SOLITARY BURN-UP WAVE SOLUTION IN A MULTI-GROUP DIFFUSION-BURNUP COUPLED SYSTEM

Transcription

1 3th Internatonal Conerence on Emergng uclear Energy Systems (ICEES 27) Istanbul Turkey June on CD-ROM Gaz Unversty Ankara Turkey (27) SOLITARY BUR-UP WAVE SOLUTIO I A MULTI-GROUP DIFFUSIO-BURUP COUPLED SYSTEM X.-. Chen W. Maschek A. Rnesk and E. Kehaber Insttute or uclear and Energy Technologes Forschungszentrum Karlsruhe P.O.B. 364 D-762 Karlsruhe Germany xue-nong.chen@ket.zk.de; werner.maschek@ket.zk.de; andre.rnesk@ket.zk.de; edgar.kehaber@ket.zk.de ABSTRACT A two-group duson model coupled wth smpled burn-up equatons s nvestgated or a one-dmensonal burn-up drt wave problem whch s related to the recently developed concept o a so-called CADLE reactor. Ths coupled system s solved successvely by usng the ntegrablty (analytcal solvablty) n the one-group theory [5] under an ntal assumpton o a constant racton o slow neutron lux. The mult-group eects are revealed by solvng the slow part o the two-group duson equatons. The method s convergent quckly practcally ater one teraton. It s shown that the racton o slow lux vares slghtly n a soltary wave soluton however t has a sgncant quanttatve mpact on the soltary wave soluton n partcular on the wave length and drt speed. Key Words: Duson model burn-up equatons mult-group soltary wave soluton CADLE concept. ITRODUCTIO The soltary burn-up wave soluton or CADLE concept becomes well-known n the seres o ICEES meetngs [ ]. The basc dea behnd ths concept s the exstence o a sel-propagatng nuclear breedng/burnng soltary wave n a ertle medum o 238 U or 232 Th where the reactvty remans almost constant or a long tme. atural thorum and uranum uel can be used or ths concept and a hgh burn-up can be acheved. Thereore uel enrchment and reprocessng are not needed and a long contnuous operaton duraton s possble. evertheless an ntal gnton s needed or ths concept as well as ether a long core or a sutable reuellng e.g. as suggested n [4 6]. Fundamental nsght nto ths new type o reactor was gven by van Dam (2) [7] and Sertz (2) [8]. An exact soltary wave soluton was ound n a -D one-group duson equaton wth artcally assumed burn-up dependent coecents and a nonlnear term o eedback eects [7]. The soltary wave soluton was ound analytcally as well n the same duson equaton wthout eedback eects but coupled by smple burn-up equatons [8] where the soltary wave s generally ore-at asymmetrc (skew). Intensve numercal studes have been made by Sekmoto Ryu and Yoshmura (2) [9]. They consdered that a nuclear gnton regon charged wth plutonum or enrched uranum was set at one end o the core and natural or possbly depleted uranum was loaded n the remanng regon. They solved the mult-group duson and burn-up equatons numercally and demonstrated the easblty o ths new concept.

