Concepts of linear nuclear power systems of slow burning with breeding are briefly discussed. The propagation of the nuclear fission wave in a long thermal fission reactor (without breeding) is considered. The evolution of the neutron density profile with time is determined. The velocity of the burning front is estimated

Akhiezer, A.

Davydov, L.

Khizhnyak, N.

Pilipenko, V.

Shulga, N.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

SLOW NUCLEAR BURNINGA.I. Akhiezer, N.A. Khizhnyak, N.F. Shulga, V.V. Pilipenko, L.N. DavydovNational Science Center “Kharkov Institute of Physics and Technology”, Kharkov, UkraineConcepts of linear nuclear power systems of slow burning with breeding are briefly discussed. The propagation of the nuclear fission wave in a long thermal fission reactor (without breeding) is considered. The evolution of the neutron density profile with time is determined. The velocity of the burning front is estimated.PACS: 28.41.Ak1. INTRODUCTIONLast two decades, and especially after the Chernobyl accident, a great interest was shown in new concepts of fission  nuclear  reactors,  as  they  were  named  safe reactors.  One  of  such  well  known proposals  was  the concept by C. Rubbia et al.[1].  The matter concerns a subcritical  nuclear  reactor  driven  to  criticality  by  a powerful accelerator system.Another one was the conceptual design by E. Teller and al.[2] of a slow burning reactor with a completely automated system of operation and power output.  The relative  safety  of  slow  burning  concept  lies  in  the necessity to produce, or breed the fissile fuel (e.g., 239Pu) from fertile (238U) in front of the moving wave of nuclear burning. Thus, the wave velocity, and the power output, depends on the sluggishness of the breeding process.But  chronologically  earlier  the  first  ideas  of  the slowly  burning  safe  reactor  were  advanced  by L. Feoktistov in 1989 [3].2. REACTOR OF FEOKTISTOVTo describe the propagation of the nuclear burning wave Feoktistov [3] maximally simplified the problem and  considered  only four  components  (neutrons,  238U, 239U,  240Pu) of the open nuclear  reaction sequences of U-Pu fuel cycle238U (n,γ) 239U239U → (β) → 239Pu239Pu (n,fission) ...These resulted in a set of diffusion-reaction equations ( )( )dndt D n vn N Na a f= + − +∆ σ σ σ8 8 Pu Pu ,dNdtvn Na8= − σ 8 8 , dNdtvn N Na99= −σ τ β8 81,( )dNdt N vn Na fPu Pu Pu= − +1 9τ σ σβ .for  the corresponding densities  of  neutrons,  238U, 239U, and 239Pu. Here v  is the mean neutron velocity and σ σa f, ,  are the absorption and fission cross-sections. It should  be  emphasized  that  the  slowness  or  the sluggishness of the burning process is ensured here by the very large time ( τ β =2.4 days for U-Pu cycle) and 24.5  days  for  Th-U cycle).  Similar  equations  can  be written for Th-U cycle with τ β =24.5 days.In the self-similar approach, when the densities of all reaction components are moving with the same velocity V  and  depend  on  the  unique  variable  z x Vt= + , Feoktistov has proved the existence of a slowly moving front  of  nuclear  burning.  To  illustrate  his  results  we obtained the exact numeric solution of above equations (in  the  self-similar  approach)  and  presented  them  in Fig. 1 (a,b) and Fig. 2 (a,b) as functions of ζ = z l/  ( l  is the neutron migration length).One can see the soliton-like moving front (from left to  right)  of  the  nuclear  reaction  where the  density of neutrons is  enhanced (Fig. 1 a).  The  raw 238-uranium material  (Fig. 1 b)  existing  in  front  of  the  reaction  is burned out behind the moving front. And just in front or the  moving  burning  wave  a  layer  of  the  newly born Plutonium-239  (Fig. 2 a)  is  formed  due  to  breeding, which is burned out to some extent after the wave has passed. The reaction intermediate product uranium-239 also can be seen as moving with the front (Fig. 2 b). The velocity  V  of  the  burning  front  depends  on  the parameters  of  the  system  through  the  ratio  of  two Plutonium concentrations, equilibrium and critical. The described  process  was  named  in  [3]  as  neutron fissioning wave.An  expansion  of  this  work  was  done  later  by V. Goldin  et  al.  [4,5].  They  have  used  the  idea  by Feoktistov  of  self-adjusting  neutron-nuclide  regime, described it,  however, in a much more comprehensive model.  They  applied  a  multi-group  description  of neutrons and an accompanying system of 16 equations 16 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 272-275.nneutronsa5 10-50.04zU238n8b5 10-50.40.60.8zFig. 1. The dependence of neutron (a) and U238 (b)  relative  densities  in  Feoktistov  reactor  on  the  self-similar coordinate ζ = z l/Pu239nPu a5 10-50.060.040.02zU239n9 b5 10-50.02zFig. 2. The  dependence  of  Pu239 (a)  and  U239 (b) relative  densities  in  Feoktistov  reactor  on  the  self-similar coordinate ζ = z l/of isotope burn-out. But they considered [5], in fact, a heterogeneous  reactor  with  alternating  layers  of differently  enriched  fissile  isotope  as  a  nuclear  fuel. Actually, in their case, several burning fronts are slowly moving during the reactor campaign through a distance not exceeding the wavelength.3. SLOW NUCLEAR BURNING IN A FULLY AUTOMATED REACTOR OF TELLERTeller and al.[2] advocated a slow burning reactor with a fully automated system of power output. The process of nuclear  burning  was  calculated  numerically  within  a cylinder geometry of the reactor. They have employed 175  neutron  energy-group model.  Reactor  design  was resolved into several hundred spatial  zones and a few hundred different materials. Sixteen isotopes are usually carried in each zone, representing both fertile and fissile isotopic  components  of  nuclear  fuel,  in  addition  to reflector and coolant elements, structural materials, and various  neutronic  poisons,  including  fission  products. The detailed results  were not  published, but  the time-running of the reactor can be seen in Fig. 3 ( z  is the distance  along the cylinder  reactor  axis).  Teller  et  al. considered,  unlike  Feoktistov,  Th-U fuel  cycle:  (232Th (n,γ) 233Pa; 233Pa → (β) → 233U; 233U (n,fission)...).2 0 1 6 0 3 0 0 4 4 0 5 6 0 7 2 0 8 8 0 1 0 0 0Z ,  c mT = 7 . 5  y e a r sTh232U233Fission productsPower outputrelative valuesa2 0 1 6 0 3 0 0 4 4 0 5 6 0 7 2 0 8 8 0 1 0 0 0Z ,  c mrelative valuesT = 3 0  y e a r sFission productsPower outputTh232U233bFig. 3. Operation of Teller fully automated reactor  in time (a) after 7.5 years; (b) at the end of campaign  after 30 yearsOne  can  see  the  same  processes  to  evolve  as described  by  the  simplified  equations  of  Feoktistov: slow movement of the wave where the density of  the fissile 233-Uranium and a power release are enhanced and a burned-out zone (depleted of Thorium) remained behind the front.174. “SLOW” BURNING IN A THERMAL REACTORAs a  first  step  to  tackle  the  problem of  the  slow burning  in  a  breeder,  we  started  with  a  hypothetical problem of  the  moving front  of  nuclear  burning in  a thermal  reactor  [6].  We  show that  a  moving front  of nuclear  burning  exists  in  this  case  also.  However,  its propagation  velocity  will  be  much higher  determined mainly  by  the  neutron  diffusion.  Contrary  to  the previous cases, the breeding will be absent, and with it the time consuming stage with β -decay also.4.1. FORMULATION OF THE PROBLEMLet’s consider fissile medium in a form of a prolate infinite  quadratic  ( a a× )  prism.  The  equation  for neutron density ( )n tr,  has a form( )∂∂ τν θ ϕpi τntD nrn trdVcfo c V= − + −−∫∆13 2 3202/ ', exp''rr r (1)Here D  is the diffusion coefficient of thermal neutrons and τ c  is the neutron lifetime in relation to the capture. The  quantities  ϕ θ, f  in  Eq. (1)  describe, correspondingly, the probability that a fast neutron will be  slowed  down  to  a  thermal  energy,  not  being resonantly  captured  by  the  nuclei  238U,  and  the probability  of  absorption  of  a  thermal  neutron  by  a nucleus of uranium. The first term in Eq. (1) is due to neutron  diffusion,  the  second  to  the  absorption  and fission losses,  an the last  one is  a  nonlocal  source of newly created fission neutrons. It describes the density of slow neutrons, produced as a result of the slowing-down  of  fast  neutrons.  