Introduction to Nuclear Reactor Kinetics

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ECOlE POLYTECHNIOUE Introduction to Nuclear Reactor Kinetics Course presented at Chulalongkorn University Bangkok, Thailand by Daniel Rozon, Ph.D., F.C.N.S. Institut de genie nucleaire Departement de genie mecanique February 1997

1 - Neutron/Nucleus Interactions and Fission- 2 Introduction Reactivity is an important reactor physics concept useful to answer common questions such as: - What is a «critical» fission reactor? - How do you «start up» a reactor? - How does power vary with time when a reactor is subjected to a perturbation? - How can we control the power in a reactor? - What is meant by «prompt critical» or «super prompt critical»? - Can the reactor «explode» like a nuclear bomb? What about Tchernobyl? Reactivity characterizes simply the dynamic state of a nuclear reactor: ~ particularily useful in reactor control Reactivity cannot be measured directly: ~ obtained from neutron field equations defined over the entire reactor domain

1 Neutron/Nucleus Interactions and Fission 3 Course Objectives The basic interactions between neutrons and nuclei will be discussed The equations governing the neutron field in a reactor will be presented The formal definition of reactivity will be provided. Its reactor physics applications and limitations will be discussed. The Point Kinetics equations will be derived and applied to typical situations in a CANDU reactor (as compared to other types of reactors). Numerical solutions with prompt temperature feedback will be presented

1 Neutron/Nucleus Interactions and Fission 4 The Fission Process when nucleons are rearranged to a more stable form, we observe: - a mass defect (L1A1) energy production (E =11M c2) Average Binding Energy per Nucleon c: o CD '0 E - e>! w C) c: ~ iii E j~ 8~M~ -1M~ ~------~~::::/ - Fission H 1 Fe 50 Atomic Number U 240:' A sum of masses of [ nucleons [ mass of ] [ total ] = l~e bound + ~ binding Isotope energy Bavg(A,Z) = I1M(A,Z) c 2 A fusion D-T: fission U-235: 1H2 + 1H3 ~ 2He4 + on 1 + energy 1'7.7 MeV

1 Neutron/Nucleus Interactions and Fission 5 Rssion Energy Produced by Fission of U-235 (MeV) Form Energy Recovered Enerav produced coolant moderator fission fragments: light 100 100 - heavy 69 69 - prompt neutrons 5-5 prompt 'Y 6 10* 5 (anti)neutrinos 11 - - fission product decay: ~ 7 7 - 'Y 6 4 2 Total 204 MeV 190 MeV 12 MeV Including ys from radiative capture. Similar values for other fissile isotopes (= 200 MeV, + 5%) Fission of 1 kg of Uranium: - 950 MWd of energy - 13500 barrels of oil equivalent - 2830 Mg coal equivalent

1 - Neutron/Nucleus Interactions and Fission 6 Fission Number of prompt neutrons per fission isotope U-233 2.438 U-235 2.430 Pu-239 2.871 Pu-241 2.969 U-238* 2.841 * Fast fission Slightly. dependent on energy of incident neutron Fission is isotropic: direction of prompt fission neutrons is independent of direction of incident neutron. v Energy. Spectrum of Prompt Fission Neutrons (U-235) 0.4-.------------... E 2 0.3 U G> Q.~ CI)~ 02 S ~. 'CiS (I) 'E 0.1 w 0.0 -I-...,...-~ -.,... r_--=;:=::;:::= I o 1 234 5 678 Energy (MeV) Average energy of prompt fission neutrons (U-235) is 1.98 MeV

1 Neutron/Nucleus Interactions and Fission 7 Neutron Interactions with Nuclei Notation zx A X Z A chemical symbol number of protons number of nucleons (mass number) The Compound Nucleus zx A + on 1 ~ (ZX A + 1 f Decay of Compound Nucleus: * Elastic Scattering (n, n) ( X A + 1 ) ~ X A + n 1 Z Z 0 Inelastic Scattering (n, n') * ( Z X A + 1 ) ~ Z X A + 0n 1 + r Radiative Capture (n, y) * (ZX A + 1 ) ~ ZX A + 1 + r Fission (n, f) * Z C 1 Z-C 2 ( X A + 1 ) ~ PF B + PF A+1-B-v + V on 1 + r (n, p) * (ZX A + 1 ) ~ Z_1 ya + 1 H 1 (n, a) * (ZX A + 1 ) ~ y A - 3 + He Z-2 2 4 (n, 2n), (n, 3n)... -

