23. Solving the Neutron Diffusion Equation, and Criticality Relations

Introduction to Reactor Problems

Overview of the Course Content

• The session is part of a Creative Commons licensed course by MIT OpenCourseWare, emphasizing the importance of donations for continued educational resources.

• The professor introduces the topic, stating that they will solve complex reactor problems, transitioning from simple one-group homogeneous reactors to more advanced two-group reactor problems.

Key Concepts in Reactor Physics

• Discussion on fission and neutron interactions including n-n reactions, photo fission, absorption, and leakage as part of the neutron balance equation.

• The focus is on understanding flux versus position (x), particularly in different reactor geometries like infinite slabs and cylinders.

Mathematical Derivation

Simplifying Complex Equations

• The professor aims to simplify equations by removing unnecessary terms related to NIN reactions and photo fission for clarity.

• Introduction of the Laplacian operator's role in different dimensions; its forms vary based on coordinate systems (1D Cartesian vs. cylindrical).

Solving for Flux

• Focus shifts to solving equations analytically starting with an infinite slab case before addressing cylindrical reactors.

• Rearrangement leads to a simplified expression involving flux and constants derived from cross-sections.

Understanding Neutron Transport

Averaging Cross Sections

• Explanation of averaging cross-sections over energy ranges; this is crucial for accurate modeling in reactor physics.

Function Solutions in Cartesian Space

• Identification that functions with their double derivative equal to a constant are typically exponential or trigonometric (sin/cos).

Flux Profile Analysis

Determining Flux Shape

• Assumption made that flux solutions take specific forms based on previous knowledge; cosine functions are favored due to symmetry considerations.

Implications for Reactor Design

• Discussion about boundary conditions at x = 0 where the sine term is eliminated due to symmetry; implications for neutron behavior at reactor edges are highlighted.

Extrapolation Length Concept

Definition and Importance

Understanding Reactor Physics and Criticality

Introduction to Cosine Function in Reactor Analysis

• The angle cosine is introduced, approximating to 2/3 times the average atomic mass of the material being scattered. This establishes a physical analogy for reactor analysis.

• The discussion emphasizes using libraries like Janis for looking up cross-section data and number densities relevant to reactor materials, making the theoretical framework more tangible.

Deriving Conditions for Reactor Geometry

• By substituting phi with cos(bx) , simplifications lead to an expression where constants are isolated, indicating that geometry does not influence certain material properties.

• A condition is established where knowledge of reactor geometry and materials allows calculation of the effective multiplication factor ( k_effective ), determining reactor criticality.

Evaluating Cosine Validity in Reactor Length

• The professor prompts students to determine what value bg must equal for the cosine function to be valid at a specific reactor length, leading to engagement from the audience.

• Clarification on how substituting values into the cosine function results in it equating to zero at defined points within the reactor's dimensions.

Establishing Criticality Condition

• The simplified expression derived shows that bg^2 = 1/k(musigma_fission + sigma_absorption)/d , setting up a criticality condition based on neutron gains versus losses.

• The final formula indicates that criticality occurs when gains from fission balance losses from absorption and leakage, establishing a foundational understanding of nuclear reactor behavior.

Exploring Effects of Absorption Changes on Criticality

• If absorption increases in a previously critical reactor, k_effective will decrease due to an increase in denominator size, illustrating how changes affect stability.

Temperature Influence on Cross Sections and Neutron Behavior

• Discussion shifts towards temperature effects; raising temperature typically decreases cross sections due to Doppler broadening, impacting neutron interactions significantly.

• As temperature rises, macroscopic cross-sections decline because atoms spread apart (lower density), affecting overall neutron behavior within the reactor environment.

Analyzing Changes in Effective Multiplication Factor

Reactor Feedback and Criticality

Understanding k Effective and Temperature Feedback

• The behavior of k effective is dependent on the materials used, leading to either positive or negative temperature feedback. This means that temperature changes can influence reactor performance in various ways.

• In some scenarios, an increase in temperature may lead to an increase in k effective, which is undesirable and prohibited in reactor design due to safety concerns.

Exploring Reactor Size Impact

• When increasing the size of a reactor without changing materials, certain parameters like nu (neutron production), sigma fission, and sigma absorption remain constant while bg decreases.

• Adding more reactor material typically results in an increase in k effective, suggesting a direct relationship between reactor size and power dynamics.

Power Dynamics and Criticality

• The power generated by a reactor is derived from the kinetic energy of neutrons; however, criticality does not directly correlate with power output. A zero-power reactor can still be critical.

• Zero-power reactors are useful for testing neutron physics knowledge as they operate at low power levels (e.g., 10 watts), allowing safe measurements of neutron flux.

Steady State vs. Transience

• If k effective deviates from one, the reactor exits steady state; gains do not equal losses anymore, leading to changes in power output.

• Manipulating control rods affects reactor power dynamically rather than statically; removing a control rod increases power but does not stabilize it immediately due to delayed neutron effects.

