GFES Reactor Theory Flashcards
What is elastic scattering?
a nucleus deflects a neutron without absorbing it; conserves energy
(“billiard ball” type of collision)
some kinetic energy of the neutron is transferred to the nucleus of the target atom; amount depends on mass
What is inelastic scattering?
kinetic energy not conserved
→neutron is incorporated into nucleus
→kinetic energy of N raises internal energy of target nucleus
→a neutron is released (with less kinetic energy than the incoming neutron had)
→target nucleus returns to ground state by emitting a gamma ray
What is radiative capture of a neutron?
→neutron is absorbed by target nucleus
→kinetic energy of N raises internal energy of new compound nucleus
→compound nucleus returns to ground state by emitting gamma rays (no neutron is released)
original target nucleus acts as neutron absorber
How does Boron-10’s microscopic cross section vary with neutron energies?
Boron-10 has a large neutron absorption cross section at low neutron energies. It gets smaller as neutron energy gets higher.
It has no resonance peaks.
The absorption cross section for thermal neutrons is high.
What is macroscopic cross section?
The probability of an incident neutron interacting with a target nucleus per unit length of travel of the incident neutron.
Σ = Nσ
Σ = macroscopic cross section
N = atomic density (atoms/cm3)
σ = microscopic cross section (barns)
What is mean free path?
how far the average neutron will travel before an interaction will take place
(inverse of macroscopic cross section)
λ = mean free path
Σ = macroscopic cross section
What is critical energy?
the minimum amount of energy for fission to occur
(different for each fuel type)
What are fissile materials?
fuel types that fission simply due to the neutron binding energy
What are fissionable materials?
fuel types that require additional energy above neutron binding energy in order to cause fission
What are fast neutrons?
have a kinetic energy greater than 0.1 MeV
(all fission neutrons are born as fast neutrons)
What are intermediate neutrons?
have a kinetic energy between 1 eV and 0.1 MeV
What are slow neutrons?
have a kinetic energy less than 1 eV
What are thermal neutrons?
neutrons that are in thermal equilibrium with their surroundings
can be fast, intermediate, or slow, depending on temperature of their surroundings
in commercial power, thermal neutrons are usually in the slow energy region
What are prompt neutrons?
Neutrons emitted within 10^-14 seconds of a fission event and are a direct result of the fission process caused by a thermal neutron
Most fission neutrons are prompt neutrons; percentage depends on type of fuel used (99.36% for U-235)
What are delayed neutrons?
neutrons emitted more than 10^-14 seconds after the fission event (avg is 12.5 seconds after; NRC usually uses 12.5 on exam)
small number (0.64% for U-235), but very important to overall reactor control
born as fast neutrons, but at a lower energy than prompt neutrons
produced well after fission event; result of the decay of a first daughter fission product from fission caused by a thermal neutron
What percentage of emitted neutrons will be prompt vs delayed in the fission of U-235?
99.36% prompt
0.64% delayed
What does the moderator do?
slows down fast neutrons to the thermal range
What is the neutron generation time?
the total time from fission to absorption for a neutron
(the lifetime of one generation)
ℓ* = prompt neutron generation time (approx 10^-4 sec)
ℓ-d = delayed neutron generation time (approx 12.5 sec)
The prompt neutron has a ______________ probability of leaking out than the delayed neutron.
(higher or lower)
higher
(more collisions required to reach thermal energies)
The ______________ neutron can cause fission of U-238 (fast fission) and the ____________ neutron cannot.
Answer options: prompt or delayed
prompt neutron can
delayed neutron can’t (typically)
What is neutron flux?
the formal measure of the number of neutrons in the core, i.e. the number of neutrons passing through a unit of area per unit of time
designated by ϕ
measured in neutrons/cm2-second
classified as either thermal flux or fast flux
What is the effective neutron multiplication factor (k-eff)?
rate of change in neutron population; factor by which the number of neutrons produced from fission in one generation is multiplied to determine the number of neutrons produced from fission in the next generation
How do we calculate k-eff?
How does k-eff relate to criticality?
k-eff = 1 is exactly critical
k-eff < 1 is subcritical
k-eff > 1 is supercritical
What is the six factor formula?
Non-leakage factors in commercial nuclear power are essentially 1, so we don’t count them.
