Reactor Kinetics Introduction

Operator Generic Fundamentals Reactor Theory - Reactor Kinetics/Neutron Sources

Reactor Kinetics Introduction
This module introduces the student to subcritical reactor operating characteristics. Topics covered are: Source neutrons Subcritical multiplication SUR/Rx period Prompt Jump/Drop/Criticality Related KAs REACTOR THEORY: Reactor Kinetics and Neutron Sources K1.01 Explain the concept of subcritical multiplication K1.02 Given the simplified formula for subcritical multiplication, perform calculations involving steady state count rate and source count rate. 2.2, 2.3 K1.03 Describe the production of delayed neutrons K1.04 Define delayed neutron fraction and effective delayed neutron fraction: state the reasons for variation. 2.4, 2.4 K1.05 Define start-up rate K1.06 Describe the factors affecting start-up rate K1.07 Explain the effect of delayed neutrons on reactor control K1.08 Explain the prompt critical, prompt jump, and prompt drop K1.09 Given the power equation, solve problems for power changes K1.10 Define doubling time and calculate it using the power equation. 1.6* 1.6* K1.11 Explain the necessity for installed neutron sources in a reactor core KA to Objective Tie on last slide. INTRO

Terminal Learning Objectives
At the completion of this training session, the trainee will demonstrate mastery of this topic by passing a written exam with a grade of ≥ 80% score on the following TLOs: Describe intrinsic and installed neutron sources and their contribution to source neutron strength over core life. Describe subcritical multiplication for a nuclear reactor and describe how subcritical multiplication affects reactor operation. Explain the factors that affect reactor period and start-up rate as well as their effect on reactor control. TLOs

Source Neutrons TLO 1 – Describe intrinsic and installed neutron sources and their contribution to source neutron strength over core life. 1.1 Describe the purpose and importance of source neutrons. 1.2 Describe, including examples, each of the following types of intrinsic neutron sources: Spontaneous fission Photo-neutron reactions Alpha-neutron reactions 1.3 Describe the primary source of intrinsic source neutrons in the reactor for the following conditions: At beginning and end of core life Immediately following a reactor shutdown Several weeks after reactor shutdown 1.4 Describe the purpose and type of installed neutron sources. Related KA K1.11 Explain the necessity for installed neutron sources in a reactor core. 2.7/2.8 TLO 1

Source Neutrons ELO 1.1 – Describe the purpose and importance of source neutrons. Source neutrons are classified as either: Intrinsic Installed Source neutrons Make up for losses with Keff < 1 (subcritical multiplication) Important during reactor shutdown and startup conditions: To confirm instrument operability Monitoring of the reactor’s neutron population changes Ensure neutron population during shutdown conditions remains high enough for indication on source range nuclear instrumentation Subcritical multiplication is discussed in later slides. Consider the six-factor discussion in previous topic: When the reactor is shutdown, Keff <1. Let’s say Keff is That means every generation 5% of the neutrons are lost due to the six factors. If 1000 neutrons start a generation, 950 end that generation. If neutron population (Counts Per Second on Source Range detectors) is constant, then source neutrons must be making up for those losses. Therefore, at the end of a generation 50 source neutrons are added to the 950 fission neutrons, bringing neutron population back up to 1000 to start the next generation. Obviously this isn’t exactly how it is happening, but this helps explain the concept of source neutrons in maintaining a constant neutron population in a shutdown reactor. ELO 1.1

Source Neutrons Knowledge Check
Source neutrons are important because they: Extend the neutron lifetime allowing for a nuclear chain reaction to occur. Allow for visible indication of neutron level in a shutdown nuclear reactor. Shorten the neutron generation time allowing for operational control of a nuclear reactor. Contribute a larger percentage of the thermal neutron population than the fast neutron population in an operating nuclear reactor. Correct answer is B. Correct Answer: B. Allow for visible indication of neutron level in a shutdown nuclear reactor. ELO 1.1

Intrinsic Source Neutrons
ELO Describe the following types of intrinsic neutron sources: spontaneous fission, photo-neutron reactions, and alpha-neutron reactions. Source Neutrons are always in core Insignificant when operating Important when shutdown Various types of intrinsic source neutrons Related KA K1.09 Describe Sources of neutrons 2.3/2.4 ELO 1.2

Intrinsic Neutron Sources
Spontaneous fission U-238 U-235 Pu-239 Cu-242, Cu-244 Photo-neutron interactions High energy gamma ray and a deuterium nucleus Alpha-Neutron interactions O-18 ELO 1.2

Spontaneous Fission Prior to a new core reactor startup
U-238 significant contributor to the source neutron population After plant operating history increases, the following contribute: curium-242 curium-244 As an example, one ton of spent nuclear fuel will contain on the order of 20 grams of curium Curium is a Transuranic element (above uranium in Chart of Nuclides). Transuranics are produced after multiple neutron absorptions and beta-minus decays beginning with U These decay schemes are not required knowledge. ELO 1.2

Photo-Neutron Reactions
In a reactor that has been operated at power, source neutrons from photo-neutron reactions become significant Hydrogen in the moderator on occasion absorbs a neutron Becomes deuterium Deuterium subjected to high energy gamma (at least 2.2 Mev) results in the following: 𝛾+ 2 1 𝐻→ 1 1 𝐻+ 1 0 𝑛 Referred to as a photo-neutron reaction because it is initiated by electromagnetic (gamma photon) radiation and results in the production of a neutron. ELO 1.2

Photo-Neutron Reactions
Deuterium (H-2) produced from hydrogen (H-1) absorbing a neutron Recall moderator has “small” cross-section for capture High-energy gammas (> 2.22 MeV) produced by fissions Based on binding energy of deuterium Interaction : 𝛾+ 2 1 𝐻→ 1 1 𝐻+ 1 0 𝑛 Photo-neutron production rate decreases after shutdown Fission products decay over time Smaller fission products, lower energy gammas This source decreases quickest after shutdown (2-3 weeks) Concept tested in NRC bank After the reactor has operated for a short time there is an abundant supply of high-energy gammas (2.22 MeV or greater) Moderator/coolant has some deuterium present because the naturally occurring atom percentage of deuterium is percent ELO 1.2

