Chemical Engineering 412

Introductory Nuclear Engineering

Final Exam Review

Spiritual Thought2

Exam 3 Performance

Part a Part b Part c Part d TotalAverage 30.7 19.2 24.4 16.5 90.8High 35.0 20.0 25.0 20.0 100.0Low 10.0 16.0 21.0 7.0 65.0Median 32.5 20.0 25.0 17.0 94.5StDev 6.0 1.3 1.1 3.4 9.0

3

Chapter 1• Units and Constants• Conservations (when apply and when don’t)

– Mass Energy Equivalence• Elementary Particles

– Bozons, Quarks, Leptons – Compositions of neutrons, protons, electrons, etc.

• Isotopes– Nomenclatures and chart of nuclides– Abundances and properties– Summary of Isotopes

• Atomic Weight– Mixtures and single element

• Traditional chemistry vs. nuclear chemistry

Chapter 2• Quantum Theory• Newtonian vs. Maxwellian physics• Special Relativity

– Time, Length, Mass alterations– Mass/Energy Equivalence + Implications– Momentum and Kinetic Energy (Classical vs. Relativistic)

• Particle Wave Duality– Schrödinger Wave Equation– Assumptions and Boundary Conditions– Solution (1D particle + Hydrogen Atom)

• Uncertainty• Implications and Conclusions

Chapter 3• Atomic Theory & Atomic Structure• Nuclear Energy Levels• Liquid Drop Model

– Repulsive vs. Attractive Forces– Calculations of total Mass

• Shell Model– Stability– Binding Energy– Calculations

• Modern Nuclear Concepts• Interesting Nucleii

Chapter 4• Nuclear Energetics (mass/energy)

– Terminology• Mass Defect vs. Binding Energy• Nuclear Reactions (1, 2, 3 particles, etc.)

– Nuclear Conservations (charge!!)– Parallel Reactions

• Q-Value– Definition– Calculations– Implications

• Z-Changes• Excited Nucleii

Chapter 5• Decay Mechanics

– Conservations– Mechanisms– Reading Chart of Nuclides

• Energy Diagrams• Decay Types & Details• Decay Constants/Half Lifes• Kinetics (single & parallel reactions)• Decay Chains

– Primary chains (4n, 4n+1, 4n+2, 4n+3)– Secular Equilibrium, Daughter product analysis

• Carbon Dating & Inorganic Dating

Chapter 6• Binary Nuclear Reactions

– Major Types & Definitions– Mechanisms– Kinematics

• Threshold Energies• Nuclear Scattering Reactions• Nuetron Interactions

– Slowing Down– Absorption

• Fission Reactions– Mechanism and products– Energies & Decay Heat– Prompt vs. Delayed

Chapter 7• Radiation Interactions with Matter

– Types– Linear Interaction Coefficient– Total Probability of Interaction

• Conceptual Interpretations• Cross Section (vs. interaction coefficient)

– Macroscopic vs. Microscopic– Interaction specific vs. total– Energy Dependence, material dependence, etc. (plots)

• Nuetron Flux & Fluence (collided vs. uncollided)• Photon Interactions (+photoelectric)• Stopping Power

Chapter 10 (I)

• Chapter 10– Criticality

• Six factor formula• Multiplication factor• Cross Sections• Neutron Life Cycle

– Moderation• Common moderators• Most effective moderators

– Bare Reactor• Flux profiles• Boundary conditions• Diffusion Equation Problems

Chapter 10 (II)

– Homogenous vs. heterogeneous– Buckling

• Geometric• Material• Constituents• How to size reactor

– Transient Reactor Behavior• Delayed neutrons• reactivity• δk• Reactor worth ($)• Reactor operation• Period and times

Chapter 10 (III)

– Poisons• Reactivity insertions• Reactivity “swing”• Reactor control methods• Long term reactivity changes and countermeasures• Changes in time

