Lectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 1. Title: Neutron Life Cycle


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1 Lectures on Nuclear Power Safety Lecture No 1 Title: Neutron Life Cycle Department of Energy Technology KTH Spring 2005 Slide No 1
2 Outline of the Lecture Infinite Multiplication Factor, k Four Factor Formula Fast Fission Factor, ε Resonance Escape Probability, p Thermal Utilization Factor, f Reproduction Factor, η Effective Multiplication Factor, k eff Fast NonLeakage Probability, P fnl Thermal NonLeakage Probability, P tnl Six Factor Formula Neutron Life Cycle of a Fast Reactor Slide No 2
3 Introduction We start with a short overview of some introductory topics, which are related to today s lecture: Neutron reactions Crosssection for neutron reactions Neutron absorption Nuclear fission Neutron moderation (or slowing down) Slide No 3
4 Neutron reactions (1) Neutrons play a very important role in nuclear reactor operations and their interactions with matter must be studied in details Reaction of neutron with nuclei fall into two broad classes: scattering absorption In scattering reactions, the final result is an exchange of energy between the colliding particles, and neutron remains free after the interaction In absorption, however, neutron is retained by the nucleus and new particles are formed Slide No 4
5 Neutron reactions (2) Inelastic scattering When a neutron undergoes inelastic scattering, it is first captured by the target nucleus to for an excited state of the compound nucleus A neutron of lower kinetic energy is then emitted, leaving the target nucleus in an excited state This excess energy is subsequently emitted as one or two photons of gamma radiation In general, inelastic scattering is limited to some high or low (depending on target mass) neutron energy ranges Slide No 5
6 Neutron reactions (3) Elastic scattering In elastic scattering the kinetic energy is conserved that is the total kinetic energy of neutron and the target nucleus is unchanged due to collision There are two types of elastic scattering: Potential scattering resembling collision of two billiard balls Resonance or compound nucleus scattering (when target nucleus is larger) where neutron is first absorbed and next expelled from nucleus, leaving the target nucleus in the ground state Slide No 6
7 Neutron reactions (4) In considering absorption reactions it is convenient to distinguish between reactions of slow neutrons and of fast neutrons There are four main kinds of slow neutron reactions: these involve capture of the neutron by the target followed by either: The emission of gamma radiation (n,γ) radiative capture The ejection of an alpha particle (n,α) The ejection of an proton (n,p) Fission (n,f) Of these the radiative capture is most common, whereas others are limited to few species. Slide No 7
8 Crosssection for neutron reactions (1) To quantify the probability of a certain reaction of a neutron with matter it is convenient to utilize the concept of crosssections The crosssection of a target nucleus for any given reaction is thus a measure of the probability of a particular neutronnucleus interaction and is a property of the nucleus and of the energy of the incident neutron Slide No 8
9 Crosssection for neutron reactions (2) Suppose a uniform, parallel beam of I monoenergetic neutrons per m 2 impinges perpendicularly, for a given time, on a thin layer δx m in thickness, of a target material containing N atoms per m 3, so that Nδx is the number of target nuclei per m 2 (see figure to the right) The nuclear cross section for a specified reaction is then defined as I I δx N R 2 σ = m / nucleus Crosssection unit: 1 b (barn) = ( Nδx) I m 2 /nucleus Where N R is the number of reactions Slide No 9
10 Crosssection for neutron reactions (3) Total cross section of uranium238 Slide No 10
11 Crosssection for neutron reactions (4) Total cross section of uranium235 Slide No 11
12 Crosssection for neutron reactions (5) Fission cross section of uranium235 Slide No 12
13 Crosssection for neutron reactions (6) The cross section σ for a given reaction applies to a single nucleus and is frequently called the microscopic cross section. Since N is the number of target nuclei per m 3, the product Nσ represents the total cross section of the nuclei per m 3 Thus, the macroscopic cross section Σ is introduced as Σ = Nσ m 1 Slide No 13
14 Crosssection for neutron reactions (7) If a target material is an element of atomic weight A, 1 mole has a mass of 103 A kg and contains the Avogadro number (N A = ) of atoms. If the element density is ρ kg/m 3, the number of atoms per m 3 N is given as 3 10 ρn = A The macroscopic cross section is thus N A 3 10 ρ Σ = A N A σ Slide No 14