Alpha-Neutron Reactions
Alpha particles from decay of heavy elements Alpha particles from capture of neutron by B-10 in moderator Alpha + Lithium Alphas interact with: O-18 (Uranium dioxide fuel) 4 2 𝐻𝑒 𝑂→ 𝑁𝑒+ 1 0 𝑛 Interactions between alpha particles and various isotopes in the reactor core also result in the production of source neutrons. There are some transuranic alpha-neutrons produced, but not tested by NRC: Curium and americium isotopes are the major transuranic elements of interest for source neutrons Curium-242 production and neutron: 𝑃𝑢+ 4 2 𝐻𝑒→ 𝐶𝑚+ 1 0 𝑛 Alpha production (163 day half-life): 𝐶𝑚→ 𝑃𝑢+ 4 2 𝐻𝑒 In typical nuclear reactor core, transuranic neutron sources produce About 100 neutrons per second for every gram of fuel in the core Approximately 1 x 107 neutrons per second core wide ELO 1.2

Intrinsic Neutron Sources
Knowledge Check Which one of the following is NOT a source of source neutrons in a shutdown core? Photo-neutron Alpha-Neutron Spontaneous fission Oxygen-16 neutron Correct answer is D. Correct answer D. ELO 1.2

Intrinsic Source Neutrons over Core Life
ELO Describe the primary source of intrinsic source neutrons in the reactor for the following conditions; at beginning and end of core life: immediately following a reactor shutdown and several weeks after reactor shutdown. Installed neutron sources provide a steady source of neutrons throughout core life Over long periods their strength does decay Intrinsic neutrons, however, do change in importance over core life Related KA K1.09 Describe Sources of neutrons 2.3/2.4 The ONLY concept tested by the NRC with regards to this ELO is the change in source neutrons with respect to time after shutdown. Nothing related to BOL vs EOL. ELO 1.3

Photo-Neutron Strength over Core Life
BOL – minor contributor to source neutrons No high energy gammas no fission products present in the core Deuterium concentration is low After some power history: Largest contributor to source neutrons when shutdown Exists for several days following shutdown from power operations Also drops off the quickest after shutdown (2-3 weeks) Only real difference between BOL and EOL (after shutdown): Spontaneous fission large source at BOL due to less alphas and gammas ELO 1.3

Intrinsic Neutron Source Strength - BOL
Immediately following reactor shutdown from power: Photo-neutron sources Spontaneous fission sources Alpha-neutron (transuranic) sources Several weeks following reactor shutdown from power: NOTE: Current bank question(s) test EOL concept on next slide! Strongest to weakest EMPHASIZE: This slide is NOT what is tested by the NRC bank! Current bank question(s) test EOL concept on next slide! ELO 1.3

Intrinsic Neutron Source Strength - EOL
Immediately following reactor shutdown from power: Photo-neutron sources Alpha-neutron (transuranic) sources Spontaneous fission sources Several weeks following reactor shutdown from power: Strongest to weakest Immediately following a trip/shutdown – PAS 2-3 weeks following a trip/shutdown - ASP ELO 1.3

Intrinsic Source Neutrons over Core Life
Knowledge Check Which one of the following neutron reactions yields the highest neutron production rate immediately following a reactor trip from extended power operations during the tenth fuel cycle? (Ignore any contribution from an installed neutron source.) Alpha-neutron reactions Beta-neutron reactions Photo-neutron reactions Spontaneous fission Correct answer is C. Correct Answer is C: NRC GFE Bank Question - P1249 ELO 1.3

Installed Neutron Sources
ELO Describe, the purpose and types of installed neutron sources. Primary (initial startup) Californium-252 Secondary Antimony-Berylium (most common type used) Installed prior to initial startup, or following extended shutdown After operating cycle, intrinsic sources provide high enough flux Installed near source range nuclear detectors Provides for good source range indication for startup Related KA K1.09 Describe Sources of neutrons 2.3/2.4; K1.11 Explain the necessity for installed neutron sources in a reactor core ELO 1.4

Primary - Installed Source Neutrons
Californium-252 Installed near source range detector Installed during initial startup Provide indication of source counts during startup Once operating, intrinsic source neutron strength sufficient ELO 1.4

Secondary - Installed Source Neutrons
Antimony-Beryllium Most common secondary installed source Antimony activated by absorbing neutron 𝑆𝑏+ 1 0 𝑛→ 𝑆𝑏+𝛾 Radioactive antimony decays to Tellurium then emits high-energy gamma (60-day half-life) 𝑆𝑏 𝛽 − 𝑇𝑒+ 0 −1 𝑒+𝛾 Gamma ray has sufficient energy to interact with the beryllium to produce a neutron 𝛾+ 9 4 𝐵𝑒→ 8 4 𝐵𝑒+ 1 0 𝑛 Note that even though this neutron source requires a neutron to start the process, the loss of these neutrons while at power are inconsequential. However, based on the half-life and production of neutrons while shutdown, they are important. ELO 1.4

Secondary - Installed Source Neutrons
Photo-Neutron Beryllium A gamma ray > 1.66 MeV can cause neutrons to be ejected by the photo-neutron reaction 𝛾+ 9 4 𝐵𝑒→ 8 4 𝐵𝑒+ 1 0 𝑛 Alpha-Neutron Beryllium Metallic beryllium with an alpha particle emitter, such as radium, polonium, or plutonium 9 4 𝐵𝑒+ 4 2 𝛼→ 𝐶 ∗ → 𝐶+ 1 0 𝑛 The various types of installed neutron sources are only tested to the extent that they – PROVIDE INDICATION, not where they come from or the chemical interactions. ELO 1.4

Installed Neutron Sources
Knowledge Check Which one of the following describes the purpose of a neutron source that is installed in a reactor during refueling for the third fuel cycle? Ensures shutdown neutron level is large enough to be detected by nuclear instrumentation Provides additional excess reactivity to increase the length of the fuel cycle Amplifies the electrical noise fluctuations observed in source range instrumentation during shutdown Supplies the only shutdown source of neutrons available to begin a reactor startup Correct answer is A. Correct Answer: A. NRC Bank Question – P3149 ANALYSIS: Installed neutron sources are designed to produce a certain number of neutrons per unit time to provide on-scale readings of source range monitors (during shutdown conditions, especially after long outages when the intrinsic neutron sources have decayed away). The most common type is Antimony-Beryllium. ELO 1.4

