FUEL TRAJECTORIES OF NUCLEAR REACTORS by JEREMIAH CHUKWUECHEFULAM OSUWA, B.Sc. (Ife) A Report Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master

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FUEL TRAJECTORIES OF NUCLEAR REACTORS by JEREMIAH CHUKWUECHEFULAM OSUWA, B.Sc. (Ife) A Report Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree Master of Engineering McMaster University 1979 MASTER OF ENGINEERING 1979 McMASTER UNIVERSITY Hamilton, Ontario TITLE: Fuel Trajectories of Nuclear Reactors AUTHOR: J.C. Osuwa, B.Sc. (Ife) SUPERVISOR: Dr. A.A. Harms NUMBER OF PAGES: vii, 47 ABSTRACT The fissile fuel inventory of an external stockpile that supplies a nuclear reactor is shown to be characterised by time dependent mean residence time of fuel in the core, the load factor for the reactor, and the breeding gains. The time dependent forms of these parameters have been considered and quantitative evaluations of the fissile fuel inventory have been carried out. The results show the substantial potentials of efficient reactors in extending the duration of fissile fuel for Nuclear industries. The inventory concept has also been extended to CANDU reactors and the result clearly depicts the effect of a net fissile fuel consuming system on the nuclear fuel reserves. ii ACKNOWLEDGEMENT The author wishes to thank Dr. A.A. Harms, Department of Engineering Physics, for his guidance in this research project. Appreciation is also extended to Miss Janet Delsey for typing this project report. iii Fig : Fig. 3.1: Fig : Fig. 4.2: Fig. 4.3: Fig. 5.1: Fig. 5.2: Fig. 6.1: LIST OF FIGURES A schematic illustration of fissile fuel flow between a nuclear reactor and its affiliated external stockpile of fissile fuel. The trajectory of fuel inventory of external stockpile for a breeder reactor obtained with asymptotic values of reactor parameters. A model showing fractions of the core fuel replaced per cycle and the duration of each fraction or 11 fuel lump., in the core. Time dependent mean residence time of fuel in a reactor core in which one-third of the core fuel is replaced per cycle. A possible time dependent load factor representation. The trajectory of the fuel inventory of external stockpile for a breeder reactor with time dependent forms of the mean residence time and the load factor incorporated. The asymptotic and quasi-static results obtained in this work compared with the discrete space-time calculation for the Clinch River Breeder Reactor by Paik et al. A CANDU calandria showing fuel channels, a section of fuel bundles, and the directions of fuel movement during a refuelling operation. Page Fig. 6.2: Fig. 6.3: Fig. 6.4: External stockpile inventory associated with a CANDU reactor for various values of mean residence time with a conversion ratio of.6. External stockpile inventory associated with a CANDU reactor for various values ofmean residence time with a conversion ratio of 1.. External stockpile inventory associated with a CANDU reactor for various values of conversion ratio with a mean residence time of 1. year v LIST OF TABLES Table I: Parametric values for the calculation of net fissile 13 fuel flows. Date corresponds to those for the Clinch River Breeder Reactor (CRBR). Table II: Fuel 11 lumps 11 used during the operation of a reactor 17 in which one-third of the total core fuel is replaced per cycle. Table III: Sample values of mean residence time. 2 ' Table IV,:. Operating and Fuel performance requirements for 29 CRBR. vi TABLE OF CONTENTS Page ABSTRACT ACKNOWLEDGEMENT LIST OF FIGURES LIST OF TABLES CHAPTER 1: CHAPTER 2: CHAPTER 3: CHAPTER 4: CHAPTER 5: Introduction Fissile Fuel Inventory Asymptotic Parameter Solution of the Fissile Fuel Trajectory Equation Time Dependent Parameters 4.1 Time Dependent Average Residence Time for a Reactor Fue Time Dependent Load Factor Mathematical Analysis of the Fuel Trajectory Equation 5.1 Detailed Integral Analysis 5.2 Quasi-Static Solution of the Fuel Trajectory Equation i i iii iv vi CHAPTER 6: Application of the Fuel Inventory Formalism to CANDU 33 Reactor 6.1 Introduction Fuel Residence Time for a CANDU Reactor External Stockpile Trajectory for a CANDU Reactor 36 CHAPTER 7: Conclusion 42 APPENDIX: 11 Equivalent Worth 11 of Reactor Fuel REFERENCES vii CHAPTER 1 INTRODUCTION The fissile fuel inventory of a nu~lear reactor can be represented as a function containing time dependent parameters which include the mean residence time of fuel in the core and blanket, the capacity or load factor, and the breeding gains. An accurate assessment of the temporal variation of the bred fissile fuel therefore rests on the establishment of the time dependent forms of these determining factors. In this analysis, the fissile fuel inventory for a nuclear reactor based on earlier work, (Ref. 1 and 4), is presented and evaluations have been made for the case of asymptotic constant parameters. The time dependent forms of the mean residence time and load factor have been considered from a practical point of view while rational