Este trabalho tem como objetivo a obtenção de soluções analíticas e semi-analíticas para o problema de transferência de calor em filmes de água líquida escoando sobre aerofólios. O problema da transferência de calor em filmes já foi abordado na literatura, no entanto, não foram encontradas, na bibliografia aberta, soluções analíticas aplicáveis ao problema da formação de gelo em filmes escoando sobre aerofólios de aeronaves. Este fato se deve à presença de um termo de transferência de calor por convecção na interface filme-escoamento de ar, que não foi considerado nas soluções analíticas encontradas na literatura. Para os problemas estudados, a obtenção da distribuição de temperaturas no filme é fundamental, pois, com ela, é possível prever a iminência da formação de gelo, ou seja, quando a temperatura de solidificação é atingida no filme. Nos caos de superfícies com proteção térmica, deve ser especificado um fluxo de calor na superfície de contato entre o filme e o aerofólio, enquanto nos casos nos quais a superfície do aerofólio está desprotegida, deve ser considerada uma temperatura fixa. Neste trabalho, para cada um destes casos, foram considerados gradientes de pressão nulos e constantes no filme dágua. Assim, ao total, quatro casos foram gerados e analisados. O método da separação de variáveis e expansões em séries de autofunções foi utilizado na obtenção das soluções analíticas e o de Galerkin na obtenção das soluções semi-analíticas. Este último método, apesar de não ser exato, não apresenta alguns dos desvios intrínsecos aos métodos numéricos usuais, pois não depende da discretização do espaço em forma de malha e das interpolações decorrentes. Foi feita uma comparação entre as soluções para gradiente de pressão nulo obtidas pelo método de Galerkin e as soluções obtidas por separação de variáveis. Desta maneira, encontraram-se os desvios da soluções semi-analíticas em relação às soluções exatas. Finalmente, foram encontradas estimativas simplificadas para a distribuição de temperaturas no filme, além de variáveis adimensionais que generalizam o problema, podendo ser traçadas, então, uma série de curvas válidas para uma extensa gama de parâmetros. / This work aims to obtain analytical and semi-analytical solutions to the airfoil film heat transfer problem. The film heat transfer problem has been already solved in the literature. Nevertheless, no consistent solution with the airfoil ice accretion problem is known. This issue is due to the presence of a convective term in the interface between the film and the airflow, which has not been taken into account in the analytical studies available in the literature. Solving the temperature distribution in the film is essential for predicting the ice growth onset, i.e., the location where the solidification temperature is reached in the film. In the cases corresponding to thermally protected surfaces, the heat flow at the airfoil surface has to be specified, while, on the other hand, for the non-protected surfaces, the surface temperature has to be specified. In this work, for each of these cases, it has been considered both zero pressure gradient and a non-zero constant pressure gradient at the water and air flow. In this way, four cases were generated and analyzed. Separation of variables and eigenfunction expansion methods were used in the analytical solutions, whereas the Galerkin method was used to obtain the semi-analytical solutions. The latter, despite being approximate, does not produce some of the numerical errors associated with the space discretization and interpolation. The zero pressure solutions were compared in order to find the deviation between the analytical and semi-analytical solutions. Finally, estimates for the film temperature distribution were found, besides dimensionless variables that generalize the problem, enabling the creation of a family of curves valid for a wide range of parameters.