Subcritical Multiplication
TLO 2 – Describe subcritical multiplication for a nuclear reactor and describe how subcritical multiplication affects reactor operation. 2.1 Explain the following: Subcritical multiplication Subcritical multiplication factor Subcritical multiplication response on a nuclear reactor startup. 2.2 For a subcritical reactor, calculate steady state neutron levels for various values of keff and reactivity additions. 2.3 State the rules of thumb for changing neutron count rate during a reactor startup. It is important for the reactor operator to have a clear understanding of reactor kinetics and how it relates to control of the reactor. TLO 2

ELO 2.1 – Explain the following: subcritical multiplication, subcritical multiplication factor, and subcritical multiplication response on a nuclear reactor startup. Subcritical multiplication is the process by which source neutrons make up for the losses when Keff < 1 to maintain a constant neutron population. If keff = 0.95, then 5% are source neutrons If keff = 0.8, then 20% are source neutrons If keff = 0, then 100% are source neutrons Related KAs K1.01 Explain the concept of subcritical multiplication (2.7, 2.8) The chain reaction in a shutdown reactor is not self-sustaining. Neutron production is less than absorption plus leakage Source neutrons ADD to the neutron generation to make up for the neutron losses. Neutron population equalizes after a few neutron generations – with keff <1. ELO 2.1

As keff approaches criticality (1.0), neutron count rate increases from subcritical multiplication Subcritical multiplication must be related to values of keff 𝐶𝑅= 𝑆 𝑜 1 1− 𝑘 𝑒𝑓𝑓 𝜂 Where: CR = neutron count rate (source range nuclear instrumentation) 𝑆 𝑜 = source strength 𝑘 𝑒𝑓𝑓 = effective neutron multiplication factor 𝜂 = detector efficiency When performing a reactor startup and determining the change in Keff or counts, it is assumed that the detector efficiency (n) and the source strength (So) do not change. We will see a derivation of this equation in a later slide (also on NRC Equation Sheet) ELO 2.1

Solving for count rate (CR) assuming: 𝜂 = 0.1 S0 = 100 cps keff = 0.9 CR would = __ cps? 𝐶𝑅= − 𝐶𝑅=100 If Keff increases to 0.95 (no other changes), then CR would = __ cps? New CR 𝐶𝑅= − 𝐶𝑅=200 Total neutron population at keff of 0.90 is 1000, but with the efficiency of 0.1, CPS are 100. Total neutron population at keff of 0.95 is 2000, but with the efficiency of 0.1, CPS are 200. ELO 2.1

Subcritical Multiplication Factor (M)
Factor by which source neutrons are multiplied by to get neutron population Subcritical Multiplication Factor formula: 𝑀= 1 1− 𝑘 𝑒𝑓𝑓 As keff approaches 1.0, M approaches infinity Recall previous count rate formula: 𝐶𝑅= 𝑆 𝑜 1 1− 𝑘 𝑒𝑓𝑓 𝜂 Since M = 1/1-keff, substituting M results in: 𝐶𝑅= 𝑆 𝑜 𝑀𝜂 ELO 2.1

Subcritical Multiplication Factor (M)
Based on the previous formulas: As keff approaches 1 M increases As M increases Count rate (or total neutron population) increases The relationship between keff and M can also be shown as: 𝑘 𝑒𝑓𝑓 =1− 1 𝑀 This formula is tested in a couple of bank questions in – Rx Operational Physics. This formula is NOT provided on the NRC Equation Sheet! ELO 2.1

Subcritical Multiplication Response to keff
Count rate comparisons (ratio) are more useful at determining the reactor response to reactivity changes Count rate ratio is a comparison of two count rates (final count rate divided by initial count rate), and can be expressed as: 𝐶𝑅 2 𝐶𝑅 1 = 1− 𝑘 𝑒𝑓𝑓1 1− 𝑘 𝑒𝑓𝑓2 Where, CR1 = initial count rate at reference time CR2 = count rate at some time later after the addition of positive reactivity keff1 = keff at initial count rate (CR1) keff2 = keff at count rate after the addition of positive reactivity As mentioned earlier, when performing a reactor startup and determining the change in Keff or counts, it is assumed that the detector efficiency (n) and the source strength (So) do not change. NOTE: This equation is on the NRC Equation Sheet There are also several questions in – Reactor Operational Physics that also use this equation. ELO 2.1

Recall the relationship of keff to reactivity (𝜌): 𝑘 𝑒𝑓𝑓 = 1 1−𝜌 Substituting this into CR ratio equation: 𝐶𝑅 2 𝐶𝑅 1 = 𝜌 1 (1− 𝜌 2 ) 𝜌 2 (1− 𝜌 1 ) If keff  1.0 and 1-  1.0, then: 𝐶𝑅 2 𝐶𝑅 1 ≈ 𝜌 1 𝜌 2 What this final equation states is that when the reactor is close to criticality there is a relationship between the final counts and reactivity in the core to the initial counts and initial reactivity in the core. FOR EXAMPLE: If the reactor is shutdown by -100 pcm and counts are 500, when the reactor is shutdown by -50 pcm the counts are approximately (-100 x 500) = (-50 x 1000) ELO 2.1

As keff approaches 1.0 Greater number of generations is required to reach equilibrium Longer time to reach equilibrium counts Reactor more and more dependent on delayed neutrons Greater change in counts for given reactivity addition Greater initial change in neutron population Called “prompt jump” (explained later) “Longer time to reach equilibrium counts” example: Assume that the reactor contains 10,000 neutrons when Keff = neutrons will be source neutrons and 9000 will be fission neutrons. Assume that the effective delayed neutron fraction is .006 (MOL). Then 54 of the neutrons will be delayed. When Keff= .9, the source neutrons way outnumber the delayed neutrons and the reactor appears to reach equilibrium without waiting for the delayed neutrons. Now keep the source strength the same and make Keff = The total number of neutrons at equilibrium is 1 E7. Source neutrons still equals 1000, but delayed neutrons equals 59,994. Now the delayed neutrons way outnumber the source neutrons and now the neutron multiplication has to wait on the delayed neutrons and their relatively long lifetime. ELO 2.1