1 - Neutron/Nucleus Interactions and Fission 8 Fission Product Distribution yield 0,1 ~~ 0,01 ~ ~ 0,001 O,OOO1'~_""''''''''''''--""''T'' ""''T --r...,.--1 60 80 100 120 140 160 180 Mass Number A stability N ".,Qf:o Fissile,,. Stability,".:.: nuclide Line,::,' 0,., X ', 1-' ;!.,',.,,.:-, Fission --,0---.;:;' fragments,' '1;;t Z _ N ~JJf~., A " ;,:Jf, ii',, " ~, z

1 Neutron/Nucleus Interactions and Fission 9 Neutron Cross Sections and Reaction Rates neutron angular density Symbol n(r,e,.o,t) units [nlcm 3 /ev/st] neutron velocity v = v.q (cmls] angular flux density 4>(r,E,.o,t) = n(r,e,.o,t).v(e) [n/cm 2 /slev/st] scalar flux density q,(r,e,t) = n(r,e,t),v(e) [nlcm 2 /s/ev] r' scalar flux (in group g) =,(r,e,t) de E g _ 1 microscopic cross section u(e) macroscopic cross section I(r,E,t) = N(r,t) u(e) reaction rate (per unit volume) R = I.q,,Ex: for a reaction of type x, cross sections are determined experimentally by measuring the attenuation of an incident neutron beam (of given E). If the volume contains many isotopes (i), simply sum over i to obtain the total reaction rate:.. Integrating drx = Gx </J N ds N (nu~es) over 1 cm3:, (neutronsj em... 2 s ~ Rx = (N Gx) </J ~ ~ds (em) = Ix </J

1 Neutron/Nucleus Interactions and Fission 10 NeutrQn cross sections Cross sections for thermal neutrons (0,025 ev), in barns (Westcott, 1960) Isotope af ay aa U233 525.0 53.0 578.0 U235 577.0 106.0 683.0 U238-2.70 2.70 Pu239 742.1 286.9 1029.1 Pu240-277.9 277.9 Pu241 1015.2 381.0 1396.2 Neutron scattering Isotope A N Hydrogime (H1) 1 18 Deuterium (H2) 2 25 Graphite (C 12 ) 12 115 Uranium (U238) 238 2172 N = number of collisions required to slow down 2 MeV neutrons t> thermal energies (0.025 ev) (Weinberg,1958).

1 Neutron/Nucleus Interactions and Fission 11 The Neutron Chain Reaction Neutron cross sectjqns...: - most isotopes possess relatively small absorption aoss sedions (O'a = Uf + ud of the order of 1-10 barns. Mean free path of neutrons in matter is large, of the order of em's. - notable exception: heavy elements (fuel), with a large absorption cross section (O'a::::: 1000 b) - only odd-number isotopes are fissile for thermal neutrons (U-233, U235, Pu-239, Pu-241,...) - even numbered isotopes allow fast fission (threshold energy::::: 1 MeV) Fission neutrons appear with an energy::::: 1 MeV, where fission cross section is small compared to thermal energies (by a factor of 1000) => advantage in slowing down neutrons (via scattering) in moderator material => thermal reactors To maintain the chain reaction in thermal reactors, neutrons must be slowed down over 8 decades, while avoiding resonance absorption in the heavy element (notably U-238, a major constituant of the fuel) => resonance capture is limited by self-shielding in fuel pins => heterogeneous reactors A controlled fission chain reaction will largely depend on a critical mass of fuel and moderator, and is vel}' sensitive to geometry an.d. the distribution ofneutrons in the reactor (space and energy distribution) => neutron field

1 Neutron/Nucleus Interactions and Fission 12 Delayed Neutrons certain fission products are precursors to the emission of delayed neutrons. For example, approximately one fission in a thousand in U-235 yields Br-87 as fission product. This isotope is radioactive and decays into an excited state of Kr-87, which stabilizes into Kr-86 by emitting a neutron: 35 8r87, fission ' product (precursor) (55 s) ) 36Kr87 + _180... ' ----.,y~~ /r radiation 36 Kr87 (emitter) 36Kr86 + on 1 '---v----' «delayed» neutron The production of this neutron in the fuel is «delayed» with regards to the fission event due to the delay associated with radioactive decay of the precursor. The half-life of precursors varies between 0.2 sand 55 s. Relatively few fission products are delayed neutron precursors. The delayed neutron fraction is defined as: f3 = vd = delayed neutrons resulting from a single fission event v total number of fission neutrons (prompt + delayed)

