## Heat Production in Fuel Elements

In nuclear reactors, there is a **direct proportionality** between the **neutron flux** and the **reactor thermal power**. This proporcionality is determined by the **fission reaction rate** per unit volume (**RR = Ф . Σ**). The fission reaction rate within a nuclear reactor is controlled by several factors. For simplicity let assume that the fissionable material is uniformly distributed in the reactor. In this case, the macroscopic cross-sections are independent of position. Multiplying the **fission reaction rate** per unit volume (**RR = Ф . Σ**) by the **total volume** of the core (V) gives us the** total number of reactions** occurring in the reactor core per unit time. But we also know the amount of energy released per one fission reaction to be about **200 MeV/fission**. Now, it is possible to determine the **rate of energy release **(power) due to the fission reaction. It is given by following equation:

**P = RR . E**_{r}** . V = Ф . Σ**_{f}** . E**_{r}** . V = Ф . N**_{U235}** . σ**_{f}^{235}** . E**_{r}** . V**

where:

**P – reactor power (MeV.s**^{-1}**)**

**Ф – neutron flux (neutrons.cm**^{-2}**.s**^{-1}**)**

**σ – microscopic cross section (cm**^{2}**)**

**N – atomic number density (atoms.cm**^{-3}**)**

**Er – the average recoverable energy per fission (MeV / fission)**

**V – total volume of the core (m**^{3}**)**

In general, the **nuclear fission** results in the release of **enormous quantities of energy**. The amount of energy **depends strongly on** the nucleus to be fissioned and also depends strongly on the kinetic energy of an incident neutron. In order to calculate the power of a reactor, it is necessary to be able precisely identify the** individual components of this energy**. At first, it is important to distinguish between** the total energy released** and **the energy that can be recovered in a**** reactor**.

**The total energy released** in fission can be calculated from binding energies of initial target nucleus to be fissioned and binding energies of fission products. But not all the total energy can be recovered in a reactor. For example, **about 10 MeV** is released in the form of neutrinos (in fact antineutrinos). Since the neutrinos are weakly interacting (with extremely low cross-section of any interaction), they do not contribute to the energy that can be recovered in a reactor.

As can be seen from the table, **the total energy released** in a reactor is **about 210 MeV** per ^{235}U fission, distributed as shown in the table. In a reactor, **the average recoverable energy** per fission is **about 200 MeV**, being the total energy minus the energy of the energy of antineutrinos that are radiated away. This means that **about 3.1****⋅****10**^{10}** fissions per second** are required to produce a power of **1 W**. Since **1 gram** of any fissile material contains about **2.5 x 10**^{21}** nuclei**, the fissioning of 1 gram of fissile material yields **about 1 megawatt-day (MWd)** of heat energy.

As can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is **significant amount of energy generated outside the ****nuclear fuel** (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated** in the coolant (****moderator****)**. This phenomena needs to be included in the nuclear calculations.

For LWR, it is generally accepted that **about 2.5%** of total energy is recovered **in the moderator**. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.

It must be also added, also the other reactor internals must be cooled sufficiently in order to prevent overheating of thir construction materials. One of most exposed components is the neutron reflector, especially the heavy reflector. While acting as a neutron shield, the heavy reflector is heated due to absorption of the gamma radiation. In order to avoid overheating, the heat in the reflector is removed by water flowing through cooling channels drilled through the reflector.

See also: Energy release per fission

See also: Residual Heat

## Temperature Profile – Nuclear Fuel

Most of PWRs use the **uranium fuel**, which is in the form of **uranium dioxide**. Uranium dioxide is a black semiconducting solid with very low thermal conductivity. On the other hand the uranium dioxide has very high melting point and has well known behavior. The UO_{2} is pressed into **cylindrical pellets**, these pellets are then sintered into the solid.

These **cylindrical pellets** are then loaded and encapsulated within a fuel rod (or fuel pin), which is made of zirconium alloys due to its very low absorption cross-section (unlike the stainless steel). The surface of the tube, which covers the pellets, is called **fuel cladding**.

See also: Thermal Conduction of Uranium Dioxide

**Thermal** and **mechanical** behavior of **fuel pellets** and **fuel rods** constitute one of three key core design disciplines. **Nuclear fuel** is operated under very inhospitable conditions (thermal, radiation, mechanical) and must withstand more than normal conditions operation. For example temperatures in the centre of fuel pellets reach more than **1000°C** (1832°F) accompanied by fission-gas releases. Therefore detailed knowledge of temperature distribution within a single fuel rod is essential for safe operation of nuclear fuel. In this section we will study **heat conduction equation** in **cylindrical coordinates** using Dirichlet boundary condition with given surface temperature (i.e. using Dirichlet boundary condition). Comprehensive analysis of fuel rod temperature profile will be studied in separate section.

**Temperature in the centerline of a fuel pellet**

Consider the fuel pellet of radius **r _{U} = 0.40 cm**, in which there is uniform and constant heat generation per unit volume,

**q**. Instead of volumetric heat rate q

_{V }[W/m^{3}]_{V }[W/m

^{3}], engineers often use the

**linear heat rate, q**, which represents the heat rate of one meter of fuel rod. The

_{L}[W/m]**linear heat rate**can be calculated from the volumetric heat rate by:

The centreline is taken as the origin for r-coordinate. Due to symmetry in z-direction and in azimuthal direction, we can separate of variables and simplify this problem to** one-dimensional problem**. Thus, we will solve for the temperature as function of radius, **T(r)**, only. For constant thermal conductivity, k, the appropriate form of the **cylindrical heat equation**, is:

The general solution of this equation is:

where C_{1} and C_{2} are the constants of integration.

**Calculate the temperature distribution, T(r)**, in this fuel pellet, if:

- the temperatures at the surface of the fuel pellet is
**T**_{U}= 420°C - the fuel pellet radius
**r**._{U}= 4 mm - the averaged material’s conductivity is
**k = 2.8 W/m.K**(corresponds to uranium dioxide at 1000°C) - the linear heat rate is
**q**and thus the volumetric heat rate is q_{L}= 300 W/cm_{V}= 597 x 10^{6}W/m^{3}

In this case, the surface is maintained at given temperatures T_{U}. This corresponds to the Dirichlet boundary condition. Moreover, this problem is thermally symmetric and therefore we may use also thermal symmetry boundary condition. The constants may be evaluated using substitution into the general solution and are of the form:

The resulting temperature distribution and the centerline (r = 0) temperature (maximum) in this cylindrical fuel pellet at these specific boundary conditions will be:

The **radial heat flux** at any radius, q_{r} [W.m^{-1}], in the cylinder may, of course, be determined by using the temperature distribution and with the Fourier’s law. Note that, with heat generation the heat flux is no longer independent of r.

The following figure shows the temperature distribution in the fuel pellet at various power levels.

______

The temperature in an operating reactor varies from point to point within the system. As a consequence, there is always **one fuel rod** and **one local volume**, that are **hotter **than all the rest. In order to limit these **hot places** the **peak power limits** must be introduced. The peak power limits are associated with a** boiling crisis** and with the conditions which could cause fuel pellet melt. However, metallurgical considerations place an **upper limits** on the temperature of the fuel cladding and the fuel pellet. Above these temperatures there is a danger that the fuel may be damaged. One of the major objectives in the design of a nuclear reactors is to provide for the removal of the heat produced at the desired power level, while assuring that the maximum fuel temperature and the maximum cladding temperature are always below these predetermined values.