STA BIL ITY CHAR AC TER IS TICS OF THE 500 MW IN DIAN PFBR

Transcription

1 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp STA BIL ITY CHAR AC TER IS TICS OF THE 500 MW IN DIAN PFBR by Vijayan L. ANURAJ *, Goverapet S. SRINIVASAN, and Kunhiraman DEVAN Indira Gan dhi Centre or Atomic Re search Kalpakkam, Tamil Nadu, India Sci en tiic pa per DOI: /NTRP A A ter the suc cess ul op er a tion o the a breeder te re ac tor or over two de cades, In dia is now near ing the com ple tion o a 500 MW (elec tri cal) pro to type a breeder re ac tor. This com mer cial scale power re ac tor is a so dium-cooled, pool-type, mixed-ox ide uelled a re ac - tor. The a bil ity char ac ter is tics o the re ac tor are an im por tant saety as pect to be ud ied. In the pres ent work, lin ear a bil ity o the pro to type a breeder re ac tor anal y sis is car ried out us ing the trans er unc tion method, while the a bil ity o the sys tem is checked via the Nyqui cri te ria. For the com plete ness o the udy, tran sient anal y sis with var i ous kinds o re ac tiv ity per tur ba tions was car ried out. The re sponse o the sys tem in both cases in di cated that the sys - tem is a ble. Key words: liq uid metal a breeder re actor, linear ability, prototype a breeder reactor, Nyqui plot INTRODUCTION In its sec ond age, In dia's three-age nu clear pro - gram in volves so dium-cooled a re ac tors. The plu to - nium re quired or these re ac tors is pro duced by pres sur - ized heavy wa ter re ac tors (PHWR) o the ir age, in op er a tion or many years. In dia's ir a re ac tor (FBTR), a te re ac tor with a power rate o 40 MW (ther - mal) has been in op er a tion at Kalpakkam since Based on the operating experience o the FBTR span ning over two de cades, a com mer cial scale pro to type re ac tor, prototype a breeder reactor (PFBR), has been de - signed and is in the i nal ages o con ruc tion. The neutronic ability o a reactor ensuring sae operation necessitates the development o a computational tool or the a bil ity anal y sis o such re ac tors. With this in mind, a mathematical model and a computer program (FOR- TRAN) based on the said model have been de vel oped at the Indira Gan dhi Cen tre or Atomic Re search. PFBR linear ability analysis using transer unc - tions and the Nyqui cri te ria is per ormed us ing the newly de vel oped pro gram. Even though lin ear a bil ity anal y sis en sures the a bil ity o such a tightly cou pled re ac tor core, a udy tak ing into con sid er ation real-time transient analysis is done or various reactivity perturba tions via a de tailed model o eed backs and heat transer encompassing the associated non-linearities. De tails o the PFBR core are dis cussed in the section De tails o the PFBR core. The meth od ol ogy * Cor re spond ing au thors; anuraj@igcar.gov.in adopted or lin ear a bil ity anal y sis is dis cussed in de - tail in the section Linear ability analysis, along with an es ti ma tion o the time con ant and the es tab lish - ment o a bil ity through the Nyqui cri te ria. In the section Re ac tiv ity per tur ba tion udy, the re sponse o the syem with various reactivity perturbations is dis - cussed. Point ki net ics equa tions and heat trans er equa tions as so ci ated with the cool ant chan nel are solved to get the time evo lu tion o re ac tor power and temperatures. It has to be noted that in ther mal re ac tors, es pe - cially boil ing wa ter re ac tors (BWR), the a bil ity o the sys tem is o pri mary con cern due to the pres ence o Xe and other poi sons. The loosely cou pled large cores o commercial thermal reactors make them vulnerable to unable oscillations. Stable limit cycle oscillations are ound in BWR at low-low con di tions. Com pared to ther mal re ac tors, a re ac tors show good a bil ity char ac ter is tics due to their tightly cou pled core with rong negative reactivity eedback eects, coolant in sin gle liq uid phase and the lack o poi son e ects in the a lux re gion. DETAILS OF THE PFBR CORE PFBR is a 500 MW (elec tri cal) MOX-u elled a re ac tor with two en rich ment zones. Im por tant core pa ram e ters per tain ing to the re ac tor are given in tab D di u sion the ory cal cu la tions are per ormed us ing a ABBN cross-section set or calculating reactivity worth dis tri bu tions and power den si ties which are es - sen tial or the a bil ity ud ies dis cussed here.