15 Crosssection for neutron reactions (8) For a compound of molecular weight M and density ρ kg/m 3, the number N i of atoms of the i th kind per m 3 is given by the following equation 3 10 ρn A N = ν ν i i M where is the number of atoms of the kind i in a molecule of the compound. The macroscopic cross section for this element in the given target material is then and for compound ρn A 10 ρn Σ i = N iσ i = ν iσ i Σ = A ( ν1σ 1 + ν 2σ 2 +L) M i M Slide No 15
16 Nuclear fission (1) Fission is caused by the absorption of neutron by a certain nuclei of high atomic number When fission takes place the nucleus breaks up into two lighter nuclei: fission fragments Only three nuclides, having sufficient stability to permit storage over a long period of time, namely uranium233, uranium235 and plutonium239, are fissionable by neutrons of all energies Of these nuclides, only uranium235 occurs in nature The other two are produced artificially from thorium232 and uranium238, respectively Slide No 16
17 Nuclear fission (2) Example of a nuclear fission of U235 Each fission of U235 is followed by a release of 2 or 3 neutrons (2.42 on average for fission of U235 with thermal neutrons) Slide No 17
18 Nuclear fission (3) Chain reaction will sustain if there is enough neutrons to cause fissions in coming generations If fission is caused by slow (thermal) neutrons, they have to be moderated (slowed down) before the next fission Slide No 18
19 Nuclear fission (4) Approximate distribution of energy per fission of 235 U: J = 1 pj MeV Kinetic energy of fission products Instantaneous gammaray energy Kinetic energy of fission neutrons Beta particles from fission products Gamma rays from fission products Neutrinos Total fission energy Slide No 19
20 Nuclear fission (5) The neutrons released in fission are of two categories: Prompt neutrons (over 99% of all neutrons are prompt) released within s Delayed neutrons emitted during several minutes after fission The average number of neutrons liberated in fission is designed as. Typical values are shown below ν thermal neutrons Fast neutrons Uranium Uranium Plutonium Slide No 20
21 Nuclear fission (6) Neutrons released after fission do not have the same energy Typical energy spectrum is shown in figure Most neutrons have energy between 1 and 2 MeV Slide No 21
22 Nuclear fission (7) Fission of uranium 235 can end up with 80 different primary fission products The range of mass numbers of products is from 72 to 161 Figure shows fission yield vs mass number Slide No 22
23 Nuclear fission (8) The fission cross section of uranium235 is very low for fast (highenergy) neutrons To increase the fission rate, it is necessary to slowdown neutrons This subject is treated in the section that follows Slide No 23
24 Slowing down of neutrons (1) After fission, neutrons move in all directions with speed up to km/s Neutrons can not move a longer time with such high speeds, which successively goes down due to collisions with nuclei; this process is called scattering After short period of time the neutron velocities approach the equilibrium velocity, which is about 2200 m/s at 20 C Slide No 24
25 Slowing down of neutrons (2) Neutron scattering can be either: Elastic (in most cases) Inelastic Classical dynamics law are used to describe the elestic scattering process Collision in laboratory system V 1 Neutron before Nucleus before A ψ V 2 Nucleus after Neutron after Collision in a centerofmass system Centrer of mass V 2 Neutron after Laboratory or Centerofmass systems are used V 1 v m Neutron before Nucleus after θ v m Nucleus before Slide No 25
26 Slowing down of neutrons (3) It can be shown that after collision, the minimum value of energy to which neutron can be reduced is αe 1 Where E 1 is the neutron energy before the collision, and A 1 α = A The maximum energy of neutron after collision is E 1 Slide No 26
27 Slowing down of neutrons (4) For hydrogen A = 1 and so And it is possible for neutron to lose all energy in one collision For carbon A = 12 and α = 0 α = The minimum energy of neutron after collision will be equal to 71.6% of that before the collision Slide No 27
28 Slowing down of neutrons (5) A useful quantity in the study of the slowing down of neutrons is the average value of the decrease in the natural logarithm of the neutron energy per collision, or the average logarithmic energy decrement per collision It is the average of all collisions of lne 1 lne 2 = ln(e 1 /E 2 ), where E 1 is the energy of the neutron before and E 2 is that after collision E ξ ln E 1 2 = 1 ln E E d d ( cosθ ) ( cosθ ) Here θ is a collision angle in the C system; integration means averaging over all possible collision angles Slide No 28
