribution ψ (z) [19], used in this work is shown in Fig. 1. The active zone for LFR is 0.6 m and 1.0 m for SFR.

5

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

2.2. Fuel heat transfer model The fuel pin in both liquid metal-cooled fast reactors is illustrated in Fig. 2. In this figure, four regions can be observed: annulus, fuel, gap and clad whose radius are r0, rf, rg and rcl, respectively. The source heat due mainly to fission reactions is concentrate in the fuel region. The heat transfer process is by conduction mechanism and natural convection, the analysis in the annulus region is neglected. The fuel heat transfer mode was derived considering the following fundamental assumptions: � The conduction in the axial direction is negligible with respect to the heat conduction in the radial direction. This results from the fact that the characteristic length of the diameter of the pin is many times smaller than the length of the pin. However, the power distribution in the axial direction is not uniform. � The volumetric heat rate generation in the fuel is uniform in the radial direction. � The gap spacing is uniform. � The annular region is in thermodynamic equilibrium with the temperature of the inner surface of the fuel. The temperature distribution in the annular fuel pin, considering each section of the rod is given by: � � ∂Tf kf ∂ ∂Tf 000 ðρCpÞf Fuel r0 � r � rf ¼ r þ q ðt; zÞ; ∂t r ∂r ∂r �

ðρCpÞg

�

∂Tg kg ∂ ∂Tg ¼ r ; ∂t r ∂r ∂r �

ðρCpÞcl

(5)

Gap

(6)

rf � r � rg

�

∂Tcl kcl ∂ ∂Tcl ¼ r ; ∂t r ∂r ∂r

Clad

(7)

rg � r � rcl

The initial condition is given by T(r,0)¼f(r), and the boundary conditions are: dTf ¼ 0; dr kg

at

dTg ¼ hg Tf dr

� Tg ;

kcl

dTcl ¼ hg Tg dr

� Tcl ;

kcl

dTcl ¼ hc ðTcl dr

Tc Þ;

at

​ at

(8)

r ¼ r0

(9)

r ¼ rf

(10)

r ¼ rg at

(11)

r ¼ rcl

In these equations ρ is the density, Cp is the specific heat capacity, k is the thermal conductivity, hc is the coolant heat transfer coefficient, hg is the gap conductance, and q’’’(t,z) is the heat source given by: 000

q ðt; zÞ ¼

Pðt; zÞ Vf

(12)

where, P(t,z) is the subchannel power given by Eq. (4), and Vf is the fuel volume. The physical properties of fuel as a function of temperature are given by Carbajo et al. [20] in SI units: Fuel density:

ρf ¼

11043:5 9:9672 � 10

1

þ 1:179 � 10 5 T

2:429 � 10 9 T 2

1:219 � 10

12 T 3

�3

Fuel specific heat � � � � 90:998 � 106 A 1:620 � 1012 111:275 � 106 B 2 Cpf ¼ 0:85 þ þ 2:9358 � 10 T þ 0:15 ðA 1ÞT 2 ðB 1ÞT 2 T2 where A¼e548.68/T and B¼e 18541.7/T. � � 1 6400e 16:35=τ ; kf ¼ 1:158 þ 4 5=2 0:1205 þ 2:6455 � 10 T τ

with

τ ¼ T = 1000

(13)

(14)

(15)

The heat transfer coefficient in the gap used in this work is hg¼5000 W m 2 K 1 [15]. Table 4 presents the physical properties of the gap and cladding used in this work. The heat transfer coefficient of Eq. (11) is calculated by hc¼kcNu/Dh, where kc is the thermal conductivity, Dh is the equivalent 6

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Table 4 Gap and clad properties [13–15]. Property

Gap

Cladding

Density (kg m 3) Specific heat (J kg 1 K 1) Thermal conductivity (W m

2.425 5191 15.8 � 10

77000 622 26

1

1

K

)

4 0.7

T

Table 5 Coolant properties [13,14]. Property

Lead

Sodium

Density (kg m 3) Specific Heat (J kg 1 K 1) Thermal conductivity (W m Viscosity (Pa s)

10520 147.3 17.1 0.001998

845 1269 68.8 0.00025

1

1

K )

diameter, and the Nusselt number for each fuel-assembly arrangement, which correlation is given by Todreas and Kazimi [8], for metal fluids: (16)

Nu ¼ 7 þ 0:025ðPrReÞ0:8

where Pr and Re are the Prandtl and Reynolds numbers, respectively. The annular fuel pellet temperature distribution is obtained considering nineteen radial nodes at each of the twenty-four axial nodes in the core. Ten nodes were considered in the fuel, four nodes in the gap, and five in the clad. The differential equations described previously are transformed into discrete equations using the control volume formulation technique in an implicit form [21]. 2.3. Thermofluid model The thermofluid in the core is modeled with mass, energy and momentum balance that considers thermal expansion effects whose physical properties for each coolant are given in Table 5.

