Development of a single-channel multi-physics model for ... · Development of a single-channel multi-physics model for lead fast reactors Manuele Auﬁero, Antonio Cammi, Carlo Fiorina,

Development of a single-channel multi-physics modelfor lead fast reactors

Manuele Aufiero, Antonio Cammi, Carlo Fiorina, Lelio Luzzi

Politecnico di Milano - Department of EnergyNuclear Engineering Division - CeSNEFVia La Masa, 34 - 20156 Milano, Italy

[email protected]

ABSTRACTIn this work, an extension of the Multi-Physics Modelling (MPM) approach developed at

the Politecnico di Milano for the nuclear reactor analysis is presented, with reference to thelead-cooled ELSY reactor. Within the same computing platform (represented by COMSOLMultiphysics R©), the Multi-Physics (MP) scheme of analysis is based on the simultaneous so-lution of the fully coupled and time-dependent partial differential equations describing heattransfer, fluid dynamics and neutronics in a fuel pin, and in the surrounding lead. The MPMapproach aims at intrinsically coupling the different phenomena occurring in a nuclear reactor,without requiring data-passing between different platforms (i.e., through the conventional cou-pled code techniques). In such scheme of analysis, the main development of present work isrepresented by the explicit consideration of the point-wise temperature and density dependenceof neutron cross-sections as well of the thermal expansion effects. Throughout the paper, thefeatures of the extended MPM approach are investigated, with reference to a single-channel rep-resentative of the active core average conditions of the ELSY reactor, at beginning of life. Theextended-MP model proves itself capable of describing the reactor behaviour both in steady-state and transient conditions, with reasonable computational requirements and good degree ofconvergence.

1 INTRODUCTION

The Lead Fast Reactor (LFR) has been selected as one of the innovative nuclear powerplants able to meet the Generation IV International Forum (GIF-IV) goals [1]: sustainability,economics, safety and reliability, proliferation-resistance and physical protection. This fissionnuclear reactor has been the subject of several studies up to now, and numerous investigationshave been carried out to assess its effective potentialities [2].

In this context, the Multi-Physics Modelling (MPM) approach [3] is a promising tool foranalysing the “reactor system”, both in operative and accidental conditions. The MPM ap-proach consists of a set of non-linear and time-dependent coupled partial differential equations,which are simultaneously solved in the same simulation environment and are descriptive of the

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different “physics” (neutronics, heat transfer, fluid dynamics, etc.) occurring in the nuclear re-actor. This approach is expected to provide a better description of the couplings among differentphenomena.

The Multi-Physics (MP) model herein presented is an extension of a previous work carriedout at the Politecnico di Milano [4], and represents a new building block in the realizationof a more complete simulation tool for the LFR core analysis. In particular, the treatment ofsolid mechanics for the evaluation of the thermal expansion effects is introduced in the MPMscheme. Moreover, a different treatment of the neutron cross-sections is considered, whichtakes into account their dependence on the local temperature and density fields inside the system(represented by a single-channel of the core).

Because of the availability of detailed core design specifications, the European Lead-cooledSYstem (ELSY) [2] is chosen as test case to assess the simulation capability of the extendedMP model, both in steady-state and transient conditions. Currently, two ELSY core layoutoptions are available [2]. The present work refers to the Open Square Fuel Assembly (OSFA)configuration, whose details can be found in Ref. [5].

The paper is organised as follows. Section 2 introduces the geometry and the parametersof the analysed single-channel, and provides a brief description of the extended MPM schemeof analysis as well as of its numerical implementation. In Section 3, the results obtained bythe present model are shown in terms of steady-state spatial distribution of some quantities ofinterest, and two case studies are presented to exemplify the MPM capabilities for simulatingthe dynamic behaviour during accidental scenarios.