approximations for the breeding gains have been made in other to provide an explicit integral form for the fuel inventory. The integrands involved however pose mathematical problems; further evaluation have therefore been based on a quasistatic scheme which incorporates the time dependent forms of the mean residence time and the load factor. The analysis begins with a specification of the fissile fuel balance for a nuclear reactor and its peripheral support facility, followed by a detailed time dependent accounting of the fissile fuel production and consumption over the reactor life. The resulting function then represents the fissile fuel trajectory from which we can obtain the net available fissile fuel to supply other reactors. The inventory concept -1- 2 has also been extended to CANDU reactors to demonstrate the effect of net fuel consuming systems on the available fissile fuel. A traditional problem encountered in the fissile fuel breeding analysis is the incorporation of various fertile and fissile nuclei into a sufficiently clear and tractable formulation, t.hi s problem has been circumvented by the use of the concept of 11 equivalent worth 11 of the various nuclei. This concept which is credited to Baker and Ross (1963), assigns some weight w., to a given nucleus to account for its contribu tion to the maintenance of fissile fuel consumption processes in a given system. The mass which appears in the balance equation thus denotes the weighted sum of all the masses of the various nuclei in the fissile material. A detailed discussion of this.. equivalent worth 11 concept is listed in the Appendix. external stockpile associated with the core and blanket processes respectively. Equation (1) is basically a statement of mass conservation which is solvable by different approaches depending on the extent of physical detail incorporated. As an initial observation, we note that the fissile fuel bred in the core or blanket is usually not instantaneously available in its desired form because of reprocessing; accounting for this will require the insertion of time-delays in the mass flow. Except for very small fractions of process losses/retentions, these considerations do not change the total availability of the fissile fuel but only its availability in a preferred form. The time delays are therefore not included in the formulation and attention is focussed on the physical processes in the reactor that effect the pro- -1- CHAPTER 2 FISSILE FUEL INVENTORY The system of interest consists of a nuclear reactor and an associated stockpile of fissile fuel (Fig. 2.1.). The fuel stockpile initially contains mi kg=of (equivalent) fissile fuel. Prior to the reactor start-up at t=o, me kg of this amount is removed to load the core. The fissile fuel inventory in the external stockpile which will be designated by m~!f is then given by t J {(dm) dt c (l) where (~~)c and (~~)b are the net fissile fuel flows to or from the 4 duction and consumption of fissile fuel over the reactor life-time. In Eq. (1) the integrand terms are obtained by the following considerations. The net fissile fuel flows for the core term, (dt)c' dm consists of two main components: a constant equilibrium component and a transient component during the early part of the reactor life; this latter part accounts for the extra fissile fuel added to the core to compensate for the initial accumulation of neutron absorbing fission products. The core term can thus be written as (2) The net fissile fuel requirements for the core at equilibrium is taken to be proportional to the fissile fuel destruction rate R(t). Hence where the core breeding gain Gc(t) is obviously a negative quantity. The transient contribution is taken to be a time dependent fraction of / the equilibrium term and can be written as (4) Here, T(t) is the transient-fraction function; it is required that T( t) -+ as t-+ t. The core term can thus be written as = Gc(t)R(t)[l + T(t)] (5) 5 Nuclear Reactor _.., )( Fue~ External stockpile.. _! Fig. 2.1: A schematic illustration of fissile fuel flow between a nuclear reactor and its affilitated external stockpile of fissile fuel. 6 The blanket is taken to consist of radial and axial regions each subject to a distinct fuel management scheme. The net fissile fuel production in each region is given in terms of a specified residence time J., by and (dm) dt b, r = mb,r(t) J.r(t) mb a ( t) (dm) = ' dt b,a J.a ( t) (6) (7) where the subscripts 11 r 11 and 11 a 11 are used to identify the radial and axial blankets respectively. The residence times J.r(t) and J.a(t) specify how long a blanket element remains in each of the blanket zones before being replaced. The blanket term in Eq. (1) is thus given by (8) The fissile fuel inventories mb,r(t) and mb,a(t) in the blankets appear with time according to their production rate by transmutation and removal rates with the specified residence times. This requires that mb,r(t) and mb,a(t) in Eq. (6) and Eq. (7) satisfy the condition dmb ( t),r dt (9) and ( 1) 7 where the net breeding gains Gb,r(t) and Gb,a(t) apply to the radial and axial blankets respectively. The initial conditions represent the initial fissile fuel mass in the blankets, usually in the form of heavily depleted fissile materials. Solutions of Eq. (9) and Eq. (1) are given by mb,j(t) = exp(- J J f t Gb,j(t)R(t)exp( dt J A.