A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016 / The blind headings created in room and pillar mining are known to be the high risk areas of the coal mine, since this is where the coal production is actually taking place and hence the liberation of maximum quantity of methane. The ventilation of this region called the localized ventilation is carried out using auxiliary ventilation devices. This ventilation may be planned and be the subject of mine standards, but it is not very well understood and implementation on a day to day basis is usually left to the first level of supervisory staff. Majority of the methane explosions have been found to occur in these working areas and blind headings. The correct use of auxiliary ventilation devices can only be carried out once the effect of the system variables associated with each device is very well understood and can be calculated mathematically. Presently, no mathematical models or empirical formulas exist to estimate the effect of the associated system variables on the flow rates close to the face of the heading. The extent of ventilation of a heading ventilated without the use of any auxiliary device is not clear. Furthermore, to design additional engineering solutions, the flow patterns inside these heading ventilated with the auxiliary ventilation devices needs to be understood. The study of the face ventilation systems and the effect of the system variables associated system with each auxiliary ventilation device can be carried out experimentally, but doing a large number of experiments underground is very difficult as it disturbs the mine production cycles. Furthermore, studying the flow patterns experimentally is even more cumbersome, and can only be done to some extent using smoke or tracer gas. Therefore, Computational Fluid Dynamic‟s (CFD) advanced numerical code ANSYS Fluent was used to study the effect of a number of system variables associated with the face ventilation systems used in blind headings. As part of the procedure, the CFD model used was validated using four validation studies, in which the numerical results were compared with the actual experimental results. The numerical results differed to a maximum of 10% for all the experimental results. The system variables associated with ventilation of a heading, without the use of any auxiliary device, with the use of Line Brattice (LB) and fan with duct were selected. A range of values was chosen for each variable, and scenarios were created using every possible combination of these variables. All the scenarios were simulated in Ansys Fluent, the air flow rates, air velocities, velocity vectors, and velocity contours were calculated and drawn at different locations inside the heading. The effect of each system variable was found using a comparative analysis. The results were represented in simple user-friendly form and can be used to estimate the air flows at the exit of the LB and face of the heading for various settings of the LB and fan and duct face ventilation systems. The analysis of the ventilation of a heading without the use of LB shows that a maximum penetration depth is found with the Last Through Road (LTR) velocity of 1.35m/s. The flow rates and the maximum axial velocities increase with the increase in the LTR velocity up to a depth of 10m (maximum air flowing into a heading of 1.26m3/s and 1.58m3/s is found for the 3m and 4m high heading using 2m/s LTR velocity). For the LB ventilation system the LTR velocities, heading height, length of the LB in the LTR and heading, angle of the LB in LTR, and distance of the LB to the wall of the heading (side wall) were varied to identify clearly the effect of these control variables, on the flow rate at the exit of the LB, and close to the face of the heading. The flow rate at the exit of the LB is found to be proportional to the product of the distance of the LB to the wall in the LTR and heading. The flow rate at the exit of the LB, face of the heading, and inside the heading is found proportional to the LTR velocity and height of the heading. It is found that a minimum length of LB is associated with each distance of the LB to the wall in the heading, to maximize the delivery of air close to the face of the heading. This length is found to be equal to 15m for 1m LB to wall distance, and 10m for 0.5m LB to wall distance. Mathematical models were developed to estimate the effect of each studied system variables on the flow rates at the exit of the LB and close to face of the heading. For the fan and duct systems the length, diameter, and the fan design flow rates were varied. It is found that for a force fan duct system only a maximum of 50% of the total air that reaches the face is fresh and the remaining 50% is recirculated air. The flow rate with the exhaust fan system is found to be much lower than the force fan duct system. It increases with the reduction in duct mouth to heading face distance, and increase in duct diameter. Mathematical models are developed to calculate the flow rates at the face of the heading using the effect of each studied system variable. The research reveals that the ANSYS numerical code is an appropriate tool to evaluate the face ventilation of a heading in a three dimensional environment using full scale models. The South African coal mining industry can benefit from the outcomes of this study, specially the mathematical models, in a number of ways. Ventilation engineers can now estimate the flow rates close to the face of the heading for different practical mining scenarios and ensure sufficient ventilation by using the appropriate auxiliary ventilation settings. The results can easily be developed into training aids using easy to use excel spread sheets to ensure that mineworkers at the coal face have a better understanding of the working of the auxiliary ventilation devices. It can also serve Academia as part of the curriculum to teach the future mining engineers how the different variables associated with the auxiliary ventilation system affect the ventilation in a heading. The research therefore, has the potential to provide a significant step toward, understanding airflow rates delivered by the auxiliary devices close to the face of the heading and the air flow patterns inside the heading as a basis for improving the working environment for underground mineworkers. / MT2017

