Nuclear Test 1

If more neutrons survive

K_eff > 1

Power increases

Super critical

If fewer neutrons survive

k_eff < 1

Power decreases

Subcritical

If exctly enough neutrons survive

K_eff = 1

Power stays constant

Critical

What does critical mean?

When each neutron generation produces exactly one subsequent generation, resulting in a constant neutron population and power.

Reactivity Equation

p = k-1/k

What is reactivity?

measure of departure from criticality.

Six Factor formula

= η∈pƒP_fP_T

Reproduction Factor

η

Neutrons produced per thermal neutron absorbed in fuel

Fast Fission Factor

∈

Extra neutrons from fast fissions

∈ >1

Resonance escape probability

p

probability neutron avoids resonance absorption

Thermal utilization Factor

ƒ

Fraction of thermal neutrons absorbed in fuel

ast non-leaking probability

P_f

Thermal non-leakage probability

P_T

What factors increase k?

Higher η, ε, p, f, PF, PT.

barns

1barn=10^-24cm²

Macroscopic Cross section

Σ=Nσ

Probability interaction per unit path length.

Total Cross Section

σt​=σs​+σa​

Prompt

99.35%

Immediate

Delayed

.65%

Seconds to minutes later

Without delayed neutrons

Power changes in milliseconds.

Reactor impossible to control.

XENON-135

I¹³⁵→ Xe¹³⁵

Huge neutron absorber: σ_a ≈ 2× 10⁶ barns

Xe after shutdown

Xe increases.

Reactivity decreases.

Restart becomes difficult.

What is xenon poisoning

Reduction in reactivity due to buildup of Xenon-135

Source Neutrons

Needed because subcritical reactors cannot sustain neutron populations

provide neutrons for startup and monitoring in subcritical conditions

Source neutrons sources

• Cf-252

• Am-Be

• Spontaneous fission

Subcritical Multiplication

M = 1/1-k

As: k→ 1

M→ ∞

What is subcritical multiplication?

Amplification of source neutrons by fission in a subcritical reactor.

1/M=0

Criticality reached

Insert rods:

decrease p

Why use a 1/M plot?

To estimate the critical rod position before reaching criticality.

Withdraw rods

p increases

Rod worth

Total reactivity change

Why is rod worth highest near core center?

Neutron flux is highest near the center

Boron: chemical shim

More boron = lower p

less boron: higher p

Moderator Temperature Coefficient

α_M = ∆p/∆T_M

Usually negative

Hotter moderator:

Less dense water.

Less moderation.

Lower reactivity.

Fuel Temp Coeffecient (doppler)

α_f = ∆p/∆T_F

Hotter fuel:

More U-238 absorption.

Lower reactivity.

Power Coefficient

α_p = ∆p/∆P

Why are negative coefficients desirable?

They provide automatic negative feedback and improve reactor stability.

Fuel Temp Profile

Centerline > Surface > clad > coolant

Largest drop often ocurs across

Fuel clad gap

Why is centerline hottest?

Heat is generated throughout the fuel and must conduct outward.

Hot Channel Factor

F_H = Max local power/ Average Power

Why do designers use hot channel factors?

evaluate worst-case fuel temperatures rather than average temperatures.

Film Boiling

Bad.

Steam blanket forms.

Heat transfer collapses.

NUCLEATE BOILING

Good.

Efficient heat transfer.

Normal operation

Why is film boiling dangerous?

Steam is a poor conductor, causing rapid fuel and cladding temperature increases.

DNBR

.1 = safe

=1 DNB

What does DNBR represent?

Margin between operating heat flux and the heat flux that causes departure from nucleate boiling.

PCM

1 pcm = 10^-5 Reactivity

Peaking Factor

F_q = Peak Power/Average Power

Higher Peaking

Higher fuel temp

Why is HALEU attractive?

t enables longer fuel cycles and advanced reactor designs.

High-Assay Low-Enriched Uranium

5% - 20%

Why does fission release energy?

Fission products have higher binding energy per nucleon than U-235, so the difference is released as energy.

Mass Defect

∆m = (Zm_p + Nm_n ) - M

Binding Energy

BE = ∆m(931.5)

Conduction

q = kA∆T/L

Thermal Resistance

R = L/kA

Convection

q = hA(T_s - T_f)

general equation for fission of U-235.

U-235 absorbs a neutron, becomes unstable, splits into two smaller nuclei, releases neutrons, and produces about 200 MeV of energy.

Why are the emitted neutrons important?

The emitted neutrons can cause additional U-235 nuclei to fission, creating a chain reaction, which is the basis of nuclear reactors and nuclear weapons.