– Reactivity Coefficients• Doppler• Void (moderator expansion)• Axial Expansion • Radial Expansion• Control Rod Drive Expansion• Calculate change in reactivity based on given coefficients

Chapter 11 (I)

– Nuclear Energy Conversion• Key Components• General layout of nuclear plant systems

– Light Water Reactors• Components• Configurations• Design• Challenges• Operation• BWR vs PWR

– Operation Perturbations• Thermal Changes• Load Changes• Fuel Changes• Accidents

Chapter 11 (II)

– Gen IV Reactors• Know types• Benefits/Disadvantages

– Evolution of Nuclear Power• Generations• Characteristics• Other Non-LWR (non Gen IV)

– Fast Reactors• Breeder vs. Burner• Key Components• Challenges• World-wide use

Chapter 12

• heat output of radioactive isotopes.• GPHS

– Characteristics– Table 12.2

• RTGs– Types– Differences & Similarities

• Electricity generation at any point in the life of an RTG.

• Space reactor concepts

Example 1: Back to the Future

• Equate Bucklings– Geometric – Easy– Material – significantly harder (why?)

• Simplest approach– Assume homogenous core– Assume single energy– Assume number density ratio based on movie– Find B2 using the 6 factor formula (keff=1)

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Example 2: Rod Ejection Accident

• A control rod is ejected from the core instantly adding $0.0005 reactivity to the core. Assuming we want a temperature increase of no more than 10 ºC, what is the minimum overall reactivity feedback coefficient (in %mil/ºC)?

• If water contributes 1 %mil/ºC of negative feedback, how much should the soluble Boron provide?

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Chapter 11.6-11.7

• Nuclear Fuel Cycle: front and back end• Enrichment Calculations

– Waste factor, feed factor, separation potentials– Use these factors to determine cost given price

• Grades and forms of Uranium• Separation Techniques• LWR Fuel compositions• Radiopharmaceuticals• Once-through cycle• Other fuel cycles – recycle, mixed oxides, etc.

Chapter 8

• Detector Types– Dead times, interaction rates, performance,

paralyzable, etc.– Examples & Diagrams, etc.– Fundamental operation principles

• Detection and Operation Modes• Spectroscopy• Efficiency• Related equipment (PMTs, SCPHA, MCA)

Chapter 9

• Know Big picture of Radiation Doses• Know various measurements, units

conversion from one to another– KERMA, exposure, Absorbed Dose, etc.– Know how to correlate to biological impacts

• Calculation of dose• Hazards of Radiation (Table usage)• Exposure limits – amounts, history, etc.• Perspective on radiation effects• Acute and latent effects/symptoms• Does model – Linear, threshold, hormesis

Chapter 13

• Beneficial Uses of Radiation +Applications– Specific isotopes and production

• Advantages/Disadvantages of radioactive• Uses of Tracers (calculate amount needed)• Uses of “Materials affecting Radiation”• Uses of “Radiation affecting Materials”• Particle Accelerators• Economics and Widespread applciations

Chapter 14

• Medical Uses of Radiation– Diagnostic vs. theraputic

• X-Rays• Mammography & Densitometry• CT Scan • SPECT• PET• MRI

New Material

• Thermal Analysis of Nuclear Fuel– Fuel pellet of all shapes– Several layers– Different materials

• Accidents– Three Mile-Island– Chernobyl– Fukushima Dai-ichi

• Nuclear Regulations

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Example 1

How long will it take a reactor operating at 100 MW to increase to 1 GW after a $0.15 increase in reactivity?

