James Madison University. Utica College of Syracuse University Introductory Nuclear. Thermal Engineering. Fundamentals of. Nuclear Engineering. Interaction and. Nuclear reactions; Q-value equation and its solution; Stacey, W. Sons, Inc, May 29, Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physi- cal systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations.
In this paper, we present an The balance of isotopes in a nuclear reactor core is key to understanding the overall performance of a given fuel cycle. This balance is in Jun 3, Reactors Appendix A. Review of Chemical Equilibria In addition, Stacey Siporin, John Murphy, and Kyle Bishop are acknowledged for their excellent assistance in compiling the solutions manual.
Such a reactor could generate electricity, produce fuel for conventional fission reactors or provide a way to transmute the long-lived actinides of nuclear waste into May 18, The control and accident anal- ysis of a nuclear reactor and the conversion of reactor per- iod into reactivity require the knowledge of the effective. All Rights Reserved. Designed by Templatic. To browse Academia. Log in with Facebook Log in with Google. Remember me on this computer.
Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. Nuclear Reactor Physics. Divek Yadav. A short summary of this paper. Nuclear Reactor Physics lecture notes AP prof. Principle of a nuclear reactor The fission process Nuclear reactions and neutron cross sections Energy dependence of neutron cross sections The neutron transport equation The diffusion equation Boundary condition One-group diffusion theory Multi-zone systems Perturbation theory Simple description of reactor kinetics Reactor kinetics with delayed neutrons Temperature effects Burn-up and conversion Fission products Reactivity and reactor control Multi-group diffusion theory Energy transfer in elastic collisions The epithermal spectrum for moderation Fermi age theory The thermal neutron spectrum Calculation of group cross sections Treatment of resonances The four-factor equation The neutron yield factor The thermal utilisation factor The fast fission factor The resonance escape probability Leakage factors Light-water reactors The fuel cycle of a light-water reactor Other reactor types The international situation with regard to nuclear energy Principle of a nuclear reactor In a nuclear reactor certain very heavy nuclei e.
If on average one of the neutrons released in a fission causes a new fission, a steady state chain reaction is initiated, whereby energy is continuously being released.
Compared with a chemical reaction, in which a few eV are released for example, compare the combustion of 1 m3 of natural gas with an energy content of 30 MJ , this is an enormous amount of energy.
To that end, one applies a coolant that can effectively remove the heat and a much larger amount of uranium is placed in the reactor core. Moreover, pure 92 U does not occur in nature. Table 1. Composition of natural uranium Isotope weight percentage mass number radioactive half-life 92 U 0.
U is much less suited for fission by neutrons, because of which the percentage U, the degree of enrichment, is very important. A reactor core is formed by placing together a large number of fuel elements in a large reactor vessel.
In order to prevent chemical reactions between the coolant and the nuclear fuel, the nuclear fuel is housed in a metal cladding. Figure 1. Fuel elements in a reactor Figure 1. The heat generated in the nuclear fuel is transferred to the cooling water, which is pumped upward along the rods. The water can start to boil by this. The steam is subsequently led to a steam turbine, in which the blades are driven and a rotation is induced.
The turbine shaft subsequently drives an electrogenerator, which generates the electric energy and supplies it to the electrical power network. The expanded steam from the turbine is condensed, after which the water is pumped through the core again. Cooling water, which is drawn from river or seawater, is used for condensation of the steam, or one applies a cooling tower.
Principle sketch of a boiling-water reactor In a so-called pressurized-water reactor see Figure 1. The heated water is led to a steam generator, in which heat is transferred to water from the secondary side, which is under a lower pressure, so that it starts boiling. The steam is then led to the turbine again. Principle sketch of a pressurized-water reactor The water in a boiling-water or pressurized-water reactor not only serves as coolant, but also for slowing down the neutrons in their energy.
The neutrons released during fission have a high energy, as we will see in the next section, whereas the chance of causing a new fission is larger if the neutrons have a low energy. By collisions with light nuclei such as hydrogen in water, the neutrons lose energy, so that the water also works as moderator. Therefore, reactors in which a moderator is applied are also referred to as thermal reactors.
In reactors in which no moderator is applied, the neutrons predominantly keep a high energy. These reactors are called fast reactors. In the next sections, first we will go into the fission process in order to understand why energy is released in nuclear fission and why only certain nuclei can be used.
Further, other possible interactions of neutrons with a nucleus will be discussed and the chance of interactions will be quantified. In the next chapter, an equation can then be derived, which describes the transport of neutrons in a reactor core. From the solution of this equation or a simplified form thereof , the conditions that a reactor core must satisfy in order to enable a self-sustaining chain reaction of fissions can be derived.