αc ρc

dTc ∂G þ ¼ 0; dt ∂z

∂Tc Pm hc ðTcl Tc Þ ¼ Af ρc Cpc ∂t ∂G ¼ ∂t

� � ξfr G2 2 ρc L

G ∂Tc ; ρc ∂z �

∂ G2 ∂z ρc

Mass ​ balance

(17)

Energy ​ balance

(18)

�

ρc g;

(19)

Momentum ​ balance

In these equations G is the mass flux, Pm is the wet perimeter (given by drod(π -4)þ4lp), Af is the flow area (cross-sectional area), L is subchannel length and g is gravitational acceleration. In the momentum balance given by Eq. (19), the friction coefficient is calculated with the following relation: �0:32 � � � 0:210 L lp ξfr ¼ 0:25 1 1þ (20) Dh drod Re where drod is the rod diameter, the rod pitch used is lp¼11.987 mm for SFR [14] and lp¼13.86 mm for LFR [13]. The hydraulic diameter for each array is given by: �pffiffi � 3 2 πd2rod 4 lp Dh ¼ (21) πdrod 2 4 To solve the balance equations of mass, energy, and momentum, the Euler method is applied. The properties of coolants are presented in Table 5 for this analysis the properties were considered constants. The coupling of the physical processes involves a complex dynamic interaction of variables among the nuclear processes, of fuel heat transfer, and thermofluid. 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, it is coupled with thermofluid through the average temperature of the sodium. The fuel heat transfer calculations require the knowledge of the nuclear heat source and the thermal properties of sodium. And the calculations of the thermofluid behavior in the core requires the cladding temperature.

7

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

3. Numerical solution In this section, are presented the methods to solve the equations of the neutronic model, heat transfer model, and thermofluid model. 3.1. Solution of neutronic model The numerical solution of the neutronic power considers two methods. The first is Runge-Kutta 4th order method, applied for the numerical solution of neutrons density, given by Eq. (1). Applying the method is obtained: ntþΔt ¼ nt þ

Δt ðm1 þ 2m2 þ 2m3 þ m4 Þ 6

(22)

where Δt is the time step, and each m term is given by: dn dn � m1 Δt� dn � m2 Δt� m1 ¼ ðnÞ; m2 ¼ nþ ; m3 ¼ nþ ; dt dt 2 dt 2

m4 ¼

dn ðn þ m3 ΔtÞ dt

(23)

The second is the Euler method, applied for the numerical solution of a concentration of the i-th neutron delayed precursor, given by Eq. (2), CtþΔt ¼ Cti þ Δt i

dCi ; dt

for

(24)

i ¼ 1; 2; :::; 6

The initial condition is given by n(0)¼n0 and Ci(0)¼βin0/(λi Λ), where n0 is the neutronic density in the stationary state. 3.2. Solution of heat transfer model The equations of heat transfer in fuel, gap and clad, given by Eqs. (5)–(7), were solved applying the finite difference method, where: � tþΔt � ∂T T T ti ρi Cpi � ρi Cpi i (25) ∂t Δt For the right side of equation: � � T tþΔt 2T tþΔt þ T tþΔt ki ∂ ∂T ki � tþΔt i i 1 þ ri � ki iþ1 T 2 ri ∂r ∂r Δr ri Δr iþ1

T tþΔt i

�

(26)

then,

ρi Cpi

� tþΔt � tþΔt � T Ti T ti ¼ ki iþ1 Δt

� 2T tþΔt þ T tþΔt ki � tþΔt i i 1 þ T 2 Δr ri Δr iþ1

� 000 þ qi T tþΔt i

(27)

