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• We present a deterministic spectral method to predict equilibrium temperature distributions, heat flux, and thermal conductivity in homogeneous and heterogeneous media. We solve the Boltzmann transport equation in a second order spatial, self-adjoint angular flux formulation. We implemented this method into the radiation transport code Rattlesnake, built using the MOOSE (Multiphysics Object Oriented Simulation Environment) framework. The spatial variable is discretized using the continuous finite element method with unstructured meshes, and the angular variable is discretized with the discrete ordinates method. We implemented the diffuse mismatch model in a general geometry to simulate phonon interfacial resistance, using the grey approximation of the Boltzmann transport equation. Using material properties generated by density functional theory and molecular dynamics methods, we were able to elucidate properties of xenon (Xe) at temperatures and pressures experienced in irradiated nuclear fuel. We found Xe to undergo phase change from liquid to solid, and were able to compute coefficients of phonon transmission and reflection at the Xe-UO$_{2}$ interface. We found $\kappa$ to decrease by about a factor of 4 with increasing temperature, agreeing with other trends and research in the open literature. We developed a new method for simulating deterministic, spectral phonon transport to predict heat flux, thermal conductivity, and equilibrium temperature distributions in homogeneous and heterogeneous materials. All the spectral phonon groups are coupled through a local average material temperature, and a new term, $\beta$, is derived and is used as a closure term in the phonon transport equation. $\beta$ acts to redistribute the fraction of total energy which is exchanged between the transport system and equilibrium distribution of phonons. This method predicts thermal conductivity trends in materials spanning geometric domain sizes from nanometers to micrometers, and exhibits the correct asymptotic heat flux behavior as domain size increases. We observed $\beta$ to be the most influential at smaller domain sizes, where equilibrium is difficult to establish due to the proximity of the boundary phonon sources; as domain size increased, $\beta$ diminished in size, and nearly vanished at the maximum domain size of 10 $\mu$m. This further makes the case to perform BTE simulations for nanometer to micrometer heat transfer, as Fourier's law will not accurately capture the heat transfer in such small domain sizes, e.g., thermoelectric devices, heat transfer around defects and heterogeneities in reactor fuel. Additionally, we developed a novel material property discretization scheme which is consistent with the discretization of the angular variable in the transport equation. We performed convergence studies to test the efficiency of the spectral method, which used a modified source iteration (MSI) to solve the linear system of equations. We compare the performance of traditional source iteration (SI) of the uncoupled self-adjoin angular flux method we previously implemented to the new method and comment on the iterative performance of each. We capture ballistic and diffusive phonon scattering, and are able to make comparisons between the accuracy and efficiency of both methods. We find that MSI outperforms SI in most cases, especially as the spatial domain becomes acoustically thick.
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