The fission process The protons and neutrons in an atomic nucleus are held together by the nuclear forces strong force. Therefore, energy is required for breaking apart the nucleus into the separate nuclear particles or nucleons. This binding energy of a nucleus is obtained by imaginary composition of the nucleus from the separate nucleons, because the mass of the whole nucleus is less than the sum of the masses of the separate nucleons.
This is shown in Figure 1. For low mass numbers it increases rapidly, with some irregularities in the case of light nuclei, reaches a maximum of 8. From this figure it becomes evident that energy can be produced by fusion of light nuclei or by fission of heavy nuclei. Seeing that the attracting forces between the nucleons mainly operate via direct neighbours, the positive first term in 1.
One can call this the volume term. Hereby it is neglected, however, that particles at the surface of the nucleus are not completely surrounded by other particles. Consequently, the binding energy has been overestimated with an amount that must be proportional to the surface area of the nucleus. By analogy with a liquid drop this effect is indicated as the surface tension effect.
The third term is connected with the coulomb interaction between the protons, which lowers the binding energy because of the repulsion between charges of equal sign. The last two terms in 1. Especially in the case of light nuclei, one sees that those with an equal number of protons and neutrons are very stable.
The heavier stable nuclei, however, contain more neutrons than protons. This excess of neutrons is necessary in order that the attractive forces between the neutrons and between the neutrons and protons can provide some compensation for the repulsion between the protons third term. At the same time, however, some instability is introduced because the surplus of neutrons occupies a number of energy levels in the nucleus, which do not contain protons.
A correction factor must be introduced for this, the so-called symmetry term, which is also important in the case of a proton surplus; therefore this term is quadratic in N-Z. When a nucleus contains an odd number of both particle types, it is nearly always unstable; the only exceptions are 21H, 63 Li, B and N. The fission process consists of splitting a nucleus into roughly equal parts.
In principle, any nucleus, if brought into sufficiently high excited state, can be split. The amount of excitation energy that is required to enable nuclear fission can be estimated from the magnitude of the electrostatic barrier and the dissociation energy of the fission in question.
In Figure 1. The height of the potential barrier is approximately given by the permittivity of the vacuum. To the left of the maximum, a bound state occurs as a result of the nuclear forces. The dissociation energy Ed is equal to the difference between the binding energy of the compound nucleus and the sum of the binding energies of the fission fragments and can thus be estimated with 1. The minimum activation energy Ea that has to be added to a nucleus to cause fission is thus EC — Ed.
Potential energy as a function of the distance between two fission fragments When the mass of a nucleus is larger than the sum of the masses of the fragments into which the nucleus can be separated, the first will show a tendency towards instability, because fission is accompanied by the release of energy.
Energy values in MeV important for fission mass number A 16 60 potential barrier EC 4 32 62 dissociation energy Ed With the aid of gamma bombardment one can determine the value of Ea. Indeed, by absorption of a neutron, both the kinetic energy and the binding energy of the neutron become available for bringing the compound nucleus into an excited state, while no coulomb forces need to be overcome such as in the case of charged particles.
If the excited state in the energy diagram of Figure 1. By applying 1. For these nuclides fission is thus a threshold reaction.
When a nucleus is excited above the potential barrier, fission need not always occur. This is detrimental to sustaining the chain reaction in a reactor. During the spontaneous fission reaction, immediately a few neutrons are emitted as a result of the large neutron surplus in the fission products. The average number v increases with increasing excitation energy, so with the kinetic energy of the absorbed neutron. The fission spectrum of U The neutrons emitted during fission have an energy distribution as sketched in Figure 1.
From this figure a preference appears for asymmetric fission. The primary fission products, which are in a highly excited state, lose their redundant energy by decay. This can happen by emission of photons and neutrons.
When the excitation energy has become smaller than the binding energy of the neutrons in the fission fragments, the prompt neutron emission stops and the remaining energy will be released in the form of photons, the so-called prompt gamma radiation. Fallen back into the ground state the fragments will still be unstable. The other part is distributed over the kinetic energy of the neutrons and the radiation energy. The total energy amounts to about MeV per fission, distributed as shown in Table 1.
In a reactor, per fission circa MeV becomes available, being the total energy minus the energy of the neutrinos. However, neutron capture processes in a reactor produce additional energy, by which the total energy becoming available amounts to about MeV per fission. This means that circa 3. Energy distribution during fission energy MeV kinetic energy fission fragments Nuclear reactions and neutron cross sections Atomic nuclei can undergo interactions with other nuclei, elemental particles protons, neutrons, electrons and electromagnetic radiation photons.