As keff gets closer to 1.0, count rate increases significantly more per rod pull and time to reach equilibrium count rate increases Time to equilibrium examples: Keff = The time to reach equilibrium is dominated by GROUP 2's mean life. GROUP 2's mean life is approximately 32 seconds, so 5 mean lives would be 160 seconds or 2 minutes and 40 seconds. Keff = At that value of Keff, the subcritical multiplication has to wait on the longest lived delayed neutron precursor group (GROUP 1). Since GROUP 1's mean life is approximately 80 seconds, and using the standard 5 mean lives to reach equilibrium, it takes 400 seconds or 6 minutes and 40 seconds to reach equilibrium. Consequently, the time to reach equilibrium is becomes more and more dependent upon the LONGER LIVED DELAYED NEUTRON PRECURSOR GROUPS as Keff approaches 1.0! Figure: Startup Trace Shows Time Versus Count Rate Increase ELO 2.1

Solving Subcritical Multiplication Problems
ELO 2.2 – For a subcritical reactor, calculate steady state neutron levels for various values of keff and reactivity additions. As criticality approached, a smaller amount of control rod withdrawal required to obtain ever increasing neutron CR change Due to increased effect of subcritical multiplication as keff approaches one 𝐶𝑅 2 𝐶𝑅 1 = 1− 𝑘 𝑒𝑓𝑓1 1− 𝑘 𝑒𝑓𝑓2 Relater KA - K1.02 Given the simplified formula for subcritical multiplication, perform calculations involving steady state count rate and source count rate. 2.2, 2.3 ELO 2.2

If keff changes from .95 to .96 and CR1 =1000 cps, what will CR2 be? 𝐶𝑅 2 𝐶𝑅 1 = 1− 𝑘 𝑒𝑓𝑓1 1− 𝑘 𝑒𝑓𝑓2 𝐶𝑅 2 1,000 = (1−0.95) (1−0.96) 𝐶𝑅 2 =1,250 𝑐𝑝𝑠 That’s a change of 250 cps, based on change in keff ELO 2.2

Knowledge Check A reactor startup is in progress. The initial count rate was 120 cps. After the first rod pull, the count rate changed to 150 cps. On the fifth rod pull, the count rate changed to 3000 cps. Assuming the initial keff was 0.9, what is the keff after the fifth rod pull? 0.995 0.996 0.92 0.9996 Correct answer is B. Correct answer is B. Show this example on the board using actual numbers: CR1 (1-Keff1) = CR2 (1-Keff2) 120 (1-.9) = 3000 (1-Keff2) Keff2 = .996 CR2/CR1 = 25, solving for Keff: Keff = (1 - .9)/25) = .004 Keff = = .996 ELO 2.2

Subcritical Reactor Rules of Thumb
ELO 2.3 – State the rules of thumb for changing neutron count rate during a reactor startup. A rule of thumb is a principle with broad application, not intended to be perfectly accurate or 100% reliable For a shutdown reactor, a number of rules of thumb are available for changes in reactivity, count rate, and keff These rules may be useful during reactor startups to verify reactor response to changes in reactivity additions Not related to any KAs ELO 2.3

Subcritical Reactor Rules of Thumb
Rule of Thumb #1 Doubling count rate (subcritical reactor) implies that enough reactivity was added to make reactor half way to criticality (keff is halfway to 1.0) Rule of Thumb #2 If enough reactivity was added to double count rate and the same amount of reactivity is added to reactor again, reactor will be supercritical Rule of Thumb # 3: With each doubling, distance to criticality is halved To again double count rate, add again half of original A less often used thumb rule related to #1 above is the following: If you add reactivity to a SD reactor and counts increase by 50%. Adding the same amount of reactivity will cause counts to double! ELO 2.3

Knowledge Check During a nuclear reactor startup, the operator adds 1.0 %Δk/k of positive reactivity by withdrawing control rods, thereby increasing equilibrium source range neutron level from 220 cps to 440 cps. To raise equilibrium source range neutron level to 880 cps, an additional ______________ of positive reactivity must be added. 4.0 percent Δk/k 2.0 percent Δk/k 1.0 percent Δk/k 0.5 percent Δk/k Correct answer is D. Correct answer is D. This is actually bank question P266 from – Rx Operational Physics There are a few different ways to answer this question: Thumbrules – When counts double you are halfway to criticality. It now takes half as much reactivity for counts to double again (since you are only shutdown by half as much) Draw a scale showing Keff = 1 and a line lower (Keff < 1). Place 200 cps on that line. Draw a line halfway between to show a doubling of counts (440 cps). If that amount added was 1.0 %Δk/k , then you were originally shutdown by 2.0%Δk/k, and by the drawing only 0.5 %Δk/k is required this time for counts to double again (880 cps). Calculation. If you added 1.0 %Δk/k to get counts to double, you were shutdown by -2.0 %Δk/k. That calculates out to a Keff of 0.98 (shutdown by 2%). When counts double to 440 you are shutdown by -1% (added 1%). To get counts again, you would need to only be shutdown by -0.5% (or add 0.5%). You can use the CR and Keff formulas to answer via calculation. ELO 2.3

Reactor Period and Startup Rate
TLO 3 – Explain the factors that affect reactor period and their effect on reactor control. 3.1 Describe the relationship between the delayed neutron fraction, average delayed neutron fraction, and average effective delayed neutron fraction and the importance delayed neutrons have on reactor control. 3.2 Describe the following equations and associated terms for: Reactor period Reactor startup rate 3.3 Given necessary reactivity variables, calculate the SUR or reactor period and other variables in the power equations. 3.4 Describe prompt critical, prompt jump, prompt drop and how reactor power is affected by a reactor trip and stepped insertion of reactivity. TLO 3