Rewriting Eq. (27) of algebraic form: tþΔt ai T tþΔt þ ci T tþΔt i 1 þ bi T i iþ1 ¼ di ;

for

i ¼ 2; 3; 4; ::: ; n

1

(28)

where, ai ¼

ki Δr2

(29a)

bi ¼

2ki Δr2

ki ri Δr

ci ¼

ki ki þ Δr2 ri Δr

8 ρ Cp 000 i i > qi ; > < Δr di ¼ > ρ Cpi > : i ;​ Δr

ρi Cpi

(29b)

Δr

(29c) for ​ for ​

i ¼ 2; 3; :::; 9 i ¼ 10; 11; :::; 18

Fuel

(29d)

Gap ​ and ​ clad

There are considering nineteen radial nodes, ten nodes were considered in the fuel, four nodes in the gap, and five in the clad. Eq. (28) has a matrix structure of the tridiagonal form, solved by the Thomas algorithm. The initial condition and boundary conditions are given by Eqs. 8–11.

8

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Fig. 3. Axial temperature profile of a) LFR and b) SFR.

3.3. Solution of thermofluid model In the case of mass balance, Eq. (17), Euler’s method was used, � � ∂Tc Gswþ1 ¼ Gsw Δzαc ρc ∂t sw

(30)

For the energy balance, given by Eq. (18), is developed a spatial approximation around to node swþ1, � ∂T �� Pm hc ðTcl Tsw Þ Gsw ðTswþ1 Tsw Þ ¼ Af ρc Cpc Δz ∂t �swþ1 ρc

(31)

Solving of analytic form with the temporal dependence for sw¼1,2,…,24, because the model has 24 axial nodes. Then, 0 1 BωTcl þ ρGc Tsw C C Tswþ1 ¼ B @ ωþ G A 1 ρ

� e ðωþG=ρc Þt þ T0 e ðωþG=ρc Þt

(32)

c

where,

ω¼

hc Pm Δz Af ρc Cpc

(33)

The initial condition is T¼T(z) at t¼0, and the boundary condition is T¼Tin at z¼0; where Tin is the inlet temperature of the thermofluid. For the momentum balance, Eq. (19), an explicit discretization is applied for sw¼1,2,…,24, given by: � ��t � ��t � � G2 ρc �swþ1 G2 ρc �sw ξfr G2sw GtþΔt Gtsw sw ρc g (34) ¼ Δt 2 ρc Δz Δz Then, the total pressure drop is: 23 � X

ΔPT ¼ sw¼1

� � ξfr G2sw 2 ρc

h

� ��t G2 ρc �swþ1

� ��t i G2 ρc �sw

�

ρc gΔz 9

(35)

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Fig. 4. Neutronic density of SFR and LFR during ULOF events.

For the mass and momentum balances, the initial condition is G¼G(z)¼Gin at t¼0, and the boundary condition is G¼Gin at z¼0; where Gin is the inlet mass flux of the thermofluid. 4. Results and discussion As mentioned above, the objective of this work is to do a comparison between the behaviour of SFR and LFR reactors during ULOF events. The parameters to analysed were: (1) neutronic density (normalized) to identify the percent of variation in the power due a loss of coolant, (2) the increment in the temperature of fuel, clad and coolant, to compare with the temperatures of security defined in the design of reactors, (3) the percentage of removal of heat after loss of coolant event, this analysis is important to observe which coolant has a better capacity to heat removal. Before to simulate ULOF events, is important to report the profiles in a stationary state of each reactor, in the next section, the temperatures in steady state are presented and are compared with the design values. 4.1. Steady state analysis In this section, the steady state of each reactor was analysed, comparing the temperatures obtained for the cladding and the coolant with the nominal temperatures (Table 1). In Fig. 3 the axial profiles of the fuel, gap, cladding and coolant in steady state are shown for the LFR and SFR respectively. The fuel profile has the same trend of the axial distribution of power (Fig. 1), and for the gap, clad and coolant, the profile increases as a consequence of the accumulation of energy, as expected. In these figures it is possible to observe the increase in the temperature in axial direction. For the LFR profiles showed in Fig. 3a, the maximum temperature of the cladding was 760.62 K, 62.53 K below the maximum temperature of 823.15 K, according to the design of the LFR [13]. In the case of the coolant outlet temperature, the obtained with the model was 748.75 K, 4.4 K below the nominal 753.15 K. For the case of SFR, Fig. 3b shows the axial profiles of each section (fuel, gap, cladding, and coolant), in this case the maximum temperature of the cladding was 841.47 K, while the nominal corresponded to 823.97 K, that is, a difference of 17.5 K [14].The coolant temperature was 825.77 K, which compared to the nominal 818.15 K indicated a difference of 6.72 K. In both reactors, the temperature obtained oscillated within the acceptable margin of error for nominal conditions of dynamic steady state. 4.2. Analysis of ULOF events For the transitory event, the behaviour of the power, fuel temperature, cladding temperature, coolant temperature, and the heat removed were analysed. To carry out the comparison, the transient values of neutronic density and heat removal were normalized with respect to themselves obtained in the steady state. In the case of temperatures, the increase in degrees is analysed, obtaining the maximum temperature in the transitory. In this way, we can observe the behavior of each reactor individually and estimate a com­ parison in terms of percentage. To simulate the ULOF, three numerical experiments were simulated to reduce the flow of coolant, leaving 90%, 70% and 50% of flow at the entrance of the core. Fig. 4 shows the behaviour of the normalized neutronic density of the LFR and the SFR during each decrease of flow. In the case of reduction to 90% of the flow, the power of the LFR decreased 0.39% with respect to the nominal power, while the reduction of the power of the SFR corresponded to 0.16%. For the 70% decrease in flow experiment, the LFR had a power decrease approximately the double that of the SFR, which was 0.72%. In the numerical experiment, it was observed, at the beginning of the transient event, that the SFR power increased by approximately 0.88%. In the case of the 50% flow decrease, the SFR showed an increase of 1.63% at the beginning of the transient, which fell fast and returned to a new steady state condition. It is worth to note that at the beginning of the transient, for the SFR the neutronic density has an increase, due to the change in the reactivity (Eq. (3)) by temperature increase and by the positive effect of the coolant expansion coefficient (Table 2). On the other hand, as the coefficient by coolant expansion is negative in LFR, the neutronic density decreases immediately, then it has a short increase but, 10