2 114 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Ta ble 1. Liq uid metal a breeder re ac tor de sign pa ram e ters Thermal output [MW] 1250 Maximum linear heat rating [Wm 1 ] Ac tive core height [m] 1 Num ber o sub-as sem blies in core1/core2 85/96 Plutonium enrichment [%] 20.7/27.7 Fuel pins per sub-as sem bly 217 Assembly pitch [m] Fuel pin diameter [mm] 6.6 Clad thick ness uel [mm] 0.45 Equivalent core diameter [m] 2 Syem pressure [MPa] To tal mass low rate [kgs 1 ] Cool ant in let tem per a ture [K] 670 Cool ant out let tem per a ture [K] 833 Cool ant in let den sity [gm 3 ] Cool ant heat ca pac ity at T [Jkg 1 K 1 ] Cool ant ther mal ex pan sion co e i cient [K 1 ] Reactor kinetics parameters or the PFBR core are given in tab. 2. Ta ble 2. Ki net ics pa ram e ters or PFBR Prompt neutron generation time, L [s] De layed neu tron rac tion, b e [pcm] 355 j b j (pcm) l j [s 1 ] For con ve nience, de layed neu tron rac tions are given in pcm units. It should be noted that 1 pcm = = Dk/k. LINEAR STABILITY ANALYSIS Point ki net ics equa tions de scrib ing re ac tor ki - net ics are non-lin ear since the re ac tiv ity de pends on the power. But or small per tur ba tions, the a bil ity o a non-lin ear sys tem could be de duced rom the a bil - ity o an as so ci ated lin ear sys tem. In the pres ent udy, the equations o reactor dynamics are linearized around the equi lib rium point and a lin ear a bil ity anal y sis is car ried out in terms o trans er unc tions. The block di a gram rep re sent ing the eed back loop in a re ac tor sys tem is given in ig. 1. Block Z(s) is the or - ward-loop trans er unc tion rep re sent ing the point ki - net ics equa tions con nect ing power and re ac tiv ity. Block K p (s) is the eed back trans er unc tion cal cu - lated by solv ing the heat trans er equa tions and re ac - tivity eedback relations. A syem transer unction is de ined as the ra tio o Laplace trans orms o the out put parameter to the input parameter. Fig ure 1. Block di a gram rep re sent ing the re ac tor sys tem here, Z(s) is known as the zero power trans er unc tion and K p (s) as the eed back trans er unc tion. Out o this, the zero power trans er unc tion can be ob tained rom point ki net ics equa tions as Z( s) 1 b j sl j s l j (1) where L is the prompt neu tron gen er a tion time, b j the de layed neu tron rac tions o the six group, l j the de - cay con ants o de layed neu tron pre cur sor, and w the cy cles re quency Mathematically, the dynamic power coeicient K p (s), also known as the re ac tiv ity eed back trans er unc tion, is de ined as the ra tio o the Laplace trans - orm o reactivity eedback r b (s) to the Laplace trans - orm o change in power DP(s) that is K p b ( s) ( s) r DP( s) (2) where s is the Laplace trans orm vari able. I s is re - placed by iw, w be ing the re quency in cy cles per sec - ond (rps), K p (iw) is the re quency re sponse unc tion. The re quency re sponse unc tion is o the orm K p ( i w) a (3) 1 iwt where a is the atic power co e i cient and t the time con ant o eed back. The dynamic power coeicient o reactivity is cal cu lated by solv ing the heat trans er equa tions ap pli - ca ble to the cool ant chan nels in the re quency do main. The lumped model o heat trans er used in the meth od - ol ogy is de scribed in de tail in res. [1, 2]. The ba sics are also well ex plained in [3]. In the re ac tor, dur ing power per tur ba tion, tran - sient tem per a ture changes are not im me di ate with the changes in re ac tor power. The tem per a ture de pends on the ra tio o en ergy-to-heat ca pac ity; hence, the change in tem per a ture and the ex pan sion e ects will lag be - hind the change in power. In ad di tion, the time re - quired or the heat trans er across the chan nel also di - rectly a ects the re ac tiv ity eed back and, thereby, the ki netic be hav ior o the a re ac tor. The eed back due to uel and clad ax ial ex pan sion, cool ant ex pan sion, spacer-pad ex pan sion and the Dopp ler e ect have also been con sid ered. The eed back con tri bu tion rom

