DYNAMICS CALCULATIONS OF
SOURCE-DRIVEN SYSTEMS IN PRESENCE OF
THERMAL FEED-BACK
G. Bianchini, M. Carta, A. D’Angelo ENEA-Casaccia
S.P. Anguillarese, 301
00060 S. Maria di Galeria (Roma), Italy [email protected]
P. Bosio, P. Ravetto, M.M. Rostagno Politecnico di Torino, Dipartimento di Energetica
Corso Duca degli Abruzzi, 24 10129 Torino, Italy
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
1. 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
non-linear 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 80 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
nite-difference implicit Euler scheme for the time variable
- a three dimensional (hex-z) model in multigroup diffusion, utilizing a quasi-static 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 ENEA solves the Fourier heat
transfer equation in cylindrical coordinates along the radial direction. Once the axial heat ux from the clad to the coolant is estimated, the temperature of the
cooling uid is evaluated. The inlet coolant velocity is assumed linearly variable
within each time step. A simpli ed 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.
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 speci c 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 con gurations. 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 con guration 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
systems.
The balance equations for neutron and delayed neutron precursors written in a general geometry take the following form
! "$#&%('*) +-,/.1032547698;:*<>=@?BADCE5F*G5H H I J KMLNPORQTSVUXWZY\[^]*_a`cbedBfhgjiMkml*nporqts s u v wMxyPzP{}|*~M}5*M r-@Xh5e$B@- M ¡¢ £P¤B¥ ¦¨§/©1ªt«¬ ®¯°P±R²r³-´Bµ$¶ ·&¸*¹\º@»½¼¿¾ÁÀ ÂÄÃÅXÆpÇÇ Ç¡ÆÉÈ (1)
2.1 TWO-DIMENSIONAL CYLINDRICAL GEOMETRY
The time discretization for the balance equations for neutrons is performed utilizing the implicit Euler scheme, which is known to be intrinsically stable. The equations for the delayed neutron precursors are formally integrated in time, as:
ÊBËÌ¡ÍÏÎÐeÎZÑÒRÓÔBÕÏÖj×XØÙ-ØÛÚÜÞÝàßcáhâäãpåÛæ-ç@èêéëeì&í(îïMðBñóò-ô\õ5öø÷Mùú*ûMüþýjÿ
(2) and then the time integral is computed by means of a discrete trapeze formula. Space is discretized utilizing a nite volume scheme. Therefore, the equations for
neutrons are integrated over a cylindrical mesh centered at point! #"$&%'#( spanning )+*-,/.10-2 3547698#:
along the radial coordinate and;=<?>A@CBDE5FHGJI?K along the axis.
Each mesh is subdivided into four parts having constant material properties and volumesLNMPOQ
R ,
form of an algebraic equation, as ]^_ `ba#cedgfih7j#kml nopqsrut7vwxyz|{~} 7#m P| - 5¡ ¢?£¤ ¥m¦§¨7©«ª ¬+®¯° ±²7³#´5µ ¶·¸P·¹|º5»½¼ ¼¿¾ ÀÁÂÃÄ Å«Æ7Ç-ÈmÉ Ê?ËÌ[Í5ÎÐÏÑ7Ò¿ÓÔ ÕeÖ?×ØÙ Ú Û«Ü ÝßÞàáâ-ã&ä7å#æ5ç èéêPéëíì î ï ðñòôóöõ ÷ßøùúû ø?üþý7ÿ "!$#%'& ()+*,-/. 0 12 354 6 7 89:5;=< >@?ABC ?DEFHGJI$KL'M
NOPQ+RTSVU$WXY+Z[\]_^a`cbed5fJgh i jk lnm o prqstu v wyxz|{}+~T (3) where superscript identi es the time instant considered.