2 MULTI-PHYSICS MODELLING APPROACH

In this section, the multi-physics scheme of analysis is presented by describing the differentequations governing fluid dynamics, heat transfer and neutronics, which are solved in the samesimulation environment. The inclusion of thermal expansion effects in such scheme is achievedby adopting the equations of solid mechanics and combining them to the other physics, bymeans of the “moving mesh” technique [6]. The work focuses on the potentialities of theMPM approach to catch the couplings between the various physics in the time scales typicalof operational/accidental transients. The details of the actual geometry of the channel, as wellas the treatment of irradiation induced and other mechanical effects, are out of the scope of thepresent paper.

2.1 Analysed geometryIn the proposed MP model, a two-dimensional, axial-symmetric representation (r, z) of a

single-channel of the ELSY OSFA reactor is considered. Reference is made to the active coreaverage conditions, at beginning of life (BOL). The main parameters of the analysed channelare listed in Table 1. Figure 1 shows the geometry of the fuel pin.

2.2 NeutronicsIn the neutronic model of the ELSY single-channel, the multi-group diffusion theory is

adopted [7]. Integrating over a finite set of energy intervals (six) the continuous neutron diffu-sion equation, along with the balance equations for six groups of precursors, the following setof partial differential equations is obtained:

Proceedings of the International Conference Nuclear Energy for New Europe 2011, Bovec, Slovenia, Sept.12-15, 2011

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Table 1: Main parameters of the analysed coresingle-channel

Average linear power 23.5 kW m−1

Inlet lead temperature 400 ◦C

Outlet lead temperature 480 ◦C

Inlet lead velocity 1.6 m s−1

Pu enrichment 17 V ol.%

Fuel density 95 %TD a

Fuel pin active height 900 mm

Fuel pellet hole diameter 2.0 mm

Fuel pellet outer diameter 9.0 mm

Cladding inner diameter 9.3 mm

Cladding outer diameter 10.5 mm

Pin-pitch 13.9 mm

aTheoretical Density

Figure 1: Analysed fuel pin and surroundinglead. Radial sizes at nominal conditions (roomtemperature), expressed in mm

1

vg

∂φg∂t

= ∇ ·Dg∇φg − Σa,gφg −∑g′ 6=g

Σs,gg′φg +∑g′ 6=g

Σs,g′gφg′ + (1)

+ (1− β)χp,g6∑

g′=1

(νΣf )g′ φg′ +6∑i=1

χd,gλici g = 1÷ 6

∂ci∂t

= −λici + βi6∑g=1

(νΣf )g φg i = 1÷ 6 (2)

The group constants (i.e., the macroscopic cross-sections and the diffusion coefficients) arecalculated solving the neutron transport equation for an infinite lattice of cells by means ofthe deterministic neutronics code ERANOS [8], using the nuclear data library JEF 2.2 [9] (theupper boundaries adopted for the six group neutron energy structure are: 20 MeV , 2.23 MeV ,0.82 MeV , 67.38 keV , 15.03 keV , 0.75 keV ). The different constants (vg, Dg, Σa,g, νΣf,g,Σs,gg′) are computed for a discrete range of temperatures for the fuel and the coolant region. Inthe case of the macroscopic cross-sections, the following functional form is adopted:

fuel, lead : Σ(T, ρ) =

(ρ

ρ0

) [Σ0 + α log

(T

T0

)](3)

representing a first attempt to allow for the heterogeneity of temperature and density fields insidethe core channel. Such relations are then introduced in the MP scheme as input parameters.

2.3 Fluid dynamics and heat transferThe model of the fluid flow (liquid lead) is based on the incompressible form of the Reynolds-

Averaged Navier-Stokes (RANS) equations, considering in particular the standard k − ε turbu-lence model (the empirical constants are given as Cε1 = 1.44, Cε2 = 1.92, Cµ = 0.9, σk = 1.0,σε = 1.3):

ρ∂v∂t

+ ρ (v · ∇) v = ∇ ·[−pI + (η + ηT )

(∇v + (∇v)T − 2

3ρkI

)](4)


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∇ · v = 0 (5)