3 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp blan ket, di er en tial con trol rod ex pan sion and re ac tor ves sel ex pan sion were ig nored be cause o their con tri - bu tion, deemed small. Cal cu la tion method Along with the ra dial and ax ial blan kets, the core re gion is mod eled in the anal y sis as well. The re ac tor is as sumed to be di vided into var i ous ra dial zones/chan - nels based on the low zon ing o the re ac tor. Since the power rating varies signiicantly axially, the core height is di vided into sev eral ax ial meshes. In the cur - rent udy, the PFBR core is as sumed to be di vided into 10 ra dial and 14 ax ial zones. Each ra dial zone con tains a cer tain num ber o sub as sem blies with iden ti cal cool - ant low rates. It is as sumed that in a par tic u lar ra dial zone all the uel pins be have iden ti cally ther mo dy - namically and, hence, heat transer calculations are done only or a rep re sen ta tive pin rom each o the ra - dial chan nels. For computational simplicity, the dynamic reac - tivity coeicient K p (iw) is writ ten as j, i j, i K p ( iw ) Dr D T ( iw ) (4) j i where Dr j,i is the isothermal temperature coeicient o reactivity and T j,i (iw) rep re sents the re quency-de - pend ent tem per a ture change per unit change in power. Su per scripts j and i rep re sent the ax ial and ra dial mesh num bers. The cal cu la tion o re quency-de pend ent temperature change T j,i (iw) is dis cussed be low. Re - ac tiv ity eed backs cor re spond ing to this tem per a ture change are cal cu lated us ing the per tur ba tion worth and ex pan sion co e i cients. Per tur ba tion worths are cal cu - lated based on a 26-group ABBN cross-sec tion set. Reactivity eedbacks rom all the prominent eedback mech a nisms are added to give the dy namic re ac tiv ity co e i cient or that re quency, eq. (4). Cal cu la tion o re quency-de pend ent temperature rise T j,i (iw) A ra di ally lumped ax i ally con tin u ous heat trans - er model is em ployed or heat trans er cal cu la tions. In this model, heat trans er equa tions or uel, clad, cool - ant and ruc tural ma te rial are C T ( z, t ) Y ( z ) P( t ) h [ T ( z, t ) Ts ( z, t )] (5) t C T s ( z, t ) s h [ T ( z, t ) Ts ( z, t )] t h [ T ( z, t ) T ( z, t )] (6) s s c C T c ( z, t ) c hs[ Ts ( z, t ) Tc ( z, t )] t Tc ( z, t ) C cu h [ Tc ( z, t ) T ( z, t )] (7) z C T ( z, t ) h [ Tc ( z, t ) T ( z, t )] (8) t where T is the tem per a ture, Y(z) the ax ial power dis - tribution, P(t) the re ac tor power per unit length, C the heat ca pac ity per unit length, h the lumped heat trans er co e i cient per unit length, v the cool ant ve - locity, t the time, and z the ax ial dis tance Sub scripts, s, c, and and or uel, ain less eel, cool ant, and ruc tural ma te rial (sheath), re spec - tively. Ba si cally, eqs. (5) to (8) are non-lin ear. To solve them in the re quency do main, these equa tions are linearized. A small per tur ba tion in re ac tiv ity is ini ti - ated which would lead to change in power and tem per - a ture, as ol lows P( t ) P DP( t ) T ( z, t ) T ( z ) DT ( z, t ) 0 c c0 DTc T ( z, t ) T ( z ) 0 ( z, t ) T ( z, t ) T ( z ) DT ( z, t ) s s0 s T ( z, t ) T 0 ( z ) DT ( z, t ) (9) where D rep re sents the small per tur ba tion. For a ep change in power at time t = 0, DP(t) = DP = P 1 P 0 or all t, where P 1 and P 0 are the i nal and ini tial pow ers. Sub i tut ing eq. (9) into eqs. (5) to (8) and solv - ing them a ter tak ing the Laplace trans orm with con - ant inlet temperature assumption we get DTc ( z, s) h tz / u DPe ( ) t ut s 1 3 c 0 z y( z ) e tz / u Y( z ) DP DTc ( z, s) DTs ( z, s) h ( ) t t DT s 1 3 Y( z ) DP DTs ( z, s) ( z, s) h ( 1 ) ( 1 ) where s is the Laplace trans orm vari able, and h h 1 t3 1 s h 1 t s hs s h 1+ c h s 1 1 t3 t = tc c cs cc c t, ts, tc, and t h h h h s s 3 dz (10) (11) (12) where t, t s, t c, and t are the time con ants o uel, eel (clad), cool ant, and ruc tural ma te rial. For su i - ciently small changes in power and re ac tiv ity, non lin - ear e ects in heat trans er equa tions can be ne glected. There ore, terms with prod ucts o small quan ti ties are ig nored. For nu mer i cal cal cu la tion, the core is di vided into ax ial and ra dial zones. Sub i tut ing s with iw, the above equa tions be come

