DYNAMICS CALCULATIONS OF SOURCE-DRIVEN SYSTEMS IN PRESENCE OF THERMAL FEED-BACK

Transcription

1 DYNAMICS CALCULATIONS OF SOURCE-DRIVEN SYSTEMS IN PRESENCE OF THERMAL FEED-BACK G. Bianchini, M. Carta, A. D Angelo ENEA-Casaccia S.P. Anguillarese, 3 6 S. Maria di Galeria (Roma), Italy carta@casaccia.enea.it P. Bosio, P. Ravetto, M.M. Rostagno Politecnico di Torino, Dipartimento di Energetica Corso Duca degli Abruzzi, Torino, Italy ravetto@polito.it ABSTRACT The paper is devoted to the development of reactor physics methods for the design and safety assessment of subcritical multiplying source-driven systems for actinide and ssion product transmutation and for energy production. The recent advances in the development and implementation of numerical techniques in the frame of a collaboration between ENEA-Casaccia (Rome) and Politecnico di Torino are described. The neutronic models and algorithms in two-dimensional cylindrical geometry and in three-dimensional hexagonal-axial geometry are presented, including full expressions for the discretization expressions. Afterwards, the thermal model is summarized and the coupling with the neutronic calculation is described. Finally, some results are presented for test calculations concerning the system currently proposed as a prototype energy ampli er. Results show the role played by thermal effects in transient situations and evidence their importance in system operation.

2 . INTRODUCTION For the design and the safety assessment of subcritical source-driven systems the development of reliable numerical tools for the solution of the neutron balance equations in multidimensional con gurations and in presence of nonlinear thermal feed-back effects is required. The Italian agency ENEA and Politecnico di Torino have been long collaborating on many aspects of the physics of these innovative neutron multiplying systems. Lately, the reactor physics research groups of these Institutions have been involved in studies on the design of an 8 MW Prototype Energy Ampli er (EAP), with hexagonal symmetry fuel elements containing U/Pu mixed oxides and cooled by liquid Lead-Bismuth eutectic. Two time dependent computational tools for the neutronics of subcritical systems have been developed for two different geometrical con gurations: - a two-dimensional (r-z) model in multigroup diffusion, utilizing a nitedifference implicit Euler scheme for the time variable - a three dimensional (hex-z) model in multigroup diffusion, utilizing a quasistatic scheme for the time evolution. These models have been numerically tested and utilized for various neutronic investigations on subcritical systems. For analyses concerning core dynamics for the safety and the monitoring procedures, non-linear thermal feed-back effects play a very important role. The two-dimensional computational tool has recently been preliminarily coupled with a simple local temperature model,which disregards spatial thermal energy transfers. Results have highlighted some interesting physical aspects. In the present work the neutronic model employed and the algorithms exploited for the numerical solution of the time-dependent multidimensional balance equations are presented. As for thermal-hydraulics a module developed at ENEAsolves the Fourier heat transfer equation in cylindrical coordinates along the radial direction. Once the axial heatux from the clad to the coolant is estimated, the temperature of the coolinguid is evaluated. The inlet coolant velocity is assumed linearly variable within each time step. A simplied heat exchanger model is implemented and either natural or forced convection can be treated. In natural convection condition the coolant velocity variation is evaluated step by step. The temperature calculation is coupled to the neutronic equations through the 2