𝑇𝑇 =𝛽𝛽𝛽𝛽𝛿𝛿𝛿𝛿

=𝛽𝛽𝛽𝛽

𝛿𝛿𝑒𝑒𝑒𝑒𝑒𝑒 − 1=

𝛽𝛽𝛽𝛽𝛿𝛿𝑒𝑒𝑒𝑒𝑒𝑒𝜌𝜌

=𝛽𝛽

𝛿𝛿𝑒𝑒𝑒𝑒𝑒𝑒𝜌𝜌 $≈

𝛽𝛽𝜌𝜌 $

=12.8𝑠𝑠0.15

= 85.3 𝑠𝑠

𝑃𝑃 = 85.3𝑠𝑠 ln1000 𝑀𝑀𝑊𝑊𝑒𝑒

100 𝑀𝑀𝑊𝑊𝑒𝑒= 196 𝑠𝑠

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Example 2The pebble bed reactor packages fuel in spheres that are coated with various structural and moderating layers. Determine the steady-state temperature profile of a spherical particle with a constant heat source. The energy transport equation for this geometry is:

1𝑟𝑟2

𝑑𝑑𝑑𝑑𝑟𝑟

𝑟𝑟2𝛿𝛿𝑑𝑑𝑇𝑇𝑑𝑑𝑟𝑟

+ 𝑞𝑞 = 0

where 𝑞𝑞 is a constant heat source (positive means heat is generated).

a) Derive an expression for the temperature profile in the fuel sphere assuming that the temperature at the edge of the fuel is 𝑇𝑇𝑟𝑟 and is known.b) If the maximum temperature in fuel (to prevent melting) is 𝑇𝑇𝑚𝑚, determine the maximum size of the sphere in terms of 𝑇𝑇𝑟𝑟 ,𝑞𝑞, 𝛿𝛿 and 𝑇𝑇𝑚𝑚.

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Example 2 Solution

BC1: �𝑑𝑑𝑑𝑑𝑑𝑑𝑟𝑟 𝑟𝑟=0

= 0 BC2: |𝑇𝑇 𝑟𝑟=𝑅𝑅 = 𝑇𝑇𝑟𝑟1𝑟𝑟2

𝑑𝑑𝑑𝑑𝑟𝑟

𝑟𝑟2𝛿𝛿𝑑𝑑𝑇𝑇𝑑𝑑𝑟𝑟

+ 𝑞𝑞 = 0

𝑑𝑑𝑑𝑑𝑟𝑟

𝑟𝑟2𝛿𝛿𝑑𝑑𝑇𝑇𝑑𝑑𝑟𝑟

= −𝑟𝑟2𝑞𝑞

𝑟𝑟2𝛿𝛿𝑑𝑑𝑇𝑇𝑑𝑑𝑟𝑟

= −𝑟𝑟3

3𝑞𝑞 + 𝐶𝐶

𝑑𝑑𝑑𝑑𝑑𝑑𝑟𝑟

= − 𝑟𝑟3𝑘𝑘𝑞𝑞 + 𝐶𝐶

𝑟𝑟2𝑘𝑘By BC1, C = 0

𝑟𝑟 = 𝑅𝑅 = 𝑇𝑇𝑟𝑟 = −𝑅𝑅2𝑞𝑞6𝑘𝑘

+ 𝐷𝐷 ⇒ 𝐷𝐷 = 𝑇𝑇𝑟𝑟 + 𝑅𝑅2𝑞𝑞6𝑘𝑘

(BC2)

𝑇𝑇 𝑟𝑟 = 𝑇𝑇𝑟𝑟 +𝑞𝑞6𝛿𝛿

𝑅𝑅2 − 𝑟𝑟2

𝑇𝑇𝑚𝑚 = 𝑇𝑇 0 = 𝑇𝑇𝑟𝑟 +𝑅𝑅2𝑞𝑞6𝛿𝛿

⇒ 𝑅𝑅𝑚𝑚𝑚𝑚𝑚𝑚 =6 𝑇𝑇𝑚𝑚 − 𝑇𝑇𝑟𝑟 𝛿𝛿

𝑞𝑞

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Example 3

The health effects from 222Rn exposure are associated primarily with its decay products, not the radon itself. Assume an air-borne radon concentration of 4 pCi/L (the EPA threshold action value) that is constantly being replenished. Also, assume radon decays only by alpha emission. If the initial concentrations of the daughter products in a lung containing 1 L of air are zero, determine the concentration of the first daughter product after 1 year assuming none of the daughter product (which is a solid) leaves with the exhaling air. The half-life data are in Appendix D of the text.