4 116 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp tz ( j) u( i) e DTc ( j, i, iw) h ( iwt ) t u( i) t s 1 3 c j tz ( j 1) u( i) i ( P2 P1 ) ( j, i) u( ) Y e e j 1 t Y( j, i)( P2 P1 ) DTs ( j, i, iw) h t ( iwt ) DT s 3 1 tz ( j ) u( i) (13) DTc ( j, i, iw) (14) t 3 ( j, i, iw) ( j, i)( P P ) h ( 1 iwt ) Y 2 1 DTc ( j, i, iw) (15) 1 iwt where w is the an gu lar re quency. The j and i are the ax ial and ra dial mesh in di ces. The eedback reactivity components are computed in the re quency do main by com put ing the ma te - rial ex pan sion with the change in tem per a ture and ther mal ex pan sion co e i cient, and then mul ti plied with reactivity worth. RESULTS The time con ant is ound to be 3.05 s [1] by it - ting the cal cu lated pro ile o the eed back re ac tiv ity trans er unc tion with the ex pres sion (3), igs. 2 and 3. A nec es sary but not su i cient con di tion or a - bil ity is that K p (0) < 0, i. e. that the atic power co e i - cient o the re ac tor mu be neg a tive. The atic power co e i cient o re ac tiv ity or PFBR is pcm/mw (0 to 100 % power) and pcm/mw (0 % to 50 % power). Thus, the nec es sary con di tion o a bil ity is satisied [3]. A nec es sary and su i cient con di tion or a bil ity is ex pressed in terms o the open-loop re ac tor trans er unc tion which is de ined as L( iw) P 0 Z( iw) K p ( iw) (16) where P 0 is the eady-ate power and Z(iw) is the zero power re quency re sponse unc tion o the re ac tor. The open-loop trans er unc tion is de rived rom the ex pres sion o the power trans er unc tion which is de ined as P0 Z( iw) H( iw) [ P Z( iw) K ( iw)] 1 0 p (17) The mag ni tude and phase o the zero power trans er unc tion and the power trans er unc tion are plot ted in igs. 4 and 5. The mag ni tude and phase o the eed back trans - er unc tion are shown in igs. 2 and 3. Fig ure 2. Mag ni tude o eed back trans er unc tion Fig ure 4. Mag ni tude o zero power trans er unc tion and power trans er unc tion (1 rps = ra di ans/sec ond) Fig ure 3. Phase o eed back trans er unc tion Fig ure 5. Phase o zero power trans er unc tion and power trans er unc tion

5 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Fig ure 6. Nyqui plot or PFBR Real and imag i nary parts o L(iw) are plot ted again each other to get the Nyqui plot. I the curve en cir cles the ( 1, 0) point, then the re ac tor is un a ble. Otherwise, the reactor is able [3]. The Nyqui plot or PFBR is given in ig. 6. It can be seen that the curve is ar away rom the ( 1, 0) point, con irm ing that the PFBR is a ble. Im por tant thermophysical pa ram e ters used or es ti mat ing the eed back re ac tiv ity trans er unc tion and ki net ics pa - ram e ters ap plied in the es ti ma tion o the zero power trans er unc tion are given in tabs. 1 and 2. REACTIVITY PERTURBATION STUDY While the Nyqui cri te ria es tab lish the in her ent ability o a reactor, a direct veriication o power evo lu tion or var i ous re ac tiv ity per tur ba tions and at var i ous power lev els o re ac tor op er a tion o ers ullproo ev i dence o re ac tor a bil ity at said in put per - tur ba tions. The power should con verge in ew sec onds and should not have ma jor os cil la tions. Re ac tiv ity pulse per tur ba tions were the in put or var i ous re ac tiv - ity pulses and at var i ous power lev els. The power co - mes back to the nom i nal value with out any os cil la - tions. To check re ac tor a bil ity when the re ac tiv ity in put is os cil la tory in na ture, re ac tor power, uel tem - per a ture and cool ant tem per a ture evo lu tion have been checked or si nu soi dal in put with var i ous re quen cies and with di er ent re ac tiv ity eed