It is worth reporting the full expression for the coef cients, 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 Internal mesh J ¡£¢¥¤¦J§¨c©aªy«a¬¯®±°²'³µ´±¶¸· ¹ º¼»¾½À¿±ÁJÂÃÅÄÆcÇaÈyÉaÊËÍ̱ÎÏJÐÑ/ÒÔÓÕ×Ö ØJÙaÚ Û ÜyÝ Þ ßàJáâ ãåäæ ç è±éê=ë¥ìíJîïHðañòaóµô±õö ÷ùøûúýüþTÿ "! # $&%'( )+*-, . /0132547698:<;=>@?7ABCEDFHG I JLKNMPO7QRTSVUW<XZY[]\^_`TacbEdeHf ghi j kml n o&pqr s+t-u v w7xyzT{}|5~" TE79 79Tm ¡£¢¥¤¦§9¨©ª «¬&®¯°"± ² ³&´µ¶ ·+¸ ¹ º&»¼½ ¾¥¿ÁÀ Â&ÃÄÅ ÆÈÇÁÉ Ê&Ë9ÌÍ ÎÐÏÒÑ ÓÕÔÖ× Ø (4)
Lower boundary ÙÚ Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Û Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú Ú ÚÜ Ý Þ&ß9àá âäãæå ç è&é9êë ìîíðï ñ ò7óõôö7÷øùúüûýþEÿ ! #"$&%(')*!+-,/.103254 6 798:<;!=>@?BAC&DFEHGJIK-L#MONQPSR3T5U V#W/X Y Z\[ ] ^_`a bcd e f!gHh#iOj5k!l#mOnpoq&rts-uOvSw3xHy#z/{ | }~ # # # ¢¡¤£¥¦#§¨ ©«ª#¬¯®/°-±³²3´-µ/¶1·!¸«¹ º »½¼¿¾ ¾ÁÀ!Â#ÃOÄpÅÇtÈ#É/ÊÌËÍÏÎtÐ-ѳÒ3Ó-ÔÕ@ÖQ×SØÙ5ÚÆ Û#Ü(Ý Þ ß\à (5) Upper boundary áâ â â â â â â â â â â â â â â â â â â â ã â â â â â â â â â â â â â â â â â â â âä å æç#èé êìë¤í î ïðñóòOôÁõ÷öø#ùúüû/ýÿþ "!#%$'&()*+-,/.1032 456 7 8"9 : ;=<>? @BADC E FGHJIKLMONPRQTSUVWYXZ []\_^=`acbd e f=ghi jlkm n o=pqr sltDu v wxyz{3|}~O T1 _ = l ¡ ¢=£¤¥ ¦¨§ª© «=¬® ¯±°ª² ³=´¶µ· ¸º¹¼»D½J¾Y¿¶ÀÁcÂà ÄÅÆÇcÈÉËÊÌ%Í'Î1ÏÐÑÒ1Ó3Ô Õ ÖØ×ÚÙ ÙÜÛÝ¶Þ ßáàåçæèéëêìOíîDïçð%ñóòôõö÷ùøúYû3üâcãä ýþÿ (6)
Right boundary !#"%$&(')*,+.-0/21.354#687:9;=< >?A@ B C D EFGH IKJML N OPQR SUTV W XYZ\[]^#_%`acb#d.e0f2g.h5i#j8k:lnmpo qrs t u v wxyz {U|} ~ n. #%c#.:n.5 ¡¢#£%¤¥¦A§¨A©.ª«:¬n.®¯°± ²³´µ¶·¹¸ º »¼½¾ ¿UÀ Á ÂÃÄÅ ÆÈÇÊÉ ËÌÍÎ ÏÑÐÊÒ ÓÔ5ÕÖ ×ÑØÚÙÛÝÜ2Þ5ß#à%áâ ã\äåæèçé8êìëîíðïòñôó:õnö÷øèùý,þÿ ú(ûü (7)
Left boundary (axis)
"! # $%&('*),+.-/012435 687.9;:<>=@?BA C DFE G HIJK LNM"O P Q@R SUTV@WXY[Z@\ ]
^`_abdcfe@ghij*kmln op`qsrutwvxzy
{ |}~ " . 8@ *>@B F ¡£¢ ¤ ¥¦§¨ ©ª « ¬®¯ °²±´³ µ¶¸·¹ º¼»¾½ ¿sÀÁ à (8)
Left lower corner ÄÅ Å Å Å Å Å Å Å Å Å Å Å Å Æ Å Å Å Å Å Å Å Å Å Å Å Å ÅÇ È ÉÊËÌ Í Î£Ï Ð ÑÒÓÔ ÕNÖ"× Ø Ù.Ú;ÛwÜ@ÝÞßáà@â ã ä`åsæç è éêëì íî"ï ð ñòóô@õö÷øùûú ü ý þ ÿ "!$# %'&() *,+-. /10,2 3 4 (9)
Upper left corner
56 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 6 6 6 6 6 68 9 :;<= >@?BA C D'EGFIHKJML'NOPIQSRUTWVUX Y Z [ \]^_ `aBb c d1egfih'jklImSnUoWpGq rts'uwvyxKz { |}~ ¡ ¢¤£¦¥¨§1©ª«I¬K ®°¯±²´³Sµ¨¶'·¹¸ º » (10)