29 Slowing down of neutrons (6) Analyzing energy change in scattering, the ratio E 1 /E 2 can be expressed in terms of mass number A and cosine of the collision angle cosθ Using this in the equation in the previous slide, the following is obtained ξ = 1+ ( A 1) 2A 2 ln A 1 A + 1 It can be seen that for Hydrogen (A=1) ξ=1 and for uranium238 ξ= Slide No 29
30 Slide No 30 Slowing down of neutrons (7) If the moderator is not a single element, but a compound containing n different nuclei, the effective or mean (weighted) value of ξ is given by For example for H 2 O we get sn s s n sn s s σ σ σ ξ σ ξ σ ξ σ ξ = ) ( ) ( ) ( ) ( ) ( O s H s O O s H H s O H σ σ ξ σ ξ σ ξ + + =
31 Slowing down of neutrons (8) An interesting application of the logarithmic energy decrement per collision is to compute the average number of collisions necessary to thermalize a fission neutron It can be shown that this number is = 14.4/ξ One also defines the moderating or slowing down power of a material as: ξσ s However, this parameter is not sufficient to describe how good a given material is as a moderator, since one also whishes the moderator to be a week absorber of neutrons That is why one use moderating ratio = ξσ s / Σ a as a figure of merit Slide No 31
32 Infinite Multiplication Factor (1) Not all neutrons produced by fission will cause new fission: Some will be absorbed by nonfissionable material Some will be absorbed parasitically in fissionable material Others will leak out of the reactor For the maintenance of a selfsustaining chain reaction it is enough that, on the average, at least one neutron produced in fission that causes fission of another nucleus Slide No 32
33 Infinite Multiplication Factor (2) The condition of a selfsustaining chain reaction is conveniently expressed in terms of a multiplication factor The number of neutrons absorbed or leaking out of the reactor will determine the value of this multiplication factor, and will also determine whether a new generation of neutrons is larger, smaller or the same size as the preceding generation Slide No 33
34 Infinite Multiplication Factor (3) Any reactor of a finite size will have neutrons leak out of it Generally, the larger the reactor the lower the fraction the neutron leakage For simplicity consider a reactor that is infinitely large, and therefore has no neutron leakage A measure of the increase or decrease in neutron flux in an infinite reactor is the infinite multiplication factor k Slide No 34
35 Infinite Multiplication Factor (4) The infinite multiplication factor is the ratio of the neutrons produced by fission in one generation to the number of neutrons lost through absorption in the preceding generation: or k = Neutron production from fission in one generation Neutron absorption in the preceding generation k = Rate Rate of of neutron neutron production absorption Slide No 35
36 Infinite Multiplication Factor (5) The condition for criticality, i.e. for a selfsustaining fission chain to be possible, in the infinite system is that the rate of neutron production should be equal to the rate of absorption in the absence of an extraneous source In other words, the requirement for criticality is: k = 1 Slide No 36
37 Four Factor Formula (1) For some thermal reactors, the infinite multiplication factor k can be evaluated with a fair degree of accuracy by means of the four factor formula The basis of this formula is the assumed division of the neutrons into three categories: Fission neutrons with energies in excess of about 1MeV which can cause fission in uranium238 as well as in uranium235 Neutrons in the resonance region which may be captured by uranium238 Thermal neutrons which cause nearly all the fission in uranium 235 and thereby generate fission neutrons Slide No 37
38 Four Factor Formula (2) A group of fast neutrons can enter into several reactions Some of these reactions reduce the size of the neutron group while other reactions allow the group to increase in size or produce a second generation There are four factors that give the inherent multiplication ability of the fuel and moderator materials: where : ε = Fast fission factor k = ε p f η p f = = Resonance escape probability Thermal utilization factor η = Reproduction factor Slide No 38
39 Fast Fission Factor (1) The first process that the neutrons of one generation may undergo is fast fission Fast fission is fission caused be neutrons that are in the fast energy range In a thermal reactor using slightly enriched or natural uranium fuel, some neutrons, before they have been slowed down appreciably, will cause fission of both uranium235 and uranium238 nuclei Slide No 39