back con di tions. Step reactivity insertions The a bil ity o the PFBR has been checked again a small re ac tiv ity per tur ba tion o 0.1 $ and large re ac tiv ity per tur ba tions o 0.3 $ and 0.5 $. It should be noted that the re ac tiv ity unit $ is a sys - tem-de pend ent unit equal to the e ec tive de layed neu - tron rac tion b (that is 1 $ = 1/b k/k). I the re ac tiv ity is greater than 1 $, the re ac tor will be super prompt crit i cal. Re ac tiv ity per tur ba tions are o one sec ond du - ra tion and an in put rom 10 s to 11 s in the eady-ate, art ing rom time zero. The udy is done or ull power and, also, part-load op er a tion o 40 % power and 20 % power with an ap pro pri ate lower cool ant low o 59.8 % and 50 %. The large but un likely re ac - tiv ity per tur ba tion o 1 $ (prompt crit i cal) is also ud - ied. Apart rom the re ac tiv ity per tur ba tion, a low per - tur ba tion o 10 % at ull power is ud ied as well. The anal y sis is car ried out with the tran sient anal y sis code PREDIS which is a part o the KALDIS code sys tem [4]. The re ac tiv ity pulse in put is be tween 10 s and 11 s, the re ac tiv ity in put re main ing zero at all other times. The ki net ics por tion o the PREDIS code has been val i dated again SEFOR ex per i ments and FBTR re ac tiv ity tran sients [5, 6]. The pro cesses mod - eled in the code are neutronics, tran sient ther mal hy - draulics, reactivity eedbacks such as Doppler, uel and clad ax ial ex pan sion, cool ant ex pan sion, spacer pad, grid plate, main ves sel and di er en tial con trol rod ex pan sion, cool ant boil ing, clad and uel melt ing and slump ing. A de tailed de scrip tion o the math e mat i cal mod el ing or re ac tor ki net ics and ther mal hy drau lics is given in [4]. In the cal cu la tions pre sented, the lumped model o heat trans er adapted or anal y sis o un pro - tected tran sients is em ployed. The same model was ap - plied or lin ear a bil ity anal y sis o the PFBR [1]. The ex act heat con duc tion model, which takes into ac count tem per a ture de pend ent ther mal con duc tiv ity, is em - ployed or the anal y sis o pro tected tran sients. In the pres ent udy, the re ac tor does not SCRAM when the re ac tiv ity per tur ba tion is in put and, thus, the plant pro - tec tion sys tem (PPS) is as sumed not to be unc tional. Reactivity perturbations input pulses are taken or the du ra tion o 1s. This is ap pro pri ate in this anal y - sis, since the time con ant o re ac tiv ity eed back or the core is 3.05 s [1] and the du ra tion o 1 s is a good com pro mise to udy eed back re ac tiv ity and power evo lu tion. A shorter per tur ba tion will give a be nign e - ect and a lon ger du ra tion will be a prob lem or saety anal y sis (not a per tur ba tion). The re ac tiv ity val ues are cho sen so as to rep re sent small and large val ues. 0.1 $ (33.7 pcm) is con sid ered a small pulse, though it is con ser va tive in that. The so dium-re moval worth o the cen tral sub as sem bly is +17 pcm. The re moval o so dium rom the en tire cen tral sub as sem bly is an un - likely oc cur rence. Higher val ues o 0.3 $ and 0.5 $ have been con sid ered in this anal y sis. These are large per tur ba tions un likely to oc cur. For the sake o il lus - tra tion, an un likely large per tur ba tion lead ing to prompt crit i cal (1 $) has also been con sid ered. The range o re ac tiv ity per tur ba tions con sid ers the en tire pos si ble range and this anal y sis is ad e quate to es tab - lish the a bil ity o the reactor. The re sults o the anal y sis o some re ac tiv ity per - tur ba tions are given graph i cally or the var i ous cases. Fig ures 7 and 8 give the evo lu tion o net re ac tiv ity and normalized power or reactor operation at ull power and or re ac tiv ity per tur ba tion in puts o 0.1 $ and 0.5 $. Fig ures 9, 10, and 11 give the evo lu tion o net re ac tiv ity and nor mal ized power or part-load op er a tion o 20 % power (50 % low) or re ac tiv ity pulses o 0.1 $, 0.3 $, and 0.5 $.