Este trabalho tem como objetivo a obtenção de soluções analíticas e semi-analíticas para o problema de transferência de calor em filmes de água líquida escoando sobre aerofólios. O problema da transferência de calor em filmes já foi abordado na literatura, no entanto, não foram encontradas, na bibliografia aberta, soluções analíticas aplicáveis ao problema da formação de gelo em filmes escoando sobre aerofólios de aeronaves. Este fato se deve à presença de um termo de transferência de calor por convecção na interface filme-escoamento de ar, que não foi considerado nas soluções analíticas encontradas na literatura. Para os problemas estudados, a obtenção da distribuição de temperaturas no filme é fundamental, pois, com ela, é possível prever a iminência da formação de gelo, ou seja, quando a temperatura de solidificação é atingida no filme. Nos caos de superfícies com proteção térmica, deve ser especificado um fluxo de calor na superfície de contato entre o filme e o aerofólio, enquanto nos casos nos quais a superfície do aerofólio está desprotegida, deve ser considerada uma temperatura fixa. Neste trabalho, para cada um destes casos, foram considerados gradientes de pressão nulos e constantes no filme dágua. Assim, ao total, quatro casos foram gerados e analisados. O método da separação de variáveis e expansões em séries de autofunções foi utilizado na obtenção das soluções analíticas e o de Galerkin na obtenção das soluções semi-analíticas. Este último método, apesar de não ser exato, não apresenta alguns dos desvios intrínsecos aos métodos numéricos usuais, pois não depende da discretização do espaço em forma de malha e das interpolações decorrentes. Foi feita uma comparação entre as soluções para gradiente de pressão nulo obtidas pelo método de Galerkin e as soluções obtidas por separação de variáveis. Desta maneira, encontraram-se os desvios da soluções semi-analíticas em relação às soluções exatas. Finalmente, foram encontradas estimativas simplificadas para a distribuição de temperaturas no filme, além de variáveis adimensionais que generalizam o problema, podendo ser traçadas, então, uma série de curvas válidas para uma extensa gama de parâmetros. / This work aims to obtain analytical and semi-analytical solutions to the airfoil film heat transfer problem. The film heat transfer problem has been already solved in the literature. Nevertheless, no consistent solution with the airfoil ice accretion problem is known. This issue is due to the presence of a convective term in the interface between the film and the airflow, which has not been taken into account in the analytical studies available in the literature. Solving the temperature distribution in the film is essential for predicting the ice growth onset, i.e., the location where the solidification temperature is reached in the film. In the cases corresponding to thermally protected surfaces, the heat flow at the airfoil surface has to be specified, while, on the other hand, for the non-protected surfaces, the surface temperature has to be specified. In this work, for each of these cases, it has been considered both zero pressure gradient and a non-zero constant pressure gradient at the water and air flow. In this way, four cases were generated and analyzed. Separation of variables and eigenfunction expansion methods were used in the analytical solutions, whereas the Galerkin method was used to obtain the semi-analytical solutions. The latter, despite being approximate, does not produce some of the numerical errors associated with the space discretization and interpolation. The zero pressure solutions were compared in order to find the deviation between the analytical and semi-analytical solutions. Finally, estimates for the film temperature distribution were found, besides dimensionless variables that generalize the problem, enabling the creation of a family of curves valid for a wide range of parameters.

Anticipating how stream water quality will respond to change, such as increased pollution or water diversions, requires knowledge of the main mechanisms controlling water and chemical constituent movement and a reasonable representation of those mechanisms. By deriving mathematical models to represent a stream system and collecting supporting field-based measurements, water quality response can be predicted. However, because each stream is unique and the movement of water and constituents is spatially and temporally complex, assessing whether the stream is appropriately represented and whether predictions are trustworthy is still a challenge within the scientific and management communities. Building on decades of stream research, this dissertation provides a step towards better representing some of the complexities found within streams and rivers to better predict water quality responses over long stream distances. First, a method is presented to assess which mechanisms are considered most important in chemical constituent predictions. Next, the number of measurements necessary to represent the general complexities of water, mass, and heat movement in streams was determined. The advancements developed in this dissertation provide a foundation to more efficiently and accurately inform water resource management.

<p>The interplay between hydraulic and chemical processes in streams and adjacent storage zones, e.g. streambed sediments, is of crucial importance for the large-scale transport of released contaminants. This thesis presents a methodology for tracer experiments and the development of mechanistic transport models. </p><p>We conducted four field tracer experiments with the reactive tracer chromium, ⁵¹Cr(III) and/or the conservative tracers potassium iodide (KI) and tritium (³H₂O), along 11 km of the Lanna Stream in Skara County, Sweden, and along a 30 km reach of the Säva Stream in Uppland County, Sweden. The field monitoring included sampling of tracer in the surface water as well as in the streambed sediments. A simultaneous injection of tritium and chromium facilitated an independent evaluation of the hydraulic transport into and out of the sub-surface storage zones. The difference in transport behaviour between the two tracers, were attributed to chemical reactivity.</p><p>In terms of idealised transport models we found that the reactivity of chromium could be characterised by simple chemical concepts. The local equilibrium assumption (LEA), the irreversible kinetics assumption (IKA) and the reversible kinetics assumption (RKA) were applied in the experimental evaluations. An independent evaluation of the streambed transport revealed that the impact of reaction kinetics was substantial. Model calculations and results from a chemical extraction procedure indicated that the chemical reactions affecting the chromium transport were to some extent irreversible.</p><p>This thesis presents a number of exact analytical solutions to the governing partial differential equations. The main theoretical contribution is the incorporation of variable coefficients for stream discharge and sediment porosity, which were measured in field.</p>