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Fig. 5. Temperature difference during ULOF events of SFR and LFR for a) fuel, b) cladding and c) coolant.

in this case, the effect predominant is a decrease in the power. To observe the impact in the temperatures of fuel, cladding ad coolant, Fig. 5 show the difference of temperature between the steady state and the transient state, thus, it is possible to observe the behaviour of temperature during ULOF. The temperatures graphed are the average temperature in each section. The increase in fuel temperature is shown in Fig. 5a, where the differences of temperature of both reactors in the three transients simulated are presented. The maximum increase is when the coolant was reduced to 50%, for SFR the temperature increases 22.7 K in the first 10 s, then, when the reactor gives a steady state, the increment stabilizes at 17.8 K. For the LFR, the maximum increment was 19.5 K in 50% of flow. The minimum increment in the fuel temperature is for LFR in 90% of flow with 1.3 K, for SFR the increment is 2.7 K. In the case of cladding, the maximum increase was of 29 K for SFR and 25.8 K for LFR, in the flow of 50%, in these cases the increment is bigger than the one of the fuel in the same conditions. Is important to note that the maximum temperature does not exceed the safety limit established in the design (Table 1). Fig. 5c shows the difference of coolant temperature, in the case of a decrease of 50% of flow, the sodium has an increase of 30.3 K in average, and the lead has 23.9 K more than its steady temperature. These increments affect the capacity of heat removal of coolants, and give an important change in the reactivity, causing changes in the power reactor. For a reduction to 70% of flow, the coolant temperature increases 15.6 K for the sodium and 11.65 K for the lead. In the case of ULOF to 90%, the sodium has an increment to

11

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Fig. 6. Removed heat during ULOF event.