6 118 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Fig ure 7. Net re ac tiv ity and nor mal ized power with time; power = 100 %, low = 100 %, per tur ba tion = 0.1 $ Nor mal ized power Net reactivity Fig ure 10. Net re ac tiv ity and nor mal ized power with time; power = 20 %, low = 50 %,per tur ba tion = 0.3 $ Normalized power Net reactivity Fig ure 8. Net re ac tiv ity and nor mal ized power with time; power =100 %, low = 100 %, per tur ba tion = 0.5 $ Nor mal ized power Net re ac tiv ity Fig ure 11. Net re ac tiv ity and nor mal ized power with time; power = 20 %, low = 50 %, per tur ba tion = 0.5 $ Normalized power Net re ac tiv ity Fig ure 9. Net re ac tiv ity and nor mal ized power with time; power = 20 %, low = 50 %, per tur ba tion = 0.1 $ Nor mal ized power Net re ac tiv ity A careul observation o reactivity inputs o 0.1 $, 0.3 $, and 0.5 $ brings out the act that power con verges to the orig i nal eady-ate value in a ew sec onds, with - out any os cil la tion. Ta ble 3 gives the times taken or the power to a bi lize to the orig i nal value. The power is eady a ter reach ing its orig i nal value. As the value o reactivity perturbation is higher, the peak power reached is also higher, as ex pected. For a 100 % power (and 100 % low), the power reaches peak value o 111 % and 200 % or 0.1 $ and 0.5 $, re spec tively, igs. 7 and 8. When the power level is low, re ac tiv ity eed - backs are lower. Hence, or lower power lev els, the neg - a tive re ac tiv ity reached is lower. As il lus tra tion, in case o a 0.5 $ perturbation, the negative reactivities reached are 0.32 $ and 0.17 $ or 100 % power (100 % low) and 20 % power (50 % low), re spec tively, igs. 8 and 11. Thw be hav iour o ma jor eed backs is il lus trated in ig. 12 or the case o 20 % power (50 % low) with 0.3 $ in put per tur ba tion. The Dopp ler, uel ex pan sion and spacer pad ex pan sion eed backs, which are the ma - jor eed backs, have been plot ted. All these are neg a tive in these per tur ba tions, with Dopp ler be ing mo neg a - tive ollowd by uel ex pan sion and spacer pad ex pan - sion. For il lus tra tion, a case o low per tur ba tion has also been ana lysed. A per tur ba tion o 10 % in crease in low value is in put or one sec ond (10 s to 11 s). At all other times low is nom i nal and re ac tiv ity is zero. The re sults or evo lu tion o nor mal ised power and net re ac - tiv ity is given in ig. 13. It can be seen that the power rises to a low value o % then alls down and re - turns to intial value at 15.5 s. The re ac tiv ity eed backs in this case are de picted in ig. 14. Sinusoidal reactivity insertions The a bil ity o the PFBR has also been checked with si nu soi dal re ac tiv ity per tur ba tions o an am pli -

7 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Ta ble 3. Time taken or power to reach orig i nal eady-ate value or the var i ous re ac tiv ity per tur ba tions No. Power [%] Flow [%] Re ac tiv ity per tur ba tion input Time at which re ac tor co mes to Time taken a ter the per tur ba tion to be tween 10 s and 11 s orig i nal eady-ate power level (s) reach orig i nal eady-ate (s) Figure 12. Input reactivity, net reactivity and reactivity eed backs; power = 20 % low = 50 %, per tur ba tion = 0.3 $ In put,... Steel expn, Spacer par, Net, _._._._ Fuel expn., _._._._ So dium expn. Dopp ler Figure 14. Input reactivity, net reactivity and reactivity eed backs, power = 100 %, low per tur ba tion 10 % Net,... Dopp ler,... Sod. expn., Spacer pad, Fuel expn Fig ure 13. Net re ac tiv ity and nor mal ized power with time; power = 100 %, low per tur ba tion 10 % Normalized power Net reactivity tude o 0.5 $ and re quen cies o 10 Hz, 1 Hz, 0.1 Hz, and 0.01 Hz, by con sid er ing all re ac tiv ity eed backs and, also, by ig nor ing all re ac tiv ity eed backs. The re - ac tor is a ble when all re ac tiv ity eed backs are in - cluded. The re ac tor is a ble when the Dopp ler eed - back alone is con sid ered, as well. When all eed backs are ex cluded, the power in creases with time and uel tem per a ture reaches the melt ing point. The in crease is greater or low re quen cies, thus in di cat ing that the re - ac tor be comes un a ble at low re quen cies, when all eed backs are sup pressed. Com puter code PREDIS [4] has been used or the tran sient anal y sis with the si - nu soi dal re ac tiv ity in put. A lumped model o heat trans er has been em ployed in the pres ent anal y sis. Some re sults o the anal y sis pre sented here are given in the orm o graphs which give the vari a tion o nor mal ized power, peak uel tem per a ture and peak clad (eel) tem per a ture with time. In all cases, the du - ra tion o 50 cy cles or the sinewave has been con