3 feed-back model. This step of the simulation requires the evaluation of average channel temperatures and the interpolation of the nuclear data of the system to obtain the values corresponding to the current temperature. Test calculations show the role played by the various time scales which are simultaneously present in the evolution of the system: the neutronic scales, which are based on the time constants for the prompt and delayed neutron populations, as well as for the shape in quasi-static schemes, and the thermal scale, based on heat transfer and capacity parameters. Some preliminary dynamic analyses are in progress on typical accelerator-current transient events in the EAP. In particular transients induced by variations of the beam-current intensity, which are specic of the Accelerator Driven Systems (ADS), have been investigated. Previous works showed that the interest in the beam-current transients is not only due to the originality of the subject (of course they do not exist in critical systems), but mainly to the important thermal effects that this kind of transients could generate. It has been recently underlined that beam-trips induce core power level drops that are analogous to those induced by scram events in critical reactors. Therefore, in analogy with the need to limit the frequency of scram events in critical fast-reactors, the interest to reduce the frequency of beam-trip events and to mitigate their thermal effects in fast ADS has been related to the risk that above reactor-core structures and intermediate heat exchangers are damaged by cyclic temperature shocks. A second kind of ADS current transient, called the proton beam-jump, deserves also due consideration. To induce these transients, the full power proton beam is assumed to be suddenly dumped into the reactor. Some preliminary tests about source transient dynamics are illustrated in the present paper. 2. THE NEUTRONIC PROBLEM The numerical solution of the time-dependent neutron problem has been achieved in the multigroup diffusion model in two geometric congurations. For the cylindrical two-dimensional geometry a direct integration in time of the balance equations has been performed, using the usual implicit differencing scheme. In full three-dimensional geometry with realistic hexagonal conguration for the fuel elements a direct integration scheme is of course computationally too heavy. Therefore, a quasi-static technique has been applied. This requires an extension of the standard method developed for initially critical reactors to source-driven 3

4 ! s v u systems. The balance equations for neutron and delayed neutron precursors written in a general geometry take the following form "$#&%('*) +-,/ ;:*<>=@?BADCE5F*G5H P B / ªt«H JI KMLNPORQTSVUXWZY\[^]*_a`cbedBfhgjiMkml*nporqts P±R²r³- Bµ$ & *¹\º@»½¼ ¾ÁÀ wmxypzp{} *~M} 5ƒ* M r ŠhŒ Ž5 e $ - š œmžÿž ÂÄà ÅXÆpÇŸÇ ž œ Ç ÆÉÈ () 2. TWO-DIMENSIONAL CYLINDRICAL GEOMETRY The time discretization for the balance equations for neutrons is performed utilizing the implicit Euler scheme, which is known to be intrinsically stable. The equations for the delayed neutron precursors are formally integrated in time, as: ÊBË Ì ÍÏÎÐeÎZÑÒRÓ ÔBÕÏÖj XØÙ-ØÛÚ ÜÞÝàßcáhâäãpåÛæ-ç@èêéëeì&í(îïMðBñóò-ô\õ5öø MùŸú*ûMüþýjÿ (2) and then the time integral is computed by means of a discrete trapeze formula. Space is discretized utilizing a nite volume scheme. Therefore, the equations for neutrons are integrated over a cylindrical mesh centered at point! #" $&%'#( spanning )+*-,/ #: along the radial coordinate and;=<?>a@cbd E5FHGJI?K along the axis. Each mesh is subdivided into four parts having constant material properties and volumeslnmpoq R,SUTWVYX?Z Z[ZX\. After integration the balance equation is written in the 4

