The effects of heat exchange and fluid production on the ignition of a porous solid

In this paper we study a system of nonlinear parabolic equations representing the evolution of small perturbations in a model describing the combustion of a porous solid. The novelty of this system rests on allowing the fluid and solid phases to assume different temperatures, as opposed to the well-studied single-temperature model in which heat is assumed to be exchanged at an infinitely rapid rate. Moreover, the underlying model incorporates fluid creation, as a result of reaction, and this property is inherited by the perturbation system. With respect to important physico-chemical parameters we look for global and blowing-up solutions, both with and without heat loss and fluid production. In this context, blowup can be identified with thermal runaway, from which ignition of the porous solid is inferred (a self-sustaining combustion wave is generated). We then proceed to study the existence and uniqueness of a particular class of steady states and examine their relationship to the corresponding class of time-dependent problems. This enables us to extend the global-existence results, and to indicate consistency between the time-independent and time-dependent analyses. In order to better understand the effects of distinct temperatures in each phase, a number of our results are then compared with those of a corresponding single-temperature model. We find that the results coincide in the appropriate limit of infinite heat-exchange rate. However, when the heat exchange is finite the blowup results can be altered substantially.     

Numerical modeling of a non-linear viscous flow in order to determine how parameters in constitutive relations influence the entropy production

Some rheological materials, such as melting polymers, cosmetic creams, ketchup, toothpaste, can be modeled as non-NEWTONian fluids by using a non-linear constitutive relation. An incompressible flow of this kind of amorphous matter can be considered as a thermodynamic process, and a solution for the pressure, velocity and temperature fields describe it fully. Since such flow processes are generally irreversible, entropy is produced leading to dissipation in the system. This energy loss can be measured indirectly in a cone/plate viscometer which is used to determine viscosity of a BINGHAM fluid. While dissipation is an observable quantity we also want to be able to calculate it. Thus the goal of this work is to explain briefly how to compute a transient flow of a viscous fluid in two-dimensional channel under a sinusoidal traction and calculate the dissipated energy for non-NEWTONian fluids. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of a two-dimensional mathematical model with variable heat exchange coefficient which arises in cryosurgery

We formulate and numerically solve a two-dimensional boundary value problem of Stefan type with nonlinear heat sources of a special kind and a variable heat exchange coefficient. The model under study arises in cryosurgery in the process of freezing some living biological tissue by a cryoinstrument of cylindrical shape placed on the surface of the tissue. The model takes into account the actually observed effect of spatial localization of heat. Some results of the computer simulation are presented.     

Mathematical modeling and the study of the heat exchange process in color kinescopes

We propose a mathematical model for computing the heat exchange in a masked color kinescope taking account of the specific geometry of its shell and the actual three-dimensional character of the heat emission in the mask. We study the temperature field in the mask and determine the size of the integral degree of blackness in the cone coating that guarantees a stable operation of the kinescope. Three figures. Bibliography: 7 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 57–63.     

Recovery of the heat equation from a single boundary measurement

We prove that we can uniquely recover the coefficient of a one-dimensional heat equation from a single boundary measurement and provide a constructive procedure for its recovery. The algorithm is based on the well-known Gelfand–Levitan–Gasymov inverse spectral theory of Sturm–Liouville operators.     

Numerical modeling of processes of heat and mass transfer in a capillary-porous material

We propose a method of solving a nonlinear heat and mass transfer boundary-value problem for a conductive drying process in a capillary-porous material. We study the effect of the characteristics of the material and the magnitude of the heat flux on the time change in temperature, bulk moisture saturation, and air density.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 149–154.     

A potential well theory for the heat equation with a nonlinear boundary condition

The present paper employs potential well arguments of a previous work by the authors [6] to obtain a sharp existence-nonexistence alternative for solutions to the linear heat equation subject to a non-linear boundary condition.     

Numerical quenching for the semilinear heat equation with a singular absorption

In this paper we study the numerical approximation for the heat equation with a singular absorption. We prove that the numerical quenching rate coincides with the continuous one. We also see that the quenching time and the quenching set converge to the continuous one. In fact, under some restriction on the initial data, the numerical quenching coincides with the continuous one. Finally, we give some numerical results to illustrate our analysis.     

Numerical calculation of salt leaching from rocks in the presence of head-driven flow

A numerical procedure is proposed for solving the problem of leaching of water-soluble salts from multilayer rocks in the presence of a head-driven flow, allowing for the interaction of seepage and desalinisation processes. The results of a numerical experiment are reported.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 47–53, 1987.     

