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 and process simulation of perlite grain expansion in a vertical electrical furnace

Expanded perlite is a lightweight material with remarkable thermal and acoustic insulation properties, rendering it widely useful in the construction and manufacturing industries. Currently applied perlite expansion technology suffers numerous technical disadvantages, which adversely affect product quality and limit the range of its applications. To overcome these established drawbacks, a new perlite expansion process has been designed on the basis of a vertical electrically heated expansion furnace. The novel furnace enables precise control of experimental conditions, in order to allow for efficient adjustment of particle residence time and internal temperature. The quality of expanded perlite strongly depends on raw material thermophysical properties as well as furnace operating conditions, and the experimental investigation of the isolated effect of each parameter on expanded product quality is technically cumbersome and extremely time-consuming and expensive.A mathematical model for perlite grain expansion has been developed in order to perform a detailed numerical investigation of process efficiency, toward the optimization of the expansion process in the novel pilot-scale furnace. The dynamic model consists of ordinary differential equations for both air and particle heat and momentum balances, as well as nonlinear algebraic equations for both air and perlite melt thermophysical and transport properties, probing the air temperature distribution within the vertical electrical furnace as well as the particle velocity, temperature and size along its trajectory inside the heating chamber. The effect of raw material physical properties (raw feed origin, initial particle size, effective water content) as well as operating parameters (air inlet temperature and flowrate, furnace wall temperature) on evolution of the particle state variables is presented and discussed. Model results indicate perlite expansion is strongly affected by raw ore feed origin, size and water content. Moreover, operating conditions affect expansion considerably, and furnace wall temperature has the strongest effect on the final particle expansion ratio attained. The new dynamic model is instrumental towards achieving a detailed comprehension of perlite expansion in the vertical electrical furnace towards multi-parametric sensitivity analysis, process optimization and efficient control.     

Mathematical modeling of the receptor-mediated endocytosis process of targeted therapeutic agents in drug delivery systems

In this paper, we propose mathematical models to describe receptor-mediated endocytosis processes. One is a stochastic differential model for the agent-target binding process. The mean extinction time and a standard variation over time profile are evaluated. The other is an age-structured model for demonstrating endocytosis and lysosome processes. A targeted drug delivery system has a complex process in how it is to deliver drug molecules in terms of administration, transportation in blood and across membranes to intracellular space, and inhibition to microtubule polymerization. In particular, receptor-mediated endocytosis of targeted therapeutic agents, such as antibody drug conjugates or ligand-targeted liposome encapsulated nanoparticles, is a key step in understanding the drug delivery mechanism. We discuss stochastic quasi steady state approximation when agent-target complex does not appreciably vary compared with the free agents. This reduces the number of the systems and the parameters; however, an initial time phase cannot be captured. In addition, we discuss the strengths and weaknesses when the age-structured model induces the reduced model compared with the full model that considers endocytosis and lysosome processes. If the total mean retention time until payload release in intracellular space is known, then the age-structured model with the Erlang distribution may fairly predict data of the released payload over time profile with far fewer parameters; however, induced compartments lose their physical meaning and describe only a delay.     

Mathematical modeling of aerosol emission from die sinking electrical discharge machining process

Emission of toxic gases and aerosol is an important hazard associated with the electrical discharge machining (EDM) process, one of the most widely used non-conventional manufacturing processes. These emissions can cause adverse health effects to the operators and has a direct impact on the environment. The emission from this process is directly related to the temperature at the process location. This paper was aimed at developing a model that quantifies the aerosol generated from the die sinking EDM process while machining steel workpiece with copper electrode. The model developed in this paper made use of energy balance and heat transfer equations. The modeling results were then validated using experimentally obtained values of the emission rate of aerosol from this process. The results showed a close correlation of +0.89 with experimental results. The model developed in this paper can predict the level of emissions at different process locations; thereby reducing the direct cost and time associated with experimentation.     

