A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents

Transcription

1 World Journal o Nuclear Scence Technology, 5, 5, 57-7 Publshed Onlne January 5 n ScRes. New Formulaton to the Pont Knetcs Equatons Consderng the Tme Varaton o the Neutron Currents nderson Lupo Nunes, qulno Senra Martnez, Ferno Carvalho da Slva, anel rtur Pnhero Palma epartment o Nuclear Engneerng, COPPE/UFRJ, Ro de Janero, Brazl Brazlan Commsson or Nuclear Energy, Ro de Janero, Brazl Emal: anunes@con.urj.br, aqulno@lmp.urj.br, erno@con.urj.br, dpalmaster@gmal.com Receved 6 January 5; accepted 7 January 5; publshed 9 January 5 Copyrght 5 by authors Scentc Research Publshng Inc. Ths work s lcensed under the Creatve Commons ttrbuton Internatonal Lcense (CC BY). bstract The system o pont knetcs equatons descrbes the tme behavour o a nuclear reactor, assumng that, durng the transent, the spatal orm o the lux o neutrons vares very lttle. Ths system has been largely used n the analyss o transents, where the numercal solutons o the equatons are lmted by the stness problem that results rom the derent tme scales o the nstantaneous delayed neutrons. Its dervaton can be done drectly rom the neutron transport equaton, rom the neutron duson equaton or through a heurstcs procedure. ll o them lead to the same unctonal orm o the system o derental equatons or pont knetcs, but wth derent coecents. However, the soluton o the neutron transport equaton s o lttle practcal use as t requres the change o the exstent core desgn systems, as used to calculate the desgn o the cores o nuclear reactors or derent operatng cycles. Several approxmatons can be made or the sad dervaton. One o them conssts o dsregardng the tme dervatve or neutron densty n comparson wth the remanng terms o the equaton resultng rom the P approxmaton o the transport equaton. In ths paper, we consder that the tme dervatve or neutron current densty s not neglgble n the P equaton. Thus beng, we obtaned a new system o equatons o pont knetcs that we named as moded. The nnovaton o the method presented n the manuscrpt conssts n adoptng arsng rom the P equatons, wthout neglectng the dervatve o the current neutrons, to derve the moded pont knetcs equatons nstead o adoptng the Fck s law whch results n the classc pont knetcs equatons. The results o the comparson between the pont knetcs equatons, moded classcal, ndcate that the tme dervatve or the neutron current densty should not be dsregarded n several o transent analyss stuatons. How to cte ths paper: Nunes,.L., Martnez,.S., da Slva, F.C. Palma,..P. (5) New Formulaton to the Pont Knetcs Equatons Consderng the Tme Varaton o the Neutron Currents. World Journal o Nuclear Scence Technology, 5,

