Introduction to Nuclear Engineering A+B

First section - General view on nuclear power plants

Brief introduction

Let us start with a brief introduction to nuclear energy and its use. First of all, it is important to underline that energy production is not an easy matter with a precise and ultimate solution. Nowadays, the global energy network, as well as the global community of consumers, is particularly interested in the cost of energy and in CO₂ and environmental concerns. In both of these, nuclear can be competitive as a cheap source of electricity—under certain conditions—and being almost CO₂-free.

From another point of view, it is well known that a nuclear reaction produces more than twenty million times the energy produced through carbon and methane combustion. This proportion value is called the mega factor and it is almost equal to 23,000,000.

Last but not least is the matter of safety of a nuclear power plant. Talking about risk, the estimated frequency of occurrence of a lethal accident, during an entire year, in a nuclear power plant, is almost four orders of magnitude less than the frequency of a lethal car accident and more than three orders of magnitude less than one of a lethal home accident. Anyway, nuclear energy is not well perceived by consumers because of three big accidents that occurred: Three Mile Island (1979), Chernobyl (1986), and Fukushima (2011).

Historical dates

• Neutron discovery 1934
• Neutron thermalization - Fermi 1934
• Fission discovery – Hahn, Meitner, Strassman 1938
• Atomic pile - Fermi 02/12/1942
• First atomic explosion 1945
• First "nuclear" electric bulb 1951
• First nuclear power plant 1954
• Non-proliferation treaty - 155 countries 1970
• Three Mile Island accident 28/03/1979
• Chernobyl accident 26/04/1986
• "Nuclear renaissance" 2008
• Fukushima accident 11/03/2011

Basic elements, classification, and key parameters

We can identify some elements present in a nuclear power plant:

• The fuel (element-bundle): contains the pellets that will be burned to produce thermal energy;
• Coolant (fluid-power channel): which will absorb the thermal energy released by the fuel;
• Moderator: used to thermalize the neutrons that will continue the nuclear chain reaction;
• Reflector: used to minimize the leakage of neutrons;
• Shielding: to reduce the radiation (gamma and beta) produced by the reactions;
• Internals structural elements;
• Control elements;
• Other Nuclear Steam Supply System (NSSS) components;
• Safety systems;
• Containment;
• Auxiliary system + balance of plant.

Some key parameters for the evaluation of a nuclear power plant are:

• Thermal power produced (MWth);
• Electric power produced (MWe);
• Net efficiency (%) = MWe / MWth;
• Load factor (%) = mean(MWe) / max(MWe);
• Availability factor (%) = total electrical energy produced / total electrical energy requested;
• Core power density (kW/L, W/cm³) = MWth / total core volume;
• Linear power density (kW/m) = MWth / total fuel or channel length;
• Specific power (kW/kg) = MWth / total fissionable material mass;
• Enrichment (%) = mass fissile material / mass fissile + fertile material;
• Burnup (MWday/tons) = total energy generated in the fuel / total fuel mass;
• U-235

The goal in a nuclear power plant is to maximize the bold type parameters: the net efficiency, which is an indicator of the capability of the system to convert the energy produced by the fuel into electric energy (can be increased by optimizing the temperature, the pressure, and the primary fuel); the load factor, that indicates the mean usage of the plant; the burnup, indicating the quantity of energy extracted each day per ton of fuel (U-235).

The fuel

It is well known that natural uranium can undergo a decay process, transforming into other nuclides, with the emission of particles and energy. In a nuclear reactor, such a process is induced by a collision with a traveling neutron, which hits the nucleus of uranium. Alas, the resulting reaction isn't always nuclear fission—the one we are interested in, because it produces energy in large quantities: other reactions are also possible:

• Nuclear fission, in which, as we said, a neutron meets a fissile isotope and splits it into two other lighter isotopes, emitting other neutrons and energy;
• Radiative capture, in which the initial isotope captures the neutron, becomes unstable, and then decays emitting a highly energetic gamma ray;
• Elastic scattering, in which the neutron hits the nucleus and bounces off of it;
• Anelastic scattering, in which the neutron does not keep its kinetic energy after bouncing off of the initial isotopes, which is now in an excited state.

When we want to analyze the chances of getting a nuclear reaction of the types described above, we shall speak about the nuclear cross section. This is a property that, for a collimated, monoenergetic, normal neutron beam directed toward non-shielded nuclei, is proportional to:

• The intensity of the neutron beam, I [neutrons cm^-2s^-1];
• The density of the target atoms, N [atoms cm^-2];

To identify this probability of reaction, we define the so-called reaction rate:

R = σ I N A

in which σ is a proportionality factor that considers the effective transverse section area of target nuclei facing the neutrons, i.e., the microscopic cross section, measured in barn (1b=10^-28 m²).

We shall also notice that the nuclear collision cross section depends on:

• Target nucleus type;
• Colliding particle type;
• Nuclear reaction type.

