Papers by Keyword: Finite Difference Method

Abstract: The design of fins needs to be optimized to ensure effective heat transfer or dissipation from the engine to the surrounding air. The choice of material for the fins is crucial to facilitate unhindered heat dissipation from the fins to the environment. This study employed a numerical simulation using the finite difference method to analyze the temperature distribution within various sections of the fins. The calculations were performed from the transient state until a steady state was achieved. The findings indicate that the fin material significantly impacts the temperature distribution across different sections of the fin. Heat propagation in the fin is predominantly through conduction, thus the thermal conductivity of the material substantially affects the temperature distribution along the fin. Fins made from conductive materials exhibit higher temperature changes over time intervals and reach a steady state more rapidly. Conversely, fins made from less conductive materials fail to transfer heat efficiently to the fin's end, resulting in a higher temperature difference between the base and the tip and a longer time to reach a steady state.

Abstract: Substantial temperature disparities between machine components and their environment frequently impede the attainment of optimal system performance. To alleviate excessive thermal loads, the utilization of fins as cooling elements is a widely adopted approach. This research endeavors to numerically examine the temperature distribution within a square fin subject to varying convective coefficients. Numerical simulations employing the finite difference method were carried out to forecast the temperature distribution within the fin during the cooling process. The simulation outcomes reveal that the convective coefficient exerts a significant influence on the temperature distribution and the temporal duration required for the fin to attain a steady state. Elevated convective coefficients facilitate accelerated heat transfer from the fin to the cooling fluid and diminished temperature differentials between the fin base and tip. Conversely, augmenting the convective coefficient also culminates in a reduction in fin efficiency as a larger proportion of the heat is unable to reach the fin tip. The findings of this study can contribute to the optimization of fin design in a myriad of mechanical engineering applications.

Abstract: The Cahn-Hilliard equation, known for describing the evolution of interfaces in multicomponent systems, can also be employed to noise reduction in mathematical functions and concentration-dependent heat transfer simulations. This work presents a finite difference method discretization of the Cahn-Hilliard equation and explores its applications. For noise reduction, three different noisy functions are simulated, demonstrating effective recovery of original functions despite significant noise levels. In heat transfer simulations, three initial temperature distributions are explored with concentration-dependent thermal diffusivity. Results show that concentration significantly affects thermal diffusivity and heat propagation, leading to non-uniform temperature distributions. Comparative simulations without concentration influence highlight the distinct impact of concentration on thermal behavior. The study underscores a reliable approach to noise reduction and insight into concentration-dependent heat transfer dynamics.

Abstract: In this paper, a novel artificial staggered grid points and under-relaxation free solution for a checkerboard pattern problem in a quasi-one-dimensional, incompressible, steady, and inviscid flow is introduced. The purpose of this numerical development is to obtain a new numerical solution, which is under-relaxation factor free scheme, more accurate, and easier to implement than a conventional staggered grid scheme. The proposed numerical solution can be described as the non-staggered grid/collocated grid central difference scheme which is free of pressure checkerboard pattern or spurious oscillation. The accuracy and convergence speed of the proposed numerical scheme is benchmarked against a conventional SIMPLE-based finite volume scheme and the exact solution for the flow problem in a convergent nozzle. The numerical analysis shows that the proposed numerical scheme outperforms the SIMPLE-based finite volume scheme in terms of accuracy, computational resource, and convergence speed. Also, the proposed numerical scheme has consistent numbers of iteration over the different grid sizes in contrast to the SIMPLE-based scheme which is iteration-grid size dependent. The proposed numerical scheme can be implemented with both uniform and non-uniform grid points and shows good agreement with the exact solution for every grid size. However, the uniform grid approach produces significantly more accurate results than the non-uniform grid approach. Hence, the choice of grid distribution is still an important factor affecting the accuracy of the proposed numerical solution. The proposed numerical technique can be further extended to solve incompressible flow problem in the complex 2D-3D domain with unstructural grids.

Abstract: The objective of this work is to study theoretically the ground state energy of a donor atom located in a two-dimensional ultra-thin cylindrical nanostructure called nanoflakes for different geometries controlled by the three geometrical parameters, in an infinite confinement potential. The solution of our equations system is based on the 2D finite difference method. Our numerical calculations show that the ground state energy of a donor atom is more important in the small area of the nanoflakes. Moreover, beyond the critical regions, the donor energy becomes stable.

