On radiative heat transfer in a plane layer of an absorbing medium

Recently there has arisen increased interest in the study of radiative heat transfer between geometrically simple systems, both as autonomous problems and as elements of more complex problems.Problems of this kind have been treated by many authors [1–111 who have considered gray, diffusely emitting and absorbing boundaries and gray nonscattering media. In most cases these investigations were restricted either to the derivation of approximate formulas for the net radiative flux, without an exact analysis of the temperature distribution in the layer [5–7], or to numerical computation [1–4], In the latter case, with the exception of [8], which contains a numerical analysis for the case of optical symmetry, no attempt was made to analyze the effect of the optical properties of the boundaries on the temperature field in the layer.These papers can be divided into two groups according to the method of analysis used. The first group includes papers based on the integral equations of radiative transfer, with the corresponding integral analytical methods [1, 2], Similar in nature are [3, 4] which use the slab method, applicable to electrical-analog computation, as well as a recent paper [8] based on probability methods.The second group of papers [5–7] is based on the so-called differential methods. Of particular interest is [7], which develops these methods to an advanced degree. In several papers the problem of radiative transfer is analyzed in conjunction with more complex problems (cf., e.g. [10, 11]).In the present work we shall attempt to carry out an approximate analytical study of problems connected with radiative heat transfer in a plane layer of an absorbing, emitting, nonscattering gray medium with temperature-independent optical properties. The layer is bounded by two parallel, diffusely emitting and diffusely reflecting, isothermal, gray planes.The paper presents the fundamental formulation of the problem, which consists in: (a) the determination of the net heat flux on the basis of given temperature distribution (direct formulation), and (b) the determination of the temperature distribution on the basis of given distribution of the net radiative heat source per unit volume and boundary temperatures (inverse formulation). The analysis is based on integral methods appropriate to the integral equations which represent the net total and hemispherical radiation flux densities [12].The author would like to thank S. S. Kutateladze for his interest in this work.     

Solution of the equations of motion for a selectively radiating gas in a shock layer

Numerical solutions are obtained for the system of integro-differential equations describing the flow of a viscous, heat-conducting, selectively radiating gas in the region between the shock wave and a blunt body. The calculations are made for bodies of radius from 0.1 to 3 m with stagnation temperature from 6000° to 15 000° K. As a result of the calculations the convective and radiative thermal fluxes in the vicinity of the stagnation point are obtained. The effect of injection on convective and radiative heat transfer is studied.The first calculations of radiative thermal fluxes in air were made about 10 years ago in [1,2]. However, the results did not take account of the effects of emission and reabsorption, nor the interaction of the convective and radiative heating processes. These effects have been studied primarily with the use of simplified models of a radiating gas. Most often the approximation used is that of a gray gas with absorption coefficient which is independent of wavelength ([3–6] and others).The appearance in the literature of quite detailed data on the selective spectral absorption coefficients of air over a wide temperature range [7,8] has made it possible to solve the direct problem of calculating the flow field of a selectively radiating gas behind a shock wave with account for all the effects mentioned above.     

Radiative and convective heat transfer during blowing into a compressed hypersonic layer of two-phase products from the ablation of a covering

The study examines the screening of the radiative heat flux in conditions of hypersonic flow around blunt bodies with ablated carbon-based coverings. In contrast to the studies already known [1–3], allowance is made for the presence of condensed microscopic particles in the products of ablation. In [4] the problem of radiative transfer is considered in a layer of two-phase ablation products with parametrically prescribed dimensions, particle temperature, and layer thickness. The present study uses a closed system of equations which describes the processes of heat and mass transfer. This gives rise to considerable differences in the numerical results, according to the degree of screening.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 161–166, November–December 1985.deceased     

Transformational-zone method of calculation of complex heat exchange during flow of optically active medium inside tube of diffuse grey surface