40 Fast Fission Factor (2) At neutron energies greater than about 1 MeV, most of the fast neutron fissions will be of uranium238, because of its larger proportions in the fuel Fast fission results in the net increase in the fast neutron population of the reactor core The crosssection for fast fission in uranium235 and uranium238 is small The fast neutron population in one generation is thus increased by a factor called the fast fission factor Slide No 40
41 Fast Fission Factor (3) The fast fission factor is defined as the ratio of the net number of fast neutrons produced by all fissions to the number of fast neutrons produced by thermal fissions ε = Number of fast neutrons produced by all fissions Number of fast neutrons produced by thermal fissions In order for a neutron to be absorbed by a fuel nucleus as a fast neutron, it must pass close to a fuel nucleus while it is a fast neutron Slide No 41
42 Fast Fission Factor (4) The value of ε will be affected by the arrangement and concentrations of the fuel and the moderator The value of ε is essentially 1.00 for a homogenous reactor where the fuel atoms are surrounded by moderator atoms However, in a heterogeneous reactor, all the fuel atoms are packed closely together in elements such as pins, rods or pellets Thus neutrons emitted from one fission can pass close to another fuel atom The arrangement in heterogeneous reactors results in ε~ Slide No 42
43 Resonance Escape Probability (1) As already discussed, neutrons increase in number as a result of fast fission After that neutrons continue to diffuse through the reactor As the neutrons move they collide with nuclei of fuel and nonfuel material and moderator in the reactor loosing part of their energy While they are slowing down there is a chance that some neutrons will be captured by uranium238 nuclei Slide No 43
44 Resonance Escape Probability (2) Absorption crosssection of uranium238 has several resonance peaks for neutron energies between 6 to 200 ev The peak values can be as high as barns, whereas below 6 ev, the absorption crosssection is as low as 10 barns The probability that the neutron will not be absorbed by a resonance peak is called the resonance escape probability, p Slide No 44
45 Resonance Escape Probability (3) the resonance escape probability, p is defined as the ratio of the number of neutrons that reach thermal energies to the number of fast neutrons that start to slow down p = Number of neutrons that reach thermal energy Number of fast neutrons that start to slow down The value of resonance escape probability is determined largely by the fuelmoderator arrangement and the amount of enrichment of uranium235 Slide No 45
46 Resonance Escape Probability (4) In a homogeneous reactor the neutrons slow down in a region close to fuel nuclei and thus the probability of being absorbed by uranium238 is high In the heterogeneous reactor neutrons slow down in the moderator where there are no atoms of uranium238 and the probability of undergoing resonance absorption is low Slide No 46
47 Resonance Escape Probability (5) The value of the resonance escape probability is not significantly affected by pressure or poison concentration In water moderated, low uranium235 enrichment reactors, raising the temperature of the fuel will rise the resonance absorption in uranium238 due to the Doppler effect (i.e.. An apparent broadening of normally narrow resonance peaks due to thermal motion of nuclei) The increase in resonance absorption lowers the resonance escape probability Slide No 47
48 Resonance Escape Probability (6) The resonance escape probability can be found from the following formula: p( E) exp N F I ξ Σ s where: ξ is the weighted average logarithmic energy decrement for both moderator and absorber, N F is the number of fuel nuclei per unit volume of the system, I is the effective resonance integral and Σ s is the total macroscopic cross section for scattering in the system Slide No 48
49 Resonance Escape Probability (7) Experimental measurements of the resonance integral for a system of isolated rods give the following formula: I = a + b A M I where a and b are constants for a given fuel material (= 2.95 and 81.5 resp for uranium and 4.45 and 84.5 for uranium dioxide, such that I will be in barns), A is the area (in m 2 ) and M is the mass (in kg) of a fuel rod A = UO + 2 M A M [] b ; I = [] b uranium 5 Slide No 49