2 X.-. Chen et al. The man purpose o ths paper s to nvestgate the eects o several energy groups nstead o only one. A neutronc model.e. two-group duson equatons coupled wth burnup equatons are proposed or obtanng a -D asymptotc soluton n a movng coordnate system. In the burn-up equatons only 238 U 239 Pu 24 Pu and a typcal sson product par (FPP) are consdered. Radoactve decay processes are neglected because the radoactve decay processes are ether too short or too long wth respect to the consdered tme scale o the order o several years. Hence as the results o the soluton o burn-up equatons macroscopc cross sectons are only unctons o the neutron luence and the ntal values o atom number densty and consequently the duson equatons become nonlnear derental-ntegral ones. In order to use the analytc solvablty o the one-group model [5] the two two-group equatons are merged nto one or the total neutron lux by summng them. The whole mult-group model s solved successvely as descrbed n the ollowng. By assumng a sutable constant racton o slow lux the merged equaton can be solved by applyng the ntegratng method [5] and thereore a soluton o the soltary burn-up wave can be ound. Based on ths soluton the slow lux can be obtaned by solvng the assocated equaton whch ncludes a known nscatter term. Consequently the racton o slow lux s derved rom the solutons whch s now luence dependent. For the next teraton step ths slow lux racton s substtuted nto the merged total lux equaton and a new soltary wave soluton o total lux s obtaned as well as a new slow lux. Ths procedure can be repeated theoretcally urther on. But practcally t s sucent to go only once through the teratve treatment. 2 EUTROIC MODEL A large energy-group number s not desrable here because the purpose o ths paper s to gan nsght nto prncpal phenomena o ths knd o reactor and to dscuss the soluton method rather than to obtan a very accurate soluton. Thereore two-group duson equatons are sutable here. The two-group duson equatons are wrtten or the neutron balance n the core v φ = ( D φ ) Σ aφ Σ 2φ + νσ φ + νσ 2φ2 (a) v 2 φ2 = ( D2 φ2 ) Σ a2φ2 + Σ 2φ. (b) where the subscrpts and 2 stand or ast and slow groups respectvely. The contrbuton to slow neutrons by the sson processes has been neglected n (b).e. sson neutrons appear only n the ast group. For the nuclde balance just or the sake o smplcty we consder only the heavy metals 238 U 239 Pu and 24 Pu characterzed by the ndces = 8 9 and two typcal knds o sson product pars (FPP).e. a burnable FPP (FPP_burn) and an nert FPP (FPP_nert). The smpled burn-up equatons read 8 = φ 8 a 8 9 = 9 φ a 9 φ + 8 c8 = φ a φ + 9 c9 ICEES 27 Istanbul Turkey 27 2/

3 Soltary burn-up wave soluton n mult-group system FPP _ burn = FPP _ burn afpp φ + = 89 FPP _ nert φ = FPP _ burn afpp φ. All symbols have ther usual meanng and all knds o φ n stands or φ + 2 φ2. Eqs. and are coupled n such a manner that the soluton o Eq. provdes φ or Eq. and the soluton o Eq. provdes needed or determnng the nuclear propertes n Eq. as Σ a n = ( a ) Σ n ν ( ) n ν Σ ( ) = n = tr n tr n D n = (3a) 3Σ tr n Σ = 2 2 where the subscrpt n stands or energy group and or sotope. (3b) 3 MATHEMATICAL SOLUTIO 3. Soluton o Burn-up Equatons The burn-up equatons n can be solved n a straghtorward manner. All atom number denstes can be expressed as a lnear combnaton o exponental unctons o the neutron luence ψ multpled by assocated ntal values.e. t = (ψ ) wth ψ = φ dt. (4) For the sake o smplcty the ollowng two smplcatons are made: () the actnde burn-up chan s delberately cut arly early namely already at 24 Pu and () the resulted onegroup mcroscopc cross secton data are approxmated to be constant.e. as same as n the one-group model. The second smplcaton s made to avod numercal ntegraton snce the mcroscopc cross secton data are n general luence dependent and explct solutons as wrtten below could then not be acheved. Although these smplcatons may cause some error n the result they are not a real restrcton or the soluton method descrbed n ths paper. The equatons or and n gve n the case o non-zero t 8 = 8 8 e a ψ. (5) 9 9ψ a c8 a8ψ a9ψ = 9 e + 8 [ e e ]. (6) a9 a8 ICEES 27 Istanbul Turkey 27 3/