The paper presents analytical and experimental investigations of influence of radiative heat transfer on complex heat exchange during flow of optically active gas inside a pipe of diffusegrey properties. It was assumed that the pipe is heated from the outside by a constant heat flux and gas flowing inside is both absorbing and emitting and of small optical density. The influence of length and radiative properties of the pipe surface and of the gas temperature distribution on the wall and in the gas were analysed. The influence of radiative energy transfer on overall heat transfer coefficient was estimated. Mathematical model of radiative convective heat exchange in a system of one-dimensional temperature field, based on zone division method of Hottel and surface transformation, was verified numerically and experimentally. The results of numerical calculations were compared with experimental results obtained during carbone dioxide (CO₂) flow inside electrically heated ceramic tube. The set of nonlinear differential equations was solved by Runge-Kutta method with Hamming modification and with the use of separable-kernel method.     

Selection of an initial approximation in an asymptotic representation of the solution of the problem of a point-source thermal explosion in a nonlinearly heat-conducting gas

Qualitative analysis shows [1] that the initial stage of a point-source thermal explosion in a homogeneous stationary gas is characterized by the predominance of radiative heat transfer. This finding is related to the nonlinear dependence of the coefficient of radiative heat transfer on the temperature of the gas T. The function (T) can be assigned in power form =/n T^n–1, where is a dimensional constant and n>1 is the nonlinearity exponent. In complete agreement with the qualitative conclusions in [1], the author of [2] found by the asymptotic method that for short periods of time after the explosion, radiative heat transfer occurs independently of the motion of the gas and completely determines it. Here, the occurrence of the shock wave in [2] is connected with the convergence of the asymptotic solution on the well-known self-similar solution for a thermal explosion in a non-heat-conducting gas [3]. Conversely, the experimental findings and qualitative analysis of the problem in [1, 4] indicate that an isothermal shock wave can occur within a finite period of time after a thermal explosion. The shock then separated from the region heated by radiation [1], while radiative heat transfer turns out to have a diminishing effect on its motion. The role of heat transfer is negligible far from the site of the explosion, and the motion of the shock becomes self-similar [3]. Using the example of a plane thermal explosion in a nonlinearly heat-conducting ideal gas for the case n1, here we propose an asymptotic representation of the solution of the above problem which will make it possible to analyze the generation of an isothermal shock wave.Translated from Zhurnal Prikladnoi Mekhanikii Tekhnicheskoi Fiziki, No. 3, pp. 84–90, May–June, 1986.We thank K. B. Pavlov for his constant attention to the work and his critical observations.     

Some physical aspects of radiative and convective heat transfer in the case of nonself-similar blowing on a plate

Heat shielding by blowing has been fairly fully studied in the neighborhood of the stagnation point of a body in a stream [1–3], but for other flow regions the investigation has barely begun [4]. It has been found that the influence of blowing on the radiative and convective fluxes and the influence of radiation on the convection on the side wall can be very different from what is obtained for the flow conditions at the stagnation point. The present paper is a study of the radiative and convective heat transfer on a plate in a H₂ + He stream for constant and self-similar blowing of carbon vapor in the form of C, C₂, and C₃.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 28–35, January–February, 1981.     

Influence of anisotropy of the permeability on the convection and heat transfer in a porous annular layer

Investigations into the convective transport of heat in porous materials are of interest for many applications in connection with the problem of increasing the efficiency of thermal insulation. In [1–5], convection in Isotropic porous media was considered. However, in many cases porous materials have an essential anisotropy of their permeability. Convective heat transfer has been inadequately studied for this case. In [6], the linearized equations were used to study the convection between infinite horizontal planes with a filling of an anisotropic material; the value of the critical Rayleigh number was found, and this agreed satisfactorily with experimental data. In the present paper, we investigate numerically convection between two infinite coaxial cylinders with an anisotropic porous filling, using the equations of convection in the Darcy—Boussinesq approximation [1–3]. The permeability tensor in the annular region is constructed from its principal values, which can be found experimentally. A method of calculation is developed and a parametric study made of the structure of the flow and of the local and averaged characteristics of the heat transfer, which are of interest for the design of thermal insulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–64, January–February, 1980.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