2 Keywords Reactor Pont-Knetcs, Neutron Current ensty, Nuclear Power ensty. Introducton In order to determne the nuclear power dstrbuton n a reactor core, one should nvestgate the neutron transport n a heterogeneous medum wth strong neutron absorpton, where these neutrons can also be scattered or escape rom the actve part o the reactor. Notwthstng that, the advances n computer processng o the countless methods to solve the neutron transport equaton, n practce the approxmaton o the neutron duson s largely used n statonary calculatons to predct the dstrbuton o neutrons o the crtcal concentraton o boron. To deal wth the movement o the neutrons n a way smlar to that o heat duson, one needs to make several approxmatons n the transport equaton, whch nclude a weak angular dependency o the angular dstrbuton o the neutrons, sotropc sources o neutrons, the dsregardng o the dervatve or neutron current densty, n comparson wth other terms that appear n the neutron transport equaton []. Once the spatal dstrbuton o the neutrons n the nuclear reactor s known, t s also mportant to predct the tme behavour o ths dstrbuton, nduced as t s by the varaton n nuclear reactvty due to the varaton o uel temperature, the varaton o the materal composton o the reactor core, the varaton n moderator densty, amongst others. The smplest way to determne the tme behavour or the nuclear power s through the soluton o pont knetcs equatons. These equatons nclude approxmatons that are added to those made to obtan the equatons or neutron duson n the structure o mult-groups o neutron energy []. Pont knetcs equatons consst on a system or the calculaton o the nuclear power the concentratons o the delayed neutrons precursors. They are rst-order derental equatons, coupled non-lnear n ther more general orm. Though qute questonable, the approxmatons made n the development o the classcal pont knetcs equatons have already been wdely analysed dscussed n the lterature. However, the nluence on these equatons rom not consderng the tme dervatve n neutron current densty dd not deserve, untl now, a systematc specc evaluaton. ue to ths, n ths paper we developed, rom the neutron transport equaton, the moded pont knetcs equatons, that are derent rom the classcal ones, as they nclude the tme dervatve o neutron current densty. In Secton, t s presented the development o the moded pont knetcs equatons. Secton presents the calculaton to obtan ther analytcal solutons. Secton 4 presents the results o the analytcal solutons o the classcal moded pont knetcs equatons. nd Secton 5 dscusses the results obtaned provdes the conclusons o ths paper.. The Moded Pont Knetcs Equatons The theory o neutron transport s the wde model to descrbe neutron dstrbuton n a nuclear reactor. It s descrbed n [] [] n terms o the angular lux o neutrons, ϕ ( r, E, Ωˆ, t) : ( ˆ ) 6 6 ϕ r, E, Ω, t + Lϕ( r, E, Ω ˆ, t) = Fϕ ( r, E, Ω ˆ, t) + λχ (, ) (,, ˆ E C r t Fϕ r E Ω, t) () v E ( r, t) where =,,, 6 the operators L, = = C χ ( ˆ E = Fϕ r, E, Ω, t) λχ ( E) C( r, t) () F, F p F are dened as ollows: t ( r ) S ( r ) L Ω ˆ +Σ, Et, Σ, E E, Ωˆ Ωˆ, t de d Ωˆ, () ( ) ( ) + ( ) p = 6 F F F, (4) 58

3 F ( ) ( β) χ ( E) υ( E) (, E, t) de dˆ Σ r Ω (5) p F ( ) βχ ( E) υ( E ) (, E, t) de d ˆ Σ r Ω. (6) The scatterng cross secton can be exped n terms o the polynomals o Legendre up to the second term, that s, the expanson s done or l = l =. It conssts o the P approxmaton Equaton () can be re-wrtten thus: ( ˆ ˆ l + E E t) ( E E t) P ( ˆ ˆ ) Σ r,, Ω Ω, Σ r,, Ω Ω. (7) S Sl l l = ˆ ˆ L Ω +Σ, Et, Σ, E Et, de d Ω Σ, E Et, Ωˆ Ωˆ de dωˆ (8) ( r ) ( r ) ( r ) t S S We apply the operator ( ) t results that: v E φ ( r, Et, ) where the operators L, F dωˆ to Equatons () () consderng the ollowng dentons: ( r Et) ϕ( r E ˆ t) φ,,,, Ω, dωˆ ( Et) ϕ ( E t) J r,, r,, Ωˆ, ΩΩ ˆ d ˆ, 6 6 (, Et, ) Lφ(, Et, ) Fφ(, Et, ) λχ ( E) C(, t) Fφ(, Et, ) + J r + r = r + r r (9) ( r, t) = = C χ( E) = Fφ( r, Et, ) λχ ( E) C( r, t) () F are dened: ( ) ΣT ΣS ( )( ) L r, Et, r, E Et, d E, () 6 F E E E E t E ( ) = ( β) χ + βχ υ( ) Σ ( r,, )( ) d () = ( ) βχ υ Σ ( ) F E E r, E, t d E. () In replacng Equaton (8) n Equaton (), multplyng the resultng equaton by Ω ater that ntegratng n the sold angle, t results that: (, Et, ) J r + φ ( r, Et, ) +Σ t ( r, Et, ) J( r, Et, ) = ΣS (, E Et, ) (, E, t) de v E r J r (4) Consderng the approxmaton descrbed n []: denng the transport cross secton: ( r, E Et, ) ( r, E, t) δ ( E E) Σ Σ S S 59