Delayed Neutrons ELO 3.1 – Describe the relationship between the delayed neutron fraction, average delayed neutron fraction, and average effective delayed neutron fraction and the importance delayed neutrons have on reactor control. Delayed neutrons play a very important role in control and stability of commercial nuclear reactors Increase neutron generation times Reactivity additions may be made without reactor power increasing at an uncontrollable rate Related KAs - K1.03 Describe the production of delayed neutrons ; K1.04 Define delayed neutron fraction and effective delayed neutron fraction: state the reasons for variation. 2.4, 2.4; K1.07 Explain the effect of delayed neutrons on reactor control Before we can look at Rx period and SUR equations, we need to revisit delayed neutrons. If you recall, it takes on average 12.7 seconds for delayed neutrons to appear (some faster, some slower). We will see that this slows down the “average” neutron generation time (considering the prompt neutron generation time was 10-4 seconds). ELO 3.1

Delayed Neutrons Delayed neutron fraction (β)
Delayed neutrons are produced from fission product daughters (following beta decay) Delayed neutron precursor half-lives vary from less than a second to about a minute after the initiating fission event Fraction of all neutrons born (prompt and delayed) as delayed neutrons for fuel isotope is called delayed neutron fraction (β) ELO 3.1

Delayed Neutron Fraction
Fraction of all neutrons born delayed for a given isotope (sometimes shown as βI, for isotope, or β235) Delayed/Prompt + Delayed Delayed neutron fractions (β) for fissile or fissionable nuclides of most interest are as follows: U-235 β = U-238 β = Pu-239 β = Pu-241 β = Majority (≈99.36%) of neutrons produced are prompt, with remaining % delayed Delayed neutrons are produced from fission product daughters (following beta decay) Delayed neutrons are “born” from a few milliseconds to about a minute after initiating fission event ELO 3.1

Average Delayed Neutron Fraction
𝜷 (beta bar) Fraction of all neutrons born delayed for ALL isotopes in the core (sometimes called βcore) Function of individual isotopes β and their fission yield For example: Clean core (1st Cycle - BOL) U %, U-238 7% ≈ 50,000 MWD/MTU (near EOL) U %; U-238 8%; Pu % Pu-241 5% As fuel mixture changes over core life, ie. plutonium-239 production from uranium-238, average delayed neutron fraction also changes. As a reminder, the DELAYED NEUTRON FRACTION does not change, since it is for a given isotope. Each atom of U-235 ALWAYS produces about delayed neutrons. The AVERAGE DELAYED NEUTRON FRACTION decreases over core life based on changes in fuel composition. The Pu-241 delayed neutron fraction slows the decrease in the average delayed neutron fraction over core life due to more fissions occurring in Pu-239 with its unusually small delayed neutron fraction. Also, since the flux in the core increases over core life, therefore, the “FAST flux increases, increasing the fission yield from U-238 (which has a large delayed neutron fraction). The interesting point is that by EOC, the majority of fissions are from atoms that started out as U-238, not U-235. NET EFFECT (as tested by the NRC) – Average Delayed Neutron Fraction decreases over core life. ELO 3.1

Average Delayed Neutron Fraction
Knowledge Check Based on the data below, what is the core delayed neutron fraction for this reactor: 0.0052 0.0054 0.0062 0.0068 Nuclide Delayed Neutron Fraction Fraction of Total Fuel Composition Fraction of Total Fission Rate U-235 0.0065 0.03 0.73 U-238 0.0148 0.96 0.07 Pu-239 0.0021 0.01 0.20 Correct answer is C. Correct answer is C. NRC Bank Question – P4425 Analysis: NOTE: The fraction of total fuel composition is not needed for this solution. It is merely a distractor. To solve this, multiply the delayed neutron fraction of each isotope by its fraction of fission rate, then add each product to determine the core delayed neutron fraction: (0.0065) x (0.73) + (0.0148) x (0.07) + (0.0021) x (0.20) = ELO 3.1

Average Effective Delayed Neutron Fraction
𝜷 eff (beta bar effective) Fraction of fissions induced (or caused) by delayed neutrons over all fissions Delayed fissions/prompt and delayed fissions Measure of effectiveness of a delayed neutron causing another fission Mathematically: where, “I” is the importance factor, or ratio of the effect that delayed and prompt neutrons have on keff It is very important to understand that this relationship compares one prompt neutron to one delayed neutron. It has NOTHING to do with the high percentage of prompt neutrons in the core versus the low percentage of delayed neutrons. This all goes back to Chapter 1 and the differences between prompt and delayed neutrons. ELO 3.1

Average Effective Delayed Neutron Fraction
𝜷 eff (beta bar effective) Two reasons why effective delayed neutron fraction is different from delayed neutron fraction Delayed neutrons are born at 0.5 MeV, versus prompt neutrons at MeV can’t cause fast fission tends to decrease effective delayed neutron fraction Delayed neutrons are less likely to leak out of core than prompt neutrons tends to increase effective delayed neutron fraction Basically, a delayed neutron is less likely than a prompt neutron at causing another fission. If necessary, review some of the questions in – Neutrons. What this means in that the value for Beta Bar Effective is slightly less than the value for Beta Bar! Now that we have reviewed differences between prompt and delayed neutrons, let’s look at why delayed neutrons are important to reactor control. ELO 3.1

Average (or Effective) Generation Time
Effect on Reactor Period Prompt neutron generation time = 10-4 seconds Delayed neutron generation time = 12.7 seconds If β is , then: Solution: Average Generation Time: =𝑇𝑖𝑚𝑒 𝑝𝑟𝑜𝑚𝑝𝑡 1−𝛽 +𝑇𝑖𝑚𝑒 𝑑𝑒𝑙𝑎𝑦𝑒𝑑 𝛽 = 1× 10 −4 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 =0.08 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 Much slower average generation time Delayed neutrons important for reactor control Average neutron generation time, rate power can rise in a nuclear reactor, is determined largely by delayed neutron generation time Mathematically: 𝑇𝑖𝑚𝑒 𝑎𝑣𝑒𝑟𝑎𝑔𝑒=𝑇𝑖𝑚𝑒 𝑃𝑟𝑜𝑚𝑝𝑡 1−𝛽 +𝑇𝑖𝑚𝑒 𝐷𝑒𝑙𝑎𝑦𝑒𝑑 𝛽 Although only a small fraction of total neutron population, delayed neutrons are extremely important ELO 3.1

