El principal objetivo de esta tesis es el desarrollo de un Método Modal para resolver dos ecuaciones: la Ecuación de la Difusión de Neutrones y la de las Ordenadas Discretas del Transporte de Neutrones. Además, este método está basado en el Método de Volúmenes Finitos para discretizar las variables espaciales. La solución de estas ecuaciones proporciona el flujo de neutrones, que está relacionado con la potencia que se produce en los reactores nucleares, por lo que es un factor fundamental para los Análisis de Seguridad Nuclear. Por una parte, la utilización del Método Modal está justificada para realizar análisis de inestabilidades en reactores. Por otra parte, el uso del Método de Volúmenes Finitos está justificado por la utilización de este método para resolver las ecuaciones termohidráulicas, que están fuertemente acopladas con la generación de energía en el combustible nuclear. En primer lugar, esta tesis incluye la definición de estas ecuaciones y los principales métodos utilizados para resolverlas. Además, se introducen los principales esquemas y características del Método de Volúmenes Finitos. También se describen los principales métodos numéricos para el Método Modal, que incluye tanto la solución de problemas de autovalores como la solución de Ecuaciones Diferenciales Ordinarias dependientes del tiempo. A continuación, se desarrollan varios algoritmos del Método de Volúmenes Finitos para el Estado Estacionario de la Ecuación de la Difusión de Neutrones. Se consigue desarrollar una formulación multigrupo, que permite resolver el problema de autovalores para cualquier número de grupos de energía, incluyendo términos de upscattering y de fisión en varios grupos de energía. Además, se desarrollan los algoritmos para realizar la computación en paralelo. La solución anterior es la condición inicial para resolver la Ecuación de Difusión de Neutrones dependiente del tiempo. En esta tesis se utiliza un Método Modal, que transforma el Sistema de Ecuaciones Diferenciales Ordinarias en uno de mucho menor tamaño, que se resuelve con el Método de la Matriz Exponencial. Además, se ha desarrollado un método rápido para estimar el flujo adjunto a partir del directo, ya que se necesita en el Método Modal. Por otra parte, se ha desarrollado un algoritmo que resuelve el problema de autovalores de la Ecuación del Transporte de Neutrones. Este algoritmo es para la formulación de Ordenadas Discretas y el Método de Volúmenes Finitos. En concreto, se han aplicado dos tipos de cuadraturas para las Ordenadas Discretas y dos esquemas de interpolación para el Método de Volúmenes Finitos. Finalmente, se han aplicado estos métodos a diferentes tipos de reactores nucleares, incluyendo reactores comerciales. Se han evaluado los valores de la constante de multiplicación y de la potencia, ya que son las variables fundamentales en los Análisis de Seguridad Nuclear. Además, se ha realizado un análisis de sensibilidad de diferentes parámetros como la malla y métodos numéricos. En conclusión, se obtienen excelentes resultados, tanto en precisión como en coste computacional. / The main objective of this thesis is the development of a Modal Method to solve two equations: the Neutron Diffusion Equation and the Discrete Ordinates Neutron Transport Equation. Moreover, this method uses the Finite Volume Method to discretize the spatial variables. The solution of these equations gives the neutron flux, which is related to the power produced in nuclear reactors; thus, the neutron flux is a paramount variable in Nuclear Safety Analyses. On the one hand, the use of Modal Methods is justified because one uses them to perform instability analyses in nuclear reactors. On the other hand, it is worth using the Finite Volume Method because one uses it to solve thermalhydraulic equations, which are strongly coupled with the energy generation in the nuclear fuel. First, this thesis defines the equations mentioned above and the main methods to solve these equations. Furthermore, the thesis describes the major schemes and features of the Finite Volume Method. In addition, the author also introduces the major methods used in the Modal Method, which include the methods used to solve the eigenvalue problem, as well as those used to solve the time dependent Ordinary Differential Equations. Next, the author develops several algorithms of the Finite Volume Method applied to the Steady State Neutron Diffusion Equation. In addition, the thesis includes an improvement of the multigroup formulation, which solves problems involving upscattering and fission terms in several energy groups. Moreover, the author optimizes the algorithms to do