4 rom Equaton (4), t s possble to wrte: (, Et, ) (, Et, ) (, Et, ) Σ r Σ r Σ r, tr t S (, Et, ) J r +Σ tr ( r, Et, ) J( r, Et, ) = φ ( r, Et, ) (5) v E There s a smlarty o Equaton (5) wth the Telegrapher Equaton, as can be seen n [4] [5]. However the methodology used to arrve at Equaton (5) s totally derent, as well as the results that ollow untl we get to the Equaton (8). In dvdng Equaton (5) by the transport cross secton, usng the denton o duson coecent, applyng J the dvergng operator dsregardng the term, t results that: ( r ), Et, J r, Et, v E where the duson coecent s dened as: ( Et) ( Et) φ ( Et) + J r,, = r,, r,,, (6),,. Σ,, ( r Et) tr ( r Et) ter that, n dervng Equaton (9) n relaton to tme, multplyng by very Equaton (9) replacng Equaton (6), one obtans: φ( r ) ( φ) ( v( E) ) ( r, Et, ) + {( L F) φ ( r, Et, )} v( E) 6 6 ( r ) = = 6 6 ( E) C( r t) Fφ( r Et) = = ( r Et),, v E ( r ) ret,,, Et, φ, Et, r, Et, r, Et, + + Lφ r, Et, v E r, Et, C r, t r, Et, = λχ E + Fφ, Et, Fφ, Et, v E t v E + λχ,,,., addng to the { ( r )} In the Equatons () (7), when n a statonary regme, all o ts tme dervatves are dsregarded. Then, t results that: ( ( r Et) φ( r Et) ) ( L F) φ( r Et) (7),,,, +,, =, (8) Consderng the adjont lux o neutrons rom Equaton (8), ths adjont equaton s: where, ( ( r Et) φ ( r Et) ) ( L + F + ) φ ( r Et),,,, +,, =. (9) + L r, Et, r, E E, t d E, () ( ) Σt ΣS ( )( ) 6 + = ( ) ( β) υ Σ χ ( ) + βυ Σ χ( ) F E r, Et, E d E E r, Et, E de () ( ) Σt ΣS ( )( ) L r, Et, r, E Et, de () 6

5 ( ) ( β) χ υ Σ ( ) + βχ υ Σ ( ) F E E r, E, t d E E E r, E, t de () = Equaton (8) s ntegrated n the volume n the energy, re-wrtten:, Et,, Et,, Et, ded r, Et, L F, Et, ded r ( r ) ( ( r ) ( r )) ( r )( ) ( r ) φ φ + φ φ = (4) V V In multplyng Equaton (7) by the adjont lux o neutrons ntegratng n the volume n the energy E, subtractng o the result by Equaton (4), one obtans: ( r, Et, ) φ ( r,, ) d d φ (,, )( ) φ(,, ) d d V v r r V = 6 ( r, Et, ) φ + Et E r+ Et L F Et E r φ ( r,, ) d d φ (,, ) { φ(,, )} d d V v r r V v V = V ( r ) φ( r ) + φ,,,, d d λχ φ Et E r Et L F Et E r Et L F Et E r C ( r, t 6 ) ( E) φ ( r, Et, ) ded r φ ( r ) { φ( r )} v = V 6,,,, d d Et F Et E r 6 6 λχ φ ( r ) ( r ) φ ( r ) φ( r ) = V = V + E, Et, C, t ded r, Et, F, Et, ded r, where, by smplcty, the unctonal dependence both o the duson coecent as o the speed was omtted. The neutron lux s wrtten as the product o an ampltude actor nt, whch s dependent on tme only, a shape actor (or shape uncton) ( r, E) ; thus, φ ( r, Et, ) nt ( r, E). (6) In wrtng the neutron lux as the product o the two actors n Equaton (6), the ntent s that the ampltude actor, nt, should descrbe most o the tme dependence whereas the shape actor, ( r, E), wll change very lttle wth tme. In denng the ntegral: 6 IF φ (, Et, ) ( β) χp( E) + βχ ( E) υ( E ) Σ (, E, t) (, E ) de ded r V = v r r r (7) one obtans, ater replacng Equaton (6) n Equaton (5) the dvson o the result by I F, that 6 6 d nt d d β nt C ( t) ( ρ + β) nt λc( t) λ = (8) = = It was consdered that the tme varaton o the duson coecent o the neutron cross sectons are neglgble, whch mmedately mples that ther dervatves are null. The several knetc parameters that appear n Equaton (8) are dened as: F V (5) (, Et, ) (, E) ded r, I v φ r r (9) I v φ (, Et, ) (, E) ded r, F r r () V ρ φ (, Et, ) {( F F) ( L L) } (, E) ded r, I r r () F V 6

