Silicon is produced in submerged arc furnaces which are heated by electric currents passing through the furnace. It is important to understand the distribution of heating within the furnace in order to accurately model the silicon production process, yet many existing studies neglect aspects of this current flow. In the present paper, we formulate a model that couples the electrical current to thermal, material flow and chemical processes in the furnace. We then exploit disparate timescales to homogenise the model over the timescale of the alternating current, deriving averaged equations for the slow evolution of the system. Our numerical simulations predict a minimum applied current that is required in order to obtain steady-state solutions of the homogenised model and show that for high enough applied currents, two spatially heterogeneous steady-state solutions exist, with distinct crater sizes. We show that the system evolves to the steady state with a larger crater radius and explain this behaviour in terms of the overall power balance typically found within a furnace. We find that the industrial practice of stoking furnaces increases the overall rate of material consumption in the furnace, thereby improving the efficiency of silicon production.

In this paper, the problem of magnetohydrodynamics (MHD) boundary layer flow of nanofluid with heat and mass transfer through a porous media in the presence of thermal radiation, viscous dissipation and chemical reaction is studied. Three types of nanofluids, namely Copper (Cu)-water, Alumina (Al 2 O 3)-water and Titanium Oxide (TiO 2)-water are considered. The governing set of partial differential equations of the problem is reduced into the coupled nonlinear system of ordinary differential equations (ODEs) by means of similarity transformations. Finite element solution of the resulting system of nonlinear differential equations is obtained using continuous Galerkin-Petrov discretization together with the well-known shooting technique. The obtained results are validated using MATLAB “bvp4c” function and with the existing results in the literature. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and the impact of various physical parameters such as the magnetic parameter M, solid volume fraction of nanoparticles 𝜙 and type of nanofluid on the flow is discussed. The results obtained in this study confirm the idea that the finite element method (FEM) is a powerful mathematical technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

The numerical modeling of a binary solidification with a mushy layer mechanism is considered in this manuscript. The nonlinear coupled system of equations describes the heat and mass diffusions of a one-dimensional spatial variable in the semi-infinite interval. Also formulated is a transformed system in a finite interval. We propose numerical methods for solving the nonlinear system using a threshold strategy based on fixed computation-domain approach. Our calculated results and those from the LeadEx field experiment are well-matched in their tendencies.

A mathematical model for the steady, mixed convection heat and mass transfer along a semi-infinite vertical plate embedded in a micropolar fluid in the presence of Soret and Dufour effects is presented. The non-linear governing equations and their associated boundary conditions are initially cast into dimensionless forms using local similarity transformations. The resulting system of equations is then solved numerically using the Keller-box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The non-dimensional velocity, microrotation, temperature and concentration profiles are displayed graphically for different values of coupling number, Soret and Dufour numbers. In addition, the skin-friction coefficient, the Nusselt number and Sherwood number are shown in a tabular form.

In this study, an analysis has been performed for heat and mass transfer of a steady laminar boundarylayer flow of an electrically conducting fluid with radiation effect of second grade subject to suction and to a transverse uniform magnetic field past a stretching sheet. Parameters Gr, Gc, Nr, M, Sc represent the dominance of the buoyant effect, radiative effect, magnetic effect and mass transfer effect which have presented in governing equations, respectively. The similar transformation and the finite-difference method have been used to analyze the present problem.

This study investigates the problem of non-Newtonian liquid evaporation from an impermeable vertical plate in a porous medium under mixed convection conditions. The analysis examines the effects of non-Darcian flow phenomena on the evaporation process and investigates the heat and mass transfer characteristics under a range of flow conditions in the aiding and opposing flow types. The parameters of interest include Rak,non, Pex,non, Pr and Sc. The data required for the calculations include a fixed wall temperature of Tw = 45°C, a free flow temperature of T∞ = 20°C and an air free flow rate of u∞ = 4m/sec and the corresponding humidity of 50% flows over fixed temperature and moist wall.

The results indicate that an increasing buoyancy force enhances the mixing effect and therefore improves the rate of heat transfer, the latent heat flux is the primary mode of heat transfer in evaporation processes and the velocity gradient at the wall increases with an increasing buoyancy effect. Furthermore, the evaporation rate increases with a reducing flow index and an increasing buoyancy effect.

This work studies the natural convection heat and mass transfer near a horizontal cylinder of elliptic cross section with constant wall temperature and concentration in a fluid-saturated porous medium. A coordinate transformation is used to obtain the nonsimilar governing boundary layer equations. The transformed governing equations are then solved by the cubic spline collocation method. Results for the local Nusselt and Sherwood numbers are presented as functions of the Lewis number, the buoyancy ratio, and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). The heat and mass transfer rates of the elliptical cylinder with slender orientation are higher than those with blunt orientation.

An analysis is performed to the study of the momentum, heat and mass transfer of a viscous fluid-saturated porous medium past an impermeable, non-isothermal stretching sheet with internal heat generation or absorption and chemical reaction. The governing partial differential equations are converted into ordinary differential equations by means of a similarity transformation. Exact solutions of velocity components together with the pressure distribution, which can not be found in the boundary layer theory, are obtained analytically; in addition, the temperature and concentration functions are given in terms of confluent hypergeometric functions. The velocity, temperature (concentration) profiles and thermal characteristics at the sheet for relevant parameters are plotted, tabulated and discussed.