4.6 K, while the lead only increases 1.8 K. The heat transfer coefficient of the refrigerant decreases as the flow decreases, generating greater resistance to heat transfer. This generates an increase in the clad temperature to remove the heat generated in the fuel. In Fig. 6, the behaviour of removed heat is shown, in all the transients, the capacity of removal is less for the sodium. For the case of reduction to 90% of flow, for SFR the removal heat is 94%, while for LFR is of 99%. In the ULOF of 70%, the sodium removes 20% less than in steady state, on the other hand, the lead still removes the 95%. In the extreme case of decrease of 50% of flow, the sodium removes only 63% of heat generated in the core and the lead removes 91%. In Fig. 6, it is possible to observe that the lead has a higher capacity to remove heat compared with the sodium, although at the beginning of transient, the removal capacity decreased quickly. In the dynamic of reactor behavior, an important parameter is the reactivity, in our case, this parameter is a function of temperature in the fuel, clad and coolant, related by a coefficient of expansion. When the reactivity is positive increases the neutronic density and when is negative, the neutronic density decreases. For sodium and lead reactors, the reactivity coefficient due to Doppler (neutronic fuel temperature effects) and reactivity due to expansion by fuel and clad have the same sign. However, the reactivity coefficient due to coolant expansion has opposite sign, in the case of sodium it is positive and negative for the lead. When the transient event starts, the flow of coolant decreases and therefore the temperature of the coolant increases, causing an increment of negative reactivity by coolant expansion in lead and a decrease of neutronic density as well as the power. For this reason, we consider that the lead reactor is more sensitive due to the neutronic density decreases faster than in the sodium reactor. On the other hand, these differences in the behavior of neutronic density during the loss of coolant generate a difference in the temperature, the sodium reactor has an increase in the power generating and an increase in the temperature. Besides, during the transient, the capacity of heat removal decreases, limiting the heat removed and causes an increase of temperature in the clad and fuel. In Fig. 6 of the manuscript, is presented the removed heat (normalized) of each reactor during the loss of coolant transient. It is possible to note that in the instant of transient starts for both reactors the capacity of heat removal decrease, but later, the capacity increase according to the stabilization of the neutronic density, due to the specific heat capacity (Cp) and density (ρ) of coolants. As it is shown in Table 5, the Cp of sodium is 1269 J kg 1 K 1 and of lead is 147.3 J kg 1 K 1, the specific heat capacity of sodium is higher than lead, but another factor important is the density ρ, in the case of sodium is 845 kg m 3 and 10520 kg m 3 for the lead, this difference of densities generate an important effect in the heat transfer process. If we analyze the energy due to the flux of fluid, given by: q ¼ ρc CpvðT

(36)

T0 ÞAf

In the transient, the velocity decrease due to the loss of flow to the equal proportion in both reactors, the flow area is constant, so, the important factor to consider is the relation (ρCp), in our analysis the (ρCp)sodium¼1.072 � 106 J m 3 K 1 and (ρCp)lead¼1.550 � 106 J m 3 K 1. For the lead reactor, the density of metal allows to have a better heat transfer process, notwith­ standing the decrease in the flow. 5. Conclusions One of the most important safety evaluations in the field of fast reactors is the unprotected response of the reactor cores under different scenarios. In this paper, a new tool for fast analysis of unprotected transients in fast reactors was presented. The implementation of the physical models in the numerical tool was done by coupling the point kinetics model with the fuel heat transfer model and the thermo fluid model for two different coolants (lead and sodium) for an average channel representing the core. The numerical experiments agreed with the expected outputs and reference temperatures, thus verifying the correct implementation of the models. For the analysed scenarios, the power response, as expected, presented an increment for the sodium coolant at the beginning of the transient due to the usually positive sodium void effect, which was counteracted by Doppler phenomenon later. For the case of lead, the trend was always negative due to the strong negative reactivity coefficients even in loss of coolant. 12

Case Studies in Thermal Engineering 16 (2019) 100556

A.-D. P�erez-Valseca et al.

Also, it was possible to compare the heat removal capacity of the two fluids, showing as expected the better performance of lead due to the higher thermal inertia and thus confirming the capability of the developed tool. An extension of the tool is under development considering several parallel average channels representing different reactor zones, for instance, different enriched radial zones but also a full parallel channel core with one channel representing one subassembly. Declaration of competing interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately in­ fluence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Reactor behavior comparisons for two liquid metal-cooled fast reactors during an event of loss of coolant”. Acknowledgments The authors acknowledge the financial support from the National Strategic Project No. 212602 (AZTLAN Platform) as part of the Sectorial Fund for Energetic Sustainability CONACYT–SENER (M� exico). Special thanks to CONACYT for the financial support given to A-D. P� erez-Valseca. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.csite.2019.100556.