6 β Fφ (, Et, ) (, E) ded r, I r r () F V 6 β β () = F V F = V 6 + φ (, Et, ) {( L F) (, E) } ded r φ (, Et, ) F (, E) ded r I r r v I r r (4) C t E, Et, C, t ded r. r r (5) χ φ V In multplyng Equaton () by the adjont lux o neutrons ater that ntegratng n the volume n the energy E, ater usng Equaton (6), one obtans that: V χ vdng Equaton (6) by φ ( r ) ( r, t) C,, d d E Et E r nt Et F E E r E Et C t E r φ ( r ) ( r ) λ χ φ ( r ) ( r ) =,,, d d,,, d d V V I F usng Equatons () (5), one obtans: ( r t) d C, (6) β = λc( r, t) + nt, (7) where =,,, 6. Equatons (8) (7) orm the new model or the pont knetcs, called moded pont knetcs.. Equvalency o the Moded Pont Knetcs Equatons wth Respect the Classc Pont Knetcs Equatons The classcal pont knetcs Equatons can be obtaned n many ways. Followng the development whch resulted n the moded pont knetc Equatons, the model or the classcal pont knetcs s obtaned by makng the approxmaton known as Fck s Law n Equaton (5), J( r, Et, ) = φ( r, Et, ) = ( r, Et, ) φ( r, Et, ). (8) Σ,, TR ( r Et) From there on, some terms o the rst second dervatve o nt the rst-order dervatve o C ( t ) appear n Equaton (7) do not appear n the correspondng equaton or the classcal knetcs. When t tends to zero, Equaton (8) results to the classcal pont knetcs equaton. The new parameters dnt ( ρ( t) β) t N λ = nt+ C t. (9) t can be called as neutron transport requency neutron absorpton requency, respectvely. When tends to zero the Equaton (8) les n Equaton (9), n other words, the pont classcal knetc equaton. Equvalent to state that the neutron transport requency s much larger than the other parameters. Consderng a homogeneous medum Equatons () (4) are smpled. Frst, the duson coecent constant the speed are consdered as the medum s homogeneous, whch gves: v I v φ = ( r, Et, ) ( r, E) ded r. (4) F V 6

7 Note that appears n the denton o Equaton (9) nto Equaton (4). = v v (4) Thereore, In Equaton (4) replace the operators L, F ( E) v =. (4) F. + φ (, Et, ) Σt (, Et, ) (, E) ded r I r r r F V v Σ ( ) φ ( ) ( ) I r r r v F V S, E Et,, E, t, E deded r ( β) χ υ Σ (,, )( φ (,, ) (, )) d d d I v r r r I F V I F V v 6 = βχ E E E t E t E E E r υ ( ) Σ ( r ) φ ( r ) ( r ) E E, t, E, t, E deded r 6 βχ ( E) υ( E ) Σ ( r, E, t) ( φ ( r, Et, ) ( r, E) ) de ded r. F = V Consderng the homogeneous medum rewrte the Equaton (4): (4) = + v t vσs φ (, Et, ) (, E) ded r v v I r r F V v + I v F 6 ( β) χ βχ υ( E ) ( r, E, t) φ ( r, E, t ) ( r, E ) de ded r. V = (44) Replacng the Equatons (7), (9) (4) n Equaton (44): once: ollows: = + v( Σ Σ ) t S (45) Σ t =Σ a +Σ S (46) = vσ (47) a 4. nalytcal Soluton o the Moded Pont Knetcs Equatons or One Group o Precursors wth Constant Reactvty The soluton or pont knetcs equatons can be obtaned n several ways, as n [6]-[8]. Pont knetcs equatons wthout the approxmaton related to the tme dervatve o neutron current densty, or one group o precursors constant reactvty, accordng to Equatons (8) (7), are: ( β ) ( ρ β) d nt dnt d λ Ct + + = nt + λct + (48) 6

