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KFK 630 SM 101/16 EUR 3674 e

Oktober nstitut fur Neutronenphysik und Reaktortechnik KFK 630 SM 101/16 EUR 3674 e The Long-Time Behaviour of Fast Power Reactors with Pu-Recycling A. Jansen Als Manuskript vervielfaltigt FUr diesen
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Oktober nstitut fur Neutronenphysik und Reaktortechnik KFK 630 SM 101/16 EUR 3674 e The Long-Time Behaviour of Fast Power Reactors with Pu-Recycling A. Jansen Als Manuskript vervielfaltigt FUr diesen Bericht behalten wir uns alle Rechte vor Gesellschaft flir Kernforschung m.b.h. Karlsruhe KERNFORSCHUNGSZENrRUM KARLSRUHE Oktober 1967 KFK 630 SM 101/16 EUR 3674 e nstitut fur Neutronenphysik und Reaktortechnik The Long-Time Behaviour of Fast Power Reactors with Pu-Recycling* A. Jansen Gesellscha:f't :t'iir Kernforschung mbh., Karlsruhe Work performed within the association in the field of fast reactors between the European Atomic Energy Community and Gesellschaft fur Kernforschung m.b.h., Karlaruhe. NTERNATONAL ATOMC ENERGY AGENCY SYMPOSUM ON FAST REACTOR PHYSCS AND RELATED SAFE'Y PROBLEMS 30 October - 3 November 1967 Kernforschungszentrum Karlsruhe Germany SM 101/16 THE LONG-TME BEHAVOUR OF FAST POWER REACTORS WTH PLUTONUM RECYLNG A. Jansen nstitut fiir Neutronenphysik und Reaktortechnik Kernforschungszentrum Karlsruhe, Germany work performed within the association in the field of fast reactors between the European Atomic Energy Community and Gesellschaft fur Kernforschung m.b.h., Karlsruhe. - 2-1, NTRODUCTON The long-time behaviour of fast power reactors with plutonium recycling generally includes the overall time behaviour caused by the fuel burn-up in the reactor and the internal and external fuel management. t is useful to distinguish between two types of problems: (a) The behaviour of the reactor between two loading events which refers to the time dependence of neutron flux. reactivity. power distribution fuel composition etc,; usually it is the object of the conventional fuel burn-up studies, (b) The behaviour of the reactor during many reloading cycles which refers to the time dependence of the isotopic composition of plutonium the uranium and plutonium content of the fuel. the breeding gain etc, n this paper we will confine ourselves to the second class of problems only, We consider a fast power reactor in which only fissionable material is consumed which is bred in the same reactor, except for the plutonium amount to start up the reactor and for a certain period a~erwards, Therefore we have to consider a closed fuel cycle including the core and the external plants for reprocessing the irradiated fuel and refabricating the fuel elements, The blank.et plutonium is introduced into the closed fuel cycle the breeding gain is drawn off this cycle. The fuel cycle for common reprocessing of the core and blanket elements is shown in Fig, 1, The time variations of the isotopic composition of the plutonium in the fuel cycle system are caused by the neutron irradiation in the reactor and by the internal and external fuel management ), The composition of the refabricated fuel elements is determined by that uranium-to-plutonium ratio which maintains the reactor critical assuming the total number of the heavy isotopes in the fuel elements to be constant. ) nternal fuel management is always done to improve the burn-up behaviour of the reactor; external fuel management refers to reprocessing and refabrication, - 3 - The problems which we are going to investigate were studied first in [1] on the basis of an idealized zero-dimensional model without taking into account the internal and external fuel management. Loading and unloading the core and blanket elements were regarded as continuous processes, n this paper we will investigate the long-time behaviour of the plutonium. composition in the fuel and the physical aspects of the fuel cycle econany under more realistic conditions, Especially loading and unloading the core and blanket elements will be treated as discontinuous processes; the external fuel management and the times for reprocessing and refabrication will be taken into account, As we will not look at the first type of problems of long-time behaviour, we can make some simplifications which refer essentially to thesecondary time variations of neutron flux and spectrum caused by the fuel burn-up. By these simplifications we are able to solve the long-time problems with sufficient accuracy by tolerable numerical efforts. The formalism developed below is applied to a sodium cooled 1 OOO MW(e) power reactor [2]. The main numerical results are given in the last section of this paper, 2, THE MATHEMATCAL FORMALSM FOR CALCULATNG THE LONG-TME BEHAVOUR 2.1. C~n~itions for Reactor Operation We consider a cylindrical reactor with few core zones, an axial and a radial blanket (Fig. 2), Exchanging the core and blanket elements takes place at each of the discrete time values ( 1 ) t = s tit s C lltc is a given constant correlated to the thermal reactor power Q and the given maximum burn-up