Combined radiative and convective heat transfer in a three-dimensional rectangular channel at different wall temperatures

 This paper deals with a numerical study of combined convective and radiative heat transfer in a three-dimensional rectangular duct with hydrodynamically and thermally developing laminar flow. The gas is assumed to be an incompressible, absorbing, emitting, isotropically scattering, gray medium. Isothermal, gray, diffuse boundary walls at different temperatures are assumed. The finite-volume method (FVM) is adopted to describe both convective and radiative heat transfer. The coupled continuity and momentum equations are solved by means of SIMPLER algorithm. Numerical results for the radiative flux show very good agreement with the available data. The effects of aspect ratio, optical thickness, scattering albedo and wall emissivity on the mean bulk temperature are also investigated. By splitting the heat flux into convective and radiative contributions, the relative importance of these components is assessed for a typical range of values of the parameters. Received on 9 February 1999     

Flow of a viscous heat-conducting gas and binary mixtures in a sink

The class of exact solutions of the one-dimensional Navier-Stokes equations corresponding to gas flows from a spherical source or sink has been investigated analytically and numerically on a number of occasions (see, for example, [1, 2]). Here, the solution for a sink is considered in the presence of heat transfer from the ambient medium. Apart from seeking the solution itself, the object of the investigation was to establish the conditions of transi tion from viscous to inviscid flow in the sink as the Reynolds number tends to infinity. As shown in [3], for zero heat flux at an infinitely remote point there is no such transition for flow in a sink. The sink flow characteristics of a binary gas mixture are investigated in detail. In the transonic flow region an asymptotic solution is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–62, January–February, 1989.     

A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

Effect of thermal radiation exchange between boundaries on convection in a gas heated from below

Natural convection problems offer many examples of branching of the solutions [1]. Usually, such branching (from the standpoint of catastrophe theory) can be described by a Whitney fold or cusp. A characteristic feature of nontrivial branching is the presence of some small but finite disturbance of the convective equilibrium conditions. In this study the perturbation disturbing the convective equilibrium of a fluid heated from below is Stefan-law thermal radiation exchange between the boundaries of the enclosure. Natural convection with lateral heating and allowance for radiative heat transfer was previously investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 47–51, September–October, 1992.     

Strong mass injection at the surface of a blunt body in a supersonic flow

The case of supersonic flow over a blunt body when another gas is injected through the surface of the body in accordance with a given law is theoretically investigated. If molecular transport processes are neglected, the flow between the shock wave and the surface of the body should be regarded as two-layer, that is, as consisting of the flow in the shock layer between the shock wave and the contact surface and the flow in the layer of injected gas. A numerical solution of the problem is obtained near the front of the body and its accuracy is estimated. Approximate analytic solutions are obtained in the injected-gas layer: a constant-density solution and a solution of the boundary-layer type in the local similarity approximation. Near the flow axis the numerical and analytic solutions are fairly close, but at a distance from the axis the assumptions made reduce the accuracy of the approximate solutions. The flow in question can serve as a gas-dynamic model of a series of problems describing the radiant heating of blunt bodies in a hypersonic flow. In the presence of intense radiative heat transfer, vaporization is so great that the thickness of the vapor layer is comparable with the thickness of the shock layer. Moreover, the thermal shielding of various kinds of obstacles in channels through which a radiating plasma flows can be organized by means of the forced injection of a strong absorber. The formulation of a similar problem was reported in [1], but the results of the solution were not given. A two-layer model of the flow of an ideal gas over a blunt body was used in [2, 3] for the analysis of radiative heat transfer. In [2] the neighborhood of the stagnation point is considered. In [3] preliminary results relating to two-layer flow over blunt cones are presented. The solution is obtained by Maslen's approximate method.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 89–97, March–April, 1972.     