What is reactivity?
the measure of the departure of a reactor from criticality; the fractional change in neutron population per generation
k-eff = 1, reactivity = 0
k-eff > 1, reactivity is positive value
k-eff < 1, reactivity is negative value
Six-Factor Variables and their effects on K-eff
What is excess reactivity?
The amount of reactivity above the level that is needed to take the reactor critical (based on k-excess, which is amount of excess fuel loading that causes k-eff >1)
k-excess = k-eff - 1
ρ-excess = k-excess/k-eff
What is shutdown margin?
the instantaneous amount of reactivity by which a nuclear reactor core is subcritical, or can be made subcritical from its present condition, assuming the most reactive control rod is stuck fully withdrawn
SDM = (1 – k-eff)/k-eff = -ρ
Xenon activity after shutdown
increases for about 10 hours (from 100% power, or square root of power before change if not 100%) (adds negative reactivity, SDM up)
decreases for the next 60-70 hours down to zero (adds positive reactivity, SDM down)
What are neutron sources?
materials that produce neutrons through processes other than neutron induced fission
two types of source neutrons: intrinsic and installed
(source neutrons don’t count in k-eff)
What is an intrinsic neutron source?
naturally occurring materials, usually fission products, which produce neutrons through naturally occurring reactions
(most important types for commercial reactors are spontaneous fission, photo-neutron reactions, alpha-neutron reactions)
What is an installed neutron source?
materials that are designed to produce a specific number of neutrons per unit time
→primary source - produces neutrons without any activation
→secondary source - must be activated in an operating core before they will produce any neutrons
What are the purposes of source neutrons?
→should be available at a high enough flux level to provide on-scale readings on startup range monitors
→help initiate fission process during startup
→help determine operability of core monitoring system
→allow operator to monitor core reactivity changes during startup and shutdown operations
What are photo-neutron reactions?
occur when a gamma causes the dissociation of deuterium into a hydrogen atom and a neutron
largest contributor of intrinsic neutron sources to the core following ONE operating cycle of a new or refueled core plus any time after MOL conditions immediately following a reactor shutdown
(quickest decreasing source after shutdown at 2-3 weeks)
What are alpha-neutron reactions?
high energy alpha particles from the decay of heavy elements in the core react with O-18 to produce neutrons
largest contributor of intrinsic neutrons several weeks after a shutdown past MOL conditions
strongest to weakest source of intrinsic neutrons approaching EOL conditions
immediately following reactor shutdown
photo-neutron sources → alpha-neutron (transuranic) sources → spontaneous fission sources
strongest to weakest source of intrinsic neutrons approaching EOL conditions
several weeks following reactor shutdown
alpha-neutron (transuranic) sources → spontaneous fission sources → photo-neutron sources
What is the relationship between k-eff and M
(M = subcritical multiplication factor)
Characteristics of subcritical multiplication
subcritical neutron level is directly proportional to the neutron source strength
utilizes source neutrons and fuel to maintain a constant neutron population while k-eff < 1
neutron population is maintained above source strength as a result of source neutrons while k-eff < 1
What is the equation for the maximum neutron population for a subcritical reactor?
N = S / (1 - k-eff)
(on NRC equation sheet)
Thumb Rule #1 for Counts vs k-eff for Startup
halving the distance to criticality doubles the counts
(doubling the count rate halves the distance to criticality)
If adding reactivity equal to 1/2 of (1 - k-eff), then count rate will double.
Thumb Rule #2 for Counts vs k-eff for Startup
criticality in 5 to 7 doublings
when initial count rate at beginning of startup has doubled 5 to 7 times, reactor will be at or near critical
Thumb Rule #3 for Counts vs k-eff for Startup
fixed reactivity additions vs count rate doubling
when enough reactivity is added to double count rate, if the same amount of reactivity is added to the reactor again, reactor will be supercritical
What is the delayed neutron precursor decay constant?
represented by λ
λ = (ln 2) / half life
the decay rate of a particular delayed neutron precursor
reciprocal of mean life
What is mean life?
represented by tau
how long, on average, a delayed neutron precursor will exist before decaying
reciprocal of delayed neutron precursor decay constant
What is the delayed neutron fraction
represented by β
fraction of all neutrons born (prompt and delayed) as delayed neutrons for a particular fuel isotope
each isotope has its own β
average delayed neutron fraction decreases over core life
What is the reactor period?