The interplay between hydraulic and chemical processes in streams and adjacent storage zones, e.g. streambed sediments, is of crucial importance for the large-scale transport of released contaminants. This thesis presents a methodology for tracer experiments and the development of mechanistic transport models. We conducted four field tracer experiments with the reactive tracer chromium, 51Cr(III) and/or the conservative tracers potassium iodide (KI) and tritium (3H2O), along 11 km of the Lanna Stream in Skara County, Sweden, and along a 30 km reach of the Säva Stream in Uppland County, Sweden. The field monitoring included sampling of tracer in the surface water as well as in the streambed sediments. A simultaneous injection of tritium and chromium facilitated an independent evaluation of the hydraulic transport into and out of the sub-surface storage zones. The difference in transport behaviour between the two tracers, were attributed to chemical reactivity. In terms of idealised transport models we found that the reactivity of chromium could be characterised by simple chemical concepts. The local equilibrium assumption (LEA), the irreversible kinetics assumption (IKA) and the reversible kinetics assumption (RKA) were applied in the experimental evaluations. An independent evaluation of the streambed transport revealed that the impact of reaction kinetics was substantial. Model calculations and results from a chemical extraction procedure indicated that the chemical reactions affecting the chromium transport were to some extent irreversible. This thesis presents a number of exact analytical solutions to the governing partial differential equations. The main theoretical contribution is the incorporation of variable coefficients for stream discharge and sediment porosity, which were measured in field.

Analytical Model of Traffic Signal Structures (TSS) is developed based on a continuous system method to observe dynamic characteristics of the structures. Conventional and basic continuous system method can show the approximate dynamic characteristics of the TSS, but the discretized continuous analytical model is proposed to get more accurate and realistic results of the TSS. In addition, the discretized continuous model can alternatively analyze the effect of the tapered cross-sectional members which are real model of TSS. For the verification of the analytical model, the dynamic characteristics of the numerical solutions by modal analysis in ABAQUS and the results of experimental measurements are provided. Compared with the numerical solutions and the experimental results, the analytical solution for each member shows its considerable accuracy. In addition, it will be also able to accurately express the effects of the linearly tapered cross-sectional member with more discretized continuous structural system. Moreover, the discretized analytical model of the TSS has the usability to observe the effects of boundary flexibility.

At many sites it has become apparent that most chemicals of concern (COCs) in groundwater are persistent and not effectively treated by conventional remediation methods. In recent years, the permeable reactive barrier (PRB) technology has proven to be more cost-efficient in the long-run and capable of rapidly reducing COC concentrations by up to several orders of magnitude. In its simplest form, the PRB is a vertically emplaced rectangular porous medium in which impacted groundwater passively enters a narrow treatment zone. In the treatment zone dissolved COCs are rapidly degraded as they come in contact with the reactive material. As a result, the effluent groundwater contains significantly lower solute concentrations as it re-enters the aquifer and flows towards the plane of compliance (POC). Effective implementation of the PRB relies on accurate site characterization to identify the existing COCs, their interactions, and their required residence time in the PRB and aquifer. Ensuring adequate residence time in the PRB-aquifer system allows COCs to react longer, hence improving the probability that regulatory concentrations are achieved at the POC. In this study, the Park and Zhan solution technique is used to derive steady-state analytical and transient semi-analytical solutions to multispecies reactive transport in a permeable reactive barrier-aquifer (dual domain) system. The advantage of the dual domain model is that it can account for the potential existence of natural degradation in the aquifer, when designing the required PRB thickness. Also, like the single-species Park and Zhan solution, the solutions presented here were derived using the total mass flux (third-type) boundary condition in PRB-aquifer system. The study focuses primarily on the steady-state analytical solutions of the tetrachloroethylene (PCE) serial degradation pathway and secondly on the analytical solutions of the parallel degradation pathway. Lastly, the solutions in this study are not restricted solely to the PRB-aquifer model. They can also be applied to other types of dual domain systems with distinct flow and transport properties, and up to four other species reacting in serial or parallel degradation pathways. Although the solutions are long, the results of this study are novel in that the solutions provide improved modeling flexibility. For example: 1) every species can have unique first-order reaction rates and unique retardation factors, 2) higher order daughter species can be modeled solely as byproducts by neglecting their input concentrations, 3) entire segments of the parallel degradation pathway can be neglected depending on the desired degradation pathway model, and 4) converging multi-parent reactions can be modeled. As part of the study, separate Excel spreadsheet programs were created to facilitate prompt application of the steady-state analytical solutions, for both the serial and parallel degradation pathways. The spreadsheet programs are included as supplementary material.