ρ∂k

∂t+ ρv · ∇k = ∇ ·

[(η +

ηTσk

)∇k

]− ρε+ ηT

[∇v :

(∇v + (∇v)T

)](6)

ρ∂ε

∂t+ ρv · ∇ε = ∇ ·

[(η +

ηTσε

)∇ε]− Cε2ρε

2

k+ Cε1

ε

kηT[∇v :

(∇v + (∇v)T

)](7)

As to the heat transfer modelling, the energy balance equations (8) and (9) are adopted:

ρCp∂T

∂t+∇ · (−K∇T ) = Q (8)

ρCp∂T

∂t+∇ · [− (K +Kt)∇T ] = Q+ ρCpv · ∇T (9)

Eq. (8) is used in the pellet hole, fuel, gap and cladding materials with the correspondingvalues of thermal conductivity, density and specific heat, while Eq. (9) is adopted in the leaddomain. The turbulent heat transfer is modelled using the Kays-Crawford model [6] for theturbulent Prandtl number. The heat source is explicitly computed, by means of the calculatedneutron fluxes:

Q =6∑g=1

(qg φg) (10)

where the coefficients qg, which take into account the energy released by events of radiativecapture, scattering and fission, are calculated by means of ERANOS [8]. The heat is consideredto be released instantaneously and locally (i.e., disregarding gamma transport and delayed nu-clear decay). In order to speed up and simplify the calculations, effective and constant valuesof the thermo-physical properties Cp and K are adopted for the considered materials (Helium,MOX, T91 steel and lead) [10].

2.4 Solid mechanics and moving meshIn order to take into account the fuel and cladding thermal expansion effects, the following

equations of linear elasticity are introduced into the MP model:

ρ∂2u∂t2

= ∇ · σ (11)

ε =1

2

[(∇u) + (∇u)T

](12)

σ = C : (ε− αth(T − Tref )I) (13)

The coefficient of thermal expansion, as well as the Young modulus and the Poisson coef-ficient (used to derive the stiffness tensor C under the isotropic material hypothesis), are keptconstant with the temperature in order to simplify the solution of the problem. The column offuel pellets is modelled as a unique continuous structure, and neglecting cracking, irradiationinduced and other mechanical (e.g., creep) effects. For the sake of simplicity, the gravitationalvolume force is disregarded.

The “moving mesh” technique offered by COMSOL [6] allows to dynamically deform themesh of the simulated domain. In the present work, the mechanical deformations are used toredefine the geometry, at each solver iteration. In this way, the different “physics” are influencedby the displacement field of the fuel and the cladding. Hence, the coupled effects due to thermalexpansion (e.g., gap thermal resistance reduction, fuel expansion feedbacks on neutronics) areexplicitly considered.


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2.5 Boundary conditionsAxial-symmetric conditions are applied to all equations, at r = 0.

Neutronics. As far as neutronics is concerned, the “albedo boundary condition” [7] is im-posed at the upper, lower and radial boundaries of the active height, in order to achieve a properspatial characterization of the neutron fluxes without burdening the computation, namely:

n · (D∇φg) = −γzφg n · (D∇φg) = −γrφg (14)

The γz coefficient is calculated from full-core ERANOS [8] simulations, while the γr coef-ficient is iterated to bring the system to criticality, verifying that the effective neutron multipli-cation factor (keff ) is equal to 1 by means of an eigenvalue calculation.

Fluid dynamics and heat transfer. A condition of thermal insulation is applied to the upperand lower boundaries of the pellet hole, fuel, gap and cladding domains:

∂T

∂z

∣∣∣∣z=0

= 0∂T

∂z

∣∣∣∣z=H

= 0 (15)

In the fluid domain, at the outer radius, a symmetry condition is considered, while at thelower boundary the inlet velocity and temperature are imposed (Tin, vin). For the sake of sim-plicity, the radial profile of the inlet lead velocity is considered flat (i.e., disregarding the effectof the fluid flow development in the core channel section below the active height) and the massflow rate is kept constant.