the length of time (usually in seconds) it takes for power to increase by a factor of e
also represented by tau, so keeping different equations straight is important
When power is increasing, reactor period is positive
When power is decreasing, reactor period is negative
What is the startup rate?
rate of change of reactor power expressed in decades per minute
(one decade is a factor of 10)
SUR = 26.06/reactor period
What is the Point of Adding Heat (POAH)?
the point at which the fission rate (and therefore heat generation rate) is high enough to raise the temperature of the fuel
(0.5 - 1.0% reactor power)
What is the prompt jump?
the very quick, almost instantaneous rise in power when reactivity is added in step fashion
(prompt neutrons can go through many generations before the delayed neutron precursor populations begin to change; prompt neutron population grows immediately upon the addition of positive reactivity until prompt neutron population reaches the point that the losses in a generation are once again made up for by the delayed neutrons that are temporarily fixed at their old rate)
What is the relationship between startup rate and reactor power?
equation provided on NRC equation sheet
Six Factor Formula
What is fast fission factor?
Fast fission factor (ε) = fast n ALL fission / fast n THERMAL fission
Affected by:
→enrichment: more U-235 means less U-238, reduces ε
→moderator temp increases, ε increases (takes longer to slow n)
→as core ages, ε goes down
Changes to ε are small; core age has largest impact
Six Factor Formula
What is fast non-leakage probability?
Fast non-leakage probability (L-f)
L-f = fast n that don’t leak out / fast n from ALL fission events
Affected by:
→moderator temp increases, L-f decreases (takes longer to slow n, more leak out)
→neutron flux closer to edges, L-f decreases slightly (more leak out)
Changes to L-f are small.
Six Factor Formula
What is resonance escape probability?
Resonance escape probability (p)
p = fast n that escape resonance capture / n that don’t leak while slowing
large effect on k-eff
Affected by:
→moderator temp increases, p decreases (takes longer to slow n, more likely to be absorbed)
→as core ages, p goes down (less U-238, but more Pu-240)
→as fuel temp goes up, p goes down (doppler broadening)
→as rods inserted, p decreases (rods have resonance absorbers)
Six Factor Formula
What is thermal non-leakage probability?
Thermal non-leakage probability (L-th)
L-th = thermal n that don’t leak / #n escaping resonance capture
Affected by:
→moderator temp increases, L-th decreases
→neutron flux moves toward edges, L-th decreases
Changes to L-th are small.
Six Factor Formula
What is thermal utilization factor?
Thermal utilization factor (f)
f = thermal n absorbed in fuel / thermal neutrons absorbed in core
(n can be absorbed by any material in the core - fuel, moderator, boron, etc.)
Affected by:
→varies during operation with density of moderator changes
→as rods inserted, f decreases (more absorption in rods)
→moderator temp increases, f increases (less dense moderator, less n absorption in moderator
→fuel enrichment increases, f increases (neutrons in fuel increase because more fuel present)
→as U-235 is burned, f decreases; however, moderator is diluted at EOL and f increases
Six Factor Formula
What is reproduction factor?
Reproduction factor (η)
η = fast n produced by thermal fission / thermal n absorbed in fuel
Affected by:
→as core ages, η decreases
→value set by fuel loading: 85% of n absorbed in U-235 result in fission and ~2.3 n releases per fission
Changes to η are small
Which factors in the six factor formula affect k-eff the most?
resonance escape (p, esp. in under-moderated region)
AND
thermal utilization (f, esp. in over-moderated region)
K-eff vs N-mod / N-fuel ratio
What are the trends in the under moderated region?
Under-Moderated Region (BOL)
(high boron, high temps)
→moderator temp goes up, k-eff goes down
→resonance escape (p) has larger effect
→resonance escape (p) goes down when moderator temp goes up
→MTC is least negative at low temps with high boron
K-eff vs N-mod / N-fuel ratio
What are the trends in the over moderated region?
Over Moderated Region (EOL, cold)
(low boron, low temps)
→moderator temp goes up, k-eff goes up
→thermal utilization (f) goes up when moderator temp goes up
→thermal utilization (f) has larger effect
→MTC is most negative with low temps and low boron
What is the effective decay constant for up-power?
0.1
What is the effective decay constant for a down power?
0.05
What is the effective decay constant for steady state power?
0.08
What is the effective decay constant for a reactor trip?
