**1. Introduction**

Nuclear reactors exhibit remarkably complicated behavior ultimately originating from the energy released through the nuclear fission process. The complicated behavior involves many phenomena including nuclear, thermal, and mechanical. Individually, these phenomena involve processes that are challenging to quantify, measure, and model. When interactions between these phenomena are considered, the quantification, measurement, and modeling challenges become daunting. This chapter describes finite element analysis (FEA) coupled with Monte Carlo analysis as a methodology for quantification of a particularly important nuclear parameter which is primarily influenced by thermal and mechanical phenomena present in nuclear reactors.

#### **1.1 Background**

The multiplication factor, k, is used to quantify the fission chain reaction in nuclear reactors. Numerous definitions exist for k, with each definition applying to a particular situation. A simple definition of k is that it represents the ratio of the number of fissions in one generation to the number of fissions in the preceding generation. Through this definition, one can see that if k is less than unity, the number of fissions declines over time, and if k is greater than unity, the number of fissions increases over time. A unique situation exists when k is exactly equal to one. In that case, the number of fissions remains constant over time and is referred to as critical.

A companion parameter to k is reactivity, ρ. Reactivity represents the deviation from the critical state, as shown in Eq. (1).

$$
\rho = \frac{\left(k - 1\right)}{k} \tag{1}
$$

The decimal form of reactivity can be converted to units of \$ by dividing the decimal value by the fraction of delayed neutrons resulting from the fission process. Delayed neutrons are those neutrons emitted during the decay of select radioactive fission products rather than being emitted at the moment of fission. For uranium-235, the delayed neutron fraction is 0.0065.

When operating a nuclear reactor, frequently, one is interested in knowing the change in reactivity resulting from various activities such as control rod movements. Other changes resulting from thermal and mechanical phenomena can produce reactivity changes. Frequently, these reactivity changes are quantified in terms of the change in reactor temperature. The result is known as a temperature coefficient of reactivity defined by Eq. (2).

$$
\alpha\_T = \frac{\Delta \rho}{\Delta T} \tag{2}
$$

The temperature coefficient of reactivity can be further subdivided into explicit subjects such as coolant temperature, fuel temperature, and even thermally driven reactor geometry changes.

From a reactor safety perspective, a negative temperature coefficient is indicative of inherently stability. If a reactor transient was initiated that results in a temperature increase, the resulting change in reactivity will necessarily be negative, which means the multiplication factor will be reduced. Eventually, the temperature increase will produce a sufficient reduction in k such that the reactor will shut down. Contrarily, a positive reactivity temperature coefficient is indicative of inherent instability. With a positive coefficient, a transient resulting in a reactor temperature increase will result in a positive reactivity change and a resulting increase in the multiplication factor. The increased multiplication factor will be accompanied by an increase in the number of fissions and resulting heat release and corresponding temperature increase which will subsequently produce an additional

*Nuclear Reactor Thermal Expansion Reactivity Effect Determination Using Finite Element… DOI: http://dx.doi.org/10.5772/intechopen.93762*

positive change in reactivity. The reactor will continue on this path until it is acted upon by a more dominate negative action or the reactor will ultimately be damaged or even destroyed.

Reactivity coefficients can be determined for numerous phenomena. For example, reactivity coefficients can be established for changes in reactor power. Thus, as the reactor power is increased, the reactivity change needed to compensate for the power change can be identified. Another interesting phenomenon that has a significant reactivity effect centers on bubble or void formation as the result of coolant boiling. Reactor designers must pay particular attention to the reactivity effect associated with coolant bubble formation because it can have a significant safety impact.

In the case of reactors that use water as a coolant, the water has a significant effect on the overall neutron energy spectrum in the reactor. Neutrons tend to be born at high energies on the order of several million electron volts. As the neutrons collide with various nuclei in the reactor, they tend to lose energy in a process called moderation. As the neutrons lose energy, they become more likely to be absorbed in uranium-235, which can then fission and release additional neutrons. Similar to the three different regimes for k, there are three regimes for moderation: undermoderation, optimum-moderation, and over-moderation. If a reactor is designed with under-moderation, the loss of coolant through bubble formation will result in a reactivity decrease because fewer neutrons will be slowed to energies where they are more likely to be absorbed in uranium-235. In the case of a reactor designed with over-moderation, the formation of bubbles will tend to result in a reactivity increase because more neutrons will be slowed to the point where they will be absorbed by uranium-235 causing an increase in the number of fissions.

The most dramatic and tragic demonstration of positive reactivity due to bubble formation was seen in the 1986 Chernobyl accident. When operated at low power, the Chernobyl reactor had a positive void reactivity coefficient. Thus, if the reactor coolant began to boil, the bubbles created by the coolant boiling led to a positive reactivity change thereby driving an increase in the multiplication factor and a corresponding increase in the number of fissions occurring in the reactor. The heat released from the additional fissions led to additional coolant boiling which drove a very rapid power increase and subsequent steam explosion and reactor destruction.