5 Þ 7 6 Ø I ï î j ]^_ `ba#cedgfih7j#kml nopqsrut7v wxyz {~} ƒ ˆ 7 #Šm Œ ŽP š œÿž-5? m 7 «ª + ø?üþý7ÿ ± ²7³# 5µ P ¹ º5»½¼ ¼ ¾ À ÁÂÃÄ Å«Æ7Ç-ÈmÉ Ê?ËÌ[Í5ÎÐÏÑ7Ò Ó Ô ÕeÖ? ØÙ Ú Û«Ü ÝßÞàáâ-ã&ä7å#æ5ç èéêpéëíì ßøùúû ðñòôóöõ form of an algebraic equation, as "!$# %'& ( )+*,-/ :5;=< >@?ABC?DEFHGJI$KL'M NOPQ+RTSVU$W XY+Z[\]_^a`cbed5fJgh jklnm i prqstu ov wyx z {}+~T ƒ where superscriptˆ identi es the time instant considered. It is worth reporting the full expression for the coefšcients, which, of course may be time-dependent owing to control operations, accident evolution or feed-back effects, for the different positions of the mesh within the spatial domain, as: Diffusion term (3) Œ Œ ŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒ ŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŽ J š Internal º¼»¾½À ±ÁJÂÃÅÄÆcÇaÈyÉaÊËÍ̱ΠÏJÐÑ/ÒÔÓ Õ Ö ØJÙaÚ mesh ß àjáâ Û ÜyÝ ãåä æ è±é ê=ë ì íjîïhðañ òaóµô±õ ö ç ùøûúýü þtÿ "! # $&% '( )+*-,. n o&p q r s+t-u œ ž Ÿ J c aªy«a ± ²'³µ ± ¹ g hi / :<;=>@?7ABCED FHG JLKNMPO7Q RTSVU W<XZY[]\ ^_ `TacbEd ehf kml v w7xy zt{} 5~ " ƒ TÊ 7Š 9ŒŽ 7 9 Ž šœ TžmŸ 9 Ž ª «& "± ³& µ ² + º&» ¼½ ¹ ¾ ÁÀ Â&Ã Ä ÆÈÇÁÉ Å Ê&Ë9Ì ÎÐÏÒÑ Í ÓÕÔ Ö (4) 5

6 ââââââââââââââââââââá ã äââââââââââââââââââââ ÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÙ Û ÜÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ ñ 6 Þ º Õ Y Lower boundary Þ&ß9à Ý âäãæå á è&é9ê ç ìîíðï ë ò7óõôö7 øù úüûýþeÿ V#W/X! #"$&%(')*!+-,/ :<;!=>@?BA C&DFEHGJI K-L#MONQPSR3T5U Z\[ ] ^_` a bcd e f!ghh#ioj5k!l#monpo q&rts-uovsw3xhy#z/{ } ~ (5) ƒ Š# Œ ˆ Ž # š #œ ž Ÿ # «ª# / -±³²3 -µ/! «¹»½¼ ¾ Û#Ü(Ý ¾ÁÀ!Â#ÃOÄpÅ ÇtÈ#É/ÊÌË ÍÏÎtÐ-ѳÒ3Ó-ÔÕ@ÖQ SØ Ù5Ú Æ ß\à Upper boundary å æç#è é êìë í ;=<>? f=gh e jlkm i o=pq n sltdu r î ï ðñóòoôáõ ö ø#ùúüû/ýÿþ E F G HJI KL MON PRQTS U VWYX Z []\_^=`acbd v w x yz{3 }~ Oƒ T ˆ ŠŒ Ž "! #%$'& ( )*+-,/ "9 _ =œ ž š Ÿl = ª «= ± ª² ³= µ º¹¼»D½J¾Y À ÄÅÆ ÇcÈÉËÊ Ì%Í'ÎÏ Ð Ñ ÒÓ3Ô Ác à ÖØ ÚÙ ýþ ÿ ÙÜÛ Ý Þ ßáà åçæè éëê ìoí îdïçð%ñóò ô õö ùø úyû3ü âcãä (6) 6

7 W t u B C C Right boundary EFG D IKJML H OPQ N SUTV R >?A@!#"%$&(') *,+.-/2.354#687:9;=< qrs XYZ\[]^#_%` acb#d.ef2g.h5i#j8k:lnmpo wxy v {U } z n. ƒ ˆ #Š% Œc #Ž. : n. 5 ~ š œž Ÿ # % A Ä.ª«: n. ± ²š³ µ ¹ º»¼½ ¾ UÀ ÂÃÄ Á ÆÈÇÊÉ Å ËÌÍ ÏÑÐÊÒ Î ÓÔ5Õ ÑØÚÙÛÝÜ2Þ5ß#à%á Ö ã\äåæèç é8êìëîíðïòñôó:õnö øèù â ý,þÿ ú(ûü (7) Left boundary (axis) "! $%&('*),+.-/ ;: <>=@?BA # DFE G HIJK LNM"O }~ { "ƒ œ žÿ Q@R SUTV@WX P Y[Z@\ ] ^`_abdcfe@gh i j*kmln o p`qsrutwv xzy ˆ.Š ŒŽ * B Fš ª «²± ³ µ º¼»¾½ ¹ sàá à  (8) 7

