NASA SP-164 THERMAL RADIATION HEAT TRANSFER Volume Ill Radiation Transfer With Absorbing, Emitting, and Scattering Media NATIONAL AERONAUTICS AND SPACE ADMINISTRATION NASA SP-164 THERMAL HEAT TRANSFER Volume III Radiation Transfer With Absorbing, Emitting, and Scattering Media Robert Siegel and John R. Howell Lewis Research Center Cleveland, Ohio Scientific and Technical I?~formationOffice 1971 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Washington, D.C. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 Price $1.75 Library of Congress Catalog Card Number 67-62877 PREFACE This is the third and final volume of the series "Thermal Radiation Heat Transfer" that is being published as the NASA Special Publication SP-164. The first and second volumes appeared in 1968 and 1969, respectively. As stated in the Preface to volume I, this publication is an outgrowth of a course in thermal radiation that has been given as part of an internal advanced study program at the NASA Lewis Research Center. This volume contains nine chapters. The first discusses some of the fundamentals of absorption of radiation along a path in amedium, and emission by the medium. The important property of the radiation intensity is developed showing that along a path in a nonattenuating nonemitting medium the intensity is invariant with position. Thus the magnitudes of any attenuation or emission can be expressed in terms of the changes produced in the intensity with distance. In chapter 2 the absorption, scattering, and emission effects along a path are combined to develop the equation of transfer. The solution of this equation gives the intensity within the medium. The intensity can then be used to obtain energy fluxes. These are related to the local temperature within the medium by means of the energy conservation equation. This provides the necessary relations to obtain energy transfers and temperature distributions in the medium. Some methods for solving these equations are given in chapter 3. These methods include various approximate techniques, a very important one being the diffusion approximation. The boundary conditions and application of the diffusion method are considered in detail. Before any radiation solution can be applied, the radiative properties must be known. Gas radiation properties vary eonsiderably with wave- length. In chapter 4 some of the fundamentals of the radiative properties are discussed. These include line broadening mechanisms and band absorption correlations. Chapter 5 considers the important situation of radiation from a gas that is well mixed so that it is isothermal. The method of analyzing heat transfer in a gas filled enclosure is developed by extending the net radiation method presented in volume I1 for enclosures containing a nonabsorbing medium. The very useful concept of radiative mean beam length is derived for use in gas energy exchange. iii iv THERMAL RADIATION HEAT TRANSFER Chapters 6, 7, and 8 provide further development by discussing the Monte Carlo method, radiation combined with conduction and convec- tion, and scattering. Finally, chapter 9 treats a number of specialized topics such as radiation in media with nonunity refractive index, and radiation from flames. CONTENTS CHAPTER PAGE 1 FUNDAMENTALS OF RADIATION IN AB- Sc3RBING. EMITTING. AND SCATTERING MEDIA ........................................................................ 1 1.1 INTRODUCTION .......................................................... I 1.2 SYMBOLS .................................................................... 3 1.3 PHYSICAL MECHANISMS OF ABSORPTION AND EMIS- SION....................................................................... 4 1.4 SOME FUNDAMENTAL PROPERTIES OF THE RADIA- TION INTENSITY ...................................................... 8 1.5 THE ATTENUATION OF ENERGY .................................. 12 1.5.1 The Extinction Coefficient ............................................ 13 1.5.2 Radiation Mean Penetration Distance ............................. 14 1.5.3 Optical Thickness ....................................................... 14 1.5.4 The Absorption Coefficient........................................... 15 1.5.5TrueAbsorptionCoefficient.......................................... 19 1.5.6 The Scattering of Energy ............................................. 20 1.6 THE EMISSION OF ENERGY ......................................... 21 1.7 DEFINITIONS USED FOR ENGINEERING GAS PROP- ERTIES ..................................................................... 24 1.7.1 Absorptance ............................................................. 24 1.7.2 Emittance ................................................................. 25 1.7.3 Transmittance............................................................ 