Mathematical modeling and study of the process of solidification in metal casting

The process of melting and solidification in metal casting is considered. The process is modeled by a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the problem and its finite-difference approximation are given. A numerical algorithm is presented for solving the direct problem. The results are described and analyzed in detail. Primary attention is given to the evolution of the solidification front and to how it is affected by the parameters of the problem. Some of the results are illustrated by plots.     

Mathematical modeling of solidification process near the inner core boundary of the Earth

Radially symmetric analytic solutions of the heat and mass transfer equations governing convection in the Earth’s fluid core are found in terms of deviations from the adiabatic reference state. We demonstrate that an increase of the convective velocity leads to a decrease of the light constituent mass fraction and specific entropy. Where fluid is rising/descending, convective motions decrease/increase the mass fraction and entropy at the inner core boundary (ICB). The influence of convective motions on the thermal fluxes at the core mantle boundary is studied. On the basis of exact solutions we demonstrate that the liquid is supercooled near the ICB. An important point is that an increase in the convective velocity directed to the ICB increases the constitutional supercooling. We show that the anelastic model (AM) can be used only at small supercoolings near the ICB. The most probable solidification scenario “constitutional supercooling and morphological instability” should be described by a mushy layer theory near the ICB and by the AM in the rest region of the fluid outer core. On the basis of dendritic theory and selection mechanisms of crystal growth the dendrite tip radius and interdendritic spacing in the mushy layer at the ICB are determined in the presence of convection.     

Mathematical modeling of laser heating of metal. Dynamical problem

Institute of Mathematics and Information, Lithuanian Academy of Sciences. Translated from Litovskii Matematicheskii Sbornik (Lietyovs Matematikovs Rinkinys), Vol. 31, No. 4, pp. 687–699, October–December, 1991.     

Mathematical modeling of laser heating of metal. Spatial problem

Institute of Mathematics and Information, Lithuanian Academy of Sciences. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 4, pp. 700–707, October–December, 1991.     

Mathematical modeling of coupled turbulent flow and solidification in a single belt caster with electromagnetic brake

A mathematical model has been developed to simulate turbulent fluid flow and solidification in the presence of a DC magnetic field in an extended nozzle for metal delivery to a single belt caster. This paper reports on predicted effects of DC magnetic field conditions in modifying flows and solidification behavior in the metal delivery system. It is shown that the application of a DC magnetic brake to the proposed system can result in a reasonably uniform feeding of melt onto the cooled moving belt. This, in turn, optimises the rate of even shell growth along the chilled substrate. In order to account for the effects of turbulence, a revised low-Reynolds k–ε turbulent model was employed. A Darcy-porosity approach was used to simulate fluid flow within the mushy solidification region. Simulations were carried out for plain carbon steel strip casting. The fully coupled transport equations were numerically solved using the finite volume method. The computed flow patterns were compared with those reported in the literature. The performance of the magnetic flow control device proposed in this work is evaluated and compared with flow modifications obtained by inserting a ceramic filter within the reservoir.     

Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields

When particles are subjected to an acoustic field particle trajectories depend on the particle and fluid compressibility and density values. Hence a combination of acoustic and flow fields on particles can be used to deflect and trap, or to segregate and/or fractionate fine particles in fluid suspensions. Using particle physics in an acoustic field, a mathematical model was developed to calculate trajectories of deflected particles due to the application of acoustic standing waves. The resulting second order ordinary differential equation was quite stiff and hence difficult to solve numerically and did not have a closed form solution. The analysis of the above equation showed that the basic problems with numerical solutions could not be ameliorated through the use of standard rescaling techniques. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. An approximate solution was derived which enabled the calculation of the particle trajectories and concentration at collection planes in the acoustic field. Analysis of the solution showed that all the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter took approximately 20 s to reach that node. Then at the bench scale, the above technology was implemented by building a flow chamber with two transducers at opposite ends to generate an acoustic standing wave. SiC particle trajectories were tracked using captured digital images from a high-resolution microscope. The displacements of SiC particles due to an acoustic force were compared with the mathematical model predictions. For input power levels between 3.0 and 5.0 W, the experimental data were comparable to mathematical model predictions. Hence it was concluded that the proposed approximate solution was both quantitatively and qualitatively closer to experimental results than the simplified form ignoring the second order term reported in the literature.     