8 ÅÅÅÅÅÅÅÅÅÅÅÅÅÄ Æ ÇÅÅÅÅÅÅÅÅÅÅÅÅÅ { Ð ð C ý ü Z Y» º ÉÊË È Í Ì Î Ï ÑÒÓ ÕNÖ" Ô éêë è í î"ï ì ÿ þ Left lower corner Ø Ù.Ú;ÛwÜ@ÝÞ ßáà@â ã ä`åsæç ñòó ô@õö øùûú "!$# %'& () *,+ -. /,2 3 4 (9) Upper left corner 9 :; < = >@?BA \] ^_ [ ` abb } ~ ƒ ˆ Š Œ D'EGFIHKJML'N O PIQSRUTWVUX c degfih'j k limsnuowpgq rts'uwvyxkz Ž š œ ž Ÿ ª«I K ± ² ³Sµ ' ¹ () 8

9 ˆ ½½½½½½½½½½½½½½¼ ¾ ½½½½½½½½½½½½½½ Á À É Ò Û ÿ é ü ý m Á' à ÅÇÆ È Ä Ê'Ë Ì ÎÐÏ Ñ Í Ó'Ô Õ Ò ØÚÙ Ö þ ÿ Lower right corner ) ù úkû Ü'ÝwÞàß'á âsãåä æèçêégëíìîgïñðóòõô ö Uø!#"%$& '( *,+ -/ :9 ; 7 =?>A@ < BDC E G?HJILKM NOQP F j kl SUT VWX Y[ZU\ R ]_^ ` a bced#f gih nporqlst uvw yuz { }~ x _ ƒ/ () Upper right corner µ Å Ê Î ŠD Œ Ž ² ³ / š œ žÿ U % ª «e # i± Ḑ¹ º» ¼¾½À ÂDÃ Ä ÆÈÇÀÉ ËDÌ Í ÏÈÐ Ñ ÓÔ Õ Ö iøù ÚÛQÜ Ý Þß àuá âã#ä å æ çè ê,ë:ìîíðïñ ò ódô õ ö Èø údû ü ù þ ý!"# $%'&)(*,+ -./ : ; <>=?A@CB DE JLK MNO PQRTS'UWV XZY\[] F GHI (2) Removal term^ _a` b cadfehg i k lnm o j qsr>thu vwx p z y {n }~ > h ƒ ˆ Š ŒŽ > h šž œ ž (3) PromptŸ ssion term ª «±a²³µ ' º¹¼»¾½À ÂÁ à ŠÆ,Ç È É Ä ËÍÌÏÎÑÐ ÒAÓÕÔÂÖ ØÙ Ê ÛªÜ Ú âäãñåæ¾çàèâé êë ñ ò ì íïî ð ó ô õ ö øúùñûü¾ýàþÿ 9 ÝnÞ ß á à (4)

10 { F É Group-to-group transfer term # $&%('*),+-.!" /32 ; < : =?>A@CBDE G3H IJLKNM OQP7R S T U(V WCXYZ [ \3] c d ^_ `Qa b e(fhgjik m l (5) Delayed emission term n o,p q rs putwv xzy ~} Q h ƒ A L u ĈŠŒ Ž? u š C œ žÿ Ž h 9 «ª j ± ³µ ² 9 ¹ ºh»9¼½ ¾«Ã Ä ÀQÁ  Å,Æ Ç È ÊË9Ì7Í ÎÐÏ ÑÒ ØÓÕÔ Ö Ù,ÚÛ ÝµÞ Ü å æ çhèêé ëaìlíïîað ñ òqóõô7öø úùüûþýlÿ ß àzá â ãä (6) "!#$ %'& () *+-,/ : ;5<>=@?ACBEDGF3H5IJ K L MN O5PQCR-SGTVU5WX Y External source term (7) The initial conditions are determined through the solution of a stationary sourcedriven problem for the initial system. This operation requires iterations with the thermal calculations, to construct a consistent equilibrium including nonlinear effects. The solution of the algebraic problem is attained by means of a classic scheme, involving an external iteration among energy groups, up to convergence. The system of equations for each group can be obtained through different algorithms, i.e., Gauss-Seidel, overrelaxationz\[or Generalized Minimal Residual (GMRES).]@] A determination of the effective multiplication constant is also foreseen. This calculation is carried out by a standard^ssion source iteration procedure. 2.2 THREE-DIMENSIONAL HEXAGONAL-AXIAL GEOMETRY In three-dimensional geometry with hexagonal elements, the spatial discretization is performed in the transverse plane on a triangular mesh structure,