0.0124
What is the approximate value for the effective delayed neutron fraction at the beginning of core life (BOL)?
0.007 at BOL
What is the approximate value for the effective delayed neutron fraction at the end of core life (EOL)?
0.0054 at EOL
What is a reactivity coefficient?
a change in reactivity (Δρ) due to unit change in some associated parameter (x)
(coefficient is derivative; defect is integral)
What is reactivity defect?
the total amount of reactivity added, positive or negative, due to changing nuclear reactor parameter by a given amount
(coefficient is derivative; defect is integral)
What is moderator temperature coefficient (MTC)?
change in reactivity per degree change in moderator temperature
can be positive or negative change
depends on the magnitude in changes of resonance escape probability (p) and thermal utilization factor (f) due to changes in moderator-to-fuel ratio (density)
MTC and Over-Moderation vs Under-Moderation
Under-Moderated: high moderator temp, high boron, more negative as temp goes up
Over-Moderated: low moderator temp, low boron, less negative as temp goes up
MTC is ____________ negative when control rods are inserted.
(more or less)
more
(neutrons survive farther distances without being absorbed so that they make it to the control rods and are absorbed there)
What is the doppler coefficient, or fuel temperature coefficient?
the amount of reactivity added into the core when the fuel temperature increases by one degree
always negative as fuel temp goes up
causes shift in resonant capture cross section peaks
What is the void coefficient?
negative reactivity inserted due to the decreased moderator density in the steam bubbles; very small effect
Fuel Temperature Coefficient Trends
larger effect per degree at lower temperatures (more negative)
smaller effect per degree at higher temperatures/higher power levels (less negative)
negative at BOL (approx -1 x 10^-5 Δk/k/°F)
more negative at EOL (approx -1.5 x 10^-5 Δk/k/°F)
What is the power coefficient?
negative reactivity added as power level increases
(a combination of MTC, FTC, and void fraction)
approx -1.5 x 10^-4 Δk/k/% power at BOL
approx -2.2 x 10^-4 Δk/k/% power at EOL
What is the boron coefficient?
approx -10 pcm/ppm or -.01 %Δk/k/ppm
becomes more negative at lower ppm due to less competition
as boron ppm increases, all coefficients become less negative (except FTC, which stays the same)
Which pair of nuclides is the most significant contributor to the total resonance capture in the core near the end of a fuel cycle?
U-238 and Pu-240
(U-238 is most significant at beginning AND end of cycle)
Which factor in the six factor formula changes the most as control rods are inserted and withdrawn?
thermal utilization factor (f)
rods withdrawn, f ↑, add +ρ
rods inserted, f ↓, add -ρ
What is a hot channel factor?
the ratio of ϕ-max / ϕ-avg
If hot channel factor = 1, flux profile is flat
If hot channel factor > 1, flux profile is peaked
High hot channel factor means there are high local power densities in the core (core power distribution within the core is proportional to the thermal neutron distribution flux).
What is differential rod worth (DRW)?
the change in reactivity per unit change in rod position
(equation provided on NRC equation sheet)
How does differential rod worth change as a rod is inserted or withdrawn?
increases, then decreases
(both directions)
What is integral rod worth (IRW)?
the reactivity inserted by moving a rod from a reference position to any other position
(DRW is the slope of the IRW curve)
How does control rod worth change with changes in core condition parameters?
What is conversion ratio with regard to fuel burnup?
the number of P-239 nuclei produced per 100 U-235 nuclei consumed (expressed in percent)
Note: buildup of Pu-239 is referenced to depletion of U-235, but it actually comes from neutron capture by U-238 and its subsequent decay.
How do burnable poisons, fission product poisons, and excess fuel affect k-eff over the span of core life?
Fission product poison buildup reduces the…
thermal utilization factor (f).
(they capture thermal neutrons so that they can’t cause fission)
What is the thumb rule for the amount of time it takes after a trip/downpower to reach peak xenon concentration?
The time in hours to reach peak xenon concentration is equal to the square root of the power before the downpower.
e.g. a trip from 100% power means a 10-hour Xe peak time, and a trip from 50% power equals approx 7 hrs for peak Xe concentration
What does it mean when a reactor is xenon precluded?
When the negative reactivity from the concentration of Xe-135 is high enough that the reactor can’t be started back up because available excess reactivity isn’t high enough.
