**3. A simple example**

The following sections demonstrate how finite element analysis can be applied to nuclear systems to calculate extremely complex phenomena. The tools used in these works were ANSYS, for the finite element analysis, and MCNP® was used for the neutron transport [5, 7].

#### **3.1 Godiva-IV**

Confirmation of the methodology described above begins with a relatively simple geometry reactor. While certainly not a homogeneous single component

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

bare cylindrical reactor, the Godiva-IV reactor provides an excellent case to apply the methodology described above and compare the modeling results to measured results. In particular, the Godiva-IV reactor temperature coefficient of reactivity is dominated by thermal expansion with limited mechanical interaction effects. Furthermore, the Godiva-IV reactor has been thoroughly characterized and detailed descriptions are available to allow construction of both the FEA model as well as the Monte Carlo model. Finally, detailed temperature measurements and the corresponding reactivity have been recorded which allow for comparison with the modeling results.

## *3.1.1 Thermal analysis*

The thermal analysis required several boundary conditions as inputs into the model. The most important was the temperature data taken from an experiment on the Godiva-IV. The temperature data provided the pulse shape and total magnitude of the temperature rise. It also conveyed the time dependency of the FEA model. Experimental measurements are important to creating accurate models. They provide a more accurate input, depending on the quality of the measurement, than necessarily calculating and input from first principles. The experiment input data were for a 1.029\$ reactivity insertion with a 68°C temperature rise over 300 seconds.

The second type of boundary conditions applied were the heat transfer coefficients, primarily the convection coefficients. These were hand calculated from heat and mass transfer equations. Hand-calculating these coefficients is normally required because these values are not normally measured for these facilities. For Godiva-IV specifically, capturing the thermal expansion temperature coefficient means modeling the thermal conduction paths as well as the convection into the room. The temperature differential of all of the components as they heat up and subsequently cool due to convection to the room is the primary driver to the complexity of the expansion. **Figure 4** shows the thermal FEA results and the temperature differentials. For more information specifically on the boundary conditions, see the reference [8].

Given the rapid structural response of the Godiva-IV pulse, the analysis type chosen was transient. The solution steps were focused on the pulse. Of the 400 solution steps, 300 steps surrounded the pulse that consisted of 10s, with the other 100 steps covering 290 seconds. The reason for this was that the model was rapidly changing during the initial nuclear heating, while the rest of the steps only contained relatively slow thermal conduction. After the temperature analysis model completed, the temperature data were exported to the structural analysis model.

#### *3.1.2 Structural analysis*

As stated previously, the structural boundary conditions are less numerous but can be more difficult to determine. For Godiva-IV, the primary structural boundary conditions were, support of the safety block, control rods, and providing a fixed support for the back side of the clamps. These boundary conditions were more straightforward than for typical models, weak springs were not required, and fixed supports were the only type of boundary conditions that were necessary. **Figure 5** shows exaggerated displacement at the end of the temperature input data. The exaggeration was required because the structural displacement on average was 0.2 mm and imperceptible to the human eye.

The structural analysis had similar timestepping as the thermal analysis. The data generated from the FEA were a set of averaged displacements on particular

#### **Figure 4.** *GODIVA-IV thermal analysis results.*

surfaces. These surfaces surrounded the curved faces of the fuel rings. The fuel rings were the focus because only the fuel movement and expansion matters to the neutronics of the reactor.

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

**Figure 6.**

*Godiva-IV temperature coefficient result.*


#### **Table 1.**

*Measured and calculated Godiva-IV temperature coefficient.*

#### *3.1.3 Neutron transport*

The exported data were applied to the neutron transport model where a series of models were created, each with a different set of displacements per an average temperature. These results are shown in **Figure 6**. The slope of the linear regression is −2E−05 Δk/k/°C. The comparison to the measured value is shown in **Table 1**. This is the temperature coefficient of the Godiva-IV reactor. The results demonstrate that a coupling method of FEA and Monte Carlo Neutron Transport has to be potential to accurately predict the temperature coefficient.