In this dissertation, we present multiscale analytical solutions, in the weak sense, to the generalized Laplace's equation in Ω ⊂ Rⁿ, subject to periodic and nonperiodic boundary conditions. They are called multiscale solutions since they depend on a coefficient which takes a wide possible range of scales. We define forms of nonseparable coefficient functions in Lᵖ(Ω) such that the solutions are valid for the periodic and nonperiodic cases. In the periodic case, one such solution corresponds to the auxiliary cell problem in homogenization theory. Based on the proposed analytical solution, we were able to write explicitly the analytical form for the upscaled equation with an effective coefficient, for linear and nonlinear cases including the one with body forces. This was done by performing the two-scale asymptotic expansion for linear and nonlinear equations in divergence form with periodic coefficient. We proved that the proposed homogenized coefficient satisfies the Voigt-Reiss inequality. By performing numerical experiments and error analyses, we were able to compare the heterogeneous equation and its homogenized approximation in order to define criteria in terms of allowable heterogeneity in the domain to obtain a good approximation. The results presented, in this dissertation, have laid mathematical groundwork to better understand and apply multiscale processes under a deterministic point of view.

The increasing concentration of CO_(2) has been linked to global warming and changes in climate. Geologic sequestration of CO_(2) in deep saline aquifers is a proposed greenhouse gas mitigation technology with potential to significantly reduce atmospheric emissions of CO_(2). Feasibility assessments of proposed sequestration sites require realistic and computationally efficient models to simulate the subsurface pressure response and monitor the injection process, and quantify the risks of leakage if there is any. This study investigates the possibility of obtaining closed form expressions for spatial distribution of CO_(2) injected in brine aquifers and gas reservoirs. Four new semi-analytical solutions for CO_(2) injection in brine aquifers and gas reservoirs are derived in this dissertation. Both infinite and closed domains are considered in the study. The first solution is an analysis of CO_(2) injection into an initially brine-filled infinite aquifer, exploiting self–similarity and matched asymptotic expansion. The second is an expanding to the first solution to account for CO_(2) injection into closed domains. The third and fourth solutions are analyzing the CO_(2) injection in infinite and closed gas reservoirs. The third and fourth solutions are derived using Laplace transform. The brine aquifer solutions accounted for both Darcyian and non-Darcyian flow, while, the gas reservoir solutions considered the gas compressibility variations with pressure changes. Existing analytical solutions assume injection under constant rate at the wellbore. This assumption is problematic because injection under constant rate is hard to maintain, especially for gases. The modeled injection processes in all aforementioned solutions are carried out under constant pressure injection at the wellbore (i.e. Dirichlet boundary condition). One major difficulty in developing an analytical or semi-analytical solution involving injection of CO_(2) under constant pressure is that the flux of CO_(2) at the wellbore is not known. The way to get around this obstacle is to solve for the pressure wave first as a function of flux, and then solve for the flux numerically, which is subsequently plugged back into the pressure formula to get a closed form solution of the pressure. While there is no simple equation for wellbore flux, our numerical solutions show that the evolution of flux is very close to a logarithmic decay with time. This is true for a large range of the reservoir and CO_(2) properties. The solution is not a formation specific, and thus is more general in nature than formation-specific empirical relationships. Additionally, the solution then can be used as the basis for designing and interpreting pressure tests to monitor the progress of CO_(2) injection process. Finally, the infinite domain solution is suitable to aquifers/reservoirs with large spatial extent and low permeability, while the closed domain solution is applicable to small aquifers/reservoirs with high permeability.

We present a novel approach for deriving analytical solutions to transport equations expressedin similarity variables. We apply a fixed-point iteration procedure to these transformedequations by formally solving for the highest derivative term and then integrating to obtainan expression for the solution in terms of a previous estimate. We are able to analyticallyobtain the Lipschitz condition for this iteration procedure and, from this (via requirements forconvergence given by the contraction mapping principle), deduce a range of values for the outerlimit of the solution domain, for which the fixed-point iteration is guaranteed to converge.