Solid mechanics. A condition of no axial displacement is applied to the fuel and claddingdomain, at z = 0. The “moving mesh” is forced to follow axially the fuel thermal expansion atthe upper boundary, while at the outer radius of the channel no radial displacement of the meshis allowed.

2.6 Numerical solutionThe set of partial differential equations described above has been simultaneously solved by

means of the “general-purpose” finite element software COMSOL Multiphysics R© [6].The geometry described in subsection 2.1 is meshed so as to achieve a good compromise

between numerical accuracy and computational requirements. In particular, as shown in Figure2, a mapped mesh is judged suitable for the gap, cladding and lead domains. A progressivemesh refinement near the wall is adopted in the lead domain (in green), while the fuel (in red) ismeshed with triangular elements in order to allow the connection to the coarser mesh chosen forthe pellet hole. The adopted elements are Lagrangian and quadratic-order. In order to reduce thecomputational cost of the simulation, the segregated solver is adopted. The equations referringto the variables of segregated groups 1 to 4 of Table 2 are solved using the direct MUMPSmethod [6]. The equations of the RANS k − ε turbulence treatment are solved by means of thePardiso direct solver [6].


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Table 2: Segregated groups

Group Variables1 φ1... φ6 neutron fluxes2 c1... c6 precursors density3 T , u temp. and displacement4 p, v pressure and velocity5 k, ε turbulence variables

Figure 2: Meshed geometry

3 RESULTS AND DISCUSSION

3.1 Steady-state behaviour of the systemIn this subsection, the capability of the proposed MP model to evaluate, in the same com-

putational environment, the most relevant variables of the coupled “physics” is exploited toanalyse the nominal steady-state behaviour of the system.

In the sequel, some results concerning neutronics, thermo-fluid dynamics and solid mechan-ics are presented. In Figure 3, the neutron fluxes (E > 0.82MeV and E < 0.82MeV ), and thespatial descriptions of temperature (Figure 3(b)) and velocity (Figure 3(c)) fields in the lead aredepicted. In Figure 4, the effects of the thermal expansions of fuel and cladding, leading axiallyto a different gap reduction, are clearly visible. This feature is typically neglected when simula-tions are performed by means of the conventional coupling of neutronic and thermo-hydrauliccodes. In Figure 5, the axial profile of the cladding surface temperature and the channel temper-ature field are reported as examples of other relevant quantities, achievable through the MPMapproach, and useful for assessing the respect of the fuel pin design limits.

(a) (b) (c)

Figure 3: Steady-state analysis: (a) neutron fluxes (E > 0.82MeV , in black, andE < 0.82MeV , in red); (b) temperature and (c) velocity field of lead


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Figure 4: Steady-state analysis: outer fuel radius and in-ner cladding radius, as a function of the axial coordinate

—— cold conditions (at room temperature)—— nominal power conditions

(a) (b)

Figure 5: Steady-state analysis: (a) cladding surface temperature, (b) channel temperature field

3.2 Dynamic behaviour of the systemIn this subsection, two accidental scenarios are considered as examples of the MPM capabil-

ities to investigate the reactor dynamic behaviour, namely: an unprotected transient overpower(UTOP) and an unprotected loss of heat sink (ULOHS).

The MPM simulations of the dynamic behaviour were able to intrinsically face the couplingof all the considered “physics”, within a computational domain featured by a “moving mesh”,showing good capabilities in terms of numerical convergence (as in the case of steady-stateanalysis), without consuming excessive computational resources.

Unprotected transient overpower. A step-wise insertion of reactivity (80 pcm) has beensimulated. Figure 6 shows the system response to the reactivity insertion in terms of power andfuel temperature. The solid lines represent the simulation without the thermal expansion effects(i.e., with dimensions at nominal conditions), while the dashed lines represent the simulation inpresence of these effects. The power rise (Figure 6(a)) has a prompt effect on the fuel temper-ature (Figure 6(b)), whose increase corresponds to a negative feedback limiting the power to amaximum value (35% greater than the nominal value). As expected, the axial expansion of thefuel acts as a further negative feedback, by lowering both the maximum peak and the stationarypower level.