Mean Free Path

λ = 1/Σ

The average distance a neutron travels before interacting.

What is a microscopic cross section?

The probability that a neutron interacts with a single nucleus.

Unit of microscopic cross section?

Barns (1 barn = 10⁻²⁴ cm²)

Equation relating microscopic and macroscopic cross sections?

Σ=Nσ

Units of macroscopic cross section?

cm⁻¹

Explain microscopic and macroscopic cross sections

Microscopic cross section is the probability of a neutron interacting with a single nucleus, while macroscopic cross section is the probability of interaction within a bulk material.

Explain all terms of the six-factor formula

The six-factor formula describes neutron production and survival using η (reproduction factor), ε (fast fission factor), p (resonance escape probability), f (thermal utilization factor), PF (fast non-leakage probability), and PT (thermal non-leakage probability)

Explain how the terms of the six-factor formula change with moderator temperature and core life

Increasing moderator temperature and fuel burnup generally reduce neutron economy and decrease several six-factor formula terms, lowering keffk_{eff}.

State the definition of reactivity and explain the units

Reactivity is a measure of departure from criticality and is commonly expressed as Δk/k or pcm (per cent mille).

Explain the mechanism of the reactivity coefficients, rod worth, and boron worth

Reactivity coefficients describe how reactivity changes with temperature or power, while rod worth and boron worth measure the reactivity change caused by control rods or dissolved boron.

Explain how the reactivity coefficients change with moderator temperature and core life

Reactivity coefficients generally become less negative as fuel burns up and moderator conditions change during core life.

Explain fission product poisons

Fission product poisons are neutron-absorbing fission products, such as Xe-135 and Sm-149, that reduce reactor reactivity.

Explain the mechanism of Xe-135 transients

Xe-135 transients occur because iodine continues decaying into xenon after power changes while xenon burnup changes simultaneously.

Explain the need for and source of source neutrons

Source neutrons are needed to monitor and start up a subcritical reactor and are supplied by neutron source materials such as Cf-252 or Am-Be.

Explain subcritical multiplication

Subcritical multiplication is the amplification of source neutrons by fission reactions in a reactor with keff<1k_{eff}<1.

Explain the purpose and construction of a 1/M plot

A 1/M plot is used during startup to predict criticality by plotting inverse multiplication versus rod position.

Explain the indications of a reactor reaching criticality during startup

Criticality is indicated when neutron count rate becomes self-sustaining and no longer depends on source neutrons.

Explain the difference between prompt and delayed neutrons and the importance of delayed neutrons

rompt neutrons are emitted immediately after fission while delayed neutrons are emitted later by fission products and allow reactors to be safely controlled.

Explain why reactor power follows steam demand while in the power range

Reactor power follows steam demand because temperature feedback changes reactivity and causes power to automatically adjust to heat removal.

Draw and explain the temperature profile across the fuel/clad

Temperature decreases from the fuel centerline through the fuel, gap, cladding, and coolant, with the centerline being hottest.

Explain the causes and consequences of a high peak centerline temperature

High peak centerline temperatures result from high local power or poor cooling and can lead to fuel cracking, fission gas release, and fuel melting.

Explain the hot channel factor and how safe peak centerline temperature is ensured

The hot channel factor accounts for local power peaking, and safe centerline temperatures are maintained through conservative design and operating limits.

Explain the causes and consequences of film boiling

Film boiling occurs when a vapor blanket forms around the fuel, drastically reducing heat transfer and increasing fuel and cladding temperatures.

Explain the Departure from Nucleate Boiling Ratio (DNBR)

DNBR is the ratio of critical heat flux to actual heat flux and indicates the margin to departure from nucleate boiling.

What is HALEU?

HALEU is uranium enriched between 5% and 20% U-235 for advanced reactor applications.

What is binding energy?

Binding energy is the energy required to separate a nucleus into individual protons and neutrons.

What is mass defect?

Mass defect is the difference between the mass of separate nucleons and the actual nuclear mass.

What is a neutron generation time?

Neutron generation time is the average time between successive neutron generations in a chain reaction.

What is rod worth?

Rod worth is the amount of reactivity added or removed by moving a control rod.

What is differential rod worth?

Differential rod worth is the reactivity change per unit control rod movement.

What is pcm?

pcm (per cent mille) is a reactivity unit equal to 10−510^{-5} Δk/k.

What is critical heat flux?

Critical heat flux is the maximum heat flux that can be removed before departure from nucleate boiling occurs.

What is the reproduction factor?

The reproduction factor is the number of neutrons produced per thermal neutron absorbed in fuel.

What is resonance escape probability?

Resonance escape probability is the chance that a neutron avoids resonance absorption while slowing down.