The  characteristic  length parameter  r0  is an average length of neutron slowing-down.The initial and edge conditions are( )n t x ay ar, .,===000 (2)Searching for solution in the form( ) ( )n t n z taxayr, , sin sin= pi pi (3)with initial condition( ) ( )( ) ( )n z t n zn z n zt,=== −0 00 0(4)one finds( ) ( ) ( ) ( )n z t A tD tdt n tz zD tc,exp /' ' exp'** *= −−− ∞∞∫τpi2 402,(5)where  D *  is  an  effective  coefficient  of  diffusion, D D rfc*≡ +1402ν θ ϕτ and  AaDf c* *≡ − −ν θ ϕ pi τ1 222  is the neutron multiplication (reproduction) factor for a reactor  of  finite  transversal  size  a .  If  A* < 0 ,  the number of neutrons, born in nuclear fission reactions, is deficient  to  sustain  a  self-maintained  chain  nuclear reaction,  i.e.  we  have  a  subcritical  regime.  With  the increase of the characteristic size  a  the parameter  A*  becomes positive (we suppose  ν θ ϕf > 1 ), and we pass into the region of overcritical  regime. Notice,  that the condition A* = 0  determines the critical transverse size of a system, where a chain reaction can be realized,( )aD rcrc f of=+−pi τ ν θ ϕν θ ϕ2 412 /(6)In the overcritical region A* > 0  the self-sustaining chain  reaction  is  possible,  that  is  manifested  in  the exponential growth with time of neutron density at every point of space.4.2. EVOLUTION OF NEUTRON DENSITY FLUCTUATIONLet us examine the evolution of fluctuation of the density of thermal neutrons that has some constant value n0  in  the  range  − ≤ ≤l z l  at  the  initial  moment  of time. In this case Eq. (5) can be simplified and takes the form  (at  large  distance  from  the  initial  fluctuation, z l> > )( ) ( )( )( )n z tn D tA tz lD tz lz lD tz lc, exp /expexp****≅−−−−−−++02244piτ(7)Eq. (7)  describes  diffusion  of  thermal  neutrons  taking into account their multiplication. It can be simplified in some important cases.  First, asymptotic expression for increasing z  and fixed t : z t→ + ∞ =, const , ( ) ( )n z t n D tzA t z lD tc, exp* **≅ −−024pi τ. (8)Second,  asymptotic  expression  for  increasing  t  and fixed z : t z→ + ∞ =, const ,( )n z t n lD tA t z l D tc, exp ,***≅± < <0 2pi τ(9)The first of the above two cases corresponds to an "instant" picture of neutron density distributed along the cylinder  axis,  and  the  second case  describes  the  time evolution  of  neutron  density  at  each  given  point  z . According  to  Eqs. (8), (9)  one  can  see,  that  at  large distance  from  the  fluctuation  the  density  ( )n z t,  exponentially decreases with distance, and at large time it grows exponentially with time due to multiplication.18A characteristic  feature  of  multiplicating  medium, according to  Eq. (8),  is  the existence of such velocity ( )v z l t v= − =/ 0  of the moving point of observation, at which the observed density of thermal neutrons  ( )n z t.  very slowly decreases with distance and is given by( )n z t nzL L DADnc, , `***≅ ≡ =0 0 0022 4piτ . (10)Such sharp change in the asymptotic behavior is the characteristic feature of multiplicating medium and may be physically explained by the compensation (along the ray  z v t l= +0 )  of  the  exponential  growth,  caused  by neutron multiplication,  by the exponential  decrease of neutron density with distance. Namely, the decrease of neutron  density  with displacement at  the  distance  δ z  will be compensated by the increase of neutron number in the multiplication process during the time δ t , needed to perform the above displacement.The velocity v0  may be named the velocity of slow nuclear  burning  in  a  system.  