Esta pesquisa tratou de investigar o papel dos registros de representação semiótica na construção do conceito de função, em particular daquelas do tipo afim e quadrática. Parte da pesquisa também foi investigar de que forma o uso de um software de matemática dinâmica, o GeoGebra, pode ajudar o processo de aprendizagem do tópico em questão. A metodologia de pesquisa utilizada é inspirada nos moldes da Engenharia Didática de Artigue. É através das atividades da sequência didática projetada e implementada em 2013, com turma de alunos do primeiro ano do Ensino Médio de uma escola estadual de Porto Alegre, que vamos mostrar como aconteceu o desenvolvimento de conexões múltiplas entre as diversas representações de uma função (algébrica, gráfica e numérica), bem como a importância do software GeoGebra como recurso pedagógico quando se quer trabalhar com múltiplas representações. As respostas apresentadas, pelos alunos, foram analisadas tendo como referência a Teoria dos Registros de Representação Semiótica de Duval. A observação de relações entre variáveis a partir da manipulação de pontos em uma construção no Geogebra propiciou a compreensão do conceito de função e gráfico, através de constante processo de conversão de registros. / This research aims to investigate the role of semiotic representation registers in the construction of the concept of function, in particular those of the affine and quadratic type. Part of the research was also to investigate how the use of a dynamic mathematics software GeoGebra can help the learning process. The research methodology is inspired by the lines of Didactic Engineering. Through a instructional sequence designed and implemented in 2013, with a group of students of the first year of high school to a state school in Porto Alegre, it is shown how they developed the connections between different registers of representations (algebraic , graphical and numerical). The experience has also shown the importance of GeoGebra as a tool to explore the different registers of representation. The production of students were analyzed taking into acount the theory of semiotic representations of Duval and it was posible to identify their understanding of the concept of variable, function and graph, through the an active process of conversion of registers.

Esta pesquisa tratou de investigar o papel dos registros de representação semiótica na construção do conceito de função, em particular daquelas do tipo afim e quadrática. Parte da pesquisa também foi investigar de que forma o uso de um software de matemática dinâmica, o GeoGebra, pode ajudar o processo de aprendizagem do tópico em questão. A metodologia de pesquisa utilizada é inspirada nos moldes da Engenharia Didática de Artigue. É através das atividades da sequência didática projetada e implementada em 2013, com turma de alunos do primeiro ano do Ensino Médio de uma escola estadual de Porto Alegre, que vamos mostrar como aconteceu o desenvolvimento de conexões múltiplas entre as diversas representações de uma função (algébrica, gráfica e numérica), bem como a importância do software GeoGebra como recurso pedagógico quando se quer trabalhar com múltiplas representações. As respostas apresentadas, pelos alunos, foram analisadas tendo como referência a Teoria dos Registros de Representação Semiótica de Duval. A observação de relações entre variáveis a partir da manipulação de pontos em uma construção no Geogebra propiciou a compreensão do conceito de função e gráfico, através de constante processo de conversão de registros. / This research aims to investigate the role of semiotic representation registers in the construction of the concept of function, in particular those of the affine and quadratic type. Part of the research was also to investigate how the use of a dynamic mathematics software GeoGebra can help the learning process. The research methodology is inspired by the lines of Didactic Engineering. Through a instructional sequence designed and implemented in 2013, with a group of students of the first year of high school to a state school in Porto Alegre, it is shown how they developed the connections between different registers of representations (algebraic , graphical and numerical). The experience has also shown the importance of GeoGebra as a tool to explore the different registers of representation. The production of students were analyzed taking into acount the theory of semiotic representations of Duval and it was posible to identify their understanding of the concept of variable, function and graph, through the an active process of conversion of registers.