Delayed Neutron Effect Over Core Life
U-235 concentration decreases, Pu-239 concentration increases β for uranium-235 is and plutonium-239 is Average effective delayed neutron fraction ( 𝜷 eff) decreases over core life Estimated values of 𝜷 eff : Cycle 1 only 0.007 (BOL), (EOL) From Cycle 3 on 0.006 (BOL), (EOL) Over core life, 𝜷 eff will decrease; therefore, for same amount of reactivity addition, reactor period will decrease and SUR will increase We will see in the future slides that the time it takes for to increase for a given amount of reactivity changes over core life due to the fact Beta-Bar-Effective decreases over core life. Amount of reactivity needed for a given reactor period decreases with decreasing value of βeff NOTE! Almost ALL NRC Bank questions relate to Beta-Bar-Effective values for Cycle 1. If a question is asked and Beta-Bar-Effective is not provided, assume Cycle 1 values (unless told otherwise). ELO 3.1

Delayed Neutrons Knowledge Check – NRC Bank
During a fuel cycle, plutonium isotopes are produced with delayed neutron fractions that are __________ than the delayed neutron fractions for uranium isotopes, thereby causing reactor power transients to be __________ near the end of a fuel cycle. larger; slower larger; faster smaller; slower smaller; faster Correct answer is D. The correct answer is: D. There is a larger fission yield from Pu-239 at the end of core life, since there are less delayed neutrons the reactor power changes faster because there are more prompt neutrons at the end of core life. NRC Bank Question P48 ELO 3.1

Delayed Neutrons Knowledge Check - NRC Bank
Delayed neutrons contribute more to reactor stability than prompt neutrons because they __________ the average neutron generation time and are born at a __________ kinetic energy. increase; lower increase; higher decrease; lower decrease; higher Correct answer is A. Correct answer is A. NRC Question P249 Analysis: While an overwhelming majority of fission neutrons are prompt neutrons (99.36% compared to 0.64% for delayed neutrons), the small percentage of delayed neutrons make the chain neutron reaction controllable due to the relatively long delayed neutron generation time (12.7 seconds for a delayed neutron compared to 10-4 seconds for a prompt neutron). In other words, delayed neutrons increase the average neutron generation time: (.9936 x 10-4) + (.0064 x 12.7) = .08 (sec) ELO 3.4

Reactor Period & Startup Rate Equations
ELO 3.2 – Describe the following equations and associated terms: reactor period and reactor startup rate. Reactor period Time (in seconds) required for reactor power to change by a factor of e Startup rate the rate of change of reactor power expressed in decades per minute (DPM) Equations provided on NRC Equation Sheet Related KA K1.05, Define Startup Rate (2.7/2.8) K1.06, Describe the factors affecting startup rate. (3.2/3.3) K1.10 Define doubling time and calculate it using the power equation. (1.6*/1.6*) NOTE: The Doubling Time equation is in the handout but not covered in the PPT because it makes no sense to memorize another equation when Powerfinal/Powerinitial = 2, is the same thing! Make sure the students have a copy of the NRC Equation Sheet. We will be introducing four main equations (calculations done in next section): Reactor Period Equation: Power Equation – Rx Period version: 𝑃= 𝑃 𝑜 𝑒 𝑡/𝜏 Power Equation – SUR version: 𝑃= 𝑃 𝑜 10 𝑆𝑈𝑅(𝑡) SUR vs Rx Period: SUR = 26.06/ ELO 3.2

Reactor Period Equation
Consists of “prompt term” and “delayed term”: Prompt term and reactivity addition rate not on NRC sheet 𝜏= Ɩ ∗ 𝜌 + 𝛽 𝑒𝑓𝑓 −𝜌 λ 𝑒𝑓𝑓 𝜌+ 𝜌 Prompt + Delayed Where: Ɩ * = prompt neutron generation time (≈ 10-4 seconds) βeff = effective delayed neutron fraction λ eff = effective delayed neutron precursor decay constant ρ = reactivity 𝜌 = reactivity addition rate (example, control rod movement) We will see that the amount of time added by the “prompt term” is so small that it isn’t really necessary. This is why it is not included on the version on the NRC Equation Sheet Also, the rate of reactivity addition by control rod movement is a lot less than available in the Navy (we don’t allow for fast scram recoveries)! Therefore, the NRC has also removed the rho-dot term from the equation. We will see, however, the impact that rho-dot has on SUR in Chapter We will also need to explain a new term - λ eff - effective delayed neutron precursor decay constant (next slide) Also note that there is a derivation of this formula that is solved for REACTIVITY! ELO 3.2

Reactor Period Equation
Reactor period formula considers: Prompt neutrons with a generation time of 10-4 seconds Delayed neutrons with effective generation time of 12.7 seconds λeff - adjusts reactor period by fraction of short and long lived delayed neutron precursors Explained on next slide Reactivity Rate of change of reactivity Delayed neutrons make reactor more controllable by increasing generation time ELO 3.2

Effective Delayed Neutron Precursor Decay Constant
Represents decay rate of particular delayed neutron precursor Symbol - 𝜆𝑒𝑓𝑓 Adjusts for balance of short-lived, medium-lived, and long-lived delayed neutron precursors 𝜆𝑒𝑓𝑓 is reciprocal of mean life () how long, on average, delayed neutron precursor will exist before decaying NRC Equation Sheet uses a 𝜆𝑒𝑓𝑓 value of .1 For small positive reactivity additions Navy had an acronym for 𝜆𝑒𝑓𝑓 .1 on the run (positive), .05 on the dive (negative), .08 steady state, on the floor (trip) This must be explained before doing any calculations. All other terms of reactor period equation have been presented already. These values are important in explaining how positive and negative reactivity additions effect time to change power. This concept will also help explain future NRC bank questions. ELO 3.2

Reactor Period Equation - Example
Given the reactor is operating at steady state power at BOL calculate the stable reactor period if .001 k/k is added to a critical reactor. Assume 𝜷 eff = .007 On NRC Equation Sheet: 𝜆𝑒𝑓𝑓 = 0.1 sec-1, Ɩ ∗ = 10-4 seconds Find : 𝜏= Ɩ ∗ 𝜌 + 𝛽 𝑒𝑓𝑓 −𝜌 λ 𝑒𝑓𝑓 𝜌 𝜏= 10 − − (.001) 𝜏=0.1 𝑠𝑒𝑐+60 𝑠𝑒𝑐 𝜏=60.1 𝑠𝑒𝑐 The “prompt” term only adds 0.1 sec This is why it is not included on the NRC Equation Sheet NOTE: The reactivity addition rate and prompt terms are not used on the NRC Equation Sheet for calculations. Show students actual formula on NRC Equation Sheet. ELO 3.2