(t) dt J A.(t))dt + mb,j(o) j = r,a ( ll.a) which can also be written as J dt I J dt + [mb,j(t)exp{ A.(t)) - Gb,j(t)R(t)exp( A (t))dt]t=o} J Substutition of Eq. (ll.b) into Eq. (1) gives the final form for the fissile fuel inventory of the external stockpile in terms of time dependent quantities to be specified; we thus have t mb r(t) mb a(t) mext(t) =mi-me+ J{Gc(t)R(t)[l+T(t)] + A;(t) + ~a(t)} dt =mi-me+ It {Gc(t)R(t)[l+T(t)] J (ll.b) 8 1. f dt J J dt + :\a(t) exp(-. yn){ Gb,a{t)R(t)exp( :\a(t))dt f dt f f dt + [mb,a(t)exp{ :\a(t)) - Gb,a(t)R(t)exp( :\a(t))dt]t=o}}dt Equation {12) is the basic equation of interest and provides for the evaluation of the amount of fissile fuel in the stockpile, on specification of the time dependent quantities. (12) CHAPTER 3 ASYMPTOTIC PARAMETER SOLUTION OF THE FISSILE FUEL TRAJECTORY EQUATION The amount of fissile fuel available in the external stockpile can be obtained from Eq. (12) providing the various integrand functions are known. Before attempting to find time dependent forms for some of the integrand functions, a numerical test for the formulation can be provided by making rational approximations for the quantities involved. The functions Ar(t) and Aa(t) represent the length of time that a fuel element remains in the radial and axial blankets respectively. These time intervals may vary from a maximum of three to a maximum of six years respectively depending on the fuel management scheme and cycle lengths. If these times are designated as mean residence times, it is obvious that after a long time or after many cycles, the mean residence time will approach the maximum time that a fuel element can reside in the blanket and we can thus make the approximations Ar(t) = Ar and Aa(t) = Aa where Ar and Aa are the asymptotic residence times. The breeding gains are the ratios of the net fissile fuel produced to the fissile fuel consumed as appropriate to the core and each of the blanket zones, and can be related to the breeding ratio BR(t) by (13) Under equilibrium condition which follows a relatively short initial transient period of the reactor life, these breeding gains are essentially constants. It is known( 3 ) that even during the pre-equilibrium period, 1 the variations of the breeding ratio do not generally exceed ~1%. Such small variations suggest that these time dependent functions can be taken as constants. Hence we will use Gc(t) = Gc, Gb,r(t) = Gb,r and Gb,a(t) = Gb,a The remaining functions to be specified are the destruction rate R(t) and the initial transient-fraction T(t). The function T(t) accounts for such nuclear effects as fission product poisioning. These effects are of very short duration when considered on the time scale of years and,. in addition, in terms of the total reactivity capacity of the core, this fraction is not predominant. The approximation to be made here is then l + T(t) ~ l. The variation in the fissile fuel destruction rate will be incorporated by the introduction of the station load factor L(t) defined by R(t) = RL(t) ( 14) where R is a constant fissile fuel destruction rate. The load factor can vary typically from a low of about.3 in the first year of operation to attain an equilibrium value of about.75 after one or two refuelling periods. Using the functions as specified above Eq. (12) reduces to ( 15) 11 For the present purpose the asymptotic value of the load factor will be used thus enabling us to write L(t) = L. Eq. (15) to the simple form This reduces m ex t(t) = m. - m + [Gc + Gb + Gb a]rlt 1 c,r, + [mb,r - ArGb,rRL][l - e-t/ar] + [mb,a - Aa Gb,aRL] [1 - e -t/aa] (16) Further evaluation requires typical values for the parameters contained in Eq. (16) and the most relevant and complete data available for this purpose are those for the Clinch River Breeder Reactor( 4 ) which are contained in Table I. If for reasons of algebraic convenience we let mi =me, the use of the data in Table I leads to t/6 ( t/3 mext(t) = 66.t- 72(1 - e ) e ) (17) Equation (17) is the asymptotic fissile fuel trajectory equation with the graphical form shown in Fig (.!) ~ - :::: ~ 176.I ~ 18.7 z H ' w ::J I.J.. I w --' H (f) (f) H -227.s I.J.. o.o TIME ( YRS J I I r. ~ f Fig. 3.1: The trajectory of fuel inventory of external stockpile for a breeder reactor obtained with asymptotic values of reactor parameters. The parametric values are based on data for the CRBR. 13 TABLE I Parametric Values for the Calculation of Net Fissile Fuel F.(lQws . corresponds to those for the Clinch River Breeder Reactor. 