Numerical Study of Forced MHD Convection Flow and Temperature Around Periodically Placed Cylinders

In this paper we consider 2D stationary boundary value problems for the system of magnetohydrodynamic (MHD) equations and the heat transfer equation. The viscous electrically conducting incompressible liquid moves between infinite cylinders with square or round sections placed periodically. We also consider similar 2D MHD channel flow with periodically placed obstacles on the channel walls. We analyse the 2D forced and free MHD convection flow and temperature around cylinders and obstacles in homogeneous external magnetic field. The cylinders, obstacles and walls of the channel with constant temperature are heated. The distributions of electromagnetic fields, forces, velocity and temperatur…

Numerical Simulation of the Problem Arising in the Gyrotron Theory

Numerical aspects for solving of certain problem arising in gyrotron theory are discussed. Particularly, finite-difference schemes using quasistationarization and method of lines were applied and the relevant results analyzed.

Conservative averaging method for solving some nonlinear heat transfer problems related to combustion

NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions …

THE MATHEMATICAL MODELING OF METALS CONTENT IN PEAT

Metals deposition in peat can aid to evaluate impact of atmospheric or wastewaters pollution and thus can be a good indicator of recent and historical changes in the pollution loading. For peat using in agriculture, industrial, heat production etc. knowledge of peat metals content is important. Experimental determination of metals in peat is very long and expensive work. Using experimental data the mathematical model for calculation of concentrations of metals in different points for different layers is developed. The values of the metals (Ca, Mg, Fe, Sr, Cu, Zn, Mn, Pb, Cr, Ni, Se, Co, Cd, V, Mo) concentrations in different layers in peat taken from Knavu peat bog from four sites are deter…

Numerical study of electrodynamic control of straw co-firing with propane

The Mathematical Modeling of Ca And Fe Distribution In Peat Layers

Bogs have been formed by an accumulation of peat - a light brown-to-black organic material, built up from partial decomposition of mosses and other bryophytes, sedges, grasses, shrubs, or trees under waterlogged conditions. The total peatlands area in Latvia covers 698 918 ha or 10.7% of the entire territory. Knowledge’s of peat metals content are important for any kind of peat using. Experimental determination of metals in peat is very long and expensive work. Using experimental data mathematical model for calculation of concentrations of metals in different points for different layers can help to very easy and fast to find approximately concentration of metals or trace elements. The resul…

Mathematical modelling and experimental study of straw co-firing with gas using electric field control of combustion characteristics

Electric Field Effect on the Thermal Decomposition and Co-combustion of Straw with Solid Fuel Pellets

The aim of this study was to provide more effective use of straw for energy production by co-firing wheat straw pellets with solid fuels (wood, peat pellets) under additional electric control of the combustion characteristics at thermo-chemical conversion of fuel mixtures. Effects of the DC electric field on the main combustion characteristics were studied experimentally using a fixed-bed experimental setup with a heat output up to 4 kW. An axisymmetric electric field was applied to the flame base between the positively charged electrode and the grounded wall of the combustion chamber. The experimental study includes local measurements of the composition of the gasification gas, flame tempe…

A Certain Mathematical Model of the Glass Fibre Material Production

There is considered the full mathematical model of chemical reactions on the surface of glass fibre material that was imbedded in the flow of acid solution and was pulled longitudionally. Self-similar forms of this model are obtained and their approximations by monotone schemes of differences are proposed. Some special cases which make possible to get the analytic solutions are underlined. The self-similar forms of the differential equations of the substances transport allow to calculate the emission of the alkaline oxide from the glass fibre material under the influence of the acid solution flow. Some conclusions with practical significance for the technological process is made up accordin…

Mathematical modelling of alternating electromagnetic and hydrodynamic fields, induced by bar type conductors in a cylinder