References [1] A. GIF, Technology roadmap for generation IV nuclear energy systems, in: Nucl. Energy Res. Advis. Comm. Gener. IV Int. Forum, Issued by the, U.S. DOE, 2014. GIF-002-00. [2] A.M. Judd, An Introduction to the Engineering of Fast Nuclear Reactors, Cambridge University Press, 2014. [3] IAEA, Comparative Assessment of Thermophysical and Thermohydraulic Characteristics of Lead, Lead-Bismuth and Sodium Coolants for Fast Reactors, IAEA, Vienna, 2002. IAEA-TECDOC-1289. [4] L. Cinotti, G. Smith, C. Artiolo, G. Grasso, G. Corsini, Lead-cooled Fast Reactor (LFR) Design, Safety, Neutronics, Thermal Hydraulics, Structural Mechanics, Fuel, Core, and Plant Design, Handbook of Nuclear Engineering, 2010. [5] J.-B. Droin, N. Marie, A. Bachrata, F. Bertrand, E. Merle, J.M. Seiler, Physical tool for unprotected loss of flow transient simulations in a sodium fast reactor, Ann. Nucl. Energy 106 (2017) 195–210. [6] M. Tesinsky, Y. Zhang, J. Wallenius, The impact of americium on the ULOF and UTOP transients of the European Load-cooed SYstem (ELSY), Ann. Nucl. Energy 47 (2012) 104–109. [7] M. Shahzad, L. Qin, S. Imam, Analysis of lead-cooled fast reactor using a core simulator, Prog. Nucl. Energy 104 (2012) 229–241. [8] N.E. Todreas, M.S. Kazimi, Nuclear Systems I: Thermal Hydraulic Fundamentals, Hemisphere Publishing Corporation, USA, 1990. [9] G. Espinosa-Paredes, J.-L. François, H. S� anchez-Mora, A.D. P�erez-Valseca, C. Martín-del-Campo, Study on the temperature distributions in fuel assemblies of lead-cooled fast reactors, Int. J. Nucl. Energy Sci. Technol. 11 (2017) 183–203. [10] J. Wang, W.X. Tian, Y.H. Tian, G.H. Su, S.Z. Qiu, A sub-channel analysis code for advanced lead bismuth fast reactor, Prog. Nucl. Energy 63 (2013) 34–48. [11] H. Chen, G. Zhou, S. Li, Z. Chen, P. Zhao, Subchannel analysis of fuel assemblies of a lead–alloy cooled fast reactor, Prog. Nucl. Energy 8 (2015) 182–189. [12] A.-D. P� erez-Valseca, G. Espinosa-Paredes, J.L. François, A. V� azquez Rodríguez, C. Martín-del-Campo, Stand-alone core sensitivity and uncertainty analysis of ALFRED from Monte Carlo simulations, Ann. Nucl. Energy 108 (2017) 113–125. [13] G. Grasso, K. Mikityuk, C. Petrovich, D. Mattioli, F. Manni, D. Gugiu, Demonstrating the Effectiveness of the European LFR Concept: the ALFRED Core Design, IAEA, Vienna, 2013. IAEA-CN-199/312. [14] V. Matuzas, L. Ammirabile, L. Cloarec, D. Lemasson, S. Perez-Martin, A. Ponomarev, Extension of ASTEC-Na capabilities for simulating reactivity effects in sodium cooled fast reactor, Ann. Nucl. Energy 119 (2018) 440–453. [15] S. Glasstone, A. Sesonske, Nuclear Reactor Engineering, Van Nostrand Reinold Company, New York, 1981. [16] J. Lepp€ anen, M. Pusa, T. Viitanen, V. Valtavirta, T. Kaltiaisenaho, The Serpent Monte Carlo code: status, development and applications in 2013, Ann. Nucl. Energy 82 (2015) 142–150. [17] G. Grasso, C. Petrovich, D. Mattioli, C. Artioli, P. Sciora, D. Gugiu, The core design of ALFRED, a demonstrator for the European lead-cooled reactors, Nucl. Eng. Des. 287 (2014) 287–301. [18] A.E. Waltar, D.R. Todd, P.V. Tsvetkov, Fast Spectrum Reactors, Springer, New York, 2014. [19] M. Aufiero, A. Cammi, C. Fiorina, L. Luzzi, A. Sartori, A multi-physics time-dependent model for the Lead Fast Reactor single-channel analysis, Nucl. Eng. Des. 256 (2013) 14–27. [20] J.J. Carbajo, G.L. Yoder, S.G. Popov, V.K. Ivanov, A review of the thermophysical properties of MOX and UO2 fuels, J. Nucl. Mater. 299 (2001) 181–198. [21] G. Espinosa-Paredes, E.-G. Espinosa-Martínez, Fuel rod model based on Non-Fourier heat conduction equation, Ann. Nucl. Energy 36 (2009) 680–693.

13