of the fuel elements as shown in equation (6), The core elements are loaded and unloaded according to the cyclic burn-up scheme described in [3], The core is subdivided into radial regions; the elements of the same radial region are combined into groups of n elements (e,g, n=3), At each point of time t only one element of each group 8 - 4 - is exchanged for a fresh fuel element, During a period of time rtitc each element of a group is exchanged once and only once. The life time of the core elements is o = natc i.e. the total irradiation time of a fuel element in the core, At each time t only the n-th part of the core elements s is exchanged for fresh elements. f Ve is the total fuel element volume of the core, the exchanged fuel element volume at t 6 is (2) 1 n tivc=-v. C The operation of the axial and radial blankets must be in agreement with the rhythm given by (1); i.e. loading and unloading the blanket elements must succeed in intervals mtitc rn may be a positive integer. The elements of the axial blanket are coupled with the core elements; therefore both are managed jointly. Contrary to this the management of the radial blanket is largely independent of the core management. We assume the re. dial blanket to be divided into NB radial zones, which are exchanged as a whole. The life-time of the j-th zone of the radial blanket may be (3) the integer m. is related to the maximum burn-up in the j-th zone, J The a.mount of uranium and plutonium discharged out of the core and the axial and radial blankets is calculated in two steps: a) the calculation of the space dependent uranium and plutonium concentrations of the discharged elements by solving the fuel burn-up equations; b) the calculation of the uranium and plutonium content of these elements by volume integration. The fuel burn-up calculations and the volume integration for the core and blanket elements are discussed in the following sections The Fuel Burn-up Equations The time behaviour of the fuel composition during neutron irradiation in the reactor is determined by the well-known fuel burn-up equations. As the main components of the fuel we regard the two uranium - 5 -, isotopes U and U and the four plutonium isotopes Pu, Pu, Pu, and 242 Pu. Therefore we have to solve a system of six first-order differential equations in time and flux time. respectively. The coefficients of these equations are the one group space dependent neutron flux and microscopic cross sections for neutron absorption and capture. which are defined in the multigroup picture as (4) ~(r 1 z) = i,i(r,z) 1 ~i(r.z) is the neutron flux in the i-th neutron energy group at r 1 z; the,i(r.z) are determined by multigroup calculations, The oi are the microscopic cross sections of the heavy nuclei in the fuel for neutron absorption and capture in the i-th energy group. n general,~ and also the o are time dependent as a consequence of the variation of the fuel composition in the reactor during neutron irradiation, But it will be possible to calculate the uranium and plutonium concentrations in the irradiated elements at the end of their life time by solving the fuel burn-up equations with sufficient accuracy 1 if we take into account time averaged, and o instead of the exact time dependent quantities. The latter, in general would require the simultaneous solution of the burn-up equations and the criticality equation of the reactor, e.g. multigroup-diffusion equations. n our case the time averaged one group quantities are calculated according to equation (4) with time averaged neutron spectra, which are determined by multigroup calculations for an average burn-up state of the reactor assuming a homogeneous distribution of the fuel. The one group neutron flux for the average burn-up state is normalized to the given reactor power Q which is regarded as constant, 2.3. The Uranium and Plutonium Output of the Core &n;d the Axial Blanket We suppose that the core is divided into N radial zones of different fuel compositions. The N. k(s.o) may be the time dependent initial con- J. 1 ) centrations of the nuclei k s 1,2,,6 in the fuel elements which are 1) The indices k = 1 1 2,,6 refer to the isotopes 23 5u, 238 u Pu,, 242 p respectively 1 which are the components of the fuel mixture. - 6 - loaded into the core zone J = 1.2, N at the time ts' We presume N. k(s.o} = N. k(o.o} for all s s ; j = 1.2,,N; k = 1.2, 6. J. J. 0 s designates the time ts at which refabricated fuel elements are loaded 0 O into the core for the first time. i.e. for closing the :fuel cycle. s is 0 determined by the external fuel management as shown below. The N. k(o,o) Jt denote the j-th initial composition of the fuel elements at t 0 = 0, For all s s the N. k(s,o) are to be calculated as functions of s, as shown - 0 Jt below. Now we have to calculate the fuel output of the core at the time ts' The N.*k(s.t.} may be the concentrations of the heavy nuclei in the J maxj fuel elements unloaded from the j-th zone at t The corresponding initial s concentrations may be given by N. k(s.o). The N. k(s,t.