29 1.8 THE CONCEPT OF LOCAL THERMODYNAMIC EQUILIB- RIUM ....................................................................... 32 1.9 CONCLUDING REMARKS ............................................. 34 REFERENCES ..................................................................... 34 2 THE EQUATIONS OF TRANSFER FOR AN AB- SORBING-EMITTING GAS .................................... 2.1 INTRODUCTION .......................................................... 2.2 SYMBOLS .................................................................... 2.3 THE EQUATION OF TRANSFER .................................... 2.3.1 Derivation ................................................................. 2.3.2 Integration by Use of Integrating Factor .......................... 2.4 ENERGY CONSERVATION WITHIN THE MEDIUM .......... 2.4.1 Radiative Equilibrium .................................................. 2.4.2 Some Mean Absorption Coefficients ............................... 2.5 EQUATION OF TRANSFER FOR PLANE LAYER ............. 2.6 THE GRAY GAS ........................................................... 2.6.1 Transfer Equations ..................................................... 2.6.2 Plane Layer Between Black Plates ................................. 2.6.3 Use of Exponential Integral Functions ............................ 2.7 ENERGY RELATIONS BY USE OF PHOTON MODEL ......... 2.8 CONCLUDING REMARKS ............................................. REFERENCES..................................................................... vi THERMAL RADIATION HEAT TRANSFER CHAPTER PAGE 3 APPROXIMATE SOLUSTIONS OF THE EQUA- TION OF TMNSFER ................................................ 3.1 INTRODUCTION .......................................................... 3.2 SYMBOLS .................................................................... 3.3 APPROXIMATE SOLUTIONS BY NEGLECTING TERMS IN THE EQUATION OF TRANSFER ............................. 3.3.1 The I'ransparent bas Approximation .............................. 3.3.2 The Emission Approxin~ation........................................ 3.3.3 The Cold Medium Approximation .................................. 3.4 DIFFUSION METHODS IN RADIATIVE TRANSFER .......... 3.4.1 Simplified Derivation of the Diffusion Equation ................ 3.4.2 The General Radiation Diffusion Equation ....................... 3.4.2.1 The Rosseland equation for local radiative flux............ 3.4.2.2 The emissive power jump as a boundary condition ......... 3.4.2.3 The emissive power jump between two absorbing-emit- ting regions ........................................................ 3.4.2.4 Summary............................................................. 3.4.3 Use of the Diffusion Solution ........................................ 3.4.3.1 Gray stagnant gas between parallel plates .................. 3.4.3.2 The discontinuity in emissive power between two gas reglons .............................................................. 3.4.3.3 Other diffusion solutions for gray gases..................... 3.4.4 Final Remarks ........................................................... 3.5 APPROXIMATIONS BY USING MEAN ABSORPTION CO- EFFICIENTS .............................................................. 3.5.1 Some Mean Absorption Coefficients ............................... 3.5.2 Approximate Solutions of the Transfer Equations Using Mean Absorption Coefficients ..................................... 3.6 APPROXIMATE SOLUTION OF THE COMPLETE EQUA- TION OF TRANSFER ................................................. 3.6.1 The Astrophysical Approximations ................................. 3.6.1.1 The Schuster-Schwarzschild approximation ............... 3.6.1.2 The Milne-Eddington approximation .......................... 3.6.2 The Differential Approximation ..................................... 5.6.2.1 Houndary conditions ............................................. 3.6.2.2 Applications of the differential approximation ............. 3.7 CONCLUDING REMARKS ............................................. REFERENCES ..................................................................... 4 AN INTRODUCTION TO THE MICROSCOPIC BASIS FOR RADIATION IN GASES AND GAS PROPERTIES ............................................................ 113 4.1 INTRODUCTION ......................................................... 113 4.2 SYMBOLS .................................................................. 113 4.3 SOME ELEMENTS OF QUANTUM THEORY .................. 115 4.3.1 Bohr Model of Hydrogen Atom ...................................... 115 4.3.2 Schriidinger Wave Equation ....................................... 117 4.4 INDUCED EMISSION AND THE PLANCK DISTRIBU- TION ........................................................................