8 The ntal conons are: dc t β = λct+ (49) dc t C nt t = ( ) n( ) =, (5) β = (5) λ n n =. (5) pplyng the conons (5), (5) (5) n Equaton (48) t s possble to wrte: One thereore chooses ( β ) nt d nt d ρ + + = n. d t t = t = dnt ρ t = (5) = n. (54) It s possble to very that the ntal conons (5), (5) (5) as appled to Equaton (8), that s, to the classcal pont knetcs, result exactly n Equaton (54). So, n replacng Equaton (54) n Equaton (5) t results: nt d t = It adds to the Equaton (48) wth Equaton (49): ρ β ρ = n. d β d d ρ λ + nt + = + d nt nt Ct t C t t dened by Equaton (49) n Equaton (48): You can also replace d d d nt ( ) dnt β ρ β λβ λ + nt + = + + λ Ct. The Equaton (57) s derved n the tme. Then, t results that: ( β ) ρ β λβ λ d nt d nt dnt dct + + = + + λ Multplyng the Equaton (56) by λ : ( β ) λ d nt d d λ nt λρ λ Ct + nt λ+ λ = + λ. It adds to the Equaton (58) wth the Equaton (59): ( β ) λ nt d nt d + + = λρ ( t) ρ β λβ λ dnt λ λ The Equaton (6) s a thrd-order homogeneous lnear derental equaton wll be solved by the Laplace Transorm technque. One can smply Equaton (6) as: nt (55) (56) (57) (58) (59) (6) 64

9 ( β ) λ nt d nt d + + λρ ( t) ρ β λβ λ dnt + + λ+ λ nt = In applyng the Laplace transorm [9] n Equaton (6), t results that: ( β ) λ { s L ( n( t) ) s n( ) s( n( ) ) ( n( ) )} + + s n t sn n ρ β λβ λ λρ { sl ( nt ) n( ) } L ( nt ), + + λ+ λ = beng, ( β ) λ + B ρ β λβ λ + + λ+ λ In Equaton (6) we solate the Laplace transorm o the neutron densty: { L ( ) ( )} (6) (6) (6) (64) n( ) s + ( n( ) ) + n( ) s + ( n( ) ) + ( n( ) ) + Bn( ) L ( nt ) = (65) λρ s + s + Bs + pplyng the reverse Laplace transorm on both sdes o Equaton (65) one have that: nt n( ) s + ( n( ) ) + n( ) s + ( n( ) ) + ( n( ) ) + Bn( ) L (66) s + s + Bs + = λρ To solve the reverse Laplace transorm one should actor the polynomals so to obtan: K K K L (67) s s s = + + nt Note that the Equaton (67) corresponds exactly to Equaton (66), n other words, can rewrte (67): nt = L ( ) K + K + K s K + + K + + K + s + K + K + K s + + s s Comparng the above equatons t s possble to deduce the number system: K ( n ( ( ) ) + n ( ) + n( ) ( + ) ) ( ) ( ) n( )( ) + ( n( ) ) + n( ) + ( n ) + ( n( ) ) + B n( ) ( ) = n + + +, (68) 65