Mathematical modeling of the processes occurring during high-temperature spray coating

We give a generalization of the mathematical model of the processes occurring in the preparation of gasthermal coatings in order to compute the radiation energy, thermal effects of turbulent flow of the gas jet of a plasmotron, spraying distance and required degree of preliminary heating of the base. The boundary conditions obtained describe the radiational/convective heat exchange of bodies with a medium through thin coatings taking account of the speed of flow of the gas jet.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 13–18.     

Mathematical modeling of temperature pulsation in a thick anode in optoelectronic systems

A numerical method is applied to analyze the one-dimensional model of temperature pulsation inside a two-layer anode with temperature-dependent thermophysical coefficients. The analysis allows for surface thermal radiation, secondary electron emission, and distribution of electron energy losses by penetration depth into the target. Penetration of pulsations deep into the target is observed.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 215–217, 1993.     

Optimal control of the solidification process in metal casting

The optimal control of the solidification process in metal casting is considered. The mathematical model is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. The mathematical formulation of the optimal control problem is given. The problem is solved numerically by direct optimization methods. The numerical results are described and analyzed. Some of the results are illustrated by plots.     

Control of phase boundary evolution in metal solidification for new thermodynamic parameters of the metal

The problem of controlling the phase boundary evolution in the course of solidification of metals with different thermodynamic properties is studied. The underlying mathematical model of the process is based on a three-dimensional nonstationary two-phase initial–boundary value problem of the Stefan type. The control functions are determined by optimal control problems, which are solved numerically with the help of gradient optimization methods. The gradient of the cost function is exactly computed by applying the fast automatic differentiation technique. The research results are described and analyzed. Some of them are illustrated.     

Mathematical modeling of the dissolving and deposition processes during solution seepage in a porous medium

A mathematical model that describes solution seepage in a porous medium and the processes of mineral dissolving and secondary deposition is proposed. Self-similar solutions describing the motion of the leading and trailing fronts, that is, the boundaries of the complete-dissolving zone, are determined. The main features of the processes under consideration are studied and numerical calculations are performed. It is shown that the model describes well the experimental data on mineral leaching by sulfate solutions. The dynamics of mineral extraction from productive solutions in a medium with a nonuniformacidity distribution are investigated. It is shown that, in the elevated-PH zones, the mineral is dissolved; whereas, in the low-acidity zones, secondary deposition of the mineral occurs. In the latter case, after the work has been completed, the bed may contain more or less considerable mineral resources, depending on the extent of the low-PH zone and its proximity to an extraction well.     

Mathematical modeling of the forward contract market

A mathematical model of a market in which transactions are performed in two steps is studied. At the first step, forward contracts are concluded and the second step represents a single-price auction, where the residual demand is satisfied. The necessary and sufficient conditions for the existence of the perfect subgame equilibrium in this model are obtained. The two-step market is compared with the Cournot auction from the standpoint of consumers.     

Mathematical modeling of plant morphogenesis

Current trends in biology call for analysis of the bulk information accumulated through mathematical simulations of biological processes aiming at revealing certain regularities and verifying hypotheses and predictions. Evolution of organisms is of special interest to mathematical modeling because it integrates a great body of different processes varying over time and over space. In this paper, models of biological processes as related to plant development are reviewed. The models are classified, and approaches to problems that are most intricate from the simulation standpoint, as well as relevant mathematical methods, are described.     

Mathematical modeling of church growth

The possibility of using mathematics to model church growth is investigated using ideas from population modeling. It is proposed that a major mechanism of growth is through contact between religious enthusiasts and unbelievers, where the enthusiasts are only enthusiastic for a limited period. After that period they remain church members but less effective in recruitment. This leads to the general epidemic model which is applied to a variety of church growth situations. Results show that even a simple model like this can help understand the way in which churches grow, particularly in times of religious revival.