Reactor Period Power Equation
Time (in seconds) for power to increase by factor of e (2.718) 𝑃= 𝑃 𝑜 𝑒 𝑡/𝜏 Where: P = Final reactor power Po = Initial reactor power τ = reactor period t = time (seconds) It is imperative to understanding how to do the Algebra to solve for Reactor Period in the exponent. Perform an example on the board, if necessary ELO 3.2

Startup Rate Power Equation
Startup rate (SUR) more commonly used at PWRs Defined as: the rate of change of reactor power expressed in decades per minute (DPM) Formula: 𝑃= 𝑃 𝑜 10 𝑆𝑈𝑅(𝑡) Where: P = Final reactor power Po = Initial reactor power SUR = startup rate t = time (minutes) It is imperative to understanding how to do the Algebra to solve for SUR in the exponent. Perform an example on the board, if necessary. ELO 3.2

Reactor Period/SUR Equations used to solve unknowns in reactor power changes 𝑡= log 𝑃 𝑃 𝑜 𝑆𝑈𝑅 𝑆𝑈𝑅= 𝜏 𝜏= 𝑡 ln 𝑃 𝑃 𝑜 𝑡=𝜏 ln 𝑃 𝑃 𝑜 𝑃= 𝑃 𝑜 𝑒 𝑡/𝜏 𝑃= 𝑃 𝑜 10 𝑆𝑈𝑅(𝑡) ELO 3.2

Reactor Period/Startup Rate Relationship
𝑃= 𝑃 𝑜 𝑒 𝑡/𝜏 and 𝑃= 𝑃 𝑜 10 𝑆𝑈𝑅(𝑡) Setting these two equations equal to each other: 𝑃 𝑜 𝑒 𝑡/𝜏 = 𝑃 𝑜 10 𝑆𝑈𝑅(𝑡) Working through the math: 𝑆𝑈𝑅= 𝜏 NOTE: the smaller the reactor period, the higher the SUR! This equation is also on the NRC Equation Sheet. There isn’t a need to do the math derivation, just understand that the unit conversion between seconds and minutes has already been done to come up with this equation. If necessary, perform the derivation on the board. ELO 3.2

Reactor Period & Startup Rate Equations
Knowledge Check A small amount of positive reactivity is added to a critical reactor in the source/startup range. The amount of reactivity added is much less than the effective delayed neutron fraction. Which one of the following will have a significant effect on the magnitude of the stable reactor period achieved for this reactivity addition? Moderator temperature coefficient Fuel temperature coefficient Prompt neutron lifetime Effective delayed neutron precursor decay constant Correct answer is D. Correct answer is D. NRC Question P47 Even though the top two terms have not been discussed yet (next chapter), it is still a good idea to remind the students that sometimes knowing the equations on the equation sheet might help them answer questions. For example, knowing the Reactor Period equation, the only terms listed in the equation are: Beta-Bar Effective (time in core life), reactivity, and Effective delayed neutron precursor decay constant. Therefore, Choice “D” is correct. ELO 3.2

Startup Rate & Reactor Period Calculations
ELO 3.3 – Given necessary reactivity variables, calculate the SUR or reactor period and other variables in the power equations. Using startup rate and reactor period equations is helpful for operators to determine power changes and rates of power change Step–by–Step Table 1. Determine the unknown variable and identify the applicable equation to solve for the unknown. 2. Solve for the unknown variable in the applicable equation. Related KAs K1.06, Describe the factors affecting startup rate. (3.2/3.3) K1.09, Given the power equation, solve problems for power changes. ((2.3/2.3) K1.10 Define doubling time and calculate it using the power ELO 3.3

Example 1 If we have a SUR of .2 DPM and we want to change power from 30% to 90% at a constant rate, how long will it take? 𝑡= log 𝑃 𝑃 𝑜 𝑆𝑈𝑅 𝑡= log 𝑡= log 𝑡= =2.39 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 Keep in mind that this is just an example and is not indicative of a power change. Reason is it states “at a constant rate”. This cannot happen in the power range because of the negative feedback of moderator and fuel temperatures adding negative reactivity. ELO 3.3

Example 2 With a constant reactor period of 100 seconds, how long will it take to change power from 7% to 100%? 𝑡=𝜏 ln 𝑃 𝑃 𝑜 𝑡=100 𝑠𝑒𝑐 ln 𝑡=100 𝑠𝑒𝑐 ln 𝑡=100 𝑠𝑒𝑐 2.66 =266 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑜𝑟 4 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 𝑎𝑛𝑑 26 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 Same note as previous slide. ELO 3.3

Example 3 A reactor period of 100 seconds equates to a SUR of ____? 𝑆𝑈𝑅= 𝜏 𝑆𝑈𝑅= 𝑆𝑈𝑅=0.26 𝐷𝑃𝑀 ELO 3.3

Knowledge Check – NRC Bank A reactor is started for the first time following a refueling outage. Reactor Engineering has determined that during the upcoming fuel cycle, β eff will range from a maximum of to a minimum of Once the reactor becomes critical, control rods are withdrawn to increase reactivity by 0.1 percent ΔK/K. Assuming no other reactivity additions, what will the stable reactor period be for this reactor until the point of adding heat is reached? 20 seconds 40 seconds 60 seconds 80 seconds Correct answer is C. Correct answer is C. NRC Question P3148 Do the math on the board if necessary. ELO 3.3

Prompt Critical and Prompt Jump/Drop
ELO 3.4 – Describe prompt critical, prompt jump, prompt drop, and how reactor power is affected by a reactor trip and stepped insertion of reactivity. Prompt jump and prompt drop are terms used to illustrate how nuclear reactor responds from large reactivity changes Delayed neutrons increase overall neutron generation time Prompt neutrons affect reactor operation immediately This section further illustrates affect of large reactivity changes and explains reactor's response Related KA - K1.08 Explain the prompt critical, prompt jump, and prompt drop ELO 3.4

