Thesis (M.Sc. (Physics))--North-West University, Potchefstroom Campus, 2008.

Thermal radiation is of significant importance in a broad range of engineering applications including high-temperature and large-scale systems. Although the governing equations of thermal radiation have been known for many years, the complexities inherent in the phenomenon, such as the multidimensionality and integro-differential nature of these equations, have made it difficult to obtain an accurate, efficient, and robust computational method. Developing the finite volume radiation method in the 1990s was a significant progress but not a panacea for computational radiation. The major drawback of this method, which is common among all methods that solve for directional intensities, is its slow convergence rate in many situations which increases the solution cost dramatically. These situations include large optical thicknesses, strongly reflecting boundaries, and any other factor that causes strong directional coupling like complex geometries. Several acceleration schemes have been developed in the heat transfer and neutron transport communities to expedite the convergence and reduce the solution cost, but none of them led to a general and reliable method. Among these available schemes, the two most promising ones, the multiplicative scheme and coupled ordinates method, suffer from failing on fine grids and being very complicated for complex scattering phase functions, respectively. In this research, a new computational method, called the QL method, has been introduced. The main idea of this method is using the phase weight concept to relate the directional and average intensities and re-arranging the Radiative Transfer Equation to find a new expression for the radiant heat flux. This results in an elliptic-type equation for the average intensity at each control volume which conserves the radiant energy in all directions in the control volume. This formulation gives the QL method a great advantage to solve for the average intensity while including the directional effects. Since the directional effects are included and the radiant energy is conserved in each control volume, this method is expected to be accurate and have a good convergence rate in all conditions. The phase weight distribution required by the QL method can be provided by a method like the finite volume method or discrete ordinates method. The QL method is applied to several 1D and 2D test cases including isotropic and anisotropic scattering, black and partially reflecting boundaries, and emitting absorbing problems; and its accuracy, convergence rate, and solution cost are studied. The method has been found to be very stable and efficient, regardless of grid size and optical thickness. This method establishes very accurate predictions on the tested coarse grids and its results approach the exact solution with grid refinement.

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Thermal radiation is of significant importance in a broad range of engineering applications including high-temperature and large-scale systems. Although the governing equations of thermal radiation have been known for many years, the complexities inherent in the phenomenon, such as the multidimensionality and integro-differential nature of these equations, have made it difficult to obtain an accurate, efficient, and robust computational method. Developing the finite volume radiation method in the 1990s was a significant progress but not a panacea for computational radiation. The major drawback of this method, which is common among all methods that solve for directional intensities, is its slow convergence rate in many situations which increases the solution cost dramatically. These situations include large optical thicknesses, strongly reflecting boundaries, and any other factor that causes strong directional coupling like complex geometries. Several acceleration schemes have been developed in the heat transfer and neutron transport communities to expedite the convergence and reduce the solution cost, but none of them led to a general and reliable method. Among these available schemes, the two most promising ones, the multiplicative scheme and coupled ordinates method, suffer from failing on fine grids and being very complicated for complex scattering phase functions, respectively. In this research, a new computational method, called the QL method, has been introduced. The main idea of this method is using the phase weight concept to relate the directional and average intensities and re-arranging the Radiative Transfer Equation to find a new expression for the radiant heat flux. This results in an elliptic-type equation for the average intensity at each control volume which conserves the radiant energy in all directions in the control volume. This formulation gives the QL method a great advantage to solve for the average intensity while including the directional effects. Since the directional effects are included and the radiant energy is conserved in each control volume, this method is expected to be accurate and have a good convergence rate in all conditions. The phase weight distribution required by the QL method can be provided by a method like the finite volume method or discrete ordinates method. The QL method is applied to several 1D and 2D test cases including isotropic and anisotropic scattering, black and partially reflecting boundaries, and emitting absorbing problems; and its accuracy, convergence rate, and solution cost are studied. The method has been found to be very stable and efficient, regardless of grid size and optical thickness. This method establishes very accurate predictions on the tested coarse grids and its results approach the exact solution with grid refinement.

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.

Borders of flow field around immersed bodies can be extended to long distances since there are no physical boundaries. In computational practice however, the flow domain must be restricted to a reasonable size by imposing appropriate boundary conditions at the edges of the computational space. In this thesis work, streamlines obtained from potential flow solution in a relatively large spatial domain are utilized to specify the boundaries and boundary conditions for a more restricted computational domain to be used for detailed viscous flow computations. A grid generation code is adopted for generation of unstructured triangular grid systems for domains involving multiple immersed bodies of any shape at arbitrary orientations such as a group of tall buildings in horizontal plane. Finite volume method is used in the solution of Laplace equation for the stream function. A deformation modulus is introduced as a probe parameter to aid locating the viscous flow boundaries. The computer code acts as a preprocessor for viscous flow computations, specifying the computational boundaries, the boundary conditions and generating the computational grid.

Coins and medallions are manufactured by using coining process which is a metal forming process. In coining of medallions, there is a strong need to shorten the production time and reduce the production cost and waste of material in conventional coining method. An alternative coining method may be considered in order to reduce the production time and the manufacturing cost. In this study, a new method has been proposed. In the proposed method, design of the medallion is performed by utilizing computer aided engineering (CAE) environment and the master die is manufactured by means of NC codes. The modular designs of blanking and coining die sets for medallions with a diameter in the range of 30-90 mm have been realized. Coining and blanking processes for production of the medallion have been simulated by using a commercial finite volume program. Moreover, a commemorative medallion for the opening ceremony of METU-BILTIR Center Forging Research and Application Laboratory has been designed. After die sets have been manufactured, the real-life experiments have been conducted by using 1000 tones mechanical forging press and 200 tones eccentric press available in Forging Research and Application Laboratory of the METU-BILTIR Center. The results have been compared with the computer simulations. After the real-life experiments, it has been observed that medallions have successfully been obtained by employing the new proposed method. Therefore, the new proposed method for coining has been verified.