Lower right corner ¼½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ¾ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½¿ À Á'ÂÃÄ ÅÇÆ È É Ê'ËÌÍ ÎÐÏ Ñ Ò Ó'ÔÕÖ ×ØÚÙ Û Ü'ÝwÞàß'áâSãåäæèçêégëíì1îgïñðóòõô¤ö1÷Uø ùúKû ü ý þ ÿ !#"%$&'( ) *,+-/. 0 1234 56 7 8:9;< =?>A@ BDCEF G?HJILKMNOQPR
SUTVWXY[ZU\ ]_^` abced#fgih jkl
m nporqLstuvwyUz{|}~ _/x
(11)
Upper right corner
D /¡ ¢¤£U¥ ¦¨§%© ª¬«e®#¯°i± ²³´ µ ¶ · ¸D¹º» ¼¾½À¿ Á ÂDÃÄÅ ÆÈÇÀÉ Ê ËDÌÍÎ ÏÈÐÑ Ò ÓÔ ÕÖ×iØÙÚÛQÜÝ¡ÞߤàUá âã#äå æçè é ê,ë:ìîíðïñ ò óDôõö ÷Èø ù úDûüý þ ÿ !"# $%'&)(*,+-.0/214365 798:
; <>=?A@CBDE FJLKMNOPQRTS'UWVXZY\[]GHI
(12) Removal term^ _a`bcAdfehgij klnmop qsr>thuvwxy z {n|}~ >h >h (13)
Prompt ssion term
¡£¢¥¤¦¨§ ¢ª© « ¬®°¯±a²³µ´'¶¸·º¹¼»¾½À¿ÂÁÃÄ Å¥Æ,Ç£ÈÉÊ ËÍÌÏÎÑÐ ÒAÓÕÔÂÖ×ØÙÚ ÛªÜ ÝnÞßà á (14) âäãÑåæ¾çÀèÂéêë ìíïî ð ñ¥ò ó£ôõö ÷ øúùÑûü¾ýÀþÿ
Group-to-group transfer term !" # $&%('*),+-. /1032 457698: ; <=?>A@CBDE F G3H IJLKNMOQP7R S TU(V WCXYZ [ \3] ^_`Qab c de(fhgjikl m (15)
Delayed emission term
n o,p qrs putwvxzy { |~}QhA LuC? uC 1¡ ¢£¥¤h¦9§¨©«ª¬ ® ¯j°±² ³µ´ ´·¶9¸¹¥ºh»9¼½¾«¿ÀQÁÂ Ã Ä Å,ÆÇÈ É ÊË9Ì7Í¥ÎÐÏÑÒÓÕÔÖ × Ø Ù,ÚÛÜ ÝµÞ ß àzá âãä å æ¡çhèêéëAìLíïîAðñòQóõô7öø÷úùüûþýLÿ (16)
External source term
"!#%'& ()*+-,/.$ 0132547 6 89:;5<>=@?ACBEDGF3H5IK J LMNO5PQCR-SGTVU5WY X (17)
The initial conditions are determined through the solution of a stationary source-driven problem for the initial system. This operation requires iterations with the thermal calculations, to construct a consistent equilibrium including non-linear 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
in order to allow the possibility of spatial re_ nement. Discrete balances are
written for each triangular prism, namely:
` ab cEd e>fhg-i jlk"monqp\rts u vwyxEz|{}h~> "V G V¡£¢¤>¥ ¦o§©¨ª (18)
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 ° the mesh average value. Using a ± nite-difference
scheme, boundary current terms are approximated with the following formulae for inner meshes:
²>³h´-µ
¶l·"¸o¹»º ¼¾½À¿Á>Âyà ÅÇÆÈÊÉÇËyÌÍÀÎÏÐ>ÑÊÒÀÓ ÔÖÕ ×Ä (19) Ø-ÙÛÚ-Ü
ÝlÞàßâá ã äæåÇçèêélë íÇîïñðóòõô\öø÷ùúüûÊýÇþÿì
while radiation boundary conditions are applied for boundary meshes. The time-integration 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: ! "$#%&(') (20) 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.