4 X.-. Chen et al. + = 8 9 a9 c8 a c9 a8 a9 a9ψ a ψ [ e e ] c9 a a 8 ψ ψ a8 a c9 a9ψ a ψ ( e e ) ( e e ) a a9 (7) FPP can be wrtten as FPP =. (8) FPP _ burn + FPP _ nert _ and can be carred out explctly smlar to [5] but the expresson or FPP burn FPP _ nert them are qute lengthy. For the sake o savng place they are omtted here. 3.2 Asymptotc Formulaton Snce we expect that there wll be an asymptotc soluton that drts at a constant speed u we ntroduce a Gallean transormaton ζ = z + u t x = x y = y ; t = t. (9) Ths means the movng coordnate system translates at the constant speed u n the negatve z- drecton and the transverse coordnates and the tme are not changed. Because o the assumpton o a tme-ndependent soluton n terms o the movng coordnates and u << v2 < v.e. the drt speed s much smaller than the mean neutron velocty o the lowest energy group the tme dervatve term n Eq. can be neglected or exactly sayng the derved convectve terms ( u / vn ) φ n ζ n = 2 can be neglected. Thus Eq. becomes quas-statc n terms o movng coordnates as ( φ ) Σ φ Σ φ + νσ φ + νσ D φ (a) a = ( φ ) Σ φ + Σ D. (b) 2 2 a2 2 2φ = where Σ a n ( ψ ) νσ n ( ψ ) D n (ψ ) and Σ 2 ( ψ ) are known unctons o the total neutron luence obtaned rom the burn-up equatons. The total neutron luence n (4) can now be expressed n terms o an ntegral n ζ or t = as t ψ = φ dt = φ( r ζ ) dζ. () u Ths equaton mples also a derental relaton between φ and ψ revealng the proportonalty between φ and u ζ ICEES 27 Istanbul Turkey 27 4/

5 Soltary burn-up wave soluton n mult-group system φ( r ζ ) = u ψ ( r ζ ). (2) ζ 3.3 Method to Solve the Duson Equatons As reported by Chen et al. [5] at the last ICEES Conerence the one-group duson equaton n a one-dmensonal case s ntegrable and the soltary wave soluton can be obtaned analytcally. Unortunately t s not so smple to ntegrate analytcally the two-group duson equatons. evertheless we can apply the ntegrablty n the one-group case and obtan an approxmate two-group soluton successvely. The ollowng procedure s carred out. At rst we merge two duson equatons nto one equaton just by summng them. By choosng a sutable constant rato o α = φ2 φ and applyng the same ntegratng procedure as n [5] a soltary wave soluton or the total neutron lux s obtaned. By solvng the second equaton (b) the slow neutron lux φ 2 s obtaned and correspondngly α (ψ ) whch s now space- or better sayng luence-dependent. Wth ths α (ψ ) one can repeat the ntegratng procedure or the total lux and the numercal soluton or the slow lux untl obtanng a new α (ψ ). I the new α (ψ ) s sucently close to the prevous one ths successve method can be nshed and the soluton s convergent to the true one or the orgnal system. Snce the luence dependence o α s qute weak at least n the current case the convergence o ths successve method s very quck whch s usually obtaned just by or 2 teratons. The whole calculaton s realzed by usng Mathematca. By summng (a) and (b) we obtan a uned sngle group equaton as ( φ) Σ φ + νσ φ = D (3) a where the macroscopc coecents are nterpreted as D φ = D φ + D2 φ2 aφ = Σ aφ + Σ a2φ2 Σ νσ φ = νσ φ + νσ 2φ2 (4a) φ = φ + φ 2. (4b) For smplcaton we may assume that D D2 and D are ndependent o space although t may not be needed or some partcular cases e.g. D = D2. Thus the expresson or D s smpled as D φ = Dφ + D2φ2. evertheless ths smplcaton s not a real restrcton or the applcaton o the method snce D can always be expressed n a bt more complcated manner as a uncton o ψ by usng the orgnal denton. ow we consder the one-dmensonal case and carry out the soluton successvely. () Choosng a sutable constant α = φ2 φ whch may be determned by tral and error ater the soluton o the equaton or the slow lux we get νσ ( ψ ) D ( ψ ) φ Σ a ( ψ ) φ + φ = (5) ζ ζ ke ICEES 27 Istanbul Turkey 27 5/