Finite volume method was employed for analyzing the three-dimensional turbulent flow structures, species distributions, and mixing behaviors of combustion flows in a rotary kiln under various operation conditions. The modified £e-£`turbulence model together with wall functions was adopted. Devolatilization of solid wastes were simulated by gaseous methane (CH4) non-uniformly distributed along the kiln bed. Combustion process was considered as a two-step reaction when primary air entered and mixed with methane gas in the first combustion chamber. Mixing-controlled eddy-dissipation model was employed for predicting the reaction rates of CH4, O2, CO2, CO and H2O. Effects of inleakage air, kiln rotation speed and methane distribution along the kiln bed were also examined. Results show that 128% excess air will get the best combustion efficiency, above which the combustion efficiency will decrease. The temperature and species are not uniformly distributed and are vertically stratified on cross-sectional plane. The combustion efficiency will also be lowered if there is inleakage airflow. Results also show rotation speed and methane distributions have little effect on combustion efficiency.

ABSTRACT Three-dimensional modeling on the aerodynamics of airflow and diffusion of air pollutants in a longitudinal-ventilated traffic tunnel was carried out. The model takes ventilation fans, traffic flow rate, speed, emission factor and piston effect of moving vehicles into consideration. Turbulent flow and dispersion of gaseous pollutants in road tunnels were solved numerically using the finite volume method. Traffic emissions were accordingly modeled as banded line sources along the tunnel floor. The effects of fan ventilation, roughness and piston effect of moving vehicles on the air flow and pollutant dilution are examined. Concentrations of gaseous pollutants CO, NOX, SO2 and THC (total hydrocarbons) at three axial locations in the tunnel, together with traffic flow rate, traffic speed and types of vehicle were measured. Case study was conducted on the Cross-Harbor Tunnel and the Chungcheng Tunnel in which on-site measurements of traffic flow were also conducted concurrently to provide traffic emission data to the tunnel environment for numerical simulation and comparisons. The aim of this study was to understand the spatial variation of air pollutants generated by traffic emissions and evaluation of ventilation performance and piston effect of moving vehicles on dilution of air pollutants in these tunnels. The results show that the major emission sources of CO are passenger cars and motorcycles, while major emission sources of NOx are trucks. Pollutants convect downstream with the wind generated either by longitudinal ventilation fans and/or moving vehicles, thus causing increasing pollutants concentrations with increasing downstream distance. The piston effect of moving vehicle alone can provide 64% ~ 85% increase of wind speed in Chungchen Tunnel and 13% ~ 20% in Cross-Harbor Tunnel. When all fans are on, showing 185% ~ 328% and 120% ~ 182% increases in Chungchen Tunnel and Cross-Harbor Tunnel, respectively. The piston effect of moving vehicle alone can provide 14% ~ 32% dilution of air pollutants in the Chungcheng Tunnel. The piston effect of moving vehicles is compounded with ventilation fans, showing a 47% ~ 66% dilution effect when all fans are on. For the Cross-Harbor Tunnel, the piston effect of moving vehicle alone can provide 9% ~ 23% dilution of air pollutants and 36% ~ 74% dilution effect when all fans are on. The results reveal that cross-sectional concentrations are non-uniformly distributed and that concentrations rise with downstream distance. When all fans were turned off, wind speed in tunnels would be considered as constant, and gaseous pollutants concentration agree with linearly alone the tunnel.

The present study considers steady laminar two-dimensional incompressible flow over both in-line and staggered flat tube bundles used in heat exchanger applications. The effects of various independent parameters, such as Reynolds number (Re), Prandtl number (Pr), length ratio (L/Da), and height ratio (H/Da), on the pressure drop and heat transfer were studied. A finite volume based FORTRAN code was developed to solve the governing equations. The scalar and velocity variables were stored at staggered grid locations. Scalar variables (pressure and temperature) and all thermophysical properties were stored at the main grid location and velocities were stored at the control volume faces. The solution to a one-dimensional convection diffusion equation was represented by the power law. The locations of grid points were generated by the algebraic grid generation technique. The curvilinear velocity and pressure fields were linked by the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm. The line-by-line method, which is a combination of the Tri-Diagonal Matrix Algorithm (TDMA) and the Gauss-Seidel procedure, was used to solve the resulting set of discretization equations. The result of the study established that the flow is observed to attain a periodically fully developed profile downstream of the fourth module. The strength increases and the size of the recirculation gets larger as the Reynolds number increases. As the height ratio increases, the strength and size of the recirculation decreases because the flow has enough space to expand through the tube passages. The increase in length ratio does not significantly impact the strength and size of the recirculation. The non-dimesionalized pressure drop monotonically decreased with an increase in the Reynolds number. In general, the module average Nusselt number increases with an increase in the Reynolds number. The results at Pr = 7.0 indicate a significant increase in the computed module average Nusselt number when compared to those for Pr = 0.7. The overall performance of in-line configuration for lower height ratio (H/Da = 2) and higher length ratio (L/Da = 6) is preferable since it provides higher heat transfer rate for all Reynolds numbers except for the lowest Re value of 25. As expected the staggered configurations perform better than the in-line configuration from the heat transfer point of view.