Unprotected loss of heat sink. A simplified simulation of an unprotected loss of heat sinkaccident has been performed by raising the inlet lead temperature to 435◦C. The results arepresented in Figure 7. Initially, a positive reactivity is inserted by the hot lead entering the


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(a) (b)

Figure 6: System response during the UTOP case study with a reactivity insertion of 80 pcm:(a) power, (b) maximum fuel temperature

channel, and causing a rise in the power level (Figure 7(a)). After few seconds, the increaseof the fuel temperature and the core axial expansion (only considered in the simulation markedwith dashed lines) smoothly reduce the reactivity and stabilise the power level to a value lowerthan the initial one.

(a) (b)

Figure 7: System response during the ULOHS case study with an inlet lead temperature increaseof 35◦C: (a) power, (b) maximum fuel temperature

4 CONCLUSIONS

In the present work, an extension of the MPM scheme of analysis for simulating the nu-clear reactor core behaviour of the GIF-IV LFR has been presented. Reference is made to asingle-channel of the ELSY OSFA design, representative of the average BOL conditions of theactive core. In particular, the capability of the MP model to reproduce the effects of thermalexpansion of fuel and cladding has been investigated. Thanks to the flexible numerical structureoffered by COMSOL, the equations related to solid mechanics have been introduced in the setof non-linear, time-dependent and coupled partial differential equations describing the neutron-ics, the heat transfer and the fluid dynamics inside the core channel. This refined MP model hasbeen applied to study the steady-state operation as well as some transients (UTOP, ULOHS)


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of importance from the safety point of view. The capacity of the MPM approach to catch im-portant coupled effects, in the same computational environment, has been shown. In particular,benefitting from the “moving mesh” technique of COMSOL, the extended MP model revealeditself able to simulate the spatial distribution as well as the time evolution of different physicalquantities, exhibiting good features in terms of numerical convergence.

Further steps are needed to achieve a better description of the LFR core behaviour, whichshould include improvements in the modelling of neutronics (e.g., through more refined eval-uation of the heterogeneity effects inside the channel and multi-group neutron energy dis-cretization) as well as in the thermo-mechanical description of the fuel pins (e.g., to handlethe contact between fuel and cladding, to incorporate the constitutive relations for other me-chanical and irradiation-induced effects). To this purpose, several efforts are currently carriedout at the Politecnico di Milano towards the development and validation of a more completeMPM simulation tool, able to offer a reliable and intrinsic evaluation of the couplings amongthe different physical phenomena in a nuclear reactor, without any claim to perform detailedneutronic, thermo-mechanical and fluid-dynamic simulations, for which dedicated and well-assessed codes already exist.

NOMENCLATURELatin symbols

C stifness tensor [Pa]ci concentration of the ith precursor group

[m−3]Cp specific heat [J kg−1 K−1]Cε1 k − ε model empirical constant [-]Cε2 k − ε model empirical constant [-]Cµ k − ε model empirical constant [-]D neutron diffusion coefficient [m]E neutron energy [MeV ]H active height [m]I identity matrix [-]k turbulent kinetic energy [m2 s−2]keff effective neutron multiplication factor [-]K thermal conductivity [W m−1 K−1]Kt lead turbulent thermal conductivity

[W m−1 K−1]n surface normal unit vector [-]p fluid pressure [Pa]Q volumetric heat source [W m−3]qg coefficients used in Eq. (10) [J m−1]r radial coordinate [m]t time [s]T temperature [K]Tin inlet lead temperature [K]Tref reference temperature used in Eq. (13) [K]T0 reference temperature used in Eq. (3) [K]u displacement vector [m]v velocity vector [m s−1]vg neutron speed of the gth group [m s−1]vin inlet lead velocity [m s−1]z axial coordinate [m]