1 Neutron/Nucleus Interactions and Fission 13 Delayed Neutrons (cont'd) We note that the delayed neutron fraction is small and is specific to the fissioning nuclide. 11 varies significantly from one isotope to the other: Vd U-233 0.0067 0.0027 U-235 0.0166 0.0068 Pu-239 0.0065 0.0023 Pu-241 0.0154 0.0052 U-238'" 0.0450 0.0158 * Fast fission Although small, the delayed neutron fraction plays an extremely impqrtant rqw in ReactQr Kinetics, as we shall see in this course. It will be impqrtant to evaluate the effective average f3 for a given reactqr, considering the fission rate in each fissiqning isqtope. For each fissile isotope, delayed neutrqn precursor prqperties can be Qbtained experimentally, with a curbe fit Qver a limited number Qf delayed grqups (generally 6 fqr the fuel): 6 Sdi(t) = n(i L Vdki A/(j e- AJcjt k=1 f3 -,... I U) 1 -Q) l:? :::] 0 en 1 c: 0... 'S Q) Z "0 Q) ~ CD 0 0 50 100 150 200 Time after Fission Pulse (s)

1 - Neutron/Nucleus Interactions and Fission 14 Decay Constants for the Delayed-Neutron Groups, l k (OU, 1985) '" HH 'nhh Group A. (S 1} Half-life {s} 1 0.0129 53.73 2 0.0311 22.29 3 0.134 5.17 4 0.331 2.09 5 1.26 0.55 6 3.21 0.22 0 ~ I I cc Ctc Ie C : c Delayed Neutrons Grou~ A single set of decay constants can be used for all isotopes: The following values were obtained from measurements: Fraction of Delayed Neutrons from the Fuel, fjkl (%) 1 0.0251 2 0.1545 3 0.1476 4 0.2663 5 0.0756 6 0.0293 Total p 0.6984 «::",,:::::::::e:::::::=««0.0087 0.0639 0.0493 0.0747 0.0235 0.0080 0.2281 0.0066 0.1234 0.0932 0.2102 0.0981 0.0086 0.5389 :H:«: :: e:::::: : 0.0206 0.2174 0.2570 0.6156 0.3570 0.1190 1.587

1 Neutron/Nucleus Interactions and Fission 15 Photoneutrons In heavy water reactors (CANDU), additional delayed neutrons appear in the moderator, called photoneutrons. They constitute only := 5% of f3 but have a much longer half-life (up to 15 days): r + 1 H2 ~ 1 H1 + on 1 ; radiation fro~ 'deuterium in ~ fission product moderator neutron (Er~2.2 Mev) The half life of the photoneutron precursors is generally longer. 9 groups are required: Fraction of Delayed Neutrons from the Moderator (Photoneutrons), IJk(%) (Kugler, 1975) TOTAL 16.7min 6.92 x 10-4 3.306 x 10-2 Jehoton,eut~~,~~~ '" '" "'''' "''' '" ""''''' Average values: A Ph = 1: (f3,/ak)/1: 13k and T 1 / 2 = In2. Api!

1 - Neutron/Nucleus Interactions and Fission 16 Delayed Neutron Source in a CANDU As a result, ther exists a slowly decreasing neutron source in a reactor after shutdown. In a CANDU, the photoneutrons dominate after only 5 minutes, eventhough at the beginning, they constitute only 5% of the delayed source. 10 0 Delayed Neutron Source in a CANDU (after shutdown from 100%FP) 10-1 10-2 -li: 10-3 TOTAL ~ "-" Q,) ~ 10-4 0 en 10-5 10-6 10.7 Delayed Neutrons (6 groups) 1 hour 10-1 10 0 10 1 102 103 104 105 10 6 10 7 10 8 Time (s)

1 Neutron/Nucleus Interactions and Fission 17 Fuel Burnug ~ irradiation (or exposure) is defined by: Q/ (t) = f>c dt' where tpc is the neutron flux in the fuel. The irradiation unit is the neutron/kilobarn n kb = 2 n.~ x 10-24cm2 x103 bam fm - sec, (T) bam kilobam (~) 10: 21 ' ~ quantifies the total fission energy produced in a fuel element per unit mass of initial heavy elements (uranium) in the fuel, up to time t: - energy produced Ek (t) = f: ' P~(t),dt' = = Pk T fission power f: IC, Et ~ Vk, dt' fission rate - units E = (kw sec) x (10-3 MWX- 1 _-!!-) TPf (f) kw 3600 sec = MWh