(most likely at EOL)
How long will a reactor remain xenon precluded?
Condition will persist until enough Xe-135 decays off that the positive reactivity of control rod withdrawal can overcome the negative reactivity of xenon.
What are the two removal processes for Xe-135?
burnout
(absorbs a thermal neutron and transforms into Xe-136, which is stable)
decay
(decays by beta emission to Cs-135, which has a very long half-life)
How is Sm-149 removed?
only burnout because isotope is stable
How long does it take for the concentration of Sm-149 to peak after a trip?
Approx 12.5 days
How long does it take for the concentration of Sm-149 to reach equilibrium after a startup?
Approx 21 days
How long does it take for Xe-135 to reach equilibrium after a startup or power increase?
40-50 hours
How much negative reactivity does Sm-149 add during operation at power (any power level)?
-1.0% Δk/k
How much negative reactivity does Sm-149 add during its peak after a shutdown from 100% power?
-1.4% Δk/k
How do Xenon-135 and Sm-149 compare?
(distribution problem means it causes changes to flux distribution, as in the oscillations caused by Xe-135)
Xe-135 is directly produced as a fission product in approximately what percentage of all fissions?
0.3%
During reactor startup, equal increments of reactivity are added, and the count rate allowed to reach equilibrium each time.
How does the time required to reach equilibrium change as the reactor starts up?
It takes longer each time.
(With each added increment, more neutron generations are produced, so it takes longer to get to equilibrium.)
What kinds of errors could make subcritical multiplication (1/M) non-conservative?
→did not wait for equilibrium
→detector too close to source resulting in CR-0 too high
→detector only sees neutrons from source and not from fuel
Reactor Startup and Approach to Criticality Summary Chart
Sufficiently review each time it’s offered.
Intermediate Range Startup Summary Chart
Sufficiently review each time it’s offered.
Reactor Heatup to 100% Summary Chart
Sufficiently review each time it’s offered.
Why are load following operations at or near the end of core life difficult?
→the amount of water needed to dilute the RCS at low boron concentrations is very large (e.g. to mitigate a large addition of positive reactivity on a down power)
→xenon transients (oscillations) are more likely to start and are more difficult to control at end of life
Power Range Operation Summary Chart
Sufficiently review each time it’s offered.
After a reactor trip and initial prompt drop, what is the reactor period and SUR for a shut down reactor as it seeks subcritical equilibrium?
-80 second period
-1/3 dpm SUR
(continues until neutron population is low enough for effect of source neutrons to be seen and subcritical equilibrium is reached)
What power level does a reactor fall to immediately after a trip?
At 10 seconds?
At 1 minute?
At 1-3 hours?
6-7%
5%
3%
1%
How do you find shutdown margin (SDM) when a reactor is critical?
SDM = ρ(rods) - ρ(power defect)
How do you find shutdown margin (SDM) when a reactor is subcritical?
SDM = (1 - Keff) / Keff
How does shutdown margin change over core life?
SDM decreases
Rod worth becomes more negative, but so does power defect (so larger positive reactivity added upon shutdown)
How do we find Keff from subcritical multiplication rate (M)?
Keff = 1 - (1/M)
What happens to irradiated Boron-10 atoms?
They undergo a neutron-alpha reaction.
After a month of operation at 100% power, what percent of rated thermal power is being produced from the decay of fission products in the reactor?
between 5% and 10%
(greater then 5%, but less than 10%)
What is the major contributor to the production of Xe-135 in a reactor that has been operating at full power for two weeks?
radioactive decay of I-135
As moderator temp increases, thermal utilization factor (f)…
increases.
an increase in moderator temperature can reduce neutron capture by the moderator sufficiently to add positive reactivity
f is dominant in over-moderated region
As moderator temp increases, resonance escape probability (p)…
decreases.
An increase in moderator temp results in lower density, which causes longer diffusion and slowing down lengths for neutrons. Because the neutrons have to travel further between collisions, the probability that a neutron might reach fuel or moderator at resonance energy rises (they’ll get absorbed instead of causing fission). Therefore, the resonance escape probability decreases.
p is dominant in under-moderated region
After shutdown (no matter how long it was operating or how long it’s shut down), the core decay heat production rate depends on…
time since shutdown
(a.k.a. time elapsed since K-eff decreased below 1.