11 in abced ` e>fhg-i order to allow the possibility of spatial re_nement. Discrete balances are jlk"monqp\rts vwyxez {}h~> u "ƒv ˆ Š Œ G Ž Œ š œ ž ŸV > o ˆª written for each triangular prism, namely: (8) where index«numbers the lateral faces of the prism and its bases,,, and indicate the triangle edge, the height of the mesh and its basis area, respectively, and subscript ²>³h -µ the mesh average value. Using a±nite-difference scheme, boundary current l " o¹»º terms ¼¾½À are Á>Âyà approximated Ä with the following formulae for inner meshes: Ø-ÙÛÚ-Ü ÝlÞàßâá ã äæåçç èêélë ì íçî ïñðóòõô\öø ùœúüûêýçþœÿ ÅÇÆ ÈÊÉÇËyÌšÍÀÎšÏ Ð>ÑÊÒÀÓ ÔÖÕ (9) while radiation boundary conditions are applied for boundary meshes. The timeintegration is performed using the generalization of the quasi-static technique, which has proved to be a very ef cient methodology for conventional systems and can be extended to source-driven systems. It is worth to acknowledge the signi cant contributions given by Jacques Devooght to the eld of quasi-static methods, for the development of effective computational tools and, especially, for establishing the procedure on consistent and sound theoretical bases. It is with great emotion and gratitude that the whole reactor physics community remembers and celebrates him at this Conference, both as a scientist and a most dear colleague and friend. In quasi-static procedures, a separation of the neutron distribution group vector in the product of an amplitude function and a shape is introduced, as:! "$#%&(') (2) The idea underlying the whole method relies in the representation of the evolution of the phenomenon along two time scales, a rapid one for the amplitude and a much slower one for the shape. Consequently, the computationally heavy calculation of the shape is performed only few times during the transient simulation. After introduction of the separation into the balance equations, a projection on a suitable weight function is carried out, in order to obtain the model for the fast-evolving amplitude, whose coef* cients depend on the shape.

12 Two problems need to be solved, when applying quasi-static techniques to ADS, i.e.: * the de+ nition of the reference system, * the de- nition of the weight function to be used for the projection of the balance equations. As far as the. rst problem is concerned, the most reasonable choice seems to be the distribution for the initial source-driven system itself, where the neutron shape is of course very different from the eigenstate of a critical system. The second problem can be solved in different ways. The converged solution of the problem is independent of the chosen weight/ however, the number of recalculations of the shape function to correctly represent the evolution of the system may be signi cantly affected by its choice. The problem connected to the extension of the quasi-static method to ADS was addressed in ref. 3 and is further discussed in the present conference THE THERMAL-HYDRAULIC MODEL Owing to the physical importance of non-linear effects in core dynamics, the neutronic calculation must be coupled with a thermal calculation. Within the on-going collaboration between ENEA and Politecnico di Torino, the neutronic codes described above are being coupled with a thermal-hydraulic code developed by ENEA.4 The thermal code can represent a single reactor channel with fuel, clad and coolant. It can consider a central fuel hole (if any) and a fuel-clad gap. The fuelclad heat exchange is taken into account by a temperature dependent coef5 cient, calculated at every time step, through the conductivity of the gas 6 lling the gap. The clad-coolant heat exchange is evaluated as well, taking into account the Nusselt number of the coolant 7 uid. The code does not consider the axial heat propagation in the fuel and in the clad assuming the temperature axial gradient negligible with respect to the radial one. All the materials properties (speci8 c heat and density) can be assigned as temperature dependent. The code solves the time dependent Fourier equation in the fuel and clad in a one-dimensional cylindrical con9 guration at any axial mesh point. The thermal source in fuel and, if signi: cant, the gamma heating, present also in the clad and in the coolant, is evaluated from the ; ssion power given by a neutronic calculation. The coolant temperature is considered as a boundary condition (in 2