50 Resonance Escape Probability (8) The integral I depends on temperature as follows: [ + β ( 300) ] I( T ) = I(300 K) 1 T Here I(300 K) is the value of the integral at T = 300 K and β is a constant which depends on the nature of the fuel and the radius of fuel rods in heterogeneous systems For UO 2 and typical fuel rods used in LWRs β = 6x103 Slide No 50
51 Resonance Escape Probability (7) Example: Calculate the resonance integral I for fuel rods containing UO 2 (density kg m 3 ); the rod diameter is 8 mm. Solution: For 1 m long fuel rod its surface area is equal to A =1* πd = m 2 and its mass is M = 1* πd 2 /4*ρ = π* /4*10200 = kg. Thus the integral I is obtained as I = = b Slide No 51
52 Resonance Escape Probability (8) Since neutrons absorbed by resonance capture in uranium238 are lost and unable to take part in sustaining the fission chain, most thermal reactors are design to maximize the resonance escape probability as far as possible In a homogeneous mixture of natural uranium fuel and carbon graphite moderator the highest value of k is hence a fission chain can not possibly be sustained Heterogeneous arrangement of the same materials can lead to k as high as 1.08 due to the increase in the resonance escape probability Slide No 52
53 Thermal Utilization Factor (1) Once thermalized, the neutrons continue to diffuse throughout the reactor and are subject to absorption by other materials in the reactor as well as the fuel The thermal utilization factor f is defined as the ratio of the number of thermal neutrons absorbed in the fuel to the number of thermal neutrons absorbed in all reactor material: f = Number of thermal neutrons absorbed in the fuel Number of thermal neutrons absorbed in all reactor materials Slide No 53
54 Thermal Utilization Factor (2) The thermal utilization factor can be expressed as follows: f = Σ U a U φ V U + Σ m a U U Σa φ V m m φ V + Σ c a U c φ V c + Σ p a p φ V p where superscripts U, m, p and c refer to uranium, moderator, poison and construction material (clad, spacers, etc), respectively. In a heterogeneous reactor the flux will be different in the fuel region than in the moderator region due to the high absorption rate by the fuel Slide No 54
55 Thermal Utilization Factor (3) In the homogenous reactor the neutron flux seen by the fuel, moderator, poisons and the construction material will be the same and the equation for f can be rewritten as f = Σ U a + Σ m a U Σa + Σ p a + Σ c a Slide No 55
56 Thermal Utilization Factor (4) The coefficient f will not in general depend on the temperature However, in heterogeneous water moderated reactors the moderator (water) expands with temperature and number of moderator atoms will decrease and this results in increase of thermal utilization Because of this effect the temperature coefficient for the thermal utilization factor is positive Slide No 56
57 Reproduction Factor (1) Most of the neutrons absorbed in the fuel cause fission, but some do not The reproduction factor is defined as the ratio of the fast neutrons produced by thermal fission to the number of thermal neutrons absorbed in the fuel η = Number of fast neutrons produced by thermal fission Number of thermal neutrons absorbed in the fuel Slide No 57
58 Reproduction Factor (2) The reproduction factor can also be stated as a ratio of rates as shown below: The rate of production of fast neutrons by thermal fission = U U fission reaction rate ( Σ f φ ) Χ the average number of neutrons produced per fission ( ν ) U U The rate of absorption of thermal neutrons by the fuel ( Σ φ ) When fuel contains several fissionable materials, it is necessary to account for each material, e.g. η U U Σ f φ ν Σ η = = ν U Σ φ Σ U a N σ ν U f U a U 235 U 235 f = U 235 U 235 U 238 U 238 N σ a + N σ a a Slide No 58
59 Effective Multiplication Factor (1) The infinite multiplication factor can fully represent only a reactor that is infinitely large To completely describe the neutron life cycle in a real, finite reactor, it is necessary to account for neutrons that leak out The multiplication factor that takes leakage into account is the effective multiplication factor k eff Slide No 59
60 Effective Multiplication Factor (2) For critical reactor the neutron population is neither increasing nor decreasing and k eff = 1 If the neutron production is grater than the absorption and leakage, the reactor is called supercritical; k eff > 1 If the neutron production is less than the absorption and leakage, the reactor is called subcritical; k eff < 1 k eff = k x P FNL x P TNL, where P FNL is the fast nonleakage probability and P FNL is the thermal nonleakage probability Slide No 60