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 signi0 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.132
3. 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 clad gap. The fuel-clad heat exchange is taken into account by a temperature dependent coef5 cient,
calculated at every time step, through the conductivity of the gas6 lling the gap.
The clad-coolant heat exchange is evaluated as well, taking into account the Nusselt number of the coolant7 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
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 (1943 K at atmospheric pressure), even higher than the stainless steel melting point (1643 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 thermal-hydraulic 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 thermal-hydraulics 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.
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 aK 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 180 cm and it is characterized by a multiplication factor hikj\jmlongprqtsuqwvux . Only one channel is considered for the
thermal-hydraulic calculation.
They 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 1 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 1 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 steady-state value are considered in the following transients: Figs. 9 through 12 show the response of the system to an oscillation around a mean value smaller than the initial steady-state value, while Figs. 13 through 16 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~Xw tu w ). In particular the effect of different temperature coef cients are
evidenced by comparing the two sets of results reported in Figs. 17 and 18 for
Vu\^`!g
and in Figs 19 and 20 for ¢¡¤£¥§¦©¨«ª¬\c®`¯!°±e²!³ . The
effect of the temperature coef´ cient on the neutron distribution can be observed
0 2 4 6 0 0.2 0.4 0.6 0.8 1 t [s] S (t )/ S 0 (a) 0 2 4 6 0 0.2 0.4 0.6 0.8 1 t [s] P (t )/ P 0 (b)
Figure 1. 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).
0 2 4 6 1020 1040 1060 1080 1100 1120 t [s] T [ K ] (a) 0 2 4 6 820 830 840 850 860 870 880 t [s] T [ K ] (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. 1.
0 2 4 6 635 640 645 650 t [s] T [ K ] (a) 0 2 4 6 620 622 624 626 628 630 632 634 636 t [s] T [ K ] (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. 1.
0 1 2 3 4 5 6 592 594 596 598 600 602 t [s] T [ K ]
Figure 4. Coolant temperature evolution on the midplane of the system for the transient of Fig. 1.
0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 t [s] S (t )/ S 0 (a) 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 t [s] P (t )/ P 0 (b)
Figure 5. Power transient (b) following a source trip as indicated (a) (switch-off followed by a step restoration to the initial value).
0 2 4 6 8 1040 1060 1080 1100 1120 1140 t [s] T [ K ] (a) 0 2 4 6 8 820 830 840 850 860 870 880 t [s] T [ K ] (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.
0 2 4 6 8 635 640 645 650 655 t [s] T [ K ] (a) 0 2 4 6 8 622 624 626 628 630 632 634 636 638 t [s] T [ K ] (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. 5.
0 1 2 3 4 5 6 7 8 9 593 594 595 596 597 598 599 600 601 602 603 t [s] T [ K ]
Figure 8. Coolant temperature evolution on the midplane of the system for the transient of Fig. 5.
0 5 10 0 0.2 0.4 0.6 0.8 1 t [s] S (t )/ S 0 (a) 0 5 10 0 0.2 0.4 0.6 0.8 1 1.2 t [s] P (t )/ P 0 (b)
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).
