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 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.     

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.     

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.     

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.     

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 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.     

Numerical simulation of the velocity distribution for a hard-sphere gas

The kinetic features of energy exchange between hard-sphere molecules are numerically simulated. The results deviate from the usual the Maxwell distribution. The high-speed tail of the molecule distribution is cut off due to the dominant deceleration of fast particles. For mixtures of gases with various molecular masses, the energy redistribution between the degrees of freedom leads to different stationary temperatures of the components.     

Simulation study on the effect of gas permeation on the hydrodynamic characteristics of membrane-assisted micro fluidized beds

Recent research has shown the potential of membrane-assisted fluidized bed reactors for various applications, and for ultra-pure hydrogen production in particular. Due to the excellent mass transfer characteristics of fluidized beds, concentration polarization (i.e. mass transfer limitation) can be overcome and the production capacity of membrane-assisted fluidized bed reactors could be further improved by maximizing the installed membrane area per unit volume, leading to the concept of a micro-structured membrane-assisted fluidized bed reactor. In this study, numerical simulations have been systematically carried out with a discrete particle model to investigate in detail the effects of gas addition and extraction through the confining porous membrane walls on the hydrodynamic characteristics of a single membrane-assisted micro fluidized bed compartment. In particular, the effect of the permeation ratio (amount of gas permeated through the membrane relative to the amount fed) and the installed membrane area on the hydrodynamics was investigated. Gas addition or extraction via the porous membrane walls confining the emulsion phase was simulated via inward or outward directed fluxes of the gas phase, which was found to have a very pronounced influence on the bed hydrodynamics. The effects of gas permeation on the solids circulation pattern, solids holdup distribution and porosity probability density function in membrane-assisted micro fluidized beds have been discussed in great detail. It has been found that gas permeation can have an adverse effect on the bed expansion caused by gas by-passing either through the bed center for the case of gas extraction or close to the membrane walls for the case of gas addition. In addition, the formation of densified zones (increased solids holdup) close to the membrane wall that was observed in case of gas extraction may increase the bed-to-membrane mass transfer resistance. These effects may strongly decrease the gas–solid contacting and the gas residence time, which may deteriorate the reactor performance. On the other hand, it is shown that these problems caused by gas permeation may be avoided by properly tuning the gas velocity through the membrane via membrane area and other design parameters and operating conditions.     

Numerical investigation of the deformation characteristics and heat generation in pneumatic aircraft tires Part II. Thermal modeling

A numerical procedure to determine the temperature rise in aircraft tires under free rolling conditions is presented in this article. Energy dissipation from cyclic inelastic deformation is considered the main heat generation source. This modeling considers the deformation process of the tire to be a steady-state problem, where all concurrent cycles are assumed to be the same as the first. The inelastic energy is determined by imposing a phase lag between the strain and the stress fields. The phase lag is assumed to be frequency independent in the range of interest, in keeping with the experimental observations in aircraft tire materials. It is further assumed that the inelastic energy is completely converted into volumetric heat input for a transient thermal conduction analysis. A conduction model is described and results are compared against thermocouple data recorded by Clark and Dodge [1].     

Viscoelastic jet drawn under conditions of heat exchange with the ambient medium

The effect of a constant or variable pull on a heated (cooled) viscoelastic jet is investigated. The problem is resolved into the subproblems of the extension of a viscoelastic rod and the motion determined by the rate of the process. The formation of polystyrene fibers is treated as an example.Mekhanika Polimerov, Vol. 4, No. 3, pp. 540–546, 1968     

A wind tunnel investigation on the local heat transfer from a sphere,including the influence of turbulence and roughness

Zusammenfassung Der Wärmeübergang von einer isotherm geheizten Messingkugel an den Luftstrom wurde bei Reynoldszahlen zwischen 4100 und 66000 lokal gemessen und mit der Theorie der laminaren Grenzschicht verglichen. Das Experiment lieferte um 10 bis 15% höhere Resultate als die Theorie. Wurde die Turbulenz der Strömung in der Grössenordnung von 0.1% durch Gitter auf 1% gesteigert, so ergab sich ein bis 40% höherer Wärmeübergang als nach der Theorie für den turbulenzfreien Fall zu erwarten ist. Druckverteilungen um die Kugel wurden gemessen um Einsicht in die realen Strömungsverhältnisse und eine Berechnungsgrundlage zu erhalten.Rauhigkeiten von 0.3 mm auf einer Kugel von 3 cm Durchmesser hatten einen vernachlässigbar kleinen Einfluss im Bereiche kleiner Reynoldszahlen. Bei hohen Reynoldszahlen führten die Rauhigkeiten lokal zu einem turbulenten Wärmeübergang. Der totale Wärmeübergang wurde dabei um 60% gesteigert.Lokale Messungen an einem Modell mit Buckeln und Vertiefungen ergaben im Mittel bei einem Buckel einen 2 bis 3mal grösseren Wärmeübergang als bei einer Vertiefung. Der gesamte Wärmeübergang des buckligen Modells war mindestens 5%, meistens 20%, aber höchstens 72% grösser als derjenige der Kugel bei gleichen Versuchsbedingungen.Die lokalen Nusseltzahlen wurden für alle Versuchsbedingungen in einer Tabelle zusammengefasst, wobei sich die maximalen Messfehler zu einer Unsicherheit von höchstens ±14% aufsummieren.

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List of symbols T Temperature difference between the heated model and the air stream - Polar angle from forward stagnation point - L Characteristic length (diameter of sphere). For the humped modelL is the diameter of the sphere with the same volume - F Surface from which the heating power is conducted to the air stream, respectively its projection on the sphere in the case of humps, hollows or small scale roughness - q Heat conducted from a heater to the air stream through the areaF, divided by T (convective thermal conductance) - Nu Nusselt number, connected withq byq=F··Nu/L. Nu can be physically interpretated as the derivative of the temperature at the surface and normal to the surface brought into a dimensionless form by means of T andL. - Molecular conductivity of heat in air. A value of 0.24 mW/(cm·C) was used for the calculations (0C). - Kinematic viscosity of air. The calculations are based on =0.18 cm²·s^–1 (0C, 740 mb). - U Velocity of the air stream (mean value). - U Fluctuation ofU in the direction ofU, characterising the free stream turbulence. - f Frequency of the fluctuations ofU - Re Reynolds number.Re=L·U/ - Thickness of the viscous boundary layer - h Height of roughness elements - Pr Prandtl number.Pr=··c/ wherec stands for the specific heat - 2p/(·U ²) Pressure at the surface of the sphere in function of , divided by the pressure at the forward stagnation point (dynamic pressures) - Wedge variable according to the notation of Merk - E ₀ First term in the expansion of the Nusselt number Symbols denoting experimental conditions N Normal sphere - BV Model with humps (germanBuckel) and hollows (germanVertiefung) - r Small scaleroughness - G1,G2 Grids made from 1 mm wires and 5 mm rods respectively, producing turbulence     

Numerical modeling of the movement of a rigid particle in viscous fluid

Modeling the movement of a rigid particle in viscous fluid is a problem physicists and smathematicians have tried to solve since the beginning of this century. A general model for an ellipsoidal particle was first published by Jeffery in the twenties. We exploit the fact that Jeffery was concerned with formulae which can be used to compute numerically the velocity field in the neighborhood of the particle during his derivation of equations of motion of the particle. This is our principal contribution to the subject. After a thorough check of Jeffery's formulae, we coded software for modeling the flow around a rigid particle based on these equations. Examples of its applications are given in conclusion. A practical example is concerned with the simulation of sigmoidal inclusion trails in porphyroblast.