The heating of buildings by ecologically clean and compact local devices is an interesting and actual problem. One of the modern areas of applications developed during last ten years is an effective usage of electrical energy by alternating current to produce heat energy. This work presents the mathematical model of one of such devices. It is a finite cylinder with viscous incompressible liquid and with metal electrodes of the form of bars placed parallel to the cylinder axis in the liquid. These conductors are connected to the alternating current. First published online: 14 Oct 2010

Simple algorithms for calculation of the axial‐symmetric heat transport problem in a cylinder

The approximation of axial‐symmetric heat transport problem in a cylinder is based on the finite volume method. In the classical formulation of the finite volume method it is assumed that the flux terms in the control volume are approximated with the finite difference expressions. Then in the 1‐D case the corresponding finite difference scheme for the given source function is not exact. There we propose the exact difference scheme. In 2‐D case the corresponding integrals are approximated using different quadrature formulae. This procedure allows one to reduce the heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary diff…

SPECIAL HPERBOLIC TYPE APPROXIMATION FOR SOLVING OF 3-D TWO LAYER STATIONARY DIFFUSION PROBLEM

In this paper we examine the conservative averaging method (CAM) along the vertical z-coordinate for solving the 3-D boundary-value 2 layers diffusion problem. The special parabolic and hyperbolic type approximation (splines), that interpolate the middle integral values of piece-wise smooth function, is investigated. With the help of these splines the problems of mathematical physics in 3-D with respect to one coordinate are reduced to problems for system of equations in 2-D in every layer. This procedure allows reduce also the 2-D problem to a 1-D problem and the solution of the approximated problem can be obtained analytically. As the practical application of the created mathematical mode…

SIMULATION OF THE HEAT TRANSPORT PROBLEMS WITH RADIATION IN PLATE

In the literature [1 -5 ] simple and effective algorithms for mathematical modelling processes of distribution of heat in multilayered spaces are created. In the given work the way of improvement o f accuracy of algorithms is considered at approximation of integrals derivatives more the supreme orders are used.

Creation of Temperature Field in a Finite Cylinder by Alternated Electromagnetic Force

One of the modern areas of applications developed during last years is effective use of electrical energy produced by alternating current in production of heat energy. This process is ecologically clean.The water is weakly electrically conducting medium (electrolyte). Devices based on this principle are developed during last ten years. Compared to classical devices with heating elements, new devices are more compact.

Numerical experiments with single mode gyrotron equations

Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrödinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was numerically examined in our earlier work. In this paper we consider the stationary and nonstationary …

CAM with special splines for solving of diffusion-convection problems with discontinuous coefficients for layered materials exposed to fire

Numerical simulation of heat transfer problem for layered gypsum products exposed to fire

On numerical simulation of electromagnetic field effects in the combustion process

This paper deals with a simplified model taking into account the interplay of compressible, laminar, axisymmetric flow and the electrodynamical effects due to Lorentz force’s action on the combustion process in a cylindrical pipe. The combustion process with Arrhenius kinetics is modelled by a single step exothermic chemical reaction of fuel and oxidant. We analyze non-stationary PDEs with 6 unknown functions: the 3 components of velocity, density, concentration of fuel and temperature. For pressure the ideal gas law is used. For the inviscid flow approximation ADI method is used. Some numerical results are presented.

Magnetic Field Control of Combustion Dynamics

Abstract Experimental studies and mathematical modelling of the effects of magnetic field on combustion dynamics at thermo-chemical conversion of biomass are carried out with the aim of providing control of the processes developing in the reaction zone of swirling flame. The joint research of the magnetic field effect on the combustion dynamics includes the estimation of this effect on the formation of the swirling flame dynamics, flame temperature and composition, providing analysis of the magnetic field effects on the flame characteristics. The results of experiments have shown that the magnetic field exerts the influence on the flow velocity components by enhancing a swirl motion in the …

On mathematical modelling of the solid-liquid mixtures transport in porous axial-symmetrical container with Henry and Langmuir sorption kinetics