Numerical inversion of heat exchange coefficient and source sink term for a thermal-fluid coupled model in porous medium

Abstract

In this paper, we focus on three inverse problems for a coupled model from temperature-seepage field in high-dimensional spaces. These inverse problems aim to determine an unknown heat transfer coefficient and a source sink term in seepage continuity equation with specified initial-boundary conditions and additional measurements. Some finite difference schemes of coupled equations are presented and analyzed.Three algorithms for these inverse problems are proposed. Some numerical experiments are provided to assert the accuracy and efficiency of proposed algorithms.     

Numerical modeling of the effect of heat and mass transfer in porous low-temperature heat insulation in composite material structures on the magnitude of stresses which develop

The stressed state of multilayer low-temperature heat insulation for a cryogenic fuel tank is considered. Account is taken of heat and mass transfer in foam plastic (the main heat insulation material) occurring at cryogenic temperatures. A method is developed for solving a set of differential equations and boundary conditions. Numerical studies of the main features of these processes are performed. It is established that below 200 K the stresses which arise in foam plastic markedly exceed the ultimate strength for this material. Stresses develop as a result of both a reduction in temperature and a drop in pressure in the foam plastic pores connected with material cooling. On the basis of the results obtained it is established that the combination of thermophysical processes which occur in foam plastic during cooling to cryogenic temperatures leads to changes in the stress-strained state of structure, which should be considered in planning aerospace technology.Scientific Research Institute of Applied Mathematics and Mechanics, Tomsk State University, Tomsk, Russia. Translated from Mekhanika Kompozitmykh Materialov, Vol. 33, No. 3, pp. 384–393, May–June, 1997.     

The effect of conjugated and radiant heat exchange on the process of non-stationary combustion of the products of intense gasification of a solid in a stream of gas

This paper develops further the results obtained in /1–4/ and uses the approximate mathematical model /2/ of the combustion of the products of intense gasification of the neighbourhood of the leading stagnation point of the body to analyse the effect of the conjugation parameters on the heat exchange, radiation and other factors on the conditions of uniqueness and stability of the stationary combustion modes. When the gasification is carried out at a constant mass flow rate, an analogy is established, depending on the relations between the parameters of the problem, between the model in question and the models of a homogeneous chemical continuous action reactor with a fluidized catalyst layer, and a reactor with a temperature regulator /5/. Simple necessary conditions for the instability of the stationary modes and the appearance of selfexcited oscillations are obtained. A strong stabilizing influence of the conjugated heat exchange and intense injection on the combustion process is established, and a destablishing influence of radiant heat exchange is found.     

Dynamic analysis of porous materials: Numerical modeling with a space-time FEM

In the current work, we investigate the dynamic analysis of a two-phase porous material using the space-time discontinuous Galerkin method. The physical model is based on the Theory of Porous Media (TPM). The finite element approximation consists of continuous approximations in space but discontinuous ones in time. The continuity condition between the adjacent time intervals is weakly enforced by the upwind flux treatment. No artificial penalty function is involved. Moreover, the Embedded Velocity Integration technique is applied to reduce the second-order equation in time into a first order one without introducing an additional constraint. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical analysis of heat and moisture transport with a finite difference method

In this article, we study a system of nonlinear parabolic partial differential equations arising from the heat and moisture transport through textile materials with phase change. A splitting finite difference method with semi‐implicit Euler scheme in time direction is proposed for solving the system of equations. We prove the existence and uniqueness of a classical positive solution to the parabolic system as well as the existence and uniqueness of a positive solution to the splitting finite difference system. We provide optimal error estimates for the splitting finite difference system under the condition that the mesh size and time step size are smaller than a positive constant which solely depends upon the physical parameters involved. Numerical results are presented to confirm our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Numerical resolution of an exact heat conduction model with a delay term

In this paper we analyze, from the numerical point of view, a dynamic thermoelastic problem. Here, the so-called exact heat conduction model with a delay term is used to obtain the heat evolution. Thus, the thermomechanical problem is written as a coupled system of partial differential equations, and its variational formulation leads to a system written in terms of the velocity and the temperature fields. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, from which the linear convergence of the algorithm could be derived under suitable additional regularity conditions. Finally, a two-dimensional numerical example is solved to show the accuracy of the approximation and the decay of the discrete energy.     

Numerical studies of a nonlinear heat equation with square root reaction term

Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Single item lot-sizing problems with backlogging on a single machine at a finite production rate

In this paper we consider a single item lot-sizing problem with backlogging on a single machine at a finite production rate. The objective is to minimize the total cost of setup, stockholding and backlogging to satisfy a sequence of discrete demands. Both varying demands over a finite planning horizon and fixed demands at regular intervals over an infinite planning horizon are considered. We have characterized the structure of an optimal production schedule for both cases. As a consequence of this characterization, a dynamic programming algorithm is proposed for the computation of an optimal production schedule for the varying demands case and a simpler one for the fixed demands case.