sid - ered to pre cisely check the de pend ence on re quency. Hence, the du ra tion o the plot is di er ent or each re - quency. Fig ure 15 gives the vari a tion o the nor mal ized power, peak uel tem per a ture and peak clad (eel) tem - per a ture, along with the time or the 10 Hz re quency, with/without reactivity eedbacks. The duration or 50 cy cles (5 s) has been shown. It can be seen that when all re ac tiv i ties are con sid ered, the trend o mean nor mal - ized power is uni orm. In the be gin ning, it goes down a lit tle over a small du ra tion o time, be cause uel and eel tem per a tures have a small buildup time and, hence, the associated reactivity eedbacks. Fuel and clad tem per a tures sat u rate around 2261 K and K and re main well be low their melt ing points (3023 K or uel and 1700 K or clad). When the re ac tiv ity eed - backs are sup pressed, we see a monotonically in creas - ing trend or nor mal ized power, uel tem per a ture and

8 120 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Fig ure 15. Nor mal ized power, uel tem per a ture and eel tem per a ture with time or si nu soi dal in put o 0.5 $ am pli tude, with (a, c, e) and with out (b, d, ) re ac tiv ity eed back; re quency = 10 Hz eel temperature. A more continuous evolution will lead to uel and clad melt ing. Within the ir 50 cy cles (5 s), they re main be low melt ing points, the peak uel tem per a ture be ing 2610 K and peak clad tem per a ture be ing 930 K. Fig ure 16 gives the vari a tion o nor mal ized power, peak uel tem per a ture, and peak clad (eel) tem per a ture with time or the 0.01 Hz re quency, with reactivity eedbacks and without reactivity eedbacks. The du ra tion or 50 cy cles (5000 s) has been shown. It can be seen that when all re ac tiv i ties are con sid ered, the trend o mean nor mal ized power is uni orm. In the be gin ning, it goes down a lit tle over a small du ra tion o time be cause ex pan sion eed backs have a build up time. But this is not vis i ble in the graph be cause it has been plot ted or a long du ra tion (5000 s). Fuel and clad tem per a tures sat u rate around 2200 K and 879 K, and re main well be low their melt ing points (3023 K or uel and 1700 K or clad). When the re ac tiv ity eed - backs are sup pressed, we see a monotonically in creas ing trend or nor mal ized power, uel tem per a - ture and eel tem per a ture. The nor mal ized power goes up, and uel reaches the melt ing point at 8.4 s. A case ex clud ing all ex pan sion eed backs and re tain ing only the Dopp ler eed back has also been ud ied. Figure 17 shows the re sult o this udy or the re quency o 10 Hz to 0.01 Hz. The du ra tion or 50 cy - cles (5 s or the re quency o 10 Hz and 5000 s or the re quency o 0.01 Hz) are shown. It can be seen that the trend o mean nor mal ized power is uni orm. There is a small buildup time or uel and eel tem per a ture and, hence, they ini tially go up and, upon this, sat u rate. Ac cord ingly, the Dopp ler eed back also ini tially changes, even tu ally to be re lected by the de creas ing pro ile o the nor mal ized power at the be gin ning o the 10 Hz re quency. As or the case o 0.01 Hz, this is

9 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Fig ure 16. Nor mal ized power, uel tem per a ture and eel tem per a ture with time or si nu soi dal in put o 0.5 $ am pli tude, with (a, c, e) and with out (b, d, ) re ac tiv ity eed back; re quency = 10 Hz not vis i ble, since the plot spans a much lon ger time (5000 s). The peak uel and clad tem per a tures lie be - low their melt ing points in both cases (2325 K or 10 Hz, 2300 K or the 0.01 Hz uel, 893 K or 10 Hz, 890 K or 0.01 Hz or clad). Hence, the re ac tor is a ble with Dopp ler eed back alone. CONCLUSION The a bil ity o a me dium-sized FBR core has been ud ied thor oughly by lin ear a bil ity anal y sis and, also, with re ac tiv ity per tur ba tions in the orm o pulses and sine waves. Lin ear a bil ity gives a neg a - tive dy namic power co e i cient with the time con ant o 3.05 s meet ing the nec es sary con di tion or a bil ity. The Nyqui plot o the loop trans er unc tion does not en cir cle ( 1, 0) point in the com plex plane, thus meet - ing the nec es sary and su i cient con di tions or a bil - ity. The anal y sis with a com plete range o re ac tiv ity per tur ba tions in the orm o pulses in di cates that re ac - tor power co mes back to the nom i nal value with out any os cil la tion, prov ing the rong a bil ity o the re ac - tor. Anal y sis with si nu soi dal pulses o var i ous re - quen cies in di cates that the re ac tor is a ble in the rong sense when all re ac tiv ity eed backs are con sid - ered and even when the Dopp ler eed back alone is con sid ered.