calculations with parallel computing. The previous solution is used as initial condition to solve the time dependent Neutron Diffusion Equation. The author uses a Modal Method to do so, which transforms the Ordinary Differential Equations System into a smaller system that is solved by using the Exponential Matrix Method. Furthermore, the author developed a computationally efficient method to estimate the adjoint flux from the forward one, because the Modal Method uses the adjoint flux. Additionally, the thesis also presents an algorithm to solve the eigenvalue problem of the Neutron Transport Equation. This algorithm uses the Discrete Ordinates formulation and the Finite Volume Method. In particular, the author uses two types of quadratures for the Discrete Ordinates and two interpolation schemes for the Finite Volume Method. Finally, the author tested the developed methods in different types of nuclear reactors, including commercial ones. The author checks the accuracy of the values of the crucial variables in Nuclear Safety Analyses, which are the multiplication factor and the power distribution. Furthermore, the thesis includes a sensitivity analysis of several parameters, such as the mesh and numerical methods. In conclusion, excellent results are reported in both accuracy and computational cost. / El principal objectiu d'esta tesi és el desenvolupament d'un Mètode Modal per a resoldre dos equacions: l'Equació de Difusió de Neutrons i la de les Ordenades Discretes del Transport de Neutrons. A més a més, este mètode està basat en el Mètode de Volums Finits per a discretitzar les variables espacials. La solució d'estes equacions proporcionen el flux de neutrons, que està relacionat amb la potència que es produïx en els reactors nuclears; per tant, el flux de neutrons és un factor fonamental en els Anàlisis de Seguretat Nuclear. Per una banda, la utilització del Mètode Modal està justificada per a realitzar anàlisis d'inestabilitats en reactors. Per altra banda, l'ús del Mètode de Volums Finits està justificat per l'ús d'este mètode per a resoldre les equacions termohidràuliques, que estan fortament acoblades amb la generació d'energia en el combustible nuclear. En primer lloc, esta tesi inclou la definició d'estes equacions i els principals mètodes utilitzats per a resoldre-les. A més d'això, s'introduïxen els principals esquemes i característiques del Mètode de Volums Finits. Endemés, es descriuen els principals mètodes numèrics per al Mètode Modal, que inclou tant la solució del problema d'autovalors com la solució d'Equacions Diferencials Ordinàries dependents del temps. A continuació, es desenvolupa diversos algoritmes del Mètode de Volums Finits per a l'Estat Estacionari de l'Equació de Difusió de Neutrons. Es conseguix desenvolupar una formulació multigrup, que permetre resoldre el problema d'autovalors per a qualsevol nombre de grups d'energia, incloent termes d' upscattering i de fissió en diversos grups d'energia. A més a més, es desenvolupen els algoritmes per a realitzar la computació en paral·lel. La solució anterior és la condició inicial per a resoldre l'Equació de Difusió de Neutrons dependent del temps. En esta tesi s'utilitza un Mètode Modal, que transforma el Sistema d'Equacions Diferencials Ordinàries en un problema de menor tamany, que es resol amb el Mètode de la Matriu Exponencial. Endemés, s'ha desenvolupat un mètode ràpid per a estimar el flux adjunt a partir del directe, perquè es necessita en el Mètode Modal. Per altra banda, s'ha desenvolupat un algoritme que resol el problema d'autovalors de l'Equació de Transport de Neutrons. Este algoritme és per a la formulació d'Ordenades Discretes i el Mètode de Volums Finits. En concret, s'han aplicat dos tipos de quadratures per a les Ordenades Discretes i dos esquemes d'interpolació per al Mètode de Volums Finits. Finalment, s'han aplicat estos mètodes a diversos tipos de reactors nuclears, incloent reactors comercials. S'han avaluat els valor de la constat de multiplicació i de la potència, perquè són variables fonamentals en els Anàlisis de Seguretat Nuclear. Endemés, s'ha realitzat un anàlisi de sensibilitat de diversos paràmetres com la malla i mètodes numèrics. En conclusió, es conseguix obtenir excel·lents resultats, tant en precisió com en cost computacional. / Bernal García, Á. (2018). Development of a 3D Modal Neutron Code with the Finite Volume Method for the Diffusion and Discrete Ordinates Transport Equations. Application to Nuclear Safety Analyses [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/112422