13 the < rst step a guess is used). Once the axial clad temperatures are known, it is easy to carry out the calculation of the clad-coolant heat = ux and of the axial coolant temperatures, at each axial position> hence, these temperatures are used as boundary conditions for the fuel-clad temperature calculation in the next time step. A high accuracy Chebyshev (Fourier like) basis is employed to approximate the exponential matrix, according to a numerical method elaborated at ENEA,?@ allowing the use of any time-step size. The code carries out only a single-phase calculationa this seems acceptable for core dynamics investigations because the code was conceived for ADS lead-bismuth cooled reactors and the boiling point of the molten lead-bismuth, is very high (943 K at atmospheric pressure), even higher than the stainless steel melting point (643 K), which constitutes the clad material. The code takes into account the forced coolant circulation condition, which means that the velocity must be given as an input datum. As an alternative, natural coolant convection can also be considered. In that case the velocity is estimated step by step as a result of the thermal conditions of the whole coolant circuit (density difference between the hot and cold leg). A simplib ed heat exchanger model is implemented, to take into account the heat losses from the primary to the secondary circuit. When the forced-circulation option is considered, the thermalhydraulic calculations are made by using a linearly coolant variable velocity inside the time step. On the contrary, if the natural convection option is chosen, the velocity is kept constant during the time step. The thermal calculation is coupled with the neutronic calculation as follows: - a thermal-hydraulics time step size is evaluatedc - a series of neutron calculations are carried out up to reach the end of the thermalhydraulics time stepd - the radial average of the axial proe le of the power density is evaluated for a F xed number of reactor zones, characterized by a thermal channelg - the axial power distributions are input to the thermal code which performs the temperature calculations for each channelh - channel average temperatures are assigned to each reactor zone, and used to modify cross sections, according to assigned interpolating functions, and a new neutronic calculation is initiated. At present, simple temperature linear functions are assumed to update cross sections. However, work is ongoing at ENEA in order to rei ne the cross sections interpolation scheme. 3

14 The steady-state conj guration is determined by starting from a tentative temperature distribution inside the system and performing an iteration sequence with the neutronic calculation at all corresponding to an effective dynamic calculation with a K xed source until an asymptotic steady condition is reached. 4. SELECTED RESULTS A few results are now illustrated in order to test the performance of the codes and to present preliminary applications to EAP conl gurations. Some test are presented for a homogeneous system in a one-group cylindrical model, considering only a linear feed-back on the capture cross section with a temperature coefm cient NPORQTSVUXWZY\[^]`_!acbedgf (hence the feed-back is negative). The system is 386 cm high and has a radius of 8 cm and it is characterized by a multiplication factor hikj\jmlongprqtsuqwvux. Only one channel is considered for the thermal-hydraulic calculation. The y rst transient concerns a source trip started by a step reduction of the source to 5% of its full-power value, followed by a ramp to restore its initial value. Figure shows the source and power transientz the following Figs. 2 through 4 report the evolutions of the temperatures in selected positions of the fuel, clad and coolant. In the following transient the source is switched off for s and then restored by a step to its initial value (see Figs. 5-8). Two different source oscillations followed by the re-establishment of the steadystate value are considered in the following transients: Figs. 9 through 2 show the response of the system to an oscillation around a mean value smaller than the initial steady-state value, while Figs. 3 through 6 consider an oscillation above the initial value. As can be seen, interesting drift phenomena appear both in the power as well as in the temperature behaviors. Results for a EAP con{ guration in the one-group model are then illustrated ( u}3~c~ X ƒw t u w ). In particular the effect of different temperature coefˆ cients are evidenced by comparing the two sets of results reported in Figs. 7 and 8 for Š ŒŽ V u \ ^ `! š œgž and in Figs 9 and 2 for Ÿ «ª \ c `! ±e²!³. The effect of the temperature coef cient on the neutron distribution can be observed in Fig. 2 through 23. 4