Esta pesquisa tratou de investigar o papel dos registros de representação semiótica na construção do conceito de função, em particular daquelas do tipo afim e quadrática. Parte da pesquisa também foi investigar de que forma o uso de um software de matemática dinâmica, o GeoGebra, pode ajudar o processo de aprendizagem do tópico em questão. A metodologia de pesquisa utilizada é inspirada nos moldes da Engenharia Didática de Artigue. É através das atividades da sequência didática projetada e implementada em 2013, com turma de alunos do primeiro ano do Ensino Médio de uma escola estadual de Porto Alegre, que vamos mostrar como aconteceu o desenvolvimento de conexões múltiplas entre as diversas representações de uma função (algébrica, gráfica e numérica), bem como a importância do software GeoGebra como recurso pedagógico quando se quer trabalhar com múltiplas representações. As respostas apresentadas, pelos alunos, foram analisadas tendo como referência a Teoria dos Registros de Representação Semiótica de Duval. A observação de relações entre variáveis a partir da manipulação de pontos em uma construção no Geogebra propiciou a compreensão do conceito de função e gráfico, através de constante processo de conversão de registros. / This research aims to investigate the role of semiotic representation registers in the construction of the concept of function, in particular those of the affine and quadratic type. Part of the research was also to investigate how the use of a dynamic mathematics software GeoGebra can help the learning process. The research methodology is inspired by the lines of Didactic Engineering. Through a instructional sequence designed and implemented in 2013, with a group of students of the first year of high school to a state school in Porto Alegre, it is shown how they developed the connections between different registers of representations (algebraic , graphical and numerical). The experience has also shown the importance of GeoGebra as a tool to explore the different registers of representation. The production of students were analyzed taking into acount the theory of semiotic representations of Duval and it was posible to identify their understanding of the concept of variable, function and graph, through the an active process of conversion of registers.

A complete formulation of the recently developed. F_N method in kinetic theory is presented and the accuracy of this advanced semi-analytical-numerical technique is demonstrated by testing the method to several classical problems in rarefied gas dynamics. The method is based on the existing analysis for the vector transport equation arising from the decomposition of the linearized BGK equation. Using full-range orthogonality, a system of singular integral equations for the distribution functions at the boundaries is established. The unknown distribution functions are then approximated by a finite expansion in terms of a set of basis functions and the coefficients of the expansion are found by requiring the set of the reduced algebraic equations to be satisfied at certain collocation points. By studying the half-space heat transfer and weak evaporation problems and the problem of heat transfer between two parallel plates it is demonstrated that the F_N method is a viable solution technique yielding results of benchmark accuracy. Two different sets of basis functions are provided for half-space and finite media problems, respectively. In all cases, highly accurate numerical results are computed and compared to existing exact solutions. The obtained numerical results help in judging the accuracy to expect of the method and indicate that the F_N method may be applied with confidence to problems for which, more exact methods of analysis do not appear possible. Then, the cylindrical Poiseuille flow and thermal creep problems, which are not amenable to exact treatment, are solved. The F_N method is formulated and tested successfully for the first time in cylindrical geometry in kinetic theory. The complete solution of the two aforementioned problems is presented with the numerical results quoted as converged being of reference-quality good for benchmark accuracy. / Ph. D. / incomplete_metadata

