*4.1.1 Thermal analysis*

EBR-II required more extensive thermal boundary conditions than Godiva-IV which was considered the simple system because the heating was simple conduction through the materials and the ultimate heat sink was convection into the room air. EBR-II was more complicated because there was forced convection using liquid sodium that flowed over the fuel elements. The ultimate heat sink was a series of heat exchangers that cooled the sodium. Heating was also not symmetric from assembly to assembly. Modeling this complex behavior would require a complex thermal-hydraulic model to simulate the various coolant channels. Creating this model would have substantially complicated the thermal FEA analysis and additional would require input information that was not measured at EBR-II. Instead, a simple cooling model was developed for each assembly. The cooling model stated

**Figure 7.** *EBR-II simplified simulation model.*

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

that the sodium inside of the duct entered the bottom of the coolant channel as cold sodium, and over the part of the channel where the fuel was located, the sodium was heated such that the outlet temperature match measurements taken at the EBR-II. Each fuel assembly type had a different cooling profile. **Figure 7** shows the different assembly types in the EBR-II FEA model.

Simplifying the coolant channels in this manner was sufficient because previous work done on the EBR-II suggested that duct-bowing was entirely driven by the temperature profile of the duct material and not by the internal structures. Thus, only the duct needed to be heated correctly.

The power input for EBR-II was derived from a linear interpolation of the ascent to power. All of the heat generation inputs were linearly scaled over timesteps. The timing did not match the real ascent to power, but that was not necessary since the model would be in thermal equilibrium for each calculated step. The more important aspect was that the thermal model would be a series of steps, each step corresponding to a different power level.

#### *4.1.2 Structural analysis*

The structural analysis required a simple boundary condition to hold the model in place, as well as a boundary condition to fix the center duct. Fixing the center duct meant that it was not allowed to thermally expand and was considered a rigid body. This was necessary to achieve convergence. Without fixing the center duct, the model could not resolve the contact overlap that existed between the ducts on the first solution step.

The structural FEA required significantly more time to solve than Godiva-IV due to the sheer size of the model (~5 million nodes) and the complexity of the thermal expansion. **Figure 8** shows an exaggerated displacement of the ducts. The southeast quadrant shows how the differences in the assembly, types, and powers can impact thermal expansion. Additionally, it demonstrates why FEA was necessary to capture all of the geometric detail of the duct-bowing temperature coefficient.

**Figure 8.** *EBR-II exaggerated structural displacement.*

## *4.1.3 Neutron transport*

Similar to the Godiva-IV model, the displacement data were exported out of ANSYS and imported into a series of MCNP® models [7]. The major difference was in the translation method. The Godiva-IV translation was averaging nodal thermal expansion and manually applying the change in radii and heights to the MCNP® input files. That approach was prohibitive for EBR-II because the resulting data exceeded 1 TB. A custom code called MCNP® Input Card and KCODE Architect (MICKA) was written to perform the node translation and MCNP® input construction. The MCNP® model for the EBR-II was itself expansive and required special data handling. More inf0rmation can be found in the reference [6].

One additional difficulty with the translation was that MCNP® cannot model a bowed-duct, only a straight hexagonal duct. To overcome this geometry limitation,

**Figure 9.** *Axial sections to simulate a bowed-duct in MCNP®.*

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

**Figure 10.** *Temperature coefficient results for duct-bowing coefficient.*

the straight duct was divided into axial sections. Each axial section was moved in space to approximate a bowed-duct. **Figure 9** shows an example of the axial slices in an assembly.

After the translation of the nodal data from the FEA to MCNP®, the analysis process was similar to that of Godiva-IV. A series of snapshots at various bulk temperatures were taken and a linear regression was performed to calculate the slope of the points. **Figure 10** shows the results of the reactivity change per degree. The coefficient was calculated to be −1.4E−03 \$/°C. While the data had a clear linear trend, some nonlinearity existed in sets of data points at lower bulk temperatures. This was consistent with historical measurements at EBR-II where lower powers exhibited a nonlinear trend in the reactivity change.

#### **5. Conclusions**

The energy released from the nuclear fission process drives complicated thermal expansion and mechanical interactions in nuclear reactors. These expansions and interactions subsequently cause changes in the neutron chain reaction balance within a reactor which results in further changes in thermal expansion and mechanical interactions. Measurement of these coupled phenomena occurring within a reactor has proven to be elusive. However, coupling finite element analysis with Monte Carlo neutron transport analysis provides a pathway to simulate the thermal expansion and mechanical interaction driven by the energy released in the neutron-induced fission process and then to subsequently determine fundamental nuclear parameters, namely, thermal expansion temperature coefficient of reactivity.

There are important safety implications associated with the thermal expansion temperature coefficient of reactivity and its relation to other temperature coefficients of reactivity. Knowing both the sign and magnitude of individual coefficients allows reactor designers to predict how a reactor will behave under transient conditions.

Using the coupling of finite element analysis and Monte Carlo neutron transport analysis, the thermal expansion temperature coefficient of reactivity was determined for the Godiva-IV reactor and found to be within 3% of the experimentally measured value.

The coupling technique was also used to determine the thermal expansion temperature coefficient of reactivity for EBR-II. The thermal expansion and mechanical interactions within EBR-II are sufficiently complex that experimentally measuring the isolated thermal expansion temperature coefficient of reactivity was not possible. However, using the coupling technique, a calculated value of −1.4E−03 \$/°C was determined for the thermal expansion temperature coefficient of reactivity. This result fits well with integral EBR-II reactivity coefficient measurements.

With the Godiva-IV comparison results and the EBR-II results, it can be concluded that coupling finite element analysis with Monte Carlo neutron transport analysis provides a powerful technique for determining important reactor safety parameters. The technique can be applied to existing reactors and reactors proceeding through the design process which gives reactor operators and designers greater confidence in reactor operating characteristics and safety margins.