15 T [K] T [K] S(t)/S (a) P(t)/P (b) Figure. Power transient (b) following a source trip as indicated (a) (step reduction to 5% followed by a ramp up to re-establishing the initial value). 2 8 (a) (b) Figure 2. Fuel temperature evolution at the channel axis (a) and at the external boundary (b) on the midplane of the system for the transient of Fig.. 5

16 T [K] T [K] T [K] (a) (b) Figure 3. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig Figure 4. Coolant temperature evolution on the midplane of the system for the transient of Fig.. 6

17 T [K] T [K] S(t)/S (a) P(t)/P (b) Figure 5. Power transient (b) following a source trip as indicated (a) (switch-off followed by a step restoration to the initial value) (a) (b) Figure 6. Fuel temperature evolution at the channel axis (a) and at the external boundary (b) on the midplane of the system for the transient of Fig. 5. 7

18 T [K] T [K] T [K] (a) (b) Figure 7. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig Figure 8. Coolant temperature evolution on the midplane of the system for the transient of Fig. 5. 8

19 T [K] T [K] S(t)/S (a) P(t)/P (b) 5 5 Figure 9. Power transient (b) following a source trip as indicated (a) (oscillation around a mean value smaller than the initial value followed by the restoration of the initial value). (a) 88 (b) Figure. Fuel temperature evolution at the channel axis (a) and at the external boundary (b) on the midplane of the system for the transient of Fig. 9. 9

20 T [K] T [K] T [K] (a) (b) 5 5 Figure. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig Figure 2. Coolant temperature evolution on the midplane of the system for the transient of Fig. 9. 2

21 T [K] T [K] (a) (b) S(t)/S P(t)/P Figure 3. Power transient (b) following a source trip as indicated (a) (oscillation around a mean value larger than the initial value followed by the restoration of the initial value). (a) (b) Figure 4. Fuel temperature evolution at the channel axis (a) and at the external boundary (b) on the midplane of the system for the transient of Fig. 3. 2

22 T [K] T [K] T [K] (a) (b) Figure 5. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig Figure 6. Coolant temperature evolution on the midplane of the system for the transient of Fig

23 Neutron Flux S(t)/S (a) P(t)/P (b) Figure 7. Power transient (b) following a source trip as indicated (a) in the EAP, assuming µp ¹»º½¼«¾u \ÀcÁeÂgÃcÄeÅ!Æ. 2.5 x z [cm] 5 r [cm] Figure 8. Steady-state Ç ux shape for the EAP, assuming ÈPÉ ÊË̻ͽÎuÏuÐ\ÑcÒÔÓÖÕc eøgù

24 Neutron Flux S(t)/S (a) P(t)/P (b) Figure 9. Power transient (b) following a source trip as indicated (a) in the EAP, assuming ÚPÛ Ü Ý޻߽à«áuâ\ãcäeågæcçeè!é. 2.5 x z [cm] 5 r [cm] Figure 2. Steady-state ê ux shape for the EAP, assuming ëpì íîï»ð½ñuòuó\ôcõôöö cøeùgú

25 Neutron Flux Neutron Flux ÿ 2.5 x 5 2 α=5. -6 α=5. -5 α= r [cm] Figure 2. Steady-state radial û ux distributions at height üþý different values of the capture temperature coef cient. for x α=5. -6 α=5. -5 α= r [cm] Figure 22. Steady-state radial ux distributions at height different values of the capture temperature coef cient. for 25