10 K K = ( n ( ( ) ) + n ( ) + n( ) ( + ) ) ( ) ( ) ( ) ( ) n n n n n B n ( ) ( ) ( ) n + n + n + n + n + B n = + +, (69) (7) = (7) ( ) + + = ( B) λρ = The Equaton (7) can be solved drectly to obtan the value o. Note that t s an equaton o the thrd de gree. Intally t replaces the varable by x : Note that n Equaton (74) the term correspondng to x B x λρ ( ) B + ( ) + λρ ( ) + ( ) + = 7 9 x p x q x s reset. Thus, t can be rewrtten: (7) (7) (74) + + = (75) The Equaton (75) was resolved accordng [] the soluton s known as Cardano-Tartagla s ormula: x q q p q q p = x = Note that Equaton (76) corresponds only one o the three solutons o the Equaton (85). The other two solu- x x whch allow us to obtan: tons can be obtaned by dvdng Equaton (75) by thereore, Follows: + + = + + x x x p x q ax b x c x + p x+ q = ax + b x+ c x x = ax b x c (76) + + = (77) The roots o Equaton (77) are the other two solutons o Equaton (76). Thus three possble solutons whch satsy the Equaton (7) are obtaned. The solutons are substtuted n Equaton (7). Equaton (7) becomes an Equaton o the second degree varable can be trvally solved. Wth solutons can be obtaned rom Equaton (7). Equatons (68), (69) (7) are all expressed only n terms o, so the values o K, K K are determned drectly. From the resoluton o the nverse Laplace transorm o Equaton (67), results that: 66

11 5. Results exp( ) exp( ) exp( ) n t = K t + K t + K t (78) Moded pont knetcs equatons or one group o precursors wth constant reactvty are solved wth the method descrbed n Secton. Numercal values or the nuclear parameters are consdered rom reerences ound n the lterature,.e., [] []-[5] as lsted n Table. The values consdered n Table or the absorpton cross secton the duson coecent correspond to the average cross sectons o a typcal core n a PWR nuclear reactor, accordng to [] []. The ntal conons (5), (5), (5) are used n the calculatons o classcal moded equatons the ntal conons (54) (55) are used only n the moded pont knetcs equatons. We consder the values o Table. The results are shown n the Fgures -6 n the Table to Table. Fgures -6 show the results o the calculatons made wth the soluton o classcal pont knetcs equaton, moded pont knetcs equaton wth transport requency equal to 4 s moded pont knetcs equaton wth transport requency equal to s. In the rst three gures the tme nterval s rom to s n the last three t goes rom to s. Note that n the graph contaned n Fgure the order o magntude or neutron densty o the classcal knetcs o the moded knetcs are qute derent or a reactvty o.7. The varaton n the neutron densty as a uncton o the reactvty s seen through a comparson between the graphs. It s possble to see that, or a reactvty equal to the racton o neutrons delayed by the total o neutrons, the neutron densty obtaned by the classcal pont knetcs equatons or a tme correspondng to s s o the Table. Parameters used n the tests. Fgure. Neutron densty as a uncton o the tme o s at s or a reactvty o.. Parameter Symbol Value ecay constant λ.8958 s Mean generaton tme. s bsorpton cross secton Σ a.4 cm uson coecent cm Neutron velocty v 6 cm/s Fracton o delayed neutrons β.7 bsorpton requency 4.67 s 67

12 Table. Calculaton o n(t) (cm ) wth pont knetcs or a group o precursors wth a neutron transport requency o 4 s. Model Reactvty t =.4 s t = s t = s t = s t = 4 s t = s Classcal knetcs ρ = Moded Knetcs ρ = Classcal knetcs ρ = Moded Knetcs ρ = Classcal knetcs ρ = Moded Knetcs ρ = Table. Calculaton o n(t) (cm ) wth pont knetcs or a group o precursors wth a neutron transport requency o s. Model Reactvty t =.4 s t = s t = s t = s t = 4 s t = s Classcal knetcs ρ = Moded knetcs ρ = Classcal knetcs ρ = Moded knetcs ρ = Classcal knetcs ρ = Moded knetcs ρ = Fgure. Neutron densty as a uncton o the tme o s at s or a reactvty o.. 68