1 Neutron/Nucleus Interactions and Fission 18 Fuel Composition in a 37 element CANDU Fuel Bundle producing 690 kw Temps irradiation burnup U-235 Pu-239 Pu-241 (jours) (nlkb) MWhlkg) (glkgu) (glkgu) (glkgu) O. o. 0.00 7.200 0.000 0.000 10..093 8.69 6.782 0.225 0.000 30..273 26.09 6.038 0.771 0.003 50..446 43.46 5.396 1.185 0.012 75..658 65.21 4.700 1.575 0.032 90..784 78.26 4.329 1.759 0.047 120. 1.040 104.33 3.672 2.043 0.085 150. 1.300 130.42 3.111 2.246 0.127 200. 1.749 173.90 2.353 2.467 0.204 250. 2.217 217.37 1.769 2.596 0.277 300. 2.692 260.85 1.325 2.672 0.345 8-.-------------------, -:::> l C) 6 - c: 0 4 ~... U-235 -c: CD 0 c: 0 Pu-239 0 2 Pu-241 ou:..=::;:::::;::~::;::::::::::;:::::;:::::;::::~~:::=:l 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Irradiation (n/kb)

ASSIGNMENT 1 Consider the two following nuclear reactions: 23SU + n ~ 13~e + 94Sr + 3n + E Using the information below: a) Calculate the energy E produced in each reaction b) Indicate which ofthese reactions produces more energy ifthe same mass ofreactants is used. Information: Atom (or particle) Mass(AMU) 3H 3.016 ~ 2.014 4He 4.002 23SU 235.044 13~e 138.918 94Sr 93.915 n 1.009 1 AMU = 931.5 MeV = 1.66 X 10-27 kg 1 MeV = 1.60 x 10-13 J

ASSIGNMENT 2 Boron and Godolinum are two soluble poisons that can be introduced in the moderator to control the reactivity in a CANDU reactor. The properties ofthese poisons are given in the attached table. a) Indicate which isotopes ofboron and gadolinium constitute the neutron poison. b) A natural boron concentration of 2.15 x 10 17 at /cm 3 is required in the moderator to introduce -28 mk of reactivity (simulating the equilibrium Xenon load). Calculate the equivalent gadolinium concentration which will produce the same reactivity effect. Bore Isotope Proportion Microscopic thermal Type of (%) neutron absorption reaction cross section (barns) B-10 19.8 3840 (n,a) B-ll 80.2.005 (n,y) Gadolinium Gd-152.2 10 (n,y) Gd-154 2.1 80 (n,y) Gd-155 14.8 6.1 x 10 4 (n,y) Gd-156 20.6 2 (n,y) Gd-157 15.7 2.5 x los (n,y) Gd-158 24.8 2.4 (n,y) Gd-160 21.8.8 (n,y) Atomic weight Boron: 10.8 Gadolinium: 157.2

ASSIGNMENT 3 A natural U0 2 fuel bundle is removed from a reactor following an exposure to an average thermal neutron flux of1.0 x 10 14 neutrons cm- 2 s 1 for 200 days. a) Calculate the bumup ofthe fuel in n/kb. b) Calculate the lf3s content that remains after this exposure, expressed in g/kgu. c) If the concentrations of Pu 239, and Pu 241 at this time are 2.39 glkgu and.221 glkgu, respectively, calculate the fraction ofthe total number offissions in the fuel that were due to plutonium (both PU 239 and Pu 241 ) just before discharge from the reactor. (Neglect the difference in atomic weights between lf3s, PU 239 and PU 241 ). The table below provides the data required for answering this question. Of 0ll,y lf3s 580.2 b 98.3 b Pu 239 741.6 b 271.3 b Pu 241 1007.3 b 368.1 b 1 bam = 10 24 cm 2 fresh fuel composition: lf3s = 7.2 g/kgu lf31 = 992.8 g/kgu

ASSIGNMENT 4 Ifon the average, 200 MeV ofenergy is produced per fission, calculate the fuel consumption per year in a CANDU-600 reactor producing 2064 MW in the primary coolant. The exit fuel burnup actually acheived is 180 MWh/kg. Is this value consistent with your previous evaluation? Ifnot, why is it different?

ASSIGNMENT 5 Assume that 3% of all fissions in a CANDU reactor are due to fast fission in U-238. Using the data provided in the class room, calculate the total delayed neutron fraction and an appropriate average decay constant for a single delayed neutron group: a) for a reactor containing only fresh fuel (natural uranium, 0.711%, U-235) b) for a reactor at equilibrium refuelling, with an average exit bumup of185 MWh/kg.

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