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Example 3 Solution (I)In this case, there is a non-changing (continuously replenished) initial amount of 222Rn, which we will call 𝑁𝑁𝑅𝑅𝑅𝑅0 that produces 218Po. The 218Po experiences first-order decay. The amount of Po at any given time is given by the solution to the following differential equation, subject to the initial condition that 𝑁𝑁𝑃𝑃𝑃𝑃 𝑡𝑡 = 0 = 0.

𝑑𝑑𝑁𝑁𝑃𝑃𝑃𝑃𝑑𝑑𝑡𝑡

= 𝜆𝜆𝑅𝑅𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅0 − 𝜆𝜆𝑃𝑃𝑃𝑃𝑁𝑁𝑃𝑃𝑃𝑃

The first term right of the equal sign is constant with respect to time while the second is not. The solution to this equation is

𝑁𝑁𝑃𝑃𝑃𝑃 =𝑁𝑁𝑅𝑅𝑅𝑅0 𝜆𝜆𝑅𝑅𝑅𝑅𝜆𝜆𝑃𝑃𝑃𝑃

+ 𝐶𝐶 exp −𝜆𝜆𝑝𝑝𝑃𝑃𝑡𝑡

Applying the boundary condition 𝑁𝑁𝑃𝑃𝑃𝑃 𝑡𝑡 = 0 = 0

𝑁𝑁𝑃𝑃𝑃𝑃 =𝑁𝑁𝑅𝑅𝑅𝑅0 𝜆𝜆𝑅𝑅𝑅𝑅𝜆𝜆𝑃𝑃𝑃𝑃

1 − exp −𝜆𝜆𝑝𝑝𝑃𝑃𝑡𝑡

=4 𝑝𝑝𝐶𝐶𝑝𝑝𝐿𝐿 2.098𝑥𝑥10−6𝑠𝑠−1

3.788𝑥𝑥10−3𝑠𝑠−11 − exp −3.788𝑥𝑥10−3𝑠𝑠−1 ∗ 365 ∗ 24 ∗ 3600 𝑠𝑠 =

2.2154𝑥𝑥10−3𝑝𝑝𝐶𝐶𝑝𝑝𝐿𝐿

=2.2154𝑥𝑥10−3𝑝𝑝𝐶𝐶𝑝𝑝

𝐿𝐿1𝐶𝐶𝑝𝑝

1012𝑝𝑝𝐶𝐶𝑝𝑝3.7𝑥𝑥1010𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑠𝑠/𝑠𝑠

𝐶𝐶𝑝𝑝=

8.19𝑥𝑥10−5𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑠𝑠𝐿𝐿

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Example 3 Solution (II)One year substantially exceeds the half-lives of both Rn and Po, so the term in square brackets above is essentially unity and this step of the system has reached dynamic equilibrium. If one recognizes this, the problem could be more easily solved as

𝑑𝑑𝑁𝑁𝑃𝑃𝑃𝑃𝑑𝑑𝑡𝑡

= 𝜆𝜆𝑅𝑅𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅0 − 𝜆𝜆𝑃𝑃𝑃𝑃𝑁𝑁𝑃𝑃𝑃𝑃 = 0

⇒ 𝑁𝑁𝑃𝑃𝑃𝑃 =𝜆𝜆𝑅𝑅𝑅𝑅𝑁𝑁𝑅𝑅𝑅𝑅0

𝜆𝜆𝑃𝑃𝑃𝑃= 2.215𝑑𝑑 − 3

𝑝𝑝𝐶𝐶𝑝𝑝𝐿𝐿

= 8.19𝑥𝑥10−5 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑠𝑠/𝐿𝐿

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