6 X.-. Chen et al. where an addtonal egenvalue k e has to be ntroduced n order to obtan a non-trval crtcal soluton. The above equaton s analytcally solvable as shown n [5]. Thereore we obtan a rst approxmate soltary wave soluton denoted as φ = φ ( ψ ) and ζ ( ψ ) wth assocate. Further we wrte Eq. (b) as k e D 2 φ2 ( Σ a2 + Σ 2 ) φ2 + Σ 2φ =. (6) ζ ζ Regardng the last term o the above equaton as a known uncton Σ 2φ ( ζ ) we solve the above equaton numercally to get φ 2 = φ2 ( ζ ) and thereore the rst approxmate α ( ζ ) = φ2 φ. Snce ζ and ψ are one-to-one correspondng α can be also wrtten as a uncton o ψ as α ( ψ ). Theoretcally we can repeat ths process several tmes to obtan (n) ( n) ( n) ( n) ( n) (n) φ rom (5) and φ2 rom (6) and consequentlyα = φ2 φ and k e. Practcally as we wll see later on one teraton s sucent to get a arly accurate result because α s qute close to α. 3.4 umercal Results A sutable normalzaton makes the ormulaton and results more clear and general. For the sake o easy recognton correspondng captal letters wll be used or the nondmensonal varables n the ollowng. The most sutable and natural normalzaton o neutron luence and lux s Ψ = a ψ Φ = φ φmax (7) The tme space and drt speed scales can be derved rom the above normalzatons as t = ( ) l D ( ) a φmax = u = l t = φmax ad (8) a where a s the average mcroscopc absorpton cross secton o the resh uel the atom number densty o heavy metal sotopes o the resh uel and D the duson coecent at the ntal state (the core wth resh uel). Thereore the spatal coordnate and the drt speed s normalzed as Ζ = ζ l u u U =. (9) 2 In the present example we have a =. 528 barn = 6.32 cm -3 and D =.556 cm. 5 We choose φmax = 3 cm -2 s - whch corresponds to a maxmal power densty o about 5 W/cm 3 whch may be too conservatve. The length and drt speed scales are derved as 8 l = 2.6 cm and u = 3.42 cm/s correspondng to about.8 cm/year. We take our materal composton to be smlar to that n a smpled standard system o sodum-cooled ast breeder reactor (FBR) wth oxde uel Wrtz (973 page 83). Twogroup data shown n Table have been generated or ths smpled standard system by applyng a transport code. The materal composton n atom number denstes gven n Table A.3. by Wrtz (973) s used or generaton o the two-group data. Just or smplcaton ICEES 27 Istanbul Turkey 27 6/

7 Soltary burn-up wave soluton n mult-group system the capture cross sectons o O a Fe Cr and have not been taken nto account or the calculatons presented n ths paper. Table. Two-group mcroscopc data n barn or the smpled standard system o sodum cooled FBR wth 5 vol% sodum 3 vol% oxde uel and 2 vol% structure (steel) or the boundary value between two energy groups s.9 kev where the upper entry s or the ast group and the lower one or the slow group. tr c ν U Pu Pu FP O a Fe Cr In the ollowng the results o our study or the smpled model wll be presented and dscussed. The burn-up soluton s shown n Fg.. Ater an ntal tral we take the constant () alpha as α = φ2 φ =. 538 or the rst step whch s nearly an average value o the rst approxmate soluton that wll be obtaned later on. We solve Eq. (5) analytcally to obtan φ and urther solve Eq. (6) numercally to obtan. Consequently we obtan φ 2 α ( ζ ) = φ2 φ or wrtten as a uncton o ψ.e. α ( ψ ). At ths step the ntal enrchment has been chosen as 9 ( ) = so that k e =. Wth α ( ψ ) we repeat the above procedure to obtan the second approxmate soluton φ and as well as assocated α ( ψ ) and k. The obtaned α ( ψ ) s shown n Fg. 2 (a) together e () wth α and α ( ψ ) whle = whch s only 29 pcm less than. k e k e φ 2 ICEES 27 Istanbul Turkey 27 7/