61 Fast NonLeakage Probability (1) In a realistic reactor of finite size some of the fast neutrons leak out of the boundaries of the reactor core before they begin the slowing down process The fast nonleakage probability P FNL is defined as the ratio of the number of fast neutrons that do not leak from the reactor to the number of fast neutrons produced by all fissions P FNL = Number of fast neutrons that do not leak from reactor Number of fast neutrons produced by all fissions Slide No 61
62 Thermal NonLeakage Probability (1) Neutrons can also leak out of a finite reactor after they reach thermal energies The thermal nonleakage probability is defined as the ratio of the number of thermal neutrons that do not leak from the reactor core to the number of neutrons that reach thermal energies P TNL = Number of thermal neutrons that do not leak from reactor Number of neutrons that reach the thermal energies Slide No 62
63 Six Factor Formula (1) k eff = k x P FNL x P TNL Inclusion of expression for k (fourfactor formula) yields k eff = ε P p P f η FNL TNL Slide No 63
64 40 neutrons net increase Fast fission Six Factor Formula (2) N0 ε = 1040 Fast nonleakage 140 fast neutrons leak N0 εp FNL = 900 Resonance escape 180 neutrons absorbed ε = 1.04 N 0 =1000 neutrons at start of generation N0 εpfnl pptnl fη = 1000 P FNL = neutrons absorbed in nonfuel p = 0.80 N0 εp FNL p = 720 reproduction η = neutrons net increase N0 εpfnl pptnl f = 495 Thermal utilization f = N0 εp FNL pp TNL = 620 Thermal nonleakage P TNL = Slide No neutrons leak
65 Neutron Life Cycle of a Fast Reactor (1) Neutron life cycle in a fast reactor is markedly different than that for a thermal reactor In a fast reactor, care is taken during the reactor design to minimize thermalization of neutrons Virtually all fissions taking place in fast reactor are caused by fast neutrons Resonance escape probability is not significant since very few neutrons exist at energies where it is significant Slide No 65
66 Exercises (1) 10:1510:30  Divide into groups of 45 students and solve the following problem: Exercise 1: The microscopic crosssection for the capture of thermal neutrons by hydrogen is 0.33 b and for oxygen b. Calculate the macroscopic capture cross section of the water molecule for thermal neutrons assuming that water density is 1000 kg/m 3 Hint: use the formula for compounds: 3 10 ρn Σ = A M ( ν σ + ν σ +L) Slide No 66
67 Exercises (2) 10:3010:45  solve the following problem: Exercise 2: Disregarding the uranium234, the natural uranium may be taken to be a homogeneous mixture of weight percent of uranium238 (absorption cross section 2.7 b) and 0.72 weight percent of uranium235 (absorption cross section 681 b). The density of natural uranium metal is 19.0 x 10 3 kg m 3. Determine the total macroscopic and microscopic absorption cross sections of this material Hint: first find mass of U235 and U238 per unit volume of mixture and then number of nuclei per cubic meter of U235 and U238 Slide No 67
68 Exercises (3) 10:4511:00 solve the following problem: Exercise 3: Calculate the thermal utilization factor for a homogenized core composed of (in % by volume): UO 2 35% and H 2 O 65%. The enrichment of the fuel is 3.2% (by weight). Microscopic cross sections [b] for absorption are as follows: water 0.66 [b], oxygen O: 2x104 [b], U 235: 681 [b], U238: 2.7 [b]. Density of UO 2 : kg/m 3 Density of water: 800 kg/m 3 Slide No 68
69 Exercises (4) Exercise 4: Calculate the moderating power and the moderating ratio for H 2 O (density 1000 kg/m 3 ) and Carbon (density 1600 kg/m 3 ). The macroscopic cross sections are given below: Isotope microscopic cross sections [b] absorption scattering Hydrogen Oxygen 27x Carbon Slide No 69
70 Exercises (5) Exercise 5: Calculate the resonance escape probability for a reactor as in Exercise 3 assuming the fuel temperature T = 1500 K and the effective resonance integral for fuel at T = 300 K equal to 25 [b]. Microscopic cross sections for scattering are as follows: water 103 [b], oxygen O: 6 [b], U235: 8 [b], U238: 8.3 [b]. Slide No 70
71 Home Assignment #1 due Description and data for problems 1 and 2 A homogenized core has the following composition (in % by volume): UO 232%, Zr 10%, H 2 O 58%. The enrichment of the fuel is 3.5% (weight). The material data are given in the Table below. Component density σ a σ kg m 3 barn barn barn f σ s ν H 2 O Zr UO U U O  2x Slide No 71
72 Home Assignment #1 due Problem 1 (5 points): Calculate the thermal utilization factor in such a core. Problem 2 (5 points): Calculate k for an identical core as in Problem 1 assuming that the resonance escape probability p is known and equal to 0.69 and the fast fission factor is 1.04 Slide No 72
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