Thermal radiation effect on non-Darcy natural convection with lateral mass transfer

 A boundary layer analysis has been presented to study the influence of thermal radiation and lateral mass flux on non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. Forchheimer extension is considered in the flow equations, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained and the combined effect of thermal radiation and fluid suction/injection on the heat transfer rate is discussed. Numerical results for the details of the velocity and temperature profiles as well as Nusselt number have been presented. Received on 7 July 1999     

Experimental investigation of supersonic flow past blunt bodies with strong distributed injection

The entry of bodies into planetary atmospheres at high supersonic velocities is accompanied by intense evaporation of the surface due to radiative heat fluxes. A series of problems involving the conduction of investigations of such kind has been proposed by Petrov. In [1], in particular, the entry of a meteorite into an atmosphere was examined. The gasdynamic aspects of this problem have been approximately simulated by many authors by intense injection of gas in theoretical, e.g., [2–5], and experimental [6, 7] studies. The theoretical studies were based on two-layer [3, 4] or three-layer [5] schemes of gas flow between the shock wave and the surface of the body. The aim of the present work was an experimental investigation of the interaction of injection with a counter supersonic flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–95, May–June, 1978.     

Calculation of inviscid radiating gas flow past a blunt body

During hypersonic gas flow past a blunt body with a velocity on the order of the escape velocity or more, the gas radiation in the disturbed region behind the shock wave becomes the primary mechanism for aerodynamic heating and has a significant effect on the distribution of the gasdynamic parameters in the shock layer. This problem has been considered from different points of view by many authors. A rather complete review of these studies is presented in [1–4].In earlier studies [5, 6] the approximation of bulk emission was used. In this approximation, in order to account for the effect of radiative heat transfer a term is added in the energy equation which is equivalent to the body efflux, whose magnitude depends on the local thermodynamic state of the gas. However, the use of this assumption to solve the problem of inviscid flow past a blunt body leads to a singularity at the body [7, 8]. To eliminate the singularity, account is taken of the radiation absorption in a narrow wall layer [7], or the concept of a viscous and heat-conductive shock layer is used [8]. A further refinement was obtained by Rumynskii, who considered radiation selectivity and studied the flow of a radiating and absorbing gas in the vicinity of the forward stagnation point of a blunt body.In the present paper we study the distribution of the gasdynamic parameters in the shock layer over the entire frontal surface of a blunt body in a hypersonic flow of a radiating and absorbing gas with account for radiation selectivity.     

Radiative and Thermal Dispersion Effects on Non-Darcy Natural Convection with Lateral Mass Flux for Non-Newtonian Fluid from a Vertical Flat Plate in a Saturated Porous Medium

The problem of natural convective heat transfer for a non-Newtonian fluid from an impermeable vertical plate embedded in a fluid-saturated porous medium has been analyzed. Non-Darcian, radiative and thermal dispersion effects have been considered in the present analysis. The governing boundary layer equations and boundary conditions are cast into a dimensionless form and simplified by using a similarity transformation. The resulting system of equations is solved by using a double shooting Runge–Kutta method. The effect of viscosity index n, the conduction–radiation parameter R, the non-Darcy parameter Gr*, the thermal dispersion parameter Ds and the suction/injection parameter f_w on the fluid velocities, temperatures and the local Nusselt number are discussed.     

Thermal Dispersion–Radiation Effects on Non-Darcy Natural Convection in a Fluid Saturated Porous Medium

The effects of thermal dispersion and thermal radiation on the non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium are studied. Forchheimer extension is considered in the flow equations. The coefficient of thermal diffusivity has been assumed to be the sum of molecular diffusivity and the dispersion thermal diffusivity due to mechanical dispersion. Rosseland approximation is used to describe the radiative heat flux in the energy equation. Similarity solution for the transformed governing equations is obtained. Numerical results for the details of the velocity and temperature profiles which are shown on graphs have been presented. The combined effect of thermal dispersion and thermal radiation, for the two cases Darcy and non-Darcy porous medium, on the heat transfer rate which are entered in tables is discussed.