Date of Award


Degree Type


Degree Name

Doctor of Philosophy (PhD)


Nuclear Engineering

Committee Chair

Won Sik Yang

Committee Member 1

Chan Choi

Committee Member 2

Hany S. Abdel-Khalik

Committee Member 3

Xiangxiong Zhang

Committee Member 4

Taek K. Kim


In order to meet the worldwide demands on sustainable energy, new generation of nuclear reactor systems has been proposed to continue the benefit of nuclear power for electricity production. Sodium-cooled fast reactor (SFR) is one of the most promising candidate among those reactor concepts for both energy production and minimizing radioactive nuclear waste and features inherent safety potential. To ensure inherent safety characteristics of SFR, accurate evaluation of reactivity feedbacks is essential. However, accurate evaluation of the negative reactivity induced by the core radial expansion and assembly bowing remains challenging. SFR has a relatively large coolant temperature rise, and thus the resulting temperature gradients lead to differential thermal expansion of the assembly duct walls that, along with the irradiation creep and swelling, result in the bowing of assembly. The deformation of assembly induces feedback reactivity that, with careful design of the core constraint system, is often a significant portion of negative feedback reactivity to guarantee inherent safety. However, during the operation of the SFR, a large number of assemblies are distorted in a complex manner, making the prediction of the bowing reactivity a very difficult task. The large uncertainty in predicting the negative assembly bowing reactivity limits the range of design space and inhibits the incorporation of the innovative features in advanced SFR design.

The objective of this doctoral work is to develop a perturbation theory method for accurate and efficient evaluation of the feedback reactivity induced by SFR core deformation. The precalculated assembly bowing reactivity coefficient could be adopted by the safety analysis tools for more realistic modeling of the SFR dynamic behavior during anticipated transient without scram (ATWS) events. The improved simulation accuracies will contribute to reducing the economic penalties due to the conservative design margins to accommodate the prediction uncertainties.

The major challenge in using perturbation theory method to calculate the assembly bowing reactivity worth is the evaluation of the bi-linearly weighted reaction rate integral for a perturbed geometry. Direct adoption of the conventional perturbation theory method based on the material property changes in the original mesh grid requires material homogenization in the meshes where discontinuous material boundaries are included due to assembly displacement. Especially for applications with realistic heterogeneous assembly model, the boundaries for discontinuous materials such as fuel, cladding, coolant, duct, etc. would intersect with millions of meshes, making numerical evaluations of the reaction integrals within such meshes extremely difficult. In this dissertation, we bypass this obstacle by formulating the first-order perturbation theory under the ‘Lagrangian frame of reference’, noticing that, in this case, the material is not changed in property but only displaced. The equivalence of such formula is shown to the conventional firstorder perturbation theory formula by applying a coordinate transformation to map the original mesh in ‘Eulerian coordinates’ to ‘Lagrangian coordinates’. The new formula provides unique convenience for modeling heterogeneous assembly displacements. The perturbation theory formulation is based on neutron transport approximation, which is more adequate than diffusion approximation for analyzing fast reactor where the anisotropic scattering effect is large.

The key practice of the proposed perturbation theory method is to convert the problem of calculating material property perturbation to the problem of evaluating the difference of the fluxes to which materials are exposed at original and perturbed locations. To determine the flux distributions, we employed the variational nodal transport code VARIANT and the unstructured geometry transport code PROTEUS-SN. Continuous forward and adjoint flux distributions in the core are determined from VARIANT calculations with a core model of homogenized assemblies. The heterogeneous assembly fluxes at both original and shifted positions are then reconstructed by combining the VARIANT solution with the finite element-based form functions of scalar flux obtained from PROTEUS-SN single assembly calculations. The discontinuous material distribution in an assembly is also represented based on the same finite element meshes adopted in the PROTEUS-SN code. Numerical tests showed that these reconstructed fluxes agree well with the full core heterogeneous solution of the PROTEUS-SN code.

We developed a computer code named RAINBOW (ReActivity INduced by assembly BOWing) applying the proposed perturbation theory method. RAINBOW reads forward and adjoint fluxes from VARIANT and PROTEUS-SN solutions and reconstructs the heterogeneous flux distributions, with which it computes assembly segment displacement worth coefficients in six directions normal to the hexagonal assembly duct walls. RAINBOW also has a capability to calculate the reactivity worth of fuel axial expansion. The fuel axial expansion module applies the material density perturbation to each axially expanded assembly segment.

The verification of the RAINBOW code was performed by comparing the perturbation results with the direct subtractions between eigenvalues of both unperturbed and perturbed cases. The reference eigenvalues were calculated by both MCNP6 Monte Carlo simulation and PROTEUS-SN deterministic calculation, both of which can model distorted heterogeneous assembly configurations. The verification tests are performed based on two-dimensional (2D) and threedimensional (3D) mini-core problems that are derived from Advanced Burner Test Reactor (ABTR). Statistical analysis indicated that the RAINBOW results are statistically consistent with the MCNP6 results. The perturbation calculation results also agree well with the PROTEUS-SN results.

In summary, this doctoral work developed a transport theory-based perturbation theory method resolving the long-standing challenge of effectively assessing feedback reactivity effect due to core geometry change. The practice of the proposed perturbation method achieves the same level of accuracy for three-dimensional heterogeneous core with several orders of magnitude smaller computational time compared to the direct subtraction of eigenvalues obtained from whole-core transport simulations with MCNP or PROTEUS-SN.