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<p>Understanding the drying of droplets is of interest for processes such as spray drying, where particulate materials are produced by evaporating moisture. Even though spray-drying is a widely used method, there are still challenges, such as undesired agglomeration or controlling the morphology and size of the final dried product. This dissertation develops a physics based model that is used to examine the droplet dynamics and drying kinetics at large and small scales.  In addition, the model simulates the internal motion of insoluble particles and  is used to better understand particle formation during spray drying type processes.</p> <p><br></p> <p>The first part of this work examines the effect of droplet-droplet collisions on evaporation and the size distribution at a large scale. Droplet collision dynamics are implemented into an Eulerian-Lagrangian framework, where droplets are tracked in the Lagrangian frame, and the background gas is modeled as a continuum. The modeling framework includes fully coupled interphase heat and momentum transfer between the droplet and gas phases. Binary collision of droplets could result in coalescence, reduction in surface area, or separation of droplets, resulting in the generation of satellite droplets and an increase in total surface area. By capturing the change in size distribution due to the collision of particles, our results show a linear relationship between the Weber number and the evaporation rate at low droplet number densities. Further, it is shown that droplet number density is a critical factor influencing the evaporation rate. At high droplet number densities, the relationship between the evaporation rate and the Weber number becomes non-linear, and at extremely high droplet number densities, the evaporation rate decreases even at high Weber numbers.</p> <p><br></p> <p>In the next part of this dissertation, the drying of a single droplet containing insoluble solid particles is investigated. Using a volume-of-fluid framework coupled with the Lagrangian phase, we study the particle transport within a droplet, and how it is affected by airflow, phase properties (e.g., viscosity and density of each phase), surface tension, and evaporation. Unlike the traditional one-dimensional modeling approach, our multi-dimensional model can capture the generation of internal flow patterns due to shear flow and the accumulation of solid particles on the surface of the drying droplet. Our results show that the surface tension effect is more pronounced at larger droplet diameters and low airflow velocities. Our approach also provides a quantitative method for modeling crust growth and formation. </p> <p>Our results show that increasing solids mass fraction, and decreasing particle diameter, slow down the internal transport of solid particles, leading to a more quick accumulation near the surface of the droplet. Further, despite the droplet undergoing a constant-rate drying stage, the accumulation of solids near the surface is non-linear. In addition, the inclusion of solids within the droplet drastically reshapes the formation of internal vortices compared to the uncoupled case, which determines solids distribution.</p>

In recent years, there has been an increasing interest in modeling crowd and evacuation dynamics. Pedestrian models are based on macroscopic or microscopic behavior. In this work, we are interested in developing models that can be used for evacuation control strategies. Hence, we use macroscopic modeling approach, where pedestrians are treated in an aggregate way and detailed interactions are overlooked. In this dissertation, we developed two-dimensional space crowd dynamic models to allow bi-directional low by modifying and enhancing various features of existing traffic and fluid dynamic models. In this work, four models based on continuum theory are developed, and conservation laws such as the continuity and momentum equations are used. The first model uses a single hyperbolic partial differential equation with a velocity-density relationship, while the other three models are systems of hyperbolic partial differential equations. For one of the system models presented, we show how it can be derived independently from a microscopic crowd model. The models are nonlinear, time-varying, hyperbolic partial differential equations, and the numerical simulation results given for the four macroscopic models were based on computational fluid dynamics schemes. We also started an initial control design that synthesizes the feedback linearization method for the one-dimensional traffic flow problem applied directly on the distributed parameter system. In addition, we suggest and discuss the information technology requirements for an evacuation system. This research was supported in part from the National Science Foundation through grant no. CMS-0428196 with Dr. S. C. Liu as the Program Director. This support is gratefully acknowledged. Any opinion, findings, and conclusions or recommendations expressed in this study are those of the writer and do not necessarily reflect the views of the National Science Foundation. / Ph. D.