} are the J. J. ma.xj solutions of the fuel burn-up equations with these initial values and the space dependent flux times ( 5) r,z is any point of the core zone j. t. is related to the maximum fuel maxj burn-up for the elements of the core zone j in M:WD/tonne, From equation (5) follows the relation between ttc the maximum burn-up for the inner core zone j = 1 expressed by the corresponding maximum flux time and the neutron flux at the core center ~(o,o): ( 6} 1 Tmax1 = - n ~(o.o) ~(o,o) is proportional to the reactor power Q, Obviously, the following relation holds: (7) N. k(s.o) = Jt for s-n s 0 for s-n s, -o - 7 - The total number tin. k ( s) of the nucleus k which is unloaded from h. Jl... ( ) t e core zone J at t fol ows by integrating N. k s,t. over the s J maxj volume of the unloaded fuel elements We obtain (8) ( = -n 2 6N. k s 1T J. H+D 2 Rj+1 H+D R. - 2 J N. k(s,t.)rdrdz, J maxj where the discharged nuclei of the axial blanket are included. H denotes the height of the core, D the total thickness of the axial blanket, R. the J inner and Rj+ 1 the outer radial boundary of the j-th zone, The division by n is necessary for only the n-th part of the elements of the zone j is discharged, n the core range we have to take into account the time dependence of the initial. composition of the fuel; in the axial blanket region we take the initial composition of the elements constant. The total output of the core and the axial blanket expressed as the total number of the discharged nuclei at the time ts denoted as 6NCk(s), is (9) 2,4. The Uranium and Plutonium Output of the Radial Blanket The uranium and :plutonium content of the radial blanket zone j at the time of discharging, which may be denoted as 6NBj:k j = 1,2,,NB, k = 1 1 2,., 6, is also calculated by a volume integral in analogy to equation ( 8), But in the case of the radial blanket n is equal to 1 and the initial values of the elements are time independent. The flux time distribution in the j-th zone for solving the burn-up equation is j :,.2. o O t~ t r,z is any point in the radial blanket zone j, - 8 - For a given value of s only that zone j of the radial blanket is unloaded for which ( 10) an integer. f no value of j exists for a given s, the total output A~k(s). * k = 1, of the radial blanket is zero: (11a) for all k. f there are t special values of j for a given s. denoted as j(t') 1 = 1,2, t, for which equation (10) is valid all the designated zones are unloaded at ts. Thus, the uranium and plutonium output of the radial blanket at t :is s ( 1 lb) The CoW?osition of the Refabricated Fuel Elements and the Breeding G~in n the case of joint reprocessing of core and blanket elements the total number of nuclei, which are unloaded from the reactor at t which are to be reprocessed is s and the ANB:(s) may vanish for certain values of s as shown before. The total numbers of uranium and plutonium nuclei are 6 (12) l A~(s). k=3 Losses during reprocessing reduce the number of nuclei by a factor of Pu and ppu, respectively. Thus. after reprocessing we have - 9 - using the same symbols for the reduced numbers. The enrichment of the uranium available af'ter reprocessing is (13) r(s) = 6N ' (s) 1 f, 6Nu(s) and the relative composition of the plutonium mixture available for the refabrication of the fuel elements is given by (14) ~(s) = i( 6Nk(s) 6NPu(s) The initial uranium and plutonium concentrations of the fuel elements for the j-th core zone which are refabricated from plutonium of the s-th discharge i.e. from 6NP (s) 1 may be denoted as n. U(s) and n. Pu(s). reu Jt J. spectively. Then. k(s), k = may be the initial concentrations Jt of the uranium and plutonium isotopes in the refabricated fuel elements. Thus 1 the following relations hold: (15) Then. ua n. Pu and n. k are space independent. Ja J, Ja The basis for calculating the initial compositions of the refabricated fuel elements are the 2N linear algebraic equations 6 l c. k n. k(s) = K.(o) k=1 J. J. J (16) 6 l n. k(s) = Z.(o) k=1 Jt J with the abbreviations j = 1t2t N C. = (vc/). k - a~ 1 k = ; J 1 k Jt J 1 k 6 (16a) K.(o)= l C. kn. k(o). n. k(o)=n. k(o,o) J k=1 Jt Jt Jt Jt 6 z.(o)= l n. k(o) J k=1 J 1 10 - Th e one group quan t ities.. o. a k an d ( vo f). k re f er to any re f erence Jt Jt point in the j-th zone (e.g. r = R z = o). As mentioned above they are J calculated for the average fuel burn-up state of the reactor assuming a homogeneous fuel distribution. The vof are defined by analogy to equation ( 4). The equations (16) are used to calculate the 2N unknown initial concentrations n. U and n. Pu with a presupposed uranium enrichment and the J. J, isotopic composition of the plutonium given by (14). The first equations (16) are given to maintain criticality. By these equations the reactivity worth of the refabricated fuel elements is fixed approximately at a constant value for each zone by adjusting the fuel composition to a constant difference between neutrons produced and absorbed in the fresh fuel at a given point of each zone. The constant differences are defined bythe fresh fuel of the initial reactor at t = o. 0 The second equations (16) take the total number of heavy nuclei in the fresh fuel elements for each zone to be constant. The constants are given again by the initial conditions at t = o. For simplification we 0 assume that the enrichment of the uranium used for fuel element refabrication is constant and equal to the initial enrichment y at t we have the relations 0 = o. Thus f only the plutonium mixture from the s-th discharge of the reactor is used for refabricating i, e, in the case of positive breeding gain, the concentrations of the plutonium. isotopes are (18) n. k(s) = Q_ (s) n. Pu' J, lt Jt With the relations (17)and (18) the equations (16) are rearranged to ( 19) n. 0 (s), C. U + n. Pu(s) c. Pu(s) = K.