Prompt Jump and Prompt Drop
Review from Earlier ELO Prompt neutrons have a very short generation time seconds Delayed neutrons have a longer generation time seconds Overall generation life time seconds Formula for a critical reactor (assumes initial reactivity is zero) Formula for a subcritical reactor can be used to show differences in prompt jump at different subcritical conditions (0.99 versus 0.95) NOTE: This formula is NOT on the NRC Equation Sheet! Even though there aren’t any questions requiring calculations based on this equation, its understanding will help answer other bank questions. ELO 3.4

Prompt Jump and Prompt Drop
Prompt Jump - immediate effect a positive reactivity addition has on prompt neutron population Prompt Drop - immediate effect a negative reactivity addition has on prompt neutron population Trip, for example Initial “Jump” of “Jump-Smile-Jump” as it relates to SUR ELO 3.4

Prompt Jump – Reactivity Addition
When positive reactivity added, prompt neutron population immediately increases Then neutron population change is delayed until delayed neutron precursor levels have increased from reactivity increase Average delayed neutron generation time of 10 seconds Based on 𝜆𝑒𝑓𝑓 of 0.1 Keep in mind that the average delayed neutron generation time of 12.7 seconds equated to steady state power. 1/12.7 = .08 (steady state). Figure: Prompt Jump ELO 3.4

Prompt Drop – Reactivity Addition
For negative reactivity addition: Prompt drop in power Immediate decrease prompt drop (see figure) After drop, rate of change slows and approaches rate from delayed term of period equation Figure: Prompt Drop ELO 3.4

Reactor Trip Response Prompt Drop (trip, for example):
Prompt neutrons are gone immediately Neutrons remaining are from delayed neutron precursors Shortest lived decay off first Longest lived remain longer Longest lived account for negative 80 second period following the trip (-1/3 DPM) When longest lived are gone, subcritical multiplication maintains neutron count rate GRAPH on next slide: The short lived groups decay very fast which produces an additional one decade drop after the prompt drop which makes it appear that the reactor FISSION power dropped two decades instantaneously. Then GROUPS 4,3,and 2 take over and for the next 4 minutes the average SUR is -.75 DPM, so the FISSION POWER drops another 2 decades. Then and only then Group 1 takes over and the SUR is -1/3 DPM until the power gets low enough for subcritical multiplication to take the SUR to zero. ELO 3.4

Figure: Reactor Trip Power Decay Response
Reactor Trip Response Figure: Reactor Trip Power Decay Response ELO 3.4

Prompt Criticality A reactor is prompt critical if it is critical on prompt neutrons alone Recall formula: 𝜏= Ɩ ∗ 𝜌 + 𝛽 𝑒𝑓𝑓 −𝜌 λ 𝑒𝑓𝑓 𝜌 If Delayed = 0 Prompt Term Delayed Term Leaves, 𝜏= Ɩ ∗ 𝜌 ELO 3.4

Prompt Critical Important Concept
If positive reactivity equals/exceeds βeff, prompt neutrons have a larger effect on reactor period Assume reactivity added at BOL = .007 Equals a startup rate of 972 DPM 𝜏= Ɩ ∗ 𝜌 𝜏= 𝜏=.0143seconds Note: this is a worst case step increase of a 800 PCM positive reactivity, something that cannot be performed in a commercial power reactor Important Concept The bottom line is that if the reactivity added is less than the effective delayed neutron fraction βeff, power cannot increase faster than the delayed neutron population increase. ELO 3.4

Prompt Jump Knowledge Check – NRC Bank
Two reactors are critical at the same power level well below the point of adding heat. The reactors are identical except that reactor A is at the beginning of a new life and a reactor B is near the end of a fuel cycle. If a step addition of positive ΔK/K is added to each reactor, the size of the prompt jump in power level observed in reactor B (EOC) will be __________ than in reactor A; and given a large reactivity step insertion reactor B would go prompt critical with __________reactivity than in reactor A. (Assume the power level in each reactor remains below the point of adding heat.) larger; less smaller; less larger; more smaller; more Correct answer is A. Reactor B has a lower Beff (smaller effective delayed neutrons fraction), so the prompt jump would be larger. Prompt critical with Reactor A requires less reactivity because its Beff is small – Prompt critical means critical on prompt neutrons only. Correct answer is A. ELO 3.4

Prompt Critical Knowledge Check - NRC Bank
A reactor is operating at steady-state 75 percent power with the following conditions: Power defect = ΔK/K Shutdown margin = ΔK/K Effective delayed neutron fraction = Effective prompt neutron fraction = How much positive reactivity must be added to make the reactor prompt critical? ΔK/K ΔK/K ΔK/K ΔK/K Correct answer is A. Correct answer is A. NRC Question P2949 HINT: A lot of extraneous information is provided. Prompt critical occurs when the reactivity added is equal to or greater than Beta-Bar-Effective. Therefore, all you need to look at is the “Effective delayed neutron fraction” value. Keep in mind that the term really should be “Average Effective Delayed Neutron Fraction”. ELO 3.4

NRC KA to ELO Tie * - Also covered in , ELO 6.1 ** - Doubling Time equation not included in PPT because Pfinal/Pinitial = 2 (is the same thing) K1.01 Explain the concept of subcritical multiplication. 2.7 2.8 1.1, 2.1 K1.02 Given the simplified formula for subcritical multiplication, perform calculations involving steady state count rate and source count rate. 2.2 2.3 K1.03 Describe the production of delayed neutrons. 2.4 3.1* K1.04 Define delayed neutron fraction and effective delayed neutron fraction: state the reasons for variation. 3.1 K1.05 Define start-up rate. 3.2 K1.06 Describe the factors affecting start-up rate. 3.3 3.2, 3.3 K1.07 Explain the effect of delayed neutrons on reactor control. 3.0 K1.08 Explain the prompt critical, prompt jump, and prompt drop. 2.9 3.4 K1.09 Given the power equation, solve problems for power changes. K1.10 Define doubling time and calculate it using the power equation. 1.6 3.3** K1.11 Explain the necessity for installed neutron sources in a reactor core. 1.4

Reactor Kinetics Introduction

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