A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations Scott W. Mosher 110 Pages Directed by Dr. Farzad Rahnema It seems very likely that the next generation of reactor analysis methods will be based largely on neutron transport theory, at both the assembly and core levels. Signifi-cant progress has been made in recent years toward the goal of developing a transport method that is applicable to large, heterogeneous coarse-meshes. Unfortunately, the ma-jor obstacle hindering a more widespread application of transport theory to large-scale calculations is still the computational cost. In this dissertation, a variational heterogeneous coarse-mesh transport method has been extended from one to two-dimensional Cartesian geometry in a practical fashion. A generalization of the angular flux expansion within a coarse-mesh was developed. This allows a far more efficient class of response functions (or basis functions) to be employed within the framework of the original variational principle. New finite element equations were derived that can be used to compute the expansion coefficients for an individual coarse-mesh given the incident fluxes on the boundary. In addition, the non-variational method previously used to converge the expansion coefficients was developed in a new and more thorough manner by considering the implications of the fission source treat-ment imposed by the response expansion. The new coarse-mesh method was implemented for both one and two-dimensional (2-D) problems in the finite-difference, multigroup, discrete ordinates approximation. An efficient set of response functions was generated using orthogonal boundary conditions constructed from the discrete Legendre polynomials. Several one and two-dimensional heterogeneous light water reactor benchmark problems were studied. Relatively low-order response expansions were used to generate highly accurate results using both the variational and non-variational methods. The expansion order was found to have a far more significant impact on the accuracy of the results than the type of method. The varia-tional techniques provide better accuracy, but at substantially higher computational costs. The non-variational method is extremely robust and was shown to achieve accurate re-sults in the 2-D problems, as long as the expansion order was not very low.

Preconditioners based upon sweeps and diffusion-synthetic acceleration (DSA) have been constructed and applied to the zeroth and first spatial moments of the 1-D transport equation using SN angular discretization and a strictly positive nonlinear spatial closure (the CSZ method). The sweep preconditioner was applied using the linear discontinuous Galerkin (LD) sweep operator and the nonlinear CSZ sweep operator. DSA preconditioning was applied using the linear LD S2 equations and the nonlinear CSZ S2 equations. These preconditioners were applied in conjunction with a Jacobian-free Newton Krylov (JFNK) method utilizing Flexible GMRES. The action of the Jacobian on the Krylov vector was difficult to evaluate numerically with a finite difference approximation because the angular flux spanned many orders of magnitude. The evaluation of the perturbed residual required constructing the nonlinear CSZ operators based upon the angular flux plus some perturbation. For cases in which the magnitude of the perturbation was comparable to the local angular flux, these nonlinear operators were very sensitive to the perturbation and were significantly different than the unperturbed operators. To resolve this shortcoming in the finite difference approximation, in these cases the residual evaluation was performed using nonlinear operators "frozen" at the unperturbed local psi. This was a Newton method with a perturbation fixup. Alternatively, an entirely frozen method always performed the Jacobian evaluation using the unperturbed nonlinear operators. This frozen JFNK method was actually a Picard iteration scheme. The perturbed Newton's method proved to be slightly less expensive than the Picard iteration scheme. The CSZ sweep preconditioner was significantly more effective than preconditioning with the LD sweep. Furthermore, the LD sweep is always more expensive to apply than the CSZ sweep. The CSZ sweep is superior to the LD sweep as a preconditioner. The DSA preconditioners were applied in conjunction with the CSZ sweep. The nonlinear CSZ DSA preconditioner did not form a more effective preconditioner than the linear DSA preconditioner in this 1-D analysis. As it is very difficult to construct a CSZ diffusion equation in more than one dimension, it will be very beneficial if the results regarding the effectiveness of the LD DSA preconditioner are applicable to multi-dimensional problems.