(o) Jt J, J, Jt J n. 0 (s) J, + n. Pu(s) J, = z.(o) J 11 with ( 19a) 6 c.kct(s) J J J J u k~3 J 'k c.u=c. 1 y+c. 2 (1-y). c.p(s)= \' f C. p (s); C. Uthe equations are solved by J. u J. (20) z.(o)c. Pu(s)-K.(o) () J,J1 J n. U s = J c. Pu(s)-c. u J. J. -z.(o)c. u+k.(o) n. (s)= J,,i J, J,Pu ( ) cj,pu s -cj,u For physical reasons n. U(s) and n. P (s) must be positive, f J. J. u (21) cj.u ~ o. i,e. y ~ 1 / (1-c. 1 /c. 2 ). n. Pu(s) is positive. f (21) is true, the Jt Jt Jt numerator of the first equation (20)!nlst be positive i.e. ( 22) K.(o) C. Pu(s) J j = 1.2,,.,,N, J, -Z.(o) J n the most interesting cases of fast power breeders with natural or depleted uranium as the fertile material the two conditions (21) and (22) are always true and the solutions (20) are :positive 2 ). n the following we will restrict ourselves to these cases, But the initial compositions for the refabricated :f'uel elements are only determined by equation (20) if the number AN~ ( s) of the available plutonium nuclei is enough for manufacturing the fuel element volume AVc i.e. in the case of positive breeding gain, The total number of plutonium nuclei from the s-th unloading of the reactor which is required for refabricating the fuel element volume AVC is given by 2 )n the cases of higher enrichments of the uranium in the fuel the equations (16} continue to be true, But the time dependence of the uranium enrichment is to be taken into account which is caused by the fuel burnup and perhaps by the fuel manageme~~~ Especially in the case of starting up the reactor as a converter Wfth U as the fissionable material we have to distinguish the time dependent enrichment of the uranium in the closed fuel cycle according to (13) and the presupposed enrichment of the uranium which is introduced into the fuel cycle from the outside to compensate for the consumed fertile material, By these facts the equations (17) and (19) are modified. But we will not discuss these more general equations and the conditions by which the solutions are positive here, 12 - fl V is J 1 N NPu ( s ) = 'ii .l 1 fl v. n. P ( s ) J= J J, u the total volume of the fuel elements of the core zone j; thus 1 flv./n is the exchanged fuel element volume. Losses during the fuel element J fabrication reduce the number of plutonium nuclei by a factor qpu therefore, the number of the required plutonium nuclei is NPu(s)/qPu instead of NPu { s) f ( 23) the initial compositions of the refabricated fuel elements, are determined by the equations {20). The breeding gain of the s th fuel cycle is defined by (24) it is positive in accordance with (23), The Parameter s 0 The time required for the external fuel management is (s 0-1 )fltc with the above definition of s Thus the plutonium unloaded fran the reactor 0 at t is reloaded into the core at t = (s+s -1)fltc Therefore, we s s+s 0-1 o have the relation {25) N. k(s+s ) = n. k(s) J, 0 J, for all values of k and j, s 0 is determined by the characteristics of the reprocessing and refabricating plants, The external fuel cycle is characterized essentially by the parameters and 6f 1 which are the minimum cooling time and the times C r - for reprocessing and refabricating 1 respectively, These quantities define - - a new parameter 6 ex = 6 c + 6 r + 6 f; furthermore, we define p = [~] flt C [x} is the maximum integer below x. 13 - For simplification we will assume in the following that the refabricated fuel element volume AV C of any fuel cycle is loaded into the core as a whole, i.e. it should not be distributed over different loading events. Now, we will express the time for loading the fuel element volume AVe of the s-th fuel cycle into the core in terms of P The time at which the first elements of the S th fuel cycle are refabricated lies between ts + ~te and ts + {p+1)atc as the case may be, whether or not 6 ex /Ate is an integer. For economical reasons we assume that the time a~er which the complete volume AVC is refabricated will not exceed Ate i.e. the last elements of the s-th cycle may be refabricated not later than ts + {p+2)atc This condition avoids unnecessary accumulations of plutonium in the external fuel cycle, n general, three values of s possible: s = p+e: e: = 1,2 or 3, respectively. 0 e:=1: if 6 ex/atc=p and if the fuel element volume AVe is managed as only one bat
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