Indeed,  in  the  case  of conventional (chemical) slow burning [7], which results in the achievement of a certain reaction temperature, the velocity  of  slow  burning  is  proportional  to  χ τ/ , where  χ  is the temperature conductivity, and τ  is the characteristic reaction time. In our case the problem is in the  achievement  of  some  fixed  neutron  density  at  a given point, caused both by multiplication and diffusion. The lifetime of neutron τ c  acts as a characteristic time of reaction, and the geometric mean of diffusivity and multiplication  parameter  A*  plays  the  role  of  the transfer coefficient. The essential role of multiplication process is supported by the fact that the velocity of slow nuclear  burning  goes  to  zero  at  A* = 0 .  This  is consistent  with  the  fact,  that  the  velocity  of  purely diffusion process tends to zero at great distance from a source ( dz dt D z z/ / ,≈ → ∞ ).The neutron density can not grow unrestrictedly as above in  linear  problem,  first  of  all  due  to  the  finite quantity of fuel. One can account for this introducing a nonlinear factor ( )1 0− N t, / N c (11)in the equation for neutron density( )( )( ) ( )∂∂∂∂piτν θ ϕpi τn z ttD nzDanrN t n z tz zrdzcfcc,, / ', exp''= − − ++ − −−− ∞∞∫222202022 11 0 N(12)where  ( )N t0,  is  proportional  to  the  total  number  of neutrons ( ) ( )N t n z t dz0 2 0, ,= ∞∫piand  N c  is  a  sufficiently  large  given  constant.  This procedure is  described in detail  in [8]  and we do not dwell on it any more.Similar, though mostly local, nonlinear equations are familiar  in  diffusion-reaction  problem  in  physical chemistry  and  other  fields.  The  most  known  is  the differential  equation  [9]  describing  the  processes  of genes  dessimination and  struggle  for  survival.  With  a nonlinear term written down in a more general form it was  thoroughly  investigated  by  A.N. Kolmogorov, I.G. Petrovsky  and  N.S. Piskunov  [10].  It  was,  in particular,  shown  that  the  asymptotic  velocity  of propagation  of  the  travelling  wave  solution  could  be also  introduced.  This  velocity  coincides  with  the velocity v0  obtained in this paper.Authors gratefully acknowledge the financial support of STCU (project №1480).REFERENCES1.C. Rubbia,  J.A. Rubio,  S. Buono,  F, Carminati, N. Fietier,  J. Galvez,  C. Geles,  Y. Kadi, R. Klapisch, P. Mandrillon, J.P. Revol, Ch. Roshe. Conceptual design of a fast neutron operated high  power  energy  amplifier. Preprint  CERN/AT/95-44 (ET), 1995, 240 p.2.E. Teller,  M. Ishikawa,  L. Wood,  R. Hyde  and J. Nuckolls. Completely automated nuclear reactors for  long-term  operation.  Int. Conf.  on  Emerging Nuclear  Energy Systems,  Obninsk,  Russia,  24-28 June 1996.3.L.P. Feoktistov.  Neutron-fissioning  wave  // Doklady  AS  USSR, 1989,  v. 309,  p. 864-867  (in Russian).4.V.Ya. Goldin,  D.Yu. Anistratov.  Fast  neutron reactor in a self-adjusting neutron-nuclide regime // Mathematical Modelling. 1995, v. 7, №. 10, p. 12-32 (in Russian).5.V.Ya. Goldin,  N.V. Sosnin  and  Yu.V. Tro-shchiev.  Fast  neutron  reactor  in  a  self-controlled regime of 2d type //  Doklady RAS. 1998,  v. 358, № 6, p. 747-748 (in Russian). 6.A.I. Akhiezer,  D.P. Belozorov,  F.S. Rofe-Beketov,  L.N. Davydov  and  Z.A. Spolnik. Propagation  of  a  nuclear  chain  reaction  in  the diffusion  approximation  //  Physics  of  Atomic  Nuclei  (Yadernaya  Fizika). 1999,  v. 62,  № 9, p. 1474-1481.7.L.D. Landau and E.M. Lifshits. Fluid Mechanics. London, 1959. 8.A.I. Akhiezer,  D.P. Belozorov,  F.S. Rofe-Beketov, L.N. Davydov and Z.A. Spolnik.  On the theory of propagation of chain nuclear reaction // Physica A. 1999, v. 273, p. 272-285.9.R.A. Fisher.  The  Genetical  Theory  of  Natural  Selection,  Oxford:  “Oxford  University  Press”, 1930.10.A.N. Kolmogorov, I.G. Petrovsky and N.S. Pis-kunov.  Investigation  of  the  equation  of  diffusion associated with the augmentation of the substance quantity, and its application to a certain biological problem //  Bulletin  of  Moscow University. 1937, v. 1, Sect. A, № 6, p. 1-26 (in Russian).19

Slow nuclear burning

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Slow nuclear burning

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