Abstract: The Langevin theory of diamagnetism is used to examine the effects of geometric confinement and hydrogenic impurity location on the diamagnetic susceptibility in a GaAs hemispherical quantum dot with an infinite confinement potential considered as vacuum. Using the finite difference approach and the effective mass approximation, the electron-donor Schrödinger equations are derived. As a function of the size of the hemi-spherical quantum dot, the mean value of the electron location and electron to ionized donor atom distance are investigated, taking into account the various impurity positions. The results show that shrinking the size of the hemi-spherical quantum dot improves the diamagnetic susceptibility by reducing the electron-donor distance. The major findings show that the donor impurity location has a significant impact on the diamagnetic susceptibility. We believe that the findings from our work into the diamagnetic susceptibility of quantum dots will be crucial in determining how well optoelectronic devices will operate.

Abstract: In this paper, we have studied the electron-donor atom diamagnetic susceptibility confined in a hemi-cylindrical quantum dot (QD). It is analyzed specifically how the impurity location affects diamagnetic susceptibility. The 3D Schrödinger equation in hemi-cylindrical QD was solved using the finite difference method within the effective mass approximation. This is accomplished by performing our system's Hamiltonian in hemi-cylindrical geometry. We have demonstrated that the hemicylindrical size and impurity position have a significant impact on the diamagnetic susceptibility. When the impurity is localized in the center of the nanostructure for the hemi-cylindrical QD, the diamagnetic susceptibility reaches its greatest value.

Abstract: Rice is one of the most consumed cereals in the world, and the cultivation of this crop has significant relevance in the southern region of Brazil. When subjected to inadequate conditions of temperature and humidity, rice becomes susceptible to attacks from pests and fungi and, therefore, care in the storage process is of paramount importance, since this is largely responsible for the quality of the harvest. Such care allows the food to arrive without harm to the consumer. In this sense, the mathematical modeling, among numerous possibilities, allows for evaluating the internal temperature of a silo and, through this, taking preventive measures so that the grains maintain their quality. The objective of this work is to model the heat transfer process in a silo prototype containing rice in husk through the explicit finite difference method for a one-dimensional and transient model considering two approaches centered on the spatial derivative: error of order 2 and 4. In addition, the thermal diffusivity of the grain with average value was analyzed. The results obtained by the solutions were analyzed through graphs and statistical indexes comparing with the experimental data of the literature, and the computer simulation was performed through the Google Colab platform. The chosen methodology proved effective for the work, and the predicted temperatures for the approximations of order 2 and 4 denote similarities both graphically and in the precision of the statistical indexes.

Abstract: The transient dynamics of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar flow of a Newtonian liquid flowing through a rectangular channel is investigated. The Boussinesq approximation is assumed for the density variation with pollutant concentration. The governing equations of mass and momentum conservation are coupled to the pollutant concentration equation as well as to the viscoelastic constitutive model for the polymer stresses. The Oldroyd-B viscoelastic constitutive model is employed to model the deformation and characteristics of the polymer stresses. The coupled system of nonlinear partial differential equations is solved numerically using robust and efficient semi-implicit finite difference methods (FDM). Solutions are presented in graphical form for various parameter values. The model can be a useful tool in understanding the dynamics of domestic and industrial pollution situations that may arise from improper discharge of long-chain hydrocarbon products into, say, water drainage systems. The novelty of this investigation is in the modelling of the long-chain hydrocarbon-product pollutants via appropriate viscoelastic (polymeric) constitutive equations. In general, it is observed that parameters which increase (decrease) the flow velocity correspondingly increase (respectively decrease) the wall shear stress. Similarly, it is observed that parameters which increase (decrease) the polymer concentration correspondingly increase (respectively decrease) the mass transfer rates. The wall shear stress and mass transfer are measurable quantities. In this respect, our work offers such measurements as predictive tools to detect the scale of contamination.

Abstract: A mathematical model of the process of gas propagation in the atmosphere and its sorption by fine flow has been developed. The use of the finite difference method in modeling allows to obtain numerical solutions of the spatial distribution of gas concentration during its deposition by a jet of arbitrary intensity and shape. The proposed method of mathematical description of the process of sorption of hazardous gases allows you to choose an arbitrary number and spatial location of nodal points that satisfy the Courant-Friedrichs-Levy condition. The developed model allows to predict the intensity of gas sorption in technological processes and in the elimination of the consequences of emergencies. The use of the developed model will increase the efficiency of emergency management and choose effective methods of sorption of hazardous gases in the atmosphere. The results of numerical calculations confirmed the efficiency of the developed model and theoretically demonstrated the effectiveness of using water curtains for the sorption of ammonia from the atmosphere. According to the simulation results, it is established that the use of fine spray jets can significantly reduce the distance of distribution of hazardous gas.