0 5 10 950 1000 1050 1100 t [s] T [ K ] (a) 0 5 10 780 800 820 840 860 880 t [s] T [ K ] (b)
Figure 10. 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.
0 5 10 625 630 635 640 645 650 655 t [s] T [ K ] (a) 0 5 10 615 620 625 630 635 t [s] T [ K ] (b)
Figure 11. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig. 9.
0 2 4 6 8 10 12 588 590 592 594 596 598 600 602 t [s] T [ K ]
Figure 12. Coolant temperature evolution on the midplane of the system for the transient of Fig. 9.
0 5 10 1 1.1 1.2 1.3 1.4 1.5 t [s] S (t )/ S 0 (a) 0 5 10 0.9 1 1.1 1.2 1.3 1.4 1.5 t [s] P (t )/ P 0 (b)
Figure 13. 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).
0 5 10 1140 1160 1180 1200 1220 t [s] T [ K ] (a) 0 5 10 880 890 900 910 920 t [s] T [ K ] (b)
Figure 14. 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. 13.
0 5 10 650 652 654 656 658 660 662 664 666 t [s] T [ K ] (a) 0 5 10 632 634 636 638 640 642 644 646 t [s] T [ K ] (b)
Figure 15. Clad temperature evolution at the inner (a) and at the outer boundary (b) on the midplane of the system for the transient of Fig. 13.
0 2 4 6 8 10 12 598 599 600 601 602 603 604 605 606 607 t [s] T [ K ]
Figure 16. Coolant temperature evolution on the midplane of the system for the transient of Fig. 13.
0 5 10 15 0 0.2 0.4 0.6 0.8 1 t [s] S (t )/ S 0 (a) 0 5 10 15 0 0.2 0.4 0.6 0.8 1 t [s] P (t )/ P 0 (b)
Figure 17. Power transient (b) following a source trip as indicated (a) in the EAP, assuming µP¶¢·¤¸¹»º½¼«¾u¿\ÀcÁeÂgÃcÄeÅ!Æ .
0 50 100 150 200 0 100 200 300 400 0 0.5 1 1.5 2 2.5 x 1015 r [cm] z [cm] N e u tr o n F lu x
Figure 18. Steady-stateÇ ux shape for the EAP, assuming ÈPÉ¢ÊË̻ͽÎuÏuÐ\ÑcÒÔÓÖÕc×eØgÙ
0 5 10 15 0 0.2 0.4 0.6 0.8 1 t [s] S (t )/ S 0 (a) 0 5 10 15 0 0.2 0.4 0.6 0.8 1 t [s] P (t )/ P 0 (b)
Figure 19. Power transient (b) following a source trip as indicated (a) in the EAP, assuming ÚPۢܤÝ޻߽à«áuâ\ãcäeågæcçeè!é .
0 50 100 150 200 0 100 200 300 4000 0.5 1 1.5 2 2.5 x 1015 r [cm] z [cm] N e u tr o n F lu x
Figure 20. Steady-stateê ux shape for the EAP, assuming ëPì¢íîï»ð½ñuòuó\ôcõÔöÖ÷cøeùgú
0 20 40 60 80 100 120 140 160 180 0 0.5 1 1.5 2 2.5x 10 15 r [cm] N e u tr o n F lu x α=5. 10-6 α=5. 10-5 α=1. 10-4
Figure 21. Steady-state radialû ux distributions at heightüþý ÿ
for different values of the capture temperature coef cient.
0 20 40 60 80 100 120 140 160 180 0 1 2 3 4 5 6 7 8 9 10x 10 14 r [cm] N e u tr o n F lu x α=5. 10-6 α=5. 10-5 α=1. 10-4
Figure 22. Steady-state radial ux distributions at height for
0 50 100 150 200 250 300 350 400 0 2 4 6 8 10 12x 10 14 z [cm] N e u tr o n F lu x α=5. 10-6 α=5. 10-5 α=1. 10-4
Figure 23. Steady-state axial ux distributions at radius , for different
values of the capture temperature coef cient.
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.
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.
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