"Safety is a value, not just a Priority" Geomechanically stable underground excavations require continuous assessment of rock mass behavior for maximizing safety. Optimizing pillar design is essential for preventing hazardous incidents and improving production in room-and-pillar mines. Maintaining regional and global stability is complicated for underground carbonate or stone deposits, where extensive fracture networks and groundwater flow become leading factors for generating unsteady ground conditions including karsts. A sudden encounter with karst cavities during mine advance may lead to safety issues, including ground collapse and outflow of unconsolidated sediments and groundwater. The presence of these eroded zones in pillars may cause their failure and poses a risk to the lives of miners apart from disrupting the pre-planned mining operations. A pervasive presence of joints and fractures plays a primary role in promoting structurally controlled failures in stone mines, which accelerates upon interaction with the karst cavities. The prevalent empirical and analytical approaches for pillar design ignore the geotechnical complexities such as the spatial density of discontinuities, karst voids, and deviation from the design during short-range mine planning. With the increasing market demand for limestone products, mining organizations, as well as enforcement agencies, are investing in research for increasing the efficiency of extracting valuable resources. While economical productivity is essential, preventing risks and ensuring the safety of miners remains the cardinal objective of mining operations. According to the Mine Safety and Health Administration (MSHA), since 2000, about 31% of occupational fatalities at all underground mines in the United States are caused due to ground collapse, which rises to 39% for underground stone mines. The objective of this study is to provide a reliable and methodological approach for pillar design in underground room-and-pillar hard rock mines for safe and efficient ore recovery. The numerical modeling techniques, implemented for a case study stone mine, could provide a pragmatic framework to assess the effect of karsts on rock mass behavior, and design future pillars detected with voids. The research uses data acquired from using remote sensing techniques, such as LiDAR and Ground-penetrating Radar surveys, to map the excavation characteristics. Discontinuum modeling was valuable for analyzing the pillar strength in the presence of discontinuities and cavities, as well as estimating a safe design standard. Discrete Fracture Networks, created using statistical information from discontinuity mapping, were employed to simulate the joints pervading the rock mass. This proposed research includes the calibration of rock mass properties to translate the effect of discontinuities to continuum models. Continuum modeling proved effective in analyzing regional stability along with characterizing the redistributed stress regime by imitating the excavation sequence. The results from pillar-scale and local-scale analyses are converged to optimize pillar design on a global scale and estimate the feasibility of secondary recovery in stone mines with a dominating discontinuity network and karst terrane. Stochastic analysis using finite volume modeling helped evaluate the performance of modified pillars to assist production while maintaining safety standards. The proposed research is valuable for improving future design parameters, excavation practices, and maintaining a balance between an approach towards increased safety while enhancing production. / Doctor of Philosophy / "The most valuable resource to come back out of a mine is a miner" – Anonymous. The United States accounted for 27% of the global limestone market share which was valued at US$58.5 billion in 2020 [148]. It is projected to reach a target of US$65.3 billion in 2027, growing even in midst of the COVID-19. As surface reserves deplete, much of the mineral demand gap is supplemented by mining underground deposits. Underground mines extract minerals from deep within the earth compared to surface mines. As a result, the miners experience a greater number of accidents in a constricted environment because of roof/tunnel collapse, fewer escape routes, ventilation, explosions, or inundation. According to the Mine Safety and Health Administration (MSHA), about 15% of all underground mine injuries in the US were caused by rockfalls since 1983. The majority of underground stone deposits are mined using the room-and-pillar mining method, which resembles a chessboard design where the light squares are mined, and the dark squares are left as rock pillars to support the tunnels. Limestone, a carbonate rock, contains a lot of fractures and joints (discontinuities). Erosion of rocks due to continuous water flow through the fractures leads to the formation of cavities known as karsts. Interaction of karsts with the prevalent fracture network increases rockfall risk during mining. The collapse of voids along with an inrush of filled rock-clay-water sludge can harm miners' lives, damage machinery, and stop further operations. Literature is scarce on topics that quantify the risk and disruption posed by these cavities in underground mines. Most rock classification systems cannot classify their effect because of the unpredictability and extensive analysis required. The objective of this research is to provide a reliable and methodological approach for designing pillars in underground hard rock mines for ensuring a safe working environment and efficient mineral recovery. This research starts with analyzing the strength of pillars, in which karst cavities were discovered while mining. The safety concerns often lead the miners to not excavate around the cavities and leave valuable resources unmined. Data from ground-penetrating radar and laser scanning surveys were used to characterize the voids and map the discontinuities. Discrete-element numerical modeling was used to simulate the pillars as an assembly of blocks jointed by the discontinuities. The simulation results help us understand the instability issues in the karst-ridden pillars and ways to improve upon the existing design. The findings were used to modulate the parameters for regional-scale models using finite volume modeling for less computationally intensive analyses and simulating rock as a continuum. The continuum models were highly effective in analyzing the instability issues due to the prevalent karstic network. This helps understand any alternative scenario that could have been implemented to optimize ore recovery while preventing risks. The results from the single pillar and regional analyses are combined to optimize pillar design on a global mine scale. This dissertation focuses on improving hazard mitigation in mines with unpredicted anomalies like karsts. Although this research is based on a specific mine site, it empowers the operators to explore the presented techniques to increase safety in all underground mines. The suggested methodology will help devise better strategies for handling instability issues without jeopardizing the mine operations. The primary motivation is to keep the underground miners safe from hazardous situations while fulfilling the secondary objective of maximizing mineral production.