5) The data Parameter Numerical Value Used Total breeding gain Gc+Gb,r+Gb,a BR-1 =.22 Equilibrium residence time for radial blanket, Ar 6 yrs Equilibrium residence time for axial blanket, Ar 3 yrs Radial blanket breeding gain, Gb,r.39 Axial blanket breeding gain, Gb,a.22 Constant fissile fuel destruction rate, R 4 kg/y Equilibrium load factor, L.75 Refuelling Interval, tc Annual Equilibrium reload fraction for axial blanket, aa 1/3 Equilibrium reload fraction for radial blanket, ar l/6 Blanket shuffling in+out Initial fissile fuel in radial blanket, mb,r ~o Initial fissile fuel in axial blanket, mb,a ~o CHAPTER 4 TIME DEPENDENT PARAMETERS 4.1 Time Dependent Average Residence Time for a Reactor Fuel The objective in this section is to obtain the average residence time for all the fissile fuel fed into a reactor core up to and including those in the core at any point in time of the reactor life. The formulation will be based on fuel management scheme which allows replacement of some fuel bundles during each of many refuelling periods. In the CRBR( 2 ) for example, l/3 of the core fuel assemblies are replaced on a yearly refuelling basis. In this case all the initial fuel load are replaced at the end of the third year. During the next refuelling period the whole one-third of the core fuel used for the first refuelling is now replaced and this sequence is maintained for the subsequent cycles. It is important to note that the result we shall obtain for the core will be generally true for average residence times in the blanket regions since the difference in the refuelling schemes are primarily due to the fraction of fuel replaced in each cycle, and the cycle lengths which ar.e specified quantities. If we now adopt a method of lumping together all fuel bundles replaced in each cycle, we shall have 3 lumps of fuel for the first core load of the above example. The number of fuel lumps then correspends to the number of refuelling periods required to replace all the initial fuel loads as illustrated schematically in Fig 15 z w :: (.) z - _.J w :::: ll 1,.. # # 4., '.,,, ,., # # NUMBER OF CYCLES, N Fig. 4.1: A model showing fractions of the core fuel replaced per cycle and the duration of each fraction or fuel lump in the core. The model assumes one-third of the core fuel is replaced at the end of each cycle, i.e. N = 3. 16 tc: N : In the formulation the following parameters will be employed: Reactor refuelling time interval specified. Number of refuelling periods or cycle required to replace all initial fuel loads; this is related to the fraction a of the core fuel assemblies replaced each time by N = 1/a. t: Point in time of the reactor life. N: Number of completed cycles at time t. i(t): Average residence time for all the core fuel used by the time t. A notion for the value of I(t) can be obtained through the following considerations. Since all initial core load remains in the reactor till the first refuelling time t, this specifies a lower bound c for the average residence time while the upper bound is determined by the number of cycles N, required to replace all the initial loads. Hence, t c I( t) N t c ( 18) A bookkeeping procedure which counts all the fuel lumps that have been used in the core at the end of each cycle and the total time they have spent in the core is shown in Table II. In Table II, the average residence time for all the fuel elements that have been used and removed from the core together with those in the core during each of the refuelling periods have been obtained from the ratio of the cumulative time to cumulative number of fuel lumps in the storage bays and core. We can thus obtain directly from the last column of Table II a general form for the average residence time I(t) and this is given by 1 each TABLE II Fuel lumps used per cycle. during the operation of a reactor in which one-third of the total core fuel is replaced No. of cycles or refuelling periods (N) Time ( tc yrs) No. of fuel lumps in core (No) I Time for various fuel lumps 1 '1 ' 1 2' 2' 1 3' 2' 1 1,3' 2 2 '1,3 3 '2' 1 1,3 '2 2 '1 '3 in core (t yrs) c No. of fuel lumps in storage bay Time for various fuel lumps 1 1 '2 1,2,3 1,2,3,3 1,2,3,3,3 1,2,3,3,3,3 1,2,3,3,3,3,3 in storage (\ yrs) I Cumulative No. of fuel lumps I 1 in core and storage bay Cumulative time (tc yrs) Average residence time of i 2.4 fue 1 1 urnp at the end of each cycle (t yrs) c I! I I I 24 N N N N-1 N +N-1 N N N N N +N-1 '-1 18 f( t) N N = =N -=-+--,-,-N ---::-1 ( 19) Since the number of cycles N are related to the time t and the cycle length tc by N = t/tc we then have I(t) (2) The average residence time f(t) thus depends on the fraction of fuel N -l replaced at the end of each cycle and the refuelling time tc. To obtain i(t) in terms of the fraction a, of the fuel assemblies replaced in each cycle we make the substitution and this leads to f( t) = lt a c t t+ (- l -l)t a (21) i.e. f( t) = tc t t + (l- a)t c (22) 19 For the case where l/3 of the fuel is replaced in each cycle on a yearly refuelling basis the average residence time of all the fissile fuel used at any time t is thus given by t ( t) = ~-=-=-=-~~-::-=.333t (23) In Table III the values of I

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