26 Neutron Flux 2 x 4 α=5. -6 α=5. -5 α= z [cm] Figure 23. Steady-state axial ux distributions at radius values of the capture temperature coef cient., for different CONCLUSIONS Two computational tools for the dynamics of source-driven systems are presented in this paper. The! rst one concerns two-dimensional cylindrical con" gurations and the solution of the neutronic equations is obtained by a direct time discretization by means of an implicit Euler scheme. The second one treats three-dimensional hexagonal-axial con# gurations, and the quasi-static technique is employed. Neutronic modules are coupled with a thermal-hydraulic channel code adapted to compute temperature distributions in a cylindrical-axial channel geometry with lead-bismuth as a coolant. These codes are suitable to be used to analyze core dynamics in ADS such as the Energy Ampli$ er Prototype. Preliminary results presented show the importance of non-linear effects in transient situations for Accelerator Driven Systems. Work is going on to improve the numerical performance of the codes. The collaboration between ENEA and Politecnico will include also systematic calculations of typical reference transients for the EAP and parametric analyses. A benchmark procedure of the computational tools developed is also foreseen as a near-future activity. 26

27 ACKNOWLEDGMENTS This work has been performed in collaboration between ENEA-Casaccia (Rome) and Politecnico di Torino and it is % nancially supported by the Italian Ministry of Research in the frame of the scienti& c program on source-driven systems. REFERENCES. C. Rubbia, et Al., Conceptual Design of a Fast Neutron Operated High Power Energy Ampli' er, CERN/AT/ 95-44, Geneva (995). 2. P. Bosio, et Al., Analysis os Some Dynamic Aspects of Subcritical Systems, International Conference on Future Nuclear Systems, GLOBAL 99, Jackson Hole (999). 3. G.G.M. Coppa, G. Lapenta, P. Ravetto, M.M. Rostagno, Three-Dimensional Neutron Analysis of Accelerator-Driven Systems, International Conference on Mathematics and Computation, Reactor Physics and Environmental Analyses in Nuclear Applications, Madrid (999). 4. M. Carta, et Al., Control of Subcriticality Level in Accelerator Driven Systems: Harmonic Modulated Source - Spatial Source Jerk Intercomparison, International Conference on Mathematics and Computation, Reactor Physics and Environmental Analyses in Nuclear Applications, Madrid (999). 5. G. Bianchini, M. Carta, A. D Angelo, TIESTE-MINOSSE a Single Channel Thermal-Hydraulics and Point Kinetics Code for ADS, ENEA Technical Note ERG/SIEC DT-SDA-8 (999). 6. W. Maschek, B. Merk, H.U. Wider, Comparison of severe Accident Behavior of Accelerator Driven Subcritical and Conventional Critical Reactors, OECD/NEA Workshop on Utilization of High Power Accelerators, 3-5 October, Mito (Japan) (998). 7. K. Kasahara, Failure Modes of Elevated Temperature Structures Due to Cyclic Thermal Transients, OECD/NEA Workshop on Utilization of High Power Accelerators, Mito (Japan) (998). 8. P.W.P.H. Ludwig, P. Wakker, A.H.M. Verkoojen, Static and Transient Thermo-Hydraulic Behaviour of a Fast Energy Ampli( er Computed with a CFD Computer Program, Ninth International Conference on Emerging Nuclear Energy Systems, Herzliya (Israel) (998). 9. P.H. Wakker, Thermal Hydraulic Simulation of the Steady State 27

28 and Transient Behaviour of the Fast Energy Ampli) er, Ninth International Conference on Emerging Nuclear Energy Systems, Herzliya (Israel) (998).. G. Monegato, Fondamenti di Calcolo Numerico, Levrotto&Bella, Torino (99).. Y. Saad, M.H. Schultz, GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems, SIAM J. Sci. Stat Comput., 7, pp (986). 2. E. Mund, this Conference (2). 3. F. Norelli, Heat Diffusion Problem Solved by the Extension Method in Cylindrical Geometry: the Code Efesto, ENEA Report, RT-WAA- (993). 28