Geometric buckling is a measure of neutron leakage and material buckling is a measure of the difference between neutron production and neutron absorption.
When
nuclear fission
Nuclear fission is a nuclear reaction, reaction in which the atomic nucleus, nucleus of an atom splits into two or more smaller atomic nucleus, nuclei. The fission process often produces gamma ray, gamma photons, and releases a very large ...
occurs inside of a
nuclear reactor
A nuclear reactor is a device used to initiate and control a fission nuclear chain reaction or nuclear fusion reactions. Nuclear reactors are used at nuclear power plants for electricity generation and in nuclear marine propulsion. Heat from nu ...
,
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
s are produced.
These neutrons then, to state it simply, either react with the fuel in the reactor or escape from the reactor.
These two processes are referred to as
neutron absorption
Neutron capture is a nuclear reaction in which an atomic nucleus and one or more neutrons collide and merge to form a heavier nucleus. Since neutrons have no electric charge, they can enter a nucleus more easily than positively charged protons ...
and
neutron leakage, and their sum is the
neutron loss.
When the rate of neutron production is equal to the rate of neutron loss, the reactor is able to sustain a chain reaction of nuclear fissions and is considered a
critical
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
reactor.
In the case of a bare, homogenous, steady-state reactor (that is, a reactor that has only one region, a
homogenous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size ...
mixture of fuel and coolant, no blanket nor reflector, and does not change over time),
the geometric and material buckling are equal to each other.
Derivation
Both buckling terms are derived from the particular
diffusion equation
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's law ...
which is valid for neutrons:
.
where k is the criticality
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
,
is the neutrons per fission,
is the
macroscopic
The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic.
Overview
When applied to physical phenomena ...
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Ab ...
for
fission
Fission, a splitting of something into two or more parts, may refer to:
* Fission (biology), the division of a single entity into two or more parts and the regeneration of those parts into separate entities resembling the original
* Nuclear fissio ...
, and from
diffusion theory Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common ...
, the
diffusion coefficient
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
is defined as:
.
In addition, the
diffusion length is defined as:
.
Rearranging the terms, the diffusion equation becomes:
.
The left side is the material buckling and the right side of the equation is the geometric buckling.
Geometric Buckling
The geometric buckling is a simple
Helmholtz eigenvalue problem that is simply solved for different
geometries
This is a list of geometry topics.
Types, methodologies, and terminologies of geometry.
* Absolute geometry
* Affine geometry
* Algebraic geometry
* Analytic geometry
* Archimedes' use of infinitesimals
* Birational geometry
* Complex geomet ...
.
The table below lists the geometric buckling for some common geometries.
Since the diffusion theory calculations overpredict the
critical dimensions, an
extrapolation distance δ must be subtracted to obtain an estimate of actual values. The buckling could also be calculated using actual dimensions and extrapolated distances using the following table.
Expressions for Geometric Buckling in Terms of Actual Dimensions and Extrapolated Distances.
Material Buckling
Materials buckling is the buckling of a
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
configuration with respect to material properties only. If we redefine
in terms of purely material properties (and assume the fundamental mode), we have:
.
As stated previously, the geometric buckling is defined as:
.
Solving for k (in the fundamental mode),
;
thus,
.
Assuming the reactor is in a critical state (k = 1),
.
This expression is in purely material properties; therefore, this is called the materials buckling:
.
Critical Reactor Dimensions
By equating the geometric and material buckling, one can determine the critical dimensions of a one region nuclear reactor.
References
{{DEFAULTSORT:Geometric And Material Buckling
Nuclear technology