RADIATION HEAT TRANSFER ANALYSIS IN TWO-PHASE MIXTURE ASSOCIATED WITH LIQUID METAL REACTOR ACCIDENTS Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Engineering By Hamza Mohamed Dayton, Ohio May 2020 RADIATION HEAT TRANSFER ANALYSIS IN TWO-PHASE MIXTURE ASSOCIATED WITH LIQUID METAL REACTOR ACCIDENTS Name: Mohammed, Hamza APPROVED BY: Jamie S. Ervin, Ph.D. Kevin P. Hallinan, Ph.D. Doctoral Committee Chair Doctoral Committee Member Chair Professor Mechanical and Aerospace Engineering Mechanical and Aerospace Engineering Andrew Chiasson, Ph.D. Elizabeth A. Ervin, Ph.D. Doctoral Committee Member Doctoral Committee Member Associate Professor Avionics Engineer Mechanical and Aerospace Engineering PE Systems Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean Professor School of Engineering School of Engineering ii ABSTRACT RADIATION HEAT TRANSFER ANALYSIS IN TWO-PHASE MIXTURE ASSOCIATED WITH LIQUID METAL REACTOR ACCIDENTS Name: Mohammed, Hamza University of Dayton Advisor: Prof. Jamie Ervin Analytical study associated with liquid-metal fast breeder reactor (LMFBR) has been investigated by using scattering and non-scattering mathematical radiation models. In the non- scattering model, the radiative transfer equation (RTE) was solved together with the continuity equations of mixture components under local thermodynamic equilibrium. A MATLAB code was used to solve these equations. This application employed a numerical integration to compute the temperature distribution within the bubble and the transient wall heat flux. First, in Rayleigh non- scattering model the particle size was 0.01 µm [6], and according to Mie theory principle, the absorption coefficient for small particle –size distribution was estimated (k = 10 m-1 was used) from reference [7] at complex refractive index of UO2 at λ = 600 µm and x = 0.0785. A MATLAB code was used to solve the radiative heat equation (RTE) in spherical coordinates. The mixture is in local thermodynamic equilibrium inside the bubble which has a black body surface boundary. The mixture in the bubble contains three components: the non-condensable gas Xenon, Uranium dioxide vapor, and fog. To simulate fuel bubble’s geometry as realistically as possible, according to experimental observation, the energy equation in a spherical coordinate system has been solved with the radiative flux heat transfer equation (RTE) to obtain the effect of fuel bubble’s geometry on the transient radiative heat flux and to predict the transient temperature iii distribution in the participating medium during a hypothetical core disruptive accident (HCDA) for liquid metal fast breeding reactor (LMFBR) for FAST. The transient temperature distribution in fog region was used to predict the amount of condensable UO2 vapor. The conclusion that can be drawn from the present study, is that the Fuel Aerosol Simulant Test (FAST) facility at Oak Ridge National Laboratory has a larger margin of safety since the bubble rising time is greater than the bubble collapse time. Second in the scattering model, the spherical harmonics method was used to solve the radiative heat transfer equation (RTE) in spherical coordinates, and the particle size was 0.07 µm [6]. The scattering coefficient of -1 UO2 particles (σ = 1.24 m ), was calculated using Mie theory at the same number of stable nuclei 3 -1 N (2.9 E15 nuclei/m ) that resulted from the absorption coefficient k = 0.082 m [7]. The P1 approximation method was used to solve the radiative transfer equation (RTE) in spherical coordinates of participating medium confined between two concentric spheres. The surfaces of the spheres are assumed to be gray, diffusely emitting and diffusely reflecting boundaries, and isothermal boundary conditions were assumed at these surfaces. Marsak’s boundary condition was used to compute the net radiative heat flux, q(τ), and the incident radiation, G(τ), to analyze and interpret CVD experiments data that were conducted in the FAST facility at ORNL [8] and Fast Flux Test Facility reactor (FFTF) at ANL. From this study, it can be concluded that there is greater margin of safety when the bubble rise time is a greater than the bubble collapse time since the bubble collapses (UO2 condenses) before it can reach the top of the vessel. In addition, the work transfer by itself can’t completely eliminate the super-heated vapor, as the bubble contains non- condensable species which hinder condensation. However, it is reasonable to assume that work transfer could decrease the amount of UO2 vapor contained in the bubble as it reached the covergas [63]. iv ACKNOWLEDGMENTS My special thanks are to Prof. Jamie Ervin, my professor and advisor, for providing the time and necessary resources for the work contained herein, and for directing this dissertation and bringing it to its conclusion with patience and expertise. I would also like to express my gratitude to everyone who has helped me with this work. This includes Prof. James Menart, who helped in the radiation heat transfer area, and Salahuddin, who offered guidance in composing and reviewing several parts of this dissertation; Shahul Hameed, who helped in formatting the document and correcting the text. I also deeply appreciate my wife, children, and my friends who motivated and supported me during the development of this dissertation. v TABLE OF CONTENTS ABSTRACT ........................................................................................................................................ iii ACKNOWLEDGMENTS ...................................................................................................................... v LIST OF FIGURES ............................................................................................................................ viii LIST OF TABLES ................................................................................................................................. x LIST OF ABBREVIATIONS AND NOTATIONS ..................................................................................... xi CHAPTER I INTRODUCTION .............................................................................................................. 1 1-1 Objectives ........................................................................................................................ 10 1-2 New Contributions ........................................................................................................... 11 CHAPTER II MATHEMATICAL MODELS ........................................................................................... 12 2-1 Non-Scattering Model ...................................................................................................... 12 2-1-1 The Conservation Equations ................................................................................. 21 2-1-2 The Energy Equation ............................................................................................. 27 2-1-3 In Fog Free Region ................................................................................................ 28 2-1-4 The Wall Heat Flux in Fog Free Region ................................................................. 30 2-1-5 Transient Temperature Profile in The Fog Region ................................................ 30 2-1-6 The Wall Heat Flux ................................................................................................ 33 2-2 Scattering Model .............................................................................................................. 33 2-2-1 Radiative Analysis in Isothermal Spherical Media ................................................ 34 2-2-2 Spherical Harmonics Method ............................................................................... 36 2-2-3 Boundary Conditions ............................................................................................ 45 2-3 Boundary Work ................................................................................................................ 49 CHAPTER III RESULTS ...................................................................................................................... 52 3-1 Rayleigh Non-Scattering Model Results ........................................................................... 52 3-1-1 Compare of Current Method by Comparison With Past Work ............................. 52 3-1-2 The Wall Heat Flux ................................................................................................ 58 3-2 The Bubble Rise Time ....................................................................................................... 61 3-3 The Bubble Collapse Time ................................................................................................ 62 3-3-1 Sample Calculations of The Bubble Collapse Time ............................................... 63 3-4 Scattering Model .............................................................................................................. 66 3-4-1 Reproduced Work for W. Li and T. W. Tong [28] .................................................. 67 3-4-2 Mie Scattering Model on Sodium Experiments .................................................... 73 vi 3-4-3 Application of Current Model to Sodium Experiments ........................................ 78 3-4-4 Bubble Collapse Time ...........................................................................................