8 X.-. Chen et al. orm. atom number densty U238 Pu239 Pu24 FPP orm. neutron luence Fgure. Burn-up soluton or constant mcroscopc data ormalzed neutron lux (a) Z Fgure 2. Comparson o rst and second approxmate solutons: (a) Fracton o slow neutron lux as uncton o neutron luence; (b) ormalzed total neutron lux where the wave drts rom rght to let. Fg. 2 (b) compares the rst and second approxmate solutons o φ. Snce the rst () approxmate soluton s based on the constant α φ tsel s actually a one-group soluton; whle the second approxmate soluton φ based on α s an approxmate twogroup soluton. The derence between them mples actually a mult-group eect. Although the varaton o α s rather small t has a sgncantly quanttatve nluence on characterstcs o the wave soluton e.g. the wave length and ts drt speed. The two-group soluton has a smaller hal-heght wave length and a smaller drt speed than the one-group soluton by 9% and 25% respectvely. The complete second approxmate soluton s shown n Fg. 3 and Fg. 4. The normalzed neutron luxes Φ and the slow lux racton α are shown n Fg. 3. The varaton Φ 2 o α ( Ζ) s smooth and qute small whereas t has a maxmum o.59 n the resh uel regon (n the ront o the wave) and a mnmum o.52 at the wave peak. The total normalzed neutron lux k and the atom number denstes o 238 U 239 Pu 24 Pu FPP are shown together n Fg. 4. As already known rom studes o the CADLE reactor e.g. [4-8] t can also be seen n the gure that k ncreases rom an ntal subcrtcal level to supercrtcal due to a net generaton o 239 Pu and then decreases to a subcrtcal level agan due to 239 Pu consumpton and FPP producton. The burn-up can be deduced rom ths gure 7 as about 3%. The drt speed u = 5.28 cm/s s derved rom U = u / u = e. (b) ICEES 27 Istanbul Turkey 27 8/

9 Soltary burn-up wave soluton n mult-group system about 6.7 cm/year and the length scale l = 2. 6 cm.e. the range o (-2 2) n Z shown n Fgs. 3 and 4 corresponds to 8.64 m and the hal-hgh wave length 2.6 m Z Fgure 3. Second approxmate soluton o Φ Φ 2 and α where the wave drts rom rght to let..8.6 U238 Wave drt drecton k.4 FPP.2 Pu239 Fgure 4. Second approxmate soluton o number denstes. Pu24-2 Z Φ k and assocated normalzed nuclde atom 4 COCLUSIOS Two-group duson equatons coupled wth smpled burn-up equatons are nvestgated or a CADLE-type reactor. In the one-dmensonal case the model s solved approxmately by a successve method based on the act that the racton o slow neutron lux changes slghtly along the reactor axs. The ntegrablty o the one-group model s used or solvng the total neutron lux and a smple numercal method or the slow lux. It has been shown that the one-group model can only provde a qualtatve soluton whch s not sucent quanttatvely. Although the approxmate soluton derved wth the descrbed smplcatons s also qute rough the method (dea) can be urther developed or a more accurate soluton o a mult-group model wth a large number o energy groups. ICEES 27 Istanbul Turkey 27 9/

10 X.-. Chen et al. 5 REFERECES. E. Teller M. Ishkawa L. Wood Completely automated nuclear reactors or long-term operaton Proceedngs o ICEES 96 Obnnsk Russa June pp.5-58 (996). 2. H. Sekmoto and K. Ryu A long le lead-bsmuth cooled reactor wth CADLE burnup Proceedngs o ICEES Petten The etherlands paper3 (2). 3. H. van Dam The sel-stablzng crtcalty wave reactor Proceedngs o ICEES Petten The etherlands pp (2). 4. H. Sekmoto and S. Myashta Startup o CADLE burnup n ast reactor rom enrched uranum core Proceedngs o ICEES 25 Brussels Belgum on CD-ROM SCK-CE Mol Belgum (25). 5. X.-. Chen E. Kehaber and W. Maschek eutronc model and ts soltary wave solutons or a candle reactor Proceedngs o ICEES 25 Brussels Belgum on CD- ROM SCK-CE Mol Belgum (25). 6. X.-. Chen E. Kehaber and W. Maschek Fundamental burn-up mode n a pebble-bed type reactor 2 nd COE-IES Int. Symposum on Innovatve uclear Energy Systems Yokohama Japan (26). 7. H. van Dam Sel-stablzng crtcalty waves Annals o uclear Energy 27 pp (2). 8. W. Sertz Soltary burn-up waves n a multplyng medum Kerntechnk 65 pp.5-6 (2). 9. H. Sekmoto K. Ryu Y. Yoshmura CADLE: The new burnup strategy uclear Scence and Technology 39 pp (2).. K. Wrtz Lectures on Fast Reactors Gesellschat ür Kernorschung m.b.h. Germany (973).. X.-. Chen W. Maschek Transverse bucklng eects on soltary burn-up waves Annals o uclear Energy 32 pp (25). ICEES 27 Istanbul Turkey 27 /