For reactor physics we can confine ourselves to interactions with neutrons, which due to their electrical neutrality do not experience coulomb repulsion and can thus become involved in interactions with nuclei already at very low energy.
According to this model two consecutive phases can be discerned in a nuclear reaction: 1 The incident particle is absorbed by the nucleus and forms a compound nucleus with it.
Because a particle is absorbed in the target nucleus, an amount of energy will be added to the compound nucleus that is equal to the binding energy of the particle plus the kinetic energy of this particle. Because conservation of momentum must be satisfied, part of the total energy is converted into kinetic energy of the compound nucleus, while the remaining energy causes the compound nucleus to attain a high-energy state. Immediately after formation of the compound nucleus, all of the energy is concentrated around the captured particle.
In consequence of the interactions between the nuclear particles, this energy will rapidly spread over all particles in the nucleus. This distribution has a statistical character, whereby it is possible that a particle gets an energy that is larger than its binding energy.
The compound nucleus then can lose its excess energy by emission of this particle. In the previous section we already saw that in the case of very heavy nuclei it is possible that the energy is distributed in such a way that two fragments arise: the nuclear fission.
The last reaction is the fission reaction. For neutron-nucleus reactions the following division is used: elastic scattering scattering inelastic scattering neutron-nucleus reactions capture absorption fission The concept of the microscopic cross section is introduced to represent the probability of a neutron-nucleus reaction. The microscopic cross section in general is dependent on the neutron energy and the type of reaction. In order to be able to determine the microscopic cross section, transmission measurements are performed on plates of materials.
Starting from the presumption that no fission or scattering occurs, the neutron attenuation by a plate with thickness x will be calculated see Figure 1. From 1. This quantity is also referred to as the relaxation length, because it is the distance in which the intensity of the neutrons that have not caused a reaction has decreased with a factor e. Energy dependence of neutron cross sections Besides a ground state, atomic nuclei also have higher energy levels, which can be excited.
The high energy levels become closer and closer to each other. For an excitation energy of about 8 MeV, as is the case after capture of a neutron with low kinetic energy, the separation between the levels is only 1 — 10 eV. If there are several decay possibilities for the compound nucleus emission of a photon, neutron, etc. If the excitation energy of the compound nucleus corresponds with one of the level energies, the probability of an interaction is large: resonance occurs.
As the excitation energy depends on the kinetic energy of the neutron, the probability of an interaction and thus the microscopic cross section varies strongly with the energy of the neutron. The microscopic cross section shows a maximum if the kinetic energy E of the neutron equals Er. As the level widths are often small, the resonance peaks can be very sharp.
Microscopic cross section at a resonance According as the neutron energy increases, the peak height of the resonance decreases and the resonances become relatively closer to each other, so that they finally cannot be distinguished anymore see Figure 1. In reality the nucleus will also have a kinetic energy as a result of heat movement.
This expresses itself in the microscopic cross section, because averaging over the energy distribution of the nuclei must take place.
Although the average energy of the nuclei is small, at room temperature 0. This leads to broadening of the resonances and lowering of the top value. This so-called Doppler effect is thus temperature dependent and plays an important role in reactors. For scattering, the microscopic cross section decreases sharply outside the resonance, and the so-called potential scattering will predominate, whereby the neutron is scattered by the potential field of the nucleus and does not form a compound nucleus.
The potential-scattering cross section is constant over a large energy range, but decreases at high energies. For light nuclides, at low energies chemical-bonding effects can occur, by which the microscopic cross section is larger than that of the free atomic nucleus e. H in H2O. In addition to elastic scattering, whereby the total kinetic energy of the particles is conserved, also inelastic scattering can occur.
In that case the nucleus remains in an excited state after emission of the neutron and rapidly decays to the ground state under emission of a photon. The incident neutron then must have sufficient energy to be able to excite this level, so that inelastic scattering is a threshold reaction. The microscopic cross sections for many nuclides and various reactions can be represented graphically by various computer programs making use of these data files.
The neutron transport equation As the free neutrons play an essential part in sustaining the fission reactions in a nuclear reactor, studying the neutron distribution in a reactor is an important part of reactor physics. Generally put the question is: how do the free neutrons distribute themselves in place, energy, time and direction of movement?
Hereby not the individual life histories of neutrons are concerned, but the statistical average behaviour of a very large number of neutrons. The mathematical description of the neutron distribution is based on a neutron balance equation, which is called the neutron transport equation. This equation is a linearized form of the Boltzmann equation, known from the kinetic gas theory.