Multi-core architectures are present throughout a large selection of computing devices from cell phones to super-computers. Parallel applications running on these devices solve bigger problems in a shorter time. Writing those applications is a difficult task for programmers. They need to deal with low-level parallel mechanisms such as data distribution, inter-processor communication, and task placement. The goal of the Standard Template Adaptive Parallel Library (STAPL) is to provide a generic high-level framework to develop parallel applications. One of the first steps of a parallel application is to partition and distribute the data throughout the system. An important data structure for parallel applications to store large amounts of data and model many types of relations is the graph. A mesh, which is a special type of graph, is often used to model a spatial domain in scientific applications. Graph and mesh partitioning has many applications such as VLSI circuit design, parallel task scheduling, and data distribution. Data distribution, significantly impacts the performance of a parallel application. In this thesis, we introduce the STAPL Parallel Graph Partitioner Framework. This framework provides a generic infrastructure to partition arbitrary graphs and meshes and to build customized partitioners. It includes the state of the art parallel k-way multilevel scheme to partition arbitrary graphs, a parallel mesh partitioner with parameterized partition shape, and a customized partitioner used for discrete ordinates particle transport computations. This framework is also part of a generic library, STAPL, allowing the partitioning of the data and development of the whole parallel application to be done in the same environment. We show the user-friendly interface of the framework and its scalability for partitioning different mesh and graph benchmarks on a Cray XE6 system. We also highlight the performance of our customized unstructured mesh partitioner for a discrete ordinates particle transport code. The developed columnar decompositions significantly reduce the execution time of simultaneous sweeps on unstructured meshes.

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional. / In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional. / In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.

Neste trabalho, o fenômeno de evaporação em gases rarefeitos e analisado, para o caso de uma espécie de gás bem como de misturas binárias. Evaporação fraca e forte são consideradas para escoamentos de gases em canal e semi-espaco. Também e investigado o fenômeno conhecido como reverso de temperatura, típico de gases em estado de rarefação. O método ADO, uma versão analítica do método de ordenadas discretas, é utilizado para construção de soluções em forma fechada para os diversos problemas e quantidades de interesse, como perfis de temperatura e fluxos de calor. Para o caso de um gás, uma solução unificada e desenvolvida para problemas formulados a partir dos modelos cinéticos, derivados da equação de Boltzmann, BGK, S, Gross- Jackson e MRS. No caso de mistura binária de gases, a formulação matemática e baseada no modelo McCormack. Particularmente, quando a evaporação forte e abordada, e aspectos não lineares devem ser incluídos, a versão não linear do modelo BGK e utilizada. Neste caso, a solução ADO do modelo linear e utilizada em um processo chamado de pós-processamento para inclusão dos termos não lineares do problema e reavaliação das quantidades de interesse, evidenciando melhoria dos resultados obtidos pela formulação linear. Uma serie de resultados numéricos são listados e é observada, de forma geral, excelente exatidão e eficiência computacional. / In this work, evaporation phenomena in rarefied gas flow, for one gas case and binary mixtures, are analyzed. Weak and strong evaporation are considered in channel and half-space problems. The reverse of temperature problem, typical in rarefied gas dynamics, is also investigated. The ADO method, an analytical version of the discrete ordinates method, is used to develop closed form solutions, to several problems and quantities of interest, as temperature profiles and heat flows. For the one gas case, an unified solution is developed for the BGK, S, Gross-Jackson and MRS models, derived from the Boltzmann equation. For binary mixtures, the mathematical formulation is based on the McCormack model. Particularly, when strong evaporation is investigated, and nonlinear aspects have to be included, the nonlinear BGK model is used. In this case, the ADO solution, provided by the linear model, is considered in a post-processing procedure which takes into account the nonlinear terms to evaluate the quantities of interest, and improved results are obtained, in comparison with the linear version. A series of numerical results are listed and, in general, an excellent accuracy and good computational efficiency are observed.