Asymptotics of the heat exchange

Let K be a compact subset in Euclidean space , and let E_K(t) denote the total amount of heat in at time t, if K is kept at fixed temperature 1 for all t?0, and if has initial temperature 0. For two disjoint compact subsets K₁ and K₂ we define the heat exchange H_K1,K2(t)=E_K1(t)+E_K2(t)−E_K1∪K2(t). We obtain the leading asymptotic behaviour of H_K1,K2(t) as t→0 under mild regularity conditions on K₁ and K₂.     

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.     

Zonal modeling of radiative heat transfer in industrial furnaces using simplified model for exchange area calculation

The zonal analysis of industrial furnaces is considered with three-dimensional radiative heat transfer, incorporated with the mathematical zone method. In this method exchange areas are determined by simplified numerical integration in three dimensions for surface-surface, surface-gas and gas–gas zones for absorbing and emitting media. By focusing on new strategies to overcome the drawbacks of evaluating direct exchange areas, it is shown that the zone method is an effective numerical method for modeling three-dimensional thermal performance of gas-filled enclosures. Also the developed method for evaluating of exchange area is presented and compared with other methods in both sides of CPU time and accuracy. The method can decrease about 70% in error of calculation of some exchange areas as compared with the other numerical methods.     

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.     

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 real-world images

The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR ₃, and exploring theHausdorff distance between these point sets, a metric spaceIM _D of functions is defined. The main purpose is to show that the functionsf∈IM _D, defined on the squareD=[0,1]², are appropriate mathematical models of real world images. The properties of the metric spaceIM _D are studied and methods of approximation for the purpose of image compression are presented. The metric spaceIM _D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM _D may be digitized and represented by a pixel functionp _n, withn pixels, in such a way that the distance betweenf andp _n is no greater than 2n ^?1/2. It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.     

Mathematical modeling of demographic processes

The article examines a mathematical model that describes the dynamics of the total population and its age structure. Time-dependent birth and death rates are assumed. The mathematical model is a first-order partial differential equation. The analytical solution makes it possible to determine the age distribution at each time instant depending on the birth and death functions and the initial distribution. The model can be used for demographic planning and forecasting. It has been applied to analyze the demographics of Russia. Translated from Prikladnaya Matematika i Informatika, No. 28, pp. 50–65, 2008.     

On heat exchange and heat transport in a geothermal plant

In the long term, the only way to address the challenging task of power supply, is to make renewable energy sources economically attractive and to use them efficiently. In particular, geothermal energy is promising to take over the base load of the power supply. Nevertheless, a lot of investigations needs to be made to use the almost inexhaustible source of thermal energy in the interior of the earth effectively. Starting from the initially isothermal state, a cold fluid is injected through a borehole into a rock. By the rising pressure gradient, the fluid flows through the porous rock and escapes through another borehole. While the fluid passes the micro cracks in the hot rock, the water is heated by the rock due to the heat exchange between the constituents. This process is simulated based on the Theory of Porous Media (TPM). The presented modelling approach of the heat transport and the flow processes in a fully saturated subsurface includes two non-isothermal constituents: an elastically deformable, materially incompressible solid skeleton where thermal expansion is neglected, and a viscous, materially incompressible fluid constituent. To solve the initial-boundary-value problem, the governing primary variables of the coupled model are spatially approximated by mixed finite elements, and the time-discretisation is carried out by an implicit Euler time-integration scheme. The aim of the presented numerical simulations is to study the heat transport and to evaluate the efficiency by varying flow rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical modeling of the distribution of galaxies in the universe

On the base of a random walk model with the transition probability given as a three-dimensional spherically symmetric stable law, a structural function is derived for the distribution of galaxies in the Universe which satisfactorily describes the scaling properties (fractality) of the distribution at small distances as well as the violation of scaling at large distances, in accordance with the data in the CfA catalogue. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III.     

Diffusion of color in the simple exclusion process

We prove a diffusion scaling limit for the macroscopic densities of colored particles performing the simply excluded random walk, and relate this to the limiting behavior of a test particle in equilibrium.