10 122 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Fig ure 17. Nor mal ized power, uel tem per a ture and eel tem per a ture with time or 10 Hz (a, c, e) and 0.01 Hz (b, d, ) with Dopp ler eed back alone AU THOR CON TRI BU TIONS Linear ability analysis using the Nyqui crite - ria was per ormed by V. L. Anuraj, while the tran sient anal y sis with re ac tiv ity per tur ba tions was done by G. S. Srinivasan. K. Devan pro vided over all guid ance dur ing the udy, in ter pret ing the re sults and pro vid ing the nec es sary in put re quired or the cal cu la tions. The anal y sis o re sults and prep a ra tion o the manu script was a joint un der tak ing o all au thors men tioned. REFERENCES [1] Anuraj, V. L., Srinivasan, G. S., Dy namic Power Co - e i cient o Reacivity Cal cu la tions and Sta bil ity Anal - y sis with Per tur ba tion Worths Based on ABBN 93 cross sec tion set, 19 th Na tional Sym po sium on Radia - tion Phys ics, Chennai, Tamil Nadu, In dia, 2012 [2] Singh, O. P., et al., Re sponse to Small Re ac tiv ity Per - tur ba tions in an Ox ide and Metal Fu elled Me - dium-sized Liq uid-metal Fa Breeder Re ac tor, An - nals o Nu clear En ergy, 20 (1993), 5, pp [3] Hummel, H. H., Okrent, D., Re ac tiv ity Co e i cients in Large Fa Power Re ac tors ANS Pub li ca tion, USA, 1979, p. 386 [4] Harish, R., et al., KALDIS: A Com puter Code Sys tem or Core Dis rup tive Ac ci dent Anal y sis in Fa Re ac - tors, IGCAR Re port, IGC-208, 1999 [5] Sharada, B., Singh, O. P., Val i da tion o the Com - puter Code, POKIN again SEFOR Ex per i ment and Analytical Results, IGCAR Report ROD/01117/SNAS-32, 1990 [6] Srinivasan, G. S., Singh, O. P., Val i da tion o the Com puter Code, PREDIS Again FBTR Ex per i men - tal Re ac tiv ity Tran sients, IGCAR Re port RPD/SAS/ FBTR/01100/CR/011, 1999 Pa per on April 27, 2015 Ac cepted on May 15, 2015

11 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 2, pp Vixajan L. ANURAX, Goverapet S. SRINIVASAN, Kunhiraman DEVAN SVOJSTVA STABILNOSTI INDIJSKOG PROTOTIPA BRZOG REAKTORA SNAGE OD 500 MW Po ieku vi{e od dve decenije uspe{nog delovawa brzog oplodnog te reaktora, Indija je danas blizu zavr{etka prototipa brzog oplodnog reaktora snage 500 MW elektri~nih. Ovaj brzi reaktor bazenskog tipa, sa snagom za komercijalnu upotrebu, hla en je natrijumom i puwen me{anim oksidnim gorivom. Svojva abilnoi reaktora predavqaju zna~ajne sigurnosne aspekte koji se prou~avaju. U ovom radu, sprovedena je linearna analiza abilnoi prototipa brzog oplodnog reaktora metodom prenosne unkcije, dok je abilno siema proverena Nikviovim kriterijumom. U ciqu kompletirawa prou~avawa, izvr{ena je analiza tranzijenata sa razli~itim vrama perturbacija reaktivnoi. Odziv siema tako e je ukazao da je siem abilan. Kqu~ne re~i: brzi oplodni reaktor sa te~nim metalom, linearna abilno, prototip...brzog oplodnog reaktora, Nikviov dijagram