# Thermal Radiation in Disperse Systems: An Engineering Approach

Leonid A. Dombrovsky
Joint Institute for High Temperatures, NCHMT, Krasnokazarmennaya 17A, 111116, Moscow, Russia; Tyumen State University, Semakov 10, Tyumen, 625003, Russia
Dominique Baillis
LaMCoS, INSA-Lyon, CNRS UMR 5259,18-20 Rue des Sciences, F69621 Villeurbanne, France
## Description

The physical basis of the majority of solutions considered in this book is the notion of radiation transfer in an absorbing and scattering medium as some macroscopic process, which can be described by a phenomenological transfer theory and radiative transfer equation for spectral radiation intensity. The book is divided into four chapters. Chapter 1 deals with computational models for radiative transfer in disperse systems. The main attention is given to simple approximate models, both traditional and modified, which have a clear physical sense and enable one to derive some useful analytical solutions to classic problems. Spectral radiative properties of single particles and fibers are considered in some detail in Chapter 2. The theoretical part of this chapter includes the Mie solution for homogeneous spherical particles and more general solutions for hollow and core-mantled spheres. Chapter 3 presents an engineering approach for both theoretical prediction and experimental determination of spectral radiative properties of quite different dispersed materials containing the morphology elements of arbitrary shape. A general theoretical basis of radiative properties determination and present-day principles of experimental characterization with identification procedure are recalled. Physical limitations of independent scattering theory are also discussed in this chapter. Some radiative and combined heat transfer problems in various disperse systems are considered in Chapter 4. For a topic that is as broad as the one considered in this book, it is very difficult to be comprehensive. However, we hope that enough key references are cited in the book to enable an interested reader to undertake a more detailed study of specific thermal radiation problems in disperse systems.