Dynamic Equilibrium in Reactor Control

• Achieving dynamic equilibrium requires continuous adjustment of control rods since static equilibrium is unattainable during operation—reactors are always adjusting based on real-time conditions.

• The concept of dynamic equilibrium is emphasized as crucial for understanding how reactors function over time rather than remaining static like dead organisms.

Two-group Balance Equations Introduction

• Transitioning into two-group balance equations involves separating gains and losses into fast and thermal regions for better analysis of neutron behavior within reactors.

Understanding Neutron Behavior in Fission Reactors

Contributions of Fast and Thermal Neutrons

• The discussion begins with the distinction between sigma thermal and sigma fast, emphasizing that both types of neutrons contribute to fission processes.

• The professor explains the need to separate variables into fast and thermal components, highlighting that thermal neutrons primarily create fast neutrons upon absorption during fission.

• Loss mechanisms for fast neutrons are introduced, specifically absorption by materials characterized by sigma af times the fast flux.

Mechanisms of Neutron Loss

• Scattering is identified as another mechanism through which neutrons can leave the fast group, necessitating an understanding of how many scatter from fast to thermal groups.

• The scattering probability is discussed, particularly in relation to hydrogen, where it is noted that scattering events do not always result in leaving the fast group.

• Leakage is mentioned as a loss mechanism for fast neutrons, represented mathematically as d_fast bg squared flux.

Transitioning Between Energy Groups

• The focus shifts to gains and losses within the thermal neutron group. It’s emphasized that only scattered neutrons from the fast group enter this region.

• A clarification on neutron behavior indicates that once they transition into the thermal group via scattering, they typically remain there without returning to faster energy states.

Loss Mechanisms in Thermal Group

• Leakage remains a significant loss mechanism for thermal neutrons; a separate diffusion coefficient applies due to varying cross-sections based on energy groups.

• Absorption is reiterated as another primary loss mechanism for thermal neutrons. The professor notes there’s no upward scattering from this group since it represents the lowest energy state.

Criticality and Reactor Geometry

Understanding Neutron Behavior in Reactors

The Concept of k Effective

• The professor explains that k effective is crucial for understanding neutron production, indicating it represents the total original sources of neutrons while everything else accounts for losses.

• Experimental data shows that neutrons are typically born with energies between 1 and 10 MeV, prompting a thought experiment about thermal neutrons.

Thermal vs. Fast Neutrons

• The discussion introduces the concept of the birth spectrum (Chi), emphasizing the need to account for both fast and thermal neutrons in calculations.

• A mathematical expression is presented to model scenarios where some neutrons are born thermal, although this situation rarely occurs in practice.

Mathematical Modeling of Neutron Flux

• The professor highlights a challenge: having two equations but three unknowns, suggesting a strategy to express everything in terms of one flux type.

• By isolating phi thermal from one equation, they aim to simplify the problem and prepare for further calculations.

Deriving Expressions for k Effective

• After manipulating equations, an expression emerges that relates various factors like fission rates and absorption cross-sections solely in terms of fast flux.

• This leads to a criticality condition formula applicable across different reactor geometries, focusing on geometric buckling as a key parameter.

Gains vs. Losses in Reactor Design

• The final expressions illustrate how gains from neutron interactions must outweigh losses, reinforcing the importance of material properties and geometry in reactor design.

• An audience question prompts clarification on whether certain scattering terms were omitted; adjustments are made to ensure accuracy in modeling.

Exploring Reactor Dynamics

• The professor invites questions about practical implications such as isotope changes or temperature variations affecting reactor behavior.

Understanding Neutron Transport and Diffusion Approximations

Discussion on Neutron Transport Equation

• The professor discusses the neutron transport equation, emphasizing the importance of understanding its components and how to derive it from memory.

• Students are encouraged to explain terms within the equation, justify simplification steps, and understand physical implications of approximations used in solving it.

Limitations of Diffusion Approximation

• The professor highlights that diffusion approximation is not valid near control rods or fuel where cross sections change abruptly.

• It is explained that diffusion describes long-distance steady-state solutions; thus, it fails in regions with drastic changes due to neutron interactions.

Criticality Conditions and Reactor Behavior

• Students may be asked to derive simple criticality conditions or predict reactor behavior under various physical changes (e.g., material properties).

• The discussion includes how geometry affects equations; for instance, cylindrical coordinates would require different mathematical approaches like Bessel functions.

Understanding Bessel Functions in Reactor Physics

• The professor introduces Bessel functions as solutions for cylindrical geometries, noting their similarity to sine and cosine functions.

• Emphasis is placed on intuitive understanding over rote memorization; students should grasp concepts like k-effective as gains over losses.

Flux Graph Representation in Cylindrical Geometry

• A question arises about flux representation in cylindrical graphs; the professor explains symmetry arguments leading to a specific form involving Bessel functions.