In this paper we study diffusion and convection filtration problem of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances. As an example we consider round cylinder with filtration process in the axial direction. The cylinder is filled with sorbent i.e. absorbent material that passed through dirty water or liquid solutions. We can derive the system of two partial differential equations (PDEs), one expressing the rate of change of concentration of water in the pores of the sorbent and the other - the rate of change of concentration in the sorbent or kinetical equation for absorption. The approximation of corresponding initial b…

Non-Isothermal Mathematical Model of Wood and Paper Drying

A mathematical model of wood or paper drying based on a detailed consideration of both heat and moisture transport phenomena is proposed. By averaging we express the model as a sequence of initial value problems for systems of two first order nonlinear ordinary differential equations. This mathematical model makes it possible to efficiently investigate the drying process of a thin wood plate or paper sheet for varying temperature and humidity conditions in the surroundings. In particular, we have considered the optimization of the heat regime over a series of steam-heated cylinders in a papermaking machine.

Self‐similar problems for modeling the surface chemical reactions with the gravitation

The mathematical model of a chemical reaction which takes place on the surface of the uniformly moving vertically imbedded glass fibre material is considered. The effect of gravitation is taken into account. Boussinesq's and boundary layer fittings allow to derive boundary value problems for self‐similar systems of ordinary differential equations. First Published Online: 14 Oct 2010

Mathematical modelling of an elongated magnetic droplet in a rotating magnetic field

Dynamics of an elongated droplet under the action of a rotating magnetic field is considered by mathematical modelling. The actual shape of a droplet is obtained by solving the initial-boundary value problem of a nonlinear singularly perturbed partial differential equation (PDE). For the discretization in space the finite difference scheme (FDS) is applied. Time evolution of numerical solutions is obtained with MATLAB by solving a large system of ordinary differential equations (ODE).

Development of combustion dynamics at co-combustion of straw with wood

SPECIAL SPLINES OF HYPERBOLIC TYPE FOR THE SOLUTIONS OF HEAT AND MASS TRANSFER 3-D PROBLEMS IN POROUS MULTI-LAYERED AXIAL SYMMETRY DOMAIN

In this paper we study the problem of the diffusion of one substance through the pores of a porous multi layered material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. As an example we consider circular cross section wood-block with two layers in the radial direction. We consider the transfer of heat process. We derive the system of two partial differential equations (PDEs) - one expressing the rate of change of concentration of water vapour in the air spaces and the other - the rate of change of temperature in every layer. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the cons…

Influence of electric field on thermo-chemical conversion of mixtures of straw pellets with coal

Reduction of a Non—Linear Parabolic Initial—Boundary Value Problem to Cauchy Problem for a System of ODEs

We consider the boundary value problem for a parabolic equation in the form $$\frac{{\partial {\text{u}}}}{{\partial t}} = \frac{1}{{p(x)}}\frac{\partial }{{\partial x}}\left( {p(x)f'(u)\frac{{\partial u}}{{\partial x}}} \right) + F(u),x \in (0,l),t0,$$ (1) $$u(0,x) = {u_0}(x),$$ (2) $$\frac{{\partial u}}{{\partial x}}{|_{x = 0}} = {f_1}\left( {{u_1}} \right),$$ (3) $$\frac{{\partial u}}{{\partial x}}{|_{x = 1}} = {f_2}\left( {{u_2}} \right),$$ (4) where u = u(t,x) is the unknown function, f 1, f 2, F, f are nonlinear functions and f′ (u) > 0, $${u_1} = {u_1}\left( t \right) \equiv u\left( {t,0} \right),{u_2} = {u_2}(t) \equiv u\left( {t,l} \right),f'\left( u \right) \equiv df(u)/du,p(x) \g…

Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains

We consider averaging methods for solving the 3-D boundary-value problem of second order in multilayer domain. The special hyperbolic and exponential type splines, with middle integral values of piece-wise smooth function interpolation are considered. With the help of these splines the problems of mathematical physics in 3-D with piece-wise coefficients are reduced with respect to one coordinate to 2-D problems. This procedure also allows to reduce the 2-D problems to 1-D problems and the solution of the approximated problemsa can be obtained analytically. In the case of constant piece-wise coefficients we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem.…

Calculation of heat and moisture distribution in the porous media layer

In this paper we study the problem of the diffusion of one substance through the pores of a porous material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. The transfer of moisture and the heat are described by the model. The system of two partial differential equations (PDEs) is derived, one equation expresses the rate of change of concentration of water vapour in the air spaces and the other the rate of change of temperature. The obtained initial‐boundary value problem is approximated by using the finite volume method. This procedure allows us to reduce the 2D transfer problem described by a system of PDEs to initial value problem…

EFFECTIVE FINITE-DIFFERENCE METHODS FOR THE SOLUTIONS OF FILTRATION PROBLEMS IN MULTILAYER DOMAINS

In papers [1,2] there were consider different assumptions for averaging methods along the vertical coordinate.These methods were applied for the mathematical simulation of the mass transfer process in multilayered underground systems. A specific feature of these problems is that it is necessity to solve the 3‐D initial‐boundary‐value problems for parabolic type partial differential equations of second order with piece‐wise parameters in multilayer domain.Therefore here an effective finite‐difference method for solving a problem of the above type is developed.This method may be considered as a generalization of the method of finite volumes [3] for the layered systems. In the case of constant…

Analysis of equations arising in gyrotron theory

The gyrotron is a microwave source whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. Powerful gyrotrons can be used to heat nuclear fusion plasma. In addition, they have found a wide utility in plasma diagnostics, plasma chemistry, radars, extra-high-resolution spectroscopy, high-temperature processing of materials, medicine, etc. However, the main application of gyrotrons is in electron cyclotron resonance heating in tokamaks and stellarators. Equations describing gyrotron operation are ordinary differential equations and Schrödinger type partial differential equations. The present paper provides a survey of the analytical a…

The exact finite‐difference scheme for vector boundary‐value problems with piece‐wise constant coefficients

We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schem…

On Mathematical Modelling of Metals Distribution in Peat Layers

In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in multilayered domain. We consider the metals Fe and Ca concentration in the layered peat blocks. Using experimental data the mathematical model for calculation of concentration of metals in different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations (PDEs) of second order with piece-wise diffusion coefficients in the layered domain. We develop here a finite-difference method for solving of a problem of one, two and three peat blocks with periodica…

Mathematical modelling and experimental study of straw co-firing with gas

The main goal of the present study is to promote a more effective use of agriculture residues (straw) as an alternative renewable fuel for cleaner energy production with reduced greenhouse gas emissions. With the aim to improve the main combustion characteristics at thermo-chemical conversion of wheat straw, complex experimental study and mathematical modelling of the processes developing when co-firing wheat straw pellets with a gaseous fuel were carried out. The effect of co-firing on the main gasification and combustion characteristics was studied experimentally by varying the propane supply and additional heat input into the pilot device, along with the estimation of the effect of co-firing on…

Numerical investigations of single mode gyrotron equation

A stationary problem with the integral boundary condition arising in the mathematical modelling of a gyrotron is numerically investigated. The Chebyshev's polynomials of the second kind are used as the tool of calculations. The main result with physical meaning is the possibility to determine the maximal value of electrons efficiency. First published online: 14 Oct 2010

Method of Lines and Finite Difference Schemes with Exact Spectrum for Solving Some Linear Problems of Mathematical Physics

In this paper linear initial-boundary-value problems of mathematical physics with different type boundary conditions BCs and periodic boundary conditions PBCs are studied. The finite difference scheme FDS and the finite difference scheme with exact spectrum FDSES are used for the space discretization. The solution in the time is obtained analytically and numerically, using the method of lines and continuous and discrete Fourier methods.