For neutrons, in particular, we have these kinds of cross sections:

• σ_F, the fission microscopic cross section;
• σ_N,γ, the radiative microscopic cross section;
• σ_S = σ_E + σ_AN, scattering microscopic cross section (sum of elastic and anelastic cross sections);
• σ_A = σ_F + σ_N,γ + σ_N,α + σ_N,P + ..., the absorption microscopic cross section;
• σ_T = σ_A + σ_S, the total microscopic cross section.

When we take into account the three-dimensionality of the matter, we have to introduce the macroscopic cross section. This quantity considers the shielding effect in a finite layer and the fact that the colliding beam's intensity decreases as it moves through the target. We define, in this case, the reaction rate as follows:

dR = σ_T(x) I N dx = dR

Observing that we shall write:

-dI = dI / dx = -σ_T(x) N I

and, upon integrating:

I(x) = I₀ e^{-N σ_T x}

We define the macroscopic cross section—here total—as the product:

Σ_T = N σ_T

Actually, nuclear cross sections also depend on:

• Energy, E, of the neutron colliding with the nuclei;
• Spatial distribution of the nuclides, r;
• Possible decay/production of nuclides in time;

so that we shall rewrite as follows:

Σ(E, r, t) = N(r, t) σ(E, r, t)

Traveling neutrons are usually emitted with a kinetic energy of about 2MeV, so they are called fast neutrons. If they are slowed down, they enter an epithermal zone, in which the quantum resonance effects with the nuclei are predominant. Further reducing the neutron speed, they enter the so-called thermal zone and are called thermalized.

As we see from the figure above, the cross sections of neutrons in respect of their energy vary from zone to zone and depend on the type of target nuclei. We shall denote the following features:

• Nuclear cross section is constant in the fast zone, independently from the type of target;
• In the same zone, U-238 presents the lowest σ_F, Pu-239 the highest;
• Upon losing energy, the neutron nuclear cross section drops for U-238—about at 1MeV;
• Below 100keV, σ_F of U-235 and Pu-239 begin rising;
• In the range of 1eV-1keV, the resonance effects give rise to fluctuations in the cross section value for both nuclides;
• In the thermal spectrum (1meV-1eV), the cross section of U-235 goes as E^-1/2, while the one of Pu-239 has a maximum around 0.5eV.

The fission reaction of U-235 is shown in the figure on the next page. We shall see that the fission gives various results:

• Around 200MeV of energy;
• Fission fragments, typically Ba-141 and Kr-92, which are unstable and highly energetic;
• 2-3 fission neutrons, which, as we will see, are fundamental for the sustainability of the chain reaction;

These produced neutrons can be exploited to produce other fission events but can also be the cause of some dangerous issues. They may, in fact, be captured by other nuclides, making them unstable and emit radiations: β and γ are the most dangerous for humans. A different issue due to the production of neutrons is that they may leak from the reactor vessel. Both these problems can be managed by building coverage and appropriate shielding around the reactor (stainless steel, concrete).

The fission chain reaction

The key parameter for a nuclear chain reaction is the mean number η of neutrons produced per neutron absorbed into the fuel. Let ν be the number of neutrons produced per fission, we have:

η = ν σ_F / σ_A

This number, as we shall see from the presence of the cross sections of fission and absorption, depends on the energy of the colliding neutrons and on the type of target nuclide. Also, other sources of neutrons might have to be taken into account—called photoneutrons—as well as other materials capable of capturing neutrons—this is called the parasitic effect.

In general, for a core made of an isotopic mixture of fuel, structural material, coolant, and moderator, we have:

η = Σ_j ν_j Σ_F,j / Σ_k Σ_A,k

In the table, there's a list of the average number of neutrons produced per fission and the thermal neutrons absorbed for the typically used fuels. We shall note that natural uranium, being composed of 99.2% U-238 that cannot be used for the thermalized neutron chain reaction, has the same average amount of neutron produced but lesser η.

So, our main goal is to sustain a stable chain reaction to have a constant production of energy by fission. In principle, one neutron from fission should suffice to propagate the fission; in practice, an excess of neutron production is required to balance the parasitic effect and the losses. It is mandatory to define and control the neutron balance of the chain.

Enrico Fermi introduced a multiplication factor k, defined as:

k = (num. of neutrons generation n + 1) / (num. of neutrons generation n)

which simply compares the quantity of neutron originated from a fission event (n generation) and the one produced by the following fission event (n + 1 generation). Easily we can say that:

• If k = 1, the number of neutrons produced does not change in time and the fission chain reaction is stable: we call this situation critical;
• If k < 1, the number of neutrons produced decreases from a generation to another: we call this situation subcritical;
• If k > 1, the number of neutrons produced increases from a generation to another: we call this situation supercritical.

To better refer to k as a function of time, we can write:

k(t) = P(t) / L(t)

where P is the production rate of neutrons in the core and L is the loss rate (absorption + leakage) in the core.