© 2010

## Table of Contents:

1. Computational Models for Radiative Transfer in Disperse Systems

1.1 The Radiative Transfer Equation

1.2 Transport Approximation

1.3 Differential Approximations

1.3.1 Two-Flux Approximation

1.3.2 P1 Approximation of the Spherical Harmonics Method

1.3.3 The Simplest Approximations of Double Spherical Harmonics

1.4 Solutions for One-Dimensional Problems

1.4.1 Radiation of an Isothermal Plane-Parallel Layer

1.4.2 Radiative Equilibrium in a Plane-Parallel Layer

1.4.3 Hemispherical Transmittance and Reflectance at Normal Incidence

1.4.4 An Estimate of P1 Approximation Error for Optically Inhomogeneous Media

1.4.5 Radiation of a Nonisothermal Layer of a Scattering Medium

1.5 Diffusion Approximation in Multi-Dimensional Problems

1.5.1 Radiation of Isothermal Volumes of a Scattering Medium

1.5.2 The Finite-Element Method for Nonisothermal and Nonhomogeneous Media

1.5.3 A More Accurate Solution Based on the Combined Computational Model

1.6 Large-Cell Model for Radiation Heat Transfer in Multiphase Systems

1.6.1 The Two-Band Model with Conventional Semi-Transparency and Opacity Regions

1.6.2 P1 Approximation and the Large-Cell Radiation Model for a Semi-Transparency Range

1.6.3 Comparison of Diffusion and Large-Cell Models for Typical Problem Parameters

1.7 Some Methods for Detailed Numerical Simulation of Radiative Transfer

1.7.1 The Discrete Ordinates Method for One-Dimensional Problems

Appendix A. Adaptive Composite Quadrature

1.7.2 Monte Carlo Simulation of Radiative Transfer

2. Radiative Properties of Particles and Fibers

2.1 Equations of the Scattering Theory

2.1.1 Spherical Particles

2.1.2 Cylindrical Particles

2.2 Limiting Cases of the General Theory

2.2.1 Rayleigh Scattering

2.2.2 Rayleigh-Gans Scattering

2.2.4 Anomalous Diffraction

2.3 Some Results for Particles of Various Types

2.3.1 Semi-Transparent Spherical Particles

2.3.2 Water Droplets in a Near-Infrared Spectral Range

2.3.3 Spectral Properties of Diesel Fuel Droplets

2.3.4 Gas Bubbles in Semi-Transparent Medium

2.3.5 Radiative Properties of Soot Particles

2.3.6 Near-Infrared Properties of Droplets of Aluminum Oxide Melt

2.3.7 Metal Particles in Infrared and Microwave Spectral Ranges

2.3.8 Water Droplets and Bubbles in a Microwave Spectral Range

2.3.9 Semi-Transparent Fibers at Arbitrary Illumination

2.3.10 Near-Infrared Properties of Quartz Fibers

2.3.11 Metal-Coated Polymer Fibers in Infrared and Microwave

2.3.12 Infrared Properties of Carbon Fibers

2.4 Thermal Radiation from Nonisothermal Spherical Particles

2.4.1 Mie Solution for Radiation Field in Isothermal Particle

2.4.2 Geometrical Optics Approximation

2.4.3 Modified Differential Approximation

2.5 Thermal Radiation from a Spherical Particle to an Absorbing Medium through a Narrow Concentric Gap

2.5.1 General Wave Solution for Radiation Flux

2.5.2 Particular Cases of Zero or Large Thickness of the Gap

2.6 Polydisperse Systems of Independent Particles

2.6.2 Monodisperse Approximation

3. Radiative Properties of Dispersed Materials: Experimental Characterization and Theoretical Modeling

3.1 Basic Principles of Experimental Characterization and Identification Procedure

3.2 Theoretical Basis of Radiative Properties Determination

3.3 Radiative Properties of Highly Porous Cellular Foams

3.3.1 Microstructure of Cellular Foams

3.4 Highly Porous Fibrous Media

3.4.1 Modeling of Radiative Properties

3.4.2 Identification of Radiative Properties

3.5 Packed Beds of Large Spherical Particles

3.5.2 Ray-Tracing Techniques

3.5.3 Computational Results for Opaque and Semi-Transparent Particles

Appendix B. Ray-Tracing Model

3.6 Near-Infrared Properties of Porous Zirconia Ceramics

3.6.1 Microstructure of Samples

3.6.2 Near-Infrared Optical Constants

3.6.3 Transmittance and Reflectance Measurements

3.6.4 Theoretical Modeling of Directional-Hemispherical Characteristics

3.6.5 Identification of Absorption and Scattering Properties

3.6.6 Theoretical Modeling of Radiative Properties

3.7 Infrared Properties of Fused Quartz Containing Bubbles

3.7.1 Theoretical Results for a Weakly Absorbing Medium Containing Bubbles

3.7.2 Experimental Data for Volume Fraction and Size Distribution of Bubbles

3.7.3 Transmittance and Reflectance Measurements

3.7.4 Inverse Problem Solution

3.8 Properties of Polymer Coatings Containing Hollow Microspheres

3.8.1 Samples of Composite Material and Experimental Procedure

3.8.2 Optical Properties of Substances

3.8.3 Experimental Results for Transmittance and Reflectance

3.8.4 Theoretical Modeling of Directional-Hemispherical Characteristics

3.8.5 Comparison of Theoretical Predictions with Experimental Data

3.9 Abnormally High Scattering by Nanoporous Silica in Visible and Near-Infrared

3.9.1 Experimental Data for Nanoporous Silica

3.9.2 Analysis of Absorption

3.9.3 Analysis of Scattering

4. Some Radiative and Combined Heat Transfer Problems

4.1 Radiation Heat Transfer in a Solid-Propellant Rocket Engine

4.1.1 Spectral and Integral Emissivity of Combustion Products

4.1.2 Radiation Heat Transfer in a Supersonic Nozzle

4.1.3 Thermal Radiation of a Two-Phase Exhaust Jet

4.2 Radiative Cooling of Particle Flow in Vacuum

4.2.1 Radiative Boundary Layer

4.2.2 Liquid Droplet Radiator for Space Applications

4.3 Combined Radiative-Convective Heat Transfer in Two-Phase Flows

4.3.1 Laminar Boundary Layer on a Flat Plate: Solution in Self-Similar Variables

4.3.2 Turbulent Boundary Layer: Solution in Physical Variables

4.3.3 Some Other Problems of Coupled Radiation and Convection

4.4 Thermal Microwave Radiation of Disperse Systems on the Sea Surface

4.5 Infrared Radiation of Weakly Absorbing Coatings Containing Hollow Microspheres

4.6 Radiative-Conductive Heat Transfer in Disperse Materials

4.6.1 Highly-Porous Fibrous Materials

4.7 Radiative Effects in a Semi-Transparent Liquid Containing Gas Bubbles

4.7.1 Radiative Transfer Model

4.7.2 Some Results for Water with Steam Bubbles

4.8 Nonuniform Absorption of Thermal Radiation in Large Semi-Transparent Particles at Arbitrary Illumination of the Polydisperse System

4.8.1 Approximate Description of Asymmetric Illumination of a Single Particles

4.8.2 Solution Based on the Mie Theory

4.8.3 Approximate Solution for Symmetric Illumination

4.8.4 Approximation of Mie Calculations for Illumination from a Hemisphere

4.8.5 Some Results for Water and Diesel Fuel Droplets

4.9 Thermal Stress in Semi-Transparent Particles under High-Flux Irradiation

4.9.1 Transient Radiative-Conductive Heat Transfer

4.9.2 Thermoelastic Stress-Strain State

4.9.3 Generalized Analysis of the Problem

4.9.4 Application to Particles of Selected Materials

4.10 Thermal Radiation from Nonisothermal Particles in Combined Heat Transfer Problems

4.10.1 Semi-Transparent Oxide Particles in Thermal Spraying

4.10.2 Cooling and Solidification of Core Melt Droplets

4.11 Thermal Radiation Modeling in Melt-Coolant Interaction

4.11.1 Model Problems of Melt-Coolant Interaction

4.11.2 Thermal Radiation Transfer and Radiative Properties of Composite Medium

4.11.3 Numerical Results Based on Simplified Radiation Models

4.11.4 Verification of the Large-Cell Radiation Model

4.11.5 Thermal Radiation from the Zone of Melt–Water Interaction