13 Fgure. Neutron densty as a uncton o the tme o s at 4 s or a reactvty o.6. Fgure 4. Neutron densty as a uncton o the tme o s at s or a reactvty o.. Fgure 5. Neutron densty as a uncton o the tme o s at s or a reactvty o.. 69

14 Fgure 6. Neutron densty as a uncton o the tme o s at s or a reactvty o.. order o, whlst the use o the moded pont knetcs equatons s o the order o 9, or a neutron transport requency o s. Thus, the derence between the results o the models s more sgncant or hghreactvty stuatons. 6. Conclusons The objectve o ths paper s to obtan a new system o equatons called equatons o pont knetcs moded n whch s consdered the eect o the tme dervatve or neutron current densty n the Equaton (5). In general, the tme dervatve o the densty o neutrons s neglected or the obtanment o the classcal model. The results presented n ths artcle show that the derence between the neutron densty obtaned rom classcal pont knetcs equatons that obtaned rom moded pont knetcs equatons s relevant. Wth the neutron transport requency equal to 4 s the derence between the neutron densty obtaned rom classcal pont knetcs equatons that obtaned rom pont knetcs equatons wthout the approxmaton or the tme dervatve o neutron current densty s relevant. Wth a neutron transport requency equal to s the derence between them t s qute sgncatve. Moded pont knetcs equatons mply a sgncant derence n results, n relaton to those obtaned wth the classcal pont knetcs. Table Table show that the results rom the classcal knetcs have an mportant derence n relaton to the model o the moded pont knetcs that ncreases when the requency s smaller. Reerences [] udersta, J.J. Hamlton, L.J. (976) Nuclear Reactor nalyss. John Wley & Sons Ltd., New York. [] Bell, G.I. Glasstone (97) Nuclear Reactor Theory. Van Nostr Renhold Ltd., New York. [] Henry,.F. (975) Nuclear Reactor nalyss. The MIT Press, Cambrdge London. [4] Hezler, S.I. () symptotc Telegrapher s Equaton (P) pproxmaton or the Transport Equaton. Nuclear Scence Engneerng, 66, [5] Espnosa-Paredes, G., Polo-Labarros, M.., Espnosa-Martnez, E.G. del Valle-Gallegos, E. () Fractonal Neutron Pont Knetcs Equatons or Nuclear Reactor ynamcs. nnals o Nuclear Energy, 8, [6] kcasu, Z., Lellouche, G. Shotkn, L.M. (97) Mathematcal Methods n Nuclear Reactor ynamcs. cademc Press, New York London. [7] Chao, Y.. ttard,. (985) Resoluton o the Stness Problem o Reactor Knetcs. Nuclear Scence En- 7

15 gneerng, 9, [8] Zhang, F., Chen, W.Z. Gu, X.W. (8) nalytc Method Study o Pont-Reactor Knetc Equaton When Cold Start-Up. nnals o Nuclear Energy, 5, [9] Hoogenboom, J.E. (985) The Laplace Transormaton o djont Transport Equatons. nnals o Nuclear Energy,, [] Fuchs,. Tabachnkov, S. () Mathematcal Omnbus: Thrty Lectures on Classc Mathematcs. mercan Mathematcal Socety, Rhode Isl. [] Hetrck,.L. (97) ynamcs o Nuclear Reactor. The Unversty o Chcago Press Ltd., Chcago London. [] Stacey, W.M. (7) Nuclear Reactor nalyss. nd Eon, Wley-VCH GmbH & CO KGa, Wenhem. [] Knard, M. llen, E.J. () Ecent Numercal Soluton o the Pont Knetcs Equatons n Nuclear Reactor ynamcs. nnals o Nuclear Energy,, [4] Palma,..P., Martnez,.S. Gonçalves,.C. (9) nalytcal Soluton o Pont Knetcs Equatons or Lnear Reactvty Varaton. nnals o Nuclear Energy, 6, [5] Jahanbn,. Malmr, H. () Knetc Parameters Evaluaton o PWRs Usng Statc Cell Core Calculaton Codes. nnals o Nuclear Energy, 4,

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