We shall now see that the neutron population in the core, N(t), at the time t may be written as a function of k. Let l, the neutron mean lifetime—typically 10^-4 s, be defined as follows:

N(t) / l = L(t)

We see that the time rate change of the neutron population is simply given by:

dN(t)/dt = P(t) - L(t)

With a bit of mathematical manipulation and by combining the above equations, we shall integrate and write down, assuming k and l constant in time:

N(t) = N₀ e^{(k-1) t / l}

To simply calculate the multiplication factor, we shall proceed following this scheme:

This selection path helps us in calculating the probability of fission given the following probabilities:

• P_NL, the probability of non-leakage of a neutron;
• P_F, the probability of absorption into the fuel, calculated as Σ_AF / Σ_A;
• P_f, the probability of fission, as σ_f / σ_A;

Hence following the events for two generations, we have:

N_n+1 = ν P_f P_AF P_NL N_n

as well as:

N_n+1 = η P_NL N_n

Accordingly, the multiplication factor is given by the so-called four factors formula:

k = η P_NL

We shall now consider the energy of the colliding neutrons. They, as we noted, generate with a mean energy of 2MeV, which means that they ought to be thermalized. Anyway, a very limited probability of inducing a fission event by a non-thermalized neutron may be present and it may not be negligible, even if, normally, it is very low, due to the presence of the moderator. Thus, we must consider the chances that "fast" fissions and "fast" leakage could happen, as well as a capture during the slowing down process (i.e., resonance). We then introduce two more parameters which consider these eventualities:

The fast fission factor:

ε = (fissions total fast + thermal) / (fissions thermal)

and the escape probability from resonance, p.

At this point, we write the so-called six-factors formula:

k_eff = η ε p P_NL

with P_NL = P_FNL P_TNL, writing the probability of fast non-leakage as P_FNL and the thermal non-leakage probability as P_TNL.

It is wise to denote that, during the process of the planning of a nuclear power plant, particularly in the selection of the material of the cladding, the geometry of the core, and every other aspect involving the nuclear chain reaction itself, the last insight has to be done upon the fuel: k_eff = 1 must be achieved, so that the chain is self-sustained.

A widespread way of reaching the critical condition, by means of the production of neutrons, is to enrich the fuel, i.e., artificially adding U-235 to natural uranium. The level of enrichment can be calculated using η for fresh fuel; 3-5% is the typical percentage for PWR and BWR; instead, 20% is the level of enrichment used for "strategic material" and it is typically seen as dangerous.

We can see from the table below that, increasing the percentage of enrichment, η does not follow linearly. This is why a lot of power plants do not use more than 3% of enrichment.

Moderation and coolant

We need a moderator to have a good process of thermalization via scattering events. We look for low mass number nuclei—gaseous hydrogen would be the best—high scattering cross section, and low absorption probability for thermal neutrons. From the table on the right, we see that light water (H₂O) has the best mass number because it contains hydrogen, and the highest scattering cross section; instead, graphite (C) has the best absorption cross section.

Light water needs enriched fuel due to its high absorption cross section, but it requires less space, can be used as a coolant, and it is easily obtainable. Graphite is less compact and requires more space, but it is solid and easier to be managed. The other two candidates, heavy water and beryllium, can be used, but they are more expensive and hardly obtainable.

Historically speaking, the first reactors were used to produce plutonium for military purposes and used natural uranium and graphite as moderators. Reactors for the production of energy use enriched fuel and light water as coolant and moderator. Some reactors used CO₂ or helium as coolant (gas reactors), to increase the thermal efficiency of the production cycle and avoid mechanical problems.

The selection of the moderator is an important step in the design of the nuclear chain reaction. In fact, with too low moderation, we have not enough thermalized neutrons to sustain the chain; on the other hand, with too much moderation, we have too much radiative captures by the moderator itself. Eventually, from the picture, we can see that the over-moderated area is more stable in case of a loss of moderator.

From a safety point of view, because the system is power-imposed (the thermal power produced depends on the fuel reactions), we cannot control the production of power if not removing the fuel. The best we can do is to control the cooling and the moderation of the reactor.

In case of a loss of coolant, we prefer to stay in an undermoderation condition because the reaction chain will turn off by itself. If the coolant becomes overheated, we lose in terms of density of the coolant and if we are working in the over-moderated area it could involve the divergence of the fission reaction. Then, in case of a depressurization of the primary or secondary system, the coolant starts boiling and to lose temperature; under these conditions, we would rather the over-moderated area.

Second section - Light water power plants

General view

About 85% of nuclear power stations use light water as a moderator, because of the presence of hydrogen and the cheapness of it. At the same time, water can be used also as a coolant. Two types of light water reactors have been developed:

• Pressurized Water Reactor (PWR), in which the water is kept liquid thanks to the high pressure (155 bar). They are very compact, but they need a secondary side in which to create steam that moves a turbine and converts the thermal power. This type of reactor is more difficult to be designed, but it is also easier to be managed with because it uses a single-phase flow;
• Boiling Water Reactor (BWR), which uses boiling water and needs higher fuel reactivity to keep up the neutron balance—for the fluid is less dense. These reactors are the most compact because they produce the steam for the turbine directly into the core. Anyway, they result more difficult to deal with.