Afin de modéliser l'absorption dans le traitement des transferts radiatifs en milieu atmosphérique, de nombreuses méthodes plus précises et plus rapides ont été développées. La modélisation de la formation du brouillard, où le rayonnement infrarouge joue un rôle très important, nécessite des méthodes numériques suffisamment précises pour calculer le taux de refroidissement. Le brouillard radatif se forme après des conditions de ciel clair, où l'absorption est le processus radiatif dominant, en raison d'un fort refroidissement nocturne. Avec l'augmentation des ressources de calcul et le développement du Calcul Haute Performance, les modèles à bandes, pour effectuer l'intégration sur la longueur des grandeurs radiométriques, sont les plus utilisés. Toutefois, le couplage entre les transferts radiatifs 3-D et la dynamique des fluides reste très coûteux en temps de calcul. Le rayonnement augmente d'environ cinquante pourcent le temps de la simulation pour la dynamique des fluides uniquement. Pour réduire le temps passé dans une itération radiative, une nouvelle paramétrization basée sur les modèles en émissivité a été développée. Cette approche nécessite seulement une résolution de l'ETR contre $N_{text{bandes}} times N_{text{gauss}}$ pour un modèle à $N_{text{bandes}}$ avec $N_{text{gauss}}$ points de quadratures sur chaque bande. Une comparaison avec des données de simulation a été effectuée et cette nouvelle paramétrisation de l'absorption infrarouge a montré sa capacité à prendre en compte les variations des concentrations gazeuses et d'eau liquide. Une étude à travers le couplage entre le modèle développé et le code de CFD Code_Saturne a été réalisée afin valider dynamiquement notre paramétrisation. Enfin une simulation exploratoire a été effectuée sur un domaine 3-D en présence de bâti idéalisé, pour capter les effets radiatifs 3-D dûs aux hétérogénéités horizontales du champ d'eau liquide et des bâtiments / The Atmospheric Radiation field has seen the development of more accurate and faster methods to take into account absorption. Modelling fog formation, where Infrared Radiation is involved, requires accurate methods to compute cooling rates. Radiative fog appears with clear sky condition due to a significant cooling during the night where absorption is the dominant processus. Thanks to High Performance Computing, multi-spectral approaches of Radiative Transfer Equation resolution are often used. Nevertheless, the coupling of three-dimensional radiative transfer with fluid dynamics is very computationally expensive. Radiation increases the computation time by around fifty percent over the pure Computational Fluid Dynamics simulation. To reduce the time spent in radiation calculations, a new method using the broadband emissivity has been developed to compute an equivalent absorption coefficient (spectrally integrated). Only one resolution of Radiative Transfer Equation is needed against $N_{text{band}} times N_{text{gauss}}$ for an $N_{text{band}}$ model with $N_{text{gauss}}$ quadrature points on each band. A comparison with simulation data has been done and the new parameterization of Radiative properties shows the ability to handle variations of gases concentrations and liquid water. A dynamical study through the coupling between the infrared radiation model and Code_Saturne has been done to validate our parametrization. Finally the model was tested on a 3-D domain with idealized buildings to catch 3-D infrared radiative effects due to horizontally inhomogenities of the liquid water content field and buildings

No description available.

This thesis discusses the development of an improved microbial inactivation model for analyzing continuous flow UV-LED air treatment systems and use of the model to evaluate the impact of several treatment system design parameters on inactivation. Model development includes three submodels: a radiation submodel, a fluid flow submodel, and an inactivation kinetics submodel. Radiation modeling defines the UV irradiance throughout the system. Fluid flow modeling provides the residence times that microbes spend exposed to the UV irradiation while passing through the system. Inactivation modeling combines irradiance and residence times with inactivation kinetics to calculate species-specific inactivation in a treatment system. The most significant development focuses on the radiation submodel as it is key to linking the UV intensity emissions to treatment system properties and inactivation rates. Various radiation transfer models previously developed by other researchers are evaluated for computational efficiency and effectiveness in modeling non-uniform LED emission and diffuse and specular wall reflections. The Discrete Ordinates Method (DOM) with Legendre-Chebyshev quadrature sets is selected for use in this research due to its ability to represent both non-uniform LED emission profiles and combined specular and diffuse surface reflection. The DOM and associated quadrature schemes are reviewed in detail and limitations in representing LED emissions discussed. Sensitivity to spatial and directional discretization is evaluated. The radiation submodel is combined with a well-accepted inactivation kinetics correlation and two simple fluid flow models: a uniform flow model and a fully-developed flow model. The use and validity of these submodels is explained and their limitations discussed. Predicted microbial inactivation from the overall model is shown to compare well with limited data from a test system. Model flexibility in evaluating several system operating and design parameters is illustrated. These analyses show that for a similar number of LEDs, highly reflective surfaces (diffuse or specular) produce higher inactivation. Other parameters are shown to impact inactivation but to a lesser degree. Square ducts result in higher inactivation than non-square ducts, a fully-developed flow profile slightly increases inactivation over a uniform flow profile, positioning LEDs on all four duct walls slightly increases inactivation when surfaces are non-reflective or diffuse, and positioning LEDs closer together results in slightly higher inactivation.