Reactivity coefficients tied to geometry changes are of interest in certain situations because they are typically fast acting and can have important safety implications. In many cases, thermally driven geometry changes are coupled with resulting mechanical interactions that severely complicate quantification and modeling approaches. While the change in geometry causes the reactivity change, typically temperature is used to quantify the reactivity coefficient since it is a change in temperature that causes thermal expansion and mechanical interaction. Thus, a reactor may have a thermal expansion temperature coefficient of reactivity or even more specifically a thermal expansion/mechanical interaction temperature coefficient of reactivity. The thermal expansion temperature coefficient of reactivity in two reactors is described below followed by a demonstration of using finite element analysis to model the thermally driven geometric changes followed by use of a Monte Carlo simulation to determine the corresponding multiplication factor value.

#### **1.2 Godiva-IV and Experimental Breeder Reactor-II**

Two reactors serve as the test bed for evaluating the analysis approach described in this chapter. One reactor a uses comparatively simple design and the other is significantly more complicated. The simple reactor design is called Godiva-IV. The Godiva-IV reactor, see **Figure 1**, is unique in that it is designed to provide a burst

**Figure 1.** *Godiva-IV reactor [1].*

of neutrons rather than being designed for extended steady state power production. The Godiva-IV reactor is very compact with a simple cylindrical shape with a 178 mm diameter and a 156 mm height. The reactor design uses a solid construction of approximately 66 kg of 93% enriched uranium alloyed with 1.5 wt.% molybdenum. No active cooling arrangement is used. The reactor construction is somewhat more complicated than a monolithic cylinder of enriched uranium. The Godiva-IV reactor uses six ostensibly equal rings. Three stacked cylinders of differing heights are located within the six rings to complete the overall cylindrical shape. Three large C-clamps are attached to the outer radius for the reactor to restrain the fuel movement during burst operations.

When the Godiva-IV reactor is operated, a large power pulse occurs and heat from the fission process is deposited in the uranium alloy. The heat causes a temperature increase and subsequent thermal expansion. As the individual components of the reactor expand, they mechanically interact. As the reactor components expand, neutron leakage from the reactor increases which leads to a decrease in the multiplication factor and subsequent termination of the reactor power pulse. Thus, Godiva-IV has a negative reactivity temperature coefficient. That is, as the reactor temperature increases, the resulting reactivity change is negative which provides an inherent shutdown mechanism.

The other reactor used to evaluate the analysis approach described in this chapter is the Experimental Breeder Reactor-II (EBR-II) [2]. The EBR-II design is significantly more complicated than Godiva-IV, see **Figure 2**. EBR-II uses liquid sodium metal as the coolant. The fuel is 67% enriched uranium metal alloyed with a collection of various metals totaling 5 wt.%. The fuel is formed into individual 3.3-mm diameter pins along with stainless steel cladding. The fuel portion of the pins is 343 mm long while the cladding portion is 638 mm long. The additional length of the cladding allows for the containment of fission product gasses. A collection of 91 fuel pins are arranged into a hexagonal configuration which is commonly referred to an assembly. The 91 fuel pins in each assembly are contained within a stainless-steel hexagonal duct. The EBR-II reactor core consists of an arrangement of 637 assemblies. The core is fundamentally divided into two regions, a driver region containing the fissile material, and a blanket region containing depleted uranium. Within the driver region there are approximately 100 assemblies including control rods, experimental assemblies, stainless steel dummy assemblies, stainless steel reflector assemblies, and assemblies that use reduced fuel content. Surrounding the driver region is a collection of approximately 500 assemblies constructed of depleted uranium. The depleted uranium assemblies absorb neutrons that leak for the driver region to transmute depleted uranium to plutonium to breed new reactor fuel.

*Nuclear Reactor Thermal Expansion Reactivity Effect Determination Using Finite Element… DOI: http://dx.doi.org/10.5772/intechopen.93762*

As EBR-II ascends to its operating power, heat from the fission process causes the fuel pins, hexagonal ducts, and all other components of the reactor to expand due to the temperature increase. The components undergo a complicated process involving thermal expansion and mechanical interaction. While the Godiva-IV thermal expansion process which leads to comparatively simple thermal expansion and mechanical interaction, the thermal expansion and mechanical interactions in EBR-II are significantly more complicated and must be subdivided into different areas. One area that is comparatively simple to understand and evaluate is the spacing of the assemblies in the hexagonal arrangement. As the reactor temperature increases during the reactor assent to power, the grid plate that holds the fuel assemblies thermally expands and the spacing between the fuel assemblies increases which results in increased neutron leakage and a decrease in the multiplication factor. A much more complicated process involves the thermal expansion and mechanical interaction of the stainlesssteel hexagonal assembly ducts. Measuring and calculating the reactivity effect of the hexagonal duct thermal expansion and mechanical interaction is particularly challenging. The analysis method described in this chapter is used to evaluate the reactivity coefficient associated with the thermal expansion driven spacing of the assemblies along with the much more complicated reactivity coefficient associated with the thermal expansion and mechanical interaction of the fuel assembly hexagonal ducts.