Linearization is possible because mutual interactions between neutrons are negligible in nuclear reactors; in other words, the neutron distribution is wholly determined by interactions between neutrons and nuclei of the medium. This quantity is thus a function of seven independent variables: three for the place, one for the energy, two for the direction and one for the time.
It can be interpreted as the number of neutrons per second moving through an imaginary small sphere with a cross section of 1 cm2. In order to develop an equation describing the transport of neutrons, we consider an arbitrary volume V, enclosed by a surface S, for which a balance is made.
Figure 2. The decrease under 4 by all possible interactions is, conformably to 2. Now integration of 2. The diffusion equation In order to arrive at a more convenient equation than 2. Then the energy-dependence disappears and 2.
Subsequently, we introduce an approximation. Multiplication of 2. If the source S is isotropic, which usually will be the case, the integral over the source strength does not contribute. In Section 5. Equation 2. Then equation 2.
The reactor-physical diffusion coefficient has the dimension of length. In other disciplines of physics it is customary to work with particle densities instead of with the flux density, so that in those disciplines a diffusion coefficient has the dimension area per time.
Substitution of 2. The diffusion equation can, therefore, not be valid at places with strongly differing properties or in strongly absorbing media. This implies that the diffusion theory may show deviations from a more accurate solution of the transport equation in the proximity of external neutron sources and interfaces. Transport quantities for thermal neutrons of 0. This distance must not be confused with the average distance traveled by the neutrons. Boundary condition In the diffusion theory.
A reasonable approximation is to demand that the inwardly directed neutron flow at the surface is zero. For a one-dimensional rectangular plate- geometry there is symmetry about the x-axis.
The neutron flow vector J will thus always be directed along this axis. As J must be continuous, the flux gradient will show a jump if the diffusion coefficients in both media differ from each other. Finally, it can be remarked that according to 2.
The scalar flux is sometimes confused with the total flux density, but equation 2. We will confine ourselves to mono-energetic neutrons or a one-group approximation, whereby one can consider all neutrons to belong to one energy group. One-group diffusion theory The one-group diffusion equation 2. The time-dependent behaviour of a neutron population in a multiplying system i. If the production term predominates, the neutron population will increase, and in the reversed case it will decrease.
When production and losses exactly balance, the neutron flux will be constant in time; in that case one speaks of a critical reactor. Only then one may set the time derivative in 3. Also for a non-critical reactor one wants to have a measure for the production, absorption and leakage of neutrons from the reactor being out of balance, without having to solve a time- dependent diffusion equation, because the exact time-dependent behaviour is the result instead of the cause for this balance.
One can obtain a steady-state equation artificially by adjusting the source term in 3. Physically this means that one makes the system seemingly critical by a fictive change in neutron production per fission. The integral thus equals the total neutron leakage from the reactor. Equation 3. The physical events in a reactor can be considered as a coming and going of successive neutron generations, whereby the fission processes are considered as moments of birth.
The multiplication factor keff then gives the ratio of the neutron population size in two successive generations. In order to arrive at an analytically solvable equation, we will assume that the reactor is homogeneous, i.
This is the case if the materials are homogeneously mixed or in the trivial case that only one material is present. In general, reactors will have a heterogeneous structure, i. For homogeneous systems 3. Infinite-plate reactor For the geometrically most simple case, a slab reactor with extrapolated width a and infinite in both other directions Figure 3.
According to 3. Therefore, B2 is also called the geometric buckling factor Bg2. Table 3. The eigenfunctions have been normalized to the value 1 at the origin. Notice that the coefficient A from 3. The result 3. The parameters for the one- group calculation are given in Table 3. The measured critical mass of such a system amounts to As the one-group model used here is a rough approximation, this accuracy is owing to the fact that the parameters given in Table 3.
There exist more solutions than 3. As the higher eigenfunctions for certain values of x are negative, these flux shapes do not have a meaning of their own.
As the eigenfunctions are mutually orthogonal, i. Also in transition phenomena in time-dependent processes the higher eigenfunctions and eigenvalues play a part due to the smaller values of the corresponding multiplication factor they die out fast and they will not be further discussed here. Multi-zone systems A reactor core may consist of various zones of different composition. How is Chegg Study better than a printed Nuclear Reactor Physics student solution manual from the bookstore?
According to Table New York: Wiley, Nuclear Reactor Physics Weston M. Stable Nuclides. Power Iteration on Fission Source. Lewis Ph. If your wanted solutions manual is not in this list, Solution manual Fundamentals of Nuclear. Noted for its accessible level and approach, the 3rd Edition of this long-time bestselling PDF etextbook provides overviews of nuclear physics, medicine, nuclear power, propulsion Fundamentals of Nuclear Reactor Physics 1 Elmer E.
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