Thesis Ph.D. ((Applied Mathematics)) -- University of Limpopo, 2022 / The world is under threat from the devastating effects of the continued depletion of the Ozone layer. Increased global warming is causing catastrophic ecological damage and imbalance due to accelerated melting of glaciers, rampant runaway veld res, widespread floods and other extreme events. The delegates to the Cop26 Climate Change Summit were reminded that the continued burning of fossil fuels is releasing carbon into the atmosphere at an unprecedented pace and scale and that the world is already in trouble. Complete substitution of fossil fuels with clean energy sources is the only solution through which the world can be saved from the deleterious effects of global warming. However, total dependence on renewable energy sources can only be possible through novel technology that enables efficient energy utilization and conservation. For instance, the evolution of advanced techniques in manufacturing processes has led to the reduction in the size of various industrial and engineering designs that consume reduced amounts of energy. Efficient energy utilization in thermo-fluid flow systems can be achieved through entropy generation minimization. Entropy is a thermodynamic quantity that represents the unavailability of a system's thermal energy for conversion into mechanical work. In this study, thermodynamic analysis of reactive variable properties third-grade fluid flow in channels with varied geometries and subjected to different physical effects was investigated with the second law of thermodynamics as the area of focus. Entropy generation and inherent irreversibility analysis were the main focus of the study where the sensitivities of these quantities to the embedded parameters were numerically and graphically described and analysed. The semi-analytic Adomian decomposition method, the semi-implicit fi nite difference scheme and the spectral quasilinearisation method were employed to solve the nonlinear differential equations modelling the flow systems. The results reveal that the effects of the parameters on flow velocity, fluid temperature, entropy generation and inherent irreversibility cannot be neglected. In particular, conditions for entropy generation minimization were successfully established and documented. / University of Limpopo

Thesis (MEng)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: This study investigates the implementation of the finite volume numerical method applicable to non-orthogonal control volumes and the application of the method to calculate the thermo-flow field within the collector area of a solar chimney power generating plant. The discretisation of the governing equations for the transient, Newtonian, incompressible and turbulent fluid flow, including heat transfer, is presented for a non-orthogonal coordinate frame. The standard k - E turbulence model, modified to include rough surfaces, is included and evaluated in the method. An implicit solution procedure (SIP-semi implicit procedure) as an alternative to a direct solution procedure for the calculation of the flow field on nonstaggered grids is investigated, presented and evaluated in this study. The Rhie and Chow interpolation practice was employed with the pressurecorrection equation to eliminate the presence of pressure oscillations on nonstaggered grids. The computer code for the solution of the three-dimensional thermo-flow fields is developed in FORTRAN 77. The code is evaluated against simple test cases for which analytical and experimental results exist. It is also applied to the analysis of the thermo-flow field of the air flow through a radial solar collector. KEYWORDS: NUMERICAL METHOD, FINITE VOLUME, NON-ORTHOGONAL, k+-e TURBULENCE MODEL, SIP / AFRIKAANSE OPSOMMING: Die studie ondersoek die implementering van 'n eindige volume numeriese metode van toepassing op nie-ortogonale kontrole volumes asook die toepassing van die metode om die termo-vloei veld binne die kollekteerder area van 'n sonskoorsteen krag aanleg te bereken. Die diskretisering van die behoudsvergelykings vir die tyd-afhanlike, Newtonse, onsamedrukbare en turbulente vloei, insluitende hitteoordrag, word beskryf vir 'n nie-ortogonale koordinaatstelsel. Die standaard k - E turbulensiemodel, aangepas om growwe oppervlakrandvoorwaardes te hanteer, is ingesluit en geevalueer in die studie. 'n Implisiete oplossings metode (SIP-semi implisiete prosedure) as alternatief vir 'n direkte oplossingsmetode is ondersoek en geimplimenteer vir die berekening van die vloeiveld met nie-verspringde roosters. 'n Rhie en Chow interpolasie metode is gebruik tesame met die drukkorreksie-vergelyking ten einde ossilasies in die drukveld in die nie-verspringde roosters te vermy. Die rekenaarkode vir die oplossing van die drie dimensionele termo-vloeiveld is ontwikkel in FORTRAN 77. Die kode is geevalueer teen eenvoudige toetsprobleme waarvoor analitiese en eksperimentele resultate bestaan. Die kode IS ook gebruik om die termo-vloeiveld binne 'n radiale son kollekteerder te analiseer. SLEUTELWOORDE: NUMERIESE METODE, EINDIGE VOLUME, NIE-ORTOGONAAL, k - E TURBULENSIE MODEL, SIP

University of Macau / Faculty of Science and Technology / Department of Mathematics