Yes / A problem with 1-D water hammer modelling is in the application of accurate unsteady friction. Moreover, investigating the time response of fluid dynamics and unsteady turbulence structures during the water hammer is not possible with a 1-D model. This review article provides a summary of 1-D modelling using the recent finite volume approach and the discussion extends to a quasi-2-D model and historical developments as well as recent advancements in 3-D CFD simulations of water hammer. The eddy viscosity model is excellent in capturing pressure profiles but it is computationally intensive and requires more computational time. This article reviews 3-D CFD simulations with sliding mesh, an immersed solid approach, and dynamic mesh approaches for modelling valve closures. Despite prediction accuracy, a huge computational time and high computer resources are required to execute 3-D flow simulations with advanced valve modelling techniques. Experimental validation shows that a 3-D CFD simulation with a flow rate reduction curve as a boundary condition predicted accurate pressure variation results. Finally, a brief overview of the transient flow turbulence structures for a rapidly accelerated and decelerated pipe flow using DNS (Direct numerical simulation) data sets is presented. Overall, this paper summarises past developments and future scope in the field of water hammer modelling using CFD.

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

On considère l'équation de convection--diffusion--réaction instationnaire. On s'intéresse à la dérivation d'estimations d'erreur a posteriori pour la discrétisation de cette équation par la méthode des volumes finis centrés par mailles en espace et un schéma d'Euler implicite en temps. Les estimations, qui sont établies dans la norme d'énergie, bornent l'erreur entre la solution exacte et une solution post-traitée à l'aide de reconstructions $Hdiv$-conformes du flux diffusif et du flux convectif, et d'une reconstruction $H^1_0(Omega)$-conforme du potentiel. On propose un algorithme adaptatif qui permet d'atteindre une précision relative fixée par l'utilisateur en raffinant les maillages adaptativement et en équilibrant les contributions en espace et en temps de l'erreur. On présente également des essais numériques. Enfin, on dérive une estimation d'erreur a posteriori dans la norme d'énergie augmentée d'une norme duale de la dérivée en temps et de la partie antisymétrique de l'opérateur différentiel. Cette nouvelle estimation est robuste dans des régimes dominés par la convection et des bornes inférieures locales en temps et globales en espace sont également obtenues / We consider the time-dependent convection--diffusion--reaction equation. We derive a posteriori error estimates for the discretization of this equation by the cell-centered finite volume scheme in space and a backward Euler scheme in time. The estimates are established in the energy norm and they bound the error between the exact solution and a locally post processed approximate solution, based on $Hdiv$-conforming diffusive and convective flux reconstructions, as well as an $H^1_0(Omega)$-conforming potential reconstruction. We propose an adaptive algorithm which ensures the control of the total error with respect to a user-defined relative precision by refining the meshes adaptively while equilibrating the time and space contributions to the error. We also present numerical experiments. Finally, we derive another a posteriori error estimate in the energy norm augmented by a dual norm of the time derivative and the skew symmetric part of the differential operator. The new estimate is robust in convective-dominated regimes and local-in-time and global-in-space lower bounds are also derived

In the present work we study the numerical solution of shallow water equations. We introduce a vectorial notation of equations laws of conservation from which we derive the shallow water equations (SWE). There is the simplify its derivation, notation and the most important features. The original contribution is to derive equations for shallow water without the using of Leibniz's formula. There we report the finite volume method with the numerical flow of Vijayasundaram type for SWE. We present a description of the linear reconstruction, quadratic reconstruction and ENO reconstruction and their using for increasing of order accuracy. We demonstrate using of linear reconstruction in finite volume method of second order accuracy. This method is programmed in Octave language and used for solving of two problems. We apply the method of the ADER type for the shallow water equations. This method was originally designed for the Euler's equation.