Greek symbols

αth linear thermal expansion coefficient [K−1]α coefficient used in Eq. (3) [-]β total delayed neutron fraction [-]βi delayed neutron fraction of the ith precursor

group [-]γr radial albedo coefficient used in Eq. (14) [-]γz axial albedo coefficient used in Eq. (14) [-]ε turbulent dissipation rate [m2 s−3]ε strain tensor [-]η lead dynamic viscosity [Pa s]ηT lead eddy viscosity [Pa s]λi dacay constant of the ith precursor group

[s−1]ν average number of neutron emitted per fis-

sion [-]ρ density [kg m−3]ρ0 reference density used in Eq. (3) [kg m−3]σ Cauchy stress tensor [Pa]σε k − ε model empirical constant [-]σk k − ε model empirical constant [-]Σ macroscopic cross-section [m−1]Σa macroscopic absorption cross-section [m−1]Σf macroscopic fission cross-section [m−1]Σs,gg′ macroscopic group transfer cross-section

(from group g to g′) [m−1]Σs,g′g macroscopic group transfer cross-section

(from group g′ to g) [m−1]Σ0 reference macroscopic cross-section used in

Eq. (3) [m−1]φ neutron flux [m−2 s−1]χd,g fraction of delayed neutrons generated in the

gth group [-]χp,g fraction of prompt neutrons generated in the

gth group [-]


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REFERENCES

[1] Generation IV International Forum, “A Technology Roadmap for Generation IV NuclearEnergy Systems”, GIF-002-00, US DOE Nuclear Energy Research Advisory Committeeand the Generation IV International Forum, 2002.

[2] A. Alemberti, J. Carlsson, E. Malambu, A. Orden, D. Struwe, P. Agostini, S. Monti, “Eu-ropean lead fast reactor-ELSY”, Nucl. Eng. Des., 241, 2011, pp. 3470-3480.

[3] A. Cammi, V. Di Marcello, L. Luzzi, V. Memoli, “The Multi-Physics Modelling ApproachOriented to Safety Analysis of Innovative Nuclear Reactors”. In: M. J. Acosta, (Ed.),Advances in Energy Research, Vol. V, Nova Science Publishers, Hauppauge, NY, 2011,pp. 171-214.

[4] L. Luzzi, A. Cammi, V. Memoli, The Multi-Physics Approach Applied to the Modellingand Analysis of the Generation IV Lead Fast Reactor, Nova Science Publishers, Haup-pauge, NY, 2011, pp. 1-43.

[5] M. Sarotto, C. Artioli, G. Grasso, D. Gugiu, “ELSY Core Design Static, Dynamic andSafety Parameters with the Open Square FA”, Technical Report, ENEA FPN-P9IX-006,2009.

[6] COMSOL Multiphysics R© 4.1, User’s Guide, COMSOL Inc., 2010.

[7] J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis, John Wiley and Sons, NewYork, NY, 1976.

[8] G. Rimpault, D. Plisson, J. Tommasi, R. Jacqmin, J. M. Rieunier, D. Verrier, D. Biron,“The ERANOS Code and Data System for Fast Reactor Neutronic Analyses”, Proc. Int.Conf. PHYSOR 2002, Seoul, Korea, October 7-10, 2002.

[9] Nuclear Energy Agency, “The JEF-2.2 Nuclear Data Library”, JEFF Report 17, 2000.

[10] F. Agosti, P. Botazzoli, V. Di Marcello, G. Pastore, L. Luzzi, “Extension of theTRANSURANUS Code to the Analysis of Cladding Materials for Liquid Metal CooledFast Reactors: a Preliminary Approach”, Politecnico di Milano, Technical Report,CESNEF-IN-11-2009, 2009.


Documents

Development of a single-channel multi-physics model for ... · Development of a single-channel multi-physics model for lead fast reactors Manuele Auﬁero, Antonio Cammi, Carlo Fiorina,